Đây là một bài báo khoa học về dây nano silic trong lĩnh vực nghiên cứu công nghệ nano dành cho những người nghiên cứu sâu về vật lý và khoa học vật liệu.Tài liệu có thể dùng tham khảo cho sinh viên các nghành vật lý và công nghệ có đam mê về khoa học
Trang 1Physica E 40 (2008) 2446–2453
Multilevel modeling of the influence of surface transport peculiarities on
growth, shaping, and doping of Si nanowires
A Efremova, A Klimovskayaa, , I Prokopenkoa, Yu Moklyaka, D Hourlierb
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
r2008 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 used 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 soludescrip-tion
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 dopants, 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
www.elsevier.com/locate/physe
1386-9477/$ - see front matter r 2008 Elsevier B.V All rights reserved.
doi: 10.1016/j.physe.2008.01.015
Corresponding author Tel.: +38 044 5257091; fax: +38 044 5258342.
E-mail address: kaignn@rambler.ru (A Klimovskaya).
Trang 2particle 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 surface 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 transport 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 considered 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]
Here /R2S 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 processes
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 simulating 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
100 100 atomic cells in size, which corresponded to a real
NW diameter of the order of 20 nm
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.
Trang 3We 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
Dloc¼1/4l2/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 Deff 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
random trajectories of all the atoms were monitored
After that, the values of the mean-square displacement
/R2S 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 (Jinp)
and the output flux of adatoms (Jout) leaving the
catalyst surface due to the surface mass transfer
relation between Jinp/Joutand the resulting 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 growth of NWs The second two MC experiments reproduce a fragment of a
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 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.
Trang 4real 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 Deffis 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 Deffexactly 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 Deff¼D0(1–Y)n, where n is some appropriate exponent and D0is 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 inFig 1in 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
sdY
dt ¼Jinpð1 YÞ sYðb þ 1=htiÞ, dh
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 Db
Rlb
, where Db is the bulk diffusion coefficient in the catalyst material, lbis the lattice spacing in the catalyst bulk, and k
is the factor of the order 0.5 The input flux Jinp 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
where n is the number of silicon (or impurity) atoms in a molecule, Eais an activation energy for chemisorption, kB
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 /tS1¼aD0(1–y)n/R2, where a ¼ m0/ (p/2)2E4.615 is a coefficient of shape in a QSSDT approximation for a hemispheric region [11], and m0 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
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.
Trang 5A steady-state (dY/dt ¼ 0) equation corresponding to a
constant axial growth rate is nonlinear The input flux Jinp
is a parameter in it The equation in a dimensionless form
can be written as follows:
Jinp
J0
¼ ð1 ErsbÞ Y
ð1 YÞþE
r
Ersb¼tb=ðtbþtsÞcorresponds to a fraction of the surface
channel within the total mass transfer, tb¼Rlb/kDbis the
characteristic time of bulk diffusion towards the catalyst
particle interface, and ts¼R2/aD0 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 tb=ts¼ ða=kÞ ðlb=RÞ ðD0=DbÞ
0:1ðD0=DbÞ Here J0¼s/teff is the characteristic flux and
teff¼tstb/(ts+tb) 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 tb/tsE100) for
an NW with the dimensions of Rp100lb(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 Jinp/J0, 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 x1, x2, and x3, 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
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
Ersb¼ 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.
Trang 6similar 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 Jout/Jinp¼f(/YS) and Jout/Jinp¼j(Jinp) on the
mean coverage /YS and on the input flux are presented,
respectively The input flux Jinp 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 YmaxE0.5 in the catalyst
central part Beginning 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 second (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 silicon Using Eq (4), we would rewrite the
rate equations for coverages with atoms of silicon Y and
impurity F in the following form:
sdY
dt ¼J1ð1 Y FÞ sYðb1þ1= th iÞ,1
sdF
dt ¼J2ð1 Y FÞ sFðb2þ1= th iÞ,2
Vh¼dh
dt ¼2sðYO1b1þFO2b2Þ,
Yð0Þ ¼ Yi and Fð0Þ ¼ Fi; x ¼ J2=J1 (5)
Here, the subscripts ‘‘1’’ and ‘‘2’’ refer to the matrix and impurity atoms, respectively The initial coverages are denoted by Yiand Fi The kinetics of NW bulk doping is determined by a relative impurity concentration near the interface during NW growth:
CI¼CIðtÞ ¼ b2F=ðb1Y þ b2FÞ ðb2=b1Þ FðtÞ=YðtÞ (6) Here from, CI¼CI(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
t1
h i1¼aD01ð1 Y FÞn=R2;
t2
We have studied system (5) numerically and calculated the impurity distribution along the NW length at different initial conditions and ratio of times tb/ts 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, CI(t)t-N-x However, the duration of this transient process strongly depends on the initial coverage Yi and on the relation between bulk and surface transport tb/tsp1/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 CI(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 Fi¼0, YiX0.5, and D0/DbX103, the transient process duration becomes comparable with the total growth period As a result, CI(t) slowly increases from
0 to x 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 J2/J1 This effect can be used to form a heterostructure 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 J2 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 D0/Dbp10 or/and Y0E0.1–0.2 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 and J )
Trang 7(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 process of the decomposition of silicon-containing
molecules at the catalyst surface, together with mobile atoms,
reaction by-products are also formed Some of them, e.g
hydrides or chlorides, can possess a small mobility and for a
long enough time occupy surface sites[2] Below we consider
this case of a two-component system, which is rather
important as affecting the growth and shaping of NWs
The MC experiments with surface random walks of a mobile
test particle in the presence of a low-mobility component
show (Fig 2) that when the coverage with a low-mobility
impurity Yslow achieves a value of 0.42pYslowp0.43,
practically complete blocking occurs in the most available
ways used to deliver adatoms from the catalyst surface
central part to the NW sidewalls (Fig 2b) The
correspond-ing dependence for Deff/D0¼F(Yslow) is shown in Fig 2a
As long as the fraction of the low-mobility atoms Yslow is
smaller than 0.35, a quasilinear relation between a
mean-square displacement /R2S and the duration of the random
walks for a mobile test particle holds However, beginning
from YslowE0.38, a nonlinear relation for /R2S ¼ 4Defftmis
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 0.38pYslowp0.43, 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 dependence on the coverage
with slowly diffusing atoms It is seen that at Yslowp0.2, slow
particles weakly influence on the situation at the surface
However, when their concentration approaches some
thresh-old value of Y E0.4, the time of the particle residence at
the surface sharply increases This 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) have no access to a NW sidewall and cannot freely move across the surface A single available transport channel for them remains in diffusion through the catalyst bulk in the direction of the catalyst/silicon growth region The latter feature seems to be rather useful 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 diffusing 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 YslowE0.4; (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 corresponding changes in the concentration of impurity at the nanocatalyst surface
Trang 8The 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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