PRL 101, 026805 (2008)
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PHYSICAL REVIEW LETTERS
Localized Magnetic States in Graphene
Bruno Uchoa,1 Valeri N. Kotov,1 N. M. R. Peres,2 and A. H. Castro Neto1
1
Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA
2
Centro de Fı́sica e Departamento de Fı́sica, Universidade do Minho, P-4710-057, Braga, Portugal
(Received 21 February 2008; published 11 July 2008; publisher error corrected 11 July 2008)
We examine the conditions necessary for the presence of localized magnetic moments on adatoms with
inner shell electrons in graphene. We show that the low density of states at the Dirac point, and the
anomalous broadening of the adatom electronic level, lead to the formation of magnetic moments for
arbitrarily small local charging energy. As a result, we obtain an anomalous scaling of the boundary
separating magnetic and nonmagnetic states. We show that, unlike any other material, the formation of
magnetic moments can be controlled by an electric field effect.
DOI: 10.1103/PhysRevLett.101.026805
PACS numbers: 73.20.Hb, 73.20.r, 73.23.b, 81.05.Uw
Graphene, a two-dimensional (2D) allotrope of carbon,
has singular spectroscopic and transport properties [1– 4]
due to its unusual electronic excitations described in terms
of massless, chiral, ‘‘relativistic’’ Dirac fermions [5]. In
addition to being a possible testbed for relativistic quantum
field theory [6], graphene has a great technological potential due to its structural robustness, allowing extreme miniaturization [7], and a flexible electronic structure that can
be controlled by an applied perpendicular electric field [8].
In this Letter we show that graphene also has potentiality
for spintronics, that is, independent control of the charge
and the spin of the charge carriers [9]. Unlike diluted
magnetically semiconductors [10] where the location of
the magnetic ions is random and hence unpredictable,
adatoms can be positioned in graphene using a scanning
tunneling microscope (STM) [11]. Furthermore, as we are
going to show, the magnetic properties of adatoms such as
size of the magnetic moment and Curie temperatures can
be controlled by an external electric field, an effect unparalleled in condensed matter systems.
The basic model for the study of magnetic moment
formation in metals is the well-known Anderson impurity
model [12]. In this model an ion with inner shell electrons
with energy 0 hybridizes, via a hopping term of energy V,
with a conduction sea of electrons. While the conduction
electrons are described by a Fermi liquid with featureless,
essentially constant, density of states (DOS), the impurity
ion is assumed to be strongly interacting. The Coulomb
energy required for double occupancy of an energy level in
the ion is given by U. Anderson showed that when 0 is below the Fermi energy and the energy of the doubly occupied states 0 U is larger than , a magnetic state is possible if U is sufficiently large and/or V sufficiently small.
Here we apply the Anderson model to graphene and
show that the energy dependence of the DOS leads to
anomalous broadening of the adatom level and strongly
favors the formation of local magnetic moments. In particular we show that, unlike the case of ordinary metals,
this anomalous broadening allows the formation of magnetic states even when 0 is above the Fermi energy at
relatively small U. We also find, in contrast with the usual
0031-9007=08=101(2)=026805(4)
metallic case, an anomalous scaling of the magnetic
boundary separating magnetic and nonmagnetic impurity
states. Finally, we establish that the local magnetic moments can be mastered by the application of an external
gate voltage, leading to a complete control of the magnetic
properties of adatoms in graphene.
We consider an impurity atom adsorbed on the surface
of the graphene sheet, on top of a carbon (see Fig. 1). The
tight-binding Hamiltonian of the electrons in graphene is
XX y
HTB t
a Ri b Rj H:c:;
(1)
hi;ji
where a Ri [b Ri ] annihilates an electron with spin
" , # on sublattice A (B) at position Ri , hi; ji stands for
summation over nearest neighbors, and t (2:7 eV) is the
nearest neighbor hopping energy. In momentum space, we
have (we use units such that @ 1):
X
HTB t kayk; bk; kbyk; ak;
(2)
k;
a)
b)
c)
FIG. 1 (color online). (a) Honeycomb lattice with an impurity
atom. Black: sublattice A; white: sublattice B. Intersection of the
Dirac cone spectrum, k vF k, with the localized level
spectrum, Ef k 0 : (b) 0 > 0; (c) 0 < 0.
026805-1
2008 The American Physical Society
PHYSICAL REVIEW LETTERS
PRL 101, 026805 (2008)
p
P
~
^ 3=2y,
^ ~ 2
where k ~ eik , with ~ 1 ax=2
p
^ 3=2y,
^ and ~ 3 ax^ the nearest neighbor vecax=2
tors. Diagonalization of the Hamiltonian (2) generates two
bands, k tjkj, which can be linearized around
the Dirac points K at the corners of the Brillouin zone:
K q vF jqj, where vF 3ta=2 (106 m=s) is
the Fermi velocity of the Dirac electrons.
The hybridization with the localized orbital of the impurity atom on a given site, say, on sublattice B, is given by
P
HV V fy b 0 H:c:, where f (fy ) annihilates
(creates) an electron with spin " , # at the impurity. In
momentum space we have
p X
(3)
HV V= Nb fy bp; byp; f ;
p;
where Nb is the number of sites on sublattice B contained
in the expanded unit cell of graphene with the impurity.
The Hamiltonian of the localized orbital is described
P
by a single level, Hf 0 fy f . The electronic correlations in the inner shell states can be described by a
Hubbard-like term: HU Uf"y f" f#y f# . Following
Anderson, we use a mean-field decoupling of the interacP
tion, HU ! Un fy f Un" n# , where n hfy f i
is the occupation for each of the two spin states. The
Hubbard term can be absorbed into the definition of the
P
local impurity energy, Hf fy f , where
0 Un is the energy of the localized electrons in a
given spin state in the presence of a charging energy U.
The formation of a magnetic moment is determined by
the occupation of the two spin states at the impurity, n . A
localized moment forms whenever n" n# The determination of n requires the self-consistent calculation of the
density of states at the impurity level, ff !, which
incorporates the broadening of the impurity level due to
hybridization with the bath of electrons in graphene. The
occupation of the impurity level is given by
Z
n
d!ff; !:
(4)
1
The Green’s function of f electrons is Gff; t
ihTf tfy 0i, and its retarded part can be written as
GRff; ! ! Rff ! i0 1 ;
(5)
where
Rff ! V 2
X
p
G0R
bb; p; !
ff; !
2
2
2
G0R
bb; p; ! !=! vF jpj i0 sign!:
In this case (6) becomes
(7)
1
j!jD j!j
1
Z !! 2 2 !2
(9)
where V 2 =D2 is the dimensionless hybridization.
Notice that, unlike the case of impurities in metals, the
impurity density of states is not a simple Lorentzian. The
impurity DOS (9) is peaked around the quasiparticle pole
at 0 and ! ! 0. We can expand Z! around the
singularity at !0 Z1 !0 for !0 ! 0, where we may
approximate Z!0 Z except for a doublelogarithmic correction that can be safely ignored [13].
The anomalous broadening gives rise to a logarithmic
divergence in the ultraviolet when the DOS of the level is
integrated in (4),
Z1
1
jj
n 2 2 arctan
Z
(10)
where Z Z and Z1
. The term contains the contribution coming from the cutoff regularization:
W E
1
D
arctan
Z ln 1
; (11)
DZ1
(6)
is the self-energy of the f electrons, which is defined in
terms of the noninteracting Green’s function of the graphene electrons in a given sublattice, G0bb; p;t
ihTbp tbyp 0i0 :
2
!
j! D2 j
j!j
ln
iV 2 2 D j!j;
D2
!2
D
(8)
where D is a high-energy cutoff of the order of the graphene bandwidth (D 7 eV). We choose the cutoff D
using the Debye prescription, i.e., conservation of the
number of states in the Brillouin zone after linearization
of the spectrum around the K point. We assume jj
D,
where band effects related to the exact definition of the
cutoff are not important.
The real part of Rff ! defines the quasiparticle residue
1
Z ! 1 V 2 =D2 lnjD2 !2 j=!2 of the f electrons, while the imaginary part gives the broadening of
the localized level due to the hybridization. As expected,
the anomalous character of the problem is explicitly manifested in the linear dependence of the broadening with the
energy, which is proportional to the electronic DOS in
graphene. Furthermore, notice that Z! vanishes at ! !
0. Replacing Eq. (8) into Eq. (5) gives the density of states
of the localized level, ff; ! 1=ImGRff; !:
where
Rff ! V 2 =Nb
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sign, and
q
2
2 2
E Z1
;
q
2
2 2
W DZ1
D :
(12)
(13)
In Fig. 2 we show the boundary between magnetic and
nonmagnetic impurity states as a function of the parameters x D=U and y 0 =U. Notice that the
magnetic boundary is not symmetric between the cases
where the impurity is above (0 > 0) or below (0 < 0
026805-2
the Dirac point. Moreover, unlike the metallic problem
[12] the boundary is not symmetric around y 0:5. This
reflects the particle-hole symmetry breaking due to the
presence of the localized level. In the case where 0 > 0
[see Fig. 2(a)], the magnetic boundary crosses the line y
0, and the level magnetizes even when the impurity is
above the Fermi energy. This is understood by the fact
that the hybridization leads to a large broadening of the
impurity level density of states (with a tail that decays like
1=!) that crosses the Fermi energy even when the bare
level energy is above it. In the opposite case of 0 < 0 a
similar effect occurs with the crossing of the magnetic
boundary along the y 1 line, something that also does
not occur in ordinary metals [14]. This implies even when
the energy of the doubly occupied state is below the Fermi
level (0 U < ), because of the large broadening, the
impurity magnetizes if U is not too large or too small. The
up turn close to y 1 and x
1 in the 0 < 0 case only
reflects that in this limit (U, 0 , for finite 0 ) the
physics of the Dirac points is irrelevant and we recover the
usual Anderson model in ordinary metals, where the transition curve approaches the point x 0, y 1 from below
[12]. This picture is physically consistent with a renormalization group calculation at fixed 0 [15].
The dependence of the scaling of the magnetic boundary
with 0 and (see Fig. 2) shows that the size of the magnetic region grows as j0 j approaches the energy of the
Dirac points. In this situation the DOS around the localized
level is suppressed, favoring the formation of a local magnetic moment. In particular, in the limit where the level is
nearly at the Dirac point (j0 j ! 0, the level nearly decouples from the bath and the impurity can magnetize in
principle for any small finite charging energy U. On the
other hand, the magnetic region shrinks in the y direction
1
a)
1.5
b)
non-magnetic
0.5
1
as the hybridization parameter grows (see Fig. 2). In the
limit of ! 0 and U finite a local magnetic moment forms
whenever 0 < y < 1, as in the case of an impurity in a metal.
The application of a potential Vg through an electric
field via a back gate [1] shifts the chemical potential and
moves the magnetic state of the impurity in the vertical
direction (y) in Fig. 2. We assume that x D=U does not
change much with applied voltage, even with screening
coming from a finite Fermi energy. Hence, the magnetization of the impurity can in principle be turned on and off,
depending only on the gate voltage applied to graphene.
This is better illustrated by looking at the behavior of the
impurity magnetic susceptibility. In the presence of a field,
the energy of the impurity spin states changes to
0 B B Un . In the zero field limit,
P the magnetic
susceptibility of the impurity, B dn =dBB0
(B is the Bohr magneton, and B is an applied magnetic
field), can be calculated straightforwardly from Eq. (10):
2B
X dn 1 U dn
d
:
dn
d 1 U2 dn
"#
d d
1
1
a)
0.5
0
0
χ ( µB2 102 )
-0.5
εo < 0
5
x = ∆D/U
10
0
0
5
c)
0.5
0.5
0
non-magnetic
εo > 0
10
0
b)
d)
10
0.1
5
0
0.1
x = ∆D/U
0.2
µ (eV)
FIG. 2 (color online). Boundary between magnetic and nonmagnetic impurity states in the scaling variables x and y for
0 > 0 (a) and 0 < 0 (b). Circles: j0 j=D 0:029, V=D
V=D 0:14;
0:14; squares:
0 =D 0:043 and
triangles:j0 j=D 0:029, V=D 0:03. The upturn close to
y 1 and x ! 0 on panel (b) is not visible in this scale when
V is very small (triangles). See details in the text.
(14)
In the lower panels of Fig. 3 we show for 0
0:2 eV, V 1 eV, and D 7 eV, for different values of
x. The corresponding magnetization line for this set of
parameters is defined by the solid curve with black circles
in Fig. 2(a). While the impurity remains nonmagnetic for
any y at x 11 (U 40 meV), as shown in Fig. 2(a), the
impurity state already crosses the magnetic boundary twice
for x 5, where U is nearly twice as large. In this case, a
large magnetic moment of 0:5 B forms below the
energy of the level, at 0:18 eV [Fig. 3(a) and 3(b)].
At x 0:45 (U 1 eV), the local moment exists for very
large 1 eV, and a strong and uniform magnetic moment of 0:9B forms in almost the whole magnetic
region [see Fig. 3(c)]. A similar qualitative behavior for
the magnetization occurs when 0 < 0 [Fig. 3(c) and 3(d)].
As U becomes large (>1 eV), the magnetic transition
becomes very sharp. For 0 0:5 eV and V 1 eV, the
impurity can magnetize for U * 0:1 eV. While the local
nσ
y = (µ- εo)/U
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PHYSICAL REVIEW LETTERS
PRL 101, 026805 (2008)
0.3
0
-0.5
0
0.5
1
1.5
µ (eV)
FIG. 3 (color online). n" , n# , and for j0 j=D
0:029 and V=D 0:14 (D 7 eV). Left panels: x 11
(dashed curves), and x 5 (solid curves). The impurity magnetizes inside the bubble (n" n# . The vertical line marks the
position of the level, 0 0:2 eV. On the right: 0 0:2 eV
(solid curve) and 0 0:2 eV (dashed curve) at x 0:45.
026805-3
PHYSICAL REVIEW LETTERS
PRL 101, 026805 (2008)
Aff(µ)
300
a)
120
200
80
100
40
0
0.1
0.2
0.3
0
b)
0
0.5
1
µ (eV)
µ (eV)
FIG. 4 (color online). Spectral function (in units of 1/eV) of
the f electrons at the Fermi energy for j0 j=D 0:029 and
V=D 0:14 (D 7 eV). (a) x 11 (solid curve) and x 5
(dashed curve) for 0 > 0 (see Fig. 2). (b) x 0:45 for 0 > 0
(solid curve) and 0 < 0 (dashed curve).
charging energy U for transition metals in a metallic matrix
is of the order of 5–10 eV [16], in graphene, where the
effective hybridization can be large due to the linear increase of the DOS, the critical U for magnetization of the
impurity can be much smaller. Hence, transition elements
and molecules that usually do not magnetize when introduced in ordinary metals can actually become magnetic in
graphene [17,18].
In order to show that the spectroscopic functions of the
magnetic impurities can also be controlled by electric field
effect we show, in Fig. 4, the spectral function of the
localized electrons
P calculated at the Fermi energy:
Aff ! 2 ff; . The solid line in Fig. 4(a)
is a nonmagnetic resonance in a situation where the impurity state does not cross the magnetic boundary of the
scaling diagram by changing y () for some fixed x. In
the other curves of Fig. 4, the spectral weight splits between two peaks located around the magnetic transitions,
near 0 and 0 U (see Fig. 3).
The dependence of the impurity density of states with ,
and hence with gate bias, allows for the identification of the
formation of local moments through ordinary transport
measurements. For finite , the hybridization of the itinerant electrons with the localized level renormalizes the
charge scattering channels and hence the carrier conductivity, 2e2 jj , where 1 is the impurity scattering
rate. Second order perturbation theory gives [16]
1
1
0
/ n0 V 2 Aff ;
interaction is purely ferromagnetic due to the vanishing of
the Fermi wave vector, kF =vF [24]. However, at finite
bias voltage the RKKY interactions display 2kF oscillations decaying like 1=r3 [25] that can couple the magnetic
moments ferromagnetically or antiferromagnetically depending on the position and geometry of the adatom lattice
(that can be conveniently chosen using a STM). Hence, by
changing the bias voltage a variety of different magnetic
phases can emerge.
In conclusion, we have examined the conditions under
which a transition metal adatom on graphene can form a
local magnetic moment. We find that due to the anomalous
broadening of the adatom local electronic states, moment
formation is much easier in graphene. Furthermore, the
magnetic properties of adatoms can be controlled by an
electric field effect allowing for the possibility of using
graphene in spintronics.
We thank V. M. Pereira, J. M. B. Lopes dos Santos,
A. Polkovnikov, and S. W. Tsai for helpful discussions.
N. M. R. P. acknowledges financial support from POCI
2010 via Project No. PTDC/FIS/64404/2006. B. U. acknowledges CNPq, Brazil, for support under Grant
No. 201007/2005-3.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
(15)
where 1
is the scattering rate of the electrons in the
0
absence of impurities and n0 is the impurity concentration.
In the limit of very large U, however, the scattering is
dominated by the spin flip channels in the Kondo regime
[19–23]. When 0 is located in the experimental range
accessible by the application of a gate voltage 0:3 to
0.3 eV [1], the shape of the dip in the conductivity produced by the impurity scattering can indicate not only the
position of the energy level but also the presence of local
magnetic moments (notice that the nonmagnetic resonance
in the spectral function is quite symmetric).
In the presence of a finite density of magnetic moments a
macroscopic magnetic state can egress due to the RKKY
interaction between them. At the Dirac point ( 0) the
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[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
026805-4
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