Zincblende-/Diamond-Structure Nanocrystals: Analytic Surface Energies by Size, Shape, and Surface Orientation

Recently, the universal analytics for the intrinsic shape of zincblende (zb) and diamond-lattice nanocrystals (NCs) were described [AIP Adv. 6, 085306 (2016)], determining exact numbers for their internal atoms N _NC, internal bonds N _bnd, and interface bonds N _IF as a function of NC size and {hkl} interface faceting for seven regular NC shapes: {001} cubes, {111} octahedra, {110} dodecahedra, {001}-{111} pyramids, {111} tetrahedra, {111}-{001} quatrodecahedra, and {001}-{111} quadrodecahedra. Here, such descriptions are updated and extended to accurately and analytically describe the density of surface free energy of NCs on a per-interface-bond basis γ_bnd,IF. These descriptions are key to chemical NC reactivity as in catalysts, chemical reactivity as occurring in etching processes, and electronic properties such as defect formation and carrier localization. In addition to accurate quantitative results demonstrated for Si- and GaAs-NCs, the new method introduced herein yields unique insights into the surface energetics hitherto unseen. Results are most striking for NC sizes of d _NC ≤ 50 nm as opposed to macroscopic values of surface free energy densities γ and NC surface areas A, allowing to accurately determine γ_bnd,IF for all NC shapes and sizes. Using this approach, some surprising NC properties are found, like certain NCs showing γ_bnd,IF decreasing with shrinking d _NC. Similar to the above reference for structural parameters, these analytic descriptions provide accurate values of the density of surface free energy, in particular for free-standing or exposed NCs, and can be used to describe the energetics of different zb- and diamond-lattice NCs.