Length Distributions of Nanowires Growing by Surface Diffusion

Article pubs.acs.org/crystal

Length Distributions of Nanowires Growing by Surface Diffusion Vladimir G. Dubrovskii,†,‡,§ Yury Berdnikov,† Jan Schmidtbauer,∥,⊥ Mattias Borg,⊥,# Kristian Storm,⊥ Knut Deppert,⊥ and Jonas Johansson*,⊥ †

St. Petersburg Academic University, Khlopina 8/3, 194021 St. Petersburg, Russia Ioffe Physical Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021 St. Petersburg, Russia § ITMO University, Kronverkskiy pr. 49, 197101 St. Petersburg, Russia ∥ Leibniz Institute for Crystal Growth, Max-Born-Strasse 2, 12489 Berlin, Germany ⊥ Solid State Physics and NanoLund, Lund University, Box 118, S-22100 Lund, Sweden # IBM ResearchZurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland ‡

ABSTRACT: We present experimental data on the time and radiusdependent length distributions of Au-catalyzed InAs nanowires grown by metal organic vapor phase epitaxy. We show that these distributions are not as sharp as commonly believed. Rather, they appear to be much broader than Poissonian from the very beginning and spread quickly as the nanowires grow. We develop a model that attributes the observed broadening to the diffusion-induced character of growth. In the initial growth stage, the nanowires are fed from their entire length, leading to a Polya-like length distribution whose standard deviation is proportional to the mean length. After the nanowire length exceeds the adatom diffusion length, the growth acquires a Poissonian character in which the standard deviation scales as a square root of the mean length. We explain why wider nanowires have smaller length dispersion and speculate on the length distributions in Au-catalyzed versus self-catalyzed growth methods.

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INTRODUCTION Nanowires (NWs) obtained by the vapor−liquid−solid (VLS) growth method1 enable a precise positioning and shaping of semiconductor nanoheterostructures, which is very attractive for scalable bottom-up nanoelectronics and nanophotonics.2,3 Organized arrays of Au4−8 or Ga9−11 droplets are generally believed to yield uniform ensembles of NWs with sharp length and diameter distributions. This size uniformity is also expected to persist in non-VLS selective area growth techniques.12 However, a detailed analysis of the NW length distributions (LDs) has not been performed so far. Broad LDs of GaAs NWs were recently reported in ref 13 in the special case of Gacatalyzed VLS growth without predeposition of gallium, where the LD spreading was attributed to difficult nucleation of Ga droplets. Otherwise, the LD broadening is thought to be at most Poissonian.14 By definition of the Poissonian growth process,14 new NW monolayers are added randomly and independently with a probability that is determined by a lengthindependent influx of the growth species. For a single NW, such growth law yields a linear time dependence of the NW length, while the statistical ensemble of NWs is described by the Poissonian LD whose variance equals the mean length. Specific nucleation antibunching in nanocatalysts14−18 leads to a narrowing of the Poissonian distribution of the nucleation probabilities within a single nanowire, as recently described by Glas.18 One could anticipate that the self-regulated pulsed nucleation with sawtooth time dependence of supersaturation © XXXX American Chemical Society

would also yield a sharp LD in the NW ensemble. However, the correlation between the nucleation statistics in a single NW and the statistical properties of the ensemble is not straightforward (this question will be considered below). As for the VLS growth kinetics, most prior works (see refs 19 and 20 for a recent review) studied only the most representative NW in the ensemble considering that all NWs emerge at the same moment of time and are not influenced by the stochastic nature of the VLS growth process. This simplified approach completely neglects the nucleation-related effects that result in different starting times for growth of different NWs13 and fluctuation-induced broadening21,22 of the NW LDs on a later growth stage. Consequently, here we present experimental data on the LDs of Au-catalyzed InAs NWs along with a rate equation model for the LD that fits and explains the data. We find that the NW LDs are always much broader than the Poissonian distribution and show that the broadening is due to a diffusion-induced character of growth. The model is not restricted to the case of Au-catalyzed NWs and describes equally well the LDs of selfcatalyzed or selective area NWs. We deduce some general features of the NW LDs under different conditions and point out the ways for improving the length uniformity. Received: December 28, 2015 Revised: February 4, 2016

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DOI: 10.1021/acs.cgd.5b01832 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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Figure 1. 30° tilted SEM images of InAs NW arrays with 1000 nm pitch, growth time of 7.5 min, and NW diameters around (a) 53 nm, (b) 75 nm, and (c) 90 nm. The scale bar is the same in all three images.

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EXPERIMENTAL RESULTS

Au-catalyzed InAs NWs were grown using metal organic vapor phase epitaxy (MOVPE). Before MOVPE growth, patterns of Au particles were prepared by means of electron beam lithography (EBL) followed by thermal evaporation of Au and lift-off. The distance between the Au particles was 1000 nm, and by varying the EBL dose the nominal NW diameter was varied between 20 and 100 nm. The NWs were grown at 450 °C, and the investigated growth times were 7.5, 15, 22.5, 30, and 60 min. Further details of the NW fabrication are given in ref 23. Scanning electron microscopy (SEM) images of the NW arrays were recorded at specific tilt angles, and NW length and diameter statistics were extracted using the SEM image analysis software nanoDim.24 Figure 1 shows typical SEM images of differently sized InAs NWs after 7.5 min of growth. It is clearly seen that the NW length decreases for larger diameters. Figure 2 shows the normalized LDs of InAs NWs with different diameters after 7.5 min of growth. Differently sized NWs were divided

Figure 3. Broadening of the LD of 100−110 nm diameter NWs with the growth time: experimental LDs (histograms) fitted by the Gaussian distributions (lines) with a = 237 and ⟨s⟩ = 1000, 2110, 4400, 5400, and 10600 for the growth times of 7.5, 15, 22.5, 30, and 60 min, respectively. Poissonian one. Interestingly, the rate of spreading varies with time: the LDs broaden faster at the beginning and slower after 22.5 min of growth. The lines in Figures 2 and 3 are the model fits to the data that will be discussed shortly.

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RATE EQUATION MODEL To understand these features of the NW LDs, we first consider the model for the NW elongation rates which is illustrated in Figure 4. We assume that the Au-catalyzed III−V NWs grow with a time-independent radius R = const under group V rich conditions such that the VLS process is limited by the kinetics Figure 2. Normalized experimental LDs (histograms) of differently sized Au-catalyzed InAs NWs grown for 7.5 min. The data are presented in terms of the dimensionless length s = L/h (with h = 0.35 nm as the height of a monolayer). The data for 52−55, 64−70, 72−80, 82−90, 90−100, and 100−110 nm diameter NWs are fitted by the Gaussian given by eq 10 (lines) with a = 140, ⟨s⟩ = 3098, a = 160, ⟨s⟩ = 2231, a = 180, ⟨s⟩ = 1915, a = 200, ⟨s⟩ = 1472, a = 220, ⟨s⟩ = 1027, and a = 237, ⟨s⟩ = 1000, respectively. into bins of 4−11 nm diameters. It is seen that the NW length gradually decreases with increasing the diameter: the mean length of the narrowest 52−55 nm diameter NWs is about 3200 monolayers and is reduced to only 1000 monolayers for 100−110 nm diameter NWs. This clearly indicates a diffusion-induced character of the VLS growth.19,20 Quite surprisingly, the measured length histograms are much broader than the Poissonian LDs in all cases and widen toward smaller diameters. Figure 3 demonstrates how the LD of 100−110 nm diameter NWs broadens with the growth time. A rather sharp LD after 7.5 min of growth is gradually transformed into a broad histogram already at 22.5 min, and this spreading is much faster than the

Figure 4. Illustration of the growth kinetics of Au-catalyzed NWs: (a) growth start due to direct impingement, (b) growth of a short NW with L < λ by direct impingement and surface diffusion from the entire NW length L, (c) growth of a long NW with L > λ collecting the group III adatoms from the top section of length λ. B



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of the group III species.19,25 The NWs are seeded by Au droplets resting on the substrate surface at t = 0. The group III vapor flux is denoted by J. For the very first NW monolayer emerging from the droplet (Figure 4a), the vertical growth rate is given by (dL/dt)L=0 = χ0JΩ. Here, Ω is the elementary volume per III−V pair in the solid state and χ0 describes the strength of the VLS process with respect to a planar layer; the NW nucleation is possible only when χ0 > 0. For a NW whose length is smaller than the diffusion length of the group III adatoms on the NW sidewalls λ, the VLS growth rate is the sum of the direct impingement rate and the diffusion flux from the sidewalls (Figure 4b). The latter is proportional to the entire NW length L and inversely proportional to the NW radius R (refs 19, 20, 25, and 26). This yields a growth rate of the form (dL/dt)L>0 = χJΩ + 2φ(L/R)JΩ. Here, the φ factor describes the efficiency of the NW sidewalls in collecting the vapor species, while χ has the same meaning as χ0 but for the droplet seated on the NW top. In this model, we ignore the influx of the group III adatoms from the substrate surface,27 which has been proven negligible for Au-catalyzed GaAs NWs.25 Assuming χ = χ0, we adopt the NW growth rate in the form dL/dt = χJΩ + 2φ(L/R)JΩ and apply it uniformly for the lengths starting from L = 0. On the other hand, for long NWs with L > λ the group III adatoms are collected only from the top part of the NW of length λ (Figure 4c), yielding (dL/dt)L>λ = χJΩ + 2φ(λ/R)JΩ (refs 25 and 26). Measuring the NW length L in number of monolayers s (L = hs, with h as the height of a monolayer and s = 0, 1, 2, ...), we can rewrite the above equations for the growth rates as ⎛ 2φh ⎞ ds s⎟v , s < smax ≡ K s = ⎜χ + ⎝ dt R ⎠

⎛ ⎞ 2φh ds smax ⎟v , s > smax = ⎜χ + ⎝ ⎠ dt R

within the monolayer growth cycle,14−18 assuming a timeindependent supersaturation in all catalyst droplets. Obviously, the REs given by eqs 3 preserve the total surface density of NWs and remaining surface droplets whose sum equals the density of initial droplets ntot: ∑s≥0ns = ntot. Therefore, the LD fs = ns/ntot is normalized to one. Using our model for Ks given by eq 1 for short NWs, eqs 3 can be put in the following dimensionless form: df0 dτ dfs dτ

= (a + s − 1)fs − 1 − (a + s)fs , s ≥ 1.

(4)

The parameter a and the dimensionless time τ are given by a=

χR 2φh χvt ; τ= vt = 2φh R a

(5)

and depend on the NW radius. These REs are equivalent to the irreversible model of heterogeneous nucleation with size-linear capture rates,28 but with a time-independent vapor influx rather than a timedependent diffusion flux from the substrate surface. The exact solutions are obtained as described in ref 29 and are given by the Polya distribution f s(a) (⟨s⟩) =

−s −a ⟨s⟩ ⎞ ⎛ Γ(a + s) ⎛ a ⎞ ⎜1 + ⎟ ⎜1 + ⎟ Γ(a) Γ(s + 1) ⎝ a ⎠ ⎝ ⟨s⟩ ⎠

(6)

with the mean length increasing exponentially with time: ⎡ ⎛ χvt ⎞ ⎤ ⟨s⟩ = a⎢exp⎜ ⎟ − 1⎥ ⎣ ⎝ a ⎠ ⎦

(1)

(7)

Here, Γ(y) denotes the gamma function. The discrete LD given by eq 6 is monotonically decreasing when a ≤ 1 and unimodal when a > 1. For a given ⟨s⟩, the LD narrows up with increasing a, remaining broader than the a-independent Poissonian at a → ∞,

(2)

for discrete s = 0, 1 ,2, .... Here, v = JΩ/h is the deposition rate in monolayers per second and smax = λ/h is the NW length after which the growth rate becomes length-independent. Equations 1 and 2 give the mean-field approximations for the NW growth rates assuming that the VLS process consists of independent uncorrelated atomic events. Solving these equations would yield the exponentially increasing NW length as long as s < smax, followed by a linear time dependence after smax, as usual in the diffusion-induced NW growth.19 To introduce the stochastic nature of the growth process, we use these deterministic growth rates in the rate equations (REs) describing the LD of statistical ensemble of NWs, which is a standard approach in macroscopic nucleation theory.20 The surface densities ns(t) of NWs having length s at time t obey the system of REs dn 0 = −K 0n0 dt dns = Ks − 1ns − 1 − Ksns , s ≥ 1 dt

= −af0

fs (⟨s⟩) = e−⟨s⟩

⟨s⟩s ≅ s!

⎡ (s − ⟨s⟩)2 ⎤ exp⎢ − ⎥ 2⟨s⟩ ⎦ ⎣ 2π ⟨s⟩ 1

(8)

with the mean length increasing linearly with time:

⟨s⟩ = χvt

(9)

The LDs tend to this Poissonian only when a is much larger than ⟨s⟩, as shown in Figure 5. The discrete LD given by eq 6 at ⟨s⟩ ≫ a ≫ 1 (corresponding to highly anisotropic NWs with ⟨L⟩ ≫ R and R ≫ h) tends to the Gaussian f (a) (s , ⟨s⟩) ≅

1 ⟨s⟩

⎡ a (s − ⟨s⟩)2 ⎤ a exp⎢ − ⎥ 2π ⎣ 2 ⟨s⟩2 ⎦

(10)

whose variance ⟨s⟩ /a is much larger than ⟨s⟩ in the Poissonian growth. Therefore, the NWs that collect the group III adatoms from their entire lengths are influenced by a much greater fluctuation-induced broadening20 than the Poissonian NWs. One interesting property of the LD given by eq 10 is the socalled scaling behavior28,29 such that the mean length times the LD is a universal function of the scaled length x = s/⟨s⟩ given by 2

(3)

The first equation means that the number of surface droplets decreases whenever the first NW monolayer has grown under it. The other equations show that the number of NWs with length s decreases by adding one monolayer to them and increases by adding one monolayer to NWs with length s − 1. These REs do not account for the supersaturation kinetics C



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Figure 5. LDs of NWs at two different mean lengths ⟨s⟩ = 30 and 90 for different a, compared to the Poissonian LDs at a → ∞.

⟨s⟩f (a) (s , ⟨s⟩) = F(x) =

⎤ ⎡ a a exp⎢ − (x − 1)2 ⎥ ⎦ ⎣ 2 2π

Figure 6. Linear diameter dependence of a, obtained from the fits shown in Figure 2.

of one. This confirms our main assumption ⟨s⟩ ≫ a ≫ 1 used in deriving the Gaussian LD given by eq 10 from the exact solution for the discrete LD. Figures 7 and 8 show the time-dependent mean lengths and standard deviations of the LDs of 100−110 nm diameter NWs

(11)

Therefore, when plotted in the scaling variables, the combination in the left-hand side should remain the same for any ⟨s⟩. After the mean NW length exceeds the diffusion length λ, the NW growth rate becomes length-independent as given by eq 2. The Green function of the NW length distribution with lengthindependent growth rates is given by the Gaussian with the mean length ⟨s⟩ and variance σ2 of the form s ⎞ ⎛ ⟨s⟩ = smax + ⎜1 + max ⎟Cχv(t − tmax ); σ 2 = ⟨s⟩−smax ⎝ a ⎠ (12)

Here, tmax is the time at which the mean NW length reaches the diffusion length and C is a constant. The solution for the NW LDs at t > tmax is then obtained by convoluting the two Gaussians representing the Green function and the LD at t = tmax. The resulting LD has the form f (a) (s , ⟨s⟩) ≅

1 2π σ

Figure 7. Transition from the exponential to linear growth regime of the 100−110 nm diameter InAs NWs (symbols); the fitting parameters for the lines are a = 237, χv = 34 min−1, and C = 0.247.

⎡ (s − ⟨s⟩)2 ⎤ exp⎣⎢ − ⎥ 2σ 2 ⎦

⎧ ⟨s⟩2 ⎪ , ⟨s⟩≤smax ⎪ a 2 σ =⎨ 2 ⎪ smax + ⟨s⟩−smax , ⟨s⟩≥smax ⎪ ⎩ a

(13)

This solution holds for any ⟨s⟩, while the mean length is given by eq 7 in the exponential stage until ⟨s⟩ < smax and by the corresponding eq 12 when ⟨s⟩ > smax.

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MODELING RESULTS AND DISCUSSION Let us now see how the model fits our experimental LDs of Aucatalyzed InAs NWs. Since our s values are very large, greater than 500, we choose to fit the data by the continuum LD in the Gaussian form given by eq 13. The solid curves in Figure 2 show the best fits for the LDs after 7.5 min of growth for NWs with different diameters. Equation 5 for a shows that this parameter scales linearly with the NW diameter, explaining why wider NWs have sharper LDs. Figure 6 demonstrates that the a values obtained from the best fits are indeed proportional to the NW diameter. It is noteworthy that the fits are obtained with rather high a ≥ 140. This is not surprising because the a parameter contains a huge value of R/h in the range of 140− 300, while the coefficient χ/(2φ) is expected to be of the order

Figure 8. Best fits by eqs 13 (lines) of the experimental standard deviations of the 100−110 diameter NW LDs (symbols) versus the mean length at different growth times.

presented in Figure 3. It is seen that the mean length increases exponentially with the growth time for t < 22.5 min and linearly for longer times. The standard deviation is proportional to the mean length for t < 22.5 but becomes sublinear for longer times. Within the model, such changes are attributed to the transition from length-linear to length-independent NW growth D



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rates. The lines in Figures 7 and 8 show the fits obtained with the model equations in the exponential and linear growth stages with a = 237 and χv = 34 min−1. The solid curves in Figure 3 show the Gaussian fits to the experimental LDs at different growth times. Very importantly, these fits are obtained with the same a = 237 for each growth time, which again confirms the validity of our model. Hence we conclude that the transition from the exponential to linear Poissonian growth occurs around 22.5 min. This corresponds to smax = 4400, and thus the diffusion length of In adatoms on the NW sidewalls can be estimated at about 1500 nm at 450 °C, in agreement with previous results on the group III adatom diffusivities.19,20,25,26 Thus, the LD width increases more rapidly in the initial growth stage and then saturates to the Poissonian broadening. This shows very clearly that feeding the NWs from their entire lengths has a very significant effect on the LDs. Finally, Figure 9 shows that the scaling hypothesis also works quite well for short NWs, while at 30 min the scaled LD narrows up as it should be in the Poissonian growth picture.

kinetics of nondiffusive group V species19 and thus should yield much narrower LDs. High surface diffusivities of Ga and In adatoms help to grow longer III−V NWs, but the length uniformity is sacrificed. If the group III supply from sidewall diffusion can be completely removed, as for the case of InAs nanowire growth within oxide templates,30 the LD may be considerably narrower. Quite importantly, purely deterministic NW growth theories should now be reconsidered from the viewpoint of the obtained results. In particular, there is a need to describe more accurately the length−time and length− diameter correlations within the ensembles of NWs. The essential step of our approach was the pre-existing growth seeds (droplets or holes), which ensure heterogeneous character of the NW nucleation. When Ga droplets are formed concomitantly with the GaAs NW growth13 or in the case of self-induced GaN NWs,31 the NW nucleation becomes homogeneous. According to the results of ref 28, this may significantly change the LD shapes. We have also neglected radial growth of NWs, which always occurs at low temperatures and influences the resulting LDs. These questions will be considered in detail elsewhere.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. phone: +46 46 2221472. Fax: +46 46 2223637. Web: http://www.nano.lu.se/jonas. johansson. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS V.G.D. thanks the Russian Science Foundation for the financial support received under Grant 14-22-00018. J.S. acknowledges financial support from the German Research Foundation (DFG). M.B., K.S., K.D., and J.J. thank the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF), the Knut and Alice Wallenberg Foundation (KAW), and NanoLund (the Center for Nanoscience at Lund University) for financial support.

Figure 9. Measured LDs of 100−110 nm diameter InAs NWs at different growth times, replotted in the scaling variables (symbols) and fitted by the scaling function given by eq 11 with a = 237 (line).

As regards the nucleation antibunching effect on the NW LDs, we note that taking the supersaturation dynamics into account, the LDs become much narrower, i.e., sub-Poissonian.15,18 We have verified this by kinetic Monte Carlo simulations on a model operating at this time scale, where emptying and refilling the particle has a self-regulating effect on the average growth rate. If we instead assume a constant supersaturation, these simulations give Gaussian LDs, in agreement with the LDs calculated and fitted to experimental results in the current work. This indicates that, for VLS growth of NWs, the supersaturation variations in the particle during layer nucleation in general can be considered small on the experimental time scale for NW growth, leading to approximately Gaussian LDs (Polya-like or Poisson-like depending on the NW length with respect to the diffusion length). The presented approach is not at all restricted to the case of Au-catalyzed VLS growth or particular growth technique and will work equally well for self-catalyzed and selective area NWs. Indeed, we only took into account the features of the material transport into the droplet without any nucleation-dependent effects. It has been shown that the NW LDs broaden much faster in systems with high adatom diffusivities, while shorter diffusion lengths or negligible surface diffusion yields the Poissonian LDs. For instance, self-catalyzed VLS growth from predeposited Ga droplets is known to be controlled by the

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REFERENCES

(1) Wagner, R. S.; Ellis, W. C. Appl. Phys. Lett. 1964, 4, 89. (2) Qian, F.; Li, Y.; Gradečak, S.; Park, H.-G.; Dong, Y.; Ding, Y.; Wang, Z. L.; Lieber, C. M. Nat. Mater. 2008, 7, 701. (3) Tian, B.; Zheng, X.; Kempa, T. J.; Fang, Y.; Yu, N.; Yu, G.; Huang, J.; Lieber, C. M. Nature 2007, 449, 885. (4) Martensson, T.; Carlberg, P.; Borgstrom, M.; Montelius, L.; Seifert, W.; Samuelson, L. Nano Lett. 2004, 4, 699. (5) Hochbaum, A. I.; Fan, R.; He, R.; Yang, P. Nano Lett. 2005, 5, 457. (6) Bryllert, T.; Wernersson, L. E.; Froberg, L. E.; Samuelson, L. IEEE Electron Device Lett. 2006, 27, 323. (7) Dayeh, S. A.; Picraux, S. T. Nano Lett. 2010, 10, 4032. (8) Kelrich, A.; Dubrovskii, V. G.; Calahorra, Y.; Cohen, S.; Ritter, D. Nanotechnology 2015, 26, 085303. (9) Plissard, S.; Larrieu, G.; Wallart, W.; Caroff, P. Nanotechnology 2011, 22, 275602. (10) Gibson, S. J.; Boulanger, J. P.; LaPierre, R. R. Semicond. Sci. Technol. 2013, 28, 105025. (11) Dubrovskii, V. G.; Xu, T.; Díaz Á lvarez, A.; Plissard, S. R.; Caroff, P.; Glas, F.; Grandidier, B. Nano Lett. 2015, 15, 5580. (12) Gao, Q.; Saxena, D.; Wang, F.; Fu, L.; Mokkapati, S.; Guo, Y.; Li, L.; Wong-Leung, J.; Caroff, P.; Tan, H. H.; Jagadish, C. Nano Lett. 2014, 14, 5206. E



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(13) Matteini, F.; Dubrovskii, V. G.; Rüffer, D.; Tütüncüoğlu, G.; Fontana, Y.; Fontcuberta i Morral, A. Nanotechnology 2015, 26, 105603. (14) Dubrovskii, V. G. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 195426. (15) Glas, F.; Harmand, J. C.; Patriarche, G. Phys. Rev. Lett. 2010, 104, 135501. (16) Wen, C. Y.; Tersoff, J.; Hillerich, K.; Reuter, M. C.; Park, J. H.; Kodambaka, S.; Stach, E. A.; Ross, F. M. Phys. Rev. Lett. 2011, 107, 025503. (17) Gamalski, A. D.; Ducati, C.; Hofmann, S. J. Phys. Chem. C 2011, 115, 4413. (18) Glas, F. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 125406. (19) Dubrovskii, V. G. Theory of VLS Growth of Compound Semiconductors. In Semiconductors and Semimetals; Fontcuberta i Morral, A., Dayeh, S. A., Jagadish, C., Eds.; Academic Press: Burlington, 2015; Vol. 93, pp 1−78. (20) Dubrovskii, V. G. Nucleation theory and growth of nanostructures; Springer: Heidelberg−New York−Dordrecht−London, 2014. (21) Dubrovskii, V. G. J. Chem. Phys. 2009, 131, 164514. (22) Dubrovskii, V. G.; Nazarenko, M. V. J. Chem. Phys. 2010, 132, 114507. (23) Borg, B. M.; Johansson, J.; Storm, K.; Deppert, K. J. Cryst. Growth 2013, 366, 15. (24) URL: www.nanodim.net. (25) Harmand, J. C.; Glas, F.; Patriarche, G. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 81, 235436. (26) Plante, M. C.; LaPierre, R. R. J. Appl. Phys. 2009, 105, 114304. (27) Fröberg, L. E.; Seifert, W.; Johansson, J. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 153401. (28) Dubrovskii, V. G.; Berdnikov, Y. S. J. Chem. Phys. 2015, 142, 124110. (29) Dubrovskii, V. G.; Sibirev, N. V. Phys. Rev. E 2015, 91, 042408. (30) Borg, M.; Schmid, H.; Moselund, K. E.; Cutaia, D.; Riel, H. J. J. Appl. Phys. 2015, 117, 144303. (31) Dubrovskii, V. G.; Consonni, V.; Trampert, A.; Geelhaar, L.; Riechert, H. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 165317.

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Length Distributions of Nanowires Growing by Surface Diffusion

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