Band-Edge Calculations for Bismuth and Bismuth-Antimony Alloys

The Lax ellipsoidal-nonparabolic (ENP) and the Cohen nonellipsoidal-nonparabolic (NENP) dispersion relations have been used to calculate the band-edge parameters for ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ ($0\le x\le 0.18$) for a number of plausible band models. These calculations provide a self-consistent picture of the band-edge configuration of the Bi-Sb alloy system which is compatible with reported experimental data. The minima which have been observed in the thermoelectric-power-vs-temperature curves for a range of Bi-Sb alloys are accounted for by our calculations. We find that the semimetal-semiconductor transition point lies in the range $0.07<x<\mathrm{}0.10\mathrm{}$ and that the magnitude of the direct $L$-point gap $E_g$ varies with temperature at a rate $0.1<\left(\frac{\partial E_g}{\partial T}\right)<\mathrm{}0.35\mathrm{}$ meV/°K, the most likely value being 0.19 meV/°K. For semimetallic alloys with $0\le x\le 0.04$, we find that $15\le \frac{m_h^{*}}{m_e^{*}}\le 26$ for the ENP model, and $9\le \frac{m_h^{*}}{m_e^{*}}\le 15$ for the NENP model, where $m_h^{*}$ and $m_e^{*}$ are, respectively, the density-of-states effective masses of holes in the $T$-point band and electrons in the $L$-point band. Corresponding ratios calculated for $T>\mathrm{}0\mathrm{}°$K and for semiconductive Bi-Sb alloys are less accurate, but are consistent with the results for the semimetallic alloys at $T=0\mathrm{}°$K.