A model of impact ionization due to the primary hole in silicon for a full band Monte Carlo simulation

The rate of impact ionization due to the primary hole in silicon is numerically derived from pseudo-wave-functions and realistic energy band structure based on a nonlocal empirical pseudopotential method including the spin-orbit interaction. The calculated impact-ionization rate SII [s−1] is well fitted to an analytical formula with a power exponent of 3.4, indicating a soft threshold of the impact ionization rate: SII [s−1]=1.14×1012 [s−1 eV−3.4]×(ε [eV]−1. 49 [eV])3.4, where ε [eV] is the energy of the primary hole relative to the valence band edge. The soft threshold originates from the complexity of the silicon band structure. The calculated impact-ionization rate shows strong anisotropy at low hole energies (ε<3 eV), while it becomes isotropic at high hole energies, indicating the isotropy of the joint density of states at high energies. Numerical calculation also makes it clear that average energies of secondary generated carriers ε̄ depend linearly on primary hole energies at the moment of their generation. The calculated average energies of secondary generated holes ε̄(hole) [eV] and electrons ε̄(electron) [eV] are well fitted to linear functions of primary hole energy ε [eV]: ε̄(hole) [eV]=3.75×10−1 ε [eV]−4.76×10−1 [eV],ε̄(electr on) [eV]=−3.14×10−1 ε [eV]−8.60×10−1 [eV]. The standard deviations of secondary generated carriers are also presented.