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

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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 , J.C. Harmand d , V.M. Ustinov a,b 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/R 2 , 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. Ó 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 SiCl 4 and H 2 on the Si(1 11) surface activated by Au drops at surface temperature T of about 1000 °C. 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 theoret ical 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 Ó 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 Surface Science 601 (2007) 4395–4401 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 11) 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 = 525 °C [10] and different III–V wires at T = 550–600 °C [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 wi re 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 (R 2 ), 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 R 2 dL/dt = AR 2 + BR * R. Dividing this over R 2 , 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/k s , where k s 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/R 2 behavior of wire length. We will present the data on the MBE grown GaAs and AlGaAs nanowires exhibiting such 1/R 2 length dependence. Secondarily, we will construct a self-consistent route to estimate the value of k s . Finally, we will consider the dependence of wire length on temperature T, flux V, drop density N W 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 U ¼ V s l 2r l X s C eq À 1, where V is the impinging flux, s l is the mean lifetime of semiconductor particles in the drop before re-evaporation, r l is the inter-atomic distance in the liquid, X s is the volume per atom in the crystal and C eq (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 V s . The wire growth rate in the steady state is given by [28] dL dH ¼ e Àc þ R à R ð1Þ 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 =1À V s /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 sidewal ls pro- vides the expression for R * in the form [28] R à ¼ 2k s b þ2k f sin a½coshðL=k f ÞÀ1 þbG sinhðL=k f Þ bG coshðL=k f ÞþsinhðL=k f Þ ð2Þ 4396 V.G. Dubrovskii et al. / Surface Science 601 (2007) 4395–4401 Here, k f ¼ ffiffiffiffiffiffiffiffiffi D f s f p ffi ffiffiffiffiffi r f p exp½ðE f A À E f D Þ=2 k B T  is the adatom diffusion length on the sidewalls (limited by desorption), E f A and E f D are the activation energies for the adatom desorption and diffusion on the sidewalls, k s ¼ ffiffiffiffiffiffiffiffiffi D s s s p is the effective adatom diffusion length on the substrate surface (normally limited by nucleation or by outgoing flux to wire sidewalls) and b ¼ðr s k s D f =r f k f D s Þffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D f s s =D s s f p . Here and below r f and r s denote 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 G ¼ I 1 ðR W =k s ÞK 0 ðR=k s ÞþK 1 ðR W =k s ÞI 0 ðR=k s Þ I 1 ðR W =k s ÞK 1 ðR=k s ÞÀK 1 ðR W =k s ÞI 1 ðR=k s Þ ð3Þ where I n and K n are the modified Bessel functions in stan- dard notations, and R W ¼ 1= ffiffiffiffiffiffiffiffiffiffi pN W 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 R min [20,23,24].AtR $ R min , 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 R min 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 k s and quantity e =1À V s /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 e 1 = X s J diff /V of adatom migration to the wire sidewall (here J diff denotes the overall diffusion flux to the wire bases per unit surface area), e 2 = V s /V of adatom incorporation to the growing surface layer and e 3 = V des /V to re-evaporate from the surface. Due to the mass conservation e 1 + e 2 + e 3 = 1 and e =1À e 2 = e 1 + e 3 by definition. Probability e 1 must be proportional to the overall diffusion flux J diff , which equals the sum of individual fluxes j diff (0) to differently sized wires: e 1 V X s ¼ N W hj diff ð0Þi ð4Þ Here and below hgi denotes the average with normalized size distribution of drops f ðRÞ ; hgi¼ R 1 0 dRf ðRÞgðRÞ: Indi- vidual diffusion flux to the wire base can be found from the solution for surface concentration of adatoms n s (r) [28] as j diff (0) = D s 2pR dn s /drj r = R in the form j diff ð0Þ¼pR V X s ½R à coshðL=k f ÞÀ2k f sin a sinhðL=k f Þ ð5Þ Inserting this into Eq. (4), upon averaging we get e 1 pN W hRi ¼ hR à Ri hRi coshðL=k f ÞÀ2k f sin a sinh ðL=k f Þð6Þ Since R * is the known function of k s , Eq. (6) allows one to find k s as function of e 1 , of the parameters of drop size dis- tribution and of the growth conditions T and V.IfR * 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 Àc þ 1 R e 1 phRiN W coshðL=k f Þ þ 2k f sin a tanhð L= k f Þ  ð7Þ If k s is considerably smaller than the radii of wires and of two-dimensional islands on the surface, we can assum e that probability e 1 is proportional to the total perimeter of wires per unit surface area P W =2pN W hRi and also that probability e 2 is proportional to the appropriately averaged perimeter of islands P I . As shown, for example, in Ref. [30], the time dependence of layer perimeter can be approxi- mated as P I ðsÞ¼p ffiffiffiffiffiffiffiffi N I s p e Às . Here, N I is the surface density of islands emerging in each layer and s is a certain relative time. Averaging this in s, we arrive at P I ffi p ffiffiffiffiffiffi N I p Z 1 0 dxx 1=2 e Àx ¼ p 3=2 2 ffiffiffiffiffiffi N I p ð8Þ The surface density of islands N I during the layer-by-layer growth can be calculated by applying the nucleation theory [22,30]. For us it is important that N I depends on temper- ature T and deposition flux V approximately as N I / V 2 exp 3K s þ 2E s D k B T  ð9Þ where K s is the specific condensation heat of two-dimen- sional ‘‘vapor’’ of adatoms and E s D is the activation energy for adatom diffusion. Finally, the probability of desorpt ion e 3 is proportional to the reverse diffusion length of single adatom on a bare substrate 1/k 0 s . Temperature dependence of k 0 s is given by the conventional exponential expression k 0 s ffi ffiffiffiffiffi r s p exp½ðE s A À E s D Þ=2k B T , where E s A is the activati on V.G. Dubrovskii et al. / Surface Science 601 (2007) 4395–4401 4397 energy for desorption. The expressions for e and e 1 follow- ing from the above analysis read e ¼ 1 þ P W k 0 s 1 þðP W þ P I Þk 0 s ; e 1 ¼ P W k 0 s 1 þðP W þ P I Þk 0 s ; k s ( R ð10Þ In the opposite case, when k s is much larger than the radii of wires and islands, but smaller than the average dis- tance between them, the probability e 1 will be proportional to surface density of wires N W , the probability e 2 will be proportional to density of islands N I and the probability e 3 will be proportional to ð2pk 0 s Þ À2 . Repeating the consider- ations described above, the expressions for e and e 1 in this case will be e ¼ 1 þ 2pN W ðk 0 s Þ 2 1 þ 2pðN W þ N I Þðk 0 s Þ 2 ; e 1 ¼ 2pN W ðk 0 s Þ 2 1 þ 2pðN W þ N I Þðk 0 s Þ 2 ; R ( k s ( R W ð11Þ 3. Results and discu ssion 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/k f ) 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] L ¼ e À c þ 2k f R  H ð12Þ 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 11)B-sur- face activated by Au drops: N W $ 10 9 cm À2 , hRi =40nm and T = 580 À 590 °C [14,15]. Different estimates for the diffusion length of Ga atoms on the GaAs(110) sidewalls at this temperature range from 3 [15] to 10 [31] lm. For wires with L <3lm we can therefore use a simplified equa- tion R * =2k s /G at L/k f ( 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 R W approximately equals 180 nm, and for the value of k s of several tens of nanometers (verified experimentally be- low in this paper) we can use the asymptote of Eq. (3) at R W /k s ) 1. In this case Eqs. (2), (3) are simplified to R à ¼ 2k s K 1 ðR=k s Þ K 0 ðR=k s Þ ! 2k s ; k s ( R 2k 2 s R lnðk s =RÞ ; k s ) R ( ð13Þ where we write explicitly two limit cases of wire growth. When the effecti ve 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/R 2 (1/R 2 -diffusion). Numerical analysis shows that the function in the right hand side of Eq. (13) can be approximated with reasonable accuracy by k 2 s =R 2 for the values of R $ k s , if we put ln (k s /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/R 2 -diffusion L ¼ e À c þ 2k s R  H; k s ( R ð14Þ k s ¼ k 0 s 1 þðP W þ P I Þk 0 s ffi 1 P W þ P I ; k s ( R ð15Þ L ffi e À c þ 2k 2 s R 2  H; k s ) R ð16Þ k s ¼ k 0 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2 pðN W þ N I Þðk 0 s Þ 2 q ffi 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðN W þ N I Þ p ; k s ) R ð17Þ Approximate expressions in Eqs. (15) and (17) apply at a low desorption from the surface. For example, when GaAs wires a re grown on the GaAs(1 11)B surface, the diffusion length of Ga atoms k 0 s approximately equals 6 lm at 590 °C [32].AtN W =10 9 cm À2 and hRi = 50 nm this corresponds to P W k 0 s % 19 and N W ½k 0 s  2 % 360. In this case the expres- sions for e and e 1 become e ¼ e 1 ¼ P W P W þ P I ; k s ( R ð18Þ e ¼ e 1 ¼ N W N W þ N I ; k s ) R ð19Þ 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 k f or k s and e À c.By measuring the average thickness of epitaxial layer on the substrate surface H s , 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 desorption from the substrate, even without measuring H s , at known N W and hRi we are able to deduce P I or N I 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 À 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 = 525 °C [10], the GaAs/GaAs(1 1 1)B–Au nanowires at grown by MBE at T = 550–600 °C [13,14,16], and also for the GaAs/GaAs(1 11)B–Au nanowires grown by MSD at T = 585 °C [17]. Below we present experimental data on the Al 0.33 Ga 0.67 As/GaAs(1 11)B–Au and the GaAs/ 4398 V.G. Dubrovskii et al. / Surface Science 601 (2007) 4395–4401 GaAs(1 11)B–Au nanowires exhibiting the 1/R 2 behavior of L(R) dependences. The Al 0.33 Ga 0.67 As wires were grown by MBE at surface temperature T = 585 °C 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 = 560 °C and H = 1000 nm. Experimental details can be found in Ref. [28]. The values of wire density N W for the both ensembles of wires are given in Table 1. From the analysis of scanning electron microscopy images of these wires [28,33], we worked out experimental length–diameter curves shown by points in Figs. 1 and 2. Solid line in Fig. 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/R 2 curve 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 in Table 1. Solid line in Fig. 2 corresponds 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 in Table 1. Analysis of data presented in Table 1 shows that wires consume abou t 32% of all adatom s 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 R min is 20 nm for Al- GaAs and 31 nm for 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 reasonabl e values for surface density of islands in the regime of complete con- densation $10 9 –10 10 cm À2 [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 À c ( R * /R). In the case of VPD at sufficiently high temperatures, when L/k f ) 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 / k f ¼ k 0 f exp G f T 0 T À 1  ð20Þ Here, the quantity G f ¼ðE f A À E f D Þ=2k B T 0 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 k s in the cases of 1/R and 1/R 2 diffusion, Eqs. (8) and (9) and the temperature dependence of k 0 s , one obtains the effective diffusion lengths as functions of N W , hRi, T and V in the form k s ¼ P W þ P I ðT 0 ; V 0 Þ V V 0 exp F s T 0 T À 1  þ 1 k 0 s ðT 0 Þ exp ÀG s T 0 T À 1  # À1 ; k s ( R ð21Þ Table 1 Parameters of lateral size distribution and theoretical characteristics of wire growth Wires N W , cm À2 k s , nm N I ,cm À2 ecU 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 °C2· 10 9 35 1.1 · 10 10 0.15 0.05 19 50 75 100 125 150 175 200 0 1000 2000 3000 4000 5000 6000 Length of nanowire [nm] 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 À c = 0.25. Dotted line is the 1/R-type theoretical curve from Ref. [33]. 0 100 200 300 400 500 0 500 1000 1500 2000 2500 3000 3500 Length of nanowire [nm] 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 À c = 0.1. Dashed line is obtained from general equations 7,8 at sin a =0,k f =10lm, e À c = 0.1, b = 0.3 and other parameters given in Table 1. V.G. Dubrovskii et al. / Surface Science 601 (2007) 4395–4401 4399 k s ¼ 2pN W þ 2pN I ðT 0 ; V 0 Þ V 2 V 2 0 exp 2F s T 0 T À 1  þ 1 ½k 0 s ðT 0  2 exp À2G s T 0 T À 1  # À1=2 ; k s ) R ð22Þ Here, T 0 and V 0 are the reference values of temperature and deposition rate, the quantities F s ¼½ð3K s =2Þþ E s D =k B T 0 and G s ¼ðE s A À E s D Þ=2k B T 0 are determined by the characteristics of adtoms on the substrate surface. It is seen that normally k s increases with the surface temper- ature and decreases with the deposition flux and is limited by the values of 1/(2phRiN W )or1= ffiffiffiffiffiffiffiffiffiffiffiffiffi 2pN W p , both decreas- ing with the density of drops N W . The temperature and flux dependence is dictated by Eq. (9) for the density of islands N I , 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 /k f ( 1, because k f exponentially decreases with T. Consider, for example, the case of 1/R 2 diffusion, when the function G in Eq. (2) is given by G =(R/k s )ln(k s /R). Direct integration of Eq. (1) at e À c = 0 with this G gives R 2 lnðk s =RÞsinhðL=k f ÞþmRk f coshðL=k f ÞÀ1½¼ 2k 2 s k f H ð23Þ where m = r f D s /r s D f .AtL/k f ! 0 and R/k s ! 0 this equa- tion is reduced to L ffi 2k 2 s R 2 lnðk s =RÞ H / k 2 s ð24Þ 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 k s , and wire length increases with the temper- ature and decreases with the deposition flux. In the opposite case of large L/k f Eq. (23) gives L ffi ln 4k 2 s H mk 2 f R ! k f / k f ð25Þ At high T the wire length therefore decreases with the tem- perature approximately as k f . 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. 3 demonstrates 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 11)B surface activated by Au drops within the tem- perature range of 460–600 °C. 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 550–580 °C 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/R 2 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. References [1] G. Zheng, W. Lu, S. Jin, C.M. Lieber, Adv. Mater. 16 (2004) 1890. [2] T. Bryllert, L E. Wernersson, L.E. Froberg, L. Samuelson, IEEE Lett. 27 (2006) 323. [3] K. Hiruma, M. Yazawa, T. Katsuyama, K. Ogawa, K. Haraguchi, M. Koguchi, J. Appl. Phys. 77 (1995) 447. 450 500 550 600 650 700 0 2 4 6 8 10 Deposition rate Average growth rate (A/s) Surface temperature T ( ° C) Fig. 3. Experimental (black circles) and theoretical (solid line) depen- dences of the length of GaAs nanowires on the surface temperature. Theoretical curve is obtained from Eqs. (22) and (23) at F s =6,G f = 25, k f = 6000 nm, V = V 0 , T 0 = 560 °C, N W =2· 10 9 cm À2 and N I (T 0 )=1· 10 10 cm À2 , m = 0.01. 4400 V.G. Dubrovskii et al. / Surface Science 601 (2007) 4395–4401 [4] M.T. Bjork, B.J. Ohlsson, T. Sass, A.I. Persson, C. Thelander, M.H. Magnusson, K. Deppert, L.R. Wallenberg, L. Samuelson, Appl. Phys. Lett. 80 (2002) 1058. [5] E. Patolsky, G. Zheng, O. Hayden, M. Lakadamyali, X. Zhuang, C.M. Lieber, Proc. Natl. Acad. Sci. USA 101 (2004) 14017. [6] R.S. Wagner, W.C. Ellis, Appl. Phys. Lett. 4 (1964) 89. [7] E.I. Givargizov, J. Cryst. Growth 31 (1975) 20. [8] A.M. Morales, C.M. Lieber, Science 278 (1998) 208. [9] F.M. Ross, J. Tersoff, M.C. Reuter, Phys. Rev. Lett. 95 (2005) 146104. [10] L. Schubert, P. Werner, N.D. Zakharov, G. Gerth, F.M. Kolb, L. Long, U. Go ¨ sele, T.Y. Tan, Appl. Phys. Lett. 84 (2004) 4968. [11] X. Duan, C.M. Lieber, Adv. Mater. 12 (2000) 298. [12] B.J. Ohlsson, M.T. Bjork, A.I. Persson, C. Thelander, L.R. Wallen- berg, M.H. Magnusson, K. Deppert, L. Samuelson, Physica E 13 (2002) 1126. [13] W. Seifert, M. Borgstrom, K. Deppert, K.A. Dick, J. Johansson, M.W. Larsson, T. Martensson, N. Skold, C.P.T. Svensson, B.A. Wacaser, L.R. Wallenberg, L. Samuelson, J. Cryst. Growth 272 (2004) 211. [14] V.G. Dubrovskii, G.E. Cirlin, I.P. Soshnikov, A.A. Tonkikh, N.V. Sibirev, Yu.B. Samsonenko, V.M. Ustinov, Phys. Rev. B 71 (2005) 205325. [15] J.C. Harmand, G. Patriarche, N. Pe ´ re ´ -Laperne, M N. Me ´ rat- Combes, L. Travers, F. Glas, Appl. Phys. Lett. 87 (2005) 203101. [16] M.C. Plante, R.R. LaPierre, J. Cryst. Growth 286 (2) (2006) 394. [17] V.G. Dubrovskii, I.P. Soshnikov, N.V. Sibirev, G.E. Cirlin, V.M. Ustinov, J.Cryst.Growth 289 (2006) 31. [18] A.A. Tonkikh, G.E. Cirlin, V.G. Dubrovskii, N.V. Sibirev, I.P. Soshnikov, Yu.B. Samsonenko, N.K. Polyakov, V.M. Ustinov, Phys. Stat. Sol. (a) 203 (2006) 1365. [19] E.I. Givargizov, A.A. Chernov, Kristallografiya 18 (1973) 147. [20] E.I. Givargizov, A.A. Chernov, J. Cryst. Growth 31 (1975) 20. [21] D. Kashchiev, Cryst. Growth Design 6 (2006) 1154. [22] D. Kashchiev, Nucleation: Basic Theory with Applications, Butter- worth Heinemann, Oxford, 2000. [23] V.G. Dubrovskii, N.V. Sibirev, Phys. Rev. E 70 (2004) 031604. [24] V.G. Dubrovskii, N.V. Sibirev, G.E. Cirlin, J.C. Harmand, V.M. Ustinov, Phys. Rev. E 73 (2006) 021603. [25] W. Dittmar, K. Neumann, in: R.H. Doremus, B.W. Roberts, D. Turnball, N.Y. John (Eds.), Growth and Perfection of Crystals, Wiley, 1958, p. 121. [26] V. Ruth, J.R. Hirth, J. Chem. Phys. 41 (1964) 31. [27] J.M. Blakely, K.A. Jackson, J. Chem. Phys. 37 (1962) 428. [28] V.G. Dubrovskii, N.V. Sibirev, R.A. Suris, G.E. Cirlin, V.M. Ustinov, M. Tchernycheva, J.C. Harmand, Semiconductors 40 (2006) 1075. [29] V.G. Dubrovskii, N.V. Sibirev, Tech. Phys. Lett. 32 (3) (2006) 185. [30] V.G. Dubrovskii, N.V. Sibirev, G.E. Cirlin, V.M. Ustinov, Semicon- ductors 40 (2006) 249. [31] T. Takebe, M. Fujii, T. Yamamoto, K. Fujita, T. Watanabe, J. Appl. Phys. 81 (1997) 7273. [32] S. Koshiba, Y. Nakamura, M. Tsuchiya, H. Noge, H. Kano, Y. Nagamune, T. Noda, H. Sakaki, J. Appl. Phys. 76 (1994) 4138. [33] G.E. Cirlin, V.G. Dubrovsii, N.V. Sibirev, I.P. Soshnikov, Yu.B. Samsonenko, A.A. Tonkikh, V.M. Ustinov, Semiconductors 39 (2005) 557. V.G. Dubrovskii et al. / Surface Science 601 (2007) 4395–4401 4401 . Diffusion-controlled growth of semiconductor nanowires: Vapor pressure versus high vacuum deposition V.G. Dubrovskii a,b, * , N.V The dependences of wire length on the drop density, surface temperature and deposition flux during vapor pressure deposition and high vacuum deposition are