ST SPAD Analytical Model

IOP Semiconductor Science and Technology Journal publishes a paper "An analytical solution for McIntyre's model of avalanche triggering probability for SPAD compact modeling and performance exploration" by Tulio Chaves de Albuquerque, Dylan Issartel, Shaochen Gao, Younes Benhammou, Dominique Golanski, Raphaël Clerc, and Francis Calmon from ST, Université Claude Bernard Lyon, and Université Jean Monnet de Saint-Etienne. 

"The modeling of two main figures of merit of SPAD, namely the Dark Count Rate (DCR) and the Photon Detection Probability (PDP) requires to calculate the avalanche triggering probability, usually obtained by solving numerically two transcendental equations (McIntyre model) as post processing of TCAD simulations. This paper proposes an analytical alternative to this approach, exploiting an approximation of the impact ionization rates, in principle valid only at high field conditions, but extended to all field by a simple fitting procedure. The proposed approximated/analytical avalanche triggering probability calculation can be efficient and relevant for SPAD compact modeling compatible with spice-like simulator. As illustration, a full analytical calculation for DCR based on both avalanche triggering probability and generation terms for a P+N abrupt diode junction is presented."

It turns out that graded junctions can achieve higher PDP than abrupt ones, and electrons generally have 2x higher probability to trigger avalanche than holes: