Piezoreflectance of Germanium from 1.9 to 2.8 eV

Abstract

The dependence of the imaginary part of the dielectric function, ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$, on a small-amplitude ac stress in the energy range from 1.9 to 2.8 eV for germanium is analyzed. The dependence of the differential $\ensuremath{\delta}{\ensuremath{\epsilon}}_{2}$ on polarization and stress direction is described in terms of three symmetry-adapted response functions: ${W}_{1}(\ensuremath{\omega})$, ${W}_{3}(\ensuremath{\omega})$, and ${W}_{5}(\ensuremath{\omega})$. The function ${W}_{1}(\ensuremath{\omega})$ characterizes the response to hydrostatic stress. [001] uniaxial stress generates ${W}_{1}(\ensuremath{\omega})$ and ${W}_{3}(\ensuremath{\omega})$ while [111] stress generates ${W}_{1}(\ensuremath{\omega})$ and ${W}_{5}(\ensuremath{\omega})$. The ${W}_{j}(\ensuremath{\omega})$ contain contributions ${{W}_{j}}^{\mathrm{shift}}(\ensuremath{\omega})$ from energy-band shifts, and also ${{W}_{j}}^{\mathrm{mvar}}(\ensuremath{\omega})$ due to optical-matrix-element variation. We find ${{W}_{3}}^{\mathrm{shift}}(\ensuremath{\omega})=0$, which implies that the critical points near 2.1 and 2.3 eV lie in the [111] direction ($\ensuremath{\Lambda}$) or at the $L$ point, in agreement with previous work. ${W}_{1}(\ensuremath{\omega})$ contains almost purely energy-shift effects, which leads to the derivative of the unstrained ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ function, since hydrostatic shifts lead to very little wave-function mixing. ${W}_{1}(\ensuremath{\omega})$ and ${W}_{3}(\ensuremath{\omega})$ give very distinct line shapes which are characteristic of energy-band shifts and matrix-element variation, respectively. Here ${W}_{5}(\ensuremath{\omega})$ can be represented as a linear combination of ${W}_{1}(\ensuremath{\omega})$ and ${W}_{3}(\ensuremath{\omega})$. We can account for the observed line shapes quantitatively on the assumption that ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ consists of two distinct contributions ${{\ensuremath{\epsilon}}_{2}}^{+}(\ensuremath{\omega})$ and ${{\ensuremath{\epsilon}}_{2}}^{\ensuremath{-}}(\ensuremath{\omega})$ from each spin-orbit-split band, which are identical except for an energy shift equal to the spin-orbit splitting. This analysis yields four deformation potential constants: ${{D}_{1}}^{1}$, ${{D}_{1}}^{5}$, ${{D}_{3}}^{3}$ and ${{D}_{3}}^{5}$. The quantities ${{D}_{1}}^{1}$ and ${{D}_{1}}^{5}$ agree with previous measurements by Zallen and Paul and by Gerhardt. ${{D}_{3}}^{3}$ agrees with the value determined by Pollak and Cardona using a dc stress method, while ${{D}_{3}}^{5}$ differs by a factor of 4. The origin of this discrepancy is not presently understood, but recent calculations by Saravia and Brust tend to support this value.