Magnetopiezotransmission Studies of the Indirect Transition in Germanium

The indirect transition in germanium at $T\ensuremath{\sim}20\ifmmode^\circ\else\textdegree\fi{}$K has been examined with magnetic fields up to 90 kG, using the high-sensitivity piezotransmission technique. Samples were in the Voigt configuration in which the Poynting vector of the radiation field is perpendicular to the applied magnetic field. The spectra recorded with light polarized either parallel or perpendicular to the magnetic field applied along a [001], [111], or [110] direction exhibit a series of sharp peaks, in contrast to the corresponding series of less distinct steps which are observed in the conventional magnetoabsorption spectra for the indirect edge. At high magnetic fields the piezotransmission peaks have been resolved over a spectral range of more than 0.1 eV which extends from the zero-field edge at 0.77 eV to photon energies at which the direct transition dominates. From the energy spacings of the peaks, we have deduced the cyclotron effective mass ${m}_{c}$ for the ${L}_{1}$ conduction band and its variation as a function of the energy $\mathcal{E}$ measured from the bottom of the band. For large $\mathcal{E}$, ${m}_{c}$ is found to vary linearly with $\mathcal{E}$, in agreement with the results of the $〈k\ifmmode\cdot\else\textperiodcentered\fi{}p〉$ perturbation analysis. But the observed slope of the ${m}_{c}$-versus-$\mathcal{E}$ curve is greater than that expected from theory. Linear extrapolation of the experimental curve gives for the effective mass at the bottom of the band ${m}_{c}(0)=(0.129\ifmmode\pm\else\textpm\fi{}0.002){m}_{0}$ for H\ensuremath{\parallel}[001]. Similarly, for H\ensuremath{\parallel}[111] and [110], we obtain for the light electron mass at the bottom of the conduction band ${m}_{\mathrm{le}}(0)=(0.079\ifmmode\pm\else\textpm\fi{}0.001){m}_{0}$ and $(0.095\ifmmode\pm\else\textpm\fi{}0.001){m}_{0}$, respectively. Assuming that the ratio of the longitudinal to the transverse effective masses is $\frac{{m}_{l}(0)}{{m}_{t}(0)}\ensuremath{\simeq}20$, the above values for ${m}_{c}(0)$ and ${m}_{\mathrm{le}}(0)$ give the transverse effective mass ${m}_{t}(0)=(0.078\ifmmode\pm\else\textpm\fi{}0.002){m}_{0}, (0.079\ifmmode\pm\else\textpm\fi{}0.001){m}_{0}, \mathrm{and} (0.079\ifmmode\pm\else\textpm\fi{}0.001){m}_{0}$ for H\ensuremath{\parallel}[001], [111], and [110], respectively. ${m}_{l}$ has been estimated from the series of peaks due to the heavy electron mass. Using an average value for $\frac{{m}_{\mathrm{he}}}{{m}_{0}}=0.352$ with H\ensuremath{\parallel}[110], we find that $\frac{{m}_{l}}{{m}_{0}}=1.54\ifmmode\pm\else\textpm\fi{}0.06$.