Piezoreflectance of Germanium from 1.9 to 2.8 eV

The dependence of the imaginary part of the dielectric function, ε₂(ω), 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 δε₂ on polarization and stress direction is described in terms of three symmetry-adapted response functions: W₁(ω), W₃(ω), and W₅(ω). The function W₁(ω) characterizes the response to hydrostatic stress. [001] uniaxial stress generates W₁(ω) and W₃(ω) while [111] stress generates W₁(ω) and W₅(ω). The Wⱼ(ω) contain contributions Wⱼˢʰⁱᶠᵗ(ω) from energy-band shifts, and also Wⱼᵐᵛᵃʳ(ω) due to optical-matrix-element variation. We find W₃ˢʰⁱᶠᵗ(ω)=0, which implies that the critical points near 2.1 and 2.3 eV lie in the [111] direction (Λ) or at the L point, in agreement with previous work. W₁(ω) contains almost purely energy-shift effects, which leads to the derivative of the unstrained ε₂(ω) function, since hydrostatic shifts lead to very little wave-function mixing. W₁(ω) and W₃(ω) give very distinct line shapes which are characteristic of energy-band shifts and matrix-element variation, respectively. Here W₅(ω) can be represented as a linear combination of W₁(ω) and W₃(ω). We can account for the observed line shapes quantitatively on the assumption that ε₂(ω) consists of two distinct contributions ε₂⁺(ω) and ε₂^-(ω) 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: D₁¹, D₁⁵, D₃³ and D₃⁵. The quantities D₁¹ and D₁⁵ agree with previous measurements by Zallen and Paul and by Gerhardt. D₃³ agrees with the value determined by Pollak and Cardona using a dc stress method, while D₃⁵ 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.