We present an analysis of carrier dynamics of the single-photon detection process, i.e., from Geiger mode pulse generation to its quenching, in a single-photon avalanche diode (SPAD). The device is modeled by a parallel circuit of a SPAD and a capacitance representing both space charge accumulation inside the SPAD and parasitic components. The carrier dynamics inside the SPAD is described by time-dependent bipolar-coupled continuity equations (BCE). Numerical solutions of BCE show that the entire process completes within a few hundreds of picoseconds. More importantly, we find that the total amount of charges stored on the series capacitance gives rise to a voltage swing of the internal bias of SPAD twice of the excess bias voltage with respect to the breakdown voltage. This, in turn, gives a design methodology to control precisely generated charges and enables one to use SPADs as conventional photodiodes (PDs) in a four transistor pixel of a complementary metal-oxide-semiconductor (CMOS) image sensor (CIS) with short exposure time and without carrier overflow. Such operation is demonstrated by experiments with a 6 µm size 400 × 400 pixels SPAD-based CIS designed with this methodology.

We have developed a direct time-of-flight (TOF) 250 m ranging Complementary Metal Oxide Semiconductor (CMOS) image sensor (CIS) based on a 688 × 384 pixels array of vertical avalanche photodiodes (VAPD). Each pixel of the CIS comprises VAPD with a standard four transistor pixel circuit equipped with an analogue capacitor to accumulate or count avalanche pulses. High power near infrared (NIR) short (<50 ns) and repetitive (6 kHz) laser pulses are illuminated through a diffusing optics. By globally gating the VAPD, each pulse is counted in the in-pixel counter enabling extraction of sub-photon level signal. Depth map imaging with a 10 cm lateral resolution is realized from 1 m to 250 m range by synthesizing subranges images of photon counts. Advantages and limitation of an in-pixel circuit are described. The developed CIS is expected to supersede insufficient resolution of the conventional light detection and ranging (LiDAR) systems and the short range of indirect CIS TOF.

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.

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).

A comprehensive scaling law for single photon avalanche diodes (SPADs) is presented through stochastic analyses of quenching mechanisms using a Monte Carlo method. By simulating random impact ionization events for individual carriers, two distinct quenching mechanisms are identified: successful quenching (SQ) and unsuccessful quenching (UQ). SQ occurs when quenching is achieved after the initial pulse of avalanche multiplication (AM), mainly attributed to the minimum average carrier number within a multiplication region (MR). In contrast, UQ involves prolonged and repetitive pulses, caused by stochastic fluctuations around the equilibrium carrier number. This study has derived an analytical expression for the probability of quenching failure (1– $\textit{P}_\text{Q}$ ) as functions of the quenching resistance and the capacitance of the MR. This analytical expression exhibits a good agreement with the simulation results. Moreover, analytical formulas for the threshold quenching resistance and the dead time have been derived as a function of the desired $\textit{P}_\text{Q}$ value. Notably, the tradeoff relationship between the dead time and the standard deviation of the voltage swing is elucidated, leading to the scaling limitation. Additionally, avalanche triggering probability (ATP), breakdown voltage, and the average voltage swing are revealed to be scale-invariant. Based on these aforementioned observations, the comprehensive scaling law is established.

We present a long range (~250 m) time-of flight (TOF) system with suppressed (nearly-free) mutual-interference. The system is based on a $1296\times960$ pixels vertical avalanche photodiodes (VAPD) CMOS image sensor (CIS). Real-time long-range 3D-imaging with 30 fps speed (450 fps for 2D imaging) is demonstrated. Designs and operation principles of the core circuits, i.e., a photon counting circuit and a global shutter driver, are fully described. The ranging method is based on a sub-range synthesis (SRS) where a range is set by the phase of exposure pulses with respect to that of the light source of the system. The depth resolution is configurable in that the minimum sub-range width limited by the light source pulse width of 10 ns or 1.5 m can be reduced to a 10 cm by introducing an indirect-TOF operation. Furthermore, by employing a random flight timing (RFT), mutual-interference in raw signal level is suppressed by 35 dB when 2 cameras are simultaneously operated. By simulation, the present system is estimated to be tolerant up to a case of 27 cameras operation.