Schottky-barrier electroreflectance of Ge: Nondegenerate and orbitally degenerate critical points

Abstract

Critical-point energies, broadening parameters, line-shape asymmetries, interband reduced masses, and polarization anisotropies are measured for the $E_0$, $E_0+{\Delta }_0$, $E_1$, $E_1+{\Delta }_1$, ${E_0}^{\prime }$, ${E_0}^{\prime }+{{\Delta }_0}^{\prime }$, ${E_0}^{\prime }+{{\Delta }_0}^{\prime }+{\Delta }_0$, $E_2$, ${E_1}^{\prime }$, $E_1+\Delta$, and ${E_1}^{\prime }+{{\Delta }_1}^{\prime }+{{\Delta }_1}^{\prime \prime }$ transitions in Ge using the Schottky-barrier electroreflectance (ER) method. The Keldysh-Konstantinov-Perel' (KKP) approximation for orbitally degenerate critical points is adapted to heavy holes and to Franz-Keldysh oscillations. Stark-shift effects are shown to be important only for energies within $\frac{4{\left(\hslash \Omega \right)}^3}{E_g^2}$ of $E_g$, negligible in this experiment. The KKP prediction that orbitally degenerate ER line shapes are represented as a linear superposition of nondegenerate line shapes is verified by separating explicitly the light-hole and heavy-hole contributions to the $E_0$ structure at intermediate fields. The polarization anisotropies are in qualitative agreement with KKP predictions, but the light-hole spectrum is much too large. Therefore, either the KKP matrix-element magnitudes or the calculated densities of states do not represent correctly the experimental conditions. Low-field line-shape asymmetries for $E_0$ and $E_0+{\Delta }_0$ are in excellent agreement with two-dimensional model density-of-states calculations. We observe a cusp less than 2 meV wide at threshold on spectra for which the intrinsic energy scale $\Omega \simeq 40$ meV, verifying remarkably well the prediction of a functional singularity at threshold in ER theory. The $E_1$ and $E_1+{\Delta }_1$ line shapes show polarization anisotropies of 1.32±0.02 and 1.20±0.02 at 300 and 10 K, respectively, for [110] fields, compared to a theoretical value of 4/3. Significant field-dependent polarization anisotropies are observed for the ${E_0}^{\prime }$ triplet, in qualitative agreement with the KKP approximation and in contrast to the predictions of nondegenerate theory. Heavy-electron-hole transitions appear to dominate the center structure of the ${E_0}^{\prime }$ triplet. The polarization anisotropy of 1.85 observed for the $E_2$ transition shows that the critical point responsible is a saddle point. The ${E_1}^{\prime }$ structures are resolved into three components. The larger energy separation, 266±10 meV, for the two lower components of points responsible is not the same as that for the $E_1$ transitions, which have a spin-orbit splitting of 184±2 meV.