A Full-Band Spherical Harmonics Expansion of the Valence Bands up to High Energies

The valence bands of silicon can be expanded exactly by spherical harmonics only up to the energy, where the bands reach the surface of the Brillouin zone (BZ). For higher energies an approximation is required. Therefore an anisotropic extension of the band structure is presented which is determined by matching the density of states (DOS) and moments of the inverse group velocity of the exact full-band (FB) structure. The Boltzmann equation (BE) based on the exact FB at low energies and the approximation at high energies is solved by the spherical harmonic expansion (SHE) method for bulk silicon including impact ionization. The results are compared to Monte Carlo (MC) data based on the exact FB structure