Band Structure of HgSe: Band Parameter Determinations from Effective-Mass Data, and Concentration Dependence and Anisotropy of Beating Effects in the Shubnikov-de Haas Oscillations

Abstract

We use an $E\ensuremath{-}\mathrm{v}\mathrm{s}\ensuremath{-}k$ relation (derived by Kane from the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ method) which includes a finite spin-orbit splitting $\ensuremath{\Delta}$ and the effects of the higher band interactions to calculate the concentration dependence of the effective mass of electrons in the ${\ensuremath{\Gamma}}_{8}$ conduction band of HgSe. The calculation employs numerical methods derived earlier by us for the analyses of Shubnikov-de Haas data relating to warping effects in the same band. We find a good fit to the observed concentration dependence of the effective mass reported by Whitsett from Shubnikov-de Haas results, using the following values of the band parameters: $P=7.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$ eV cm, ${E}_{g}=\ensuremath{-}0.22$ eV, $\ensuremath{\Delta}=0.45$ eV and ${A}^{\ensuremath{'}}=0$, ${B}^{\ensuremath{'}}=0$, $M=\ensuremath{-}5$, ${L}^{\ensuremath{'}}=\ensuremath{-}2$, and $L\ensuremath{-}M\ensuremath{-}N=4$ in units of $\frac{{\ensuremath{\hbar}}^{2}}{2{m}_{0}}$. The close agreement between the theory and experiment indicates that a finite $\ensuremath{\Delta}$ and the inclusion of higherband interactions should be considered in the evaluation of band parameters from experimental results in HgSe. Beating patterns in the Shubnikov-de Haas oscillations in $n\ensuremath{-}\mathrm{H}\mathrm{g}\mathrm{S}\mathrm{e}$ were initially reported by Whitsett. In this paper, we present the results of an experimental investigation which examines in detail the dependence of the beating patterns on electron concentration and magnetic field direction. The concentration range studied extended from 4.69 \ifmmode\times\else\texttimes\fi{} ${10}^{17}$ to 3.73 \ifmmode\times\else\texttimes\fi{} ${10}^{18}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. Magnetic field strengths up to 80 kOe were employed in the measurements. The results are similar to the earlier observations of Whitsett; namely, two nodes or minima in the oscillatory amplitude occur for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}〈111〉$, one minimum for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}〈100〉$, and no minima for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}〈110〉$. For the orientations $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}〈111〉$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}〈100〉$, the Landau-level numbers of the nodal positions are found to be independent of concentration. The variations of the nodal positions with magnetic field direction have been examined for the orientations $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\perp}\stackrel{\ensuremath{\rightarrow}}{\mathrm{I}}\ensuremath{\parallel}〈110〉$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\perp}\stackrel{\ensuremath{\rightarrow}}{\mathrm{I}}\ensuremath{\parallel}〈100〉$. Comparison of the results with published data on the III-V semiconductor GaSb indicates that the orientational effects may be a general feature of the beating in zinc-blende structures. The results are analyzed in terms of a semiclassical model of two Fermi surfaces split by inversion asymmetry, using the band parameters derived from the effective-mass analysis. This model predicts the observed concentration dependence of the beating frequency for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}\ensuremath{\parallel}〈111〉$, assuming that the two nodal points seen for this orientation are related by simple beating. The calculated splitting of the surfaces, derived from this model, is compared with the predictions of Roth's nonclassical model which involves interaction between electron spin and external magnetic fields.