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Scaling relation of domain competition on (2+1)dimensional ballistic deposition model with surface
To cite this article: Kenyu Osada et al 2016 J. Semicond. 37 092001
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Vol. 37, No. 9
Journal of Semiconductors
September 2016
Scaling relation of domain competition on (2C1)-dimensional ballistic deposition
model with surface diffusion
Kenyu Osada1 , Hiroyasu Katsuno2; 3; Ž , Toshiharu Irisawa2 , and Yukio Saito4
1 Department
of Physics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
Centre, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
3 Department of Physical Science, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan
4 Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan
2 Computer
Abstract: During heteroepitaxial overlayer growth multiple crystal domains nucleated on a substrate surface compete with each other in such a manner that a domain covered by neighboring ones stops growing. The number
density of active domains decreases as the height h increases. A simple scaling argument leads to a scaling law of
h with a coarsening exponent D d /z, where d is the dimension of the substrate surface and z the dynamic
exponent of a growth front. This scaling relation is confirmed by performing kinetic Monte Carlo simulations of the
ballistic deposition model on a two-dimensional (d D 2) surface, even when an isolated deposited particle diffuses
on a crystal surface.
Key words: domain competition; ballistic deposition model; Kardar-Parisi-Zhang universality class; surface diffusion
DOI: 10.1088/1674-4926/37/9/092001
PACS: 68.43.Jk
1. Introduction
In the field of heteroepitaxial crystal growth, much research has been devoted to reduce voids, dislocations and
cracks which are introduced due to the lattice mismatch between the deposited crystal and the substrate. These defects
degrade a deposited crystal quality and lower its performance
as electronic and optoelectronic devicesŒ1 . To avoid these defects and to improve adsorbate crystal quality, some measures
are proposed such as to introduce a low-temperature buffer
layerŒ2; 3 , to apply an epitaxial lateral overgrowth methodŒ4
or to grow adsorbate crystal on nano-patterned substrateŒ5 . All
these methods are expected to select only a few crystal domains
to survive and to improve an adsorbate crystal quality.
Recently, Saito et al. have studied the domain competitionŒ6 during the ballistic deposition (BD) on a onedimensional (1D) substrate surface patterned with nano pillarsŒ5 . They investigated the crystal domain shape growing
on top of a nano pillar and the effect of the surface diffusion of particles on domain competition. By assuming that the
growth front is self-affineŒ7 , the new scaling relation of domain competition is proposed, and it is confirmed by means of
kinetic Monte Carlo (KMC) simulations of an extended 1D-BD
model. However, since the real system is three-dimensional,
it is also intriguing to check the validity of the scaling relation for the deposition on a two-dimensional substrate surface.
It is also known that the BD model belongs to the universality class of the Kardar-Parisi-Zhang (KPZ) equationŒ8 11 , and
some experimental realizations of the KPZ interface have recently been reportedŒ12; 13 . Therefore, the present study may
also have some relevance to the KPZ systems.
In this paper, we study the domain competition problem
by using a two-dimensional (2D) BD model; substrate surface
is two dimensional and deposited particles grow in the third
dimension. In general, adsorbed crystals in the BD model contain many voids and the crystal density is very low. In order
to grow compact adsorbate crystals with a high density, we
extend the BD model by including a diffusion process such
that an isolated particle diffuses on the adsorbate crystal surfaceŒ6 . By performing KMC simulations of the extended BD
model, we investigate dynamic exponents of the growth front
of the BD model in (2C1)-dimensions. Using obtained exponents, the scaling relation of domain competition for (2C1)dimensions is analyzed. Our result connects the growth kinetics
at the growth front and the crystal domain size in real systems.
2. Scaling relation of domain competition
The scaling relation of the domain competition was previously studied theoretically and numerically in a 1D systemŒ6 .
Here, we extend the theory to a d -dimensional system.
Adsorbate particles are deposited vertically to the substrate
surface of a size Ld . The position of the growth front of an
adosrbate crystal at a time t is written as hi .t/, where i is a position of the substrate surface. The average height of the growth
front is defined as:
1 X
h.t / D d
hi .t /;
L i
and it is proportional to time under a constant deposition flux.
The width of the growth front is:
u 1 X
Œhi .t / h.t/2 :
W .t / D t d
L i
† Corresponding author. Email:
Received 16 March 2016, revised manuscript received 19 April 2016
© 2016 Chinese Institute of Electronics
J. Semicond. 2016, 37(9)
Kenyu Osada et al.
When the average height h.t/ is small, W .t/ increases in proportion to hˇ with the growth exponent ˇ. When h.t/ is large,
W .t/ is saturated to a finite value which depends on a linear
size of the system L as W L˛ with the roughness exponent
˛. The crossover between the two behaviors occurs at hLz
with the dynamic exponent z D ˛/ˇ Œ7 .
We adopt the ballistic deposition rule for a deposited particleŒ8 ; it solidifies when it touches the substrate or other solidified adsorbates. In addition, a deposited particle carries a domain index of a substrate or a solidified particle that it touches.
If a deposited particle touches many solid particles, the domain
index of the underlying solid is selected. If there is no underlying solid, the domain index is selected among horizontally
neighboring solid particles with equal probabilities.
When a domain is covered by neighboring ones, it loses
supply of deposited particles from above and stops growing: it
becomes inactive. Instead, those domains which are active can
be identified in the top view of the growth front. The domain
number density is defined as the number of active domains
divided by the system size Ld . It is expected to decrease as
height h increases in a scaling form as:
From Equations (3) and (4), we obtain the scaling relation of
the domain competition:
z D d:
This scaling relation is found valid for the BD model for d D 1
by Saito et al.Œ6 . Here we study if this relation holds in d D 2
case even with a surface diffusion of isolated particles.
3. BD model with surface diffusion
To test the validity of the scaling relation of the domain
competition Equation (5) in three dimensions, we introduce the
(2C1)-dimensional BD model. The substrate surface is initially
flat, and particles are deposited to it vertically. The lateral position of a deposited particle r D (i , j ) is selected randomly,
and its height hi;j .t/ is changed in the following manner:
The growth exponent ˇ is obtained from the slope of the front
width W versus height h. The exponent ˛ is conventionally obtained by comparing saturation values of front width for various system sizes. However, there is another method to estimate
the exponent ˛ for a given system size by using the heightdifference correlation functionŒ14 , G.t , r). It is defined by:
G.t; r/ D fŒh.t; r 0 C r/
with a coarsening exponent . The density decrease suggests a
coarsening such that average domain size from the top view becomes large. Domain boundaries fluctuate on the growth front
at random, and when they merge, an enclosed domain disappears. Thus the domain size is expected to be related to the
lateral correlation length of the growth front. By assuming that
the growth front is self-affine such that every lateral length is
characterized by a single scale of a correlation length , which
is proportional to h1=z , then the average domain size 1=d
scales as:
1=d h1=z :
hi;j .t C 1/ D maxfhi;j .t/ C 1; hi
Figure 1. Surface diffusion of an isolated particle at a site O. It can
move to either a site A or B. If an isolated particle moves to B, then
it stops motion by forming a lateral bond.
1;j .t/; hi C1;j .t/;
1 .t/; hi;j C1 .t/g;
where the function max provides the maximum value in the
To investigate the scaling relation of the domain growth,
values of exponents ˛ and ˇ of the growth front are necessary.
h.t; r 0 /2 gr 0 ;
where r represents the distance between two points on the substrate, and < >r 0 represents an average over all the lattice
points r . Assuming the self-affine form of the growth front,
the height-difference correlation function varies as:
G.t; r/ D r 2˛ G.r=t
The scaling function G.x/
approaches a constant G.0/
asymptotically as x! 0, and Equation (8) reduces to G.r/ r 2˛ asymptotically as t! 1. From the distance dependence
of the asymptotics of the height-difference correlation function
G.r/, we can obtain the exponent ˛ without varing a system
In general, an aggregate produced by BD contains many
voids in it. To the contrary, the real crystal is dense. In order
to resolve the discrepancy, we extend the BD model to incorporate surface diffusion of isolated particles. The surface diffusion makes the growth front smooth and the crystal density
large. An isolated particle without any lateral bonds, as a particle at a site O in Figure 1, can move to the neighboring site
laterally. When an isolated particle comes in lateral contact
with another particle, it solidifies and stops moving. The dimensionless diffusion rate is defined as DQ D D=f a4 , where
D is the diffusion constant of an isolated atom on the surface,
f is a deposition flux of particles and a is a lattice constant.
The flux and the lattice constant are set to unity in this paper.
In general, the motion of an isolated particle to a terrace of a
different height, for instance, from the site O to B in Figure 1,
is disfavored compared to the lateral motion in the same level,
from O to A. This effect is represented by an extra energy barrier for the surface diffusion, the so-called Ehrlich-Schwoebel
(ES) barrierŒ15; 16 . To promote surface smoothening, we simply
ignore this additional ES barrier so that the rate of the diffusion
Q On a site B, a partoward a lower terrace is the same rate D.
ticle makes a lateral bond and stops motion for ever. By the
surface diffusion so defined, the growth front becomes flat and
the crystal density increases.
J. Semicond. 2016, 37(9)
Kenyu Osada et al.
To simulate the processes stated above, we use a rejectionfree algorithm. The transition rate related to the diffusion process is given by D=a2 times the number of isolated particles,
and the transition rate of deposition with a flux f is given by
f L2 . The total transition rate is the sum of the above two
processes, and the next process carried out is determined by the
n-fold wayŒ17 . With this Monte Carlo step, the time increases
by the inverse of the total transition probability.
4. Simulation result
To discuss the scaling behavior in the domain competition,
it is necessary to know exponents of the growth front in the 2D
BD model. In the following subsections, we first study kinetic
roughening of the growth front of the BD model without and
with surface diffusion. Then, the process of domain coarsening
is studied. Specifically, our interest lies in if the scaling relation
Equation (5) between the coarsening exponent and the dynamic
exponent holds even under the existence of surface diffusion
4.1. Growth front dynamics of BD model without surface
Figure 2(a) shows the variation of the growth front width
W of a BD model without surface diffusion as the average
height h increases in a log-log plot. The system sizes shown are
L D 256 (circles), L D 512 (boxes) and L D 1024 (crosses),
respectively. A data point is obtained by averaging 100 samples. The initial rapid increase of a front width collapses on a
single line irrespective of the system size, and is well fitted to
a power law W h1=2 , represented by a dashed line, up to h 10. This behavior is similar to that of the random deposition,
since the substrate is flat and deposited crystal branches do not
mutually interfere initially.
When h > 10, the front width crosses over to the BD
behavior, and eventually saturates at values depending on
the system size. For a given system size L, we assign
the maximum value of the local slope in the log-log plot,
max(log[W .h1 //W .h2 /]/log[h1 /h2 ]), to the growth exponent
ˇ.L/. We thus obtain ˇ.L D 1024) D 0.16. A solid line in
Figure 2(a) represents a power law behavior W h0:16 , and it
fits quite well with the front width behavior of a system size
L D 1024. The obtained value of the exponent ˇ D 0.16 is
close to the value reported previously for small systemsŒ18 .
However, the exponent of (2C1)-dimensional BD model with
a very large system size up to 214 214 is recently reported
to be ˇ D 0.24Œ11 , the same value as other models belonging to the KPZ universality classŒ19 and with the (2C1) KPZ
modelŒ20 . Our value ˇ.L D 1024) D 0.16 is smaller than that
because of the smallness of our simulated size.
We now consider the evaluation of another exponent ˛.
Figure 2(b) shows the distance dependence of the asymptotics
of the height-difference correlation function G.r/ in a log-log
plot. G.r/ is calculated by using data of the front height configuration when the front width is saturated. The roughness exponent ˛.L/ is obtained from the maximum local slope of the
logG.r/ versus log r for a given system size L. From Figure
2(b), we obtain the roughness exponent ˛.L D 1024) D 0:27,
for example. The solid line which represents a power law be-
Figure 2. (Color online) Kinetic roughening of the growth front of the
BD model without diffusion. (a) Log-log plot of growth front width
W versus average height h for various system sizes L. (b) Log-log
plot of asymptotic height-difference correlation function G.r/ versus
distance r for various system sizes. (c) Sum of scaling exponents ˛ Cz
versus system size L. The value asymptotically approaches a constant
2. Every data point is obtained by averaging 100 samples.
havior G.r/r 0:54 agrees quite well with the height correlation G.r/ for a large system size L D 1024. The value ˛.L D
1024) D 0.27 is compatible with the value ˛ D 0.28 obtained
previously for small systemsŒ18 , but is smaller than the value
˛ D 0.39 for large systemsŒ11 as well in the case of the growth
exponent ˇ. From the values of exponents ˛ and z obtained
for various system sizes L, we calculate the dynamic exponent
z.L/ D ˛/ˇ, and plot the sum ˛ C z in Figure 2(c) for various
system sizes. It is larger than 1.8 for L > 256, and tends to saturate to a constant value 2. This result is compatible with the
J. Semicond. 2016, 37(9)
Kenyu Osada et al.
Figure 3. (Color online) Kinetic roughening exponents, (a) ˛ (circle) and ˇ (triangle), and (b) z (circle) and ˛C z (triangle) of the growth front
Q The system size is set L D 256.
versus surface diffusion rate D.
KPZ relation ˛KPZ C zKPZ D 2, even though each exponent, ˛
or ˇ, differs from the value of (2C1) KPZ universality class.
As a whole, our simulations of BD model without surface diffusion well reproduce the results by Saberi et al.Œ18 , even ˛ is
differently estimated.
4.2. Effect of surface diffusion on the growth front dynamics of the BD model
We extend a 2D BD model to include an effect of surface
diffusion of isolated particles. Scaling exponents of the growth
front are evaluated in the same manner in the previous subsection. Since we have confirmed in the previous subsection
without surface diffusion that the system size L > 256 is large
enough to estimate the scaling exponents, we here choose a system size L D 256 to study surface diffusion effect on dynamics
of kinetic roughening.
Figure 3(a) shows the scaling exponents, ˛ and ˇ at various
Q If the diffusion is absent, the values of expodiffusion rate D.
nents are ˛ D 0:19 and ˇ D 0:12 for the size L D 256. With a
Q both exponents ˛ and ˇ weakly depend
finite diffusion rate D,
on D with broad maxima around DQ 100. At
p this value of the
diffusion rate, the diffusion length is about 4DQ 20, much
smaller than the system size L D 256. Therefore, the finite
size effect cannot be the origin of this exponent’s increase. For
Q both exponents start to decrease, and even becomes
larger D’s,
smaller than the values obtained in the case without surface
diffusion. In an extreme case when the diffusion length 4DQ
becomes of the order of the system size, or DQ 104 , the effect of the periodic boundary condition comes into play. Since
the periodic boundary condition is known to lead an underestimation of scaling exponentsŒ8 , the decrease of the estimated
exponents at large DQ might reflect this finite size effect.
In the case of a solid-on-solid model, simulations with a
surface diffusion show a decrease of ˇ with an intermediate
small plateau on increasing the diffusion rateŒ8; 21 . There, the
plateau value is interpreted to be the value of an exponent of
the growth front dynamics with surface diffusion. In case of
our BD simulation with surface diffusion, variation of exponents ˛ and ˇ is more complicated as shown in Figure 3(a), and
one cannot rely on the same interpretation straightforwardly.
Even though we lack a clear physical picture of this complex
behavior of exponents, ˛ and ˇ, they increase or decrease simultaneously. The ratio z D ˛=ˇ is almost constant except for
Q as shown in Figure 3(b). The summation ˛ C z stays
large D,
larger than 1.8, and is close to 2 except for a very large DQ as
DQ D 104 as shown in Figure 3(b). We thus confirm that the
KPZ scaling relation ˛KPZ C zKPZ D 2 holds irrespective of the
surface diffusion.
4.3. Scaling relation of domain competition in (2C1)dimension
We are now ready to study dynamics of domain competition. We investigate the height dependence of the domain denQ The initial substrate is flat,
sity at various diffusion rates D.
and we assume that each lattice site on a substrate surface defines a different domain. When a deposited particle touches the
substrate, it inherits the domain index of the underlying substrate site. A domain increases its size by incorporating further
deposited particles from above. We have to keep track of the
domain indices of deposited particles in all the domains during
growth. Therefore, the system size is set rather small as L D
256 due to the constraint of storage space and of CPU time.
Figure 4 shows the vertical cross sections of domain competition during the BD (a) without and (b) with surface diffusion. The diffusion rate is DQ D 1024 in Figure 4(b). The crystal
density increases from 0.3 with DQ D 0 to 0.85 with DQ D 1024.
The surface diffusion makes crystal dense as expected. Different domains carry different colors, and domain competition and
coarsening is obvious in both cases.
We shall now quantitatively study domain coarsening.
Height dependence of the domain number density from the
top view is shown in Figure 5(a) for various diffusion rates
Q Circles, crosses and triangles correspond to data for DQ D
0, 256 and 1024, respectively. The domain number density decreases monotonically as the deposited crystal grows. The density variation is classified into three stages. In the initial stage,
the density remains close to unity, since neighboring domains
grow independently and do not yet compete with each other.
This initial stage corresponds to that of the growth front where
the front width W increases in proportion to h1=2 . In the intermediate stage, neighboring domains start to compete for deposited particles, and the number density starts to decrease in
J. Semicond. 2016, 37(9)
Kenyu Osada et al.
Figure 4. (Color online) Vertical cross sections of the domain coarsening during the 2D BD (a) without and (b) with surface diffusion with a
surface diffusion rate DQ D 1024. Colors differentiate domains. Crystals have a width L D 256 and grow up to the height h D 234 in this figure.
Figure 5. (Color online) Domain coarsening. (a) Log-log plot of domain density versus height h. The system size is L D 256. Circles, squares
and triangles are for DQ D 0, 256 and 1024. (b) Coarsening exponent versus the diffusion rate DQ of isolated particles. Each data point is
obtained by averaging 100 samples.
power laws, as expected in Equation (3). In the final stage,
only one domain survives after the competition and the density
eventually reaches the final value determined by the system
size, D 1=2562 1.5 10 5 . From Figure 5(a), the height
before the domain density starts to decrease is proportional to
Q However, the
the magnitude of the surface diffusion rate D.
height necessary to reach a single domain state is not proportional to the magnitude of the diffusion rate. The number density for the diffusion rate DQ D 256, for instance, decreases
more rapidly than that for DQ D 0. The rapid decrease of the
domain number density may be related to the enhancement in
the kinetic roughening exponents of the growth front when the
surface diffusion is included; the exponents ˛ and ˇ have a
broad maximum around DQ D 256, for a small size L D 256,
as shown in Figure 3(a).
From the intermediate-stage decay of the domain density
, the maximum of the local slope of log versus log h is assigned to be a value of a coarsening exponent , and it is plotted
for various diffusion rates DQ in Figure 5(b). initially remains
constant, then increasespwith a maximum at DQ 500. At this
DQ the diffusion length 4DQ 45 is smaller than the system
size L D 256, and the effect of the periodic boundary condition cannot be the origin of this maximum. The large value of
means speeding-up of coarsening, and it may be related to
the enhanced kinetic roughening, since roughening exponents
˛ and ˇ also increases around the same DQ as shown in Figure
3(a). Closer comparison of behaviors of and ˛ or ˇ, however,
reveals a little difference that remains constant up to DQ 6
100, whereas ˛ or ˇ are more susceptible even for a small D.
Finally, we examine the validity of the scaling relation,
Equation (5), of the domain competition. In Figure 6, the product z is shown for different values of the diffusion rate DQ
from results of Figure 3(b) and Figure 5(b). Error bars show
the standard deviation obtained from the law of propagation of
errors. The solid horizontal line shows the substrate surface dimension d D 2. The product zremains close to 2 for the wide
Q except a region around DQ 500,
range of the diffusion rate D,
where has a maximum. For still larger DQ >103 , the product
z again recovers to the universal value d D 2. Thus, we may
tentatively conclude that the product z takes a universal value
of space dimension d D 2 in the BD model irrespective of the
surface diffusion.
J. Semicond. 2016, 37(9)
Kenyu Osada et al.
Figure 6. (Color online) The product z with the coarsening expoQ Error bars represent the standard
nent for various diffusion rates D.
deviation. The system size is L D 256. Solid line represents d D 2.
5. Summary
We study domain competition during the deposition
growth of a three-dimensional crystal on a flat substrate. Kinetic Monte Carlo simulations of the 2D BD model are extended so as to include surface diffusion of isolated particles.
Surface diffusion makes the growth front smooth, and also
makes crystal dense such that a density about 0.3 of the model
without surface diffusion increases to the density more than
0.85 when a surface diffusion rate is enhanced to DQ D 1024.
A domain or a crystal grain is defined for each substrate
site, and deposited particles inherit domain index by touching
to the substrate or to already crystallized particles. A domain
stops growing when it is covered by neighboring ones, and the
number density of domains decreases as the deposited crystal
grows. This domain coarsening is characterized by the coarsening exponent .
From the simulations, exponents, ˛ and ˇ, of the kinetic
roughening of the growth front and of the domain coarsening
are obtained. Scaling exponents and z D ˛/ˇ depend weakly
on the diffusion rate DQ of isolated particles, but the scaling
Q With
relation z D d seems robust for a wide range of D.
a strong surface diffusion, the density of a grown crystal is as
high as crystal densities in real experiments. We expect that the
scaling relation z D d will be examined for real experimental
systems under deposition growth.
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