Refinement of Nucleation Theory for VaporâLiquidâSolid Nanowires

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Refinement of nucleation theory for vapor-liquid-solid nanowires V. G. Dubrovskii Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b00124 • Publication Date (Web): 30 Mar 2017 Downloaded from http://pubs.acs.org on April 17, 2017

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Crystal Growth & Design

Refinement of nucleation theory for vapor-liquid-solid nanowires V. G. Dubrovskii1,2,3* 1

2

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 3

ITMO University, Kronverkskiy pr. 49, 197101 St. Petersburg, Russia

ABSTRACT: Features of nucleation of two-dimensional islands in vapor-liquid-solid nanowires determine the nanowire growth rates, morphologies and crystal phases and thus are central for controlling the physical properties of such structures. Herein, we present a development of nucleation theory that accurately accounts for the droplet depletion effect on the island formation energy. We show that the latter modifies itself to a curve with a maximum at a critical size and a minimum at a stationary size which determines the size to which the island can actually grow without a refill from vapor. This effect is general and should occur in any nucleation from an isolated nanoparticle. For III-V nanowires, the depletion is extremely sensitive to the concentration of the group V atoms in the droplet and is more pronounced for smaller nanowire radii and lower concentrations. Whenever the stationary size is smaller than that of the facet, the droplet depletion decreases the amplitude of the chemical potential oscillations. These results refine our picture of how nanowires grow and have potential impact on their resulting properties.

*Electronic mail: [email protected]

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INTRODUCTION

The vapor-liquid-solid growth (VLS) of semiconductor nanowires (NWs)1-3 often proceeds in a nucleation-limited regime whereby two-dimensional (2D) solid islands containing many semiconductor atoms or pairs are formed from a supersaturated liquid alloy of the growth constituencies with a metal catalyst. This process is described within the classical nucleation theory3-5 using a macroscopic approximation for the free energy of forming an island (the formation energy for brevity). A confined volume of the catalyst droplet and the top NW facet where the islands emerge leads to several important features of nucleation in VLS NWs. First, the VLS growth usually has a mononuclear character in which only one island nucleates in each layer and then rapidly spreads almost instantaneously (compared to the typical NW growth rates) to fill a complete ML slice of the NW5-15. Second, the droplet surface area changes due to instantaneous nucleation, leading to the so-called Gibbs-Thomson correction to chemical potential of the liquid phase10,12,14. Third, chemical potential oscillates in synchronization with the island nucleation, growth and the droplet refill from vapor, yielding a sub-Poissonian nucleation antibunching and an oscillatory behavior of the growth interface with truncated edges16-20. Tremendous progress achieved in understanding and controlling the NW growth rates7,8,10, morphologies17,21 and even crystal structures9-13,21-23 (see Refs. [5] and [6] for a detailed review) largely relies upon this picture. Surprisingly, however, one important effect in the VLS growth seems to be overlooked in the past. While the surface-induced Gibbs-Thomson contribution into chemical potential was previously studied with great care (leading nevertheless to a minor correction)10,11, a much larger composition-induced contribution due to depleting the droplet with its semiconductor constituencies as the island nucleates was not included at all. This effect is generally well-known for nucleation in an isolated particle24-27 and can strongly modify the overall growth behavior. Glas and coauthors16 studied only the influence of the droplet depletion on the nucleation probability for the next island, assuming that any island that has emerged will instantaneously 2 ACS Paragon Plus Environment

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grow to the size of the facet. This view has gained much support and is widely used in modeling of nucleation statistics in VLS NWs19,20. On the other hand, decreasing supersaturation in the liquid phase must have a similar effect on the growth kinetics of a single island within each nucleation pulse and may considerably change our picture of how NWs grow. Consequently, here we refine the conventional expression for the island formation energy in VLS NWs10 in the most general case of Au-catalyzed III-V NWs in order to see how the earlier results are modified by the concentration-induced correction for chemical potential. We show that, generally, any island nucleated from an isolated droplet cannot grow infinitely. Rather, it will evolve only to a finite stationary size at which the energetically favorable decrease of chemical potential in the liquid-solid phase transition is compensated by the energetically unfavorable decrease of chemical potential in the liquid phase itself. This growth limitation is general and should occur in any nucleation from an isolated nanoparticle. For VLS III-V NWs, the effect is negligible in situations where the stationary size is larger than the size of the facet, corresponding to high group V concentrations and large NW radii. For low group V concentrations and small NW radii, the stationary size is smaller than that of the facet and the island cannot cover the top facet without refill.

MODEL

Figure 1 shows the VLS growth model in the case of the triple phase line (TPL) nucleation in Au-catalyzed III-V VLS NWs9. Let us first consider the surface energy change upon the island nucleation. By creating an island at the TPL, we build the solid-liquid interface with the surface energy (1 − x ) phγ SL and the solid-vapor interface with the surface energy xphγ SV , where p is the island perimeter, x is its fraction along the TPL and h is the height of a III-V monolayer. The surface energy of the droplet changes due to elimination of its part − xphγ LV sin β and the surface energy change ∆S = ( 2 sin β )∆V / R due to the volume variation ∆V = (Ω S − Ω L )i in the instantaneous liquid-solid phase transition of i III-V pairs having the elementary volumes Ω S 3 ACS Paragon Plus Environment


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and Ω L in the solid and liquid phases, respectively. This gives the additional Gibbs-Thomson term γ LV sin β (Ω S − Ω L )i / R which is positive when Ω S > Ω L , as in the Au-Ga-As system10, and is proportional to the island size i (which is why it modifies chemical potential). The total surface energy change thus equals10

∆Gsurf = [(1 − x )γ SL + x (γ SV − γ LV sin β )] ph +

2γ LV sin β (Ω S − Ω L )i . R

(1)

For x = 1 , p = 2πR and i = (πR 2 h ) / Ω S , this yields the exact result for the complete monolayer slice in the non-wetting growth mode ∆Gsurf = [γ SV − (Ω L / Ω S )γ LV sin β ]2πRh (Ref. [5]). Therefore, we do not see any justification of introducing the Ω L / Ω S multiplying factor in the

γ LV sin β term of Eq. (1), as suggested in Ref. [12], although the resulting difference is small due to Ω L being close to Ω S .

Figure 1. III-V VLS system with an Au-III-V droplet having the atomic concentrations c3 of the group III and c5 of the group V atoms before nucleation, decreasing to c3' and c5' , respectively, after nucleation of a 2D island at the TPL. For highly volatile arsenic or phosphorous species, the group V concentration is always much lower than the group III one. The island has the monolayer height h and consists of i III-V pairs (the island size for brevity). The contact angle of the droplet with respect to horizontal equals β . The droplet base radius equals the NW radius R . The surface energies of interest are those of the liquidvapor interface ( γ LV ), solid-vapor lateral interface for one third of the island perimeter ( γ SV ) and solidliquid lateral interface for two thirds of the island perimeter ( γ SL ). 4 ACS Paragon Plus Environment

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Equation (1) accurately accounts for the position-dependent surface energy change as well as the surface-induced modification of chemical potential, which appears to be small and cancels when Ω L = Ω S (because the droplet surface area in this case does not change after nucleation). For the volume term in the formation energy, all prior works5-11,13-16,19,20,22,23 used the simple expression − ∆µi , with ∆µ (c3 , c5 ) being the chemical potential difference for III-V pairs in infinitely large liquid and solid phases, given in Ref. [28] for different Au-III-V VLS systems. The chemical potential difference depends on the two atomic concentrations defined as c X = N X /( N 3 + N 5 + N Au ) = N X / N tot for X = 3, 5 and Au (with c Au = 1 − c3 − c5 ), where N X is the number of this type atoms in the droplet and N tot is the total number of all atoms in the droplet. However, the − ∆µi expression becomes inaccurate when the droplet size is finite and the island growth process proceeds much faster than the refill from vapor (which is required for the mononuclear VLS growth10). Indeed, the chemical potential difference after and before nucleation should relate to the entire VLS system including the droplet. Before the island nucleation, the volume energy of a ternary Au-III-V alloy was µ 3 N 3 + µ 5 N 5 + µ Au N Au , where the liquid chemical potentials µ X = µ X (c3 , c5 ) for X = 3, 5, Au are taken at the group III and V concentrations c3 and c5 before nucleation. After i III-V pairs are removed from the catalyst to ' form the island, the volume energy becomes µ3' ( N 3 − i ) + µ5' ( N 5 − i ) + µ Au N Au + iµ35 , with µ 35 as

the reference chemical potential of a III-V pair in solid. The liquid chemical potentials

µ X' = µ X (c3' , c5' ) should now be taken at the group III and V concentrations c3' and c5' after nucleation. Taking the difference, we obtain ' ∆Gvol = −i ( µ3' + µ5' − µ35 ) − ( µ3 − µ3' ) N 3 − ( µ5 − µ5' ) N 5 − ( µ Au − µ Au ) N Au .

(2)

Assuming that i is much smaller than N tot , we expand the chemical potential differences to the first order in i / N tot , similarly to Ref. [16] 5 ACS Paragon Plus Environment


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 ∂µ  i ∂µ µ X − µ X' =  X (1 − 2c3 ) + X (1 − 2c5 ) . ∂c5  ∂c3  N tot

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(3)

Due to the mononuclear character of nucleation and follow-up growth of a single island in NWs of interest, the expansion given by Eq. (3) actually applies for any i between zero and the size of the entire facet16. Using this in Eq. (2), we arrive at  ∂∆µ (1 − 2c3 ) ∂∆µ (1 − 2c5 )  i 2 ∆Gvol = − ∆µi + c3 + c5 − ∂c5 c3 c5  N tot  ∂c3   ∂µ   ∂µ ∂µ ∂µ  ∂µ ∂µ  −  c3 3 + c5 5 + (1 − c3 − c5 ) Au (1 − 2c3 ) + c3 3 + c5 5 + (1 − c3 − c5 ) Au (1 − 2c5 ) i ∂c3 ∂c3  ∂c5 ∂c5   ∂c5   ∂c3  (4) Here, ∆µ ( c3 , c5 ) = µ 3 ( c3 , c5 ) + µ 5 ( c3 , c5 ) − µ 35 is the liquid-solid chemical potential difference per III-V pair, taken at the initial group III and V concentrations before nucleation. It is clear that both bracket terms in the last expression in the right hand side of Eq. (4) (the i correction term hereinafter) equal zero for ideal solution with µ X = k BT ln c X + const , T as the surface temperature and k B as the Boltzmann constant. Indeed, differentiating, for example,

µ3 and µ Au with respect to c3 and multiplying the results to c3 and 1 − c3 − c5 , respectively, results in the terms k BT and − k B T whose sum equals zero, while µ5 is independent of c3 . In the regular solution model, the liquid chemical potentials are given by28

µ3 = µ3P + k BT ln c3 + ω35c52 + ω3 Au (1 − c3 − c5 ) 2 + (ω35 + ω3 Au − ω5 Au )c5 (1 − c3 − c5 ) ,

µ5 = µ5P + k BT ln c5 + ω35c32 + ω5 Au (1 − c3 − c5 ) 2 + (ω35 + ω5 Au − ω3 Au )c3 (1 − c3 − c5 ) , 2 P µ Au = µ Au + k BT ln(1 − c3 − c5 ) + ω3 Au c32 + ω5 Au c5 + (ω35 + ω5 Au − ω3 Au )c3c5 ,

(5)

with µ XP as the chemical potentials of pure liquids and ω XY as the interaction parameters between X and Y atoms for X , Y = 3, 5 and Au. It is easy to show that both bracket terms in the i correction equal zero also for this model. Actually, the i correction should be zero for any system to avoid a paradox of introducing a finite free energy change by nucleating a negligibly small 6 ACS Paragon Plus Environment

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island in infinite system (because the i correction contains no size-dependent parameter that vanishes when R → ∞ ). Dividing both ∆Gsurf and ∆Gvol to k BT and expressing all terms as functions of i , we obtain our final result for the formation energy F (i ) = ∆G (i ) / k BT 3

F (i ) = − ∆µ i + ai 1 / 2 +

RSL  R  (1 − 2c5 ) 2 i+ 5 i . R c5  R

(6)

The parameters are given by

a=

k2 Γ (Ω S h )1 / 2 , Γ = (1 − x )γ SL + x (γ SV − γ LV sin β ) , 1/ 2 k1 k BT

RSL =

2γ LV sin β (Ω S − Ω L ) , k BT

(7)

(8)

~ 1/ 3  1 ∂∆µ 3Ω L  R5 =  c5 . ∂c5 πf ( β )   k BT

(9)

Here, ∆µ = ∆µ /(k BT ) is the chemical potential difference for infinite liquid and solid phases in

~ thermal units and Ω L is the elementary atomic volume in the liquid phase. The total number of atoms

in

the

droplet

is

given

~ by N tot = (πR 3 / 3) f ( β ) / Ω L ,

where

f ( β ) = (1 − cos β )( 2 + cos β ) /[(1 + cos β ) sin β ] is the geometrical function relating the volume of spherical cap to the cube of its base. The shape constants k1 and k 2 determine the surface area s = k1r 2 , and perimeter p = k 2 r of the island depending on linear size r . The last term in Eq. (6) can equivalently be presented as αi 2 (1 − 2c5 ) /(c5 N tot ) with α = (1 / k BT )c5 (∂∆µ / ∂c5 ) , which is reduced to the result of Ref. [16] when i = (πR 2 h ) / Ω S equals the size of the facet. Using the arguments of Ref. [16], we leave only the leading dependence of the last term in Eq. (6) on c5 and neglect its dependence on c3 . This can be done due to a much higher sensitivity of the depletion effect to the group V element that is present in the droplet at a much lower concentration. The ∆µ ( c3 , c5 ) and c5 values in Eqs. (6) and (9) correspond to the initial state of 7 ACS Paragon Plus Environment


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the droplet before nucleation and hence are fixed throughout the growth cycle of each monolayer. According to Ref. [23], the formation energy should be renormalized according to

∆F (i ) = F (i ) − F (1) , which ensures that ∆F (1) = 0 .

RESULTS AND DISCUSSION The obtained formation energy contains two R -dependent terms: a small Gibbs-

Thomson term with RSL which is inversely proportional to R and scales as i , and a positive composition-induced correction due to the depletion effect which is inversely proportional to R 3 and scales as i 2 . Both terms vanish when R → ∞ . However, for finite R , the i 2 term in Eq. (6) always dominates for large enough i . This changes the shape of the ∆F (i ) function which has not only a maximum at a small critical size ic (as in infinite systems4,5), but also a minimum at a certain stationary size i s , determines the size to which the island can grow without refill. The ic and i s values become closer for smaller ∆µ . For very low ∆µ (or c5 ), the maximum can disappear completely meaning that nucleation is no longer possible, as demonstrated in Fig. 2 for the parameters shown in the insert and ∆µ varying with c5 as ∆µ = 4 + 2 ln( c5 / 0.06) .

Figure 2. Possible shapes of the formation energy given by Eq. (6) for the fixed a = 28.7, RSL = 0.83 nm,

R5 = 0.21 nm, R = 15 nm, and different c5 , with the chemical potential value varying with the arsenic concentration as ∆µ = 4 + 2 ln( c5 / 0.06) . The corresponding values of the total numbers of arsenic atoms N 5 are shown in the legend for each c5 . 8 ACS Paragon Plus Environment

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Figure 3 shows the formation energies ∆F (i ) versus i at different arsenic concentrations c5 for the following model parameters of 15 nm radius Au-catalyzed GaAs NW29: Ω S = 0.0452

~ nm3, Ω L = 0.04 nm3, h = 0.326 nm, Ω L = 0.02 nm3, γ LV = 1.0 J/m2, γ SL = 0.35 J/m2, γ SV = 1.2 J/m2, β = 115o, equilateral triangle island with side r ( x = 1/3, k1 = 3 / 4 , k 2 = 3 ), T = 550 oC and c5 ( ∂∆µ / ∂c5 ) = 2 for all c5 . Now we assume that the ∆µ value is fixed at 2.0 for any c5 , which can be ensured, for example, by increasing the gallium concentrations c3 for lower c5 (Ref. [28]). For simplicity, we also do not account for variation of the surface energies22,30 and the contact angle31 with the droplet composition. These values yield Γ = 0.33 J/m2, a = 17.8, RSL = 0.83 nm and a constant R5 = 0.21 nm. The TPL nucleation is preferred due to

γ SL + γ LV sin β > γ SV (Ref. [9]). The size of the entire facet equals ~ 5000 GaAs pairs for R = 15 nm. The critical size ic equals 20 GaAs pairs. As expected, the curves in Fig. 3 show that there is no growth limitation for high arsenic concentrations above ~ 0.011, while at c5 = 0.01 the island cannot immediately fill the entire facet for energetic reasons and a refill from vapor is required to complete the NW monolayer. Analyzing Eq. (6) in the most realistic case where the critical size ic and stationary size i s are well separated ( is >> ic ), the critical size is given by the conventional expression

ic = a 2 /( 4∆µ 2 ) , while the nucleation barrier equals Fc = a 2 /( 4 ∆µ eff ) − F (1) . Here, we write ∆µ eff ≡ ∆µ − RSL / R for brevity. The stationary size equals 3

c ∆µ N c ∆µ  R is = 5 eff   = tot 5 eff . 2(1 − 2c5 )  R5  2α (1 − 2c5 )

(10)

{

}

For a given NW radius, any island will grow to the maximum size imax = min is , (πR 2 h) / Ω S , which is either the stationary size of the size of the facet. Purely instantaneous mononuclear



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growth without refill occurs when the stationary size is larger than that of the facet. The condition is > (πR 2 h ) / Ω S is equivalent to R > Rmin , where Rmin =

~ 2πhR53 (1 − 2c5 ) 6Ω L h α (1 − 2c5 ) = ΩS c5 ∆ µ Ω S f ( β ) c5 ∆µ

(11)

with neglect of the RSL / R Gibbs-Thomson correction term.

This value determines the

minimum NW radius for the VLS growth without refill at a given set of conditions ( c5 and ∆µ ) and geometry.

Figure 3. Formation energy of equilateral triangle island in 15 nm radius Au-catalyzed GaAs NW versus the number of GaAs pairs in the island for different arsenic concentrations c5 at a fixed ∆µ of 2. The corresponding values of the total numbers of arsenic atoms N 5 , shown in the legend for each c5 , are greater than the facet size (5000 GaAs pairs) for all c5 except for the lowest concentration of 0.004. The critical size ic equals 20 GaAs pairs. For the highest c5 of 0.05, the island grows to the size of the facet and is not influenced by the depletion effect. This feature pertains for the arsenic concentrations higher than 0.011, while at c5 = 0.01 the island growth stops at is ≅ 3000 . Further decrease of the arsenic concentration leads to decreasing i s to approximately 2100 GaAs pairs for c5 = 0.007, where the droplet still contains enough arsenic atoms to complete the NW monolayer. Whenever the stationary size is smaller than the size of the facet, the VLS growth requires a refill from vapor.

We now briefly consider how the earlier picture of the sawtooth oscillations of chemical potential16,17 changes with the new formation energy containing the stationary point. As in Ref. [9,16,17], we assume that the island growth is instantaneous between ic and imax . Let 10 ACS Paragon Plus Environment

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∆µ max = ∆µ eff ( c3 , c5 ) be the initial chemical potential at which nucleation occurs. For the minimum

chemical

potential

at

the

maximum

island

size

we

can

write

∆µ min = ∆µ max − α (1 − 2c5 )imax /( c5 N tot ) , as in Ref. [16]. Using Eq. (10), we then obtain ∆µ min (imax = is ) = ∆µ max / 2 . The amplitude of the chemical potential oscillation thus equals ∆µ max / 2 when the stationary size is smaller than that of the facet. This value is smaller than it would be for a complete monolayer at imax = (πR 2 h ) / Ω S . Therefore, for is < (πR 2 h ) / Ω s , the oscillations are suppressed, as shown schematically in Fig. 4. (This picture assumes that the chemical potential increases linearly with c5 and c5 increases linearly with time, which, strictly speaking, occurs only in systems without desorption16,17,19,20. The period of oscillations is determined by the rate of refill from vapor). Looking at Eq. (11), the minimum NW radius below which the amplitude of oscillations starts decreasing, is larger for lower supersaturations, arsenic concentrations and contact angles.

Figure 4. Illustration of the chemical potential oscillations whose amplitude is determined by the size of the facet when is > (πR 2 h ) / Ω S (complete monolayer cycle) or by the stationary size when

is < (πR 2 h ) / Ω S (incomplete monolayer cycle). In the former case, any island that nucleates at the maximum chemical potential ∆µ max immediately covers the entire facet and the chemical potential instantaneously decreases to ∆µ min . A slow follow-up refill to ∆µ max yields the sawtooth shape of the curve. In the latter case, the droplet depletion effect stops the island growth at a stationary size



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corresponding to ∆µ max / 2 .After that, new arsenic atoms should be brought from vapor to solid without changing the chemical potential of the liquid phase. This effect decreases the amplitude of the chemical potential oscillations and changes the shape of the curve.

In conclusion, we have shown that a restricted amount of building material available in a small nanoparticle renders the growth to a self-limiting regime with a stationary size. This is explained by the fact that we lose more energy by emptying a confined mother phase than gain by growing the island of a new phase. For III-V VLS NWs, the droplet depletion effect is anticipated for smaller NW radii and group V concentrations. When the stationary size is smaller than that of the facet, the instantaneous VLS growth picture is modified to a combined mode in which only the initial growth step is instantaneous, while the follow-up growth continues at the rate of refill. The depletion effect occurs earlier than the obvious lack of the group V atoms in the droplet required to build a monolayer. This refined behavior is important for understanding and controlling the structural properties and nucleation statistics of VLS NWs. In particular, it may suppress the nucleation antibunching16 and oscillations of the growth interface17 under the conditions where both effects are most anticipated, i.e., for small NW radii, low growth rates and group V concentrations.

ACKNOWLEDGMENT The author gratefully acknowledges financial support of the Russian Science Foundation under the Grant 14-22-00018.

REFERENCES

(1) Wagner, R. S.; Ellis, W. C. Appl. Phys. Lett. 1964, 4, 89. (2) Zhang, A.; Zheng, G.; Lieber, C. M. Nanowires: Building blocks for nanoscience and nanotechnology, Springer, 2016. (3) Givargizov, E. I. Highly anisotropic crystals, Springer, 1987. (4) Kashchiev, D. Nucleation: Basic Theory with Applications, Butterworth Heinemann, 2000. 12 ACS Paragon Plus Environment

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TOC figure Refinement of nucleation theory for vapor-liquid-solid nanowires

V. G. Dubrovskii

Brief Synopsis Refined nucleation theory for vapor-liquid-solid nanowires accounts for the droplet depletion effect on the island formation energy. The latter modifies itself to a curve with a maximum at a critical size and a minimum at a stationary size to which the island can grow without a refill from vapor. This effect should occur in any mononuclear growth from a nanophase. Whenever the stationary size is smaller than that of the facet, the droplet depletion decreases the amplitude of the chemical potential oscillations.


Refinement of Nucleation Theory for VaporâLiquidâSolid Nanowires

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Refinement of Nucleation Theory for VaporâLiquidâSolid Nanowires