A computer program to simulate the response of SiPMs
AA computer program to simulate the response of SiPMs
E. Garutti, R. Klanner ∗ , J. Rolph, J. Schwandt Institute for Experimental Physics, University of Hamburg,Luruper Chaussee 147, D 22761, Hamburg, Germany.
Abstract
A Monte Carlo program which simulates the response of SiPMs is presented. Input to the program arethe mean number and the time distribution of Geiger discharges from light, as well as the dark-count rate.For every primary Geiger discharge from light and dark counts in an event, correlated Geiger dischargesdue to prompt and delayed cross-talk and after-pulses are simulated, and a table of the amplitudes and timesof all Geiger discharges in a specified time window generated.A number of different physics-based models and statistical treatments for the simulation of correlatedGeiger discharges can be selected. These lists for many events together with different options for the pulseshapes of single Geiger discharges are used to simulate charge spectra, as measured by a current-integratingcharge-to-digital converter, or current transients convolved with an electronics response function, as recordedby a digital oscilloscope. The program can be used to compare simulations with different assumptions toexperimental data, and thus find out which models are most appropriate for a given SiPM, optimise theoperating conditions and readout for a given application or test programs which are used to extract SiPMparameters from experimental data.
Keywords:
Silicon photo-multiplier, simulation program, current transients, charge spectra.
Contents1 Introduction 12 The SiPM response model 23 Selected results 5
SiPMs (Silicon Photo-MultiPliers), matrices of photo-diodes operated above breakdown voltage, arethe photo-detectors of choice for many applications. They are robust, have single photon resolution, highphoton-detection efficiency, operate at voltages below 100 V and are not affected by magnetic fields.However, their detection area is limited, their response is non-linear for high light intensities and theirperformance is affected by dark counts, prompt and delayed cross-talk and after-pulses.The simulation program which is described in this paper can be used to ∗ Corresponding author. Email address: [email protected], Tel.: +49 40 8998 2558
Preprint submitted to Elsevier June 22, 2020 a r X i v : . [ phy s i c s . i n s - d e t ] J un . test and verify analysis programs used to extract SiPM parameters from experimental data,2. determine SiPM parameters by comparing measurement and simulation results, and3. evaluate and understand the impact on the measurement of photons of different SiPM parameters,probability distributions for cross-talk and after-pulses and readout schemes.A number of simulation programs are documented in the literature. An incomplete list follows. AMonte Carlo program to simulates the multiplication process which is responsible for Geiger dischargesis presented in Ref. [1]. Programs which simulate the shape of the transients for different options for thereadout electronics are discussed in Refs. [2, 3] and references therein. Monte Carlo programs addressingthe readout of light from scintillators with SiPMs are documented in Refs. [4, 5, 6], where the last onehas been implemented in the GEANT4 framework [7]. The simulation discussed in Ref. [8] puts the mainemphasis on the optimisation of the time resolution using SiPMs for PET scanners.The Monte Carlo program described in this paper is less complete than some of the programs mentionedabove, as neither the SiPM gain and photon-detection efficiency, nor the SiPM current response based onan electrical model are simulated. However, it is more flexible, as it allows to change• the shape of the light pulses illuminating the SiPM,• the pulse shape of single Geiger discharges,• the probabilities and probability distributions for prompt and delayed cross-talk and after-pulses, and• the readout mode and its electronic response function.A number of different options are implemented in the program, and extensions are straight-forward.The paper is structured in the following way. First, the parameters used to describe the SiPM performanceand the different models implemented so far are discussed. Then, examples of the usage of the programare given: A comparison of simulation results with experimental data and studies of the impact of differentassumption on measured charge spectra. Finally, the main results are summarized.
2. The SiPM response model
First, an overview over the program flow is presented. The program simulates individual events. Forevery event primary Geiger discharges induced by light and by dark counts are generated, and their times andamplitudes are stored in an array, which is called
Geiger Array . Next, for every primary Geiger dischargetime-correlated discharges due to prompt and delayed cross-talk and after-pulses are generated and theiramplitudes and times are appended to the
Geiger Array . Electronics noise and random fluctuations forthe individual Geiger discharges are taken into account. To every entry in the
Geiger Array a normalisedcurrent pulse with the shape of the current from a single Geiger discharge is assigned.The charge for an event is obtained by the sum of the integrals of the current pulses in the readoutgate. For the current transients the time interval of the simulation is subdivided into time bins. The currenttransient of an event is obtained by the sum of the individual current transients in the time bins convolvedwith the response function of the readout. The 2-D distribution, pulse amplitude versus time difference forpulses exceeding a specified time interval, ∆ t min , is obtained by time ordering the elements of the GeigerArray , summing the pulses occurring in the time interval ∆ t min and calculating the time distance to thefollowing pulse. As discussed in Ref. [9], this 2-D distribution is a powerful tool to characterise SiPMs.In the following are presented the model parameters used, the assumptions for the different optionsimplemented in the program and their physics motivation . More details on the functioning of SiPMs canbe found in Refs. [2, 3, 9, 10]. Table 1 summarises the parameters and variables used. Time interval for the simulation and gate: t , t start , t gate , n t . The gate during which the currenttransient is integrated for the charge measurement, starts at t start and has the length t gate . The simulationof Geiger discharges is performed in the time interval − t ≤ t ≤ t start + t gate . The time interval t + t start ,which precedes the start of the gate, has to be chosen sufficiently long so that the current integral of aGeiger discharge from a dark count at t = − t is sufficiently small so that it can be ignored. The parameter t start is introduced so that delay curves can be simulated, which can be used to determine pulse shapesfrom charge measurements as discussed in Ref. [11]. For the simulation of the current transients the time2nterval between t and t start + t gate is divided in n t equal time bins. For n t a power of 2 is chosen so thatthe FFT (Fast Fourier Transform) algorithm with the number of operations increasing only ∝ n t · log ( n t ) asopposed to ∝ n t can be used. Gain fluctuations: σ G . The gain of a SiPM is given by G = C · ( V bias − V of f ) , where C is the sum ofthe capacitance of a single pixel and the capacitance parallel to the quenching resistor, and V bias the biasvoltage and V of f the voltage at which the Geiger discharge stops. It has been observed that differencesin V of f dominate the gain fluctuations [11]. The Geiger discharge stops when the voltage is too low tomaintain the discharge. This is a statistical process which causes fluctuations of V of f . In addition, theaverage of V of f can depend on the position of the Geiger discharge in a pixel because of variations of theelectric field and may also be different from pixel to pixel. Last but not least also the capacitance C mayvary from pixel to pixel. In the simulation the mean amplitude of a primary Geiger discharge is set toone, and gain fluctuations are taken into account by multiplying the amplitudes of Geiger discharges byGaussian-distributed random numbers with mean one and rms spread σ G . Number, times and amplitudes of primary Geiger discharges from photons: N γ G , n γ G , A i γ G , t i γ G . N γ G is the mean number of primary Geiger discharges induced by photons, and n γ G the actual numberfor a given event. Implemented in the program are a fixed number, n γ G = N γ G , which assumes that N γ G is an integer, or random n γ G values generated according to a Poisson- or a Gauss-distribution. ThePoisson distribution describes the statistics of a light pulser, like an LED or a laser, whereas the Gaussdistribution is more appropriate for a calorimetric measurement. For the occurrence of the individualdischarges, i γ G = ... n γ G , random times, t i γ G , are generated. The choices offered by the program are aGauss distribution, which approximates the light from an LED or a laser, the sum of two exponentials andthe distribution ( e − λ · t − e − λ · t ) · λ /( λ − λ ) , which approximates the time distribution of photons froma scintillator, and from a scintillator read out via wave-length shifters. For the amplitudes, A i γ G , randomnumbers according to a Gauss distribution with mean one and rms width σ G are generated. In this wayfluctuations in the amplitudes of the Geiger discharges are taken into account. The values of A i γ G and t i γ G are stored in the Geiger array , for the further analysis.
Number, times and amplitudes of primary Geiger discharges from dark counts:
DCR , n DC , A iDC , t iDC . Dark counts result from the thermal generation of electron-hole pairs which initiate Geiger discharges.They occur randomly in time, and their mean number in the time window of the simulation N DC = DCR · ( t + t start + t gate ) for the dark-count rate DCR . The number of dark pulses in a given event, n DC ,is obtained as a Poisson-distributed random number with the mean N DC . The time of an individual darkcount, t iDC , is obtained from a random number uniformly distributed between − t and t start + t gate . Asfor the Geiger discharges from light, for the amplitudes, A iDC , random numbers according to a Gaussdistribution with mean one and rms width σ G are generated. The list of A iDC and t iDC is appended to the Geiger Array . Prompt cross-talk: p pXT , A ipXT , t ipXT . Prompt cross-talk is caused by photons produced in the Geigerdischarge which convert in the amplification region of a different pixel and cause a Geiger discharge there.Given the short light path, they can be considered prompt. The light path can be directly through the siliconbulk, via reflection on the back or the front surface of the SiPM, or via a detector coupled to the SiPM.The introduction of trenches filled with light-absorbing material has significantly reduced the probability ofprompt cross-talk [2]. Three probability distributions for prompt cross-talk with the characteristic parameter p pXT are implemented: Binomial, Poisson and Borel These distributions are used to generate for everyprimary Geiger discharge a random number, n pXT , the number of prompt cross-talk Geiger discharges. If n pXT >
0, the
Geiger Array is extended for every prompt Geiger discharge by n pXT entries with the timesof the primary Geiger discharge and the amplitudes drawn from Gaussian-distributed random numbers withmean one and rms width σ G . Delayed cross-talk probability and time constant: p dXT , τ dXT , A idXT , t idXT . Discharges which aredelayed with respect to the primary Geiger discharge are called delayed cross-talk. The dominant source ofdelayed cross-talk are photons from the primary Geiger discharge which convert to an electron-hole pair in The Borel probability density function [12] e − p · n · ( p · n ) n − / n ! describes the occurrence of n events including the primaryoccurrence for the probability parameters p . Therefore the number of prompt cross-talk discharges using the Borel distribution is n pXT = n −
1. In Ref. [13] it is shown that the Borel distribution can be used to describe the cross-talk distribution of SiPMs. t dXT , is added to the time of the primary Geiger discharge. For t dXT , a random number withthe time distribution e − t / τ dXT · Θ ( t ) is generated, where Θ ( t ) is the Heavyside step function. As the originof the delayed cross-talk are photons generating electron-hole pairs in the non-depleted bulk with minoritycarriers diffusing into the multiplication region and producing Geiger discharges there, the exponential timedependence is certainly only a crude approximation. A more realistic model for the time dependence, ifavailable, should be used. After-pulse probability, time constant and signal-recovery time: p AP , τ AP , τ rec , A iAP , t iAP . After-pulses are delayed Geiger discharges in the same pixel in which the primary Geiger discharge has occurred.They can be caused by charges which are trapped by states in the silicon band gap and released aftersome time. Compared to delayed cross-talk, the simulation of after-pulses is more complicated as theprobability as well as the amplitude are influenced by the preceding Geiger discharge. To simplify thesimulation only a binomial distribution is used, which results in either no or one after-pulse. The time of apossible after-pulse, t AP , is obtained from a random number with the time distribution e − t / τ AP · Θ ( t ) , andthe probability from p AP · ( − e − t AP / τ rec ) . The term ( − e − t AP / τ rec ) which takes into account that theGeiger-breakdown probability is reduced as long as the pixel is recharging, results in the reduced after-pulseprobability p AP · τ rec /( τ rec + τ AP ) . For a similar reason, the average amplitude of the after-pulse is notone but ( − e − t AP / τ s ) , which is multiplied with a Gaussian-distributed random number with mean ona and rms spread σ G to account for the gain fluctuations. Geiger Array: n G , A iG , t iG . At this stage of the simulation the Geiger Array contains for all n G Geigerdischarges of an event the amplitudes, A iG , and the time stamps, t iG . This information is used in thefollowing to calculate for a given pulse shape charge spectra, transients and time difference distributions. Pulse shape of a single Geiger discharge: τ s , r f , τ f . Typical pulses from a SiPM have two components:A slow one from the recharging of the pixel, and a fast one if there is a capacitance in parallel with thequenching resistor [9]. For the pulse of a single Geiger discharge at t = I ( t ) = (cid:32) − r f τ s · e − t / τ s + r f τ f · e − t / τ f (cid:33) · Θ ( t ) , (1)is implemented. The time constants of the fast and slow component are τ f and τ s , respectively, and r f isthe contribution of the fast component As the integral ∫ ∞ I ( t ) d t =
1, current amplitudes and charges usethe same normalisation.
Total charge and electronics noise: Q , σ . The total charge of an event, Q , is obtained from the GeigerArray by adding to Q (cid:48) = n G (cid:213) iG = A iG · ∫ t start + t gate t start I ( t − t iG ) d t (2)a Gaussian-distributed random number with the mean 0 and the rms width σ , which takes into accountthe electronics noise. Looping over many events and entering the Q values into a histogram, gives the finalsimulated charge spectrum. Current transient, electronics noise and response function: n t , I it , t it , σ I , R j , σ R . Similar to Eq. 2, thecurrent transient for an event without electronics noise is I (cid:48) ( t ) = n G (cid:213) iG = A iG · I ( t − t iG ) , (3)from which I (cid:48) it = I (cid:48) ( t it ) , the current values at the times t it , which have been introduced before, are obtained.Electronics noise is introduced by adding to every I (cid:48) it -value a Gaussian-distributed random number with4ean 0 and rms width σ I . The result convolved with the electronics response function R j using the FFT(Fast Fourier Transform) gives I it . Presently a Gauss function with an rms width σ R is used for R j . Otherresponse functions including electronics filters can be easily implemented.Group Symbol Description τ s time constant slow componentpulse shape τ f time constant fast component r f fraction fast componentgate t start gate startand t gate gate lengthtime − t start time simulationparameters n t number time bins transient simulation N γ G mean number dischargesprimary Geiger n γ G actual number dischargesdischarges light A i γ G amplitudes of discharges t i γ G times of discharges DCR dark-count ratedark n DC actual number dark countscounts A iDC amplitudes of discharges t iDC times of discharges p pXT probabilityprompt n pXT actual number dischargescross-talk A ipXT amplitudes of discharges t ipXT times of discharges p dXT probabilitydelayed τ dXT time constantcross- n dXT actual number dischargestalk A idXT amplitudes of discharges t idXT times of discharges p AP probability τ AP time constantafter- τ rec Geiger probability recovery timepulses n AP actual number discharges A iAP amplitudes of discharges t iAP times of dischargesnoise σ G gain fluctuationsand σ electronics noise Q-measurementelectronic σ I transient current noiseresponse R ( t ) electronic response functionfunction σ R rms width for Gaussian R ( t ) n G total number Geiger dischargesGeiger array A iG amplitudes of Geiger discharges t iG times of Geiger discharges Table 1: Parameters and variables used in the simulation program.
3. Selected results
The 2D distribution of the amplitude, A , versus the time difference, ∆ t , for dark counts is a powerful toolto characterise noise and correlated Geiger discharges of SiPMs. The time difference between consecutive5eiger discharges with a separation exceeding ∆ t min is denoted ∆ t . For A the amplitudes of Geigerdischarges with ∆ t < ∆ t min are added. The A value of the later discharge or group of discharges is plotted. Figure 1: 2D-distribution of simulated dark-count events in log ( ∆ t ) bins. x axis: time difference, ∆ t , between consecutive Geigerdischarges; y axis: current amplitude, A , of the later Geiger discharge. For more details see text (colour on-line). Fig. 1 shows for 5 × events in a time interval of 50 µ s for DCR = p pXT = . p dXT = . p Ap = . , the A versus ∆ t distribution in log ( ∆ t ) bins. The horizontal bands for A = ∆ t min . The rising band between A = ∆ t (cid:46)
50 ns is due to after-pulses with reduced amplitudes because of the recharging of the pixel. (a) (b)
Figure 2: Distribution of the time difference, ∆ t , between consecutive Geiger discharges separated by more than 1 ns for the 5 × simulated dark-count events presented in Fig. 1. (a) All Geiger discharges in log ( ∆ t ) bins. (b) Geiger discharges for different selectionsof the amplitudes, A , in linear ∆ t bins. Fig. 2a shows the projection of the A vs ∆ t distribution on the ∆ t axis in log ( ∆ t ) bins. In the absence ofcorrelated Geiger discharges the expected distribution for random dark counts is ∝ e − DCR · ∆ t , which resultsfor log ( ∆ t ) bins in the dependence ∆ t · e − DCR · ∆ t . This explains the linear rise for ∆ t (cid:38)
200 ns and the peakclose to 1 / DCR = µ s. The deviation from the linear dependence for ∆ t (cid:46)
200 ns is due to correlatedGeiger discharges from cross-talk and after-pulses. These values of the parameters are significantly bigger than what is achieved by present day SiPMs. They have been chosen tobetter illustrate the different effects ∆ t dependence in equal size bins for all A values, and for the A =
1, 2, ... bands. For ∆ t (cid:38)
200 ns the expected dependence for dark counts ∝ e − DCR · ∆ t is observed for all A values. For small ∆ t values the curves deviate from the exponential dependence because of correlated Geiger discharges. Theeffect is strongest for all A , where the increase towards ∆ t min is due to delayed cross-talk and after-pulses.The A = A ≥ In Ref. [11] detailed measurements for a KETEK SiPM with 4382 15 µ m × µ m pixels are presentedand a fit program is described and used to extract the SiPM parameters as a function of bias voltage. For thisSiPM the turn-off voltage at room temperature is V of f = .
64 V, and the pulse shape is a single exponentialwith the decay time τ ≈
20 ns. For voltages between 29.5 V and 35.0 V in steps of 0.5 V, charge spectrawith and without illumination by a pulsed LED and a gate of width t gate =
100 ns were recorded. For afixed light intensity the average number of primary Geiger discharges, N γ G , increases with voltage becauseof the increase of the Geiger probability. The values found from the fit program are N γ G = .
79 at 29.5 Vincreasing to 1.64 at 35 V. The dark-count rate,
DCR , increases from 85 kHz to 210 kHz. For the voltagedependence of the other SiPM parameters we refer to Ref. [11].Fig. 3 compares at 31 V the transient recorded with a CAEN setup (Ref. [14]) with 250 simulatedtransients. The comparison allows to estimate the pulse shape and the band width of the readout. (a) (b)
Figure 3: Comparison of (a) measured and (b) simulated transients at a 31.0 V bias voltage. The bands corresponding to zero, oneand more Geiger discharges can be well distinguished. In addition, dark counts and delayed pulses are seen. The experimental datawere recorded with the CAEN setup [14] which has a bandwidth of 125 MHz.
Figs. 4a and 4c show for the different voltages the simulated and measured charge spectra with illumina-tion, and Figs. 4b and 4d without illumination. In order to allow a comparison between experimental dataand simulations, the charge scale of the simulated data is scaled by the gain and shifted by the pedestal whichwere determined in Ref. [11]. A satisfactory qualitative agreement between measurements and simulationsis found. Figs. 4e and 4f show 50 simulated transients at 29.5 V and at 35 V, respectively. An increase inphoton detection efficiency, in
DCR , as seen from the pulses preceding the light pulse, and an increase indelayed cross-talk and after-pulses is clearly observed.For a more quantitative validation of the simulations, standard methods are used to extract SiPMparameters from the simulated charge spectra and compare them to the input values.The mean number of primary Geiger discharges from the illumination, N γ G , can be obtained from N γ G = − ln (cid:16) N γ ped / N γ tot (cid:17) + ln (cid:16) N darkped / N darktot (cid:17) , (4)with N γ ped and N darkped the number of events in the pedestal with and without illumination, respectively, and N γ tot and N darktot the corresponding total number of events. Fits of Gauss functions to the charge spectra in7 a) (b)(c) (d)(e) (f) Figure 4: Comparison of simulated charge spectra with the experimental results of Ref. [11] for voltages between 29.5 V and 35.0 V:(a) Simulated charge spectra with illumination, shifted vertically by increasing multiples of 10 . (b) Simulated charge spectra withoutillumination, multiplied by an increasing power of 4. (c) Measured charge spectra with illumination, shifted by an increasing multipleof 2500. (c) Measured charge spectra without illumination, multiplied by an increasing power of 4. (d) 50 simulated transients at29.5 V. (e) 50 simulated transients at 35.0 V. the pedestal regions are used to determine N ped values. Eq. 4 uses the fact that in the absence of a primaryGeiger discharge there is also no correlated noise, and that the probability of zero occurrences for a Poissondistribution with mean N is e − N . The second term in Eq. 4 takes into account that dark counts reduce8 γ ped . Fig. 5a compares as a function of voltage the N γ G values assumed in the simulation with the valuesobtained from the spectra of Fig. 4 using Eq. 4. For all voltages the values agree within 1 %. (a) (b) Figure 5: Comparison of the values of (a) the mean number of primary Geiger discharges from light, N γ G , and (b) the dark-count rate, DCR , extracted from the simulated charge spectra (Figs. 4a, 4b) with the input values obtained from the measured spectra (Figs. 4c,4d) in Ref. [11]. For more details see text.
The dark-count rate,
DCR , can be obtained from the spectra without illumination using
DCR = N dark > . /( N darktot · t gate ) (5)with N dark > . , the number of events in the dark spectrum above half the total charge of a single Geigerdischarge. Fig. 5b compares as a function of voltage the DCR values assumed in the simulation with thevalues obtained from the spectra of Fig. 4b. For U <
30 V the peaks of 0 and 1 Geiger discharge are not wellseparated. As a result, the
DCR values determined are too high. In the region 30 V ≤ U ≤
32 V the assumedand the extracted values agree within 1 %. For higher voltages the extracted values are systematically higherthan the input values. The deviation increases approximately linearly with voltage, and reaches a value of ≈ Q < . Q > . Figs. 6 and 7 show the simulated charge spectra for the SiPM with and without illumination for r f , thefraction of the fast component, between 0 and 50 % in 5 % steps and t gate values of 100 ns and 20 ns. Thetime constants of the fast and slow component of the signal are 1.5 ns and 20 ns, respectively. The charge Q is normalised so that the integral of the current of a single Geiger discharge (Eq. 1), ∫ ∞ I ( t ) d t = DCR =
200 kHz for the dark-count rate, and N γ G = t gate =
100 ns with illumination are notaffected by a change of r f . The reason is that both the fast and the slow component of the prompt Geigerdischarges are fully integrated. However, the region in between the peaks is affected by r f . The number ofentries at Q = . r f = r f =
50 %. This region is populated by darkcounts occurring approximately one time constants before the start and before the end of the gate, resultingin Q ≈ .
5. For τ f (cid:28) τ s , the fast component hardly contributes and the fraction of events at Q ≈ . ∝ ( − r f ) .As expected the results for N γ G , as obtained from the fraction of events in the pedestal peak, are notaffected by the change of r f . The same holds for the extraction of the dark-count rate from the spectra ofFig. 6b. Similar to Sect. 3.2 DCR =
205 kHz is obtained, which is again a few percent higher than the inputvalue of 200 kHz. 9ig. 7a shows that the peak positions of the charge spectra for t gate =
20 ns with illumination increaseas a function of r f . The peak from a single Geiger discharge moves from ≈ . r f = r f =
50 %. The reason is that the entire fast component but only a fraction of the slow component areintegrated by the short gate. The determination of N γ G is not affected by r f . The dark spectra of Fig. 7b for t gate =
20 ns show that for r f = r f . As a result the determination of DCR using the methoddescribed in Sect. 3.2 becomes unreliable for shorter gates. (a) (b)
Figure 6: Dependence of the simulated charge spectra on r f , the contribution of the fast signal component, for t gate =
100 ns and
DCR =
200 kHz. (a) Illuminated SiPM with an average of 5 primary Geiger discharges. For clarity the spectra are shifted verticallyby increasing multiples of 5000. (b) SiPM without illumination. For clarity the spectra are multiplied by increasing powers of 4. (a) (b)
Figure 7: Same as Fig. 6, however for t gate =
20 ns.
In this section the influence of the dark-count rate,
DCR , on the charge spectra for two different gatelengths, t gate , is investigated. This is relevant for the use of SiPMs in high radiation environments, whichcauses an increase in DCR by radiation damage (Ref. [15], [16]), and for applications in the presence ofbackground light. 10igs. 8 and 9 show for gate lengths t gate =
100 ns and 20 ns simulated charge spectra with and withoutillumination for
DCR values between 100 kHz and 50 MHz. For the spectra with illumination the numberof light-induced primary Geiger discharges N γ G =
5. For the pulse shape a single exponential with thetime constant τ s =
20 ns, and for the probabilities of after-pulses, and of prompt and delayed cross-talk thevalues 15 %, 10 % and 10 % are assumed. (a) (b)
Figure 8: Simulated charge spectra as a function of the dark-count rate,
DCR , between 100 kHz and 50 MHz for pulses with the timeconstant of τ s =
20 ns and a gate length t gate =
100 ns. (a) Illuminated SiPM with a mean number of primary Geiger discharge N γ G =
5. For clarity the spectra are shifted vertically by an increasing multiple of 10 . (b) SiPM without illumination. For claritythe spectra are multiplied by an increasing power of 4. (a) (b) Figure 9: Same as Fig. 8 with t gate =
20 ns.
Up to
DCR = DCR the mean chargeand the background below the peaks increase, and the amplitudes of the individual peaks decrease. For
DCR =
50 MHz, which corresponds to an average of 5 dark counts in a 100 ns gate, the individual peakshave essentially disappeared and the mean charge is shifted by approximately
DCR · t gate · ∫ t gate I ( t ) d t .Comparing the charge spectra with illumination for t gate =
100 ns (Fig. 8a) with the ones for t gate =
20 ns(Fig. 9a) shows that the degradation of the spectra with
DCR is worse for the longer t gate . For t gate =
20 ns11eaks corresponding to different number of Geiger discharges are still visible at 50 MHz. This reason isthat more dark counts are integrated for the longer gate.Similar effects are observed when comparing the dark spectra for t gate =
100 ns (Fig. 8b) and 20 ns(Fig. 9b): The longer the gate the more dark counts are integrated which results in an increase of the meancharge. It can also be noted that for t gate =
100 ns up to 20 MHz pedestal and single Geiger discharge peakare well separated, and
DCR can be determined using Eq. 5.
4. Summary and outlook
A relatively simple and flexible Monte Carlo program for the simulation of the response of SiPMs ispresented. For a given dark-count rate and the mean number, frequency and time distribution of primaryGeiger discharges from a light source, correlated Geiger discharges corresponding to after-pulses andprompt and delayed cross-talk are generated. A number of different physics-based models and statisticaltreatments for the simulation of correlated Geiger discharges are implemented and can be selected.To illustrate the usefulness of the simulation program, a number of examples for its use are given:• Simulation of amplitude versus time difference distributions of consecutive Geiger discharges fordark counts.• Comparison of the current transients measured with a transient recorder with simulated transients.• Comparison of the charge spectra measured with a gated QDC (charge to digital convertor) withsimulated spectra.• Simulation of the influence of a fast component of the SiPM pulses on the charge spectra as recordedwith a QDC.• Simulation of the influence of dark-count rates up to values of 50 MHz on the charge spectra asrecorded with a QDC with different gate lengths.These examples demonstrate that such a simple but flexible simulation program can be used to test and verifyanalysis programs used to extract SiPM parameters from experimental data, determine SiPM parameters bycomparing measurement and simulation results, and evaluate and understand the impact on the measurementof photons of different SiPM parameters, probability distributions for cross-talk and after-pulses and readoutschemes.
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