# Self-Consistent Advection-Diffusion Monte Carlo Simulation of Quenching in Single-Photon Avalanche Diodes

Rémi Helleboid, Kenzo Morel-Handa, Isobel Nicholson, Gabriel Mugny, Denis

Rideau, Marco G. Pala, Philippe Dollfus, Jérôme Saint-Martin

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Rémi Helleboid, Kenzo Morel-Handa, Isobel Nicholson, Gabriel Mugny, Denis Rideau, et al. Self-Consistent Advection-Diffusion Monte Carlo Simulation of Quenching in Single-Photon Avalanche Diodes. International Workshop on Computational Nanotechnology, May 2025, Salt Lake City, United States. hal-05418456

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Submitted on 16 Dec 2025

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# Self-Consistent Advection-Diffusion Monte Carlo Simulation of Quenching in Single-Photon Avalanche Diodes

R. Helleboid*, K. Morel-Handa*, I. Nicholson†, G. Mugny†, D. Rideau†

M. Pala*, P. Dollfus*, J. Saint-Martin*,

*C2N, CNRS and Univ. Paris-Saclay, France

†STMicroelectronics, France and United Kingdom

**Abstract**—This paper introduces a self-consistent Advection-Diffusion Monte Carlo (ADMC) method for transient modeling of Single-Photon Avalanche Diodes (SPADs), coupling carrier transport with electrostatics via the Poisson equation to simulate avalanche and quenching dynamics. The approach employs a Finite Element Method (FEM) framework for 3D field calculations and explores key phenomena such as the build-up field effect, quenching mechanisms, and quenching time distributions.

## I. INTRODUCTION

Single-Photon Avalanche Diodes (SPADs) are highly sensitive photodetectors capable of detecting individual photons by leveraging avalanche multiplication in a reverse-biased p-n junction. Their ability to provide precise timing resolution and single-photon sensitivity makes them indispensable in applications such as quantum computing, LiDAR, and biomedical imaging. Efficient simulation of SPAD transient behavior, including avalanche dynamics and quenching processes, is crucial for optimizing device performance, as these phenomena directly influence timing accuracy, detection efficiency, and noise characteristics [1]. Accurate modeling tools are therefore essential to advance SPAD technology and enable their integration into next-generation systems. Standard SPAD simulation tools, such as TCAD software, typically rely on drift-diffusion models to simulate carrier transport and avalanche dynamics. While these models are computationally efficient, they lack the features needed to capture the stochastic nature of the avalanche and thus quenching dynamics. To address this limitation, self-consistent Monte Carlo, based on solving the Boltzmann equation with the Monte Carlo method (MC), has been proposed to model SPAD dynamics [2]. Those approaches are computationally intensive and require significant computational resources, which limits their practicality for simulating large-scale SPAD and limits the amount of statistics that can be gathered, which is crucial for the study of quenching. In [3] an efficient MC method was proposed to simulate the transport and impact ionization of carriers in SPADs, based on the Fokker-Planck equation. The method was shown to accurately capture the avalanche multiplication process. In this work, we couple the MC method with a Finite Element Method (FEM) framework to solve the Poisson equation self-consistently, enabling the simulation of avalanche and quenching dynamics in SPADs. The proposed methodology is summarized hereafter.

## II. MODELING AND SIMULATION

The transport of carriers is modeled using an advection-diffusion MC method, where carriers are driven according to the drift velocity and diffused according to the Einstein's law. It results in a Fokker-Planck-like equation for the probability density function of the carriers that is solved using a particle MC approach with the following dynamics [3]:

$$ \begin{cases} \mathbf{X}^0 = \mathbf{x}_0 \\ \mathbf{X}^{n+1} = \mathbf{X}^n + [-\mathbf{v}(\mathbf{X}^n, t_n) + \text{div}(\mathbb{D}(\mathbf{X}^n, t_n))] dt + \sigma(\mathbf{X}^n, t_n) \mathbf{W}^{n2} \end{cases} $$

where $\mathbf{X}$ is the position of the carrier, $\mathbf{v}$ is the drift velocity, $\mathbb{D}$ is the diffusion tensor, $\sigma$ is the square root of the diffusion tensor, and $\mathbf{W}$ is a vector of independent standard normal

random variables. The impact ionization and particle generation are simulated concurrently with the carrier transport, using the method proposed in [3].

The Poisson equation:

$$ \text{div}(\varepsilon \nabla \Phi) = -q(p - n + N_D - N_A) $$

where $\varepsilon$ is the permittivity, $\Phi$ is the electrostatic potential, $q$ is the elementary charge, $p$ and $n$ are the hole and electron densities, and $N_D$ and $N_A$ are the donor and acceptor densities is solved using a FEM framework. The carrier densities are calculated from the particle positions using a point-in-cell method, and smoothed when setting the element quantities over the nodes. We use a simple external R-C quenching circuit (see Fig. 5), which enables us to compute the voltage across the SPAD directly from the SPAD current, calculated using the Ramo-Shockley theorem.

## III. RESULTS AND DISCUSSION

To illustrate the appearance of a build up field effect during the avalanche, we show in Fig. 2 an avalanche in a SPAD at three different times: when the particle is injected, after 20 ps and after 100 ps. We observe that the depletion region is flooded with carriers, which creates an important change in the density and thus on the electrostatics. This impact is shown in Fig. 3 where the carriers densities, the electrostatic potential and the electric field are plotted before and during the avalanche along the x direction. We observe an important drop in the electric field in the avalanche region, which we call the avalanche build-up field effect (ABUE). This effect strength is highly dependent on the device geometry, as illustrated in Fig. 4.

Quenching is a crucial process in SPADs, as it determines the timing resolution and the afterpulsing probability. Fig. 6 shows the quenching process in a SPAD quenched by a R-C circuit. The figure shows that in some cases, the quenching fails, leading to multiple avalanches. Fig. 7 shows the impact of the ABUE on the quenching process, it lowers the peak current and the voltage sweep, hence the time during which the voltage is below the diode breakdown voltage. This results in poor quenching efficiency. The ABUE is desactivated in the simulation by assigning a zero 'Poisson-weight' to carriers generated by impact ionization.

Finally, the quenching statistics are investigated. Fig. 8 shows 1000 quenching events. We observe that some simulations take up to 5 ns to quench, in contrast with the 200 ps standard quenching time. The detection of such events was made possible by the statistically significant amount of data provided by the MC method, itself enabled by the efficient implementation of the method.

## REFERENCES

1. Cova, S. et al., "Avalanche photodiodes and quenching circuits for single-photon detection," Applied Optics, vol. 35, no. 12, pp. 1956-1976, Apr. 1996, number: 12 Publisher: Optica Publishing Group.

2. Dollfus, P. et al., "Avalanche breakdown and quenching in Ge SPAD using 3D Monte Carlo simulation," Solid-State Electronics, vol. 194, p. 108361, 2022, publisher: Elsevier.

3. Helleboid, R. et al., "A Fokker-Planck-based Monte Carlo method for electronic transport and avalanche simulation in single-photon avalanche diodes," Journal of Physics D: Applied Physics, vol. 55, Oct. 2022.
<figure><img src="image_2.png" /><figcaption>Fig. 1. Flowchart of the self-consistent Monte Carlo method for SPAD simulation.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Avalanche in a SPAD at three different times: when the particle is injected, after 20 ps and after 100 ps. Electrons and holes are shown in blue and red, respectively.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Longitudinal cut in the center of the 3D SPAD structure showing the carrier density (top), electrostatic potential (middle), and electric field (bottom) at equilibrium and during the avalanche.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Top: current from multiple simulations of a SPAD with different widths (latter dimension y/z of the diode structure). Bottom: avalanche saturation current as a function of the SPAD width</figcaption></figure>

<figure><img src="image_6.png" /></figure>

$$V_{\text{anode}}^{i+1} = V_{\text{anode}}^{i} + \frac{\Delta t}{R_{q}C_{q}} [-R_{q}I + (V_{\text{bias}} - V_{\text{anode}})]$$

<figure><img src="image_7.png" /><figcaption>Fig. 6. Voltage and current evolution during three different types of simulation outcomes: no avalanche, single avalanche with successful quenching, and failed quenching.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Voltage and current evolution during a quenching event with and without the avalanche build-up field effect.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Voltage evolution during 1000 quenching events (top) and the corresponding quenching time distribution (bottom), extracted from the time when the voltage reaches 99% of its initial value.</figcaption></figure>