Single  cell  is  an  integrated  circuit  with  a  diode  and  a  resistor  combined  as  shown  above.    The  bias  voltage  is  applied  so  as  to  drive  current  in  the  “wrong”  direction  through  the  diode.    The  diode  function  as  an  APD,  avalanche  photodiode.  Any  light  initiates  a  “lightning  bolt”  of  current.    The  characteristics  of  the  “lightning  bolt”  is  that  Vbr,  the  breakdown  voltage  is  used  to  keep  the  process  going  so  the  effective  voltage  across  the  diode  in  terms  of  Ohm’s  law  is  Vbias-­‐Vbr.    The  active  region  is  similar  to  a  plasma  and  can  be  characterized  by  a  resistance  Rd.  During  the  intial  phase  the  potential  energy  is  used  to  sustain  the  avalanche  and  the  amount  of  current  can  be  calculated  based  on  the  available  voltage  and  an  effective  resistance.    This  specific  behavior  seems  reasonable  but  not  obvious.    In  any  case  this  is  the  model.  

 When  modeling  the  device  one  must  include  sources  of  capacitance.  The  quenching  and  diode  elements  do  have  some  associated  capacitance.  As  discussed  below  the  diode  capacitance  partially  determines  the  voltage  applied  across  the  APV.    If  this  were  zero  the  current  would  instantaneously  drop  to  just  above  the  breakdown  voltage  i.e.  the  steady  state  discussed  below.  

 Above  circuit  represents  the  situation  when  no  light  is  present  and  therefore  no  avalanche  started.    The  blue  values  show  the  voltages  at  various  points  in  the  circuit.  

 When  light  hits  the  circuit  is  connected  as  above.  Some  observations  Cd  is  ignored  to  simplify  the  calculation.    In  the  article  the  author  recognizes  that  as  the  voltage  at  the  middle  point  drops  Cq  capacitor  will  need  to  be  chared  to  the  difference  in  voltage  between  the  bias  supply  Vbias  and  the  the  middle  voltage  value.  The  sum  of  the  cap  voltages  will  always  equal  Vbias.    The  CD  cap.  will  drop  2.3  mV  out  of  71.1  so  we  sill  neglect  its  action  on  the  circuit.  To  get  a  handle  on  the  problem  let  us  assume  that  100  V  was  applied  across  each  capacitor  and  compute  the  charge  stored  and  the  current  required  to  deliver  this  charge  in  a  ns:  Qq=0.42  pC  which  could  be  delivered  in  1ns  with  0.42mA  Qd=  1.5  pC  which  could  be  delivered  in  1ns  with  1.5  mA    These  do  not  require  large  current  capacity  for  the  PS  and  indeed  the  voltages  across  Cq  will  be  at  a  maximum  of  2.3  V=  Vbias-­‐Vbr.    Within  the  model  with  Vbr  as  a  PS  the  maximum  voltage  available  is  the  difference  between  the  two  PS.    At  t=0  the  capacitor  Cd  is  fully  charged  to  the  Vbias  and  no  current  is  flowing.  The  voltage  across  the  Rd  is  then  2.3V  resulting  in  a  current  of  Id(t=0)=2.3mA.  This  current  is  a  combination  of  the  current  charging  Cq  and  the  current  from  the  

Vbias  or  Iq.    At  t=0  this  is  0  but  grows  as  the  Cq  charges  (CD  discharges).    At  t=∞  a  steady  state  will  be  reached  where  the  current  is  based  on  V=Vbias-­‐Vbr=  2.3  V  and  the  resitance  is  R=Rd+Rq=180kΩ  with  most  of  the  voltage  drop  across  the  quenching  R.    The  current  at  this  point  is  13μA  [eq.  7:    VOB/Rq]1  much  less  than  the  original  2.3  mA  at  t=0.  Of  course  Vbr  is  not  a  real  power  supply  but  just  replicates  the  action  of  the  energy  losses  in  the  avalanche  generation.    The  voltage  across  the  diode  will  continuously  drop  until  the  process  resets  and  the  diode  resumes  its  nonconductive  state.  The  relevant  time  constant  is  the  movement  of  charge  for  Cq  is  RDCq=4  ps.  Capacitors  charge  and  discharge  exponentially.    𝑉 𝑡 = 𝑉! 1− 𝑒!

!!    where  τ=RC  the  time  constant.      

 x-­‐axis  is  time  in  terms  of    τ  an  the  y-­‐axis  is  the  fractional  voltage    V/Vo.    To  charge  up  to  99%  requires  about  4.5  τ    and  in  one  τ  you  reach  about  67%.    In  this  problem  the  primary  limit  to  the  current  flow  is  the  quenching  resistor  and  the  capacitors  charge  and  discharge  through  the  diode  RD.  So  the  diode  capacitor  drops  quickly  to  the  point  where  it  is  approximately  the  break  down  voltage  and  the  quenching  capacitor  charges  up.    The  time  scale  is  a  few  ps  an  this  current  is  much  larger  than  the  current  delivered  by  the  bias  supply.  At  t=large    the  current  driven  through  the  diode  resistor  is  based  primarily  on  how  much  current  passes  through  the  quenching  resistor  and  because  the  resistances  are  so  different  there  is  very  little  voltage  drop  across  the  diode  resistor.    Again,  the  breakdown  voltage  is  not  a  real  PS  but  is  treated  as  a  PS  to  model  the  system.        opens if ID drops below a predefined threshold This approach does not follow the basic circuit operation in that the current in the quenching resistor should increase until the avalanche turns off and this is implemented not by looking at Iq but ID and not by

0  

0.2  

0.4  

0.6  

0.8  

1  

1.2  

0   2   4   6   8   10   12  

Discharging  Cap  

Charging  

seeing a high threshold but rather a low threshold. One must understand the circuit to see that this is indeed an equivalent or at least reasonable way to model the process.

 In  the  model  the  switch  is  opened  when  ID  drops  below  Iq  (Iq=30μA).    If  we  attribute  the  current  to  the  charging  Cq  only  then  the  charge  is  a  few  percent.  This  seems  reasonable  since  the  max  current  from  the  bias  supply  is  about  13  μA.    However  the  article  points  out  that  the  RD  doesn’t  influence  the  signal  rise  time.    This  is  due  to  the  fact  the  measuring  circuit  figure  2  in  reference  1.    Not  clear  to  me  why  the  signal  rise  time  is  at  ns  rather  than  ps  level.        

 

Similar  idea  but  a  different  model  for  the  avalanche.  The  other  basic  elements  the  same2    

 

 [over  voltage]  VOB  Avalanche3    ==  article  Corsi2  has  a  thorough  description  of  the  model.  

     aside  discharge  with  PS  Vbr=I(t)R+Q(t)C  

𝑉!" = 𝐼 𝑡 𝑅 +𝑄 𝑡𝐶 ;        𝑍 = 𝑄 𝑡 − 𝑉!"𝐶;              

𝑑𝑍𝑑𝑡 = 𝐼;            

𝑍𝐶 =

𝑑𝑍𝑑𝑡 𝑅  

𝑍 = 𝑍!𝑒! !!";      𝑄 𝑡 − 𝑉!"𝐶 = 𝑉!"𝐶  𝑒

! !!"      

resistor Rq, a small parasitic capacitance Cq, placed inparallel to Rq, and a current source, which models the totalcharge delivered by the microcell during the Geigerdischarge caused by an event. A further small capacitanceCp must be also considered in parallel to each microcell, toaccount for the parasitics between the substrate of thedevice and the contact of the quenching resistor. Fig. 1shows the equivalent circuit of the SiPM resulting from alarge number N of parallel-connected Geiger-mode (micro-cells) photodiodes. The figure highlights the case in whichonly one microcell at a time fires, as happens when a singledark current pulse is considered [4].

Rs represents the input resistance of the front-endelectronics, usually very small (few tens of ohms), whereasCg models the lumped contributions of the parasitics Cp.The value of the quenching resistors can be easily extractedfrom the characteristic of the SiPM in forward bias, in theregion where the voltage on the diodes is almost constantand the ratio between the variations of voltage and currentin the device is approximately equal to Rq/N. The value ofthe Vbr can be also easily extracted with a curve tracer. Ameasurement of the impedance of the SiPM biased in theproximity of the breakdown voltage allows evaluating theequivalent parallel lumped values of the capacitance andconductance of the device, respectively, Cm and Gm. At thesignal frequency used to perform this measurement(1MHz), the overall contribution of the Cq capacitancesis very small compared to Rq, and can be neglected. Usinga suitable preamplifier of known gain, a typical darkcurrent pulse can be read-out and exploited to evaluate thetotal charge Q delivered in the Geiger discharge of a singlemicrocell, equal to Q ! (Cq+Cd)DV, where DV is the

ARTICLE IN PRESS

Table 1Model parameters extracted for two different devices

SiPM IRST,N ! 625,Vbias ! 35V

SiPM Photonique,N ! 516,Vbias ! 63.8V

Rq (kO) 393.75 774Vbr (V) 31.2 61Q (fC) 148.5 127.1Cd (fF) 34.13 40.3Cs (fF) 4.95 5.40Cg (pF) 27.34 16.53

Rs

V out

CqRq

Rq/(N-1)

(N-1)Cq

(N-1)Cd

Cg

Cdlav

V bias

Fig. 1. Model of the whole SiPM.

0.025

0.02

0.015

0.01

0.005

0

-0.005

Puls

e A

mpl

itude

[V]

0 0.2 0.4 0.6 0.8 1 1.4 1.61.2Time [s]

Measured Single PulseMeasured Single Pulses - averagedSimulated Pulse with an Ideal AmplifierSimulated Pulse with the finite Bandwidth Amplifier

Fig. 2. Fitting of real data with the simulation results on the device model.

F. Corsi et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 416–418 417

resistor Rq, a small parasitic capacitance Cq, placed inparallel to Rq, and a current source, which models the totalcharge delivered by the microcell during the Geigerdischarge caused by an event. A further small capacitanceCp must be also considered in parallel to each microcell, toaccount for the parasitics between the substrate of thedevice and the contact of the quenching resistor. Fig. 1shows the equivalent circuit of the SiPM resulting from alarge number N of parallel-connected Geiger-mode (micro-cells) photodiodes. The figure highlights the case in whichonly one microcell at a time fires, as happens when a singledark current pulse is considered [4].

Rs represents the input resistance of the front-endelectronics, usually very small (few tens of ohms), whereasCg models the lumped contributions of the parasitics Cp.The value of the quenching resistors can be easily extractedfrom the characteristic of the SiPM in forward bias, in theregion where the voltage on the diodes is almost constantand the ratio between the variations of voltage and currentin the device is approximately equal to Rq/N. The value ofthe Vbr can be also easily extracted with a curve tracer. Ameasurement of the impedance of the SiPM biased in theproximity of the breakdown voltage allows evaluating theequivalent parallel lumped values of the capacitance andconductance of the device, respectively, Cm and Gm. At thesignal frequency used to perform this measurement(1MHz), the overall contribution of the Cq capacitancesis very small compared to Rq, and can be neglected. Usinga suitable preamplifier of known gain, a typical darkcurrent pulse can be read-out and exploited to evaluate thetotal charge Q delivered in the Geiger discharge of a singlemicrocell, equal to Q ! (Cq+Cd)DV, where DV is the

ARTICLE IN PRESS

Table 1Model parameters extracted for two different devices

SiPM IRST,N ! 625,Vbias ! 35V

SiPM Photonique,N ! 516,Vbias ! 63.8V

Rq (kO) 393.75 774Vbr (V) 31.2 61Q (fC) 148.5 127.1Cd (fF) 34.13 40.3Cs (fF) 4.95 5.40Cg (pF) 27.34 16.53

Rs

V out

CqRq

Rq/(N-1)

(N-1)Cq

(N-1)Cd

Cg

Cdlav

V bias

Fig. 1. Model of the whole SiPM.

0.025

0.02

0.015

0.01

0.005

0

-0.005

Puls

e A

mpl

itude

[V]

0 0.2 0.4 0.6 0.8 1 1.4 1.61.2Time [s]

Measured Single PulseMeasured Single Pulses - averagedSimulated Pulse with an Ideal AmplifierSimulated Pulse with the finite Bandwidth Amplifier

Fig. 2. Fitting of real data with the simulation results on the device model.

F. Corsi et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 416–418 417

resistor Rq, a small parasitic capacitance Cq, placed inparallel to Rq, and a current source, which models the totalcharge delivered by the microcell during the Geigerdischarge caused by an event. A further small capacitanceCp must be also considered in parallel to each microcell, toaccount for the parasitics between the substrate of thedevice and the contact of the quenching resistor. Fig. 1shows the equivalent circuit of the SiPM resulting from alarge number N of parallel-connected Geiger-mode (micro-cells) photodiodes. The figure highlights the case in whichonly one microcell at a time fires, as happens when a singledark current pulse is considered [4].

Rs represents the input resistance of the front-endelectronics, usually very small (few tens of ohms), whereasCg models the lumped contributions of the parasitics Cp.The value of the quenching resistors can be easily extractedfrom the characteristic of the SiPM in forward bias, in theregion where the voltage on the diodes is almost constantand the ratio between the variations of voltage and currentin the device is approximately equal to Rq/N. The value ofthe Vbr can be also easily extracted with a curve tracer. Ameasurement of the impedance of the SiPM biased in theproximity of the breakdown voltage allows evaluating theequivalent parallel lumped values of the capacitance andconductance of the device, respectively, Cm and Gm. At thesignal frequency used to perform this measurement(1MHz), the overall contribution of the Cq capacitancesis very small compared to Rq, and can be neglected. Usinga suitable preamplifier of known gain, a typical darkcurrent pulse can be read-out and exploited to evaluate thetotal charge Q delivered in the Geiger discharge of a singlemicrocell, equal to Q ! (Cq+Cd)DV, where DV is the

ARTICLE IN PRESS

Table 1Model parameters extracted for two different devices

SiPM IRST,N ! 625,Vbias ! 35V

SiPM Photonique,N ! 516,Vbias ! 63.8V

Rq (kO) 393.75 774Vbr (V) 31.2 61Q (fC) 148.5 127.1Cd (fF) 34.13 40.3Cs (fF) 4.95 5.40Cg (pF) 27.34 16.53

Rs

V out

CqRq

Rq/(N-1)

(N-1)Cq

(N-1)Cd

Cg

Cdlav

V bias

Fig. 1. Model of the whole SiPM.

0.025

0.02

0.015

0.01

0.005

0

-0.005

Puls

e A

mpl

itude

[V]

0 0.2 0.4 0.6 0.8 1 1.4 1.61.2Time [s]

Measured Single PulseMeasured Single Pulses - averagedSimulated Pulse with an Ideal AmplifierSimulated Pulse with the finite Bandwidth Amplifier

Fig. 2. Fitting of real data with the simulation results on the device model.

F. Corsi et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 416–418 417

architecture have been identified, in terms of dynamic range, frequency response, compactness and simplicity. A simplified analytical expression of the signal achieved by the SiPM coupled to the front-end electronics has been derived for some of the considered front-end structures and SPICE simulation have been found in good agreement with this expression. Finally the model has been successfully validated by comparing SPICE simulations to measured waveforms obtained with a real detector coupled to different preamplifiers with known characteristics.

II. MODEL AND PARAMETER EXTRACTION As well known, the SiPM structure is composed by the

parallel connection of hundreds of micro-cells, each consisting of a Geiger-mode operated photodiode passively quenched by a large series resistor [1,2]. The model of the single micro-cell, other than the diode capacitance Cd and the quenching resistor Rq, contains also a small parasitic capacitor in parallel to Rq, which works as a fast path for the charge delivered during the avalanche [7]. A metal routing is used to connect in parallel hundreds of these micro-cells sharing the same substrate, as depicted in Fig. 1 for a SiPM produced by ITC-irst [8].

Fig. 1. Microphotograph of a SiPM detector showing the metal routing grid used to connect in parallel the micro-cells.

A further parasitic capacitance Cg between the terminal of

the whole device is introduced, due to the presence of this metal grid which spans over the entire surface of the SiPM. As an example, if a SiPM of 1mm2 area is considered, assuming that the metal routing grid covers a reasonable 35% of the total surface, a value of about 11pF can be estimated for the capacitance Cg, considering only the contribution due to a typical metal-to-substrate capacitance per unit area of 0.03 fF/Pm2. A greater value can be envisaged for Cg, due to the fringe capacitance of the metal lines, the bonding pad etc. These considerations suggest that an accurate electrical model of the SiPM, able to reproduce its behavior when an event triggers the avalanche in one or more micro-cells, must

include this capacitance Cg. Fig. 2 represents the linearized, small-signal equivalent circuit of the whole device, in case only one out of the total N micro-cells is interested by a Geiger discharge, as happens, for instance, when a single dark count event is generated.

Fig. 2. Equivalent circuit of the SiPM, including the grid parasitic

capacitance Cg. In this model the waveform of the current source IAV can be

considered a Dirac’s delta pulse QG(t), where: Q='V(Cd+Cq) (1)

is the charge delivered by a fired micro-cell and 'V is the applied overvoltage, i.e. the difference between the bias voltage Vbias and the breakdown voltage Vbr. This assumption holds true as long as all the time constants introduced by the circuit are much larger than the ones associated to the avalanche phenomenon, which is a realistic hypothesis. Thanks to the superposition principle, the same circuit of Fig. 2 can be also used to model the case in which more than one micro-cell is interested by an avalanche event, simply considering IAV formed by more Dirac’s delta pulses, distributed in time accordingly to the arrival of the events.

To evaluate the parameter values of the model in Fig. 2 which fit a real device, an effective characterization procedure must be devised.

The first step of the procedure is the measure of the quenching resistor Rq. This can be easily carried out by biasing the SiPM so that the diodes which compose the detector operate in the forward region [8]. In this way all the variations of the bias voltage on the detector are absorbed by the quenching resistors, since the voltage drop on the forward-biased diodes exhibits only very small variations. Thus the slope of the forward I-V static characteristic of the device provides directly the value of Rqtot=Rq/N. Fig. 3 shows such a characteristic for the SiPM depicted in Fig. 1. The linear fit of the curve is also reported, showing that the non-linear contribution of the diodes to the voltage drop is definitely negligible. From this curve a slope of about 1.6 mS has been extracted, which results in a value of about 393k: for Rq, since N=625.

Equation (1) can be used to extract the value of both the sum Cd+Cq and the breakdown voltage Vbr. The charge

metal routing grid

1277

Authorized licensed use limited to: UNIVERSITA PISA S ANNA. Downloaded on November 21, 2008 at 03:32 from IEEE Xplore. Restrictions apply.

𝑄 𝑡 = 𝑉!"𝐶  𝑒! !!" + 𝑉!"𝐶  

define    𝑄!" = 𝑄! − 𝑉!"𝐶  as  the  amount  of  charge  on  the  capacitor  above  the  that  which  balances  the  breakdown.  

𝑄!" = 𝑄!"!𝑒! !!"  

 So  the  Cd  discharge  is  as  if  there  it  was  charged  just  to  Vob      Herman  modeling4  VE=VOB    uses  VE  as  the  over  breakdown  claims  that  relevant  parameter  os  VE  

 

 

     

better than 100 ps FWHM. In particular, deviceswith a small active area 1,10-µm diameter2 attainbetter than 30 ps FWHM at room temperature andbetter than 20 ps when cooled to 265 °C.5,6,34–37 Forthick-junction silicon SPAD’s 1i2 the breakdown volt-age, VB is from 200 to 500 V; 1ii2 the active area isfairly wide, with a diameter from 100 to 500 µm; 1iii2photon detection efficiency is very high in the visibleregion, remarkably better than 50% over all the540–850-nm wavelength range and declines in thenear IR but is still ,3% at 1064 nm; 1iv2 resolution inphoton timing is fairly good, better than 350 psFWHM for reach-through types33 and around 150 psFWHM for devices having a smoother field profile.4,12

In recent years deeper insight has been gained in thephysical phenomena that underlies detector opera-tion, and ultimate limits of the performance inphoton timing have been understood for both thinand thick detectors.6,36–39

With germanium SPAD’s, photon detection effi-ciency greater than 15% at the 1300-nm wavelengthand photon timing with 85-ps FWHM resolutionhave been experimentally verified.9,26,40–43 With re-gard to III–V devices, photon detection efficiencyabove 10% at the 1550-nm wavelength and photontiming with 250-ps FWHM resolution have beenverified for InGaAsP SPAD’s.44,45 In both cases,devices specifically designed to work in the Geigermode have not yet been reported, and the behavior ofcommercially available photodiodes is plagued bystrong afterpulsing effects because of carrier trap-ping 1see Section 22. The situation bears some simi-larity to that of silicon devices 25 years ago,46

meaning that there is much room for improvementin the material technology.

Essentially, SPAD’s are p–n junctions that operatebiased at voltage VA above breakdown voltage VB.At this bias, the electric field is so high that a singlecharge carrier injected into the depletion layer cantrigger a self-sustaining avalanche.46–49 The cur-rent rises swiftly 1nanoseconds or subnanosecondrise time2 to a macroscopic steady level in the milliam-pere range. If the primary carrier is photogener-ated, the leading edge of the avalanche pulse marksthe arrival time of the detected photon. The cur-rent continues to flow until the avalanche can bequenched by lowering the bias voltage to VB or below.The bias voltage is then restored, in order to be ableto detect another photon. This operation requires asuitable circuit that must 1i2 sense the leading edge ofthe avalanche current, 1ii2 generate a standard out-put pulse that is well synchronized to the avalancherise, 1iii2 quench the avalanche by lowering the biasto the breakdown voltage, 1iv2 restore the photodiodevoltage to the operating level. This circuit is usu-ally referred to as the quenching circuit. As dis-cussed below, the features of the quenching circuitdramatically affect the operating conditions of thedetector and, therefore, its actual performance.Our aim in this paper is to discuss different quench-ing strategies, comparing simple passive-quenching

arrangements and more elaborate active-quenchingcircuits. We also discuss the suitability of the vari-ous quenching circuits for operation with a remoteSPAD connected by a coaxial cable, with reference todetectors mounted in receptacles within an appara-tus or a cryostat, where the circuit cannot be mountedunless it is integrated with the detector.

A brief review of the main features of SPAD’s,which must be taken into account in the design orselection of the quenching circuit, is given in Section2. The operation of SPAD’s in passive-quenchingcircuits is analyzed in Section 3. The operatingprinciple and the essential features of the active-quenching circuits are dealt with in Section 4.Gated detector operation is analyzed in Section 5 forboth passive and active circuits. Circuits with mixedpassive–active features are discussed in Section 6.In conclusion, the main advantages offered by SPADdetectors and the role of active and passive circuitsin their application and development are highlightedin Section 7. All the experimental data reportedhave been obtained in our laboratory unless other-wise specifically quoted.

2. Single-Photon Avalanche Diode OperatingConditions and PerformanceBias supply voltage VA exceeds breakdown voltageVB of the junction by an amount called excess biasvoltage VE 5 1VA 2 VB2, which has a determininginfluence on detector performance. It is worthstressing that the value of ratio VE@VB matters, notthe VE value alone, because the performance isrelated to the excess electric field above the break-down level.4,6,36–39 Since VB ranges from 10 to 500 Vin the different available SPAD’s, the VE values to beconsidered are from ,1 to 50 V and more.

A. Photon Detection EfficiencyFor a photon to be detected, not only must it beabsorbed in the detector active volume and generatea primary carrier 1more precisely, an electron–holepair2, it is also necessary that the primary carriersucceeds in triggering an avalanche. The efficiencyof photon detection thus increases with excess biasvoltage VE, since a higher electric field enhances thetriggering probability.4,6 Typical data obtained withthin-junction and thick-junction SPAD’s are shownin Fig. 1.

B. Time-ResolutionThe resolution in single photon timing also improvesat a higher electric field and hence at higher VE,6,36–39

as illustrated in Fig. 2.

C. Dark-Count RateAs it happens in PMT’s, thermal generation effectsproduce current pulses even in the absence of illumi-nation, and the Poissonian fluctuation of these darkcounts represents the internal noise source of thedetector. As illustrated in Fig. 3, the SPAD dark-count rate increases with excess bias voltage.

20 April 1996 @ Vol. 35, No. 12 @ APPLIED OPTICS 1957

The dark-count rate includes primary and second-ary pulses.46 Primary dark pulses are due to carri-ers thermally generated in the SPAD junction, sothat the count rate increases with the temperatureas does the dark current in ordinary photodiodes.The rate also increases with VE because of twoeffects, namely, field-assisted enhancement of theemission rate from generation centers and an in-crease of the avalanche triggering probability.

Secondary dark pulses are due to afterpulsingeffects that may strongly enhance the total dark-count rate. During the avalanche some carriers arecaptured by deep levels in the junction depletionlayer and subsequently released with a statisticallyfluctuating delay, whose mean value depends on thedeep levels actually involved.46,47 Released carrierscan retrigger the avalanche, generating afterpulsescorrelated with a previous avalanche pulse. Thenumber of carriers captured during an avalanchepulse increases with the total number of carrierscrossing the junction, that is, with the total charge of

the avalanche pulse. Therefore afterpulsing in-creases with the delay of avalanche quenching andwith the current intensity, which is proportional toexcess bias voltage VE. The value of VE is usuallydictated by photon detection efficiency or time resolu-tion requirements, or both, so that the trappedcharge per pulse first has to be minimized by minimiz-ing the quenching delay. If the trapped chargecannot be reduced to a sufficiently low level, afeature of the quenching circuit can be exploited forreduction of the afterpulsing rate to a negligible or atleast an acceptable level. By deliberately maintain-ing the voltage at the quenching level 1see Section 32,during a hold-off time after quenching, the carriersreleased are prevented from retriggering the ava-lanche.47 As shown in Fig. 31a2, for silicon SPAD’s atroom temperature a few hundred nanoseconds holdoff can reduce by orders of magnitude the totaldark-count rate at higher excess bias voltage, since itcovers most of the release transient and practicallyeliminates afterpulsing. For SPAD’s that work atcryogenic temperatures the method is less effective,since the release transient becomes much slower andthe hold-off time required to cover it may be much

Fig. 1. Dependence of the photon detection efficiency of SPAD’son excess bias voltage VE: 1a2 detection efficiency for photons at830-nm wavelength versus VE for a thin SPAD developed in ourlaboratory6,34 11-µm junction width, breakdown voltage VB 5 16 V,10-µm active area diameter2; 1b2 detection efficiency versus wave-length with parameter VE for a thick SPAD, the EG&G Slik4

125-µm junction width, breakdown voltage VB 5 420 V, 250-µmactive area diameter2. Experimental data are from our labora-tory.

Fig. 2. Dependence of the FWHM resolution in photon timing onexcess bias voltage VE: 1a2 thin-junction SPAD of Fig. 11a2 at roomtemperature 1filled circles2 and cooled to 265 °C 1filled squares2,1b2 thick-junction SPAD of Fig. 11b2 at room temperature.Experimental data are from our laboratory.

1958 APPLIED OPTICS @ Vol. 35, No. 12 @ 20 April 1996

The dark-count rate includes primary and second-ary pulses.46 Primary dark pulses are due to carri-ers thermally generated in the SPAD junction, sothat the count rate increases with the temperatureas does the dark current in ordinary photodiodes.The rate also increases with VE because of twoeffects, namely, field-assisted enhancement of theemission rate from generation centers and an in-crease of the avalanche triggering probability.

Secondary dark pulses are due to afterpulsingeffects that may strongly enhance the total dark-count rate. During the avalanche some carriers arecaptured by deep levels in the junction depletionlayer and subsequently released with a statisticallyfluctuating delay, whose mean value depends on thedeep levels actually involved.46,47 Released carrierscan retrigger the avalanche, generating afterpulsescorrelated with a previous avalanche pulse. Thenumber of carriers captured during an avalanchepulse increases with the total number of carrierscrossing the junction, that is, with the total charge of

the avalanche pulse. Therefore afterpulsing in-creases with the delay of avalanche quenching andwith the current intensity, which is proportional toexcess bias voltage VE. The value of VE is usuallydictated by photon detection efficiency or time resolu-tion requirements, or both, so that the trappedcharge per pulse first has to be minimized by minimiz-ing the quenching delay. If the trapped chargecannot be reduced to a sufficiently low level, afeature of the quenching circuit can be exploited forreduction of the afterpulsing rate to a negligible or atleast an acceptable level. By deliberately maintain-ing the voltage at the quenching level 1see Section 32,during a hold-off time after quenching, the carriersreleased are prevented from retriggering the ava-lanche.47 As shown in Fig. 31a2, for silicon SPAD’s atroom temperature a few hundred nanoseconds holdoff can reduce by orders of magnitude the totaldark-count rate at higher excess bias voltage, since itcovers most of the release transient and practicallyeliminates afterpulsing. For SPAD’s that work atcryogenic temperatures the method is less effective,since the release transient becomes much slower andthe hold-off time required to cover it may be much

Fig. 1. Dependence of the photon detection efficiency of SPAD’son excess bias voltage VE: 1a2 detection efficiency for photons at830-nm wavelength versus VE for a thin SPAD developed in ourlaboratory6,34 11-µm junction width, breakdown voltage VB 5 16 V,10-µm active area diameter2; 1b2 detection efficiency versus wave-length with parameter VE for a thick SPAD, the EG&G Slik4

125-µm junction width, breakdown voltage VB 5 420 V, 250-µmactive area diameter2. Experimental data are from our labora-tory.

Fig. 2. Dependence of the FWHM resolution in photon timing onexcess bias voltage VE: 1a2 thin-junction SPAD of Fig. 11a2 at roomtemperature 1filled circles2 and cooled to 265 °C 1filled squares2,1b2 thick-junction SPAD of Fig. 11b2 at room temperature.Experimental data are from our laboratory.

1958 APPLIED OPTICS @ Vol. 35, No. 12 @ 20 April 1996

 

 

longer and hence seriously limit the dynamic rangein photon-counting measurements.43

The key factor for attaining a low dark-count rateis detector fabrication technology. In silicon technol-ogy, efficient gettering processes minimize both theconcentration of generation centers that are respon-sible for the primary dark-current pulses and of deeplevels that act as traps of avalanche carriers. Asillustrated in Fig. 31b2, silicon SPAD’s have beenrecently produced with a very low dark-count rate,that is, with an extremely low generation rate andalmost negligible trapping 1extremely weak trappingwith very short release, ,10 ns at room tempera-ture4,482.

D. Thermal EffectsBreakdown voltage VB strongly depends on junctiontemperature. The thermal coefficient value de-pends on the SPAD device structure and is fairlyhigh, typically around 0.3%@K.4,46,49 At constantsupply voltage VA, the increase of VB causes adecrease of excess bias voltage VE, which in percent-age terms is greater than that of VB by the factorVB@VE. The resulting percent variation of VE is very

strong at a low VE level, ,30%@K, and fairly highalso at a high VE level, ,3%@K. The effects ondevice performance are significant. The avalanchecurrent itself dissipates considerable energy in thedevice: the instantaneous pulse can attain watts ofpower. The thermal resistance from the diode junc-tion to the heat sink strongly depends on the type ofmounting 1packaged device, chip on carrier, etc.2 andcan range from less than 0.1 to 1 °C@mW. At a highcounting rate, the mean power dissipation causes asignificant temperature increase, particularly inSPAD’s with high VB 1see Sections 3 and 42. Remark-able effects are observed in detector performance,particularly in cases in which the photodiode chip isnot mounted on an efficient heat sink and the meancount rate of the avalanche pulses varies.4 It istherefore important to stabilize accurately the junc-tion temperature in working conditions. It is alsopossible to stabilize VE directly by increasing thesupply voltage VA as the junction temperature rises.However, this introduces a positive feedback withmoderate loop gain, since it slightly increases thepower dissipation 1see Sections 3 and 42. For SPAD’shaving high VB, an upper limit or a coarse stabiliza-tion of the temperature must be associated with thebias voltage feedback.

3. Passive-Quenching CircuitsIn the experimental setups employed in the earlystudies on avalanche breakdown in junctions,46,49 theavalanche current quenched itself simply by develop-ing a voltage drop on a high impedance load. Thesesimple circuits, illustrated in Fig. 4, are still cur-rently employed and have been called50,51 passive-quenching circuits 1PQC’s2. The SPAD is reversebiased through a high ballast resistor RL of 100 kV ormore, Cd is the junction capacitance 1typically ,1pF2, and Cs is the stray capacitance 1capacitance toground of the diode terminal connected to RL, typi-cally a few picofarads2. The diode resistance Rd isgiven by the series of space–charge resistance of theavalanche junction and of the ohmic resistance of theneutral semiconductor crossed by the current. TheRd value depends on the semiconductor device struc-ture: it is lower than 500 V for types with a widearea and thick depletion layer 3Figs. 11b2, 21b2, and31b24 and from a few hundred ohms to various kiloohmsfor devices with a small area and a thin junction3Figs. 11a2, 21a2, and 31a24.

Avalanche triggering corresponds to closing theswitch in the diode equivalent circuit. Figure 5shows the typicaly waveforms of diode current Id anddiode voltage Vd, or of the transient excess voltageVex 5 Vd 2 VB:

Id1t2 5Vd1t2 2 VB

Rd5

Vex1t2Rd

. 112

A. Quenching TransitionThe avalanche current discharges the capacitancesso that Vd and Id exponentially fall toward the

Fig. 3. Dependence of the dark-count rate on excess bias voltageVE: 1a2 thin SPAD of Fig. 11a2 at room temperature; the parameterquoted is the hold-off time after each avalanche pulse 1see text2;1b2 thick SPAD of Fig. 11b2 operated at room temperature with40-ns hold-off time; substantially equal results are obtained withlonger hold off, indicating that trapping effects are almost negli-gible in this device. Experimental data are from our laboratory.

20 April 1996 @ Vol. 35, No. 12 @ APPLIED OPTICS 1959

longer and hence seriously limit the dynamic rangein photon-counting measurements.43

The key factor for attaining a low dark-count rateis detector fabrication technology. In silicon technol-ogy, efficient gettering processes minimize both theconcentration of generation centers that are respon-sible for the primary dark-current pulses and of deeplevels that act as traps of avalanche carriers. Asillustrated in Fig. 31b2, silicon SPAD’s have beenrecently produced with a very low dark-count rate,that is, with an extremely low generation rate andalmost negligible trapping 1extremely weak trappingwith very short release, ,10 ns at room tempera-ture4,482.

D. Thermal EffectsBreakdown voltage VB strongly depends on junctiontemperature. The thermal coefficient value de-pends on the SPAD device structure and is fairlyhigh, typically around 0.3%@K.4,46,49 At constantsupply voltage VA, the increase of VB causes adecrease of excess bias voltage VE, which in percent-age terms is greater than that of VB by the factorVB@VE. The resulting percent variation of VE is very

strong at a low VE level, ,30%@K, and fairly highalso at a high VE level, ,3%@K. The effects ondevice performance are significant. The avalanchecurrent itself dissipates considerable energy in thedevice: the instantaneous pulse can attain watts ofpower. The thermal resistance from the diode junc-tion to the heat sink strongly depends on the type ofmounting 1packaged device, chip on carrier, etc.2 andcan range from less than 0.1 to 1 °C@mW. At a highcounting rate, the mean power dissipation causes asignificant temperature increase, particularly inSPAD’s with high VB 1see Sections 3 and 42. Remark-able effects are observed in detector performance,particularly in cases in which the photodiode chip isnot mounted on an efficient heat sink and the meancount rate of the avalanche pulses varies.4 It istherefore important to stabilize accurately the junc-tion temperature in working conditions. It is alsopossible to stabilize VE directly by increasing thesupply voltage VA as the junction temperature rises.However, this introduces a positive feedback withmoderate loop gain, since it slightly increases thepower dissipation 1see Sections 3 and 42. For SPAD’shaving high VB, an upper limit or a coarse stabiliza-tion of the temperature must be associated with thebias voltage feedback.

3. Passive-Quenching CircuitsIn the experimental setups employed in the earlystudies on avalanche breakdown in junctions,46,49 theavalanche current quenched itself simply by develop-ing a voltage drop on a high impedance load. Thesesimple circuits, illustrated in Fig. 4, are still cur-rently employed and have been called50,51 passive-quenching circuits 1PQC’s2. The SPAD is reversebiased through a high ballast resistor RL of 100 kV ormore, Cd is the junction capacitance 1typically ,1pF2, and Cs is the stray capacitance 1capacitance toground of the diode terminal connected to RL, typi-cally a few picofarads2. The diode resistance Rd isgiven by the series of space–charge resistance of theavalanche junction and of the ohmic resistance of theneutral semiconductor crossed by the current. TheRd value depends on the semiconductor device struc-ture: it is lower than 500 V for types with a widearea and thick depletion layer 3Figs. 11b2, 21b2, and31b24 and from a few hundred ohms to various kiloohmsfor devices with a small area and a thin junction3Figs. 11a2, 21a2, and 31a24.

Avalanche triggering corresponds to closing theswitch in the diode equivalent circuit. Figure 5shows the typicaly waveforms of diode current Id anddiode voltage Vd, or of the transient excess voltageVex 5 Vd 2 VB:

Id1t2 5Vd1t2 2 VB

Rd5

Vex1t2Rd

. 112

A. Quenching TransitionThe avalanche current discharges the capacitancesso that Vd and Id exponentially fall toward the

Fig. 3. Dependence of the dark-count rate on excess bias voltageVE: 1a2 thin SPAD of Fig. 11a2 at room temperature; the parameterquoted is the hold-off time after each avalanche pulse 1see text2;1b2 thick SPAD of Fig. 11b2 operated at room temperature with40-ns hold-off time; substantially equal results are obtained withlonger hold off, indicating that trapping effects are almost negli-gible in this device. Experimental data are from our laboratory.

20 April 1996 @ Vol. 35, No. 12 @ APPLIED OPTICS 1959

 

 

                                                                                                                   1  Simulation  of  SiPM  Signals,    Seifert  et.  al.  (Basic  Paper)  2  Modeling  …,  Corsi    :  Paper  just  extracts  the  parms  in  table  3  Corsi  2:  Another  ref.  Corsi  approach  to  SiPM  modeling  4  Another  model  by  Herman  

asymptotic steady-state values of Vf and If:

If 5VA 2 VB

Rd 1 RL>

VE

RL

, 122

Vf 5 VB 1 RdIf. 132

The approximation is justified since it must be RL :Rd, as shown in the following. The quenching timeconstant Tq is set by the total capacitance Cd 1 Csand by Rd and RL in parallel, i.e., in practice simplyby Rd,

Tq 5 1Cd 1 Cs2RdRL

Rd 1 RL> 1Cd 1 Cs2Rd. 142

If If is very small, Vf is very near to VB. When thedeclining voltage Vd1t2 approaches VB, the intensity ofId1t2 becomes low and the number of carriers thattraverse the avalanche region is then small. Sincethe avalanche process is statistical, it can happenthat none of the carriers that cross the high fieldregion may impact ionize. The probability of such afluctuation to zero multiplied carriers becomes signifi-cant when the diode current Id falls below <100 µA,and rapidly increases as Id further decreases.49 Theavalanche is self-sustaining above a latching currentlevel Iq , 100 µA and is self-quenching below it.The Iq value is not sharply defined, as is evidenced bya jitter of the quenching time with respect to theavalanche onset and by a corresponding jitter ofdiode voltage Vq at which quenching occurs. Inmost computations Vq can be assumed practicallyequal to VB although it is slightly higher:

Vq 5 VB 1 IqRd. 152

The total charge Qpc in the avalanche pulse, animportant parameter for evaluating the trappingeffects 1see Section 12, can thus be evaluated, settingin evidence its relation to asymptotic current If andcharacteristic time constant Tr of the voltage recov-ery

Qpc 5 1VA 2 Vq21Cd 1 Cs2 > VE1Cd 1 Cs2 > IfTr, 162

Tr 5 RL1Cd 1 Cs2. 172

Fig. 4. Basic PQC’s: 1a2 configuration with voltage-mode out-put, 1b2 configuration with current-mode output, 1c2 equivalentcircuit of the current-mode output configuration. The avalanchesignal is sensed by the comparator that produces a standardsignal for pulse counting and timing.

Fig. 5. Pulse waveforms of a SPAD of the type in Fig. 1 thatoperates in the PQC of Fig. 41b2, displayed on a digital oscilloscope:a, avalanche current Id; b, diode voltage Vd.

1960 APPLIED OPTICS @ Vol. 35, No. 12 @ 20 April 1996

asymptotic steady-state values of Vf and If:

If 5VA 2 VB

Rd 1 RL>

VE

RL

, 122

Vf 5 VB 1 RdIf. 132

The approximation is justified since it must be RL :Rd, as shown in the following. The quenching timeconstant Tq is set by the total capacitance Cd 1 Csand by Rd and RL in parallel, i.e., in practice simplyby Rd,

Tq 5 1Cd 1 Cs2RdRL

Rd 1 RL> 1Cd 1 Cs2Rd. 142

If If is very small, Vf is very near to VB. When thedeclining voltage Vd1t2 approaches VB, the intensity ofId1t2 becomes low and the number of carriers thattraverse the avalanche region is then small. Sincethe avalanche process is statistical, it can happenthat none of the carriers that cross the high fieldregion may impact ionize. The probability of such afluctuation to zero multiplied carriers becomes signifi-cant when the diode current Id falls below <100 µA,and rapidly increases as Id further decreases.49 Theavalanche is self-sustaining above a latching currentlevel Iq , 100 µA and is self-quenching below it.The Iq value is not sharply defined, as is evidenced bya jitter of the quenching time with respect to theavalanche onset and by a corresponding jitter ofdiode voltage Vq at which quenching occurs. Inmost computations Vq can be assumed practicallyequal to VB although it is slightly higher:

Vq 5 VB 1 IqRd. 152

The total charge Qpc in the avalanche pulse, animportant parameter for evaluating the trappingeffects 1see Section 12, can thus be evaluated, settingin evidence its relation to asymptotic current If andcharacteristic time constant Tr of the voltage recov-ery

Qpc 5 1VA 2 Vq21Cd 1 Cs2 > VE1Cd 1 Cs2 > IfTr, 162

Tr 5 RL1Cd 1 Cs2. 172

Fig. 4. Basic PQC’s: 1a2 configuration with voltage-mode out-put, 1b2 configuration with current-mode output, 1c2 equivalentcircuit of the current-mode output configuration. The avalanchesignal is sensed by the comparator that produces a standardsignal for pulse counting and timing.

Fig. 5. Pulse waveforms of a SPAD of the type in Fig. 1 thatoperates in the PQC of Fig. 41b2, displayed on a digital oscilloscope:a, avalanche current Id; b, diode voltage Vd.

1960 APPLIED OPTICS @ Vol. 35, No. 12 @ 20 April 1996

asymptotic steady-state values of Vf and If:

If 5VA 2 VB

Rd 1 RL>

VE

RL

, 122

Vf 5 VB 1 RdIf. 132

The approximation is justified since it must be RL :Rd, as shown in the following. The quenching timeconstant Tq is set by the total capacitance Cd 1 Csand by Rd and RL in parallel, i.e., in practice simplyby Rd,

Tq 5 1Cd 1 Cs2RdRL

Rd 1 RL> 1Cd 1 Cs2Rd. 142

If If is very small, Vf is very near to VB. When thedeclining voltage Vd1t2 approaches VB, the intensity ofId1t2 becomes low and the number of carriers thattraverse the avalanche region is then small. Sincethe avalanche process is statistical, it can happenthat none of the carriers that cross the high fieldregion may impact ionize. The probability of such afluctuation to zero multiplied carriers becomes signifi-cant when the diode current Id falls below <100 µA,and rapidly increases as Id further decreases.49 Theavalanche is self-sustaining above a latching currentlevel Iq , 100 µA and is self-quenching below it.The Iq value is not sharply defined, as is evidenced bya jitter of the quenching time with respect to theavalanche onset and by a corresponding jitter ofdiode voltage Vq at which quenching occurs. Inmost computations Vq can be assumed practicallyequal to VB although it is slightly higher:

Vq 5 VB 1 IqRd. 152

The total charge Qpc in the avalanche pulse, animportant parameter for evaluating the trappingeffects 1see Section 12, can thus be evaluated, settingin evidence its relation to asymptotic current If andcharacteristic time constant Tr of the voltage recov-ery

Qpc 5 1VA 2 Vq21Cd 1 Cs2 > VE1Cd 1 Cs2 > IfTr, 162

Tr 5 RL1Cd 1 Cs2. 172

Fig. 4. Basic PQC’s: 1a2 configuration with voltage-mode out-put, 1b2 configuration with current-mode output, 1c2 equivalentcircuit of the current-mode output configuration. The avalanchesignal is sensed by the comparator that produces a standardsignal for pulse counting and timing.

Fig. 5. Pulse waveforms of a SPAD of the type in Fig. 1 thatoperates in the PQC of Fig. 41b2, displayed on a digital oscilloscope:a, avalanche current Id; b, diode voltage Vd.

1960 APPLIED OPTICS @ Vol. 35, No. 12 @ 20 April 1996