Galvanomagnetic Studies of Degenerate Gallium-Doped Germanium: Nonparabolic Energy Bands with Variable Warping

Abstract

Galvanomagnetic studies of degenerate $p$-type germanium are utilized to investigate the valence band structure at Fermi level penetrations up to 0.5 eV. Extensive resistivity and Hall measurements on a series of gallium-doped samples yield, for impurity concentrations >5\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, apparent free-hole concentrations significantly in excess of the gallium concentrations as determined directly using several independent techniques. The experimentally observed disparity between the electrical and direct determinations of the gallium content can be accounted for by a detailed consideration of the valence band structure of germanium. Lax and Mavroides' treatment of the conductivity and Hall effect for parabolic, warped energy surfaces has been extended to include nonparabolic surfaces with variable warping. Application of this extended treatment to Kane's model of the multiple valence band structure of germanium satisfactorily predicts the observed departure of the Hall coefficient factor from unity over the impurity concentration range studied. The scattering relaxation time $\ensuremath{\tau}$ is deduced from a comparison of the measured and calculated values of the conductivity. An analysis of $\ensuremath{\tau}$ on the basis of screened impurity scattering in the Born approximation yields the observed energy dependence for $E<0.29$ eV. For $E>0.29$ eV a marked decrease in $\ensuremath{\tau}$ occurs, which is attributed to interband impurity scattering involving the split-off band. It is suggested that a pronounced temperature dependence observed in the resistivity for $T>80\ifmmode^\circ\else\textdegree\fi{}$K arises from a corresponding temperature dependence of the electronic screening radius.