On the convergence of the recurrence solution of McIntyre's local and non-local avalanche triggering probability equations for SPAD compact models

McIntyre's local and non-local equations are used to calculate the avalanche triggering probability, useful in the modeling of Dark Count Rates and Probability Detection Efficiency of Single Photon Avalanche Detectors (SPAD). Although non-linear, the local equations have a closed-form solution, while the more rigorous non-local history-dependent equations can be solved only by an iterative approach. However, the convergence of the latest approach is fairly slow, making the modeling of SPAD properties relatively time consuming. An alternative and numerically efficient method is proposed to solve the local equations, avoiding the use of the closed-form solution and its time-consuming root finding step. Several approaches are then discussed to improve the convergence of the non-local history-dependent equations.