Piezoreflectance of Germanium from 1.9 to 2.8 eV

The dependence of the imaginary part of the dielectric function, ${\epsilon }_2\left(\omega \right)$, 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 $\delta {\epsilon }_2$ on polarization and stress direction is described in terms of three symmetry-adapted response functions: $W_1\left(\omega \right)$, $W_3\left(\omega \right)$, and $W_5\left(\omega \right)$. The function $W_1\left(\omega \right)$ characterizes the response to hydrostatic stress. [001] uniaxial stress generates $W_1\left(\omega \right)$ and $W_3\left(\omega \right)$ while [111] stress generates $W_1\left(\omega \right)$ and $W_5\left(\omega \right)$. The $W_j\left(\omega \right)$ contain contributions ${W_j}^{\mathrm{shift}}\left(\omega \right)$ from energy-band shifts, and also ${W_j}^{\mathrm{mvar}}\left(\omega \right)$ due to optical-matrix-element variation. We find ${W_3}^{\mathrm{shift}}\left(\omega \right)=0\mathrm{}$, which implies that the critical points near 2.1 and 2.3 eV lie in the [111] direction ($\Lambda$) or at the $L$ point, in agreement with previous work. $W_1\left(\omega \right)$ contains almost purely energy-shift effects, which leads to the derivative of the unstrained ${\epsilon }_2\left(\omega \right)$ function, since hydrostatic shifts lead to very little wave-function mixing. $W_1\left(\omega \right)$ and $W_3\left(\omega \right)$ give very distinct line shapes which are characteristic of energy-band shifts and matrix-element variation, respectively. Here $W_5\left(\omega \right)$ can be represented as a linear combination of $W_1\left(\omega \right)$ and $W_3\left(\omega \right)$. We can account for the observed line shapes quantitatively on the assumption that ${\epsilon }_2\left(\omega \right)$ consists of two distinct contributions ${{\epsilon }_2}^{+}\left(\omega \right)$ and ${{\epsilon }_2}^{-}\left(\omega \right)$ 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.