On the effect of non local impact ionization on avalanche multiplication   for uniform electric fields
  An alternative definition for the non local impact ionization coefficient is given and its relationship to the ionization pathlength probability distribution function (PDF) is determined. A model for the ionization pathlength PDF is proposed and the resultant non local ionization coefficient is derived. An expression for the avalanche multiplication is derived. Results are calculated using non local ionization coefficients for two electric field values using fits to ionization pathlength PDFs calculated using Monte Carlo techniques. These results show two features: the well known dead space effect and at the higher field value a resonance effect. The resonance effect gives rise to level sections in the multiplication curve for multiplications of 2, 4, 8 and 16 if the ionization coefficient ratio is small. 
John Stephen Marsland
physics.ins-det