Normalized theory of impact ionization and velocity saturation in nonpolar semiconductors via a Markov chain approach

This paper presents an investigation of the velocity, energy, and impact ionization distributions in nonpolar semiconductors at very high fields. The treatment uses a finite Markov chain formulation. When optical phonon collisions and impact ionization are the major scattering mechanisms in the semiconductor, a transition matrix which characterizes the transition probabilities between virtual states defined by small discrete energy intervals can be easily computed. The resulting matrix provides the means not only to study the impact ionization phenomenon but also the steady state transport velocity and energy distribution of the charge carriers at high electrical fields and a given lattice temperature. In addition, the effects on the transport properties due to either an abrupt infinite (AI) or a finite energy dependent (FED) ionization cross-section above the ionization threshold energy are examined. The calculated avalanche transport velocity shows excellent agreement with the experimental data in Si obtained by Duh and Moll. The resulting calculations when extrapolated to a lower field also agree favorably with existing saturation drift velocity data in n and p type Si and p type Ge. The energy distribution is shown to be strongly affected by the choice of the model for the energy dependence of the ionization cross-section. One of the main applications of the results is to assist investigation of the non-localized nature of electron and hole avalanche ionization coefficients previously noted by Okuto and Crowell (O-C). The present results for this spatial distribution can replace O-C's intuitively chosen exponential approximation. The spatial ionization distribution generated by the present calculation is essentially exponential with a threshold energy dark space. This result provides a useful kernel for a more precise formulation in studies that relate impact ionization coefficients to charge multiplication data. The normalized ionization coefficients obtained from the AI model are very similar to Baraff's calculation as are the FED model results after appropriate normalization. Simple analytical expressions with meaningful asymptotic results for the average ionization energy and the ionization coefficient are also derived from the present data. These results are applicable for a range of different energy dependence of the ionization cross section provided that the average energy for pair production is used as the effective threshold parameter.