Bandgap Dependence and Related Features of Radiation Ionization Energies in Semiconductors

The problems dealt with concern the production of electron-hole pairs in a semiconductor exposed to high-energy radiation. The goal is to develop a simple phenomenological model capable of describing the present experimental situation from the standpoint of yield, variance, and bandgap dependence. We proceed on the premise that ε, the average amount of radiation energy consumed per pair, can be accounted for by a sum of three contributions: the intrinsic bandgap (EG), optical phonon losses r(ℏωR), and the residual kinetic energy (9/5) EG. The approach differs from prior treatments in the sense that the residual kinetic energy relates to a threshold for impact ionization taken to be 32EG in accordance with indications stemming from studies of avalanching in p-n junctions. This model is subjected to three quantitative tests: (a) Fano-factor variations are found to reflect the relative weight of phonon losses K=r(ℏωR)/EG, but residual energy fluctuations govern the statistical behavior for K2 ≲0.3. An application to Ge yields good agreement with the best measurements available (F=0.13±0.02 at 77°K). (b) The bandgap dependence of pair-creation energies conforms to the model ε= (14/5) EG+r(ℏωR) and suggests that optical phonon losses remain essentially constant 0.5≤r(ℏωR)≤1.0 eV. This would imply that the mean-free-path ratio for pair production and phonon emission (r=λ̄I/λR) is of the order of 10 or 20 for most semiconductors. (c) A detailed assessment of the situation in Si leads to the conclusion that, in this material, λ̄I is approx 400 Å. The figure accords, roughly, with inferences made from the spectral distribution of hot electrons emitted by shallow junctions and thus points to ``average'' impacts occurring at about 5 eV; by the same token, it substantiates the conception of pairs originating either through plasmon decay or in the final stages of a branching process.