Lucky drift model for non-local impact ionisation incorporating a soft threshold energy

The lucky drift model of impact ionisation is extended to consider non-local effects. Analytical expressions are developed for the ionisation pathlength probability distribution function (PDF) and good agreement is found with a numerical method based on lucky drift. The lucky drift expressions agree reasonably well with ionisation pathlength PDFs calculated using the Monte Carlo technique at moderately high electric fields (3×10⁷ V m^-1) but not at very high fields (10⁸ V m^-1) and a reason for this is proposed. Expressions for the non-local ionisation coefficient are found based on the lucky drift model and the expressions are evaluated numerically. Lucky drift is also used to find the probability of injection across a potential barrier and comparison with measurements shows good agreement except at very high electric field values. Reasons for the discrepancy at high fields are proposed.