Lucky-drift model for nonlocal impact ionisation

The probability-distribution function (PDF) for impact-ionisation path length is a crucial quantity for understanding and modelling the low-noise behaviour of avalanche photodiodes with short multiplication regions. In these devices the high electric fields needed to produce avalanche narrow the PDF 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 corresponding Monte Carlo calculations employing a parabolic energy band deformation-potential optical-phonon scattering and hard-threshold impact ionisation. The overall behaviour expected for the PDF is reproduced by the simple model. However the unphysical, catastrophic assumption made for energy-relaxing collisions in lucky drift results in an unphysical delta function in the PDF which significant weight at high electric fields