Mean gain of avalanche photodiodes in a dead space model

Based on a first order expansion of the recursive equations, we derive approximate analytical expressions for the mean gain of avalanche photodiodes accounting for dead space effects. The analytical solutions are similar to the popular formula first obtained in local approximation, provided that the ionization coefficients, α and β, are replaced with suitable effective ionization coefficients depending on dead space. The approximate solutions are in good agreement with the exact numerical solutions of the recursive equations for p-i-n devices as well as for photodiodes with nonconstant electric field profile. We also show that dead space causes non negligible differences between the values of the effective ionization coefficients entering in carrier continuity equations, the carrier ionization probability per unit length and the ionization coefficients derived by experimenters from multiplication measurements