arXiv:1108.4655v1 [cond-mat.stat-mech] 23 Aug 2011
Kinetic modelling of epitaxial film growth with upand downward step barriers
F.F. Leal1,2 , T. J. Oliveira1 and S. C. Ferreira1 ‡
1
Departamento de Fı́sica - Universidade Federal de Viçosa, 36571-000, Viçosa, Minas
Gerais, Brazil
2
Instituto Federal de Ciência, Educação e Tecnologia - Campus Itaperuna, Rodovia
BR 356 - KM 03, 28300-000, Itaperuna, Rio de Janeiro, Brazil.
E-mail: ffleal@iff.edu.br,silviojr@ufv.br,tiago@ufv.br
Abstract. The formation of three-dimensional structures during the epitaxial growth
of films is associated to the reflection of diffusing particles in descending terraces
due to the presence of the so-called Ehrlich-Schwoebel (ES) barrier. We generalize
this concept in a solid-on-solid growth model, in which a barrier dependent on the
particle coordination (number of lateral bonds) exists whenever the particle performs
an interlayer diffusion. The rules do not distinguish explicitly if the particle is executing
a descending or an ascending interlayer diffusion. We show that the usual model, with
a step barrier in descending steps, produces spurious, columnar, and highly unstable
morphologies if the growth temperature is varied in a usual range of mound formation
experiments. Our model generates well-behaved mounded morphologies for the same
ES barriers that produce anomalous morphologies in the standard model. Moreover,
mounds are also obtained when the step barrier has an equal value for all particles
independently if they are free or bonded. Kinetic roughening is observed at long
times, when the surface roughness w and the characteristic length ξ scale as w ∼ tβ
and ξ ∼ tζ where β ≈ 0.31 and ζ ≈ 0.22, independently of the growth temperature.
PACS numbers: 05.70.Ln, 89.75.Da, 89.75.Hc, 05.70.Jk
Submitted to: J. Stat. Mech.: Theor. Exp.
1. Introduction
Since epitaxial techniques became available, a large number of experimental works
reporting on the production of films with three-dimensional patterned structures, called
generically of mounds, as well as the analytical and computational modelling devoted
to explain these patterns have been found in literature (see Ref. [1] for a review).
Mounds have been observed during the growth of a wide diversity of films, ranging
from metals [2, 3, 4] to inorganic [5, 6] and organic [7, 8] semiconductor materials.
The morphological properties, one of the most important features in film production,
‡ On leave at Departament de Fı́sica i Enginyeria Nuclear, Universitat Politécnica de Catalunya, Spain.
Leal et al., Kinetic modelling of epitaxial film growth
2
depend of distinct physical mechanisms as diffusion rate [9, 1], deposition rate [10],
growth temperature [11, 6], deposition shadowing [12], among others.
During the film production, the surface may follow distinct growth regimes [1]. The
so-called kinetic roughening, where the surface dynamics is described by scale invariance,
has significant interest. The power law behaviours of the surface rms width w ∼ tβ
and the characteristic correlation length ξ ∼ tζ are the basic scaling laws involved
in the kinetic roughening, defining the growth and coarsening exponents, respectively.
There is a third, not independent exponent related to the scaling of the global surface
fluctuations, the roughness exponent α = β/ζ [13]. Two scaling regimes are of special
interest in mound formation: The first one involves slope selection [14, 15, 16], where
the ratio between the characteristic length and the surface width approaches a constant
value. This condition is satisfied for β = ζ. The second one is the super-roughness
scaling regime [17, 13], characterized by α > 1. The latter implies that the surface is
locally smooth, having different scaling exponents for global and local fluctuations [17].
The mound formation during the epitaxial growth is frequently associated to the
reflection of particles (atoms or molecules) in descending terraces due to presence of
the so-called Ehrlich-Schwoebel (ES) barrier [1]. The explanation for this step barrier
is founded in the experimental observation that particles are reflected backwards in
the terrace edges more frequently than move to a lower layer [18, 19]. Although
other mechanisms, as short-range attraction towards ascending steps [20] and fast edge
diffusion [21, 22, 23] can lead to mound formation, they are weaker than the ES barrier.
Kinetic Monte Carlo (KMC) is a standard simulation method for atomistic
modelling of epitaxial growth. The basic idea was formerly introduced by Clarke and
Vvedensky [24, 25] and generalized in many other frameworks [1, 9]. In the ClarkeVvedensky model, diffusion is a thermally activated process described by an Arrhenius
law D ∼ exp(−E/kB T ) [9], where E is a diffusion activation energy, T the temperature
and kB the Boltzmann constant. The ES barrier is usually modelled with an additional
energy barrier Eb to particles moving downwards in the edge of a terrace [1]. In a
simplified picture, Eb is the additional energy needed when a particle crosses a step edge.
Therefore, a step barrier must also exist for particles performing upward diffusion (see
Fig. 1). However, the ES barrier is generally considered only for particles diffusing from
an upper to a lower terrace and barriers for ascending steps were considered in a limited
number of models. Šmilauer and Vvedensky [14] proposed a step barrier proportional
to the difference between next-nearest neighbour (NNN) bonds before and after the
particle hop only if the number of NNN bonds decreases after the hop, independently
if the adatom performs an up- or downward diffusion. Recently, a model with a barrier
dependent on the step height was investigated [26]. Besides the barrier in descending
steps, this model also assumes an ascending barrier for multilayer steps. However, the
barrier in monolayers was implemented as usual and the model is not fully symmetric
in to relation down- and upward step diffusion. Barriers in ascending steps were also
considered in the limited mobility model of Wolf-Villain [27], where probabilities of
moving to lower or upper terraces after a deposition step were introduced. Mound
Leal et al., Kinetic modelling of epitaxial film growth
3
formation is obtained when the probability of upward diffusion is larger than downward
diffusion.
In the present work, we revisit the effects of ES barriers in KMC simulations by
comparing the standard model, having a barrier only in descending steps, with the
case where a barrier for upward diffusion is also activated. Moreover, we consider
different barriers depending if the particle has or not lateral bonds. The model allows
to investigated the interesting symmetric case where bonded and free (with no lateral
bonds) particles undergo the same barrier. We show that the standard bond counting
model produces unrealistic columnar mounds when the diffusion of particles with lateral
bonds cannot be neglected. This feature limits the applicability of this model to a short
range of temperature. On the other hand, the model with a bond dependent step barrier
produces well-behaved mounded surfaces in a wide range of temperature, consistent with
typical experimental conditions.
We have organized the paper as follows. In Sec. 2 we present the models and KMC
simulation strategies. Comparison between a barrier exclusively to downward diffusion
and the bond dependent step barrier is done in Sec. 3. Further analyses of our model,
including the symmetric case, are presented and discussed in Sec. 4. We summarize our
results and conclusions in Sec. 5.
2. Kinetic Monte Carlo model
We model homoepitaxy on a substrate at a temperature T represented by a triangular
lattice with periodic boundary conditions. Particles are deposited at a constant rate F
under the solid-on-solid restriction, implying that voids and overhangs are absent. A
deposition event consists in increasing the height by hj → hj +1, where hj is the number
of particles deposited in the site j. The choice of the deposition site j is done in two
steps: Firstly a site is chosen at random, and secondly a new particle is deposited in the
most energetically favourable (largest bond number) site among the chosen one and its
nearest neighbours (NN). This transient mobility is justifiable if we consider that the
particle arrives at the substrate with a kinetic energy higher than the typical activation
energy in the substrate (kB T ). This deposition rule corresponds to the classical WolfVillain (WV) model [28]. The transient mobility in the deposition rule plays an import
role in the surface morphology at low temperatures, as we will show in Sec. 4. It is
important to mention that a transient mobility was used in previous models of epitaxial
growth with thermally activated diffusion [1, 14].
Intra and interlayer thermally activated diffusion are allowed for any surface
particle. Diffusion happens between NN sites and depends on the initial and final
position of the particle. The diffusion rate from site a j to a neighbour j ′ is given by
the Arrhenius law
E(j, j ′ )
′
D(j, j ; T ) = ν exp −
,
(1)
kB T
Leal et al., Kinetic modelling of epitaxial film growth
4
where ν is an attempt frequency and E the diffusion activation energy given by
E(j, j ′ ) = E0 + nj EN + Eb (j, j ′ ).
(2)
The energy E0 represents the interaction with the substrate and/or bulk of the film, EN
is the contribution of each one of nj lateral bonds, and Eb is the step barrier present
for interlayer diffusion. Apart of the step-edge barrier, the model is equivalent to the
thermal activation model proposed by Šmilauer and Vvedenski [14]. Algorithmically,
the step barrier is present only if ∆h(j, j ′ ) 6= 0, where ∆h(j, j ′ ) ≡ hj ′ + 1 − hj .
Apart from the details of the barrier rules, which are described below, the kinetic
Monte Carlo simulations are implemented as follows. A particle is deposited and then
N diffusion attempts are sequentially implemented. In a diffusion attempt, a particle
of the surface is selected at random and its diffusion is implemented accordingly to the
ratio D/F given by Eq. (1). The diffusion is efficiently implemented by keeping updated
lists containing the positions of the particles with the same number of lateral bonds.
Let ρn , n = 0, 1, . . . , 6 be the fraction of particles with n lateral bonds. Between two
deposition events, the average number of diffusion attempts of the particles having n
bonds is Nn = ρn D(n; T )/F where
E0 + nEN
D(n; T ) = ν exp −
(3)
kB T
is the diffusion rate excluding the step barrier contribution. In the implementation, we
P
sequentially choose N = 6n=0 Nn † particles, with the coordination number obeying the
ratio N0 : N1 : · · · : N6 . For each selected particle (site j), a neighbour (site j ′ ) is chosen
with equal chance and the particle hops to this site with probability
Eb (j, j ′ )
′
,
(4)
Ph (j, j ; T ) = exp −
kB T
which includes the effect of the step barrier. With the complementary probability the
particle stays in the same site. The process is repeated until the runs of N diffusion
attempts are finished and a new particle deposited.
In the present work, we are interested in the effect of an uphill in addition to the
usual downhill step-edge barrier. Figure 1 shows some important situations involving
interlayer diffusion. Let us firstly consider the particles A and C lying in monosteps. In
both cases, it is intuitive that staying in the same layer is favoured since, otherwise, the
particle has to detach and rebind in an adjacent layer. So, an additional step barrier
(not necessarily equal) must be present in both configurations. The kink particle B,
however, should diffuse up- or downwards with equal chance due to the local symmetry
of this configuration. In a simple description, particles A and B may be subject to a
similar step barrier that may possibly be different from the barrier for particle C.
Our model has descending and ascending interlayer diffusions that depend if the
particles are bonded or not. By simplicity, we assume two kind of step barriers: Eb0
† More precisely, the number of attempts is increased by 1 with probability N − N̄ , were N̄ is truncation
to the integer part of N
5
Leal et al., Kinetic modelling of epitaxial film growth
C
E
B
D
A
Figure 1. Illustration of important situations involving interlayer diffusion: Diffusion
through monolayer (left) and multilayer (right) steps. Both, particles D and E have
to overcome a barrier when moving through the steps.
for free particles (n = 0) and Ebn for bonded particles (n ≥ 1). Notice that rules
do not concern explicitly if the particles are moving down- or upwardly, but, free
particles will never be in an ascending step since this configuration requires n ≥ 1.
In principle, configurations D and E are apparently equivalent but the situation is
much more complicated in 2+1 dimensions due to the bonds not shown in an 1+1
representation. However, it is clear that a step barrier must exist for both particles and
our model is, therefore, an improvement for these configurations. Actually, as shown
in Ref. [26], although the presence of a barrier in a multilayer step is important, the
details of the barrier dependence with the step height do not alter significantly the final
surface morphology. We refer to our rules as BDSB (bond dependent step barrier) while
the standard model we call of DSB (downward step barrier).
3. Results: BDSB versus DSB models
Figure 2. Surface morphologies for the BDSB (top) and the DSB (bottom) models
for distinct temperatures after a deposition of 100 ML. The step barriers in the
BDSB model are Ebn = 0 and Eb0 = 0.05 eV while the barrier in the DSB model
is Eb = 0.05 eV. See supplementary material for surface evolutions.
Leal et al., Kinetic modelling of epitaxial film growth
6
The KMC simulations were performed with the fixed parameters E0 = 1 eV,
EN = 0.11 eV, F = 1 monolayers (ML) per second, and ν = 1013 s−1 . Temperature
was varied in the interval T = 473 − 623 K corresponding to a range of temperature
∆T = 150 K compatible with several semiconductor and metal film growths [1]. We
simulated substrates sizes L = 256, 512 and 1024, and no significant difference was
observed in the results. Therefore, except if explicitly mentioned, the results correspond
to L = 512.
In order to analyse the effects of the step barrier for kink particles, we compare
the DSB and BDSB models using a null barrier for bonded particles (Ebn = 0). In this
case, BDSB includes an ES barrier only for free particles in descending steps while in
DSB model the ES barrier is always present in downward movements. Figure 2 shows
the surface morphologies for three temperatures after a deposition of 100 monolayers. A
fixed step barrier of 0.05 eV is used for free particles in BDSB model and for downward
diffusion in DSB model. Both models exhibit mounded morphologies, qualitatively
similar at low temperatures (see T = 473 K, for example). However, when temperature
is increased, the surfaces of DSB model become anomalously mounded, with unrealistic
columnar structures and huge steps while regular mounded morphologies are observed
for BDSB. If the deposition time is increased, the surface for T = 523 K also develops
columnar mounds in DSB model, but does not for BDSB, as can be seen in the videos 1
and 2 of the supplementary material. Increasing step barrier, representing more realistic
values in many systems, strongly enhances the anomaly in the DSB whereas BDSB
remains well-behaved (data not shown).
The DSB unrealistic behaviour at high temperatures has a simple explanation
formerly realized by Villain [29]: For a particle in a descending step (particle C, Fig. 1),
the chances of attachment in an ascending step or of a new terrace nucleation in the
same layer are greater than the chances of attachment in the subjacent step due to ES
barrier. However, at high temperatures, when a particle in an ascending step (particle
A, Fig. 1) has a non-negligible mobility, the detachment and climbing chances are equal
and the asymmetry of the rule for kink particles (particle B, Fig. 1) produces a large
destabilizing uphill current. This result shows that the step barrier in the kink particles
plays a central role in the determination of the mound morphology.
A basic quantity used to characterize the surface morphology is the height-height
correlation function defined as [21, 22]
Γ(r) = hh(x)h(x + r)ix ,
(5)
where h(x) is the surface in the mean height reference andp
h· · ·ix is the average over the
surface. The interface width (rms roughness) w is given by Γ(0) and the first maximum
of Γ(r) measures an average distance between mounds δ. Figure 3(a) shows the heightheight correlation function against distance for systems at different temperatures. The
curves were rescaled by Γ(0) to improve visibility. The oscillating correlation function
is the hallmark of mounded surfaces [1, 21]. Figure 3(b) shows the surface width and
mound separation against temperature for both models. The interface width grows
7
Leal et al., Kinetic modelling of epitaxial film growth
2
1.2
10
1.2
36
0.8
δ (ML)
Γ/Γ(0)
0.8
473 K
523 K
573 K
0.4
24
-0.4
0
10
20
30
40
50
60
70
w (ML)
Γ/Γ(0)
0.0
0.4
30
80
18
480 510 540 570 600 630
T (K)
1
10
r (ML)
BDSB
DSB
0
-0.4
0
0
10
20
30
40
50
60
70
80
r (ML)
10400
450
500
550
600
650
T (K)
(a)
(b)
Figure 3. Surface morphological characterization for BDSB and DSB models.
(a) Height-height correlation function after the deposition of 100 ML at distinct
temperatures. Results for BDSB and DSB models are shown in the main plot and
inset, respectively. (b) Interface width (main plot) and characteristic mound separation
(inset) against temperature. The fixed parameters are the same used in Fig. 2.
exponentially with temperature in both cases meaning a temperature driven instability,
but the DSB produces anomalously huge surface widths in the investigated temperature
interval. The characteristic mound distance is closely constant for DSB and grows
monotonically with temperature for BDSB, as shown in the inset of Fig. 3(b).
It is important noticing that DSB model produces a closely constant mound
separation and a fast increase of the surface width with temperature. Morphologically,
we have a strong increase of the mound heights and grooves. At higher temperatures
(not shown), the uphill currents are so intense that parts of the initial substrate remain
uncovered, and highly columnar structures are completely formed upon the template of
sub-monolayer islands.
The growth of three-dimensional structures involves a complex relation between the
down- and upward fluxes through the steps in addition to the adsorption and nucleation
mechanisms. We quantify the net flux though steps using an out-of-plane current defined
by
1 X
1 X
Jz =
D(n
;
T
)
Θ[∆(j, j ′ )]Ph (j, j ′ ; T ),
(6)
j
2L2 j
q ′
j ∈V(j)
in which the sum over j runs over all sites while the sum over j ′ runs over all q nearest
neighbours of j. The factor 1/2 accounts for the double counting of bonds (j, j ′ ). The
factor D(nj ; T ) is given by Eq. (3), Ph (j, j ′ ; T ) by Eq. (4), and Θ(x) returns the sign
of x and Θ(0) = 0. This quantity is the average interlayer diffusion rate per site since
the intralayer diffusion is not counted. In order to reduce statistical fluctuations, the
current at a time t is the average over a short interval δt around t. The currents for
the BDSB and DSB models are compared in Fig. 4. For low temperatures, the currents
show a downward (negative) flux in both models that is attenuated in the DSB due to
8
Leal et al., Kinetic modelling of epitaxial film growth
0.00
0.3
T = 473 K
Jz (arb. unit.)
-0.02
2.0
BDSB
DSB
0.2
1.5
0.1
-0.04
1.0
T = 523 K
T = 573 K
0.0
-0.06
0.5
-0.1
-0.08
-0.10
0.0
-0.2
20
40
60
t (ML)
80
100
20
40
60
t (ML)
80
100
-0.5
20
40
60
80
100
t (ML)
Figure 4. Out-of-plane currents against time for BDSB and DSB models at distinct
temperatures. Step barriers are the same used in simulations shown in Fig. 2. Symbols
represent simulations and solid line sigmoid regressions as guides to the eyes. The data
correspond to averages over 50 samples and the error bars smaller than symbols were
omitted.
the absence of barrier to upward diffusion. The flux tends monotonically and slowly to
zero. Apart of a short initial transient, the data is very well fitted by a Hill function
tη
(7)
Jz = −j0 + j0
A + tη
with exponents η ≈ 0.34 and η ≈ 0.48 for BDSB and DSB, respectively. For intermediate
temperatures, the BDSB current also grows monotonically to a zero flux while a very
large positive current is observed in DSB. For high temperatures, both models have
positive currents, but the currents in BDSB are much smaller than in DSB.
Slope selection is an important feature in mound formation [14, 15, 16], where the
ratio between characteristic width and height tends to a constant value at long times. In
terms of continuous equation approaches, the slope selection involves a balance between
up- and downhill currents [15, 16] that is strongly violated in DSB model at intermediate
and high temperatures, as shown in Fig. 4. The current in the BDSB model at low and
intermediate temperatures, on other hand, grows monotonically to a balance steadystate implying slope selection. The decaying and positive current observed at high
temperatures also states slope selection for t → ∞, but the time required to observe
it is very large due to a very slow decay (Fig. 5(b) shows a quantitative analysis for
other parameters). The absence of a characteristic slope in DSB model is also evident
in Fig. 2 due to the anomalous columnar morphology. Video 2 of the Supplementary
Material further illustrates the absence of slope selection in DSB model.
4. KMC simulations of the BDSB model
Now, we concentrate in the effects of the uphill step barrier. The imbalance that
promotes the uphill destabilizing current and mound formation can be controlled with
the step barriers for free and bonded particles. Fig. 5(a) shows the surface width and
characteristic length for the interesting symmetric case, in which the barrier is the
same for all particles, Eb0 = Ebn = 0.06 eV. The interface width is strongly reduced
9
Leal et al., Kinetic modelling of epitaxial film growth
ξ
1
ζ = 0.22
w, ξ (ML)
10
β = 0.31
473 K
523 K
553 K
10
573 K
0
0.6
w
w
0.4
0.2
623 K
0
10
5
t
-1
10
10
1
2
15
3
10
473 K
20
4
10
10
t (ML)
(a)
10
0
523 K
−Jz (arb. unities)
473 K
523 K
553 K
-1
10
553 K
(c)
10
-2
1
10
2
3
10
10
4
10
t (ML)
(b)
Figure 5. Surface dynamics for symmetric BDSB model with Eb0 = Ebn = 0.06 eV.
(a) Interface width and characteristic length against time. Inset shows the interface
width at short times. (b) Out-off plane current against time. Solid lines are power
regressions and error bars smaller than symbols were omitted. (c) Surface morphologies
after deposition of 103 ML.
if compared with the asymmetric barrier case shown in Figs. 2 and 3. Therefore, an
important role of the uphill step barrier is the stabilization and smoothing of surfaces
implying reduction of the three-dimensional structure amplitudes.
At high temperatures, the interface width presents an initial oscillating behaviour
of a layer-by-layer growth regime as shown in the inset of Fig. 5(a). At longer times, the
layer-by-layer growth is replaced by a kinetic roughening featured by asymptotic power
laws w ∼ tβ and ξ ∼ tζ , with exponents β ≈ 0.31 and ζ = 0.22. Here, the characteristic
length is defined as the first zero of height-height correlation function [21, 22]. The
coarsening exponent obtained in our simulations is close to that observed in a model
with a barrier dependent on the next nearest neighbours [14]. Our simulations point out
that the exponents are independent of temperature and barrier strength, even though the
asymptotic scaling regime of the characteristic length was not reached in our simulations
Leal et al., Kinetic modelling of epitaxial film growth
10
at high temperatures. The layer-by-layer phase becomes longer for higher temperatures
since, in these situations, the particles have enough energy to overcome the edge barrier
and to bind to highly coordinated sites at the adjacent layers. The kinetic roughening
was formerly explained by Villain [29] in terms of the particle edge reflection caused by
the ES barrier to the downward movement, as discussed in Sec. 3. For the symmetric
barrier model, the origin of the kinetic roughening is similar but the imbalance between
downward and upward currents is additionally because of particles in ascending steps
(bonded by definition) are, in an average, less diffusing than those in descending ones.
Notice that symmetric barriers hinder mound formation in the WV model in d = 1 + 1
dimensions [27]. Mound formation was observed in the WV model in d = 2 + 1 without
step-barrier, but the artefact of noise reduction was necessary to overcome the very slow
crossover to the asymptotic limit [22].
Figure 5(a) also shows a re-entrant behaviour of the interface width as function of
temperature for intermediate growth times t ≃ 100 − 300 ML. Re-entrant behaviours,
which mean an interface width firstly increasing and then decreasing with temperature
(no monotonic dependence), were reported in a classical experiment [30] and the
corresponding kinetic modelling [31, 32] of the growth of Ag/Ag(100), among other
systems [1]. While specific models were devoted to describe this complex behaviour [1,
31, 32], it has appeared spontaneously in our simulations during the crossover between
layer-by-layer and kinetic roughening growth regimes.
Interlayer currents against time for the symmetric barrier simulations are shown
in Fig. 5(b). The results differ from the asymmetric barrier model since the current
is always negative and approaches zero as a power law Jz ∼ −t−η with exponents
η = 0.1 − 0.25. This algebraic decay means that the balance between down- and
uphill diffusion, theoretically required for slope selection, will be reached only at
infinite times when the interface widths are already saturated. In fact, the growth and
coarsening exponents obtained in our simulations obey the relation β > ζ that implies
an aspect ratio of the mounds (height/width) increasing with time and, consequently,
no slope selection is strictly observed. A growth exponent greater than the coarsening
exponents also means that the surface is described by the so-called super-roughness
scaling regime [13] for which the roughness exponent α = β/ζ > 1. Generic scaling
theory states that α > 1 implies in locally smooth surface where the interface width in
a scale of size ǫ scales linearly as w ∼ ǫ [13].
Surface morphologies obtained with symmetric barriers at distinct temperatures are
shown in Figure 5(c), where one can clearly see three-dimensional structures and mounds
for the investigated temperature range. The mound shape is approximately pyramidal
for the lower temperature (T = 473 K) and becomes dome-like for the higher ones.
These finds are consistent with the theoretical claim of a null current requirement for
slope selection, since the pyramidal shape with barely well-defined slopes was observed
for the smallest current intensity (T = 473 K) while for the largest current intensity
(T = 553 K), a signature of slope selection is not evident.
The symmetric barrier is the simplest and, for this reason, the most interesting
11
Leal et al., Kinetic modelling of epitaxial film growth
2
10
w (ML)
1.0
w (ML)
1
0.6
0.4
0.2
10
∆Eb = −∞
∆Eb=−0.02 eV
∆Eb=0.00
∆Eb=+0.02 eV
∆Eb=+0.04 eV
0.8
0
5
10
β = 0.31
20
15
t (ML)
0
10
10
1
10
2
3
10
t (ML)
(a)
(b)
Figure 6. (a) Interface width against time for fixed T = 573 K and Eb0 = 0.06 eV
and distinct relative step barriers, ∆Eb = Eb0 − Ebn , decreasing from top to bottom.
The curves represent averages over 10 independent samples. (b) Surface morphologies
after the deposition of 100 ML.
case that produces mound formation. However, we can go beyond this result and
investigate the role of other mechanisms by exploring different barriers for bonded
and free particles, since there is no restriction imposing equal barriers to both kinds
of particles. The interlayer diffusion of bonded particles also plays an essential role
for mound formation. In Fig. 6(a), we investigate the relation between step barriers
for bonded and free particle by analysing the interface width against time at a growth
temperature of T = 573 K and a fixed step barrier for free particles Eb0 = 0.06 eV.
We define a relative step barrier as ∆Eb = Eb0 − Ebn . When the relative barrier is
positive, mounded morphologies with a large interface width are obtained as can be
seen in the bottom snapshots of Fig. 6(b). For a negative relative barrier, the surface
has a long phase, lasting for several monolayers, with an almost layer-by-layer growth
but the kinetic roughening is still observed for sufficiently long times. Videos 3 and 4 of
the supplementary material show the surface evolution for negative and positive relative
Leal et al., Kinetic modelling of epitaxial film growth
12
T = 473 K
T = 523 K
T = 573 K
Figure 7. Surfaces at different temperatures obtained after a deposition of 100 ML
for the simple SOS (left panels) and WV (right panels) deposition rule.
barriers. The kinetic roughening regime is absent in the extreme case of a forbidden
interlayer diffusion of the bonded particles (Ebn = ∞) as shown in the most top left
snapshot of Fig. 6(b) and in Fig. 6(a). Notice that varying the relative step barrier from
-0.02 to 0.04 eV, we were able to reproduce a rich variety of spatio-temporal behaviours
ranging from layer-by-layer to self-assembled three-dimensional structures. Finally, it
is worth noticing that the morphology dynamics does not change considerably for low
temperatures when the relative step barrier is varied (data not shown) since, in theses
cases, the diffusion of bonded particles is itself negligible.
In principle, our model has two competing mechanisms for mound formation: the
deposition with a transient mobility and the thermally activated diffusion. However,
the transient mobility contribution to mound formation is very weak as shown for the
WV model [22] and becomes negligible when compared with the diffusion contribution
at temperatures investigated in the present work. However, the transient mobility
mechanism is still important to prevent anomalous grooves in surface, particularly,
at the lower investigated temperatures. In Fig. 7, we compare surface morphologies
generated using a transient mobility deposition rule with simulations using a simple
SOS deposition, where the particles are deposited directly in a randomly selected site.
Notice that both models generate mounded surfaces even for a small amount of deposited
material. As one can see, the surfaces are equivalent at higher temperatures, but at lower
temperatures, the surfaces for a simple SOS deposition are featured by grooves that do
not correspond to real epitaxial systems. In summary, mound formation is controlled
Leal et al., Kinetic modelling of epitaxial film growth
13
by the step diffusion rules, but the transient mobility in the deposition works as an
important smoothing mechanism at lower temperatures.
5. Conclusions
We have investigated the mound formation in kinetic Monte Carlo simulations of a
model for epitaxial film growth with thermally activated diffusion. The model has a step
barrier, generically known as Ehrlich-Schwoebel barrier [19, 18], for interlayer diffusion
depending on the number of bonds of the adsorbed particle. The step barriers are
present for interlayer diffusion when a particle is moving up- or downwardly. Moreover,
the rules does not explicitly distinguish between up- and downward diffusion.
Simulations of our bond dependent step barrier (BDSB) model were compared
with the standard model having only a downward step barrier (DSB). We have shown
that DSB and BDSB are equivalent at low temperatures, when the interlayer diffusion
of bonded particles is small. However, if we increase the substrate temperature in
a range compatible with several experimental situations (typically ∆T & 100 K),
the surfaces obtained with the standard model become unrealistic, forming columnar
structures with anomalously huge interface widths and step heights. Our model, on
other hand, generates well-behaved mounded morphologies, considering a null step
barrier for bonded particles. The analysis of the interlayer current shows that DSB has
an anomalously large positive (upward) current responsible by the unstable columnar
growth, whereas the current in the BDSB model tends to a steady state of balance
theoretically required for mound slope selection [15, 16]. When we consider our model
with barriers only in descending steps, we observed regular mounded morphology
whenever the barrier for bonded particles is smaller than that for free particles. This
important result suggests that different barriers for bonded and free particles may be
present in real systems.
The case of equal barriers for bonded and free particles in BDSB model was also
investigated. This situation is particularly interesting due to the symmetry between
down- and upward step barriers (Fig. 1). In odds with previous reports for WolfVillain model with an ES barrier in d = 1 + 1 dimensions [27], the BDSB model
also produces mounds for a symmetric step barrier. A negative interlayer current
algebraically approaching a null value (power law decay) was observed, meaning that
the balance between down- and uphill diffusion will be reached only at infinite times
when the interface widths are already saturated. We have determined the evolution
of the interface width and characteristic length in order to characterize the kinetic
roughening regimes. The growth and coarsening exponents observed were β ≈ 0.31 and
ζ ≈ 0.22, respectively. These exponents correspond to the so-called super-roughness
scaling regime where α = β/ζ > 1.
In the asymmetric case, distinct relative barriers ∆Eb = Eb0 − Ebn were used. For
a negative relative barrier, a long phase with a layer-by-layer growth is observed at
early times and a crossover to a kinetic roughening emerges at long times. Nearby this
Leal et al., Kinetic modelling of epitaxial film growth
14
crossover, a re-entrant dependence of the interface width with temperature is observed.
This behaviour was also experimentally observed in the growth of Ag/Ag(100) [30],
among other systems [1].
Acknowledgments
This work was partially supported by the Brazilian agencies CNPq and FAPEMIG. SCF
thanks the kind hospitality at the Departament de Fı́sica i Enginyeria Nuclear/UPC.
References
[1] Evans J W, Thiel P A and Bartelt M 2006 Surf. Sci. Rep. 61 1–128
[2] Jorritsma L C, Bijnagte M, Rosenfeld G and Poelsema B 1997 Phys. Rev. Lett. 78 911–914
[3] Caspersen K J, Layson A R, Stoldt C R, Fournee V, Thiel P A and Evans J W 2002 Phys. Rev.
B 65 193407
[4] Han Y, Ünal B, Jing D, Qin F, Jenks C J, Liu D J, Thiel P A and Evans J W 2010 Phys. Rev. B
81 115462
[5] Johnson M D, Orme C, Hunt A W, Graff D, Sudijono J, Sander L M and Orr B G 1994 Phys.
Rev. Lett. 72 116–119
[6] Tadayyon-Eslami T, Kan H C, Calhoun L C and Phaneuf R J 2006 Phys. Rev. Lett. 97 126101
[7] Zorba S, Shapir Y and Gao Y 2006 Phys. Rev. B 74 245410
[8] Hlawacek G, Puschnig P, Frank P, Winkler A, Ambrosch-Draxl C and Teichert C 2008 Science
321 108–111
[9] Michely T and Krug J 2003 Islands, Mounds, and Atoms: Patterns and Processes in Crystal
Growth Far from Equilibrium (Berlin: Springer–Verlag)
[10] Hamouda A B H, Pimpinelli A and Phaneuf R J 2008 Surface Science 602 2819 – 2827
[11] Ferreira S O, Bueno I R B, Suela J, Menezes-Sobrinho I L, Ferreira S C and Alves S G 2006 Appl.
Phys. Lett. 88 244102
[12] Shim Y, Borovikov V and Amar J G 2008 Phys. Rev. B 77 235423
[13] Ramasco J J, López J M and Rodrı́guez M A 2000 Phys. Rev. Lett. 84 2199–2202
[14] Šmilauer P and Vvedensky D D 1995 Phys. Rev. B 52 14263–14272
[15] Siegert M and Plischke M 1994 Phys. Rev. Lett. 73 1517–1520
[16] Schinzer S, Köhler S and Reents G 2000 Eur. Phys. J. B 15 161–168
[17] López J M, Rodrı́guez M A and Cuerno R 1997 Phys. Rev. E 56 3993–3998
[18] Schwoebel R L and Shipsey E J 1966 J. Appl. Phys. 37 3682–3686
[19] Ehrlich G and Hudda F G 1966 J.Chem. Phys. 44 1039–1049
[20] Amar J G and Family F 1996 Phys. Rev. Lett. 77 4584–4587
[21] Murty M R and Cooper B 2003 Surf. Sci. 539 91–98
[22] Chatraphorn P P, Toroczkai Z and Das Sarma S 2001 Phys. Rev. B 64 205407
[23] Pierre-Louis O, D’Orsogna M R and Einstein T L 1999 Phys. Rev. Lett. 82 3661–3664
[24] Clarke S and Vvedensky D D 1987 Phys. Rev. Lett. 58 2235–2238
[25] Clarke S and Vvedensky D D 1988 J. Appl. Phys. 63 2272–2283
[26] Leal F F, Ferreira S C and Ferreira S O 2011 J. Phys.: Condens. Matter 23 292201
[27] Rangdee R and Chatraphorn P 2006 Surf. Sci. 600 914 – 920
[28] Wolf D E and Villain J 1990 Eur. Lett. 13 389
[29] J Villain 1991 J. Phys. I France 1 19–42
[30] Stoldt C R, Caspersen K J, Bartelt M C, Jenks C J, Evans J W and Thiel P A 2000 Phys. Rev.
Lett. 85 800–803
[31] Bartelt M C and Evans J W 1999 Surface Science 423 189–207
Leal et al., Kinetic modelling of epitaxial film growth
[32] Shim Y and Amar J G 2010 Phys. Rev. B 81 045416
15