Band-Edge Calculations for Bismuth and Bismuth-Antimony Alloys

Abstract

The Lax ellipsoidal-nonparabolic (ENP) and the Cohen nonellipsoidal-nonparabolic (NENP) dispersion relations have been used to calculate the band-edge parameters for ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$ ($0\ensuremath{\le}x\ensuremath{\le}0.18$) 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 $0.07<x<0.10$ and that the magnitude of the direct $L$-point gap ${E}_{g}$ varies with temperature at a rate $0.1<(\frac{\ensuremath{\partial}{E}_{g}}{\ensuremath{\partial}T})<0.35$ meV/\ifmmode^\circ\else\textdegree\fi{}K, the most likely value being 0.19 meV/\ifmmode^\circ\else\textdegree\fi{}K. For semimetallic alloys with $0\ensuremath{\le}x\ensuremath{\le}0.04$, we find that $15\ensuremath{\le}\frac{{m}_{h}^{*}}{{m}_{e}^{*}}\ensuremath{\le}26$ for the ENP model, and $9\ensuremath{\le}\frac{{m}_{h}^{*}}{{m}_{e}^{*}}\ensuremath{\le}15$ 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 $T>0\ifmmode^\circ\else\textdegree\fi{}$K and for semiconductive Bi-Sb alloys are less accurate, but are consistent with the results for the semimetallic alloys at $T=0\ifmmode^\circ\else\textdegree\fi{}$K.