Ionization coefficients in semiconductors: A nonlocalized property

In the light of energy-conservation considerations a nonlocalized concept (NLC) has been introduced to describe the avalanche ionization effect in semiconductors. From this formulation it appears that the intrinsic electron and hole ionization coefficients cannot be directly compared with experimental data. We show the coupled relationships between these parameters and experimentally obtainable apparent ionization coefficients. The conventional definition of ionization coefficient is modified to maintain its relevance for high-electric-field situations via a pseudolocal approximation. With the use of a previous estimate of the average distance for the ionization scattering given by Okuto and Crowell, apparent ionization coefficients have been calculated for Si, Ge, GaAs and GaP. The agreement with the existing experimental results for high electric fields is especially satisfactory for Si in which an anomalously large high-field threshold energy had previously been reported. We also show that for any material at a given electric field there exists a critical multiplication factor beyond which the pseudolocal approximation does not hold. To examine the importance of this limit an exact formulation was solved for selected $p-i-n$ junctions. The results indicate the importance of boundary corrections associated with the threshold energies for ionization scattering but indicate that the pseudolocal approximation can be appropriately modified for use even at high multiplications.