Multilevel modeling of the influence of surface transport peculiarities on growth, shaping, and doping of si nanowires

Physica E 40 (2008) 2446–2453
Multilevel modeling of the influence of surface transport peculiarities on
growth, shaping, and doping of Si nanowires
A. Efremov
a
, A. Klimovskaya
a,Ã
, I. Prokopenko
a
, Yu. Moklyak
a
, D. Hourlier
b
a
Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, 45 Nauki Avenue, 03028 Kyiv, Ukraine
b
Institute d’Electronique, de Microe
´
lectronique et de Nanotechnologies, ISEN, UMR-CNRS 8520, F-59652 Villeneuve d’Ascq, France
Available online 15 February 2008
Abstract
The growth, shaping, and doping of silicon nanowires in a catalyst-mediated CVD process are analyzed within the framework of a
multilevel modeling procedure. At an atomistic level, surface transport processes and adsorption are considered by MC simulations. At
the macroscopic level, numerical solutions of chemical kinetics equations are used to describe nanowire elongation growth and doping.
Both atomistic and kinetic considerations complementing each other reveal the importance of surface transport and the role of low-
mobility impurities present on the catalyst surface in the nanowire growth process. In particular, a controllable shaping and selective
doping of nanowires is possible by means of well-directed effects on the surface transport of both silicon and impurity adatoms. Some
nonlinear effects in the growth and doping caused by percolation-related phenomena are demonstrated.
r 2008 Elsevier B.V. All rights reserved.
PACS: 62.23.Hj; 81.10.Aj; 82.20.Wt; 05.10.Ln; 68.35.Fx; 66.30.Pa
Keywords: Nanowire growth; MC simulations; Kinetic modeling; Nanowire doping; Controllable shaping

1. Introduction
The semiconductor industry nowadays decidedly ap-
proaches the era of nanotechnologies. In view of prospects
for further decreasing sizes of circuitries and their elements,
a vital need arises in low-dimensional materials and objects
with strictly controlled electronic properties determined by
a nanostructure.
Silicon nanowires (NWs), grown in a catalyst-mediated
CVD process, are still considered to be the most promising
type of nano-objects. This fact is associated with a
significant amount of device applications in addition to
those already realized. However, a selective synthesis of
NWs possessing a desired structure and properties remains
a rather difficult problem so far.
To determine the best conditions for the growth of NWs
and for a better understanding of the role played by
different macro- and micro-parameters of the synthesis
(temperature, pressure, catalyst surface properties, growth-
limiting factors), it is necessary that numerical and
statistical simulation techniques be us ed more extensively.
This will enable us to predict the NW structure, shape and
size evolution, distribution of impurities, and other
relevant properties determined at the growth stage.
In this work, we consider some essential mechanisms of a
catalyst-mediated synthesis of NWs involving a surface
transport across the catalyst particle surface. We use
multilevel (combined) modeling as a tool. At an atomistic
level, it includes an Monte- Carlo (MC)-simulation of
selected key stages of the process. A macroscopic descrip-
tion of the growth process is based on a numerical solution

of a system of rate equations for chemical reactions and
mass transfer.
We will consider the influence of the mass transfer of
adatoms with different local mobilities (silicon itself,
conventional dopa nts, products of surface reactions, etc.)
across the catalyst surface on the growth, shape, and
doping of NWs.
Using the MC technique, we have obtained the
dependence of an effective coefficient of silicon adatom
surface diffusion on the surface coverage and catalyst
ARTICLE IN PRESS
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1386-9477/$ - see front matter r 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physe.2008.01.015
Ã
Corresponding author. Tel.: +38 044 5257091; fax: +38 044 5258342.
E-mail address: (A. Klimovskaya).
particle size for a single- and two-component system. For
the case where the second component is a low-mobility
impurity, we have simulated the kinetics of adatom
spreading across the catalyst surface and their approaching
sidewalls of NWs.
It has been shown that when the surfa ce coverage with a
low-mobility impurity exceeds some critical value, the mass
transfer of mobile adatoms is passed from diffusion to sub-
diffusion [1]. When the coverage increases further, a full
blocking of the surface trans port occurs.
The results obtained allow us to explain a number of
experiments where some unusual shape of an NW was
observed [2–7] and to predict several nontrivial effects

when NWs are doped from a gaseous phase during the
growth process.
2. Modeling
We will restrict our consideration to the processes at the
catalyst surface (Fig. 1 ). As compared to other parts of a
nano-object, here most intensive decomposition of an
active gas takes place and the highest concentration of
mobile silicon atoms or dopants is maintained. The control
of further redistribution of these atoms, while they are
moving towards the interface and sidewalls of a NW, seems
to be just the key, which controls the nano-object shape
and its doping quality.
The NW growth process in general and some of its stages
can be considered within the kinetic approach framework
(the mean field model) by a numerical solution of the
corresponding balance equations [2]; however, an adequate
consideration of the surface transport of adatoms across
the catalyst requires another approach to be applied.
Here, the concentrations of different components can be
high enough and their mobilities can differ from each other
in an arbitrary way. In this specific case, the surface
transport cannot be consider ed as a usual diffusion, but
represents the so-called ‘‘strange process’’. A kinetic
description of such processes is possible within the frame-
work of the so-called fractional dynamics or strange
kinetics [1]. In particular, the Einstein relation transforms
into [1]
R
2


¼ 4D
eff
t
m
. (1)
Here /R
2
S is the mean-square displacement of a randomly
walking particle during the time t, and in a general case
m6¼1. For a multicomponent nonlinear system and also
when the mass transfer is accompanied by other processes
(adsorption from a gaseous phase, chemical reactions, etc.),
this formalism meets great difficulties. At the same time,
the above peculiarities, including a detailed description of
the space distribution evolution of adatoms across the
catalyst surface, can be directly studied by applying the
MC techniques without using the ‘‘strange kinetics’’
formalism.
We will deliberately restrict the application of the MC-
simulation to this part of a growing nano-object alone and
only to the above-mentioned key stage of the process. We
will not use an atomistic approach to simulate the whole
growth process. Such a consideration will be used to show
how these results can be included into a more comprehen-
sive macroscopic model based on a numerical solution of
chemical kinetics rate equations for an axial or radial
growth, shaping, doping, and other related pr ocesses.
A local application of the MC-simulation to the
transport processes in only this area allows us: (i) to
estimate correctly the values of some physical parameters

(the dependence of the transport coefficient on the surface
concentration) with their subsequent utilization in a
chemical kinetics simulation by numerical solution of the
rate equations, (ii) to reproduce some key stages of a real
physical experiment. Besides, we will demonstrate a
number of advantages of ‘‘the multilevel simulation’’. This
approach includes both atomistic and macroscopic descrip-
tion within a single model framework and allows us to
avoid the above-mentioned difficulties of the approach
based on strange kinetics.
While simulat ing atomic transport across the catalyst
surface and analyzing the accompanying chemical pro-
cesses, the following elementary events were selected:
(i) The adsorption from a gaseous phase onto a single,
randomly selected empty surface site [8]. (ii) A diffusion
jump of a randomly selected adatom in a random direction
for some distance l (the distance between the nearest sites)
during a time step t if there is a free site. In a general case,
an irreversible diffusion of the selected adatom into the
catalyst bulk may be considered. This will be done in the
framework of a growth kinetic model in the following
sections. The MC-simulation that included the above
processes was carried out on a two-dimensional lattice of
100 Â 100 atomic cells in size, which corresponded to a real
NW diameter of the order of 20 nm.
ARTICLE IN PRESS
Fig. 1. Distribution of molecular and atomic fluxes assumed in the kinetic
NW growth model: (1) a liquid eutectic drop/solid metal catalyst particle,
(2) the NW body, (3) adsorption of molecules from a gaseous phase, (4)
surface transport of adatoms towards the NW sidewalls, (5) bulk transport

of silicon (impurity) adatoms towards the catalyst/silicon interface.
A. Efremov et al. / Physica E 40 (2008) 2446–2453 2447
We have carried out four types of numerical experi-
ments, which complemented each other:
(i) A migration of a test particle across the catalyst
surface from the center of a round region until the first
contact with its boundary is achieved. This was
simulated for various coverages. Then its path time
was averaged [9]. In this case, other atoms represented
some kind of a background for the test particle
movement. They jumped with a local mobility of
D
loc
¼ 1/4l
2
/t, which, in a general case, did not
coincide with the local mobility of the test particle
itself. The adsorption was ignored in the experiment.
This MC-simulation experiment allows us to estimate
an effective transport coefficient D
eff
as a function of
the coverage and the exponent m, which describes the
state of such a surface (Fig. 2a).
(ii) A classical MC-simulation experiment was also carried
out [9,10] where (at periodic boundary conditio ns)
random trajectories of all the atoms were monitored.
After that, the values of the mean-square displacement
/R
2

S were calculated for a given process duration of
t. These results are compared with the previous case.
(iii) We simulated the behavior of a system consisting of
two kinds of atoms with strongly differing mobilities.
We observed a non-steady-state kinetics of mobile
atoms, which left the catalyst surface and diffused
towards the NW sidewalls at various initial concentra-
tions of components (Fig. 3).
(iv) We calculated a steady-state relation between the input
flux of molecules adsorbed from a gaseous phase (J
inp
)
and the output flux of adatoms (J
out
) leaving the
catalyst surface due to the surface mass transfer
(Fig. 1). In a similar man ner, we determined the
relation between J
inp
/J
out
and the resul ting steady-state
average concentration of adatoms /YS at the catalyst
surface (Fig. 4a and b).
The first two versions of simulations enable us to obtain
transport coefficients and to use them in a numerical
solution of rate equations describing the grow th of NWs.
The second tw o MC experiments reproduce a fragment of a
ARTICLE IN PRESS
Fig. 2. Results of MC simulation of the mass transfer at the catalyst particle surface. (a) The dependence of a normalized effective diffusion coefficient on

the surface coverage and its analytical approximation. It is obtained from the relationship between the mean path time of a test particle /tS and the
corresponding mean-square displacement /R
2
S ¼ R
2
. Here D
0
corresponds to a surface diffusivity value at zero coverage. The MC experiment
conditions: (1) migration of a test particle at the surface covered with the mobile atoms, which are identical to the test particle itself. Periodic boundary
conditions are applied to the background adatoms. The test particle moves along a random trajectory from the center of the region with a given radius of
R until the first contact with its boundary is achieved. (2) Migration of a mobile test particle across the surface previously covered with a low-mobility
impurity. Here, the mobilities of the test and background particles differ by more than three orders of magnitude. (b) Space distribution of allowed
trajectories of a test particle at the surface covered with a slow background impurity at Y
slow
¼ 0.38, 0.41, 0.42, and 0.43, respectively.
Fig. 3. The MC simulations of the kinetics for spreading of the mobile
atoms across the surface in a two-component system. At t ¼ 0, both mobile
and slow background impurity atoms are randomly and uniformly
distributed over the surface. The curves, smoothed by a median filter,
describe the time dependence of mean coverage with the mobile atoms
/Y
mob
S corresponding to different initial coverages with slow impurity
Y
slow
of 0, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, and 0.69 (1–8). The initial
coverage with the mobile atoms for all the cases is identical and equals 0.3.
A. Efremov et al. / Physica E 40 (2008) 2446–24532448
real physical process and allow us to compare the results of
MC and kinetic simulations.

3. Results and discussion
3.1. Effective transport coefficients and their use in an NW-
growth kinetic model
The experiments with a test particle migration on a
lattice covered with the mobile atoms of the same kind
have shown that the Einstein relation remains valid within
a wide range of surface concentrations.
An effective transport coefficient D
eff
is a monotonically
decreasing function of Y shown in Fig. 2a. Here, in our
case, the mean residence time of an adatom at the surface
/tS depends on the system size R and the diffusion
coefficient D
eff
exactly like the quasi-steady-state diffusion
time (QSSDT) does [11] in an ordinary diffusion process.
Let us remember that QSSDT is the time in which about
two-thirds of all the atoms leave a given region. Using this
analogy and applying some analytical approximation of
the obtained dependence for an effective diffusion coeffi-
cient (e.g. D
eff
¼ D
0
(1–Y)
n
, where n is some app ropriate
exponent and D
0

is the surface diffusivity on a free surface),
we can obtain a simple balance equation for the mean
concentration of adatoms at the catalyst particle surface
(shown in Fig. 1 in the form of a hemisphere of radius R).
We note that the NW axial growth (elongation) rate is also
expressed as a function of mean concentration of adatoms
at the surface. Therefore, without considering the fine
details of the adatom space distribution (that we even
cannot find within the framework of an ordinary kinetics),
we will write the following system of NW-growth rate
equations in the form
s
dY
dt
¼ J
inp
ð1 À YÞÀsYðb þ 1=htiÞ,
dh
dt
¼ 2 bsYO, (2)
where s is the surface density of sites available both to
diffusion and to adsorption, b is the rate constant for bulk
transport of adatoms while they are passing through the
catalyst particle bulk, Y is the mean coverage. For b, the
following estimate can be taken:
b % k
D
b
Rl
b

,
where D
b
is the bulk diffusion coefficient in the catalyst
material, l
b
is the lattice spacing in the catalyst bulk, and k
is the factor of the order 0.5. The input flux J
inp
is
determined by the number of molecules, which impinged
and would be really adsorbed on the surface (if the surface
was free), per unit area and unit of time
J
inp
¼ nJ
mol
expðÀE
a
=k
B
TÞ, (3)
where n is the number of silicon (or impurity) atoms in a
molecule, E
a
is an activation energy for chemisorption, k
B
is the Boltzmann constant, and T is the temperature of the
surface.
The surface transport intensity is given by a reciprocal

characteristic particle residence time /tS at the hemisphere
surface. In our case /tS
À1
¼ a D
0
(1–y)
n
/R
2
, where a ¼ m
0
2
/
(p/2)
2
E4.615 is a coefficient of shape in a QSSDT
approximation for a hemispheric region [11],andm
0
is
the first root of the zero-order Bessel function.
The second equation of the system describes the NW
axial growth due to bulk diffusion of silicon atoms from
the surface to the catalyst/NW interface.
Here, h is the height of the NW and O is the atomic
volume of a silicon atom in a silicon lattice, the factor 2 is
the ratio of the catalyst external surface area to that of the
interface for a hemispheric catalyst particle.
ARTICLE IN PRESS
Fig. 4. The MC simulations of the surface steady-state characteristics
under the adsorption of silicon-containing molecules together with the

surface transport of silicon adatoms towards the sidewalls. (a) The
dependencies of a normalized steady-state output flux J
out
/J
inp
(1) and an
averaged steady-state surface coverage /YS (2) on the input flux of
molecules to the surface J
inp
. (b) The normalized steady-state output flux
as a function of the averaged steady-state surface coverage.
A. Efremov et al. / Physica E 40 (2008) 2446–2453 2449
A steady-state (dY/dt ¼ 0) equation corresponding to a
constant axial growth rate is nonlinear. The input flux J
inp
is a parameter in it. The equation in a dimensionless form
can be written as follows:
J
inp
J
0
¼ð1 À E
r
sb
Þ
Y
ð1 À YÞ
þ E
r
sb

Yð1 À yÞ
nÀ1
. (4)
E
r
sb
¼ t
b
=ðt
b
þ t
s
Þ corresponds to a fraction of the surface
channel within the total mass transfer, t
b
¼ Rl
b
/kD
b
is the
characteristic time of bulk diffusion towards the catalyst
particle interface, and t
s
¼ R
2
/aD
0
is the corresponding
characteristic time of the surface mass transfer towards the
boundary line between the catalyst particle and the NW

sidewall.
A relative role of the surface transport can be character-
ized by the ratio of times t
b
=t
s
¼ða=kÞðl
b
=RÞðD
0
=D
b
Þ%
0:1ðD
0
=D
b
Þ.HereJ
0
¼ s/t
eff
is the characteristic flux and
t
eff
¼ t
s
t
b
/(t
s

+t
b
) is the effective time of adatom residence at
the catalyst surface. It is seen that the surface-transport
relative role increases with the decreasing in catalyst particle
size. It is possible to show that the surface transport will
surely dominate over that of the bulk (e.g. at t
b
/t
s
E100) for
an NW with the dimensions of Rp100l
b
(Rp20 nm) in the
case when the surface diffusion coefficient (for zero cover-
age) exceeds that of the bulk by about three orders of
magnitude. The last condition is met in excess for many
materials and diffusants, although there are also exclusions.
A steady-state coverage Y, as a function of the
dimensionless input flux J
inp
/J
0
, was numerically calculated
in agreement with Eq. (4) for n ¼ 1.5 (Fig. 2a). The
dependences obtained are shown in Fig. 5. It is seen that
some narrow interval of critical values of the input flux
exists for the system. Just here a sharp step is observed on
the dependence. In this case, the system is located in the
vicinity of the percolation threshold Y ¼ 1/2.

Within this critical interval, the steady-state equation has
three solutions x
1
, x
2
, and x
3
, corresponding to a given
input flux (Fig. 5a). And even small input flux fluctuations
in the critical region will cause great oscillations in the
value of Y and all parameters associated with this value
including axial and radial growth rates.
As the input flux increases, the concentration of adatoms
at the outset increases gradually. However, when passing
the percolation threshold, the system sharply switches to
the state with a high surface concentration of adatoms and
high axial growth rate. Any reverse changes in the input
flux against the background of an intensive bulk transport
will cause reactivation of the surface mass transfer, a gain
in shell growth, a sharp decrease in the surface concentra-
tion, and the corresponding slowdown of the axial growth.
As a result of such fluctuations in the input flux and the
surface coverage during the growth process , variations in
the shape become possible (including the appearance of
periodic oscillations [3,4]). Besides, it should be emphasized
that in the critical region of the fluxes, depending on the
initial conditions on the catalyst particle surface (which
generally are weakly controllable), the surface may come to
any of the three above-mentioned steady states, which
correspond to Eq. (4) roots. Thus , at given conditions of

temperature and pressure in the growth chamber, three
different axial growth rates may be observed for the NWs.
As a result, this can cause an unexpected scattering in NW
lengths for a given ensemble.
The dependences between the input and output fluxes
obtained using the MC technique (Fig. 3) have shown that
in a system of a finite size at a high-input flux, the surface
transport is suppressed only in the central part of the nano-
object surface. Here, the local surface concentration a easily
overcomes the percolation threshold, and the situation is
ARTICLE IN PRESS
Fig. 5. The solutions of a steady-state Equation (4). (a) A graphic
illustration of the solution at different E
r
sb
. Curves 1–5 correspond to the
right-hand part of Eq. (4) denoted by Y(Y) and Curve 6 is the straight line
corresponding to the given value of the normalized flux in the left-hand
part of this equation. The points of intersection of the two curves give the
steady-state solutions (coverages) for a given input flux. Curves 1–5 are
obtained at E
r
sb
¼ 0.95, 0.983, 0.99, 0.995, and 1.00, respectively. (b) The
steady-state coverage as a function of the input flux at different values of
E
r
sb
obtained as the first (least) root of Eq. (4). This solution is realized for
a zero initial condition at the surface. Curves 1–5 correspond to

E
r
sb
¼ 0.90, 0.95, 0.983, 0.99, and 0.995, respectively. Corresponding
values of the ratio of times t
b
/t
s
are 9, 19, 57.8, 99, and 199.
A. Efremov et al. / Physica E 40 (2008) 2446–24532450
similar to that predicted by a kinetic model Eq. (4). At the
same time, due to the presence of a near-boundary sink for
atoms towards the NW sidewall, a narrow strip containing
sufficient number of vacant sites always exists at the catalyst
particle periphery. Not only a rapid surface transport is
possible through these sites, but also local adsorption from a
gaseous phase goes on. The main contribution to the output
surface flux is made not by impeded diffusion delivery
from the central part to the periphery, but by the direct
adsorption of molecules to the vacant sites located in the
very peripheral region of a catalyst particle. The width of the
region gradually decreases with the increase in the input flux.
This feature is demonstrated in Fig. 4 where the depen-
dences of J
out
/J
inp
¼ f(/YS)andJ
out
/J

inp
¼ j(J
inp
)onthe
mean coverage /YS and on the input flux are presented,
respectively. The input flux J
inp
is determined similarly to
Eq. (3), but here it is measured as the number of adatoms
incoming on a free surface (due to chemisorption and
dissociation of impinging molecules) per one site, per one
MC time step. In a like manner, the output flux is the
number of adatoms leaving the given region of the surface
(due to the surface transport), per one MC time step and per
unit of the boundary length.
The surface transport ceases coping with the flux that
comes to the surface already beginning from /YSE0.25,
which corresponds to reaching Y
max
E0.5 in the catalyst
central part. B eginning from this moment, the initially
homogeneous system is split into the central part, over-
saturated with adatoms and a periphery, relatively free of
adatoms. A rough-and-ready kinetic model, being incapable
of describing these fine details in the surface distribution
of atoms, nevertheless correctly predicts the dependence of
the concentration on the flux in the catalyst’s central part,
where the adatoms are distributed almost homogeneously.
Thus, a kinetic simulation used in combination with the
MC simulation of test particle random walks may be rather

promising while studying a real growth.
3.2. Peculiarities of NW doping according to the kinetic
model
Let us add a seco nd (doping) component to the gaseous
phase. We will consider that the parameters characterizing
this doping impurity with regard to the molecular
adsorption, bulk diffusion, and surface transport at the
catalyst do not strongly differ from the corresponding
parameters of sil icon. Using Eq. (4), we would rewrite the
rate equations for coverages with atoms of silicon Y and
impurity F in the followi ng form:
s
dY
dt
¼ J
1
ð1 À Y À FÞÀsYðb
1
þ 1 = t
1
hiÞ
,
s
dF
dt
¼ J
2
ð1 À Y À FÞÀsFðb
2
þ 1 = t

2
hiÞ
,
V
h
¼
dh
dt
¼ 2sðYO
1
b
1
þ FO
2
b
2
Þ,
Yð0Þ¼Y
i
and Fð0Þ¼F
i
; x ¼ J
2
=J
1
. (5)
Here, the subscripts ‘‘1’’ and ‘‘2’’ refer to the matrix and
impurity atoms, respectively. The initial coverages are
denoted by Y
i

and F
i
. The kinetics of NW bulk doping is
determined by a relative impurity concentration near the
interface during NW growth:
C
I
¼ C
I
ðtÞ¼b
2
F=ðb
1
Y þ b
2
FÞ%ðb
2
=b
1
ÞÁFðtÞ=YðtÞ. (6)
Here from, C
I
¼ C
I
(z),z ¼ h(t) will be the impurity
distribution along the NW length after the growth process
accomplishment. In the case when local mobilities of
matrix and impurity atoms are close to each other and
F/Y51, then a factor that retards diffusion will just be a
summary coverage of the surface. The MC simulation

results allow us to write the characteristic surface mass
transfer times as
t
1
hi
À1
¼ aD
01
ð1 À Y À FÞ
n
=R
2
;
t
2
hi
À1
¼ aD
02
ð1 À Y À FÞ
n
=R
2
:
(7)
We have studied system (5) numerically and calculated the
impurity distribution along the NW length at different
initial conditions and ratio of times t
b
/t

s
. The results
obtained can be summarized as follows:
(i) After a long enough period of time, the ratio of
concentrations at the interface approaches that of
fluxes in a gaseous phase, C
I
(t)
t-N
-x. However, the
duration of this transient process strongly depends on
the initial coverage Y
i
and on the relation between
bulk and surface transport t
b
/t
s
p1/R. This effect is
thus size dependent.
(ii) In the case when the initial concentrations of both the
components at the catalyst surface are equal to zero,
the transient process is absent and C
I
(t) ¼ x already at
the very beginning of the growth. As a result, a
homogeneous bulk doping of an NW is achieved.
(iii) If F
i
¼ 0, Y

i
X0.5, and D
0
/D
b
X10
3
, the transient
process duration becomes comparable with the total
growth period. As a result, C
I
(t) slowly increases from
0tox. The base and middle parts of an NW turn out
to be low-doped and only the upper part will be doped
according to the relation x ¼ J
2
/J
1
. This effect can be
used to form a heterostruct ure in the NW bulk.
(iv) Any manipulations with the input flux of molecules
containing a doping impurity will be inefficient until
YX0.5. For example, changes of J
2
in time do not
allow us to achieve the corresponding impurity
distribution along the NW length due to the existing
time-lag between changes in the gaseous phase and
those at the nano-object surface.
(v) The retardation of the doping level from a given law

for changes of the input fluxes x ¼ x(t) practically
disappears if D
0
/D
b
p10 or/and Y
0
E0.1–0.2. There-
fore, to achieve more or less homogeneous or
controlled inhomogeneous doping of an NW along
its length, one should use the mode of small input
fluxes (J
1
and J
2
).
ARTICLE IN PRESS
A. Efremov et al. / Physica E 40 (2008) 2446–2453 2451
(vi) The above-mentioned narrow peripheral region at the
catalyst particle surface, which is saturated with the
vacant sites, is just that place where small coverages
are maintained during the whole growth process. This
makes it possible (at great input fluxes of the silicon-
containing molecules) to realize a selective and
homogeneous coaxial doping of the NW external shell.
3.3. The influence of a slowly diffusing impurity at the
catalyst surface on the NW growth processes
In the pro cess of the deco mposition of silico n-containing
molecules at the catalyst surface, together with mobile at oms,
reaction by-products are also fo rmed. Some o f them, e.g.

hydrides or chlorides, can possess a small mobility and for a
long enough time occupy surfac e sites [2]. Below we consider
this case of a two-component system, which is rather
important as affecting the gr owth an d shaping of NWs.
The MC exp eriments with surface random walks of a mobile
test particl e in the prese nce of a low-mobility component
show (Fig. 2) that w hen t he covera ge with a low-mobility
impurity Y
slow
achieves a va lue of 0.42pY
slow
p0.43,
practically comp lete blocking occurs in the most available
ways used to deliver adatoms from the catalyst surface
central part to the NW sidewalls (Fig. 2b). The co rrespond-
ing dependence for D
eff
/D
0
¼ F(Y
slow
)isshowninFig. 2a.
As long as the fraction of the low-mobility at oms Y
slow
is
smaller than 0.35, a quasilinear rel ation between a mean-
square displacement /R
2
S and the duration of the random
walks for a mobile test particle holds. However, beginning

from Y
slow
E0.38, a nonlinear relation for /R
2
S ¼ 4D
eff
t
m
is
realized. The so-called sub-diffusion (mo1) [1] is observed,
and m continues to decrease while the coverage with slow
atoms approaches 0.43 .
In this critical interval of coverages, slowly diffusing
atoms start forming individual coherent aggregations. In the
range of 0.38pY
slow
p0.43, most of the diffusion ways
gradually become temporarily blocked. As a result, the
surface flux towards the periphery of the catalyst particle
becomes increasingly more anisotropic. This effect of
anisotropy can be responsible for the appearance of some
unusual forms in the world of nano-objects, such as springs.
Thus, the MC experiments on random walks of a mobile
test particle have shown that the growth of the slow
component concentration retards and then completely
blocks the surface transport. The kinetics of the
mobile adatom spreading from the catalyst surface that
we observed in another MC-experiment (Fig. 3 ) is in full
agreement with this statement.
The kinetics allows us to estimate a characteristic time of

adatom residence at the surface, for a fixed initial concentra-
tion of the mobile particles, in d ependence on the coverage
with slowly diffusing atoms. It is seen that at Y
slow
p0.2, slow
particles weakly influence o n the situation at the surface.
However, when their concentration approaches some thresh-
old value of Y
slow
E0.4, the time of the particle residence at
the surface sharply increases. T his is in agreement with the
results of the above-mentioned series of MC experiments. At
even greater coverages with a slow component ( up to 0.6),
most mobile atoms (excluding those adsorbed directly at the
catalyst periphery) h ave no access to a N W sidewall a nd
cannot freely mo ve across t he surface. A single available
transport channel for them remains in diffusion through the
catalyst bulk i n t he direction of the ca ta lyst /silicon growth
region. The latter feature seems to be rather u seful for the
creation of NWs in the form of ideal cylinders although
growth rate in this case is about twice as low.
In particular, introduction of chlorine-containing com-
plexes into a gaseous medium, used in Ref. [12], made it
possible to suppress the formation of conical NWs and to
grow practically ideal silicon cylinders by employing
titanium silicide as a catalyst. A mechanism of the action
of this additive on the NW shape may just be the surface
transport suppression described above.
4. Conclusion
An atomic transport at the catalyst particle surface has

been studied using the MC technique. To estimate an
effective transport coefficient and reproduce some aspects
of a real situation, we have used four different scenarios for
the MC experiment.
In addition, we have carried out a kinetic simulation of
the NW growth and doping by a numerical solution of the
corresponding rate equations of chemical kinetics using the
dependences obtained in the MC experiment. This enabled
us to compare atomistic and macroscopic approaches.
Such a multilevel simulation proved to be rather fruitful in
view of improving our understanding of the general
features of the process. The results obtained within the
framework of a kinetic approach have been confirmed and
amended by the MC simulation.
These results have revealed an important role of surface
transport in NW growth and doping. It has been shown
that the presence of low-mobility impurities at the surface
allows us to block the surface transport, efficiently suppress
the flux from the catalyst surface towards the NW
sidewalls, and control the NW shape. Similarly, by
changing the relationship between the surface and bulk
fluxes, we can obtain various versions of the doping
impurity distribution along the NW length and radius. We
have also observed several nonlinear effects: (1) total or
partial blocking of the surface transport in the presence
of slowly diffu sing additives at the surface; (2) appearance
of the surface flux anisotropy and transfer to the mode of
sub-diffusion, when the coverage with slowly diffusing
atoms reaches the value of Y
slow

E0.4; (3) essential
suppression of the surface transport at great input fluxes
for a single-component system; (4) the presence in
some cases of a significant time lag between a change
in the concentration of impurity-containing molecules in
the gaseous phase and the corres ponding changes in the
concentration of impurity at the nanocatalyst surface.
ARTICLE IN PRESS
A. Efremov et al. / Physica E 40 (2008) 2446–24532452
The results of the simulation allow us to predict some
approaches to control shape and impurity distribution
during the NW growth. In particular, it becomes possible
to realize controllable cylindrical or conical shapes,
homogeneous or coaxial doping.
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