sensors Article Modeling and Analysis of Capacitive Relaxation Quenching in a Single Photon Avalanche Diode (SPAD) Applied to a CMOS Image Sensor Akito Inoue *, Toru Okino, Shinzo Koyama and Yutaka Hirose Panasonic Corporation, 1 Kotari-yakemachi, Nagaokakyo City, Kyoto 617-8520, Japan; email@example.com (T.O.); firstname.lastname@example.org (S.K.); email@example.com (Y.H.) * Correspondence: firstname.lastname@example.org Received: 23 March 2020; Accepted: 22 May 2020; Published: 25 May 2020 Abstract: We present an analysis of carrier dynamics of the single-photon detection process, i.e., from Geiger mode pulse generation to its quenching, in a single-photon avalanche diode (SPAD). The device is modeled by a parallel circuit of a SPAD and a capacitance representing both space charge accumulation inside the SPAD and parasitic components. The carrier dynamics inside the SPAD is described by time-dependent bipolar-coupled continuity equations (BCE). Numerical solutions of BCE show that the entire process completes within a few hundreds of picoseconds. More importantly, we find that the total amount of charges stored on the series capacitance gives rise to a voltage swing of the internal bias of SPAD twice of the excess bias voltage with respect to the breakdown voltage. This, in turn, gives a design methodology to control precisely generated charges and enables one to use SPADs as conventional photodiodes (PDs) in a four transistor pixel of a complementary metal-oxide-semiconductor (CMOS) image sensor (CIS) with short exposure time and without carrier overflow. Such operation is demonstrated by experiments with a 6 µm size 400 × 400 pixels SPAD-based CIS designed with this methodology. Keywords: avalanche breakdown; avalanche photodiodes; CMOS image sensor (CIS); quenching; single photon avalanche diode (SPAD) 1. Introduction Single-photon avalanche diodes (SPADs) are devices capable of detecting an individual photon by generating a large current pulse due to the Geiger mode (GM) avalanche multiplication of an electron-hole pair created on initial detection of a single photon. [1–4]. Rapid progress has been made in the development of SPAD-based complementary metal-oxide-semiconductor (CMOS) image sensors (CIS). Targeted applications include photon-counting imagers [5–7], scientific applications [8,9], and time-of-flight ranging sensors for autonomous driving systems [10–12]. In general, operation of a SPAD is comprised of three stages; (i) an idling stage; (ii) single-photon detection followed by a GM pulse generation; (iii) quenching the GM pulse. In the idling stage, a SPAD is biased with a voltage (|Vex |) in “excess” of the breakdown voltage (|VBD |). During the quenching process, generated charges accumulate on any capacitive component in parallel with the SPAD. This, in turn, brings the voltage across the terminals of the SPAD down to or below |VBD | where the quenching is completed [2,13,14]. Then, the accumulated charges are drained out through a series resistance in order to recover (recharge) the SPAD to the idling state. Conventional SPAD-based CISs have pixel circuits with many components for quenching and counting GM pulses resulting in relatively large pixel sizes and low fill factors [2–4,15]. However, if a SPAD-based CIS with a standard four-transistors Sensors 2020, 20, 3007; doi:10.3390/s20103007 www.mdpi.com/journal/sensors Sensors 2020, 20, 3007 2 of 14 (4T) pixel architecture could be applicable, then one could expect scaling advantages in system size and high resolution.  To this end, it is necessary to quench the triggered avalanche process under two conditions, i.e., (i) within a specified period of exposure and (ii) with a controlled amount of charges avoiding overflow. One attractive and probably the only solution, which satisfies the above two conditions simultaneously and what the authors use in this paper, called “capacitive quenching” is a kind of passive quenching (because it is not active quenching, so it has to be a kind of passive quenching, even though it is not resistive), similar to the resistive passive quenching (for example, [1,2,4]) where the resistance is set very high, thus making the quenching possible via the intrinsic diode capacitance and parasitic capacitances connected to the quench node. Then, a methodology to design the quenching period and the total amount of charges is required. The purpose of this paper is to present a quantitative design methodology of a SPAD-based CIS pixel incorporated into the 4T pixel circuit with the capacitive quenching capability. We present an analytical model of a SPAD, based on bipolar-coupled continuity equations (BCE) as used in references [2,13]. A model circuit is a parallel combination of a SPAD and a capacitor describing any parasitic and space charge components in series with a switch and a voltage source. We numerically solve the BCE of this model circuit from an idling stage to a complete quenching in the time domain. With a representative capacitance value, it is shown that generated charges can be stored on the capacitor within 200 ps order. More importantly, the total amount of the stored charges is found to be highly controllable as a function of the excess voltage, ~2|Vex |, comprising the initial bias voltage |Vex | plus the reverse-excess voltage |Vre | which is nearly equal to |Vex | pulled below the breakdown voltage. Thus, the amount of charges in the multiplication region is symmetric with respect to the breakdown voltage reaching the maximum when the internal bias returns exactly to |VBD |. Thus, it is possible to design a potential profile of a SPAD to accommodate all the charges by setting up a barrier larger than 2|Vex |. Experimentally, operation of a device, i.e., 400 × 400 6 µm pixels SPAD-based CIS, designed by using this method is demonstrated . We note that the results obtained in the present work (both simulations and experiments) are different from the common sense of quenching of SPADs; “avalanche multiplication stops at the breakdown voltage.” This could be because; (1) this is the first time that the avalanche is analyzed with such equations, without assuming that avalanche stops at breakdown, (2) because in other works, SPAD was typically readout by oscilloscope or by any other component connected to it which introduces an additional stray capacitance, of few pF, and this can change the dynamics, (3) because instead in this case the SPAD is very small (6 µm) and with very small capacitance and no attached or very small parasitic capacitances, during avalanche build-up and quenching. Finally, the analysis is extended to a resistive quenching circuit and it is found that the driving force of the quenching is the same as that of the capacitive quenching, i.e., a field due to accumulated charges on the parallel capacitor. 2. Modeling of Capacitive Relaxation Quenching As a representative model device, we consider a one-dimensional SPAD constituting a p-i-n structure biased into a Geiger mode regime, above the breakdown voltage, |VBD |, and with the excess voltage of |Vex |. Symbols describing physical quantities are listed in Table 1. Important assumptions are; (1) the length of the i-region or the width of the depletion region is constant (=W); (2) carriers in the i-region transport with their saturation velocities, i.e., vs,e for electrons and vs,h for holes. As for condition (1), many SPADs are formed with a simple p-n junction, i.e., not the p-i-n structure. However, considering a large bias voltage and its small change after quenching, modulation of depletion width is estimated within ~5%, so that the assumption is reasonable. On the other hand, as for condition (2), since the field strength of the avalanche region is 3.5 × 105 V/cm to 4.0 × 105 V/cm and the material is silicon, constancy of saturation velocities is also reasonable . Sensors 2020, 20, 3007 3 of 14 Table 1. Symbols and values of the physical constant quantities. Meaning Symbol Value Width of depletion region W 0.80 µm Saturation velocity of electron  vs,e 1.02 × 107 cm /s Saturation velocity of hole  vs,h 8.31 × 106 cm/s α0 3.80 × 106 cm−1 β0 2.25 × 107 cm−1 a 1.75 × 106 V/cm b 3.26 × 106 V/cm Coefficients of impact ionization ratio  With the definitions of field dependent ionization rates of electrons (= α(t)) and holes (= β(t)), time dependent numbers of electrons (= n(t)) and holes (= p(t)) in the entire i-region are described as follows. p(t) n(t) n(t) dn(t) + − (1) = dt τi,e (t) τi,h (t) τd,e dp(t) p(t) p(t) n(t) = + − dt τi,e (t) τi,h (t) τd,h (2) 1 = α(t)vs,e τi,e (t) (3) 1 = β(t)vs,h τi,h (t) (4) 2vs,e 1 = W τd,e (t) (5) 2vs,h 1 = W τd,h (t) (6) where τi,e (t) and τi,h (t) are effective lifetimes or average inverse of ionization rates for electrons and holes. τd,e (t) and τd,h (t) are, respectively, the effective lifetime of electrons and that of holes in the i-region escaping out with the saturation velocities. The factor 2 derives from an assumption that carriers are generated in the middle of the i-region on the average. Equations (1) and (2) are represented by a matrix form as follows. d N = TN (7) dt where ! n(t) N= (8) p(t) and T = 1 τi,e − 1 τi,e 1 τd,e 1 τi,h 1 τi,h − 1 τd,h (9) are a carrier number vector and a transition matrix. We consider generation and quenching of a Geiger mode pulse assuming a SPAD applied to a CMOS image sensor (CIS) with a common four transistors circuit  as shown in Figure 1a. A fixed voltage is applied to the anode of the SPAD. By turning on the reset transistor (RST) and transfer transistor (TRN), the total SPAD bias is fixed and after turning off RST and TRN, the SPAD enters the exposure mode during which the cathode and the floating diffusion node (FD) are left floating. A simplified equivalent circuit and a band diagram of a SPAD after the reset are illustrated in Figure 1b. Sensors 2020, 20, 3007 4 of 14 Sensors 2020, 20, x FOR PEER REVIEW 4 of 14 The capacitance is is actually, actually, i.e., physically speaking, speaking, the the capacitance capacitance intrinsic intrinsic of of the the SPAD SPAD itself, itself, The series series capacitance i.e., physically i.e., of the depleted region. Upon a single photon detection, the SPAD generates a large current pulse i.e., of the depleted region. Upon a single photon detection, the SPAD generates a large current pulse by effect (Figure With the the increase increase of of carrier carrier accumulation accumulation in in the the series series capacitor, capacitor, by the the avalanche avalanche effect (Figure 1c). 1c). With the voltage of the floating node decreases, and the internal electric field of the SPAD is reduced. the voltage of the floating node decreases, and the internal electric field of the SPAD is reduced. Eventually, the avalanche effect stops or is quenched as shown in Figure 1d. Eventually, the avalanche effect stops or is quenched as shown in Figure 1d. Figure pixel circuit. Abbreviations TRN, RST, FD,FD, SF, and Figure 1. 1. (a) (a) Circuit Circuitdiagram diagramofofa afour fourtransistors transistors pixel circuit. Abbreviations TRN, RST, SF, SEL and denote transfer transistor, reset transistor, floating diffusion, source SEL denote transfer transistor, reset transistor, floating diffusion, sourcefollower followertransistor, transistor,and and select select transistor. Simplified equivalent circuitcircuit models and band of single-photon avalanche transistor. (b–d) (b)–(d) Simplified equivalent models anddiagrams band diagrams of single-photon diode (SPAD), (b) after the reset process, (c) during avalanche multiplication, and (d) after It is avalanche diode (SPAD), (b) after the reset process, (c) during avalanche multiplication, and RQ. (d) after noted that the series capacitances in (b–d) are a summation of the diode capacitance and the stray RQ. It is noted that the series capacitances in (b)–(d) are a summation of the diode capacitance and components. (e) A typical ofchart the pixel The notations, “H” and high the stray components. (e) Atiming typicalchart timing of thecircuit. pixel circuit. The notations, “H”“L”, andmean “L”, mean voltage and low voltage is applied to gates of the transistors, respectively. The arrow denoted as high voltage and low voltage is applied to gates of the transistors, respectively. The arrow denotedhν as indicates an arrival of a photon during an exposure periodperiod resulting in voltage drop ofdrop the node h indicates an arrival of a photon during an exposure resulting in voltage of theSPAD. node SPAD. The time dependence of the reverse voltage of the SPAD, V (t) , due to both the space charge and accumulated charge on the total capacitance of the SPAD cathode C is described as; The time dependence of the reverse voltage of the SPAD, |𝑉(𝑡)|, due to both the space charge and accumulated charge on the total capacitance of n the 𝐶 is described as; (t)SPAD ) + Nc (tcathode V (t) = V0 − q· (10) 𝑛(𝑡)C+ 𝑁 (𝑡) (10) |𝑉(𝑡)| = 𝑉 − 𝑞 ∙ 𝐶 where |V0 | is an initial reverse bias of the SPAD (= V (t = 0 s) ), q is a unit electronic charge, and Nc (t) where |𝑉 | is an reverse bias of the (= |𝑉(𝑡 = 0satisfying s)|), q is athe unitfollowing electronicrelation charge, and 𝑁 (𝑡) is the number of initial charges accumulated onSPAD the capacitance is the number of charges accumulated on the capacitance satisfying the following relation (t) dN t) n𝑛(𝑡) 𝑑𝑁c ((𝑡) = . (11) = (11) dt τ𝜏d,e . 𝑑𝑡 , Sensors 2020, 20, 3007 5 of 14 V (t) gives rise to reduction of the ionization field, (E(t)), in the i-region of the APD, and is related to the time-dependent ionization rates of electrons (α(E(t))) and of holes (β(E(t))) given according to the Chynoweth’s law . ! a ( ) (12a) α t = α0 · exp − E(t) ! b β(t) = β0 · exp − (12b) E(t) where the values of α0 , β0 , a and b due to Ref.  are listed in Table 1. Although more accurate expressions of impact ionization rates exist, for example references [21,22], our modeling at room temperature and electric field of 3.5 × 105 V/cm to 4.0 × 105 V/cm is covered by this choice of the ionization formula. E(t) is then given by E(t) = V (t) W . (13) 3. Results 3.1. Numerical Calculation The calculated results with the device parameters in Table 1 and initial biases |V0 | = 29–32 V are plotted in Figure 2. An initial condition is set as a situation in which an electron-hole pair is generated in the middle of the i-region assuming single-photon detection. At the beginning, the amount of charge in the i-region, n, increases with time as shown in Figure 2a. The carriers escaped from the depletion region are accumulated on the series capacitance (i.e., Nc in Figure 2b) which, in turn, decreases the reverse voltage of SPAD as shown in Figure 2c. The time needed for this quenching process is short and is shown to be within 150 ps when |V0 | is above 30 V. It should be noted that the time scale of the process is governed by the time-dependent time constants constituting the elements of matrix T of Equation (9), i.e., the rates of impact ionization and the escape velocities of carriers. It is also interesting to note that, independent of |V0 |, when the reverse bias voltage drops to |VBD |, the number of charge, n, reaches the maximum and correspondingly its time derivative equals zero as shown in Figure 2a (indicated by dashed vertical lines). This behavior reflects the nature of |VBD | as a point where the multiplication factor is infinite. After that point, n decreases and finally becomes zero. The above analysis means the ionization continues even after the device voltage returns to the breakdown voltage because both electrons and holes still exist in the i-region and the impact ionization rates are still finite. Therefore, in order to completely quench the process, the reverse voltage must go below the breakdown voltage as V (t → ∞) = |VBD | − |Vre |, where |Vre | expresses the voltage amount that should drop below the breakdown voltage. In order to accommodate all generated charges or, in other words, to prevent carrier overflow, the potential barrier of the storage capacitance |Vbarrier | must be higher than the voltage swing after the quenching, VQ = |Vex | + |Vre |. The condition is expressed as qNc (t → ∞) = C · ∆VQ ≤ C · (|Vbarrier |). (14) We calculated VQ as a function of initial bias voltage and plotted as a blue line in Figure 3. As explained above, VQ is found to be larger than |Vex | (the black dashed line in Figure 3 indicating that VQ |=|Vex ). The slope of the calculated VQ is about 2. This can be explained by the fact that, in Figure 2a, all the curves are nearly symmetric with respect to |VBD |. This means that the same number of carriers are generated before and after the moment when the reverse bias reaches |VBD | resulting in the total voltage change of “2” × |Vex |. Sensors Sensors 2020, 2020, 20, 20, 3007 x FOR PEER REVIEW (a) 6 V0 = 32 V 4 n (104) of 14 66 of 14 31 V 30 V 2 29 V 0 (b) 0 100 t (ps) 200 0 100 t (ps) 200 4 Nc (105) 3 2 1 0 (c) 35 30 |V| (V) |VBD| = 27.5 V 25 20 0 100 t (ps) 200 Figure 2. Calculated parameters in time domain. The time increment is 0.1 ps. (a) Time evolutions of Figure 2. Calculated parameters in time domain. time increment is 0.1 ps. of the electron number in the i-region, (b) the numberThe of accumulated electrons in (a) theTime seriesevolutions capacitance, the electron number in SPAD. the i-region, (b) green, the number of accumulated electrons the series capacitance, and (c) reverse bias of The red, purple, and blue lines denote, in respectively, the results and (c) reverse bias SPAD. red,32green, purple, and blue lines denote, respectively, results with initial biases 29 of V, 30 V, 31The V, and V. A horizontal dashed line in (c) indicates and |VBD | thevertical | and with initial biases 29 V, 30 V, 31 V, and 32 V. A horizontal dashed line in (c) indicates |𝑉 dashed lines show the times when dn/dt = 0 or V (t)|=|VBD . vertical dashed lines show the times when dn/dt = 0 or |𝑉(𝑡)| = |𝑉 |. It is noted that the above results are in contrast with a common quenching condition of It is that There the above arepossible in contrast withfora such common quenching condition of VQ ~|V [1,23,24]. couldresults be three reasons apparent difference; (1) use ex | noted |∆𝑉 | [1,23,24]. could be three possible reasons for such apparent difference; (1) rather use of of the|~|𝑉 set of EquationsThere (1)–(6) without assuming the cease of avalanching is scarcely done and the set of Equations (1)–(6) of avalanching is scarcely and rather unnoticed. In fact, other rare without examplesassuming also withthe the cease introduction of a stochastic modeldone of ionization at unnoticed. In fact, otherthey rareindeed examples alsoVwith the introduction of a stochastic model of ionization the breakdown voltage, found > . (2) because in other previous works, SPAD |V | ex Q | . (2)connected at the breakdown voltage, they indeed or found ∆𝑉other because into other previous works, > |𝑉 was typically readout by oscilloscope by any component it which introduces SPAD was typically readout byofoscilloscope bycan anychange other the component connected to itinstead, which an additional stray capacitance, few pF, andor this dynamics. (3) because introduces straysmall capacitance, of few andsmall this can change the dynamics. (3) because in this case,an theadditional SPAD is very (6 µm) and haspF, very capacitance and no attached or very instead, in thiscapacitances case, the SPAD very small (6 µm) and has build-up very small capacitance and noisattached small parasitic andisresistance, during avalanche and quenching. This in high or very small capacitances duringbeing avalanche This contrast to theparasitic assumption regardingand theresistance, cathode voltage nearlybuild-up constantand duequenching. to a large stray is in high contrast capacitance [24,25].to the assumption regarding the cathode voltage being nearly constant due to a large stray capacitance [24,25]. Sensors Sensors2020, 2020,20, 20,3007 x FOR PEER REVIEW 77 of of 14 14 3.0 2.5 ΔVQ (V) 2.0 1.5 1.0 0.5 0.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 Vex (V) Figure 3. The voltage drop after the quenching (= ∆VQ ) with respect to the initial bias (= V0 ). The blue dots connected by line, the black dashed line, and the red circles connected by dashed line denote, Figure 3. The voltage drop after the quenching (= 𝛥𝛥𝑉𝑉𝑄𝑄 ) with respect to the initial bias (= 𝑉𝑉0 ). The blue respectively, the calculated results of ∆VQ , ∆VQ = Vex , and the experimental result. A red hatched area dots connected by line, the black dashed line, and the red circles connected by dashed line denote, is a region between calculated curves; ∆VQ and ∆VQ = Vex , where experimental results fall within. respectively, the calculated results of 𝛥𝛥𝑉𝑉𝑄𝑄 , 𝛥𝛥𝑉𝑉𝑄𝑄 = 𝑉𝑉𝑒𝑒𝑒𝑒 , and the experimental result. A red hatched area It is noted that the measured results of ∆VQ are converted from the actually measured voltage of a is a region between calculated curves; 𝛥𝛥𝑉𝑉𝑄𝑄 and 𝛥𝛥𝑉𝑉𝑄𝑄 = 𝑉𝑉𝑒𝑒𝑒𝑒 , where experimental results fall within. It sensing node or a floating diffusion (FD) by taking account of the capacitance values of a SPAD and FD. is noted that the measured results of 𝛥𝛥𝑉𝑉𝑄𝑄 are converted from the actually measured voltage of a sensing nodeResults or a floating diffusion (FD) by taking account of the capacitance values of a SPAD and 3.2. Experimental FD. To confirm the calculated results in the previous section, we measured the output voltage amplitude of the fabricated 3.2. Experimental Results 6 µm 400 × 400 pixels SPAD-based CIS . The specification of the device is summarized in Table 2. A cross-sectional view of the pixel and the simulated potential profiles in To confirm results are in the previous voltage the horizontal andthe thecalculated vertical direction illustrated in section, Figure 4.we Formeasured this pixel,the we output set the barrier amplitude of the| =fabricated 6 µm 400 × 400 pixels SPAD-based CIS| . The specification of the device potential 3.8 V to prevent overflow for the maximum |V |Vbarrier ex = 1.5 V and voltage swing of 2.7 V. is summarized in Table 2. A cross-sectional view of the pixel and the simulated potential profiles in the horizontal andTable the vertical directionofare Figure 4. For thisimage pixel,sensor we set(CIS). the barrier potential 2. Specifications theillustrated fabricated in SPAD-based CMOS |𝑉𝑉𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 | = 3.8 V to prevent overflow for the maximum |𝑉𝑉𝑒𝑒𝑒𝑒 | = 1.5 V and voltage swing of 2.7 V. CMOS Technology 65 nm 1P4M Pixel Size 6 µmimage sensor (CIS). Table 2. Specifications of the fabricated SPAD-based CMOS Array size CMOS Technology Physical Signal Pixel Size Type ArrayQuenching size Physical Signal Fill Factor Quenching Type Voltage Operation Fill Factor DCR(@RT) Operation Voltage Frame Rate DCR(@RT) Frame Rate 400 × 400 65 nm 1P4M Photo-Charge6 µm Capacitive quenching 400 × 400 70%Photo-Charge Capacitive −3.3 V~−29 V quenching 70% 100 cps −3.3 V~−29 V 60 fps 100 cps 60 fps Sensors 2020, 20, 3007 Sensors 2020, 20, x FOR PEER REVIEW 8 of 14 8 of 14 Figure A cross cross sectional sectional view SPAD with avalanche photodiode photodiode structure structure (VAPD) (VAPD) and Figure 4. 4. A view of of aa SPAD with aa vertical vertical avalanche and the designed potential profiles in the horizontal (A-A’) and the vertical (B-B’) direction. The the designed potential profiles in the horizontal (A-A’) and the vertical (B-B’) direction. The transistor transistor shown on the the cross cross section section represents representsthe thereset resettransistor transistor(RST), (RST),ininFigure Figure1a. 1a. shown on A block diagram of the developed CIS is shown in Figure 5. The substrate voltage is externally A block diagram of the developed CIS is shown in Figure 5. The substrate voltage is externally controlled from 0 V for the normal imaging mode to −30 V for the Geiger mode. The shutter scheme is controlled from 0 V for the normal imaging mode to −30 V for the Geiger mode. The shutter scheme switchable between a rolling mode and a global mode. The readout path consists of the gradation is switchable between a rolling mode and a global mode. The readout path consists of the gradation image readout circuit for low gain modes and of the binary image readout circuit for the Geiger mode. image readout circuit for low gain modes and of the binary image readout circuit for the Geiger mode. The gradation image readout circuit comprises a column amplifier (COLAMP), correlated double The gradation image readout circuit comprises a column amplifier (COLAMP), correlated double sampling (CDS), and the horizontal shift registers (HSR). The Geiger mode binary signal is directly fed sampling (CDS), and the horizontal shift registers (HSR). The Geiger mode binary signal is directly into the same HSR operated both as a binarizing circuit and as data-transferring flip-flops. fed into the same HSR operated both as a binarizing circuit and as data-transferring flip-flops. Figure Figure 5. 5. A block diagram of the developed CMOS image sensor (CIS). Oscilloscope thethe voltage amplitude at FDat nodes low illumination condition Oscilloscope waveforms waveformsofof voltage amplitude FD under nodes aunder a low illumination 2 ) and 10 ns laser pulse (shown as a light blue line) with different bias conditions are plotted (<1 nW/cm 2 condition (<1 nW/cm ) and 10 ns laser pulse (shown as a light blue line) with different bias conditions in 6a–c. Histograms the voltageofamplitude at amplitude FD nodes with different are areFigure plotted in Figure 6a–c. of Histograms the voltage at FD nodes bias withconditions different bias 2 intensity at plotted in Figure 6d–f. The object is illuminated by a near-infrared LED with 10 nW/cm conditions are plotted in Figure 6d–f. The object is illuminated by a near-infrared LED with 10 nW/cm2 the objectatplane. Withplane. normal mode conditions (Figure 6a), the signal amplitude is primarily due to intensity the object With normal mode conditions (Figure 6a), the signal amplitude is primarily random noise. In contrast, under the Geiger mode condition, |Vex | > 0.5 due to random noise. In contrast, under the Geiger mode condition, |𝑉 V, | >the 0.5output V, the signal outputclearly signal shows binary feature exhibiting a large amount of carriers, i.e., a large amplitude of oscilloscope output clearly shows binary feature exhibiting a large amount of carriers, i.e., a large amplitude of and clearly separated peaks in the histogram. highest signal does not exceed SF oscilloscope output and clearly separated peaksSince in thethe histogram. Since level the highest signal levelthe does saturation plotted aslevel a dashed in each carrier multiplication is quenched with not exceedlevel the(1.3 SF V) saturation (1.3 red V) line plotted asplot, a dashed red line in each plot, carrier amultiplication finite signal level or without overflow. The pictures taken with the same bias conditions are shown in is quenched with a finite signal level or without overflow. The pictures taken with the same bias conditions are shown in Figure 6g–i. The object can be seen only in Geiger mode pictures. It is noted that the output voltage is slightly less than |𝑉 |, e.g., 1.1 V with |𝑉 | = 1.2 V. Considering Sensors 2020, 20, 3007 Sensors 2020, 20, x FOR PEER REVIEW 9 of 14 9 of 14 Figure 6g–i. The object can be seen only in Geiger mode pictures. It is noted that the output voltage the gain of source follower (0.8),1.1 theVinput-referred voltage swing should be 1.4 about follower 1.2–1.3 times is slightly less than with |Vex | = 1.2 V. Considering the gain ofVsource (0.8), |Vex |, e.g., larger than |𝑉 |. Thus, we not only observe that the output signal level dependence on |𝑉 |, but also the input-referred voltage swing should be 1.4 V about 1.2–1.3 times larger than |Vex |. Thus, we not only that output voltage swingsignal indeed is estimated to be than |𝑉 that |, especially clearlyswing observed in observe that the output level dependence onlarger also output voltage indeed |Vex |, but Figure 6c,f. It to is be alsolarger notedthan that|Vthe voltage drop period of 120 ns is longer than calculated 200 ps or is estimated ex |, especially clearly observed in Figure 6c,f. It is also noted that the less. This is due to a capacitance of the probe. during the and quenching, voltage drop period of 120 ns is longer oscilloscope than calculated 200 Since ps or less. This is exposure due to a capacitance of the the reset transistor turned off and currentand flows out of thethe FDreset node, this effect is not aoff problem oscilloscope probe.isSince during theno exposure quenching, transistor is turned and no for determining theofoutput value. current flows out the FDvoltage node, this effect is not a problem for determining the output voltage value. Figure 6. (a–c) Oscilloscope waveforms of output signals (Yellow: SF output, Blue: Light pulse) of Figure (a)–(c) Oscilloscope of output signals (Yellow: pulse) ofof pixels 6. measured at |Vex | = (a)waveforms N.A. (non-avalanche region), (b) 0.7 SF V, output, (c) 1.2 V.Blue: (d–f)Light Histograms = (a) N.A.at (non-avalanche (b) 0.7 V, region), (c) 1.2 V.(e) (d)–(f) of pixels measured |𝑉 | measured output signal ofatpixels (non-avalanche 0.7 V,Histograms (f) 1.2 V. A red |Vex | = (d) N.A.region), (d) N.A. (non-avalanche region), (e) 0.7taken V, (f)at1.2 red output of pixels measured at |𝑉 | =voltage line insignal the graph indicates SF saturation (1.3 V). (g–i) Pictures of a zebra equals |VV. ex | A line the(non-avalanche graph indicates region), SF saturation voltage (1.3V.V). (g)–(i) Pictures ofoutput a zebravoltage taken atis|𝑉 | equals (g) in N.A. (h) 0.7 V, (i) 1.2 It is noted that the slightly less (g) N.A. (non-avalanche region), (h) 0.7 V, (i) 1.2 V. It is noted that the output voltage is slightly less than |Vex |, e.g., 1.1 V with |Vex | = 1.2 V. Considering the gain of source follower (0.8), the input referred |, e.g.,is1.1 |𝑉 swing = 1.2than V. Considering the gain of source follower (0.8), the input than voltage 1.4VVwith which|𝑉is |larger |Vex |. referred voltage swing is 1.4 V which is larger than |𝑉 |. The measured ∆VQ , is plotted in Figure 3 as red circles in comparison to calculated results. The The amplitude ∆VQ|Δ𝑉 of |,experiment exceeds ∆VQ |=in1.3|V voltage swing | and measured is plotted in Figure|V3exas red circles comparison to calculated results. Theis ex | . The maximum 1.3 V within the measurable range and the designed potential barrier is enough to avoid the overflow. amplitude |Δ𝑉 | of experiment exceeds |𝑉 | and |Δ𝑉 | = 1.3|𝑉 |. The maximum voltage swing is The experimental result confirms thethe calculation the voltage of quenching the 1.3 V within the measurable range and designed that potential barrier swing is enough to avoid theexceeds overflow. excess bias. We discuss reasonthe thatcalculation the experimental ∆V is smaller than the calculation. The experimental result the confirms that the voltage swing of quenching exceeds the Q excessThe bias. We discuss the reason that the∆V experimental |Δ𝑉 | by is smaller than the(Figure calculation. standard deviations of ∆V the histograms 6a–c) and are Q (σ Q ) are analyzed The standard deviations of |Δ𝑉 | (σ |Δ𝑉 | ) are analyzed by the histograms (Figure 6a–c) and shown as error bars in Figures 3 and 7a as a function of |Vex |. Interestingly, σ ∆VQ decreases with Interestingly, σ |Δ𝑉 | are in Figurethis 3 and 7aa as function of |𝑉 |.variation theshown increaseasoferror We consider resultFigure reflects facta that the standard is affected more |Vex |.bars decreases with the increase of |𝑉 |. We consider this result reflects a fact that the standard variation significantly with a situation where the carrier number is small or |Vex | is low. We also analyzed photon isdetection affected efficiency more significantly a situation where the carrier number is small or |𝑉 is low. We (PDE) of with the SPAD as shown in Figure 7b (with arbitrary unit). PDE| increases as also analyzed photon detection efficiency (PDE) of the SPAD as shown in Figure 7b (with arbitrary because of the rise of avalanche triggering probability . |Vex | unit). PDE increases as |𝑉 | because of the rise of avalanche triggering probability . Sensors 2020, 20, 3007 Sensors 2020, 20, x FOR PEER REVIEW 10 of 14 10 of 14 (a) (b) 35 25 PDE (a.u.) σ (|ΔVQ|) (%) 30 20 15 10 5 0 -0.5 0.0 0.5 1.0 1.5 2.0 -0.5 Vex (V) 0.0 0.5 1.0 1.5 2.0 Vex (V) Figure 7. (a) The standard deviation of ∆VQ . (b) Photon detection efficient (PDE). Figure 7. (a) The standard deviation of |Δ𝑉 |. (b) Photon detection efficient (PDE). 4. Discussion 4. Discussion As mentioned above, avalanche multiplication doesn’t stop and the carrier number reaches the maximum value at the breakdown voltage. This is duedoesn’t to the character thecarrier breakdown voltage where As mentioned above, avalanche multiplication stop andofthe number reaches the carrier generation and escape processes balance, i.e., equilibrium point. In this section, we discuss maximum value at the breakdown voltage. This is due to the character of the breakdown voltage the breakdown voltage as and an equilibrium point and derive which universally hold at the where carrier generation escape processes balance, i.e.,formulas equilibrium point. In this section, we breakdown voltage. We also discuss the difference between the formulas experiment anduniversally the calculation. discuss the breakdown voltage as an equilibrium point and derive which hold Furthermore, at thevoltage. end of We this also section, we show the generality of the modeland andthe consideration at the breakdown discuss the difference between the above experiment calculation. by analyzing BCE inend a resistive quenching Furthermore, at the of this section, we circuit. show the generality of the above model and consideration by analyzing BCE in a resistive quenching circuit. 4.1. Breakdown Voltage as an Equilibrium Point of the Dynamical System 4.1. Breakdown as an Equilibrium of the Dynamical System When the Voltage bias voltage reaches thePoint breakdown voltage |V BD |, the time derivative of the carrier number is zero carrier number the maximum value, i.e.,time derivative of the carrier When the and bias the voltage reaches thereaches breakdown voltage |𝑉 |, the number is zero and the carrier number reachesd the maximum value, i.e., N = 0. (15) 𝑑 dt (15) 𝑵 = 𝟎. 𝑑𝑡 The condition indicates balancing of impact ionization and the carrier drain processes. The bias Themeeting condition indicates balancing ionization and the carrier drain processes. The bias voltage this condition shouldofbeimpact the breakdown voltage. Therefore, the state with the voltage meeting this condition should be the breakdown voltage. Therefore, the state with the breakdown voltage is an equilibrium point which separates unstable (Geiger mode) and stable breakdown voltage is an equilibrium point which separates unstable (Geiger mode) and stable (linear (linear mode) regimes. mode) regimes. We investigate the condition for Equation (16) to have a nontrivial solution. Equating the We investigate the condition for(9)) Equation (16)i.e., to det have a 0, nontrivial solution. Equating the determinant of the matrix T (Equation to be zero, and by using Equations (3)–(6), |T| = determinant of the matrix 𝑻 (Equation (9)) to be zero, i.e., det |𝑻| = 0, and by using Equations (3)–(6), we obtain the following simple expression we obtain the following simple expression W W α 𝑊+ β 𝑊= 1. (16) (16) 2 𝛼 + 𝛽2 = 1. 2 2 It that thethe breakdown occurs only when an electron and a hole cause one cause ionization It simply simplymeans means that breakdown occurs only when an electron and a hole one event during their transport through the average distance of the i-region (= W/2). This is identical ionization event during their transport through the average distance of the i-region (= W/2). Thisto is the avalanche breakdown of the references for a case[26, where ratesthe of identical to the avalanchecondition breakdown condition of [26,27] the references 27] the for ionization a case where electrons holes are different. the ratio of the electron numbers in and the ionizationand rates of electrons and Correspondingly, holes are different. Correspondingly, the and ratiohole of the electron i-region under in this condition calculated to be is calculated to be hole numbers the i-region is under this condition τ𝜏d,e, n 𝑛 v𝑣s,h, == == ,, p 𝑝 v𝑣s,e, τ𝜏d,h, (17) (17) meaning that that the the ratio ratio is is inversely inversely proportional proportional to to their their velocities velocities or or proportional proportional to to the the transport transport meaning lifetimes. It is noted that the ratio does not depend on the ionization rates of electrons and holes. lifetimes. It is noted that the ratio does not depend on the ionization rates of electrons and holes. Sensors Sensors 2020, 2020, 20, 20, 3007 x FOR PEER REVIEW 11 of of 14 14 11 4.2. Difference of 𝛥𝑉 between Simulation and Experiment 4.2. Difference of ∆VQ between Simulation and Experiment As shown in Section 3.2, the experimental |∆𝑉 | (the red circles in Figure 3) exceeds |𝑉 | but it As shown in Section 3.2, the experimental V (the red circles in Figure 3) exceeds |Vex | but it due does not reach the calculation result, Δ𝑉 ≈ 2|𝑉 Q| (the blue line in Figure 3). This is probably does not reach the calculation result, ∆VQ ≈ 2|Vex | (the blue line in Figure 3). This is probably due to to spatial inhomogeneity of the field in the real device. Near toward the edge of a SPAD (hatched spatial inhomogeneity of the field in the real device. Near toward the edge of a SPAD (hatched area in area in Figure 8a), the electric field is relatively low by the potential barrier, ϵ|𝑉 |, where 0 ≤ Figure 8a), the electric field is relatively low by the potential barrier, |Vbarrier |, where 0 ≤ ≤ 1. This is ϵ ≤ 1 . This is due to extension of depletion region from the surrounding isolation. Avalanche due to extension of depletion region from the surrounding isolation. Avalanche multiplication rarely multiplication rarely occurs near the edge of the SPAD (Figure 8c), while Δ𝑉 ≈ 2|𝑉 | consists occurs near the edge of the SPAD (Figure 8c), while ∆VQ ≈ 2|Vex | consists around the center (Figure 8b). around the center (Figure 8b). Because carriers diffuse perpendicular to the field direction during Because carriers diffuse perpendicular to the field direction during and/or after multiplication , and/or after multiplication , the total value of |∆𝑉 | should fall between |𝑉 | and 2 |𝑉 | as the total value of VQ should fall between |Vex | and 2|Vex | as indicated by a red hatched area in Figure 3. indicated by a red hatched area in Figure 3. Because of the experimental result, ∆𝑉 ≃ 1.3|𝑉 | (shown Because of the experimental result, VQ ' 1.3|Vex | (shown in Section 3.2), we consider the effective in Section 3.2), we consider the effective avalanche region is about 70% of the SPAD area reasonably avalanche region is about 70% of the SPAD area reasonably agreeing with an effective fill factor (75%) agreeing with an effective fill factor (75%) estimated from a TCAD simulation. estimated from a TCAD simulation. (a) (b) Center －－ P －－ －－ + －－ －－ + N (c) 𝑉(𝑡 = 0s) Edge －－ － － － 𝑉 𝑡 = 0s −𝑉 Figure 8. (a) A top view of a SPAD. The edge of the SPAD is hatched. (b,c) Simplified band diagrams Figure 8. (a) A top view of a SPAD. The edge of the SPAD is hatched. (b), (c) Simplified band diagrams at the center of the SPAD (b) and at the edge of the SPAD (c). at the center of the SPAD (b) and at the edge of the SPAD (c). 4.3. The Mechanism of Resistive Quenching 4.3. The Mechanism of Resistive Quenching In order to comprehend the mechanism of quenching, we extend the BCE model to the resistive In order to comprehend the mechanism quenching, weamount extend of thethe BCE model to thecarriers resistive quenching circuit as schematically shown inof Figure 9a. The accumulated is quenching circuit as schematically shown in Figure 9a. The amount of the accumulated carriers is rewritten from Equation (11) to rewritten from Equation (11) to n ( t ) Nc dNc = − (18) dt τd,e RC 𝑑𝑁 𝑛(𝑡) 𝑁 = − (18) 𝜏 current 𝑑𝑡 𝑅𝐶flowing through resistance, R. , where the second term of the right hand expresses The wasofmade on the same SPAD with V. The calculated device |VBDflowing | of 27.5 through where thecalculation second term the right hand expresses current resistance, R. terminal voltage the number carriers depletion are |plotted Figure 9b,c, respectively, of 27.5inV. The calculated device Theand calculation wasofmade on in thethe same SPAD region with |𝑉 for three quenching resistances (R = 100 kΩ (blue curves) R = 30 kΩ (green curves) R curves)). terminal voltage and the number of carriers in the depletion region are plotted= ∞ in (red Figure 9b,c, R = ∞ corresponds to the capacitive quenching. be(blue notedcurves) is that,Rregardless of the values respectively, for three quenching resistances (R First = 100tokΩ = 30 kΩ (green curves)ofRR, = from initial photo-carrier generation to quenching process gives nearly identical carrier evolution as ∞ (red curves)). R = ∞ corresponds to the capacitive quenching. First to be noted is that, regardless of indicated curves below ~200 psto(Figure 9c). This is explained by discussions the valuesbyofthe R, overlapped from initialpulse photo-carrier generation quenching process gives nearly identical in Section 3.1; the as overall time scale of the process ispulse governed bybelow the time constants given in This matrix carrier evolution indicated by the overlapped curves ~200 ps (Figure 9c). is elements T of Equation (9). It is emphasized that the Geiger mode multiplication is completed or explained by discussions in Section 3.1; the overall time scale of the process is governed by the time quenched the completion of this carrier completion. driving force the Geiger process mode is the constants by given in matrix elements T of pulse Equation (9). It isThe emphasized thatofthe bias field whichisiscompleted determinedorbyquenched the balance the supply and pulse the field created byThe the multiplication bybetween the completion of voltage this carrier completion. carriers stored on the capacitance C. After the quenching, the bias recovers to the initial value with driving force of the process is the bias field which is determined by the balance between the supply time constant RC. It is noted by that the maximum swing ofcapacitance the terminalC.voltage with finite quenching voltage and the field created the carriers stored on the After the quenching, the bias resistance is reduced from that of the pure capacitive quenching, i.e., 2|V (Figure 9b). This is due | ex recovers to the initial value with time constant RC. It is noted that the maximum swing of the terminal voltage with finite quenching resistance is reduced from that of the pure capacitive quenching, i.e., Sensors 2020, 20, x FOR PEER REVIEW Sensors 2020, 20, 3007 12 of 14 12 of 14 2|𝑉 | (Figure 9b). This is due to the fact that some of the generated carriers are lost through the resistance. The voltage of the capacitive i.e., 2|𝑉 is considered be theswing worst to the fact that some ofswing the generated carriers quenching, are lost through the |,resistance. The to voltage case voltage drop for a CISi.e., pixel. for the purpose of preventing carrier overflow, the design of theofcapacitive quenching, 2|VThus, is considered to be the worst case of voltage drop for a CIS |, ex criterion of the barrier potential (Equation (15)) is still applicable. When the resistance is R = 30 kΩ, pixel. Thus, for the purpose of preventing carrier overflow, the design criterion of the barrier potential the accumulated charge is drained to the source thethe completion of the ionization due (Equation (15)) is still applicable. When thevoltage resistance is R before = 30 kΩ, accumulated charge is drained carrier loss from thethe capacitance. And voltage beyondloss |𝑉 from | giving to too the fast voltage source before completion of the terminal ionization due togoes too back fast carrier the rise to another Geiger mode pulse. Thus, the carrier number oscillates with time. After the oscillation, capacitance. And the terminal voltage goes back beyond |VBD | giving rise to another Geiger mode |𝑉 |. Thissteady corresponds typically steady Thus, current and the terminal voltage only pulse. theflows carrier number oscillates with recovers time. After thetooscillation, currenttoflows and called “bad quenching” phenomenon and in line with the rule-of-thumb given in Ref. , that it is the terminal voltage recovers only to |VBD |. This corresponds to typically called “bad quenching” necessary to have at line leastwith 50 kΩ quenching resistor each, voltthat of excess bias of the phenomenon and in theofrule-of-thumb given for in Ref. it is necessary to SPAD. have at least in order completely quench the bias Geiger mode multiplication and to recover to the 50 kΩTherefore, of quenching resistor for each to volt of excess of the SPAD. initial state, recharge time constant, to RC, shouldthe beGeiger longer mode than the time necessary quenching Therefore, in order completely quench multiplication andfor to the recover to the process (quenching In the present calculation where thethe quenching time isfor about ps and initial state, rechargetime). time constant, RC, should be longer than time necessary the 200 quenching the capacitance is 6 time). fF, a threshold resistance value is estimated to be about 33iskΩ in agreement process (quenching In the present calculation where the quenching time about 200 ps andwith the the results of Figure 9; successful quenching with 100 kΩ, and almost quenched but failed with 30 capacitance is 6 fF, a threshold resistance value is estimated to be about 33 kΩ in agreement with the kΩ. of Figure 9; successful quenching with 100 kΩ, and almost quenched but failed with 30 kΩ. results R (a) |V| (b) 30 29 |V| C R = 30kΩ R = 100 kΩ 28 27 R=∞ 26 25 0 500 1000 t (ps) 1500 2000 0 500 1000 t (ps) 1500 2000 (c) 10000 8000 n 6000 4000 2000 0 Figure 9. (a) A model circuit for a resistive quenching. (b) Calculated reverse biases for R = 100 kΩ (blue) 30 kΩ (green)circuit R = ∞for (red). (c) Carrier numbers(b) forCalculated R = 100 kΩreverse (blue) Rbiases = 30 kΩ = FigureR9.= (a) A model a resistive quenching. for (green) R = 100RkΩ ∞ (red). Capacitance is 6fF for all conditions. (blue) R = 30 kΩ (green) R = ∞ (red). (c) Carrier numbers for R = 100 kΩ (blue) R = 30 kΩ (green) R = ∞ (red). Capacitance is 6fF for all conditions. 5. Conclusions 5. Conclusions In conclusion, we have presented modeling and analysis of a Geiger mode pulse generation and of the capacitive relaxation quenching in a SPAD-based CIS. The carrier dynamics are modeled by In conclusion, we have presented modeling and analysis of a Geiger mode pulse generation and BCEs. Numerical calculations of BCE show that the carrier number in a SPAD depletion region is of the capacitive relaxation quenching in a SPAD-based CIS. The carrier dynamics are modeled by maximum just at the breakdown voltage which reassures the role of the breakdown voltage as an BCEs. Numerical calculations of BCE show that the carrier number in a SPAD depletion region is maximum just at the breakdown voltage which reassures the role of the breakdown voltage as an Sensors 2020, 20, 3007 13 of 14 equilibrium point. This, in turn, specified a necessary condition for successful quenching setting the final state being below |VBD | and the voltage swing is found to be nearly as twice large as |Vex | for a purely capacitive quenching. These analyses are essential to a design of SPAD-based CIS to avoid overflow. This design methodology is confirmed by experiments with the 400 × 400 6 µm pixels SPAD-based CIS. There is no overflow as confirmed by histograms of the images. This means that the signal level of images falls below the saturation level of the pixel output. We also show the driving force of the quenching is the same, i.e., reduction of the bias field due to carriers stored on the parallel capacitance, both in a resistive quenching circuit and in a capacitive quenching circuit demonstrating the generality of the presented modeling and analysis regardless of circuit types. Author Contributions: Conceptualization, Y.H.; Investigation, A.I. and Y.H.; Methodology, A.I., T.O., S.K. and Y.H.; Validation, A.I. and T.O.; Visualization, A.I. and T.O.; Writing – original draft, A.I.; Writing – review & editing, Y.H. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Cova, S.; Longoni, A.; Andreoni, A. Toward picosecond resolution with single-photon avalanche diodes. Rev. Sci. Instr. 1981, 52, 408–412. [CrossRef] Charbon, E. 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