Framework

Algra et al.24, 35 modelled the relationship between nanowire diameter and twin spacing in InP and GaP TSL nanowires by considering individual nucleation events. Relative nucleation probabilities were determined by finding the free energy difference between a facet conserving nucleus and its twin. For the point at which nucleation of either orientation is equally probable a linear relationship was again derived;

…(5-1)

with being the axial distance from the point where the nanowire cross section is hexagonal, a geometrical constant previously calculated to be 1.16,24 the planar bilayer spacing in the direction, , , and the surface energies per

unit area of a twin plane, the solid-liquid, solid-vapour and liquid-vapour interfaces respectively, the angle of the droplet with respect to a {111}B facet at a hexagonal cross section (preferred nucleation sites were analysed for this geometry by Liu et al.1053), the effective interfacial energy barrier to nucleus formation, the supersaturation in the Au seed as defined by the difference in chemical potential between III-V pairs in solution relative to those in the solid and the nanowire diameter Figure 5-15 schematically illustrates these various geometrical and physical parameters.

The length may be considered something of a balance between an increasingly unfavourable contact angle due to the distortion of the seed particle and the energetic barrier to twin formation. As the distortion of the seed particle will be an increasing function of it is perhaps intuitive that an increased resistance to this distortion, , will reduce while an increased barrier to twin formation, , will increase it. The barrier to twin formation is furthermore not only a function of but also the interfacial area between nanowire and nucleus. As will act to increase and reduce the critical nucleus size 583, 591 an increase in their ratio, , will also act to increase the barrier to twin formation reducing the expected value of .

While gives the point beyond which twin formation is energetically more favourable for an individual nucleation event, the axial distance between twin planes is ultimately the product of multiple nucleation events. Algra et al.24, 35 found a good fit to their non- linear experimental result by calculating the most probable number of uninterrupted facet conserving nucleation events ending in a twin, the total segment length being less than two times the critical distance ;

…(5-2)

…(5-3)

where and have their usual meanings and is another geometrical constant

Figure 5-15 | Geometry and physical parameters; , , and are the surface energies per unit area of a twin plane, the solid-liquid, solid-vapour and liquid-vapour interfaces respectively, the supersaturation in the Au seed as defined by the difference in chemical potential between III-V pairs in solution relative to those in the solid, the angle of the droplet with respect to a {111}B facet at a hexagonal cross section, the planar bilayer spacing in the direction,

Section 5.4 - Modelling previously calculated to be 1.98.24 In Equation 5-2 the term , which represents the energetic barrier to twin formation,35 is seen to control the deviation of the probabilistic segment length from that calculated geometrical in Equation 5-1. Factors that increase the probability of a twin event (reduce ) such as a reduction in the twin plane surface energy, , or the interfacial area between nanowire and nucleus [proportional to for a hexagonal nanowire cross section]35 will act to reduce the linearity of the relationship between TSL segment length, , and diameter, . Figure 5-16 investigates this behaviour by plotting as a function of for various different values of the pertinent physical parameters.

The values of these physical parameters and the ranges they describe were chosen here to be both physically relevant to the current work and illustrative of the relevant trends. The supersaturation of the seed particle is firstly seen to have a highly significant effect on , with higher supersaturation increasing the probability of twinning event and therefore reducing the expected length [Figure 5-16(a)]. Increasing twin plane and solid-liquid surface energies both reduce the probability of twinning and produce a more linear relationship between and [Figure 5-16(b,c)]. The observed significance of the solid-liquid surface energy further reinforces the potential importance of any liquid ordering as was discussed by Algra et al.35 Also reinforced is the importance of correctly defining the equilibrium contact angle which is currently a fitted parameter as both in situ measurement and calculation have proven difficult.243 Finally, the liquid-vapour and solid-vapour surface energies are seen to have an opposite influence on . The effect of the surface-vapour surface energy is, in particular, relatively less significant than some of the other parameters considered. With a significant barrier to twin formation, , the probability of a twin or facet changing nucleation event is reduced and the likelihood of an uninterrupted sequence of facet conserving nuclei approaching the length predicted through energetics is increased. In the case of GaAs, the twin plane formation energy has been reported1054 as 2.75x10-2 J/m2 which is higher than the corresponding values of 2.10x10-2 J/m2 and 9.00x10-3 J/m2 reported for GaP1054 and InP1054 respectively. Turning to , the dominating contribution to this term is which for Au on planar GaAs can be approximated by .35, 591 For GaAs, of a {111}B surface under relatively low V/III ratios has been calculated1048 to be 1.11 J/m2 which is again higher than the values

a) b)

c)

d)

e) f)

Figure 5-16 | Numerical study of the influence of selected physical parameters on the relationship between twin spacing and nanowire diameter. a) Supersaturation in the seed; b) twin plane energy interfacial energy; c) solid liquid interfacial energy; d) contact angle e) liquid vapour interfacial energy; f) solid vapour interfacial energy. The parameters used for the common curve shown in black are the mid-range values in each case; , ,

Section 5.4 - Modelling of 0.96 J/m2 and 0.8 J/m2 reported for GaP24 and InP35, 1055 respectively. With higher surface energies related to both twin plane formation and the generation of solid liquid interface it may be thus expected that, as observed experimentally, the relationship between twin spacing and nanowire diameter will better approximate the linear behaviour found for in GaAs relative to both GaP and InP.