- Email: [email protected]
Electrical characterization and simulation of SiPMs
Nuclear Instruments and Methods in Physics Research A 787 (2015) 340–343
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Electrical characterization and simulation of SiPMs Florian Scheuch n, Daniel Führen, Thomas Hebbeker, Carsten Heidemann, Markus Merschmeyer III. Physikalisches Institut A, RWTH Aachen University, D-52074 Aachen, Germany
art ic l e i nf o
a b s t r a c t
Available online 28 January 2015
Silicon Photomultipliers (SiPMs) are versatile and sensitive photon detectors that experience a fast growing variety of use in particle physics and related fields of application. These photo detectors have a very promising photon detection efficiency and are therefore interesting for very low light flux applications such as scintillation and fluorescence light detection. As a semiconductor device the SiPM's gain and time response strongly depend on the operating temperature and voltage. Thus they have to be understood for a proper use of the SiPM. Therefore, accurate electrical simulations of the SiPM's behavior involving electrical readout and front-end electronics help to improve the design of experimental setups, since several different designs can be tested and simulated with a manageable amount of effort. To perform these simulations, a detailed equivalent circuit of the SiPM has to be used containing a set of well-defined parameters. For this purpose, SPICE simulations of SiPMs and readout electronics have been performed. These simulations utilize an improved SiPM model consisting of resistors, capacitances and inductances. The SiPM parameters for these simulations have been determined by measuring the impedance over a wide frequency range while applying a DC voltage in forward direction and various DC voltages from zero up to the SiPM breakdown voltage in order to determine the behavior under operating conditions. The impedance measurements, the electrical model and the resulting simulations are presented. The impact of different setups and the electrical properties of the SiPM is discussed. & 2015 Elsevier B.V. All rights reserved.
Keywords: SiPM Single photon detection Equivalent circuit Impedance measurement
1. Introduction
2. Electrical equivalent circuit
Silicon Photomultipliers (SiPMs) are tiny and compact semiconductor devices for light detection at single-photon level. These devices have a growing field of use in particle and medical physics. Because of their semiconductor nature, SiPMs are very sensitive to temperature variations and their applied bias voltage. An electrical equivalent circuit has been introduced including the voltage dependency of the pn-junction. Impedance measurements have been performed to determine the circuit's parameters. These measurements have been done with different instruments to cover a huge frequency range. Using these measurements, SPICE simulations can be performed to investigate the signal output of different electronic designs for front-end electronics. This allows for an assessment of the electrical performance of amplifiers and power supply with less prototyping.
The equivalent electrical circuit is shown in Fig. 1. The circuit consists of SiPM cells and additionally parasitic components that are used for the whole SiPM. Every SiPM cell consists of a diode capacitance ðC D Þ that represents the p–n-junction within the SiPM as it is assumed as a parallel plate capacitor. The diode capacitance is discharged during a cell breakdown. A quenching resistor (RQ ) is placed in series to the diode capacitance as it is done in the real SiPM. Every real resistor has a parasitic capacitance that is called quenching capacitance ðC Q Þ. These three devices form a SiPM cell. To simulate parasitic effects of the routing grid and connection pins on the SiPM a grid capacitance ðC G Þ is introduced. This electrical equivalent circuit is based on work of Corsi et al. [1]. Related publications are [2–4]. In addition to this model a bulk inductance ðLB Þ and a bulk resistance ðRB Þ are used. These devices help to model the impedance behavior at high frequencies (see Section 3). Furthermore, the bulk inductance is the only device within the circuit that limits the slope of the rising edge in the SiPM output signal.
n
Corresponding author. Tel.: þ 49 241 80 27338. E-mail address: [email protected] (F. Scheuch).
http://dx.doi.org/10.1016/j.nima.2015.01.066 0168-9002/& 2015 Elsevier B.V. All rights reserved.
F. Scheuch et al. / Nuclear Instruments and Methods in Physics Research A 787 (2015) 340–343
341
Fig. 2. Impedance measurement and applied fit for an S10362-33-050C Hamamatsu SiPM with 3600 cells at zero bias voltage in the frequency range from 20 Hz to 500 MHz
Fig. 1. Electrical equivalent circuit of a SiPM.
The absolute impedance depending on the measurement frequency can be calculated using this model and yields 0 ! 1 1 1 1 1 @ A : j ZðωÞj ¼ iωLB þ RB þ iωC G þ n þ 1 i ω C D RQ þ iωC Q ð1Þ 3. Measurements of electrical parameters The values of the electrical parameters of the SiPM's equivalent circuit can be determined by measuring the impedance over a wide frequency range and fitting formula (1) to the measured data. These measurements were taken with an Agilent E4980A LCR meter in the frequency range from 20 Hz to 2 MHz and an Agilent E5061B-3LF LF-RF network analyzer in the frequency range from 100 kHz to 500 MHz. A DC voltage of up to 40 V can be applied when using the LCR meter. The measurement voltage amplitude is kept on a low level (50 mV) to reduce a variation of the p–njunction depth during the measurement. In a first step, the quenching resistance is measured. During this measurement a DC voltage of 2 V is applied to the SiPM in forward direction. The measurement frequency is set to 20 Hz. Thus, the diode capacitance of the p–n-junction vanishes. The quenching capacitance and the grid capacitance have very high impedances due to the low frequency and do hardly contribute to the total impedance. The inductance has a very low impedance due to the measurement frequency. The (ohmic) impedance is then calculated by RQ ¼
j Z measured j n
ð2Þ
with the measured impedance value Z measured and the number of SiPM cells n. In a next step, the impedance is measured without applied bias voltage over the whole frequency range. The remaining electrical parameters are determined by a fit to this measurement. A typical impedance curve including the fit is shown in Fig. 2. In the last step, the impedance curve is measured for applied bias voltages up to 40 V with the LCR meter. A new fit is done while all parameters but the diode capacitance are kept constant. Thus the diode capacitance can be measured up to 40 V (see Fig. 3). To determine the diode capacitance at the SiPM's operating voltage of 70 V the diode capacitance is extrapolated using 1 C D ðU bias Þ p U bias
ð3Þ
as the expected behavior of a parallel plate capacitor [5]. Using this procedure yields the values for an S10362-33-050C Hamamatsu MPPC shown in Table 1.
Fig. 3. Impedance measurement of an S10362-33-100C Hamamatsu MPPC with 400 cells at bias voltages from 0 V to 34 V in the frequency range from 20 Hz to 500 MHz
Table 1 Electrical parameters for an S10362-33-050C Hamamatsu MPPC obtained with impedance measurements. The uncertainties are the statistical uncertainty obtained by the fit. Parameter
Value
Uncertainty
Unit
RQ CD CQ CG LB RB
143.7 774.1 6.2 27.4 4.71 3.27
0.2 0.3 0.2 0.8 0.02 0.03
kΩ fF fF pF nH Ω
4. A new measurement adapter for bias voltages up to 100 V Standard commercial devices for the measurement of impedance do not allow for a connection mechanism for SiPMs and the possibility to connect an external bias voltage source simultaneously. For this reason a custom version of a measurement adapter has been designed that is capable of these functions. This adapter should be useable with the GWInstek LCR-8110G meter since this device is going to be used for further measurements instead of the Agilent LCR meter. The LCR-8110G has a frequency range from 20 Hz to 10 MHz. The main challenge is to design a device that can decouple voltages of up to 100 V and has very low parasitic characteristics. The decoupling of the bias voltage from the measurement device is done with ceramic capacitors with a capacity of 10 μF. A few diodes are used to reduce the risk of high voltages applied to the LCR meter. A circuit diagram is shown in Fig. 4. Three different printed circuit boards were produced according to the diagram. To characterize them an open-circuit measurement and a closed-circuit measurement were conducted. These measurements were then compared to the same measurement
342
F. Scheuch et al. / Nuclear Instruments and Methods in Physics Research A 787 (2015) 340–343
Fig. 6. Closed-circuit measurement for three different versions of a printed circuit board and a commercial device as benchmark.
Fig. 4. Circuit diagram of the measurement adapter with decoupling of the bias voltage via 10 μF ceramic capacitors and connectors for the SiPM and the external voltage source.
Fig. 7. Simulated voltage trace of a SiPM including readout electronics and the corresponding measurement with an oscilloscope.
Fig. 5. Open-circuit measurement for three different versions of a printed circuit board and a commercial device as benchmark.
that was done with a commercial device by Hewlett Packard that had no possibility to plug in an external bias voltage source. The measurement without a plugged device (see Fig. 5) shows that the parasitic capacitance has an influence in this frequency region. However, the impedance does not fall below 5 k Ω. This is sufficient because the expected impedance of the SiPMs under test will be of Oð100 ΩÞ. The parasitic inductance can be obtained by measuring the impedance of the adapter with a wire in place of the SiPM (see Fig. 6). This impedance ideally should be zero. These impedance values are also more than one order of magnitude smaller than the expected SiPM's impedance. Nevertheless, another iteration of the adapter will be produced to reduce parasitic effects. This will further increase the measurement precision.
5. SPICE model creation and simulation SPICE simulations can be performed with the measured values for the electrical parameters. The following simulations were done using LTSpice1 [6]. For this simulation the whole SiPM is integrated into one circuit. This includes all measured parameters and a list of all cell breakdowns 1
Linear Technology SPICE v. 4.10.
that are going to be simulated. Every line of this list includes the number of the cell that experiences the breakdown and the time of the breakdown. The list can be in arbitrary order and every cell can experience a breakdown unlimited times. To create the SiPM subcircuit a java program was written that uses the list of cell breakdowns and the electrical parameters and outputs the sub-circuit. The cell breakdown is simulated via a voltage dependent resistor ðRswitch Þ that is connected in parallel to the diode capacitance. During the cell breakdown the resistor is powered (via V breakdown ) for 100 ps which corresponds to the duration of the breakdown process including photon absorption and charge multiplication [7]. Due to the powering the resistance has a value of Rswitch o 1 Ω. So the whole charge on the diode capacitance can compensate. After this period of time, the resistor is switched back to Rswitch ¼ 400 GΩ which simulates the small dark current in the cell. The generated sub-circuit can then be placed everywhere in other SPICE circuits. Thus it is easily included in a simulation of the SiPM's readout electronics. The list of cell breakdowns might for instance be obtained by GEANT 4 simulations of SIPMs (G4SiPM) that provide time stamp and the hit cell [8]. An integration of the SPICE simulation into G4SIPM is planned. A typical generated voltage trace is shown in Fig. 7. The timing of each pulse was measured. These times were used as an input for the simulation. A good agreement of simulation and measurement can be observed.
6. Summary and outlook An enhanced SiPM model has been presented that is able to reproduce the impedance behavior over a very wide frequency range. A measurement method to determine the model parameter
F. Scheuch et al. / Nuclear Instruments and Methods in Physics Research A 787 (2015) 340–343
was described that uses two impedance measurement devices specialized for the different frequency ranges. A measurement adapter was developed that can apply up to 100 V to the SiPM during the impedance measurement to obtain the diode capacitance at the SiPM's working point. Precise SPICE simulations can be done using a sub-circuit that contains the whole information about the SiPM and is easy to use with SPICE. In a next step the measurement adapter will be improved to allow for better measurement accuracy. A large variety of SiPMs will be measured to provide a comparison between the devices in different simulated setups. The SPICE simulation will be integrated into a GEANT 4 package for simulations of SiPM to directly obtain an electrically generated voltage trace out of the GEANT simulation. Acknowledgments This work was funded by the Federal Ministry of Education and Research of the Federal Republic of Germany under Contract no. 05H12PA1. The author thanks the electronics and mechanics workshop of the Physics Institute III A of RWTH Aachen University.
343
References [1] F. Corsi, C. Marzocca, A. Perrotta, A. Dragone, M. Foresta, A. Del Guerra, S. Marcatili, G. Llosa, G. Collazzuol, G.-F. Dalla Betta, N. Dinu, C. Piemonte, G. Pignatel, G. Levi, Electrical characterization of silicon photo-multiplier detectors for optimal front-end design, in: Nuclear Science Symposium Conference Record, 2006. IEEE, New York, NY, USA, volume 2, pp. 1276–1280. [2] S. Seifert, H.T. van Dam, J. Huizenga, R. Vinke, P. Dendooven, H. Löhner, D.R. Schaart, IEEE Transactions on Nuclear Sciences 56 (2009) 3726. [3] N. Pavlov, G. Mæhlum, D. Meier, Gamma spectroscopy using a silicon photomultiplier and a scintillator, in: Nuclear Science Symposium Conference Record, IEEE, New York, NY, USA, 2005, pp. 173–180. [4] K.A. Wangerin, G.-C. Wang, C. Kim, V. Danon, Passive electrical model of silicon photomultipliers, in: Nuclear Science Symposium Conference Record, IEEE, New York, NY, USA, 2008, pp. 4906–4913. [5] S.M. Sze, K.K. Ng, Physics of Semiconductor Devices, eq. 24, 3rd ed., John Wiley & Sons, Inc., 2007, p. 85. [6] L.W. Nagel, D. Pederson, SPICE (Simulation Program with Integrated Circuit Emphasis), Technical Report UCB/ERL M382, EECS Department, University of California, Berkeley, 1973. [7] A. Lacaita, M. Ghioni, F. Zappa, G. Ripamonti, S. Cova, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 326 (1993) 290. [8] T. Niggemann, E. Dietz-Laursonn, T. Hebbeker, A. Künsken, M. Lauscher, M. Merschmeyer, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, http://dx.doi.org/10.1016/j.nima.2015.01.067, this issue.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Electrical characterization and simulation of SiPMs Florian Scheuch n, Daniel Führen, Thomas Hebbeker, Carsten Heidemann, Markus Merschmeyer III. Physikalisches Institut A, RWTH Aachen University, D-52074 Aachen, Germany
art ic l e i nf o
a b s t r a c t
Available online 28 January 2015
Silicon Photomultipliers (SiPMs) are versatile and sensitive photon detectors that experience a fast growing variety of use in particle physics and related fields of application. These photo detectors have a very promising photon detection efficiency and are therefore interesting for very low light flux applications such as scintillation and fluorescence light detection. As a semiconductor device the SiPM's gain and time response strongly depend on the operating temperature and voltage. Thus they have to be understood for a proper use of the SiPM. Therefore, accurate electrical simulations of the SiPM's behavior involving electrical readout and front-end electronics help to improve the design of experimental setups, since several different designs can be tested and simulated with a manageable amount of effort. To perform these simulations, a detailed equivalent circuit of the SiPM has to be used containing a set of well-defined parameters. For this purpose, SPICE simulations of SiPMs and readout electronics have been performed. These simulations utilize an improved SiPM model consisting of resistors, capacitances and inductances. The SiPM parameters for these simulations have been determined by measuring the impedance over a wide frequency range while applying a DC voltage in forward direction and various DC voltages from zero up to the SiPM breakdown voltage in order to determine the behavior under operating conditions. The impedance measurements, the electrical model and the resulting simulations are presented. The impact of different setups and the electrical properties of the SiPM is discussed. & 2015 Elsevier B.V. All rights reserved.
Keywords: SiPM Single photon detection Equivalent circuit Impedance measurement
1. Introduction
2. Electrical equivalent circuit
Silicon Photomultipliers (SiPMs) are tiny and compact semiconductor devices for light detection at single-photon level. These devices have a growing field of use in particle and medical physics. Because of their semiconductor nature, SiPMs are very sensitive to temperature variations and their applied bias voltage. An electrical equivalent circuit has been introduced including the voltage dependency of the pn-junction. Impedance measurements have been performed to determine the circuit's parameters. These measurements have been done with different instruments to cover a huge frequency range. Using these measurements, SPICE simulations can be performed to investigate the signal output of different electronic designs for front-end electronics. This allows for an assessment of the electrical performance of amplifiers and power supply with less prototyping.
The equivalent electrical circuit is shown in Fig. 1. The circuit consists of SiPM cells and additionally parasitic components that are used for the whole SiPM. Every SiPM cell consists of a diode capacitance ðC D Þ that represents the p–n-junction within the SiPM as it is assumed as a parallel plate capacitor. The diode capacitance is discharged during a cell breakdown. A quenching resistor (RQ ) is placed in series to the diode capacitance as it is done in the real SiPM. Every real resistor has a parasitic capacitance that is called quenching capacitance ðC Q Þ. These three devices form a SiPM cell. To simulate parasitic effects of the routing grid and connection pins on the SiPM a grid capacitance ðC G Þ is introduced. This electrical equivalent circuit is based on work of Corsi et al. [1]. Related publications are [2–4]. In addition to this model a bulk inductance ðLB Þ and a bulk resistance ðRB Þ are used. These devices help to model the impedance behavior at high frequencies (see Section 3). Furthermore, the bulk inductance is the only device within the circuit that limits the slope of the rising edge in the SiPM output signal.
n
Corresponding author. Tel.: þ 49 241 80 27338. E-mail address: [email protected] (F. Scheuch).
http://dx.doi.org/10.1016/j.nima.2015.01.066 0168-9002/& 2015 Elsevier B.V. All rights reserved.
F. Scheuch et al. / Nuclear Instruments and Methods in Physics Research A 787 (2015) 340–343
341
Fig. 2. Impedance measurement and applied fit for an S10362-33-050C Hamamatsu SiPM with 3600 cells at zero bias voltage in the frequency range from 20 Hz to 500 MHz
Fig. 1. Electrical equivalent circuit of a SiPM.
The absolute impedance depending on the measurement frequency can be calculated using this model and yields 0 ! 1 1 1 1 1 @ A : j ZðωÞj ¼ iωLB þ RB þ iωC G þ n þ 1 i ω C D RQ þ iωC Q ð1Þ 3. Measurements of electrical parameters The values of the electrical parameters of the SiPM's equivalent circuit can be determined by measuring the impedance over a wide frequency range and fitting formula (1) to the measured data. These measurements were taken with an Agilent E4980A LCR meter in the frequency range from 20 Hz to 2 MHz and an Agilent E5061B-3LF LF-RF network analyzer in the frequency range from 100 kHz to 500 MHz. A DC voltage of up to 40 V can be applied when using the LCR meter. The measurement voltage amplitude is kept on a low level (50 mV) to reduce a variation of the p–njunction depth during the measurement. In a first step, the quenching resistance is measured. During this measurement a DC voltage of 2 V is applied to the SiPM in forward direction. The measurement frequency is set to 20 Hz. Thus, the diode capacitance of the p–n-junction vanishes. The quenching capacitance and the grid capacitance have very high impedances due to the low frequency and do hardly contribute to the total impedance. The inductance has a very low impedance due to the measurement frequency. The (ohmic) impedance is then calculated by RQ ¼
j Z measured j n
ð2Þ
with the measured impedance value Z measured and the number of SiPM cells n. In a next step, the impedance is measured without applied bias voltage over the whole frequency range. The remaining electrical parameters are determined by a fit to this measurement. A typical impedance curve including the fit is shown in Fig. 2. In the last step, the impedance curve is measured for applied bias voltages up to 40 V with the LCR meter. A new fit is done while all parameters but the diode capacitance are kept constant. Thus the diode capacitance can be measured up to 40 V (see Fig. 3). To determine the diode capacitance at the SiPM's operating voltage of 70 V the diode capacitance is extrapolated using 1 C D ðU bias Þ p U bias
ð3Þ
as the expected behavior of a parallel plate capacitor [5]. Using this procedure yields the values for an S10362-33-050C Hamamatsu MPPC shown in Table 1.
Fig. 3. Impedance measurement of an S10362-33-100C Hamamatsu MPPC with 400 cells at bias voltages from 0 V to 34 V in the frequency range from 20 Hz to 500 MHz
Table 1 Electrical parameters for an S10362-33-050C Hamamatsu MPPC obtained with impedance measurements. The uncertainties are the statistical uncertainty obtained by the fit. Parameter
Value
Uncertainty
Unit
RQ CD CQ CG LB RB
143.7 774.1 6.2 27.4 4.71 3.27
0.2 0.3 0.2 0.8 0.02 0.03
kΩ fF fF pF nH Ω
4. A new measurement adapter for bias voltages up to 100 V Standard commercial devices for the measurement of impedance do not allow for a connection mechanism for SiPMs and the possibility to connect an external bias voltage source simultaneously. For this reason a custom version of a measurement adapter has been designed that is capable of these functions. This adapter should be useable with the GWInstek LCR-8110G meter since this device is going to be used for further measurements instead of the Agilent LCR meter. The LCR-8110G has a frequency range from 20 Hz to 10 MHz. The main challenge is to design a device that can decouple voltages of up to 100 V and has very low parasitic characteristics. The decoupling of the bias voltage from the measurement device is done with ceramic capacitors with a capacity of 10 μF. A few diodes are used to reduce the risk of high voltages applied to the LCR meter. A circuit diagram is shown in Fig. 4. Three different printed circuit boards were produced according to the diagram. To characterize them an open-circuit measurement and a closed-circuit measurement were conducted. These measurements were then compared to the same measurement
342
F. Scheuch et al. / Nuclear Instruments and Methods in Physics Research A 787 (2015) 340–343
Fig. 6. Closed-circuit measurement for three different versions of a printed circuit board and a commercial device as benchmark.
Fig. 4. Circuit diagram of the measurement adapter with decoupling of the bias voltage via 10 μF ceramic capacitors and connectors for the SiPM and the external voltage source.
Fig. 7. Simulated voltage trace of a SiPM including readout electronics and the corresponding measurement with an oscilloscope.
Fig. 5. Open-circuit measurement for three different versions of a printed circuit board and a commercial device as benchmark.
that was done with a commercial device by Hewlett Packard that had no possibility to plug in an external bias voltage source. The measurement without a plugged device (see Fig. 5) shows that the parasitic capacitance has an influence in this frequency region. However, the impedance does not fall below 5 k Ω. This is sufficient because the expected impedance of the SiPMs under test will be of Oð100 ΩÞ. The parasitic inductance can be obtained by measuring the impedance of the adapter with a wire in place of the SiPM (see Fig. 6). This impedance ideally should be zero. These impedance values are also more than one order of magnitude smaller than the expected SiPM's impedance. Nevertheless, another iteration of the adapter will be produced to reduce parasitic effects. This will further increase the measurement precision.
5. SPICE model creation and simulation SPICE simulations can be performed with the measured values for the electrical parameters. The following simulations were done using LTSpice1 [6]. For this simulation the whole SiPM is integrated into one circuit. This includes all measured parameters and a list of all cell breakdowns 1
Linear Technology SPICE v. 4.10.
that are going to be simulated. Every line of this list includes the number of the cell that experiences the breakdown and the time of the breakdown. The list can be in arbitrary order and every cell can experience a breakdown unlimited times. To create the SiPM subcircuit a java program was written that uses the list of cell breakdowns and the electrical parameters and outputs the sub-circuit. The cell breakdown is simulated via a voltage dependent resistor ðRswitch Þ that is connected in parallel to the diode capacitance. During the cell breakdown the resistor is powered (via V breakdown ) for 100 ps which corresponds to the duration of the breakdown process including photon absorption and charge multiplication [7]. Due to the powering the resistance has a value of Rswitch o 1 Ω. So the whole charge on the diode capacitance can compensate. After this period of time, the resistor is switched back to Rswitch ¼ 400 GΩ which simulates the small dark current in the cell. The generated sub-circuit can then be placed everywhere in other SPICE circuits. Thus it is easily included in a simulation of the SiPM's readout electronics. The list of cell breakdowns might for instance be obtained by GEANT 4 simulations of SIPMs (G4SiPM) that provide time stamp and the hit cell [8]. An integration of the SPICE simulation into G4SIPM is planned. A typical generated voltage trace is shown in Fig. 7. The timing of each pulse was measured. These times were used as an input for the simulation. A good agreement of simulation and measurement can be observed.
6. Summary and outlook An enhanced SiPM model has been presented that is able to reproduce the impedance behavior over a very wide frequency range. A measurement method to determine the model parameter
F. Scheuch et al. / Nuclear Instruments and Methods in Physics Research A 787 (2015) 340–343
was described that uses two impedance measurement devices specialized for the different frequency ranges. A measurement adapter was developed that can apply up to 100 V to the SiPM during the impedance measurement to obtain the diode capacitance at the SiPM's working point. Precise SPICE simulations can be done using a sub-circuit that contains the whole information about the SiPM and is easy to use with SPICE. In a next step the measurement adapter will be improved to allow for better measurement accuracy. A large variety of SiPMs will be measured to provide a comparison between the devices in different simulated setups. The SPICE simulation will be integrated into a GEANT 4 package for simulations of SiPM to directly obtain an electrically generated voltage trace out of the GEANT simulation. Acknowledgments This work was funded by the Federal Ministry of Education and Research of the Federal Republic of Germany under Contract no. 05H12PA1. The author thanks the electronics and mechanics workshop of the Physics Institute III A of RWTH Aachen University.
343
References [1] F. Corsi, C. Marzocca, A. Perrotta, A. Dragone, M. Foresta, A. Del Guerra, S. Marcatili, G. Llosa, G. Collazzuol, G.-F. Dalla Betta, N. Dinu, C. Piemonte, G. Pignatel, G. Levi, Electrical characterization of silicon photo-multiplier detectors for optimal front-end design, in: Nuclear Science Symposium Conference Record, 2006. IEEE, New York, NY, USA, volume 2, pp. 1276–1280. [2] S. Seifert, H.T. van Dam, J. Huizenga, R. Vinke, P. Dendooven, H. Löhner, D.R. Schaart, IEEE Transactions on Nuclear Sciences 56 (2009) 3726. [3] N. Pavlov, G. Mæhlum, D. Meier, Gamma spectroscopy using a silicon photomultiplier and a scintillator, in: Nuclear Science Symposium Conference Record, IEEE, New York, NY, USA, 2005, pp. 173–180. [4] K.A. Wangerin, G.-C. Wang, C. Kim, V. Danon, Passive electrical model of silicon photomultipliers, in: Nuclear Science Symposium Conference Record, IEEE, New York, NY, USA, 2008, pp. 4906–4913. [5] S.M. Sze, K.K. Ng, Physics of Semiconductor Devices, eq. 24, 3rd ed., John Wiley & Sons, Inc., 2007, p. 85. [6] L.W. Nagel, D. Pederson, SPICE (Simulation Program with Integrated Circuit Emphasis), Technical Report UCB/ERL M382, EECS Department, University of California, Berkeley, 1973. [7] A. Lacaita, M. Ghioni, F. Zappa, G. Ripamonti, S. Cova, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 326 (1993) 290. [8] T. Niggemann, E. Dietz-Laursonn, T. Hebbeker, A. Künsken, M. Lauscher, M. Merschmeyer, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, http://dx.doi.org/10.1016/j.nima.2015.01.067, this issue.