# Analytic Description of Cross Sections for Regular Forms of Zincblende- and Diamond-Lattice Nanowires

## TL;DR

This paper derives geometric formulas for zinc-blende and diamond-lattice nanowires, enabling better understanding of their structure, stress response, and impurity behavior through analytical descriptions and comparison with experimental data.

## Contribution

It introduces universal geometric number series for zinc-blende and diamond-lattice nanowires, facilitating improved data interpretation and structural analysis.

## Key findings

- Derived formulas for atom and bond counts in nanowires
- Identified impurity self-purification threshold at specific bond-to-atom ratio
- Validated models with experimental cross-section data

## Abstract

Semiconductor nanowires (NWires) experience stress and charge transfer by their environment and impurity atoms. In return, the environment of NWires experiences a NWire stress response which may lead to propagated strain and change in shape and size of NWire cross sections. We deduce geometrical number series for zinc-blende- (zb-) and diamond-lattice NWires of diameter $d_{\mathrm{Wire}}$ to obtain the number of NWire atoms $N_{\mathrm{Wire}}(d_{\mathrm{Wire}}[i])$, bonds between NWire atoms $N\mathrm{_{bnd}}(d_{\mathrm{Wire}}[i])$ and interface bonds $N\mathrm{_{IF}}(d_{\mathrm{Wire}}[i])$ for six high symmetry zb NWires with low-index faceting frequently ocurring in bottom-up and top-down approaches of NWire processing. Along with these primary parameters, we present specific length of interface facets, cross section widths and heights as well as the cross section area. The fundamental insights into NWire structures revealed here offer a universal gauge and thus enable major advancements in data interpretation and understanding of all zb- and diamond-lattice based NWires. We underpin this statement with results from the literature on cross-section images from III-V core-shell NWire growth and on Si-NWires undergoing self-limiting oxidation and etching. The massive breakdown of impurity doping due to self-purification is shown to occur for both, Si NWires and Si nanocrystals (NCs) for a ratio of $N\mathrm{_{bnd}}/N_{\mathrm{Wire}}=1.94\pm0.01$ using published experimental data.