Extended moment method for electrons in semiconductors

The semiclassical Boltzmann equation for electrons in semiconductors with the Kane dispersion law or the parabolic band approximation is considered and systems of moment equations with an arbitrary number of moments are derived. First, the paper deals with spherical harmonics in the formalism of symmetric trace-free tensors. The collision frequencies are carefully studied for the physical properties of silicon. Then, for the parabolic band approximation, the hierarchy of equations for full moments of the phase density and the corresponding closure problem is discussed. In particular, a set of 2R scalar and vectorial moments is considered. To answer the question which number R one has to chose in order to retain the physical contents of the Boltzmann equation, the moment equations are examined in the drift-diffusion limit and in an infinite crystal in a homogeneous electric field (transient and stationary cases) for increasing number of moments R. The number R must be considered to be sufficient, if its further increase does not change the result considerably and the appropriate numbers for the processes are given.