Functions Represent SiPM Response Especially Linear Behavior After Saturation

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1 1 2 Functions Represent SiPM Response Especially Linear Behavior After Saturation 3 Katsushige Kotera, Weonseok Choi, and Tohru Takeshita 4 October 2, Abstract We developed functions to represent wide ranges of SiPM responses The functions model reactivation of pixels of the SiPM during each incident event Because the number of detected photons by reactivation increases rationally and linearly in our model, the linear behavior of SiPM response after saturation, for which the reason was unknown, can be represented with our functions From 72 samples of SiPM responses, the functions were tested They showed their high performance with one additional correction to the simple function 1 Introduction The pixelated photon detector (PPD, also known as SiPM)[1] is being developed rapidly for many fields that require detection of photons in a small space, and even in magnetic fields With SiPM, high gain of O(1 5 ) is obtained with lower bias voltage of less than 1 volts One complicated aspect of this device is that the response is nonlinear As the first order of the expression of such a SiPM behavior,equation 1 is frequently used, ( Nfire LO = N pix 1 e ) ϵnin/npix (1) Therein, N fire is the number of fired pixels, N pix is the number of pixels this SiPM has, N in is the number of photons that arrived at the sensor, and ϵ is the detection efficiency This equation, leading order (LO), explains the saturation phenomenon of SiPM: the charge of each pixel is released only once in an event by arriving photons so that some multiple hits on a pixel are counted as one hit However, when the recovery time of each cell of SiMP is faster than the duration of one event, as it is for MPPC produced by Hamamatsu K K, [2] some probability exists of each cell contributing to an increasing signal more than once [3] Then it makes the N pix greater than the real number of pixels effectively In such cases, Equation 2, LO, in which N pix of Equation 1 is replaced with Npix eff is used to represent experimentally obtained results of SiPM response more realistically [4]: N LO fire ) = Npix (1 eff e ϵn in/n eff pix, (2) 13 where Npix eff functions as a fitting parameter when this equation is fitted to real data 14 However, Equation 2 is applicable in a limited region of the number of incident photons, even 15 with Npix eff We often meet a rather linear response with numerous incident photons [5] Some 16 efforts at implementing the mechanism of the crosstalk and the after pulse failed to explain this 17 phenomenon Therefore, we design some functions for this phenomenon and demonstrate how 18 those functions represent the response in this article 1

2 Empirical functions 21 The next leading order Figure 1 presents the response of an SiPM according to Equation 1 as a function of the number of photons arriving at the SiPM pixels (solid) together with a line representing the response without saturation (dashed) We can recognize that the number of photons which can contribute 5 Number of fired pixels Incident photons ( = 1) Figure 1: Ideal response of a 16 pixel SiPM (solid curve) and response without saturation (dashed line) with parameter ϵ = 1 the multiple hits on a cell increases as the number of incident photons increases The number of photons between the dashed line and solid curve in Fig 1 can contribute to the multiple hits Considering this affect, Equation 1 is modified in the following, next leading order (NLO): ( )}] Nfire NLO = N pix 1 e )[1 ϵnin/npix + α {ϵn in N pix (1 e ϵnin/npix, (3) where α is only one additional parameter representing the ratio of the multiple hits on a pixel depending on the relation between the pixel recovery time and the signal duration As the simplest correction we do not care about the complex time structure of the pixel recovery 22 Variance of the decay time constant When we use Equation 2, LO, we assume that the decay time constant of the signal, τ, of SiPM is constant, and that the constant is determined with the resistor and capacitor of each pixel, τ = RC The resistor is called the quenching resistor, which finishes the avalanche that occurred in each pixel Although the variance of R and C contributes to the width of the photoelectron peaks of the single photon spectrum, they are negligibly small for our discussion However, additional R C exists when we consider the resistor coupled with the whole of pixels in series as a parasitic resistor Figure 2 presents a cartoon explaining such a case Figure 2 right demonstrates that the sum of the fired capacitors of pixels works toward the parasitic resistor In this case, the decay constant has a part, τ parasitic contributed according to Equation 4: τ parasitic = R parasitic C n fired, (4) 3 31 where R parasitic is the resistor coupled with all pixels, C is the capacitor of each pixel, and n fired is the number of fired pixels 2

3 Figure 2: Set of cartoons explaining the effect of a parasitic resistor left Usual model right Model where the decay time depends on the number of fired pixels Considering this effect, Equation 3 is modified as the following: NLO : Nfire NLO = Nfire (1 NLO e γ ) 1+βN fire NLO, (5) where Nfire NLO is the same as in Equation 3, γ is the integration time measured in term of τ = RC, and β is the parasitic resistor measured in term of R, the quenching resistor of each pixel We fit this Nfire NLO to 72 data of MPPCs described in section 3 23 Crosstalk and After pulse The SiPM responses are influenced by other effects, crosstalk and after pulse A crosstalk phenomenon occurs when a photon is created by the primary avalanche It subsequently makes a second avalanche at the surrounding pixels [1] An after pulse occurs when a second avalanche is seeded by the release of trapped electron in some lattice defect of silicon crystal or a diffused hole, charge carrier, generated in the bulk of SiPM occurs some time after the primary avalanche It then increases the signal charge The function to represent the number of fired pixels considering those effects is the following: ( ) Nfire C A = LO ( ) 1 + P cross e ϵn in/n pix (1 + P after ) (6) 36 Therein, where P cross and P after respectively represent the probabilities of the crosstalk and the 37 after pulse The effect of the after pulse simply increases the apparent number of fired pixels It 38 is mostly absorbed in the Npix eff in Equation 2, LO Therefore, we do not add this effect into our 39 previous equations However, the effect of the crosstalk changes the shape of response curve, 4 although it is small as discussed in section 43 One additional parameter for a small effect serves 41 only to produce confusion Therefore, we test Equation 6 separately from other functions Data fitted The 72 samples of responses of 16 pixel (25 25 µm 2 pitch) MPPCs, MPPC-11-25M [2] attached on a 1 mm diameter wavelength shifting (WLS) fiber Y-11 2 [6] inserted into individual scintillator strip of mm 3 were measured Figure 3 shows the setup of the measurement Those units consisting of the scintillator strip enveloped into a reflecting film, the WLS 1 For ILC-ScECAL provided by Hamamatsu KK 28, corresponding to S P 2 Kuraray Co Ltd 3

4 fiber and the MPPC are developed for the CALICE Scintillator strip electromagnetic calorimeter (ScECAL) for the international linear collider experiment [4, 7] The scintillator strips were obtained by cutting from 3 1 mm 2 cross-section scintillator bars to 45 mm lengths The scintillator bars were produced using the exclusion method [8] having a hole to install the WLS fiber at the centroid along the scintillator bar length The WLS fiber has 1 ns decay time [9] Therefore, the effect of multiple hits on a pixel is clarified with this system A picosecond laser pulse 3 was injected into the scintillator strip via a hole of a few millimeters diameter on the reflector film: the laser intensity was controlled by rotating a polaroid film between the laser source and another polaroid film; a half miller was put between a optical lens and the sample scintillator strip to separate a light way to monitor the laser light intensity The laser light frequency was 48 nm Figure 3: Setup of the Npix eff measurement: a) target scintillator enveloped in reflector (left, top view; right, side view); b) WLS fiber; c) irradiation position with a small hole in reflector; d) MPPC; e) half mirror: f) photomultiplier tube; g) lens; h) polaroid (fixed); and i) polaroid (rotatable) A bias voltage of 3 V greater than the breakdown voltage was provided to individual MPPCs The voltage of output pulses from the MPPC was integrated into the charge during the 15 ns synchronized with a signal from the laser pulser Each measured signal in a charge using an analog-digital-converter 4 was converted to the number of fired pixels by division by the signal strength corresponding to a fired pixel, which was preliminarily measured For this procedure, spectra of a few photo electron (pe) peaks are expanded to have separation in pe peaks using an amplifier 5 A ratio to interpret the ADC counts with the amplifier to the ADC counts without the amplifier was measured with a laser light intensity whose spectrum ranged in full scale of the ADC counts in both with and without the amplifier Although the laboratory temperature was controlled using a common air conditioner, the measured number of photons was corrected using a preliminarily measured linear relation between the response and the temperature The temperature was measured in one-minute intervals when data were obtained The laser light intensity in the same timing as the MPPC signal was measured using a photo-multiplier tube (PMT) that had confirmed linearity in the range of this experiment The integrated charge of PMT pulse was converted into a digital number using an ADC as well as measurement of the MPPC signal strength The ratio between the ADC counts of PMT and the number of incident photons is absorbed in the parameter ϵ for all functions Figure 4 presents plots of the responses from nine MPPCs as functions of the ADC counts of the PMT: 72 MPPCs are grouped in eight readout cables Other groups have the same behavior as that shown in Fig 4 Note that the ADC counts of PMT corresponding to the order O(2) times of the number of incident photons on MPPC 3 PiL4X (Head) + EIG2DX (Controller) provided by Advanced Laser Diode System ALS GmbH 4 Charge sensitive ADC, C9 provided by Hoshin electronics Co LTD 5 Amplifier-shaper-discriminator (ASD) IC and 16-ch ASD board for ATLAS [1] 4

5 Number of fired pixels ADC counts of PMT Figure 4: MPPC responses as nine examples The responses come closer to the linear behavior as the incident light intensities increase to more than1 ADC counts of PMT Fitting to the data 41 Fitting with the LO To represent the MPPC response shown in Fig 4 with Equation 2, LO, it is necessary to limit the range of the fitting Otherwise, the deterioration of fitting produces a meaningless result The upper limit of the fitting range is given as the inflection point of difference of the N fired ( N fired ) with respect to the ADC counts of the PMT: a logarithmic plot of this shows a kink of the linear behavior around 8 ADC counts Figure 5 shows such a fitting result as an example Number of fired pixels χ 2 /NDF = 146 / 12 eff N pix = 226 ± 11du ADC counts of PMT 85 Figure 5: Example of fitting result of Equation 2 (LO) to MPPC responses as a function of the incident photon measurement using PMT The range of fitting was limited up to 8 ADC counts of PMT 86 Large Npix eff eff, of 226 corresponds to reactivation of the pixels: The average of Npix of MPPCs is 2428 with 245 standard deviation Although constant correction for the reactivation 88 phenomenon such as Npix eff works in the limited range shown in Fig 5, rather linear like behavior 89 is apparent after saturation Fitting with NLO function Figure 6 left presents a typical result of fitting with the Equation 5 (NLO ) to data: the solid line shows the fitting result The vertical dashed line represents the limit of application of the 5

6 LO function, enabling us to recognize great improvement of the NLO function According to our model, the reactivation of pixels is represented with the term having coefficient α instead of Npix eff ; N pix was fixed to 16 in this fitting 4 25 Number of fired pixels cable: 3, channel: 9 Npix: 16 ± PMT PDE scale: 658 ± 6 Ratio whole C: 22 ± 6 Retio recovory: 13 ± 1 Gate: (126 ± 36) tau χ 2 /NDF = 22 Entries : N pix = free : N pix = ADC counts of PMT χ 2 /NDF Figure 6: left Example of fitting result of Equation 5 (NLO ) to an MPPC responds as a function of the incident photon measured with PMT The full range of our data was taken as the fitting range The dashed vertical line shows the limit of LO fitting right Distribution of the reduced χ 2 of MPPCs The hatched histogram shows that the case N pix is a fitting parameter, whereas the open one shows case N pix is fixed at Figure 6 right shows that the distribution of the reduced χ 2 of 71 (72) samples in N pix is fixed at 16 (a free parameter): one datum with N pix = 16 exceeds the range of this plot Although the fitting result with fixed N pix slightly degrades more than that with the free N pix = Npix eff, those are comparable to demonstrate that our model of reactivation of pixels works β: parasitic-resist(quenching resist) γ: integration time o(τ) Figure 7: Correlation between fitting parameter β and γ in Equation 5 Solid line is the fitting result with a linear line: β = (165 ± 6) 1 4 γ + (13 ± 45) 1 5 Figure 7 shows the correlation between the parameter β and γ in NLO, which are introduced respectively to model the parasitic resistor measured in terms of the quenching resistor and the integration time A strong correlation indicates that the ratio between β and γ is a specific property of our measuring system: we did not change the conditions used for taking the data The properties of MPPCs are well unified Although the integration time greater than hundreds of τ is unrealistic and a certain value must exist for our system, β and γ varied in a large range 6

7 16 with large uncertainty, as shown in Fig 7 This phenomenon means that the ratio of β/γ is a 17 stronger parameter to ascertain the figure of the function than β and γ individually Therefore, 18 we test a fixed value 3 τ for γ leaving β as a fitting parameter: the decay time, τ of µm 2 19 pitch MPPC is 5 ns [11]and we set the integration time as 15 ns 11 The distribution of reduced χ 2 as a result of fitting with γ = 3τ and free N pix does not 111 show a marked difference from the case with both β and γ as fitting parameters The mean 112 of Npix eff is 1776 ±264{root mean square (RMS)} This increase is consistent with the increase 113 of the charge of signals according to the after pulse phenomena The after pulse probability 114 of µm 2 pitch MPPC was measured as 1% with a 3 V over voltage [12] Agreement 115 between the increase of the N pix and the after pulse probability is discussed in subsection together with crosstalk phenomena Model considering the crosstalk and the after pulse Equation 6 correcting to Equation 2 does not improve the situation from Fig 5 Therefore, the main effect to change the response curve from LO ( ) does not derive from the crosstalk effect and the after pulse effect at least When the crosstalk term in Equation 6 is implemented in NLO 121 with fixed γ = 3 τ and free N pix = Npix eff, the crosstalk ratio is extracted as 11± 12(RMS) Al- 122 though this values is consistent with the crosstalk ratio of our MPPCs, the influence on the other 123 parameters are small; recovery ratio, α: {14 ± 17(RMS)} 1 4 {12 ± 3(RMS)} parasitic resister, β = {48 ± 4(RMS)} 1 3 {47 ± 4(RMS)} 1 3, and the scale fac- 125 tor, ϵ: 76 ± 27(RMS) 71 ± 27(RMS) without significant change of the reduced χ 2 s Discussion 51 Evaluation of the parasitic resistor The fitting with γ = 3 τ and free N pix gives the parasitic resistor β = {48 ± 4(RMS)} 1 3 R, where R is the quenching resistor on a pixel This value agrees with that estimated from the correlation between β and γ in Fig 7: β/γ = (165 ± 6) 1 4 The quenching resistor of our MPPC was measured as 25 MΩ The parasitic resistor then becomes 12 kω The substrate of MPPC and our readout circuit are some candidates of the source of the parasitic resistor We are preparing the next step to investigate their potential and to test with addition of some artificial parasitic resistor Additionally, we are seeking another mechanism which follows the same equation Need for decay time correction 137 Figure 8 left shows the fitting result obtained with Equation 3, NLO, with fixed N pix = 16, 138 demonstrating that the correction in Equation 5, NLO, is mandatory When the N pix is set as a fitting parameter, N pix = Npix eff eff 139, the fitting result is improved as shown in Fig 8 right The Npix 14 compensates apart instead of the correction of Equation 5 However, the fitting still degrades 141 more than in the case with NLO The Npix eff becomes extremely large Some readers think that 142 their integration time of data taking is sufficiently long However, the shift of charge by the 143 parasitic resistor on each pixel is very small and we cannot directly confirm whether the shift of 144 charge exists or not, because we cannot see individual pe peaks in spectrum of such a numerous 145 fired pixels Other parameters We discussed Npix eff, β, and γ Here, we discuss other fitting parameters α: ratio by which a photon can be detected by the reactivation of pixels The parameter is not greatly varied among analytical methods: {14 ± 17(RMS)} 1 4 with 7

8 Number of fitted pixels cable: 3, channel: 9 Npix: 16 + PMT PDE scale: Retio recovory: 6 + χ 2 /NDF = 1677 Number of fitted pixels cable: 3, channel: 9 Npix: PMT PDE scale: Retio recovory: 4 + χ 2 /NDF = ADC counts of PMT ADC counts of PMT Figure 8: left Example of fitting result of Equation 3 (NLO) with a fixed parameter N pix = 16 to an MPPC responds as a function of the incident photon measured with PMT The fitting line shows marked discrepancy from the data right In addition, a fitting result of NLO to the same data is shown with free parameter N pix = Npix eff The discrepancy is moderated by the fact that N eff pix absorbs it free Npix eff and fixed integration time γ = 3 τ, {117 ± 28(RMS)} 1 4 with N pix = 16 and free γ We confirmed that α became with an MPPC attached directly on the same size scintillator strip of Kuraray SCSN38 without WLS fiber The decay time of SCSN38 is 25 ns [6] 6 That of the WLS fiber Y11 is 12 ns [9] Although roughly speaking, the ratio of α quantitatively agrees with the ratio of those decay times, we need to consider the fact that 25 ns was shorter than the recovery time as in subsection ϵ: scale factor of MPT amplitude to interpret the number of incident photons 157 including the factor of the photon detection efficiency No great difference exists 158 among analysis conditions: 7 ± 28(RMS) with LO, 79 ± 27(RMS) with free Npix eff 159 fixed integration time of 3 τ, 8 ± 27(RMS) with N pix = 16 and free γ Number of fitting parameters A function with numerous fitting parameters naturally produces better fitting results Therefore, we discuss the number of fitting parameters of our functions Equation 2, LO, has two fitting parameters, Npix eff and ϵ When we go to NLO with fixed N pix the number of fitting parameters is four: α, β, γ and ϵ so that the increase is two When we take a fixed γ, even roughly estimated from our experiment, we can reduce one more fitting parameter without degradation and without sensitivity of the setting value of γ Although the term γ/(1 + βnfire NLO ) in NLO is affected from 1 in the denominator when Nfire NLO is small, the effect is not large because γ is usually set sufficiently larger than one In fact, Equation 7 provides fitting results without much degradation, N NLO fire = N NLO fire (1 e η N NLO fire ) (7) Therein, λ is a new parameter instead of β and γ Consequently, the increase of the number of fitting parameters is only one from LO, although its performance is markedly higher than that of LO 6 pulse shape from SCSN38 corresponds to SCSN61 in their catalogue 8

9 Additional tests of models with other types of SiPM Our new functions were tested to assess the responses of MPPCs attached on the WLS fiber/ scintillator systems The systems are suitable for our purposes because the rather long decay time of WLS fiber clearly produces the reactivation effect However, recent products that are achieved lower after pulse and lower crosstalk more clearly show that such a linear behavior begins at a rather lower number of fired pixels than the mechanical number of pixels even with direct injection of the picosecond laser light Although one of potential is the fact that even very small ration of α makes a linear behavior as we have shown so that we need only very short recovery time for this phenomenon, we are preparing measurements to test with our function on such new SiPMs for later studies By comparing those MPPCs with the old MPPCs, which have high crosstalk and high after pulse, we can also test Equation 6 even implemented into Equation 5, NLO Although the models represent the SiPM response well, this fact does not prove the correctness of the model We are preparing further investigations into which various sizes of parasitic resistor are added artificially to the readout circuit Simultaneously, we are considering potentials of other mechanisms following the same equation as that introduced in this report 6 Summary 183 The next leading order of the functions to represent SiPM behavior was developed and tested 184 with 72 MPPCs having 16 pixels in 1 1 mm 2 ; an MPPC was attached on the cross-section 185 of WLS fiber; the WLS fiber was inserted into the center of the scintillator strip, the dimensions 186 of which were mm 3 The functions fit well when an additional correction by intro- 187 ducing a change of decay time of pulses is implemented The applicable range is more than five 188 times larger than the LO, in which an effective N pix (Npix eff ) is implemented Those wide ranges 189 of response have rather linear behavior, which was not clearly explained in the mechanism to 19 date The number of fitting parameters can be reduced to one additional to the LO without 191 large degradation of fitting Acknowledgments The authors would like to thank the CALICE collaboration In particular, the CALICE-ASIA group provided a layer of the prototype of scintillator electromagnetic calorimeter and have contributed to numerous important discussions The authors extend special appreciation to Toru Iijima of Nagoya University, who allowed us to use the laser device This work is supported in part by a Grant-in-aid for Specially Promoted Research: a Global Research and Development Program of a State-of-the-Art Detector System for the ILC of the Japan Society for the Promotion of Science (JSPS, Grant No 232) References [1] S Gomi et al, NIM, A581, 427 (27) [2] Hamamatsu Photonics KK, [3] H Oide, et al, PoS(PD7), 8, (27) [4] K Francis et al, NIM, A763, 278 (214) [5] L Gruber, S E Brunner, J Marton, and K Suzuki, arxiv: , physicsins-det (213) [6] Kuraray Co Ltd, [7] K Kotera, Performance of the CALICE Scintillator-Based ECAL Depending on the Temperature Proceedings of LCWS211, Granada arxiv: (212) 9

10 [8] Sung Hyun Chang et al, JKPS, Vol53, No 6, 3178 (28) [9] O Mineev, etal, arxiv: (211) [1] O Sasaki, et al, Proceedings of 5th Workshop on Electronics for the LHC Experiments, LEB 99 (1999) [11] H Otono, et al,, PoS(PD7), 7, (27) [12] P Eckert, et al, NIM, A62, 217 (21) 1