Scattering by ionization and phonon emission in semiconductors

Calculations of the pair-creation energy $\ensuremath{\epsilon}$, the Fano factor $F$, and the quantum yield for semiconductors are done for the assumptions that in each scattering event all possible sets of product particles are equally probable, that the energy bands are those of free particles separated by a band gap ${E}_{g}$, and that there is a single phonon energy $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$. A new method of calculating these quantities is advanced. In it a pair-number probability distribution ${p}_{n}(E)$, the probability that a particle with energy $E$ ultimately creates $n$ pairs, is calculated recursively with increasing $E$. The first and second moments of the ${p}_{n}(E)$ distribution yield $\ensuremath{\epsilon}$, $F$, and the quantum yield as functions of $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$, ${E}_{g}$, and a parameter $A$, proportional to the ratio of the matrix elements for electron scattering by phonon emission and by ionization. We find that a single value of $A$, which fits the $\ensuremath{\epsilon}$ observed for Si, gives values for these quantities in good accord with experiments for many semiconductors. The calculated $\ensuremath{\epsilon}$ is found insensitive in many semiconductors to electron-energy loss to plasmons and to differences in the threshold energy for ionization representing real band-structure features. The assumption that all possible sets of product particles are equally probable in each scattering event leads to ultimate nonuniform population of the states with energies below the threshold for ionization, in contrast to the uniform population assumed in some earlier approaches. Results of other existing approaches in which the final-state distribution is calculated, an alternate method, were duplicated using this new method. A simple paradigm is used to illustrate these methods and assumptions.