Stress-Modulated Magnetoreflectance for the Direct Transitions ${\Gamma }_{{25}^{\prime }}^{\frac{3}{2}}\to {\Gamma }_{2^{\prime }}$ and ${\Gamma }_{{25}^{\prime }}^{\frac{1}{2}}\to {\Gamma }_{2^{\prime }}$ in Germanium

We have measured the stress-modulated magnetoreflectance in germanium at ∼ 30°K over the energy range 0.9-1.3 eV with magnetic fields up to 85 kG applied along a [110] direction. The measurements were made in the Faraday configuration with right and left circularly polarized light. The structure in piezoreflectance observed at zero magnetic field is correlated with transitions from the $J=\frac{3}{2}$ valence-band edge (${\Gamma }_{{25}^{\prime }}^{\frac{3}{2}}={\Gamma }_8^{+}$) and the spin-orbit split-off valence-band edge (${\Gamma }_{{25}^{\prime }}^{\frac{1}{2}}={\Gamma }_7^{+}$) to the conduction-band edge (${\Gamma }_{2^{\prime }}={\Gamma }_7^{-}$) at the center of the Brillouin zone. The structure in the magnetopiezoreflectance spectra is analyzed in terms of interband transitions between Landau levels for the valence and conduction bands. Considering the split-off valence-to-conduction-band transitions, we obtain the reduced mass for these transitions, $m_r\equiv \frac{m_c{m}_{\mathrm{so}}}{(m_{\mathrm{so}}-m_c)}=(27.\mathrm{}2\pm 0.3)\times {10}^{-3\mathrm{}}m$, the sum of the $g$ factors ${g_c}^{+}{g}_{\mathrm{so}}=-13\mathrm{}\pm 3$, and the spin-orbit splitting $\Delta =0.\mathrm{}296\pm 0.002$ eV. The light- and heavy-mass valence-to-conduction-band transitions have been analyzed using the coupled-band scheme of Pidgeon and Brown in which the valence bands are treated together with the conduction band. With the valence-band parameters of Luttinger ${\gamma }_1^L=13.\mathrm{}38$, ${\gamma }_2^L=4.\mathrm{}30$, ${\gamma }_3^L=5.\mathrm{}68$, and ${\kappa }^L=3.\mathrm{}41$ as determined by Hensel and Suzuki from combined resonances in uniaxially stressed $p$-type germanium, the best fit to the experimental data for the light- and heavy-mass transitions is obtained with the following parameters: the interaction energy between the conduction and valence bands $E_p=26.\mathrm{}8\pm 0.4$ eV, the parameter $F=(-1.\mathrm{}1\pm 0.2)(\frac{{\hslash }^2}{m}$) which represents the conduction-band interaction with the other bands. These parameters give $m_c=(0.\mathrm{}0380\pm 0.0005)m$, and $g_c=-3.\mathrm{}0\pm 0.2$, which are in excellent agreement with those obtained from the conventional magnetoabsorption measurements. With these values of $m_c$ and $g_c$, we deduce from our results for the split-off transitions $m_{\mathrm{so}}=-(0.\mathrm{}095\pm 0.007)m$ and $g_{\mathrm{so}}=-10\mathrm{}\pm 3$.