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

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: [email protected] (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

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

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

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.

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

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

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

+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.

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

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

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

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

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

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

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

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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