Advances in the theory of III–V nanowire growth dynamics

Within each of the main types of states (figure 3(a)), a 'local state' p is characterized by the mean intrinsic properties of some local surrounding (the 'local ensemble'), see figure 3(b), which is large enough to represent the thermodynamic characteristics and small enough to represent the local environment when the global system is out of equilibrium. At interfaces between two main types of states, a single interface is typically chosen, which means that one distinguish between particles on each side of the interface with local state properties depending only on the local environment of the main state to which they belong (figure 3(c)). A 'single Gibbs interface' is usually introduced to attach interface excess quantities to an assumed infinite sharp interface between two phases, i.e. no atoms belong to the interface, only excesses. To describe the growth dynamics we need to treat the b and a states as separate states, but only consider interface excesses between the classical v, l and s phases. To insure a consistent treatment of the kinetics in terms of the intrinsic thermodynamic parameters, it is convenient to measure the chemical potentials of the all various states with respect to the chemical potential in the ERS,

$$\begin{equation} \delta \mu_{p{\rm -ERS},i} =\mu_{p,i} -\mu_{i}^{{\rm ERS}}, \end{equation} \tag{ 2 }$$

where μ_p,i are the chemical potential of the state p. The chemical potentials with respect to the four ERS states are

$$\begin{eqnarray} &&\fl \delta \mu_{{\rm v-ERS},i} (p_{i} ,T)=k_{{\rm B}} T\ln \left( {\frac{p_{i}}{p_{i}^{{\rm ERS}}}} \right), \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&\fl \delta \mu_{a_{j} {\rm -ERS,III(V})} (\rho_{j,{\rm III}} ,\rho_{j,{\rm V}} ,T)=k_{{\rm B}} T\nonumber\\ &&\times \ln \left( {\frac{\bar{{\rho}}_{j,{\rm III(V})} (1-\bar{{\rho}}_{j,{\rm III}}^{{\rm ERS}} -\bar{{\rho}}_{j,{\rm V}}^{{\rm ERS}} )}{\bar{{\rho}}_{j,{\rm III}({\rm V})}^{{\rm ERS}} (1-\bar{{\rho}}_{j,{\rm III}} -\bar{{\rho}}_{j,{\rm V}} )}} \right), \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} &&\fl \delta \mu_{{\rm l-ERS},i} (x_{{\rm III}} ,x_{{\rm V}} ,T)=\mu_{{\rm l},i}^{\infty} (x_{{\rm III}} ,x_{{\rm V}} ,T)+\gamma_{{\rm vl}} \frac{\partial A_{{\rm vl}}}{\partial N_{{\rm l},i}}-\mu_{i}^{{\rm ERS}},\nonumber\\ \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\fl \delta \mu_{{\rm s-ERS},{\rm III\mbox{--}V}}^{X} =\sum\limits_j {\gamma_{j} \frac{\partial A_{j}}{\partial X}} \frac{\partial X}{\partial N_{{\rm s,III\mbox{--}V}}}+\Delta \varepsilon_{{\rm s}}, \end{eqnarray} \tag{ 6 }$$

where ρ_j,i is the adatom density on the jth facet and A_j and γ_j are the area and interface energy of the jth interface respectively. The form of equation (4) is a simplified version which stems from a detailed calculation of the partition function, see [50]. For the full expression, the following two terms should be added to equation (4): $-\bar{{Z}}_{j,{\rm aa}} (B_{j,{\rm III(V})} (\bar{\rho}_{j,{\rm III(V})} -\bar{\rho}_{j{\rm ,III(V)}}^{\rm ERS})$ $+B_{j,{\rm III\mbox{--}V}} (\bar{\rho}_{j,{\rm V(III})} -\bar{\rho}_{j,{\rm V(III})}^{\rm ERS}))$. $\bar{{Z}}_{{\rm aa}}$ is a reaction constant (including coordination number) for facet j, and B_j,III(V) and B_j,III–V are the binding free energies for III–III(V–V) and III–V bonds on the j surface, respectively. If the adatom concentrations and binding energies are low, equation (4) can be approximated by an ideal behaviour, $\delta \mu_{a_{j} {\rm -ERS},i} (\rho_{j,i} ,T)\cong k_{{\rm B}} T\ln ({\rho_{j,i}}/{\rho_{j,i}^{{\rm ERS}}})$, which strongly reduces computation time. The relative chemical potential of the solid $\delta \mu _{{\rm s-ERS,III\mbox{--}V}}^{X}$ (equation (6)) in terms of a given parameter X, is the change in Gibbs free energy per pair due to a corresponding change in X, such as a length or an angle. In this continuum approach it describes the mean thermodynamic properties for the chosen parameter X. For a full description of the NW crystal an complete set of independent parameters, {X}, is needed. That is, adding matter to a nanosize crystal will change not only its volume, but also its shape, and therefore the interface excesses. In addition to its volume, the crystal must thus be defined by a set of parameters {X} such as facet areas, projected facet heights, facet angles, edge lengths or local interface curvature. It is important to notice that {X} is a chosen set of independent parameters that fully define the choice of crystal geometry (several choices are possible; an example is given in section 3.5). Then, the change of energy of the crystal when matter is added to it comprise a first term, associated to its change of shape, and a second term associated to its change of volume (which is simply related to the chemical potential of the reference infinite solid, as introduced in equation (1)). In the first term of equation (6), the independence of the X parameters and the effect of the changes of these parameters on the areas of the interfaces to which excess energies are associated, are taken into account. The second term is the difference in bulk cohesive energy between the standard reference of the ERS (typically ZB) and the actual formation structure s. The liquid–solid system will tend towards the equilibrium shape which is the one where the sum of all chemical potentials of the set are equal (See for example [51] for a treatment of a fully facetted solid in two dimensions using the concept of weighted curvature [52]). Note that if the crystal structure s is the same as the ERS, $\delta \mu_{{\rm s-ERS,III\mbox{--}V}}^{X}$ is only a size effect as the bulk chemical potential is the same as the ERS (i.e. Δε_s = 0). See section 2.2 for more details. In addition to these interface size effects, it was suggested by Schmidt et al [53] and Schwartz and Tersoff [45] that an excess TL energy, which may arise from an in-balance of capillary forces at the TL, plays an important role on the dynamics of NW growth. See section A.6 in the appendix for a discussion.

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As mentioned in the introduction, the general average rate at which a given p → q transition takes place depends exponentially on the Gibbs free energy of activation for reaching the transition state (TS), as $P_{pq,i} \propto \exp (-({\delta g_{pq,i}^{{\rm TS}}}/{k_{{\rm B}} T}))$. $\delta g_{pq,i}^{{\rm TS}}$ is taken as the difference in free energy per atom between the state p (calculated from the thermodynamic parameters describing this state) and the transition state of the particle between p and q. If a given transition requires a bond-dissociation of molecules into single atoms (e.g. As₂ → 2As), the dissociation enthalpy and entropy should be added to $\delta g_{pq,i}^{{\rm TS}}$.

The activation energy for reaching the TS can be written as $\delta g_{pq,i}^{{\rm TS}} =\delta g_{pq,i}^{{\rm TS,ERS}} -\delta \mu_{p{\rm -ERS},i}$, where $\delta g_{pq,i}^{{\rm TS,ERS}}$ is the activation energy for a p to q transition, and δμ_p−ERS,i is the chemical potential with respect to the ERS (see figure 4). The mean flux of atoms in the state p crossing the pq boundary per unit area (or length) is given by

$$\begin{eqnarray} \fl \Gamma_{pq,i} =\left\{{{\begin{array}{@{}l@{\tqs}l@{}} {\Xi_{pq,i} \bar{{c}}_{p,i} \exp \left( {-\frac{\delta g_{pq,i}^{{\rm TS,ERS}} -\delta \mu_{p{\rm -ERS},i}}{k_{{\rm B}} T}} \right)} &\\\ms {{\rm if}\ \delta g_{pq,i}^{{\rm TS,ERS}} \ge \delta \mu_{p{\rm -ERS},i} ,} \\\ms {\Xi_{pq,i} \bar{{c}}_{p,i}} &\\\ms {{\rm if}\ \delta g_{pq,i}^{{\rm TS,ERS}} \le \delta \mu_{p{\rm -ERS},i},} \end{array}}} \right. \end{eqnarray} \tag{ 7 }$$

where Ξ_pq,i is a 'single atom flux' prefactor accounting for the number of attempts per atom to pass from the p state to the TS between p and q per unit time and unit area. $\bar{{c}}_{p,i}$ is the normalized density of group i atoms in state p, i.e. the probability of having an atom in the state. When $\delta g_{pq,i}^{{\rm TS,ERS}} <\delta \mu_{p{\rm -ERS},i}$, the transition is considered to be barrier-free. The form of Ξ_pq,i can be very different depending on the type of transition. If the p state is part of a condensed state (a, l or s), the prefactor can be written as, Ξ_pq,i = Z_pq,iν_p,i, where Z_pq,i is the steric factor⁵ of the p to q transition per unit area and ν_p,i is a vibration frequency. For the gas states (b or v), we are only interested in the transitions to condensed states, and the prefactor can be written as, $\Xi_{{\rm b}({\rm v}),i} ={S_{{\rm b}({\rm v})q,i} f_{{\rm b}({\rm v}),i}^{\bot}}/{\bar{{c}}_{{\rm b}({\rm v}),i}}$. Here $f_{{\rm b}({\rm v}),i}^{\bot}$ is the effective flux of atoms/molecules from the b (or v) states impinging normal to the interface of the q = l (or s) states. In order to calculate the effective flux across a pq boundary, the backward q to p flux needs to be subtracted from the forward p to q flux, ΔΓ_pq,i = Γ_pq,i − Γ_qp,i. Under ERS conditions, we can apply an equation of detailed balance (i.e. the net fluxes of material across a boundary equal zero, $\Delta \Gamma_{pq,i}^{{\rm ERS}} =0$), which implies that

$$\begin{equation} \Xi_{qp,i} =\Xi_{pq,i} \frac{\bar{{c}}_{p,i}^{{\rm ERS}}}{\bar{{c}}_{q,i}^{{\rm ERS}}}\exp \left( {-\frac{\Delta g_{pq,i}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right) \end{equation} \tag{ 8 }$$

with $\Delta g_{pq,i}^{{\rm TS,ERS}} =\delta g_{pq,i}^{{\rm TS,ERS}} -\delta g_{qp,i}^{{\rm TS,ERS}}$ (Note that if the ERS transition state barrier is symmetric the exponential simply vanishes, as would be the case for a reversible state transition without requirements for dissociation/formation of bonds only one way). This is a general consequence of the detailed balance assumption when merging thermodynamics and transition state kinetics. The detailed balance provides an equilibrium relation between the ratios of coordination factors, attempt frequencies, possibly asymmetries for transition barriers at fixed ERS compositions. Finally, using equation (7) and equation (8), the net transition flux across the pq boundary is given as

$$\begin{eqnarray} &&\fl \Delta \Gamma_{pq,i} =\Xi_{pq,i} \exp \left({-\frac{\delta g_{pq,i}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right)\left(\bar{{c}}_{p,i} \exp \left( {\frac{\delta \mu_{p{\rm -ERS},i}}{k_{{\rm B}} T}} \right)\right.\nonumber\\ &&\left.-\,\frac{\bar{{c}}_{p,i}^{{\rm ERS}}}{\bar{{c}}_{q,i}^{{\rm ERS}}}\bar{{c}}_{q,i} \exp \left(\frac{\delta \mu_{q{\rm -ERS},i}}{k_{{\rm B}} T}\right)\right). \end{eqnarray} \tag{ 9 }$$

As in equation (7), if $\delta g_{pq,i}^{{\rm T,ERS}} \le \delta \mu_{p{\rm -ERS},i}$, i.e. $\exp (-({(\delta g_{pq,i}^{{\rm TS,ERS}} -\delta \mu _{p{\rm -ERS},i} )}/{k_{{\rm B}} T_{p}}))$ is set to one. The entropy in the first exponential can be put into a new prefactor, $\Xi'_{pq,i} =\Xi _{pq,i} \exp ({\Delta s_{pq,i}^{{\rm TS,ERS}}}/{k_{{\rm B}}})$, that can be used as a temperature independent fitting parameter⁶.

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To keep track of the atomic movements involved in the axial NW growth, a mass transfer equation are used to describe the atomic flow to and from the liquid phase [22],

$$\begin{equation} \frac{{\rm d}}{{\rm d}t}N_{{\rm l}} =I_{{\rm III}} +I_{{\rm V}} -I_{{\rm inc}}. \end{equation} \tag{ 10 }$$

Here the liquid sorption currents I_i of group i atoms,

$$\begin{equation} I_{i} =\int {\Delta \Gamma_{{\rm al},i} \,{\rm d}l_{{\rm TL}}} +\int {\Delta \Gamma_{({\rm vb}){\rm l},i} \,{\rm d}A_{{\rm vl}}} \end{equation} \tag{ 11 }$$

describe the effective 'adatom to liquid' and 'gas to liquid' currents. I_inc is the effective atomic incorporation current from the liquid into the solid, N_l is the number of atoms in the liquid, l_TL is the triple line (TL) length and A_vl is the projected liquid–vapour surface area [54]. If the equilibrium vapour pressure, $p_{i}^{{\rm eq}}$, of a large liquid phase with a given composition is known (see section 3.1), the liquid to vapour transition rate from a liquid in such a state must fulfil the criteria, $\Gamma_{{\rm lv}} \le {p_{i}^{{\rm eq}}}/{\sqrt {2\pi m_{i} k_{{\rm B}} T}}$, simply due to mass conservation. However, this criteria may be violated when size effects play an important role. Following the transition state approach, a simple version (sufficient in most cases) would be to assume no transition state barrier for sorption and a single vapour species for each element:

$$\begin{eqnarray} &&\fl \Delta \Gamma_{({\rm vb}){\rm l},i} \cong f_{i,\bot} -\frac{x_{i}}{x_{i}^{{\rm ERS}}}\frac{p_{i}^{{\rm ERS}}}{\sqrt {2\pi m_{i} k_{{\rm B}} T}}\exp \left( {\frac{\delta \mu_{{\rm l-ERS},i}}{k_{{\rm B}} T}} \right) \end{eqnarray} \tag{ 12 }$$

where f_i,⊥ = f_b,i,⊥ + f_v,i,⊥ is the effective impinging flux of group i. For typical growth conditions where f_V > f_III, the vapour pressure of group V can be assumed to be proportional to the incoming flux, f_v,V ∝ f_b,V. This is because a huge contribution of the excess As species must come from secondary adsorption, see [23]. Secondary adsorption of group III can typically be neglected, although for growth on substrates covered with a thermally grown oxide layer it can play a significant role, as shown by Rieger et al [55]. The va and al transition flux can be written, respectively, as

$$\begin{eqnarray} &&\fl \Delta \Gamma_{({\rm vb}){\rm a},i} \cong f_{i,\bot} -\frac{\rho_{i}}{\rho_{i}^{{\rm ERS}}}\frac{p_{i_{n}}^{{\rm ERS}}}{\sqrt {2\pi m_{i_{n}} k_{{\rm B}} T}}\exp \left( {\frac{\delta \mu_{a{\rm -ERS},i}}{k_{{\rm B}} T}} \right),\nonumber\\ \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} &&\fl \Delta \Gamma_{{\rm al},i} =\Xi_{{\rm la},i}^{\prime} \exp \left(\!{-\frac{\delta h_{{\rm la},i}^{{\rm TS,ERS}}}{k_{{\rm B}} T}}\!\right)\nonumber\\ &&\times\,\left(\!{\frac{\bar{{\rho}}_{i}}{\bar{{\rho}}_{i}^{{\rm ERS}}}x_{i}^{{\rm ERS}} \exp \left(\!{\frac{\delta \mu_{a{\rm -ERS},i}}{k_{{\rm B}} T}}\!\right)-x_{i} \exp \left(\!{\frac{\delta \mu_{{\rm l-ERS},i}}{k_{{\rm B}} T}}\!\right)}\!\right).\nonumber\\ \end{eqnarray} \tag{ 14 }$$

Finally, the net sorption currents (equation (11)) are given as

$$\begin{eqnarray} &&\fl I_{({\rm vb}){\rm l},i} =\int {\Delta \Gamma_{({\rm vb}){\rm l},i} \,{\rm d}A_{{\rm vl}}}\nonumber\\ &&\cong A_{{\rm vl}} \left( {f_{{\rm v},\bot} -\frac{x_{i}}{x_{i}^{{\rm ERS}}}\frac{p_{i_{n}}^{{\rm ERS}}}{\sqrt {2\pi m_{i_{n}} k_{{\rm B}} T}}\exp \left( {\frac{\delta \mu_{{\rm l-ERS,}i}}{k_{{\rm B}} T}} \right)} \right)\nonumber\\ &&+\,A_{{\rm vl}}^{\prime} f_{{\rm b},i},\nonumber\\ &&\fl I_{{\rm al},i} =\int {\Delta \Gamma_{{\rm al},i} \,{\rm d}A_{{\rm vl}}}\nonumber\\ &&\cong L_{{\rm TL}} \Xi_{{\rm al},i} \left( {\bar{{\rho}}_{i} -\bar{{\rho}}_{i}^{{\rm ERS}} \frac{x_{i}}{x_{i}^{{\rm ERS}}}\exp \left( {\frac{\delta \mu_{{\rm l-ERS},i}}{k_{{\rm B}} T}} \right)} \right). \end{eqnarray} \tag{ 15 }$$

In equation (15) all information about the transition state barriers from the l to the v or a states is stored in the ERS parameters, due to the detailed balance assumption at equilibrium. Only a given projection of the liquid surface A'_vl is exposed to the incident beam flux, depending on the beam direction and droplet geometry [57]. L_TL is the length of the TL. Note that if $\delta g_{lq,i}^{{\rm TS,ERS}} <\delta \mu _{{\rm l-ERS},i}$, the exponentials vanish in equation (15) according to equation (7).

To get a more intuitive feeling of the effect of growth conditions on the adatom kinetics in terms of an effective diffusion length, adatom migration on a large homogeneous planar interface serves as a good example [56]. Even though this approach is not accurate for modelling the growth dynamics, it is instructive and intuitive, and sufficient to understand many overall growth phenomena as function of growth conditions. There are three main transition paths for an adatom, namely surface diffusion (aa), desorption (av) and incorporation (as). Using the TS approach, See section A.1 in the appendix, the diffusion length of an adatom on a surface j can be written as

$$\begin{eqnarray} &&\fl\lambda_{j,i} \cong \left[\bar{{Z}}_{{\rm aa},i}^{\prime} l_{{\rm a},i}^{2} \exp \left( {-\frac{\delta h_{{\rm aa},i}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right)\left( \bar{{Z}}_{{\rm as},i}^{\prime} \exp \left( {\frac{\delta \mu _{{\rm a}-{\rm ERS},i}}{k_{{\rm B}} T}} \right)\right.\right.\nonumber\\ &&\quad\left.\left.+\,\bar{{Z}}_{{\rm av},i}^{\prime} \exp \left( {-\frac{\delta h_{{\rm av},i}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right) \right)^{-1}\right]^{1/2}, \end{eqnarray} \tag{ 16 }$$

where we assume that $1-\bar{{\rho}}_{j,i} \approx 1$, and that the density of incorporation sites is given as $\bar{{c}}_{{\rm inc},{\rm III}({\rm V})} \propto \exp ({\delta \mu_{{\rm a-ERS},i}}/{k_{{\rm B}} T})$. l_a,i is the lattice site spacing and the entropy change is included in the prefactors, $\bar{{Z}}_{pq,i}^{\prime} =\bar{{Z}}_{pq,i} \exp ({\delta s_{pq.i}}/{k_{{\rm B}}})$. In figure 5 we show estimations of diffusion lengths as a function of growth conditions, using parameters given in section A.3 in the appendix.

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Above we have treated the static case of adatom diffusion. An approach to treat the dynamics of adatom diffusion and the adatom collection to the liquid phase is discussed in detail in section A.2 in the appendix, where we show how to merge a 'Dubrovskii/Johansson' static diffusion scheme into the dynamic formalism using the TS kinetics with a uniform diffusivity along NW and substrate, a method which is used for the modelling in section 3.

We now turn to the actual crystal formation at the ls interface. Here, we will for simplicity assume that the liquid diffusion is fast on the time scale of NW growth, and that the liquid phase is homogeneous. The dynamic treatment including non-homogeneous liquids can be carried out if a reference composition and liquid diffusivity are known. See for example [57, 58] for treatment of non-homogeneous liquids. The possibility of fast diffusion along the growth interface during VLS is assumed negligible, as indicated by a study by Dick et al [59]. As shown by Schwarz and Tersoff [45], if the solid were isotropic, the equilibrium shape would be with a curved ls interface. But as the authors also pointed out in a later publication, in the anisotropic case (which is relevant for III–V NW growth), the morphology is strongly faceted [46]. It is complicated to treat the dynamical evolution if the solid is partially wetted by a droplet which at the same time is changing in size during growth. For such a system the preferential orientations of the facets depends on the liquid phase size and it is necessary to describe the crystal growth in terms of both facet sizes and facet orientations (and therefore an independent parameter set {X} of both 'areas' and 'orientations', as explained in section 2.1). An additional complication affects the evolution of the crystal shape if the facets are limited in their growth rate by the formation of a small nucleus [60], see section 2.3 for a treatment of the nucleation limited axial growth at the topfacet. For VLS growth one only considers growth at the ls interface and distinguishes between two types of ls transitions:

Nucleation free growth. Facets which are limited in their growth rate or in their change of orientation by the transfer of single pairs to the growth front, as described by equation (9).

Nucleation limited growth. Facets which are limited in their growth rate by the formation of a small nucleus, or more generally limited in their change of X due to an energy barrier which is larger than the single pair transition state barrier.

As in [26], the ls growth system will be divided into two main regimes (mainly due to traditional reasons as explained below).

Regime I. The TL stays in contact with the topfacet.

Regime II. The TL is not in contact with the topfacet, or possibly only for a short time during a nucleation event at the topfacet.

The vast majority of literature on the nucleation at the topfacet has assumed an ideal regime I, where the ls interface is perfectly flat, see for example [25]. However, it is very uncertain under which material systems and growth conditions ideal regime I conditions applies. It is likely that it is only relevant under non-steady-state conditions where the liquid decreases significantly in size, such as immediately after closing the shutter of the group III source or upon during cool down where the nucleation barrier is lowered [27]. But as recent in-situ TEM experiments [40–42] strongly suggests and as shown in the modelling examples in section 3.5, regime II may be a dominant VLS steady-state growth mode.

Many studies suggest that the dominating type of growth at the topfacet is strongly nucleation limited (see for example [62]) while small truncation facets at the edges of the growth interface might be nearly nucleation free [40, 41]. The chemical potential of the solid depends on the stacking type of the crystal structure (e.g. s: WZ(2H), ZB(3C), 4H, 6H, etc), with ZB and WZ being the most common sequences, where the ZB structure has the lowest cohesive energy for most III–V's and are therefore favoured in bulk materials [61, 62]. The liquid needs to reach a critical level of supersaturation (typically of the order of a few hundred meV per III–V pair) [25] before the nucleation barrier at the topfacet can be overcome. Under this constraint other facets which are not nucleation limited will reshape in respond to the elevated liquid chemical potential at a rate determined by equation (9), and the whole growth system is therefore in a configuration far from equilibrium. For VLS growth, group V is typically the less abundant specie in the liquid, i.e. x_III > x_V. For a fixed solid stoichiometry, the activation energy for the nucleation free single pair ls transitions, equation (9) can be written as

$$\begin{eqnarray} &&\fl \Delta \Gamma_{{\rm ls,III\mbox{--}V}}^{X} =\Xi_{{\rm ls,III\mbox{--}V}} \exp \left( {-\frac{\delta g_{{\rm ls,III\mbox{--}V}}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right)\nonumber\\ &&\times\,\left(x_{{\rm V}} \exp \left({\frac{\delta \mu_{{\rm l-ERS,III\mbox{--}V}}}{k_{{\rm B}} T}} \right)-x_{{\rm V}}^{{\rm ERS}}\right.\nonumber\\ &&\left.\times\,\exp \left({\frac{\delta \mu_{{\rm s-ERS},{\rm III\mbox{--}V}}^{X}}{k_{{\rm B}} T}} \right) \right). \end{eqnarray} \tag{ 17 }$$

As the liquid chemical potential δμ_{l−ERS,III–V} is an oscillating function due to the nucleation limited growth at the topfacet [62], the parameter X (describing nucleation free facet size or angle) will therefore oscillate accordingly. Because the chemical potentials depend on location and system morphology, so do the transition fluxes, and the free energy minimization needs to be described with respect to an appropriate set of independent parameters, {X(ω)}. Generally speaking, the larger the parameter set the more accurately the modelling, but also the more computations are needed. In three dimensions, the chosen set of parameters {X(ω)} will depend on ω which is defined to be the angle between the middle of the sidefacet and position as measured from the centre of the top facet, see [26] for clarification.

As shown in the stereographic projection in figure 6(a), if only considering ZB and WZ stacking, it is sufficient to divide the ω -dependence of the crystal into three sections because the ZB crystal structure has three-fold symmetry. The WZ crystal structure has six-fold symmetry around the growth axis and is therefore also described completely within this region. In table 1 in section A.3 in the appendix, we give the interfaces with lowest energies for the ZB and WZ structure (we restrict ourselves to the upper half hemisphere with polar (1 1 1) ZB or [0 0 0 1] WZ directions). To describe the NW diameter as a function of ω in terms of the cross sectional Wulff shape⁷, we need to look at the energies of the facets in the θ = 90° plane (the outer ring) in the stereographic projection in figure 6(a). For a cross sectional six-fold symmetric NW it is enough to describe the NW diameter at the growth interface in the range ω = [−30° : 30°] as

$$\begin{equation} d_{{\rm NW}} (\omega )=\frac{d_{{\rm NW}} (\omega =0^{\circ})}{(1+\omega \eta (\omega ))\cos (\omega )}, \end{equation} \tag{ 18 }$$

where the function η(ω) determines the cross sectional shape of the growth interface. η(ω) is a complicated function that depends on many factors. We can simplify it as η(ω) = η₀(cos ⁻¹(ω) − 1)ω⁻¹, where η₀ = 0 for complete hexagonal facetting and η₀ = 1 in the isotropic case (complete axi-symmetric cross section). In the case of the ZB structure which has a three-fold symmetric crystal structure, it is very likely that the NW cross section does not have a perfect six-fold geometrical symmetry. In this case we need to take account of the possibility of a three-fold symmetric cross section where the diameter is given by d_NW(ω) = r₊(ω) + r₋(ω) with r₊(ω) and r₋(ω) = r₊(ω + 180°) being the radius as measured from the centre of the NW crystal. For complete facetting (η₀ = 0) and a constant NW volume, the relation between r₋ and r₊ is:

$$\begin{equation} r_{+} =2r_{-} -\sqrt {3r_{-}^{2}-2\sqrt 3 \cdot A_{{\rm c}}} , \end{equation} \tag{ 19 }$$

where A_c is the cross sectional area. According to Wulff, in the absence of a liquid phase, the cross sectional equilibrium shape of the NW crystal would be given by γ_A/γ_B = r₊/r₋, where γ_A(γ_B) is the effective vertical surface energy of the facet normal to the r₋(r₊) vector.

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For a more complete description of the dynamics we need the values of the anisotropic surface and interface energies for the different crystal structures. To carry out the iterative minimization of the free energy are desirable. To this end, we need a γ plot with rounded cusps that can approach arbitrarily close to the sharp cusps of faceted orientations. This can be realized by summing a set of 2D Lorentzian functions centred on the facets of high symmetry, which have the lowest interface energies. The angular dependence of the interface energy is then described, in angular coordinates (θ, ω), by

$$\begin{eqnarray} &&\fl \gamma_{{\rm vs},j} (\theta ,\omega )=\gamma_{{\rm vs}0} -\sum\limits_{h\,k\,l} {c_{h\,k\,l} \frac{{\rm I}_{h\,k\,l}}{1+({\phi _{h\,k\,l} (\theta ,\omega )}/{w_{h\,k\,l}})^{2}}} , \end{eqnarray} \tag{ 20 }$$

where $\phi_{h\,k\,l} (\theta ,\omega )=\arccos (\cos (\theta -\theta _{h\,k\,l} )+\sin (\theta )\sin (\theta_{h\,k\,l}) (\cos (\omega -\omega _{h\,k\,l} )-1))$ is the angle between the facet h k l (see table 1) and direction (θ, ω), where θ = 0 corresponds to the growth direction (see figures 6(b) and (c) for ZB and WZ structure).

In equation (20), the maximum interface energy is noted γ_vs0, and the decrease in interface energy at each high symmetry facet is given by the 'intensity' I_hkl = γ_vs0 − γ_vs,j. The values of γ_vs,j for the main orientations can either be found in the literature or obtained from density functional theory calculations. w_hkl is a scale parameter which specifies the half-width at half maximum of the energy increase around the (h k l) facet. c_hkl is a constant close to unity, but if w_hkl is large the interface energy may have to be adjusted to a value slightly lower than unity because the contributions from adjacent facets may overlap. We will simply assume that the ls interface energy is given by γ_ls(θ, ω) = σγ_vs(θ, ω), where σ is typically assumed to be a constant of the order 0.3–0.5.

We will here treat the nucleation limited growth which takes place at the ls top facet separately because this is where the axial growth and where the final crystal structure of the NW is formed. Many recent experimental studies have indicated that growth on the dominating ls (1 1 1)/(0 0 0 1) top facet is limited by the formation of a nucleus, which means that the liquid supersaturation needs to exceed a certain critical value before a new monolayer can be formed, see for example [39, 63]. This implies that the topfacet is stabilized as long as the difference in chemical potentials between the liquid and topfacet is smaller than a critical value, due to large activation energies both ways. Because the mother phase (the liquid) is small, the liquid supersaturation drops far below the critical level after a ML formation and probability of having a subsequent second nucleation is unlikely. We are therefore only interested in single nucleation events. To describe the probability of forming a critical nucleus we need to take account of the stochastic nature of the phase fluctuations which causes nucleation. But first, we need an expression for the mean nucleation rate.

If the movement of atoms in and out of clusters of various sizes (smaller than the critical nucleus) at the growth interface, takes place on a timescale much smaller than the time between nucleation events, the nucleation probability can be derived assuming steady-state nucleation rate conditions [64–66], which is the typical assumption in NW growth theory [10, 19, 20, 25]. It is reasonable to assume that the attachment/detachment frequency of III–V pairs to and from the clusters on the (1 1 1)B topfacet is limited by the group V elements. This is not only because the concentration of group V is low in the liquid but also because the group III elements are attached with only one covalent bond on average in the 'B' terminated surface when group V is absent. Once group V is present, the pair is stabilized leaving only one free covalent bond per pair on average (reconstruction is not considered in the continuum formalism). With this, the mean nucleation rate at given site with coordinates (r, ω) at the topfacet (r measured from the centre) can then be written as

$$\begin{eqnarray} &&\fl j_{{\rm s}(r,\omega )} =A_{n^{\ast}} Z_{{\rm s}(r,\omega )} c_{1} \Xi _{{\rm ls},{\rm III\mbox{--}V}} x_{{\rm V}} \exp \left( {-\frac{\delta g_{{\rm ls,int}}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right)\nonumber\\ &&\times\,\exp \left( {-\frac{\Delta G_{n^\ast ,{\rm s}(r,\omega )}}{k_{{\rm B}} T}} \right), \end{eqnarray} \tag{ 21 }$$

where $A_{n^{\ast}}$ is the step area of the critical nucleus of n^* pairs, $Z_{{\rm s}(r,\omega )} =1/{n^{\ast}}\sqrt {{\Delta G_{n\ast ,{\rm s}(r,\omega )}}/{4\pi kT}}$ is the 2D Zeldovich factor and $\Delta G_{n\ast ,{\rm s}(r,\omega )} =-\sum\nolimits_{i=2}^{n^{\ast}} {(\delta \mu_{{\rm l-ERS},{\rm III\mbox{--}V}} -\delta \mu_{{\rm s}(r,\omega ){\rm -ERS,III\mbox{--}V}}^{i} )}$ is the formation free energy of the nucleus, with $\delta \mu_{{\rm s}(r,\omega ){\rm -ERS,III\mbox{--}V}}^{i}$ being the chemical potential of a cluster of i pairs at (r, ω). s denotes solid structure described by its stacking type (ZB (3C), WZ (2H), 4H, etc). Consistent with transition state approach described above, the forward flux from the liquid to the cluster, Γ_ls,III–V, is assumed independent of the size of the cluster, (the backward flux from the clusters depends on the cluster size but cancels out in the derivation). $\delta g_{{\rm ls,}int}^{{\rm TS,ERS}}$ is the transition state barrier for attachment of a single pair to the clusters at the interface. The detailed kinetics at the interface is unknown; we thus simply assume that the concentration of single III–V pairs attached to the interface c₁ (single pair clusters) is equal to the concentration of the group V in the liquid, c₁ ≈ x_V. Once the nucleation event has occurred, the ML is completed in a non-nucleation limited manner, at a rate given by equation (17), and the liquid supersaturation builds up slowly again until the next nucleation event takes place.

The nucleus formation free energy can be written as in a more familiar form,

$$\begin{equation} \Delta G_{n^\ast ,{\rm s}(r,\omega )} =-\Delta \mu_{{\rm ls},{\rm III\mbox{--}V}}^{\infty} n^{\ast}+h\sum\limits_{k=1}^m {l_{k} \gamma_{{\rm step}(r,\omega ),k}}, \end{equation} \tag{ 22 }$$

where the first term is the formation free energy required to form the volume part of the nucleus. The second term is the excess free energy due to the formation of a dividing step. γ_step(r,ω),k and l_k are the free energy and length of the kth step facet, respectively. As the nucleation takes place when the number of pairs in the cluster exceeds the critical value, n ⩾ n^*, which is associated with the maximum free energy increase given by the condition,

$$\begin{eqnarray} &&\fl \frac{{\rm d}\Delta G_{n^\ast ,s(r,\omega )}}{{\rm d}n}=-\Delta \mu _{\rm ls,III\mbox{-}V}^{\infty}\nonumber\\ &&+\,h\sum\limits_{k=1..m} {\left( {\frac{{\rm d}l_{k}}{{\rm d}n}\gamma_{{\rm step}\left( {r,\omega} \right),k} +l_{k} \frac{{\rm d}\gamma_{{\rm step}\left( {r,\omega} \right),k}}{{\rm d}n}} \right)} =0 \end{eqnarray} \tag{ 23 }$$

we can derive an explicit expression for the nucleation barrier $\Delta G_{n}^{\ast}$ by extracting n^* from equation (23) (l_k depends on n^*) and insert it into equation (22). The last term in the summation of equation (23) is typically neglected in continuum models, as the interface energies are assumed constant as a function of interface area.

For regime II we will divide all the possible nucleation sites into three main classes (see figure 7).

• (A)  
  At the edge between the topfacet and truncated facet, see [42]. Here the nucleus forms an extension to the truncated facet a crystal structure different from the equilibrium bulk structure can be dictated by the orientation of this facet (similarly to what was proposed in the case of a nucleus in contact with a vapour by Glas et al [25]).
• (B)  
  Nucleation at the centre of the top facet, see for example [19]. The preferential crystal structure here is the structure with the lowest cohesive energy which is typically ZB [67].
• (C)  
  It is possible that the truncation size becomes positive at a given ω, before one to the other types of nucleation events takes place. Then a TL nucleation event will be induced at the topfacet and the necessary step for step flow is formed. A fast completion of the monolayer will lower the supersaturation and move the truncation back to negative values (provided that the barrier of forming the truncation facet is small enough). For a six-fold crystal geometry it is likely that such an event will take place at the corners, i.e. ω = 30°. If the liquid size is decreasing TL nucleation becomes more and more dominant and the system will eventually move into regime I.

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Under conditions where the time needed to reach steady-state composition in the liquid is smaller than the time between the formation of two consecutive MLs, the total centre-nucleation rate can be written as $J_{{\rm c}} \cong 2\pi j_{{\rm c}} \int_{{\rm Center}} {(({d_{{\rm NW}} (\omega )}/2)-\Delta z(\omega )\tan (\theta_{{\rm T}} (\omega ))-l_{m}^{\ast} )\,{\rm d}\omega}$, where Δz(ω) tan(θ_T(ω)) is the decrease in topfacet length at ω due to the truncation. For truncation edge nucleation we will integrate over the part with a negative truncation, $J_{T} \cong \pi \int_{l_{{\rm T}}} {l_{m}^{\ast} (\omega )(d_{{\rm NW}} (\omega )-\Delta z(\omega )\tan (\theta_{{\rm T}} (\omega )))j_{{\rm T}} (\omega )\,{\rm d}\omega}$. In order to carry out a more realistic modelling of the s-stacking probabilities we can account for the stochastic nature of nucleation by multiplying equation (21) by a random number between 0 and 1, ℜ(0, 1), at each time step, and define a normalized value δ above which nucleation will take place

$$\begin{equation} \int_{{\rm Topfacet}} {j_{{\rm s}(r,\omega )}} \,{\rm d}A_{{\rm topfacet}} \cdot \Re (0,1)\ge \delta. \end{equation} \tag{ 24 }$$

Finally, whenever one or more sites fulfil equation (24) the rate of the subsequent step flow and completion of a ML are determined by equation (17). However it is possible that the truncation (Δz(ω) tan(θ_T(ω))) under certain conditions goes to zero and at certain positions becomes positive before equation (24) is fulfilled (figure 7(c)). In this case a step is naturally provided at the triple line (TL) and completion of a monolayer will take place at the same time as the truncation most likely goes to negative again due to a lowering of the liquid supersaturation.

To simulate a specific process such as self-catalysed GaAs NW growth on Si (1 1 1) substrates, requires the relevant ERS parameters and size effects based on the assumptions made for the simulation. Thus, before giving detailed simulation examples of the overall NW growth, we will first go through the specific calculations needed for this system. As mentioned, modelling the overall morphology does not require detailed information on the shape of the ls interface, and in this section we will therefore assume an axi-symmetric cross section (ω dependence can be neglected) and an ideal regime I with a single flat ls interface. In this case we do not need to define an independent parameter set, but only use the liquid size evolution and nucleation at the topfacet to determine the evolution of the crystal morphology in terms of the diameter d_NW at the growth interface and the vl contact angle θ with respect to the topfacet. The size effect terms in equation (5) and (6) for the chemical potential can be found using the trigonometric relations,

$$\begin{eqnarray*}&&\fl A_{{\rm ls}} =\frac{\pi d_{{\rm NW}}^{2}}{4},\tqs A_{{\rm vl}} (\theta )=\frac{\pi d_{{\rm NW}}^{2} (1-\cos (\theta ))}{2\sin^{2}(\theta )} \end{eqnarray*}$$

and

$$\begin{eqnarray*}&&N_{{\rm l}} (d_{{\rm NW}} ,\theta )=\frac{\pi d_{{\rm NW}}^{3}}{24\Omega_{{\rm l}}}\frac{(1-\cos (\theta ))^{2}(2+\cos (\theta ))}{\sin^{3}(\theta )}, \end{eqnarray*}$$

where Ω_l is the atomic volume in the liquid. A change in d_NW implies not only a change in the ls and vl areas but also the formation of a new vs area corresponding to the absolute change in the ls area. We have not taken account here of the possibility of wetting the sidefacets for a cylindrical shaped cross section, as described in [68, 69]. For a detailed analysis of the wetting in regime I (i.e. on a flat hexagonal top facet) see [26]. To calculate the chemical potentials of the ERS (equation (1)), we need to calculate the liquid chemical potentials when the liquid phase is in equilibrium with the solid. For liquid binaries (self-assisted growth), the chemical potential is given by the tangent method, or correspondingly;

$$\begin{equation} \mu_{{\rm l},i}^{\infty} (x_{{\rm V}} ,T)=g_{{\rm l}}^{\infty} (x_{{\rm V}} ,T)+(1-x_{i} )\frac{\partial g_{{\rm l}}^{\infty} (x_{{\rm V}} ,T)}{\partial x_{i}}. \end{equation} \tag{ 25 }$$

Here the liquid free energy per atom of an infinitely large binary alloy is given by $g_{{\rm l}}^{\infty} (x_{{\rm V}} ,T)=(1-x_{{\rm V}} )g_{{\rm l,III}} (T)+x_{{\rm V}} g_{{\rm l,V}} (T)+g_{{\rm l,mix}} (x_{{\rm V}} ,T)$ where g_l,mix(x_V, T) = (1 − x_V)x_V[L₀(T) − L₁(T)(1 − 2x_V)] + RT[(1 − x_V) ln(1 − x_V) + x_V ln(x_V)] accounts for the asymmetry in the compositional effect on the free energy by using the Redlich–Kister formalism [70] as in [71] with two liquid interaction parameters L₀ and L₁. These parameters together with the free energy values of the pure components g_l,i are given for $g_{{\rm Ga}_{{\rm 1}-x_{{\rm V}}} {\rm As}_{x_{{\rm V}}}}$ and $g_{{\rm In}_{1-x_{{\rm V}}} {\rm As}_{x_{{\rm V}}}}$ in table 2 of section of A.3 in the appendix, where the equilibrium concentrations are estimated from fitting the liquidus values reported in [87]. All Gibbs free energies and chemical potentials are relative to the enthalpy of the standard element reference (HSER_i)⁷⁵, and denoted g'_l(T) and μ'_l,i(T), respectively. Using these data, the ERS chemical potential $\mu_{i}^{{\rm ERS}'}=\mu_{{\rm l},i}^{\infty'}(x_{{\rm V}}^{{\rm ERS}} ,T)$ is calculated using equation (25) and the relative chemical potential is simply $\delta \mu _{{\rm l},i} (x_{{\rm V}} ,T)=\mu_{{\rm l},i}^{\prime} (x_{{\rm V}} ,T)-\mu _{i}^{{\rm ERS}'}$.

To calculate the partial vapour pressures over a liquid of given composition, we note that $\mu_{{\rm v},i_{n}}^{'n} =n\mu_{{\rm l},i}^{\prime}$, where n is the number of atoms in the molecule considered and 'ⁿ denotes that the value is given with respect to n times the standard reference. Using the thermodynamic data from appendix 2 in [75], we find an expression for the Gibbs free energy of a pure i_n species, $g_{{\rm v},i_{n}}^{{\rm pure}'^{n}}(T)=\Psi_{i}^{'n} (T)+RT\ln (P)$, where P is the total pressure and $\Psi_{i}^{'n} (T)$ is a function of temperature only (see table 3 in section A.3 in the appendix for the thermodynamic data). Now, since $\mu_{{\rm v},i_{n}}^{'n} -\mu_{{\rm v},i_{n}}^{{\rm pure}'^{n}}=RT\ln ({p_{i_{n}}}/P)$, where $\mu_{{\rm v},i_{n}}^{{\rm pure}'^{n}}=g_{{\rm v},i_{n}}^{{\rm pure}'^{n}}$, we can write the following expression for the vapour pressure of element i_n:

$$\begin{equation} p_{i_{n}} (x_{{\rm V}} ,T)=\exp \left( {\frac{n\mu_{{\rm l},i}^{\prime} (x_{{\rm V}} ,T)-\Psi_{i}^{'n} (T)}{RT}} \right). \end{equation} \tag{ 26 }$$

The corresponding ERS pressures are then found by setting $\mu_{{\rm l},i}^{\prime} (x_{{\rm V}} ,T)=\mu_{i}^{{\rm ERS}'}$.

From figure 8(a) we see that the only species that may have significant partial pressures are the Ga and As₂ species. As the liquid supersaturation increases (increasing x_As), the vapour pressure of Ga remains almost constant, which means that the desorption flux of Ga form the liquid is almost constant. This means that the vl and al transition fluxes for the Ga species are roughly independent of the supersaturation. On the other hand, as the supersaturation increases the desorption of the As species increases very strongly (note the log scale).

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To complete the ERS description, we need to calculate the adatom densities, $\rho_{{\rm NW},i}^{{\rm ERS}}$ and $\rho_{{\rm sub},i}^{{\rm ERS}}$, which we do by using kinetics. For the adatom collection we follow the approach outlined in section A.2 in the appendix, and the ERS adatom densities are found using equation (36) under ERS conditions, where $\rho_{{\rm NW},i}^{{\rm ERS}}$ is calculated by setting L_NW → ∞ and ΔΓ_al,i → 0, and $\rho_{{\rm sub},i}^{{\rm ERS}}$ is found by setting r → ∞, both under conditions of the calculated ERS beam fluxes found above. Using the parameters listed in section A.2 in the appendix, the ERS adatoms densities are $\rho_{{\rm NW,Ga}}^{{\rm ERS}} (T=630\,^{\circ}{\rm C})=5.3\times 10^{17}\,{\rm m}^{-2}$ and $\rho_{{\rm NW,As}}^{{\rm ERS}} (T=630\,^{\circ}{\rm C})=0.16\times 10^{17}\,{\rm m}^{-2}$.

Tuning the fitting parameters can be time consuming. The fitting values of the relevant prefactors and activation free energies for adatom desorption and incorporation used in the simulations presented below are given in section A.3 in the appendix. In order to use the diffusion lengths given by equation (16), we need estimates of the activation energies $\Delta g_{pq,i}^{{\rm TS,ERS}} =\Delta h_{pq,i}^{{\rm TS,ERS}} -T\Delta s_{pq,i}^{{\rm TS,ERS}}$. As the entropy change as a function of temperature is negligible compared to the enthalpy change, we include the entropy contribution into the temperature independent prefactors as $\bar{{Z}}_{{\rm as},i}^{\prime} =\bar{{Z}}_{{\rm as},i} \exp ({\Delta S_{pq,i}^{{\rm ERS}}}/{k_{{\rm B}}})$. This leaves us with enthalpy barriers which can be estimated from zero temperature ab initio calculations such as density functional theory methods [88]. After having built up the simulation framework, it can be used to analyse a variety of features and systems. Here, we will only give a few examples.

For typical MBE growth of self-assisted GaAs NWs on a Si (1 1 1) covered with a thin native SiO_x layer, Ga beam fluxes corresponding to planar growth rates of 0.1–0.3 μ m h⁻¹ are commonly used, with a V/III flux ratio in the range 5–100 and a substrate temperature around T = 630 °C [72, 73]. There exists a certain 'growth parameter window', namely ranges of values for the basic growth parameters (temperature and beam fluxes), where it is possible to obtain NW growth (as a rule of thumb, the higher the temperature the higher the V/III ratio [22]). A general feature of the simulations is that there are sharp and well-defined boundaries for the growth parameter window. As the critical liquid supersaturation needed for nucleation at the topfacet is almost independent of the applied pressures (beam fluxes) [26], the axial growth rate is simply dictated by the time it takes for the liquid to reach the critical concentration of As, $\delta \mu_{{\rm l-ERS,As}}^{{\rm crit}}$, after being lowered upon a nucleation event and subsequent ML formation. If we neglect for simplicity the surface diffusion of As species and account for the impinging v states by simply using that the beam flux hits the total vl interface, the minimum As flux needed to obtain growth is roughly given as

$$\begin{equation} f_{({\rm bv}){\rm l,As},\bot}^{{\rm crit}} \approx \frac{x_{{\rm As}}^{{\rm crit}}}{x_{{\rm As}}^{{\rm ERS}}}f_{{\rm lv,As}}^{{\rm ERS}} \exp \left( {\frac{\delta \mu_{{\rm l-ERS,As}}^{{\rm crit}}}{k_{{\rm B}} T}} \right). \end{equation} \tag{ 27 }$$

Here $x_{{\rm As}}^{{\rm crit}}$ is the critical concentration of As needed for a nucleation event and $f_{{\rm lv,As}}^{{\rm ERS}} \approx \sum\nolimits_n {{p_{{\rm As}_{n}}^{{\rm ERS}}}/{\sqrt {2\pi m_{{\rm As}_{n}} k_{{\rm B}} T}}}$ is the flux of material evaporating from the liquid under ERS conditions. This means that the critical As flux is strongly dependent on the nucleation barrier and is only very little dependent on the Ga flux as long as there is a large liquid Ga phase. For the simulation shown in figure 9(a), the critical impinging As flux needed to overcome the nucleation barrier is roughly $f_{({\rm bv)l},{\rm As},\bot}^{{\rm crit}} \approx 100\cdot f_{{\rm lv,As}}^{{\rm ERS}}$. To examine how the axial growth rate depends on the incoming fluxes, we need to look at the time it takes to refill the liquid phase after ML formation in order to recover the critical level. The outgoing lv flux of As depends on the liquid chemical potential roughly as $\Gamma_{{\rm lv}} \propto ({x_{{\rm As}}}/{x_{{\rm As}}^{{\rm ERS}}})^{2}$ (because δμ_l−ERS,i depends on the As concentration roughly as $\ln ({x_{{\rm As}}}/{x_{{\rm As}}^{{\rm ERS}}}))$. Now, because a small (large) droplet size will lead to a large (small) decrease in the As concentration immediately after a ML formation, the time needed to refill the liquid to the critical concentration depends on the droplet size. Thus, especially in the regions of the growth parameter window where the droplet size changes during growth, the incoming flux of Ga may also play an important role on the growth rate. In figure 9(b), it is seen that the droplet size increase at low V/III ratios, but as the V/III is increased the expansion of the droplet slows down as growth accelerates and Ga is incorporated faster into the NW. For moderate V/III ratios, where the droplet stays in a steady-state regime, the growth rate becomes more or less linear with the As flux until it reaches a limit where the droplet gets small and eventually gets consumed [74]. The apparent linear relation between NW length and As flux at moderate V/III ratios is consistent with previous reports [75]. At very high incoming As fluxes, As just consumes the droplet and NW growth becomes impossible.

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In figure 10(b), a series of 6 min simulations shows an example of the huge change in morphology when changing the As₂ flux around the lower limit of the growth window. The NW diameter increase when the droplet reaches a size where the contact angle exceeds the wetting angle on the side walls. A higher V/III ratio implies less tapered NWs, because the droplet does not increase in size at the same speed as for lower V/III ratios (figure 9).

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It is well known that it is generally possible to affect the crystal structure adopted by the NWs by tuning the growth conditions (for a review see [76]). In the case of self-catalysed GaAs NWs the preferential structure under quasi-steady-state growth conditions is typically ZB [77]. However, as shown by Jabeen et al [78] and Spirkoska et al [79] and many others, the density of twin planes (TPs) is generally observed to be highest at the beginning and at the end of the growth. This is another indication that changes in the growth conditions change the probabilities of forming ZB and WZ. However, there can be a wide variety in the distribution of crystal phases and defects along the NW length, since these depend on the complicated interplay between the various growth parameters. In particular, it is difficult to obtain a perfect crystal structure throughout the whole NW because the effective V/III ratio, I_V/I_III, changes as the NW grows. This is seen in a typical TEM image of a self-catalysed GaAs NW (figure 11), where the temperature and beam fluxes are kept constant during growth. To explain this, we have to use dynamics.

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As proposed by Ramdani et al [23], secondary adsorption is to a good approximation proportional to the beam flux of the material in excess (i.e. As), and such contributions are simply taken account of by assuming that the beam impinges on the total liquid surface. This gives effectively a higher collection from the gas states than if we only had considered direct impingement from the beam states. In these simulations, the NW diameter typically stays constant because the contact angle stays between the wetting angles on the topfacet and sidefacet, but the evolution of the liquid size is not monotonous. Relating the typical structural distribution seen in figure 11 to the typical evolution of relative size of the liquid predicted from the simulations (shown in figure 12), shows good agreement with theoretical predictions by Krogstrup et al [25] using the flat topfacet assumption (regime I).

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The crystal structure with the highest formation probability depends on the size of the liquid phase relative to the growth interface area. This match apparently well with the present simulations, which are done in regime I. However, it should be noted that whether the overall modelling it is done in regime I or II, the evolution of the droplet size seems to be qualitatively the same. As will be seen for regime II modelling in the next sections, truncation edge nucleation might also favour WZ at relative small droplets.

As shown by Bauer et al [80] and Plissard et al [81] it is possible grow positioned self-catalysed GaAs NW arrays using a Si/SiO_x template, and when growing the wires using a hole array in a SiO_x layer thermally grown on the Si substrate, approximately the same growth temperatures as above is used, but the Ga flux needs to be equivalent to a planar growth rate of 0.8–1.2 µm h⁻¹ and the V/III flux ratios need to be in the range 1–5 (see [83, 17] for details). This is a much higher Ga flux than for growth on untreated substrates with native oxide and is an indication that the av transition rate from the thick thermally grown oxide layer is dominant for the adatom state, as also seen in. Thus, for growth on a patterned oxide layer of approx. 20–30 nm of SiO_x, the diffusion length is strongly limited by desorption for both Ga and As species. As it has not been possible to find activation enthalpies for av transitions on oxide surfaces in the literature, we simply take it to be half the value on a corresponding crystalline surface. The density of incorporation sites at the oxide surface is set to zero. For the growths on Si (1 1 1) wafers with both the native oxide layer and the thermal oxide layer, it is assumed that the low energy pathway of diffusion is one dimensional on the NW sidefacets (along the NW growth axis) and isotropic on the substrate. (See figure 13 for experimental and theoretical results on a sharp upper temperature limit for this type of growth.)

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As mentioned in sections 2.2 and 2.3, and as indicated by many recent experiments [40–42], the assumption of a perfect flat liquid–solid interface is in most cases not a good assumption when analysing the details of the liquid–solid dynamics and structural formation probabilities. To analyse the liquid–solid dynamics at the growth region in more detail we need to define a more complete parameter set {X}. To do this, we first write the Gibbs free energy of the total ls NW growth system in the form,

$$\begin{equation} G_{{\rm sys},j} =\int_0^{180^{\circ}} {\Delta G_{{\rm sys},j} (\omega )\,{\rm d}\omega}, \end{equation} \tag{ 28 }$$

where ΔG_sys,j(ω) is the free energy of a representative thin 'double cake piece' throughout the whole ls system, as shown in the top view illustration of a NW with a typical six-fold axial symmetry in figure 14(a).

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The construction of the ls growth system at a given ω considered in this section is sketched in (b). For a single faceted solid crystal the equilibrium shape (called the 'Wulff shape') and can be calculated exactly if the surface energy function in equation (20) is known [82]. But the equilibrium shape of a liquid–solid system is extremely complex to derive and we will make simplifying assumptions in order to make qualitative predictions of a corresponding liquid–solid 'Wulff shape'. In section A.7 in the appendix we discuss the complete three-dimensional ls system of constant liquid curvature and complete facetting.

In equation (28), the integration over ω using analytical equations is difficult to carry out, thus we will start by looking at a single slice construction for which the liquid curvature stays constant. The choice of the parameter set {X(ω)} used with equation (6) when describing the dynamics of the NW growth system is obviously crucial for the overall evolution of the structure and morphology in the simulation. Recent in situ growth experiments have suggested that a truncated morphology at the growth interface edge is a general phenomenon, and it will therefore be taken into account here. Thus, we will choose, {X} ∈ {d_NW, Δz₊, Δz₋, θ_T+, θ_T−, ξ} as our parameter set (see figure 15), where θ_T+ and θ_T− are the truncation angles at ω = 0° and ω = 180° respectively. Note that all these parameters are functions of, but only considering a single cut of a finite thickness (of say dω = 1°) can give us an idea of the mean properties of the total three-dimensional growth region. The Gibbs free energy for a single slice, ΔG_sys,ZB(WZ)(ω = 0°) dω, can be found using basic trigonometric relations (see section A.4 in the appendix).

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The ls system is continuously adjusting towards $\Delta \mu_{{\rm ls}}^{X} =\delta \mu_{{\rm l-ERS}} -\delta \mu_{{\rm s-ERS}}^{X} =0$ conditions, but the input of free energy from the beam fluxes, vapour and adatoms and the interplay with anisotropic solid and the nucleation limited growth on the top facet keep the system out of equilibrium. Under certain conditions, the solid can enter a regime where undesired facets are locked in because a free energy barrier has to be overcome in order to form a facet lowering the total free energy of the system. The liquid–solid driving forces of liquid Ga_1−xAs_x assisted GaAs NW growth is plotted as a function of truncation height Δz for a certain set of fixed parameters in using equation (6) and the single slice construction around ω = 0°. In figure 16(a) it is seen that the equilibrium value of Δz_± is larger for smaller systems. For the single slice construction the equilibrium morphology will always have a negative truncation. However a non-steady-state evolution of the growth system can force the system into regime I and to get back to regime II will require nucleation of a truncated facet which requires a certain formation free energy. Figure 16(b) shows that, in the single slice construction, varying the truncation on one side have a small effect on the truncation on the opposite side. Figure 16(c) shows an important general trend, namely that truncation heights are generally smallest at smallest droplet sizes. This means that relatively small droplets has higher tendency of going into regime I than larger droplets, in accordance with [26]. In figure 16(d), it is seen that a strong dependence of the liquid concentration on the truncation size indicates that it is the composition which plays a dominating role on the oscillating morphology.

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Figures 16(e) and (f) show the driving forces around certain facets in the case of ZB and WZ structure, respectively. It is seen that for certain sets of orientations and parameters, the system needs to form another facet orientation to reach a quasi-equilibrium state. If the potential barrier to form such a facet is large, the system can enter an unstable growth mode.

In figures 17(b) and (c) it is shown that the truncation angles affect the driving forces in a more complicated way than the other parameters which are considered here. This implies that the liquid–solid growth region can stay in a dynamical metastable and still steady-state regime (see section A.5 in the appendix).

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The surface energies and the interface energy function given by equation (20) play a crucial role on the NW growth simulations in general and on the truncation dynamics in particular. The interface energy function is plotted for the single slice construction in figure 17(a). w_hkl, which specifies the half-width half maximum of the energy decreases around the (h k l) facet is an important parameter for the dynamical system. If w_hkl is small, the corresponding truncation facet orientation is locked to a low energy facet orientation and it is unlikely that the facet can overcome the energy barrier $\Delta G_{{\rm ls,III\mbox{--}V}}^{\theta_{{\rm T}1} -\theta _{{\rm T}2}}$ needed to form another facet and a more preferable configuration. In figures 17(b) and (c) shows how the ls driving force for truncation angle (i.e. the free energy change per pair due to a change in θ_T) depends on the orientation for a given parameter set, where Δz = 2 nm is closer to equilibrium than Δz = 1 nm. We emphasize that in this continuum approach with single truncation facets it has not been taken into account that facet orientations becomes discrete when Δz becomes small. The formation of a new facet orientation can be nucleation limited if the barrier is larger than the single transition state barrier $\Delta G_{{\rm ls,III\mbox{--}V}}^{\theta_{{\rm T}1} -\theta_{{\rm T}2}} >\delta g_{{\rm ls,III\mbox{--}V}}^{{\rm TS,ERS}}$ and such a transition has to be treated in a framework similar to that of section 2.3. However these transitions will not be treated in detail here; instead, in the simulations the probability of forming another truncation facet orientation simply depends on the evolution of the system morphology. For large values of w_hkl the angle of the truncation facet can change more or less freely and it will oscillate in accordance with the oscillations of the growth system. However, to make qualitative predictions about a given growth process and the structural formation probabilities it is necessary to have reasonably good estimates of the parameters describing the surface energy functions. This is indicated in figure 18 where each set of simulation parameters gives different results.

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For the single slice construction, it is possible to predict the impact of the growth interface size and morphology on the relative formation rates of the ZB and WZ stacking's, given a set of simulation parameters. In figures 18(a), (b), (c) three different simulations S1, S2, S3 of the truncation dynamics show some general trends as a function of the growth interface diameter, even though huge quantitative differences are seen due to changes in the parameters determining the shape of the interface energy functions. The quantities which are plotted here are average values after reaching a quasi-steady state; see section A.5 in the appendix for examples of the truncation dynamics. Parameters used for the simulations are listed in section A.3 in the appendix. If truncation edge nucleation dominates at the topfacet, WZ would be favoured at small diameters and ZB at larger diameters (figure 18(a)) if considering the interface energy function as plotted in. In figure 18(b) it is seen that the truncation facet seems to be smaller with increasing size of the liquid–solid growth region, even though it does not necessarily have a monotonic dependence due to the anisotropic interface energy. In figure 18(c) it is shown that the axial growth rate is strongly dependent on the size of the growth region. However, the actual dependence is not simple depends on the simulation parameters used, where the relative liquid size does also play an important role on the truncation angle as seen in figure 18(d). Thus, because values for the surface energy function, the various energy barriers and kinetic reaction constants are not well known, the modelling of NW growth is still far from being a supplement to NW growth experiments.

For the condensed adatom regime, it can be shown (using mass conservation) that the general equation for steady-state adatom collection can be written in a relatively compact form,

$$\begin{eqnarray} &&\fl \Delta \Gamma_{{\rm al},i} =\frac{2\pi}{l_{j}}\sum\limits_{r'=\frac{d_{{\rm NW}}}{2}}^\infty {r'(\Delta \Gamma_{({\rm vb})a_{r'} ,i} -\Delta \Gamma_{a_{r'} {\rm s},i} )}\nonumber\\ &&+\,\sum\limits_{l=0}^{\frac{L_{NW}}{h_{ML}}} {(\Delta \Gamma_{({\rm vb})a_{l} ,i} -\Delta \Gamma_{a_{l} {\rm s},i} )} , \end{eqnarray} \tag{ 29 }$$

where the first summation accounts for the net transition flux from the substrate to the sidefacets and the second summation for the net generation of adatoms from the beam and the vapour along the NW sidewalls. a_r'(l) is the adatom site at r' (or l) along the substrate (or NW) surface as measured from the NW foot. l_j is the distance between the two adjacent adatom sites. The substrate diffusion is assumed to be isotropic, which is a reasonable for growths carried out on substrates with a (1 1 1) orientation or substrates covered with an amorphous oxide layer. A simple approach is to assume that there exist certain effective collection areas characterized by corresponding effective diffusion lengths [56]. Even though this approach is not very accurate for modelling the growth dynamics, it is instructive and intuitive. To get a more intuitive feeling of the adatom kinetics in terms of a diffusion length in the transition state approach, adatom migration on a large homogeneous planar interface serves as a good example. Then all parameters are translation invariant and there is no net diffusion. Since in this case we do not have to distinguish between the adatoms as all states are independent of position in the continuum approach we will just label all adatoms with an 'a'. There are three main transition paths for an adatom; surface diffusion (aa), desorption (av) and incorporation (as). The as mechanism can be further divided into two types of incorporation mechanisms: incorporation at a favourable site (such as a kink) leading to radial growth, or by interdiffusion which can take place at all sites. Incorporation by interdiffusion is only relevant for impurities such as dopants since exchange of group III and V element will not have a net effect on the adatom state, and will therefore be neglected here. In such conditions, an adatom diffusion length is a well-defined quantity. The mean length displacement (i.e. the mean distance between the location of 'birth' and 'death' events, where 'death' is determined by either an 'as' or 'av' transition) is $\lambda_{j,i} =\sqrt {D_{j,i} \tau_{j,i}}$, where $D_{j,i} =Z_{{\rm aa},i} \upsilon_{{\rm a},i} l_{j}^{2}(1-\bar{{\rho}}_{j,i} )\exp (-((\delta g_{{\rm aa},i}^{{\rm TS,ERS}}-\delta \mu_{{\rm a-ERS},i})/{k_{{\rm B}} T}))$, is the mean adatom diffusivity, and $\tau _{j,i} =(\tau_{j,i,{\rm as}}^{-1} +\tau_{j,i,{\rm av}}^{-1} )^{-1}$ is the average adatom lifetime. Here l_j is the distance between the two adjacent adatom sites along the lowest energy direction(s), with activation free energy $\delta g_{{\rm aa},i}^{{\rm TS}}$. For simplicity, higher energy directions are ignored. If an adatom occupies a given site, it is impossible for another adatom to jump into the same site. Thus, the concentration of free sites, $1-\bar{{\rho}}_{j,i}$, is included in the diffusivity, $\bar{{\rho}}_{j,i}$ being the normalized adatom density. The lifetimes ended by an 'as' or 'av' state transition are inversely proportional to the respective transition rates,

$$\begin{eqnarray*}&&\tau_{j,i,{\rm as}} =\frac{1}{\bar{{Z}}_{{\rm as},i} \bar{{c}}_{{\rm inc},i} \nu_{{\rm a},i}}\exp \left( {\frac{\delta g_{{\rm as},i}^{{\rm TS,ERS}} -\delta \mu_{{\rm a-ERS},i}}{k_{{\rm B}} T}} \right) \end{eqnarray*}$$

and

$$\begin{eqnarray*}&&\tau_{j,i,{\rm av}} =\frac{1}{\bar{{Z}}_{{\rm av},i} \nu_{{\rm a},i}}\exp \left( {\frac{\delta g_{{\rm av},i}^{{\rm TS,ERS}} -\delta \mu_{{\rm a-ERS},i}}{k_{{\rm B}} T}} \right), \end{eqnarray*}$$

respectively. $\bar{{c}}_{{\rm inc},i}$ is the normalized density of probable incorporation sites (kinks or possibly steps at high adatom densities and/or low temperatures). This important factor illustrates a major difference between the av and as transitions, namely that desorption can take place everywhere, which is not the case for incorporation. τ_j,i,as is conditioned by the incorporation of both a group III and a group V element because of the fixed 1 : 1 stoichiometry of the III–V solid. For the modelling, τ_j,i,as is limited by incorporation of an adatom at a kink site, which means that the solid chemical potential equals the ERS potential, and possible sidefacet nucleation events will not be considered. For desorption of adatoms, the intrinsic activation barrier⁸ $\delta g_{{\rm av},i}^{{\rm TS,ERS}}$ is independent of the other components.

The general equation for the adatom diffusion length at a given point at a homogeneous interface is therefore:

$$\begin{eqnarray} &&\fl\lambda_{j,i} =\left[\bar{{Z}}_{{\rm aa},i} l_{{\rm a},i}^{2} (1-\bar{{\rho}}_{j,i} )\exp \left( {-\frac{\delta g_{{\rm aa},i}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right)\right.\nonumber\\ &&\quad\times\left( \bar{{Z}}_{{\rm as},i} \bar{{c}}_{{\rm inc},i} \exp \left( {-\frac{\delta g_{{\rm as},i}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right)\right.\nonumber\\ &&\quad\left.\left.+\,\bar{{Z}}_{{\rm av},i} \exp \left( {-\frac{\delta g_{{\rm av},i}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right) \right)^{-1}\right]^{1/2}, \end{eqnarray} \tag{ 30 }$$

which is apparently independent of chemical potential and vibration frequency of the adatoms. However the number of incorporation sites $\bar{{c}}_{{\rm inc},i}$ depends on the local adatom densities of both components and on the orientation of the local facet, and is therefore also dependent on the chemical potential of the local adatom state, see equation (4).

Calculating the adatom density distribution using the general adatom diffusion equation, equation (29), in terms of growth conditions has proved to be difficult. Here we will show a simplified approach by using the transition state fluxes in a classical Fickian diffusion scheme to find the adatom density distribution in terms of the basic growth parameters. As in previous studies [83, 17] we only distinguish between two types of facets, the NW sidefacets (NW) and a planar substrate facet (sub). If we for simplicity assume that; $1-\bar{{\rho}}_{j,i} \approx 1$, and that $\bar{{c}}_{{\rm inc},i}$ is a constant along the length, meaning that the diffusion length only varies with time and does not vary along a given facet. Thus two coupled diffusion equations,

$$\begin{eqnarray*}&&\fl D_{{\rm NW},i} \frac{{\rm d}^{2}}{{\rm d}z^{2}}\rho_{{\rm NW},i} (z)=D_{{\rm NW},i} \frac{\rho_{{\rm NW},i} (z)}{\lambda_{{\rm NW},i}^{2}}-f_{{\rm NW},i,\bot}\\ &&-\,\Gamma_{{\rm sa(NW),}i} -\Gamma_{{\rm va}({\rm NW}),i} \end{eqnarray*}$$

and

$$\begin{eqnarray*}&&\fl D_{{\rm sub},i} \frac{1}{r}\frac{{\rm d}}{{\rm d}r}\left( {r\frac{{\rm d}}{{\rm d}r}\rho_{{\rm sub},i} (r)} \right)=D_{{\rm sub},i} \frac{\rho_{{\rm sub},i} (r)}{\lambda_{{\rm sub},i}^{2}}-f_{{\rm sub},i,\bot}\\ &&-\,\Gamma_{{\rm sa(sub}),i} -\Gamma_{{\rm va(NW}),i} \end{eqnarray*}$$

need to be solved. Here the diffusivity will be assumed uniform, $D_{j,i} =Z_{{\rm aa},i}^{\prime} \upsilon_{{\rm a},i} l_{j}^{2}\exp ( {-({\delta h_{{\rm aa},i}^{{\rm TS,ERS}}}/{k_{{\rm B}} T})} )$. If shadowing effects and influence from other NWs on the substrate are ignored we can assume that dρ_sub,i/dr|_r → ∞ = 0. The average incoming beam fluxes are given as f_NW,i,⊥ = f_i sin(φ_i)/π and f_sub,i,⊥ = f_i cos(φ_i), where φ_i is the angle of the incoming beam of group i with respect to the substrate normal. 1/π is the fraction of the NW facets which is exposed to the beam which is perfectly consistent with the transition state approach where transitions are independent of the state they are moving into. Solutions are then of the form,

$$\begin{eqnarray} &&\fl \rho_{{\rm NW},i} (z)=C_{1} \exp \left( {\frac{z}{\lambda_{{\rm NW},i}}} \right)+C_{2} \exp \left( {-\frac{z}{\lambda_{{\rm NW},i}}} \right)\nonumber\\ &&+\,\frac{\lambda_{{\rm NW},i}^{2} \left( {\frac{f_{i} \sin (\varphi _{i} )}{\pi}+\Gamma_{{\rm sa},i} +\sum_n {\Gamma_{{\rm va},i_{n}}}} \right)}{D_{{\rm NW},i}}, \end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray} &&\fl \rho_{{\rm sub},i} (z)=C_{3} K_{0,\frac{r}{\lambda_{{\rm NW},i}}} +\frac{\lambda_{{\rm sub},i}^{2} \left(\!{f_{i} \cos (\varphi_{i} )\!+\!\Gamma _{{\rm sa},i} \!+\!\sum_n {\Gamma_{{\rm va},i_{n}}}}\!\right)}{D_{{\rm sub},i}},\nonumber\\ \end{eqnarray} \tag{ 32 }$$

where K_h,x is the modified Bessel function of order h evaluated at x. To solve for the constants (C_i) we need three boundary conditions;

$$\begin{equation} D_{{\rm NW},i} \left. {\frac{{\rm d}\rho_{{\rm NW},i}}{{\rm d}z}} \right|_{z=L_{{\rm NW}}} =\Delta \Gamma_{{\rm al},i}, \end{equation} \tag{ 33 }$$

$$\begin{equation} D_{{\rm NW},i} \left. {\frac{{\rm d}\rho_{{\rm NW},i}}{{\rm d}z}} \right|_{z=0} \left. {=-D_{{\rm sub},i} \frac{{\rm d}\rho_{{\rm sub},i}}{{\rm d}r}} \right|_{r=\frac{d_{{\rm NW}}}{2}} , \end{equation} \tag{ 34 }$$

$$\begin{equation} \rho_{{\rm NW},i} (z=0)=\rho_{{\rm sub},i} \left( {r=\frac{d_{{\rm NW}} (z=0)}{2}} \right). \end{equation} \tag{ 35 }$$

Equation (33) assumes quasi-steady-state growth, combining the adatom to adatom state transition flux at z = L_NW with the net adatom to liquid state transition flux, which is driven primarily by the thermodynamic driving force. Because the ΔΓ_al,i depends on the adatom density, ρ_NW,i(L_NW, δμ_a−ERS,i), it needs to be isolated in equation (31) before it is put into equation (4). Using equation (7) for Γ_sa,i and Γ_va,i, with c_v,i = p_i/RT (the ideal gas law), δμ_a−ERS,i (which depends on the adatom densities) is solved numerically at every time step and before being put back into ΔΓ_al,i. Equation (34) combines the adatom fluxes at the NW root, whereas equation (35) assumes a continuous adatom density function across the substrate–nanowire interface, see [86, 87]. Equation (35) requires that the transition state barriers across the NW–substrate interface are symmetric.

Solving the coupled adatom diffusion equations for diffusion along the NW facets and on an isotropic substrate with the boundary conditions, equation (33)–(35), leads to the following expression for the adatom density on the NW sidewall,

$$\begin{eqnarray} &&\fl\rho_{{\rm NW,}i} (z)=\frac{\lambda_{{\rm NW},i}}{D_{{\rm NW},i}}\nonumber\\ &&\times\left[\left( {\begin{array}{@{}l@{}} -\cosh \left( {\frac{z}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} D_{{\rm NW},i} \lambda_{{\rm sub},i} \Delta \Gamma_{{\rm al},i}\\ \quad +\cosh \left( {\frac{z-L_{{\rm NW}}}{\lambda _{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} (\Gamma_{{\rm sub},i} D_{{\rm NW},i} \lambda_{{\rm sub},i}^{2}\\ \quad-\,\Gamma _{{\rm NW},i} D_{{\rm sub},i} \lambda_{{\rm NW},i}^{2} )- \sinh \left( {\frac{z}{\lambda_{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} \\ \quad\times\, D_{{\rm sub},i} \lambda_{{\rm NW},i} \Delta \Gamma_{{\rm al},i} +\cosh \left( {\frac{L_{{\rm NW}}}{\lambda _{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}}\\ \quad\times\, \Gamma_{{\rm NW},i} D_{{\rm sub},i} \lambda_{{\rm NW},i}^{2} +\sinh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} \\ \quad\times\,\Gamma_{{\rm NW},i} \lambda_{{\rm sub},i} \lambda_{{\rm NW},i} \\ \end{array}} \right)\right]\nonumber\\ &&\times\bigg[\begin{array}{@{}l@{}}\cosh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} D_{{\rm sub},i} \lambda_{{\rm NW},i} \\ \quad+\,\sinh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} D_{{\rm NW},i} \lambda_{{\rm sub},i} \end{array}\bigg]^{-1}, \end{eqnarray} \tag{ 36 }$$

where Γ_j,i = f_j,i,⊥ + Γ_va,i + Γ_sa,i is the generation flux of i adatoms of the j'th surface. As ΔΓ_al,i is a function of ρ_NW,i(z = L_NW), ρ_NW,i is isolated in equation (36), without isolating ρ_NW,i from δμ_a−ERS,i we get,

$$\begin{eqnarray} &&\fl \rho_{{\rm NW},i} (L_{{\rm NW}} )=\nonumber\\ &&\left[\left({\begin{array}{@{}l@{\!\!}} \cosh \left({\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub,}i}}} D_{{\rm NW},i} \lambda_{{\rm sub,}i} \Xi_{{\rm al},i}\\ \quad\times\,\exp \left( {-\frac{\Delta g_{{\rm al},i}^{{\rm ERS}} -\delta \mu_{{\rm l-ERS},i}}{k_{{\rm B}} T}} \right)\frac{\bar{{\rho}}_{i}^{{\rm ERS}}}{x_{i}^{{\rm ERS}}}x_{i}\\ \quad+\,K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} (\Gamma_{{\rm sub},i} D_{{\rm NW},i} \lambda_{{\rm sub},i}^{2}\\ \quad-\Gamma_{{\rm NW},i} D_{{\rm sub},i} \lambda_{{\rm NW},i}^{2} ) \\ \quad-\,\sinh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} D_{{\rm sub},i} \lambda_{{\rm NW},i} \Xi_{{\rm al},i}\\ \quad\times\,\exp \left( {-\frac{\Delta g_{{\rm al},i}^{{\rm ERS}} -\delta \mu_{{\rm l-ERS},i}}{k_{{\rm B}} T}} \right)\frac{\bar{{\rho}}_{i}^{{\rm ERS}}}{x_{i}^{{\rm ERS}}}x_{i}\\ \quad +\,\cosh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} \Gamma_{{\rm NW},i} D_{{\rm sub},i} \lambda_{{\rm NW},i}^{2}\\ \quad +\,\sinh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} \Gamma_{{\rm NW},i} \lambda_{{\rm sub},i} \lambda_{{\rm NW},i} \\ \end{array}} \right)\right] \nonumber\\ &&\times\left[\left(\begin{array}{@{}l@{}} \frac{D_{{\rm NW},i}}{\lambda_{{\rm NW},i}}\left(\cosh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} D_{{\rm sub},i} \lambda_{{\rm NW},i}\right.\\ \left.\quad+\,\sinh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub,}i}}} D_{{\rm NW},i} \lambda_{{\rm sub},i} \right)\\ \quad+\,\frac{1}{\rho}\cosh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda _{{\rm sub},i}}} D_{{\rm NW,}i} \lambda_{{\rm sub},i} \Xi_{{\rm al},i}\\ \quad\times\,\exp \left( {-\frac{\Delta g_{{\rm al},i}^{{\rm ERS}} -\delta \mu_{{\rm a-ERS},i}}{k_{{\rm B}} T}} \right) \\ -\frac{1}{\rho}\sinh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} D_{{\rm sub},i} \lambda_{{\rm NW},i} \Xi_{{\rm al},i}\\ \quad\times\exp \left( {-\frac{\Delta g_{{\rm al},i}^{{\rm ERS}} -\delta \mu_{{\rm a-ERS},i}}{k_{{\rm B}} T}} \right) \\ \end{array} \right)\right]^{-1}.\nonumber\\ \end{eqnarray} \tag{ 37 }$$

Note that Ξ_al,i is a triple line flux, i.e. a particle transfer per length per time. If we assume a barrier free al transition, the exponentials vanish in equation (37) and the only dependence on δμ_{aj−ERS, III(V)} is through the diffusion lengths. δμ_{aj−ERS,III(V)}(ρ_j,III, ρ_j,V, T) at z = L_NW can now be solved numerically at every step time in a double iterative process for both $\bar{{\rho}}_{j,{\rm III}} (z=L_{{\rm NW}} )$ and $\bar{{\rho}}_{j,{\rm V}} (z=L_{{\rm NW}} )$ choosing certain initial values, step size and acceptable error values depending on the computation time available and accuracy needed. The principle of a single numerical computation loop in a typical math language (here Mathcad) is shown below,

$$\begin{eqnarray} &&\fl\delta \mu_{a_{j} {\rm -ERS},{\rm III(V})} (\rho_{{\rm NW},i} ,T,\delta \mu_{{\rm quess,III}} ,\delta \mu_{{\rm quess,V}} ,{\rm step,error})_{i}\nonumber\\ &&=\left| \!{\overline {\, {\begin{array}{@{}l@{}} \delta \mu_{{\rm III(V})} \leftarrow \delta \mu_{{\rm quess,III(V})} \\ \delta \mu_{{\rm V(III})} \leftarrow \delta \mu_{{\rm quess,V(III})} \\ \Delta_{{\rm III(V})} \leftarrow 1\,{\rm eV} \\ \mbox{while}\vert \Delta_{{\rm III(V})} \vert >{\rm error} \\\ms \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| {{\begin{array}{@{}l@{\!}} {\begin{array}{@{}l@{}} F_{{\rm III}} \leftarrow \delta \mu_{a_{j} {\rm -ERS},{\rm III}} (\rho _{j,{\rm III}} (L_{{\rm NW}} ,\delta \mu_{{\rm III}} ),\\ \qquad \rho_{j,{\rm V}} (L_{{\rm NW}} ,\delta \mu_{{\rm V}} ),T) \\ F_{{\rm V}} \leftarrow \delta \mu_{a_{j} {\rm -ERS,V}} (\rho_{j,{\rm III}} (L_{{\rm NW}} ,\delta \mu_{{\rm III}} ),\\ \qquad\rho_{j,{\rm V}} (L_{{\rm NW}} ,\delta \mu_{{\rm V}} ),T) \\ \end{array}} \\ {\begin{array}{@{}l@{}} G_{{\rm III}} \leftarrow \delta \mu_{{\rm III}} \\ G_{{\rm V}} \leftarrow \delta \mu_{{\rm V}} \\ \end{array}} \nonumber\\ {\begin{array}{@{}l@{}} \Delta_{{\rm III}} \leftarrow F_{{\rm III}} -G_{{\rm III}} \\ \Delta_{{\rm V}} \leftarrow F_{{\rm V}} -G_{{\rm V}} \\ \end{array}} \nonumber\\ \delta \mu_{{\rm III(V})} \leftarrow \delta \mu_{{\rm III(V})} +{\rm step}\cdot {\rm sign}(\Delta_{{\rm III(V})} )\nonumber\\ \qquad\mbox{if}\left| {\Delta_{{\rm III(V)}}} \right|>{\rm error}\wedge \left| {\Delta_{{\rm V(III)}}} \right|>{\rm error} \nonumber\\ \end{array}}} \right.\nonumber\\\ms \left( {\begin{array}{@{}l@{}} \delta \mu_{{\rm III}} \\ \delta \mu_{{\rm V}} \\ \end{array}} \right) \nonumber\\ \end{array}}}} \right. \end{eqnarray} \tag{ 38 }$$

where δμ_{aj−ERS, III(V)}(ρ_j,III(L_NW, δμ_III), ρ_j,V(L_NW, δμ_V), T) is given by equation (4) with $\bar{{\rho}}_{j,{\rm III(V})} (L_{{\rm NW}} ,\delta \mu_{{\rm III(V})} )$ being the value from equation (37). The calculated value of δμ_{aNW−ERS, III(V)} is a 1 × 2 matrix with δμ_{aNW−ERS, III} and δμ_{aNW−ERS, V} on each position. Note that much computation time is saved by choosing the simplest version $\delta \mu_{a_{{\rm NW}} {\rm -ERS},i} (\rho_{{\rm NW},i} ,T)\cong k_{{\rm B}} T\ln ({\bar{{\rho}}_{{\rm NW},i} (z=L_{{\rm NW}} )}/{\bar{{\rho}}_{{\rm NW},i}^{{\rm ERS}}})$, which only requires one iteration loop for each element at each time step, which may be a rough but fairly reasonable simplification at low total fluxes and if only looking at axial growth. After this step, δμ_aNW−ERS,i(ρ_NW,i, T), is finally put into equation (37) which is again put into equations (14) and (36).

Solving for the adatom density on the isotropic substrate (which is a reasonable approximation on (1 1 1) surfaces and amorphous oxide layers), leads to the following solution,

$$\begin{eqnarray} &&\fl \rho_{{\rm sub},i} (r)=\frac{\lambda_{{\rm sub},i}}{D_{{\rm sub},i}}\nonumber\\ &&\times \left[\left(\begin{array}{@{}l@{}}\cosh \left({\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} \Gamma_{{\rm sub},i} D_{{\rm sub},i} \lambda_{{\rm sub},i} \lambda_{{\rm NW},i} \\ \quad+ \sinh \left({\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} \Gamma_{{\rm sub},i} D_{{\rm NW},i} \lambda_{{\rm sub},i}^{2}\\ \quad+\left((\Gamma_{{\rm NW},i} \lambda_{{\rm NW},i}^{2} -\Gamma_{{\rm sub},i} D_{{\rm NW},i} \lambda_{{\rm sub},i}^{2} )\right.\\ \quad\times\cosh \left( {\frac{L_{{\rm NW}}}{\lambda _{{\rm NW},i}}} \right)+\Delta \Gamma_{{\rm al},i} \lambda_{{\rm NW},i} D_{{\rm sub},i} \Big)K_{0,\frac{r}{\lambda_{{\rm sub},i}}} \end{array}\right)\right]\nonumber\\ &&\times \left[\begin{array}{@{}l@{}}\cosh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{1,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} D_{{\rm sub},i} \lambda_{{\rm NW},i}\\ \quad+\,\sinh \left( {\frac{L_{{\rm NW}}}{\lambda_{{\rm NW},i}}} \right)K_{0,\frac{d_{{\rm NW}}}{2\lambda_{{\rm sub},i}}} D_{{\rm NW},i} \lambda_{{\rm sub},i}\end{array}\right]^{-1}. \end{eqnarray} \tag{ 39 }$$

Values without references are fitting parameters or estimated values.

. 

Parameters	Values	Ref.
Ξ'al,Ga	1 × 104 nm−1 s−1	—
Ξ'al,As	1 nm−1 s−1	—
Ξls,III–V	1 × 103 nm−2 s−1	at
$\times\,\exp \left( {-\frac{\delta g_{{\rm ls,III\mbox{--}V}}^{{\rm TS,ERS}}}{k_{{\rm B}} T}} \right)$		T = 630 °C
$\bar{{Z}}_{{\rm aa,III}}^{\prime}, \bar{{Z}}_{{\rm aa,V}}^{{\rm '}}$	1 × 10−3	—
$\bar{{Z}}_{{\rm as,III}}^{\prime} \bar{{Z}}_{{\rm as,V}}^{\prime}$	1 × 10−15	—
$\bar{{Z}}_{{\rm av,III}}^{\prime}, \bar{{Z}}_{{\rm av,V}}^{\prime}$	1 × 10−2	—
$\bar{{Z}}_{{\rm av,III,sub}}^{{\rm '}}, \bar{{Z}}_{{\rm av,V,sub}}^{\prime}$	1 × 10−3	—
$\Delta h_{{\rm aa,\{1}\,{\rm \bar{{1}}}\,{\rm 0\},Ga}}^{{\rm ERS}}$	0.3 eV	[84]
$\Delta h_{{\rm aa},\{1\,\bar{{1}}\,0\},{\rm As}}^{{\rm ERS}}$	0.65 eV	[84]
$\Delta h_{{\rm aa},\{1\,1\,1\},{\rm Ga}}^{{\rm ERS}}$	0.3 eV	[84]
$\Delta h_{{\rm av,\{1}\,{\rm \bar{{1}}}\,{\rm 0\},Ga}}^{{\rm ERS}}$	2.3 eV	[84]
$\Delta h_{{\rm av},\{1\,\bar{{1}}\,0\},{\rm As}_{2}}^{{\rm ERS}}$	2 eV	[84]
$\Delta h_{{\rm av,SiO}_{x} {\rm ,As}}^{{\rm ERS}}//\Delta h_{{\rm av,SiO}_{x} {\rm ,Ga}}^{{\rm ERS}}$	1 eV//1.5 eV	—
$\Delta h_{{\rm as,SiO}_{x} ,i}^{{\rm ERS}}$	0 eV	—
$\gamma_{{\rm vs,ZB}\{1\,\bar{{1}}\,0\}}$	4.98 eV nm−2	[85]
$\gamma_{{\rm vs,WZ}\{\bar{{1}}\,2\,\bar{{1}}\,0\}}$	4.42 eV nm−2	[85]
$\gamma_{{\rm vs,ZB}\{3\,1\,1\}} /\gamma_{{\rm vs,WZ}\{\bar{{1}}\,2\,\bar{{1}}\,2\}}$	7 eV nm−2	—
$\gamma_{{\rm vs,ZB}\{2\,0\,1\}} //\gamma_{{\rm vs,WZ}\{\bar{{1}}\,2\,\bar{{1}}\,1\}}$	6 eV nm−2	—
$\gamma_{{\rm vs,ZB}\{3\,1\,2\}} /\gamma_{{\rm vs,WZ}\{\bar{{1}}\,2\,\bar{{1}}\,3\}}$	7 eV nm−2	—
γvs,ZB{111B}/γvs,WZ{1000B}	5 eV nm−2	—
γvl (liquid Ga)	4.2 eV nm−2	[77]

Following the single slice construction shown in figure 1(b), the associated trigonometric quantities are given by:

$$\begin{eqnarray*} &&\fl \vartheta (\Delta z_{+} ,\Delta z_{-} ,d_{{\rm NW}} )={\rm arctan}\left( {\frac{\Delta z_{+} -\Delta z_{-}}{d_{{\rm NW}}}} \right),\nonumber\\ &&\fl d_{{\rm eff}} (\Delta z_{+} ,\Delta z_{-} ,d_{{\rm NW}} )=\frac{d_{{\rm NW}}}{\cos (\vartheta )},\nonumber\\ &&\fl d_{{\rm top}} (\Delta z_{+} ,\Delta z_{-} ,\theta_{{\rm T}}^{+} ,\theta _{{\rm T}}^{-} ,d_{{\rm NW}} )=d_{{\rm NW}} -\Delta z_{-} \tan (\theta _{{\rm T}-} )\nonumber\\ &&-\,\Delta z_{+} \tan (\theta_{{\rm T}+} ),\nonumber\\ &&\fl R_{{\rm l}} (\Delta z_{+} ,\Delta z_{-} ,{\rm d}_{{\rm NW}} ,\xi )=\frac{d_{{\rm NW}}}{2\cos (\vartheta )\sin (\xi )},\nonumber\\ &&\fl H_{{\rm l}}^{\prime} (\Delta z_{+} ,\Delta z_{-} ,d_{{\rm NW}} ,\xi )=R_{{\rm l}} +\Xi (\xi )\sqrt {R_{{\rm l}}^{2} -\left( {\frac{d_{{\rm eff}}}{2}}\right)^{2}},\nonumber\\ \fl \Xi (\xi )={\begin{array}{@{}lll@{}} 1 & {{\rm for}} & {\xi > \pi /2,} \\ {-1} & {{\rm for}} & {\xi < \pi /2.} \\ \end{array}} \end{eqnarray*}$$

The volumes of the slice shown in figure 1(c) are given by

$$\begin{eqnarray*} &&\fl \Delta V_{{\rm s,low}{\rm \pm}} (\Delta z_{\pm} ,d_{{\rm NW}} )=\frac{d_{{\rm NW}}^{2} {\rm d}\omega^{2}}{8}(\Delta z_{{\rm ref}} -\Delta z_{\pm} ),\nonumber\\ &&\fl \Delta V_{{\rm s,top}\pm} (\Delta z_{\pm} ,\theta_{{\rm T}\pm} ,d_{{\rm NW}} )=\frac{{\rm d}\omega}{6}\Delta z_{\pm} \left[3\left( {\frac{d_{{\rm NW}}}{2}} \right)^{2}\right.\nonumber\\ &&\left.+\,(\Delta z_{\pm} \tan (\theta_{{\rm T}\pm} ))^{2}-\frac{3}{2}d_{{\rm NW}} \Delta z_{\pm} \tan (\theta_{{\rm T}\pm})\right],\nonumber\\ &&\fl \Delta V_{{\rm s,T}\pm} (\Delta z_{\pm} ,\theta_{{\rm T}\pm} ,d_{{\rm NW}} )=\frac{1}{12}\,{\rm d}\omega (\Delta z_{\pm} )^{2}\tan (\theta_{{\rm T}\pm})\nonumber\\ &&\times\,(3d_{{\rm NW}} -2\Delta z_{\pm} \tan (\theta_{{\rm T}\pm} )), \end{eqnarray*}$$

and the total liquid and solid volumes are therefore given by

$$\begin{eqnarray*} &&\fl \Delta V_{{\rm l}} (\Delta z_{+} ,\Delta z_{-} ,\theta_{{\rm T}+} ,\theta _{{\rm T}-} ,d_{{\rm NW}} ,\xi )=\frac{2d\omega}{2\pi}\nonumber\\ &&\times\,\frac{1}{3}\pi H_{{\rm l}}^{'2}(3R_{{\rm l}} -H_{{\rm l}}^{\prime} )-(\Delta V_{{\rm s,top}-} +\Delta V_{{\rm s,top}+} ),\nonumber\\ &&\fl \Delta V_{{\rm S}} (\Delta z_{+} ,\Delta z_{-} ,\theta_{{\rm T}+} ,\theta _{{\rm T}-} ,d_{{\rm NW}} )=\Delta V_{{\rm s,low}-}\nonumber\\ &&+\,\Delta V_{{\rm s,top}-} +\Delta V_{{\rm s,low}+} +\Delta V_{{\rm s,top}+}, \end{eqnarray*}$$

respectively. The corresponding number of atoms in the respective phases are given by, ΔN_l(s) = ΔV_l(s)/Ω_l(s), with Ω_l(s) being the atomic volumes. The areas of the side-, truncation-, and top-facet (see figure 2(a)) are given by

$$\begin{eqnarray*} &&\fl \Delta A_{{\rm vl}} (\Delta z_{+} ,\Delta z_{-} ,d_{{\rm NW}} ,\xi )=\frac{2d\omega}{2\pi}2\pi R_{{\rm l}} H_{{\rm l}}^{\prime} ,\nonumber\\ &&\fl \Delta A_{{\rm ls}1\,1\,1} (\Delta z_{+} ,\Delta z_{-} ,\theta_{{\rm T}+} ,\theta_{{\rm T}-} ,d_{{\rm NW}})\nonumber\\ &&=\,\frac{1}{2}d\omega [(d_{{\rm NW}} -\Delta z_{+} \tan (\theta_{{\rm T}+} ))^{2}\nonumber\\ &&\quad +\,(d_{{\rm NW}} -\Delta z_{-} \tan (\theta_{{\rm T}-} ))^{2}],\nonumber\\ &&\fl \Delta A_{{\rm ls}\_{\rm T}\pm} (\Delta z_{\pm} ,\theta_{{\rm T}\pm} ,d_{{\rm NW}})\nonumber\\ &&=\,\frac{1}{2}d\omega \frac{\Delta z_{\pm}}{\cos (\theta _{{\rm T}\pm} )}(d_{{\rm NW}} -\tan (\theta_{{\rm T}\pm} )\Delta z_{\pm} ),\nonumber\\ &&\fl \Delta A_{{\rm vs}} (\Delta z_{+} ,\Delta z_{^{-}} ,d_{{\rm NW}} )=(2\Delta z_{{\rm ref}} -\Delta z_{-} -\Delta z_{+} )\frac{d_{{\rm NW}}}{2}\,{\rm d}\omega. \end{eqnarray*}$$

Parameters for the simulations of the truncation dynamics are shown in table 4.

An additional contribution to the free energy of the system may come from an in-balance of capillary forces meeting at the TL [45]. A change in one of the involved interface orientations implies a change in such a TL energy, and in order to reach mechanical equilibrium, an increase in strain per unit TL length in the solid and/or by a local change in the vl curvature on the cost of more vl interface is induced. Both effects alter the chemical potentials and can therefore have an influence on the growth dynamics. The effect of TL forces on the growth dynamics of NWs was introduced by Schwarz and Tersoff [45], who used the tangential component of the TL force on a locally smooth solid surface to describe the TL motion, and the normal component altering the solid chemical potential at the TL. We will here take a slightly different approach and let the TL equilibration allow take part in the total free energy minimization process in all dimensions. Because changes in the liquid volume induce changes in the TL excess, we assign the TL excess to the liquid phase for convenience and add an extra term to the liquid chemical potential as

$$\begin{eqnarray*} &&\fl \delta \mu_{{\rm l},i} (x_{{\rm III}} ,x_{{\rm V}} ,T,\omega )=\mu_{{\rm l},i}^{\infty} (x_{{\rm III}} ,x_{{\rm V}} ,T)+\gamma_{{\rm vl}} \frac{\partial A_{{\rm vl}}}{\partial N_{{\rm l},i}}\nonumber\\ &&+\,\frac{{\rm d}\Upsilon (\omega )}{{\rm d}N_{{\rm l},i}}-\mu_{i}^{{\rm ERS}} . \end{eqnarray*}$$

Here dϒ(ω) is the TL excess free energy per length at ω and ϒ is the total TL excess. The effect of the TL force on crystal growth is difficult to quantify mainly because it has been difficult to measure experimentally. Nevertheless if we as in [45] define an effective width of the TL, w_eff, the TL force along the pq interfacial component can be written as,

$$\begin{eqnarray*} &&\fl f_{pq} =w_{{\rm eff}} (\gamma_{pq} +\gamma_{qw} \cos (\theta_{q} )+\gamma _{pw} \cos (\theta_{p} )+\tau \kappa_{pq} ), \end{eqnarray*}$$

where pqw is any cyclic permutation of vls. τ is the line excess free energy depending on f_pq itself and κ_pq the line curvature at ω projected on the pq component (see figure 22 for a cross sectional illustration of the TL). Assuming the TL curvature is negligible, the net force along all interfaces at the TL in equation (39) vanish at equilibrium, and the surface energies and corresponding contact angles are given by

$$\begin{eqnarray*}&&\fl \gamma_{pq} =\frac{\sin (\theta_{w,eq} )}{\sin (\theta_{p,eq} )}\gamma _{qw}\ {\rm and}\ \cos (\theta_{p,eq} )=\frac{\gamma_{qw}^{2} -\gamma_{pq}^{2} -\gamma_{pw}^{2}}{2\gamma_{pq} \gamma_{pw}}. \end{eqnarray*}$$

Away from equilibrium, we will describe the TL excess per length as,

$$\begin{equation*} {\rm d}\Upsilon (\omega )=\frac{d_{{\rm NW}} (\omega )}{2}\,{\rm d}\omega \vert f_{{\rm TL}} (\theta_{{\rm l}} (\omega ),\theta_{{\rm s}} (\omega ))\vert \end{equation*}$$

with $\vert f_{{\rm TL}} (\theta_{{\rm l}} ,\theta_{{\rm s}} )\vert =\sqrt {f_{{\rm ls}\vert \vert}^{2} +f_{{\rm ls}\bot}^{2} +\tau^{2}\kappa^{2}} =w_{{\rm eff}} [(\gamma_{{\rm ls}} +\gamma_{{\rm vs}} \cos (\theta _{{\rm s}} )+\gamma_{{\rm vl}} \cos (\theta_{{\rm l}} ))^{2}+(\gamma _{{\rm vs}} \sin (\theta_{{\rm s}} )-\gamma_{{\rm lv}} \sin (\theta_{{\rm l}} ))^{2}+\tau^{2}\kappa^{2}]^{1/2}$ being the net force per length at ω. To see the effect induced by the TL tension around the TL, evolution of the local morphology may be described by a local curvature dependent driving force as in [45].

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Table 1. Facets for ZB and WZ structure for the upper hemisphere with the lowest predicted surface energies are described with a set of angles (ω, θ) as shown in figure 6. Here 90° − θ = θ_T = 90° is defined to be the growth axis. See figure 19 for examples of truncation facets of the WZ crystal.

ZB facet normals			WZ facet normals
ω = {−30°, 90° ...}	ω = {0°, 60° ...}	ω = {30°, 150° ...}	ω = {−30°, 30° ...}	ω = {0°, 60° ...}
$\{2\,\bar{{1}}\,\bar{{1}}\}A$	$\{1\,0\,\bar{{1}}\}$	$\{1\,1\,\bar{{2}}\}B$	$\{0\,\bar{{1}}\,1\,0\}$	$\{\bar{{1}}\,2\,\bar{{1}}\,0\}$
θT = 0°	θT = 0°	θT = 0°	θT = 0°	θT = 0°
$\{3\,\bar{{1}}\,\bar{{1}}\}$	$\{3\,1\,\bar{{1}}\}$	$\{1\,1\,\bar{{1}}\}$	$\{0\,\bar{{2}}\,2\,1\}$	$\{\bar{{1}}\,2\,\bar{{1}}\,1\}$
θT = 10°	θT = 31.5°	θT = 19.5°	θT = 15.0°	θT = 17.1°
{1 0 0}	{2 1 0}	$\{2\,2\,\bar{{1}}\}$	$\{0\,\bar{{1}}\,1\,1\}$	$\{\bar{{1}}\,2\,\bar{{1}}\,2\}$
θT = 35.3°	θT = 50.8°	θT = 35.3°	θT = 28.1°	θT = 31.7°
{2 1 1}	{3 2 1}	{1 1 0}	$\{0\,\bar{{1}}\,1\,2\}$	$\{\bar{{1}}\,2\,\bar{{1}}\,3\}$
θT = 70.5°	θT = 67.8°	θT = 54.7°	θT = 46.9°	θT = 42.8°

Table 2. The coefficients of the free energy expressions of the pure elements in the case of InAs and GaAs are taken from the SGTE database [86], and are relative to the to the enthalpy of the standard element reference (HSER). The interaction parameters are taken from Ansara et al [75]. T is the corresponding Kelvin temperature and all values are in Joule per mole. The equilibrium As mole fraction x_V,eq is found from fitting liquidus values from [87], in the range T = 400–800 °C for ZB GaAs and T = 350–550 °C for ZB InAs. All equilibrium data are found from experimental measurements and are relying on thermodynamical parameters which therefore should coexist in kinetic equilibrium.

Liquid	$g_{{\rm Ga}_{1-x{\rm V}} {\rm As}_{x{\rm V}}} ({\rm J}\,{\rm mole}^{-1})$	$g_{{\rm In}_{1-x{\rm V}} {\rm As}_{x{\rm V}}} ({\rm J}\,{\rm mole}^{-1})$
g'l,III(T)	−1389.2 + 114.049T − 26.069 299T ln(T) + 1.0506 × 10−4T2 − 4.0173 × 10−8T3 − 118 332T−1	−3479.81 + 116.8358T − 27.4562T ln(T) − 5.4607 × 10−4T2 − 8.367 × 10−8T3−211 708T−1
g'l,V(T)	1.717 245 × 104 + 99.786 39T − 23.3144T ln(T) − 0.002 716 13T2 + 11 600T−1	1.717 245 × 104 + 99.786 39T − 23.3144T ln(T) − 0.002 716 13T2 + 11 600T−1
L0(T)	−25 503.6 − 4.3109 · T	−15 851 − 11.270 53 · T
L1(T)	−5174.7	−1219.5
$x_{V}^{ERS} (T)$	6.752 × 10−7 exp(0.0141 · T)/100	(9.9 × 10−4 exp(0.009 72 · T) − 0.3)/100

Table 3. Thermodynamic data taking from Ansara et al [75], where $g_{{\rm v},i_{n}}^{{\rm pure}'^{n}}(T)=\Psi_{i}^{'n} (T)+RT\ln (P)$, with P being the total vapour pressure in units of 0.1 MPa. T is the corresponding Kelvin temperature and all values are in Joule per mole.

Gas	$\Psi_{i}^{'n}(T) ({\rm J}\,{\rm mole}^{-1})$
Ga	263 612.519 + 33.487 1429T − 30.750 07T ln(T) + 0.005 377 45T2 − 5.465 34 × 10−7T3 − 150 942.65T−1
In	237 868.024 − 110.524 313T − 8.405 227T ln T − 0.015 6847T2 + 2.2119 6333 × 10−6T3 − 110 674.05T−1
As	272 027.85 − 32.253 3338T − 21.215 51T ln T + 4.389 1495 × 10−4T2 − 7.393 995 × 10−8T3 + 9666.555T−1
As2	179 351.548 + 10.551 9715T − 37.359 66T ln T − 5.618 06 × 10−5T2 − 2.130 98 × 10−8T3 + 104 881.15T−1
As4	129 731.745 + 230.754 352T − 83.044 65T ln T − 2.514 8475 × 10−5T2 + 1.044 4733 × 10−9T3 + 252 728.45T−1

To analyse the total liquid–solid dynamics we need to include the ω -dependence on all parameters in the parameter set, {X(ω)}. However this is as mentioned a very complex problem and we will here only make some rough simplifications in order to get qualitative ideas and better understanding about the three-dimensional system. As compared to the single slice construction above at least one additional parameter, namely η(ω) (see equation (18)), needs to be added to the parameter set. If the liquid is assumed to have a constant curvature an example of a choice of parameter set could be; {X(ω)} ∈ {d_NW, η, Δz₊, Δz₋, θ_T+, θ_T−, ξ}. The form of η(ω) which describes the cross sectional shape of the growth interface, is very important for the total configuration but it is also a very complex parameter to include. It has not been possible in the time of writing to find a consistent method to solve this system. In this section we will instead show some implications of typical assumptions used for modelling NW growth, which will serve as instructive and informative insight to the three-dimensional anisotropic ls system. Such as under which configurations and conditions the NW growth system will move in and out of regimes I and II. The free energy minimization process of the ls system during growth is complex mainly due to the interplay between the isotropic liquid and the anisotropic solid. If we imagine that the cusps of the gamma plot shown in figures 6(b) or (c) are very sharp and deep, then the system will choose total sidewall facetting even at the TL, and the liquid phase will 'adjust' to this as long as the system is regime II. In an ideal regime I (a single planar ls topfacet) the nucleation statistics can be treated in the framework proposed in [26], which is mainly a relevant regime during changes in growth conditions where the relative size of the liquid is decreasing. If we furthermore assume that the vl interface tension is strong, the isotropic (and assumed homogeneous) liquid prefers a constant curvature due to a strong Laplace pressure. To describe such a system we will first choose a single slice construction which is oriented in such a way that ω = 0° is in the direction of the liquid–solid displacement, Δr (see figure 23(a)). Using cylindrical coordinates, (r, ω, z), we can write two intersections between the wire and liquid as (r₋, 180°, z₋) and (r₊, 0°, z₊). For a given radius of curvature there exist two solutions, one for ξ ⩾ 90° and one for ξ < 90°, as seen in figure 23(b). Here

$$\begin{eqnarray*}&&z_{-} =-\frac{\Delta}{2}-\frac{d_{{\rm NW}0}}{2}\sqrt {\frac{4R_{{\rm l}}^{{\rm 2}} -\Delta^{2}-d_{{\rm NW}0}^{2}}{\Delta^{2}+d_{{\rm NW}0}^{2}}}, \end{eqnarray*}$$

and the two intersections are given by

$$\begin{eqnarray*} &&\fl \begin{array}{@{}l@{}} \left. {\begin{array}{@{}l@{}} (r_{-} ,180^{\circ},z_{-} )=\left(-\sqrt {R_{{\rm l}}^{2} -z_{-}^{2}} ,180^{\circ},z_{-}\right) \\ (r_{+} ,0^{\circ},z_{+} )=\left(-\sqrt {R_{{\rm l}}^{2} -z_{+}^{2}} ,0^{\circ},\Delta +z_{-}\right) \\ \end{array}} \right\}\quad {\rm for}\ \xi \ge 90, \\ \left. {\begin{array}{@{}l@{}} (r_{-} ,180^{\circ},z_{-} )'=(-r_{+} ,0^{\circ},-z_{+} ) \\ (r_{+} ,0^{\circ},z_{+} )'=(-r_{-} ,180^{\circ},-z_{-} ) \\ \end{array}} \right\}\quad {\rm for}\ \xi \le 90, \\ \end{array} \end{eqnarray*}$$

where Δ = Δz₋ − Δz₊ is difference in truncation in the two sides and d_NW0 is the diameter at ω = 0°. The z-coordinate for intersection between wire and liquid as a function of ω are then given by, $z(\omega )=-\sqrt {R_{{\rm l}}^{2}\!-\!(\cos (\omega )d_{{\rm NW}} (\omega )\!+\!\Delta r)^{2}\!-\!(\sin (\omega )d_{{\rm NW}} (\omega ))^{2}}$, where $\Delta r=\sin (\vartheta )\sqrt {R_{{\rm l}}^{2} -({d_{{\rm NW}0}^{2}}/{4\cos (\vartheta )^{2}})}$ is the displacement between of centre of the NW crystal and the liquid centre at ω = 0°. Now analysing the wetting consequences when assuming total sidefacetting (η(ω) = 0 in equation (18)) will give us some qualitative ideas about the real system and under which conditions TL nucleation can take place, as shown in C.

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In figure 24 the truncation heights are plotted for different relative sizes of the droplets and for six-fold and three-fold facetting. It is obvious that a relative large droplet will have a smaller probability of inducing positive truncations and it is also obvious that a hexagonal shape is the most convenient shape in relation with a liquid. If we allow for a three-fold facetting we need to define two lengths, r₋ and r₊, to describe the diameter, see figure 24(c). It is very likely that for real systems that the truncation is negative all the way around the TL and only becomes positive at a given location when the liquid supersaturation is high and induce either TL nucleation or move the TL into regime I. A TEM image along the $[1\,\bar{{1}}\,0]$ zone axis of a GaAs NW with a Ga droplet on top is shown in figure 21(d). The red circle on the enlarged view is a perfect circle, which fits almost perfect to the shape of the Ga droplet. Thus if the liquid curvature is constant also as a function of ω, we can say that if the NW crystal completely faceted (η₀ = 0), the truncation height would vary as a function of ω as shown in. In the other extreme if the system is completely axi-symmetric η₀ = 1 the truncation height would be independent of ω. It is important to note that for real liquid–solid growth systems using the η₀ parameter to describe the system would give a value somewhere in between 0 and 1, and the amplitude of the curves in figure 24 will be smaller. See the figure text for a discussion of different cases of total facetting and constant liquid curvature.

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In figure 25 we see the relationship between parameters; d_NW, ξ, Δz and ω under six-fold and three-fold symmetric sidewall facetting in the case of constant liquid curvature.

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