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).

Highlights

  • D EVELOPMENT of single-photon avalanche diodes (SPADs)-based CMOS image sensors (CISs) is in continual progress reaching a mega-pixel generation [1], [2]

  • In an resistive quenching (RQ) regime, the breakdown voltage is identified as a stable spiral sink

  • Two necessary conditions for a successful quenching are found as Vmin < VBD (28) and CE at a local minimum (32) giving a necessary quenching resistance (QR) condition for a successful quenching (34)

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Summary

INTRODUCTION

D EVELOPMENT of single-photon avalanche diodes (SPADs)-based CMOS image sensors (CISs) is in continual progress reaching a mega-pixel generation [1], [2]. An RQ circuit has to simultaneously fulfill two functions in tradeoff, i.e., stopping or quenching the avalanche current and recharging to an idling stage [6], [7]. To balance these two opposite effects, the value of the quenching resistance (QR) is critical. Analytical expressions for two characteristic time constants are derived; an impact ionization lifetime and a fall-time for the bias voltage to drop to the breakdown voltage The former is shown to be equivalent to the inverse of the “avalanche frequency” of an impact ionization avalanche transit-time diode (IMPATT) [14].

Model Circuit and Basic Equations
Derivation of an LNDE
Stability Analysis of the Quenching System
Numerical Analysis of Time-Dependent Behavior
Quenching Condition
Capacitive Quenching
Time Constants of Quenching
CONCLUSION
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