Number Series of Atoms, Interatomic Bonds and Interface Bonds Defining Zinc-Blende Nanocrystals as Function of Size, Shape and Surface Orientation: Analytic Tools to Interpret Solid State Spectroscopy Data

Semiconductor nanocrystals (NCs) experience stress and charge transfer by embedding materials or ligands and impurity atoms. In return, the environment of NCs experiences a NC stress response which may lead to matrix deformation and propagated strain. Up to now, there is no universal gauge to evaluate the stress impact on NCs and their response as a function of NC size $d_{\mathrm{NC}}$. I deduce geometrical number series as analytical tools to obtain the number of NC atoms $N_{\mathrm{NC}}(d_{\mathrm{NC}}[i])$, bonds between NC atoms $N\mathrm{_{bnd}}(d_{\mathrm{NC}}[i])$ and interface bonds $N\mathrm{_{IF}}(d_{\mathrm{NC}}[i])$ for seven high symmetry zinc-blende (zb) NCs with low-index faceting: {001} cubes, {111} octahedra, {110} dodecahedra, {001}-{111} pyramids, {111} tetrahedra, {111}-{001} quatrodecahedra and {001}-{111} quadrodecahedra. The fundamental insights into NC structures revealed here allow for major advancements in data interpretation and understanding of zb- and diamond-lattice based nanomaterials. The analytical number series can serve as a standard procedure for stress evaluation in solid state spectroscopy due to their deterministic nature, easy use and general applicability over a wide range of spectroscopy methods as well as NC sizes, forms and materials.