Variational Treatment of Warm Electrons in Nonpolar Crystals

Deviations from Ohm's law in nonpolar crystals are treated for weak fields by the variational method. A simple band structure is assumed. Scattering by both acoustical and optical phonons, and ionized impurities is included. It is shown that the influence of optical phonons on the field dependent mobility ($E^2$ term where $E$ is the electrostatic field) is maximum for a temperature which corresponds approximately to the optical phonon energy. The field dependent mobility is highly sensitive to ionized impurity scattering as in the case of acoustical phonons alone. Finally, the convergence of the variational method is established in limiting cases using as a representation a set of polynomials which are orthonormal with respect to the collision operator. Extensive calculations are given for electrons in germanium and comparison with experiment is discussed.