2008 IEEE Nuclear Science Symposium Conference Record M10·202

Passive Electrical Model of Silicon PhotomultipliersKristen A. Wangerin, Gin-Chung Wang, Chang Kim, Varon Danon

Abstract- An electrical model is developed to simulate,characterize, and predict the response of SSPM detectors fordifferent device geometries and measurement circuitconfigurations. In particular, the model allows investigation ofthe effects of increasing parasitic capacitance with increasingdiode area on the timing and magnitude of the readout signal.Passive components in the model are extracted frommeasurements and then used in the model to understand andpredict device performance. The avalanche is represented with aswitch in series with a voltage source and diode resistor, insteadof a current source. This approach allows the change inpotential, current through the diode, and timing of the avalancheto be simulated. Experimental and modeled pulses are comparedfor two different size devices. The model is first developed andvalidated using the txt mm2 device. Predictive capability isdemonstrated with the 3x3 mm2 device; in the scaling-up of thedevices, only expected model parameters are changed, and theexperimental and modeled pulses are in good agreement. Thecurrent through the diode and voltage change across the diode asfunctions of time agree with expectations.

I. INTRODUCTION

Solid-state photomultipliers (SSPMs) are a rapidly

developing detector technology, with the potential to advancea range of applications, ranging from high-energy physics,biological sensors, nuclear medicine, DNA. sequencing,andhomeland security [1]. Currently, detectors are composed of ascintillator coupled to a photomultiplier tube (PMT). Thescintillator converts gamma radiation into optical photons, andthe PMT converts the optical photons to an electronic signaland amplifies it. The detector, however, is limited by thequantum efficiency of the photocathode of the PMT, and thebulky, fragile vacuum tube requires high voltage. StandardPMTs are also sensitive to magnetic fields, which distortelectron trajectories to the first and second dynodes, and canbe damaged by excess ambient light [2,3].

SSPMs overcome many of the limitations of PMTs whilecombining advantages of PMTs and silicon detectors. Theyare compact, robust, stable, and low power devices. UnlikePMTs, SSPMs are unaffected by magnetic fields, makingthem an attractive option for applications such as PET-MRI.SSPMs have high detection efficiency, high gain, and good

Manuscript received November 14,2008.K. A. Wangerin is with GE Global Research, Niskayuna, NY 12189 USA

(telephone: 518-387-5414, e-mail: wangerin@research.ge.com).G-C. Wang is with GE Global Research, Niskayuna, NY 12189 USA

(telephone: 518-387-7971, e-mail: wang@research.ge.com).C. Kim is with GE Global Research, Niskayuna, NY 12189 USA

(telephone: 518-387-4683, e-mail: kim@research.ge.com).Y. Danon is with Rensselaer Polytechnic Institute, Troy, NY 12180 USA

(telephone: 518-276-4008, e-mail: danony@rpi.edu).

• . . . .i;:.: :;';l~;'t ~;. i.~,\:••! ~:;(.::-n't~11 '(l:~t. :~.::.; ~ ..,I ~ '"

energy resolution. Disadvantages are thafthey are currentlysmall and expensive.

A SSPM is a photosensor consisting of an array ofphotodiodes, or cells, that are connected in parallel andoperated above their breakdown voltage in Geiger mode [4,5].When an optical photon strikes one of these cells, the cellundergoes an electrical breakdown, and its charge is collectedonto a common electrode. The diode is in series with aquenching resistor, and the voltage across the diode dropsduring the avalanche. The decreasing potential slows theavalanche until the current is quenched and the cell begins torecharge. The output signal of the SSPM is proportional to thenumber of cells that are struck by optical photons.

An electrical model of an SSPM can enhance understandingof the design and behavior of SSPMs. The model in this workis developed from two previous electrical models that havebeen developed to simulate the behavior of solid-statephotomultipliers [6,7]. Fig 1 shows the model developed byPavlov et al [6], and Fig 2 shows the model developed byCorsi et al [7]. In both models, the cells of the SSPM aredivided into one cell that undergoes breakdown and theremaining cells represented as an equivalent circuit. The timeconstants of the response, of a cell undergoing breakdown andrecharging are modeled with passive components. Theavalanche of a Geiger cell is simulated using a current pulse inthe diode.

The model by Pavlov et al. simulated the pixel current overtime in a 1.1 x 1.1 mm2 SSPM to characterize the timing of thedevices and define dead time and maximum count rate. Usingcircuit parameters, the charge released from the diode wascalculated, and the timing was estimated based on timeconstants of the circuit. The work simulated two readout

resistors of 1000 and 50 .Q and concludes that the largerresistor slows down both the release of charge and recoverytime of the pixel. There was a small inductor with a value of10 nR/cm, presumably to account for stray inductances of thedevice.

The purpose of the model developed by Corsi et al. was tooptimize the front-end readout with an SSPM coupled to anideal and finite bandwidth preamplifier. A current source wasagain used to simulate the diode avalanche, and the chargewas calculated based on model parameters of overvoltage andcapacitance. There was an additional parasitic gridcapacitance to correctly account for all time constants thatcharacterize the shape of the signal waveform. A method forextraction of model parameters was described, and parametersare extracted for two unknown size SSPMs, one from ITC-irstand one from Photonique. The governing equations, whichdefine the influence of parameters on the circuit timeconstants, were developed. The time constants of the circuit

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Fig 1. SSPM model with a current pulse simulating the avalanche [6].

Fig 2. SSPM model with a current pulse simulating the avalanche and anadditional parasitic grid capacitance [8].

SgnalOut

I",

Fig 4. SSPM array of pixels connected in parallel [11].

VB•••_--_••i:. ~~

Tv, 0 -t·Fig 3. Voltage and current across the diode as a function of time [9].

toward the cathode and anode. As these carriers gain kineticenergy, they interact, liberating additional carriers. Acascading avalanche amplifies what would otherwise be asmall signal from a single detected photon. The final outputsignal is large enough to be digitized by electronics with ahigh signal-to-noise ratio. The avalanche is stopped by aquenching resistor; the large resistance limits the currentavailable to flow through the diode to sustain the avalanchecurrent. The output signal is thus limited by the discharge ofthe potential built up across the capacitance of the silicon.

The potential and current across the diode as a function oftime is shown in Fig 3. As the charge built up across the diodedischarges, this potential decreases, and the probability ofelectron-hole pair generation decreases. The avalanchecurrent stops increasing when the space charge in thedepletion region from generated electron-hole pairs collapsesthe voltage such that the internal field is below the avalanchefield. The potential drops to the breakdown voltage, Vbr,

shown in Fig 3, which is the voltage when an avalanche ofzero current through the diode can be sustained, or themultiplication factor of electron-hole pair generation equalsunity [9]. The breakdown voltage is better described as a bandor region, rather than a rigid voltage point. The diode currentdrops, and when the current is on the order of 10-20 J.lA, thefluctuation of carriers will drop to zero, and the avalanche willbe quenched [10].

The full discharge of one APD does not provide anyinformation on the incoming energy of the photon, so the APDis subdivided into an APD cell array. The output detectorsignal is proportional to the number of APD cells that fire as aresult of absorption of an optical photon. Fig 4 shows anequivalent circuit of the diodes and quenching resistorsconnected in parallel. The diodes are reversed biased, and theoutput signal is read as a voltage across a resistor to ground.

Parasitic"SJrildP

CtlpaCitallCE

1II

(N.I)Cq :11II11I

othermlcrocctllr.

----------, ---,IIIII11

Cg :

11111___I

FiringmlCroceli

------------ .. .----

II. THEORY

SSPMs are composed of an array of avalanche photodiode(APD) cells operated in Geiger mode. APDs convert lightdirectly into charge. When an optical photon is absorbed inthe semiconductor material, an electron-hole pair is created.The detector is biased above breakdown, and the voltagecreates a large electric field that accelerates the charge carriers

j'biu .~

~Lit

one pixel

In order to better relate the model to the physics, previousmodels are further developed in this work to simulate theavalanche with a switch, diode resistor, and voltage source,shown in Fig 14. The switch allows the potential across andcurrent through the diode to be simulated based on the circuitand not the definition of the current source. Closing theswitch initiates the avalanche. The potential decreases acrossthe cell to the breakdown voltage. The current rises within ananosecond and then decreases to the avalanche quenchingcurrent when the switch opens.

Experimental and modeled pulses are compared for IxImm2 and 3x3 mm2 25 J.lm devices over a range of biasvoltages. Parameters in the model are extracted fromexperimental data for the two devices, and only theseparameters are modified when simulating different devices.Modeled results are in good agreement in experimental results.The model will be able to predict the response of SSPMs fordifferent device geometries and measurement circuitconfigurations.

were explored for readout resistor values of 20, 50, and 75 Qas voltage versus time. Finally, the responses of theexperimental data and model for one of the SSPMs werecompared using two different readout amplifiers, and thecomparison plots are shown.

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A. Pulse Time Constants

The generation and transport of the charge carriers from thedepletion region to the terminals of the device define the risetime of the current pulse. Details on the timing of theavalanche current can be found in [10,12,13,14,].

The resistances and capacitances of the diode determine therecovery time. After the avalanche, the quenching capacitancerecharges almost instantaneously; the potential across thediode is very close to zero, this capacitance is small, and it isin parallel with the very large quenching resistance. Thus, therecharging of the diode is first dominated by that of thequenching RC. The diode capacitance then begins torecharge, defining the pulse tail. As the voltage builds upacross the diode, current flows through the quenching resistor,and the quenching capacitor will slowly discharge as thevoltage across it changes.

B. Equivalent Electrical Model

An SSPM array can be represented as an electrical circuit[6]. The main electrical components are the quenchingresistor (Rq), silicon resistor (Rt), and capacitor (Cd) of thediode. There is also a parasitic capacitance associated with thequenching resistor (Cq). Other stray inductance andcapacitance of the diode are not explicitly considered, as thesevalues are small [15].

Circuit parameters, such as the collected charge, diodecapacitances, breakdown voltage, and quenching resistance,can be extracted. The current pulse from the diode can beconsidered a Dirac delta pulse in time with charge, Q, that canbe expressed as

Q =(Vbias - Vbr )(Cd +Cq ) • (1)

This relationship holds true because the time constants of thecircuit, dominated by the capacitance, are longer than those ofthe avalanche. The charge associated with the discharge of asingle cell is proportional to the gain as a function of biasvoltage. The charge generated in an avalanche is the timeintegral of the current, and current is the voltage measuredover a resistor. Plotting bias voltage versus the charge, theslope of the curve provides Cd + Cq• Extrapolating to the y­intercept provides the breakdown voltage, the point where thegain would equal zero [16]. These capacitances act in parallel.The quenching capacitance is immediately recharged due tothe large Rq. Once Cq is fully recharged, Cd begins torecharge. As the voltage across Cq changes, Cq will dischargein parallel with Cd.

The quenching resistor value can be extracted by forwardbiasing the diode [7]. The slope of the forward IV curve isequal to the transconductance. Taking the reciprocal gives thetotal resistance, Rq_toh which is equal to

_ RqRCLtot

- - (2)n

where Rc is the quenching resistance of one cell, and n is thenumber of cells.

III. METHODS

Experimental and modeled pulses are compared over arange of bias voltages for lxl mm2 and 3x3 mm2 HamamatsuMulti-Pixel Photon Counter (MPPC) devices with 25 mm cellsize, shown in Fig 5. A NanoLED 05A laser initiates theavalanche in the SSPM cells, and the intensity is filtered suchthat only a few cells fire for each event. Pulse waveforms areacquired with a Tektronix TDS51 04B digital phosphoroscilloscope. The bandwidth is 1 GHz with a sampling rate of1 GS/s. It is assumed that the laser pulse is significantlyshorter than the time response of the SSPM and that thefrequency is low enough that the SSPM recovers fullybetween events. All measurements are performed with theSSPM in a light-tight aluminum box to reduce noise. A I-mmdiameter hole in the box allows the laser to shine on theSSPM. All data is taken at room temperature.

The lxl mm2 devices, with pulse amplitudes on the order ofmillivolts, are characterized without a preamplifier. The 3x3mm2 devices are characterized with a Mini-Circuits ZX-604016E preamplifier. Due to their larger diode area and highercapacitance, these devices have more dark noise and smallercell pulse amplitudes, in the range of 50 to 200 J-lV. Theintrinsic noise of the oscilloscope is 1 mV. The true signalamplitude for the bigger devices is obtained by dividing thedata taken with the preamplifier by the preamplifier gain.Because the preamplifier slightly shapes the signal, the pulseshape is obtained by increasing the laser intensity so that thesignal is above the baseline noise without using thepreamplifier. The measurement circuit connection diagram isshown in Fig 6.

The maximum amplitude of each pulse is histogrammed.The histogram has peaks and valleys according to the numberof cells that fired for each event. The spacing of the peaksdefines the cell gain. Data is taken over a range of voltagesfrom near the breakdown voltage to a voltage whenspontaneously breakdown begins to dominate. The optimumbias voltage is defined as that with the maximum signal-to­noise ratio and before excessive spontaneous breakdown.

Experimental data are used to extract model parameters aswell as to help develop and validate the electrical model. Theelectrical model is developed in SPICE to first simulate the1xl mm2 SSPM response in both pulse shape and amplitude.Each cell is composed of resistance, capacitance, and a switch.The quenching resistor is in parallel with the quenchingcapacitance. A capacitor simulates the charge buildup acrossthe silicon diode and is in parallel with the diode resistor. Theavalanche is modeled using a switch; the switch closes toinitiate the avalanche and opens to end it and allow the cell torecharge. When the switch closes, the diode capacitor acts inseries with the diode resistance and a voltage source thatrepresents the breakdown potential. The time when the switchcloses is determined by the current through the diode. Themodel is then compared to 3x3 mm2 SSPM measurements toboth validate the model and understand the effects ofincreasing SSPM size on pulse shape.

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72.3 vii71.7 v~

~-I!~i

(a) (b) (c)Fig 5. SSPM 25 urn devices: (a) 1 mm (b) 1 mm cell array (c) 3 mm.

7 cells~ cells1 cell

~ '\"

i2i.~1

~~~ i___~__~__.L--J20 30 ~O 50 60 70

time (os)

Fig 9. Averaging ofwavefonnsbased on number of cells

firing (71.9 V).

2.512~

1.5f j~:~'

allOW~ ~ 40 ~ 00 m

TIme (ns)

Fig 10. Pulses for three cellsfiring as a function of

applied voltage.

Laser

Fig 6. Connection diagram.

2) Parameter ExtractionThe breakdown voltage is extracted by plotting the gain of

the diode versus voltage; it is the x-intercept of a linear fit tothe data, when the gain of the diode would theoretically bezero. The gain or charge is calculated by integrating the signalover time and dividing by the equivalent resistance. Thecharge is first integrated over the entire signal, and results areshown in Fig 11; the calculated breakdown voltage of 69.9 Vis overestimated due to afterpulsing. Fig 12 shows thebreakdown voltage is 68.3 V when extracted integrating onlythe charge over the rise time of the pulse.

The quenching and diode capacitances are extracted using

C = Q 69.7 =19.4 pF'Vbias - Vbreakdown 71.9 - 68.3

where Q is 69.7 ± 33.1 C for a bias voltage of 71.9 V, andVbreakdo\\TI is taken to be 68.3 V. The large uncertainty is due tothe large pulse-to-pulse variation for small numbers of cellsfiring. The capacitance of Cquenching + C diode is calculated to be19.4 pF. These parameters can not be extracted as readilyindependently, so there is a degree of freedom on what theirvalues are in the model. The diode capacitance is taken to be25 pF, shown when validating the model, and the quotedterminal capacitance is 35 fF [18]. Some variation is expectedbetween devices due to variations in silicon wafers.

The SSPM is forward biased to extract the quenchingresistance. The current versus the voltage is plotted in Fig 11.The quenching resistance is equal to the slope of the linear

region. For the 1 mm2 diode and 25 Jlm cell size, there are1600 cells. Using

R =~=}(;.0088=113.6Q,~total n 1600

where ~total = 1/0.0088 = 113.6 Q and n = 1600, thequenching resistance, Rq, of one cell is 181.8 ill.

2 4 6Ampaude (mV)

BootII

6001

~~() I

200r

6040 50Time (ns)

30

IV. RESULTS

VoltageSupply

A. lxl mm2 SSPM Devices

1) Experimental ResultsExperimental data is acquired over a range of applied

voltages. For each bias voltage, digital waveforms arecaptured, shown in Fig 7 at a voltage of 71.9 V. Thehistogram of the maximum pulse amplitudes is shown in Fig8. The histogram shows the number of cells firing for eachevent. As the bias voltage is increased, the pulse amplitudeincreases both because more cells are firing and the cell gain isincreasing. From the histogram, events with the same numberof firing cells are grouped and averaged to form a single pulse.Fig 9 shows the averaged pulses for each group. The tails ofthe pulses appear to increase for increasing bias voltage, andthis is an effect of afterpulsing in the waveforms, a release oftraps in the multiplication region that can retrigger avalanchebreakdown [17]. Fig 10 shows the averaged pulses over arange of bias voltages.

Fig 7. Pulse wavefonns acquired(71.9 V).

Fig 8. Histogram of pulseamplitudes (71.9 V).

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+--I,R oUl 1C 0

~ 50 ! 30pi :

1501

I c = 34.96 * V - 2443I Vbr =69.9V

G100~.e: IQ) I

i 50~U I

I

ot i69.9 70.3 70.7 71.1 71.5 71.9 72.3

Voltage (V)

Fig 11. Extracting the breakdown voltage by plotting bias voltage versusthe gain.

V_Applied ~

71.9.V

VbAval anchi ng Cell

Rq c _ Cq

180k~J5f

..~SWItch 'I

t~ln'l:A=?no;:/, Cd ItOpen=2.2ns I 25f f

Rd ;1 Vbr I1k I 67.7 I

~--::~II~

Sandby Cells

C~eq

8p

SWItch."IClose=N/A 1< Cd eq I

IOpen=N1A ~ 40P iRd eq;

O~07 ~ 6v:.~ :

'---;+I~

ReadoUl

I' ~72~ V=O.1607*C + 68.31, Vbr =68.3 ~ ,.3<-" ~

i x

>71~- I

j70~

g 69~6SL---L----'-----'------'-:--______::'

o 10 15 20 25Charge (pC)

Fig 12. Extracting the breakdown voltage by integrating only over the firsthalf of the pulse.

::::r- · 1~ 0.008 j I = 0.0088 * V - 0.0067

J~~L...-. _/ .. I0.0 0.4 0.8 1.2 1.6 2.0

Voltage (V)

Fig 13. Extraction of diode equivalent resistance by forward biasing thediode.

3) Electrical Model ValidationExtracted parameters for quenching resistance and

capacitance and diode capacitance are input into the model,shown in Fig 14. The breakdown voltage is taken to be 67.7V. The model is validated by comparing the modeled pulseshapes to experimental data for events with three avalanchingcells. The pulse shapes are compared through rise time, peakamplitude, and fall time, which are determined by the timeconstants of the circuit.

Fig 14. Electrical model for lxl mm2 SSPM with calculated parametersfor one firing cell and the remaining cells represented as an equivalent circuit.

The change of voltage or potential across the diode isplotted, along with the current through the diode. During anavalanche, the voltage across the diode drops to thebreakdown voltage, shown in Fig 15. As the potentialdecreases to the breakdown voltage, the number of chargecarriers generated in the avalanche decreases, so the currentalso decreases, shown in Fig 16. When the probability togenerate new carriers becomes small, the avalanche is nolonger self-sustaining, and the current drops to zero. Theswitch then closes, and the cell begins to recharge. It ispossible that the voltage across the diode may drop below thebreakdown voltage, due to the space charge in the depletionregion. This effect is not included in the modeling.

The effect of overvoltage on pulse shape and magnitude isinvestigated. The gain of the devices increases withovervoltage, shown in Fig 17(a). The voltage and current areshown in Fig 17(b). The modeled and experimental pulses arein good agreement at lower bias voltages, as shown for 71.1V. As the voltage is increased, the tail of the experimentalpulse is much higher; this is the result of after pulses in someof the pulses averaged and is not from the original avalancheevent.

With constant switch timing, it is found that the amplitudeof the modeled pulse does not increase as much as theexperimental pulses do with increasing bias voltage. As thebias voltage increases, the potential across the diode is initiallygreater, so it takes longer for the current to drop to thequenching current. The switch timing is, therefore, modifiedso that the switch closes when the current reaches a constantvalue for all bias voltages. The quenching current is taken tobe 25 uA, near previously reported quenching current values[10]. At higher bias voltages the total charge collected fromthe avalanche is greater, so the amplitude of the modeledpulses is increased and in better agreement with theexperimental pulses, shown in Fig 18(a). The voltage acrossand current through the diode as a function of time are shownin Fig 18(b).

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Fig 19. Pulse waveforms acquired Fig 20. Pulse amplitude histogramwith the preamplifier (70.0 V). with the preamplifier (70.0 V).

I peak arll'iib:;~inoise

246arll'litude (mY)

120011000

1'800

§ 600~

u :101.

4000:

Fig 16. Current (LIiode) throughdiode as a function of time.

3OOO~

1 '~ 2000[

8 1ooo~ I,

i \.

Ol_~_"::"~_L -"-_.__ ...L _,__-'

2 2.2 2.4 2.6 2.8 3Time (ns)

Fig 15. Voltage (Vm) across thediode as a function of time.

72rI

7H

~70lQl I

~ 69f I> I' I I~-=;;Itagei

68r .' I 71.7Y-voltagel!Ii. 71.1 Y-volt~~

67}~-W- c~==30-40Time (ns)

(a) (b)Fig 17. For increasing bias voltage and constant switch timing,

comparison of experimental and modeled (a) pulses for increasing biasvoltage and (b) voltage across (Vm) and current through the diode.

1~5cells

I 3 cells1 cell i

~~

Fig 21. Averaged pulses with the preamplifier for 0 to 5 cells firing for abias voltage of71.0 V.

721h

711..\

~70r'~ !

~69i \68~ "'-...... ---.,

67~---~--,-~---~---·,J-1002 2.1 2.2 2.3 2.4

Time (ns)

r-·--- -,i 71.1 Y-exp I

: 71.7 Y-exp I

I 72.3 Y-exp I

! -71.1 Y-modell-71.7 Y-modell

:~--..., • ~....·1fIo '>'+., ,,, ."'-

1---72.-3 Y-e,q>l! 71.7 Y-exp I

! 71.1 Y-exp I

1,

-,. 7.2',3 Y-model-71.7 Y-model

'~_.~" ,-",model

~~o20- 03-4'0-50----60

Time (ns)

Fig 24. Averaged pulses (raw data) Fig 25. Averag~d'pulses ac~uired

acquired without the preamplifier without the preamphfl~r normah~ed toand with a strong laser pulse. gain-corrected preamphfier amplItudes.

Fig 23. Gain-corrected averagedpulses with the preamplifier over a

range of bias voltages.

Fig 22. Averaged pulses with thepreamplifier over a range of

bias voltages.

15r----{\ -~ --- c-n~

, \ i 70.0 YII 0.5~~O..YJji, ;> i

~ 10~ § OAr

~ .g 0.3~

.- , ~0.21·~ sf ~<C ! 0.1 ~ ~

O~ ~ ~ ,_80 100 20 40 60 80

Time (nsl

2) Parameter ExtractionIntegration of the current gives estimates of the breakdown

voltage as well as the total cell capacitance. The amplitudecorrected data that was taken without the preamplifier isintegrated as voltage over time to get the total charge of theevent. The variation in charge is smaller than what it was forthe lxl mm2 device because the original acquired wavefonnsare well above the noise. The afterpulsing is also too small tobe seen. The charge versus the bias voltage is plotted in Fig26, and the breakdown voltage is calculated to be 67.9 V.

(~ (b)Fig 18. For increasing bias voltage and modified constant switch timing,

comparison of experimental and modeled (a) pulses for increasing biasvoltage and (b) voltage across (Vm) and current through the diode.

B. 3x3 mm2 SSPM Devices

1) Experimental ResultsThe 3x3 mm2 SSPMs are tested at bias voltages from 69.0

to 71.0 V. Data is first taken with the preamplifier. The laserpulse is attenuated so that only a few cells fire for each event,shown in Fig 19. Fig 20 shows the histogram of the peakpulse amplitudes. Pulses are grouped and averaged in Fig 21according to the number of cells firing for the event. Fig 22shows averaged pulses with three cells firing for a range ofbias voltages. The pulses are corrected by accounting for thegain of the preamplifier in Fig 23. The amplitude of theresponse from a single cell is on the order of hundreds ofmillivolts and is less for the larger device due to the parasiticcapacitance of the other cells and a larger quenching resistor.

The preamplifier distorts the pulse shape, so data is alsoacquired without the preamplifier and with a stronger laserpulse. Fig 24 shows pulse waveforms for a range of biasvoltages. The number of cells firing is not determinable. Thewaveforms for each bias voltage are averaged to obtain onepulse. The pulses are scaled in Fig 25 to the pulse amplitudeof the gain-corrected preamplifier data.

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701 -'--1

~:r C=~23~-1m,VM~6:'9V < ~I

~30r A< .11<5 20r y

10~

oL68 68.5 69 69.5 70 70.5 71

Voltage (V)

Fig 26. Total charge integrated for the averaged pulses over a range ofbias voltages. Extrapolating the data to zero estimates the diode breakdown

voltage.

Sandby cells

SWItch ,T----l------·:,IClose=N/A '",' Cd_eq IIOpen=N/A '1 360p 1

Rd_eq i Vbr :

0.071 67.6 \

L,-jJ!t----i

Cq5f

L_c220nH

l---.- Readout

t-;:;-lc 0

<- 50 T31l'p

1

Avalanching cell

Swttch ."'1IClose=2ns";'

tOpen=2.15ns

Rd !lk I 6V:~

Rq250k

Vb

V_Applied

71.QV

Fig 28. Electrical model for 3x3 mm2 SSPM with calculated parameters forone firing cell and the remaining cells represented as an equivalent circuit.

3) Electrical Model ValidationThe SSPM is simulated with one cell firing, and the

remaining 14399 cells are grouped as an equivalent circuit.The experimental data to which the modeled results arecompared is the gain-normalized data acquired without thepreamplifier. The modeled and experimental data arecompared in Fig 29. The modeled pulse follows the shape ofthe experimental pulse except for some small differences inpulse shape that shift with bias voltage. For a voltage of 69.0V, the pulses follow closely over the rise time and first part ofthe tail. The second part of the modeled pulse tail is slowerthan the experimental. As the bias voltage increases, thedifferences between the pulse shapes shift to the first parts ofthe pulse. For a voltage of 71.0 V, the modeled pulse has aslower rise time and a faster decay over the first part of the tailin comparison to experimental pulse. The second part of thepulse tails match well for all voltages. Unlike for the lxl mm2

data, the 3x3 mm2 does not show afterpulsing; there are manycells firing in each event so the after pulses from a few cellsare not visible in the pulse.

The switch timing is varied so that it always closes whenthe current reaches the quenching current of 25 JlA. Thevoltage across the diode and the current through the diode areshown in Fig 30. The current reaches the quenching valuebefore the asymptotical part of the curve, so the switch timingdoes not change much over the range of bias voltages; theswitch opens at 2.12 ns at 69.0 V and 2.16 ns at 71.0 V.

0.8 1.2 2.0Voltage (V)

,.00577..0_ . ~R2=O.9994/ I

/ I

/ I

0.08

0.07

0.06

~ 0.05

~ 0.04

a0.03

0.02

0.010.00 ~ ..•..._._~ ..•

0.0 0.4

The quenching and diode capacitances are extracted. Thecharge, Q, is 52.3 ± 2.2 C for a bias voltage of 70.5 V. Thebreakdown voltage is taken to be 67.6 V. The capacitance ofCq + Cd is calculated to be 18.0 pF, compared to 19.4 pF forthe IxI mm2 device. The capacitances input into the modelare kept equal to 5 pF for Cq and 25 pF for Cd, from the Ixlmm2 model, as it is expected that these values would besimilar.

The diode is forward biased to extract the quenchingresistance. Fig 27 shows the collected IV data and the curvefit to the linear portion. Multiplying the inverse slope by thenumber of cells gives the quenching resistance,

(I ) (diOde size)2 ( 1 )( 3mm )2

R q = slope cell size = 0.0577 25,urn

resulting in a value of249.6 kn.The electrical model is then tested against the 3x3 mm2

device. Relevant model parameters are modified to representthe values extracted from 3x3 mm2 data, shown in Fig 28.The quenching resistor is taken to be 250 kn, and thebreakdown voltage is taken to be 67.6 V. The diode andquenching capacitances are kept constant. The pulse shapesand voltage and current of the diode are compared.Differences in pulse shape are considered and explored.

Fig 27. Extraction of diode equivalent resistance by forward biasing thediode.

15T~-~- ~I!~f::: I

_ 69 V - exp

~ 100r [;7.1 v - modelQ) I -70V-model~ i ~~9V·modE!1~ 501« I

°l~~~20 40 60 80 100

Time (ns)

Fig 29. Comparison of 3x3 mm2 diodeexperimental and modeled pulses.

Fig 30. Voltage across the diodeand current through the diode.

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The differences in pulse shape with bias voltage areexplored. It is hypothesized that the bigger diode behavesdifferently due to extra capacitance and inductance. Theexperimental pulse shape may also be different from themodel because the model is simulating one cell firing, whereas the experimental data is actually a pulse with many cellsfiring that is scaled to the amplitude of one cell. Therefore,the equivalent electrical model of the SSPM is valid for smallnumbers of cells firing, and the modeled pulse shape isrepresentative of experimental results. As the number of cellsfiring increases to hundreds or thousands, the differences inpulse shape increase. Although this is a limitation of themodel, the differences found here are small.

v. CONCLUSIONS

An electrical model is developed in SPICE to simulate,validate, and predict the response of SSPMs and enable furtherunderstanding and development of these detectors. Passivecomponents are used to simulate the time constants of theresponse. The avalanche is modeled using a voltage sourceand switch, which allows for more accurate and intuitivemodeling of the device behavior. Model parameters arerelated to the physics of the device, and values are extractedbased on experimental measurements. An RC circuit shieldsthe effects of the voltage cable, and an LC circuit at thereadout accounts for effects of the measurement setup.

The model is developed and validated using a 1x1 mm2

device. Experimental and modeled pulses are compared. Theswitch timing in the model is adjusted with increasing biasvoltage so that the switch always closes when the diodecurrent reaches the latching current. The pulse amplitudes arein agreement for pulses at different bias voltages and withdifferent numbers of cells firing. The pulse rise time anddecay times are also in agreement. The decay times of theexperimental pulses appear slower for higher bias voltages,but this is a result of afterpulsing in the diode and not from theresponse of a single cell.

A 3x3 mm2 device is used to demonstrate the predictivecapabilities of the model. The model parameters are updatedto reflect those extracted from the 3x3 mm2 device. All otherparameters are kept constant. The experimental and modeledpulses are mostly in agreement. Small differences in pulserise time, amplitude, and fall time are hypothesized to be theeffect of larger diode area as well as the result of the variationin number of cells firing on time constants. With the validatedmodel, the properties of the readout signal of an SSPM can beinvestigated as a function of bias voltage, readout circuit, anddevice design and geometry.

REFERENCES

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[2] C.1. Stapels, 1. F. Christian, G. Entine, M. R. Squillante,F. L. Augustine, and W. G. Lawrence. The Solid-State Photomultiplier

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[3] K. S. Shah, 1. Glodo, M. Klugerman, W. M. Higgins, T. Gupta, and P.Wong. High Energy Resolution Scintillation Spectrometers. IEEETrans. Nucl. Sci., 51 (5) (2004) 2395-2399.

[4] P. Buzhan, et al. An Advanced Study of Silicon Photomultiplier. ICFAInstrumentation Bulletin. 2001. Available at:http://www.slac.stanford.edu/pubs/icfa/fallOl/paper3/paper3.pdf.Accessed June 23, 2008.

[5] G. Bondarenko, et al. Limited Geiger-mode Silicon Photodiode withVery High Gain. Nuclear Physics B (Proc. Supp!.) 61B (1998) 347-352.

[6] N. Pavlov, G. Mrehlum, and D. Meier. Gamma Spectroscopy using aSilicon Photomultiplier and a Scintillator. IEEE Nuclear ScienceSymposium Conference Record (2005) 173-180.

[7] F. Corsi, et al. Modeling a Silicon Photomultiplier (SiPM) as a SignalSource for Optimum Front-End Design. Nuclear Instruments andMethods in Physics Research A 572 (2007) 416-418.

[8] F. Corsi, et al. Electrical Characterization of Silicon Photo-MultiplierDetectors for Optimal Front-End Design. IEEE Nuclear ScienceSymposium Conference Record (2006) 1276-1280.

[9] S. Cova, et al. Avalanche Photodiodes and Quenching Circuits forSingle-Photon Detection. Applied Optics 35 (12) (1996) 1956-1976.

[10] C. Piemonte. A New Silicon Photomultiplier Structure for Blue LightDetection. Nuclear Instruments and Methods in Physics Research A 568(2006) 224-232.

[11] A. G. Stewart, et al. Study of the Properties of New SPM Detectors.Proceedings ofSPIE 6119 (2006) 84-93.

[12] 1. Ninkovic. Recent Developments in Silicon Photomultipliers. NuclearInstruments and Methods in Physics Research A 580 (2007) 1020-1022.

[13] M. Razeghi and A. Rogalski. Semiconductor Ultraviolet Detectors.Applied Physics Reviews. Journal of Applied Physics 79 (10) (1996)7433-7473.

[14] A. Lacaita, M Ghioni, and S Cova.. Double Epitaxy Improves Single­Photon Avalanche Diode Performance. Electronics Letters 25 (13)(1989) 841-843.

[15] Henry W. Ott. Noise Reduction Techniques in Electronic Systems.Second Edition. John Wiley & Sons, New York: 1988.

[16] R. Haitz. Model for the Electrical Behavior of a Microplasma. Journalof Applied Physics 35 (5) (1964) 1370-1376.

[17] Y. Kang, et al. Afterpulsing of Single-Photon AvalanchePhotodetectors. Lasers and Electro-Optics Society. The 16th AnnualMeeting of the IEEE (2003) 775-776.

[18] Hamamatsu. MPPC (Multi-Pixel Photon Counter). Catalog No.KAPD0002E02. July 2007.

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