A test of the lucky-drift theory of the impact ionisation coefficient using Monte Carlo simulation

Abstract

A new formulation, due to Burt (1985), of the lucky-drift theory of the impact ionisation coefficient is tested against a Monte Carlo simulation for a model semiconductor. The model is especially tailored to ensure that the electrons have an energy-independent mean free path against the emission or absorption of a phonon. Lucky-drift theory is used to derive simple analytic expressions for the steady-state drift velocity, the steady-state average energy and the impact ionisation coefficient. The lucky-drift predictions for these quantities are compared with estimates derived from Monte Carlo simulation for a wide range of values for the applied electric field. The agreement between the lucky-drift theory predictions and the Monte Carlo estimates is impressive considering the simplicity of the theory. The lucky-drift theory prediction for the impact ionisation coefficient agrees well with the Monte Carlo estimates for a range of field strengths for which the impact ionisation coefficient changes by more than three orders of magnitude. No fitted parameters are used in the lucky-drift theory. Ridley's prediction (1983) for the impact ionisation coefficient also agrees well with the authors' Monte Carlo estimates. Simple asymptotic expressions for the impact ionisation coefficient are derived for high and low fields. These asymptotic expressions are found to have surprisingly large ranges of applicability. The reasons for this are given. The relationship between the expressions due to Burt and Ridley for the impact ionisation coefficient is discussed.