# Quenching Statistics of Silicon Single Photon Avalanche Diodes

Thibauld Cazimajou, Marco G. Pala, Jerome Saint-Martin, Remi Helleboid, Jeremy Grebot, Denis Rideau, Philippe Dollfus

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Thibauld Cazimajou, Marco G. Pala, Jerome Saint-Martin, Remi Helleboid, Jeremy Grebot, et al. Quenching Statistics of Silicon Single Photon Avalanche Diodes. IEEE Journal of the Electron Devices Society, 2021, pp.1-1. 10.1109/JEDS.2021.3127013. [hal-03439663](https://hal.science/hal-03439663)

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# Quenching Statistics of Silicon Single Photon Avalanche Diodes

Thibauld Cazimajou, Marco Pala, Member, IEEE, Jérôme Saint-Martin, Rémi Helleboid, Jérémy Grebot, Denis Rideau, Philippe Dollfus, Member, IEEE

**Abstract**—The statistical behavior of silicon-based single-photon-avalanche-diodes (SPADs) is investigated by using self-consistent 3-D Monte Carlo simulations. The coupling of Poisson and Boltzmann transport equations allows us to go beyond the analysis of avalanche breakdown and its timing and to extend the investigation to the quenching of the photodetector circuit. We find out that the quenching of SPADs is probabilistic and strongly depends on the surrounding circuit, in particular on the so-called quenching resistance. Independently of the SPAD deadtime, it appears that the extinction time needed to suppress any avalanche event may vary over a very large range.

**Index Terms**—Avalanche breakdown, Monte Carlo methods, Avalanche photodiodes

## I. Introduction

Single Photon Avalanche Diodes are fast and sensible photodetectors that can be used for a wide range of applications, including quantum computing [1], quantum cryptography [2], 3D laser detection [3] and imaging, fluorescence lifetime imaging [4]. A SPAD is essentially a reverse biased p-n or p-i-n junction, with an applied voltage $V_D$ higher than the breakdown voltage $V_B$. Under such conditions, an electron-hole (e-h) pair generated by photon absorption can be accelerated by the electric field, triggering impact ionization and the generation of new e-h pairs. This avalanche phenomenon leads to a strong increase of the diode current, which makes possible the detection of photon absorption at the circuit level.

By using particle Monte Carlo (MC) method for solving the Boltzmann transport equation (BTE), different aspects of SPAD operation have been studied using either simplified [5,6] or full-band MC approach [7]. These works have allowed the detailed statistical analysis of avalanche breakdown probability, excess noise factor, time to avalanche (delay between photon absorption and onset of measurable current), jitter (due to statistical spreading of transit times and avalanche build up) [7].

Another important stage of operation is the quenching of SPAD, needed to detect a new photon. It is possible to design a passive quenching circuit with a simple RC circuit connected in

series with the SPAD [8]. To accurately describe this quenching stage within numerical simulation, it is obviously necessary to consider the self-consistent time evolution of current, bias voltage, carrier density and electric field. To the best of our knowledge, the full SPAD operation in the presence of quenching circuit has mainly been discussed so far on the basis of compact modeling [9] in which the quench is considered as successful if the current becomes smaller than a threshold current [10]. Computationally efficient Verilog-A model and TCAD mixed-mode simulation can be used too [11]. However, these approaches suffer from strong approximations and require extensive calibration. In this paper we go further to consider the intrinsic stochasticity of quenching that has received low attention in the literature [10,12] and has never been addressed by means of self-consistent MC simulation within the mixed-mode approach. At the cost of high computational time, this approach includes all stochastic features of carrier transport, allowing us to investigate the statistical aspects of SPAD quenching, like the quenching probability and the distribution of quench duration.

In this letter, a typical silicon SPAD including passive quenching circuit is investigated on the basis of 3-D particle MC simulation self-consistently coupled to the Poisson's equation.

## II. MODEL AND SIMULATED DEVICE

The simulated device consists in a silicon PN junction with length $L = 1300$ nm and square cross section of width $W = 300$ nm. The device is connected in series with a passive quenching circuit, here a RC circuit. The diode has a realistic doping profile with doping concentration of $6 \times 10^{17}$ cm$^{-3}$ and $6.35 \times 10^{17}$ cm$^{-3}$ for the P- and N-type region, respectively, with a quasi-linear transition region of about 900 nm long. A voltage $V_{bias}$ is applied between the terminals of the circuit composed by the SPAD and quenching circuit in series.

The 3-D particle Monte Carlo code used here has been developed to investigate a large range of devices, including the effect of random distribution of discrete doping impurities [13]. To demonstrate the stochastic aspects of quenching, analytic bands have been considered here. Full band simulations could be used to improve the physical description of hot carrier

Manuscript received xxxxxx, 2020; revised xxxx xx, xxxx. Date of publication xxxxxx xx, xxxx; date of current version xxxxxx xx, xxxx. This work was supported in part by the French Industry Ministry (Nano2022 project under IPCEI program) and by the French ANR with the project GeSPAD (ANR-20-CE24-0004).

T. Cazimajou, P. Dollfus, M. Pala, J. Saint-Martin are with Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies,

F-91120 Palaiseau, France (e-mail: thibauld.cazimajou@universite-paris-saclay.fr).

D. Rideau, R. Helleboid and J. Grebot are with STMicroelectronics, F-38920 Crolles, France (e-mail: denis.rideau@st.com).
effects. In this model the conduction band is made of six Δ valleys [14] and four L valleys while the heavy and light hole bands are spherical and non-parabolic, with room-temperature effective masses determined in [15] (see supplementary materials for more details on band structure and scattering parameters). For the SPAD investigation, it is of prime importance to implement a correct description of impact ionization processes for both electrons and holes. For the band structure used here, we implemented Keldysh-like formulations of ionization rates [16] with parameters adjusted, including the power exponent [17], to fit well the experimental field dependence of ionization coefficients [18,19]. The impact ionization rates implemented here are:

$$ \frac{1}{\tau_{ii}}[s^{-1}] = 1.2 \times 10^{13} \left( \frac{E - E_G}{E_G} \right)^{2.5} \quad (1) $$

$$ \frac{1}{\tau_{ii}}[s^{-1}] = 1.0 \times 10^{9} \left( \frac{E - E_G}{E_G} \right)^{2.5} + 2.8 \times 10^{13} \left( \frac{E - 2E_G}{2E_G} \right)^2 \quad (2) $$

for electrons and holes, respectively, where $E_G = 1.12$ eV is the bandgap in Si. The resulting ionization coefficients are in good agreement with experimental data, as shown in Fig. 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Ionization coefficient in Si as a function of inverse electric field. Comparison between experimental data [18,19] and simulation.</figcaption></figure>

For the device simulation, the MC algorithm is coupled with a finite-element Poisson solver. The connection of the SPAD with the passive ($R_Q, C_Q$) quenching circuit is made in a mixed-mode approach, as follows. At time step $i + 1$ the voltage $V_D^{i+1}$ is updated with respect to $V_D^i$ according to

$$ V_D^{i+1} = V_D^i + \frac{\Delta t}{R_Q C_Q} [-R_Q I^i + (V_{bias} - V_D^i)] \quad (3) $$

where $I^i$, the current in the device at time step $i$, is calculated with the Ramo-Shockley theorem [20]. The initial single photon absorption is simulated by generating the first electron-hole (e-h) pair with energy and momentum randomly selected assuming thermal equilibrium distribution.

### III. RESULTS AND DISCUSSION

The internal capacitance $C_{SPAD} = 20.8$ aF and resistance $R_{SPAD} = 29.8$ kΩ are deduced from the I-V and Q-V characteristics (Figure 2), together with the breakdown voltage $V_B = 15.28$ V. These values of $C_{SPAD}$ and $R_{SPAD}$ are consistent with the limited cross section area of the simulated diode, i.e. smaller and higher, respectively, than the values commonly

measured in real devices. Accordingly, for the quenching circuit, we have chosen values of $R_Q$ and $C_Q$ higher and smaller, respectively, than that used in real systems. Throughout the paper, a single value of $C_Q$ of 0.1 fF has been used while values of $R_Q$ ranging from 300 kΩ to 1300 kΩ were considered. For $V_{bias} = 18$ V without quenching circuit ($R_Q = 0$ Ω, i.e. for a constant applied voltage $V_D = V_{bias}$) the depletion region of the junction extends from $X_{DEP1} = 360$ nm to $X_{DEP2} = 790$ nm with a maximum electric field of about 600 kV/cm (see Fig. 3). After the avalanche triggering, the current builds up in the device. The depletion region becomes wider and occupied by electrons and holes. Consistently, the electric field spans a wider zone and its maximum value is reduced down to 500 kV/cm (see Fig. 3).

<figure><img src="image_3.png" /><figcaption>Fig. 2. Avalanche current (blue squares) and Charge of electrons (red squares) inside the SPAD in steady state function of Bias Voltage. Blue lines and red lines are the linear model used to extract equivalent electric parameters (breakdown voltage $V_B$, equivalent SPAD resistance $R_{SPAD}$, equivalent SPAD capacitance $C_{SPAD}$).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Carrier concentration for electrons (red lines) and holes (blue lines) and electric field profiles (green lines) for a constant voltage ($V_{bias} = 18$ V). These profiles are measured before (solid lines) and after breakdown (dotted lines).</figcaption></figure>

In Fig. 4 are plotted (a) the number of impact ionization events, the time evolution of both (b) the current and (c) the voltage across the SPAD connected to a quenching circuit with $C_Q = 0.1$ fF and $R_Q = 500$ kΩ. The first e-h pair (resulting from photon absorption) is always generated at $t = 0$. Four different random choices of wave vectors of the initial e-h pair are presented. Green curves correspond to a successful quench, red curves show the case of successful quench, but after secondary avalanche breakdown onset, whereas blue and pink curves are typical examples of a non-successful quench.
The different behaviors can be separated in different phases marked by vertical lines in the figures. During the first phase, the first generated e-h pairs are accelerated by the electric field, which remains nearly frozen, and generates other e-h pairs, triggering an increase of current. During the second phase, after avalanche breakdown, through the effect of the quenching circuit, the strong increase of current leads to a decrease of voltage $V_D$ and of electric field in the junction. This reduces the number of impact ionizations per free e-h pair.

<figure><img src="image_5.png" /><figcaption>Fig. 4. (a) Time evolution of the cumulated number of impact ionization events (II) for initial e-h pair generated at $t = 0$ s with a quenching circuit ($R_Q = 500$ kΩ, $C_Q = 0.1$ fF, $V_{bias} = 18$ V, $X_{GEN} = 500$ nm). Each color corresponds to a random set of wave vector coordinates of initial e-h pair. Vertical bars delimitate 4 different phases of operation, described in the text. (b) Corresponding time evolution of terminal current. (c) Corresponding time evolution of the SPAD voltage $V_D$.</figcaption></figure>

During the third phase, the voltage across the SPAD becomes lower than the breakdown voltage, leading to a mean number of ionizations per generated carrier less than 1, which tends to break the avalanche process. Though it has been rarely reported, this undershoot voltage has been observed experimentally and analyzed by Inoue et al.[21]. We thus see a plateau in the cumulated number of impact ionization events and a strong current decrease. This current fall-off allows $V_D$ to re-increase above $V_B$. We then enter the fourth phase that is critical for the success of the quench.

If all e-h pairs have left the high-field region of SPAD the number of impact ionizations remains at the same level, the current vanishes, and $V_D$ continues to increase up to $V_{bias}$ and the quench is thus successful (green lines). In contrast, if some carriers remain in the high-field region during this fourth phase (when $V_D > V_{bias}$), a new breakdown may occur. In some cases, this secondary avalanche breakdown can eventually quench (red lines). It leads to an afterpulsing-like effect. However, this is not related to any trap assisted mechanism [22], but to a

competing effect of deadtime and extinction time. When impact ionization events continue to occur, the current tends to oscillate around a finite value and the voltage across the SPAD cannot recover its initial value $V_{bias}$ (blue and pink lines): the quench is not successful. It should be noted that when additional avalanches occur, the initial SPAD voltage may not be fully recovered within the typical RC circuit dead time. Thus, the effective dead time may increase statistically, without the presence of defects.

The simulation approach presented here is therefore very useful to explore the statistical features of the quench operation. To this goal, in addition to the avalanche breakdown probability, $P_{BD}$, previously discussed in other works [6,7], we analyze the quenching probability, $P_Q$, as a function of the quenching resistance $R_Q$, together with the relevant characteristic times.

In some cases, although few e-h pairs have been generated, the cumulative number of ionization events quickly stops to increase instead of continuing its exponential increase. An avalanche breakdown can thus be characterized by two characteristic times. First, $T_1$ is the time interval between the creation of the initial e-h pair by photon absorption and the second e-h pair generation by impact ionization. For this specific SPAD with $V_{bias} = 18$ V and $X_{GEN} = 500$ nm, $T_1$ typically varies between 0.05 and 4.25 ps. Secondly, $T_A$, the time to avalanche, is the time when the number of impact ionizations is sufficient to self-maintain the exponential increase of the current. Here, $T_A$ is defined as the time between the photon absorption and the time when the SPAD current (smoothed by rolling average over 2 ps) reaches a threshold value of 2 µA characterizing the breakdown regime. This time to avalanche may vary over a broad range in accordance with the statistical nature of the avalanche process and should be properly evaluated in order to accurately estimate the jitter.

Figure 5(a) shows the time to avalanche $T_A$ and the breakdown probability as a function of $R_Q$ varying from 300 kΩ to 1.3 MΩ (with $C_Q = 0.1$ fF). We observe that both quantities remain essentially independent of $R_Q$. This can be explained by the fact they are measured just before to onset of avalanche breakdown, i.e. when the current is low and the quenching circuit has no influence on the internal SPAD state. The statistical distribution of the computed $T_A$ is plotted in the histogram of Fig. 5(b) for 503 different simulations (346 breakdowns). This distribution can be well fitted by a gamma distribution with parameters $k$ and $\theta$ of 7.71 and 1.64 ps, respectively. It confirms the large spreading of this quantity around the mean value of 12.6 ps with a standard deviation of 4.55 ps.

Figure 6(a) shows the quenching probability and the extinction time $T_{EXT}$ as a function of $R_Q$. $T_{EXT}$ is defined as the time between $T_A$ and the last ionization event when the SPAD quenches. By considering a total time of 260 ps after photon absorption, i.e. about four times greater than the mean value of $T_{EXT}$, a quench is considered as successful if the last ionization event occurs before 230 ps. We observe that the extinction time $T_{EXT}$ tends to reduce when increasing the resistance $R_Q$, while $P_Q$ increases. Remarkably, this probability changes from 0% for $R_Q = 300$ kΩ to 100% for $R_Q = 1.3$ MΩ. This agrees with the analysis of [23] that explained why the quenching resistance must be high enough to obtain a successful quench. The
histogram of the extinction time is plotted in Fig. 6(b) for $R_Q = 700$ kΩ in the cases of successful quenches (65 values have been used). Again, this distribution can be fitted with a gamma distribution, with parameters $k = 23.6$ and $\theta = 2.66$ ps. A large spreading of $T_{EXT}$ is observed, as the standard deviation is 12.9 ps, for a mean value of 62.8 ps. It should be noted that the values of $T_{EXT}$ greater than 150 ps have been excluded from the fitting procedure. They correspond to quenches after secondary avalanche and cannot be captured by the gamma distribution of single avalanche cases. As shown in Fig. 6(a), the quenching probability depends on the RC time constant of the moving node. Hence, a SPAD exhibiting a low quenching probability will not only increase its charge per pulse consumption, but also the recharge dead time. Experimentally, this may be tested by measuring pulse-width and inter-pulse time distributions [24].

<figure><img src="image_6.png" /><figcaption>Fig. 5. (a) Avalanche time $T_A$ (black squares) and breakdown probability (red stars) versus quenching resistance $R_Q$ for a constant value of $C_Q = 0.1$ fF. Black squares are mean values of $T_A$, while vertical bars indicate minimum and maximum values. Half of the values are in the closed area. (b) Avalanche time histogram for all values of $R_Q$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. (a) Extinction time (black squares) and quenching probability (red stars) versus $R_Q$ for a constant value of $C_Q = 0.1$ fF. Black squares are mean values, vertical bars indicate minimum and maximum values. Half of the values are in the closed area. (b) Extinction time histogram for $R_Q = 700$ kΩ and $C_Q = 0.1$ fF.</figcaption></figure>

IV. CONCLUSION

We have shown that, thanks to self-consistent 3-D MC simulation, together with a mixed-mode approach to include the passive quenching circuit, it is possible to capture the statistical properties of both avalanche and quenching stages of the SPAD electrical behavior. This study provides insights into SPAD operation in terms of avalanche and quenching probabilities as well as avalanche and extinction times. Such features are of high importance to optimally design this class of photodetectors.

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