Nonlinear Carrier Dynamics in a Single Photon Avalanche Diode: Stability, Bifurcation, and Quenching Condition

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Abstract
Nonlinear carrier dynamics during quenching of a single photon avalanche diode (SPAD) are investigated. A Lienard type differential equation is derived from carrier continuity equations and it is solved analytically and numerically. Universal characteristics of the quenching carrier dynamics, i.e., stability, bifurcation, and quenching conditions, are analyzed on a trace-determinant plane of the Jacobian matrix and on a phase plane as functions of the quenching resistance (QR). With a finite QR or a resistive quenching (RQ), the breakdown voltage is found to be an attractor leading to a nonquenched final state. In order to produce a successful quenching, i.e., a status identified as an unstable fixed point (FP) with a zero-carrier state, two conditions are found to be necessary: 1) bias voltage dropping below breakdown voltage to ensure carrier decrease and 2) carrier extinction (CE) in this carrier decreasing period. The two conditions together lead to a threshold of QR. On the other hand, a capacitive quenching (CQ) appearing as a special case of RQ with an infinite resistance is found to show a completely different bifurcation character. CQ is derived to be equivalent to a logistic equation giving a transcritical bifurcation at the breakdown voltage and a final state identified as a stable FP with natural CE. Finally, two time constants both governed by an excess voltage are derived. In particular, one of them, a lifetime of impact ionizations, is found to be equivalent to the ``avalanche frequency'' of an impact ionization avalanche transit-time diode (IMPATT).

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