New Perspective on Passively Quenched Single Photon Avalanche Diodes: Effect of Feedback on Impact Ionization

Single-photon avalanche diodes (SPADs) are primary devices in photon counting systems used in quantum cryptography, time resolved spectroscopy and photon counting optical communication. SPADs convert each photo-generated electron hole pair to a measurable current via an avalanche of impact ionizations. In this paper, a stochastically self-regulating avalanche model for passively quenched SPADs is presented. The model predicts, in qualitative agreement with experiments, three important phenomena that traditional models are unable to predict. These are: (1) an oscillatory behavior of the persistent avalanche current; (2) an exponential (memoryless) decay of the probability density function of the stochastic quenching time of the persistent avalanche current; and (3) a fast collapse of the avalanche current, under strong feedback conditions, preventing the development of a persistent avalanche current. The model specifically captures the effect of the load’s feedback on the stochastic avalanche multiplication, an effect believed to be key in breaking today’s counting rate barrier in the 1.55–μm detection window.


Introduction
Recent growth of applications such as quantum cryptography [1], photon-counting optical communication [2], time resolved spectroscopy, time resolved reflectometry [3], quantum imaging [4] and three dimensional laser radar [5] has fueled considerable interest in single photon avalanche diodes (SPADs) [7]. SPADs operate by converting each photogenerated electron hole pair to a large number of carriers via an avalanche of impact ionizations. When the applied bias is above breakdown the number of impact ionizations may increase indefinitely, yielding, in principle, an infinite multiplication factor. However, a ballast resistor may be introduced in series with the SPAD to provide negative feedback and prevent runaway avalanche. The avalanche current then saturates at a level governed by the power supply and the resistance [6,7]. This current is referred to as the self sustaining or persistent avalanche current. The persistent current may terminate owing to fluctuations in the carrier production at a stochastic time, known as the quenching time [7], after which a new incoming photon may be detected. This mode of operation is referred to as the passive quenching mode and it offers a considerable simplification over the active-quenching mode [7], which requires a complex bias circuitry.
Presently, many photon-counting applications at 1.55-μm are constrained by after-pulsing. Afterpulses are dark counts triggered by the release of trapped carriers from earlier avalanches. These prolong the SPAD's recovery time as they prohibit error-free counting of subsequent incoming photons. Many practitioners [13] in the field believe that a key to breaking the present-day counting rate barrier (10 MHz) for free-running or non-periodic gating at the 1.55-μm wavelength rests in exploiting negative feedback in SPADs to minimize the charge flow after an avalanche trigger and hence reduce afterpulsing. Increasing the achievable free-running photon counting rate to 100 MHz or 1GHz in a simple and cost effective way is essential in applications, e.g., the maximum data transmission rate in photon-counting pulse-position modulation is directly proportional the maximum achievable counting rate. Another example would be photon-counting imaging, in which the flux arrival time at each pixel is random and can involve very short interarrival times requiring high free-running counting rates. To elevate the photon-counting rate, a new generation of fast-quenching SPADs, the negative feedback avalanche diode (NFAD) [8] and the self-quenching and self-recovery avalanche detector [9], have been reported that can potentially transform SPAD technology in providing a solid-state equivalent of the ideal vacuum-tube microchannel plate multipliers. Interestingly, the physical structure of NFADs with a monolithically integrated quench resistor, has revealed a new phenomenon neither observed nor predicted before: the persistent avalanche current in NFADs has an oscillatory behavior. This phenomenon, which is key to understanding the observed memoryless property of the quenching time, cannot be explained by conventional theory, which fails to account for the effect of feedback on the stochastic impact ionization process.
Here we present a stochastically self-regulating avalanche model that fully captures the effect of negative feedback on the stochastic nature of the impact ionization process. It is the first significant expansion beyond the original SPAD device description by Haitz in 1964 [6] and predicts three important behaviors that cannot be addressed by the traditional modeling methods. First, it predicts the oscillatory behavior of the persistent avalanche current. Second, it predicts that the probability density function of the stochastic quenching time of the persistent avalanche current has an exponential (memoryless) decay. Third, under conditions that lead to strong feedback, the stochastic avalanche current can collapse before a persistent avalanche current can be established. All three behaviors are in qualitative agreement with recent experimental studies of NFADs that have not yet been theoretically explained.

Limitations of the traditional model for passively quenched SPADs
We have used a stochastic approach to calculate the probability density function (pdf) of the quenching time under a key assumption implicit in the traditional model. As shown in the following the results are unrealistic, demonstrating the inadequacy of this model to capture the effect of feedback on impact ionization. Figure  1a shows the traditional model of a passively quenched SPAD [6,7]. In this model the SPAD is represented by its depletion capacitance, Cd, in parallel with a series combination of a switch, sw, a dynamic resistance, Rd, and a DC bias source, Vb, representing the breakdown voltage of the SPAD. In the absence of an avalanche trigger the switch is open and the bias across the diode is Va, which is set slightly above the breakdown voltage, Vb by the excess voltage, Vex. When an avalanche is triggered the model assumes that the switch is instantly closed and the capacitance Cd discharges through the diode's dynamic resistance Rd, which reduces the voltage across the SPAD to a value that depends on the ratio of Rd and RL. In steady state, the voltage across the SPAD is given by VSPAD= Va -VexRL/(RL + Rd) ≈ Vb, for RL ≫ Rd. In addition, the steady state avalanche current is given by Iss ≈ Vex/RL, and the voltage across the resistor, RL, is VRL ≈ Vex, for RL ≫ Rd. (a) Traditional model for a passively quenching SPAD circuit. id represents the self sustaining current through the multiplication region of the SPAD; Rd is its equivalent dynamic resistance; Cd is its junction capacitance; RL is the load resistor and CL is its parasitic capacitance. The traditional model neglects the effect of feedback on the impact ionization process; it assumes that after the trigger of an avalanche, the electric field remains constant at the breakdown threshold, so that the core of the device is represented by a voltage generator, Vb. (b) Stochastically self-regulating avalanche model for passively quenched SPADs. The circuit represents a series combination of a SPAD and a negative feedback load. The load is described as a parallel combination of a resistance, RL and a capacitance, CL. The SPAD is modeled as two parallel branches; one branch consists of the diode depletion capacitance, Cd, the other includes the Monte Carlo simulator, which is represented by the stochastic voltage controlled current source (VCCS) id. The resistor Rd, in series with the VCCS, accounts for the resistance of the bulk regions.
The presence of the DC source, Vb, in the traditional model reflects the assumption that after an avalanche event is triggered the electric field in the avalanche region, responsible for the persistence of impact ionization, remains precisely at breakdown until the persistent current collapses owing to the stochastic fluctuations inherent in the impact ionization process, that is when all carriers chance to exit the multiplication region without ionizing. We shall show that this constant field assumption implies the unrealistic consequence that the quenching time, Tq, has memory.
Following the constant field assumption and building upon the recursive technique for avalanche multiplication developed by Hayat et al. [10,11], we have shown (see the Appendix) that the probability that the avalanche current, I self quenches before time t has elapsed is given by ), q is the electronic charge, C is a dimensionless constant of order unity, τ0 is the average of the electron and hole transit times across the multiplication region and k = β/α is the hole/electron ionization coefficient ratio. Equation (1) was first pointed out by the authors in [12] without proof and here we provide its derivation in the Appendix.
The problem with this formula, Eq. (1), arising from the constant field assumption, is that it predicts that the quenching time should have memory, since the form of FI(t) (not being exponential in t) implies that the probability rate of quenching diminishes in time. More precisely, if we assume that quenching has not occurred by time t, then the probability that quenching should occur between times t and t + Δt is P{t ≤ Tq ≤ t + Δt | Tq > t} = Δt · (T/t 2 ) exp(−T/t), instead of being simply proportional to Δt as in the memoryless case.
Indeed, memory is not observed in the quenching process of self quenched SPADs. Figure 2ashows measurements of the pdf of the quenching time for an NFAD SPAD, showing exponential decay, implying that the decay process is memoryless. The data was provided by Princeton Lightwave Inc. The traditional model also fails to predict the oscillatory behavior in persistent current also observed by Itzler et al. [8,13] and shown in Fig. 2b. Fig. 2 Experimental results. (a) Measured pdf of the quenching time [13]. The exponential decay of the pdf implies that the quenching time is memoryless. (b) Measured voltage across the SPAD for an excess bias of Vex ≈ 1.7 V [13]. The current shows oscillatory behavior about the steady state before it quenches spontaneously. The complete structure of the device can be found elsewhere [14].
3. Stochastically self-regulating avalanche model Figure 1b shows our proposed stochastically self-regulating avalanche model of a passively quenched SPAD [15]. The main difference between this and the traditional model of Fig. 1a is that the switch and voltage generator Vb in Fig. 1a, which represented the on/off state of the SPAD, are now replaced by a stochastic voltage controlled current source (VCCS) id. A Monte Carlo simulator of the dynamics of the avalanche multiplication is used to produce the current in the VCCS. Moreover, as the voltage across RL changes so does the bias on the SPAD, and hence also the stochastic avalanche current id, since the ionization coefficients, α and β used by the Monte Carlo simulator depend on the instantaneous electric field through the junction capacitance, Cd. As the carriers multiply stochastically their resulting current is calculated using Ramo's theorem [16] from the number of carriers inside the multiplication region. Hence, by contrast with the traditional model, our stochastically selfregulating avalanche model captures the effect of feedback on the stochastic evolution of carrier multiplication associated with the persistent avalanche current.

Monte Carlo simulator of avalanche multiplication under dynamic electric field
In Fig. 3 we illustrate the operation of the Monte Carlo simulator used to produce the current in the VCCS by mimicking a SPAD with a dynamic and stochastic bias. In the simulator, the multiplication region extends from x = 0 to x = w, and this region is divided into L small increments each of width Δw, representing L bins, each terminating at xk = kΔw, where k = 1,2,3,..., L. The total number of bins L, which determines the spatial increment Δw, is chosen so that the product α · Δw (β · Δw) is small. The binomial model described here is a good approximation of the continuous-space ionization process provided that α · Δw(β · Δw) ≪ 1. Indeed, in our simulations with L = 1600 and Δw = 1 nm. The maximum value for β · Δw ≈ 9 × 10 −4 , which is much less than one. The total simulation time, from t = 0 to t = Tmax, is divided into M small increments, each of duration Δt, where Δt represents the time taken for a carrier to travel a distance Δw. For simplicity we have assumed that electrons and holes have the same drift velocity, v. A particular time in the simulation is described as tj = jΔt, where j = 1,2,3,...,M. We assume that holes (electrons) move in the positive (negative) direction of x. In addition, we employ the common rule that at any time interval [t,t + Δt] the probability that an electron will impact ionize is given by α(ECd(t))Δw, where ECd(t) is the instantaneous electric field through Cd. Similarly, the probability that a hole will impact ionize is given by β(ECd(t))Δw. To track the stochastic evolution of the total number of carriers at each instant we define Xe(tj, xk) and Xh(tj, xk) as the number of electrons and holes, respectively, at bin location xk and time tj. The effect of the dead space is ignored in the carrier multiplication process since we consider SPADs with thick multiplication regions (> 1μm), which are preferred for Geiger mode operation [17]. It is well known that for thick multiplication regions the effect of the dead space does not play a relevant role in the carrier multiplication process. Therefore, in thick multiplication regions the dead space can be ignored. On the other hand, for thin multiplication regions (< 500 nm) the dead space becomes important and to accurately describe the impact ionization process the dead space must be taken into account. Considering the transport and ionization properties of the carriers, and by ignoring their dead space, we can write the following stochastic dynamical equations: (2) ( +1 , ) = ( , +1 ) + ( ( , +1 ), ( ( )) ) + ( ℎ ( , −1 ), ( ( )) ) and (3) ℎ ( +1 , ) = ℎ ( , −1 ) + ( ℎ ( , −1 ), ( ( )) ) + ( ( , +1 ), ( ( )) ) .
In the above equations the notation b(n, p) stands for a binomial random variable of size nand success probability p; thus b(n, p) represents the total number of successful ionization events resulting from n independent attempts, each with success probability p. The boundary conditions at k = 1 and L must clearly be handled separately in Eqs. (2) and (3).
To trigger an avalanche the multiplication region is reverse biased above breakdown and a carrier is injected at the start of the multiplication region. Equations (2) and (3) are implemented at every time increment and samples of the required binomial random variables are generated. Figure 3 shows a fictitious example which illustrates the total number of carriers in the multiplication region at each time, the direction of motion of the carriers and the impact ionization events generated, during 5 intervals of time Δt, by a hole injected at time t = 0 and at location x = 0. After time tj has elapsed the instantaneous stochastic current id(tj) is calculated using Ramo's theorem: (4) ( ) = ∑ ( ( , ) + ℎ ( , )) .

=1
All other currents and voltages in Fig. 1 are calculated by solving the standard circuit equations. The instantaneous values of the electric field dependent ionization coefficients are recalculated at every time increment to allow for the change of voltage across the SPAD as a result of the instantaneous feedback from the load.

Results
We next apply our stochastically self-regulating avalanche model to simulate the unique attributes of the new generation of self quenched SPADs. In particular, we are interested in predicting the statistics of the quenching time and the observed oscillatory behavior of the persistent current. We have simulated a passively quenched InP SPAD using the following values of the circuit parameters: junction capacitance: Cd = 0.1 pF, load resistor: RL = 22 kΩ and load capacitance: CL = 0.001 pF. The resistor Rd is not included in the equations that describe the model because its effect is absorbed by the voltage controlled current source, id. In the simulations it is assumed that the electric field in the multiplication region is spatially uniform, which corresponds to a multiplication region without doping. To start the simulation a hole is injected at the edge of the multiplication region of width w = 1600 nm. The circuit is biased by the power supply at a voltage Va = Vb + Vex so that the SPAD is reverse biased beyond its breakdown voltage, Vb by the excess voltage, Vex. The theoretical breakdown voltage was calculated from the divergence of McIntyre's multiplication expression [18] (5) By using the electric-field dependent expressions for the electron and hole ionization coefficients for InP [19], the breakdown voltage is found to be Vb = 64.61 V. Figure 4 shows the calculated avalanche current, ia, the feedback voltage, VRL and the voltage across the SPAD, VCd as a function of time, displayed in terms of both the carrier transit time, w/v, and the RC time constant of the circuit, RL(CL + Cd). We have assumed that both holes and electrons travel at the velocity v = 6.7 × 10 6 cm/s. In the simulations the value of the excess voltage is 0.39 V. Once an avalanche is triggered, then when the diode is biased above breakdown the mean avalanche current grows exponentially, after a brief transient of the order of the transit time, according to the theory of mean impulse response of APDs above breakdown [11,20]. This growth discharges the capacitor Cd and therefore reduce the junction voltage VCd, which in turn causes the avalanche current to increase more slowly. Equivalently, from a feedback perspective the large avalanche current flowing through the junction increases the Ohmic drop across RL, causing a drop in the junction voltage VCd. The avalanche current eventually falls until the junction bias falls below the breakdown voltage. This is a significant outcome of the stochastically selfregulating avalanche model and it is contrary to the traditional model, which dictates that the junction voltage never drops below Vb. The DC source then begins to recharge the capacitor with a time constant τr ≈ RLCd, causing the avalanche current to increase once again. The repetition of these discharge and recharge cycles yields the oscillatory behavior seen in Fig. 4, where the current through the diode oscillates about Iss ≈ 18 μA, the feedback voltage oscillates around the excess bias voltage Vex ≈ 0.39 V, and the voltage across the SPAD oscillates above and below the breakdown threshold. This repetition continues until the stochastic fluctuations inherent in the impact ionization process cause the spontaneous quenching of the avalanche current. In the simulation shown in Fig. 4 quenching occurs after about 2340 transit times.

Circuit behavior after an avalanche trigger
To better illustrate the cycles in the oscillatory behavior described above we plot in Fig. 5 the calculated voltage across the junction capacitor, VCd (red curve) together with the current, id(blue curve). In particular, we mark four successive stages of behavior, from the onset of the avalanche until the spontaneous quenching of the persistent current. (For clarity the curve idwas truncated and its first peak is not shown.) A key point that we have noted in our simulations is that spontaneous quenching invariably occurs during the recharge cycle of the junction capacitor, where the persistent current is at its lowest and the number of ionizing carriers is at a minimum. This observation is critical in understanding the exponential decay of the pdf of the quenching time, which is discussed later.

Quenching behavior
In our simulations we have considered an observation window of 8000 transit times (∼ 200 ns). The observation window is the interval of time during which the persistent current was observed when the quenching time was determined. The quenching time was measured within the observation window. A realization that shows a persistent current that does not spontaneously quench within the observation window is considered to be self sustaining. We have found that within the considered observation window the probability of spontaneous quenching increases as the current Iss decreases. This is because when the current is reduced so is the number of ionizing carriers, increasing the chance that all carriers present in the multiplication region exit without impact ionizing.

Figure 6a
shows representative examples of the persistent current regime without quenching (red curve), the case where spontaneous quenching occurs after a period of persistent current flow (blue curve) and the case when quenching occurs immediately following the first current peak, shown in the black curve. In this example the excess bias voltage is about 0.39 V and RLwas varied to achieve the different values of Iss. It should be mentioned that the three quenching behaviors described above have been observed on NFAD devices by Princeton Lightwave, Inc., with appropriate variations in the feedback resistor RL. Moreover, a similar fast selfcollapse of the avalanche current was reported by Shushakov et al. [21,22] and Zhao et al. [9] in devices where the feedback was provided by means of a charge-accumulation effect due to a potential barrier outside the multiplication region. The work of Shushakov et al. [21,22] also included a Monte-Carlo simulation of the stochastic avalanche process in the presence of feedback, which was used to calculate the distribution of the gain. The quenching behavior described above was reproduced consistently for excess voltages below 0.8 V.
However, for higher excess bias voltages we did not observe the behavior shown in the blue curve of Fig. 6a, in which we have a period of persistent current flow followed by spontaneous quenching. Instead, in the higher excess voltage case the system goes from the regime of persistent current flow that does not quench to that of quenching immediately following the first peak of the current for higher RL. This may be because with higher excess voltages the resulting stronger feedback causes the current id to overshoot as it falls, making it difficult for the system to execute even a single period of persistent oscillating current.

Probability density function of the quenching time
The pdf of the quenching time, Tq, for an excess bias voltage of 0.39 V was estimated by repeating the simulation of the persistent current (from trigger instant to quenching instant) 2267 times. The quenching time, Tq, is the interval of time measured from the start of the avalanche until its spontaneous quenching. The result is shown in Fig. 6b, revealing that the decay of the tail of the pdf is exponential, implying, in turn, that the quenching process is memoryless. This observation is consistent with the measurements performed on NFAD devices shown in Fig. 2a.
The memoryless property of the quenching process can be understood from the fluctuating behavior of the voltage across the SPAD and the persistent current. Recall that these quantities oscillate about Vb and Iss, respectively. We have learned from our simulations that quenching invariably occurs only during the recharge cycle of the junction capacitance, as shown in Fig. 5, and that the probability that quenching occurs during the discharge stage of the capacitance is negligible. We now make two key observations. The first is that the probability that quenching occurs during the recharge cycle of the junction capacitance is the same for all recharge cycles. (The beginning of each recharge cycle of the junction capacitance starts right after the voltage across Cd reaches a local minimum.) This is due to two factors: (a) the electric field profile is almost identical in all recharge cycles. The point here is that the electric field remains above breakdown in half cycle and then remains below breakdown in the second half. (b) The number of carriers at the beginning of each recharge cycle is almost the same for all recharge cycles, owing to the periodicity of the persistent current. The number of carriers at the beginning of each recharge cycle is almost the same in a statistical sense, meaning that the probability distribution of this number is approximately the same from cycle to cycle but the actual numbers can be different. Hence, prior to quenching, both the electric field and carrier number conditions are almost reproduced periodically at the beginning of each recharge cycle. This, in turn, implies that the probability of quenching is approximately the same for all recharge cycles. It is emphasized that the probability of quenching is approximately the same on average (in a statistical sense), although the actual values may vary from cycle to cycle and from experiment to experiment. The second observation is that quenching events over different recharge cycles are statistically independent. We can assume that the quenching events are statistically independent because different recharge cycles involve different carriers, since the duration of the cycle is much greater than the carrier transit time across the multiplication region. Thus, if Prepresents the probability that quenching occurs in a specific recharge cycle, given that quenching has not occurred earlier, then by using the two observations made above we can write the probability that quenching occurs at the nth recharge cycle as P(1 -P) n−1 . This is exponential in form and thus satisfies the memoryless property.

Conclusions
In conclusion, we have developed a model to calculate the response of a passively quenched SPAD, reverse biased above breakdown. Our model considers a closed loop system, capturing the effect of the feedback introduced by the resistive load on the stochastic nature of the avalanche multiplication. This approach differs from the conventional traditional model [6,7], which captures the deterministic feedback, maintaining the device at breakdown, but neglects the dynamic coupling between the voltage across the SPAD, the feedback from the load and the impact ionization process. As a consequence the traditional model provides no way of determining the oscillatory behavior of the persistent avalanche current and the statistics of the quenching time. Moreover, we have shown that the traditional model leads to unrealistic predictions of the pdf of the quenching time. By contrast the stochastically self-regulating avalanche model enables us to predict the stochastic current-voltage evolution and quenching characteristics in passively quenched SPAD circuits. Our model predicts key attributes of the stochastic avalanche current seen in experiments performed on the new generation of SPAD structures that rely on negative feedback. To the best of our knowledge, the proposed stochastically self-regulating avalanche model is the only one capable of predicting the oscillatory behavior of the persistent current and the statistics of the quenching time. Our model therefore constitutes a reliable simulation framework to aid the design and optimal operation of an emerging generation of SPAD devices that rely on negative feedback.

Appendix
Consider an electron (hole) generated at position z in a multiplication region with uniform electric field, which spans 0 < z < w, and assume that it will ionize for the first time at a distance ζ downstream from its generation point and at a time t later with probability density function he(h) (ζ,t). A carrier injected at z in a multiplication region will then give rise to an avalanche current which terminates before time t has elapsed with probability Fe(z,t) for an injected electron, and Fh(z,t) for an injected hole. Along the lines of the recursive multiplication theory [11,20,23], these probabilities are given by are the probabilities that the injected electron and hole avoid ionizing within the multiplication region before time t and Qe,h(z,t) are the probabilities that these carriers drift and diffuse out of the multiplication region before this time.
Equations (6) and (7) hold above, below and precisely at breakdown and may be solved numerically to follow the temporal evolution of Fe(h)(z,t). The behavior below breakdown has been studied numerically by Hayat and Dong [11] and by Ng et al. [23] and the Malthusian behavior was pointed out by Hayat et al. [10] and examined further by Groves et al. [20]. A brief exposition of the behavior at breakdown was first reported in [12] without proof and we provide it here.
To obtain the result in Eq. (1) it is necessary to determine the asymptotic behavior of the avalanche duration distribution function at breakdown. To do this we write the avalanche duration probabilities in terms of the fe,h(z,t) = 1 -Fe,h(z,t), since at breakdown we expect fe,h(z,t) → 0 at long times. Without loss of generality we can write Eqs. (6) and ( At times t earlier than a transit time the inhomogeneous "source" terms, Re,h(z,t) (1 -Qe,h(z,t)) in Eqs. (8) and (9) are non-zero. The effects of carrier diffusion and the gradual collapse of the electric field at the edges of the multiplication region in a real device will ensure that these terms fall to zero with time in a continuous and smooth manner, so that the only discontinuity is at time t = 0, when the system is switched on. The simple model used here, where diffusion is absent and the field falls abruptly to zero at the edges of the multiplication region, may be considered a limiting case of the real system. Since the integral terms on the right-hand side of Eqs. (8) and (9) are well behaved and we expect fe,h(z,t) to approach zero at long times and to be analytic away from t = 0, we attempt a Laurent expansion [24] of the solutions in powers of inverse time (10) ,ℎ ( , ) = ∑ ,ℎ; ( ) × ( ) − . ∞ =1 Terms of only negative power in t are included because Malthusian behavior [25,20], of the form exp(−γt), and requiring positive powers, would lead to an inconsistently finite and constant breakdown probability, since γ approaches zero at breakdown [10]. In fact numerical simulations of Eqs. (6) and (7) (not shown here) confirm that the coefficient of the Malthusian exponential approaches zero at breakdown, consistent with n > 0.
Inserting the expansion in Eq. (10) into Eqs. (8) and (9) and equating coefficients of equal powers of t we obtain a coupled series of equations for the coefficients, fe,h;n(z). At sufficiently long times the lowest order terms, in 1/t will dominate. The equations for the coefficients, fe,h;n(z) are homogeneous (since the inhomogeneous source terms involving the R and Q vanish at times longer than a transit time) and so these quantities are given only to within constant factors (although the factors for fe;1(z) and fh;1(z) are related). The values of these factors are determined by the initial conditions contained in the source terms, which do not appear in this asymptotic analysis. However, Eqs. (8) and (9) are nonlinear and the equations for the higher order coefficients, fe,h;n(z), with n = 2,3,..., involve the lower order coefficients so that these can be determined successively after the fe,h;1(z) are known.
The dynamical equations for the avalanche carrier densities in a uniform multiplication region are given, e.g., by Emmons [26]. In terms of the electron and hole concentrations per unit length, n(z) and p(z). They are given by , where I is the current carried by these distributions.
We now determine the probability distribution function of the quenching time in a passively quenched SPAD under constant electric field at breakdown. To calculate the statistics of the duration of such avalanche pulses we observe that the mean avalanche current is generated by electrons and holes distributed throughout the multiplication region. In the local model we can regard these as primary carriers, each generating its own individual avalanche current, all of which flow in parallel to generate a total mean current, I. For the avalanche pulse to quench each of these individual avalanche currents must terminate independently. The probability that this happens before time t elapses is given by FI(t) = Πi Fe,h(zi,t), where the product is over all electrons and holes, situated at zi in the multiplication region, and Fe,h(zi,t) is the probability that an electron (hole) injected at zi will give rise to an avalanche pulse which terminates before time t has elapsed. When the SPAD is biased precisely at breakdown (i.e., prior to the avalanche current collapsing) Fe,h(zi,t) = 1 -fe,h(zi)/t. Interestingly, we note that this asymptotic behavior is different from those corresponding to below or above breakdown, for which the asymptotic behavior is exponential [11,20]. Thus, at times long compared with the carrier transit times we find (21) ln( ( )) = ∑ ln (1 − ,ℎ ( ) ) ≈ − 1 ∑ ,ℎ ( ).
By using the expressions for the fe,h;1(z) given by Eq. (19), and the expressions for the n(z) and p(z) given by Eq. (20), respectively, we arrive at Eq. (1).