Droplet Bulge Effect on the Formation of Nanowire Side Facets

Detailed calculations demonstrate that this effect can promote otherwise unfavorable nanowire side facets when high-surface-energy droplet materials l...
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Droplet Bulge Effect on the Formation of Nanowire Side Facets Steffen Breuer,*,†,‡ Lou-Fé Feiner,§ and Lutz Geelhaar† †

Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5−7, 10117 Berlin, Germany Department of Electronic Materials Engineering, Australian National University, Canberra ACT 0200, Australia § Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands ‡

ABSTRACT: The process of lateral facet formation during vapor−liquid−solid nanowire growth is modeled to explain why the crystalline orientation of III−V nanowire side facets is affected by the droplet material. We find that the energy required for facet formation depends on the composition and shape of the liquid droplet. A droplet bulge effect that favors tilted nanofacets is identified: the lateral surface energy is reduced if the angle between droplet and facet is small. Detailed calculations demonstrate that this effect can promote otherwise unfavorable nanowire side facets when high-surface-energy droplet materials like Au are employed.

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INTRODUCTION It is hard to overstate the importance of surface effects for the physics of nanowires and other nanoscale objects.1 Such effects include surface-bound charge carrier accumulation or depletion, internal electric fields, and localized carrier recombination, which can be employed for sensing applications,2,3 but more generally can strongly influence the performance of any device. The strength of these effects commonly depends on the crystallographic orientation of the surface. Thus, it is imperative to study the orientation of nanowire surfaces, most importantly their side facets, and to understand how facet formation is controlled. The shape of crystals grown directly from the vapor phase is well understood to be governed by the minimum of the solid− vapor surface energy, γSV (equilibrium crystal shape).4 However, this thermodynamic criterion cannot explain why for vapor−liquid−solid (VLS) nanowire growth many observations5−18 indicate an effect of the choice of droplet material on the side facet configuration. Here, we investigate what is the most favorable side facet depending on droplet material and growth conditions. Going beyond thermodynamic energy minimization, we describe the kinetics of VLS facet formation. Thereby, we identify an effect of the liquid droplet on facet formation kinetics. We show that the nucleation of outward-tilted facets is supported by the outward bulge of the droplet and that this effect is proportional to the droplet’s liquid−vapor surface energy, γLV . We use our © XXXX American Chemical Society

model to calculate the most favorable GaAs nanowire side facet for Ga- and Au-assisted GaAs nanowire growth under various growth conditions and thus explain the experimental observations.

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REVIEW OF OBSERVED SIDE FACETS

Considering side facet orientations, a particularly well-studied case is that of [111]-oriented III−As nanowires in the zincblende structure, which have been grown using several droplet materials.5−19 It was found that {110} side facets develop if the growth is either droplet-free13−15 or mediated by group-III droplets,15−18 see Figure 1a,c. (As one noteworthy exception from this trend, there is one report of {112} facets for Ga-assisted GaAs nanowires, which however results from incorrectly referring to {110} instead of {111} as the cleavage planes of the Si substrate.20 When corrected, {110} side facets result once more.) These observations should not be startling, since {110} facets constitute the typical cleavage planes for III− V compounds and have particularly low surface energy.4 Moreover, the orientation of the nonpolar {110} facets parallel to the growth direction allows atomically flat sidewalls and thus an ideal prismatic nanowire shape. Received: December 3, 2012 Revised: May 27, 2013

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Figure 1. Nanowire facet orientations (bold) in relation to bulk-truncated zincblende crystal structure, showing (a) extended {110} facets and (b) {112} macrofacets from the top. Along the wire length, extended {110} facets are vertical (c), while {112} macrofacets can be composed from segments with (d) {111}A and {001}B tilted nanofacets (composing {112}A), as well as {1̅1̅1̅}B and {001̅}A tilted nanofacets (composing {1̅1̅2̅}B) or alternatively (e) from {111}A and {1̅1̅1̅}B tilted nanofacets requiring twin defects (dashed).

growth modes that lead to the tilted, polar and vertical, nonpolar side facets, respectively. We thus envisage layer-by-layer growth in which the addition of each III−V monolayer consists of three stages: nucleation at the vapor−liquid−solid triple phase line (TPL) at one edge of the hexagonal (1̅1̅1̅)B top facet of the nanowire, rapid exergonic expansion of the nucleus after it has reached critical size, and completion at the other edges, the layer thus finally covering the entire nanowire−liquid interface. In addition, we will later discuss briefly facet formation in the case of nucleation in the center. While there is agreement among researchers that VLS growth happens in a layer-by-layer fashion by the lateral expansion of a single nucleus, there are relevant details in the picture sketched above that are currently debated or even unknown. In particular, there is currently no consensus about the exact geometry of the liquid−solid growth interface. High-resolution transmission electron microscopy (TEM) images obtained after growth demonstrate an atomically flat growth interface, for example, for Au/GaAs.27 However, recent in situ TEM investigations showed for Al2O3, Ge, Si, and GaP nanowires that the liquid−solid interface can exhibit a pulsating truncated region near the triple phase line,28−30 and an accompanying theoretical analysis suggested that nucleation does not take place at the TPL.30 While there is no doubt about the validity of these results, it is at present unclear how general this phenomenon is, both with respect to growth conditions and material system. In particular, for III−As nanowires, as considered here, a truncated liquid−solid interface has never been reported. Moreover, for axial InAs/GaAs nanowire heterostructures a smooth, atomically sharp interface has been observed,31 and such an interface structure would be difficult to explain if the liquid−solid interface had been truncated all along the TPL during growth. Interestingly, in all reported cases the truncated region periodically grows and shrinks, and at least in one case it appears to vanish just in the moment of nucleation.28 Most importantly, in many cases where truncation of the nanowire top facet has been observed, the edge opposite to the truncated edge apparently remains flat,

Yet, whenever Au droplets are employed for VLS growth of zincblende III−As nanowires, tilted {111} and {001} nanofacets are found, which compose {112} macrofacets, see Figure 1b,d,e.5−12 This is surprising, since the (112) surface has been observed to be unstable as a macroscopic planar surface, which has been attributed to its relatively large surface energy.21 One possible composition is illustrated in Figure 1d, where {1̅1̅2̅}B ({112}A) macrofacets are composed from {11̅ 1̅ }̅ B and {001}̅ A ({111}A and {001}B) nanofacets occurring in the ratio 2:1. (Here X in {hkl}X denotes the actual (predominant) termination of polar {hkl} nanofacets (macrofacets). This convention is different from the one used in ref 22, where X always refers to the relevant macrofacet.) Similar although typically more irregular compositions have frequently been observed.5−8 Figure 1e depicts an alternative composition of nanowire side facets with overall {112} orientation from alternating {111}A and {1̅1̅1̅}B nanofacets only, which however necessitates the formation of one planar twin defect at each boundary between segments.9−12 In general, such alternation of nanofacets along the nanowire is understood to be caused by alternation of nucleation position during VLS growth between A and B side facets of the nanowire.23 Furthermore, it has been observed that under identical growth conditions,24,25 different nanowire facets develop during axial VLS and during radial droplet-free growth, which also indicates that the droplet strongly influences facet formation.

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VLS MONOLAYER GROWTH In our analysis, we will focus on the following question: why are polar, tilted ({111} and {001}) facets found for nanowire growth from Au droplets, while nonpolar, vertical ({110}) facets result when element III droplets are employed instead? To answer this, we will not focus on the details of the evolution of each specific facet, such as how the twin plane arises in the structure with an overall {112} side facet (Figure 1e), which have already been the subject of independent studies in their own right.22,23,26 Instead, we want to take a more general approach and compare the kinetics of the two different facet B

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thus presenting an unperturbed TPL.29,30 This indicates that even when truncation of the growth front does occur it is still possible for nucleation to take place at the TPL, provided the truncation is not complete and permanent. Finally, such TPL nucleation has been essential for the successful theoretical description of several important nanowire growth phenomena.22,23,26,27,32 We will therefore discuss facet formation for nucleation at the TPL in some detail and then describe briefly how this relates to facet formation for central nucleation. There is even less experimental knowledge about the cross sectional shape of the growing nucleus. Nevertheless, we can infer from the condition of neutrality that opposite lateral facets of the nucleus necessarily have opposite polarities and from the characteristics of the zincblende structure that they are necessarily parallel, see Figure 1c−e. Thus, any B-polar lateral facet ({1̅1̅1̅}B or {001}B) must be opposed by a corresponding A-polar ({111}A or {001}̅ A, respectively) lateral facet, while a nonpolar ({110}) lateral facet cannot be opposite to any polar one. Strictly, this does not fully determine the shape of the nucleus, because it leaves the lengths of the lateral facets undefined, but it leads quite naturally to the assumption of nuclei with hexagonal shape, with their six lateral facets (both the external one, in contact with the vapor, and the five internal ones, in contact with the liquid) either being all {110} or alternating between {111}A and {11̅ 1̅ }̅ B (or between {001}̅ A and {001}B). As regards completion of a monolayer, we note that when a lateral facet of an expanding nucleus arrives at an edge of the top surface of the nanowire, the lateral facet necessarily retains its character when it becomes part of the side facet of the nanowire, again because of neutrality combined with the zincblende structure. For example, a {111}A lateral facet arriving at a {112}A nanowire side facet cannot even turn into a neutrality conserving {001̅}A nanofacet, because the structure of the already present part of the nanowire near the {112}A side facet permits only {111}A and {001}B nanofacets, whereas a change into a {001}̅ A nanofacet would require a repositioning of many atoms, which is prohibited by a huge free energy barrier.33 Thus we consider each of the two facet growth modes to involve characteristically only a single type of side facets, either tilted or vertical, in all three stages of monolayer growth, namely as nanowire side facet at the edge where nucleation occurs, as lateral facets of the nucleus, and as a new piece of a nanowire side facet when the monolayer is completed. We notice that both {110} and {111} edges will always be present at the triple phase line simultaneously (though not in equal proportion) because of fluctuations, especially during the initial stage of nanowire growth, and thus be available as starting points for nucleation. The facet growth mode with favored kinetics, that is, with larger probabilities of the relevant processes, should then rapidly become dominant, and stationary growth in this mode will establish itself, thus determining the nature of the nanowire side facets.

compare the formation energies of critical nuclei with different facets under different droplet materials. The creation of a nucleus involves a change in Gibbs free energy, ΔG, which has two contributions: First, the process is driven by the energy yielded in the reaction of Ga and As in the liquid droplet to GaAs in the solid nanowire, which is described by the difference Δμ (expressed in J/m3) in the chemical potentials of supersaturated liquid phase and solid phase. Second, lateral sidewalls of the nucleus are formed, which costs a lateral surface energy, Γ (expressed in J/m2). These two contributions make up the volume and surface terms of ΔG = −AhΔμ + PhΓ, where A is the nucleus base area, h its height, and P its perimeter.27 The total lateral surface energy depends on the fraction x of the nucleus perimeter in contact with the vapor (for a regular hexagonal nucleus, x = 1/6) and on the orientation i of the formed side facet. It can be expressed as22,26,27 Γi = (1 − x)γLS + x Γext, i

(1)

where γLS is the solid−liquid interface energy (assumed to be isotropic) and Γext,i is the ef fective surface energy of the created external solid−vapor facet. The resulting critical nucleation barrier, ΔG*i = fπhΓi2/Δμ, where f = P2/(4πA) is a geometrical constant close to 1, determines the nucleation probability pi ∝ exp(−ΔGi*/(kBT)), which implies that the nucleus with smallest ΔGi* is kinetically favored. Thus, for a comparison between nuclei with different side facets but identical crystal structure (and therefore identical Δμ), it suffices to consider the respective Γext,i. Note that if the perimeter fraction x would depend on the magnitude of Γext,i such as to reduce ΔGi*, this could not invert but would just enhance any existing differences between nucleation barriers. The effective surface energy of the external facet, Γext,i, is obtained as follows. Figure 2 shows the geometry during

Figure 2. Model of nucleation at the triple phase line during VLS growth. The facet tilt, θi, the droplet−nanowire contact angle, βL, and the droplet−facet contact angle, δi, are indicated.

creation at the TPL (with contact angle βL between the liquid surface and the top surface of the nanowire) of a nucleus with an external facet with tilt angle θi. The formation of the external facet requires an upward shift of the TPL, and the resultant changes in area of the adjacent surfaces imply a change in Gibbs free energy.27,34 The solid surface (i.e., the side facet of the nanowire in contact with the vapor) becomes longer by the length of the external facet, that is, by l = h/(cos θi), and the surface of the liquid (in contact with vapor) becomes shorter by l cos δi, where δi = βL − θi − π/2 is the contact angle with respect to the external facet. Note that we must ignore the apparent size change of the nucleus−liquid (top) interface by l sin θi, which would result from comparing the shape of the actual nucleus having its actual external facet with that of a hypothetical reference nucleus with a vertical lateral facet.27,34 This is because in the present situation the nucleus−liquid

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THEORETICAL MODEL The decisive role of the liquid droplet in the kinetics can only be explained by a theoretical model that includes the droplet explicitly, and particularly the vapor−liquid−solid TPL, where the external facets of the nanowire are formed. A VLS growth model that fulfils this purpose has been formulated by Glas et al.,27 which compares the formation energies of critical nuclei with different crystal structure. Here, we adapt this model to C

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interface area is the same independent of the tilt θi of the external facet, since each nucleus is considered to have the appropriate opposite facet with identical tilt, as discussed above. Thus, the free energy (per unit area) Γext,i associated with the size changes of the adjacent surfaces (or equivalently the effective surface energy of the external facet of the nucleus) is Γext, i =

variation due to these two separate causes remains sufficiently small where it affects formation of a nucleus with a particular external facet, such that it can be neglected for the purpose of the present paper and that βL may be considered the timeaveraged value at the center of a particular edge. Finally, we stress that the same effective surface energy, Γext,i, or equivalently, Γext,i ′ , that determines the probability for TPL nucleation with external facet i also determines the probability and therefore the rate of monolayer completion with nanofacet i where structurally allowed, as discussed by Voronkov.34 While this is not important for TPL nucleation where the faceting of the nucleus is determined at nucleation, it becomes especially relevant in the case where nucleation takes place in the central part of the nanowire−droplet interface and the preferential facet growth mode is necessarily decided by the monolayer completion process. Our analysis below, based upon eq 3, thus is generally valid for VLS growth, both for TPL nucleation and for central nucleation.

γSV, i − γLV cos δi cos θi

(2)

This equation establishes that the contribution from the external facet to the lateral surface energy is reduced whenever the angle δi between droplet and facet is small. Because it is our aim to compare the effective surface energy of different external facets, we can neglect terms that do not depend on the facet parameters θi and γSV,i. Doing so after expanding cos δi = sin(βL−θi) = sin βL cos θi − sin θi cos βL, we thus arrive at the effective surface energy γSV, i Γ′ext, i = + γLV cos βL tan θi cos θi (3)

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PARAMETERS Detailed calculations are required to demonstrate that the droplet bulge effect indeed explains the different observations for Au- and Ga-assisted GaAs nanowires. The required parameters are briefly discussed in the following. The solid−vapor surface energy, γSV,i, can be calculated using density functional theory (DFT), taking into account intrinsic reconstructions (with Ga, As, or both). Which reconstruction is the most stable depends on the Ga and As content of the ambient vapor phase, characterized by the chemical potential of As in the vapor, μAs, relative to that of As in the bulk, μAs(bulk).4 The very presence of Ga droplets during self-assisted VLS growth indicates that Ga-rich conditions prevail, i.e. μAs ≃ −0.64 eV.4 For Au-assisted VLS growth, there is no similar indicator. While the growth experiments are generally performed under As-rich conditions, it is not clear how strong the corresponding increase in μAs is. Surely, the upper boundary, where the As vapor is in equilibrium with solid As, that is, μAs = 0, is not reached. For the Au-assisted case, we will therefore consider Ga-rich, as well as moderately As-rich, conditions, for which we choose μAs = −0.2 eV. For both ambient conditions, the surface energy, γSV,i, of the lowestenergy reconstruction of each possible GaAs facet is presented in Table 1 as extracted from ref 4.

which depends on the facet orientation i (while we dropped the term −γLV sin βL). We examine eq 3 in order to understand the lateral surface energy qualitatively. The first term is simply the surface energy of the side facet corrected for a possible area increase due to facet tilt. This term concerns only the solid and would be the only one present for droplet-free growth. By contrast, the second term in eq 3 includes parameters of the liquid. Since the liquid droplets are generally observed to be outward-bulged during nanowire growth (βL > π/2),5,27,35,36 outward-tilted facets (θi > 0) such as {11̅ 1̅ }̅ B have a reduced lateral surface energy and thus a higher probability to occur. We call this the droplet bulge effect. Its strength scales with the surface energy, γLV, of the liquid droplet. Yet, if each possible nanowire side facet has the same tilt angle, for example, θi = 0 as is the case for the wurtzite crystal structure, the droplet bulge effect has equal strength for each facet and thus cannot influence which facet is favored. For zincblende nanowires, side facets with different tilt angles are possible, and the droplet bulge effect can become important. Next, we briefly address a potential complication of the above description of TPL nucleation, namely that, strictly, the contact angle βL at a specific edge of the nanowire top facet is not uniquely defined, in two different ways. First, because the droplet sticks to the top facet edges, except near the corners, it is not axisymmetric, and this makes βL vary along the TPL.37 This implies that strictly the optimum angle (and thus the optimum position along an edge of the hexagonal top facet of the nanowire) for TPL nucleation may be different for each of the different external lateral facets of the various nuclei (e.g., a {11̅ 1̅ }̅ B external nanofacet will occur preferably at a position where βL − π/2 is close to 19.5° and a {001}B nanofacet at a position where it is close to 35.5°). However, the situation is considerably less severe than might appear from the results in ref 37. According to our computations of the shape of the droplet with the Surface Evolver package,38 the contact angle remains nearly constant over most of that part of the edge where the droplet sticks to it and changes only, rather rapidly, close to the point where it detaches from the edge. Second, for nanowires with tilted side facets, the shape of the top facet itself changes slightly upon addition of each monolayer, and this causes βL to vary in time.23 We have verified that the combined

Table 1. Reconstructed GaAs Surface Energies under GaRich and Moderately As-Rich Conditions (from Ref 4) and Tilt Angles with Respect to the Nanowire Growth Axis facet i

tilt θi (deg)

γSV,i (Ga-rich) [J/m2]

γSV,i (As-rich) [J/m2]

{110} {111}A {1̅1̅1̅}B {001̅}A {001}B

0 −19.5 19.5 −35.3 35.3

0.83 0.87 1.11 1.04 1.04

0.83 0.87 0.90 0.93 0.93

As regards the surface energy of the Ga droplets, γGa LV, pure liquid Ga is considered, although the Ga is alloyed to a small fraction with As.39,40 Experimentally determined values for Ga droplets in vacuum at 550 °C are 0.67 and 0.71 J/m2.41,42 The droplet−nanowire contact angle, βL, can be measured after growth, provided the droplets are not consumed. Angles between 90° and 140° are visible for Ga droplets on selfD

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Figure 3. Diagrams indicating the most favorable facet as a function of βL and γLV for (a) Ga droplets under Ga-rich conditions, (b) Au droplets under Ga-rich conditions, (c) Au droplets under Ga-rich conditions including extrinsic reconstruction with Au, and (d) Au droplets under As-rich conditions. Dotted rectangles indicate the respective most probable parameter ranges.

γSV,i. This is taken as an indication that in fact a different set of γSV,i applies for the case of Au-assisted VLS growth. We consider two alternatives: an extrinsic reconstruction of the GaAs facets with Au atoms from the droplets (Figure 3c) and a change in external conditions toward moderately As-rich reconstructions (Figure 3d). Experimental evidence for an extrinsic reconstruction of GaAs induced by Au under Ga-rich conditions has been found by scanning tunnelling microscopy (STM): for planar GaAs(111)B decorated with Au droplets and subjected to annealing a (√3 × √3)R30°Au reconstruction was reported.48 As a result, the surface energy of GaAs(111)B should be reduced, and this reduction was calculated by DFT to be 0.13 J/m2 with respect to the value of γSV,i under Ga-rich conditions.49 Figure 3c presents the result of the calculation using this Aureconstructed γSV,i for the {1̅1̅1̅}B facets and the intrinsically reconstructed γSV,i for the remaining facets. Now, the diagram indeed shows a narrow region where the {1̅1̅1̅}B facets are most favorable, which would be sufficient for the formation of alternating {111}A and {1̅1̅1̅}B facets or of alternating {111} and {001} facets. Both configurations compose overall {112} facets, and the detailed formation mechanism has been understood in each case.22,23 Could thus the experimental observation of tilted facets for Au-assisted nanowire growth be caused by an extrinsic reconstruction of GaAs(111)B with Au? An estimation of the amount of Au required for a (√3 × √3) R30° reconstruction of the whole nanowire side surface makes this appear unlikely. All Au atoms contained in a sphere of 20 nm diameter would be consumed by the reconstruction of the side surface of a nanowire with identical diameter and less than 1 μm length and then VLS growth would have to stop. This does not correspond to observations. Nevertheless, we present this case here to point out possible effects of extrinsic reconstructions. Figure 3d shows that alternatively, under moderately As-rich growth conditions, tilted polar {11̅ 1̅ }̅ B facets become favored in an even wider region, caused by the stabilization of Asreconstructed surfaces. Again, in accordance with experimental observations,5−12 tilted {1̅1̅1̅}B and {001}B facets are the most favorable over a considerable range of liquid parameters. We stress that the considered change toward moderately As-rich ambient conditions alone does not suffice to make {11̅ 1̅ }̅ B most stable thermodynamically, since it is still the facet with {110} orientation that has the smallest surface energy (see also Table 1). Instead, the major contribution to the formation of tilted facets during Au-assisted VLS growth is the droplet bulge

assisted nanowires grown by ourselves and by other groups.19,35,43 For Au-assisted nanowires, the droplets strictly consist of a Au−Ga−As ternary alloy. Again, the As content is very low,44 but the Ga content is substantial. Thus, the Au−Ga−As droplet can be approximated as an Au1−xGax alloy, and atomic Ga fractions x up to 50% have been found after growth.45 Following Algra et al., a range for γAu−Ga between 1.2 and LV 1.6 J/m2 is considered.46 Typical contact angles of Au−Ga droplets on top of GaAs nanowires were between 90° and 140° as found by high-resolution transmission electron microscopy after growth.5,27,47

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RESULTS AND DISCUSSION In principle, the effective surface energy, Γ′ext,i, for each GaAs facet can now be calculated using eq 3. However, particularly the liquid parameters βL and γLV are not very accurately known. Therefore, we present in Figure 3 facet favorability diagrams, which indicate the most favorable facet (smallest Γext,i ′ ) for a wide range of the liquid parameters. In this way, the droplet bulge effect is separated from that of the better known solid parameters. Four different diagrams are shown corresponding to four sets of assumptions concerning the surface energies of the solid facets. In each case, the most likely values for γLV and βL are indicated by a dotted rectangle. Figure 3a depicts the diagram for Ga-assisted VLS growth under Ga-rich ambient conditions. In the wide range of all presented values, favorability regions for {110} and {001}B facets are found, while nuclei with {001} and {11̅ 1̅ }̅ B side facets are not favorable at all. For liquid Ga droplets during nanowire 2 growth, γLV is expected to be relatively small (γGa LV ≈ 0.7 J/m ). Thus, we find that {110} facets should be formed during selfassisted GaAs nanowire growth. In other words, because of the low γLV, the droplet bulge effect is rather weak and {110} facets, which have the smallest surface energy, are most favorable. This finding agrees with the experimental observations.13−18 In Figure 3b, the case for Au-assisted growth is presented, when essentially Ga-rich conditions prevail. Since the same γSV,i values were used as before, the favorability regions remain unchanged. However, γLV is considerably larger for Au droplets such that a different section of the diagram is relevant (dotted rectangle): The droplet bulge effect leads to the prediction of outward-tilted polar {001}B facets or {110} extended facets. Since {111} facets are not kinetically favorable at all under these conditions, the common observation of {111} and {001} side facets cannot be explained by this set of solid parameters E

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effect, and its strength arises from the large surface energy of liquid Au. The presented comparison of the possible conditions for Auassisted VLS growth makes it clear that surface reconstructions can have a decisive effect on the side facets. While a reconstruction of GaAs(111)B with Au would consume more Au than is observed, it is clear that the growth conditions for Au-assisted VLS growth of GaAs nanowires are substantially more As-rich than for the Ga-assisted case. Therefore, we propose that {1̅1̅1̅}B facets, which are main constituents of the often observed {112} side facets, are actually As-reconstructed. It would be interesting if this could be confirmed experimentally, for example, by STM. From our calculations and the absence of experimental observations of {110} facets for Au-VLS-grown GaAs nanowires, we can deduce that the contact angle βL during growth must have been at least 105°. Smaller contact angles measured after growth may be due to the well-known effect that Ga and As leave the droplet during cooldown.50

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CONCLUSION In summary, we have modeled the kinetics of side facet formation during VLS nanowire growth and calculated the most favorable facet for different droplet materials and growth conditions. We found that side facet formation can be understood in detail by considering the droplet bulge effect, which relates to the reduction in lateral surface energy whenever the angle between droplet and facet is small. This energy reduction is proportional to the surface energy of the liquid droplet, which explains why high-surface-energy polar {112} side facets do occur in Au-assisted VLS growth, while Ga-assisted growth invariably produces nanowires with lowsurface-energy nonpolar {110} facets. In general, this understanding can be used to control the side facet orientation of nanowires and optimize in turn the performance of devices based on them.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: steff[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are indebted to Peter Kratzer for sharing his surface energy calculations of Au-reconstructed GaAs(111)B. Furthermore, we thank him, Frank Grosse, and Rudolf Hey for stimulating discussions. The critical reading of the manuscript by Erik P. A. M. Bakkers and Vladimir Kaganer is greatly appreciated. S.B. acknowledges support from the Australian Research Council. L.-F.F. would like to express his thanks to the PDI and its director, Henning Riechert, for their hospitality and support during his stays at the research institute.

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REFERENCES

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