Galvanomagnetic Theory for Electrons in Germanium and Silicon: Magnetoresistance in the High-Field Saturation Limit

For constant scattering time τ and ellipsoidal energy surfaces, the Boltzmann transport equation reduces to a phenomenological equation of motion for electrons from which a conductivity tensor is derived. The calculations for germanium and silicon differ in the orientation of the ellipsoids. The resistivity tensor is evaluated in the saturation limit, and explicit expressions for the angular dependence of the magnetoresistance are elaborated for certain high-symmetry combinations. The theoretical findings are in qualitative agreement with experiment, thus providing confirmation of the 4- or 8-ellipsoid [111] and the 3- or 6-ellipsoid [100] models of the energy surfaces in n germanium and n silicon, respectively. Essential agreement with energy-dependent τ theory is also established.