Optimum Cross‐Sections for Wurtzite‐Structure Nanowires From Analytic Thermodynamic and Crystallographic Arguments

Crystalline nanowire (NWire) cross-section shapes are known to vary considerably in experiment; an accurate analytic evaluation of NWire cross-sections from a thermodynamic perspective is still missing. Building on previous work, analytic descriptions are given for wurtzite (w) NWire cross-section morphing to arbitrary convex hexagonal shapes in order to evaluate the crystallographic-structural stability of experimental NWire cross-sections. The maximum of the ratio of NWire-internal bonds per NWire atom $$N_{\rm {bnd}}/N_{\rm {Wire}}$$ describes the most stable crystallographic and thermodynamic shape of the NWire as far as internal forces are concerned. This maximum $$Max.(N_{\rm {bnd}}/N_{\rm {Wire}})$$ occurs when the first derivative of the above ratio $$\partial [N_{\rm {bnd}}({\mathcal {V}_{\rm opt}})/N_{\rm {Wire}}({\mathcal {V}_{\rm opt}})]/\partial {\mathcal {V}_{\rm opt}}=0$$, with $${\mathcal {V}_{\rm opt}}$$ presenting respective optimum morphing indices to obtain $$Max.(N_{\rm {bnd}}/N_{\rm {Wire}})$$. The stability evaluation is then carried out by comparing $$N_{\rm {bnd}}/N_{\rm {Wire}}$$ derived from the experimental image over the complete morphing range of the NWire cross-section represented by $${\mathcal {V}}$$. This user-friendly analytic approach allows to calculate the optimum morphing indices and thus all follow-on parameters, such as $$N_{\mathrm{Wire}}$$, $$N_{\mathrm{bnd}}$$, the number of interface bonds $$N\mathrm{_{IF}}$$ and the NWire cross-section area $$A$$. The normalization of analytic results enables a direct comparison of cross-sections with different shape, size, and even material. General results of the analytic decriptions explain the preferred growth of w-NWires with growth vector $$\mathbf {G}=[0001]$$ and six $$\lbrace 1\bar{1}0\rbrace 0$$ facets by their highest ratio $$N_{\rm {bnd}}/N_{\rm {Wire}}$$ for the same cross-section area of all three NWire types considered. Examples with experimental data demonstrate the versatility and detailed insight the stability evaluation provides to any w-lattice NWire cross-section.