Intrinsic Optimum Thermodynamic Shapes of Zincblende- and Diamond-Structure Nanowire Cross-Sections

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 zincblende (zb) 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 Nbnd/NWire$N_{\rm {bnd}}/N_{\rm {Wire}}$ describes the most stable crystallographic and thermodynamic shape of the NWire as far as internal forces are concerned. This maximum Max.(Nbnd/NWire)$Max.(N_{\rm {bnd}}/N_{\rm {Wire}})$ occurs when the first derivative of the above ratio partial derivative[Nbnd(Vopt)/NWire(Vopt)]/partial derivative Vopt=0$\partial [N_{\rm {bnd}}({\mathcal {V}_{\rm opt}})/N_{\rm {Wire}}({\mathcal {V}_{\rm opt}})]/\partial {\mathcal {V}_{\rm opt}}=0$, with Vopt${\mathcal {V}_{\rm opt}}$ presenting respective optimum morphing indices to obtain Max.(Nbnd/NWire)$Max.(N_{\rm {bnd}}/N_{\rm {Wire}})$. The stability evaluation is then carried out by comparing Nbnd/NWire$N_{\rm {bnd}}/N_{\rm {Wire}}$ derived from the experimental image over the complete morphing range of the NWire cross-section represented by V${\mathcal {V}}$. This user-friendly analytic approach allows to calculate the optimum morphing indices and thus all follow-on parameters, such as NWire$N_{\mathrm{Wire}}$, Nbnd$N_{\mathrm{bnd}}$, the number of interface bonds NIF$N\mathrm{_{IF}}$ and the NWire cross-section area A$A$. Three examples with experimental data demonstrate the versatility and detailed insight the stability evaluation provides to any zb- and diamond-lattice NWire cross-section.Using the analytic cross-section morphing of zb-nanowires to arbitrary shapes (K & ouml;nig & Smith, Acta Cryst. B 2022, 78 643), the cross-section shape with the maximum number of nanowire-internal bonds per nanowire atom is derived, representing the most stable intrinsic crystalline form (structural-thermodynamic optimum). The associated parameter space serves as structural quality gauge, with said maximum at its apex, for evaluating the structural stability and any structure-related properties of experimental nanowire cross-sections as demonstrated by examples. image