Calculation of properties of the electron-hole liquid in uniaxially stressed Ge and Si

We present a detailed theoretical study of the stress dependence of properties of the electron-hole liquid, both at zero and finite temperatures, in $〈111〉$-stressed Ge and $〈100〉$-stressed Si. These properties include the ground-state equilibrium density, pair energy, electron and hole Fermi energies, sign of the electron-hole drop charge, luminescence linewidth, and liquid compressibility. The results are compared at $T=0\mathrm{}$ to the calculations of Kirczenow and Singwi and at $T\approx$2 K to the available data. We discuss the possibility of a phase transition associated with the depopulation of the upper electron valleys in Ge. We also discuss methods of extrapolating from finite to infinite stress. The importance of the nonparabolicity of the valence bands is emphasized throughout. We discuss ranges of validity for a low-temperature expansion of the free energy. Results are presented for the systematic low-temperature variation of the liquid density, Fermi energy, and chemical potential and for the critical temperature and density. These theoretical results are found to be in reasonably good agreement with available data. Finally, we discuss scaling relations for combinations of electron-hole—liquid properties.