Lucky-drift model for nonlocal impact ionisation

The probability-distribution function (PDF) for impact-ionisation path leng th is a crucial quantity for understanding and modelling the low-noise beha viour of avalanche photodiodes with short multiplication regions. In these devices the high electric fields needed to produce avalanche narrow the PDE reducing the randomness in ionisation position and hence the noise in the multiplication. A simple method is presented for calculating PDFs using the 'lucky-drift' model. The results are compared with those predicted by corr esponding Monte Carlo calculations employing a parabolic energy band deform ation-potential optical-phonon scattering and hard-threshold impact ionisat ion. The overall behaviour expected for the PDF is reproduced by the simple model. However the unphysical, catastrophic assumption made for energy-rel axing collisions in lucky drift results in an unphysical delta function in the PDF which carries a significant weight at high electric fields.