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Giovanni MORELLORelated papers
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Nanoparticle research disciplines—chemical synthesis, applied physics and devices based on their physical-chemical properties, and computational physics—have been very active fields for the last 15 years or so, because of the potential and current applications in medicine, catalysis, energy storage, environment and electronics applications. This wide spectrum of disciplines and their applications keep metallic nanoparticles as one of the most promising nanostructures and their research as one of the cornerstones of nanotechnology. In this contribution we present a comprehensive and extended geometrical description for the most common shapes and structures for metallic nanoparticles, as well as experimental results for these geometries with some variations given by truncations.
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Chemistry and Properties of Nanocrystals of Different Shapes
Chemical Reviews, 2005
Review on Inorganic Nano crystals unique benchmark of Nanotechnology
2014
Most generally, "Nanocrystal" is used to mean a crystalline nanoparticle. Some sources define a nanocrystal as any singlecrystalline nanomaterial with at least one dimension ≤ 100 nm., while others define it more generally to be a nanoparticle with any kind of crystalline structure including e.g. multiply twinned configurations. A general material object with all dimensions less than 1 micrometre, i.e., 1000 nanometers, should be referred to as a nanoparticle, not a nanocrystal. In case it is single-crystalline or polycrystalline, it may be referred to as a nanocrystal. Crystalline nanoparticles are of interest because of many reasons. Silicon nanocrystals can provide efficient light emission even while bulk Si does not and be used as for memory components. Nanocrystals embedded in solids can exhibit much more complex melting behaviour than conventional solids [ and can form the basis of a special class of solids. They can also provide single-domain crystalline systems tha...
Some properties of spherical and rod-shaped semiconductor and metal nanocrystals
Pure and Applied Chemistry, 2000
Physics Reports 501 (2011) 75–221
Contents lists available at ScienceDirect
Physics Reports
journal homepage: www.elsevier.com/locate/physrep
Physical properties of elongated inorganic nanoparticles
Roman Krahne a , Giovanni Morello b , Albert Figuerola a , Chandramohan George a ,
Sasanka Deka a , Liberato Manna a,c,∗
a
Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy
b
Nanoscience Institute of CNR, Via Arnesano, 73100, Lecce, Italy
c
Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
article
info
Article history:
Accepted 29 December 2010
Available online 11 January 2011
editor: G.E.W. Bauer
Keywords:
Nanoparticles
Nanocrystals
Nanorods
Elongated nanostructures
Optics
Electronics
Plasmonics
Magnetism
Catalysis
Mechanical properties
abstract
Inorganic nanoparticles are among the most investigated nano-objects nowadays, both in
fundamental science and in various technological applications. In this paper, we describe
the physical properties of nanoparticles when their shape is elongated, i.e. rod-like or wirelike. In most cases, we will analyze the transition in the physical properties when the shape
of the nanoparticles evolves from spherical to rod-like.
© 2011 Elsevier B.V. All rights reserved.
Contents
1.
2.
∗
Introduction.............................................................................................................................................................................................
Optical properties of elongated semiconductor nanoparticles ............................................................................................................
2.1.
Introduction ................................................................................................................................................................................
2.2.
Quantum confinement in low-dimensional systems ...............................................................................................................
2.3.
Excitons and quantum confinement regimes ...........................................................................................................................
2.4.
Optical properties of elongated nanoparticles: overview ........................................................................................................
2.4.1.
Electronic structure of CdSe nanorods: a case study.................................................................................................
2.5.
Relaxation mechanisms in nanorods.........................................................................................................................................
2.5.1.
1D excitons in homostructures...................................................................................................................................
2.5.2.
Temperature and size dependence of the exciton relaxation processes .................................................................
2.5.3.
Exciton–phonon interaction: homogeneous broadening .........................................................................................
2.5.4.
Auger effects in nanorods ...........................................................................................................................................
2.6.
Single nanorod properties ..........................................................................................................................................................
2.6.1.
Charged/neutral excitons and phonon contributions ...............................................................................................
2.6.2.
Influence of electric fields on the optical properties of single nanorods .................................................................
2.6.3.
Polarization properties ................................................................................................................................................
2.7.
Nanorod heterostructures ..........................................................................................................................................................
Corresponding author at: Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy. Tel.: +39 010 71781502.
E-mail address: liberato.manna@iit.it (L. Manna).
0370-1573/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.physrep.2011.01.001
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90
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76
3.
4.
5.
6.
7.
R. Krahne et al. / Physics Reports 501 (2011) 75–221
2.7.1.
Semiconductor dot/rod heterostructures .................................................................................................................. 94
2.7.2.
Electron–hole dynamics in core–shell nanorods....................................................................................................... 95
2.7.3.
Photoluminescence studies on single core–shell nanorods...................................................................................... 97
2.7.4.
Type-I and type-II transitions in CdSe/CdS nanorods................................................................................................ 99
2.7.5.
Optical properties of ordered nanorod assemblies ................................................................................................... 102
2.7.6.
Collinear nanorods and nanobarbells......................................................................................................................... 103
2.7.7.
Multifunctional nanorods ........................................................................................................................................... 105
Electrical properties of elongated nanoparticles .................................................................................................................................. 107
3.1.
Introduction ................................................................................................................................................................................ 107
3.2.
Electrical experiments on single nanorods ............................................................................................................................... 108
3.2.1.
Single nanorod conductance probed by STS .............................................................................................................. 109
3.2.2.
Core–shell nanorods fabricated by seeded growth ................................................................................................... 111
3.2.3.
Single nanorod conductance probed in planar nano-junctions................................................................................ 111
3.2.4.
Gold tipped nanorods (nanodumbbells) .................................................................................................................... 114
3.3.
Electrical and photoconduction properties of nanorod assemblies ........................................................................................ 116
3.4.
Nanorod assemblies for photovoltaic applications .................................................................................................................. 118
3.5.
Thermoelectric properties of elongated nanoparticles ............................................................................................................ 119
Optical properties of elongated metal nanoparticles ........................................................................................................................... 122
4.1.
Overview ..................................................................................................................................................................................... 122
4.2.
Dielectric function of metal nanoparticles................................................................................................................................ 123
4.3.
Plasmonic properties of small spherical metal nanoparticles ................................................................................................. 124
4.4.
Plasmonic properties of large spherical metal nanoparticles .................................................................................................. 126
4.5.
Plasmonic properties of ellipsoidal-shaped metal nanoparticles............................................................................................ 128
4.6.
Plasmons in elongated nanoparticles of arbitrary shape ......................................................................................................... 130
4.7.
Dephasing of surface plasmons: spherical vs. elongated particles.......................................................................................... 132
4.8.
Ultrafast electron dynamics in metal nanoparticles: overview............................................................................................... 134
4.9.
Ultrafast dynamics in elongated nanoparticles ........................................................................................................................ 136
4.10.
Near field optical properties of metal nanoparticles: spheres and rods ................................................................................. 139
4.10.1.
Near field optical properties: classical electromagnetic theory and quantum mechanical calculations .............. 139
4.10.2.
Surface enhanced fluorescence................................................................................................................................... 142
4.10.3.
Surface enhanced Raman scattering........................................................................................................................... 143
4.10.4.
Photoluminescence enhancement from metal nanorods ......................................................................................... 144
Magnetic properties of elongated nanoparticles .................................................................................................................................. 146
5.1.
Introduction ................................................................................................................................................................................ 146
5.2.
General considerations on magnetic nanocrystals ................................................................................................................... 146
5.3.
Anisotropy considerations in magnetic nanocrystals .............................................................................................................. 149
5.4.
Iron oxide-based 1D nanostructures ......................................................................................................................................... 152
5.4.1.
Magnetite-based 1D nanostructures .......................................................................................................................... 153
5.4.2.
Hematite-based 1D nanostructures ........................................................................................................................... 154
5.4.3.
Maghemite-based 1D nanostructures ........................................................................................................................ 157
5.4.4.
ε-Fe2 O3 -based 1D nanostructures .............................................................................................................................. 158
5.4.5.
Assembly of oxide-based 1D nanostructures ............................................................................................................ 158
5.5.
Metallic 1D nanostructures ........................................................................................................................................................ 161
5.5.1.
Co 1D nanostructures .................................................................................................................................................. 161
5.5.2.
Fe 1D nanostructures................................................................................................................................................... 166
5.5.3.
Ni 1D nanostructures................................................................................................................................................... 168
5.5.4.
Alloys of special technological interest ...................................................................................................................... 170
5.5.5.
Magnetization reversal studies in 1D metallic nanostructures ................................................................................ 173
5.5.6.
Domain Wall dynamics in 1D metallic nanostructures ............................................................................................ 178
Catalytic properties of elongated inorganic nanoparticles................................................................................................................... 184
6.1.
Introduction ................................................................................................................................................................................ 184
6.2.
Rod-like morphology and catalytic properties of inorganic nanoparticles ............................................................................ 185
6.2.1.
Metals, core/shell metals, segmented metals and metal alloy nanorods................................................................. 186
6.2.2.
Metal oxide, semiconductor, and mixed nanorods ................................................................................................... 189
Mechanical properties of elongated nanoparticles and melting studies............................................................................................. 196
7.1.
Introduction ................................................................................................................................................................................ 196
7.2.
Mechanical properties of materials: a few useful definitions.................................................................................................. 198
7.3.
Studies on elastic properties of nanorods and nanowires ....................................................................................................... 199
7.4.
Plastic behavior of nanocrystalline materials ........................................................................................................................... 200
7.5.
Experiments on plastic behavior of single micro- and nanocrystalline samples ................................................................... 201
7.5.1.
Metallic whiskers......................................................................................................................................................... 201
7.5.2.
Pillars ............................................................................................................................................................................ 202
7.6.
Models of plastic behavior of single micro- and nanocrystalline samples ............................................................................. 203
7.6.1.
Strain gradients............................................................................................................................................................ 203
7.6.2.
Dislocation starvation.................................................................................................................................................. 204
7.6.3.
Source truncation hardening ...................................................................................................................................... 204
R. Krahne et al. / Physics Reports 501 (2011) 75–221
8.
77
7.6.4.
Surface-controlled dislocation multiplication in metallic BCC micro/nanopillars.................................................. 207
7.6.5.
General remarks .......................................................................................................................................................... 208
7.7.
Buckling ....................................................................................................................................................................................... 208
7.8.
Melting studies of elongated inorganic nanoparticles ............................................................................................................. 210
Conclusions and outlook......................................................................................................................................................................... 211
Acknowledgements................................................................................................................................................................................. 213
References................................................................................................................................................................................................ 213
1. Introduction
Understanding the size and shape dependence of physical properties in nanoscale particles is a fundamental step
towards the design, fabrication and assembly of materials and devices with predictable behavior. In recent years, there
has been a remarkable advancement in the ability to fabricate shape-controlled nanoparticles, for example rods, wires,
and nanoparticles with branched shapes, especially via synthetic approaches in solution. Shape-controlled inorganic
nanoparticles are among the most promising candidates as building blocks in nanoscale materials and devices, both because
their physical properties are modified considerably compared to those of spherical nanoparticles and because their intrinsic
geometry opens many new opportunities for their assembly into organized superstructures. In this work, we have decided
to review the physical properties of elongated inorganic nanoparticles, with particular emphasis on the transition in these
properties when the shape of the nanoparticles evolves from a sphere to a rod, but we will consider in many cases also
nanowires. From the point of view of specific properties and materials, we have decided to cover the optical properties of
semiconductors and noble metals, the electrical properties of semiconductors, the magnetic properties of various metals
and metal oxides, the catalytic properties of various classes of materials and the mechanical properties of metals and metal
alloys.
Sections 2 and 3 of this review will be devoted to the optical and electrical properties of elongated semiconductor
nanocrystals. Semiconductor nanocrystals are among the most studied materials in nanoscience nowadays, due to the
large number of potential applications employing these materials, for example in optical devices (lasers [1–3], light
emitting diodes [4,5], photodetectors [6], solar cells [7–9]) or biological labeling [10,11], to cite a few. Elongated, rodshaped semiconductor nanocrystals possess interesting physical properties which depend on their size, aspect ratio and
chemical composition, and these nanoparticles have been proposed as active materials in light emitting devices [12],
photocatalysis [13], optically induced light modulation [14], photovoltaics [7–9,15], wavefunction engineering [16–18],
and optical memory elements exploiting the exciton storage process [19]. More in general, these nanoparticles have been
considered as replacement for spherical nanocrystals (the so-called ‘‘quantum dots’’) in all those studies in which the
elongated shape could in principle add new or improved properties.
Section 4 will deal with optical properties of elongated metal nanocrystals. Metallic nanocrystals have been proposed
in a wide range of applications in various fields, among them sensing, biosensing, photodynamic therapy, photovoltaics,
optics (light emitting diodes, photodetectors, lasers, imaging techniques beyond the diffraction limit), nano-optics and
nano-electronics (for example plasmonic waveguides) [20–32]. Metal nanostructures can interact strongly with light in
the visible and near infrared region of the spectrum, due to the presence of free electrons, which can be promoted both
to empty energy levels in the same band or to levels of an empty overlapping band. An incident electromagnetic field can
elicit collective oscillations of these free electrons [20–23], which cause a displacement of the electrons from the nuclei,
leading to the formation of various possible distributions in the surface charges. This creates Coulomb interactions between
positive and negative charges, which induce restoring oscillating forces acting on free electrons. Each type of surface charge
distribution is characterized by a collective oscillation mode, also termed as localized surface plasmon resonance. Various
factors influence the possible types of SPRs in nanostructures and the frequencies at which they are observed and the shape
of metal nanoparticles is certainly one of them. As an example, in rod-shaped nanoparticles the plasmon mode is split into
two modes, a longitudinal one and a transverse one. There are many other physical effects connected with an elongated
shape which differ from the spherical case, and these will be covered in detail in Section 4.
In Section 5 we will review the magnetic properties of elongated nanoparticles. Many of the application of magnetic
nanoparticles are in life sciences and biomedicine [33–35]. Superparamagnetism is the term used for describing the absence
of coercivity and remanent magnetization in particles that still maintain a considerable amount of polarizable spins under
the effect of an external magnetic field [36,37]. These magnetic nanocrystals, also known as superparamagnets, combine
their reduced sizes with their magnetic field responsive character even if no residual magnetization is observed in the
absence of an external magnetic field. For this reason they have been proposed as vectors for both in vitro and in vivo transport
of different drugs or biomolecules attached to their surfaces, thereby providing selective access to cellular or molecular levels
which are inaccessible to conventional therapeutic approaches. In the same way, they can also be used in biodetection and
bioseparation techniques since once the target molecule has been attached to the nanocrystals, the application of an external
magnetic field will allow their recovery [38,39]. Iron oxide is clearly the most suitable material for such purposes due to its
high chemical stability, biocompatibility and superparamagnetic properties, and iron oxide nanoparticles are being already
used in several diagnostic and therapeutic techniques [40].
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
The achievement of higher coercivity values in particles with reduced size for information storage devices, or faster
magnetic responses for smaller biomedical vectors, could be possible if one considers not only the finite size effects of
spherical nanocrystals but also the additional phenomena arising from the shape anisotropy of particles such as nanorods.
Nanorods or other one-dimensional nanosystems could also be capable of widening the temperature range of applications
of a certain magnetic material compared to its bulk form (as will be shown later). The uniaxial shape anisotropy of metallic
and oxide nano-objects will probably become a key factor for the development of improved devices. This section will also
present an overview of various classes of magnetic materials that have been synthesized in nanorod shapes.
Section 6 will deal with the catalytic properties of elongated nanoparticles. Today, there is an increasing demand
for catalytic materials, in terms of catalytic efficiency, cost of production, specificity, durability and environmental
sustainability [41–44]. This demand is driving research towards the exploitation of new nanoscale catalyst particles, in
which the individual components have specific size, shape, exposure of specific reactive surfaces, and suitable combination
of materials [45,46]. Micro- and nanoparticles of various materials have been used as catalysts for many years [47–50], and
experimental evidence has been collected so far demonstrating that the catalytic activity of particles is strongly related to
their size, and in particular that nanosized particles exhibit increased catalytic activity with respect to larger particles, due to
their higher surface to volume ratio [51,52]. With recent advances in the synthesis of inorganic nanoparticles with controlled
size and morphology [53–56], interest has grown towards the understanding of how the catalytic performance of these
materials is dependent on shape. In terms of catalytic properties, there are several reasons why an elongated morphology
is often preferable over a spherical morphology, and these will be described in Section 6. Several case studies of nanorodshaped catalysts will be discussed in this section.
Section 7 deals mainly with the mechanical properties of elongated nanoparticles. The miniaturization of micro electromechanical devices and the fabrication of thin films in the electronic industry have started to raise questions already
decades ago about the mechanical behavior of confined systems. Early experiments on tensile testing of metal whiskers
with micrometer transverse sides have evidenced strengths much higher than the bulk value [57], and recently pure metals
and alloys with at least one dimension in the micro- and nanoscale range have been investigated, thanks to advances in the
fabrication of new generations of samples suitable for mechanical testing (for example micropillars prepared by focused ion
beam) and in various techniques for studying their stress and deformation properties. Those studies have revealed a marked
deviation in the mechanical properties of samples from bulk-like behavior already when their size is of the order of a few
micrometers, which is comparable to the length scale of many plasticity mechanisms based on dislocation nucleation and
propagation. The increased strength of single nanocrystals could be useful for applications of these materials as active probes
in nanoindentation, scanning probe microscopy and field emission [58–60], to cite a few. Section 7 ends with a paragraph
on melting studies on nanorods.
Finally, we will conclude the present review article with some remarks and an outlook on future directions in this field.
2. Optical properties of elongated semiconductor nanoparticles
2.1. Introduction
The optical properties of nanocrystals are dictated by their electronic structure, and we start this section with a short
overview on the main quantum effects in confined systems. We then describe the optical peculiarities of semiconductor
nanorods by highlighting the main characteristics that distinguish them from the more traditional quantum dots. The
first part of this section will focus on the comprehension of the general properties of nanorods made of a single material
(henceforth referred to as ‘‘homostructures’’), while in the second part of the section we will highlight the properties of
nanorods made of sections of different materials (henceforth referred to as ‘‘heterostructures’’), which have been synthesized
by various groups thanks to the latest developments in colloidal nanocrystal synthesis.
2.2. Quantum confinement in low-dimensional systems
A system in which the motion of electrons or other particles (holes, excitons, etc.) is restricted in one or more dimensions,
due to some potential profile, is usually referred to as a ‘‘low-dimensional’’ system and shows quantum confinement effects.
Due to their dual wave–particle nature, electrons in a solid are treated as particles having an effective mass m∗ (accounting
⃗ = h̄⃗k. Here, h̄ is
for the periodicity of the crystal potential) and a linear momentum arising from their wave-like nature p
⃗
the Planck’s constant divided by 2π , and k represents the wavenumber of the associated wave of wavelength λ = 2π /k.
The behavior of the electrons is strongly sensitive to the dimensions of the solid in which they move. In the bulk, the infinite
extension of the solid is imposed by assigning the so-called periodic boundary conditions, such that the electrons are not
affected by the borders of solid in terms of wavefunction and energy. If however the dimensions of the solid are reduced, the
electrons start to ‘‘feel’’ the borders and the assumption of infinite extension of the solid in all the three spatial coordinates
does not hold any more. In such a case the system is considered as ‘‘quantized’’.
The simplest way to understand what happens to electrons in the case of a quantized system (in terms of wavefunctions
and energies) is to consider the classical textbook case of an electron in a box, which is approached by solving the Schrödinger
R. Krahne et al. / Physics Reports 501 (2011) 75–221
79
Fig. 2.1. Electrons in a one-dimensional quantum box: energy levels and wavefunctions.
equation in one dimension (Fig. 2.1) [61]:
d2 ψ
dx2
+
2mE
b2
= V with V =
We seek for a solution of the form:
0,
∞,
−a/2 < x < a/2
.
x ≤ −a/2, x ≥ a/2
(2.1)
ψ(x) = Aeikx + Be−ikx .
(2.2)
ψ(x) = A eikx − e−ikx = 2iA sin(kx)
(2.3)
Since the potential is infinitely high at the border regions, we need to impose the conditions that the wavefunctions have
to vanish at the borders (i.e. ψ(−a/2) = 0 and ψ(a/2) = 0, see Fig. 2.1). Upon imposing ψ(−a/2) = 0 we obtain:
and, after substitution in the Schrödinger equation we obtain the following expression for the energy:
E=
h̄2 k2
2m
(2.4)
.
Since ψ(a/2) = 0, the following identity must hold:
ψ(a/2) = 2iA sin(ka) = 0.
(2.5)
The identity is verified only when ka = nπ , with n = 1, 2, 3. Therefore the parameter k is quantized, and the separation
between two consecutive values of k is ∆k = ∆nπ /a. The quantization energy is expressed as:
En =
h̄2 π
2ma2
n.
(2.6)
In the expression above, the confinement energy is inversely proportional to the square of the size a of the box. As a result,
a large value of a corresponds to small spacing of the values in k-space. In the bulk, for example, in which a is practically
infinite, there will be a continuous distribution of states in the k-space, in all spatial directions. In a quantum well only one
spatial dimension is reduced (down to a few nm), and the electrons can freely move only in a plane, i.e. they behave as a
two-dimensional electron gas (2DEG). The wavenumbers have a quasi-continuous distribution along that plane, whereas
they are ‘‘quantized’’ along the reduced dimension, following the relation ∆k = ∆nπ /a. Several quantum well systems
have been fabricated and studied to date (for an overview of some of the pioneering works see for example [62–66]). If
the movement is restricted in two directions, the system is referred to as a ‘‘quantum wire’’, and the electrons can move
freely only along the non-reduced dimension. Examples of 1D systems are inorganic semiconductor and metallic nanowires,
some organic molecules and carbon nanotubes [67–79]. In the case of quantization in all directions, the electrons possess
discrete wavenumbers in the k-space and the system is called a ‘‘quantum dot’’. One way to discriminate among the different
quantized systems in terms of electronic states is via the density of states function D(E), which represents the number of
electronic states in a unitary interval of energy (Fig. 2.2).
In a bulk material (a three-dimensional solid) the density of states follows the well-known dependence on the square
root of the energy (3D case in Fig. 2.2). This situation corresponds to a quasi-continuous distribution of states for a freeelectron gas. In the 2D case (i.e. a two-dimensional solid, 2D case in Fig. 2.2), the global density of states function preserves
the quasi-continuous character but becomes a step function framed by the initial square root function. As a consequence, a
zero-point energy is introduced and the step spacing is dictated by the energy spacing along the quantized direction. In the
1D case (the ‘‘quantum wire’’) the wavenumbers are quantized in two directions and the density of states follows a saw-tooth
function (1D case in Fig. 2.2). The latter still describes a quasi-continuous distribution of states, but is now characterized by
series of singularities corresponding to the quantization introduced in two dimensions, while the hyperbolas in between two
singularities are well described by the reverse of square root of the energy. The 0D sketch of Fig. 2.2 shows what happens in
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 2.2. Evolution of the ‘‘density of states’’ function from a bulk (3D solid) to a 2D, 1D and 0D system.
the case of quantization in all three dimensions. The quasi-continuous distribution of states has now collapsed into a series
of discrete levels, represented by several Dirac delta functions. In the case of a flat potential with infinite boundaries, the
level spacing increases with energy, as described by Eq. (2.6). Such atomic-like structure in the density of states makes these
0D systems fascinating objects, for what concerns both fundamental physics and technological applications [80–85].
2.3. Excitons and quantum confinement regimes
The energy gap of a semiconductor is the energetic separation between the lowest unoccupied electronic state and the
highest occupied state at T = 0 K. The exciton band gap is written as:
(2.7)
Eg = Eg0 + Eq + EC
where Eg0 is the energy gap of the corresponding bulk solid, Eq is the contribution introduced by the quantization and
corresponds to the zero-point energy for each system, and EC is the contribution of the Coulomb attraction between electrons
and holes. Their relative contribution is basically related to the degree of confinement to which a system is subjected. In
order to better define the role of the quantum confinement on each term of the above expression, we need to introduce the
concept of ‘‘exciton’’, which is a ‘‘quasi-particle’’. An exciton is a bound system, composed of an electron (e) and a hole (h) that
experience a mutual Coulomb interaction. Here, we will be dealing with a particular type of exciton, the ‘‘Wannier–Mott’’
exciton, in which the e–h distance is much larger than the lattice constant of the crystal in which it is formed [86]. One key
parameter associated with the exciton is its binding energy, which is defined as:
Eb =
h̄2
2µrB2
(2.8)
.
In the expression above, µ is the reduced mass of the exciton, defined as 1/µ = (1/me + 1/mh ), me and mh being the
effective masses of electrons and holes, respectively. rB represents instead the Bohr radius of the exciton, defined as:
rB =
h̄2 ε 2
(2.9)
µe2
with ε being the dielectric constant of the semiconductor. The Bohr radius of the exciton represents its natural
spatial extension after its creation (i.e. upon electron–hole pair generation) until its annihilation (upon electron–hole
recombination). In bulk semiconductors such radius ranges from a few nm to some tens of nm, depending on the material.
The quantization energy is related to the zero-point energy of electrons and holes in the system considered, and it has always
a positive value. In a spherical nanocrystal, for instance, it is defined as [87]:
Eqe,h =
h̄2 π 2
2me,h a2
(2.10)
.
The Coulomb contribution is in general negative, but in some cases it can also be positive (see later in this section). It is
proportional to e2 /(ε 2 a) and in a spherical quantum dot it can be estimated (as a first approximation) as being equal to [87]:
EC = −1.8
e2
ε2 a
(2.11)
.
Therefore, by considering all contributions, the energy gap can be derived from the following expression:
Eg = Eg0 +
h̄2 π 2
2me a2
+
h̄2 π 2
2mh a2
− 1.8
h̄2
µrB a
.
(2.12)
The optical properties of a semiconductor nanocrystal are dictated by the ratio between the spatial confinement, i.e. the
size of the nanocrystal (a), and the Bohr radius rB of the exciton. Depending on this ratio, three different regimes of
quantization can be defined: weak (a ≫ rB ), intermediate (a ≈ rB ) and strong (a < rB ). In the weak confinement regime,
the Coulomb term dominates. Electrons and holes can form the exciton and do not actually ‘‘feel’’ the ‘‘restricted’’ size of
R. Krahne et al. / Physics Reports 501 (2011) 75–221
81
the semiconductor. In the intermediate regime, the system can behave in different ways, depending on the ratio between
a and the Bohr radii of electrons (re ) and of the holes (rh ), respectively. In the case that rh is smaller than re , the hole will
stay confined at the center of the nanocrystal with the electron orbiting around it, similarly to the case of a donor impurity
[88,89], if a falls in between rh and re . In the strong confinement regime the quantization term dominates over the binding
energy of exciton.
2.4. Optical properties of elongated nanoparticles: overview
The dependence of the energy gap on the size of the nanocrystals is used as a powerful tool for designing materials
with well-controlled optical properties. The continuous progress in the development of novel and sophisticated synthesis
techniques has opened the possibility to exploit other key parameters for engineering the electronic structure of
nanocrystals, such as the shape and the chemical composition. In this respect chemical approaches that employ the socalled ‘‘high temperature’’ thermal decomposition of precursors in surfactants have become a popular route to colloidal
nanoparticles [10,20]. This method is so powerful and versatile that a large fraction of nanocrystals discussed in this review
have been fabricated in this way. It is also important to note that this method, as many others, yields nanocrystals that are
stabilized in the liquid phase by means of a monolayer of organic molecules (i.e. the surfactants) bound to their surface.
These molecules in our view need to be considered as a fundamental component of the nanocrystal as they play important
roles in the various physical properties of the nanocrystal, as will be discussed extensively in this review.
Elongated (i.e. rod-shaped) nanocrystals, also known as nanorods, are probably the most studied nanocrystal systems
after the spherical ones. Other more complex shapes have been investigated too, but they will not be discussed here [90].
The typical lengths of the nanorods span from tens to some hundreds of nm, while their diameters are generally in the range
of a few nm. These nanorods can be considered as intermediate systems between quantum dots and nanowires, since the
nanorod length is typically larger than the Bohr radius. Therefore the carriers experience strong confinement only along
two dimensions, whereas they can delocalize along the long axis of the rod, giving rise to the so-called 1D exciton [91]. This
confinement leads to a variety of new properties with respect to spherical nanocrystals, in terms of electronic structure,
symmetry, polarization and carrier dynamics.
2.4.1. Electronic structure of CdSe nanorods: a case study
Among rod-shaped semiconductor nanocrystals, CdSe nanorods represent probably the most investigated samples, for
what concerns both their optical and their electronic properties. Here, we will give an overview of their electronic structure,
by highlighting some of the pioneering works in the field (for example those by Efros et al. [92–94] and by Woggon et al.
[95–98]). The concepts discussed here can be considered as of general validity for nanocrystals having hexagonal
crystallographic structure and are applicable to nanorods of a wide range of diameters and lengths. The starting point
is to consider a nanocrystal with dimensions that are much larger than its lattice constant, such that the effective mass
approximation [92] is applicable. This condition is practically fulfilled in all cases, since the nanocrystal diameter is hardly
smaller than 2–3 nm. In the case of quantum dots, the notation used to name the quantum states follows closely that of an
atomic system. We define the total angular momentum J = (L + S ) as the sum of the total orbital angular momentum L and
the multiplicity term S, the latter accounting for the electron spin. The relative momentum projections are j, l, and s. The
electron ground state has s-symmetry and presents a double degeneracy, which is due exclusively to the spin momentum.
Thus J = 0 + 1/2, its projections are j = ±1/2, and the state is conventionally indicated as 1Se . On the other hand, the
first hole level, having a p-symmetry, is fourfold degenerate, having J = 1 + 1/2 = 3/2 (j = 3/2, 1/2, −1/2, −3/2), and is
named 1S3/2 . The composition of the two ground states yields the eightfold degenerate exciton ground state 1S3/2 1Se .
Various effects intervene to lift the degeneracy of the states, namely the internal crystal field, shape effects and the
electron–hole exchange interaction. The first effect arises from an intrinsic property of semiconductors that have hexagonal
lattice structure and therefore manifests itself in both bulk and nanoscale materials. The second effect takes into account
the deviation from the ideal spherical shape of nanocrystals, while the third effect accounts for mixing of electron and hole
spins. The first two effects can be grouped together, as they arise from the intrinsic asymmetry of the material/nanocrystal
(see Fig. 2.3). The intrinsic crystal field produces a first splitting of the valence band, i.e. the lowest hole state, in the socalled Kramers doublet, which consists of two doubly degenerate states with j = ±1/2 and j = ±3/2 [92]. Let us define a
parameter β as the ratio between the mass of the light hole mlh and the mass of the heavy hole mhh (hence β = mlh /mhh ).
The energetic splitting due to the intrinsic crystal field is then expressed as [92]:
∆1 = ∆CF f (β)
(2.13)
where ∆CF is a parameter related to the crystal field (CF) splitting in a crystal with hexagonal structure, contributing to
determine the hole ground state as that having |j| = 3/2, while f (β) is a function which is unique for each material (see
Ref. [92] for details). It is worth to stress that ∆1 does not depend on the size of nanocrystal. Since f (β) is always positive,
the lowest hole level is fixed with the heavy hole state with |j| = 3/2.
In order to take the shape anisotropy into account, let us model an elongated nanocrystal as an axially symmetric
ellipsoidal particle (i.e. an ellipsoid with principal axes a = b < c), and let us define the ratio of the major to minor axes as
c /b = 1 + ε , ε being the ellipticity. The induced splitting in this case is:
∆2 = 2u(β)E1 (β)ε.
(2.14)
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 2.3. Fine structure splitting of the lowest excitonic states F = 1 and F = 2 in CdSe nanocrystals due to crystal field, shape anisotropy and exchange
interactions.
Fig. 2.4. (a) Dependence of the hole ground state function on the light-to-heavy hole mass ratio β ; (b)–(d) Dimensionless functions associated with:
(b) hole level splitting due to hexagonal lattice structure, (c) the hole splitting due to the crystal shape asymmetry, (d) the exciton splitting due to the
electron–hole-interaction.
Source: Taken with permission from Ref. [92].
Here, u(β) is a dimensionless function associated with the hole level splitting due to the crystal shape (see Fig. 2.4, for details
see Ref. [92]) and E1 (β) is the hole ground state energy which can be written as:
E1 (β) =
h̄2 ϕ 2 (β)
2mhh a2
(2.15)
where ϕ 2 (β) is a term related to the spherical Bessel functions and a is related to the nanocrystal size. For quasi-spherical
nanocrystals a = (b2 c )1/3 .
Another important point is the trend in the function u (β), in particular for what concerns its sign [92]. As shown in
Fig. 2.4(c), u (β) reverses its sign past a certain value of β , meaning that for some materials the shape anisotropy induces a
negative splitting. Since the global energy splitting is the sum of the single asymmetry contributions (∆t = ∆1 + ∆2 ), the
final result can reverse the hole ground state between |j| = 3/2 and |j| = 1/2. A negative ∆2 is found for example in CdSe
nanorods, for which β = 0.28, where a possible inversion would depend on the hole ground state energy and on the radius
of rods.
We now discuss the excitons, formed by composition of these electron–hole states, which result in two-excitonic fourfold
degenerate states, having total angular momentum F = 1 and F = 2 (see right side of Fig. 2.3). The exchange interaction
contributes to an increase of the splitting of the remaining states, defining a fine structure for nanocrystals for a series of
possible shapes, as depicted in Fig. 2.5 [92].
The relevant quantum number, in this case, is the projection f of the total angular momentum F . The fine structure
of the lowest excitonic level is composed by a distribution of states having different values of f : one state with f = ±2,
two states with f = ±1 (named Upper and Lower, depending on the branch they originate) and two others with f = 0
R. Krahne et al. / Physics Reports 501 (2011) 75–221
83
Fig. 2.5. Evolution of the band edge exciton as a function of the nanocrystal size, as calculated for four different shapes by Efros and co-workers [92].
Source: Reprinted with permission from Ref. [92].
(Upper and Lower). Only three of them are optically active, namely the states 0U , ±1U and ±1L , whereas the remaining ±2
and 0L states are passive [92]. The ±2 state is optically forbidden because photons cannot have an angular momentum ±2.
The 0L has zero optical transition probability because of an interference phenomenon between the two indistinguishable
states with zero angular momentum [87], due to the influence of the electron–hole exchange interaction.
The shape of the nanocrystal can have significant influence on which of the above states represents the exciton ground
state. For perfectly spherical nanocrystals the ±2 is the ground state, whereas in prolate nanocrystals an inversion of the
±2 with the 0L state can occur, because the state ±2 originates from the hole state with |j| = 3/2, whilst 0L arises from
the state |j| = 1/2. When the conditions for the sign change of ∆t are met, the ground state is inverted. Nanorods can be
approximated by axially symmetric prolate ellipsoids with ellipticity ε defined as ε = (2rB /b) − 1, with b being the ellipsoid
diameter and rB the Bohr radius. In the case of strong lateral confinement (b < 2rB ), the ellipsoids are subject to a possible
inversion of the ground state between ±2 and 0L . Following Eq. (2.14), the shape asymmetry can lead to a negative splitting
value ∆2 and then to a negative net splitting ∆t = ∆1 + ∆2 < 0, if the ellipsoid radius is smaller than a critical value.
This can happen because ∆2 becomes increasingly important in the strong confinement regime, and at some point it would
cause the light hole state with j = ±1/2 to become the hole ground state. The coupling with the electron state 1Se yields a
fourfold degenerate state, with angular momentum 0 (two states) and ±1. The hole state with j = ±3/2 yields the second
doubly degenerate state with momentum ±1 and ±2. In practice, the new lowest exciton level would be the state 0L and
the exciton fine structure resembles that of Fig. 2.5(c).
Woggon and co-workers [95] calculated the electronic fine structure of a single CdSe nanorod, taking into account the
typical electron–hole exchange interaction for their system (see Fig. 2.6). As expected, by varying the rod radius, two regimes
could be identified: for a radius larger than about 3.7 nm the arrangement of states reflects the well-known fine structure
of a spherical nanocrystal, i.e. the lowest energy exciton has angular momentum ±2: for smaller radii on the other hand
the exciton ground state has zero angular momentum projection along the major rod axis. This would induce a strongly
polarized emission of light, as opposed to the quasi-circularly polarized emission predicted for the ±1 excitons. Due to
the small energetic separation among these states, thermal population processes give rise to novel phenomena in terms of
polarization and carriers relaxation (as discussed later on).
2.5. Relaxation mechanisms in nanorods
The carrier mobility in nanorods has a 1D character [91], and this leads to optical properties in nanorods that are
remarkably different from those of spherical nanocrystals, among them photoluminescence (PL) lineshape and polarization,
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 2.6. Exciton fine structure of CdSe nanorods as calculated by Woggon et al. [95].
Source: Taken with permission from Ref. [95].
a
b
0.8
1.0
Absorptance (arb.u.)
Absorptance (arb.u.)
1.2
0.8
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.0
400
450
500 550 600
wavelength (nm)
650
700
400
450
500 550 600
wavelength (nm)
650
700
Fig. 2.7. (a) Optical absorption spectra of a series of CdSe nanorods with 5 nm width having different lengths of 12.5, 17.5, 25, 30, 35 nm (form (a) to (e));
(b) Optical absorption of a series of CdSe nanorods with 20 nm length having different widths of 3.5, 4, 4.25, 4.5, 5 nm (form (a) to (e)).
Source: Reprinted with permission from Ref. [99].
radiative and non-radiative transitions, and their dependence on nanocrystal size and temperature. Their understanding
represents a first step towards the exploitation of such nanostructures in optical devices. In this section we will discuss
the basic optical properties of nanorods in terms of relaxation processes and spectral features, by stressing the differences
between nanorods that are entirely made of one material (homostructures) and those made by sections of different materials
(heterostructures). We will distinguish the case of electron–hole relaxation inside the same material from that of relaxation
concerning carriers localized in two different materials. In the first case, the possible optical transition is named of type-I,
whereas in the second case is of type-II.
2.5.1. 1D excitons in homostructures
In the previous section we have described the fine structure of the ground state exciton in nanorods having a wurtzite
structure. Here we will summarize the experimental evidences collected by optical spectroscopy measurements, carried
out on single nanorods and on ensembles of nanorods, and will highlight the differences of nanocrystals with respect to the
more traditional spherical nanocrystals. In nanorods, two confinement regimes can coexist, causing the transition energy
to vary mainly as a function of the lateral confinement instead of the rod length. In Fig. 2.7 the effect is clearly visible from
the absorption spectra of a series of CdSe nanorods of different diameters and lengths [99]. Two sets of five samples were
analyzed in the study (Fig. 2.7). In one set (Fig. 2.7(a)), the nanorods had different lengths (12.5, 17.5, 25, 30 and 35 nm, from
a to e) and comparable diameters (around 5 nm), while in the other set (Fig. 2.7(a)), the nanorods had different diameters
(3.5, 4, 4.25, 4.5 and 5 nm from (a) to (e)) and comparable lengths (around 20 nm). By changing the nanorod length there
was no appreciable shift of the optical transitions, whereas by increasing the radius an evident red-shift of the absorption
peaks was observable.
Optical spectroscopy combined with scanning tunneling spectroscopy (STS) measurements, performed by Banin, Millo
and co-workers [100,101] and by El-Sayed and co-workers [99] have revealed a manifold of degenerate states which are
split by crystal and shape anisotropy and by the electron–hole exchange interaction (see Fig. 2.8). These works anticipated
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
a
b
5
4
Energy (eV)
Intensity (arb. units)
3
2
0.0
–0.2
–0.4
–0.6
–0.8
0.0
0.4
0.7
1.1
E-Edet(eV)
1.4
1.5
2.0 2.5 3.0
Radius (nm)
3.5
Fig. 2.8. (a) Photoluminescence excitation (PLE) spectra of CdSe nanorods of different sizes (corresponding length × diameter is indicated in brackets),
showing peaks related to the first exciton transitions. It is evident that the diameter has a strong influence on the exciton level structure, whereas nanorods
with different lengths but comparable diameter show similar spectra. The detection energy Edet was 2.25 eV for spectra a and b, and 2.03 eV for spectra c
and d. (b) Conduction and valence band states distribution as function of the rod radius.
Source: Taken with permission from Ref. [100].
the subsequent theoretical and experimental results on the exciton fine structure reported by Woggon and co-workers [95]
(the reader can refer to Section 3 of this review for a more detailed discussion on the STS experiments).
Nanorod homostructures often show only one emission peak [139], corresponding to carrier relaxation from the lowest
exciton state, similar to spherical nanocrystals. At room temperature, the PL broadening is around 70–100 meV for a sample
of rods with narrow distributions of lengths and diameters. Such broadening, as in spherical quantum dots, is the result
of two contributions, namely inhomogeneous and homogeneous broadening [92]. Inhomogeneous broadening is mainly
affected by the general quality of the sample and is related to the size dispersion, thickness fluctuations in single rods,
quality of the rod surface, and it can be controlled (in principle) by improving these parameters. This type of broadening
represents often an obstacle for the study of the intrinsic physical properties of semiconductor nanorods, and it can be
bypassed only by performing experiments on single rods. Homogeneous broadening refers instead to coupling of charge
carriers to phonons, is an intrinsic process and cannot be overcome by any improvement in synthesis. It will be discussed
in detail in the following.
2.5.2. Temperature and size dependence of the exciton relaxation processes
After excitation, charge carriers inside a crystal tend to restore the initial state of equilibrium by losing their excess energy.
As in all semiconductor systems, semiconductor nanorods undergo several possible relaxation pathways. In the present
discussion, we separate such processes in two main categories, namely radiative processes (i.e. loss of energy by means of
the emission of photons) and non-radiative processes (i.e. loss of energy via any other mechanism). The most common nonradiative relaxation process is the creation of phonons (i.e. lattice vibrations) via carrier–phonon interaction, which leads
to heating of the material. This coupling does not influence exclusively the decay of the carries towards lower energetic
states, since in some particular cases it can promote the carriers into higher energy states, allowing them to escape from
the material, by means of a process known as ‘‘thermal escape’’ [102–104]. Multicarrier scattering processes are additional
non-radiative relaxation processes, and they are all generally grouped in a class, the so-called ‘‘Auger-type’’ mechanisms
[105–114]. They manifest when two or more electron–hole (e–h) pairs are present in the semiconductor, and consist in the
transfer of the energy of an e–h pair to a third particle (electron, hole or exciton).
2.5.3. Exciton–phonon interaction: homogeneous broadening
The main features that distinguish nanostructures from bulk materials originate from the localized character of the
electron and hole wavefunction and the discrete nature of their optical transitions. In the bulk, the broadening of the exciton
line is mainly determined by the polar coupling of both electrons and holes to optical phonons [115]. The piezoelectric and
deformation potential coupling of both carriers to acoustic phonons is usually not very important [116]. This situation is
different in nanostructures, where the local charge neutrality character of the exciton becomes predominant, producing an
ideal null polar coupling of the exciton to optical phonons [117]. This holds true for infinite barriers, such that the electron
and hole wavefunctions are practically identical. In general, in real systems (finite barrier) a decrease of the polar coupling
with increasing barrier is expected [117–119]. On the other hand, since the deformation potential coupling is proportional
to 1/R2 (R being the radius of a spherical dot), the coupling strength to acoustic phonons is increased as the dimensions
are reduced below the Bohr radius [116,120]. Also the temperature affects each of these contributions. The temperature
dependence of the spectral line width can be expressed as [121]:
Γ (T ) = Γ0 + σ T + γ NLO (T )
(2.16)
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 2.9. Size dependence of the ratio between the second and the first order phonon replica of CdSe/ZnS nanorods. The solid line is a guide to the eye.
Source: Taken with permission from Ref. [122].
where Γ0 is the inhomogeneous broadening, σ is the exciton–acoustic phonons coupling coefficient, γ is the coefficient
accounting for the exciton–optical (LO) phonon coupling and NLO represents the Bose function for LO phonon occupation:
1
.
(2.17)
eELO /kB T − 1
In the expression above ELO is the energy of the longitudinal optical phonon with momentum k = 0 (i.e. the phonon that
preferably couples to the lowest exciton state) and kB is the Boltzmann’s constant. Due to the different energetic dispersions
curves of acoustic and optical phonons, the two couplings dominate at different temperature ranges. The acoustic phonons,
having smaller energies (a few meV), contribute heavily to the broadening at low temperature (until 50–70 K [103]), whereas
the optical phonons (with energies of a few tens of meV) dominate at higher temperature [103]. Nanorods having wurtzite
structure, like CdS and CdSe, present a pronounced polar character, where the Fröhlich interaction (with optical phonons) is
the predominant coupling action between charge carriers and optical phonons [122]. In the study of the exciton–LO phonon
interaction, the fundamental quantity considered is the so-called Huang–Rhys factor [118]. Experimentally, an estimation of
the Huang–Rhys factor is possible by Raman spectroscopy, by simply considering the ratio of the signal intensity of the first
to the zero order phonon replica [122]. Lange and co-workers [122] have performed Raman studies on CdSe nanorods and
demonstrated an increase of the exciton–phonon coupling with decreasing rod radius, and the effect could be enhanced by
growing a ZnS shell on the surface of nanocrystals acting as passivating layer (see Fig. 2.9). The Huang–Rhys factor remained
below the bulk value, according to the model of Nomura et al. [118].
Quantum confinement affects not only the coupling strength of charge carriers with phonons, but also the energy of
the LO phonons. This happens in quantum dots as well as in nanorods. Cardona and co-workers [123] have determined
the dispersion of the LO phonons as function of the spherical nanocrystals size on the basis of the asymmetric lineshape of
Raman spectra (Fig. 2.10). They found a decrease of the energy of the optical modes with decreasing size, and explained the
low energy asymmetry of Raman spectra as due to the contribution of off-resonance vibronic modes.
In nanorods, the red-shift of the phonon modes becomes more pronounced, for example due to the loss of spherical
symmetry which leads to the detection of vibrational modes with non-zero angular momentum. Lange et al. [122,124] and
Nobile et al. [125] have conducted independent studies (summarized in Fig. 2.11) on the role of quantum confinement on
the optical phonon energies. They found a red-shift of the LO phonon energy with decreasing rod diameter, accompanied
by a substantial independence on the aspect ratio. This red-shift can be understood in a simple picture by considering the
negative dispersion of the optical phonon branches with respect to the transferred wave vector k, where k = 2π /d, d being
the diameter of the nanorod. Detailed theoretical calculations of the confined phonon modes in high aspect ratio ellipsoids,
supported by Raman experiments on CdSe nanorods, have resolved a fine structure of the phonon modes perpendicular and
parallel to the long axis of the nanocrystals (Fig. 2.11). The energy of surface optical (SO) phonon modes can be approximated
following the approach by Gupta et al. [126]:
NLO =
2
2
ωSO
= ωTO
+
with:
ϖp2
ε + εm f (x)
2
2
ϖp2 = ε∞ (ωLO
− ωTO
),
;
x = q · d/2
q = 2π /L,
(2.18)
f (x) = (I0 (x) · K1 (x))/(I1 (x) · K0 (x))
(2.19)
where I and K are Bessel functions. Therefore the SO phonon energy depends on both the length L and the diameter d of the
nanorods [125–127].
R. Krahne et al. / Physics Reports 501 (2011) 75–221
87
Fig. 2.10. Dependence of the first four optical-vibronic lp = 0 modes on the CdSe dot radius.
Source: Taken with permission from Ref. [123].
2.5.4. Auger effects in nanorods
Multicarrier Auger effects play a fundamental role in the relaxation processes of nanostructures. Auger recombination
represents a limitation for the use of nanostructures in optical devices requiring optical gain, since non-radiative multiparticle scattering hinders the desired population inversion. However, the Auger efficiency can be significantly reduced
by a wise choice of materials and via engineering of the band alignments and wavefunction localization [2,130,131]. The
physical interest on the Auger processes resides in the role played by the quantum confinement on the assignment of
the new rules and the Auger efficiency in nanoparticles with respect to the bulk. Quantum confinement increases the
Coulomb interaction between the charge carriers inside the nanomaterial with respect to the bulk counterpart. Moreover,
the translation momentum conservation acting in bulk materials is broken in nanoparticles, allowing for more efficient
Auger processes [105,132,133]. On the other hand, the discreteness of electronic states in nanocrystals should imply a
reduced availability of states that can satisfy the energy conservation condition, whereas in the bulk a continuum of states
is always guaranteed (in this sense, Auger recombination would be more efficient in the bulk). The interplay of both
contributions determines the net effect in each system. In this regard, several differences emerge between nanorods and
spherical nanoparticles.
Since the Auger recombination rate is strongly sensitive to the carrier density and consists in e–h pair annihilation, such
a mechanism affects the temporal trend of the relaxation processes. Therefore, the general procedure for studying Auger
mechanisms occurring in nanoparticles consists in time resolved (TR) measurements of both the PL (TRPL) and the transient
absorption (TA) as function of the photogenerated carrier density. The simplest way to analytically describe an Auger process
is to consider the following equation [114]:
dn
= −CA (M )nM .
(2.20)
dt
This equation expresses the decay rate of the population density n in the presence of M interacting particles (intended as
electrons, holes or excitons). The key parameter is the Auger constant CA (M ) which corresponds to the actual probability
for the Auger process to occur. It is dependent on the number of interacting carriers in the system considered. The general
solution of Eq. (2.20) is:
n( t ) =
1+
n(0)
1
M −1
CA (M )n(0)M −1 t
M −1 .
(2.21)
This expression is employed to extract the Auger constant by means of a fitting procedure of n(t ), which is given by the
measured temporal decay; the quantity n(0) is usually known, while M is often kept as parameter. M becomes a key
parameter if one wants to determine in which regime Auger mechanisms take place. In quantum confined systems, there are
two main types of mechanisms: three-carrier relaxation and biexcitonic Auger relaxation. The first occurs is the strong 0D
confinement regime and provides M = 3, whereas the second gives M = 2 and occurs in the case of a 1D system. In strongly
confined 0D nanocrystals the Coulomb contribution to the exciton energy is negligible with respect to the confinement
term (see discussion at the beginning of this section). This makes it possible to consider the charge carriers in the dot as
independent particles, thus producing a cubic behavior in Eq. (2.20) [105]. The minimum number of carriers allowing Auger
relaxation is then M = 3, i.e. the energy possessed by an e–h pair is transferred to a third carrier. In the case of nanorods,
e–h pairs give rise to 1D excitons, since the exciton extends along the long axis of the nanorods (see Fig. 2.12). Along that
direction, the confinement energy can have a smaller contribution with respect to the Coulomb interaction, allowing for
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 2.11. (a) Raman spectra showing the LO phonon mode excited with light polarized parallel and perpendicularly with respect to the long axis of the
nanorods. (b) Cross sections of the calculated phonon potential for the LO Raman-active modes with M = 0 and different angular momentum L. (a) and (b)
are taken for Ref. [128]. (c) Calculated energy of the different phonon modes vs. the rod aspect ratio. The red and blue lines are calculated from the model
in Ref. [128], the black line shows the dispersion of the SO modes according to the model of Gupta et al., and the green dot are data points from Ref. [129].
(d) Frequency of the LO phonons as a function of rod diameter (from Ref. [124]).
Fig. 2.12. Sketch of the exciton extension in spherical (left image) and rod-shaped (right image) nanocrystals and relative implications in Auger processes.
In spherical nanocrystals the interacting charge carriers can be considered independent, giving M = 3, whereas in nanorods four carriers (two e–h pairs)
can be considered as two interacting particles (two excitons), giving M = 2.
the formation of a 1D exciton. Therefore, in terms of Auger mechanism, one must consider two excitons as the minimum
number of interacting particles, thus M = 2, as one exciton can transfer its energy only to a second exciton.
Comparative studies carried out on spherical nanocrystals and on nanorods with various aspect ratios have provided
an estimate of the size ranges of the two regimes and of the relevant Auger constants. Klimov and co-workers have
demonstrated that below an aspect ratio AR = 8 charge carriers in nanorods behave as uncorrelated particles, whereas
above this value e–h pairs organize themselves in 1D excitons and recombine via a biexcitonic Auger process [113]. The
Auger constant that gives a measure of the Auger efficiency in systems with the same volume was found to be larger in
nanorods than in spherical nanocrystals of equal volume. In Fig. 2.13 we show this result in terms of Auger lifetime (which
is inversely proportional to the Auger constant), as derived from Ref. [113]. This behavior has been ascribed to the major
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
τ2 (ps)
400
300
200
Dots
100
Rods
100
200
300
400
500
Nanocrystal volume (nm3)
600
700
Fig. 2.13. Auger time constant of two e–h pairs recombining in dots (blue symbols) and rods (red symbols), as a function of the nanocrystal volume. (For
interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Source: Taken with permission from Ref. [113].
Fig. 2.14. Representation of two different rods grown along a parallel (a) and a tilted (b) direction with respect to the c-axis of the wurtzite structure. In
the first case, the rod is a perfect cylinder, and no polar facets are created at the lateral sides of the rod. In the second case, the rod has a more irregular
shape, which can lead to exposure of tiny polar facets at the lateral sides of the rod.
surface contribution to the matrix element of the Auger transition: nanorods have a larger surface area when compared to
spherical nanocrystals of equal volume, and this should lead to a larger Auger constant [113].
The surface plays also a nontrivial role in defining the optical properties of nanorods, especially for what concerns the
existence of surface states and their influence on the carrier dynamics. The efficiency of the Auger processes may be affected
by means of trapping at such states [133]. This has important implications on the carrier cooling from higher excited states
to the band edge, which depends on the Auger efficiency, as demonstrated by Klimov and co-workers [112]. In their study,
they showed that in nanorods multicarrier generation can cause an Auger heating, consisting in the increase of the energy
of an exciton, as a consequence of an energy transfer event deriving from a second exciton recombination. The consequent
increase in carrier temperature competes with the more classic phonon emission as a relaxation channel, and decreases the
carrier cooling in the high excitation density regime.
2.6. Single nanorod properties
As said before, CdSe represents perhaps the most studied system among the spherical and elongated semiconductor
nanoparticles. They are commonly considered as prototype materials in the study of the optical properties of single
nanoparticles with wurtzite structure. Lorentzian lineshape, spectral diffusion, and intermittent emission (called ‘‘blinking’’)
are some of the properties shared by single spherical and elongated nanocrystals [96,134,135]. Defects states can induce
charge accumulation by classical carrier trapping at defects lying at energies lower than the band gap, which occurs following
an excitation. Irregularities in the nanocrystal shape can lead to charge storing in absence of an excitation, if a distribution of
a permanent surface charge can be created. This happens when the rod does not extend perfectly along the wurtzite c-axis,
but is twisted with respect to such axis. This can be realized if some stacks of unit cells along the c axis do not span the
whole length of the rods, thus creating tiny polar facets along the lateral sides of the rod (see Fig. 2.14). In such a situation,
as observed by Krishnan et al. [135], a net surface charge density is found at the surface of the rod. The exposed cells are
characterized by uncompensated charges, giving rise to an accumulation of a density of charge. The propensity of charges
to distribute easily on the rod surface influences strongly the optical properties of single nanorods, especially for what
concerns the emission energy and its intensity. For single nanorod experiments, major efforts have been made towards the
comprehension of both the role of the surface on the localization and recombination of carriers, and on its impact on the
polarization of emission and on the exciton fine structure.
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Fig. 2.15. Two single CdSe rod spectra detected at 15 K. The upper part corresponds to a charged exciton spectrum, showing fine structure due to excess
charge, whereas the bottom spectrum refers to a neutral exciton, showing peaks (labeled as S0 and S1 ) belonging to the rod fine structure. Inset: statistics
on 30 single nanorods.
Source: Taken with permission from Ref. [95].
Single nanorod spectra present more complicated dynamics in the temporal evolution of the emission energy, line shape,
and intensity, when compared to spherical nanocrystals. Under continuous wave excitation, nanorods show the typical
intermittent behavior, as observed in other well-studied systems. At low temperature, single rod spectra are characterized
by a broadening of the order of a few meV, and have in general an asymmetric line shape [95,96,136]. This asymmetry, which
is observable only at temperatures around 10–15 K, can be ascribed to several reasons. These include charged excitons,
fine structure of the lowest excitonic state, acoustic phonon contributions and influence of internal electric fields. In the
following, we will discuss the single contributions of phonons, charged exciton, as well as internal and external electric
fields, on the spectral features of single rods.
2.6.1. Charged/neutral excitons and phonon contributions
The role of charged excitons on the spectral features has been studied extensively by means of PL measurements at
low temperature, which were performed as a function of the excitation density [96]. The fundamental difference between
charged and neutral exciton spectra lies on the effect of the excess of charge on the fine structure of the lowest excitonic
state. For neutral excitons a double peak emission usually arises from single nanorods at low temperature (∼5 K), and is
assigned to intrinsic bright states. This feature vanishes at higher temperature, as thermal population contributes to a state
mixing (see Fig. 2.15). The spectral line shape of Fig. 2.15 can be explained by the theoretical fine structure, as discussed
by Woggon et al. in Ref. [95]. As shown in Section 2.4 (Fig. 2.6) nanorods experience an inversion of the ground state fine
structure below a critical rod radius of ∼3.7 nm, which induces a change in the effective ground state of the rod.
Nanocrystals with spherical symmetry have a fixed dark ground state that is characterized by a momentum projection of
±2. Nanorods with radius smaller than 3.7 nm have instead a ground state with projection 0 (dark, labeled 0L , contrary to the
higher energy, bright 0U state). Despite the dark nature of the state 0L , its proximity to the nearest bright state (labeled ±1)
is of the order of some µeV, which allows the thermal population of the bright state at very low temperature. Consequently,
it is common to assign the dual nature of the single rod spectra (the peaks S1 and S0 of Fig. 2.15) to the emission of the lowest
bright states ±1 and 0U . PL spectra showing the fine structured exciton ground state are considered a fingerprint of neutral
exciton emission. For a charged exciton (trion) the fine structure is expected to vanish due to the excess charge present in
the rod [96]. The charged exciton peak is shifted with respect to the neutral exciton due to Coulomb interaction in the trion
system. In addition, at high excitation density also the recombination from biexcitons (XX) can be observed (see Fig. 2.16).
2.6.2. Influence of electric fields on the optical properties of single nanorods
Stark effect experiments, i.e. the study of the impact of an external electric field on the electronic structure, represent
the most common approach to study and control the carrier localization along the rod. Fluctuations in the emission energy,
PL intensity and broadening are among the main parameters that characterize the optical properties of nanorods [137].
Fig. 2.17 shows typical time traces from two different nanorods belonging to the same sample of CdSe nanorods, measured
at 10 K, under CW excitation and without external electric field [137].
At first glance, the two rods have dissimilar emission energies, broadening and relative temporal evolutions. In particular,
rod A shows a very stable behavior, consisting of an unmodified PL spectrum over time, whereas rod B shows considerable
instability, as if it underwent mutating conditions during its emission process. Systematic studies carried out on large sets of
nanorods [17] and by continuous monitoring of the same nanorod, have indicated a close correlation between the emission
energy, the broadening and the intensity of the PL signal. As a general trend, a red-shift of the emission energy corresponds
to an increase of the full width at half maximum (FWHM) and a decrease of the emission intensity (Fig. 2.18) [137].
R. Krahne et al. / Physics Reports 501 (2011) 75–221
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Fig. 2.16. Power dependence of a single rod emission for a neutral exciton (a) and a charged exciton (b).
Source: Taken with permission from Ref. [96].
Fig. 2.17. Temporal evolution of the PL spectra of two representative CdSe nanorods.
Source: Taken with permission from Ref. [137].
Interestingly, such a behavior can also be induced by applying an external electric field, as shown in Fig. 2.19. Here, a
bimodal switching effect is seen when an electric field is applied to the nanorods, which either turns off the PL of some rods
or turns on the PL of some others. This behavior can be understood by considering the effect of the surface and the elongated
shape of the rods, which can lead to a varying degree of overlap of the electron–hole wavefunctions. Such variations can have
either extrinsic or intrinsic sources. Extrinsic sources are for example external electric fields, whereas intrinsic sources could
be related to the dynamics of carriers from/to surface trap states, which induce the formation of varying local fields. If on one
hand the elongated shape offers the possibility to localize the e–h wave functions in several configurations, the extended
surface of the nanorods contributes heavily to the creation of numerous trap sites, where carriers can be localized.
When nanocrystals are subject to an external electric field, they experience the so-called quantum confined Stark effect
(QCSE), which induces a red-shift of the optical transitions (both in absorption and emission), a decrease of the transition
probabilities [137] and an increase of the spectral broadening. As an example, Fig. 2.20 provides a qualitative description of
the exciton distortion induced by the application of an electric field. For nanorods aligned along the electric field direction,
electrons and holes are pulled in opposite directions, towards the ends of the rods. The result is that their wavefunction overlap is reduced, and the optical transition red-shifts and decreases in intensity (due to a decrease in the oscillator strength).
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Fig. 2.18. Statistical analysis of the emission properties of three single nanorods subject only to local field effects. Left panels show the integrated PL
intensity reduction with increasing red-shift. Right panels correlate the spectral broadening to the energetic shift: a red-shift corresponds to an increased
FWHM.
Source: Taken with permission from Ref. [137].
Electric Field (kV/cm)
0
-175
-350
-175
0
2.00 2.02 2.04 2.06
2.00 2.02 2.04 2.06 2.08
Energy (eV)
1.95
1.98
2.01
2.04
Fig. 2.19. Switching of the PL in single nanorods due to external electric fields.
Source: Taken with permission from Ref. [137].
The process depends strongly on the mutual orientations of the electric field and of the long axis of the rod: if they are
parallel, the effect is maximized, whereas if they are perpendicular the influence of the electric field is negligible.
For what concerns the correlation of the broadening of the emission peak with the PL intensity and the emission energy
we refer to the treatment of Ref. [137]. The starting point is to consider the energetic shift caused by an electric field up to
the second order:
∆E = µε +
1
2
αε 2 .
(2.22)
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Fig. 2.20. Effects of the local electric field on the exciton extension along the rod, leading to a reduction of e–h overlap.
Source: Taken with permission from Ref. [137].
Fig. 2.21. Dependence of the polarization factor (P) and of the optical transition strengths along the nanorod axis (Iz ) and perpendicular to the nanorod
axis (Ix ) on the aspect ratio of the nanorods (e).
Source: Taken with permission from Ref. [138].
Here, ∆E is the energy shift, ε is the electric field (due to external or local fields), µ the dipole moment and α the
polarizability. By differentiating Eq. (2.22) one can obtain the expression for the predicted change in the spectral width:
δ(∆E ) = (µ + αε)δε
(2.23)
µ and α can be found experimentally by Stark effect measurements. δε, if referred to fluctuations of local charges, is related
to the distance of such charges from the exciton and then to the spatial fluctuation [137].
2.6.3. Polarization properties
The polarization properties of nanorods represent one of their distinctive characteristics with respect to other colloidal
nanocrystals, like for example spherical quantum dots, and have received much attention recently (both from the theoretical
and experimental point of view). As previously discussed, the electronic structure of rods is strongly affected by their size
and shape, in particular by their aspect ratio (AR). The fine structure of the lowest exciton state is the dominating factor in
defining not only the character of the emitting state (whether dark or bright) but also the polarization properties. Since we
deal here with nanostructures with AR > 1, only the component of the angular momentum having projections along the
z long axis of the rod should be considered [95,138]. In nanorods, an inversion of the hole ground state occurs at a critical
radius R = 2.1 nm according to Li et al. [138], and at 3.7 nm following Woggon et al. [95], as discussed before.
Here, the optical transition should be strongly polarized along the wurtzite c-axis (axis z in Fig. 2.21), with an increasing
polarization degree with increasing AR. Polarization spectroscopy on single nanorods provides experimental proof of the
theoretical predictions. Polarized emission up to 80% has been demonstrated for CdSe nanorods with AR > 2 [139] whereas
for spherical nanocrystals only values up to 10% were obtained, see Fig. 2.22.
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Fig. 2.22. (A) Imaging of the emission of a single rod recorded along mutually perpendicular polarization directions simultaneously; (B) Plot of the
polarization degree as a function of the rotation angle of the nanorods in (A) (solid circles) and the polarization for a single spherical nanocrystal (triangles);
(C) Experimental polarization factor as a function of the aspect ratio of the rods.
Source: Taken with permission from Ref. [139].
Excitation polarization measurements have demonstrated the optical transition polarization of nanorods both in
one [140] and two-photon excitation [141]. Fig. 2.23 shows results of one- and two-photon excitation experiments. In the
one-photon experiment, the polarization curve followed the typical cos2 (Θ ) function, whereas a dependence on cos4 (Θ )
was found in two-photon excitation, according to the nonlinear character of the two-photon absorption process. The
level crossover occurring in nanorods and proposed by various theoretical approaches [95,139] allows to understand the
polarization dependence of the nanorods excitation. Indeed, the ground state becomes bright in rods, allowing for the
absorption of polarized light along the major c-axis.
Experimental [142] and theoretical [143] studies on free-standing InP nanowires indicated that the polarization
properties of the band edge transition in one-dimensional semiconductor nanostructures depend both on the intrinsic
level structure, which shows p-type wavefunction characteristics, and on the geometry of the dielectric environment of
a cylindrical structure where the diameter is much smaller than the wavelength of the light. In this case, the light wave
component along the wire is not reduced, while the component perpendicular to the wire is strongly attenuated. The
combination of these two effects can explain the giant polarization of the band edge transition, of up to 96%, that was
observed in InP nanowires. Temperature dependent polarization measurements can help to distinguish between the band
structure and the geometrical contributions, as the first should depend on temperature, while the second should not.
2.7. Nanorod heterostructures
We will now discuss different types of nanorod heterostructures and their respective optical properties, as well as their
potential for device applications. Our main focus will be on semiconductor heterostructures consisting of materials with
different band gaps, and we will briefly mention also semiconductor–metal heterostructures. Many excellent reviews have
appeared recently on this topic [144–147], and the reader is directed to those reviews for more in-depth coverage. Fig. 2.24(a)
shows some of the nanorod heterostructures that have been reported so far by various groups, while Fig. 2.24(b) shows two
possibilities of band alignment for semiconductor heterostructures, namely type-I and type-II. In type-I heterostructures, the
optically excited carriers are confined in the lower band gap material, whereas in type-II heterostructures the electrons are
localized in one material, and the holes in the other material. As we will see shortly, for some materials the band alignment
can be tuned from a type-I to quasi-type-II, where the holes are localized in one material and the electrons are delocalized
over the whole nanostructure, via quantum confinement effects. In many cases, the appropriate choice of materials and
the geometry allows to obtain suitable energy gaps for the targeted application. Among the rod-like type-II nanostructures
that have been fabricated so far, we find dot-in-a-rod structures [17,148,149], collinear nanorods [150–156], nanobarbells
[157,158] and magnetic-semiconductor nanorods [159,160].
2.7.1. Semiconductor dot/rod heterostructures
This particular type of nanorod heterostructure is characterized by a spherical core of material A and a rod-shaped shell of
material B, as illustrated in Fig. 2.25 for two different material combinations that result in type-I and type-II band alignment,
respectively. In the type-I structure, electrons and holes are localized in the same spatial region, namely the core region,
before recombination, because both the lowest electron state and the highest hole state reside in the same material. In
several reported cases, however, the electrons are delocalized over the whole rod volume due to the small conduction band
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Fig. 2.23. (a) Polarized excitation measurements performed on two single nanorods (symbols) and relative best fit curves (continuous lines) to the cos2 (Θ )
function of the excitation angle. Taken with permission from Ref. [140]; (b) Polarized excitation experiment carried out by two-photon excitation. Symbols
represent the experimental data, whereas continuous lines are the best fit curves to the cos4 (Θ ) function. Taken with permission from Ref. [141]; (c) AFM
image of InP nanowires (scale bar = 5 µm) (A), and PL emission intensity of a single InP nanowire recorded with exciting light polarization parallel (B) and
perpendicular (C) to the nanowire. The scale bar is 3 µm long; (d) Dielectric constant model for light polarization parallel and perpendicular to a cylinder.
(c) and (d) are taken with permission from Ref. [142].
offset [16–18]. The situation is different for type-II structures, where the minimum of the conduction band and the maximum
of the valence band are located in different materials regions of the system. This makes a type-II transition from the minimum
of the conduction band in material A (rod-shell) to the maximum of the valence band in material B (core) possible.
In the following, we will focus on the case of CdSe/CdS dot/rod system sketched in Fig. 2.25 (left panel), where a spherical
CdSe core is embedded in an outer CdS rod. Typical absorption and emission spectra of such nanorods are displayed in
Fig. 2.26 [161,162]. We find that the absorption is dominated by the CdS shell, whereas the emission originates from states
related to the CdSe core. This behavior is clearly advantageous for the optical emission properties, because a large 1D system
(i.e. the entire rod) is capable of harvesting the excitation light, whereas the emission comes from a small 0D system with a
well-defined level structure, i.e. the CdSe core.
In order to better interpret the experimental data, theoretical absorption spectra were calculated using the envelope
function approximation [149], which allowed to assign the strong onset in absorption at higher energy to the lowest
excitonic transition of CdS, while the much smaller peak at lower energy could be assigned to the transition occurring in the
CdSe core. The emission peak was slightly red-shifted with respect to the lowest CdSe absorption, suggesting a direct origin
from the core. The photoluminescence excitation (PLE) measurements on two representative samples of nanorods having
different aspect ratios (6 and 19, respectively, see Fig. 2.27) confirmed this result [149]. The narrow line width of the peaks
in PLE at low temperature made it possible to resolve the phonon replica of the PL emission. Here, the energy shift of the
phonon replica was 26 meV, which corresponds to the LO phonon energy in CdSe and therefore provided a clear proof that
the emission originated from the core material.
2.7.2. Electron–hole dynamics in core–shell nanorods
Pump–probe transient absorption measurements have been performed to study the carrier dynamics in core–shell
nanorods [163]. In such experiments the sample is excited by a short pulse at a specific wavelength, and then the relative
change in transmission ∆T /T is measured at defined delay intervals (normalized by the transmission without prior
excitation). The normalized relative transmission reflects the population of the electronic levels caused by the excitation
pulse. Such pump–probe experiments have revealed the role of surface and interface states in core–shell nanorods [163]. As
an example, a recent work studied the ultrafast electron–hole dynamics of CdSe (spherical core)/CdS (rod-shell) nanocrystals
and pointed to electron delocalization even for the low energy transitions in these systems [161]. Fig. 2.28 shows ∆T /T
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Fig. 2.24. (a) Examples of heterostructured nanorods: dot/rod, Co-tipped, collinear nanorods and nanobarbells; (b) Band alignment of type-I and type-II
heterostructures.
Fig. 2.25. Sketches of two possible nanorods heterostructures that have been actually synthesized by wet-chemical approaches, and relative band
alignment and e–h wavefunction localization.
Source: Taken with permission from Ref. [148].
spectra at different time delays for two excitation energies: high energy at 390 nm exciting carrier in the CdS shell, and low
energy at 590 nm exciting carriers in the CdSe core. At high excitation energy, a strong instantaneous photobleaching for
the levels associated with the shell is observed (at around 460 nm), followed by a much slower build up of a peak at 590 nm
due to carrier relaxation into the core. However, in the case of excitation of carriers in the core levels, also an instantaneous
photobleaching at 460 nm is observed. This indicates bleaching of the electronic states of the shell, due to the delocalization
of the electrons in the conduction band. As we will see later on, many optical experiments can be explained assuming
electron delocalization in such nanostructures.
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Fig. 2.26. Absorption (blue) and photoluminescence (red) spectra of core–shell nanorods. The absorption is dominated by the nanorod shell due to its
much larger volume with respect to the core. The emission peak is slightly red-shifted with respect to the first absorption peak related to transitions of the
core. The dashed line shows the absorption of CdS only nanorods, and the inset displays the absorption and emission of spherical CdSe nanocrystals that
were used as seeds for the core–shell rod synthesis. (For interpretation of the references to colour in this figure legend, the reader is referred to the web
version of this article.)
Source: Taken with permission from Ref. [161].
Fig. 2.27. PL (black lines) and PLE (red lines) spectra of two representative samples of CdSe/CdS nanorods having aspect ratios equal to 19 (top) and
9 (bottom), as recorded at 10 K. The detection energies are indicated by the arrows on the blue side of the PL. The lowest resonances in the PLE signal
correspond to the ground state resonance (a) and its phonon replica (b). (For interpretation of the references to colour in this figure legend, the reader is
referred to the web version of this article.)
Source: Reprinted with permission from Ref. [149].
2.7.3. Photoluminescence studies on single core–shell nanorods
Mueller et al. studied the optical properties of single nanorods prepared via the classical wet-synthesis method and
determined the role of surface charges on the spectral position, intensity and broadening [16]. Fig. 2.29(a) shows two PL
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Fig. 2.28. Normalized transient absorption spectra recorded from core–shell nanorods fabricated by the seeded growth method; (a) Excitation energy
corresponding to transitions in the CdS shell, (b) and to transitions in the CdSe core.
Source: Taken with permission from Ref. [161].
spectra recorded from the same core–shell nanorod at different times, manifesting fluctuations of the PL peak over time.
They could correlate this spectral jitter to fluctuations of the charges on the surface of the nanorods [16]. In nanorods, in
contrast to spherically symmetric nanostructures, these fluctuations can strongly influence the electronic level structure
via the quantum confined Stark effect (QCSE). In detail, when the surface charges are close to the core, where the emission
occurs, the PL peak is red-shifted and broadened due to the Stark effect, and when the surface charges are localized at the
far end with respect to the core, the PL occurs at higher energy and with a sharper line width (Fig. 2.29(b)–(c)).
These findings point directly to the possibility to tailor the wavefunction distribution by the aspect ratio of the nanorods,
which was further discussed by Mueller et al. [17]. In addition to the aspect ratio, they demonstrated that also external
electric fields can be used to tune the nanorod emission intensity and wavelength [17,19]. Also the Förster energy transfer
between such nanorods and organic dye molecules (Fig. 2.29(d)–(e)) could be tuned by external electric fields in a way that
the emission originated from the nanorod at zero electric field, and from the organic dye when an electric field was applied.
Controlled switching in between these two states is shown in Fig. 2.29(f)–(g). In these core–shell nanorods, the wavefunction
distributions of the electrons and holes could be modified strongly by an external electric field, from a configuration with
large overlap of electrons and holes in the core region, to a spatial separation where the holes were localized in the core
and the electrons forced to the opposite tip by the electric field. In the latter case, the emission intensity was largely
reduced due to the small spatial overlap of the electron and hole wavefunctions, and the emission was red-shifted due
to the band bending, as shown in Fig. 2.30(a). Indeed, Kraus et al. [19] showed that exciton emission could be quenched by
the electric field and that it could be recovered after microsecond time scales by switching the electric field off, as shown in
Fig. 2.30(c)–(e).
The screening of the Stark effect by high intensity excitation was studied by Morello et al. [165]. Consecutive PL
acquisitions on single core–shell nanorods fabricated by the seeded growth method showed only small fluctuations of the
energetic position (of the order of a few meV) due to the spectral diffusion, which was indicative of a low level of surface
charge fluctuations. However, an increase in the excitation density resulted in a systematic blue-shift of the emission peak,
which varied in the range from 15 to 35 meV for different rods (see Fig. 2.31(c)). The reversibility of the experiment implied
that this shift could be considered as an intrinsic property, rather than due to some photodegradation process. Fig. 2.31(d)
shows power-dependent PL positions of an ensemble of nanorods taken at different temporal delays. Here, 2 µJ/cm2 refers
to an excitation density corresponding to a low excitation regime (less than one photogenerated e–h pair per rod), whereas
at higher densities the system is in the multiexciton generation regime. A noticeable dynamical red-shift occurred in the
multiexcitonic regime (with a maximum shift of 25 meV measured at excitation density of 107 µJ/cm2 ), in contrast to the
almost constant temporal PL position featuring the low density excitation measurement.
A transient red-shift resulting from the photo-induced dynamical de-screening of an internal electric field is a commonly
observed process in a number of systems exhibiting internal electric fields. For example, in GaN-based systems [166,167]
the piezoelectric nature of the material gives rise to a polarization that originates from the lattice strain at the GaN/AlN
interface. We note that the lattice mismatch at the GaN/AlN interface is comparable to that of CdSe/CdS heterostructures
(of the order of 4.1% [151,161,165]). In addition to the screening of an internal field, another possible reason for the blueshifted high power spectra could be of biexcitonic origin. However, theoretical calculations allowed to rule out the biexciton
emission as cause of the observed nonlinearity [165].
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Fig. 2.29. (a) Fluctuations in the emission energy recorded from single CdSe/CdS nanorods. Taken with permission from Ref. [16]; (b) Illustration of the
origin of these fluctuations due to surface charges from Ref. [18]; (d)–(e) Mechanism of Foerster energy transfer from nanorods to organic dyes, which is
controlled by an external electric field; (f) Switching of the PL of single nanorods by an external electric field; (g)–(h) Switching the emission of nanocrystaldye couples on and off by an external electric field for two different nanocrystal-dye combinations. Taken with permission from Ref. [164].
Another appealing property of CdSe/CdS core–shell nanorods is that they can act as single photon sources. An
experimental proof for single photon emission is to measure the coincidence of the emitted photons in a high sensitivity
Hanbury Brown and Twiss setup based on two avalanche photodetectors [168]. For single photon emitters the coincidence
histogram shows negligible counts at zero delay time, an effect that is called photon antibunching [169,170], in contrast to
the photon bunching that occurs in the emission of thermal light sources due to the bosonic nature of photons. Pisanello
et al. [171] investigated the potential of single core–shell nanorods as fully polarized single photon sources operating at
room temperature, and confirmed also the spatial directionality of the nanorod emission, which resembles the emission
geometry of a dipole antenna (Fig. 2.32).
2.7.4. Type-I and type-II transitions in CdSe/CdS nanorods
For bulk and superlattices structures the band alignment at the CdSe/CdS interface is of type-I [172,173]. However, for
nanostructures this might be different due to quantum confinement effects. For holes the bulk valence band offset is of about
0.8 eV, which is too large to be inverted by quantization effects, especially if we consider the higher effective mass of the
holes with respect to the electrons. In the conduction band, on the other hand, it is possible to have a positive or negative
band offset, depending on the quantization energy in the core. Indeed, the reported values span the large range from −0.25
to +0.27 eV [148]. So far the most practical way to describe CdSe/CdS dot/rod heterostructures has been to assume a zero
conduction band offset [17,18], thus implying a quasi-type-II structure, provided that the experimental results could be well
explained. The possibility of tuning the core size in these nanostructures, using the so-called ‘‘seeded growth’’ approach [149]
has allowed to study the conduction band offsets in these core/shell nanorods in more detail.
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Fig. 2.30. (a)–(b) Energy shift of the emission peak of a single CdSe/CdS nanorod induced via the Stark effect by an external electric field; In (a) the e-field
is oriented parallel to the long axis of the rod, and in (b) is perpendicular. (a)–(b) are taken with permission from Ref. [17]; (c)–(d) Electrical exciton storage
in semiconductor nanocrystals at room temperature; (c) Prompt emission spectrum recorded in a 50 ns window overlaying excitation with (solid line)
and without (dashed line) an electric field pulse; (d) As in (c) but with the detection placed 1.03 µs after excitation following the 1 µs field pulse. In this
case the stored excitons result in a luminescence flash; (e) Time resolved PL decay with (full circles) and without (hollow squares) an electric field pulse of
1 µs duration. The arrows show the times of recording the spectra in (c) and (d). The sketches indicate the separation of electron and hole wave functions
during and after application of the field. (c)–(e) are taken with permission from Ref. [19].
Fig. 2.31. (a) Image of a random field of single nanorods acquired by scanning the sample across a diffraction-limited laser spot of a confocal microscope.
The sample was prepared by drop casting a nanomolar nanorod solution onto a cover glass and by rapidly blowing off the solution by means of a strong
nitrogen flow; (b) Spectra collected consecutively from a single rod (black lines) (identified by the arrow in panel (a)) obtained at the same laser power,
and relative Lorentzian best fit curves (green lines); (c) PL spectra from a single rod (black lines) (indicated by arrow in panel (a)) obtained by varying the
excitation power, from 1% to 20% of the maximum available power (1 mW), and relative Lorentzian best fit curves (green lines). Taken with permission from
Ref. [165]; (d) Transient emission energies measured at different delays after the excitation of an ensemble of nanorods, recorded at excitation densities
ranging from 2 µJ/cm2 , at which no shift was observed, to the maximum value (107 µJ/cm2 ) allowed by the experimental setup. Taken with permission
from Ref. [165].
As was discussed earlier in this section, many optical experiments can be well explained by a flat conduction band profile,
whereas electrical studies by scanning probe spectroscopy have revealed a pronounced confinement of the electrons in the
core region (see also Section 3 of this review on electrical properties). This contradiction has been resolved recently by a
work of Banin et al. [174], in which a systematic PL and absorption study on nanorods with different core and shell sizes
R. Krahne et al. / Physics Reports 501 (2011) 75–221
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Fig. 2.32. (a) Antibunching of the single nanorod emission; (b) Normalized PL amplitude vs. polarization angle of a single nanorod; (c)–(e) Antibunching
traces relative to three different emission polarizations; (f)–(i) Spatially resolved emission of single nanorods by defocalized emission measurements in a
confocal microscope.
Source: Taken with permission from Ref. [171].
has shown that the electron delocalization is strongly sensitive to the core size. In detail, the lowest level in the conduction
band is localized in the core for large core sizes (in that work this meant diameters above 2.8 nm) and the system behaves
like a type-I structure, whereas for smaller cores the quantum confinement raises the energy of this level above the conduction band minimum of the shell. The experimental evidence of such behavior was obtained by varying the excitation
density of an ensemble of rods. In general, at low excitation density (corresponding to a generation of less than one e–h pair
per rod) only one PL peak was observed and was associated to single exciton emission. With increasing excitation power
density (up to three orders of magnitude) additional peaks gradually appeared, which were ascribed to biexciton and triexciton emission (in order of apparition). Banin et al. found that depending on the core size the biexciton PL may be red- or
blue-shifted with respect to the single exciton energy. This shift provides a measure of the biexciton binding energy, which
is different in type-I and type-II systems. The biexciton is composed of four carriers (two electrons and two holes), which
in a type-I structure are distributed in the same spatial region, whereas in type-II structure electrons and holes are spatially
separated into different regions. The result is depicted in Fig. 2.33. When the four carriers reside in the same material, they
can distribute themselves in a manner that minimizes the Coulomb interaction among them, resulting in an energetically
favorable bound biexcitonic system. The binding energy is thus negative and the consequent emission is red-shifted with
respect to the single exciton PL (Fig. 2.33, left panel).
On the contrary, in a type-II configuration only carriers having the same charge reside in the same spatial region, as
they are obliged to do so by the particular conduction and valence band offset (Fig. 2.33, right panel). The result is that the
Coulomb interaction between the charge carriers is not minimized, thus a blue-shift with respect to the exciton energy is
predicted and is actually measured in several nanostructures [2,130].
Fig. 2.34 shows the optical spectra for nanorods with a large (a) and small (b) core size, and summarizes the results
obtained by Banin and co-workers on CdSe/CdS nanorods having different core diameters [174]. For core diameters larger
than 2.8 nm the biexcitonic peak was red-shifted with respect to the single exciton peak and for diameters smaller than
2.8 nm the relative positions of the exciton and biexciton were inverted, indicating the shift from a type-I to a type-II system
(see Fig. 2.34(c)).
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 2.33. Sketch of the biexciton system in type-I and type-II heterostructures. In type-I structures the four photogenerated charges reside in the same
material in a quadrupole configuration. The type-II materials constrain just the charges having the same sign to stay in the same materials, giving rise to a
unfavorable energetic configuration for the biexciton.
b
c
∆BX-X [meV]
a
40
20
0
–20
–40
d
2.6
Peak position [eV]
2.5
2.4
2.3
2.2
2.1
2
500
550
600
650
Wavelength [nm]
700
500
550
600
650
Wavelength [nm]
700
2.2
2.6
3.0
3.4
Core diameter [nm]
3.8
Fig. 2.34. (a) Transient emission spectra (solid black line) measured at the peak of the excitation pulse at increasing pulse energies ranging from 0.01
to 150 photon per dot (ppd) (a) for 4 nm diameter CdSe cores embedded in 45 nm × 6 nm CdS rods, (b) for 2.2 nm diameter CdSe cores embedded in
114 nm × 4.7 nm CdS rods; (c) Plot of the difference between the biexciton and exciton energetic positions for CdSe/CdS nanorods as function of the seed
diameter. The crossover from negative values (attractive biexciton binding energy) to positive energies (repulsive biexciton binding energy) represents a
fingerprint of the transformation of the system from type-I to type-II; (d) multiexciton peaks as function of the core dimensions: exciton (circles), biexciton
(squares), triexciton (downward-pointing triangles) and CdS exciton (triangles). The crossover involving exciton and biexciton is noticeable.
Source: Taken with permission from Ref. [174].
2.7.5. Optical properties of ordered nanorod assemblies
As discussed earlier in this section, the advent of the seeded growth approach for the fabrication of core–shell nanorods
has significantly improved the quality of the nanorod samples in terms of monodispersity, line width of the emission,
and quantum yield [149]. This new synthesis technique, where the seed is fabricated and size selected in a prior step,
has boosted the progress on assembly of anisotropic nanocrystals, on the study of optical properties of such assemblies,
and additionally on the fabrication of branched nanocrystals [174–176]. An interesting feature of nanorods with uniform
lengths and diameters consists in the possibility to align them over large areas [149,177,178], as shown in Fig. 2.33. Confocal
spectroscopy made it possible to optically characterize regions of nanorods aligned according to various geometries with
respect to a substrate, namely lateral, vertical and random. As one example, arrays of laterally aligned CdSe/CdSe nanorods
exhibited linearly polarized emission approaching a degree of polarization of 80% [148,171]. Fig. 2.35 shows scanning
electron microscopy (SEM) images of laterally aligned nanorods, and the polarized signal originating from those areas.
Fig. 2.36 reports the spatially resolved PL spectra collected from regions showing different types of nanorod
alignment [149]. Measurements on regions of rods vertically oriented with respect to the substrate were almost nonemissive and the weak PL was slightly blue-shifted (a few meV) with respect to the more intense signal originating from
disordered nanorods. A more pronounced blue-shift (up to 30 meV) and much higher intensity of the PL were noticeable
on lateral assemblies. The difference in PL intensity between laterally- and vertically aligned nanorods could be ascribed to
R. Krahne et al. / Physics Reports 501 (2011) 75–221
103
Fig. 2.35. (a)–(c) SEM images of nanorods aligned by means of electric fields; (d) Schematic illustration of the setup (i) employed for the polarization
measurements and relative results (ii, iii). Spectra in (ii) show the emission polarized parallel (red signal) and perpendicularly (blue signal) to the alignment
direction. Panel (iii) show the PL vs. the polarization angle (red symbols) fitted to a cos2 function (black line).
Source: Taken with permission from Ref. [149].
the spatial anisotropy of the optical dipole emission in nanorods. The transition dipole in such nanorods is parallel to the
rod long axis, thus the PL intensity is expected to be maximal in the plane perpendicular to the dipole and minimal in the
direction of the long axis. The unexpected behavior consists in the energy shift, and a possible explanation of this effect
is the screening of an internal (piezo) electric field present in the individual nanorods due to the photogenerated charge
distributions in neighboring rods (as discussed in detail earlier).
The possibility to organize nanorods into ordered arrays on micrometer and millimeter length scales makes them an
interesting material for macroscopic optical applications. As an example, Rizzo et al. fabricated a polarized light emitting
diode based on an array of laterally oriented nanorods [179]. Also, nanorods can be employed as active material for lasing
devices. Here, the emitting medium is typically embedded in an external resonator for optical feedback [180–182]. A
novel approach for self-assembled micro-lasers was proposed recently by Zavelani et al. [183]. Here the coffee stain effect
that governs the microfluidic dynamics in an evaporating droplet [184,185] led to the formation of well-defined stripes
that consisted of densely packed, laterally aligned nanorods. The borders of these stripes were sharp enough to form a
Fabry–Perot cavity for the emission of the nanorods (Fig. 2.37).
2.7.6. Collinear nanorods and nanobarbells
Collinear heterostructures exhibit a direct exposure of both materials at their surface and are interesting candidates
for photovoltaic [150–155] and photocatalytic [156–158] applications (see also Section 6 of this review). Among the typeII rod-like systems, CdSe/CdTe has the advantages of a well-controlled synthesis and small lattice mismatch between the
two materials. Therefore it is the most studied heterostructure with type-II band alignment [153–155,157,186,187]. Optical
investigation techniques including steady state (absorption and PL measurements) and time-resolved spectroscopy allow
to monitor effectively the carrier dynamics and localization in these nanostructures. Fig. 2.38 shows typical absorption and
PL spectra of collinear CdSe/CdTe nanorods. The absorption spectra show features arising from a superposition of the single
sections that constitute the heterostructure, whereas the PL signal occurs at an energy that is well below the band gap of
the single material sections, if these are considered separately.
In collinear nanorods the emission from the CdSe material is often observed as well, but with a strongly reduced intensity
with respect to the core–shell systems [153,175,187]. The presence of such visible emission has been ascribed to ultrafast
hole transfer from the core to the outer material and/or to resonant energy transfer from seeds having band gap larger than
the shell [187]. Further evidence of the real nature of the detected PL signal is usually obtained by analyzing the lifetime
of both emission peaks. Due to the spatial separation of e–h wavefunctions, the radiative rate (proportional to the e–h
wavefunctions overlap) is strongly reduced with respect to a type-I optical transition. This is reflected in time-resolved
measurements that show type-II PL lifetime ranging from few hundreds of nanoseconds to some microseconds, as opposed
to the more classical PL decay on the nanosecond timescale observed in type-I transitions [186–188].
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 2.36. (Main panel): PL from region of laterally-, vertically aligned and disordered nanorods with relative Gaussian best fit curves (green lines). Upper
panels show TEM, PL and backscattering images from the same regions.
Source: Taken with permission from Ref. [149].
Fig. 2.37. Lasing from self-assembled nanorod cavities. (a) Optical dark field microscope image of the coffee stain deposit formed by spray deposition of
a solution of nanorods dissolved in toluene. The borders of the coffee stain ring are well defined enough to function as a resonator and thus to promote
lasing; (b) Emission from a coffee stain nanorod deposit for three different pump excitation intensities: below, near and above lasing threshold.
Source: Taken with permission from Ref. [183].
Collinear nanorods are preferred in the study of the fundamental physical properties, since it is possible to discern the
role of core and shell in the absorption process. This allowed for example to determine precisely the dynamics of charge
carriers after generation by means of pump–probe technique. Transient absorption measurements performed by Dooley
and co-workers disclosed the real carrier pathways and found an ultrafast charge separation on the 500 fs timescale [155].
The noteworthy characteristic of these systems, apart from the type-II transition, is the presence of a PL peak falling in the
visible range and referable to radiative emission of the outer material (see for instance Fig. 2.38) [153,154,187].
R. Krahne et al. / Physics Reports 501 (2011) 75–221
105
Fig. 2.38. Absorption and PL spectra of a representative coaxial CdSe/CdTe heterostructure. Inset: sketch of the band alignment and labeling of the
transitions indicated on the rod absorption (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the
web version of this article.)
Source: Taken with permission from Ref. [155].
Fig. 2.39. (a) Collapse of the CdSe/CdTe nanobarbells PL with respect to the starting CdSe nanorods. Taken with permission from Ref. [157]; (b) Comparison
between PL coming from CdS nanorods (blue line) and ZnSe/CdS nanobarbells (red line) showing increased efficiency.
Source: Taken with permission from Ref. [158].
Differently from some collinear and dot/rod systems, where an internal seed material is always buried in an outer
material having the role of both charge separator and passivating agent, nanobarbells are characterized by having a second
component deposited only on the tips of the rod (see Fig. 2.24) [157,158]. Apart from the geometry, a fundamental difference
with respect to the structures previously described lies in the role of the second material. Nanobarbells are of potential
interest for photocatalysis, where both the separated charges have to be readily available for chemical reactions, since both
material components are directly exposed to the ambient conditions.
This specific geometry has also strong implications on the optical properties of the overall nanocrystal, since the second
material does not grow as a passivating uniform layer on top of the starting rod. Indeed, the most common behavior in these
structures consists in a drastic change of the luminescence performances, mainly due to trapping processes occurring at the
surface of both exposed components. Depending on the material, the effect may correspond to an increase or decrease of the
performances. Fig. 2.39 shows an example of what happens for two different systems, CdSe/CdTe and ZnSe/CdS [157,158]. In
the CdSe/CdTe nanobarbells the PL is quenched after deposition of CdTe on the CdSe rod, which makes them ideal structures
in photocatalytic applications (see Fig. 2.39(a)). On the contrary, the PL signal coming from ZnSe/CdS system evolves from a
broad, trap related emission of CdS rods to a sharper, more intense signal falling in the same spectral range (Fig. 2.39(b)).
2.7.7. Multifunctional nanorods
The advent of synthesis approaches to colloidal nanocrystals with narrow distributions of their geometric parameters
has directed the research towards the fabrication of more elaborate multi-component nanostructures, which in principle
would be able to fulfill multiple tasks simultaneously. In particular, the combination of magnetic and optical properties
is attractive for application of such nanocrystals as bio-labeling and analytical separation, drug delivery, multimodal
diagnostics and therapeutics. Among the several colloidal nanostructures that have been reported in the last years
(and not limited to nanorods), we mention Co/CdSe core–shell nanocrystals [189], FePt–CdX (X = S, Se) heterodimers
[190–193], γ -Fe2 O3 –MeX (Me = Cd, Zn, Hg and X = S, Se) heterodimers [194–196], and CdSe/CdS and CdSe nanorods
decorated with PtNi/PtCo or Co domains [160,197]. All these structures have been fabricated with the initial goal to
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 2.40. (a) Elastic zero-loss EFTEM image of a sample of Co-tipped nanorods; (b) Composite false-color image obtained from (a) showing the different
domains of the sample. CdS rod (green), cobalt domains (blue spheres on the tips of the rods), CdSe cores (blue zones inside the CdS rods).
Source: Taken with permission from Ref. [159].
Fig. 2.41. (a) Series of absorption (continuous lines) and PL spectra (dashed lines) of starting CdSe cores (black lines), CdSe/CdS nanorods (red lines), a
physical mixture of Co nanoparticles and CdSe/CdS nanorods (green lines), Co-tipped nanorods (blue lines); (b) Time traces of the PL recorded at room
temperature on starting CdSe/CdS nanorods (red line), physical mixture of Co nanoparticles and CdSe/CdS nanorods (green line), Co-tipped nanorods (blue
line).
Source: Taken with permission from Ref. [159].
‘‘pack’’ into a single nanostructure magnetic and optical properties, although the final outcome has not always been very
successful.
For what concerns nanorods for example, the basic idea in this direction is to grow other materials with optical/magnetic
properties selectively at their tip regions. Unfortunately a common fate of metal-tipped nanorods is the drastic drop of
their PL quantum yield (PL QY), due to fast semiconductor–metal charge transfer. However, the last generation of seeded
grown CdSe/CdS nanorods (having a starting QY approaching 70% at room temperature) offers the opportunity to preserve
higher efficiency after semiconductor–metal coupling. Recently, fluorescent Co-tipped CdSe/CdS dot/rod heterostructures
exhibiting ferromagnetic behavior at room temperature have been successfully reported by Deka et al. [159]. In Fig. 2.40
we show an image of these Co-tipped nanorods, obtained by energy filtered TEM (EFTEM). In the sample, which has been
imaged before separation of the metal-tipped nanorods from unreacted rods, one can see clearly the selective growth of Co
domains only on one tip region of individual rods, and in a few cases the analysis allowed to identify this tip region as being
far from the region of the CdS rod in which the initial CdSe was buried (for more details, see Ref. [159]). Fig. 2.41(a) shows
the optical absorption and emission spectra of the initial CdSe cores (i.e. the ‘‘seeds’’) of the CdSe/CdS nanorods, of physical
mixtures of Co nanocrystals and CdSe/CdS nanorods, and finally of Co-tipped nanorods (the latter obtained after magnetic
separation, hence containing a negligible amount of non-reacted nanorods). As discussed previously, the absorption from
R. Krahne et al. / Physics Reports 501 (2011) 75–221
107
CdSe/CdS nanorods presents a sharp profile in correspondence of the CdS excitonic absorption (about 495 nm) and a smooth
behavior at longer wavelengths, where a weak CdSe absorption can be detected.
The relative PL emission clearly falls in the CdSe spectral domain, indicating a type-I recombination in the cores, and is
slightly red-shifted (together with the lowest absorption peak) with respect to the starting seeds counterpart. This behavior,
largely observed in heterostructures, is ascribed in this case to spreading of the electron wavefunction, which leads to the
observed red-shifts. The physical mixture of rods and Co particles has an absorption spectrum that reproduces a linear
combination of the two systems considered, and a PL typical of the nanorods, with a PL quantum yield of about 25%.
On the contrary, Co-tipped nanorods present an almost flat absorption as a consequence of the effective electronic
coupling between the two components. Their PL QY, despite being only around 3%, is nevertheless the highest efficiency
reported so far for semiconductor–magnetic nanocrystals in which the two domains are directly bound to each other
(i.e. without any molecule/polymer acting as a spacing between them). The reduction of the PL QY is explained in terms of fast
charge transfer from the semiconductor to the metal particles and has been investigated by time-resolved spectroscopy (see
Fig. 2.41(b)). The PL from the CdSe/CdS nanorods decays in about 12 ns, whereas in a physical mixture of CdSe/CdS nanorods
and Co nanoparticles the lifetime is about 7 ns. On the other hand, the Co-tipped rods exhibit a time constant of only 350 ps.
Considering that these three systems undergo a progressive decrease of their PL QY, the reduction of the lifetime cannot
be ascribed to an increased radiative recombination rate (which would lead to a progressive increase of the PL efficiency).
Instead, the reduction in lifetime in Co-tipped nanorods can be attributed to an increase of the non-radiative rate, due to
the rapid charge transfer induced by the electronic communication between the semiconductor material and the cobalt
domain attached to it. For the mixture of CdSe/CdS nanorods and Co nanoparticles, in which no direct Co nanorod interface
is present, the moderate decrease in the decay time can be ascribed to the activation of non-radiative channels different
from charge transfer. Most likely, these are due to energy transfer from semiconductor to metal, as already observed in
similar systems [198]. Despite the large reduction of the emission efficiency in Co-tipped nanorods, some detectable PL
signal remains, which makes such heterostructures promising candidates as active media in optical/magnetic applications.
3. Electrical properties of elongated nanoparticles
3.1. Introduction
In this section we will discuss electrical properties of elongated nanoparticles (nanorods) both on a single particle level
and for thin films consisting of densely aggregated nanorods. We will deal only with semiconductor nanorods, and, like in the
previous section, we will focus mostly on the CdSe material system as an example to explain the physical properties, since
CdSe nanorods have been studied in great detail. We will not discuss explicitly the electrical properties of metallic nanorods,
but we treat in general the double barrier tunnel junction and Coulomb blockade effects, which are the major factors that
dominate the conductive behavior of metal nanostructures. As highlighted in the previous section, the finite size of the
nanorods leads to quantum confinement effects and consequently affects their energy level structure. The optical, electrical,
and mechanical properties of the nanorods can be discussed in terms of the related quasi-particles like excitons, electrons,
holes, and phonons confined to a cylinder with length L and diameter D. Quantum size effects will have significant impact
if the length L, or the diameter D, or both, are of the order or smaller than the Bohr radius associated with the respective
quasi-particle. The Bohr radius is defined as ab = ε (m/m∗ ) a0 , where m and a0 are the electron mass and Bohr radius,
respectively, m∗ is the effective mass of the quasi-particle in the bulk material of which the nanocrystal is composed and
ε is the dielectric constant of the nanocrystal material in the bulk. The energy spectrum of a quasi-particle confined to a
cylinder can be derived from the three-dimensional Schrödinger equation in cylindrical coordinates r, φ , and z, and towards
that scope we will briefly sketch the solutions in these coordinates. In the simplest case, we can discuss a free particle with
effective mass m∗ that is confined by infinitely high potential barriers within the cylinder with radius R = D/2 and length
L, i.e. the potential V (r ) = 0 for r < R and infinite otherwise. In this case independent solutions for the three coordinates
can be found of the form [199]:
(3.1)
ψ (r , φ, z ) = ρ (r ) Φ (φ) Z (z )
where:
d2 Z
dz 2
= k2zk Z
d2 Φ (φ)
dφ 2
r2
d2 ρ
dr 2
(3.2)
= −l2 Φ
+r
dρ
dr
(3.3)
+ (k21 rn2 − l)ρ = 0.
(3.4)
With the relation K 2 = k2zk + k2ln for the momentum vector. The solution in z, along the c axis of the rod, is of the form
Z (z ) = sin(kzk ) = sin
kπ
L
z
,
k = 1, 2, 3 . . . .
(3.5)
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 3.1. (a) Sketch of a nanorod, and description of the cylindrical coordinates; (b) Schematic illustration of the charge tunneling process related to a
single nanorod in between two electrodes, in the weak coupling regime. The electronic levels of the nanorod that are within the window of the bias voltage
contribute to the conductance; (c) Circuit diagram describing such a junction. Sketch illustrating the contact geometry in scanning tunneling spectroscopy
(d) and for planar electrode junctions (e). In the case of the planar device geometry a highly doped Si substrate coated with a thin oxide layer can be used
as a back gate electrode.
In the radial direction we obtain solutions to the Bessel equation:
ρ(r ) = AJl (k1 rn ) + BYl (k1 rn ),
(3.6)
with A and B numbers and Jl and Yl Bessel functions of the order l of the first and second kind. Due to the radial symmetry
this leads to ring like distribution of the wave functions around the z axis of the nanorod.
The electrical characterization of single nanorods can provide detailed information on the electronic level structure of the
electrons and the holes. The most common contact configuration for electrical experiments on single nanorods (and single
nanoparticles in general) is represented by a double-barrier-tunnel-junction (DBTJ) [200], which is sketched in Fig. 3.1(a).
In this geometry the nanorod is connected to two electrodes (source and drain) via tunnel barriers that can be characterized
by a capacitance and a resistance. The tunnel barriers originate in most cases from the organic molecules that are needed to
stabilize the surface of the nanorods, and/or from the specific contact scheme of the experiment, for example from the gap
between the tip and nanorod in scanning probe experiments. Electrical experiments on thin films of nanorods are interesting
with respect to possible device applications of this class of nanomaterials, for example in photovoltaics [7,201–203] or in
light emitting devices [179]. The thin film can either be deposited on top of the electrodes that were previously fabricated
on a planar, isolating substrate (usually in a stripe geometry with some microns in electrode distance), or vice versa the
electrodes can be deposited on top of the nanorod film, for example by metal evaporation, or the nanorod film can be
deposited on a conducting substrate and contacted in a vertical configuration with the second electrode on top of the film,
for example by a metallic layer or a conductive scanning probe tip.
In general, current transport measurements provide complementary information with respect to optical experiments,
where the exciton level structure is probed. A comparison of the results of optical experiments and current transport
is interesting in order to gain more insight into the intrinsic properties of the nanorods. Here we have to consider
on one hand the Coulomb interaction in between the electron and the hole, which decreases the exciton energy with
respect to the electron and hole level structure, and on the other hand the Coulomb charging of the nanoparticle and the
leverage (originating from the ratio of the contact resistances), which affects the observed energy gap in current transport
measurements [101,204]. As a matter of fact, the way in which the nanorod is connected to the electrical contacts has a
significant influence on its electrical response, and therefore we have to consider always the combined system of electrodes
and nanocrystal.
3.2. Electrical experiments on single nanorods
Here, we will discuss the specific case of nanorods that are weakly coupled to the external electrodes via contacts that
exceed the quantum resistance h/e2 = 25.8 k. This condition is fulfilled for STS experiments and for most of the planar
contact configurations. In this case the nanorod is connected to two electrodes in a DBTJ configuration, and we assign to
each of the tunnel junctions a capacitance C and a resistance R. We can expect that the electric response is governed by a
superposition of the electronic level structure of the nanorod, and the Coulomb charging effects, if these energies exceed
R. Krahne et al. / Physics Reports 501 (2011) 75–221
109
the thermal energy. In order to charge the nanorod with a single electron we require a finite charging energy ∆E that can
be evaluated by the circuit diagram displayed in Fig. 3.1(c) to be e2 /[2(C1 + C2 )]. The capacitance values determine also the
voltage division between the junctions, V1 /V2 = C2 /C1 . This voltage division leads to the above mentioned leverage in the
electrical circuit:
Vbias
V1
=1+
C1
C2
(3.7)
.
This leverage must be considered in order to obtain correct values for the electronic level structure of the nanorods. In
scanning tunneling spectroscopy (STS) the tip–nanorod distance is usually chosen in such a way that the nanorod substrate
capacitance is much larger than the nanorod–tip capacitance and therefore the applied bias drops almost entirely on the
tip–nanorod junction. In planar devices, however, the leverage depends on the fixed contacts of source and drain electrodes
to the nanorod, which in this case is an intrinsic property of the device. In most cases both electrodes are fabricated by
the same process step (for example by EBL and thermal metal evaporation), which leads to an approximately even voltage
division between the contacts. In such a symmetric configuration it is, for example, much harder to assign if the conductivity
arises from the electrons or from the holes, since the bias window extends over both positive and negative polarity. We
remark that on a planar device geometry three-terminal measurements are possible, in which the charge tunneling can be
controlled, for example via a planar backgate electrode. So far, however, only Coulomb charging of the nanocrystals has
been observed in this way, and no signature of the semiconductor level structure was detected. The precise control over the
contact properties in a fixed device configuration remains still a challenge. For the electronic level structure of a nanorod in
a DBTJ one should expect a superposition of the Coulomb charging levels with the quantized energy levels of the nanorod.
Based on the approach of Hanna and Tinkham [205] for the conductivity of spherical metallic particles, Porath et al. [200]
calculated the tunneling rate from the electrode i = 1, 2 onto a nanoparticle with a discrete level spectrum in a DBTJ using
the following expression:
Γi+ (n) =
2π
h
∫
|Ti (E )|2 Di (E − Ei )f (E − Ei )Dd (E − Ed )[1 − f (E − Ed )]dE .
(3.8)
Here n is the number of excess electrons on the nanoparticle, Ti (E ) is the tunneling matrix element across the junction i, Di
and Dd are the density of states on the electrode i and the nanoparticle, Ei and Ed are the corresponding Fermi levels whose
relative positions after tunneling [Ed (n ± 1) − Ei (n)] depend on n, and f (E ) is the Fermi function. For practical purposes,
Ti (E ) can be replaced by a phenomenological tunneling resistance parameter Ri ∝ Ti (E )−1 . In order to gain some insight
into the charging dynamics of a single nanoparticle, we will look at some limiting cases that are of practical interest. Let us
first assume an asymmetric DBTJ with C 1 ≪ C 2, therefore the onset in tunneling occurs at junction 1. In this configuration,
we will observe charge transport at positive bias due to the electrons, while at negative bias the charge transport will be
due to the holes. If R1 ≫ R2, then the tunneling rate off the nanoparticle is much larger than the tunneling rate onto the
nanoparticle, i.e. the statistically the charges will tunnel into an ‘‘empty’’ nanoparticle, and therefore no Coulomb charging
effects can be observed. In the opposite case, with R1 ≪ R2, the charges tunneling onto the nanoparticle remain there long
enough such that the following charge has to overcome the Coulomb energy in order to tunnel onto the nanoparticle, and
the Coulomb charging staircase can be observed in the I–V curve.
In the case of a more symmetric DBTJ, on one hand the leverage factor increases, and on the other hand simultaneous
tunneling of electrons and holes can occur, which leads to much more complicated dynamics. Plots simulating tunneling
spectra of this model for different limiting cases, and a detailed discussion on the charging dynamics can be found in
Ref. [206]. For completeness, we will describe now the condition under which a tunneling current through a DBTJ can be
observed. The condition for resonant tunneling is the line-up of the Fermi level of the nanorod with the Fermi level of
one of the electrodes. Following Ref. [200], the probability distribution of n, P (n), can be determined from the steady state
condition:
P (n)[Γi+ (n) + Γ2+ (n)] = P (n + 1)[Γi− (n + 1) + Γ2− (n + 1)].
(3.9)
The tunneling current can then be calculated self-consistently from:
I (V ) = e
−
P (n)[Γ2+ (n) − Γ2− (n)] = e
−
p(n)[Γ1− (n) − Γ1+ (n)].
(3.10)
3.2.1. Single nanorod conductance probed by STS
Scanning tunneling spectroscopy (STS) is a powerful tool to probe the electronic level structure of single nanocrystals
and their ensembles. For STS experiments, the nanorods are typically deposited from a highly diluted solution onto a
conductive substrate, for example highly oriented pyrolytic graphite (HOPG), or onto a planar substrate covered with
a gold film. The nanocrystal–substrate interface provides one electrical contact. The second contact is realized by the
scanning tunneling tip, where the contact properties like resistance and capacitance can be tuned by the nanorod–tip
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 3.2. (a) Scanning tunneling spectroscopy spectra of CdSe nanorods of various lengths and diameters; (b) Energy gap vs. nanorod radius derived from
optical (full circles) and tunneling (open circles) data. The optical data was corrected by the Coulomb interaction term. The solid line shows the result of
a calculation based on a multiband envelope function approximation. The inset illustrates the energy level structure and the first optical transition. The
derivative dI /dV of the current voltage spectrum is often plotted vs. the voltage because peaks in the dI /dV vs. V correlate to the electronic level structure
of the system. Taken with permission from Ref. [100]. (c) STS spectra of CdSe nanorods coupled via dithiol molecules to a Au substrate show Coulomb
charging, while CdSe nanorods on HOPG exhibit only resonant tunneling (d). Taken with permission from Ref. [101].
distance. Fig. 3.1(d) illustrates the experimental configuration and the related circuit diagram is shown in Fig. 3.1(c). In
particular, the distance in between the tip and the nanorod can be tuned such that almost all of the applied voltage
drops on the tip–nanorod contact. This asymmetry allows to probe the conduction and valence band levels separately by
applying positive and negative bias voltage, respectively. The contact resistance in between the nanorod and the conductive
surface can be tuned to a certain extent, for example by surface functionalization with specific linker molecules, such as
dithiols.
The experiments shown in Fig. 3.2(a) on single CdSe nanorods revealed that for CdSe, where the Bohr radius (5 nm) is
about one order of magnitude smaller than the nanorod length, the confinement depends almost entirely on the nanorod
diameter. This observation is supported by calculations based on a multiband envelope function approximation that takes
the coupling between the light hole and heavy hole valence bands into account. Here the electrons and holes are treated as
free particles with an effective mass confined by a cylindrical potential, with barrier height of 5 eV. If the CdSe nanorods are
sufficiently decoupled from the conductive substrate, for example by organic molecules like dithiols, also Coulomb charging
signatures can be observed in the STS spectra. The Coulomb charging energy can be expressed as E = e2 /2CSUM , where e
is the electron charge, and CSUM accounts for the capacitive coupling of the nanorod to the leads. For a single nanorod in
the experiment described above the contact capacitances were of the order of 10−19 F, which lead to a charging energy of
around 0.4 eV [101]. Electron–phonon coupling has been observed in STS experiments at low temperature on CdSe dots and
rods by Sun et al. [207], which led to phonon replicas in the differential tunneling current spectrum.
Fig. 3.3 shows STS spectra recorded on InAs nanorods, where the Bohr radius (34 nm) is larger than the length of the
nanorods. In this case also the nanorod length did influence the confinement energy, as the increase in the energy gap with
decreasing rod length demonstrated. Furthermore, the optical spectroscopy and STS data recorded on InAs dots revealed
a very strong discrepancy between the values for the band gap derived by the two techniques (the band gap obtained by
STS is much larger than that observed in optical spectroscopy). This difference cannot be accounted for by simply correcting
the optical data, which reflects the band gap of the exciton, i.e. of a bound electron–hole pair, with the Coulomb interaction
term. Although this difference can be partly accounted for by voltage division effects that enlarge the tunneling gap, it seems
that contributions arise from an intrinsic enlargement of the band gap observed in tunneling experiments. The latter could
also be affected by dielectric confinement [208], which is a signature of 1D confinement.
R. Krahne et al. / Physics Reports 501 (2011) 75–221
111
Fig. 3.3. (a) Tunneling spectra recorded on InAs nanorods on graphite. (b) Energy gap vs. nanorod radius derived from optical and tunneling experiments.
The optical data was corrected by the Coulomb interaction term.
Source: Taken with permission from Ref. [204].
Fig. 3.4. (a) STS spectra of a single CdSe/CdS NR acquired at 4.2 K and on three different regions along the rod (black lines) compared with the corresponding
theoretically predicted curves (red lines); (b) 3D STM image of an individual NR; (c) Cross sections of the tunneling current at a sample bias of 1.2 V (upper
green curve) that shows the larger tunneling current over the core region, and the calculated electron ground-state probability density assuming a band
offset of 0.3 V (blue curve), that shows the localization of the electron ground state in the CdSe core. (For interpretation of the references to colour in this
figure legend, the reader is referred to the web version of this article.)
Source: From Ref. [148].
3.2.2. Core–shell nanorods fabricated by seeded growth
The band structure of CdSe/CdS core–shell nanorods has been discussed in detail in Section 2. Complementary results
were obtained by Steiner et al. [148] who were able to spatially resolve the electronic level structure of the electrons and
holes by STS over the nanorod volume. The results are displayed in Fig. 3.4(a), which reports the dI /dV spectra of different
positions of the scanning probe tip with respect to the nanorod. Here positions 1 and 2 are over the CdS shell section of the
nanorod, whereas at position 3 the tip was on top of the nanorod CdSe core (Fig. 3.4(b)). We observe clearly the onset of
tunneling current at lower bias on position 3, both for the valence band states (at negative bias), and for the conduction band
states (at positive bias). Since the tunneling current gives a direct measure of the density of states below the probe tip, we
can conclude that the lowest energetic levels of both electrons and holes are localized in the core for this nanorod sample.
A scan along the long axis of the nanorod at constant positive bias voltage of 1.2 V, thus at the onset of conduction, clearly
reveals the smaller band gap over the core region, and the data could be well fitted assuming a band offset at the core–shell
interface of 0.3 V in the conduction band. This offset is much larger than the values deducted from optical pump–probe
experiments, where the data could be well explained by assuming no band offset in the conduction band. A recent work of
Banin et al. [174] unveiled this contradiction by showing the conduction band offset depends very sensitively on the core
size (see Section 2).
3.2.3. Single nanorod conductance probed in planar nano-junctions
A different strategy to contact single nanorods is to fabricate fixed electrical contacts by lithography or metal deposition
techniques, which leads to structures that can be of interest for practical device applications. Here the challenge is to achieve
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
a
b
c
e
d
Fig. 3.5. Energy levels of bulk metal and semiconductor structures before (a) and after (b) being in contact. Φm and Φs are the work functions of the metal
and semiconductor, respectively, χ is the electron affinity and Eg is the band gap of the semiconductor.
Source: Taken with permission from Ref. [214].
the nanoscale resolution needed to address single nanorods that have a length of around 50 nm or less, and to obtain
control over the properties of the nanoscale contact. The tools of choice are in this respect electron-beam lithography
(EBL) and electron-beam or focused-ion-beam induced deposition techniques. In general two strategies can be followed:
(i) the electrodes are prepared prior to the deposition of the nanocrystals [209], and in this case the difficulty is the precise
positioning of single nanorods into the electrode gap, or (ii) the nanorods are deposited from a highly diluted solution on a
planar substrate, mostly a silicon wafer covered with a thin insulating layer, and their position is mapped with respect to
previously fabricated markers. This second case is so far the more successful approach and will be discussed in the following.
In this EBL approach the sample is spin coated with PMMA, then the electrode pattern is aligned with respect to the nanorod
that shall be contacted, and the PMMA is exposed with the electron beam [210–212]. In the case of electron or focused ion
beam induced metal deposition no PMMA deposition is needed since the metal is directly deposited by the beam. Hence
mapping and deposition can be done in a single run [213].
We will now discuss nanorods contacted by EBL and subsequent metal deposition in more detail. Alivisatos et al.
[210,212] used the above described EBL overlayer approach to contact single CdSe and CdTe nanorods in planar electrode
devices where the Si substrate was used as a back gate. One interesting question with respect to metal contacts to nanorods
is if a Schottky barrier is formed in between the metal and the semiconductor material. For bulk materials the alignment of
the Fermi level and the continuity of the vacuum level leads to a band bending in the metal–semiconductor interface region,
as depicted in Fig. 3.5. The band alignment is made possible by charge transfer between the semiconductor and the metal
and leads to a depletion region of typically tens to hundreds of nanometer in width. This depletion region is referred to as the
Schottky barrier. If we consider laterally fabricated metal contacts to nanorods that typically have diameters smaller than
10 nm, the interesting question is if such charge transfer, and consequently the band alignment is possible and a Schottky
barrier can be formed.
Leonard and Talin [214] studied the properties of metal contacts fabricated laterally on semiconductor nanowires. They
performed a self-consistent calculation of the charge and potential, by which they obtained the charge from the onedimensional density of states:
DNR (E ) =
√
2m∗
πh
1
(E − EG /2)− 2 .
(3.11)
In the expression above m∗ is the effective mass and EG is the energy gap of the semiconductor nanowire. The potential
was derived by solving the Poisson equation for the nanowire numerically. The result of this calculation is that for silicon
nanowires with diameters smaller than 10 nm the band bending is weak due to the limited available depletion width (see
Fig. 3.6). However, for larger diameters, for which according to the work function difference between the metal and the
semiconductor an ohmic contact would be expected, the band bending leads to a Schottky barrier, for example in silicon
nanowires with 10 nm diameter (Fig. 3.6(b)).
Demchenko and Wang [215] calculated the effect of lateral electrodes fabricated on CdSe rods on the energy gap and
the wave function distribution of the carriers by an atomistic pseudopotential method. Since the nanorod diameter is in
the range of a few nanometers, charge transfer between the rod and the metal electrode is limited due to the limited
available depletion length, as discussed above. Therefore a Schottky contact between the metal and the nanorod is assumed.
Demchenko and Wang [215] showed that the surface polarization potential is surprisingly large, especially for small
nanostructures, where it alters the familiar electronic structure and gives rise to a nanocontact phenomenon: electrodeinduced wave function localization. In particular, the surface polarization potential is around 0.5 V lower in the region of
the metallic contact (see Fig. 3.7(a)), and the nanorod bang gap is significantly reduced. Also, the shape of the wave function
in the conduction and valence band is considerably altered. While for the conduction band states the wave functions are
R. Krahne et al. / Physics Reports 501 (2011) 75–221
113
Fig. 3.6. Band bending across nanowires with diameters of 2 nm (a) and 10 nm (b). The dotted line shows the Fermi level.
Source: Taken with permission from Ref. [214].
0.8
(b)
Pcenter axis [eV]
0.6
0.4
0.2
(c)
0
– 0.2
– 0.4
0
5
10
15
Z [nm]
20
25
Fig. 3.7. (a) Polarization potential vs. the z axis of a nanorod with different diameters. The sketch illustrates the metal–nanorod contact geometry;
(b) Contour plots of the conduction band minimum (CBM) and Valence-Band-Maximum (VBM) wave functions of nanorods with 3 nm diameter. The
upper panel shows the case of a free-standing rod, the lower panel the case of a nanorod with a metal electrode covering one quarter of its length.
Source: Taken with permission from Ref. [215].
just shrunk in the z-direction, the valence band states show a different nodal structure in comparison with their free rod
counterparts. The continuity of the valence band states in Fig. 3.7(c) could favor charge transport. Therefore the assumption
that for laterally contacted nanorods the conduction is mediated by the valence band states could be reasonable. For charge
transport, the wave function localization has to be overcome by an external electric field, which can be estimated to be
around 0.5 V (for CdSe nanorods with 23 nm length that are covered for 1/4 of their length by each contact).
The above described scheme of laterally fabricated Pd metal electrodes (see Fig. 3.8(a)–(b)) was realized experimentally
by Cui et al. [212], in order to contact CdSe nanorods and by Trudeau et al. to contact CdTe nanorods [210]. In contrast
to the STS experiments, the spectra obtained from the planar junctions did not reveal the semiconductor band gap, but
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 3.8. (a) Schematic illustration of the lateral contact scheme; (b) SEM image of a CdTe nanorod contacted by Pd electrodes; (c) Color map of the
conductance of a single nanorod vs. source–drain and gate voltage. The color map shows the Coulomb blockade diamonds, which are a fingerprint of single
electron tunneling (SET).
Source: Taken with permission from Ref. [210].
showed a Coulomb charging gap (see Fig. 3.8(c)) of the order of some tens of meV. This finding could be the signature of the
above described wave function localization that is induced by the metallic contacts. In a symmetric junction, the applied
voltage bias will drop equally at the two barriers and therefore will result in a zero current plateau which spans a voltage
range V = 2ECB /q. The Coulomb charging energy in such a planar junction is about one order of magnitude smaller than
in the STS experiments, due to the much larger contact capacitances. The deposition of electrodes by metal evaporation
on the nanorod surface can lead to the migration of Au atoms towards the central region of the rod that was intended to
be left uncoated by the metal. Trudeau et al. have investigated this effect for CdTe nanorods that were partly overcoated
with Pd [210]. They observed the presence of Au atoms on the nanorod surface at a distance of tens of nanometers from the
intended contact region. Short nanorods, with a length that did not exceed much the migration distance of the Au atoms,
showed a significantly reduced charging energy, compared to the charging energy of nanorods that were slightly longer.
3.2.4. Gold tipped nanorods (nanodumbbells)
Metal–semiconductor hybrid nanocrystal structures provide a different approach towards electrical contacts to
semiconductor nanorods. Recent advances in chemical synthesis enabled the controlled growth of small gold domains at
the tips of CdS, CdSe or PbTe nanorods [144,216]. In these nanodumbbells the nanoscale metal–semiconductor interface is
realized as an intrinsic property of the nanocrystal. Spatially resolved investigations by scanning tunneling spectroscopy
(STS) of Au–CdSe nanodumbbells revealed a Coulomb blockade gap in the vicinity of the metal domain, and the electronic
structure of the semiconductor nanorod at the center of the dumbbell, as shown in Fig. 3.9(b). At position 1, above the
metal domain, the series of peaks can be associated to the Coulomb charging of the Au cluster at the tip. Here the Coulomb
charging energy can be approximated by the charging energy of a sphere, namely E = e2 /2 (4π ε0 r ), which yields a value of
700 meV for a sphere with radius of 1 nm. The I–V response can be modeled more accurately by a single electron tunneling
(SET) process in a DTBJ as Steiner et al. described in their work [217]. The response at the center of the nanorod was very
similar to the previously discussed spectra of bare CdSe nanorods (see Fig. 3.2). Interestingly, additional peaks appeared in
the spectra taken above the metal–semiconductor interface region, at the positions 2 and 3. These additional structures
could be associated to localized metal–semiconductor interface states. Furthermore, an overall shift of the spectrum 3
with respect to spectrum 4 was observed which could be due to a voltage offset from a Schottky contact. Particularly
interesting were the spectra recorded in the vicinity of the metal–semiconductor interface, which revealed localized metal
induced gap states. In this region (with the tip in the vicinity of the metal–semiconductor interface) negative differential
conductance was observed, which could be ascribed to resonant alignment of the Coulomb blockade levels with the
states that originate from the semiconductor level structure. One advantage of Au nanorod dumbbells compared to bare
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
b
0.08
Current (nA)
dl/dV (arb. u.)
a
c
0.04
0.00
–0.04
d
–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5
Sample Bias (V)
e
f
–0.08
–1.0
–0.5
0.0
0.5
1.0
Sample Bias (V)
g
Fig. 3.9. (a) Scanning tunneling microscopy image of a CdSe nanorod with Au domains at the tips (a so-called dumbbell structure); (b) Scanning tunneling
spectroscopy spectra recorded at different positions of the tip with respect to the dumbbell, as illustrated in the inset; (c) Current voltage characteristics
that show negative differential conductance recorded with the tip in the vicinity of the metal–semiconductor interface. (a)–(c) are taken with permission
from Ref. [217]; (d) Illustration of the charge transfer dynamics after photoexcitation of nanodumbbells in aqueous solution from [218]; (e)–(g) Scanning
probe images of CdSe–Au nanodumbbells on HOPG; (e) Topography recorded in tapping mode; (f)–(g) electric force signal derived from the AC signal in
interleaved mode. The circles are guides to the eye that indicate the positions of individual nanodumbbells. Taken with permission from Ref. [219].
nanorods towards applications in electric circuits could be their improved conductivity due to good contact formation of
the Au domains with the metal electrodes. Indeed, Sheldon and co-workers showed a significantly increased conductivity
of Au–CdSe dumbbells that were positioned into planar electrode junction by electrophoresis [209]. Nanodumbbells can
also be interesting for visible light induced photocatalysis. Banin and co-workers introduced a novel synthesis method to
obtain CdSe–Au dumbbells dissolved in water and showed that irradiation of the solution leads to charge transfer of the
photoexcited electrons from the CdSe rods to Au domains (as illustrated in the scheme in Fig. 3.9(d)), whereas the holes
can be scavenged by ethanol present in the solution [218] (see Section 6 for additional details). More recently, detailed EFM
studies of the light induced charge separation that occurs in such nanodumbbells confirmed the above described charge
transfer mechanism. Fig. 3.9(f)–(g) shows the charge images of nanodumbbells on HOPG recorded in interleaved mode
before (f) and after irradiation (g). In contrast to what was observed from the individual components, i.e. the CdSe nanorods
and Au spheres which show positive charging, the nanodumbbells showed negative charging after illumination. Experiments
on insulating substrates indicated no charging effects, therefore the charging of the dumbbells depended critically on the
escape pathways of the photogenerated holes from the CdSe nanorod to the substrate [219].
We need to remark here a major drawback of CdSe–Au nanodumbbells in which the Au domain is grown on the tips
of CdSe nanorods at room temperature: namely there appears to be no preferential epitaxial relationship between Au and
CdSe [220]. The lack of preferential crystallographic alignment will probably represent a limitation in nano-electronics,
because each metal–semiconductor nano-interface would behave in a different way [217]. A recent study by Figuerola
et al. has demonstrated however that annealing of these nanocrystals yields well-defined CdSe//Au interfaces, located at
the CdSe polar surfaces [221]. The higher quality of these interfaces should make the heterostructures more suitable for use
in nanoscale electronic devices.
Electron-beam induced deposition (EBID) provides a single step approach to fabricate electrical contacts to single
nanorods, with no need of a lithography process to define the electrode pattern. In this method the material for the electrodes
is deposited from organometallic precursors injected from the gas phase into the electron beam, and the pattern of the
scanning electron beam defines the electrode shape, allowing for nanoscale resolution. Moreover, the conductive behavior
of single nanorods contacted by this method is different from the characteristics of the DBTJ, which suggests that the nanorod
is in the strong coupling regime to the electrodes. Steinberg et al. used EBID to contact single CdSe and CdS nanorods [213]. In
their measurements they observed smooth current voltage curves that could be fitted using the WKB approximation [222],
which results in the Fowler–Nordheim formula [223] for current density. This finding suggests that the tunneling barrier
typically formed at the contacts (which results from surface contamination or surfactants on the nanorod surface) was absent
if EIBD was used for the fabrication of the electrode contacts. Instead, it seems that the electrons were injected through a
triangular barrier, which was formed by the offset of the work functions at the interface (Φmetal = 4.5 V for tungsten and
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Φnanorod around 4.3–4.5 eV) and the tilted band structure resulting from the bias voltage. The electron affinity of the nanorods
depends on the quantum confinement (which leads to a significant deviation from the electron affinity of the bulk material)
and on the temperature. The first term can be estimated from the band gap values obtained by optical experiments, and
the temperature dependence can be described by the Varshni equation [224]. Single nanorods contacted by EBID showed
bi- and multi-stable conductive behavior [213], and this could be correlated to the charging and discharging dynamics of
surface traps in the vicinity of the contacts and the nanorod surface.
The bi-stable conductivity could be modeled by considering only a single surface trap, which makes this behavior
interesting for charge sensor applications. The electrical injection of charge carriers into valence or conduction band levels
is possible for a single nanorod connected in an electrode junction, if the applied bias potential exceeds the band gap of
the nanorod. Radiative recombination of the injected charges would lead to the emission of photons and consequently
to electrically driven luminescence of the nanorods. Gudiksen et al. [211] used the EBL overlayer approach to contact
single CdSe nanorods and employed the doped silicon substrate as a gate electrode (Fig. 3.10(a)). At low bias voltage
the Coulomb charging behavior described above was observed. However, the nanorods started to emit light when the
bias voltage exceeded a certain threshold value that was comparable to the band gap, as can be seen in Fig. 3.10(b). The
interesting question is if this electroluminescence originated indeed from the radiative recombination of carriers that were
injected into the semiconductor nanorod. For this to occur, the Fermi levels of the source and drain electrodes would need
to align with conduction and valence band levels of the nanorod, respectively, which would be possible only for highly
symmetric devices. Fits to the Coulomb charging of the nanorods at low bias voltage showed that electroluminescence was
observed from devices with asymmetric tunnel resistances for the two contacts. Also the magnitude of the gap around
zero bias varied considerably for the light emitting devices. These observations make resonant electron/hole injection
very unlikely. The authors therefore suggested that the nanorod emission results from inelastic scattering of the injected
carriers with excitons (Fig. 3.10(c)) or directly from inelastic tunneling which is resonantly enhanced by the exciton
level structure of the nanorod (see Fig. 3.10(d)). Gudiksen et al. estimated the efficiency of the electroluminescence as
10−5 photons per electron passage, which is about a factor of 100 smaller than the best efficiency reported from a
nanocrystal–organic hybrid light emitting device [225]. In a different work on electroluminescence of CdSe nanowires
(which were considerably longer than the nanorods discussed so far) it was shown that the emission occurs in the vicinity of
the metal electrode–semiconductor nanorod contacts, which confirmed that inelastic scattering of the tunneling electrons
is at the origin of the electroluminescence [226].
3.3. Electrical and photoconduction properties of nanorod assemblies
The electrical properties of assemblies of nanorods are of great interest for applications in photovoltaics and in light
emitting devices. The more complex shape of nanorods with respect to spherical nanoparticles can lead to improved
properties and performances. Defined geometrical arrangements that are coupled to distinctive physical properties of
individual nanorods (for example, linearly polarized absorption and emission) could be translated into unique and
predictable macroscopic properties of the ensembles. Close-packed arrays of nanorods could lead to collective effects that
would result, for example, from the structure of the superlattice that is formed. The conductive properties of an oriented
ensemble of nanorods could be expected to show signatures of mini-band formation and charge delocalization. Steiner
et al. [227,228] investigated ensembles of CdSe and InAs nanorods and found a significant reduction of the band gap
measured on close-packed ensembles of rods with respect to isolated nanorods. This reduction was more pronounced in
InAs nanorods than in CdSe nanorods due to the larger Bohr radius and consequently stronger coupling in InAs. Also the
geometry and orientation of the nanorod ensemble on the band gap have been measured by STS. A vertically packed cluster
of CdSe nanorods demonstrated a stronger reduction of the band gap than an ensemble of laterally aligned nanorods [227].
One major issue in devices based on layers of nanorods is the electrical coupling in between the individual nanorods.
Typically, the rod-to-rod charge transfer is very poor due to the insulating nature of the surfactants that cover the nanorod
surface. One successful strategy to overcome this problem is to exchange the long chain surfactants that were used to
stabilize the nanocrystals during the nanocrystal growth with shorter chain molecules, which might even supply charge
carriers to the nanocrystals. In this respect surfactant exchanges with hydrazine and pyridine have lead to a significant
increase in film conductivity, and hydrazine treatment resulted in n-type conductivity of PbSe nanocrystal films [229].
Another strategy to increase the conductivity of the nanorod layer is to anneal the device under vacuum or under inert
atmosphere, which leads to a carbonization of the surfactants (at sufficiently high temperature, around 300–350 °C) [230].
Planar devices with micrometer spaced electrodes represent a useful tool for probing the photoelectrical properties of
nanorod arrays, where the nanorods can be deposited onto the electrode structures from solution either by spin coating,
slow solvent evaporation under controlled atmosphere or by coffee stain evaporation processes [202,203,231]. Randomly
oriented nanorod arrays were fabricated for example by Romero et al. [232] by drop casting the nanorod solution on
interdigitated electrode structures. However, even by simple drop casting the nanorods tended to self assemble side-byside into track like structures. These nanorod films demonstrated hysteresis and switching between high and low conductive
states and also showed peculiar current oscillations. Considering that the conductive path between the electrodes is formed
by nanorod tracks, which should provide the best electrical coupling, the origin of the film properties could be assigned
to superlattice effects and tunneling through interface states related to the surfactant molecules. In the low conductive
regime the current is dominated by charge tunneling through the insulating barriers between the rods. When the voltage
R. Krahne et al. / Physics Reports 501 (2011) 75–221
117
Fig. 3.10. (a) Scanning electron microscopy image in false color of a single nanorod trapped between source and drain electrodes. The inset shows a
TEM image of the nanorods; (b) Electroluminescence intensity plotted together with a current voltage spectrum. The single nanorod transistor design is
sketched in the inset; (c)–(d) Illustrations of the processes that can lead to electroluminescence. In (c) a tunneling charge inelastically scatters and creates
an exciton that recombines under the emission of a photon. In (d) the photon is created directly via the inelastic scattering of the tunneling charge.
Source: Taken with permission from Ref. [211].
bias exceeds a certain threshold value (that differs for each sample) the energy levels of the nanorods and the interface state
can align and charges can resonantly tunnel through the barrier, as illustrated in Fig. 3.11, which leads to the switching into
the higher conductive state. Following this model, the current oscillations that are observed in the higher conductive state
result from subsequent alignment of the nanorod and of the interface state levels with respect to the bias voltage. The sheet
resistance of such randomly oriented nanorods is of the order of 1015 /cm, which is a considerably high value and is most
likely due to the poor electrical coupling between the nanorods.
The photoconductive properties of ordered nanorod arrays in which the long axis of the rods is oriented parallel to the
electric field has been studied by Millo and co-workers [202] (Fig. 3.12). Hydrazine treatment was employed to reduce the
inter-rod distance and improve the photoconductivity. The nonlinear current voltage response could be modeled by a fieldionization process [233], according to which the current (I (v)) dependence of the site-to-site potential ν can be expressed
as:
exp
I (v) ∝
√
−4 2 h̄2 d
3m(ev−γ )
[
mΦ
h̄2
3/2
m(Φ +γ −ev)
−
γ −ev
1 + exp − a
h̄2
]
.
(3.12)
Here m is the rest mass of the electron, e is the elementary charge, γ is the energy cost between the initial and final
states that arises from the energy required to overcome the Coulomb interaction of the photoexcited electron–hole pair
and the interaction of each charge with its respective image charges. A phenomenological parameter a describes the
effective broadening of γ due to variations in the nanocrystal size and inter-nanocrystal separation. Φ and d are the height
and the width of the tunnel barrier, respectively. The site-to-site potential can be evaluated by dividing the applied bias
voltage by the number of rods that are needed to form a conductive path between the electrodes for the parallel and
perpendicular configuration. Millo et al. [202] observed a nonlinearity in the photoconductivity of the nanorod arrays at the
transition when the site-to-site potential exceeds the energy threshold that is needed to separate the photoexcited charges.
Below this threshold the photoconductivity is dominated by the increasing effectiveness of the electric field in exciton
ionization, whereas when the external field is larger than γ the photoconductivity is dominated by tunneling through the
inter-nanocrystal barrier characterized by Φ and d. Photoconductivity and photoluminescence should be complementary
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
a
E2
E1
c
b
100
80
I (pA)
60
40
20
localized states
10
0
5
10
15
20
Vbias (V)
Fig. 3.11. Charge transport in rod arrays in which the long axis of the rods is oriented perpendicular to the external electric field. (a) Illustration of the
nanorod array orientation with respect to the electrodes; (b) Current voltage characteristics of parallel tracks of nanorods showing hysteresis and periodic
oscillations. The upper inset shows the oscillation on a magnified scale, the lower inset depicts the temperature dependence.
Source: Taken with permission from Ref. [232].
processes and indeed the quenching of the PL with increasing voltage bias and thus increasing photocurrent has been
demonstrated [202,203]. Persano et al. [203] compared the photoconductivity and PL quenching of core–shell CdSe/CdS
nanorods arrays oriented parallel and perpendicular to the external electric field direction. They found that PL quenching
is more effective for nanorod arrays where the long axis of the rods is oriented parallel to the electric field direction, as
depicted in Fig. 3.13(b). This behavior can be explained by the more effective ionization along the long axis of the nanorods
where the confinement length is much larger than the exciton Bohr radius. For type-I core–shell nanorods (see Section 2)
the PL quenching depends also on the energy of the light excitation. For a fixed external electric field, the exciton ionization
is more efficient when the charges are created in the shell (see Fig. 3.13(b)). If the exciting light energy is smaller than the
band gap of the shell, then the photoexcited charges are localized at the core and an additional energy barrier has to be
overcome for exciton ionization. The spectral response of the photocurrent of the nanorods corresponds very well to their
absorption spectra. Only at high energies above the band gap the photocurrent is reduced, due to more effective trapping of
the photogenerated carriers at surface states [163,233].
3.4. Nanorod assemblies for photovoltaic applications
Colloidal nanocrystals are an interesting material as active layers in photovoltaics, and nanorods should have
advantageous properties regarding the absorption cross section due to their higher surface area compared to spherical
nanocrystals, and because their elongated shape should improve electron transport. Such an idealized all-inorganic solar
cell based on nanorods is sketched in Fig. 3.14(a). All-inorganic nanocrystal solar cells of this type have been fabricated
with spherical nanocrystals [234], but in the case of nanorods it was not possible, so far, to achieve homogeneous, ordered
layers of nanorods on areas large enough to be interesting for solar cells. For example, the elongated rod-shape makes it
more difficult to obtain films without any voids and electrical shortcuts. Another option is to fabricate hybrid solar cell
devices consisting of semiconductor nanorods and conjugated polymers (see Fig. 3.14(b) for an idealized design), which
were investigated, for example by the group of Alivisatos [7,201]. Here, nanorods and the polymer poly (3-hexylthiophene)
(P3HT) were dissolved in pyridine and chloroform to form an organic/inorganic blend that could be spin-coated onto the
anode, in which the P3HT promoted the hole conduction pathway, and the rods the harvesting of the electrons. However,
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8
I (nA)
4
0
–4
–8
–40
–20
0
VSD (V)
20
40
Fig. 3.12. Charge transport in rod arrays with rod orientation parallel to the external electric field. (a)–(b) Scanning electron microscopy image of aligned
nanorod arrays in between planar electrodes. Treatment of the nanorod layer with hydrazine reduces the inter-rod distance and leads to cracks in the film;
(c) Illustration of the nanorod alignment; (d) Nonlinear current voltage response of nanorod arrays under blue laser illumination and in the dark.
Source: Taken with permission from Ref. [202].
in such a blend the nanorods were mostly disordered (Fig. 3.14(c)). These works confirmed the higher external quantum
efficiency of nanorods as compared to spherical nanoparticles, which lead a power conversion efficiency of 1.7% under air
mass 1.5 global solar conditions.
3.5. Thermoelectric properties of elongated nanoparticles
Thermoelectric materials allow for the direct conversion of temperature differences into electric voltage and vice versa
and as such they are used for energy generation, as temperature sensors and as controllers and cooling elements. In principle,
the thermoelectric properties originate from the difference in Fermi levels of two materials, combined with a low thermal
conductivity that helps to maintain the temperature difference. If these materials are put into contact, the Fermi level
alignment leads to charge diffusion over the interface and to an electric field that compensates the Fermi level difference.
Assuming Boltzmann distributions for the charge carriers in both materials (n1 , n2 ), the contact voltage can be expressed
as:
∆U =
kT
e
ln
n2
n1
(3.13)
.
We see that this contact voltage at the material interface is temperature-dependent, and therefore a difference in
temperature at the two contact interfaces leads to a thermovoltage, defined as:
Uth =
k
e
ln
n2
n1
∆T .
(3.14)
The thermoelectric power, also called Seebeck coefficient, S is defined from the open circuit electric field that is created by
the temperature gradient:
∆V
= S,
∆T
or E = S grad T .
(3.15)
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 3.13. (a) Normalized PL amplitude, PLBIAS /PL0 , and photocurrent vs. bias voltage for rod arrays oriented parallel and perpendicular to the external
electric field; (b) Normalized PL amplitude vs. laser excitation energy recorded at a bias voltage of 120 V for nanorod arrays oriented parallel to the external
electric field; (c) Absorption and photocurrent spectra of CdSe/CdS core–shell nanorods recorded at a voltage bias of 64 V under ambient conditions. The
inset shows absorption and photocurrent of the core on a magnified scale.
Source: Taken with permission from Ref. [203].
Fig. 3.14. (a) Idealized scheme of an all-inorganic solar cell based on a layer of nanorods. (b) Optimal design of a hybrid nanorod–polymer solar cell.
(c) Cross-sectional TEM image from a solar cell based on a blend of CdSe nanorods and conjugated polymer Poly(3-hexylthiophene) (P3HT). (d) Structure
of P3HT, illustration of the charge transfer and the design of the solar cell. (e) External quantum efficiency of solar cells based on CdSe nanorods with 7, 30
and 60 nm length. (c)–(e) From Ref. [7].
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Fig. 3.15. Thermoelectric power of some bulk metals at T = 0 °C, with reference to Pb which is set to zero.
Source: From Ref. [235].
a 15.0
b
0.6
–40
0.5
–60
S (10 V/K)
–6
ZT
5.0
–80
0.4
–100
0.3
–120
0.0
0
20
40
60
a (Å)
80
100
–140
50
κ (W/m-K)
10.0
100
150
200
T (K)
250
300
0.2
Fig. 3.16. (a) Thermoelectric figure of merit in dependence of wire width calculated for Bi2 Te3 nanowires. Taken with permission from Ref. [236];
(b) Seebeck coefficient and thermal conductivity of Bi/Te core–shell nanowires. Taken with permission from Ref. [238].
The sign of the thermovoltage allows to identify the type of majority carriers (positive is p-type), as can be seen from
Eq. (3.14). From the ratio of k/e in Eq. (3.14) we expect the thermoelectric power to be in the µV /K range. Another
characteristic parameter is the Peltier coefficient Π that is defined by the heat absorbed by the lower junction per unit
time P and the current I on the junction:
P = Π I,
(3.16)
which leads to Π = ST . Fig. 3.15 lists the thermoelectric power of some bulk metals at T = 0 °C with reference to Pb which
is set to zero. The thermoelectric figure of merit is defined as ZT ≡ S 2 σ T /κ , where σ and κ are the electrical and thermal
conductivity, respectively.
For bulk materials it is difficult to improve the ZT, because electrical and thermal conductivity are related via the
Wiedemann–Franz-law, κ/σ = LT (where L is the Lorenz number), and thus an increase in electrical conductivity results
also in an increase in thermal conductivity. In this respect, nanostructured materials promise novel alternatives to increase
the thermoelectric figure of merit, since size and quantum effects have different impact on the thermal and electrical
conductivity. Nanowires are an especially appealing structure because the electrical conductivity is enhanced due to a
reduced number of channels for electron scattering. At the same time, the electronic contribution to the thermal conductivity
(electron scattering on phonons) is reduced, and for very small diameters surface scattering will further reduce the thermal
conductivity. As an example, Dresselhaus and co-workers calculated the thermoelectric figure of merit (ZT) for Bi and Bi2 Te3
nanowires in dependence of the nanowire diameter, and found a strong increase of ZT when the wire diameter was decreased
below the mean free path of the electrons [236,237] (Fig. 3.16).
Materials for thermoelectric applications should have a small electron effective mass and a highly anisotropic Fermi
surface. A promising material system for thermoelectric experiments can be described by Pn2 X3 , where Pn stands for Bi,
Sb, Pb, and X for S, Se, Te. We will now discuss briefly some thermoelectric experiments on nanorods from the above
mentioned materials. Doped Bi2 Te3 nanorods with 30–130 nm diameter were fabricated in a one-step, aqueous phase
synthesis approach, and films obtained by drop casting of nanorod solution were annealed at 350 °C under vacuum. The
Seebeck coefficient of these films was −100 µV/K, revealing the n-type doping, but was significantly lower than the Seebeck
coefficient of undoped bulk Bi2 Te3 , which behaves as a p-type semiconductor [239]. Films of cross-linked Bi2 S3 nanorods
have been fabricated on Si substrates and showed a high Seebeck coefficient of −755 µV/K [240]. High aspect ratio PbTe
nanorods were assembled into thin films onto glass substrates prepatterned with Au electrodes by dip coating, and a
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
subsequent hydrazine treatment was applied in order to enhance the film conductivity [241]. Such films showed a band
gap of 0.27 eV, and a Seebeck coefficient of 263 µV/K, which is comparable to bulk PbTe. Te/Bi core–shell nanowires were
fabricated by a two-step solution phase method, and the characterization of thin films of this 1D nanocomposite material
yielded a high Seebeck coefficient of −128 µV/K at T = 300 K, but due to the comparatively low electrical conductivity the
obtained ZT was only 0.05 [238].
An enhancement of the thermopower has also been observed by Zuev et al. in single nanowires of Sb2 Te3 when the
diameter became smaller than 40 nm [242]. This work showed also a linear increase in thermopower with increasing
temperature according to the classical Mott relation for diffusive thermopower S, which is defined as [243]:
S=−
π 2 k2B T d ln σ (E )
3e
dE
(3.17)
4. Optical properties of elongated metal nanoparticles
4.1. Overview
Metal nanostructures are well known to produce colors, due to their strong light absorption and scattering in the visible
region of the spectrum (see Fig. 4.1). This property has been exploited in glass artworks like, for example, the rose window
of the Notre Dame cathedral in Paris where silver and gold nanoparticles make the coloring of the glass. This effect is
caused by one of the most important types of interactions of metal nanoparticles with the electromagnetic field. Metals
are characterized by the presence of free electrons, which can be promoted via ‘‘intraband’’ transitions to empty energy
levels in the same band or to empty levels of an empty overlapping band, via absorption of low energy photons. An incident
electromagnetic field can elicit collective oscillations of these free electrons [20–23] and when the frequency of the incoming
radiation matches the frequency of an intrinsic electron oscillation, energy from the incoming radiation is strongly absorbed
(i.e. a ‘‘resonance’’ occurs). Such energy is then transformed into heat and into light (scattering). These collective and
coherent oscillations of electrons cause a displacement of the electrons from the nuclei, leading to the formation of various
possible distributions in the surface charges (Fig. 4.1). Each type of surface charge distribution is characterized by a collective
oscillation mode, also termed as localized surface plasmon (SP). There are various factors that influence the possible types
of SPs in nanostructures and the resonance frequencies at which they are observed. Other types of interactions of metal
nanoparticles with the electromagnetic radiation are:
(i) Absorption processes due to electronic transitions of bound electrons from occupied to empty bands (interband
transitions);
(ii) Scattering processes due to accelerations of electrons in the nanostructures caused by the electromagnetic radiation,
which leads to isotropic radiation.
Interband electronic transitions in metal nanoparticles are rather insensitive to particle size (except in the case of sub2 nm metal clusters, which are made of a few atoms), and are located at high energy (for example above 3.8 eV in silver
nanoparticles). For nanoparticles with diameters between 10 and 30 nm, the dominant effect in the visible region is the
excitation of plasmon modes. In this size regime, and in the simple case of spherical nanoparticles, basically a single plasmon
mode of dipolar character is excited and its wavelength is independent of particle size, but is strongly dependent on the
dielectric function of the surrounding medium, and also the width of the plasmon resonance band remains poorly correlated
with size. For particles smaller than 10 nm, a size that is significantly smaller than the mean free path of electrons, free
electrons collide frequently with the surface of the nanoparticles, and this leads to broadening of the plasmon resonance
band due to dephasing effects. At sizes above 30 nm other effects become non-negligible, like retardation, which lead to
broadening and loss in intensity of the plasmon band, as well as to a spectral shift in the red. At sizes above 100 nm, the
optical properties are dominated by scattering of light. In addition, higher order modes (i.e. quadrupolar, octupolar) start
contributing to absorption.
The shape of metal nanoparticles has perhaps the most striking influence on their optical properties. As an example,
in rod-shaped nanoparticles the plasmon mode is split into two modes, a longitudinal one and a transverse one. In the
following, our discussion will be as much as possible general and will cover all metal nanoparticles, but in some occasions
we will focus our attention on gold nanoparticles, for several reasons: first of all gold nanoparticles do not undergo oxidation
processes, hence their theoretical description does not need to take into account the presence of a surface oxide layer
(which complicates considerably the theoretical treatment); secondly, gold nanoparticles are now fabricated in a variety
of sizes and shapes, hence the testing of the various computational approaches on real samples is directly feasible; lastly,
gold nanoparticles have been employed and/or proposed in a variety of applications. In this section, we will give a brief
overview of the optical properties of metal nanoparticles in various size regimes, and we will focus then the attention on
the effect of shape, with particular emphasis on elongated nanoparticles. Many excellent research articles, reviews and
books on this topic have been published in the last years, covering many aspects of localized surface plasmons in metal
nanostructures [20–32].
In the following, we will be concerned mainly with the far field behavior of metal nanoparticles, hence their interaction
with the electromagnetic field via absorption and scattering processes (each characterized by a specific single particle cross
R. Krahne et al. / Physics Reports 501 (2011) 75–221
123
Fig. 4.1. Left: vials containing gold nanoparticles (vial on the far left) and hollow nanoparticles (all the other vials) of various diameters and wall thicknesses.
Taken with permission from Ref. [32]. Right: sketches showing surface plasmon resonance in spherical nanoparticles and nanorods inside a polarized light
field. Taken with permission from Ref. [26].
section, σabs and σsc ). We will not deal here with the near field properties, which govern the behavior of nanoparticles in
close proximity with other particles/molecules. These will be discussed at that end of this section. Let us assume that we
are dealing with a dilute sample of nanoparticles, hence each particle behaves independently from the others with respect
to the incident radiation. Under these circumstances, the intensity of light that is transmitted through this sample is given
by the expression:
I = I0 e[−(σabs +σsc )NL] .
(4.1)
In the above expression I0 is the intensity of the incoming light, N is the number of particles per unit volume, and L is the
length of the path travelled by the light in the sample. The quantity σabs + σsc is also known as the extinction cross section.
4.2. Dielectric function of metal nanoparticles
We follow here the approach described by Noguez [244]. Let us suppose, as a first approximation, that we can treat
nanoparticles as objects with the same dielectric function ϵ as the bulk metal. In a metal, the dielectric function can
be considered as formed by a contribution from the electrons near the Fermi level, which are essentially ‘‘free’’, and a
contribution from bound electrons. We call the contribution of free electrons to ε as εintra . The bound electrons, populating
fully occupied bands, interact only with higher energy photons and can be promoted to empty bands separated from the
occupied band by an energy gap (therefore via interband transitions). We call the contribution of these electrons to ε as
εinter . If εexp is the measured bulk dielectric constant, then we can write εexp = εinter + εintra .
The dependence of εintra on the frequency ω can be described satisfactorily by the Drude model [243,245]. This model
approximates the metal as a plasma, i.e. a medium composed of an equal number of conduction band electrons and of
positively charged ion cores, in which the conduction electrons are not bound to atoms. The contribution of these free
electrons to the dielectric function is approximated to:
εintra (ω) = 1 −
ωp2
ω (ω + i/τ )
(4.2)
.
In the expression above, ωp is the plasma frequency, defined by ωp2 = 4π ne2 /m, with n the density of conduction electrons,
while m and e are the mass and the charge of the electron, respectively. The physical meaning of the plasma frequency can
be easily understood in the following way. In the plasma, if all the free electrons are slightly displaced in one direction with
respect to the ions by application of an external electric field, a Coulomb force is created due to macroscopic displacement
of the electrons with the respect to the positive ions lattice. This force tries to ‘‘pull’’ the electrons back, and, if the field is
suddenly turned off (hence no other forces are acting on the system), the force causes the whole plasma to oscillate at a
frequency equal to ωp [243]. The ‘‘damping factor’’ 1/τ in the Drude model arises instead from various scattering processes
of electrons (mainly electron–electron and electron–phonon scattering) and is the inverse of its relaxation time τ .
In small particles (i.e. radii smaller than 10 nm), there is an additional size-dependent damping term which is related to
electron scattering from the boundary of the particle. In this size regime the free-electron contribution to dielectric function,
which takes into account the surface damping term, is [246]:
εintra (ω, R) = 1 −
ωp2
ω (ω + i/τ + iωD )
.
(4.3)
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 4.2. Real (left) and imaginary (right) part of the dielectric function of silver, gold and copper. In each panel, the interband and intraband contributions
are shown. Worth of notice is that εim is in general small compared to εreal and that, above the short wavelength range that is dominated by interband
transitions (325 nm for silver, 550 nm for gold and 600 nm for copper), εim does not vary much with wavelength.
Source: Taken with permission from Ref. [24].
In the expression above ωD = A · v F /R is a frequency damping term that is directly proportional to the Fermi velocity vF
(i.e. the Fermi velocity associated to the Fermi energy EF via EF = 21 mvF2 ) and is inversely proportional to the particle radius
R through the constant A.
Since the dielectric function εexp can be measured, and the εintra part can be modeled using the Drude model, one can
extract the behavior of εinter at varying wavelength for various metals. Fig. 4.2 shows the real part (εreal ) and the imaginary
part (εim ) of the dielectric function for Ag, Au, and Cu [24]. In the panels, both the real and the imaginary components are
decomposed in their interband and intraband contribution. One can see for example that for silver the major contribution of
interband transitions to the imaginary part is below 325 nm, while in gold is below 550 nm, and for copper is below 600 nm.
On the other hand, the contribution of interband transitions to the real part is only a positive, slowly varying background
in all cases. Interband transitions contribute to absorption, but do not contribute to plasmon modes. Therefore, as we shall
see in the next section, in metal nanoparticles the optical absorption spectra are complicated by these interband transitions
effects.
4.3. Plasmonic properties of small spherical metal nanoparticles
In principle, one can model the behavior of metal nanoparticles by solving the Maxwell’s equations for various sizes and
shapes of the nanoparticle, and assuming for the metal nanoparticle the dielectric function of the bulk metal, based only on
the Drude model, as done by Mie [247], or considering also the contribution from interband transitions. The simplest case
to treat is that of a spherical particle, in a uniform medium of dielectric function εmedium , and immersed in a monochromatic
radiation field having a plane wave form. The analytical solution of this problem exists, and yields an extinction cross section
that is a series whose terms contain the various multipolar contributions for both extinction and scattering:
σext =
σsc =
∞
2π −
|k|2
l=1
∞
2π −
2
|k|
l=1
(2l + 1) Re {Al + Bl }
(2l + 1) |Al |2 + |Bl |2 .
2π
√
ε
(4.4)
(4.5)
medium
In the expression above k =
is the magnitude of the wave vector of the incoming radiation, and Al and Bl are the
λ
electric and magnetic scattering coefficients for the multipolar mode defined by l (l = 1 for dipole, l = 2 for quadrupole,
l = 3 for octupole, and so on). They are expressed in terms of Riccati–Bessel functions that depend both on the particle radius
R and on λ. The analytical solutions above give a satisfactory description of the optical behavior of metal nanoparticles in a
broad range of sizes. There are also many approximate solutions to the problem, since one can distinguish various limiting
cases. In one limiting case for example the size of the particle is much smaller than the wavelength λ of the radiation
R. Krahne et al. / Physics Reports 501 (2011) 75–221
125
(by much smaller here we mean smaller than one tenth of λ). Then the field felt by the particle can be assumed to be
constant throughout the particle (albeit it is still time-dependent). This is called the ‘‘quasi-static’’ or ‘‘electrostatic’’, or also
‘‘non-retarded’’ regime. This is equivalent to assume that the speed of light can be considered infinite, so that the Maxwell
equation correlating electric field E and magnetic inductance B:
∇ ×E =−
1 ∂B
(4.6)
c ∂t
and the equation correlating magnetic field H , electric displacement field D and the current density Jj :
∇ ×H =
1 ∂D
c ∂t
+
4π
c
(4.7)
Jf
are effectively reduced to ∇ × E = 0 and ∇ × H = 0. Under these approximations, there is no coupling between electric
and magnetic fields, and the optical response of the nanoparticle is basically dictated by the electric field. From the terms in
the expressions above, one can derive a simple expression for the multipolar polarizability, which is the following:
αl =
3λ2
4π
2
Al = 4π R3 εmedium
l (ε − εmedium )
lε + (l + 1) εmedium
.
(4.8)
Hence the resonance conditions occur for ε = −εmedium (l + 1) /l. Let us consider for now only the case of a dipole mode.
Then the polarizability is given by the formula:
α1 = 4π R3 εmedium
ε − εmedium
ε + 2εmedium
(4.9)
and the resonance is reached for ε = −2εmedium . It turns out that for particle radii smaller than 30 nm the charges are
essentially displaced homogeneously inside the particle (i.e. all free electrons are moving in phase), such that mainly an
electrical dipolar distribution of charges is generated on the surface of the particle. This is equivalent to say that the incident
light polarizes uniformly the nanoparticles, hence the only significant contribution in the interaction of the small particle
with the radiation is of electrical dipolar type, i.e. high order electric multipoles are negligible. Furthermore, for small
particles, say around 10 nm or smaller, their interaction with the incoming light is dominated by non-radiative processes
(i.e. light is absorbed but is not emitted again). We can then assume that, at these small sizes, scattering, which is instead
a radiative process and which indeed requires coupling between electric and magnetic field, is negligible (i.e. σsc is close
to zero). Then the extinction cross section of a single particle is the same as the absorption cross section and can be
approximated by [245]:
3/2
σext ≈ σabs =
24π 2 R3 εmedium
λ
εim
.
2
(2ϵmedium + εreal )2 + εim
(4.10)
Here R is the radius of the particle, and εreal and εim are the real and the imaginary parts of the dielectric function of the metal,
respectively. An important aspect to notice is that εim is in general small compared to εreal and does not vary much with the
frequency ω (or with the wavelength), as shown in Fig. 4.2. Therefore, if we plot σabs vs. ω, the maximum in absorption is
reached for εreal = Re (ε (ω)) = −2εmedium , (this is also known as the Fröhlich condition) while the peak width is dependent
on εim and εmedium . So, as a first approximation, the size of the particle has little influence on the peak position and on the
width of the plasmon band. For gold nanoparticles in water, for example, the resonance peak is around 520 nm.
In the case of particle radii smaller than about 10 nm, however, the dielectric function of the particles becomes sizedependent, as described above. It turns out that this so-called ‘‘non-local’’ or ‘‘surface dispersion’’, or ‘‘surface damping’’
effect does not influence the peak position of the resonance band. Instead, it leads to a broadening of the plasmon band,
because the electrons lose coherence in their motion at each scattering event with the surface of the nanoparticle. Since the
plasmon bandwidth Γ can be correlated to the dephasing time T2 of the plasmon oscillation via the relation Γ = 2h̄/T2 (see
also Section 4.7), a larger bandwidth means a faster dephasing of the oscillation. In small particles then the collision of the
electrons with the surface introduces an additional term that contributes to the loss of coherence of the electron oscillation,
therefore to a shortening of the overall dephasing time (and equivalently, a broadening of the band). The broadening of
the plasmon band in small particles is so relevant that the band practically disappears for particle sizes smaller than 2 nm.
Fig. 4.3(a) and (b) reports the calculated extinction efficiency for two spherical silver particles, of radii equal to 5 nm and
50 nm, respectively, with and without contribution from surface damping. For the 5 nm particles the surface damping
contribution is significant.
In a recent study, Baida et al. were able to record the optical extinction coefficient of individual silver nanoparticles in
the size range between 10 and 50 nm, coated with a silica shell [248]. The authors found that already for particles smaller
than 25 nm the plasmon band started broadening, an effect that could be indeed interpreted classically due to surface
damping. More in general, the authors found that the full width at half maximum of the plasmon resonance was inversely
proportional to the size of the particles [248]. In the study, the authors pointed out that the surface contribution can be
also interpreted quantum mechanically as due to a breakdown of the translational invariance, which reflects an increasing
probability of dipolar transitions between electron eigenstates when the size of the particle is reduced, as the momentum
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
is not a good quantum number any more. The study evidenced that, especially for small metal clusters, one often needs to
go beyond the simple and generic ‘‘classical’’ models that we have been describing above, and has to study more in detail
the electronic structure of the specific metal. Various models have been developed in this respect. Two important reviews
covering the physics of simple metal clusters (alkali metals and noble metals) are the work of de Heer [249], which compares
the predictions of various theoretical models with experiments for spherical clusters, also covering non-spherical clusters
(behavior of the mass-abundance spectra, polarizabilities, ionization potentials, photoelectron spectra, optical spectra, and
fragmentation phenomena) and the work of Brack [250], which treats in detail the jellium model of simple metal clusters.
From the bulk side (i.e. bulk metal surfaces) on the other hand, a broad overview of the various methods for calculating the
electronic structure of metals surfaces (above all density functional theory) and electronic excitations at metal surfaces is
the work of Liebsch [251]. In the Section 4.10.1, which describes the near field optical properties of metal nanoparticles, we
will consider the case in which ‘‘classical’’ calculations and quantum mechanical calculations yield substantially different
results.
4.4. Plasmonic properties of large spherical metal nanoparticles
Many effects need to be taken into account for large nanoparticles. Let us consider first higher order multipolar modes.
In spherical nanoparticles, higher order multipolar modes become non-negligible at particle radii around 30 nm and higher,
where quadrupole modes start appearing. According to the Eq. (4.8), the resonance condition for a quadrupolar mode is
ε = − (2/3) εmedium , so it is shifted at lower energies with respect to the dipolar mode. The quadrupolar mode is first
manifested as a small shoulder at the shorter wavelength side in the optical absorption spectrum that is still dominated by
the dipolar plasmon mode, but as nanoparticle size increases the contribution from the quadrupolar mode increases (see
Fig. 4.3(b) and (d)). In some cases, for example in Au nanoparticles, the quadrupolar mode, for small particle sizes, being
weak in intensity, is hidden by the interband transitions [24]. In Ag nanoparticles instead, such overlap does not occur and
therefore the quadrupolar mode is already evident at smaller sizes compared to gold [24]. Octupolar modes are relevant
only for much bigger particles, and for even higher order modes, the resonance conditions approach ε = −εmedium , which is
indeed the resonance condition of the planar surface plasmon (i.e. plasmon resonance for a flat metal surface in contact with
a dielectric medium of dielectric function εmedium ). This is because in multipolar modes fast charge oscillations practically
cancel out the interaction between distant charges, therefore each small region on the surface of the metal behaves as in
a planar bulk metal. These considerations however need to be taken with care, since at the large sizes at which multipolar
modes are active, the optical behavior is additionally influenced by retardation effects. Indeed, in addition to multipolar
modes, when the nanoparticles radius is about 30 nm or larger, retardation effects influence the optical response (see
Fig. 4.3(e) and (f)). One of this retardation effects (the so-called ‘‘energy-shifting’’ effect) arises because the particle size
is not any more negligible with respect to the wavelength of the radiation, or equivalently stated, the wavelength of the
incoming radiation cannot be considered as infinite, hence the field is not homogeneous everywhere inside the particle.
More precisely, these effects appear already when the diameter 2R of√the particle is around 1/10 of the mode wavelength
λm of the radiation in the medium surrounding the particle (λm = λ/ εmedium ).
An elementary explanation of this effect can be sought by considering that when a dipole mode is excited in a spherical
particle, the distance between opposite charges can be approximated to the nanoparticle diameter. Hence one side of the
particle will ‘‘feel’’ any change occurring at the opposite side at the particle with a phase retardation equal to 2π · 2R /λm .
The oscillation period of the dipole mode increases in order to take such retardation into account, which is equivalent to
say that energy of the plasmon peak associated to a dipole mode shifts to the red as the particle size increases. When higher
order multipolar modes are involved, the distance between opposite charges on the surface of the particle is smaller than the
particle diameter, and the phase retardation is smaller than in the dipolar mode. Indeed, the phase retardation in a spherical
metal particle scales roughly as 2π · 2R/(l · λm ) with l = 1, 2, 3, . . . for dipole, quadrupole, octupole mode, and so on. Dipole
modes therefore are influenced more by the particle size, due to retardation effects, than higher order modes.
Another retardation effect is due to radiation scattering, which becomes already important for particles radii bigger than
20 nm, and is the dominant effect for particles radii above 100 nm. In these larger particles the electrons are accelerated
as a consequence of the electromagnetic field generated by the incident radiation. Therefore, they also start radiating in
all directions, which causes them to lose energy, i.e. part of the energy of the plasmonic oscillations is converted into
photons. This leads to a broadening in the surface plasmon resonance and to a decrease in its intensity. Hence, as opposed
to small particles, in large particles their interaction with the incoming light is dominated by radiative processes (i.e. energy
is absorbed by the incoming light but is transformed again into photons).
In order to understand more quantitatively how the dipolar resonance is affected by retardation effects for large particle
sizes, hence beyond the purely electrostatic regime, a modified approach is required, which takes into account the finiteness
of the wavelength of the incoming radiation. Let us consider the approach of Meier and Wokaun [252]. If we still stick to
the approximation that the external field E0 is spatially uniform around the particle, then also the field that is generated
inside the particle is uniform and equal to E0 + Edep , with Edep being the so-called depolarization field [253]. Let us call P the
polarization of a sphere of radius immersed in the external field E0 . Then also P can be considered at a first approximation
as homogeneous over the volume of the sphere and given by the expression:
4π P = (ε − εmedium ) E0 + Edep .
(4.11)
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
5
9
a
4
b
8
7
6
3
5
4
2
3
2
1
1
0
250
300
350
400
450
500
550
7
0
300
400
500
600
700
12
c
6
10
5
d
8
4
6
3
4
2
2
1
0
250
300
350
400
450
0
250
300
350
400
450
500
40
20
e
18
16
35
f
30
14
25
12
10
20
8
15
6
10
4
5
2
0
250
300
350
400
0
250
Wavelength (nm)
450
300
350
400
450
Fig. 4.3. (a)–(b) Extinction efficiency for Ag spherical nanoparticles with radius equal to 5 nm and 50 nm, respectively, calculated using the analytical solution of the Mie theory, excluding surface damping effects (black line) or including them (green dashed line). One can see that surface damping is relevant
for the 5 nm particles, while it is negligible for the 50 nm ones. In (b), the additional peak at shorter wavelengths is due to the quadrupolar mode. The extinction at wavelengths shorter than 325 nm is due to absorption caused by interband transitions. The shape of the curve in this region is insensitive to size.
(c)–(d) Absorption efficiency for spherical silver nanoparticles with radius equal to 7 nm and 40 nm, respectively, using the analytical solution of the Mie
theory (black curves). The green dashed curve represents instead the contribution of the quadrupolar plasmon mode to extinction. This is negligible for the
7 nm particle, but can be already appreciated for the 40 nm particle (see also panel b); (e)–(f) Extinction efficiencies for spherical silver nanoparticles with
radius equal to 20 nm and 40 nm, respectively, using the analytical solution of the Mie theory (black lines), considering only the dipolar approximation
(green dashed lines), dipolar + retardation effects (blue dotted lines) for the 20 nm particle, dipolar + retardation effects + quadrupolar (blue dotted lines)
for the 40 nm particle, and only the quadrupolar contribution for the 40 nm particle.
Source: Taken with permission from Ref. [21].
Meier and Wokaun [252] determined the depolarization field at the center of a spherical dielectric nanoparticle by
assigning a dipole moment dp = PdV to each volume element dV of the particle, by calculating the dipolar field generated
by each of these dipoles at the center of the particle, and by integrating over the whole volume of the sphere (i.e. over all the
dipoles). This approach is known as the ‘‘Modified Long Wavelength Approximation’’. The depolarization field they found
was related to the polarization through the following expression:
Edep =
−
4π
3
24
+k
π
3
2
R + ik
32
4π
3 3
R
2
P
(4.12)
where k = 2π /λ. The first term in the parenthesis is the contribution to the depolarization in the quasi-static regime. The
second term contributes to what is called ‘‘dynamic depolarization’’ [252,254] and is interpreted as ‘‘dephasing’’ between
radiation emitted by different regions of the particle [254]. It has also been interpreted as the depolarization field created
by the oscillating dipole [244]. The third term is instead a purely imaginary term and is related to radiation emitted from
the dipole. If we keep only the first term of Eq. (4.12) for determining the polarization in the expression (4.11), then we use
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
this value of the polarization in the expression linking the dipolar polarizability α to the polarization P and to the external
field E0 :
4
3
π R3 P = εmedium α E0
(4.13)
then we get the same expression for the dipolar polarizability as described by Eq. (4.9). A more complicated expression for
the polarizability is obtained instead if we include also the second and the third term of expression (4.12) in the calculation. In
addition, Meier and Wokaun considered the more elaborate case in which the polarization inside the particle is not uniform.
In this case the exciting field felt by different regions of the particles is not the same, as discussed above. An expression for
the dipolar polarizability of a sphere of radius R and volume V, which embeds all the effects described so far, is the following
[252,255]:
α=
1−
1
3
+
εmedium
ε−εmedium
−
1
10
1
30
(ε + εmedium ) x2 + O x4
(ε + 10ε medium ) x2 − i
3/2
4π 2 εmedium V
3
λ3
V.
+ O x4
(4.14)
In the expression above x is the so-called ‘‘size parameter’’ (x = π R/λ). In the denominator, the quadratic term is actually
the ‘‘energy-shifting’’ effect of the depolarization field inside the nanoparticle, which causes a shift of the resonance peak,
and is related to the second term of expression (4.12). Meier [28] gave an intuitive explanation of this term: simply stated,
the bigger is the particle, the farther apart are the charges in the oscillating dipole and the smaller is the restoring force.
Therefore, the resonance condition shifts to lower energies. The imaginary term in the denominator is related to radiation
damping. The quadratic term in the numerator accounts instead for the ‘‘non-uniform’’ exciting field inside the particle,
hence for the fact that
the polarization throughout the particle has a varying phase, and it also causes a shift in the red of the
resonance. The O x4 terms can be safely ignored. One can easily see that if all the former terms are neglected, we get back
the expression of the dipolar polarizability as described by Eq. (4.9). The retardation effects on the extinction efficiency of
nanoparticles can be seen clearly in Fig. 4.3(e) and (f).
4.5. Plasmonic properties of ellipsoidal-shaped metal nanoparticles
In metal nanoparticles whose shape deviates from that of a sphere, analytical solutions of the extinction cross section
based on the Mie theory have not been found, and only approximations are available. A particularly fortunate case is
that of ellipsoidal particles, for which analytical solutions are found in the quasi-static limit. For an ellipsoid with three
principal axes of radii R1 , R2 , R3 the dipolar polarizability along its i-th axis in the quasi-static approximation is given by the
expression [256]:
αi = V εmedium
ε − εmedium
.
εmedium + Ni (ε − ε medium )
(4.15)
In the expression above, V is the volume of the ellipsoid (V = 4π R1 R2 R3 /3) and the term Ni is the ‘‘geometrical factor’’ along
the i-th axis, which is given by the formula:
Ni =
R1 R2 R3
2
∫
∞
0
ds
s+
R2i
s+
R21
s + R22
s + R23
.
(4.16)
The three geometrical factors obey the relation N1 + N2 + N3 = 1, hence in the case of a sphere N1 = N2 = N3 = 1/3. From
the equation above it can be seen that in ellipsoidal particles the resonance conditions are set by:
1
.
ε = εmedium 1 −
Ni
(4.17)
In the simple case of R1 = R2 ̸= R3 , one can define the aspect ratio of the ellipsoid as R3 /R1 . As soon as the aspect ratio
increases (so we have a prolate ellipsoid), the N3 polarizability along the 3rd axis decreases, and, according to the expression
above, the resonance conditions for the polarizability along the 3rd axis (the longer axis) require larger negative values of
ε. This is usually referred to as the ‘‘longitudinal plasmon resonance’’ and occurs at considerably longer wavelength than
in spherical nanoparticles. As an example, for gold ellipsoids with an aspect ratio equal to 6, N3 is roughly equal to 0.05, so
that the resonance condition requires ε ≈ −20εmedium . This is achieved at a wavelength of around 760 nm for nanoparticles
in air (if one assumes for the dielectric function of gold that measured for bulk gold films [257]). On the other hand, the
resonance condition for the polarizability along the short axes occurs at shorter wavelengths than in spheres. Again, in
the previous case of an ellipsoid with aspect ratio equal to 6, N1 = N2 ≈ 0.5, hence the resonance conditions for the
‘‘transverse’’ polarizability require ε ≈ −εmedium , which is achieved at a wavelengths shorted than that of a sphere (for
which ε = −2εmedium ). Summarizing, in axially symmetric ellipsoidal particles (R1 = R2 ̸= R3 ) the dipolar resonance mode
is split into one mode at longer wavelength (the longitudinal mode) and one mode at shorter wavelength (the transverse
mode). This can be seen clearly in Fig. 4.4(L).
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
A
B
C
D
E
F
G
H
I
J
L
K
0
ε/ε medium
-4
-8
-12
0.0
0.2
0.4
N
0.6
0.8
1.0
Fig. 4.4. (A–I) Transmission electron microscopy (TEM) images of gold nanorods of various aspect ratios synthesized by colloidal wet-chemical approaches.
The black scale bar is the same in all the images and is equal to 100 nm. In going from (A) to (I) the aspect ratio of the particles increases from 1.5 to 3.5.
Taken with permission from Ref. [258]. (J) UV–visible spectra of the samples displayed in (A–I). Taken with permission from Ref. [258]. (K) Geometrical
factors for axially symmetric ellipsoids as a function of the aspect ratio. Taken with permission from Ref. [258]. (L) Curve correlating the plasmon resonance
conditions as a function of the geometrical factor in a spheroidal particle. Taken with permission from Ref. [23]. The inset displays the dielectric function
of gold [257] vs. the wavelength, which indicates that the imaginary part of the dielectric function does not vary much with the wavelength. Therefore
most of the variation in ε , at least above 550 nm, is due to variation in the real part. In a sphere, for which there is only one geometrical factor (N = 1/3),
the resonance condition is met for ε/εmedium = −2. In an axially symmetric ellipsoid, for which there are two geometrical factors, there are two resonance
conditions, one is met for ε/εmedium > −2, and corresponds to the transverse plasmon resonance, and the other is met for ε/εmedium < −2, and corresponds
to the longitudinal plasmon resonance. As can be seen from the inset that reports the dielectric function of gold, this condition is met at shorter wavelength
for the transverse resonance mode (i.e. smaller negative values of ε ) and at longer wavelength for the longitudinal resonance mode (i.e. larger negative
values of ε ).
Individual plasmon modes can be excited using light that is polarized such that its electric field is oscillating along one of
the axes of the rods. This has been demonstrated for example by embedding gold nanorods in a polymer film, and then by
stretching the film, which induced a preferential alignment of the rods along the stretching direction [23,259]. Absorption
by this film of light polarized either along the stretching direction or perpendicular to it caused selective excitation of the
longitudinal and of the transverse plasmon modes, respectively. Individual plasmon modes were also excited on single
nanorods, in a confocal microscope setup (see Section 4.7) [260].
In analogy with the case of spherical nanoparticles, if the overall aspect ratio of the ellipsoid is kept constant, but its overall
size is increased, retardation effects start becoming relevant, so that in bigger rods the longitudinal plasmon resonances
shifts to longer wavelengths and the plasmon band broadens. An expression for the dipolar polarizability of the longitudinal
mode in ellipsoids (i.e. along the principal axis with geometrical factor N) which takes into account these effects, and that
is similar to that discussed above for spherical nanoparticles, was found recently by Kuwata et al. [255]:
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
α≈
V
N+
εmedium
ε−εmedium
+ Aεmedium x2 + Bεmedium x4 − i
3/2
4π 2 εmedium V
3
λ3
(4.18)
with A and B being polynomial expressions of the variable N (i.e. the geometrical factor). This is an empirical formula,
which was derived by fitting data extracted from dark field optical spectroscopy of individual nanorods of various sizes
and aspect rations. In a recent work, Au nanorods with very large aspect ratios (around 50), which were synthesized via an
electrochemical approach, exhibited a shift in the longitudinal plasmon resonance that was far bigger than that predicted
by the Gans’ theory [261]. The maximum of the band was at 12 µm, while Gans’ theory predicts only 4 µm. This is a clear
case in which retardation effects are extremely important.
Finally, it must be noticed that the non-local effects discussed earlier for small spherical nanoparticles do not influence
the longitudinal resonance in ellipsoids, since their long axis is usually longer than the mean free path of the electrons.
4.6. Plasmons in elongated nanoparticles of arbitrary shape
For nanoparticles of arbitrary shapes, it is not possible to solve analytically the Maxwell equations, therefore numerical
approximation methods have to be sought. Many methods have been developed so far, each of them characterized by its
peculiarities, advantages, and disadvantages. For a coverage of them, see for example [25,27].
One popular computational approach is based on the boundary element method (BEM) [262]. This method has been
used in many problems of physics and engineering since it can solve differential equations in a space made of regions of
different materials [263,264]. Recently, it has been exploited to calculate the plasmonic response of metallic nanoparticles
of arbitrary shapes [264–269]. With respect to other methods, its computational requests are not so high, because it only
deals with the interfacial region of the particles and not with the volume (hence there is a reduction of one dimension in the
computational problem). In practice, the electromagnetic field inside a region of a given material is completely defined by
the fields and the derivatives of the fields at the boundary of the region. In the case of a metal nanoparticle immersed in a
medium of another material, one can formulate the electromagnetic field scattered by the particle as boundary charges and
currents. Boundary conditions are then imposed such that the parallel components of the electric and magnetic fields are
continuous at the interfacial region between the particle and the surrounding medium, and this leads to a system of surfaceintegral equations. These surface integrals are then discretized, i.e. they are turned into summations over surface elements
in which the overall interface is tessellated. An additional advantage of this method is that when the system presents an
overall cylindrical symmetry, the calculations are considerably simplified.
Perhaps one of the most popular numerical approaches for calculating the optical response of metal nanoparticles is the
discrete dipole approximation (DDA) [23,27,270–272]. In this approach the particle is immersed in a cubic box made of N
polarizable cubic ‘‘elements’’. Each element can be occupied by a portion of the nanoparticle, or it can be ‘‘external’’ to the
nanoparticle: therefore any arbitrary shape for the nanoparticle can be modeled. Each i-th element, positioned at distance
ri from the origin of the box, has dipolar polarizability αi . In each element, the polarization induced by the interaction with
the local electric Eloc field is:
(4.19)
Pi = αi Eloc (ri ) .
The local field in each element (we call it simply Eloc ,i ) is the sum of the incident field in the element (Einc ,i ) with the fields
generated by all the other dipoles inside the particle (Edipole,i ):
Eloc (ri ) = Einc , i + Edipole, i = E0 exp (ik ·r i ) −
−
j̸=i
Ai,j · Pj .
(4.20)
Each term in the summation above can be considered as the scattered field produced by the element j at the position of the
element i, so it also contains the polarizability αj of the j-th element. Expressions for Ai,j can be found in many reviews [23,27,
270–272]. Therefore, one is faced with a 3N ×∑
3N ‘‘interaction’’ matrix A, and needs to solve 3N complex linear equations in
order to identify the polarization vector P =
Pi from which the optical properties of the particle (including the extinction
cross section) can be derived. There are actually many ways of solving the equation above, via various approaches, like for
example the iterative approach using the conjugate gradient method [270]. It is clear that to model a nanoparticle accurately,
a three-dimensional grid with a large number of elements is required, which can be computationally demanding, therefore
methods have been developed by which the summation over the dipole fields in the expression (4.20) are performed using
code based on Fast Fourier transforms [270].
Based on the DDA approach, the extinction coefficients for particles of a wide variety of shapes have been calculated,
including disks, prisms, spheroids, cubes, icosahedra, and so on. Even only focusing on metallic nanorods, it is worth to note
that in most cases their resemblance to spheroids is very poor, so that the Gans’ approach described above does not give
a detailed picture of their optical response. Chemically synthesized gold nanorods, for example, can have pentagonal cross
section, resulting from their multiple-twinned structure [273]. Some of these chemical approaches yield instead rods that are
single crystals with octagonal cross section [274]. Also, in some cases the rods have rounded end-caps, while in other cases
the end-caps are irregular. With recent advances in chemical synthesis, in optical microscopy of individual nanoparticles,
and in computational approaches, it has been possible to study in detail the plasmonic properties of metal nanorods as a
R. Krahne et al. / Physics Reports 501 (2011) 75–221
131
Fig. 4.5. (A) Longitudinal plasmon energy for ellipsoidal Au nanoparticles (red crosses) and rod-shaped Au nanoparticles with hemispherical end-caps
(yellow circles) as a function of the particle length and aspect ratio. The blue squares are referred instead to spherical nanoparticles (all particles were
considered as being immersed in air); (B) Extinction cross section at the peak maximum; (C) Full width at half maximum for the various series of samples.
Source: Taken with permission from Ref. [25].
function of their geometrical parameters [24,258,275]. Especially in the case of nanoparticles synthesized in solution, the
samples are basically in the form of diluted colloids, so that particle–particle and particle–substrate coupling effects are not
present. Therefore the investigation of their size- and shape-dependent optical properties is much facilitated. Indeed, the
random orientation of nanoparticles dispersed in a medium leads to an absorption spectrum that represents the average
contribution of all the plasmon resonances.
Myroshnychenko et al. [25] reported a systematic computational study on the evolution of longitudinal plasmon energy
and full width at half maximum (FWHM) of the plasmon band for gold nanoparticles having either ellipsoidal shapes or rod
shapes with hemispherical end-caps. These properties were monitored as a function of the length and the aspect ratio of the
particles, and were compared to the plasmonic properties of spherical nanoparticles. The calculations were done using the
BEM method. The results are summarized in Fig. 4.5. For all the particles considered in the calculations, the plasmon mode is
of dipolar character. For rods and ellipsoids, the calculations are referred to the longitudinal plasmon mode, and the exciting
radiation has its electric field oscillating along the long axis of the particles. It can be seen from Fig. 4.5 that, for a fixed aspect
ratio (therefore moving along one of the lines connecting the symbols), in both rods and ellipsoids the plasmon resonance
shifts to the red, as in spheres. This is basically due to retardation effects, as discussed above. The plasmon resonance in
rods and ellipsoids shifts to the red also when the length is fixed, but the aspect ratio is increased (therefore in the graph
we are moving from right to left along an horizontal line), but now for a different reason. As we recall from Gans’ theory
for spherical ellipsoids, an increase in aspect ratio at constant length means a decrease in polarizability along the long axis,
which shifts the resonance condition to longer wavelengths.
A further interesting aspect to consider is the decrease of the FWHM of the longitudinal plasmon band for rods and
ellipsoids of increasing aspect ratios, at fixed length (Fig. 4.5(C)). This can be explained by considering that the absorption
in mainly dictated by the ratio εim / |ε|, which drops at increasing wavelength. Therefore a shift to the red in the plasmon
resonance is accompanied by a decrease in the absorption. Finally, if the aspect ratio is increased, but the length is increased,
the FWHM increases, due to radiation damping effects, which we have discussed before (see also Section 4.7 for more details
on the plasmon linewidth).
Another systematic study, in this case on the evolution of optical properties in silver, gold and copper nanoparticles
of various elongated shapes, was recently carried out by Noguez and co-workers [24]. In their calculations, based on the
discrete dipole approximation, they first modeled non-elongated shapes, namely a sphere, a dodecahedron, a decahedron, an
octahedron and a cube. Then they elongated these shapes along one direction, and obtained respectively a prolate ellipsoid,
an hexagonal prism, a pentagonal prim, a tetrahedral prism and a rectangular prism, all with aspect ratio equal to 2.8 (see
Fig. 4.6(A)). In all the calculations, the volume of the particles was kept constant and equal to that of a sphere of 40 nm radius.
The main aim of the study was to highlight the change in the plasmonic properties in a prism when the number of lateral
facets goes from practically infinite (the prolate ellipsoid) down to four (the rectangular prism). The extinction efficiencies
were calculated by averaging over all the possible orientations of the particles with respect to the incoming radiation, so as
to simulate the extinction spectrum of nanoparticles in a colloidal solution.
Fig. 4.6(B) reports the optical extinction spectra, along with the contribution from absorption and scattering, of
decahedral-shaped particles and of the corresponding pentagonal prisms, for Ag, Au and Cu nanoparticles. In general, at
short wavelengths, the contribution from absorption is predominant, while scattering dominates at longer wavelengths.
Also, we notice that the contribution to scattering is higher in Ag than in Au and in Cu. In the spectra of the prisms, one clearly
sees the splitting of the plasmon peak into a longitudinal peak at longer wavelength, and into a transverse peak at shorter
wavelength, with respect to the peak of the ‘‘parent’’ decahedral particle. This transverse plasmon peak is clearly evident
in the Ag nanoprisms, less evident in Au, and even less in Cu, because in the two latter materials they start overlapping
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 4.6. (A) Evolution from sphere, dodecahedron, decahedron, octahedron and cube, to their corresponding elongated shapes. (B) Optical extinction
spectra, along with the contribution from absorption and scattering, for decahedral shape particles and the corresponding decahedral prism for Ag, Au and
Cu nanoparticles.
Source: Taken with permission from Ref. [24].
with absorption due to interband transitions (below 600 nm for copper and below 550 nm for gold). In silver, interband
transitions are found below 325 nm, so they do not interfere with the transverse plasmon peaks (the reader is invited to
look again at Fig. 4.2, which displays the dependence of the real and imaginary component of the dielectric function on the
wavelength). Especially for silver, the transverse plasmon peak is in reality the sum of a multitude of peaks, and this is due
to its pentagonal cross section, which strongly deviates from a circular shape: the single transverse mode in the prolate
ellipsoid is split into several transverse modes for the pentagonal prism case.
Fig. 4.7 reports the extinction spectra for all the shapes studied by the authors. In the case of the sphere, for Ag two peaks
can be distinguished clearly, the one at shorter wavelengths being due to the quadrupolar mode, which for Au and Cu is much
less intense and practically hidden by interband transitions. In the ellipsoids, the dipolar mode splits into a longitudinal one
and a transverse one, and again this is much clearer in Ag than in Au and in Cu. In all the other non-elongated shapes different
from the sphere (dodecahedron, decahedron, octahedron and tetrahedron), the plasmon band (as anticipated before for the
decahedron) is not a single peak, but is the sum of a multitude of peaks. The smaller the number of facets in these shapes,
the more distinct and sharper become the individual peaks, each related to a different charge distribution mode, and more
spaced they are from each other. The most striking case is that of the silver cube, in which four peaks are clearly visible.
Likewise, in the corresponding silver prisms, for the transverse plasmon band, the smaller the number of lateral facets in
the prisms, the more distinct and sharper become the peaks related to the various transverse modes. Again, in the rectangular
prism, four distinct and well separated transverse modes are visible.
In a recent work by Schmucker et al., Au nanorods of various diameters (from 35 to 100 nm) and lengths were synthesized
electrochemically using anodic aluminum oxide templates [276]. The authors then recorded their optical extinction spectra
and carried out a series of calculations based on the discrete dipole approximation. The peculiarity of this work stands in
the study of the dependence of the dipole plasmon wavelength on both the length and the diameter of the rods. The results
of the calculations, which agreed well with the experimental data, were then used by the authors to derive an equation that
correlates the longitudinal plasmon resonance in rods to both their length and diameter, and which is valid even beyond
the quasi-static regime [276].
4.7. Dephasing of surface plasmons: spherical vs. elongated particles
Surface plasmons in metal nanoparticles dephase (i.e. decay) through both inelastic processes (characterized by a time
constant T1 ) and elastic processes (characterized by a time constant T ∗ ) [260], and the overall time constant of plasmon
decay T2 is defined as:
1
T2
=
1
2T1
+
1
T∗
.
(4.21)
R. Krahne et al. / Physics Reports 501 (2011) 75–221
133
Fig. 4.7. (A) Extinction efficiencies for various nanocrystal shapes in the case of Ag, Au, and Cu.
Source: Taken with permission from Ref. [24].
Inelastic processes, which dominate the plasmon decay, can be further classified into radiative and non-radiative decay
processes. Radiative processes (i.e. radiation damping) are relevant only in large particles, as discussed above. Non-radiative
processes instead are important also in small particles and are due to scattering of individual electrons into electron–hole
pairs, i.e. electrons that participate in the collective oscillation can give off part of their energy via a scattering event
and form an electron–hole pair. The latter can occur either via intraband excitation (i.e. electron–hole pairs created
within the conduction band), or via interband excitation (electrons from the d band are promoted in the conduction
band). All the various processes are sketched in Fig. 4.8(a). In some applications (for example in surface enhanced Raman
scattering, see later), it is desirable for the dephasing time to be as long as possible, and furthermore it is important
to minimize the non-radiative decay, for example in order to minimize sample heating. The dephasing time is directly
correlated to the homogeneous linewidth Γ = 2h̄/T2 of the surface plasmon, which can be measured for individual
particles.
Sönnichsen et al. [260] recorded light-scattering spectra from individual gold nanoparticles (both nanorods with various
aspect ratios and spherical nanoparticles with various diameters), they estimated Γ for every particle and from it they
derived the dephasing time T2 . Typical scattering spectra of individual spherical nanoparticles and nanorods, both excited
with linearly polarized light, are reported in Fig. 4.8(b). In the case of nanorods, the resonance was excited by light that
was polarized along the long axis of the rod, such as to excite only the longitudinal plasmon mode. The full width at
half maximum Γ and the dephasing time T2 for the various samples analyzed are reported in Fig. 4.8(c), top graph, as a
function of the plasmon resonance energy. In spherical nanoparticles, the shift towards the red of the resonance energy is
due to retardation effects (as discussed earlier), while in nanorods is due to increase in the aspect ratio (the experiment
probed the longitudinal plasmon mode in rods). In spheres of the same nominal diameter, there was a distribution of both
resonance energies and Γ (gray triangles in the graph of Fig. 4.8(c)), most likely due to local fluctuations in the environment
close to each particle, but averaging both resonance energy and Γ for the samples with the same nominal radius gave
values (black triangles in the same graph) that were well in line with the theory (the solid lines in the graph). In spheres,
the dephasing time decreased for the particles with the more red-shifted resonance, hence the larger particles. This can
be explained by an increase in the radiation damping, which both shifts the plasmon resonance to lower energies and
broadens the peak, as discussed earlier in this section. In rods, however, the dephasing time increased significantly for
particles with more red-shifted resonance, hence for larger aspect ratio rods (black circles in the graph). Such increase in
dephasing time could not be ascribed to radiative processes, since these processes are negligible in particles with small
volumes (contrary to spheres, the rods studied could have large aspect ratios but their overall volume was small). On the
other hand, the increase could be explained easily by considering that interband excitations require a threshold energy of
1.8 eV. In large aspect ratio rods, longitudinal plasmon modes can have resonance energies comparable to or even smaller
than 1.8 eV, therefore for such rods the non-radiative decay channel via interband transition is precluded. Also in spheres
with plasmon resonance energy below 1.8 this decay channel is precluded, but these spheres have such a large volume that
in them radiative damping effects dominate the decay process. Hence in such spheres the overall plasmon decay time is
short.
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 4.8. (a) Sketch of the radiative and non-radiative decay processes of surface plasmon modes in metal nanoparticles; (b) Typical scattering spectra
of individual spherical gold nanoparticles and nanorods, both excited with linearly polarized light. In the case of nanorods, the resonance was excited by
light that was polarized along the long axis of the rod, such as to excite only the longitudinal plasmon mode. On the left: polar plot of the intensity from
the longitudinal plasmon mode in an individual gold nanorod as a function of the angle between the long axis of the rod and the polarization direction
of the exciting light; (c) Top graph: full width at half maximum Γ and dephasing time T2 for the various samples analyzed, as a function of the plasmon
resonance energy. Black circles are referred to rods, gray triangles are referred to spheres, and black triangles are referred to averages over several spheres
having the same nominal diameter. The lines in the graph are the results of exact electrodynamic calculations using the Mie theory. Bottom graph: same
data as in the top graph but plotted as quality factor (resonance energy over plasmon linewidth).
Source: Taken with permission from Ref. [260].
Another important parameter involved in these measurements is the quality factor Q, which is the ratio of the resonance
energy over the plasmon linewidth. This is reported in the bottom graph of Fig. 4.8(c) for the various samples analyzed.
It represents the enhancement of the oscillation amplitude of an oscillating system with respect to the driving amplitude,
and corresponds to the field enhancement in the case of surface plasmons. This is an important figure of merit in surface
plasmon resonance (SERS, see later). Hence one conclusion is that large aspect ratio rods are more suitable than spheres for
SERS.
A consequence of the reduction of plasmon dephasing in rods with respect to spheres is that the scattering quantum yield
at resonance conditions, defined as the ratio between the scattering efficiency Qsca and the total extinction efficiency Qext is
higher in rods than in spheres [277]. Also, for rods having the same aspect ratio, but different volumes, larger rods have a
higher scattering efficiency than smaller rods (the same is true for spheres of different diameters, as discussed above) [277].
Hence large particles are in general more suitable for imaging applications, since they are good light scatterers, while smaller
particles, in which a larger fraction of the absorbed light is not re-emitted but instead is converted into heat, are more suitable
in applications like for example photothermal therapy [30].
4.8. Ultrafast electron dynamics in metal nanoparticles: overview
Ultrafast techniques measure the transient change in the optical properties of a sample. When a femtosecond laser pulse
excites a metal, the electron distribution, which is initially in equilibrium and is given by the Fermi distribution, is perturbed,
such that ‘‘hot electrons’’ are created. Electrons then scatter elastically which each other, until a new Fermi distribution is
reached, corresponding to a higher temperature of the electron gas. The pulse can be in the infrared region, which perturbs
only the electrons in the conduction band, or it can be in the UV–visible region, in which case interband transitions might
also occur [278–280].
In order to understand the relaxation dynamics of these hot electrons in metals, the two-temperature model is often
invoked, which models how energy is exchanged between electrons and phonons [278–280]. Due to the large population
of electrons, electron–electron scattering is fast (it is actually of the order of hundreds of fs), so that the hot electron gas is
assumed to thermalize quickly to a temperature Tel , which differs from that of the lattice Tlat . Also, we can assume that the
duration of the laser pulse is much shorter than the electron–phonon relaxation time, the latter being of the order of a few
(1–4) ps. Based on these considerations, the dynamics leading to electron thermalization due to exchange of energy with
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
b
Lattice Temperature (K)
Electron Temperature (K)
a
Delay time (ps)
Delay time (ps)
Fig. 4.9. Time evolution of the electron and lattice temperature in a Cu film 200 Å thick, as modeled by solving numerically the coupled Eqs. (4.22) and
(4.23), for two different laser pulse energies (15 nJ (a) and 65 nJ (b), laser pulse modeled as Gaussian with FWHM = 0.3 ps and peaking at 1 ps), and for
various values of the electron–phonon coupling constant G. Top panels refer to electron temperature, bottom panels refer to lattice temperature.
Source: Taken with permission from Ref. [279].
the lattice can be described by a set of two coupled equations [279,280]:
Cel (Te )
∂ Tel
= κ ∆2 Te − Ge−ph (Tel − Tlat ) + LP (z , t )
∂t
(4.22)
∂ Tlat
= Ge−ph (Tel − Tlat ) .
(4.23)
∂t
In the expressions above, Cel (Te ) = GTe is the heat capacity of the electrons, and is proportional to the electron temperature,
Clat
while Clat is the heat capacity of the lattice, and can be approximated as 3kB n, where kB is the Boltzmann’s constant and n
is the atomic density of the metal. G is the electron–phonon coupling constant [243]. In the first equation, the first term is
related to thermal conductivity losses, the second term describes electron–phonon coupling, and the third term describes
the temporal evolution of the heating laser source [279,280]. The maximum temperature Te reached by the electron gas is
basically dictated by the energy of the laser pulse. El-Sayed et al. [279], starting from different values of G and by modeling
a laser heating pulse as a Gaussian with FWHM = 0.3 ps and peaking at 1 ps, solved numerically the two coupled equations
above for a film of Cu metal under a specific set of conditions. The evolution of Te and Tlat over time is shown in Fig. 4.9(a) and
(b). The difference between (a) and (b) is basically the intensity of the laser pulse. From the figure it can be clearly seen that
an increase in the coupling constant leads to a faster decay of Te and faster increase of Tlat , therefore to a shorter relaxation
time, while a more intense laser pulse leads to a higher value of Te and to a longer relaxation time. The latter effect can
be easily correlated with the increase in the electronic heat capacity at higher temperature. In a more elaborate approach,
which takes into account the surrounding of the metal as a third thermal reservoir at a constant Tsur temperature, the second
equation is replaced by the following one [281]:
Clat
∂ Tlat
= Ge−ph (Tel − Tlat ) − Gph−ph (Tlat − Tsur ) .
∂t
(4.24)
In the expression above Gph−ph is the coupling constant of the metal with the surrounding medium.
What said so far applies to metals in general. In nanoparticles too, the electron–phonon relaxation time was found to
depend on the intensity of the laser pump [282] and is longer at higher pump intensities, for the same reason discussed
above (namely the temperature dependence of the electron heat capacity). The questions that we want to address now are
first of all if the relaxation dynamics in metal particles depends on particle size, and then how the shape of the nanoparticles
influences the electron–phonon interactions. We address the first question here, and the second question in paragraph 4.9.
In principle, electron–phonon interactions should decrease with decreasing particle size, which should lead to an increasing
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lifetime of electron–phonon relaxation. An intuitive guess for this is that the density of states for both electrons and holes
decreases with decreasing particle size, which leads to lower spectral overlap, hence less coupling. However, the bulk band
structure in metals like Au is already developed for sizes above 2 nm, and the spacing between the electronic levels is already
larger than the thermal energy, so that there is still good spectral overlap between electron and phonon densities of states.
At very small sizes, for example less than 1 nm for Au nanoparticles, the relaxation time increases, indicating that in this
size regime the nanoparticles behave like molecular clusters, the spacing between electronic levels increases, and therefore
electron–phonon coupling is definitely weaker [283,284].
On the other hand, the electron free path in metals is in the 30–50 nm range [243], therefore effects related to surface
scattering of electrons should contribute to shorten the electron–phonon relaxation time, in this case via coupling with
surface phonon modes. This seems reasonable too, since electron-surface scattering was found to be responsible for
dephasing of the coherent plasmon oscillations (see above). It turns out however that coupling of electrons with surface
phonons depends on the ratio between the number of valence band electrons and the atomic mass of the metal [285],
which influences the electron-surface scattering cross section. This ratio in gold is small (large atomic mass, only one valence
electron). Various studies [286,287] have shown that the electron–phonon relaxation time in gold nanoparticles down to
2–2.5 nm diameter is of the order of 1 ps, which is practically the same as in bulk gold. Therefore the electron–phonon
coupling constant, at least for Au, remains practically unchanged down to 2 nm. The ratio is instead larger in other materials,
like for example gallium, silver and tin (see [261] and references cited therein). As an example, for tin nanoparticles (tin has
4 valence electrons and much lighter mass) decreasing relaxation times with decreasing particle size (from 6 down to 2 nm)
were measured [288].
Another point that deserves further attention is that the energy exchange of the metal nanoparticles with the surrounding
medium should be quite fast, due to their high surface to volume ratio. The time scale for phonon–phonon coupling is of
the order of hundreds of picoseconds, and indeed the relevant parameter is proportional to the square of the nanoparticle
radius [29,289]. Various reports (see for example [29]) have shown that the matrix in which the nanoparticles are embedded
has also an influence on the electron–phonon constant, so on the relaxation time for this process. For the same sample of
nanoparticles, longer relaxation times are found if the matrix (for example the solvent) has a low thermal conductivity or
if there is poor thermal contact between the nanoparticle and the matrix. The latter effect is also dependent on the type of
stabilizing molecules that coat the surface of the nanoparticles.
Electron–phonon and phonon–phonon relaxation processes can be followed by monitoring the transient absorption
spectra of a solution of nanoparticles at increasing delay times following excitation by a laser pulse (this is done by a second
broad band probe pulse with a specific delay time). The plotted signal is the difference between the probe pulse and a
reference pulse that is recorded before the pump pulse, which in practice corresponds to the optical absorption spectrum
of the unperturbed system (the top spectrum in Fig. 4.10(a)). The negative transient absorption (∆A) (a ‘‘bleach’’) observed
in the region of the maximum plasmon absorption means that in the perturbed system, following the laser pulse, there
is a decrease and broadening in plasmon absorption. This bleach recovers over time, due to electron–phonon and then to
phonon–phonon scattering processes (Fig. 4.10(a), inset). The bleach in the plasmon absorption can be explained as due to
dependence of the dielectric function of the metal on the electron temperature [285]. The higher the electron temperature,
the less intense and more broadened is the plasmon absorption, and this can be verified if the modified value of the dielectric
function is entered in the Mie equations and the equations are solved. A transient absorption can also be positive in certain
regions of the spectrum. This can be ascribed to broadening and shifting of the plasmon band. In Fig. 4.10(b), for example, the
time trace of the signal at 480 nm, on the blue side of the plasmon band, shows first a positive absorption, then a bleaching.
The initial absorption can be ascribed to broadening of the band. Then, as the energy is transferred from the electron gas to
the lattice, the particle heats and expands, leading to a decrease in electron density (i.e. the particle has still same number
of electrons, but now has a bigger volume) and as such to weaker plasmon absorption. In the 520 nm trace, at the maximum
of the plasmon absorption, instead only bleaching is observed, consistent with what said earlier. At 550 nm, the first bleach,
like for the 520 nm trace, is due to lower plasmon absorption and broadening due to higher electron temperature. Then, as
the lattice heats, the plasmon narrows and shifts to the red, leading to a positive absorption.
One interesting effect concerns the lattice vibrations observed in the electron relaxation dynamics of some metal
nanoparticles (see Fig. 4.10(c)–(d)). Femtosecond transient absorption spectra have shown modulation of the order of tens
to hundred picoseconds. This modulation is explained as due to excitation of a ‘‘breathing mode’’ of the particle, basically a
coherent expansion and contraction in the volume of the particle, following a fast heating of the particle (Fig. 4.10(d)). Since
the position of the plasmon depends on particle size, such coherent variation in volume over time leads to an oscillation in
the transient absorption. The frequencies of these acoustic modes are found to be inversely proportional to the nanoparticle
radius (see [22,290] and references therein). Apparently, these frequencies agree well with those calculated from continuum
elastic theory [291].
4.9. Ultrafast dynamics in elongated nanoparticles
Various experiments have indicated that the shape of metal nanoparticle does not influence the electron–phonon
relaxation time. As an example, Link and El-Sayed [22] recorded the bleach recovery of the plasmon band of two spherical
gold nanoparticles having different sizes. These were then compared to the bleach recovery of the longitudinal and
transverse plasmon signal of a sample of rod-shaped gold nanoparticles. In the experiments, the optical density of the
R. Krahne et al. / Physics Reports 501 (2011) 75–221
a
b
c
d
137
Fig. 4.10. (a) Transient absorption spectra of gold nanoparticles with a diameter of 15 nm at various delay times following a 400 nm femtosecond pulse.
The top spectrum corresponds to the reference spectrum, i.e. the optical absorption spectrum of the unperturbed system. The two positive absorption
bands that appear in the transient spectra at the opposite sides of the bleach signal are due to the initial broadening of the plasmon band at higher electron
temperature. The inset shows the recovery of the bleach signal (at the wavelength corresponding to the maximum of the peak) over time, which can
be modeled with two characteristic decay times, a short 3.1 ps time ascribed to electron–phonon coupling, and a much longer time corresponding to
phonon–phonon coupling (i.e. thermalization of the particles with the matrix). Electron–electron coupling is beyond the instrument resolution in the
present case. Taken with permission from Ref. [29]; (b) Temporal evolution of the transient absorption at various wavelengths (namely 480, 520, and
550 nm). Taken with permission from Ref. [22]; (c) The evolution of the transient absorption signal at 590 nm during the first picoseconds. The oscillation
of the signal is ascribed to acoustic phonon breathing mode. Taken with permission from Ref. [22]; (d) Sketch showing the acoustic breathing mode in
spherical nanoparticles.
samples and the intensity of the laser pulse were adjusted so as to reach the same electron temperature in all the samples. In
all the experiments, comparable electron–phonon relaxation times were found, thus indicating that neither size nor shape
influence electron–phonon coupling significantly (Fig. 4.11(a)). Other experiments have confirmed these results. In a recent
study [261] the surface plasmon behavior of nanorods with very large aspect ratios (around 50) was studied, for which the
longitudinal plasmon resonance was found to fall in the mid-infrared (12 µm, as discussed in Section 4.5). Also in that case,
electron–phonon coupling constants similar to that of the spheres and of shorter rods were measured.
Several groups have found that some aspects of the plasmon dynamics in rods differ from those of spherical
nanoparticles [261,286,292]. As we have seen before, following a laser pulse, the time trace on the blue side of the plasmon
band in spherical nanoparticles shows first an absorption (due to plasmon broadening) and then a bleach (due to plasmon
narrowing and shifting to the red). In rods of various aspect ratios, instead, first an initial bleach is observed, which later
recovers to an absorption [261,286,292]. The initial dynamics is consistent with the usual broadening and decrease in
intensity of the plasmon band, immediately after the pulse, due to high electron temperature. Then, the bleach recovery
to an absorption is consistent with a blue-shift of the plasmon band, upon heating of the lattice, which is opposite to what
Si found in spheres. An explanation for this behavior was ascribed to a radial expansion of the rod, due to lattice heating,
with a consequent decrease in the aspect ratio of the rod and therefore to a blue-shift of the longitudinal plasmon band. An
example of such behavior is shown in Fig. 4.11(c) for long aspect ratio Au nanorods (scanning electron microscope images
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 4.11. (a) Bleach recovery of the plasmon absorption band for gold nanoparticles with diameters equal to 15 and 48 nm, and for the longitudinal
and transverse plasmon mode in a sample of gold nanorods with aspect ratio equal to 3.1. Taken with permission from Ref. [22]; (b) Scanning electron
microscope images of Au rods with large aspect ratios (50) and (c) normalized time traces at different frequencies/wavelengths. The sample was excited at
400 nm. The longitudinal plasmon resonance was peaked at 830 cm−1 , which means that in all the experiments the sample was probed on the blue side of
the plasmon band. As the probe frequency is increased (see for example the traces at 2150 and 2660 cm−1 ), the bleach recovery is followed by absorption.
Time traces at 500 nm and 570 nm probe instead both sides of the transverse plasmon band. (b) and (c) are taken with permission from Ref. [261].
of the rods are shown in Fig. 4.11(b)). Here, the more the time trace is shifted to the blue side of the plasmon band (peaking
at 830 cm−1 ), the more evident is the recovery of the initial bleaching to an absorption.
Also in Ag ellipsoids and Au nanorods, oscillations in the transient absorption signal, following femtosecond excitation,
were detected [293–295]. In silver ellipsoids, for example, depending on whether the laser probe was resonant to the
longitudinal or to the transverse plasmon band, the oscillation was found to have a period equal to 2L/c1 and 2d/c1 ,
respectively, with L being the length of the rod and d its diameter, and c1 is the longitudinal speed of light in bulk
gold [293]. On the other hand, from continuum mechanics it can be calculated that in a rod-shaped object made of an
elastic isotropic material, in addition to a fundamental breathing mode (i.e. a mode in which both the diameter and the
length are simultaneously expanded and compressed), a fundamental extensional mode exists, in which an axial expansion
is accompanied with a radial compression and the other way around (Fig. 4.12(a)). In the first case, the bulk modulus of the
material is involved, while in the second case the Young modulus is involved [23,292].
Hu et al. calculated the natural resonant frequencies for the extensional and breathing modes for a cylindrical rod in the
limit of high aspect ratio (L/d ≫ 1) and found:
(n)
=
ωext
(n)
ωbr =
2n + 1
L
τn
d
π
E
ρ
E (1 − ν)
.
ρ (1 + ν) (1 − 2ν)
(4.25)
(4.26)
R. Krahne et al. / Physics Reports 501 (2011) 75–221
139
In the expression above τn is an eigenvalue, E is the Young’s modulus (defined as the ratio of the stress to the strain of the
material, see also Section 7 of this review article), ρ is the density of the rod and ν is the Poisson’s ratio of the rod. The
Poisson’s ratio is the ratio of the relative contraction strain, or transverse strain normal to the applied load, to the relative
extension strain, or axial strain in the direction of the applied load. It varies from material to material, and in bulk gold is
equal to 0.42. Fig. 4.12(b) shows the results of finite element analysis on the temporal response of gold rods with aspect
ratio equal to 5. In the figure, the Fourier components of the temporal response of the change in volume are reported vs. the
normalized frequency:
L
.
(4.27)
2π
From the figure it can be seen that the fundamental extensional mode has much lower amplitude than the fundamental
breathing mode and it occurs at much lower frequency. These two modes are those that contribute the most to the
vibrational dynamics of the rods. From the two equations above, one can derive the ratio of the resonant frequencies of
the fundamental extensional and breathing modes:
ω=ω
(0)
ωbr
(1 − ν)
.
(4.28)
(1 + ν) (1 − 2ν)
Therefore, in principle the aspect ratio L/d of the rod can be derived by studying the temporal response of the rod. Fig. 4.12(c)
(0)
ωext
τ0
=
π
L
d
reports the transient absorption decays for Au nanorods of two different aspect ratios, as measured by Hu [292] for two
different wavelengths of the probe laser, one at each side of the plasmon absorption band. All the traces show a fast (few ps)
component related to the electron–phonon relaxation process, followed by a long component (phonon–phonon relaxation)
that is modulated by the coherent vibrational modes of the rods. In the various time traces, the oscillation period was in the
40–120 ps range, depending on the aspect ratio of the sample (Fig. 4.12(e)).
For each sample, on the other hand, the oscillation period in both traces was independent from the probe wavelength
(Fig. 4.12(e)): this is because these particles, among those analyzed in the work, were monodisperse in length, and therefore
all wavelengths probed the same nanocrystals. In other words, there was no size selective excitation of nanorods according to
their aspect ratio in the sample, because basically all nanoparticles in each sample had the same aspect ratio. These oscillation
periods were found to be in good agreement with those calculated using Eq. (4.25) for the fundamental extensional mode of
rods of corresponding aspect ratio. The time trace of some samples exhibited in addition a faster modulation (Fig. 4.12(d)),
which the authors attributed to the breathing mode of the rods, and which (according to the authors) should be observed
mainly in rods with large diameters.
In principle, from the experimental data, since the expression relating the frequency of the fundamental extension mode
to the Young’s modulus is known (see above), one should be able to extract the Young’s modulus. The value extracted by
the authors from the gold nanorods were however much lower than the bulk value [292]. A possible explanation for this
discrepancy was that the axis along which the rod stretches in the extensional mode was the pentagonal twinning axis
(these rods were multiple twins with an overall pentagonal cross section along their long axis). This has no equivalent in
the bulk, so that possibly different mechanical properties are to be found along this direction in nanorods. The reader will
find additional details on mechanical properties of elongated nanoparticles in Section 7 of this review.
4.10. Near field optical properties of metal nanoparticles: spheres and rods
4.10.1. Near field optical properties: classical electromagnetic theory and quantum mechanical calculations
When surface plasmons in metal nanoparticles are excited by the incident light, the electromagnetic field close to the
particle surface is significantly enhanced. This can be understood in simple terms by considering a plasmon oscillation as
a photon confined in the volume of the metal nanoparticle, which is much smaller than the space occupied by the same
photon in a bulk medium. As a consequence, the electromagnetic field associated with such confined photon is also quite
intense. One can therefore conclude that the confinement of a photon in a nanoparticle leads to a remarkable focusing
of the electromagnetic field in the nanoparticle and in the region immediately surrounding it. The nanoparticle therefore
behaves as an antenna, actually as an ‘‘optical antenna’’ (as opposed to the standard antennas which interact with microwave
radiation and which are used in telecommunications). Indeed, the field decays within a distance from the nanoparticle’s
surface that is comparable to the nanoparticle size [296]. The strength of this field depends on a variety of factors [272,297].
In a spherical nanoparticle, for example, it increases with the particle size, but above a certain size (for example around
60 nm in gold nanoparticles), the field decreases since the particle starts experiencing radiative damping [296]. The field is
also strongly dependent on the shape of the particle. The field enhancement can be defined as |E |2 / |E0 |2 , with E0 being the
incident field and E the field in the vicinity of the nanoparticle. Particles which possess regions with high curvature, like for
example prisms and rods, show a field enhancement in these regions that is much stronger than in the case of spheres [272,
297]. This is also called the ‘‘lightening rod effect’’ [298]. Calculations on rods, spheroids and spheres using the discrete
dipole approximation [297] have indicated that in all these particles ‘‘hot’’ sites at the regions of high curvature exhibit field
enhancements of the order 103 , while spherical particles show enhancements only of the order of 102 (see Fig. 4.13).
One major limitation of calculations based on classical electromagnetic theory is that the dielectric function is assumed to
change abruptly at the interface between the nanoparticle and the external environment. This assumption however neglects
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
c
∆ A (arb. units)
a
0
50
100 150 200 250 0
Delay Time (ps)
0
50
100 150 200 250 0
Delay Time (ps)
50
100 150 200 250
Delay Time (ps)
b
∆ A (arb. units)
Fourier Components
1.5
1
0.5
0
0
2
4
6
ω–
8
10
12
d
14
e
140
120
Period (ps)
∆ A (arb. units)
100
200
300
Delay Time (ps)
(a)
100
80
60
40
600
0
50
100
Delay Time (ps)
700
800
900
Wavelength (nm)
1000
150
Fig. 4.12. (a) Sketch showing the extensional and breathing modes in a rod; (b) Fourier components for temporal response of change in volume normalized
by initial strain, vs. normalized frequency for a gold rod of aspect ratio 5, calculated using the Finite Element approach; (c) Transient absorption spectra of
gold nanorods with average lengths of (a) 46 nm and (b) 75 nm, at two different probe laser wavelengths. (d) Transient absorption spectra of gold nanorods
with average lengths of (a) 55 nm, showing a fast modulation with a period of around 11 ps, which was assigned to a vibrational mode corresponding to
the breathing mode of the rods; (e) Oscillation period vs. probe wavelength for rod samples of different average lengths: 46 nm (filled triangles), 61 nm
(open circles), 73 nm (filled rhombs), 75 nm (open rhombs), 89 nm (filled circles), and 108 nm (open squares).
Source: All images are taken with permission from Ref. [292].
the tunneling of electrons outside the nanoparticles, i.e. the spill out of their wavefunction, which indeed leads to a gradual
variation of the dielectric function at the interface. A recent work by Zuloaga et al. used first principles time-dependent
density functional theory (TDDFT) to study the optical properties of silver nanorods [299]. While the TDDFT calculations
yielded longitudinal and transverse plasmon resonances that were in accordance with those found by classical calculations
(see above), the local enhancement of the electric field induced by the plasmon as estimated by TDDFT was considerably
smaller than that obtained using the classical approach, especially at distances shorter than 0.5 nm from the surface of the
nanorod. This effect was attributed by the authors to the spill out of the electron density. Fig. 4.14, left panel, A–D, reports the
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
A
0
50
B
40
40
30
30
20
20
0
E
F
0
20
10
10
10
C
0
50
D
0
G
0
H
Fig. 4.13. (A)–(D) Electric field contours around silver nanoparticles with radii equal to 30 and 60 nm, respectively, immersed in vacuum, calculated
using the discrete dipole approximation; (A) and (B) represent the cross sections passing through the center of the spheres, along the plane containing
both the propagation direction (k) and the polarization direction of the incident field, while C and D represent the cross sections along the plane along
the polarization direction of the incident field (indicated with ‘‘E’’) and perpendicular to the propagation direction. For the 30 nm sphere, the contours
(C) are referred to the 369 nm light, which corresponds to the maximum in the dipolar plasmon resonance mode. For the 60 nm spheres, the contours
(D) refer to the 358 nm light, which corresponds to the maximum in the quadrupolar plasmon resonance mode; (E)–(F) Electric field contours, at 775
nm (corresponding to the maximum of the longitudinal plasmon mode), around a silver ellipsoid whose volume is equivalent to that of a sphere of 80
nm radius; (E) is the cross-sectional view passing through the center of the ellipsoid, along the plane containing the polarization direction of the incident
field and perpendicular to the propagation direction; (F) is the cross-sectional view along the plane containing both the propagation direction and the
polarization direction of the incident field; (G)–(H) Electric field contours around a silver trigonal prism. Both images are cross-sectional views passing
through the center of the prism, along the plane containing the polarization direction of the incident field and perpendicular to the propagation direction,
but for two different wavelengths, namely 770 nm (G) and 460 nm (H).
Source: Taken with permission from Ref. [27].
|E | / |E0 | maps for four nanorods of four different aspect ratios ξ (1, 1.5, 2.5 and 3 respectively) and overall length ranging
from 3.5 to 5.3 nm, calculated at the corresponding resonant plasmon frequencies [299]. The top four maps refer to classical
calculations, while the bottom four maps refer to TDDFT calculations. A much smoother density profile in the TDDFT, with
lower peak intensities, can be already appreciated with respect to the classical map. The plot on the top left panel reports
the maximum value of the electromagnetic field enhancement for the rods using both classical electromagnetic theory (red
line) and TDDFT (blue line), as a function of aspect ratio ξ of the rod. In both approaches, the enhancement increases with
increasing aspect ratio, but the classical calculations yield always a higher enhancement, and the difference with respect to
the quantum mechanical calculations becomes more striking at higher aspect ratios.
The central panel of Fig. 4.14, Plot A, shows the equilibrium electronic density n(d) as a function of the distance d from
the surface of a rod with aspect ratio 3, both in the longitudinal direction (red line) and in the transverse direction (blue line),
and also the electronic density for a sphere (black line), as found by TDDFT (negative values in d are referred to positions
inside the nanoparticle). Apart from the oscillatory behavior of the electron density, in all cases there is a similar spill out of
the electronic density for about 1.6 Å (3 Bohr), as this depends only on the Fermi level of the particle [299]. However it can
be seen from Plot B of the central panel, which reports the induced screening charge density δ n(d) along the same directions
as above (at the plasmon frequency), that δ n increases for the nanorod along the long axis (red line). This increased electron
charge density is responsible for the stronger electric field enhancement at the tip of the nanorod with respect to a sphere.
So in this respect quantum mechanical calculations give a result in agreement with that of classical calculations. The main
difference is that while classical calculations also predict a higher charge density at the tip of the rods, but this density does
not spill out of the rod surface, quantum mechanical calculations show that such charge density can spill out of the rod
considerably.
The plot A in the right panel of Fig. 4.14 reports the electric field enhancement as a function of the distance d from the
surface of the particle, for a spherical silver nanoparticle, according to classical calculations (red line) and to TDDFT (black
line). Plot B, still on the right panel, reports the same data but for the aspect ratio 3 rod. In both cases, one can see that there
is agreement between the two types of calculations, i.e. the same field enhancements are obtained, but only for distances
larger than around 5 Å (10 Bohr). Below this distance, electric field enhancement is lower according to TDDFT, which predicts
that such enhancement should go through a maximum, at about 3 Bohr from the surface, and then decline again for shorter
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 4.14. Left panel: top: maximum value of the electromagnetic field enhancement for silver nanorods using both classical electromagnetic theory (red
line) and TDDFT (blue line), as a function of aspect ratio ξ . Bottom: |E | / |E0 | maps for four nanorods of four different aspect ratios ξ (1, 1.5, 2.5 and 3
respectively) and overall length ranging from 3.5 to 5.3 nm, calculated at the corresponding resonant plasmon frequencies. The top four maps refer to
classical calculations, while the bottom four maps refer to TDDFT calculations. Central panel: Plot A: equilibrium electronic density n(d) as a function of the
distance d from the surface of a rod with aspect ratio 3, both in the longitudinal direction (red line) and in the transverse direction (blue line), and also
the electronic density for a sphere (black line), as found by TDDFT. Plot B: induced screening charge density δ n(d) along the same directions in Plot A (at
the plasmon frequency). Right Panel: Plot A: electric field enhancement as a function of the distance d from the surface of the particle, for a spherical silver
nanoparticle, according to classical calculations (red line) and to TDDFT (black line). Plot B: same data as in Plot A but for the aspect ratio 3 rod.
Source: All images are taken with permission from Ref. [299].
distances. This is in striking contrast with respect to classical calculations, according to which the maximum enhancement
is right at the surface of the particle. The discrepancy can be explained by considering that in classical calculations the
surface charge can be represented by a delta function peaked at the surface, which results in maximum enhancement right
at the surface, while according to quantum mechanics the charge induced by the plasmon resonance is spatially smeared.
Such smearing is therefore responsible for both a lower field enhancement and for a shift in the position of the maximum
enhancement.
Near field optical properties of individual nanoparticles can now be accessed experimentally. Advances in the fabrication
of transmission electron microscopes with monochromatic electron beams (with energy resolution of 0.1 eV), in spectral
imaging and in data processing have made possible to map plasmon modes in individual nanoparticles using a technique
known as electron energy loss spectroscopy (EELS) [300–302]. In this experiment, high energy electrons from the beam
interact with a sample and lose part of their energy by causing excitations of electron core levels and plasmon oscillations
in the sample. The electrons from the beam that have interacted with the sample have therefore less energy than that of
the incident electrons. A suitable spectrometer in the microscope records then the EELS spectrum. This technique in general
is used to retrieve information on the compositional features of the sample: type of elements, their oxidation state and
chemical environment. Recently, several reports have appeared in which plasmon maps of metal nanoparticles of various
materials and shapes have been retrieved (via EELS in scanning TEM mode) from the intensities of electrons that have lost
an energy corresponding to the specific plasmon mode of the nanoparticles [300–302]. For gold nanorods for example, both
longitudinal and transverse plasmon modes of a single nanoparticle have been imaged and the results have been shown to
be in good agreement with calculated electron density maps. This near field imaging technique has by far higher resolution
than other approaches, such as scanning near field optical microscopy.
One problem with EELS is however that the acquisition of a spectral image may take several hours, during which
contamination and radiation damage may occur in the specimen exposed to the high intensity electron beam. In addition to
this, during such a long experiment the sample may also drift in position, blurring the image and compromising the spatial
resolution. To limit these effects, another technique known as Energy Filtered TEM (EFTEM) is used. Since an EELS spectrum
consists of electrons that have passed the specimen and the optical system of the microscope, the latter can be used to form
images by using the inelastically scattered electrons. Thus, by a specific device called ‘‘energy filter’’, an image is formed
with electrons that have undergone selected energy losses in the sample. A comparison of EFTEM and STEM EELS plasmon
imaging of gold nanoparticles in a monochromated TEM was reported by Schaffer et al. in 2010 [303]. Finally, recent studies
have even shown the possibility to image coupling of plasmon modes in interacting metal nanoparticles [304,305].
4.10.2. Surface enhanced fluorescence
When a fluorophore molecule or a quantum dot is placed near the surface of a nanoparticle, its fluorescence can be
quenched or enhanced. If the fluorophore is directly bound/adsorbed to the particle surface, its fluorescence is usually
quenched, while if it is positioned a few nanometers away from the particle surface, there is an enhancement in the
R. Krahne et al. / Physics Reports 501 (2011) 75–221
143
fluorescence. This effect is attributed to several reasons, which include increased absorption by the fluorophore, faster
radiative decay rate of the fluorophore, and better coupling of the emission to the far field. Hence the metal nanoparticle
is practically behaving as a nanoantenna, which collects the incident light, concentrates it spatially, due to the field
enhancement effect described above, and conveys it to the fluorophore. This effect is maximized when the plasmon
resonance of the particle is tuned such as to maximize the degree of spectral overlap with the dye. Since in metal
nanoparticles, especially in nanorods, the frequency of the plasmon resonance can be tuned over a wide spectral range,
interesting applications have been proposed in sensing/biosensing based on this mechanism [306,307]. When an isolated
fluorophore is subject to a weak excitation by an electric field E0 (measured at the fluorophore position), its fluorescence S0
is given by:
S0 = ξ η0 |d · E0 |2
(4.29)
In the expression above d is the dipole moment, ξ is the collection efficiency and η0 is the quantum efficiency of the
fluorophore, defined as:
0
Γrad
η0 =
0
0
Γrad
+Γnrad
(4.30)
0
0
Γrad
and Γnrad
are the radiative and non-radiative decay rates, respectively. When the fluorophore is in the proximity of a
metal nanoparticle, it will feel a stronger local field E. In addition, it will be characterized by a new radiative decay rate Γrad ,
due to coupling of the fluorophore and the plasmon, and an additional non-radiative decay channel will be present, with
decay rate Γnrad , due to absorption by the metal (i.e. metal losses). The fluorescence signal enhancement will then be:
S
S0
=
η |d · E |2
η0 |d · E0 |2
(4.31)
The modified quantum efficiency η is given by:
η=
η0
(1−η0 )
F
+
η0
ηa
(4.32)
0
= Γrad /Γrad
is the so-called Purcell factor, and describes the enhancement in radiative decay, while ηa =
Γrad / (Γrad + Γnrad ) is the ‘‘antenna efficiency’’ [308,309]. Depending on the relative magnitudes of the various parameters,
various cases are found, as described below. If the quantum efficiency η0 of the fluorophore is 1, then η = ηa in any
F
case, and there can only be reduction in quantum efficiency when the fluorophore is close to the particle. However, when
η0 < 1, which is always the case, then η depends both on F and on ηa . When both these two parameters are large, then η
is maximized. Therefore, by looking at the two equations above, one can state that in order to have a strong improvement
in the fluorescence, the system has to create a strong electric field locally and it must have a large antenna efficiency and a
large Purcell factor.
In principle, one advantage of using elongated metal nanoparticles with respect to spheres is that their near field and
far field optical properties can be tuned by tuning the aspect ratio of the rods, which regulates the resonance conditions for
the longitudinal plasmon mode. In this way, metal enhancement of the fluorophores emitting in the near infrared region of
the spectrum can be achieved [310]. Another potential advantage, as discusses earlier, is that in nanorods the dephasing of
the longitudinal plasmon mode is slower than that of spherical nanoparticles, which translates into a higher quality factor
(see Section 4.7) [260]. Also, elongated nanoparticles and prisms can have sharp corners, near which the field is strongly
increased, as discussed above. However, various aspects need to be taken into account in these situations. One of them is the
spectral range where the fluorescence enhancement effect is desired. For example, in the short wavelength range spherical
particles behave better than nanorods/spheroids [308]. Also, depending on the spectral range, one needs to choose the
material of the particle (i.e. aluminum, silver, gold, copper), in order to optimize the Purcell factor. Another important point is
that the resonance needs to be in a spectral region where the dissipation in the metal is minimized [311]. This is a very active
area of research at the moment: a recent computational study [309] for example has investigated how gold ‘‘nanocones’’
perform as optical antennas for metal enhanced fluorescence with respect to spheres and rods, highlighting how in these
nanocones the longitudinal plasmon resonance can be tuned to shorter wavelengths with respect to rods (by increasing
the cone angle), without compromising the Purcell Factor and the antenna efficiency. Furthermore, nanocones with large
basal area can be assembled vertically on a substrate with less effort than for nanorods, which should be useful in a variety
of applications (the field enhancement is highest at the region near the apex of the cone, and this region is well exposed
in vertically assembled nanocones). For an in-depth coverage of optical antennas, also from an historical perspective, the
reader can consult Ref. [312].
4.10.3. Surface enhanced Raman scattering
Also the surface enhanced Raman scattering (SERS) effect is related to a molecule in close proximity to a metal
nanoparticle or to a roughened metal surface. The effect was discovered in 1977 [313,314], rationalized in 1978 [315],
and since then has been exploited in various applications, until the detection of single molecules was demonstrated. In
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analogy to metal enhanced fluorescence, the enhancement of the Raman scattering is mainly due to the enhancement of
the electromagnetic field near the surface of metal nanoparticles. There are various other effects involved in SERS, most of
them playing minor roles (for a coverage of them see for example [316]). In the case of a spherical nanoparticle immersed in
a medium of dielectric constant εmedium , assuming that light is incident on the particle, with an electromagnetic field vector
E0 aligned along one of the axes of the particle (say the z-axis), the absolute square of the field outside the sphere E is [317]:
|E |2 = |E0 |2 |1 − g |2 + 3 cos2 Θ 2 Re (g ) + |g |2
g =
ε − εmedium
.
ε + 2εmedium
(4.33)
(4.34)
In the expression above, Θ is the angle between the excitation field vector and the molecule on the surface of the particle.
The Raman intensity depends on |E |2 . The term g becomes large when resonance conditions are met, and in such case the
expression above becomes:
|E |2 = |E0 |2 |g |2 1 + 3 cos2 Θ .
(4.35)
Then the Raman intensity is highest when the molecule on the surface of the particle is along the z-axis of the particle, so
either at 0° or at 180°. In addition to enhancement of the field, therefore to an increase in the probability that the molecule
emits a Raman photon, there is a probability that the metal nanoparticle absorbs Raman photons (i.e. surface plasmon of the
particle are excited by the Raman photons) and re-emits them. Therefore, the intensity of the Raman signal can be enhanced
by two processes, implying the incident and the scattered field, and the overall enhancement can be written as:
Er =
2
|E |2 E ′
4
|E0 |
2
16 |g |2 g ′ .
(4.36)
In the expression above the primed variables refer to the re-radiated fields. In the case of a prolate spheroid, assuming
that the incident electric field is aligned with the long axis of the spheroid, the enhancement is maximal at the tip regions of
the spheroids (where the curvature is highest), and is bigger for more prolate spheroids (for an analytical treatment see for
example [316]). However, as discussed above for the case of a sphere, this maximum Raman enhancement would be true
only if all the molecules were located at these apex regions. In a real case, however, molecules would be located all over the
surface of the particles, therefore one needs to perform an average of the field enhancement over the whole surface of the
nanoparticle [316]. It turns out that the averaged enhancement is only 2 for a sphere, and it actually decreases to zero for
infinitely prolate ellipsoids. On the other hand, for infinitely oblate ellipsoids, the averaged enhancement is 2.44 [316]. These
calculations however need to be taken with care, since in them the dielectric constant of the metal was considered as that of
the Drude model. When considering other terms in the dielectric constant, the maximum averaged enhancement was found
for moderately oblate ellipsoids [316]. In analogy with what discussed before on the field enhancement in nanoparticles and
on the metal enhanced fluorescence, the magnitude of SERS depends not only on the shape of the particle, but also on its
overall size. Factors that tend to reduce field enhancement, hence also SERS, are scattering of electron from the surface, for
small particles, and radiation damping for big particles. Hence, depending on the composition of the particles, there is a
range of nanoparticle sizes for which SERS is maximized [316].
A recent work by Rodriguez-Fernandez et al. [318] studied the effect of nanoscale surface corrugation on both far and
near field optical properties of spherical Au nanoparticles of various sizes, ranging from 100 to 200 nm. Especially for what
concerns the near field, the authors were able to probe the optical amplification via ensemble SERS measurements (using
1-naphthalenethiol as a model analyte) and found that the surface corrugation plays an important role in the plasmonic
response of the particles: namely, SERS amplification was always higher when working with surface corrugated particles.
4.10.4. Photoluminescence enhancement from metal nanorods
Another important effect related to near field properties of metal nanoparticles is the enhancement in the
photoluminescence (PL) from metal nanorods with increasing aspect ratios [319]. PL from bulk metals (for example copper
and gold) is centered near the interband transition edge of the metals and is ascribed to recombination of the electrons
in the conduction bands with holes in the d-band. The quantum yield of this PL is relatively low (of the order of 10−10 ).
In Au nanoparticles, excitation at 480 nm for example leads to excitation of surface plasmon modes, involving electrons
in the sp band but also to intraband transitions involving d-electrons (see Section 4.2), which create holes in the d-band.
Recombination of the sp electrons with the d-holes leads to PL. In rod-shaped nanoparticles, due to local field enhancement
in the regions of higher curvature, an enhancement in the PL signal is actually observed [319]. An important parameter
involved in the estimation of the enhancement in photoluminescence is the so-called ‘‘local field enhancement factor’’,
which is generally defined as the ratio of the field acting on an individual emitter to the average field in the medium. It can
be shown that for a rotational spheroid of a material with dielectric function ε(ω), axes a and b, and volume V, the local field
correction factor L(ω) at an optical wavelength λ is defined by [319]:
L (ω) =
LLR
ε(ω) − 1 + LLR 1 +
4π 2 iV [1−ε(ω)]
3λ3
.
(4.37)
R. Krahne et al. / Physics Reports 501 (2011) 75–221
145
Fig. 4.15. (a) Experimental PL spectra of gold nanorods of average aspect ratios equal to 2.0 (i), 2.6 (ii), 3.3 (iii), 4.3 (iv) and 5.4 (v) and diameters all around
20 nm; (b) The peak in the emission wavelengths scales linearly with the aspect ratios of the rods; (c) The quantum efficiency of the PL emission scales
according to the square of the length of the rods; (d) Plot displaying P1 /β1 |E0 |2 as a function of the gold ellipsoid length for various emission wavelengths.
The inset shows how P1 /β1 |E0 |2 depends on the square of the length of the ellipsoid, in the range of lengths comprised between 20 and 80 nm; (e) Plot
of P1 /β1 |E0 |2 as a function of the emission wavelength for ellipsoids having the same aspect ratios as the rods used in the experiments (see panel a). The
inset shows how the emission maximum from these ellipsoids scales with the length of the rods, in analogy with the experimental data shown in (b).
Source: Taken with permission from Ref. [319].
The term LLR above is the ‘‘lighting rod factor’’ and is defined by:
LLR = 1 −
ξ Q1′ (ξ )
Q1 (ξ )
(4.38)
with:
ξ=
1
1−
ξ +1
Q1 (ξ ) = ln
2
ξ −1
ξ
b 2 ;
a
Q1′ (ξ ) =
− 1;
dQ1 (ξ )
dξ
.
(4.39)
Based on the expressions above, the single photon luminescence power P1 for excitation and emission energies hωex and
hωem , for an incident field E0 can be calculated as:
P1 = 24 β1 |E0 |2 V L2 (hωex ) L2 (hωem )
(4.40)
2
where β1 is a proportionality constant. The ratio P1 /β1 |E0 | represents actually the enhancement of the single photon PL
from the ellipsoid with respect to the PL of bulk gold, and can be calculated. Mohamed et al. [319] calculated the above ratio
for a series of gold ellipsoids with constant diameter of 20 nm and increasing lengths. They also synthesized gold nanorods
of various aspect ratios and measured their optical emission spectra and PL quantum yields. Their calculations were in
agreement with the experimental results, namely the single photon luminescence power increased with the square of the
nanorod lengths, in the range between 20 and 80 nm (Fig. 4.15), and also the PL red-shifted with increasing rod/ellipsoid
length.
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5. Magnetic properties of elongated nanoparticles
5.1. Introduction
Bulk magnetic materials have found many applications in several technological fields since centuries. But nanoscale
magnetic materials arose only in the last two decades. Interest in such finely engineered systems is due to the new
unconventional functionalities that they may add to already well-known materials and hence for their capacity to expand
their range of application to fields where they would have never been foreseen, like in biomedicine [320,321].
Magnetic materials are commonly divided based on their magnetic hardness, i.e. on the external energy required for
the reversal of their magnetization, or more precisely, the magnetic field required to overcome the energy barrier for
the inversion of their magnetic moments. For example, hard magnets such as SmCo5 , Nd2 Fe14 B, CoPt and FePt show high
values of coercivity, which is useful for the design of magnetic recording media in magnetic storage devices [38,322–325].
Hard magnetic nanomaterials have already showed improvements in storage capacities compared to commercial devices,
fulfilling the increasing demand by information and communication industry [326]. Additionally they are expected to reduce
the size of portable devices as well as their cost and to increase design flexibility without compromising their performance.
On the other hand, materials such as metallic iron or cobalt, permalloys and spinel-type ferrites MFe2 O4 (M = Mn, Fe, Ni, Zn)
are called soft magnets due to their low coercive fields. Their high resistivity and high magnetic permeability make them
ideal for the design of non-reciprocal electronic components in miniature microwave devices, reading head sensors and
recording head cores in hard drives [327].
One of the main breakthroughs in the application of magnetic materials offered by nanostructured systems is found in
life sciences and biomedicine. Superparamagnetism is the term used for describing the absence of coercivity and remanent
magnetization in particles that still maintain a considerable amount of polarizable spins under the effect of an external
magnetic field [36,37]. Due to their small sizes and their zero net magnetization at zero field, they can be safely exploited
in biomedicine, for instance as nanovectors for specific targets that are not accessible via other conventional approaches
[33–35]. In the same way, they can also be used in biodetection and bioseparation techniques since once the target molecule
has been attached to the nanocrystals, the application of an external magnetic field allows for their recovery [38,39].
Iron oxide is clearly the most suitable material for such purposes, due to its high chemical stability, biocompatibility and
superparamagnetic properties. As a matter of fact, iron oxide nanoparticles are being already used in some diagnostic and
therapeutic techniques [40]. On the other hand, magnetic nanocrystals with higher magnetic moment and higher anisotropy
would be desirable in order to improve their performance and reduce the required doses. Some metallic nanoparticles fulfill
these characteristics but unfortunately they also show a much lower degree of chemical stability and an increased toxicity
compared to their oxide analogues.
Targets of current research in materials sciences, such as the achievement of higher coercivity values in reduced size
particles for information storage devices, or faster magnetic response for smaller biomedical vectors, could be achieved
if one considers not only the finite size effects of spherical nanocrystals but also the additional phenomena derived from
the shape anisotropy of particles such as nanorods. Nanorods or other one-dimensional nanosystems could also widen the
temperature range of application of a certain magnetic material compared to its bulk form (as will be shown later). The
uniaxial shape anisotropy of metallic and oxide nano-objects will probably become a key factor for the development of
improved devices.
This section of the review is organized as follows: initially, an introduction on nanoscale related magnetic properties is
presented. Then, the main examples of elongated, rod-like primary magnetic nanocrystals will be discussed, first for oxidebased materials and later for metals. The study of the magnetization reversal processes in 1D nanostructures is of crucial
importance for the development of spintronics technology. The main results obtained in this field will also be reviewed.
Preparation methods for the anisotropic growth of nanocrystals have been extensively reviewed previously and several
papers and book chapters deal with them [328–333]. Here we will focus mainly on the description of the structural and
magnetic properties of magnetic nanoparticles. Since the assembly of anisotropic nanoparticles is crucial for the exploitation
of their properties, special attention will be paid to works in which some degree of organization of elongated nanoparticles
has been achieved.
5.2. General considerations on magnetic nanocrystals
Magnetism originates from the magnetic dipoles associated to the electron spin and orbital moment. Each of these
magnetic dipoles can couple with the neighboring ones in the crystal lattice by exchange interaction leading to a collective
alignment that causes the macroscopic magnetic effect that we can measure. As shown in Fig. 5.1, three main parameters
can describe the magnetic strength and magnetization of the material and are:
1. The coercive field, HC , which is the external field required to reduce the magnetization back to zero. HC is strictly related
to the anisotropic energy barrier that must be overcome to invert the direction of the magnetic dipoles of the material. This
anisotropic energy barrier can have different sources that will be discussed in detail in Section 5.3.
2. The saturation magnetization, MS , which is the maximum value of magnetization that the material can reach with
increasing field.
R. Krahne et al. / Physics Reports 501 (2011) 75–221
147
Fig. 5.1. (a) Spin configuration of the 4 types of magnetic exchange observable in magnetic materials without the effect of an external magnetic field;
(b) Parameters describing the strength and magnetization of superparamagnetic, ferromagnetic and ferrimagnetic materials: coercive field, HC ; saturation
magnetization, MS ; and remanent magnetization, MR ; (c) Temperature at which a magnetic transition between ordered and randomly oriented phases
takes place, TB .
3. The remanent magnetization, MR , which indicates the residual magnetization of the material at zero applied field; the
squareness of the material is defined as MR /MS .
All three parameters can be easily identified in the hysteresis loop produced in field-dependent magnetization
measurements. Depending on the magnetic response observed, we can classify magnetic materials as diamagnets,
paramagnets, ferromagnets, ferrimagnets or antiferromagnets [334,335] (see Fig. 5.1). A material that creates a magnetic
field in opposition to an externally applied magnetic field, thus causing a repulsive effect, is called a diamagnet. They
show weak and negative magnetic susceptibility values, typically ranging from −10−6 to −10−8 . The perfect diamagnet
with a susceptibility value of −1 is what is known as a superconductor. Oppositely to diamagnets, the other types of
magnetic materials are attracted by the applied magnetic field: a material in a paramagnetic phase shows randomly oriented
magnetic dipoles that can only be aligned in the presence of an external magnetic field and along its direction. Paramagnets
show weak but positive susceptibility values, typically ranging from 10−3 to 10−5 . Without the effect of a magnetic field,
the thermal energy is high enough to decouple all the spins. Such behavior is characterized by a complete absence of
coercivity and remanence, which means that, when the external magnetic field is switched off, the internal magnetic dipoles
randomize again and no energy is required to demagnetize the material. In other words, thermal energy (kT ) is higher
than the exchange energy between the spins and hence they are not able to maintain co-linearity. However, when the spin
exchange energy is sufficiently high as to overcome kT, the spins interact with each other giving rise to different situations,
depending on the nature of the exchange and the degree of intrinsic anisotropy of the material. These types of materials
are schematized in Fig. 5.1. Superparamagnetism, for instance, describes the magnetic behavior of a material in which all the
spins are always collinear to each other but its anisotropy energy is too low to keep them aligned along a particularly stable
magnetocrystalline direction (easy magnetization axis). As a consequence, a superparamagnetic material exhibits a constant
and collective fluctuation of its spins, meaning that they are always aligned parallel one to the other, but they collectively
move in a random fashion. Only when an external magnetic field is applied along a certain direction, the spins align along
the same direction. A superparamagnetic material shows zero coercivity and remanence. However, superparamagnetism
is strictly a time-dependent property: as it will be explained later, the superparamagnetic response depends strongly on
the relaxation time of the material and on the measuring time of the technique chosen. Alternatively, when the anisotropy
energy of the material is higher than kT, the aligned spins are blocked along a certain direction and cannot fluctuate as in
the superparamagnetic case, even in the absence of an external magnetic field. When this happens, depending on the nature
of the spin exchange, different behaviors can be observed. If the spins align parallel to each other, an enhanced collective
response is observed. This is what is known as ferromagnetism. Ferromagnets exhibit large and positive susceptibility values,
typically ranging from 103 to 104 .
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In contrast to the ferromagnetic situation, neighboring magnetic dipoles can align antiparallel in the lattice, which means
that they cancel each other. This type of magnetic exchange can lead to two different situations: antiferromagnetism, when
the magnetic dipoles or spins interacting present the same value and hence the material exhibits a net zero magnetization,
and ferrimagnetism when the two coupled spins show different values and in that case a net magnetic dipole different
than zero still magnetizes the material, even in the absence of an external magnetic field. It is important to keep in mind
that we only describe magnetic interactions between dipoles that belong to a single isolated magnetic nanoparticle in a
magnetically diluted sample, without taking into account further interactions between nanoparticles that would lead to
enhanced collective properties. The reader is redirected to more specialized works on such phenomena [336].
The alignment of the different magnetic dipoles in a superparamagnetic, ferromagnetic or ferrimagnetic material creates
a magnetic field in space that is associated to an energy value. This energy can be minimized by creating oppositely faced
domains in the material, so that the field, and consequently also the energy, are decreased. The so-called ‘‘domain walls’’
that are thus created carry a certain value of energy associated as well. The magnetostatic energy due to the alignment of
the dipoles increases proportionally to the volume of the particle, while the domain wall energy increases proportionally
to the surface area of the domains. When the decreasing dimensions of a magnetic particle approach a critical radius, it
may be more energetically favorable for the particle to form a single domain (SD) instead of maintaining its multidomain
character. This defines the critical radius, rc , of the particle. As shown in Fig. 5.2, the competition between magnetostatic and
domain wall energies is strongly dependent on the dimensions of the particles and obviously on the intrinsic nature of the
material itself, in particular on different anisotropy energy terms [334,335]. The rc value below which a particle acts as a
single domain particle can be deduced from equating the energies of single and multidomain configurations, which yields:
rc ∼
= 9(AKu )1/2 /µ0 Ms2 .
(5.1)
rSP = (6kB T /Ku )1/3
(5.2)
TB = Ku V /(kB ln(τm /τ0 )).
(5.3)
In the expression above, A represents the exchange stiffness, Ku is the magnetocrystalline anisotropy, µ0 is the vacuum
permeability. Values of rc can range from a few nanometers for soft magnets (15 nm for Fe, 35 nm for Co or 30 nm for γ Fe2 O3 ) to some hundreds of nanometers for harder ones (750 nm for SmCo5 ). The high stiffness of hard magnets is capable of
stabilizing large single domain particles. In the SD regime, a particle is considered as a single uniformly magnetized domain.
It is important not to confuse the single crystal particle with its single or multidomain magnetic behavior: even a single
crystal particle can show multidomain (MD) magnetic features if its dimensions are above rc .
Let us assume that, for a given constant temperature T, a spherical SD particle shows a permanent ferromagnetic
alignment of its magnetic dipoles. The radius of the particle may then be reduced down to a value at which the thermal
energy kB T (with kB being the Boltzmann constant) at a given temperature T is large enough to overcome the anisotropy
barrier (see Fig. 5.2). While the modulus of the magnetization remains unchanged, the fluctuations of the magnetization
direction in time would result in a complete randomization and hence in an averaged magnetization value of zero. In
other words, the particle would switch to a superparamagnetic state. The radius that defines the superparamagnetic limit,
rSP , below which a spherical particle behaves as a superparamagnet, is determined by a magnetic stability with a flipping
probability of the magnetic moment equal to 10% over one second [334,335] and can be calculated using Eq. (5.2):
where kB is the Boltzmann constant. It is easy to understand that a hard magnetic material with a high value of Ku is able
to maintain its ferromagnetic alignment down to smaller dimensions (rSP values of 3–4 nm) at room temperature, while
softer magnets with lower values of Ku switch to the superparamagnetic regime already at larger dimensions (rSP values of
ca. 20 nm) at the same temperature. The same phenomenon can be described from another point of view: for a fixed radius
and increasing temperature, and for a ferromagnetic SD spherical particle, a value of temperature T is reached at which
again the thermal energy kB T is large enough to overcome the anisotropy barrier. Above that temperature the particle shows
random fluctuations of its coupled spins and hence it enters the superparamagnetic regime. This temperature is known as the
blocking temperature, TB , and can be easily identified by the so-called Zero Field Cooling-Field Cooling measurement, ZFCFC (see Fig. 5.1), where the magnetization of the sample is measured at increasing temperature under a weak external field
under two different conditions: (i) by cooling the sample without external field before measuring; (ii) by cooling the sample
under the effect of a magnetic field. The temperature below which the two curves diverge is known as the irreversibility
temperature, Tirr , while the maximum of the ZFC curve indicates the TB of the particles. While for isolated ferromagnetic
nanoparticles both temperatures are equal, effects such as inter-particle interactions, spin disorder, spin-glass and other
phenomena cause these values to be different. The blocking temperature of a particle can be expressed as [337]:
In the expression above V is the volume of the nanoparticle, τ0 is a time constant characteristic of the material (usually in
the 10−12 –10−9 s range) and τm is the characteristic measuring time of the technique. If τm is assumed to be 1 min, then
Eq. (5.3) becomes TB = Ku V /25kB . Again it appears clear that, for equal dimensions, hard magnetic materials with high
values of Ku are able to maintain their ferromagnetic alignment up to higher temperatures (more energy is required to
thermally agitate the spins), while softer magnets with lower values of Ku switch to the superparamagnetic regime already
at low temperatures. However, the TB measured depends strongly on the measuring time characteristic of the technique
(τm ) chosen for its determination, as shown in Eq. (5.3). Magnetometry techniques usually require quite long measuring
R. Krahne et al. / Physics Reports 501 (2011) 75–221
149
Fig. 5.2. Size dependence of the coercivity of magnetic nanoparticles. The size-dependent transitions between magnetic single and multidomains, as well
as those between magnetically ordered and randomly oriented phases, are also indicated.
times (∼102 s) and hence lower TB values are measured compared to other faster techniques. For example τm as measured
from Mössbauer spectroscopy is around 10 orders of magnitude shorter. As a consequence, for the same sample, the TB as
measured by Mössbauer spectra is significantly higher than the one measured by magnetometry [338]. It is obvious that the
techniques and the conditions used for the TB determination must be carefully considered before comparing results.
The term blocking temperature applies to the transition from a ferromagnetic to a superparamagnetic state when the
temperature is increased. It is often used for small nanoparticles and it can range in value from a few degrees Kelvin to
temperatures above room temperature, depending on the volume and anisotropy of the particle, as shown in Eq. (5.3). On
the other hand, the transition from a ferro- or ferrimagnetic state to a paramagnetic state when increasing temperature in
bulk materials is known as the Curie temperature, TC . TC values are usually well above room temperature. The transition
from an antiferromagnetic to a paramagnetic state in bulk materials when increasing temperature is determined by the
so-called Néel temperature, TN . The reader is redirected to Ref. [339] for further details on such magnetic transitions.
5.3. Anisotropy considerations in magnetic nanocrystals
Until now we have discussed the dependence of the transition between a ferro-, ferri- or antiferromagnetic state on one
hand, and a superparamagnetic state on the other hand, on the temperature for a constant particle radius, and on the size
when temperature is kept constant. Let us now focus on the particle size dependence of the main magnetic parameters that
characterize the ordered state of ferro- and ferrimagnetic nanoparticles; these are the coercive field, HC , and the saturation
magnetization, MS . For this, some considerations about the different anisotropy contributions taking place in the crystal
should first be clarified. The magnetic properties of the particles can be described by the anisotropic Heisenberg model:
H= −
1
2
J
−
ij
si · sj + Hanis
(5.4)
where J is the nearest neighbor exchange interaction and Hanis is the Hamiltonian that contains all different sources of
magnetic anisotropy. From such Hamiltonian, the whole energy spectrum of the system can be calculated. The total energy
of a magnetic material can be considered as the sum of various terms [340]:
ET = EZ + Ek + ED + Eλ + Eexch + Esurf + Eshape
(5.5)
The terms are described as follows: EZ is the Zeeman energy arising from the interaction of the magnetic dipoles of the
material with the external magnetic field; Ek is the magnetocrystalline energy; ED is the magnetostatic dipole–dipole energy;
Eλ is the magnetoelastic energy due to the effect of magnetostriction (i.e. a slight change in the dimensions of the crystal
when magnetized). The latter induces elastic strains in the lattice, favoring directions of magnetization that minimize these
strain energies; Eexch is the exchange energy; Esurf is the surface anisotropy energy and Eshape is the energy arising from the
shape of the particle.
The term magnetic anisotropy is used to describe the dependence of the internal energy on the direction of the
spontaneous magnetization. As a consequence, easy and hard directions of magnetization appear, which are able to minimize
and maximize respectively the total energy of the material. The most energetically stable configuration is the one in which
the total magnetization vector lies along the easy direction/s. The energy difference between the easy and the hard directions
arises from two microscopic interactions: the spin–orbit interaction and the long-range dipolar coupling of magnetic
moments. In general terms, the spin–orbit coupling is responsible for the magnetocrystalline, magnetoelastic and surface
anisotropies, while the shape anisotropy is a dipolar contribution [341].
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When only bulk samples are taken in consideration, magnetocrystalline, magnetostatic and magnetoelastic energies are
usually the main sources of anisotropy in the material. On the other hand, in thin films, fine particles and nanocrystals,
surface and shape anisotropies acquire a significant weight, and, even if they cannot overcome the bulk anisotropic
contribution in all cases, at least they can strongly compete with it. It is worth to point out that shape anisotropy can be the
dominant effect even for bulk materials, when those show large values of MS and low magnetocrystalline anisotropy [339].
The main effects of the different anisotropy contributions on the direction of the spontaneous magnetization of nanoparticles
are shown in Fig. 5.3 and are also discussed below:
Magnetocrystalline anisotropy: such anisotropy derives from the intrinsic crystalline symmetry of the material (crystal
field) and hence reflects the same symmetry [342]. One or more easy magnetocrystalline axes can be defined based on
the symmetry of such anisotropy; when the magnetization lies along them, the total energy of the material is minimized
and the material itself is in the most stable energetic configuration. One of the simplest but also most common forms of
magnetocrystalline anisotropy is the uniaxial anisotropy. This type of anisotropy is associated to the hexagonal crystal
symmetry and it makes the c axis of its hexagonal crystal lattice become the easy axis for its internal magnetization as
well. In cubic structures, the three [111] directions of the crystal act as easy axes of magnetization. The magnetocrystalline
energy for single domain crystals with uniaxial anisotropy can be calculated from the following expression:
Ekuni = Ku V sin2 θ
(5.6)
where Ku is the uniaxial magnetocrystalline anisotropy constant, V is the particle volume and θ is the angle between the easy
magnetization axis and the direction of the magnetization. Hence, for such a particle, Ku V would be the magnetocrystalline
energy barrier necessary to switch the direction of its magnetization. The higher is the energy barrier, the higher is also the
coercivity of a single domain particle. In the single domain regime, and for particles with dimensions above rSP , the coercive
field, which strongly depends on Ek , increases with increasing dimensions. When the radius of the particle approaches the
critical radius rc , the energy of the whole system reaches high values, but the energy required for the formation of domain
walls is still too large. In this range of sizes, particles develop other mechanisms for reducing the total energy without forming
multiple domains [343–345]. When these mechanisms are operative, the spins do not align themselves homogeneously
along the external magnetic field. Instead, they are now homogeneously varying along one crystal direction, (i.e. buckling), or
instead rotating in circular symmetry creating a vortex (i.e. curling). Buckling is the preferred mechanism for thin nanorods
and nanowires, while curling is mostly observed in thin disks. The effect of the activation of such mechanisms in single
domain particles is a continuous decrease of the HC with increasing dimensions. Finally, once the radius of the particle
clearly exceeds rc , the particle decreases its total energy by the formation of several magnetic domains and the inversion
of its magnetization takes place by the displacement of such domain walls, further decreasing the coercivity of the particle.
The whole picture is described in Fig. 5.2. The magnetocrystalline anisotropy is a completely intrinsic characteristic of the
material and does not depend on the presence of an external field, as it is instead the case for the magnetoelastic anisotropy.
Surface anisotropy: compared to bulk samples, nanocrystals show a very high surface to volume ratio and hence all effects
arising from the contribution of surface atoms are strongly enhanced. Surface anisotropy can increase by more than one
order of magnitude the total anisotropy of the sample with decreasing size from bulk to nanocrystals. As a consequence,
it can become the dominant contribution to the anisotropy, with values above those arising from the magnetocrystalline
anisotropy. Atoms located at the surface have an incomplete coordination sphere and this leads to a broken symmetry
compared to the core atoms. Such decrease in the crystalline symmetry at the surface induces a different orientation of
the surface spins with respect to the magnetization direction, spin pinning and surface-core strains that reduce the surface
energy [346]. The surface anisotropy energy can be expressed as follows:
Esurf = Ksurf
∫
cos2 θ · dS .
(5.7)
Here, Ksurf is the surface anisotropy density constant, θ is the angle between the magnetization and the normal to the surface
and dS is the surface element. The sign of the surface energy depends on the nature of the crystal lattice and on the surface
orientation. Surface anisotropy has one main effect on the ferro- and ferrimagnetic parameters of nanocrystals compared to
the bulk. As previously said, the increased surface area gives rise to a higher amount of pinned spins in the crystal that cannot
follow the spontaneous magnetization direction. This is translated into higher values of coercivity, since the anisotropy
term has been increased, but also into lower values of saturation magnetization, since not all spins are able to align along
the direction of the magnetization under magnetic fields of usual magnitude. The presence of spin-glass shells and the
observation of spin-canting phenomena are common surface effects in magnetic nanoparticles [347]. With increasing size
of the nanocrystal, and with the consequent decrease in surface to volume ratio, one should expect lower values of HC
and higher values of MS . However, the final trend of the coercivity with size depends on the fine equilibrium between
magnetocrystalline and surface contributions, since they have opposite effects.
The situation can become even more difficult to predict if one takes into account the possible stabilization of magnetic
nanocrystals with capping molecules. Surfactant molecules are mainly used in bottom-up synthetic approaches where their
presence allows a high level of control on the size and the shape of nanocrystals, which are hardly achievable by traditional
top-down approaches. Depending on the nature of these capping molecules, a relaxation of the surface atoms can be possible
[348]: the coordination sphere of the surface atoms is partially recovered, their broken symmetry is reduced and hence the
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Fig. 5.3. Direction of the spontaneous magnetization of nanoparticles depending on various independent contributions: magnetocrystalline, surface and
shape anisotropy.
surface spins are more easily aligned with the magnetization direction, thus increasing MS . Surface effects are hence largely
dependent on the synthesis method as well as on the quality of the crystals.
Shape anisotropy. This arises from the demagnetizing field associated with the surface magnetic charge distribution
(magnetic dipoles) of a magnetic object [336]. The demagnetizing field generated in non-spherical nanoparticles is a
consequence of the long-range dipolar interactions in the particle. The energy associated with the shape anisotropy of a
particle can be expressed as the summation of dipolar interactions over the whole cluster [349]:
−
→
Eshape m = −
−
→
−
→ −
→2
3µ2at − m · rij
2
i̸=j
−
5
→
rij
(5.8)
−
→
where m is a unit vector along the cluster magnetization direction, µat is the atomic magnetic moment and rij is the vector
connecting any two atoms at sites i and j. A uniformly magnetized single domain spherical particle has no shape anisotropy
because the demagnetizing factors are isotropic in all directions. However, in the case of a non-spherical particle, it is easier
to magnetize the particle along a long axis than along a short one. This is due to the demagnetizing field, which is smaller in
the long direction because the induced magnetic poles at the surface are further apart. If we define the magnetization and
demagnetization factors relative to the x, y and z directions of the crystal as Mx , My , Mz and Nx , Ny , Nz respectively, the shape
anisotropy energy of a uniform magnetized ellipsoid can be expressed as [350]:
Eshape = 1/2µ0 V (Nx Mx2 + Ny My2 + Nz Mz2 )
(5.9)
where the demagnetization factors satisfy the normalization relation Nx + Ny + Nz = 1. Then, the magnetostatic energy for
an ellipsoid of revolution around the z-axis is equal to:
Eshape = 1/2µ0 VMS 2 (Nz cos2 θ + Nx sin2 θ)
(5.10)
where θ is defined as the angle between the direction of magnetization and the long axis z of the particle. From Eq. (5.10),
where only shape features have been considered, it is easy to see that an elongated nanocrystal in the z direction minimizes
its shape energy by aligning the magnetization direction with the long axis of the particle, while its shape energy is highest
for a magnetization direction perpendicular to it.
Shape anisotropy becomes particularly interesting when its competition with the magnetocrystalline anisotropy is
considered. Both contributions define easy and hard magnetic axes of magnetization along which the energy of the particle is
minimized or maximized, respectively. Two extreme situations can be described: let us consider an elongated single domain
magnetic nanocrystal, with uniaxial magnetocrystalline anisotropy, such as a Co nanorod with a hexagonal crystal lattice.
The easy magnetization axes of this nanorod, arising from magnetocrystalline and shape contributions, can be either parallel,
when the nanorod growth direction corresponds to the easy magnetocrystalline axis (c axis), or perpendicular, when the
growth of the crystal is induced in a perpendicular direction with respect to the c axis of the crystal. The combination of these
two easy axes of magnetization decides the effective easy magnetization direction in the nanorod and its effective anisotropy
energy barrier for the magnetization reversal, as shown in Fig. 5.4. In an isolated single domain elongated particle, in which
magnetocrystalline and shape anisotropy are cooperatively coupled, the anisotropy energy is enhanced and accordingly
both the TB and HC are expected to increase with respect to its spherical analogue.
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Fig. 5.4. Cooperation/competition between crystalline and shape magnetic anisotropies for the direction of the effective easy magnetic axis of the
nanoparticle. The variation of the effective anisotropy energy barrier for the magnetization reversal derived from such competition is also described. θ
represents the angle between the magnetocrystalline easy axis and the magnetization direction.
Shape anisotropy can therefore be employed to tune or modify several magnetic parameters characterizing the behavior
of mainly ferro- and ferrimagnets. Therefore, the control over the magnetic properties by the fine engineering of the
morphology of nanocrystals could become a key factor for the use of nanostructured magnetic particles with improved
performance in several fields, such as information storage and biomedicine. However, since so many structural, size, shape
and surface stabilization factors influence the final magnetic behavior of nanocrystals, clear conclusions or trends cannot be
extracted by simple inspection of experimental results.
5.4. Iron oxide-based 1D nanostructures
Iron oxides, being the oldest known magnetic materials in history, are by far the most studied ones, both in bulk systems
and in their nanoscale counterparts. Iron oxide can grow in different structures, depending on formation conditions. Hence,
clearly different magnetic features can be observed depending on their chemical nature and crystal ordering. Before starting
with the review of elongated iron oxide nanocrystals, we therefore briefly describe the basic crystallographic and magnetic
features of the most important varieties.
The wüstite mineral, FeO, is the most reduced form of iron oxide, with all cations being in the +2 oxidation state. For this
reason it is strongly susceptible to disproportion into metallic iron, α -Fe, and magnetite, Fe3 O4 . It crystallizes in the cubic
crystal system and shows antiferromagnetic order below the bulk TN of 195 K. Due to its chemical metastability and low
values of magnetization and critical temperature, it has not attracted much interest up to now.
On the other hand, the magnetite mineral, Fe3 O4 , has a chemical stability that is far beyond that of wüstite. Fe3 O4 can
be understood as a mixture of FeO·Fe2 O3 crystallized together in the same lattice, not to be confused with a solid solution.
It belongs to the inverse spinel group of materials, which have a face-centered-cubic lattice. Its structure can be described
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as a cubic close-packed lattice of O atoms where A2+ and B3+ cations partially occupy the octahedral and tetrahedral sites,
forming AB2 O4 . More precisely, all Fe2+ ions are in Oh sites together with half of the Fe3+ ions, in total half of the available
Oh sites are occupied; the other half of the Fe3+ ions are in the Td sites, meaning that just 1/8 of the available Td sites are
filled. Below its TC of 858 K it shows a ferrimagnetic order, in which spins of Fe2+ and Fe3+ ions, even if antiferromagnetically
coupled, do not cancel in the crystal and a net magnetic polarization is still present in the material. Magnetite has a permanent
magnetic moment with nearly full spin polarization at room temperature. For this reason, it has become one of the most
studied and used iron oxides. Due to the cubic symmetry of its crystal lattice, it has a low magnetocrystalline anisotropy,
with easy axes along the three [111] directions, while the [100] direction is the magnetically hard one. The MS for bulk Fe3 O4
is 92–98 emu/g. Magnetite can also show a spontaneous intercorrelated change of both lattice symmetry and electrical
conductivity at 125 K for the bulk. This is what is known as the Verwey transition. Below the Verwey temperature, TV ,
crystallographic changes are often accompanied by further anomalies in the electric and magnetic properties. A proper
description of such phenomena can be found in the review article of Walz [351]. However, the observation of such transition
in nanostructured materials is quite complicated because it seems to be very sensitive to the stoichiometry and crystallinity
of the particles. Only a few works pointed out the evidence of the Verwey transition in nanocrystals [352].
Hematite (α -Fe2 O3 ) is the most stable iron oxide under ambient conditions. Because of its low cost, high resistance to
corrosion and environmentally friendly properties, hematite has been used in many applications as catalyst [353], pigment
[354], gas-sensor [355] or electrode material [356]. It is characterized by a rhombohedrically centered hexagonal structure
of corundum-type, with a close-packed oxygen lattice in which two thirds of the octahedral sites are occupied by Fe3+ ions.
Hematite is characterized by two temperature-dependent magnetic transitions: Below its TN of 948 K, the low symmetry
of the cation sites allows spin–orbit coupling to cause canting of the antiferromagnetically coupled spins when they are in
the plane perpendicular to the c axis of the lattice. This condition gives rise to what is known as a ‘‘canted antiferromagnet’’
or ‘‘weak ferromagnet’’ with a low net magnetic moment along the c direction. However, when lowering the temperature,
hematite shows a transition from weak ferromagnetic to antiferromagnetic: the Morin temperature, TM , is the temperature
below which the low magnetic moment vanishes due to a change in anisotropy that forces the alignment of the spins along
the c direction. For bulk hematite TM is 263 K.
Maghemite, γ -Fe2 O3 , can be regarded as the fully oxidized form of magnetite, with an identical cubic spinel structure
where the Fe2+ ions of the octahedral sites have been completely removed. The loss of magnetic ions results in a slight
decrease of the room temperature spontaneous magnetization (∼80 emu/g for the bulk), compared to its mixed valence
analogue Fe3 O4 . Maghemite has ferrimagnetic behavior below its TC of ca. 918 K due to the non-equivalent crystallographic
sites of its Fe3+ ions; however, this TC is difficult to determine experimentally since at high enough temperatures maghemite
undergoes an irreversible crystallographic change to α -Fe2 O3 , with the consequent dramatic loss of magnetization. This
phase transition has been observed to occur over a wide range of temperatures and seems to be highly dependent on the
grain size and the presence of impurities.
Fe3 O4 and γ -Fe2 O3 are difficult to distinguish by X-ray diffraction or electron microscopy techniques due to their identical
crystal symmetry and almost identical lattice parameters. However, oxidation state sensitive techniques such as X-ray
Absorption Near Edge Structure (XANES) or Mössbauer spectroscopy can easily solve this problem.
It is worth mentioning the existence of the highly metastable ε -Fe2 O3 phase of iron oxide. This phase has a very narrow
temperature range of existence and actually it has only been observed in the nanoscale regime [357]. It can be regarded as an
intermediate phase between α -Fe2 O3 and γ -Fe2 O3 and it crystallizes in a noncentrosymmetric orthorhombic structure that
opens the possibility to use this oxide as an interesting material for piezo-, magneto- and pyroelectric applications. From a
magnetic point of view, ε -Fe2 O3 shows a ferrimagnetic behavior below its TC temperature of 510 K and more interestingly,
due to the low symmetry of its crystal lattice it presents high values of magnetocrystalline anisotropy compared to the
previous iron oxide allotropes.
5.4.1. Magnetite-based 1D nanostructures
A few examples of single crystalline 1D Fe3 O4 nanostructures have been reported in the literature. Nanorods with average
diameters of 40–50 nm and lengths up to 1 µm were synthesized by Lian et al. through hydrolysis of FeCl3 and FeSO4
solutions containing urea. The reaction was carried out under reflux conditions and in the absence of surfactants and it
was followed by an aging process [358]. Selected area electron diffraction (SAED) measurements confirmed the single
crystalline nature of the rods even though the growth direction could not be discerned. Field-dependent magnetization
measurements on randomly oriented samples indicated a value of MS of 67.55 emu/g, which is significantly lower than the
corresponding value for the bulk Fe3 O4 , 92 emu/g. On the contrary, under hydrothermal conditions, and with the assistance
of polyethylene glycol as surfactant, Wang et al. obtained uniform nanorods with diameters of 20–25 nm and lengths up to
200–300 nm (Fig. 5.5(a)) [359]. High resolution transmission electron microscopy (HRTEM) analysis confirmed the single
crystalline character of the rods and further identified the [110] direction as their growth direction (Fig. 5.5(b)). This direction
interestingly matches with one intermediate easy magnetization axis of the cubic structure of magnetite. The authors claimed
that this could be one reason for the high value of MS observed, 90.5 emu/g, which is only slightly lower than that of
bulk samples. In another work, the same authors used a similar synthetic approach for the synthesis of the first single
crystalline Fe3 O4 nanowires, the main difference being that the reaction was carried out under the effect of an external
magnetic field with the hope that this would induce an oriented growth of the material [360]. Depending on the strength
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of the applied magnetic field, they successfully synthesized polyhedral-shaped nanoparticles, nanowires of 35–100 nm of
diameter and 480–2700 nm of length, as well as nanofibers of 20 nm diameter and 800 nm length approximately (Fig. 5.5(c)
and (d)). Also in this case, HRTEM allowed to identify the [110] easy magnetocrystalline axis as the growth direction of
the wires. In Fig. 5.5(e), magnetic properties of powder samples of nanowires and nanofibers are compared with those
of cubic or hexagonally-shaped Fe3 O4 nanocrystals, obtained under the same conditions but in the absence of magnetic
field. For 1D systems, MS values were found to be in the range between 35 and 40 emu/g, less than the half of the value
attributed to the bulk and lower than the value observed for cubic or hexagonal nanocrystals (68.7 emu/g). The projection
of the magnetization vectors along the field direction is lower for a randomly oriented sample of nanowires than that for a
collection of nanoparticles without shape anisotropy. As a consequence the authors claimed that the total magnetization of
the sample could be significantly reduced due to the randomization of the wires. Similar observations were made by Wan
et al. for both Fe3 O4 nanorod and nanowire samples compared to analogous spherical nanoparticles [361]. Additionally, in
that case the authors studied the differences in coercivity: higher values of HC for 1D samples were attributed to the shape
anisotropy of the nanocrystals, in agreement with what theory predicts.
Recently, the first preliminary experimental results concerning the performance of rod-shaped magnetic nanocrystals in
biomedicine have emerged [362]. Perez and co-workers reported on the synthesis of water-soluble dextran-coated Fe3 O4
nanorods and their activity as magnetic sensors for bacterial detection. The authors first cross-linked and aminated the
dextran coating of the nanorods in order to conjugate them to the antibodies of the test bacterium. They chose Mycobacterium
avium spp. paratuberculosis (MAP) as their model organism because the identification of MAP using current methods is
difficult, due to the very slow growth of the bacterium. The detection was made by monitoring the T2 relaxation times using a
0.47 T relaxometer. A significant improvement was observed in the detection of MAP bacteria in milk samples when nanorod
Fe3 O4 structures were used instead of spherical ones, as compared to their previous publication [363]. Improvements were
found both in detection limit and in detection kinetics and they were attributed to the high magnetic water relaxivity
associated with nanorods and to the large number of antibodies conjugated per nanorod.
5.4.2. Hematite-based 1D nanostructures
Hematite is by far the most studied form of iron oxide due to its high stability and wide range of applications. Several
authors reported on the synthesis of hematite nanorods [358,364–367], nanowires or other elongated nanostructures [368–
371]. However, just a few of them performed their magnetic analysis. As mentioned before, bulk crystalline α -Fe2 O3
undergoes two different magnetic transitions with temperature: below its TN of 960 K, the material shows a weak
ferromagnetism where the spins are antiferromagnetically coupled in the basal plane of the lattice but slightly canted along
the c direction giving rise to a low magnetic moment. Below the TM of 263 K, the spins are antiferromagnetically aligned
aligned along the c axis of the crystal and its magnetic moment is cancelled. It is known that both transition temperatures are
significantly affected by the size of the nanocrystal, decreasing with decreasing size [372], in a way that small particles still
behave as weak ferromagnets with a non-negligible magnetic moment at low temperatures below the TM of bulk hematite.
In the last decade several works have provided experimental evidences of their shape dependence as well. In 1999, Fiorani
et al. reported on the low value of the TM for acicular-shaped α -Fe2 O3 nanoparticles (length × diameter: 350 × 85 nm)
which was observed to be 170 K, a temperature at which the authors observed a sharp increase of the magnetization with
increasing temperature due to the appearance of a weakly ferromagnetic phase [373]. They assigned the decrease in TM
with respect to the bulk value to the size reduction and shape anisotropy of the particles. A similar value of TM (ca. 166
K) was also found by Liu et al. for single crystal nanorods of 500–1100 nm in length and 30–50 nm in diameter [374]. The
nanorods showed a magnetic hysteresis loop at room temperature (HC = 280 Oe), which confirmed the high temperature
weak ferromagnetic state of the sample.
Woo et al. observed a constant increase of the magnetization from 5 to 300 K for single crystalline hematite nanorods
(16 × 5 nm), which is representative of a ferromagnetic behavior already at 5 K with an MR and HC of 0.28 emu/g and 53
Oe, respectively [375]. No clear TM value was observed for that sample. It is worth noting that these rods are much thinner
than the ones of the previous examples [373,374]. The Morin transition was found to be strongly dependent on the particle
size for spherical nanocrystals and it was usually not observable anymore for particle diameters in the range from 8 to 20
nm or below [372]. For this reason TM was also expected to follow a similar trend with decreasing diameter of the rod. Some
groups made an attempt to study the evolution of the Morin transition with differently sized α -Fe2 O3 nanorods. Following
an hydrothermal synthetic approach with an additional thermal treatment in air, differently sized nanorods with or without
a porous morphology were obtained, depending on the presence of particular inorganic salts in the reaction mixture, which
did not alter the chemical nature of the hematite phase, but only its morphology [376]. Their TM was correlated with the
diameter of the rod-shaped nanoparticles. The magnetic measurements are presented in Fig. 5.6.
These results indicated a classical Morin transition for particles with larger diameters (300–500 nm in one sample and
60–90 nm in the second; the second sample was characterized by a porous morphology with pore diameters in the range
between 20 and 50 nm) in which the high temperature weak magnetization dropped drastically upon cooling below the TM ,
for both ZFC and FC curves. In both cases TM was seen to be 255 K, irrespective of the rod diameter and very close to the TM
value for the bulk. On the other hand, samples of nanorods with diameters between 2 and 16 nm had either lower values
of TM (235 K), or even no apparent Morin transition down to 5 K. In both samples the phenomenon was associated with the
decrease in diameter of the rods. The authors claimed that the presence of nanosize internal pores in one sample could be
responsible for the disappearance of the antiferromagnetic phase in that case, but no experimental proofs were reported.
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Fig. 5.5. (a) Transmission Electron Microscope (TEM) micrograph of the rod-like nanocrystals from Ref. [359]; (b) High resolution TEM micrograph of
one single nanorod of the sample in (a) confirming its single crystalline character; the inset is the electron diffraction pattern obtained from a single
Fe3 O4 nanorod; TEM micrographs of nanocrystals prepared in Ref. [360] in the absence, (c), and presence, (d) of external magnetic field; and (e) magnetic
hysteresis curves measured at room temperature for polyhedral-(A), nanowire-(B) and nanofiber-(C) like particles of Ref. [360].
Source: Taken with permission from Refs. [359,360].
Zhao et al. compared the magnetic properties of two single crystal nanorod samples obtained using different synthetic
approaches, i.e. iron–water vapour reactions and hydrothermal techniques: needle-shaped rods of sample 1 had average
lengths between 1.5 and 2 µm and diameters of 20 nm at the tips and 100 nm at the base. Sample 2 was formed by
ellipsoidal rods with long axis of 400–500 nm and short axis of 50–60 nm [377]. Magnetic measurements are reported
in Fig. 5.7. At 200 K, the hysteresis loops of samples 1 and 2 showed different values of HC , those being 198 Oe and 688 Oe,
respectively. This observation was in accordance with the multidomain character of sample 1 and the pseudo-single domain
character of sample 2, based on the criteria given by Dunlop [378]. Such different magnetic domain features are responsible,
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Fig. 5.6. Temperature dependence of ZFC and FC magnetization for an applied field of 500 Oe for samples from Ref. [376]. Insets are their corresponding
differential ZFC curves.
Source: Taken with permission from Ref. [376].
as described in Section 5.2, for lower coercivity values in the case of multidomain samples, due to the lower energy required
for domain wall motion compared to the coherent magnetization rotation of a single domain. An interesting and unexpected
phenomenon was observed for sample 1 in the temperature-dependent FC/ZFC measurements recorded at low fields, which
was assumed to be strictly related to the shape anisotropy of the nanocrystal. A decrease in the hematite magnetization value
is expected with decreasing temperature when TM is reached, due to the weak ferromagnetic to antiferromagnetic transition.
The FC/ZFC curves were recorded for both samples at a fixed field value of 100 Oe. The expected profile of the curves was
observed for both samples, from which a value of 122 K was determined for TM of sample 1, while no apparent TM was seen
in sample 2, in accordance with the reduction of size. However, when FC/ZFC experiments were performed under a weak
external magnetic field of 10 Oe, sample 1 exhibited an unexpected sharp increase of magnetization at the same TM observed
in the measurement performed at 100 Oe.
The authors calculated shape anisotropies that were very similar for both samples and very close to the value of
magnetocrystalline anisotropy of rhombohedral hematite. This strongly suggests a competition of both kinds of anisotropy
in the effective magnetization direction in the samples. Below TM , the magnetocrystalline easy axis of magnetization for
α -Fe2 O3 is found along the [111] direction of the crystal; on the other hand, at temperatures above TM there are three easy
magnetization axes in the basal plane of the structure and each axis is separated from the others by 60°. The shape anisotropy,
however, tends to align the magnetization, independently of the temperature, along the growth direction of the rods which,
for sample 1 is the [118] direction, while for sample 2 it is the [111] one. Hence, the easy axis of magnetization lies in a
direction resulting from the coherence of the shape and the magnetocrystalline anisotropies, which is clearly different for
samples 1 and 2 and also above and below TM . When weak magnetic fields are applied, the magnetization of the sample is
mainly directed by the intrinsic anisotropy of the nanorods and different magnetic behaviors can be expected for samples
1 and 2. In this case, the authors concluded that for sample 1 the magnetic susceptibility along the resultant direction has
a larger value for temperatures below TM than for those above TM . Alternatively, when higher magnetic fields are applied,
e.g. 100 Oe, the pinning of the spins due to shape anisotropy is overcome and hence the expected behavior at TM is restored.
It is interesting to notice that experimental results are usually explained by magnetocrystalline anisotropy considerations,
since these dominate in the bulk material. However, in the case of highly anisotropic nanocrystals, where important shape
contributions also take place, new phenomena can emerge and conventional arguments need to be reconsidered.
Shape anisotropy, as discussed before, is expected to increase the total anisotropy of the particle, and so higher values
of coercivity should arise from higher aspect ratio nanorods. Zeng et al. reported an increase of the value of HC from low
(ca. 1.8) to very high (ca. 21) aspect ratio hematite nanorods [379]. The expected trend was observed with HC values at
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Fig. 5.7. Temperature-dependent magnetic susceptibility measurements and hysteresis loops of hematite nanorods from Ref. [377]: (a), (b) nanorods
prepared by iron–water vapour reactions; (c), (d) nanorods prepared by hydrothermal techniques. The inset in (b) is the dχ/dT vs. T curve for the first
sample when an external magnetic field of 100 Oe was applied.
Source: Taken with permission from Ref. [377].
5 K running from 67 Oe, for the lowest aspect ratio rods, to 584 Oe for the highest aspect ratio ones, with an intermediate
value of 146 for nanorods with an aspect ratio of 18. Other authors reported similar coercivities for nanorod ensembles
of different dimensions but similar aspect ratios, confirming the previously described trend [380]. In the case of Zeng
et al., the three samples showed a constant increase in magnetization with decreasing temperature and no apparent Morin
transition was observed for any of them. This could mean that all samples are in a weak ferromagnetic state even at low
temperatures, which was supported by the presence of coercivity at 5 K in all cases. The authors believe that the increased
surface contribution arising from the high shape anisotropy of the crystals is responsible for the absence of TM . Contrary
to Zeng’s observations, Zhang et al. did not observe any difference in magnetic behavior in two samples of nanorods with
significantly different aspect ratios [381]. The samples were synthesized according to the same hydrothermal procedure,
but at two different temperatures. The observed low temperature hysteresis loop was assigned to a ferromagnetic state that
was still present at 5 K, even though no FC/ZFC measurements were reported to check the presence or absence of a Morin
transition. Experimentally, by means of temperature-dependent magnetic measurements, the value of TM can be determined
by the sharp peak in the dχ/dT vs. T curve. When such peak is not observed, which means that the susceptibilities of the FC
curve do not decrease with decreasing temperature, the sample is said not to show any apparent TM , and hence coercivity is
still expected at low temperatures. However, even when a value of TM can be determined and consequently the formation
of an antiferromagnetic phase is confirmed at T < TM , often an hysteresis can still be observed at low temperatures.
Sorescu and co-workers have studied different hematite particles with polyhedral, plate-like, disk-shaped and needle-like
morphologies, prepared by hydrothermal methods, using Mössbauer spectroscopy [382]. Their results have proven that the
weak ferromagnetic and antiferromagnetic phases can coexist below TM , but the population of the first strongly depends on
the morphology of the particle, being the largest for elongated needle-like nanocrystals, as shown in Fig. 5.8.
5.4.3. Maghemite-based 1D nanostructures
As described above, maghemite (γ -Fe2 O3 ) is an oxidized form of Fe3 O4 . Although ferrimagnetic properties are still
present in γ -Fe2 O3 , its magnetization is lower than that of the mixed valence analogue. Special attention should be paid
during magnetic characterization: at high enough temperatures, γ -Fe2 O3 can undergo a phase transformation to the
antiferromagnetic or weak ferromagnetic α -Fe2 O3 . This transformation can occur in a wide range of temperatures depending
on grain size and type of impurities. The literature concerning nanorods or elongated nanostructures of the γ phase of
iron oxide is limited and is focused mainly on the synthetic aspects [366,383]. In a few reports, the magnetic study of the
nanorods was undertaken to some extent. For example, in 2004 Woo et al. prepared single crystalline maghemite nanorods
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Fig. 5.8. Percentage population of the weak ferromagnetic phase, with respect to the antiferromagnetic one, as a function of temperature, for hematite
particles of different morphologies [382].
Source: Taken with permission from Ref. [382].
with dimensions 5 × 17 nm [375]. The authors attributed the maximum in the ZFC magnetization vs. temperature curve to
a value of TB of 130 K, lower than the TC value of the bulk, but higher than the value measured for maghemite nanospheres
with similar diameter [384].
5.4.4. ε -Fe2 O3 -based 1D nanostructures
The highly metastable ε -Fe2 O3 phase of iron oxide has a very narrow temperature range of existence and actually it has
only been observed in the nanoscale [357]. In 2004, Ohkoshi and co-workers reported for the first time a ε -Fe2 O3 single
phase in the form of nanorods, covered by a thin silica shell (lengths in the range between 100 and 140 nm and diameters
of 20–40 nm) [385,386]. Pure ε -Fe2 O3 nanorods possessed good crystallinity and were observed to grow along the a axis
of the orthorhombic lattice, as shown in Fig. 5.9(a) and (b). Due to its low symmetry crystal structure, it presents high
values of the magnetocrystalline anisotropy compared to previous iron oxides. ε -Fe2 O3 is a ferrimagnetic material with a
TC of 510 K. However, following the usual trend with reducing size of the crystal, nanorods reported by Ohkoshi had a TB
of 480 K, i.e. between the high temperature superparamagnetic and the low temperature ferrimagnetic states. This means
that such small nanorods were ferrimagnetic at room temperature and, moreover, they had giant values of coercivity: HC
at room temperature was observed to be 2 T, that is 20,000 Oe, as shown in Fig. 5.9(c). The stability of the ε phase in the
nanorods is believed to be due not only to the growth of a stabilizing silica shell, but also to the presence of Ba2+ ions in
the reaction mixture which, by means of the formation of a thin BaO layer, creates an additional space confinement on
the surface of the rods [387]. However, the removal of these two components after the synthesis does not seem to alter
the magnetic properties of the nanorods. The authors evaluated the different anisotropy contributions taking place in the
nanocrystal and they explained the high value of HC by (i) the single magnetic domain character of the nanorods, due to
their small size and high degree of crystallinity and (ii) their high value of magnetocrystalline anisotropy energy (more
than one order of magnitude higher than in α - or γ -Fe2 O3 ). However, the calculated shape anisotropy contribution in the
rods is suggested to be responsible for 0.53 T of the total HC of the sample, and hence the shape contribution should not be
neglected.
5.4.5. Assembly of oxide-based 1D nanostructures
The regular alignment of 1D magnetic nanoparticles (i.e. rods, wires) is the only configuration that allows for the
observation of anisotropic magnetic features that depend on the relative direction of the applied magnetic field with respect
to the assembly orientation. Such angular dependence of the magnetic properties of well-aligned magnetic 1D systems is
expected to introduce unique characteristics, which could be useful for several devices, and has been matter of study for
many decades. The presence of dipolar interactions between nanoparticles in the assembly should also be taken into account:
they can significantly modify the magnetic behavior of the single particle, as observed by several groups [388]. Already in
1961 Bate studied the variation of magnetic properties with increasing angles between the applied field and the alignment
direction of γ -Fe2 O3 acicular particles of aspect ratio close to 7, dispersed in a suitable non-ferromagnetic matrix, with
different coverage degrees. The results showed clear evidence of different magnetic behavior due to the shape anisotropy
of the particles [389].
The synthesis of iron oxides has been usually carried out in solution, either by sol–gel or by hydrothermal techniques,
leading to isolated particles that are randomly oriented in the sample. The effect of the shape anisotropy in such systems
could be observed in single particle studies, which are generally difficult to perform. Otherwise, the controlled alignment
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Fig. 5.9. Low (a) and high (b) resolution TEM micrographs of ε -Fe2 O3 nanorods from Ref. [385]. The insets are a high resolution TEM image of the particle
showing its lattice planes and a selected area of the Fourier transform pattern of this image (crystal planes identified). (c) Magnetization vs. field plots at
room temperature for ε -Fe2 O3 nanorods.
Source: Taken with permission from Ref. [385].
of the particles is strictly required if one wants to study the angular-dependent magnetic properties of ensembles of
particles. However, the experimental difficulty of aligning the as-prepared 1D oxide nanostructures has represented a
clear limitation compared to metal 1D systems for which, as will be discussed later in Section 5.5, several experimental
procedures have been developed to grow regularly oriented 1D nanorod or nanowire assemblies. Nevertheless, some
synthetic approaches have also been reported for the growth of well-aligned oxide 1D nanoparticles. Some of these
approaches include: the controlled precipitation of Fe precursors on substrates [390,391], the thermal oxidation of
metal foils under well-controlled conditions [392], pulsed laser deposition techniques [393], metal–organic chemical
vapour deposition on substrates (MOCVD) [394] or other vapour–solid routes [395]. As reported previously for randomly
oriented 1D nanostructures, α -Fe2 O3 is also the most widely studied form of iron oxide for what concerns oriented
assemblies.
α -Fe2 O3 nanowires of lengths up to 15 µm with a sharp tip and wider base of 50 nm and 100–300 nm length, respectively,
were grown on Fe foils by Kim et al. (Fig.
[396]. They clearly confirmed their single crystalline character and their
5.10(a)–(c))
rhombohedral structure where the 1120 was identified as the only growth direction. Temperature and field-dependent
magnetic measurements on randomly oriented wires that were detached from the substrate were used to determine a
TM of 125 K (far below the bulk value) and a ferromagnetic state both at room temperature and at 5 K, with a coercive
field of 150 Oe at 5 K. The authors also described the results obtained from magnetic measurements performed under an
external magnetic field oriented parallel and perpendicular to the nanowire axis (Fig. 5.10(d) and (e)). The same TM value was
extracted from data obtained in both configurations. Interestingly, magnetic hysteresis was only observed when the field
was applied perpendicular to the growth direction of the nanowires, while zero coercivity was observed when the field was
parallel to it. The authors suggested that this was a clear evidence of the
of magnetocrystalline anisotropy over
dominance
shape anisotropy. While the shape anisotropy easy axis is parallel to the 1120 direction, the magnetocrystalline anisotropy
easy axis is parallel to the [0001] direction and hence perpendicular to the growth direction. Consequently, when the field
is applied in the direction of the magnetocrystalline anisotropy, more square-shaped cycles with non-zero coercivities were
observed.
Wu et al. performed similar measurements on α -Fe2 O3 nanorods vertically aligned on Si substrates with the [104] as
the growth direction [394]. In that case, both parallel and perpendicular orientations of the field with the axis of the rods
led to similar values of coercivity of 200 Oe. However, the saturation magnetization of the sample was observed to be
significantly higher when the field was applied perpendicular to the rod axis, probably indicating the dominance of the
magnetocrystalline anisotropy over the shape contribution. Several factors such as the synthetic procedure followed for the
sample preparation, and the different size and aspect ratios, compared to the nanowires described before and reported by
Kim et al. [396], could be some of the reasons for the different observed behavior. In any case, the resulting easy axis of
magnetization depends on the directions of both the magnetocrystalline and the shape easy magnetic axes, as described
earlier (Fig. 5.4). Consequently, the different growth directions of the 1D structures in both cases led to effective easy
magnetic axes with different orientations. This is probably responsible for the observation of coercivity in the nanorod
sample of Ref. [394], independent of the magnetic field direction, while in the nanowire sample of Ref. [396] coercivity was
only observed in the perpendicular direction.
Chueh et al. described the synthesis of single crystalline α -Fe2 O3 nanowires grown on metallic substrates with growth
along the [110] direction [395]. A thermal treatment of the as-synthesized α -Fe2 O3 nanowires was conducted under
reducing atmosphere in order to obtain single crystalline
Fe3O4 nanowires which maintained the original dimensions and
shape of the starting wires, but with growth along the 110 direction. Nanowires of both types were separated from the
original substrates and deposited independently on Si substrates before performing magnetic characterization, in order to
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Fig. 5.10. (a) Scanning Electron Microscope (SEM) micrographs showing the high-density α -Fe2 O3 nanowire arrays grown on substrates in Ref. [396];
(b) Side view reveals the vertical alignment of the nanowires; (c) A magnified view reveals the sharpness of the wire tips; M–H curves of α -Fe2 O3 nanowire
arrays, under a magnetic field applied parallel and perpendicular to the nanowire axis. The measurements were carried out at 5 K (d) and 300 K (e) for
these two directions of the magnetic field.
Source: Taken with permission from Ref. [396].
avoid the magnetic contribution from the metallic substrates. The authors studied the effect of the applied magnetic field on
the nanowires lying on the Si substrate. They found that fields up to 5 kOe were not sufficient to align α -Fe2 O3 nanowires,
due to their extremely weak ferromagnetism in the canted-spin state between the TN and TM . On the contrary, magnetic
fields of the same magnitude were sufficient to induce a complete alignment of Fe3 O4 nanowires on the Si substrate. A
coercive field of about 220 Oe and high remanent magnetization values were measured when the field was applied parallel
to the substrate, i.e. parallel to the wire axis. Instead, when the direction of the external field was applied perpendicular
to the substrate, i.e. perpendicular to the wire axes, the coercivity decreased to a value of 140 Oe and also the remanent
magnetization suffered a significant decrease with respect to the previous situation. These observations were rationalized
by the authors in terms of shape anisotropy.
In the case of CoFe2 O4 1D structures, the high magnetocrystalline anisotropy is usually predominant over the shape
anisotropy [397]. The global anisotropy of CoFe2 O4 elongated nanostructures can be better enhanced by magnetostrictive
phenomena than by shape control. This is the case of CoFe2 O4 nanopillars grown along the [110] direction and embedded
in a BaTiO3 matrix that induces significant lattice stress on the ferrite material, as reported by Zheng et al. [398].
Other works have reported the assembly of rod-shaped α -Fe2 O3 nanocrystals in solution to form 3D superstructures.
In these cases, however, the orientation-dependent magnetic characterization could not be carried out since the 3D
organization of rod-shaped building blocks in the superstructure entailed several different orientations of the same. Zhu
et al. described the preparation and characterization of 3D urchin-like α -Fe2 O3 superstructures [399]. Oriented attachment
of isolated nanorods seemed to be the preferred mechanism for the formation of the superstructures. Higher remanent
magnetization and coercivity values were measured for such structures compared to the ones observed for the isolated α Fe2 O3 nanorods, although the reasons for this were not clarified in the article. The oriented attachment is also the mechanism
by which An et al. prepared three different types of 3D α -Fe2 O3 -based superstructures [400]. Depending on the reaction
conditions the authors were able to prepare elliptic superstructures made by assembled rod-shaped α -Fe2 O3 building blocks,
spherical superstructures composed of several nanoplatelet crystals, and spindle superstructures with a much integrated
nature of the building block components, all shown in Fig. 5.11.
The magnetic characterization of the three final products indicated higher values of coercivity for the superstructures
with an elliptical shape. The enhanced shape anisotropy of both the superstructure and the initial building blocks seem to be
responsible for this. However, saturation magnetization values were lower for the elliptically-shaped assemblies, probably
due to the smaller size of the individual rod-shaped nanocrystals.
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Fig. 5.11. SEM micrographs showing different α -Fe2 O3 superstructures from Ref. [400]: (a) Elliptic superstructures made by assembled nanorods;
(b) Spherical superstructures composed of several nanoplatelet crystals; and (c) Spindle superstructures with a much integrated nature of the building
block components.
Source: Taken with permission from Ref. [400].
5.5. Metallic 1D nanostructures
Magnetic metals such as Fe, Co or Ni have higher magnetic moments than their corresponding oxides. This property can
be exploited in both biomedical and magnetic recording media applications. The synthesis of 1D metallic nanostructures
has been classically carried out by template or membrane-assisted methods [401], metal evaporation [402], pyrolysis [403],
molecular beam epitaxy [404], lithography [333] and surfactant-assisted colloidal methods [405]. All these methods have
been proved to successfully lead to well-defined magnetic metal 1D nanostructures. Contrary to oxide-based nanorods or
wires, in the case of pure metals synthetic approaches in which 1D structures are grown already regularly oriented on
substrates are much more developed. It is important to keep in mind that metals are very sensitive to oxygen and can easily
evolve to core–shell or hybrid structures in which the magnetic properties have been modified. Reduced magnetization
values or exchange bias phenomena are common fingerprints of oxide-passivated metallic nanostructures [406,407]. The
study of magnetization reversal processes and domain wall dynamics in metallic nanowires is of crucial importance for the
development of spintronics. The last part of this section will therefore be devoted to the description of the state-of-the-art
concerning this field.
5.5.1. Co 1D nanostructures
The hexagonal closed-packed structure of Co is the most studied one due to its intrinsic high anisotropy along the c
axis, which in terms of magnetism obviously defines its magnetocrystalline easy axis. It is also the thermodynamically
preferred phase at low temperatures. From what described in the introduction section, the anisotropy of ferromagnetic Co
1D structures grown along the [0001] direction is expected to be significantly large due to cooperating magnetocrystalline
and shape contributions. Co nanowires were grown by electrochemical deposition techniques inside the porous structure
of polycarbonate membranes by Whitney et al. [408]. This early work led to parallel Co nanowires perpendicularly oriented
to the substrate with variable diameters, depending on the size of the pores of the original membrane. The distribution
of the nanowires on the substrate was somehow broad, with interwire distances ranging from 5 to 50 nm or more. The
wires exhibited surface roughness and a polycrystalline character. Their preferred growth direction coincided with the
magnetocrystalline easy axis, [0001], even if other growth directions were also seen, especially for narrower wires. For Co
nanowires with a diameter of 60 nm, several conclusions were drawn from the comparison of the magnetic data obtained
with magnetic external fields applied parallel and perpendicular to the wire axis. First, larger and square-shaped hysteresis
loops were obtained when the field was applied parallel to the wire axis, indicating the resistance of the magnetic moments
to be misaligned from the growth direction of the wire. A further proof of preferred orientation of the magnetic moments is
given by the significantly higher fields required to reach saturation magnetization when the field was applied perpendicular
to the axis. Coercivity values in the range between 150 and 680 Oe were observed depending on the diameter of the wire,
increasing with decreasing dimensions but, in any case, always higher than the few tens of Oe measured for bulk Co. The
opposite trend between coercivity and diameter was rationalized by the magnetic multidomain character associated to
larger nanowires which is responsible for the degradation of their coercivities. The authors highlighted the importance
of such oriented ferromagnetic systems for high-density perpendicular recording media, due to the enhanced coercivities
observed and to the capacity to retain high values of magnetization in the absence of the magnetic field. The results reported
by Whitney et al. were further confirmed by Henry and co-workers some years later [409]. Co nanowires were prepared in
this case by analogous synthetic procedures. By means of magnetization and magnetic torque measurements, as well as
magnetic force microscopy (MFM), the authors were able to study the cooperative effect of magnetocrystalline and shape
anisotropy contributions for nanowires with a single domain character when diameters were below a critical size of 50
nm. When the field was applied parallel to the revolution axis of the wire, a single magnetic domain was observed, with the
magnetic moment aligned parallel to the axis, as shown in Fig. 5.12(a)–(d). Instead, when the field was applied perpendicular
to it, the magnetic configuration consisted of a succession of longitudinally magnetized antiparallel domains, as shown in
Fig. 5.12(e)–(g). Such a situation arises from the competition between magnetocrystalline and shape anisotropy. On the
other hand, for nanowires with diameters larger than 50 nm, complex multidomain patterns were observed in which the
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Fig. 5.12. (a) MFM image of a nominal 10 µm long, 35 nm thick Co nanowire after applying an axial field of 1.3 T (longitudinal remanent state).
(b) Experimental (open circle) and theoretical (solid line) MFM profiles along the wire axis. (c) Simulated MFM contrasts assuming, as for the calculated
profile shown in (b), the magnetic configuration sketched in (d), i.e. a homogeneous longitudinal magnetization in the wire. (e) MFM images, (f)
densitometer traces along the wire axis and (g) schematic magnetization distributions for a 10 µm long, 35 nm thick Co nanowire after applying a transverse
field of 1.3 T (transverse remanent state).
Source: Taken with permission from Ref. [409].
magnetization had a large transverse component. However, the origin of the change of crystallographic texture of Co, as the
wire became wider, was not clarified.
The electrochemical deposition of metals inside the pores of anodized aluminum oxide templates (AAO) is a simple and
inexpensive technique for the growth of aligned polycrystalline metallic nanowires [410]. Kahn et al. grew Co nanowires
by exploiting this kind of templates [411]. XRD spectra indicated that the preferred growth direction of the wires was not
the one parallel to the c axis of the hcp Co structure. However, greatly enhanced coercivities were measured when the
field was applied parallel to the revolution axis of the wires while very low HC values were observed in the perpendicular
direction. The variation of the coercive field and squareness (MR /MS ) of the sample with the rotation of the external magnetic
field featured a clear gradual increase when the field was approaching a parallel direction with respect to the wire axis,
confirming the uniaxial anisotropy of the sample. Based on XRD results, the authors concluded that the magnetocrystalline
anisotropy was not the predominant contribution in the structures. The large anisotropy appeared to be related to surface
and shape effects. The large scale fabrication of arrays of cobalt nanorods was also achieved by Aslam et al. via a similar AAO
template approach, using a modified electrodeposition method [412]. The length of the nanorods could be tuned by simply
varying the electrodeposition time. The rod diameter and inter-rod separation in the vertically aligned arrays were 100 and
40 nm, respectively. More homogeneous samples, in terms of nanorod lengths, were obtained when a magnetic field was
applied perpendicular to the substrate during the electrodeposition process. XRD and TEM measurements indicated the
single crystalline character of the rods and the face-centered-cubic symmetry of their lattice. The fcc structure is usually
obtained at high temperatures, compared to the usual hcp one, but it becomes stable also at room temperature or below
when nanoscale objects are considered [413]. With respect to the hcp phase, the cubic phase of cobalt is characterized by
higher saturation magnetization values but a lower anisotropy. Magnetic measurements were performed for the samples of
well-aligned cobalt nanorods of different lengths with the field applied parallel and perpendicular to the growth direction
of the rods, see Fig. 5.13, which was parallel to the (220) texture. When the field was applied parallel to the rods, a clear
increase in the saturation magnetization was observed for longer rods: actually rods of 14 µm exhibited a MS which was
one order of magnitude larger than the smaller rods of 750 nm. The saturation of all the samples decreased when the field
was applied perpendicular to the growth direction of the rods. This observation was used by the authors to conclude that
the effective easy magnetic axis lay along the nanorod axis.
Other template-based methods have exploited the equilibrium self-assembled morphology of asymmetric di-block
copolymers in order to create nanopores of high aspect ratios, in which metals can be grown by the previously mentioned
electrodeposition techniques [414]. Co nanowires of lengths of about 500 nm, with a high aspect ratio of 36, were prepared
R. Krahne et al. / Physics Reports 501 (2011) 75–221
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Fig. 5.13. (a) Hysteresis curves measured at 300 K of Co nanorods of different length prepared during an electrodeposition period of (i) 30 min (nanorod
length ∼14 µm), (ii) 8 min (nanorod length ∼3.8 µm), and (iii) 3 min (nanorod length ∼0.75 µm). The applied field is oriented along the elongation
direction of the nanorods. (b) Hysteresis loop for a 30 min electrodeposited Co nanorods array, the substrate is tilted parallel and perpendicular to the
magnetic field.
Source: Taken with permission from Ref. [412].
Fig. 5.14. (a) SEM image of a fracture surface of an array of Co nanowires grown within an array of nanopores formed from block copolymers. The growth
of the nanowires was terminated before the template was completely filled, and above the nanowires of Co is the unfilled array of nanopores. Magnetic
response of Co nanowires: (b) Magnetic hysteresis at 300 K of an array of Co nanowires with diameters of 14 nm, wire lengths of 500 nm, and array period
of 24 nm. Curves for the field both parallel to and perpendicular to the wire axis are shown. Each curve was normalized to full saturation. (c) Magnetic
hysteresis loops of a continuous Co film (shown for comparison) show considerably smaller coercivity. The electrochemically grown film is 500 nm thick.
Source: Taken with permission from Ref. [414].
(Fig. 5.14(a)) and magnetically characterized. They had large coercivities of about 800 Oe at room temperature, much higher
than those associated with Co films, which were of the order of 10 Oe (Fig. 5.14(b) and (c)). However, the difference in
coercivity between the magnetic field applied along the wire or in the transverse direction was not as high as one could
expect considering their high aspect ratio values. The enhanced coercivity was mainly associated to the small diameter of
the wires (ca. 14 nm), which is well below the critical single domain diameter of approximately 50 nm. Smaller diameters
would in principle allow access to the single domain behavior, increasing in this way the coercivity of the material. These
highly anisotropic vertical arrays of cobalt nanowires with densities in excess of 1.9 × 1011 wires per square centimeter were
considered by the authors as good candidates for the future fabrication of ultrahigh density information storage devices.
Colloidal synthetic methods are another type of approach that allows for a precise control over the size of the
nanocrystals. Surfactant molecules present in the reaction mixture attach to the nanocrystal surface and limit the further
growth of the latter. Moreover, the introduction of a mixture of surfactant molecules in the reaction environment can easily
induce a preferential growth of the material leading to well-defined 1D nanostructures. This is usually associated to the
different bonding affinities of the surfactants to selective crystallographic facets of the nanocrystals. A precise and clear
example to show how the shape anisotropy can overcome the superparamagnetic limit and hence lead to ferromagnetic
nano-objects even at room temperature was reported by Dumestre et al. [415]. Pure hcp cobalt nanorods and nanowires
were prepared by the thermal decomposition of organometallic precursors under a H2 atmosphere in organic solvents
and in the presence of a suitable mixture of long alkyl chain surfactant molecules. All samples were monocrystalline and
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adopted the c axis of the hcp structure as the growth axis of the 1D nanoparticle. Nanorods and wires were well protected
by capping ligands that stabilized them in solution. Hence, only randomly oriented samples were obtained in this case. The
magnetic properties of 3 nm diameter spherical nanoparticles obtained by analogous procedures were compared to the
properties of 9 × 40 nm2 nanorods. The magnetocrystalline anisotropy of hcp Co is calculated to be Ku = 4 × 105 J m−3
but the total anisotropy of a rod-shaped nanocrystal can be increased by a factor of 2 when estimations on shape anisotropy
are considered [334]. The comparison between spherical and rod-like nanocrystals immediately pointed out the increase
in anisotropy due to shape effects. Field-dependent magnetization measurements at room temperature showed a clear
hysteresis loop representative of a ferromagnetic state for the elongated particles, while a superparamagnetic state with
absence of coercivity was observed for the spherical ones. Similarly, at 2 K, where both samples showed ferromagnetism, the
HC was observed to increase from a value of 1100 Oe for the spherical nanoparticles to a maximum value of 8900 Oe for the
rod-shaped ones. An important magnetic feature of these nanoparticles, independently of their shape, is that they maintain
a saturation magnetization at low temperature, which is almost identical to that found in bulk cobalt. This behavior was
directly linked to the σ -donor character of the amine and acid ligands protecting the nanocrystals. Its effect could somehow
decrease the broken symmetry of the surface atoms of the nanocrystals, and consequently their spins would be able to
coherently align and rotate with the spins of the internal atoms, further increasing the magnetization of the material.
One year later, the same authors described a procedure to control the length of the rods by using carboxylic acids with
different chain lengths [415]. Nanorods of different aspect ratios were synthesized in solution (Fig. 5.15(a)–(c) and (f)). The
formation of crystalline 3D superlattices of the previous cobalt nanorods on the TEM copper grid by a crystallization process
was also reported (Fig. 5.15(d)–(e)). The rods were organized in a hexagonal arrangement with inter-particle distances
of approximately 2 nm. The magnetic characterization was performed on all samples and is reported in Fig. 5.15(g)–(i).
However, at 2 K, both magnetization and coercivity values were observed to be lower than the ones reported in the previous
work, where rods showed no sign of organization. The authors claimed that the source of this magnetization reduction could
be found in the expected antiferromagnetic coupling between interacting and laterally aligned nanorods [416,417]. However
they suggested that a remanent ferromagnetic order would be still stabilized through the shape anisotropy of the building
blocks in the assembly.
Recently, Viau et al. used the solution-based polyol process for the preparation of highly monodisperse Co nanorods [418].
Well-crystallized hcp-Co nanorods of different aspect ratios were obtained, and the c axis of the lattice was observed to be
parallel to the growth direction of the nanorods (Fig. 5.16(a)). The mean diameter of the particles could be varied between
8 and 35 nm while the mean length was tuned between 100 and 350 nm, resulting in different samples with definite aspect
ratios in the range from 4 to 30. The samples showed a reduction of the saturation magnetization between 30% and 50%
compared to the bulk value, which was assigned mainly to the presence of a thin oxide-passivating layer, as detected
by XRD measurements. However, all samples were ferromagnetic at room temperature. In order to study the magnetic
anisotropy features of aligned nanorods, the authors measured the field-dependent magnetization of the samples in two
different ways: first, a solution of cobalt nanorods was deposited on a flat substrate with an external magnetic field applied
during the evaporation of the solvent, and second, a toluene suspension of the same cobalt nanorod sample was cooled
under a magnetic field and frozen at 140 K. The alignment of the nanorods was achieved in both cases and high values of
coercivity were observed, clearly exceeding previously reported data of nanorods obtained by diverse methods, as reported
in Fig. 5.16(b).
Actually, a value of HC of 5500 Oe was measured at room temperature. Under these conditions also the saturation
magnetization was maintained to a high degree in the absence of the magnetic field (remanence to saturation ratio
MR /MS = 0.81). At 140 K, the temperature of the frozen solution, an even more square-shaped hysteresis loop was measured
with a value of HC of 9000 Oe (note, however, that the measuring temperature in this case was lower than that of the previous
measurement). A fit of M vs. H was carried out for the experimental data recorded at 140 K using a Stoner–Wohlfarth model
in which both the magnetocrystalline and shape anisotropies were considered together with a standard deviation of the
rods orientation of 15° [419]. From the fit, comparable contributions for the two types of anisotropy in the nanorods were
extracted: a coercive field of 4200 Oe was attributed to magnetocrystalline anisotropy while one of 4700 Oe was associated
to the shape anisotropy contribution.
Recently, the same authors suggested the strong effect of the nanorod–tip shape on the magnetic hardness of the
nanomaterial [420]. They simulated the hysteresis loops for several types of nanorods with the same length but with
tips of different shapes, such as simple ellipsoids, cylinders, cylinders with rounded edges, dumbbells (cylinders with
spherical ends) and diabolos (cylinders with conical ends). The geometrical models considered for the study are sketched
in Fig. 5.17(a). All calculations were done for Co nanorods. The authors did not consider the magnetocrystalline anisotropy
in order to avoid over-parameterization. Ellipsoids seemed to provide the highest coercive fields that can be achieved by
using the shape anisotropy, followed by cylinders (Fig. 5.17(c)). On the other hand, dumbbells and diabolo-like nanorods
were magnetically softer, and in them coercive field values were reduced by about a factor of 2 with respect to ellipsoids.
The calculations confirmed that HC increases with the aspect ratio of the nanorod independently of the shape of the tips.
Interestingly, they also led to conclusions about the evolution of the coercivity in nanorods as a function of the misalignment
from the applied magnetic field. Indeed, for ellipsoids and cylinders, remarkable changes of HC were predicted when
the magnetic field direction was tilted at an angle comprised between 0° and 30° from the revolution axis, as shown in
Fig. 5.17(b). Further deviations did not modify considerably the value of HC but mainly changed the shape of the hysteresis
loop. Instead, for dumbbells and diabolos, the coercivity should not suffer drastic changes at angles between 0° and 60°.
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Fig. 5.15. TEM micrographs of Co nanorods from Ref. [415] synthesized by solution methods with dimensions: (a) 9 × 33 nm; (b) 5 × 85 nm and
(c) 8 × 128 nm. Scale bar: 30 nm; (d) and (e) TEM micrographs of self-organized nanorods of 8 × 128 nm (scale bar: 10 nm); (f) High resolution TEM
micrograph and electron diffraction pattern of some aligned rods; (g)–(i) Magnetization curves recorded at 2 K of cobalt nanorods in correspondence with
TEM images (a)–(c).
Source: Taken with permission from Ref. [415].
Fig. 5.16. (a) SEM micrograph of cobalt wire assembly, prepared by depositing the wires on a Si substrate while applying a magnetic field; (b) Magnetization
curves of cobalt nanorods (L = 100 nm; d = 12.5 nm), (1) deposited on a flat substrate with an external magnetic field (T = 300 K); (2) frozen in toluene
under an external magnetic field (T = 150 K); (3) simulation of the frozen sample magnetization curve using the Stoner–Wohlfarth model for an assembly
of non-interacting wires.
Source: Taken with permission from Ref. [418].
As mentioned earlier, single domain magnets are expected to reverse their magnetization coherently, meaning that all
individual spins of the particle rotate in unison. However, particular shapes of the particles, such as rods, could induce a
slight deviation of the single domain magnetization reversal mechanism. The results obtained by the authors on differently
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Fig. 5.17. (a) Different types of particles models: (1) Ellipsoids, (2) cylinders, (3) capped cylinders, (4) dumbbells with small spherical ending, (5) dumbbells
with larger spherical endings, (6) cylinders with small cone endings, (7) cylinders with larger cone endings, and (8) small diabolos; (b) Evolution of the
hysteresis loop as calculated in the Stoner–Wohlfarth model for an ellipsoid with an aspect ratio of 3.6 for different orientations of the applied field with
respect to the easy axis; (c) Evolution of the coercive field as a function of the aspect ratio for the different types of objects. The magnetic field is applied at
5.7° with respect to the object easy axis.
Source: Taken with permission from Ref. [420].
shaped nanorods were based on the non-coherent magnetization reversal process of mainly dumbbells and diabolos. These
particular types of nanorods, due to the shape of their tips, were able to develop vortex-like magnetic states for which the
reversal of the magnetization of the whole particle was much less energy demanding. Consequently they were characterized
by a drastic drop in coercive field with respect to an ellipsoid, for which calculations indicated that the reversal proceeded
in a perfectly coherent way.
5.5.2. Fe 1D nanostructures
The body-centered-cubic structure, bcc, of metallic Fe is the most stable one at temperatures up to about 900 °C; this
crystal phase is called α -Fe and it has a TC of 770 °C, below which the metal becomes ferromagnetic. Due to the high
symmetry of this phase, the enhancement of the magnetic anisotropy of α -Fe nanoparticles by the anisotropic growth of
the objects becomes a key advantage in high-density magnetic recording applications. Both physical and chemical methods
have been developed for the synthesis of Fe 1D nanostructures and often template-assisted methods have been of great
help. As an example, a clean faceted NaCl(110) surface passivated with amorphous SiO2 has been used as a template for
the preparation of Fe nanowires [421]. The deposition of Fe under ultrahigh vacuum conditions was performed using a
collimated electron-beam source aligned 65° from the template normal. The so-called shadow deposition technique allowed
for the selective deposition of Fe on the substrate planes exposed to the beam. As a result, nanowires with an approximate
length of 10 µm and diameters of about 30 nm were obtained. The average distance between wires on the substrate was
90 nm. Electron diffraction patterns confirmed that the nanowires were polycrystalline and no preferential orientation of
the Fe grains was observed; consequently the authors did not consider the contribution of magnetocrystalline anisotropy as
the predominant factor in the magnetic characterization of the sample. Field-dependent magneto-optical Kerr effect (MOKE)
measurements indicated that saturation magnetization is reached already at low fields when the external field was applied
longitudinal to the wire growth axis. Moreover, the higher coercivity values of about 2 kOe observed in the same direction
led to the conclusion that the easy magnetization axis of the nanowires was parallel to the wire growth axis. Hysteresis
loops recorded at different angles between the sample and the external magnetic field seemed to indicate that the reversal
process of the magnetization is not dominated by domain wall motion but by a coherent rotation of the spins of the wire.
Alternatively, vertically aligned Fe nanowires were prepared via electrodeposition into the holes of porous anodic
aluminum oxide (AAO) templates [422]. Nanowires with diameters of about 35 nm and aspect ratios above 1000 were
observed by SEM measurements (Fig. 5.18(a)). XRD patterns confirmed that the α -Fe phase was the only one present in
the arrays. Even though no preferential growth direction was ascertained, the authors claimed that the shape anisotropy
was responsible for the high magnetic anisotropy observed in the field-dependent magnetization measurements: hysteresis
loops recorded with the external field parallel to the nanowire growth axis were characterized by an almost square shape
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Fig. 5.18. (a) Morphology of a Fe nanowire; (b) Hysteresis loops of Fe nanowire array. Perpendicular (parallel) corresponds to the applied field
perpendicular (parallel) to the sample plane.
Source: Taken with permission from Ref. [422].
with high values of squareness (remanent to saturation magnetization ratio), as shown in Fig. 5.18(b). The coercive field
measured under these conditions was 1832 Oe, which is much higher compared to the low value observed when the field was
applied parallel to the substrate, 118 Oe. In the latter case, the sample had high values of saturation fields and low remanence,
indicating that a hard magnetic axis was found in the transverse direction of the wires, while the easy magnetization axis
was parallel to the growth direction. The authors claimed that if each of the nanowires was decoupled from the neighboring
ones and consequently could act as a recording unit, the high anisotropy observed in the sample together with the high
density of wires on the substrate would make this kind of assembly useful for ultrahigh density perpendicular recording
media.
In 2001, Cao et al. described the preparation of arrays of pure metallic Fe nanowires by electrodeposition into the pores
of polyaniline nanotubules [423]. Nanowires had lengths of about 60 µm and were coated with polyaniline tubes of external
diameter of about 200 nm. Pure α -Fe was confirmed by XRD and EDX spectra. However, the authors did not comment on
the direction of growth of the wire, which probably could not be ascertained. Some degree of anisotropy was also observed
when the external magnetic field was applied parallel or perpendicular to the wire axis, being HC equal to 119 and 210 Oe,
respectively. Contrary to what was reported by Yang and co-workers in the previously cited work, here the easy axis seems
to be perpendicular to the growth direction of the wires.
Single-wall carbon nanotubes can also be exploited as templates for the growth of extremely thin Fe nanowires, as
reported by Borowiak-Palen et al. [424] FeCl3 was introduced inside the holes of the nanotubes by simply immersing them
in an over-saturated FeCl3 solution. After several washing cycles, the thermal treatment of the filled tubes at 593 K led
to the reduction of the Fe3+ ions forming long continuous and thin (1 nm) metallic Fe nanowires. The sample exhibited
a ferromagnetic behavior with a room temperature coercivity of 180 Oe. The authors suggested that large aspect ratios,
high surface areas and significant interwire dipolar interactions could all contribute to the room temperature magnetic
stabilization of the wires. Magnetic measurements after several months showed identical results compared to the asprepared samples, confirming the role of the carbon shell as efficient anti-oxidant layers protecting the metallic nanowires
contained inside the nanotubes.
Wet-chemical routes for the fabrication of 1D iron nanostructures have been explored since years. Oppegard et al.
already studied in 1961 the shape anisotropy of axially aligned Fe and FeCo nanoparticles prepared by the reduction in
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Fig. 5.19. Magnetization normalized by mass vs. temperature for the 2 nm spherical iron nanoparticles and the 2 nm × 11 nm iron nanorods at the applied
magnetic field of 100 Oe.
Source: Taken with permission from Ref. [426].
solution of Fe(II) salts under the effect of a magnetic field [425]. However, in this case, the shape anisotropy contribution
to coercivity was strongly hindered by agglomeration phenomena. In 2000, Park et al. reported the synthesis of α -Fe
nanorods via an oriented attachment mechanism of spherical iron nanoparticles induced by the ligand passivation of certain
crystal facets [426]. 2 nm spherical iron nanoparticles were prepared in organic solvents by the thermal decomposition of
organometallic precursors in the presence of stabilizing surfactant molecules. The formation of the rods was achieved by
the solubilization of previous spherical nanoparticles in pyridine in the presence of an excess of surfactant molecules that
were different in nature from the ones that were previously used. The mixture was refluxed until a precipitate appeared.
The product was composed of nearly dispersed bcc Fe nanorods. The width of the rods was observed to be constant and close
to 2 nm, while the length could be varied depending on the concentration of surfactants in the reaction mixture. Nanorods
of 11, 22 and 27 nm were formed. Magnetic properties of spherical 2 nm iron nanoparticles and 2 × 11 nm iron nanorods
were compared. As observed in Fig. 5.19, nanorod samples showed a much higher TB compared to their initial spherical
particles (110 K and 12 K, respectively). The contribution of the larger size and change of shape in the nanorod sample was
evaluated in order to rationalize such an increase. Indeed, global magnetic anisotropy constants were estimated for both
spherical and rod-shaped nanoparticles and were found to be 9.1 × 106 ergs/cm3 and 1.6 × 107 ergs/cm3 , respectively.
Moreover, in the case of the rod-like particles, the shape contribution to the total anisotropy constant was estimated to
be 7.9 × 106 ergs/cm3 and hence of the same order as for the pure magnetocrystalline one. This explained the significant
increase in blocking temperature in the nanorod sample, even if no values were reported for the coercivity.
Porous magnetic nanoparticles are interesting materials for magnetic transportation [427] and sensing applications [428]. However the synthesis has been mainly optimized for oxide-based materials which, as said earlier, possess
a much lower value of magnetization compared to metals. Recently, the preparation of acicular Fe nanoparticles with a
porous structure was described [429]. Elongated α -Fe2 O3 nanoparticles were prepared by hydrothermal synthesis. Further
annealing of the oxide structures under H2 led to α -Fe porous nanoparticles that maintained the original shape. XPS measurements indicated the presence of iron oxide and phosphate passivation layers, which probably where responsible for the
decrease in magnetization with respect to the other nanostructured particles. The large surface areas of the particles and the
phosphate molecules attached to them were considered by the authors as reasons why such elongated porous structures
had higher coercivity values compared to other nanostructured particles of similar sizes.
In the last two examples, as well as in other wet-chemical procedures for the synthesis of rod-like nanoparticles,
randomly oriented nanorods have been obtained for which only average magnetic properties could be measured. It is then
worth mentioning an innovative approach described by Vayssieres and co-workers, in which the synthesis of vertically
aligned α -Fe nanowires was carried out on a substrate from an aqueous solution where no template, surfactant or applied
electric or magnetic field was required [391]. The elongated structures were 0.8–1 µm long and had diameters of about
30–40 nm. Each of these 1D structures was observed to be formed by stacking of columns and bundles of nanorods 5–10 nm
wide and 15–30 nm long. These small nanorods displayed no preferential relative orientation in the nanowires and probably
because of this, no significant difference was found in the coercivity and in the shape of the hysteresis loops recorded with
the field parallel and perpendicular to the wires. However, the direction of the easy magnetization axis was suggested to be
parallel to the growth direction of the wires, based on the observation that slightly higher magnetization values had been
observed with the field applied in this direction, independently of the temperature.
5.5.3. Ni 1D nanostructures
Metallic nickel crystallizes with a face-centered-cubic lattice and exhibits lower values of magnetocrystalline anisotropy
than Fe and Co. For this reason works based on the synthesis of metallic Ni are not so abundant as for the other two
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metals. However, its high permeability and magnetoresistant properties have made it ideal for magnetization reversal and
domain wall studies (which are reviewed in Sections 5.5.5 and 5.5.6, respectively). Some examples of the synthesis of nickel
elongated nanostructures have also been reported. Cordente et al. described the synthesis of monocrystalline Ni nanorods
in organic solvents by the thermal decomposition of organonickel compounds in the presence of long chain hydrocarbon
amines [405]. The amine played a double role in the reaction mixture: (i) it selectively coordinated particular facets of the
colloidal Ni particles, confining their further growth in one single direction and (ii) its electron σ -donor character avoided
the magnetization fading of the magnetic material, which is usually the case when strong π -acceptor ligands are used in the
synthesis. TEM analysis indicated homogeneous sizes for the nanorods of about 4 × 15 nm. Nanorods tended to organize
on the TEM grids with their long axes parallel to one another, probably because by such arrangement their magnetostatic
repulsion is minimized. High resolution TEM confirmed that the growth axis of the nanorods was parallel to the (111) plane.
As expected, the MS value of the sample at 2 K was very close to the bulk value for Ni due to the presence of amines as capping
ligands of the nanocrystals. The ZFC-FC magnetization measurements indicated a TB of 100 K. Based on this value of TB [430],
the authors estimated a shape anisotropy constant of 7.7 × 105 erg/cm3 , comparable in strength to the magnetocrystalline
anisotropy constant for bulk Ni of 7.0(±0.5)× 105 erg/cm3 . These results demonstrated how the rod-like shape of magnetic
nanocrystals can increase the effective anisotropy of the material by a factor of 2. Ni nanorods were also prepared by the
reduction of nickel(II) salts in microemulsion systems in the presence of a suitable surfactant [431]. The products isolated
at the first stages of the synthesis were composed of 10 nm spherical nanocrystals that further evolved with time until
rod-shaped nanocrystals with aspect ratios of 10–20 were formed. Field-dependent magnetization measurements were
performed for both the spherical and rod-shaped samples at room temperature. The nanorods had a coercivity of 332 Oe,
which was higher than that of bulk Ni (100 Oe) and than that of the initial 10 nm spherical particles (173 Oe).
Also in the case of Ni, AAO templates have been used for the formation of well-aligned 1D nanostructures. Several
synthetic and structural parameters can strongly affect magnetization processes. Consequently it is difficult to conclude
about the real effect of the aspect ratio of the particles if one considers different samples reported in the literature that have
not been prepared and studied under equivalent conditions. A representative example that shows the effect of the aspect
ratio on the magnetic behavior for 1D nanostructures was reported by Oh et al. [432]. The authors used the electrodeposition
technique in AAO templates to form Ni nanostructures. All synthetic parameters were kept constant during the synthesis,
and just by simply controlling the deposition time in the range from 5 to 15 min, 100 nm spherical nanodots, 400 nm long
nanorods and 3.4 µm long nanowires could be synthesized (Fig. 5.20(a)–(c)). The diameter of the last two samples was
100 nm, equal to the diameter of the nanospheres. Hysteresis loops were recorded at 5 K for the three samples. The field
was applied both parallel and perpendicular to the substrate. When the field was applied perpendicular to the substrate, or
in other words parallel to the elongated structures, significant changes were observed in coercivity by varying the aspect
ratio of the particles (Fig. 5.20(d)–(f)): while Ni nanospheres had a HC of 200 Oe, the nanorods and nanowires showed HC
values of 630 Oe and 730 Oe respectively. Also the squareness of the hysteresis loops was increased following the same trend.
On the other hand, and confirming the high magnetic anisotropy of the high aspect ratio systems, hysteresis loops recorded
with the magnetic field applied parallel to the substrate of the samples were characterized by very similar curves, with very
low coercivities and remanent magnetizations. It seems then clear that the magnetic properties of the three systems were
dominated by their shape anisotropy and, in particular, were associated to the aspect ratio of the nanostructures.
Metallic Ni is known for its negative magnetostriction coefficient, meaning that under the effect of a magnetic field
the material undergoes compression. Alternatively one can say that an external stress is able to modulate the anisotropy
of the material. Since the magnetocrystalline anisotropy of Ni is lower than in Fe or Co metals, the effects of shape and
magnetoelastic anisotropies become more apparent in Ni. Kumar et al. reported on the competition between shape and
magnetoelastic anisotropies in Ni nanowire arrays fabricated by electrodeposition within the porous of AAO templates [433].
Arrays of nanowires with typical diameters of 30 nm and lengths between 1 and 2 µm were prepared in which the interwire
distance was observed to be around 100 nm (Fig. 5.21(a)). XRD patterns indicated the cubic nature of Ni, which appeared
to grow along the [111] direction. This means that the growth direction of the wire coincides with one of the three easy
magnetization axes of the lattice. Magnetic measurements were carried out for the Ni nanowires embedded in the alumina
template and supported on aluminum substrates. The magnetic data recorded at room temperature indicated a clear outof-plane anisotropy, confirmed by the high value of coercivity (800 Oe) and squareness (MR /MS = 0.82) obtained when the
field was applied perpendicular to the Al substrate, and compared to the narrow hysteresis loops observed when the field
was applied parallel to it. These observations are in agreement with the large shape anisotropy of the sample. However,
the MR and HC values at 50 K, compared to the ones at room temperature, decreased in the out-of-plane measurement and
increased in the in-plane one. Such a trend cannot be simply explained by shape anisotropy, which predicts continuous
temperature dependence. The authors estimated the temperature dependence compression of the Ni nanowires taking in
consideration the different thermal expansion coefficients of the Al substrate, the alumina template and the Ni nanowires
themselves. The magnetoelastic anisotropy constants obtained were of the same order of magnitude as the shape anisotropy
constants but acting in the perpendicular direction. This fact could well explain the presence of a maximum in the HC
and MR vs. temperature representations shown in Fig. 5.21(b) and (c). The maximum was observed to be at around room
temperature. Below this temperature, magnetostrictive anisotropy overcame the shape anisotropy. Hence the array of
nanowires gradually increased the component of the effective anisotropy in the direction perpendicular to the growth axis
of the wires. Other factors were evaluated in this work, namely the width of the Al substrate and the adhesion ability of the
wires to the Al substrate and the alumina template. These could explain the trends observed also by other authors [434].
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Fig. 5.20. SEM images of nanostructured nickel: (a) nanodots, (b) nanorods and (c) nanowires after removal of the AAO templates. The diameter, length
and the distance between the pores of the AAO template are 100 nm, 40 mm, and 210 nm, respectively; (d)–(f) Hysteresis loops for nickel nanostructures in
correspondence with SEM images (a)–(c) measured at 5 K with the field applied parallel (black squares) and perpendicular (white circles) to the membrane
plane.
Source: Taken with permission from Ref. [432].
5.5.4. Alloys of special technological interest
Permanent magnets are defined as those magnetic materials that, once magnetized to saturation, create their own
persistent magnetic field even when the external magnetic field is switched off. They are all characterized by very high
values of coercivity as well as remanent magnetization. Hence, the ideal permanent magnet is characterized by a squareshaped hysteresis loop, in which MS is kept constant in the demagnetizing process until a high value of magnetic field of
opposite direction is able to induce magnetization reversal. The parameter that better describes their performance is the
BHmax product, which represents the maximum achievable energy density of the magnetic field into the air gap surrounding
the magnet. These types of materials have special interest in the data storage industry since they are used as magnetic
recording media or as head actuators in computer hard disks, where non-volatile data storage systems are required. Apart
from this, permanent magnets are used in other applications where strong magnetic fields are necessary: examples are the
fabrication of magnetic resonance instruments for research activities or biomedical imaging techniques (MRI). Moreover,
permanent magnets can act as transducers for the conversion of electrical signals into sound, and this is why they have
become essential components in loudspeakers and headphones, as well as bearings for supporting a load using magnetic
levitation, and as a consequence they can be found as components of turbines, pumps, motors and generators. Permanent
magnet alloys include SmCo5 ; Nd2 Fe14 B; aluminum, nickel and cobalt alloys with iron (Alnico); titanium, cobalt, nickel, and
aluminum alloys with iron (Ticonal) and platinum-based alloys such as FePt and CoPt.
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Fig. 5.21. (a) Morphology of a broken alumina template filled with Ni nanowires; (b) Temperature dependence of spontaneous magnetization MS , in-plane
and out-of-plane remanence MR ; (c) Influence of temperature on the in-plane and out-of-plane coercivity HC .
Source: Taken with permission from Ref. [433].
Industrial approaches to these materials have always preferred the classical powder metallurgy process, whereby
magnets are prepared by pulverizing an ingot precursor and liquid-phase sintering the magnetically aligned powder into
dense blocks, which are then thermally treated and finally magnetized. Obviously, such industrial procedures are far from
fabricating size and shape-controlled nanostructured permanent magnets. However, the possibility to obtain nanoscale
permanent magnets, either with a single or multidomain behavior, is highly attractive due to the enhanced magnetic
properties expected thereof, as described in the introductory part of this section. Unfortunately, the synthesis of permanent
nanomagnets has encountered several difficulties and has been addressed only in the last years, albeit with modest results.
Two main types of rare-earth-based alloys are currently known: Nd2 Fe14 B and SmCo5 . Nd2 Fe14 B is nowadays
the strongest permanent magnet available. It has a tetragonal structure and an exceptionally high value of uniaxial
magnetocrystalline anisotropy with a maximum coercivity of 2.5 T and a maximum value of MS of 1270 emu/cm3 . Hence,
it is highly resistant to demagnetization and has the potential for storing large amounts of magnetic energy, which makes
them ideal for permanent magnet applications. Hexagonally close-packed SmCo5 magnets are softer and are not able to
store an amount of magnetic energy as large as that of Nd2 Fe14 B. They were the first rare-earth containing magnets to be
commercialized and, even if softer than Nd2 Fe14 B, they have higher Curie temperatures, which makes them magnetically
stable up to at least 700 °C. In practice, the magnetic properties of both types of magnets highly depend on the alloy
composition, microstructure, and manufacturing technique employed.
Rare-earth metals are specially known for their high reactivity and oxygen sensitivity, which actually increase in
nanostructured materials due to their high surface to volume ratio. This makes the production of nanoparticles of these
materials a nontrivial task. Moreover, their equilibrium phase diagrams contain several compounds with different crystalline
structures and compositions, among which only a few possess the desired hard magnetic properties. In general, thermal
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annealing treatments can be effectively used to improve the crystallinity of the material and to obtain its pure desired phase.
However, when such treatments are performed on nanosized materials, deterioration rather than improvement of magnetic
properties is usually observed, mainly due to oxidation side reactions or sintering processes. A few examples of the synthesis
of spherical SmCo5 and Nd2 Fe14 B nanoparticles exist in the literature [435,436], even if they rarely have significant values of
coercivity at room temperature, due to one of the reasons described above [437–441]. For what concerns the preparation of
one-dimensional nanoparticles, such as rods or wires, only a couple of examples can be found in the literature. Chinnasamy
et al. reported the synthesis of SmCo5 nanoblades prepared by the high temperature reduction of the appropriate metal salts
in a liquid polyol medium that acts both as the solvent and as the reducing agent [442]. The particles were characterized
by XRD and high resolution TEM: both the SmCo5 hexagonal structure and the single crystal character of the particles were
confirmed. They presented a blade-like structure with an average width of 10 nm and length of 100 nm. The particles were
ferromagnetic at room temperature and had a HC of 6.1 kOe and a MS of 40 emu/g. The authors suggested both the shape
and the magnetocrystalline anisotropies as responsible for the high values of coercivity observed. In a different publication,
Nd5 Fe95−x Bx nanowires were fabricated using electrodeposition techniques in AAO templates [443]. The wires were 40 nm
wide in average and had aspect ratios of approximately 75. However, both SAED and XRD measurements indicated that the
structures were amorphous. This could explain the very low values of coercivity measured, together with the fact that the
sample was not exhibiting the stoichiometry associated with the Nd2 Fe14 B hard magnetic material.
Better results have been obtained in the preparation of magnetic Pt-based one-dimensional nanostructures, partially
due to their higher chemical stability and resistance to oxidation. FePt and CoPt alloys can form in different stoichiometries
and crystalline structures, like the previous rare-earth-based magnets. Equiatomic compositions are the optimal ones for
enhancing their magnetic properties. Their fcc structure presents soft magnetic properties, while their fct lattice is the
one responsible for their hard behavior. The ordered tetragonal phase is easily synthesized by a thermal annealing of
the disordered cubic structure. Most works on FePt or CoPt 1D nanostructures, dealth only with synthetic conditions and
strategies for their preparation [444–447], while only a few of them reported on their magnetic characterization, which
however was always quite superficial [448–452]. The preferred synthetic method for the fabrication of arrays of Pt-based
magnetic nanowires is the electrodeposition of the metals in AAO templates [445,453–460]. However, the synthesis has
also been achieved by other methods such as hydrogen reduction of Co and Pt salts inside templates [449], solvothermal
methods [444,450], thermal decomposition of organometallic precursors [447,451], electron-beam induced deposition
techniques [461] and virus-based scaffold methods [446].
A few authors undertook an in-depth study of the magnetic properties of either fcc or fct FePt or CoPt elongated structures.
Yasui et al. developed for instance a fabrication technique by which it was possible to control the crystal orientation
of nanometer size fct CoPt nanowires [453]. By substituting the W underlayer (which was used as electrode during the
electrodeposition process) with a [1] oriented Pt one, the authors were able to induce the growth of the wires with a clear
c-axis orientation. This orientation is actually the easy magnetocrystalline axis of the tetragonal structure, hence high values
of HC up to 7400 Oe and squareness up to 0.96 were observed when the applied magnetic field followed the wire direction.
For the samples grown over W underlayers, no anisotropic magnetic behavior was observed depending on the direction
of the magnetic field, which was assumed to be due to the random c axis orientations of the wires in that sample. Similar
results for polycrystalline fct CoPt nanoparticles have been observed: the absence of anisotropic magnetic properties in such
wires has been associated in all cases to a random orientation of the c-axis between the wires or between the grains within
each wire [456–458].
Mallet et al. studied fct CoPt nanowires grown with a preferred [111] direction [455]. No difference was found in the
hysteresis loops when comparing the measurements done with the magnetic field applied parallel to the wire axis with
those done with the field applied perpendicular to the wire axis. In that case, the growth direction of the wires makes an
angle of 55.45° with the [001] magnetocrystalline axis and based on this, the measurements confirmed the higher strength of
the magnetocrystalline anisotropy with respect to the shape anisotropy of the wires. The authors also studied the anisotropic
soft magnetic behavior of the fcc CoPt nanowires previous to their annealing. The easy axis of magnetization perpendicular
to the wire axis seemed to be clearly dictated by strong dipolar coupling interactions between ferromagnetic wires in the
sample, despite the fact that both magnetocrystalline and shape easy axes were collinear in that case.
One year before, the same authors had studied the magnetic anisotropy observed in fcc CoPt nanowire arrays fabricated
within a polycarbonate membrane [454] instead of an alumina membrane, as was done in the publication discussed
above [455]. When polycarbonate membranes were used, lower values of wire packing were observed, and hence also
weaker dipolar couplings were expected. Nanowires were also grown along the [111] direction which, in this case, was
observed to be the easy magnetization axis. Magnetic measurements with a magnetic field applied parallel to the wires axes
yielded a value of HC of 2320 Oe and a squareness of 0.98. In the same article, the diameter dependence of the coercivity
and squareness for CoPt nanowires was also addressed.
An anisotropic magnetic behavior has been observed also in fcc CoPt nanowires by Gao et al. [459]. The wires were
polycrystalline and had a preferential [110] texture along the growth direction. Magnetocrystalline and shape contributions
to anisotropy are expected to be comparable in magnitude in fcc CoPt alloys. The authors studied the hysteresis loops
observed in the sample by means of micromagnetic simulations, considering that the [110] direction makes an angle of
35° with the [111] magnetocrystalline easy axis and that, due to the polycrystallinity of the wires, the [111] directions could
be randomly distributed with an azimuthal angle in the cross section perpendicular to the wire axis. The calculated data was
in agreement with the experimental results and confirmed the random distribution of the [111] directions. The geometrical
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dependence of the magnetic behavior in polycrystalline fcc CoPt nanowires was studied by Shamaila et al. [460]. A crossover
of effective easy axis of magnetization was observed as a function of wire diameter and length.
Besides magnetometric measurements, other alternative techniques have been rarely used to magnetically characterize
Pt-based monodimensional nanostructures. In 2005 Che et al. used off-axis electron holography measurements to evaluate
the magnetic properties of a single nanorod [461]. Each nanorod was seen to be composed of a chain of crystalline FePt
nanoparticles. The measurements indicated that the magnetization of the rod was parallel to their growth direction, and
a value of its remanent magnetization of 1.53 T was measured, revealing a hard magnetic nature. Studies on the orbital
and spin contribution to the magnetic moment in CoPt nanowires by X-ray Magnetic Circular Dichroism (XMCD) have been
undertaken by Li et al. [462].
In contrast to hard magnets, soft magnets are characterized by very low values of coercivity and can reach the
magnetically saturated state with relatively low magnetic field values. They are usually characterized by very weak
magnetostrictive effects and very high magnetic permeability. The nearly zero values of magnetostriction is of crucial
importance in industry since variable stresses derived from their use could strongly affect their magnetic properties. On
the other hand, their high magnetic permeabilities make them ideal for magnetic switching applications, for example as
magnetic recording head sensors for the magnetic data storage industry.
Most soft magnets are Fe-based materials, such as metallic iron, silicon–iron alloys or iron phosphides. However,
nickel–iron alloys are the most representative examples of this type of materials. NiFe alloys are commonly known as
Permalloys and the name usually refers to a composition close to Ni80 Fe20 . Several reasons have made of permalloys
one of the most investigated soft materials, namely they have the highest values of permeability and their electrical
resistivity varies within a 5% range, depending on the strength and direction of an applied magnetic field. Additionally,
they are characterized by very low values of magnetocrystalline anisotropy, which makes them ideal for studying physical
phenomena arising from their shape anisotropy. Bulk permalloys are usually fabricated in the metallurgical industry
by melting and shaping of the precursor ingots. Nanoscale fabrication methods for permalloys and other magnetic
nanostructures are also available, and they have been extensively reviewed by Schuller and co-workers [333]. Due to
their variable electrical resistance and their high magnetic permeabilities, permalloys have been the subject of study
in many publications dealing with magnetoresistant properties and magnetization reversal studies in one-dimensional
nanostructures. Some of the most important works published in these fields are reviewed in Sections 5.5.5 and 5.5.6.
5.5.5. Magnetization reversal studies in 1D metallic nanostructures
By the end of the last century, thanks to the increasing technological demand focused on the miniaturization of sensors
and on higher density magnetic storage devices, the emerging field of spintronics has been intensively addressed and
developed [322]. This innovative technology exploits the intrinsic spin of the electron and its associated magnetic moment,
in addition to its fundamental electronic charge, in solid state devices. They use magnetic moment to carry information and
this, compared to nowadays technology, could introduce several improvements in terms of low power dissipation, nonvolatile data retention, radiation hardness and high integration densities.
From an applied physics point of view, the continuous increase in the density of magnetic recording media must entail a
deep knowledge of the behavior of size-confined ferromagnetic structures, thin films and elongated particles. In particular,
the understanding of the mechanisms and dynamics of their magnetization reversal processes is crucial for going beyond
the state-of-the-art: the two stable magnetization directions of nanostructured ferromagnetic metals are strictly associated
to the two Boolean logic states, 1 and 0, necessary to encode information. They must be stable over time but they also must
be reversible. Hence, in the last 20 years, a considerable effort has been made to elucidate the complex physical phenomena
occurring during spin reversal and transportation in isolated and size-controlled magnetic particles, with a special attention
to nanowire systems.
Based on micromagnetic theories [337,463,464], three magnetization reversal modes can be observed for single domain
ferromagnetic crystals with a cylindrical or spheroidal shape: the coherent rotation, the curling mode and the buckling mode.
The observed mode mainly depends on the radius of the studied wire. The three reversal modes are shown in Fig. 5.22.
In the coherent rotation mechanism, the individual spins of the elongated particle homogeneously rotate following
the direction of the external magnetic field until the complete magnetization reversal is accomplished. As assumed by
the Stoner–Wohlfarth model [337], such mechanism is only operative in very small particles and the magnitude of the
magnetization remains constant along the direction of the magnetic field during the reversal process. Alternatively, for larger
particles, two other incoherent reversal modes can take place, which actually are less energy demanding and consequently
lead to lower coercive forces. These two incoherent reversal modes rely on the homogeneous variation of the spin vectors
along certain crystal directions (buckling) [465] or on their rotation in circular symmetry creating a vortex (curling) [464].
Both of them entail a non constant value of magnetization during the reversal process. The field value at which the reversal
takes place is known as the switching field (HSW ).
Two main factors have made possible to follow magnetization reversal processes in elongated ferromagnetic systems:
on the one hand, the development of new nanofabrication techniques opened the possibility to prepare samples with a
high degree of control over their dimensions down to a few nanometers in diameter or thickness. Such structures might
display a real 1D behavior compared to previous samples of larger sizes. On the other hand, more sensitive magnetic
characterization techniques have been developed with which new insights on the magnetic properties of individual and
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Fig. 5.22. Magnetization reversal modes in elongated nanoparticles. Coherent rotation (top) is characterized by an homogeneous rotation of all individual
spins following the external magnetic field. Buckling mode (middle) is characterized by the homogeneous variation of the spins along certain crystal
directions. Curling mode (bottom) is characterized by the rotation of the spins in the crystal creating a vortex.
isolated magnetic particles were obtained. The magnetization of one single magnetic nanowire is of the order of 10−11
emu and hence slight changes in its magnetization value during reversal processes cannot be detected by conventional
magnetometry techniques. Previous experiments were performed, in general, on large assemblies of theoretically
identical particles, even if the real dispersion of the sample in terms of size, morphology, composition, orientation and
separation between the different magnetic entities was clearly hiding several factors and limiting the interpretation of
the results. Nowadays, electron holography [466], micro-SQUID (Superconducting Quantum Interference Device) [467]
and Magnetic Force Microscopy techniques [468,469] have made possible the study of magnetization switching in single
nanowires.
The measurement of the magnetoresistive properties of metals has also been proved to be an appropriate method for
the detection of very small magnetization variations. Several authors have used Anisotropic Magnetoresistance (AMR)
measurements for the study of magnetic reversal processes. The anisotropic magnetoresistive effect is due to the anisotropy
of spin–orbit scattering in transition ferromagnetic metals and leads to changes in the resistivity as the angle between the
current and the magnetization is modified. Resistance measurements can be performed both at room and low temperature
with high accuracy. Therefore, AMR techniques can be used as a probe of the magnetization orientation with respect to
the current, even if the total change in resistivity between the state where the magnetization is parallel to the current
and the state where it is perpendicular is usually very small. The main results reported in the literature concerning the
study of the magnetization reversal mechanisms by using previously mentioned techniques are summarized in the next
paragraphs.
The first study of an isolated nanowire with a diameter smaller than 100 nm was reported in 1996 by Wernsdorfer
et al. [467]. Two Ni nanowires with diameters of 92 ± 4 nm and 50 ± 5 nm and aspect ratio 100:1 or larger were studied by
micro-SQUID techniques at 0.13 and 6 K. Ni nanowires were obtained by electrochemical filling of the pores of commercial
nanoporous polycarbonate membranes of thicknesses between 6 and 10 µm. After the dissolution of the membrane with
chloroform, the samples were drop-cast on a chip containing a few hundreds of SQUIDs. SEM observations helped to localize
the microbridge dc SQUID where a single nanowire had been trapped, as shown in Fig. 5.23(a). TEM measurements confirmed
the polycrystalline nature of the wires with a typical crystallite size of 10 nm. The polycrystallinity of the wires makes
one assume that its magnetocrystalline anisotropy is negligible, as hypothesized both in the Stoner–Wohlfarth and in the
curling theories. A variable external magnetic field was applied in the plane of the SQUID at several θ angles, where θ is the
angle between the magnetic field direction and the wire axis (easy anisotropy direction). The flux induced by the wires was
detected by the SQUID. Hysteresis loops of the magnetic flux of the sample under slowly varying applied fields (typically a
few mT/s) were recorded for several θ angles and are shown in Fig. 5.23(b). They were all characterized by an abrupt change
of the signal at a certain value of field. Such sharp transition is associated to the magnetization reversal of the wire, which
occurs at the switching field. In the case of the wider nanowires, the experimental data were compared to the predictions
of the curling mode of magnetization reversal in an infinite cylinder [463]. Based on this theory, the angular variation of the
switching field can be expressed as:
HSW =
MS
2
a2
a(1 + a)
+ (1 + 2a) cos2 θ
(5.11)
√
where a = −1.08(d0 /d)2 . The exchange length d0 = 2 A/MS (A is the exchange constant) defines the transition from
uniform rotation to curling. From the fit shown in Fig. 5.23(c), a curling magnetization reversal mode was confirmed and a
value of d0 (Ni) of 34 ± 4 nm was extracted, in accordance with previous data reported in literature [470]. The histogram of
the HSW values depending on the θ angle (also shown in the inset of Fig. 5.23(c)) revealed that different spin configurations
with different energy barriers of nucleation were contributing to the reversal process.
R. Krahne et al. / Physics Reports 501 (2011) 75–221
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Fig. 5.23. (a) SEM micrograph of a microbridge dc SQUID and a Ni wire of diameter of 65 nm; (b) Hysteresis loops of the wire in (a) at several values of
the angle between the applied field and the wire axis; (c) Angular variation of the switching field of a Ni wire, 92 ± 4 nm in diameter, 5 µm in length.
Bars: width of the histograms. Line: prediction of the curling model. Inset: histogram of the switching field at the angle θ = −9°; (d) Angular variation of
the switching field of a Ni wire, 50 ± 5 nm in diameter, 3.5 µm in length. The width of the switching field distribution is smaller than the dot size. Inset:
histogram of the switching field at the angle θ = 5°.
Source: Taken with permission from Ref. [467].
In contrast to these observations, the angular variation of the HSW for nanowires with diameters smaller than 75 nm
had a local maximum at θ = 0, revealing the presence of coherent rotation reversal mechanisms as predicted by the
Stoner–Wohlfarth model (see Fig. 5.23(d)). This appeared to be due to the small diameters of the wires which were actually
close to the d0 value. The authors explained such behavior by considering a predominant curling mode reversal at small
θ angles (with some reminiscence of coherent rotation that gives rise to a moderate maximum at θ = 0) while at larger
θ angles a complete coherent rotation was taking place. The narrow histograms of HSW with θ , displayed in Fig. 5.23(d),
suggested in this case that a single energy barrier is responsible for the reversal. Based on calculations by means of
micromagnetic equations, the authors claimed that the magnetization reversal by the curling mode was triggered by the
nucleation of a vortex at one end of the nanowire that later propagated along the structure. This concept had been treated
analytically before [471]. To confirm this mechanism, switching time measurements at 0.13 K were fitted to an exponential
β
formula of the nonswitching probability P (t ) = e−(t /τ ) where τ defines the mean waiting time and β is a factor that
takes into account the distribution of energy barriers. The field and temperature dependence of τ followed an Arrhenius
law from which a value of 15,000 K for the energy barrier for magnetization reversal was extracted. This value could be
roughly correlated with an activation volume around 20 nm3 , which was actually more than 200 times smaller than the
total volume of the nanowire, confirming the previous nucleation–propagation mechanism. These observations diverge
from the ones reported by the same authors for single nanoparticles, which indicated a magnetization reversal by uniform
rotation [472]. In a following work, and by means of similar micro-SQUID measurements, the authors studied the dynamics
of pinning and depinning of the propagation of the magnetization reversal occurring in some samples as well as the negative
effect of chemical oxidation in the reversal process [473]. Both phenomena introduced deviations in the single exponential
law for the previously mentioned nonswitching probability.
Compared to micro-SQUID measurements, AMR techniques offer the possibility to study the same systems at room
temperature. This represents a significant advantage since the magnetization reversal processes can be studied at
temperatures close to working temperatures of potential devices where nanostructures could be integrated. Several
examples of the use of this technique have been reported in the last 10 years. The angular dependence of HSW is
monitored experimentally by observing the discontinuity of the magnetoresistive hysteresis in this case. The AMR of bulk
polycrystalline samples is proportional to cos2 θ , where θ is the angle between the current (parallel to the wire axis) and the
magnetization M(H) (which in turn is a function of the applied field H) [474,475]. This law originates from the high symmetry
of the resistivity tensor in bulk materials. Nevertheless, the finite size effects of the wire lead to diffusive scattering at its
surface which is expected to introduce some deviations from this law [475]. Neglecting such deviations, the magnetoresistive
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curve R(H) is related to the magnetization M(H) by:
R (H ) = R0 + (∆R)max
M (H )
MS
2
.
(5.12)
In Eq. (5.12), the magnetization is assumed to be uniform and must be measured along the wire axis. In that case, M (H ) is
proportional to MS following the equation M (H ) = MS cos[θ(H )]. The ratio (∆R)max /R0 is known as the AMR ratio.
Wegrowe et al. reported in 1998 the study of the magnetization reversal mechanisms of individual non-interacting Ni
nanowires fabricated by electrodeposition techniques in a porous template membrane [476]. The electrical contact for
the resistivity measurements was performed in situ by potentiometric techniques, whereby a patterned thin layer gold
film electrode was formed on the top of the membrane by sputtering through a mask. The average pore diameter of the
membrane was 30 nm and the obtained nanowires had a polycrystalline character. The magnetoresistance hysteresis loops
were recorded at room temperature with a slowly varying applied field of about 0.1 mT/s, for applied fields oriented at
θ = 10° and 80° with respect to the wire axis. Irreversible jumps in the loops were observed at the HSW strictly correlated
with curling reversal modes, while the rest of the loops was characterized by a reversible trend in accordance with a uniform
rotation mechanism. The angular dependence of the HSW for the two θ angles shared many similarities with the graphics
reported in Ref. [467] for nanowires with small diameters, in that case measured by micro-SQUID techniques. The validity
of both synthetic and AMR techniques for the study of magnetization reversal processes in single ferromagnetic nanowires
was hence confirmed.
The same authors performed later more accurate studies on similar Ni nanowires by AMR measurements, using more
elaborated theoretical models of the angular dependence of the HSW [477]. AMR measurements at different angles are
reported in Fig. 5.24(a), where the irreversible resistance jumps are clearly observed. The HSW (θ ) curve derived from these
measurements was characterized by a significant hump at low angles, centered at θ = 0°, as shown in Fig. 5.24(b). HSW (θ)
curves for curling reversal modes in infinite cylinders are predicted to have a U-shape, while they present a sharp peak at
θ = 0° if the magnetization reversal occurs by coherent rotation. However, in this case the shape of the curve could not be
clearly associated to any of these models. The classical curling theory assumes the absence of magnetocrystalline anisotropy
in the cylinder [463]. Nonetheless, a revised version of this model was suggested by Aharoni in which a magnetocrystalline
anisotropy parallel to the wire axis was also taken into account in the curling rotational mode [464].
The fit of the experimental data by this new model was able to reproduce the shape of the experimental curve to a
high extent, especially for what concerned the low angle values (see Fig. 5.24(b)). However, the fit overestimated the HSW
values by about 0.5 kOe. In other words, an important contribution had been omitted in the model which actually should
facilitate the magnetization reversal, as deduced from the lower experimental values. Several contributions were mentioned
by the authors as possible reasons to explain the low HSW values measured, such as the effect of surface anisotropy or the
presence of a perpendicular magnetocrystalline anisotropy (not considered in the theoretical model). Among the possible
explanations the authors claimed that most likely the presence of structural defects in the polycrystalline Ni nanowires
could be responsible for the observed phenomenon. The presence of highly defective crystal regions could easily facilitate
the nucleation of magnetization reversal domain walls in some small parts of the wire. In this sense, the authors modified
the original demagnetizing factors that defined the shape anisotropy of a wire with aspect ratio 100:1, and limited them to
those corresponding to a volume with aspect ratio 2:1, in accordance with what they considered to be the activation volume,
or the volume in which the magnetization reversal domain wall nucleated. Setting the demagnetizing factors to the values
corresponding to a volume 50 times smaller than that of the entire wire, the fitted values were precisely rescaled to the
experimental ones. This confirmed the fact that the curling reversal was not acting simultaneously in the whole length of
the wire but instead the mechanism is better described by a nucleation–propagation process, as also ascertained by previous
micro-SQUID studies [467]. The authors suggested that the small discrepancies still present at large angles could be due to
the pinning of the magnetization by surface defects during propagation or by a deviation from the AMR law (Eq. (5.12)), as
a consequence of the finite size effects in the wire.
Other studies were later published by the same group [478] and by Pignard et al. [345] which confirmed these findings.
In the last case, the authors observed two irreversible jumps of the resistance as the magnetization was reversed (see
Fig. 5.25(a)). While the first one represented the nucleation of the magnetization reversal as in all previous examples, the
second was associated to the trapping of this domain wall at a pinning center during its propagation. The authors claimed
that the pinned domain wall could only be released at higher fields leading to the observed second jump, after which the
magnetization of the wire was completely reversed. In this work, the temperature dependence of the switching field was
also addressed: at θ = 0°, the HSW increased from 600 Oe to 2300 Oe when cooling the sample from room temperature down
to 10 K as seen in Fig. 5.25(b). That means that a decrease in temperature leads to an additional anisotropy contribution that
reinforces the wire axis as the easy direction for the magnetization. SQUID magnetometry measurements were performed
at different temperatures to extract the anisotropy constants (K ) in each case. At room temperature a value of K equal to
7.2 × 105 erg/cm3 was obtained, which corresponded to the one associated with the shape anisotropy of an infinite cylinder.
It was then clear that magnetocrystalline anisotropy was negligible in the Ni nanowire. Values of K of 11.5 × 105 and 12.3 ×
105 erg/cm3 were obtained at 100 K and 35 K respectively. Assuming a temperature-independent shape anisotropy, the
authors stated that the additional uniaxial anisotropy measured had its origin in the well-known magnetoelastic properties
of Ni. In the experiments, Ni nanowires were measured as prepared inside the polycarbonate membrane template. Due to the
large mismatch of the thermal expansion coefficients between the two materials, Ni nanowires were subjected to variable
R. Krahne et al. / Physics Reports 501 (2011) 75–221
177
Fig. 5.24. (a) Ni magnetoresistive hysteresis at 15° and 45°. Bias current 0.3 mA. The continuous lines are predictions based on the curling model of
magnetization reversal and the AMR quadratic dependence of the projection of the magnetization in the direction of the current. Inset: zoom of the
magnetoresistive discontinuity at the switching field HSW , for θ = 15°. The points correspond to increasing (full circles) and decreasing (empty circles) field
measurements; (b) Switching field of Ni nanowires vs. angle between wire and field. Dotted line: values of the switching field of the curling reversal mode
in the cylinder 100:1, with magnetocrystalline anisotropy K as adjustable parameter. Solid line: curling with the previous K and adjustable demagnetizing
factors Dz and Dx .
Source: Taken with permission from Ref. [477].
mechanical stresses depending on temperature, that were studied by the authors by means of XRD spectra. Such stresses
were able to increase the whole anisotropy of the system at low temperatures. However, the additional magnetoelastic
anisotropy was found not to modify the magnetization reversal mechanism observed in Ni nanowires at room
temperature.
AMR techniques have also been used to study magnetization reversal processes in Permalloy nanowires. Oliveira et al.
recently reported on the reversal mode of Ni81 Fe19 nanowires fabricated by means of mechanical lithography using atomic
force microscopy [479]. The wires had lengths between 5 and 10 µm, widths between 300 and 400 nm and thicknesses in
the range of 3–10 nm. One electrically connected wire is shown in the SEM micrograph of Fig. 5.26(a). As evidenced by the
dimensions reported, the nanowire did not display a cylindrical shape as in all previously reported examples; instead a long
thin narrow tape nanowire had been fabricated this time. Magnetoresistance hysteresis loops were recorded at different
angles and usual irreversible jumps were observed in all cases, indicating the values of HSW . The angular dependence of
HSW , displayed in Fig. 5.26(b) and (c) for two nanowires of different dimensions, could be well fitted by the equations
derived from the curling nucleation mode, which would indicate that the magnetization reversal could evolve through
similar mechanisms for thin narrow tape-like or cylinder-like nanowires.
Interestingly, the three demagnetizing factors used in the fit had to be almost equal in value (Dx = Dy = Dz = 1/3) in
order to reproduce the experimental curve. These demagnetizing factors have no physical meaning for long nanowires of
high aspect ratio in which Dx = Dy ≫ Dz should be expected. Actually the values of Di used in the previous fit correspond
approximately to a spherical sample and hence they are unrealistic for the sample. These results indicated that a mechanism
of magnetization reversal other than curling was acting in such tape-like permalloy nanowires. The shape of the HSW (θ)
curve clearly indicated that coherent rotation was not the predominant mechanism. From the micromagnetic theories, only
the buckling reversal mechanism was left, which so far had only been considered in theoretical numerical analysis [465].
The fit of the data using the buckling reversal model was able to reproduce the experimental curves to a high extent by
using in this case realistic values of the demagnetizing factors for a tape-like wire. The authors stated that the shape of the
wire was responsible for the buckling reversal mechanism in opposition to the well-known Stoner–Wohlfarth and curling
models for infinite cylinders, and highlighted that in previous studies the data extracted from other long narrow thin tape
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 5.25. (a) AMR characteristics at 300 K of a single wire for ϕ = 10° and 75° (d = 75 nm). Inset: zoom showing the irreversible jumps of the resistance
as the magnetization is reversed; (b) nucleation field vs. temperature of a single Ni wire (d = 575 nm). The applied field is parallel to the wire axis (ϕ = 0°).
Source: Taken with permission from Ref. [345].
wires had been interpreted by means of micromagnetic theories developed for ellipsoidal or cylindrical geometries, due to
the absence of appropriate models for tape-like geometries.
5.5.6. Domain Wall dynamics in 1D metallic nanostructures
We have discussed so far some representative works dealing with the study of the different magnetization reversal
mechanisms acting in size-confined 1D magnetic systems. The non-uniform magnetization reversal processes are
characterized by the nucleation of a domain wall (DW). The DWs are ten- to hundred-nanometer-size transition regions
in which the orientation of local magnetizations M gradually changes between neighboring magnetic domains of different
M orientations in an unsaturated magnetization state [480]. When the dimensions of the magnetic particles approach
those of the DW regions, as it is the case for sub-micrometer nanowires, it becomes possible to study the motion or
dynamics of a single DW along one direction, i.e. the axis of the wire. Some magnetic devices have been proposed, in which
DWs encode information and this is further transmitted along the wire length by field- or current-induced DW motion
[481,482]. The working speed of the device is obviously correlated to the DW velocity. To compete with current technologies,
high-speed operation, and hence fast wall propagation, is absolutely required. However, nanofabrication technologies and
highly sensitive time- and space-resolved measuring techniques have made possible the study of the DW motion only in
the last ten years. In this subsection we will highlight the most significant achievements made in this field.
The Walker’s theory [483] was modeled for the study of one-dimensional DW motion in thin films. Due to the lack of other
models accounting for further dimensional confinements, the propagation of DWs in sub-micrometer metallic nanowires
has been studied and interpreted based on this model. In the Walker model, the DW velocity, ν , increases linearly with the
value of the applied magnetic field H according to the relation:
ν = µH
(5.13)
µ = γ ∆/α
(5.14)
ν = (γ ∆/α)H
(5.15)
where µ is the DW mobility, which can be expressed as:
where γ is the gyromagnetic ratio, ∆ is the DW width and α is the phenomenological Gilbert damping parameter (the
physical mechanisms behind the latter have not been well-understood yet [484]). Hence, Eq. (5.13) can be expressed as:
R. Krahne et al. / Physics Reports 501 (2011) 75–221
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Fig. 5.26. (a) SEM image of a ferromagnetic nanostructure of permalloy consisting of two rectangular pads measuring 70 × 80 mm2 connected by a
permalloy nanowire having length of 12 µm, width of 290 nm, and thickness of 12 nm; (b) and (c) switching field of permalloy nanowires as a function
of the in-plane angle. The open circles represent the experimental data measured by AMR and the solid lines are the fits with the theoretical model
for the nucleation field using the magnetization buckling mode. The calculations for the two samples were carried out using the following parameters:
(b) length = 10 µm, width = 460 nm and thickness = 3.5 nm; (c) length = 6 µm, width = 290 nm and thickness = 2 nm.
Source: Taken with permission from Ref. [479].
This seems to be true for low magnetic field values, also known as viscous regime. However, the Walker model predicts
that a magnetic field threshold value exists, above which the DW propagation velocity suffers a sudden drop and enters the
turbulent regime. In this regime, the propagation of DWs evolves from a simple translation to more complex precessional
modes. The magnetic field value at which the maximum ν is observed is known as the Walker field (HW ). Several authors
have observed experimentally this discontinuity and some of them have studied the morphological, geometrical and
physical parameters underlying the DW dynamics in both regimes, and the possible ways to avoid the Walker breakdown
phenomenon [485–488].
One of the first in-depth studies concerning DW motion was reported by Ono et al. [489]. They investigated the
magnetization reversal dynamics of a single NiFe/Cu/NiFe trilayer wire with thicknesses of 40 nm, 10 nm and 5 nm,
respectively, and width of 500 nm. An initial magnetic field of 500 Oe was applied in order to align M in one direction parallel
to the wire axis. The magnetization states of the wire were monitored by giant magnetoresistance (GMR) measurements,
where changes in the resistivity of the material were detected as the magnetic field was swept in the opposite direction,
while a 100 µA current was applied through the long axis of the wire. The time variation of the resistance during the M
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Fig. 5.27. (a) Time variation of the resistance during the M reversal of the 40 nm NiFe layer at 77 K, which was collected at 40 ns intervals. The applied
magnetic field simultaneously monitored by digital oscilloscope was 121 Oe. Because the sweeping rate of the applied magnetic field was 20 Oe/s, the
variation of the applied magnetic field during M reversal was less than 2 × 10−5 Oe. Thus, the applied magnetic field is regarded as constant during the
measurements; (b) dependence of domain wall velocity v on amplitude H of the applied magnetic field at 100 K.
Source: Taken with permission from Ref. [489].
reversal revealed how the magnetic DW propagated in the wire. The reversal of the 5 nm thin NiFe layer took place at 80 Oe
by the pinning and depinning of the DW, as indicated by the gradual increase of resistivity with increasing magnetic field.
On the other hand, the 40 nm thick NiFe layer reversed abruptly at 120 Oe within a time shorter than the 10 ms measuring
time used in the experiment. When the measuring time was decreased to 40 ns intervals, the changes in resistivity with
time at 77 K and 121 Oe were measured. The 40 nm thick NiFe layer showed a constant DW propagation velocity of 182 m/s
during its magnetization reversal as seen from the linear variation of resistance with time (Fig. 5.27(a)). The constant velocity
suggested that the M reversal takes place by the propagation of a single DW.
The same experiment was performed at different values of the magnetic field and the values of DW velocity ν measured as
a function of H (between 100 and 150 Oe) were fitted using the Walker equation ((5.13)). The results are shown in Fig. 5.27(b).
A value of mobility µ equal to 2.6 ± 0.2 m/sOe was obtained, which was seen to be constant in the temperature range
between 100 and 160 K. This value of µ was much lower than values observed in NiFe thin films of similar thickness [490].
The authors concluded that the DW mobility in the wire was mainly limited by Gilbert damping, since a value of α of 0.63
was calculated using the equation µ = γ ∆/α and assuming a DW width of 100 nm. Such a large value of α was attributed
to the presence of defects at the surface and interface of the wire.
A controversial work was published in 2003 by Atkinson et al., who claimed that lateral constrainment of the DW does
not necessarily affect the intrinsic DW propagation velocity with respect to thin films [491]. A 5 nm thick and 200 nm wide
permalloy wire was chosen as the object of study. Hence, the wire was 300 nm narrower and 35 nm thinner than the one
studied in the previous report [489]. The magnetic switching dynamics at room temperature were monitored in this case by
means of continuous wave laser magneto-optic Kerr effect measurements (MOKE) with nanosecond temporal resolution.
The magnetic fields, at which DW propagation velocities were measured, were much weaker compared to the previous
work, due to the lower coercivity of the wire, and ranged from 10 to 49 Oe. Very high values of DW velocity were observed
in this case: over 500 m/s above 40 Oe and a maximum of 1500 m/s at 49 Oe. The authors suggested a thermally activated
DW motion still within the viscous regime defined in the Walker model. In this regime, the mobility, defined as the slope of
the field dependence of the wall velocity, should be constant in the absence of pinning sites, as stated in Eq. (5.14). However,
experimental results indicated a field-dependent mobility in this case, which was attributed to the presence of defects in
the wire that were pinning the DW propagation, until thermal activation overcame the pinning energy, allowing the DW
to move. The authors assumed that, at the highest magnetic field measured, the defects no longer pinned the wall and
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181
Fig. 5.28. Simulated time-resolved domain wall (DW) motion in a permalloy wire. Wire thickness 5 nm, width 200 nm, damping constant α = 0.01. (a)
Comparison of wall velocities for perfect (defect-free) edges (field H = 12 and 20 Oe applied at time t = 0 to a DW in equilibrium) and rough edges
(H = 20 Oe, grain diameter D = 7 nm); (b) pictures at different times of the moving DW at 20 Oe with perfect edges (the picture height is the wire width;
the right edge of the pictures at t > 0 corresponds to the same physical position; color coding of the magnetization orientation uses the color wheel for
the in-plane directions, and goes to white or black when the magnetization becomes out of plane). White arrows depict the DW rest structure. The inset is
a representation of the antivortex core structure in the square box; (c) comparison of experimental values (the two data sets of Ref. [491]) and numerical
simulations (α = 0.02, D = 0 and 20 nm). The error bars span twice the standard deviation of the velocity (D = 20 nm). For the rough wire, the wall can
travel 20 µm without stopping for fields larger than 45 Oe, whereas at 30 Oe it stops within 1 µm (zero temperature calculations). Fast wall motion is seen
to extend at least up to 100 Oe.
Source: Taken with permission from Ref. [494].
thus the velocity was defined by the intrinsic mobility of the wire. A value of mobility of 31 m/sOe and of Gilbert damping
coefficient of 0.053 were estimated by using Eqs. (5.13) and (5.14) between the origin and the data point measured at the
highest field. These values were much closer to the ones previously observed in films [492] and also to the ones predicted
by theory [493]. The results were regarded as a proof that the operational speed of potential spintronic devices involving
DW motion in magnetic nanostructures would not be affected by structural confinement.
From that moment on, several theoretical and experimental studies have tried to shed light on the reasons for the
divergence in results. Nakatani and co-workers [494] performed a set of micromagnetic simulations on a nanowire of
dimensions that were identical to the nanowire used in the work of Ref. [491]. Initially the authors simulated the time
dependence of the DW velocity fixing the nanowire thickness, width and damping constant to 5 nm, 200 nm and 0.01,
respectively. The simulations considering an ideal defect-free wire reversing at applied magnetic fields of 12 or 20 Oe yielded
the expected results based on the Walker theory. The results of the simulation are displayed in Fig. 5.28(a). At low fields, a
stationary velocity of 608 m/s was reached in agreement with the viscous regime, while at higher fields the turbulent regime
became apparent and the average velocity of propagation decreased to a value of 60 m/s. The nature of the DW motion above
the Walker breakdown was explained by the authors as an alternation of transverse magnetization- and antivortex-like walls
as shown in Fig. 5.28(b).
The transverse magnetization wall is energetically more stable than a vortex and hence propagates at higher velocities.
However, this type of wall presented a V-shape and, due to exchange interactions, the precession around the applied field
was enhanced at the corner of the V domain. This resulted in the nucleation of an antivortex that absorbed most of the
torque supplied by the magnetic field and as a consequence it moved slowly, dramatically decreasing the velocity of the DW
motion. This antivortex moved to one edge of the wire where it finally disappeared and the initial transverse magnetization
wall and DW velocity were restored. This type of motion explained the inhomogeneous velocity values simulated with time
and the low average velocities observed in most experimental cases when measurements are performed in the turbulent
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Fig. 5.29. Average domain wall velocity vs. field step amplitude. Arrows mark the domain wall injection field, Hi , and velocity peak, Hp . Straight solid lines
are linear fits to the data below Hp and above Hi , respectively; the curved line is a visual guide. Filled symbols show velocities obtained for a square wave
drive field and open symbols those obtained using injection pulse waveforms. The inset shows the detail around the velocity peak.
Source: Taken with permission from Ref. [487].
regime. Subsequent theoretical [495] and experimental [496] studies have also addressed the DW propagation mechanisms
above the Walker field in similar nanowires. Such studies confirmed the periodicity observed in the wall transformation
processes during propagation, in contrast to stochastic propagation modes. In the work from Nakatani, the authors also
studied the effect of the wire edge roughness on the DW propagation velocity above the Walker field. Results are displayed
in Fig. 5.28(c). Interestingly, simulations showed that the presence of sufficiently rough edges suppressed the antivortex
nucleation at the edges, allowing the DW to move at the highest speed above the Walker field, as the ideal wire was doing
in the viscous regime. The authors used a new model for their simulations that took in consideration the grain size (D) of
polycrystalline wires as the main source of roughness. They then compared their calculations with experimental results of
Atkinson et al. [491]. Calculations considering ideal defect-free wires were only able to reproduce experimental velocity
values in the low field region (below 35 Oe). However, when a mean grain diameter of D = 20 nm was considered in the
simulations, the whole set of experimental velocities at low and high fields in Ref. [491] could be reproduced, as reported in
Fig. 5.28(c). The authors claimed that all the experimental points were in the roughness-stabilized regime of wall motion. It
is assumed by the authors that the roughness must be larger than the exchange length of the wire in that case. The exchange
length of soft magnetic materials like permalloys is their characteristic length of micromagnetics and can be expressed as
1/2
2A/µ0 MS2
, where A is the exchange constant and MS is the saturation magnetization. Since nucleation of antivortexes
occurs at the edges of the wires, roughness values larger than the exchange length are optimal to suppress their nucleation.
In contrast, in a thicker wire such as the one studied by Ono et al. in Ref. [489], similar values of roughness do not play the
same role and the DW motion is not in the roughness-stabilized regime. By means of theoretical calculations, Nakatani and
co-workers suggested the role of the wire edge roughness as a stabilizer of the fast DW motion up to fields much larger
than those associated with the Walker breakdown in ideal nanowires. They also stated that roughness should rather be
engineered than avoided when fabricating nanostructures for DW propagation. Alternatively, Glathe et al. demonstrated
experimentally that the application of a transverse magnetic field can also be used as a way to suppress Walker breakdown
events [497].
It was so far clear that the field-induced DW motion along nanowires could be divided in different propagation regimes
depending on the amplitude of the applied magnetic field H, the so-called viscous and turbulent regimes. Therefore, it is
obviously necessary to study a wide range of magnetic fields in order to clearly distinguish between one regime and another
without uncertainty. However, depending on the intrinsic characteristics of the wire, the fields at which the viscous regime
would be observed are lower than the ones required for the domain wall injection into the wire, Hi . In this sense, Beach
et al. were able to devise new experimental conditions that allowed for the dynamic measurement of DW propagation in a
ferromagnetic permalloy nanowire over a broad field range [487]. A 20 nm thick and 600 nm wide Ni80 Fe20 wire was studied
by longitudinal magneto-optic Kerr effect measurements. In their experiments, a minimum Hi of 24 Oe was required to inject
a DW into the wire and induce the magnetization reversal of the latter. However, once the DW had been injected, constant
fields down to 1.5 Oe were observed to be sufficient to propagate it. Hence, using specially tailored drive field waveforms,
the authors were able to study the DW dynamics along the wire from 1.5 until 70 Oe. The field dependence of the velocity
measured for the permalloy wire is reported in Fig. 5.29.
Three different regimes were clearly discerned, each of them characterized by a different value of the DW mobility, µ. At
fields below 4 Oe, the velocity increased linearly with field with mobility close to 25 m/sOe. Above 4 Oe, ν began to decrease
with increasing H, exhibiting a negative DW mobility in this regime. However the negative µ started increasing gradually
with increasing magnetic field and became positive again above 20 Oe. For H values larger than 30 Oe, the field dependence
of the DW velocity recovered the linearity, but with a ten-fold reduction in µ compared to the low field regime. These results
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were in complete agreement with the Walker model for 1D DW systems [483]. At low fields below the Walker transition,
the DW motion followed the well-known linear mobility relation of Eq. (5.13) where µ = γ ∆/α . After a regime in which
the plane of transverse wall magnetization canted increasingly away from the direction of wall motion, the field-dependent
variation of the velocity showed an asymptotically linear relation with a slope defined again by the mobility. However, at
large fields the mobility was better expressed as
µ = γ ∆/(α + α −1 )
(5.16)
This equation was able to explain from a phenomenological point of view the ten-fold drop of mobility above the Walker
field. The authors claimed that the slope of ν vs. H over a limited field range is insufficient to draw conclusions about the
intrinsic damping unless one knows the relation of that field range to the Walker breakdown field. In this framework, the
authors were able to compare the a priori contradictory results in Ref. [489,491]. The measurements of Ref. [489] for a
permalloy nanowire of similar width and thickness were only performed at high magnetic fields. They revealed very similar
values of ν and µ when compared to the ones obtained with the measurements of Beach et al. in the high field regime. The
authors suggested that the data presented by Ono et al. in Ref. [489] probably belonged to the linear ν = f (H ) regime at
high fields where µ had to be expressed as in Eq. (5.16). If this was true, the low ν and µ values presented would make more
sense since the measurements would have been performed within the turbulent regime and not in the viscous one, where the
classical equation (5.15) can be applied. On the other hand, the wire studied by Atkinson et al. in Ref. [491] was only measured
at low H values. The wire, in that case, was characterized by a high mobility value comparable to the one measured by Beach
et al. in the low field regime previous to the Walker breakdown phenomenon. Moreover, the value of the Gilbert damping
parameter calculated by using Eq. (5.14) was completely reasonable for a permalloy nanowire. Hence, Beach suggested that
Atkinson’s results had been obtained by measuring the nanowire within its viscous regime. However, that would imply a
Walker field in that system much larger than the 4 Oe observed by Beach et al. The roughness effect on thin nanowires
predicted by Nakatani et al. [494] was assumed to be responsible for the shift of the Walker breakdown field to higher
values.
As said at the beginning of this subsection, the Walker’s theory [483] was modeled for the study of one-dimensional
DW motion in thin films. Generally speaking, also nanostructured wires follow the trends marked by the Walker model.
However, it should be noted that discrepancies between the Walker model and the experimental results for size constrained
magnetic nanowires are often observed. Hence in recent years theoretical studies have been undertaken in order to evaluate
these differences, suggest improved micromagnetic models, and determine the effect of several geometrical or damping
parameters on the main parameters describing the DW dynamics of magnetic nanowires, i.e. DW velocity, DW mobility and
Walker breakdown field. Most of them are limited to the interpretation of the dynamics in the viscous regime below the
Walker field. A. Kunz performed several micromagnetic simulations by using the three-dimensional Landau–Lifshitz–Gilbert
(LLG) equation of motion of each magnetic moment in the wire [498]. For nanowires of identical dimensions, the effect of
the varying Gilbert damping parameter was evaluated from 0.005 to 0.1. Calculations suggested that the maximum DW
speed was independent from the α value, while instead the Walker field increased linearly with increasing α values. High
values of the damping parameter are thought to suppress the precessional motions and vortex nucleations within the wall,
shifting in this way the Walker breakdown phenomenon to higher field values. In a set of subsequent simulations, the α
parameter was maintained constant while the width of the wire was gradually increased. It was shown in that case that
the value of the Walker field decreased, and the average speed increased as the width of the wire was increased. The new
micromagnetic calculations indicated that the classical Walker model overestimates both the DW velocities (by a factor of
3) and the Walker fields. The authors suggested that these discrepancies with the Walker model could be the reason for the
need of unphysical DW width and damping parameter values, in order to force experimental data agreement with theory.
Recently, Bryan et al. used the same LLG equation in micromagnetic simulations to study the relationship between the wire
dimensions and the DW dynamics [499]. In classical 1D models, the magnetization is assumed to be uniform throughout
the wire cross section and, based on this, the value of the Walker field, HW , and DW mobility, µ, can be expressed as [500]:
HW =
µ=
1
2
v
Hx
α MS Nz − Ny
=
γ∆
α
(5.17)
(5.18)
where Ny and Nz are the demagnetizing factors along the wire width and thickness, respectively. The ∆ wall width depends
on Ny and Nz as well as on the applied field. However, the demagnetizing factors do not depend on the absolute dimensions
of the wire, but on the relative ones. Hence in the 1D classical model wires with the same width-to-thickness ratios (w/T )
should present identical values of HW and µ, regardless of their absolute dimensions. However, calculations performed by
the authors suggested that nanowires with identical w/T but different absolute dimensions were not behaving in the same
way. The calculated w/T dependence of the DW mobilities for wires of different widths is shown in Fig. 5.30(a). The wall
mobility clearly increases with increasing w/T , even if agreement with the 1D model was only observed for the narrowest
wires, while the wider ones diverged significantly from predictions. Moreover, the dependence of µ on w/T changes as the
wire width increases, suggesting that the DW mobility is determined by the absolute, rather than the relative, values of the
wire width and thickness. It was also observed in the calculations that as either wire width or thickness is increased, HW
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Fig. 5.30. (a) The domain wall mobility, µ, at Hx (below HW ) for permalloy wires of various widths, w, and thicknesses, T, as a function of w/T . The solid
line shows wall mobility calculated using a 1D model. The effect of (b) the wire width, w, and (c) the wire thickness, T, on the Walker breakdown field, HW ,
for several wire dimensions.
Source: Taken with permission from Ref. [499].
decreases, as shown in Fig. 5.30(b) and (c). New equations were proposed by the authors, which experimentally related the
width and thickness of the wire with its DW mobility and Walker field with a better agreement than the equations included
in 1D models.
All previous examples regarded the magnetic field-induced DW propagation in 1D nanostructured systems. However,
in recent years experimental evidences have confirmed the possibility to drive a DW along nanowires by the application of
an electrical current. The pioneering theoretical studies predicting the current-induced DW displacement were reported by
Berger [501,502]. When a spin-polarized conduction electron crosses a DW, it changes its spin direction trying to follow the
local magnetization of the material. In this process, there is a transfer of angular momentum from the electron to the DW,
which is responsible for the physical displacement of the latter. This phenomenon is also known as the spin–torque effect.
The main advantages arising from such current-induced DW propagation is the simplicity in the design, as it eliminates
magnetic field-generating circuits in the device and is seen as a promising alternative for writing in ‘‘one bit/one structure
devices’’, such as the spin momentum transfer magnetic random access memory [503]. However, while the field-induced
DW motion is quite well-established, current-induced DW dynamics have not been well-understood yet. Several authors
have been addressing this phenomenon from both a theoretical [504–508] and experimental [488,509–514] point of view,
in order to unveil and understand the several effects occurring when large electrical currents flow across a DW. For an indepth coverage of the current-induced DW propagation phenomena and its state-of-the-art, the reader is redirected to the
previously cited works and the references therein included.
6. Catalytic properties of elongated inorganic nanoparticles
6.1. Introduction
In this section, we will confine our attention to the studies reported so far on elongated nanoparticles of metals, metal
oxides, semiconductors and mixed materials as catalysts for a series of chemical reactions. For the various systems, we will
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highlight, whenever possible, the relationships between shape anisotropy and catalytic behavior. It is worth recalling here
that a catalyst is a substance that influences the rate of a given chemical reaction, but which remains unchanged at the end
of the reaction. If the catalysts species are in the same phase as the reactant species (if for example they are all in the liquid
phase), then we speak of ‘‘homogeneous catalysis’’, otherwise we will speak of ‘‘heterogeneous catalysis’’ (if for example the
catalysts are a solid and the reaction species are either in the liquid or in the gas phase). Fig. 6.1(a) sketches the capability
of a catalyst to lower the energy barrier associated with a given chemical reaction. Often, another desirable property of a
catalyst is to increase the rate of a specific reaction pathways over others (i.e. B → C instead of B → D or B → E), in order to
yield a lower (or even the least) energetically favored product over many other possible products [515,516]. In this section
we will refer to a catalyst as a solid support, since inorganic nanoparticles are indeed nanosized solids. The specificity of a
catalyst depends largely on catalyst’s surface, on the type and distribution of surface active sites, onto which reactive species
are adsorbed and undergo the following steps of dissociative adsorption and desorption, to yield the final or the targeted
products (Fig. 6.1(b), (c)).
Nowadays, there is a huge demand for novel catalytic materials, in terms of catalytic efficiency, cost of production,
specificity, durability and environmental sustainability [41–44]. This demand is constantly driving research towards the
exploitation of new nanoscale materials, in which the individual components have tailored size, shape, exposure of specific
reactive surfaces, and suitable combination of materials [45,46].
6.2. Rod-like morphology and catalytic properties of inorganic nanoparticles
Micro- and nanoparticles of various materials have been conventionally used as catalysts since many years [47–50], and
experimental evidences have been collected so far demonstrating that the catalytic activity of particles is strongly related to
their size, and in particular that nanosized particles exhibit increased catalytic activity with respect to larger particles, due
to their higher surface to volume ratio [51,52]. Moreover, in the last decades extensive surface characterization of catalysts
surfaces (via electron microscopy, scanning probe microscopy and various surface spectroscopy techniques) has helped to
understand the reasons behind specific reactivity, selectivity and reaction pathways on different crystal facets of various
catalyst materials in a wide range of chemical reactions. As an example, Fig. 6.2 illustrates how different crystal facets of a
Pt3 Ni alloy crystal exhibit different catalytic activities towards the oxygen reduction reaction (O2 + 4H+ + 4e− = 2H2 O)
(ORR) [518].
Subsequently, recent advances in the synthesis of inorganic nanoparticles with controlled size and morphology [53–56]
have sparked research towards the understanding of how the catalytic performance of these materials is dependent on their
shape. Despite the shape dependence of nanocatalysts is not fully understood, some landmark reports have started to draw
clear correlations between morphology and catalytic activity in nanoparticles [519–523]. Many studies have highlighted for
example how cube-, rod-, branched and multi-pod shaped nanoparticles can have improved catalytic activity with respect
to the corresponding samples with more heterogeneous distribution of shapes, when their specific shape allows for some
sets of particularly active facets to be exposed to the external environment.
Fig. 6.1. (a) A sketch on the role of a catalytic material in a given chemical reaction; (b) The variation of the relative coverage of N atoms (y) chemisorbed
at 693 K on various Fe single crystal facets, upon exposure to nitrogen gas. (c) Catalytic oxidation of CO on a Platinum (Pt) surface, reaction pathways and
energy diagram.
Source: Taken with permission from Ref. [517].
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Fig. 6.2. Histograms showing the effect of surface morphology (i.e. crystal facet that is exposed) and electronic surface properties on oxygen reduction
reaction kinetics (ORR) on Pt3 Ni (hkl) surfaces as compared to the corresponding Pt (hkl) ones [(using rotating ring disk electrodes (RRDE) in HClO4 (0.1
M) at 333 K with 1600 rotations per minute). Specific activity is given as a kinetic current density ik , measured at 0.9 V vs. RHE. Values of d-band center
position obtained from ultraviolet photoemission spectra (UPS) are listed for each surface morphology and compared between corresponding Pt3 Ni (hkl)
and Pt (hkl) surfaces].
Source: Taken with permission from Ref. [518].
Here, we will discuss mainly about elongated nanoparticles. However, we will not confine the discussion only to rodshaped particles, but we will also consider wire-, belt-, and net-like morphologies. In terms of catalytic properties, the main
reasons why an elongated morphology would be preferable over a spherical morphology can be summarized as:
I At constant volume, the surface to volume ratio increases when the shape evolves from a sphere to an elongated one
and as the aspect ratio of the nanoparticle increases further. This leads to a better catalyst if the facets that become more
extended in rods are indeed those exhibiting higher catalytic activity. In addition, an increase in surface to volume ratio
might lead to an increase in the number of surface active sites. Also, anisotropic crystals could act as efficient charge
separation systems, for example in photocatalytic reactions (see later in this section).
II The mechanical stability of a particle acting as catalyst might increase as a consequence of its elongation along a specific
crystallographic direction [524,23] (see Section 7 of this review for more insights into mechanical properties of elongated
nanoparticles).
III Nanorods can be assembled with specific geometries with respect to a substrate or with respect to each other
(vertical, horizontal, end-to-end) [178,185,525], which might favor the exposure of a given set of facets to the external
environment.
IV Easy identification of the nanorod morphology among other irregularly shaped support materials used (for example
binders, fillers and other diluents (nearly spherical or irregularly shaped)) over number of cycles for recovery.
In this section we will discuss first examples of catalysts based on metals, core/shell metals and metal alloy nanoparticles,
highlighting the importance of shape in those reports. We will then report examples of metal oxide and semiconductor
nanorods, as well as nanorods of mixed materials.
6.2.1. Metals, core/shell metals, segmented metals and metal alloy nanorods
Habas et al. [526] have studied the distinct catalytic activity of metallic Pt/Pd core@shell nanoparticles of various
shapes, namely cubes, octahedra and cubo-octahedra (Fig. 6.3(a)–(c)), towards the electrochemical oxidation of formic
acid (HCOOH + 1/2O2 → CO2 + H2 O). The oxidation rate on nanocubes was found to be five times higher than that
on octahedral and cubo-octahedral shaped particles. The increased activity in case of nanocubes was attributed to their
specific crystallographic habit, which exposed Pd (100) facets instead of the Pd (111) ones. This clearly demonstrates how
one particular set of crystal facets of nanostructures selectively catalyses the reactions than other sets. In addition, by an
epitaxial overgrowth of gold (Au) on cubic Pd seeds, the formation of Au nanorods containing or embedding Pt cubes was also
achieved, see Fig. 6.3 (d)–(e). The Au rods grown on Pt cubes were found to be pentagonally twinned crystals, growing along
the (220) direction. Such nanostructures (nanorods with embedded Pt domains) were proposed for the shape-dependent
catalytic studies.
Similarly, Berhault et al. reported the selective catalytic activity of Pd nanorods for the hydrogenation of buta-1, 3-diene to
but-2-ene, which was rationalized as due to the exposure of the (110) facets in these rods [527]. On these facets the effective
adsorption of reactive species has appeared to be favored in the reaction. This characteristic behavior was also confirmed
in palladium (Pd) rods and in branched nanostructures by Chen et al. [528] (Fig. 6.4(a)–(c)). In analogy with the Au rods
morphology described previously, these Pd nanorods were multiple twins with pentagonal cross section, with selective
growth along the (110) direction. These nanorods were used as catalysts for the Suzuki coupling reaction (carbon–carbon
bond formation) [529], with about 90% yield of the product (biphenyl) in the first time use. Again, the observed activity and
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Fig. 6.3. (a) From left to right: SEM and TEM images of cubic platinum nanoparticles, high angle annular dark field scanning TEM (HAADF-STEM) image
of an individual cubic particle, sketch model showing the axes projection along the (100) direction; (b) Same sequence of images as in (a), but referred to
cubo-octahedral particles. Here the sketch model shows the axes projection of the cube along the (100) and the (111) directions; (c) Same sequence of
images as in (a), but referred to octahedral particles, the sketch model shows the axes projection of the Pt cube (the seed) along the (111) direction; (d)
SEM image (left) and TEM image (right) of gold nanorods; (e) High resolution TEM image showing the Pt cube on the surface of gold nanorods (left) and
nanorods viewed from (112) and (100) directions with continuous (111) lattice fringes (right).
Source: Taken with permission from Ref. [526].
Fig. 6.4. The table on the top panel reports the cyclability of the Pd catalysts in the Suzuki coupling reaction. (a) TEM image of Pd nanorods with a sketch
showing their fivefold twinned structure; (b) Selected area electron diffraction (SAED) pattern demonstrating that the nanorods grow preferentially along
the (110) direction, with an overlap of the (001) and (11 2) reflexes of the face-centered-cubic (FCC) crystal structure; (c) High resolution TEM view of
the panel from (a) showing d spacing of 2.28 °A, which corresponds to the (111) lattice planes of Pd. (d) SEM image of Pd nanorods with fivefold twinned
structure.
Source: Taken with permission from Ref. [528].
selectivity for the reaction were attributed to specific facets that defined the habit of the nanocrystals. The nanorods were
recovered and tested for the second time, and resulted in a yield of about 85%, indicating the recyclability of the catalyst
nanorods yield, see Fig. 6.4. Therefore, the nanorods seemed efficient also for multiple uses.
A similar trend was found in bimetallic PtPb nanorods by Maksimuk et al. [530] (Fig. 6.5(a)–(c)). The nanorods were
elongated along the (110) direction, and the incorporation of Pb atoms into the Pt lattice was reported to take place in sites
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Fig. 6.5. (a) TEM image of PtPb nanorods. (b) Selected are electron diffraction patterns from a single PtPb nanorod. (c) Power X-ray diffraction pattern of an
ensemble of PtPb nanorods. (d) Model sketch of four proposed crystal structures of PtPb nanorods. (e) Cyclic voltammogram showing the electrocatalytic
oxidation of methanol on PtPb nanorods and on PtRu/C commercial catalyst.
Source: Taken with permission from Ref. [530].
with high coordination numbers due to the stronger binding energy of such sites (Fig. 6.5(d)). The electrochemical oxidation
of methanol (CH3 OH → CO2 + H2 O) was carried out in order to test nanorods vs. a commercially availed PtRu/C catalyst
(from Johnson-Matthey, Pt 30 wt%: Pd 15 wt%). From the anodic peak current (ipa) in cyclic voltammetry, the activity of
the nanorods was found to be higher than the reference catalyst with respect to the normalized unit mass of Pt (Fig. 6.5(e)),
which clearly indicates higher rate of methanol oxidation in the case of nanorods.
Phase segregated bimetallic nanorods have also been used for catalytic applications. Using electrodeposition in
sequences, Liu et al. [531] have prepared multi-segmented Pt–Ru nanorods (Fig. 6.6(a)–(e)). At room temperature, the
nanorods were seen to electrocatalyse the methanol oxidation reaction under acidic media (Fig. 6.6(f)). Interestingly,
the catalytic activity of the nanorods was found to increase upon increasing the number of Pt–Ru segments, and was
explained on the basis on an increased number of Pt–Ru interfaces. Also CuPt metal alloy nanoparticles, as reported by
Liu et al. [532], showed shape-dependent catalytic activity. Spherical nanoparticles and nanorods of CuPt were tested
for the CO oxidation reaction (CO + 1/2O2 → CO2 ). The nanostructures were dispersed on a γ -alumina (Al2 O3 )
support and were loaded into a CO feedstock. Under identical experimental conditions, the average light-off temperature
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Fig. 6.6. Field emission SEM images of segmented nanorods (a) Pt–Ru, (b) Pt–Ru-Pt, (c) Pt–Ru-Pt–Ru, (d) Pt–Ru-Pt–Ru-Pt, (e) Pt–Ru-Pt–Ru-Pt–Ru. The
diameter of the rods was 200 nm in all the samples; (f) Cyclic voltammograms of nanorods, at 0.1 M HClO4 and 0.5 M CH3 OH, polarized at a constant
potential vs. Ag/AgCl (3 M KCl) at room temperature.
Source: Taken with permission from Ref. [531].
values (Tav g ) for both catalysts (∼556 K for nanorods and ∼490 K for spherical nanoparticles) were found to be lower
than the commercially available Pt (2–3 nm)/Alumina catalyst (700 K). Although the spherical nanoparticles (2.5 nm)
were extremely reactive in such size range (∼2 nm), nanorods with an aspect ratio range ∼5:1–25:1 appeared to be
comparatively efficient, see Fig. 6.7(a)–(g). In addition to the high surface area of nanostructures used, the surface of the
CuPt alloy particles appeared to be less poisoned by CO adsorption, due to the more oxophilic character of the active
sites.
6.2.2. Metal oxide, semiconductor, and mixed nanorods
We will consider first selected examples of metal oxide and metal/metal oxide supported catalytic systems. Xie et al. [533]
reported the synthesis of Co3 O4 nanorods, which according to HRTEM were elongated along the (110) direction and were
preferentially exposing the (110) facets (Fig. 6.8(a)–(h). These facets accounted for about 41% of the total surface area of
the rods. These nanorods were seen to catalyse the CO oxidation reaction even at temperatures as low as −77 °C and at
room temperature 100% conversion for 65 h was found, therefore the rods exhibited much higher activity than the more
traditionally used metal oxide supported Au nanoparticles for this type of reaction. From structure reconstruction analysis
via HRTEM, it was proposed that the high abundance of Co3+ cations on the (110) facets provided effective sites for CO
adsorption. Similar results were also found for the oxidation of propane at high temperature (100% at 350 °C) and such
oxidizing ability of Co3 O4 nanorods was found to be comparable to that of Pt and Pd catalysts. The higher activity and
selectivity were therefore directly attributed to the nanorod morphology, which exposed preferentially the catalytically
active sites. Also, these nanorods were found to possess good stability even under high moisture conditions, and they
retained the rod-shape even upon repeated use (Fig. 6.8(i)).
Copper oxide (CuO) nanostructures too display strong shape-dependent catalytic behavior. Zhou et al. [534] have studied
the relative differences in catalytic activity of anisotropic CuO nanostructures, including belts and platelets, vs. spherical
nanoparticles (Fig. 6.9(a)–(c)). The most developed facets in the various nanostructures were the (111) ones for spherical
nanoparticles, (001) for nanobelts, and (011) for nanoplatelets. Upon performing CO oxidation, the conversion rate was
found to be in the order of nanoplatelets > nanobelts > nanoparticles (Fig. 6.9(d)–(e)). Here, the increased activity of
nanoplatelets was attributed to the exposure of the (011) facets, which appeared to facilitate desorption of O2 from the
surface, as determined by CO temperature programmed reduction (TPR). In case of nanobelts, the activity was even higher,
due to the exposure of the (001) facets. As the close-packed facets (111) of the oxides (which are the most exposed facets in
spherical nanoparticles) possess lower surface energy, they are more stable than the (011) and (001) open facets.
The shape-dependent catalytic activity is significantly pronounced in the case of CeO2 nanorods. Si et al. [535] prepared
CeO2 nanostructures with various crystal habits, namely rods with exposed (110) and (100) facets, cubes with exposed (100)
facets and polyhedrons with exposed (111) and (100) facets (Fig. 6.10(a)–(d)). Upon performing the water gas shift (WGS)
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Fig. 6.7. (A) TEM image of CuPt nanorods. Left inset: selected area electron diffraction pattern. Right inset: TEM image of spherical CuPt nanoparticles.
(B)–(C) Histograms of diameter and length distributions of the nanorods. (D)–(E) HRTEM of nanorods. (F) X-ray diffraction patterns. (G) Energy dispersive
spectroscopy analysis.
Source: Taken with permission from Ref. [532].
reaction (CO + H2 O ↔ CO2 + H2 ) the catalytic activity from light-off curves followed the order: rods > polyhedrons > cubes.
Almost 100% CO conversion efficiency was reported in the case of nanorods (Fig. 6.10(e)). The higher activity of the nanorods
was attributed to their exposed (110) and (100) facets, which appeared to be energetically more favorable for introducing
oxygen vacancies that stabilize the metallic phase during catalytic reactions (due to the oxygen buffer effect). Besides, when
considering the formation energy of anion vacancies for the various CeO2 facets, the exposed (110) and (100) facets of
nanorods were found to be a more favorable support for embedding or dispersing metal domains such as Au, which increases
the catalytic activity due to the metal/metal oxide support effect [536]. This also has an implication on the mechanical
properties of the nanorods: after growing gold domains, the lattice micro-strain measurements for all three samples revealed
that the nanorods were characterized by the larges lattice strain, which correlated with the increased catalytic activity of
the nanorods, see Fig. 6.10, top panel. Sayle et al. [537] too have pointed out that ceria nanorods with exposed (110) facets
exhibit higher elastic deformation than the brittle bulk phase.
Han et al. [538] have also reported similar results, highlighting that in CeO2 rods the preferentially exposed (110) and
(100) facets play a key role in their catalytic activity. First of all, it is important to note that the reduction of CeO2 nanorods
in H2 started at lower temperature (150 °C) with respect to both spherical nanoparticles and bulk ceria. In the WGS reaction,
while the spherical nanoparticles displayed almost no activity, nanorods possessed increased catalytic activity as a function
of the temperature, up to about ∼350 °C (Fig. 6.11(a)–(b)). As discussed earlier in the case of ceria nanorods, the amount of
oxygen vacancies that formed on the CeO2 nanorods (during the reduction process) due to the exposed facets, improved the
catalytic activity. Furthermore, by dispersing Pd nanoparticles onto the nanorods, the activity increased further, as expected
due to the metal/metal oxide support effect (Fig. 6.11(c)).
Not only the shape but also the phase of the nanoparticle can influence its catalytic activity. Liang et al. [539] have
synthesized MnO2 nanorods of different crystal phases (namely the α , β , γ and δ phases of MnO2 ) via an hydrothermal
method (Fig. 6.12a–d). All the nanorods were found to grow along their (002) and (001) crystallographic axes. The surface
parameters of the nanorod samples before and after CO oxidation from various characterizations are listed in Fig. 6.12, top
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Fig. 6.8. (a)–(b) TEM images of Co3 O4 nanorods; (c)–(f) HRTEM images of the nanorods, showing (1-10), (100), (001) and (110) lattice fringes; (g) Sketch
highlighting the habit of the nanorods. The red shaded facets are the catalytically active ones; (h) Possible reaction pathway for CO oxidation on Co3 O4
nanorods, with a ball and stick model for CO adsorption and oxidation on the active C03+ site; (i) Effects of moisture content, regeneration and temperature
on the oxidation of CO over Co3 O4 nanorods.
Source: Taken with permission from Ref. [533].
panel. Interestingly, the activity of various samples of nanorods towards the CO oxidation was found to rank according to
the following order: α = δ > γ > β (Fig. 6.12e–f). The higher activity of the δ and α -MnO2 nanorods for CO oxidation was
found to be strongly influenced by their crystal phase. From a catalytic performance and stability test (CTL), the activity order
(α = δ > γ > β ) was found to be the same. CO adsorption on MnO2 was also found to be in the order δ > α > γ > β MnO2
by CO temperature programmed desorption (TPD). On the other hand, Sui et al. [540] reported that the β -phase of MnO2
nanorods elongated along the (002) direction exhibited the highest photocatalytic activity towards the decomposition of
rhodamine-B than other MnO2 phases (α , δ , γ ), showing the specificity of the nanorods in their catalytic behavior.
Similarly, Lin et al. [541] have reported the fabrication of ZnO nanorods/Cu nanoparticles composite catalysts and
tested them for the oxidation of methanol (Fig. 6.13(a)–(j)). The composites prepared from these nanorods displayed lower
reduction potential than commercial catalysts (such as Cu–ZnO–Al2 O3 , MDC-3). The composite electrodes used, in addition
to exhibiting higher rate for methanol oxidation (93%), had a higher H2 production rate (183 mmol gcat−1 h−1 at 250 °C)
than commercial ones. From the Arrhenius plot, the reaction activation energy for the nanorod catalysts was found to be
lower (27.5 kJ mol−1 ) than the commercial catalyst (36.8 kJ mol−1 ), showing that the methanol oxidation rate could be
faster on nanorod-shaped catalysts. The high surface area of the nanorods, the metal nanorod interfacial compatibility and
the stability of the electrodes were attributed to such higher activity.
Elmalem et al. [542] have reported the distinct photocatalytic properties of Pt-decorated cadmium CdSe nanorods
(Fig. 6.14(a)–(d)). Under irradiation, efficient charge separation takes place at the metal/semiconductor interface, with the
electrons moving to the metal tip region, from where they can induce faradic reactions. Methylene blue (MB) was used as
model dye in order to test the visible light photocatalytic activity of the nanorods. Two kinds of Pt–CdSe nanostructures,
namely nanorods and nanonets, were synthesized, and were compared in terms of their catalytic activity. The nanonets
were better photocatalysts than the nanorods, due to the smaller Pt domains (1.9 ± 0.5 nm), which were found to
be more densely located at their surface (in the samples of nanorods tested the Pt domains had an average size equal
to 3.3 ± 1.1 nm).
The incorporation of metal nanoparticles onto metal oxide nanorods was also found to lead to efficient photocatalysts
due to the electron–hole separation at the metal/semiconductor interface. Zinc tungstate (ZnWO4 ) nanorods containing
silver (Ag) nanoparticles were prepared by Yu et al. [543] via a sonochemical synthesis approach. The nanorods had good
crystallinity (Fig. 6.15(a)–(d)) and were stable in solution. The photocatalytic properties of these rods were tested on the
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Fig. 6.9. TEM images of (a) CuO nanoparticles; (b) CuO nanobelts and (c) CuO nanoplatelets; (d) Specific rate of CO conversion over CuO nanoparticles,
nanobelts and nanoplatelets at 110 °C with respect to exposed crystal facets of each nanostructure, (111) for nanoparticles, (001) for nanobelts and (011
for nanoplatelets); (e) CO temperature programmed reduction profiles of each nanostructure.
Source: Taken with permission from Ref. [534].
degradation of rhodamine-B (RhB) under irradiation at λ = 365 nm, using a UV lamp. The bare ZnWO4 rods exhibited
comparatively lower activity than the Ag/ZnWO4 sample. By increasing the number of Ag domains (∼3 nm) on the nanorods
the activity was shown to increase (Fig. 6.15(e)). The catalytic activity had a strong dependence on the high surface area,
crystallinity and aspect ratio of nanorods.
Metal-doped semiconductor nanorods have also been tested. For example Zhu et al. [544] exploited a thin film of Crdoped TiO2 nanorods in Na2 SO4 electrolyte solution for photoelectrochemical cells. The Cr-doped TiO2 nanorods showed
high photocatalytic activity for the water-splitting reaction (H2O ↔ H2 + O2 ) (Fig. 6.16(a)–(d)). Here, another favorable
role of the nanorod morphology was that it increased the electron transport of the electrodes. The morphology of the samples
after many cycles of photocatalytic reaction was found to be preserved, which is important for maintaining a well-defined
electrode/electrolyte interface over cycling.
Sun et al. [545] synthesized Sb2 S3 semiconductor nanorods (band gap 1.64 eV) with diameters around 50–100 nm and
lengths around 100–1000 nm, which possessed photocatalytic activity under visible light, see Fig. 6.17(a)–(f). In the presence
of nanorods, under visible light irradiation the absorption peak at λ = 464 nm for methyl orange (MO) was found to decrease
over 30 min at a rate faster than when using reference catalysts such as TiO2 –N2 or CdS. Also, the photocatalytic stability test
showed that the nanorods retained their crystal structures, as XRD peak positions and relative intensities were identified
as those of the pristine samples. However, loss of activity was observed after the first cycle. The conversion ratio from MO
photodegradation using these nanorods was estimated to be up to 97%.
Recently, Amirav et al. [546] have reported a significantly high photocatalytic activity of metal/semiconductor nanorod
heterostructures. The starting materials were CdS nanorods containing a spherical CdSe seed (see also Sections 2 and 3 of
this review). Pt nanoparticles were then grown selectively at the tips of the nanorods (Fig. 6.18a–h). As a comparison, the
authors synthesized also ‘‘CdS-only’’ nanorods (i.e. not carrying a CdSe seed buried inside) and grew Pt domains at their tip
regions. The CdSe/CdS-Pt nanorods had a higher catalytic activity than the ‘‘CdS-only’’–Pt nanorods, due to the more efficient
charge separation in the former rods: upon photoexcitation, holes were localized quickly in the CdSe region, while electrons
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193
e)
Fig. 6.10. Top panel: Lattice micro-strain (ϵ ) of different crystal planes of ceria and gold/ceria nanorods, cubes and polyhedra; (a)–(c) TEM and (d) HRTEM
images of gold nanoparticles on ceria (a) nanorods (b, d) cubes and (c) polyhedral; (e) Water gas shift reaction (WGS) light-off profiles for 1% gold on ceria
(•) nanorods () cubes and (N) polyhedra.
Source: Taken with permission from Ref. [535].
Fig. 6.11. (a) H2 and CO2 relative pressure during the water gas shift reaction (WGS) on CeO2 nanorods; (b) TEM image of the samples after the WGS
reaction; (c) Variation of the CeO2 lattice parameters during the temperature ramping for Pd loaded CeO2 nanorods and bulk CeO2 .
Source: Reprinted with permission from Ref. [538].
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Fig. 6.12. Top panel: Surface area, porous volume, crystal size and aspect ratio of the MnO2 nanorod samples before and after CO oxidation; (a)–(d)
TEM images of the MnO2 nanorods: α -MnO2 , β -MnO2 , γ -MnO2 and δ -MnO2 respectively, with insets showing selected area electron diffraction patterns
of nanorods; (e) CO oxidation activities of MnO2 nanorods under 2 vol.% CO, 98 vol.% air and flow rate of 70 mL min−1 ; (f) T10 , T50 , T90 , T100 are the
temperatures of CO conversion 10%, 50%, 90% and 100%, respectively.
Source: Taken with permission from Ref. [539].
Fig. 6.13. (a) Cross-sectional SEM image of vertically aligned ZnO nanorods; (b) TEM image of a single ZnO nanorod showing stacking faults; (c) HRTEM
image of ZnO nanorods; (d) TEM image of ZnO NR@ Cu nanoparticles; (e) HRTEM image of Cu nanoparticles on the surface of a single ZnO nanorod;
(f) Histogram of Cu nanoparticles from TEM analysis. The catalytic activity of • nanorods vs. commercial catalysts; (g) Methanol reforming profiles;
(h) Hydrogen production rate as a function of the temperature; (i) Kinetic constant as a function of the temperature; (j) Arrhenius plot for the methanol
reforming reaction.
Source: Taken with permission from Ref. [541].
were localized in the Pt domains, hence far from the holes. This type of charge separation was not possible for the ‘‘CdSonly’’–Pt nanorods. The hydrogen evolution rate on the nanorods surface was shown to be a function of the light intensity.
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Fig. 6.14. (a) HRTEM image of a single CdSe–Pt hybrid nanostructure grown at pH 10. Inset: fast Fourier transform image showing the (111) reflexes of the
Pt nanocrystals; (b) Scheme of a light induced charge separation mechanism in a CdSe–Pt nanorod; (c) Time trace of the normalized concentration of the
MB dye in a sol of CdSe–Pt nanonets (dashed lines) and nanorods (solid lines), measured at pH 7; (d) Sequential photocatalysis experiments with CdSe–Pt
nanonets showing the catalytic activity during three injections of MB at various times.
Source: Taken with permission from Ref. [542].
1.0
0.9
0.8
C/C0
0.7
0.6
0.5
0.4
0.3
0.2
0
20
60
40
Time (min)
80
100
Fig. 6.15. TEM images of (a) ZnWO4 nanorods; (b) Ag 4% ZnWO4 nanorods; (c–d) HRTEM image of ZnWO4 nanorods; (e) Degradation of rhodamine-B
(RhB) during irradiation under UV light (365 nm). The profiles show RhB concentration changes.
Source: Taken with permission from Ref. [543].
Interestingly, the longer nanorods (∼70 nm) were 4 times more photocatalytically active for hydrogen evolution than the
27 nm long nanorods. Nanorod samples of various other lengths were tested, and this trend was confirmed. Therefore, apart
from the enhanced charge separation (in case of CdSe/CdS-Pt nanorods), it is also important to note that the number of
active sites suitable for adsorption of the reactive species increased in the longer nanorods, which also helped to enhance
the activity of the catalysts.
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Current Density/(A/cm2)
8.0 × 10-4
6.0 × 10-4
4.0 × 10-4
2.0 × 10-4
0.0
20
40
60
80
100
120
Time/s
2.5× 10-4
Current Density/(A/cm2)
2.0 × 10-4
1.5× 10-4
1.0× 10-4
5.0× 10-4
0.0
0.0
0.5
1.0
1.5
Potential/V
Fig. 6.16. TEM images of (a) undoped TiO2 nanorods; (b) TiO2 nanorods doped with 3 mol% Cr. (c) Photocurrent response curves for the sample at different
applied electrode potentials in 0.5 M Na2 SO4 under Hg lamp pulsed irradiation (∼8 mW/cm2 ) [(a) commercial anatase; (b) undoped TiO2 ; (c) 3 mol Cr%
doped TiO2 ; (d) commercial TiO2 1.5 V, (e) undoped TiO2 1.5 V, (f) 3 mol% Cr-doped TiO2 1.5 V] and (d) variation of photocurrent density vs. applied
electrode potential for the samples in 0.5 M N2 SO4 , under Hg lamp irradiation (∼8 mW/cm2 ): [(a) commercial anatase, (b) undoped TiO2 , (c) 3 mol Cr%
doped].
Source: Taken with permission from Ref. [544].
7. Mechanical properties of elongated nanoparticles and melting studies
7.1. Introduction
The synthesis of highly performing materials, the constant miniaturization of micro-electromechanical devices (MEMS),
the fabrication of thin films for the micro- and nano-electronic industry, the development of tribological applications,
the deposition of hard coatings on cutting tools, have all raised questions related to the mechanical behavior (and its
predictability) at the micro- and nanoscale, both in ‘‘bulk’’ nanostructured materials (i.e. polycrystalline materials composed
of grains in the nanosize range), and in individual nanocrystals. The study of mechanical properties in materials of
physical geometries in the micron and sub-micron (i.e. nanometer) size range is still in its infancy. ‘‘Bulk’’ nanostructured
materials are known to possess increased strength and the early experiments already carried out several decades ago
on tensile testing of ‘‘single crystalline’’ metal whiskers with micrometer transverse sides have evidenced strengths
much higher than the bulk value [57]. The increased strength of single nanocrystals could be useful for applications
of these materials as active probes in nanoindentation, scanning probe microscopy and field emission [58–60], among
others.
In the last decade, pure metals and alloys with at least one dimension in the micro- and nanoscale range have been
investigated, thanks to advances in the fabrication of new generations of samples suitable for mechanical testing (for
example micropillars prepared by focused ion beam) and in various techniques for studying their stress and deformation
properties, under both tensile and compressive deformation, the latter for instance via micro/nanoindentation [547]. All
these studies have revealed a marked deviation in the mechanical properties of samples from bulk-like behavior already
when their size is of the order of a few micrometers, which is comparable to the length scale of many plasticity mechanisms
based on dislocation nucleation and propagation (see next sections).
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Fig. 6.17. (a) TEM image of Sb2 S3 nanorods; (b) HRTEM image of a Sb2 S3 nanorod; (c) Photocatalytic properties of different catalysts: [ Sb2 S3 nanorods
under visible light, • Sb2 S3 nanorods in the dark; N CdS under visible light; H P25-N under visible light; P25-N under UV light]; (d) Cyclic runs in the
photodegradation of MO in the presence of Sb2 S3 nanorods under visible light irradiation; (e) TEM image and (f) HRTEM image of the Sb2 S3 nanorods after
the catalytic reaction.
Source: Taken with permission from Ref. [545].
e
Normalized H2 Production Rate
Relative QE for H2 Production
a
60nm
20nm
27nm
40nm
70nm
f
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0
60nm
200
400
600
800
Time [min]
Absorbance
Wavelength [nm]
H2 [µmol/h]
h
Apparent Quantum Yield
g
Normalized Power
Fig. 6.18. (a) A sketch of a CdSe/CdS-Pt nanorod, with energy band diagram indicating that holes are confined to the CdSe region, while the electrons
are transferred to Pt nanoparticles; (b) HRTEM (left) and HAADF images (right) of a CdSe/CdS-Pt nanorod; (c–d) TEM images of Pt-tipped of two samples
of CdSe/CdS-Pt nanorods, with lengths equal to 27 nm and 70 nm, respectively; (e) Relative quantum efficiency of hydrogen production; (f) Hydrogen
production rate from CdSe/CdS-Pt nanorods and from ‘‘CdS-only’’–Pt nanorods; (g) Influence of the incident light wavelength on the quantum yield for
hydrogen production; (h) Influence of light intensity (white light 0–1.3 W) on the hydrogen production rate.
Source: Taken with permission from Ref. [546].
In the following, after a brief introduction on mechanical properties in general and on the features of ‘‘bulk’’
nanostructured materials (henceforth referred to as ‘‘nanocrystalline’’ materials) we will focus our attention on single microand nanocrystals. In the discussion, there will be no particular reason why we will take into consideration only elongated
micro- and nanoparticles (i.e. whiskers and pillars), except for the fact that these shapes are those that have facilitated
mechanical testing up to now.
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Fig. 7.1. (a) A cylinder-shaped solid under tensile stress and (b) under shear stress deforms by an amount ∆x; (c) A typical stress–strain curve, in present
case is referred to a steel specimen.
7.2. Mechanical properties of materials: a few useful definitions
In this section we will recall a few definitions pertinent to the mechanical properties of materials that will be used in the
other sections. For a more scholarly (and comprehensive) coverage of the topic, the reader is referred to various textbooks
(for example [548–551]).
All solids are elastic up to some extent. It is possible in other words to identify a set of conditions under which a
force is applied to a rigid body, such that the force will cause a slight change in the dimension/shape of the body, due
to its twisting, compressing, or pulling, but then the body will return to its initial state once the deforming force is
removed. The deforming force (F ) per unit area (A) is called stress, or equivalently stated F is the magnitude of the force
applied perpendicular to the area A on the solid object. Hence stress produces strain on a physical body. Fig. 7.1(a) for
example shows a cylinder undergoing a tensile stress. Here the strain produced due to tensile stress is a dimensionless
quantity (∆l/l). The opposite situation, i.e. the cylinder being compressed uniaxially, is referred to as of ‘‘compressive
stress’’.
Under the conditions of perfect elastic behavior of a body, stress and strain are proportional to each other and the
proportionality constant is called modulus of elasticity or Young’s modulus (E ). The Young’s modulus of an isotropic elastic
material is a parameter that is related to its stiffness, and is defined as:
F
A
=E
∆l
l
.
(7.1)
In case of shear stress (G) the force vector is applied parallel to one face of the material, rather than perpendicular to
it (in Fig. 7.1(b) the force is applied parallel to the basal planes of the cylinder). Here the corresponding modulus is called
shear modulus and is written as:
F
A
=G
∆x
l
(7.2)
where ∆x/l is the dimensionless strain. Shear stress plays for example a crucial role in the buckling of shafts that rotate
under a given load.
A stress–strain curve of a material is generated by applying an increasing stress to a sample and by measuring its strain.
An example of stress–strain curve is shown in Fig. 7.1(c). The stress–strain curve is generally composed of two regions: a
region of elastic behavior, in which, as said before, the material returns to its initial states once the stress is removed, and a
region of plastic behavior, in which the deformation in the material is irreversible. The value of the stress corresponding to
the change from elastic to plastic behavior is defined instead as the ‘‘yield stress’’: it is the stress past which the crystal lattice
of the material becomes unstable and does not respond elastically any more. The corresponding point in the stress–strain
curve is called the ‘‘yield point’’. The ideal strength of a material is defined as the critical stress at which a perfect, defectfree crystal at 0 K yields under uniform deformation [552]. The ‘‘ideal’’ strength sets an upper limit to the strength that the
material can attain under given loading conditions. A real bulk material yields at a stress that is below its ideal strength, due
to the presence of lattice defects such as dislocations and grain boundaries.
Work hardening, also known as ‘‘strain hardening’’, is referred to when a metal, at a temperature well below its melting
point, is strained past the yield point. What is commonly observed is that an increased stress is needed, past the yield
point, to produce a further plastic deformation. Hence the material is hardened, or in other words it becomes more
difficult to deform. The behavior can be seen in the plastic region of stress–strain curve of Fig. 7.1(c): the curve, past
R. Krahne et al. / Physics Reports 501 (2011) 75–221
199
Fig. 7.2. Definition of a Burgers vector. A ‘‘Burgers circuit’’ is drawn in a real crystal (a) and in a perfect crystal (b). In both cases the loop spans the same
number of lattice vectors. The vector that is required to close the loop in the ideal crystal is the Burgers vector.
the yield point has a different slope than in the elastic region, but such slope is generally greater than zero (as would
be for a perfectly plastic material), and is known as the ‘‘work (or strain) hardening coefficient’’. Work hardening can
be explained by considering that a plastic deformation is associated with the movement and multiplication of a large
number of dislocations in the crystal. During plastic deformation, the mutual distance between dislocations decreases and
they start blocking the movement of each other, i.e. they start forming dislocation junctions (a mechanism also known as
‘‘dislocation forest hardening’’), and additionally they start piling up at grain boundaries. Past this point, a higher stress is
required to make the dislocations move further, i.e. to induce a further plastic deformation, up to the ultimate strength,
which is the point of maximum stress in the curve. The shape of the stress–strain curve in the region of plastic behavior
region is strongly dependent on the material but also on the conditions under which the test is carried out. For example,
the ultimate strength might not correspond to the strength at which the sample ruptures. Often, stress–strain curves are
plotted in terms of engineering stress and engineering strain, meaning that the stress and strain are calculated based on the
original dimensions of the sample and not based on their values during the measurement. As another useful definition, the
ductility of a material can be defined as its ability to undergo a plastic deformation (usually under tensile stress) without
fracturing.
It is useful, for the discussion that will follow, to define the concept of dislocation–displacement vector, also known as
Burgers vector, associated to a dislocation [551]. To identify this vector in a simple cubic lattice, we draw a closed loop in
the crystal once a dislocation has been inserted, making sure that the loops encircles the region where the dislocation line
is present and where the displacement from the ideal reference crystal is large (Fig. 7.2(a)). We then record the number of
lattice vectors travelled on each side of the loop. Finally, we trace the same loop in a real crystal, made by the same number
of lattice vectors as before. The loop this time will not be closed (Fig. 7.2(b)). The vector that is required to close the loop
is the Burgers vector. In practice, the Burgers vector identifies the direction and the magnitude of the lattice dislocation
introduced by the movement of a dislocation. Depending on the type of dislocation, the Burgers vector is parallel (i.e. for a
screw dislocation) or perpendicular (i.e. for an edge dislocation) to the dislocation line. For a detailed discussion on Burgers
vectors, see for example Ref. [551].
As said before, the motion of dislocations is what produces plastic deformation in a crystal. This plastic deformation is
manifested itself by a ‘‘slip’’, i.e. crystal planes glide along each other. Slip in a crystal in general occurs along some specific
planes, those that contain the shortest Burgers vectors. For face-centered-cubic (fcc) metals, these are the 111 planes (i.e. the
close-packed ones) and the directions of slip are any of the ⟨110⟩ ones. For body-centered-cubic (bcc) metals instead, since
there are no close-packed planes, any slip along a specific plane type and direction type requires an activation energy. As we
will see later, this difference between fcc and bcc metals translates into a mechanical behavior that is completely different
for the two types of lattices, also at the nanoscale.
7.3. Studies on elastic properties of nanorods and nanowires
Several methods have been proposed to determine the Young’s modulus and more in general to study the
elastic properties of nanorods, nanowires and nanotubes, among them AFM bending experiments for metal nanowires
[553,554] time-resolved spectroscopy approaches for nanorods [555,556] (see also Section 4), and thermal vibrations using
TEM [557,558], often coupled with theoretical calculations. In an AFM bending experiment, the force mapping function of the
instrument is used to perform spatially resolved deformations over the full length of a suspended one-dimensional sample,
while in time-resolved spectroscopy studies an ultrafast pump laser is used in order to heat the particles and a tunable pulse
is used as a probe. Heating by laser irradiation coherently excites the phonon modes of the particles, which correlate with
the expansion coordinates of the particle [524] (see also Section 4). One has to keep in mind however that both classes of
experiments give results that are affected by fairly large uncertainties, due both to intrinsic limitations of the techniques
and to the heterogeneity of the samples.
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Fig. 7.3. Measured values of Young’s modulus (as extracted from optical experiments) for different samples of Au nanorods grown along the [100] direction
are plotted against the surface to volume ratio of the nanorods (calculated from the average dimensions determined by TEM). The solid horizontal line is
the bulk value of E[100] and the dashed line is the average value determined from the experimental data. The open symbols correspond to samples prepared
via photochemical reduction samples, whereas the closed symbols correspond to the samples prepared via chemical reduction.
Source: Taken with permission from Ref. [524].
Not always however the measured value of Young’s modulus of the nanowires/nanorods is found to be larger than
the bulk value. As an example, samples of gold nanorods elongated along their [100] and [110] crystallographic direction
exhibited a decrease (∼20%–30%) in the value of Young’s modulus with respect to the bulk E[100] and E[110] values
[292,524,559]. Measured values of Young’s modulus for the different Au nanorod samples (grown along the [100] direction)
plotted against the surface to volume ratio are shown in Fig. 7.3 [524]. Similar results were observed by Hu et al. in the
case of gold nanorods synthesized by wet-chemical approaches [559]. This observation was generally explained as due to
the high surface to volume ratio of the nanomaterials, since the ratio has a significant influence on their physical properties.
Also, as discussed in Section 4, metal nanorods in some cases are made of multiple twins sharing an edge which corresponds
to the elongation direction of the rods. Since such peculiar structure is not found in the corresponding bulk solid, the Young
modulus along the elongation direction could differ from that of the bulk.
7.4. Plastic behavior of nanocrystalline materials
Various studies have tried to extract an empirical scaling law effect relating the mechanical properties of a material with a
length scale in the micro/nanometer range. As already mentioned before, nanocrystalline materials exhibit increased yield
strength compared to conventional materials, in some cases even approaching the theoretical value [560]. One generally
accepted equation that describes this behavior is the so-called ‘‘modified Hall–Petch relation’’ [561] which correlates
the yield strength σy of the nano/microstructure to its characteristic length scale d (for example the grain size in a
microcrystalline material):
σy = σ0 + kdn .
(7.3)
In the expression above σ0 is the bulk yield strength and the terms k and n are materials-related constants. For metals n
is usually found to be around −0.5, but in a more general expression this exponent can vary considerably (i.e. from −0.5
to −1). One needs to stress that the above expression is an empirical one, hence it does not reflect a universal mechanism.
Indeed various physical mechanisms are involved in the plastic deformation of nanomaterials, and the relevance of each
mechanism is dependent on a series of parameters (for example the specific material, the aspect ratio of the sample, its
initial defect structure and even the type of mechanical test that is performed).
A major cause of plastic deformation in materials is the nucleation, propagation and multiplication of dislocations.
Therefore, one commonly accepted reason why in nanocrystalline materials an increase in strength is observed at decreasing
grain size is that dislocations cannot propagate freely throughout the material, due to the presence of a large number of
two-dimensional obstacles, such as grain and interphase boundaries and stacking faults [560]. Stated simply, the smaller
the size of the domains, the more obstacles dislocations encounter during their propagation, hence they more likely undergo
piling up at such obstacles before they can multiply (Fig. 7.4(a)). Here, accumulation of stress builds up until dislocations
are nucleated at the other side of the boundary, thus releasing stress at the boundary.
The same mechanism that obstructs dislocation motion reduces however the ductility of nanomaterials. Research in
this area is very active, since it would be ideal to have nanocomposites for structural applications in which both strength
and ductility are improved. Recent studies in the field have shown that carefully engineered coherent internal boundaries in
nanoscale materials could achieve this goal [562]. As said before, common strengthening strategies rely on grain boundaries.
These however are incoherent, i.e. there are no defined crystallographic relationships between the grains at the boundary,
where voids and broken bonds are present. Grain boundaries have therefore high excess energy, and high stresses can deform
R. Krahne et al. / Physics Reports 501 (2011) 75–221
a
GB strengthening
b
Nanoscale TB strengthening
201
Fig. 7.4. (a) The propagation of a dislocation stops at a grain boundary. A grain boundary is however an incoherent, high energy interface; (b) When a twin
plane is present, the dislocation can glide along the twin plane, which helps improving ductility of the material.
Source: Taken with permission from Ref. [562].
and rupture the interface between the grains, so that the grains have a limited ability to accommodate dislocations before
rupturing. Coherent internal boundaries on the other hand have lower excess energy and among them twin boundaries
have high thermal and mechanical stability. Twin boundaries offer an additional pathway for dislocations to propagate:
apart from propagation in neighboring domains, i.e. across the twin boundary, dislocations can glide along a twin boundary.
This is possible because of the coherent interface between the two domains along the boundary, which makes it possible
for the material to undergo locally substantial shear deformation along the twin boundary, without rupturing (Fig. 7.4(b)).
By such a mechanism, both higher strength and ductility can be achieved. Studies have revealed that higher strength and
ductility are achieved at smaller twin thickness λ [562], with λ in the nanometer range.
7.5. Experiments on plastic behavior of single micro- and nanocrystalline samples
7.5.1. Metallic whiskers
It is known since many decades that micron and sub-micron whiskers of pure metals, which are fibrous (i.e. ‘‘elongated’’)
single crystals with a high crystallinity, have tensile strengths that are much higher than the bulk value. Cu whiskers for
instance have shown significantly high yield strengths, of the order of a few GPa [563]. Tensile tests of dislocation and microcrack free ⟨110⟩-oriented single crystalline elongated molybdenum nanoparticles, with diameters ranging from 25 to 90 nm
(at the apex region) were conducted inside a field-ion microscope (FIM) [564]. The experimental values of tensile strength
varied between 6.3 and 19.8 GPa and increased with decreasing nanorod diameter. Also SiC nano-whiskers are known
to have strengths considerably greater than those observed in corresponding bulk single crystals, actually approaching
theoretical values [565,566], and the greatest strengths were found for whiskers that fractured at the pinning site where the
strain was the greatest, whereas lower values were found for SiC whiskers that fractured at some distance from the pinning
site [566]. These observations suggested that defects can limit the strengths of the whiskers.
Thanks to recent advances in synthesis, manipulation and testing of materials, high-quality whiskers with submicrometer diameters have been fabricated and their mechanical behavior has been studied under tensile stress, evidencing
yield stresses close to the theoretical value. As an example, single crystalline Cu whiskers, fabricated by physical vapour
deposition, of several micrometers in length and diameters below 300 nm, were tested using a tensile stage mounted in
a scanning electron microscope [567]. The whiskers exhibited an elastic behavior up to very large strain (2%) and a yield
strength as high as 6 GPa, after which either sudden failure or ‘‘strain bursts’’ were observed (i.e. large increases in strain at
almost constant stress) [567]. Although no clear trend with size could be identified, due to the somehow stochastic behavior
of the samples, samples with the smallest diameters exhibited higher yield stress, close to the theoretical one (Fig. 7.5). Many
other examples exist in the literature of a similar behavior.
A simple way of explaining the behavior of whiskers is by considering that they are generally defect-free before loading,
hence a higher stress is required than in the corresponding bulk materials to reach the onset of plastic deformation, because
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Fig. 7.5. Stress–strain curves from tensile testing of Cu whiskers (single crystals) with several micrometers in length and diameters below 300 nm.
Source: Taken with permission from Ref. [567].
defects need to be generated in them. However, this does not set a basis for rationalizing and quantifying the observed trend
in increasing yield strength with decreasing whisker size.
A relation linking yield strength and size can be found via Weibull statistics, which has been used to analyze fracture
data in solids, in particular to correlate the probability of fracture as a function of the applied stress [568]. Weibull statistics
correlates the number of critical defects in a solid to its volume or its surface area. Simply stated, the smaller the volume V
of the sample or of its surface area A, the lower the probability of critical defects, the higher will be its yield strength σy :
σy = σ0
V0
V
1/m
or equivalently
σy = σ0
A0
A
1/m
.
(7.4)
In the expression above m is the Weibull parameter, which is an index of the dispersion of the distribution and is material
dependent, while σ0 is the yield strength for a reference volume V0 or a reference surface A0 . The Weibull statistics however
cannot be properly used on nanoscale materials like whiskers, because these are nearly defect-free or have only a small
number of critical defects. Indeed, when applied to the analysis of fracture strengths of various nanostructures, it shows a
poor correlation coefficient. A modified version of the Weibull statistics has been proposed by Pugno and Ruoff in 2004 [569]
and later termed as ‘‘nanoscale Weibull statistics’’ [570]. The new distribution correlated the yield strength not to the volume
or to the surface area of the nanostructure, but to the number of critical defects, and introduced the concept of a ‘‘fracture
quantum’’, namely the possibility that even a very small defect could cause the failure of a nearly defect-free structure. This
new statistics provided a better correlation with experimental data [570] and was applied to a variety of nanostructures,
including for example carbon nanofibers and WS2 nanotubes [571].
7.5.2. Pillars
In recent years, several reports on deformation of small single crystalline pillars (both micron- and sub-micron-sized),
prepared by focused ion beam, have appeared. Also in these studies an increased strength with decreasing size was observed,
and stresses corresponding to a significant fraction of the theoretical strength were recorded in both uniaxial tension
and compression tests of nano/micropillars [58,572–575]. These pillars however were different from the defect-free metal
whiskers discussed earlier, as they were characterized by the presence of dislocations already before the loading, therefore
they were not defect-free. On the other hand they were single crystalline, therefore the increase in strength in these samples
could not be explained by the presence of obstacles (like grain boundaries) to the propagation of dislocations, as for the
polycrystalline materials discussed earlier. Therefore, a major conclusion of these works was that the decrease in sample size
must somehow start affecting the mechanism connected with the generation, multiplication and accumulation of defects.
In this respect it is worth recalling here some of the results of Uchic et al. [572], for example those on pure Ni and Ni3 Al–Ta
micropillars. The authors prepared micropillars of various diameters, from 40 µm down to 500 nm, by focused ion beam,
and tested them under compressive stress using a conventional nanoindenter equipped with a flat-punch indentation tip.
For Ni, the 40 and 20 µm diameter samples behaved more or less like the bulk in terms of stress–strain curves, i.e. similar
yield strength as for the bulk, a smooth transition from elastic to plastic behavior, and a steady state work hardening (see
Fig. 7.6, top panel). SEM imaging after the loading tests evidenced the presence of slips bands, coherent with a deformation
of the pillars along specific sets of planes. Pillars of 5 µm or below had a markedly different behavior, namely they were
characterized by a much higher yield stress, then past the yield point they exhibited ‘‘strain bursts’’: the strain in the samples
rapidly increased to values up to 19%, so that large flat regions appeared in the stress–strain curve. Equivalently stated, a
low work hardening rate was seen. Both the value of yield stress and of the strain burst depended on the sample (several
5 µm pillars were tested, all of them yielding at higher stress than the bulk, but each of them yielding at a different value and
R. Krahne et al. / Physics Reports 501 (2011) 75–221
203
Fig. 7.6. Top panel: (A) Stress–strain curves at room temperature for Ni micropillars with ⟨134⟩ crystallographic orientation and with various diameters
(from 4 to 40 µm) and for a bulk crystal sample. (B) is a scanning electron microscopy image of a single pillar (20 µm diameter) before mechanical testing.
The same sample is imaged in (C) after testing. This sample underwent a fast 19% ‘‘strain burst’’ during testing. Large single-slip plane displacements can be
seen in this sample. Bottom panel: (A) Stress–strain curves at room temperature for Ni3 Al–Ta alloy micropillars with ⟨123⟩ crystallographic orientation and
with various diameters (from 0.5 to 20 µm) and for a bulk crystal sample. (B) is a scanning electron microscopy image of a single pillar (20 µm diameter)
after mechanical testing. This sample underwent a fast 10% ‘‘strain burst’’ during testing. (C) This is another pillar, of 1 µm diameter which was completely
sheared off during testing as a consequence of a large strain burst.
Source: All panels have been adapted with permission from Ref. [572].
exhibiting a varying extent of strain burst). The overall behavior of these samples was therefore different from both larger
samples (i.e. bulk-like) and from defect-free whiskers: while their yield strength were not in the high range of defect-free
whiskers, they nonetheless were characterized by higher yield strength than the bulk, and also much lower work hardening
rate than in the bulk. For Ni3 Al–Ta samples, the transition from bulk-like behavior was seen already for samples of 20 µm
size (Fig. 7.6, bottom panel), so that the influence of sample size was observed at comparatively larger sizes than for Ni
pillars.
Strain burst in these samples, as well as in the whiskers discussed earlier, are symptomatic of a single catastrophic event
that sets the transition from the elastic to the plastic regime. They suggest a mechanism of plastic deformation that does not
proceed through multiplication and accumulation of dislocations, which would show up in an appreciable work hardening
rate, most likely because these dislocation-related processes are inhibited by a size effect.
7.6. Models of plastic behavior of single micro- and nanocrystalline samples
7.6.1. Strain gradients
The outcome of experiments like the ones described above has started several studies aiming at elucidating the
mechanical behavior in confined systems and numerous computational models have been developed which go beyond the
more traditional continuum models [547,574,576]. One critical rethinking on plasticity in confined dimensions concerns the
common belief about the presence of strain gradients as the major reason of strain hardening in micro- and nanomaterials.
Indeed, most mechanical testing on these materials in the past had consisted of nanoindentation experiments, in which
the plastic deformation was confined in a small region of a given sample. A relationship was found experimentally
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which correlated the square of the measured hardness of the material to the indentation depth, and is known as the
‘‘indentation size effect’’ [577]. This behavior was explained in the following way: especially when probing shallow depths
with indentation, non-uniform stresses and strains are generated in the material, while much smaller stress and strain
gradients are present in the material when the indenter penetrates at greater depths. Such gradients generate so-called
‘‘geometrically necessary’’ dislocations which contribute to strain hardening of the material. A model was developed then
that tried to correlate strain gradients with the indentation strain effect [578]. This model however suffers from the major
drawback that it is not applicable to indentations depths shallower than 100 nm (where continuum mechanical models
are not appropriate). Also, more recent experiments, such as the compression tests on nanopillars discussed earlier, have
involved the uniaxial compression of samples in a way that no strain gradients were generated inside the pillars [573]. These
experiments have revealed that pillars with smaller diameters were characterized by larger compressive stresses. Therefore,
additional mechanisms should be operative in the plastic deformation of nanomaterials.
7.6.2. Dislocation starvation
A simple mechanism that explains the increase in strain in single crystal micro- and nanocrystalline samples is the socalled ‘‘dislocation starvation’’ [573]. In general, dislocation motion leads at some point to dislocation multiplication. This
process is characterized by a length scale δ , namely a dislocation can travel on average up to a distance δ before it can
replicate itself. If however such length scale is comparable or smaller than the size of the crystal, the dislocation reaches
the surface of the crystal, where it annihilates, before actually replicating itself. In this crystal, as soon as the ‘‘dislocation
starvation’’ condition is met, higher stresses would be required in order to generate dislocations with smaller critical lengths,
hence the yield strength of the material is increased. Length scales associated with dislocation multiplication are known for
several materials (for example in silver such length scale is around 1 µm) [579], and they set a critical size for the physical
dimension of a material below which it can exhibit a plastic behavior that is different from the bulk. The mechanism of
dislocation starvation has been proved to be operative for example in in situ analysis of Ni micropillars under compressive
strain [580].
We must recall that sub-micrometer crystals with diameters of a few hundreds of nanometers (<500 nm) might be
dislocation-free, because their overall volume can be so small that even with a large density of dislocations (1012 /m2 ) no
dislocation is found in them. In these cases, the critical stress required to initiate the plastic behavior is the one needed to
nucleate a dislocation from the surface. For such samples with ‘‘small’’ diameters the quality of the surface plays a key role.
Nanopillars, which are produced by focused ion beam, have a rather defective surface and in addition their basal plane is
still connected to the bulk. Whiskers and nanowires, which are prepared by other approaches, on the other hand tend to
have smoother surfaces and they shown indeed higher strengths [581,582] than pillars of a comparable size.
7.6.3. Source truncation hardening
More recent studies have challenged the simple model of dislocation starvation [574,576,582]. The above model assumes
that the initial density of dislocations in a material is independent of its size, and additionally that the number of dislocations
in a deformed sample is equal to or even less than that in the initial non-deformed sample. Recent reports, for example
on Nickel pillars with diameters in the 1–10 µm range [574], have shown that instead the density of dislocations in the
deformed sample could even increase as the dimensions of the sample decrease, and that dislocation density can be higher
than the initial density, thus in contradiction with the dislocation starvation mechanism. Therefore, more elaborate models
of plasticity in confined dimensions have been proposed. Among them it is worth citing the work of Parthasarathy [582],
which is in reality a further elaboration of the dislocation starvation mechanism, but which introduced the so-called ‘‘source
truncation hardening’’ mechanism for rationalizing plastic behavior in confined dimensions. The model correlated the
limited volume of a sample, the distribution of dislocation source lengths and the truncations of dislocation sources at the
surface to the increase in strength in microcrystals and sub-micrometer crystals.
In order to understand this mechanism, it is important to introduce a few concepts. First of all, we need to define the
‘‘critical resolved shear stress’’ (CRSS), which is the stress required for the first percolation event of a dislocation through
the whole sample. In a bulk sample this can be correlated to the stress required to initiate and maintain the processes
of dislocation nucleation and multiplication, keeping in mind that these occur in the presence of a dislocation forest. An
additional important concept is the image stress. In an infinite, elastic medium, a dislocation generates a stress (tensor) field
σij∞ . In a finite medium, for example a thin film, if one assumes that dislocations generate the same field as in a bulk infinite
solid, traction forces then develop at the surface of the film, due to interaction of dislocations with the surface (Fig. 7.7(a)).
img
In order to cancel such traction forces, an image stress field σij needs to be invoked (Fig. 7.7(b)). Therefore, the stress field
σijtot of a finite, traction-free solid can be considered as the superposition of a bulk stress field σij∞ (i.e. like the one generated
img
if dislocations would propagate to infinity beyond the boundaries of the solid) and of an image stress field σij
σijtot = σij∞ + σijimg .
(Fig. 7.7(c)):
(7.5)
There are different approaches to calculate such image stress fields [583]. Image stresses, by exerting a force on
dislocations, can modify the orientation of dislocations with respect to the surface.
R. Krahne et al. / Physics Reports 501 (2011) 75–221
205
Fig. 7.7. (a) Traction forces developing at the surface of a film when assuming that the dislocations in the film generate the same stress field σij∞ as in a
img
bulk infinite solid; (b) Such traction forces can be cancelled by invoking an image stress field σij
; (c) The sum of these two fields yields the stress field
tot
ij
inside the film.
Source: Taken with permission from Ref. [583].
σ
Fig. 7.8. (a) Operation of a Frank–Read source of dislocation. It starts as a straight dislocation in a slip plane. The two ends of the dislocation, named as ‘‘A’’
and ‘‘B’’, stay pinned. As soon as a shear stress is applied on the slip plane, the dislocation lengthens and curves to an arc, while its ends stay pinned. The
dislocation bends further and then separates into a loop encircling the pins and into a new dislocation joining the pinning points. While the loop expands
further, a new loop is being formed; (b) In a sample of finite dimensions, the dislocation separates into two ‘‘arms’’ before it can form a loop, i.e. the source
of dislocation is ‘‘truncated’’. The original double ended source then splits into two single-ended sources. The longer arm, i.e. the one whose pin on average
has the longer distance from the free surface (λmax in the figure) is the one that determines the critical resolved shear stress; (c) Single-ended sources in a
family of slip planes (i.e. where the ‘‘slip’’ will occur during plastic deformation) in an elongated micro/nanoparticle.
Source: Panel (c) has been adapted with permission from Ref. [582].
We also need to describe the behavior of a typical source of dislocation, namely the Frank–Read source (see
Fig. 7.8(a)) [550,551], which is among the weakest sources of dislocation. This is a ‘‘double ended’’ source and starts as a
straight dislocation in a crystal slip plane with its two ends pinned [551]. The details of the mechanism are not relevant for
the present discussion, while it is sufficient to know that when a shear stress is applied on the slip plane, the dislocation
lengthens and curves to an arc, while its ends stay pinned (the two ends will be called the ‘‘pins’’ from now on). Eventually,
the dislocation bends further until it separates into a loop encircling the original pins and a new dislocation joining the
pinning points. The loop expands further, while a new loop is being formed, and so on (Fig. 7.8(a)).
The ‘‘operation’’ of such a source in a sample of confined dimensions is shown in Fig. 7.8(b), for a single source in a
slip plane, and in Fig. 7.8(c) for an elongated, cylinder-shaped particle. In a sample of finite dimension, the dislocation will
separate into two ‘‘arms’’ before being able to form a loop, because it will meet the surface and will be truncated there. There
will be now two single-ended sources, each with one end pinned. The interaction of each source with the surface is hard to
model, but most likely the image stress due to the surface will tend to rotate the dislocation line until it is perpendicular to
the surface (as also found for dislocations in free-standing films [583]).
The stress required to move each of these arms around its pinning point will depend on the shortest distance of its
respective pin from the surface. In particular, the longer arm is the one that will determine the CRSS, because less stress
will be required to move this arm. In a sample with a distribution of several single-ended (or equivalently, single-armed)
sources, the arm with the longest length will determine the CRSS. The model of Parthasarathy et al. [582] starts with a
cylinder of radius R and length L, by considering n pins (i.e. a n number of sources, located randomly in the sample) and
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Mean source length
Sandard deviation of
source length
a
# of sources
# of sources
CRSS, MPa
c
CRSS, MPa
b
Sample diameter, microns
Sample diameter, microns
CRSS, MPa
d
Sample diameter, microns
Fig. 7.9. (a) Mean source length and standard deviation, as a function of the number n of sources, using the model of Parthasarathy et al. [582], for a cylinder
of unit radius and with orientation of slip planes equal to 45°. All the other parameters used in model are referred to a fcc crystal, and the dislocation density
is taken as 1012 /m2 (as measured in nickel). Dots are the results of numerical simulations, lines are the analytical equations; (b) CSSR vs. sample diameter
for nickel micropillars. Blue squares represent experimental data from Refs. [572,584]. The continuous line reports the CRSS as calculated by the model
using the mean source length as extracted from the distribution, while the dotted lines report the corresponding CRSS calculated using the upper and lower
standard deviations from average source length, respectively. The horizontal dotted line is the maximum value of CSSR attainable according to the model;
(c) CSSR vs. sample diameter for gold micropillars. Blue and red symbols refer to experimental data from Refs. [573,585] while the various lines refer to
the model as described in panel (b). (d) This panel reports the same data as in panel (c), but in addition it displays data from gold nanowires (gray dots
interpolated with a black dotted line), taken from Ref. [553]. All panels have been adapted with permission from Ref. [582].
evaluates the distribution of these single-armed source lengths. The number of sources is proportional to the dislocation
density in the material (assumed as 1012 /m2 in nickel microcrystals [584]) and to the volume of the sample: the bigger the
volume, the higher the number of sources. The results of both analytical solutions and numerical simulations of their model,
for a cylinder of unit radius and with orientation of slip planes equal to 45°, are reported in Fig. 7.9(a). They show that the
mean source length decreases when decreasing the number of sources, hence with decreasing sample volume. A decrease in
source length corresponds to an increase in CRSS. The authors also compared their results with various sets of experimental
data, and found a good agreement (Fig. 7.9(b)). Therefore, this simple model which considered a statistical variation in the
source length of dislocations, due to the finite size of the crystal, could rationalize the increase in yield strength at decreasing
sample size.
More elaborate models, combining together some of the mechanisms described above, have been also proposed. Recently,
Norfleet et al. [574] combined traditional mechanisms of interactions of dislocations during work hardening (‘‘dislocation
forest hardening’’), with source truncation hardening. Another work, by Malygin [547], presented a theoretical analysis of
plasticity in microcrystals and nanocrystals that takes into account also the generation of dislocations from surface sources
and escape of dislocations from the surface. While generation of defects from surface leads to strain hardening, their escape
R. Krahne et al. / Physics Reports 501 (2011) 75–221
207
Fig. 7.10. (A)–(E) Mechanism of dislocation multiplication in a micropillar of a bcc metal. (G) Three-dimensional geometry of the dislocation.
Source: Taken with permission from Ref. [591].
from surface tends to counterbalance it, and equilibration between the two processes is predicted by the model to occur
at the early stages of plastic deformation, a mechanism termed as ‘‘exhaustion’’ of strain hardening, which indeed matches
with results from experiments. The detailed description of these models is however beyond the scope of this review, and
the interested reader is directed to those articles for more insights.
7.6.4. Surface-controlled dislocation multiplication in metallic BCC micro/nanopillars
The mechanisms of dislocation starvation and source truncation hardening have been proven able to explain the plastic
behavior of micro/nanopillars of various metals, especially the fcc ones. However, none of those mechanisms have in reality
clarified at a microscopic level how a single dislocation moves inside a micropillar once it is formed, or even why dislocations
junctions form. In order to answer all these questions, a more refined modeling is needed in addition to experiments. Several
numerical simulations of mechanical behavior of materials are now available, which, coupled with experimental results on
a wide variety of samples, are starting to shed light on the mechanisms of plasticity in crystals ranging from micron to submicron to nanometer scale [586]. Among them, discrete dislocation dynamics and its implementations (for example with
finite element approaches) clearly stands out [576,587–590].
A recent molecular dynamics study has shown for example that a different mechanism of plasticity can be operative in
fcc metal micropillars [591]. Various effects have been found to concur to this different behavior, namely image stresses
and the different structure of dislocations in bcc metals. In brief, simulations have found that a dislocation generated from
the surface of a micropillar, before exiting the pillar is able to generate one or more dislocations that move in a direction
opposite to it. Since this process can be repeated, a single dislocation can generate several dislocations, hence can cause a
considerable amount of plastic deformation, which is remarkably different from the case of fcc metal micropillars [591].
The mostrelevant
results of the simulations are shown in Fig. 7.10. A dislocation with Burgers vector [111] /2 is initially
present in a 011 glide plane (Fig. 7.10(A)). The dislocation quickly reorients with the Burgers vector and becomes a screw
dislocation (see Fig. 7.10(B)). It is also important to point out that any non-screw dislocation in a bcc crystal would have
instead moved fast and left already the crystal. While continuing to move, this dislocation generates a ‘‘cusp’’ (Fig. 7.10(C)),
which later evolves into a loop (Fig. 7.10(D)). The loop expands further (Fig. 7.10(E)), until it meets the surface, thus
generating two additional dislocations (Fig. 7.10(F)). Of these two new dislocations, one travels in the same direction as
the original dislocation, while the other moves in the opposite direction (i.e. opposite Burgers vector).
These simulations show clearly that a micro/nanopillar of a bcc metal is unlikely to be found in a dislocation starved state,
because a well-defined mechanism exists in it for dislocation multiplication, therefore a bcc micropillar should have a higher
number of dislocations than a fcc micropillar of the same diameter. Results of both molecular dynamics and dislocation
dynamics simulations have indicated that there exists a critical stress σc , below which this mechanism is however not
operative, namely the cusp is able to escape the pillar without generating a loop. Interestingly enough, the larger the pillar,
the smaller is σc , meaning that in smaller pillars such dislocation multiplication mechanism requires a large stress (for
example around 5 GPa in a Mo pillar of 36 nm diameter, according to simulations [591]).
Weinberger et al. [591] pointed out that these simulations help to understand the so-called ‘‘strain-softening’’ effect seen
in bcc Mo micropillars that were initially defect-free [592] and which behaved elastically up to 10 GPa, and then suddenly
collapsed. Indeed, as soon as a critical stress is reached in small micropillars, a large number of dislocations is formed, i.e. a
sort of avalanche effect is generated which suddenly leads to a large plastic deformation. Things are more complicate in
the case of large micropillars, because in these cases self-multiplication of dislocations creates rapidly a large number of
dislocation junctions, which act to stop the further movement of dislocations (i.e. strain hardening occurs).
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7.6.5. General remarks
More in general, by taking into account the various mechanisms, one can identify broadly three size regimes in which
plastic deformation follows three different pathways [561]: (i) the nano-regime, below a few hundreds of nm, in which
materials exhibit a considerably high strength, because they are mainly dislocation-free. In this regime, the plastic behavior
is often stochastic, i.e. is characterized by the ‘‘catastrophic’’ nucleation of a single defect, which can occur at a different
stress from sample to sample, and which initiates various events that lead to a sudden deformation of the material, via a
‘‘strain burst’’. So no hardening is usually observed in this size regime and the sample undergoes a brittle failure; (ii) In the
size range from a few hundred of nm to a few microns, the plastic behavior is governed by effects such dislocation starvation,
source truncation, dislocation multiplication (in bcc crystals) and other effects as discussed earlier. Broadly speaking, this
regime sets in as soon as the average distance between dislocations in of the same order of the size of the sample. Also in
this regime plasticity can be governed by stochastic events that can lead to strain bursts, as shown earlier; (iii) Above a few
microns, the plasticity is basically as in the bulk, hence is dictated by dislocation forest hardening.
7.7. Buckling
Buckling (i.e. lateral bending or bowing) is a failure mode of a column or pillar subject to compressive strain, which is
due to mechanical instability. This type of column failure usually happens to columns that are very long and thin. There is
another buckling process that is often observed, which is called self-buckling. Here a free-standing, vertical column, with
density ρ , Young’s modulus E, and radius r, will buckle under its own weight if its height exceeds a certain critical value.
The Euler–Bernoulli theory is often used for calculating the maximum axial load that a long, thin, ideal column can carry
without buckling [593]. An ideal column is considered as perfectly straight, homogeneous, and free from any initial stress.
The maximum load (critical load), causes the column to be in a state of unstable equilibrium, that is, any increase in the load,
or the introduction of the slightest lateral force, will cause the column to fail by buckling. The Bernoulli–Euler equation for
a straight column under uniaxial compression is given by:
Pcrt =
C π 2 EI
L2
(7.6)
.
where C is a parameter that depends on the end condition of the column, Pcrt is the critical load, E is Young’s modulus of
elasticity, L is the length, and I is the moment of inertia. In the case of a rod of diameter d, I is equal to π d4 /64, while in the
case of a tube of inner diameter di and outer diameter do I is equal to π (d40 − d4i )/64. The starting point in the formulation
of the Euler bending model is to consider the column as an elastic medium, obeying the linear form of the Hooke’s law, and
consisting of ‘‘fibers’’ under uniaxial compression and under distributed lateral pressure. A more elaborate version of this
theory was developed by Timoshenko, who took into account also the possibility for the beam to be deformed by shear, and
which was found to describe more accurately the behavior of low aspect ratio beams [594–596].
Both the Bernoulli–Euler and the Timoshenko models are based on linear elasticity theory. One major over-simplification
of this theory however is that the stress at a given point in a material is dependent linearly on the strain at that same
point (and only at that point) in the material, while no information is included about the long-range interatomic and
intermolecular cohesive forces. This lack of length scale information in the ‘‘local’’ elasticity theory makes its application
questionable to nanostructures. These cannot be considered any more as a continuum but instead their lattice parameter
and/or interatomic distances are not any more negligible with respect to one or more of their dimensions (they should be
rather seen as a ‘‘granular’’ medium than as a continuum). An example of behavior that cannot be predicted by the standard
‘‘local’’ Bernoulli–Euler beam model is the decrease in phase velocity of wave propagation in carbon nanotubes at high
frequencies, due to a strong contribution of the peculiar nanotube microstructure to the flexural wave dispersion [597].
A major advancement in this field has been first of all the development by Eringen of the theory of non-local continuum
mechanics [598,599], in which the stress at each point of a given body is a function of the strain at all the other points of
the body, and as such this theory can take into account the ‘‘internal’’ length scale of the forces for that specific material.
The equations of Eringen contain the so-called scale coefficient e0 a, in which a defines the internal characteristic length (for
example the lattice parameter or an interatomic distance), while the parameter e0 is related to the specific material and can
be estimated by fitting experimental data. The e0 a term has the dimensions of a length and can range from a fraction of a
nm up to a few nm.
Following the works of Eringen, many reports have appeared on the application of the model of non-continuum
mechanics to both the Bernoulli–Euler pillar (see Ref. [600] and references cited therein) and the Timoshenko pillar
[597,601]. Wang et al. for example [601] derived an expression relating the buckling load for a non-local Timoshenko
rod/tube P to the buckling load PE for a local Bernoulli–Euler rod/tube:
P =
PE
1+
PE
Ks GA
+ (e0 a)2
PE
EI
(7.7)
in which A is the cross-sectional area of the pillar, Ks is the shear correction factor of the Timoshenko theory, E and I are
again Young’s modulus of elasticity and the moment of inertia of the pillar, and G is the shear modulus. According to the
expression above, the local Euler buckling load is obtained by setting Ks → ∞ and a = 0, the non-local Euler buckling load
R. Krahne et al. / Physics Reports 501 (2011) 75–221
209
Fig. 7.11. (a–b) Load vs. displacement curve of the as grown ZnO nanorods from indentation experiment: (a) buckled ZnO nanorods, and (b) bending
flexibility of ZnO nanorods curve. (c–d) Load vs. displacement curve of the etched ZnO nanotubes from the nanoindentation experiment, (c) buckled ZnO
nanotubes, and (d) Loading–unloading curves probing the bending flexibility of ZnO nanotubes.
Source: Taken with permission from Ref. [606].
when setting Ks → ∞ and a ̸= 0, and the local Timoshenko load when a = 0. The authors compared the various loads
with each other for different aspect ratios of the rods and for different values of the scale coefficient e0 a. They found that the
buckling load is reduced when considering the non-local models, especially when the pillars have short aspect ratios (from
the expression above one can already deduce that the buckling load P decreases when the scale coefficient e0 a increases).
Therefore, the small scale effect has the same influence on the buckling load as the transverse shear deformation.
Especially in the case of piezoelectric nanodevices, their mechanical characterization and most notably the study of
the buckling behavior of a given nanostructure are extremely important. As of today, buckling studies have focused
mainly on carbon nanotubes and on ZnO nanorods/nanowires [566,601–607]. Most notably wurtzite ZnO, due to its
noncentrosymmetric nature, has been used as piezoelectric material, for example in piezoelectric thin films for highfrequency bulk acoustic wave and surface acoustic wave transducers and resonators. We will mention here, as a
representative experiment, the work of Riaz et al., who studied the buckling behavior of ZnO nanorod arrays and of nanotube
arrays obtained by the same nanorods via chemical etching [606]. Each rod on average was 208 nm in diameter and 2219 nm
in length, while in the corresponding nanotubes obtained via chemical etching the inner diameter was 125 nm on average. In
their experiments, carried out with a nanoindenter, an area containing a large number of rods (or tubes) was loaded to a given
force and then unloaded in a force controlled mode [606]. The mechanical behavior of the two samples is shown in Fig. 7.11.
In both samples (Fig. 7.11(a) and (c)), the loading portion of the curve can be divided in three zones, namely a prebuckling
zone, a buckled zone and a postbuckled zone. In the prebuckling zone the force increased linearly with displacement, so
both nanorods and nanotubes were stable in this region, up to the buckled zone, in which the rods and the tubes became
unstable. The load at which the rods were unstable (2890 µN) was considerably larger than that at which the nanotubes
were unstable (697 µN). The postbuckled zone in both samples had many critical points (indicated by 1, 2, 3, . . . in the
curves). At each of this point, there was a buckling event that deformed the samples more and more and also stiffened them
more, so that more force was required to generate another buckling event.
It can be clearly seen that the nanotubes array sample was more fragile than the nanorod array sample, first because the
buckling load in nanotubes is smaller than in the corresponding ‘‘non-hollow’’ nanorods (i.e. lower moment of inertia), and
also because the chemical etching most likely produced structural defects that should be sensitive to buckling. The authors
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R. Krahne et al. / Physics Reports 501 (2011) 75–221
Fig. 7.12. Top views (a, c, e, g, i, k, m) and cross-sectional SEM views (b, d, f, h, j, l, n) of copper nanorod arrays annealed at various temperatures. A different
sample was used for each annealing temperature. Pre-melting starts at ∼400 °C and a continuous film is formed at T > 550 °C. The scale bars shown are
300 nm long.
Source: Taken with permission from Ref. [619].
performed also loading–unloading experiments on fresh regions of both samples, but in this case the force applied was only
up to the first critical point (Fig. 7.11(b) and (d)). These experiments probed in practice the bending flexibility of the sample.
No critical point was observed in these experiments, and the nanotube array sample exhibited higher displacement that the
nanorods array sample, such that the nanotubes were found to be roughly five times more flexible than the nanorods, which
the authors claimed could be useful for piezoelectric devices.
7.8. Melting studies of elongated inorganic nanoparticles
We close this section with a brief overview of the melting studies in elongated inorganic nanoparticles. At the melting
point, a material undergoes a transition from the solid state to the liquid state. The melting point depends both on the
specific composition of the material and on its size. Nanoparticles exhibit lower melting point temperatures as compared to
their bulk counterpart, because of the large fraction of (more reactive) surface atoms. Such behavior can be explained by the
simple capillary theory of equilibrium of finite phases. The melting temperature in nanoparticles is inversely proportional to
the radius of the nanoparticles [608]. Generally, melting of nanoparticles starts at their surfaces. Experimental techniques
used to observe the melting of nanoparticles include for example X-ray [609,610] and electron diffraction [608,611,612],
calorimetry [613], femtosecond laser spectroscopy [614] and surface Brillouin scattering [615]. Several reports have shown
that samples of metal nanorods made of same material, but differing from each other in terms of average diameters and
lengths, have dissimilar melting points. More specifically, shortening of the rod length while keeping the diameter constant
results in a lowering of the melting temperature.
Of particular relevance are melting studies involving ultrafast lasers pulses. When gold nanorods in colloidal solution are
irradiated with femtosecond pulses having energies in the mJ range, they are transformed into spherical nanoparticles [614].
The femtosecond laser pulse heats the electrons of the gold nanorods, which then couple to the lattice phonons [616,617]. As
the nanorod lattice becomes hot, the nanorod melts and isomerizes into the more thermodynamic stable spherical particle
shape. This can be followed optically, by monitoring the change in absorption intensity of the longitudinal surface plasmon
band of an aqueous solution of gold nanorods as a function of the number of laser pulses: indeed when complete shape
transformation of nanorods to nanospheres takes place, the longitudinal surface plasmon band disappears (see also Section 4
of this review). An important parameter in these studies is the energy required to melt a nanoparticle. El-Sayed et al. derived
this energy in the case of a gold nanorod, based on thermodynamic reasoning, spectroscopic measurements and using bulk
R. Krahne et al. / Physics Reports 501 (2011) 75–221
211
values (i.e. the melting temperature Tm and the enthalpy of fusion ∆Hfus of gold (63.7 J/g) were used) [618]. The energy ENR
required to melt a single gold nanorod was calculated experimentally as:
ENR = cp × m × ∆T + m × ∆Hfus
(7.8)
where ∆T is the temperature difference between room temperature (25 °C) and the melting temperature Tm (1064.18 °C)
of bulk gold; cp is the specific heat capacity of metallic gold (0.129 J/g K) and m is the mass of a gold nanorod. This equation
could be further used for other metallic nanorod systems too. The energy threshold for the complete melting of the nanorod
distribution into spherical nanoparticles was determined spectroscopically by Link et al. [618]. It was found for example
that it takes about 60 femtojoules (fJ) to melt a gold nanorod with an aspect ratio of 4.1 (44 nm × 11 nm) [618].
The melting temperature of copper nanorod arrays synthesized via oblique angle sputter deposition was found to be
much lower than the bulk melting point (1083 °C) [619]. Different melting stages of Cu nanorods, as observed under SEM,
are displayed in Fig. 7.12. The as-deposited nanorods were ∼2300 nm in length, ∼100 nm in diameter, and were separated
from each other by gaps varying between ∼10 and ∼30 nm. The melting process was investigated via analysis of SEM, TEM,
and XRD measurements. A pre-melting stage (or surface melting) was observed to occur at an annealing temperature of
∼400 °C under vacuum. As the temperature was raised, the arrays of Cu nanorods started to coalesce and formed a dense
continuous film with a (111) texture at ∼550 °C. The enhanced surface diffusion and low temperature pre-melting in the
present Cu nanorods are believed to be mainly due to their high surface/volume ratio. The authors speculated that the defects
in the rods may also play a role in the pre-melting.
8. Conclusions and outlook
We have covered in this review article several aspects connected with the physical properties of elongated inorganic
nanoparticles. In the following, we will give an outlook on what, in our opinion, will be the most significant challenges
laying ahead in this fascinating area of research.
Semiconductor nanorods have demonstrated extremely promising properties for optical applications both on the single
nanoparticle level and for large assemblies. The shape anisotropy of nanorods leads to several interesting features in their
optical behavior, such as linearly polarized emission, tunable emission wavelength via the Stark effect, and large absorption
cross sections paired with strong quantum confinement. These properties make semiconductor nanorods superior to
spherical nanocrystals in many aspects and consequently current research makes a strong effort to improve nanorod
synthesis and the techniques to implement them into device structures. One critical issue for light emitting applications
is the number of defect states, since such states provide channels for non-radiative recombination. In this direction various
core–shell types of nanorods have emerged that show superior performance with respect to rod homostructures. In order to
achieve brighter emission, one or two shells of higher band gap materials can be grown around the emitting core material.
One particularly promising approach for light emitting devices is the core–shell structure that is fabricated via the seeded
growth method, as discussed extensively in Sections 2 and 3 of this review. This geometry, consisting of a spherical low band
gap core embedded in a higher band gap rod-shaped shell, has several advantages: (i) the synthesis yields samples with a
narrow size distribution, which leads to small FWHM of the emission peak and favors ordered assembly; (ii) the emission
wavelength and the electron (de)localization can be tailored, within certain limits, via the core size, and (iii) non-radiative
exciton recombination is greatly reduced compared to homostructures. Such core–shell nanorods can find applications on
a single particle level in biological tagging, as single photon emitters (in combination with photonic crystals or cavities),
or as light modulators. Self-assembly of nanorods into large scale ordered array enables the exploitation of some of the
anisotropy-related properties also in macroscopic devices. For example, the polarized emission and the dipole antenna-like
light field distribution have been demonstrated for micron to millimeter scale assemblies [149]. Moreover, large ensembles
of nanorods show collective properties like amplified stimulated emission and lasing. Also here research efforts are in
progress aiming at increasing the control over the nanorod assemblies and their optical properties. One promising trend
focuses on hybrid structures consisting of nanorod layers and organic materials. For example, films of dense nanorod
assemblies have already demonstrated their functionality in macroscale devices such as LEDs [179]. Another promising
route is the use of collinear nanorod heterostructures, where the band offset at the interface can be tailored for the targeted
application. Since such nanorods expose both materials on the surface they will be good candidates for applications as
sensors or in photovoltaics.
Films of nanocrystals have been studied extensively with respect to their charge transport properties [620]. For what
concerns the exploitation of semiconductor nanorods in electrical devices, one critical aspect at present is the contact
fabrication on a single nanorod level. This remains very demanding and costly, with little control over the contact resistance,
and in most cases with no possibility of up-scaling of the device fabrication. Furthermore, the approaches described in
Section 3.2 are not compatible with integrated CMOS technology. Therefore, the realization of practical electrical devices
with single nanorods as active elements appears unlikely at present. In this sense the studies on single nanorods find their
justification more from the basic research point of view, since they provide valuable information on the electronic structure
of these novel nanomaterials (for a general overview on charge transport and single-electron effects in nanoscale systems,
the reader can consult Ref. [621]). The current trend goes towards nanorod heterostructures (we discussed the CdSe/CdS
core–shell rods, and the Au-sphere/nanorod dumbbells), which allow to tailor the band gap, enhance or reduce charge
separation within the nanorod, or to combine electrical, magnetic, and optical functionality within a single nanostructure.
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We expect that basic research will make great advances in the next years in this direction. The situation is different for
large scale assemblies of nanorods, where the challenge is to control the uniformity of the nanorod arrays in order to
exploit the advantageous properties that result from their anisotropic shape. Efforts have been made in this respect in the
last years that yielded promising results. Centimeter-scale assemblies of laterally and vertically oriented nanorods have
been demonstrated. Such large scale assemblies will make nanorods interesting as active materials for photovoltaics, light
emitting diodes, and sensors.
Plasmonic nanoparticles have been among the most investigated materials in nanoparticle research and have found so
far application in many fields, like optics, sensing and biosensing, hyperthermia treatment in medicine, photovoltaics, and
others. The possibility to tune the shape of nanoparticles, hence to manipulate the optical response both in the near field and
in the far field, has opened new scenarios in both fundamental research and technological applications. One very exciting
area of research in this field at present is the fabrication of ordered architectures of nanorods/nanowires, as these assemblies
have shown coupling between plasmon modes of the individual components that leads to extended and guides plasmonic
modes, similar to photonic waveguides. All these architectures exhibit also tunable values of permittivity depending on the
polarization directions, which will be useful for applications in metamaterials. As another example, precisely positioned
metallic nanorods have been used as dipole nanoantennas, being capable of receiving optical power from a waveguide,
localizing it and thus allowing it to be converted into an electrical signal by means of a plasmonic-to-electronic converter.
This has extremely interesting implications for the integration of plasmonics into nanoelectronic circuits. A recent review
covering this topic and other recent advances in plasmonics is the one by Gramotnev et al. [622]. Plasmonic nanoparticles,
due to their capability of acting as nanoantennas, are finding increasing applications in photovoltaics and in photocatalysis,
and certainly the ability to control the shape and the type of assembly of the nanoparticles will play a key role in all those
fields.
A considerable effort has been made so far in the study of one-dimensional magnetic nanoparticles, such as ellipsoids,
rods and wires. Even though basic science has driven this field of research so far, the work has been strongly motivated
by potential but crucial technological improvements derived from them. The significance of shape anisotropy in magnetic
nanoparticles has been confirmed and supported by a large number of experimental and theoretical works. Shape
anisotropy can significantly enhance the global magnetic anisotropy of small particles: the room temperature stabilization
of magnetically ordered states in nanostructured materials has been often attributed to shape effects, in contrast to spherical
particles which often show random fluctuations of their magnetic moment at that temperature. Such enhanced anisotropy
is mainly translated into increased coercivities without compromising the small sizes of the single particles. Data storage
industry can greatly beneficiate from it by designing new magnetic data storage devices with higher densities, reduced
dimensions and faster operational speeds.
Various synthetic and nanofabrication procedures have been developed in the last decades for both oxide- and metalbased materials, with the exception of permanent nanomagnets, for which still several synthetic difficulties have to be
overcome. Consequently, it is now possible to obtain 1D magnetic nanomaterials with controlled sizes down to which
magnetic single domain phenomena can be studied. Moreover, high resolution magnetic characterization techniques are
nowadays also available. The exploitation of anisotropic magnetic particles in current industry strictly requires the regular
and homogeneous organization of the elongated magnetic building blocks in a device. Several strategies have been shown to
be useful for this aim, especially for metals, on which research has been much more focused due to their higher technological
importance compared to oxides. However, this issue has been only partially addressed and will probably become a key aspect
for the future implementation of anisotropic nanoparticles in technology.
Theory has always played an important role in the study of magnetic systems. Despite this, there is still a lack of models
accounting for finite size effects present in 1D nanostructures. With the recent experimental capabilities for studying
magnetization reversal processes and domain wall dynamics of 1D nanostructured systems, the doors for the development
of a new generation of spintronic devices are open. However, classical models seem to be insufficient to predict all physical
phenomena occurring in these nanoparticles. Hence, the formulation of specific micromagnetic theories that can be applied
to most of the technologically interesting magnetic materials, will continue to be a very active research field.
It is also worth mentioning the increasing role of magnetic nanoparticles in biomedicine. Unfortunately, toxicity and
chemical stability concerns have significantly limited their study. Therefore, still very few experimental evidences are
available to support the use of 1D magnetic nanostructures in this field. In principle, the increased magnetic anisotropies of
1D systems should be crucial for the development or the improvement of innovative diagnostic and therapeutic techniques.
The potential of elongated nanoparticles in medicine will probably encourage their study in the coming years.
It is now increasingly evident that the catalytic properties of anisotropic nanocrystals (mainly rod-shaped) are different
from their bulk and isotropic nanostructures in terms of their activity and specificity for various reactions. In particular, in
the rod-like or elongated morphology their high surface areas and the exposure of clearly defined crystals facets are the
main factors that can enhance their catalytic properties. However, the shape-dependent catalytic function of anisotropic
nanocrystals needs to be investigated further in terms of their surface and electronic properties. This will help not only
to draw clear relationships between the shape and catalytic properties but also to develop the application methods of
catalysts that are appropriate for different reaction conditions. So far metal nanorods and metal NPs incorporated metal
oxide nanorod heterostructures were shown to be efficient catalysts for water-splitting, methanol oxidation, CO oxidation
and obviously in organic synthesis. As their optical absorption sweeps across ultraviolet, visible and infrared regions, they
could be employed as photocatalysts for a wide range of photo conversion reactions. Thus, such catalytic systems based on
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nanorods or elongated nanocrystals can offer energy-efficiency and precise scalability of materials conversion into targeted
products, and their exploitation in cells and electrochemical super capacitors is expected to grow in the years to come.
For what concerns the mechanical properties of elongated nanoparticles, we can certainly say that research in this area is
still in its infancy. Only recently, thanks to advances in synthesis and micro/nanofabrication, as well as in the development
of more sophisticated setups for mechanical testing, have reliable data become available, but not always the results are
straightforward to interpret. Various studies have shown that mechanical behavior in confined dimensions is dependent
on a wide range of parameters, including the conditions of the experiment itself, and various open questions will require
increasing attention in the future. Also, modeling of mechanical properties at nanoscale is still a formidable challenge and
only in the last few years, thanks to innovations in modeling and in increased computing power, it has been possible to
carry out large scale simulations in 3 dimensions. Issues in this area are particularly critical as devices are being increasingly
miniaturized and their mechanical response needs to be addressed. Further exciting directions in this field are for example
the study of phase transitions in confined systems, as well as shock-induced amorphization, as it has been observed that
metallic one-dimensional structures are transformed into amorphous structures when subject to high strain rates [586]. All
these transformations need to be carefully studied, especially considering that nanoscale materials and devices in the future
are expected to be used in applications involving extreme conditions.
Acknowledgements
The authors acknowledge financial support from the European Union through the FP7 starting ERC grant NANO-ARCH
(contract number 240111) and the FP7 grant MAGNIFYCO (contract number NMP4-SL-2009-228622). We would like to
thank the following persons for their many useful suggestions and comments: Prof. Uri Banin and Prof. Oded Millo (Hebrew
University of Jerusalem, Israel), Prof. Stefan Maier (Imperial College, London, UK), prof. Luis Liz-Marzán (University of Vigo,
Spain), Dr. Josep Nogués Sanmiquel (Institut Catalá de Nanotecnologia, Barcelona, Spain), Prof. Pietro Ballone (Queen’s
University Belfast, Northern Ireland), Dr. Luca Ceseracciu and Dr. Marco Allione (Istituto Italiano di Tecnologia, Genova,
Italy) and Prof. Herre Van Der Zant (Kavli Institute of Nanoscience, Delft University of Technology, The Netherlands).
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