# Energy response of X-rays under high flux conditions using a thin APD for
the energy range of 6–33 keV

T. Masuda<EMAIL_ADDRESS>T. Hiraki  H. Kaino  S. Kishimoto  Y.
Miyamoto  K. Okai  S. Okubo  R. Ozaki  N. Sasao  K. Suzuki  S. Uetake  A.
Yoshimi  K. Yoshimura  Research Institute for Interdisciplinary Science,
Okayama University, Okayama, Japan  High Energy Accelerator Research
Organization, Tsukuba, Ibaraki, Japan

###### Abstract

This paper reports on the demonstration of a high-rate energy measurement
technique using a thin depletion layer silicon avalanche photodiode (Si-APD).
A dedicated amplitude-to-time converter is developed to realize simultaneous
energy and timing measurement in a high rate condition. The energy response of
the system is systematically studied by using monochromatic X-ray beam with an
incident energy ranging from 6 to $33\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$.
The obtained energy spectra contain clear peaks and tail distributions. The
peak fraction monotonously decreases as the incident photon energy increases.
This phenomenon can be explained by considering the distribution of the energy
deposit in silicon, which is investigated by using a Monte Carlo simulation.

###### keywords:

avalanche photodiode , X-ray

##  1 Introduction

Silicon avalanche photodiodes (Si-APDs) are extensively used in a variety of
experiments because of their single photon sensitivity, magnetic field
insensitiveness, robustness, and fast time response[1, 2, 3, 4, 5, 6].
Although most of the applications use Si-APDs as highly sensitive visible or
near-infrared light sensors, they are able to detect X-rays at a single photon
level with good signal-to-noise ratio.

Si-APDs are used both with and without scintillators for X-ray detection. In
particular, direct X-ray detection without any scintillators is used for
experiments that need fast time response for a single photon[7, 8, 9]. Si-APD
without a scintillator can realize satisfactory time resolution of typically
$\sim$ns in the full width at half maximum (FWHM), and higher than
$100\text{\,}\mathrm{p}\mathrm{s}$ was also reported[10]. A high count rate
capability of more than $10^{6}$–$10^{8}$ counts per second (cps) can also be
easily achieved using APDs. It is difficult to sustain such high counting rate
by using scintillators. The energy resolution of Si-APD itself is another
advantageous characteristic. Si-APDs operated in the linear region present an
energy resolution of approximately 10–20% in FWHM. Operation involving both
the fast time response and energy sensitivity is one of the advantages of Si-
APD for direct X-ray detection.

It is well known that the internal structure of Si-APD strongly affects the
spectra of both time and energy. The photon absorption position widely
distributes along the thickness because the attenuation length of photons in
silicon is of nearly the same order as that of the Si-APD thickness in the
energy region of a few to tens of keV. The drift velocity of electrons in
silicon is approximately
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$
at $10\text{\,}\mathrm{V}\mathrm{/}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and
room temperature[11], thereby limiting the time resolution. The thickness of
the drift region therefore, should be less from the viewpoint of timing
performance. The energy response has a different dependence on the structure.
The output pulse height of Si-APD irradiated by monochromatic X-rays generally
has a low-energy tail component as well as a full energy peak. The presence of
the tail component is potentially due to the photons absorbed not at the drift
region but those at the avalanche region[12]; thus, a thicker drift region is
better for energy measurement. Many studies have focused on the energy
response of various Si-APDs for X-rays[13, 14];however, because of the
incompatibility between the energy and timing response, there are few reports
on the energy response of the Si-APDs optimized for fast time response.

We recently developed a fast single X-ray photon detection system by using
thin Si-APDs[15]. It was developed for the synchrotron-radiation based nuclear
resonant scattering (NRS) measurement of thorium-229[16]. The aim of the
experiment is population transfer from the nuclear ground state to the nuclear
first excited state whose energy is extremely low[17, 18]. The process that we
use is nuclear resonant excitation to the second excited state followed by
deexcitation to the first excited state. The NRS signals are detected to
confirm the population transfer. The NRS measurement is carried out in the
time domain as is usually done with general NRS measurements. The problem is
the expected short lifetime and the narrow partial width for the resonant
excitation of the objective state — only
$\sim$$150\text{\,}\mathrm{p}\mathrm{s}$ and
$\sim$$1\text{\,}\mathrm{n}\mathrm{e}\mathrm{V}$, respectively. To overcome
this problem, the detection system needs quite fast time response and high
rate tolerance. In addition, it needs to measure the energy of a single photon
simultaneously to suppress the background due to the radioactivity of
thorium-229 and its daughter nuclei. The developed system had an overall time
resolution of $120\text{\,}\mathrm{p}\mathrm{s}$ in FWHM and a short tail of
the order of $10^{-9}$ at $1\text{\,}\mathrm{n}\mathrm{s}$ apart from the peak
in the time spectrum. It could also measure both the energy and timing of a
single photon simultaneously, even at a high rate of more than
${10}^{6}\text{\,}\mathrm{c}\mathrm{p}\mathrm{s}$ for one Si-APD.

In the previous work, we focused mainly on the timing performance of the
system by using actual NRS data. In this paper, however, we describe the
method of both the timing and energy measurement in high count rate, and the
obtained performance by using dedicated monochromatic X-ray beam whose energy
is tuned from 6 to 33 keV. A Monte Carlo simulation is also performed to
explain the obtained energy spectra.

##  2 Energy measurement method

A technique commonly used to measure the energy is to integrate the charge of
the detector signal by a gated integrator followed by processing by an analog-
to-digital (AD) converter or a multichannel analyzer. AD conversion of shaped
pulse height, which is proportional to the charge, is also used. For cases in
which both the energy and timing need to be recorded, amplitude-to-time
converters (ATCs) or charge-to-time converters followed by TDC are also widely
used[19, 20, 21, 22, 23]. This method can further simplify a method compared
to a system built with both ADCs and TDCs. The pulse shape recording technique
is impractical for such high rate measurement of around or above
$1\text{\times}{10}^{6}\text{\,}\mathrm{c}\mathrm{p}\mathrm{s}$ per channel
owing to a large amount of data. The ATC method is a practical solution.

The developed ATC consists of a peak-hold circuit, a constant discharge
circuit, and peripheral logic circuits. Figure 1 shows an image of the printed
circuit board of ATC for a channel. Figure 2 shows the schematic diagram of
the ATC. The converter operates as follows. The transistor T1 is open in the
initial position and the constant current flows from the
+$5\text{\,}\mathrm{V}$ source to the $-5\text{\,}\mathrm{V}$ sink. A start
signal and a negative analog pulse are input from the input connectors;
subsequently, the analog switch M7 changes the connection and the positive
pulse converted by the operational amplifier M1 goes to T1\. During the
increase in the positive pulse, T1 is open and the capacitor C1 is immediately
charged by the current from the +$5\text{\,}\mathrm{V}$ source. When the pulse
is decreasing, C1 maintains the voltage at its peak value and T1 closes. Then,
C1 constantly discharges through the resistor R1, and returns to the initial
condition. One of the comparators, M8, toggles the output logic state to high
when the end point of the positive pulse arrives at T1, and the second
comparator M9 toggles the output logic state to low when the conversion is
completed by the following logic circuit. The state of the analog switch M7 is
changed immediately after the analog pulse passes through it, and it is
maintained during the conversion to ignore the next signal coming before the
conversion is completed. The timing sequence of the ATC is shown in Fig. 3.

The ATC demonstrated a conversion time of
$10\text{\,}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{k}\mathrm{e}\mathrm{V}$.
This value can be reduced by decreasing the value of C1 or R1 by a certain
degree. We selected $390\text{\,}\mathrm{p}\mathrm{F}$ as C1 because we noted
a deterioration of the energy response when C1 was
$100\text{\,}\mathrm{p}\mathrm{F}$.

The energy spectra obtained in the actual NRS measurement at high count rate
of more than ${10}^{6}\text{\,}\mathrm{c}\mathrm{p}\mathrm{s}$ have been
reported in our previous paper[15]. This is potentially useful for other types
of measurements as well as the 229Th NRS measurement, for example, depth
selective Mössbauer spectroscopy by electron detection proposed in [24].

Figure 1: (color online) Printed circuit board of one channel of ATC. Figure
2: Simplified ATC circuit diagram. The power supply lines, pull-up (down)
resisters, bypass capacitors, electrostatic discharge protection components,
and other less relevant components are omitted. M1–3: operational amplifiers
(AD8000), M4–5: Logic converters (LTC6957), M6: CMOS D-type Flip-Flop
(TC74LCX74), M7: Analog switch (MAX4644), M8–9: ECL comparators (MAX9693),
M10-11: ECL D-type Flip-Flops (MC100LVEL51), M12: ECL buffer (MC100LVEL16),
M13: Schmitt buffer (NL27WZ17), and T1: NPN transistor (2SD2656). H and L
represent logic high and low, respectively. Figure 3: Time sequence of the ATC
operation demonstrating two cases for pulses with different amplitudes. The
solid (dashed) lines represent a higher (lower) pulse height case.

##  3 Energy response study

This section presents the energy responses of the Si-APD as a direct X-ray
sensor. The absolute efficiency was measured at an energy of
$13\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$, which is our research object in
the thorium-229 NRS experiment. The internal structure can be estimated by
using the efficiency data. The incident energy dependence of the energy
spectra was considered for values from 6 to
$33\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$.

###  3.1 Setup

The experiments described in this section were performed at the BL-14A beam
line of KEK Photon Factory (KEK-PF)[25]. The incident X-ray beam from the
storage ring was monochromatized by a Si(111) double-crystal monochromator.
The typical relative bandwidth $\Delta E/E$ of the beam is $2\times 10^{-4}$.
A Si(553) monochromator was used above
$24\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$. The beam energy was tuned from 6
to $33\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$. Pin-holes located at the upper
stream of the Si-APD defined the beam size. Several filters made of pure
metals were used to moderate the beam intensity. The beam was injected
directly into the Si-APD perpendicular to the surface.

The Si-APD (Hamamatsu Photonics S12053-05) whose sensitive area was
$0.5\text{\,}\mathrm{m}\mathrm{m}$ in diameter, is a reach-through type Si-
APD. The detector system was almost the same as that used in the previous work
but the number of Si-APD was changed from six to seven, and an associated
minor modification of a preamplifier circuit layout was carried out. One out
of the seven Si-APDs was tested in this measurement because they are
identical, including the circuit. The device was operated at room temperature.
The applied reverse bias voltage was $150\text{\,}\mathrm{V}$ and the nominal
gain was 50.

The output signal was amplified by a preamplifier (Mini-circuits, RAM-8A+)
whose gain was $32\text{\,}\mathrm{d}\mathrm{B}$, and it was sent to the CFD
and a pulse shaping circuit. The CFD determined the pulse timing within a
deviation of $\pm 25\text{\,}\mathrm{p}\mathrm{s}$, and also provided a start
signal to the following ATC described in Sect. 2. Both the CFD and ATC output
and an accelerator reference signal were digitized by a multi-stop TDC (FAST
ComTec, MCS6) at a sampling rate of 100-ps. The recorded data were converted
to timing and energy information for each signal in the offline analysis. The
energy information was converted from the time difference between an ATC
output pulse and the previous CFD output pulse.

###  3.2 Efficiency at $13\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$

The absolute efficiency of the Si-APD was measured by limiting the beam size
to $0.4\text{\,}\mathrm{m}\mathrm{m}$ in diameter by a pin-hole so that the
sensitive area of the Si-APD covered the whole beam. The absolute photon flux
was measured by using a NaI scintillator with a
150-$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-thick beryllium window.
Because the NaI scintillator is a low count rate detector, we placed
additional zirconium filters in front of the NaI scintillator to reduce the
counting rate to less than ${10}^{4}\text{\,}\mathrm{c}\mathrm{p}\mathrm{s}$.
The transmittance of the filters was estimated by changing the filter
combination. The scintillation lights were read by a photomultiplier tube and
were digitized by a multi-channel analyzer.

The energy spectrum measured by the Si-APD is shown in Fig. 4. The counting
rate was $3.7\text{\times}{10}^{5}\text{\,}\mathrm{c}\mathrm{p}\mathrm{s}$ and
the measuring time was $100\text{\,}\mathrm{s}$ for each spectrum. There was a
clear peak at $13\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$; the FWHM was 21%.
The low energy tail was also observed. The fluctuation at the 3–4 keV region
was caused by digital electric noise in the ATC circuit board.

The absolute efficiency was measured by counting the count rate of the Si-APD.
The efficiency of the whole region including the tail component was
approximately 1.7%, which corresponded to a silicon thickness of
$4.8\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$; this was calculated based
on the efficiency and the linear attenuation coefficient of silicon[26]. If
energy selection is required, the efficiency in the full energy peak is also
important. The efficiency within the $\pm 20\%$ region from the peak was
1.0$\pm$0.3%, and it corresponded to a silicon thickness of
$2.9\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.

Figure 4: Observed energy spectrum for
$13\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ X-rays.

###  3.3 Energy spectra

Energy spectra were obtained for various incident X-ray energies: 6.0, 9.0,
12.0, 13.0, 15.0, 18.0, 24.0, 27.0, 29.2, 30.0, and 33.0 keV. The beam size
was $0.8\text{\,}\mathrm{m}\mathrm{m}$ in diameter in these measurements and
thus, the sensitive area was fully irradiated similar to an actual situation.
Figure 5 shows the energy spectra. The peak-to-tail ratio decreases in a
pronounced manner with increase in the incident energy. On the other hand, the
relative widths which are the ratio of the peaks width and the peak values are
almost constant, between 20–30%. Despite this system being optimized for
timing performance, it can still obtain energy information at this level.

We noted a slight nonlinearity in the energy spectra. Figure 6 shows the
relation between the measured values at the peaks and the actual incident
X-ray energy. The values at the peaks were obtained by Gaussian fit at the
peaks. We checked the preamplifier output pulse shapes for each incident
energy and they did not demonstrate such degradation. This nonlinearity was
therefore primarily owing to the shaper and the ATC saturation and was not
caused by the Si-APD itself. Note that the nonlinearity of the spectra shown
in Fig. 5 was already corrected. We used an empirical function of
$p_{0}+p_{1}x+p_{2}\exp(x/p_{3})$ for the correction. The fit results are
shown in Fig. 6.

Figure 5: (color online) Energy spectra with linearity correction. All
histograms are normalized so that their integration is 1. Each label
represents the incident X-ray energy for each histogram. Figure 6: Incident
energy vs. pulse height. The horizontal axis represents the peak centers of
the ATC spectra. One count corresponds to $0.1\text{\,}\mathrm{n}\mathrm{s}$.
The vertical axis indicates the energy of the incident X-ray beam. The solid
points denote data for each incident energy. The curve shows the fit function
described in the text, and the parameters in the box are the fit results.

##  4 Discussion

###  4.1 Peak width

The peak width is defined as the half width at the half maximum (HWHM), which
indicates the interval from the peak center to the half maximum point in the
higher energy side. This is done to avoid the low-energy tail effect. The peak
width for each incident energy is shown in Fig. 7, and it exhibits a roughly
linear behavior. The FWHM energy resolution, which is twice the peak width, is
approximately 20–30%.

Figure 7: Peak width for each incident energy. The solid circles denote the
data. The straight line represents the linear fit crossing the origin.

The FWHM of a full energy peak $\Delta E$ of a Si-APD in X-ray detection can
be written as follows[13]:

$\displaystyle\Delta E$ $\displaystyle=$ $\displaystyle
2.35\left[E\varepsilon(f+F-1)+\left(\frac{\sigma_{MU}}{M}\right)^{2}E^{2}+(\Delta
E_{\mathrm{n}})^{2}\right]^{1/2},$

where $E$ is the energy of the peak, $\varepsilon=$
$3.65\text{\,}\mathrm{e}\mathrm{V}$ is the unit energy generating one
electron-hole pair in silicon, and $f$ is the Fano factor, which can be
assumed to be $\sim 0.1$ for a silicon detector[27]. $F$ is the excess noise
factor; it depends on the gain and it is of the order of one[12]. The term
$\sigma_{MU}/M$ refers to the gain deviation which is caused by non-uniformity
of the avalanche gain and the system. $\Delta E_{\mathrm{n}}$ is the noise
contribution. The first term in the square bracket, which is the statistical
term, contributed only less than 5% in our measurement and the noise term does
not depend on the energy; thus, the observed energy resolution was dominated
by the gain deviation.

###  4.2 Peak fraction

Figure 8 shows the incident energy dependence of the peak fraction. Here, the
peak fraction is defined as the number of events in the energy region within
two times of the half width from the center to both sides divided by the
number of events corresponding to more than
$2\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$. A monotonically decreasing trend
is noted. This behavior cannot be simply explained on the basis of the X-ray
attenuation length and the position dependence of the avalanche gain along the
thickness. The absorbed positions of irradiated photons uniformly distribute
in the depletion layer because the thickness of the depletion layer is less
than $5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and the attenuation
length in silicon is, for example,
$280\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ for
$13\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ photons[26]. That means the peak
fraction would be constant in the high energy region.

Figure 8: Incident energy dependence of the peak fraction. The solid circles
denote the data and the straight line represents the linear fit.

This tendency may be explained by considering the travel of initial
photoelectrons in silicon, as shown in Fig. 9. The photoelectron originating
at the photoelectric absorption point has energy equal to the energy
difference between the incident photon energy and the binding energy of the
K-shell electron of silicon. This photoelectron moves in the silicon and
creates many electron-hole pairs along its track. In the usual case in which
the energy is low and a Si-APD is thick, there is no need to consider the
spread due to the photoelectron travel; however, the spread becomes important
in case the thickness of the Si-APD is low and the photon energy is high such
that the photoelectron range is comparable with the Si-APD thickness.
Photoelectrons that originate even in the drift region can move to the
avalanche region and create electron-hole pairs; then, they contribute not to
the full energy peak but to the tail component. The probability of occurrence
of either depends on the range of electrons in silicon.

Figure 9: (color online) Schematic of the microscopic view of the reach-
through type Si-APD operation for an X-ray photon. The X-ray photon comes in
from the left side and is absorbed in the Si-APD. The photoelectron that has
sufficient kinetic energy moves inside randomly owing to the multiple
scattering process. The track of the photoelectron is represented by the dark
yellow zig-zag line. Many electron-hole pairs are created along the track. The
red (white) circles represent electrons (holes). The electrons are transferred
to the cathode side as indicated by the red arrows and multiplied due to the
high electric field in the avalanche region.

A Monte Carlo simulation was performed to demonstrate the phenomenon. The
simulation was based on the Geant4 (version 10.2) package[28, 29, 30] and its
MuElec extension[31]. The extension can simulate electrons in silicon down to
$16.7\text{\,}\mathrm{e}\mathrm{V}$ and the validated energy range is
$50\text{\,}\mathrm{e}\mathrm{V}$–$50\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$.

The simulation procedure is as follows: An absorption position along the depth
was randomly generated for a photon based on the attenuation length at its
incident energy. The energy deposit distribution along the photoelectron
tracks was simulated by using the MuElec model. The output signal was
generated by integrating the local energy deposits multiplied by the avalanche
gain at the corresponding local points. Finally, the output signal was smeared
by a gaussian distribution which corresponds to the gain deviation of
$\sigma_{MU}/M$. We accumulated $10^{5}$ events for one energy set by
repeating the procedure.

The avalanche gain along the thickness $g(z)$ was simply assumed as the
complementary error function, i.e., $g(z)\propto\mathrm{erfc}[(z-t)/d]$. Here,
$z$ is the depth from the irradiated surface, $t$ corresponds to the depth at
the center of the avalanche region, and $d$ corresponds to the width of the
avalanche region. We assumed the parameters as
$t=3.95\,$$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, $d=0.85$
$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, and $\sigma_{MU}/M=0.07$ to
reproduce the absolute efficiency and the peak width at
$13\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ described in Sect. 3.2.

Figure 10 shows the simulated energy spectra. The upper figure shows the
spectra taking into account only the position dependence of the gain, the
photon absorption position distribution, and the gain deviation. The spectra
shown in the lower figure include the energy deposit distribution due to the
photoelectron travel. A higher incident energy corresponds to a larger
difference in the spectra. The peak fraction decreases owing to the energy
deposit distribution, as shown in Fig. 11. This simulation can explain the
rapid decrease in the peak fraction. Because the internal gain distribution
was simply assumed as to be a one-dimensional complementary error function in
this simulation, knowledge of the three-dimensional internal structure may be
useful to reduce the quantitative disagreement between the data and the
simulation in the full energy region. In addition, it is possible that the
imperfection of the simulation model might be the issue.

The Si-APD in this study is thinner than those used in previous studies for
energy measurement; therefore, the energy deposit distribution is relatively
important for the consideration of the energy response especially for higher
energy. The standard deviation of the energy deposit distribution is only
$0.09\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ along the depth in our
simulation for the incident energy of
$6\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$; however, the corresponding value
is $2.67\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ for the incident
energy of $30\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$, which is comparable
with the thickness of the depletion layer and thus cannot be ignored.

Figure 10: (color online) Simulated energy spectra. a) Energy deposit spread
not included. b) Energy deposit included. The color of each histogram is the
same as that in Fig. 5. Figure 11: Simulated peak fractions. The open circles
correspond to the simulation including the energy deposit distribution in
silicon and the open boxes correspond to the one without it. The dashed line
represents the linear fit to the open circles. The solid circles and the solid
lines are respectively the experimental data and the linear fit, as in Fig. 8.
They are drawn for comparison.

##  5 Conclusion

We developed an X-ray photon counting system that equips thin Si-APDs. The
system can accumulate both timing and energy of single X-ray photon
simultaneously in high rate conditions. We developed a dedicated fast ATC that
demonstrates a conversion time of only
$10\text{\,}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{k}\mathrm{e}\mathrm{V}$ to
realize this requirement. Although the system was developed to maximize the
timing performance, it exhibits reasonably good energy resolution using the
ATC and a commercial TDC. The absolute efficiency for
$13\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ photons was 1–2%. The FWHM energy
resolution was 20–30% in the energy region from 6 to
$33\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$. The peak fraction shows
monotonous decreasing with respect to the incident photon energy. This
phenomenon can be explained by considering the energy deposit distribution in
silicon owing to the travel of the photoelectrons; this was observed by using
a Monte Carlo simulation based on the Geant4 MuElec process.

This work demonstrates the high rate energy measurement and the availability
of the energy information with a Si-APD optimized for fast time response. It
is seen that the energy deposit spread in silicon due to the travel of
photoelectrons as well as the absorption position distribution of irradiated
photons is important to understand the energy spectra obtained with such thin
devices.

## Acknowledgments

We thank M. Seto and S. Kitao for providing MCA6. This work was performed
under the approval of the Photon Factory Program Advisory Committee (Proposal
No. 2017G085). T.H. is supported by Grant-in-Aid for Japan Society for the
Promotion of Science (JSPS) Research Fellow. This work was supported by JSPS
KAKENHI Grant Numbers JP15H03661, JP17K14291, JP18H01230, and JP18H04353. A.Y.
acknowledges the MATSUO foundation.

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