Anisotropy of the Gray-TinΓ8+Conduction Band

The angular dependences of the Shubnikov-de Haas periods and effective masses of the ${{\Gamma }_8}^{+}$ conductionband electrons in gray tin are analyzed in terms of a mathematical model similar to that of Dresselhaus, Kip, and Kittel for the ${{\Gamma }_8}^{+}$ valence bands of Ge. The observed 5% variations in periods and masses are accounted for by $〈111〉$ protrusions on a nearly spherical Fermi surface. The degree of nonsphericity, as measured by the nonsphericity parameter $F$, is independent of the ${{\Gamma }_8}^{+}$ electron concentration. The absolute value of $F$ for gray tin agrees closely with that for the ${{\Gamma }_8}^{+}$ bands of Ge. The $A$, $B$, and $C^2$ coefficients of Dresselhaus et al. and the related matrix-element sums $F^{\prime }$, $H_1$, and $G$ are evaluated. They are generally consistent in both sign and magnitude with the Groves-Paul band-structure model. The one exception is the positive matrix-element sum involving the ${\Gamma }_{12}$ band.