A Dynamic Nonlinear Impedance Model of a Single Photon Avalanche Diode

A nonlinear LCR parallel circuit model of a single photon avalanche diode (SPAD) is derived from a Lienard-type nonlinear differential equation. The resistance and the inductance associated with avalanche multiplication (AM) are time-dependent and governed by the avalanche time constant due to the impact ionization ratio. Time dependences of current, voltage, resistance, and inductance in the model are analyzed by numerically solving the equation. During the initial generation of a Geiger-mode pulse, when the voltage reaches the breakdown voltage, the resistance diverges to limit the maximum current and the inductance reduces to give the maximum speed of the voltage variation with the avalanche time constant. In the frequency domain, avalanche impedance (AI) spectra are obtained as a ratio of voltage and current spectra calculated as Fourier transforms of the time domain signals. The AI spectra exhibit a negative-resistance and an inductance. An analytic expression for the impedance is derived and found to comprise only a quenching resistance and a stray capacitance as indicated by a constant radius of a Nyquist plot. Finally, the present model is shown to incorporate the models of impact-ionization-avalanche transit-time (IMPATT) diodes when a very small-signal limit is assumed.