Diffusion controlled growth of semiconductor nanowires vapor pressure versus high vacuum deposition

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

Diffusion-controlled growth of semiconductor nanowires:

Vapor pressure versus high vacuum deposition

V.G Dubrovskii a,b,*, N.V Sibirev c, R.A Suris a,b, G.E Cirlin a,b,c,

a St Petersburg Physical Technical Centre of the Russian Academy of Sciences for Research and Education, Khlopina 8/3, 195220 St Petersburg, Russia

b Ioffe Physical Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021 St Petersburg, Russia

c Institute for Analytical Instrumentation of the Russian Academy of Sciences, Rizhsky 26, 190103 St Petersburg, Russia

d CNRS-LPN, Route de Nozay, 91460 Marcoussis, France

Available online 20 April 2007

Abstract

Theoretical model of nanowire formation is presented, that accounts for the adatom diffusion from the sidewalls and from the sub-strate surface to the wire top Exact solution for the adatom diffusion flux from the surface to the wires is analyzed in different growth regimes It is shown theoretically that, within the range of growth conditions, the growth rate depends on wire radius R approximately as 1/R2, which is principally different from the conventional 1/R performance The effect is verified experimentally for the MBE grown GaAs and AlGaAs wires The dependences of wire length on the drop density, surface temperature and deposition flux during vapor pressure deposition and high vacuum deposition are analyzed and the differences between these two growth techniques are discussed

 2007 Elsevier B.V All rights reserved

Keywords: Nanowires; Kinetic growth model

1 Introduction

Semiconductor nanowires perpendicular to a substrate

have recently attracted an increasingly growing interest

as nanostructured materials with applicability to

nano-electronics[1,2], nanooptics[3,4] and nanosensors[5]

Sil-icon wires with micrometer diameters were first fabricated

more than 40 years ago[6] These wires were grown by the

so-called vapor–liquid–solid mechanism [6,7]from vapors

SiCl4 and H2 on the Si(1 1 1) surface activated by Au

drops at surface temperature T of about 1000C Modern

epitaxial techniques enable to fabricate Si [8–10], III–V

and II–VI[11–18]semiconductor wires by the same

mech-anism but with diameters reduced typically to 10–100 nm

Different growth technologies of nanowire formation can

be divided in two groups In the first group, material is deposited onto a substrate from a vapor phase [1–8], for example in chemical vapor deposition (CVD) In the sec-ond group, material is delivered from a particle beam un-der high vacuum conditions [10,14–18], form example in molecular beam epitaxy (MBE) We shall call these groups, for briefness, as vapor phase deposition (VPD) and high vacuum deposition (HVD) Studying the forma-tion mechanisms of nanowires is important from the view-point of fundamental physics of growth processes as well

as for fabrication of controllably structures nanowires for various device applications These investigations require the development of relevant theoretical models Also, there have been relatively few systematic studies of the dependence of nanowire morphology on the growth con-ditions Among these, we would like to mention recent re-sults on the length–radius dependences of MBE grown Si

[10]and GaAs[14,16]wires, CVD grown wires of different

0039-6028/$ - see front matter  2007 Elsevier B.V All rights reserved.

doi:10.1016/j.susc.2007.04.122

*

Corresponding author Address: Ioffe Physical Technical Institute of

the Russian Academy of Sciences, Politekhnicheskaya 26, 194021 St.

Petersburg, Russia Tel.: +7 812 448 6982; fax: +7 812 297 3178.

E-mail address: dubrovskii@mail.ioffe.ru (V.G Dubrovskii).

www.elsevier.com/locate/susc

III–V compounds[13], and GaAs wires grown by

magne-tron sputtering depositon (MSD) [17]

The well known Givargizov–Chernov model of wire

for-mation[19]accounts for the Gibbs–Thomson effect of

ele-vation of chemical potential in the cylindrical wire of

radius R Their formula for the wire growth rate dL/dt

was found to provide a good fit with experimental

length–radius curves of Si wires grown by VPD on the

Si(1 1 1) surface activated by Au at T 1000 C for radii

in the micrometer range[19,20] Kashchiev[21]recently

ap-plied the formula for the growth rate of crystal face of finite

radius R [22] for the description of nanowire formation

This model does not consider the Gibbs–Thomson effect,

but accounts for the transition from mononuclear to

poly-nuclear mode of nucleation at the wire top as the radius of

wire increases Some of the authors of this paper developed

a more detailed model of wire growth[23]that handles the

both effects simultaneously All models described predict

an increasing dependence of wire length on R and correlate

with some MBE experiments for comparatively short and

thick GaAs wires[21,23]

However, many experimental results on L(R)

depen-dence in modern VPD and HVD techniques demonstrate

that Si wires at T = 525C [10] and different III–V wires

at T = 550–600C [13,14,17] exhibit the decreasing L(R)

dependencies in the range of diameters typically from 40

to 200 nm Experimental curves are usually described by

the formula dL/dt = A + BR*/R, where A and B are

cer-tain R-independent parameters and R*is the characteristic

diffusion radius The 1/R behavior is typical when the wire

growth is controlled by the adatom diffusion from the

substrate surface [14,17] and/or the wire sidewalls [13]

Adatom diffusion flux supplies semiconductor material

to the drop and provides supersaturation sufficient to

drive the nucleation When wire length exceeds the

ada-tom diffusion length on the sidewalls, the supplying flux

decreases, the alloy concentration in the drop and the

drop radius itself diminishes, that leads to wire tapering

[15] Generalized kinetic approach of Ref [24] is capable

of semi-quantitative description of such growth behavior

It is now well understood that the size-dependent Gibbs–

Thomson effect [19] and the decrease of

nucleation-medi-ated growth rate on a small face [21,23] are important

when the growth is controlled by the direct impingement

of material to the drop It normally happens at high

sur-face temperatures and, consequently, small diffusion

lengths of adatoms When the growth temperature is

low-ered, the diffusion lengths increase and the wire growth is

controlled primarily by the adatom diffusion Since the

impingement flux is proportional to drop surface area

(R2), and diffusion flux from the sidewalls is proportional

to wire perimeter (R), the rate of particle sink at the

li-quid–solid interface will be R2 dL/dt = AR2+ BR*R

Dividing this over R2, one ends up with the equation of

wire growth predicted theoretically in earlier works, for

example, by Dittmar and Neumann [25], Ruth and Hirth

[26] and Blakely and Jackson[27]

While in a given growth experiment the wire lengths and diameters are anyway dictated by the size distribution of catalyst drops f(R), it is important to consider the effect

of growth conditions on the morphology of wires grown

at fixed f(R) but at different temperatures T and fluxes V Some data and on the L(T) and L(V) dependencies of MBE grown Ga(Al)As nanowires have been recently ob-tained and modeled in Refs.[24,18] This paper continues the study of diffusion-controlled nanowire growth, with a closer look at the kinetic processes on a substrate surface First of all, we will show that the 1/R dependence of the wire length is not the general case, because R* depends

on R via the ratio R/ks, where ks is the effective diffusion length on the substrate surface Under typical conditions during HVD this leads to two main growth modes, one with conventional 1/R and another with 1/R2behavior of wire length We will present the data on the MBE grown GaAs and AlGaAs nanowires exhibiting such 1/R2length dependence Secondarily, we will construct a self-consistent route to estimate the value of ks Finally, we will consider the dependence of wire length on temperature T, flux V, drop density NW and drop size distribution f(R) in VPD and HVD growth techniques and discuss the differences be-tween them

2 Theoretical model

Thermodynamic driving force for the wire growth is the supersaturation of gaseous phase, defined as

U¼ V sl 2rlX s C eq 1, where V is the impinging flux, sl is the mean lifetime of semiconductor particles in the drop before re-evaporation, rlis the inter-atomic distance in the liquid,

Xs is the volume per atom in the crystal and Ceq(T) is the equilibrium concentration of alloy As shown in Ref.[24],

if U is much higher than the alloy supersaturation f, the wire growth is mainly controlled by the transport of semi-conductor particles to the drop Our growth model, de-scribed in more detail in Ref.[28], takes into account (i) the direct impingement at rate V; (ii) the desorption from the drop; (iii) the diffusion flux of adatoms to the drop and (iv) the growth of non-activated surface at rate Vs The wire growth rate in the steady state is given by[28]

dL

dH¼ e  c þR

Here, H = Vt is the deposition thickness, the R-indepen-dent term accounts for the direct impingement on the drop surface, desorption from the drop (c = 1/(U + 1)) and the growth of non-activated surface (e = 1 Vs/V) The diffu-sion-induced contribution is described by the characteristic radius R* The exact solution of system of diffusion equa-tions at the substrate surface and at the wire sidewalls pro-vides the expression for R*in the form[28]

R¼2ksbþ 2kfsin a½coshðL=kfÞ  1 þ bG sinhðL=kfÞ

bGcoshðL=kfÞ þ sinhðL=kfÞ ð2Þ

Here, kf ¼ ffiffiffiffiffiffiffiffiffi

Dfsf

p

ffi ffiffiffiffiffir

f

p exp½ðEf

A Ef

DÞ=2 kBT is the adatom diffusion length on the sidewalls (limited by

desorption), Ef

A and Ef

D are the activation energies for the adatom desorption and diffusion on the sidewalls,

ks¼ ffiffiffiffiffiffiffiffiffi

Dsss

p

is the effective adatom diffusion length on

the substrate surface (normally limited by nucleation

or by outgoing flux to wire sidewalls) and

b¼ ðrsksDf=rfkfDsÞ ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dfss=Dssf

p

Here and below rf

and rsdenote the areas of adsorption site on the wire

side-walls (f) and on the substrate surface (s) The effective

inci-dent angle of impinging flux to the wire a is used to treat

simultaneously the cases of VPD and HVD In VPD

tech-niques sina = 1, because the vapors surround the drop and

the wire In HVD techniques with particle beam

perpendic-ular to the surface one can assume that sina 0, so that

particles impinge only the substrate surface The function

G is given by

G¼I1ðRW=ksÞK0ðR=ksÞ þ K1ðRW=ksÞI0ðR=ksÞ

I1ðRW=ksÞK1ðR=ksÞ  K1ðRW=ksÞI1ðR=ksÞ ð3Þ

where Inand Knare the modified Bessel functions in

stan-dard notations, and RW¼ 1= ffiffiffiffiffiffiffiffiffiffi

pNW

p

is the average half-dis-tance between the wires At R*= const Eq (1) is

immediately reduced to the simplest growth equation

dis-cussed in the Introduction While deriving Eqs (1)–(3) it

has been assumed that the wire is the cylinder of a fixed

ra-dius R The same model can be also applied to the case of

wire in the form of a regular prism or hexagon, in the latter

case R is the radius of a circle inscribed to this prism or the

hexagon However, it is important to point out that the

solutions given by Eqs (1)–(3) apply only when the wire

shape is independent on the polar angle in the substrate

plane If it is not the case, one has to consider the X and

Y dependence of the diffusion flux to the wire base and

the solutions become more complex Also, there is a lower

bound for radius R, below which the wire growth can not

be described by the diffusion Eqs (1)–(3) Namely, R

should be larger than the Givargizov–Chernov minimum

radius Rmin [20,23,24] At R Rmin, the Gibbs–Thomson

effect considerably lowers the growth rate of very thin wires

and should be taken into consideration Using the

esti-mates of Ref [24], the value of Rmin decreases at higher

supersaturations U and is smaller than 10 nm during the

MBE growth of GaAs wires at typical growth conditions

of T 500–600 C and V  1 ML/s, so that the diffusion

growth model must be valid at least for the wires with

R > 20 nm

To this end, diffusion length ksand quantity e = 1 Vs/V

serve as two external parameters of the model Let us now

show how these two parameters can be determined

self-con-sistently First of all, we introduce the probabilities

e1= XsJdiff/V of adatom migration to the wire sidewall (here

Jdiffdenotes the overall diffusion flux to the wire bases per

unit surface area), e2= Vs/V of adatom incorporation to

the growing surface layer and e3= Vdes/V to re-evaporate

from the surface Due to the mass conservation e1+ e2+

e3= 1 and e = 1 e2= e1+ e3by definition Probability e1

must be proportional to the overall diffusion flux Jdiff, which equals the sum of individual fluxes jdiff(0) to differently sized wires:

e1

V

Xs

Here and below hgi denotes the average with normalized size distribution of drops fðRÞ; hgi ¼R1

0 dRfðRÞgðRÞ: Indi-vidual diffusion flux to the wire base can be found from the solution for surface concentration of adatoms ns(r)[28] as

jdiff(0) = Ds2pR dns/drjr = Rin the form

jdiffð0Þ ¼ pRV

Xs

½RcoshðL=kfÞ  2kfsin a sinhðL=kfÞ ð5Þ Inserting this into Eq.(4), upon averaging we get

e1

pNWhRi¼

hRRi hRi coshðL=kfÞ  2kfsin a sinhðL=kfÞ ð6Þ Since R*is the known function of ks, Eq.(6)allows one to find ksas function of e1, of the parameters of drop size dis-tribution and of the growth conditions T and V If R*can

be treated as R-independent, Eq.(6) determines R* explic-itly and Eq.(1) is reduced to the formula of Ref.[29]

dL

dH ¼ e  c

þ1 R

e1

phRiNWcoshðL=kfÞþ 2kfsin a tanhðL=kfÞ

ð7Þ

If ksis considerably smaller than the radii of wires and

of two-dimensional islands on the surface, we can assume that probability e1 is proportional to the total perimeter

of wires per unit surface area PW= 2pNWhRi and also that probability e2is proportional to the appropriately averaged perimeter of islands PI As shown, for example, in Ref.[30], the time dependence of layer perimeter can be approxi-mated as PIðsÞ ¼ p ffiffiffiffiffiffiffiffiN

Is

p

es Here, NIis the surface density

of islands emerging in each layer and s is a certain relative time Averaging this in s, we arrive at

PIffi p ffiffiffiffiffiffi

NI

p Z 1 0

dx x1=2ex¼p

3=2

2

ffiffiffiffiffiffi

NI

p

ð8Þ The surface density of islands NIduring the layer-by-layer growth can be calculated by applying the nucleation theory

[22,30] For us it is important that NIdepends on temper-ature T and deposition flux V approximately as

NI/ V2

exp 3Ksþ 2Es

D

kBT

ð9Þ

where Ks is the specific condensation heat of two-dimen-sional ‘‘vapor’’ of adatoms and Es

Dis the activation energy for adatom diffusion Finally, the probability of desorption

e3is proportional to the reverse diffusion length of single adatom on a bare substrate 1/k0s Temperature dependence

of k0s is given by the conventional exponential expression

k0ffi ffiffiffiffiffir

s

p exp½ðEs  EsÞ=2kBT, where Es is the activation

energy for desorption The expressions for e and e1

follow-ing from the above analysis read

e¼ 1þ PWk

0 s

1þ ðPWþ PIÞk0

s

; e1¼ PWk

0 s

1þ ðPWþ PIÞk0

s

; ks R

ð10Þ

In the opposite case, when ks is much larger than the

radii of wires and islands, but smaller than the average

dis-tance between them, the probability e1will be proportional

to surface density of wires NW, the probability e2 will be

proportional to density of islands NI and the probability

e3will be proportional toð2pk0

sÞ2 Repeating the consider-ations described above, the expressions for e and e1in this

case will be

e¼ 1þ 2pNWðk

0

sÞ2

1þ 2pðNWþ NIÞðk0sÞ2;

e1¼ 2pNWðk

0

sÞ2

1þ 2pðNWþ NIÞðk0sÞ2; R ks RW ð11Þ

3 Results and discussion

Let us now see how the above model describes the most

important limit regimes of nanowire growth In the case of

VPD (sin a = 1) under the assumption L/kf 1 the wire

growth is determined entirely by the adatoms adsorbed

on the sidewalls and Eqs (1)–(3) are reduced to the well

known expression[13,20]

L¼ e c þ2kf

R

In the case of HVD the adsorption on the sidewalls is

small (sina 0) and the wire growth is mainly controlled

by the adatoms arriving from the substrate surface [14]

For numerical estimates, consider typical conditions of

MBE growth of GaAs nanowires on the GaAs(1 1

1)B-sur-face activated by Au drops: NW 109

cm2,hRi = 40 nm and T = 580 590 C [14,15] Different estimates for the

diffusion length of Ga atoms on the GaAs(1 1 0) sidewalls

at this temperature range from 3 [15] to 10 [31] lm For

wires with L < 3 lm we can therefore use a simplified

equa-tion R*= 2ks/G at L/kf 1 instead of general equation(2),

and even for longer wires with L up to 10 lm it could still be

a reasonable approximation Further, the distance between

wires RWapproximately equals 180 nm, and for the value of

ksof several tens of nanometers (verified experimentally

be-low in this paper) we can use the asymptote of Eq.(3) at

RW/ks 1 In this case Eqs.(2), (3)are simplified to

R¼ 2ks

K1ðR=ksÞ

K0ðR=ksÞ!

2ks;ks R

2k 2 s

Rlnðk s =RÞ;ks R

(

ð13Þ

where we write explicitly two limit cases of wire growth

When the effective diffusion length on the surface is much

smaller than the wire radius, R*is R-independent and the

wire length depends on R as 1/R (1/R-diffusion) In the

opposite case, when the diffusion length is much larger than the wire radius, R*is reverse proportional to R with a weak logarithmic correction, and the wire length depends on R approximately as 1/R2(1/R2-diffusion) Numerical analysis shows that the function in the right hand side of Eq.(13)

can be approximated with reasonable accuracy by k2s=R2 for the values of R ks, if we put ln (ks/R) 1 Now we note that the cases of R*= const and R*/ 1/R correspond

to a trivial averaging in self-consistent Eq (6) Thus, Eq

(13)together with Eqs.(6) and (10), (11)gives the following results for the wire lengths and effective diffusion lengths in the cases of 1/R- and 1/R2-diffusion

L¼ e c þ2ks

R

ks¼ k

0 s

1þ ðPWþ PIÞk0s ffi

1

PWþ PI

; ks R ð15Þ

Lffi e c þ2k

2 s

R2

0 s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2pðNWþ NIÞðk0sÞ2

q ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2pðNWþ NIÞ

ð17Þ Approximate expressions in Eqs.(15) and (17) apply at a low desorption from the surface For example, when GaAs wires are grown on the GaAs(1 1 1)B surface, the diffusion length of Ga atoms k0sapproximately equals 6 lm at 590C

[32] At NW= 109cm2andhRi = 50 nm this corresponds

to PWk0s  19 and NW½k0

s2 360 In this case the expres-sions for e and e1become

e¼ e1¼ PW

PWþ PI

e¼ e1¼ NW

NWþ NI

Presented formulas give the possibility to derive some information concerning the nanowire growth mechanisms from the experimental L(R) curves Namely, by fitting these curves by Eq (12) in VPD or one of Eqs (14) or (16)in HVD, we find the values of kf or ksand e c By measuring the average thickness of epitaxial layer on the substrate surface Hs, we calculate e and obtain the desorp-tion coefficient c = 1/(U + 1), and, hence, the supersatura-tion of gaseous phase U In absence of desorpsupersatura-tion from the substrate, even without measuring Hs, at known NW

andhRi we are able to deduce PI or NI from Eq (15) or (17), then calculate e by means of Eq (18) or (19) and get the values of c and U from measured e c

As mentioned already, the 1/R diffusion law has been ver-ified experimentally for the Si/Si(1 1 1)–Au nanowires grown

by MBE at T = 525C [10], the GaAs/GaAs(1 1 1)B–Au nanowires at grown by MBE at T = 550–600C[13,14,16], and also for the GaAs/GaAs(1 1 1)B–Au nanowires grown

by MSD at T = 585C[17] Below we present experimental data on the Al Ga As/GaAs(1 1 1)B–Au and the GaAs/

GaAs(1 1 1)B–Au nanowires exhibiting the 1/R2behavior of

L(R) dependences The Al0.33Ga0.67As wires were grown by

MBE at surface temperature T = 585C and deposition

thickness of AlGaAs H = 725 nm Experimental details

can be found in Ref.[33] The GaAs wires were grown by

MBE at T = 560C and H = 1000 nm Experimental details

can be found in Ref.[28] The values of wire density NWfor

the both ensembles of wires are given inTable 1 From the

analysis of scanning electron microscopy images of these

wires[28,33], we worked out experimental length–diameter

curves shown by points inFigs 1 and 2 Solid line inFig 1

is the best fit of theoretical length–diameter curve given by

simplified equation(16) Also for comparison, the dashed

line corresponds to the best fit of general equation(12)at

sina = 0 in the 1/R diffusion mode, modeled in Ref.[33] It

is seen that the 1/R2curve provides considerably better fit

to the experimental results Numeric values of growth

char-acteristics, obtained from fitting the experimental L(R)

curves as described above, are summarized inTable 1 Solid

line inFig 2corresponds to the best fit obtained from

sim-plified equation(16)and dashed line relates to R*given by

general equations(1)–(3)at sina = 0 Theoretical values of

parameters for GaAs are also presented inTable 1

Analysis of data presented inTable 1shows that wires

consume about 32% of all adatoms in the case of AlGaAs

and about 15% in the case of GaAs The effective diffusion

length on the substrate surface is 1.23 times higher for

Al-GaAs because of lower surface density of islands, although

diffusivity of Al atoms itself is much lower than that of Ga

Since the minimum radius of drop Rmin is 20 nm for Al-GaAs and 31 nm for Al-GaAs wires, the maximum ratio of wire length to the deposition thickness (L/H)max is 7.3 times for AlGaAs and only 3.2 times for GaAs We also note that the above estimations provide reasonable values for surface density of islands in the regime of complete con-densation109–1010cm2[30]

Let us now consider the drop density, temperature and flux behavior of nanowire length during VPD and HVD Assume, for semi-quantitative analysis, that the R-indepen-dent term in Eq.(6)is negligible (e c  R*/R) In the case

of VPD at sufficiently high temperatures, when L/kf 1 and the wires grow due to the adatoms directly impinging their sidewalls, Eq (12)gives the temperature dependence

of wire length in the form

L/ kf ¼ k0fexp Gf

T0

T  1

ð20Þ Here, the quantity Gf ¼ ðEf

A Ef

DÞ=2kBT0 is determined entirely by the adatom characteristics on the sidewalls Therefore, we can ignore completely the processes on the substrate surface [13] The length of wires is density and flux independent and decreases at higher T, because the dif-fusion length on the sidewalls becomes smaller

In the case of HVD, making use of Eqs.(15) and (17)for

ksin the cases of 1/R and 1/R2diffusion, Eqs.(8) and (9)

and the temperature dependence of k0s, one obtains the effective diffusion lengths as functions of NW, hRi, T and

V in the form

ks¼ PWþ PIðT0; V0ÞV

V0

exp Fs

T0

T  1



þ 1

k0sðT0Þexp Gs

T0

T  1

; ks R ð21Þ

Table 1

Parameters of lateral size distribution and theoretical characteristics of

wire growth

cm2

k s , nm

N I , cm2 e c U

Al 0.33 Ga 0.67 As at

585 C

2.7 · 10 9

43 5.9 · 10 9

0.32 0.07 13 GaAs at 560 C 2 · 10 9 35 1.1 · 10 10 0.15 0.05 19

0

1000

2000

3000

4000

5000

6000

Diameter of nanowire [nm]

Fig 1 Experimental (black squares) and theoretical (solid line) length–

diameter dependences of AlGaAs nanowires Theoretical curve is obtained

from simplified equation (21) at k s = 43 nm and e  c = 0.25 Dotted line

is the 1/R-type theoretical curve from Ref [33]

0 500 1000 1500 2000 2500 3000 3500

Diameter of nanowire [nm]

Fig 2 Experimental (points) and theoretical (lines) length–diameter dependences of GaAs nanowires Squares, triangles and circles represent experimental data from different parts of the substrate Solid line – simplified equation (21) at k s = 35 nm and e  c = 0.1 Dashed line is obtained from general equations 7,8 at sin a = 0, k f = 10 lm, e  c = 0.1,

b = 0.3 and other parameters given in Table 1

ks¼ 2pNWþ 2pNIðT0; V0ÞV

2

V20exp 2Fs

T0

T  1



þ 1

½k0

sðT02 exp 2Gs

T0

T  1

; ks R ð22Þ

Here, T0 and V0 are the reference values of temperature

and deposition rate, the quantities Fs¼ ½ð3Ks=2Þþ

Es

D=kBT0 and Gs¼ ðEs

A Es

DÞ=2kBT0 are determined by the characteristics of adtoms on the substrate surface It

is seen that normally ksincreases with the surface

temper-ature and decreases with the deposition flux and is limited

by the values of 1/(2phRiNW) or 1= ffiffiffiffiffiffiffiffiffiffiffiffiffi

2pNW

p , both decreas-ing with the density of drops NW The temperature and flux

dependence is dictated by Eq.(9)for the density of islands

NI, decreasing at higher T and lower V[30]

While considering the temperature dependence of wire

length during HVD, we should consider the growth

equa-tion beyond the limit L/kf 1, because kf exponentially

decreases with T Consider, for example, the case of 1/R2

diffusion, when the function G in Eq (2) is given by

G = (R/ks)ln(ks/R) Direct integration of Eq (1) at

e c = 0 with this G gives

R2lnðks=RÞ sinhðL=kfÞ þ mRkf½coshðL=kfÞ  1 ¼2k

2 s

kf

H

ð23Þ where m = rfDs/rsDf At L/kf! 0 and R/ks! 0 this

equa-tion is reduced to

Lffi 2k

2

s

R2lnðks=RÞH/ k

2

This is the case of zero desorption from the sidewalls,

tak-ing place at sufficiently low substrate temperatures

There-fore, at low T the dependence of L on the growth

conditions is dictated by the diffusion length on the

sub-strate surface ks, and wire length increases with the

temper-ature and decreases with the deposition flux In the

opposite case of large L/kfEq.(23)gives

Lffi ln 4k

2

sH

mk2fR

!

At high T the wire length therefore decreases with the

tem-perature approximately as kf This simple analysis shows

that the length of wire of given radius at otherwise same

conditions must have a maximum at a certain optimal

tem-perature.Fig 3demonstrates the correlation between

the-oretical L(T) dependence obtained from Eqs.(22) and (23)

at fixed R = 25 nm with the experimental data on the

length of GaAs nanowires grown by MBE on the

GaAs(1 1 1)B surface activated by Au drops within the

tem-perature range of 460–600C It is seen that the average

growth rate of nanowires is always higher than the

deposi-tion rate (2 A˚ /s) in the interval of T from 460 to 590 C, has

a maximum around 550–580C and rapidly decreases at

higher T due to adatom desorption from the sidewalls

To conclude, we have developed theoretical model of nanowire formation that can systematically handle the description of the wire length depending on its radius and technologically controlled growth conditions Within the range of growth conditions, L(R) curves obey the 1/R2 law, and this dependence is verified experimentally for Ga(Al)As wires The drop density, temperature and flux behavior of wire length has been also studied It has been shown that this behavior is principally different during high temperature VPD, when the drop is fed by the particles ad-sorbed on the sidewalls, and during HVD, when the wire growth is mainly stimulated by the adatom diffusion from the substrate surface Finally, we have shown that the anal-ysis of experimental L(R) curves provides important infor-mation concerning the characteristics of growth processes such as the effective diffusion length on the substrate sur-face, the diffusion length on the sidewalls and the diffusion flux to the wires

Acknowledgements

The authors are grateful to the financial support re-ceived from SANDiE program, different RFBR, Russian Federal Agency for Science and Innovations within the frame of Contract No 02.513.11.3042 and different scien-tific programs of RAS N.V.S wishes to thank the Dynasty Foundation for the financial support

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