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A simple yet accurate equivalent circuit model of the SiPM is of crucial importance to the design and optimization of the readout electronics. Although the SPAD pixel is well-modeled in Figure 3.2(a), the SiPM electrical model is much more complicated especially when multiple pixels are fired. In practice, the avalanche breakdown in different pixels certainly do not turn on and off synchronously, and the loading effect of the readout electronics should also be taken into consideration, adding complexity to the modeling.

By assuming that pixels are fired and quenched synchronously, the SiPM equivalent circuit model could be significantly simplified. Figure 3.4 shows an example of the simplified mode, where N_f pixels in the SiPM with N pixels are fired while the remaining N − N_f pixels stay unfired. Similar to the SPAD case, a switch is implemented in such a way that its closing and opening mark the turn on and off of the avalanche breakdown in all N_f fired pixels. Since the diodes of the unfired pixels are not triggered, they are modeled as a capacitance solely. C_s is added to model the parasitic capacitances contributions from the metal grids and bonding pads of the SiPM. The capacitance of the readout electronics can also be merged into Cs. The

loading effect of the readout electronics is presented as Z(s), which is a small resistance for most cases.

It is apparent that the SiPM response with respect to a pixel firing has to be carried out for the avalanche breakdown turn-on and turn-off phases separately. Considering a resistive readout impedance (denoted as R_L) and assuming that the avalanche breakdown is triggered at t = 0 and quenched at t = T , the overall current presented to the readout electronics can be expressed by io(t) = ( i1(t), 0 ≤ t ≤ T, i2(t), t ≥ T (3.7) where i₁(t) represents the current flow into the load during the avalanche, and i2(t) describes

the current behavior after the avalanche is quenched. A comprehensive analytical analysis of these two currents has been performed in [54].

Following the analysis described in [45, 52, 54], the SiPM output current during the rising and quenching transient operations can be expressed by

i1(t) = I0·  1 − τq− τi τd− τi · exp  −t τi  +τq− τd τd− τi · exp  −t τd  (3.8) where I0 ≈ Nf · If = NfVov/Rq is the asymptotic steady-state current of the SiPM with Nf

VBD Rd/Nf NfCd S Rq/Nf NfCq (N − Nf)Cd (N − Nf)Cq Rq N −Nf Cs Vbias io Z(s) vi vd

Nffired pixels N − Nffired pixels

Figure 3.4: Equivalent circuit of the SiPM with Nf pixels fired. The loading effect of the

readout electronics is modeled as Z(s). In the following analysis, Z(s) = RL is adopted.

constants are given by

τi≈ Rd· (Cd+ Cq), τq ≈ Rq· Cq, τd≈ RL· [Cs+ N (Cd||Cq)] (3.9)

with τ_i τ_d τ_q. Equation (3.8) reveals that the output current pulse reaches its maximum rapidly with a time constant of τ_i, and then decays relatively slowly to the asymptotic value I0. While the decay time constant τd shows a dependency on the load resistance, the rising

time constant τ_i is essentially an intrinsic parameter. Both τ_i and τ_d are negligibly dependent on N_f, which means the current pulses have the same shape regardless of the number of fired pixels.

After the avalanches are quenched for t > T , the SiPM pixels are in recovery to the initial bias conditions and all the capacitances entirely charge or discharge to their quiescent values. Two processes can be distinguished during the recovery phase. The first component is the discharging of the unfired pixels because of the non-zero value for vi(T ) = io(T )RL. This

process is fast because of the small resistive load RL, giving a time constant approximated to

τd given in equation (3.9). Another process is the recharging of the fired pixels through the

large quenching resistor, which is slow. The overall recovery current i2(t) is the superposition

of these two components. The slow component is dominant and much more important so that i2(t) can be approximated to be i2(t) ≈ NfIf · exp  − t τr  , whent > T + 3τ_d (3.10) with the recovery time constant τr given by

τr≈ Rq(Cq+ Cd) + RL· [Cs+ N (Cd||Cq)] ≈ Rq(Cq+ Cd) (3.11)

The above two equations are valid because in most cases R_L[Cs+ N (Cd||Cq)]  Rq(Cq+ Cd)

with a small RL.

Compared to the SPAD model described in the previous section, there is one extra time constant τdprovided by the SiPM. This time constant originates from the non-zero impedance

fully described with the SPAD model. In this case, all the SPAD pixels inside SiPM work independently.

The general requirements for the design of SiPM readout electronics can be benefited from the insights provided by equation (3.8) and (3.10). Considering the applications where pre- cise charge measurement is important, the total amount of charge collected by the readout electronics is the integration of i_o(t) by

Qtotal= Z ∞ 0 io(t) dt ≈ Z ∞ T i2(t) dt ≈ Nf · Vov(Cq+ Cd) (3.12)

where the charge contribution from I₁(t) is neglected because T is a small number in the order of few hundreds picosecond. This equation reveals that the output charge is dominantly determined by the slowest components of the circuit. The SiPM gain (G), which defined as the output charge divided by the number of fired pixels, is given by

G = Vov(Cq+ Cd)/e (3.13)

where e is the electron charge. In this case, both the gain and the recovery time constant τr are

almost unaffected by the load impedance, which is desirable for most sensor applications. By requiring R_L[Cs+ N (Cd||Cq)]  Rq(Cq+ Cd), the readout electronics should have low input

impedance and small input capacitance, which would be endorsed into Cs. Besides, because

i2(t) decays with a time constant of τr, the bandwidth for the charge readout electronics should

be larger than 1/(2πτ_r), which is in the order of few tens of MHz.

For the applications where timing information is critical, it is important for the low-jitter fast timing readout electronics to preserve the fast rising edge of the sensor pulses, which is characterized by i1(t). The signal slope at the avalanche breakdown starting point (t = 0) can

be used to evaluate the speed [52] v0(0) = RLNf Vov Rq · τq τiτd ≈ NfVov N RdCd+ RdCs(1 + Cd/Cq) (3.14) In general, small-sized devices with small N C_d, or small-pitch sensors with small C_d and Cs

give a fast slope and thus better jitter performance. Surprisingly, a relative larger C_q is also beneficial even though providing a larger τi. This is because Cq works as a fast forward

discharging path for the charge delivered during the avalanche breakdown, contributing a zero at −1/τ_q to the output current transfer function. Given the fast rising slope, the bandwidth of the timing readout electronics should be as high as possible, usually above several hundreds of MHz.