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;
okino.toru@jp.panasonic.com (T.O.); koyama.shinzo@jp.panasonic.com (S.K.);
hirose.yutaka@jp.panasonic.com (Y.H.)
* Correspondence: inoue.akito@jp.panasonic.com
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
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(4T) pixel architecture could be applicable, then one could expect scaling advantages in system size and
high resolution. [16] 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 [17]. 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 [18].
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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 [19]
vs,e
1.02 × 107 cm /s
Saturation velocity of hole [19]
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 [18]
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 [17] 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.
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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
,
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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 [20].
!
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. [19] 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 |.
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(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
>
[14].
(2)
because
in
other
previous
works,
SPAD
|V
|
ex
Q
| [14]. (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].
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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 [17]. 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| [17].
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
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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
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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
[25].
|Vex |
unit). PDE increases as |𝑉 | because of the rise of avalanche triggering probability [25].
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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
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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 [28],
and/or after multiplication [28], 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
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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.
[24],
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[24],
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
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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.
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