# Large area photon calorimeter with Ir-Pt bilayer transition-edge sensor for
the CUPID experiment

V. Singh<EMAIL_ADDRESS>Department of Physics, University of California,
Berkeley, Berkeley-94720, CA  G. Benato  GSSI-Gran Sasso Science Institute,
I-67100, L’Aquila, Italy  Department of Physics, University of California,
Berkeley, Berkeley-94720, CA  M. Beretta  Department of Physics, University of
California, Berkeley, Berkeley-94720, CA  C. Capelli  Nuclear Science
Division, Lawrence Berkeley National Laboratory, Berkeley-94720, CA  C.L.
Chang  High Energy Physics Division, Argonne National Laboratory,
Lemont-60439, IL  Kavli Institute for Cosmological Physics, University of
Chicago, Chicago-60637, IL  B.K. Fujikawa  Nuclear Science Division, Lawrence
Berkeley National Laboratory, Berkeley-94720, CA  E.V. Hansen  Department of
Physics, University of California, Berkeley, Berkeley-94720, CA  Yu.G.
Kolomensky  Department of Physics, University of California, Berkeley,
Berkeley-94720, CA  Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley-94720, CA  M. Lisovenko  High Energy Physics Division,
Argonne National Laboratory, Lemont-60439, IL  L. Marini  INFN, Laboratori
Nazionali del Gran Sasso, I-67100, Assergi, Italy  Department of Physics,
University of California, Berkeley, Berkeley-94720, CA  WK. Kwok  Materials
Science Division, Argonne National Laboratory, Lemont-60439, IL  V. Novosad
Materials Science Division, Argonne National Laboratory, Lemont-60439, IL
Physics Division, Argonne National Laboratory, Lemont-60439, IL  J. Pearson
Materials Science Division, Argonne National Laboratory, Lemont-60439, IL  B.
Schmidt  Nuclear Science Division, Lawrence Berkeley National Laboratory,
Berkeley-94720, CA  K.J. Vetter  Department of Physics, University of
California, Berkeley, Berkeley-94720, CA  G. Wang  High Energy Physics
Division, Argonne National Laboratory, Lemont-60439, IL  B. Welliver
Department of Physics, University of California, Berkeley, Berkeley-94720, CA
Nuclear Science Division, Lawrence Berkeley National Laboratory,
Berkeley-94720, CA  U. Welp  Materials Science Division, Argonne National
Laboratory, Lemont-60439, IL  V. Yefremenko  High Energy Physics Division,
Argonne National Laboratory, Lemont-60439, IL  J. Zhang  High Energy Physics
Division, Argonne National Laboratory, Lemont-60439, IL

###### Abstract

CUPID is a next-generation neutrinoless double-beta decay experiment that will
require cryogenic light detectors to improve background suppression, via the
simultaneous readout of heat and light channels from its scintillating
crystals. In this work we showcase light detectors based on a novel Ir-Pt
bilayer transition edge sensor. We have performed a systematic study to
improve the thermal coupling between the photon absorber and the sensor, and
thereby its responsivity. Our first devices meet CUPID’s baseline noise
requirement of <100 eV rms. Our detectors have risetimes of $\sim$180 $\mu$s
and measured timing jitter of <20 $\mu$s for the expected signal-to-noise at
the Q-value of the decay, which achieves the CUPID’s criterion of rejecting
two-neutrino double-beta decay pileup events. The current work will inform the
fabrication of future devices, culminating in the final TES design and a
scaleable readout scheme for CUPID.

††preprint: APS/123-QED

##  I Introduction

CUPID (CUORE Upgrade with Particle IDentification) [1] is the proposed next
generation upgrade of CUORE (Cryogenic Underground Observatory for Rare
Events) [2] that will primarily search for the lepton number violating process
of neutrinoless double beta (0$\nu\beta\beta$) decay. The discovery of a
lepton number violation would be a clear signature beyond the Standard Model
of particle physics [3, 4]. Moreover, depending on the decay’s underlying
mechanism [4, 5, 6, 7, 8, 9, 10], the experiment can shed light on the
neutrino’s absolute mass scale, which is still unmeasured. CUPID will build on
the experience of CUORE, an experiment that uses about a tonne of low
temperature calorimeters operating close to $\sim$10 mK temperatures. CUPID
will be housed in CUORE’s state-of-the-art cryogenic infrastructure [11] when
the latter stops data collection. However, it will depart significantly from
CUORE in its choice of detector. CUORE currently uses natTeO2 crystals
(50$\times$50$\times$50 mm3) as low-temperature calorimeters, which are only
sensitive to heat signals and therefore do not offer particle identification
capabilities. In CUORE, the dominant background is from energy degraded
$\alpha$ particle events in the crystal that can mimic the $\beta^{-}$ signals
that we expect from the 0$\nu\beta\beta$ process. CUPID aims to improve this
using scintillating Li2100MoO4 (LMO) crystals (45$\times$45$\times$45 mm3),
where Mo will be enriched to contain > 95% 100Mo. By simultaneously measuring
thermal and scintillation signals, $\alpha$ events can be efficiently
identified, as their light yield is a factor of five lower than that of
$\beta$/$\gamma$ particles [12, 13].

If a 0$\nu\beta\beta$ decay event occurs, the two electrons emitted in the
process deposit the transition energy of 3034 keV ($Q_{\beta\beta}$) in an LMO
crystal. This results in a measurable heat signal and emission of sizeable
scintillation light. In the CUPID detector, each LMO will be paired with a
low-temperature auxiliary calorimeter with a wide area ($\sim$20 cm2) to
detect light with high photon collection efficiency. The energy of total light
expected to hit an auxiliary calorimeter facing one of the LMO crystal
surfaces is $\mathcal{O}(1~{}\text{keV})$ for $\sim$3 MeV electrons or gamma-
rays [14]. As a consequence, this device must have a very low energy threshold
of $\mathcal{O}(100~{}\text{eV})$ rms. The detector should also have a good
timing resolution to differentiate a pile-up event from two-neutrino double
beta (2$\nu\beta\beta$) decays, which occur at a rate several orders of
magnitude higher than 0$\nu\beta\beta$ decay [15]. The slow response time of
massive LMO calorimeters can cause an accidental pileup of 0$\nu\beta\beta$
events and/or background events that will limit the sensitivity of the
searches. The pileup effect can be severe for 100Mo, which is known to have a
short half-life of 7.1$\times$1018 yr amongst all 2$\nu\beta\beta$ decaying
isotopes [16]. The CUPID goal of keeping the 2$\nu\beta\beta$ decay pileup
rate below 5$\times$10-4 (counts/kg/keV/yr) requires an effective timing
resolution of <170 $\mu$s on the light detectors [1]. The light detectors
should also have a dynamic range of 10–15 keV to calibrate energy response
with X-rays from Fe and Cu if needed. Other requirements include high
radiopurity and low heat capacity of the materials used for the detector’s
construction. We have summarized the requirements of light detectors for CUPID
baseline design in Table 1.

Table 1: Light detector requirements for the baseline design of the CUPID experiment. Coverage area | 45 mm $\times$ 45 mm
---|---
Number of detectors | 1710
Light absorption | > 90 %
Baseline rms | < 100 eV
Pileup resolution | < 170 $\mu$s
Dynamic energy range |  $\mathcal{O}$(10 eV) to $\mathcal{O}$(10 keV)
238U contamination | < 20 mBq/kg
232Th contamination | < 10 mBq/kg

Developing a high-resolution and high-efficiency detector coupled with a fast
transition-edge sensor (TES) is promising. Schäffner et al. [17] used silicon-
on-sapphire discs to achieve a baseline resolution of $\sim$23 eV over a
coverage area of $\sim$13 cm2. Recently, Fink et al. [18] demonstrated an
excellent baseline energy resolution of $\sim$4 eV using Quasiparticle-trap-
assisted Electrothermal feedback Transition-edge sensors (QETs) [19] on a
surface area of $\sim$46 cm2. While both works used tungsten (W)-TES, there
are inherent challenges in producing thin films at high throughput [20]. TES
production requires a highly controlled environment with reproducible
crystallinity and minimal structural and radioactive impurities in thin films.
Furthermore, since the transition temperature is an inherent property of the
material, it is difficult to tune the transition temperature ($T_{c}$) of
W-TES by controlling its crystalline structure or introducing ferromagnetic
impurities. Tuning is critical for applications where lowering $T_{c}$ can
improve energy resolution and better match the operating temperature of the
cryogenic apparatus.

In our previous work [21], we have detailed the fabrication recipe for
producing Ir-based films with predictable and reproducible critical
temperatures. We used a bilayer TES where we altered the $T_{c}$ of a
superconducting Ir film using a normal Pt film on top of it. We use the
superconducting proximity effect [22, 23] to tune the $T_{c}$ by varying the
thickness of the Pt film. The thermal characteristics of the Ir-Pt bilayer
have been presented in Ref. [ 24] where we reported the electron-phonon (e-ph)
coupling strengths for the Ir-Pt bilayer and demonstrated that the addition of
normal metal gold pads overlapping the TES and the photoabsorber improves the
thermal coupling between them.

In this paper, we present one of the first light detector characterizations
using Ir-Pt bilayers and showcase their potential as a suitable detector
technology for CUPID.

##  II Detectors

We use high resistivity (>10000 $\Omega\cdot$cm) intrinsic silicon wafers
(diameter = 50.8 mm and thickness = 280 $\mu$m) as calorimeters (Fig. 1).
Silicon can be produced with high radiopurity and has a relatively high Debye
temperature $\Theta_{D}\sim$640 K [25], which ensures a marginal contribution
to the total heat capacity of the calorimeter. We have fabricated several
devices but selected two devices, Detector-I and Detector-II, for detailed
characterization reported here. Both detectors are similar, with the
difference being that Detector-II has a layer of Si3N4 anti-reflective (AR)
coating deposited on the side of the wafer opposite the TES. P. Chen et al.
[26] have measured the luminescence spectra of LMO crystals down to a 20 K
temperatures and found a broad peak around 551 nm with the luminescence
intensity increasing with decreasing temperature. We chose Si3N4 as an anti-
reflecting coating material due to its excellent optical properties, chemical
stability, and natural compatibility with Si substrates. An added benefit is
that thin films of this material are widely used by the microfabrication
community and can be patterned via Reactive Ion Etching, which may be required
in the future stages of the project. We sputtered the films reactively in a
UHV-grade deposition chamber (base vacuum of 1$\times$10-8 Pa) using a DC
power supply and a high-purity (99.999%, undoped) Si target. We applied low
power (20 W) RF bias for in situ substrate cleaning, as a part of the pre-
sputtering protocol, and during the deposition process. As-grown films are
very smooth; we found that the average roughness measured by an Atomic Force
Microscope was <1 nm. Furthermore, we confirmed excellent adhesion properties
by inspecting the structural integrity of the samples after sonication in IPA
and water for a prolonged time (30 min). We optimized the composition of thin
films by varying the flow of N2, while keeping the Ar flow, the deposition
power, and the working pressure fixed at 10 sccm, 150 W and 3 mTorr,
respectively. We measured the reflectance spectrum in the UV-VIS range using
the Filmetrics F20 tool to determine the optical properties of the deposition
films. As shown in the inset of Fig. 2, we can fine-tune the optical constant
of the sputtered films in a wide range by adjusting the nitrogen flow. Based
on these data, we have chosen to grow nitride films with a N2 flow of 10 sccm,
resulting in a refractive index n= $\sim$2.05 (at 632 nm), which is close to
the values typical for stoichiometric Si3N4 composition. As shown in Fig. 2,
$\sim$68 nm of thin material provides reflectivity of <1 % in the entire range
of interest (500–600 nm), a significant improvement compared to (35–40)% for
bare (uncoated) Si.

Figure 1:  Low-temperature photon calorimeter. (a) Top view of Detector-I: The
Ir-Pt bilayer sensor (TES) is located in the center of the Si wafer. The
superconducting Nb traces connect the sensor to the bonding pads at the edge
of the wafer. Single-crystal sapphire plates are used to provide a weak
thermal link to the copper bath. (b) Bottom view of Detector-II where 68 nm
Si3N4 has been deposited as an AR coating. Detector-II is identical to
Detector-I except for the AR coating (see the text for details). Figure 2:
Measured reflectivity of a Si wafer with 68 nm Si3N4 on top of it (solid blue
line). The reflectivity in the spectral region of interest is below 1% (dotted
red line). For comparison, we also show the reflectivity of the bare Si wafer
(dotted blue line).

We instrumented the Si absorber with an Ir-Pt bilayer TES close to the center
of the Si wafer. The bilayer TES is a 45 nm Ir film deposited atop a 20 nm Pt
film, both with an area of 330 $\mu$m $\times$ 300 $\mu$m. We then deposit Au
pads (700 $\mu$m $\times$ 700 $\mu$m) on both sides of the bilayer to increase
the phonon coupling between the sensor and the absorber. The Au pads have a
thickness of 200 nm and cover an area of 120 $\mu$m $\times$ 300 $\mu$m on
each side of the bilayer. The choice of Au pad size was reached after studying
its effect on the sensor-absorber coupling as described in Appendix A. The
remaining area of 90 $\mu$m $\times$ 300 $\mu$m was not covered with Au to
obtain a normal resistance (Rn) between $\sim$(0.5 – 1) $\Omega$. The expected
critical temperature of the sensor is $\sim$50 mK. Fig. 3 shows a schematic of
the TES patterned on the Si absorber.

Figure 3:  Schematic of the Ir(45 nm)-Pt(20 nm) bilayer sensor on the wafer.
The Au pads overlap a portion of the TES and Si photoabsorber in order to
improve the thermal coupling of the TES to the absorber.

We use superconducting Nb traces (200 nm thick and 20 $\mu$m wide) as
electrical leads that run from the sensor edges and terminate as wire bonding
pads near the edge of the wafer (Fig. 1). The Nb leads are on top of the Au
pads overlapping the sensor. We use 32 $\mu$m diameter Al-1%Si wire bonds to
connect the TES to the electrical circuit. A thin gold strip (1000 $\mu$m
$\times$ 100 $\mu$m $\times$ 150 nm) is deposited some distance away from the
TES, for use as a Joule-heater. This heater has a measured residual resistance
of 1 $\Omega$ below 1 K temperature.

We use sapphire (Al2O3) crystals as the support (weak) link between the Si
wafer and the Cu holder which is coupled to the thermal bath. Bachmann et al.
[27] have shown that 1/3 of the weak-link heat capacity contributes to the
total heat capacity of the calorimeter. Therefore, we compute the additional
heat capacity of the sapphire ($\Theta_{D}\sim$1042 K [28]) to be <10% of the
Si absorber. Next, we glue the wafer to the sapphire plates using a tiny dot
of UV-cured glue to restrict its horizontal plane movement. The sapphire, in
turn, is held to the Cu holder using copper clamps, as shown in Fig. 1. We
expect the thermal boundary impedance between Cu and sapphire to be much
larger than the bonded contact between Si and sapphire [29]. This ensures that
heat does not leak out to the thermal bath before reaching the sensor.

##  III Measurements and Data Analysis

We performed the measurements in an Oxford Instruments Triton 400 dilution
refrigerator with a cryogen-free pulse tube cooler that reaches temperatures
down to $\sim$10 mK. Each TES is operated with a near constant voltage bias to
stabilize the device in the superconducting transition through negative
electrothermal feedback [19]. To achieve a voltage biased state, we utilize
shunt resistors that have a much lower resistance than the TES at its typical
operating point ($R_{sh}$ $\sim$ 20 m$\Omega$, $R_{TES}$ $\sim$ 50% $R_{n}$).
We acquire the TES signal after amplifying it using a single-stage DC-
Superconducting Quantum Interference Device (SQUID) array and readout
electronics from Magnicon [30]. The DC-SQUID and the shunt resistor are
mounted in the Still stage of the dilution refrigerator, which operates at
$\sim$ 700 mK. We use superconducting NbTi wires from the 700 mK stage to the
10 mK stage to bias the TESs. We monitor the temperature of the experimental
stage using a Johnson-Noise thermometer [31].

We measured a Tc of $\sim$33 mK for both of our detectors and a $R_{n}$ of
$\sim$550 m$\Omega$. The Tc’s are substantially lower than the expected value
of $\sim$50 mK. This is not surprising because the TES is very weakly
thermally coupled to the thermal bath. Therefore, any parasitic power, through
RF/EMI or vibrations, can keep the wafer at a higher temperature than the
measured bath temperature. As CUPID is designed to operate at ($\sim$ 10–15
mK), we operate our detectors at a base temperature of 15 mK.

We used two 200 $\mu$m core multimode fiber optic cables ($\lambda$ $\approx$
400 – 2200 nm), one for each detector, inside the cryostat to send light
pulses from outside to calibrate the detector. The fiber ends face the
opposite side of the wafer where the TES is deposited and have approximately
normal angle of incidence. We use an off-the-shelf commercial LED with a
wavelength of $\sim$600 nm and an emission width (FWHM) of $\sim$12 nm. The
LED optical fiber source is uncalibrated; however, it provides valuable
information regarding timing and amplitude-dependent pulse shapes and can be
used for energy calibration based on Poisson statistics of the photons
impinging on the detectors. We do not have an infrared filter for the fibers
inside the refrigerator. Although we have taken the utmost precaution to
thermalize the fibers at different stages of the dilution refrigerator, some
infrared radiation may leak into our detector through the fiber. The Au-heater
is prone to injecting excess heat due to ambient RF/EMI interference and
possible ground loops. This makes utilization of the fibers to excite the
detector a much better option (see Appendix B).

We generate a train of LED light pulses of varying amplitudes with a known
separation time to test the detector response to the light signal. We choose
the separation time between the generated LED pulses so that they are far
enough in time to allow the detector to return to its quiescent state. We keep
the voltage across the LED constant and vary the voltage pulse width across
the LED to get a different number of photons for each pulse. As a result, the
number of photons emitted from the LED has a linear dependence on its pulse
width. The width of the LED voltage is varied between 40 – 400 ns. The pulse
widths of excitation are much smaller than the typical risetime of the
detectors (see Fig. 4), and, therefore, we treat all the LED excitations as
instantaneous. In order to accurately identify these LED induced events, we
acquire a synchronous signal from our LED voltage generator in order to
provide a tag for these events in our data stream. This allows us to reject
events from ambient background radioactivity (x-rays, muons, etc).

Figure 4:  Average pulse shapes for different LED excitations on Detector-I in
the 20 ms event window. The inset shows the average pulse shapes for
Detector-I and Detector-II for similar amplitudes. Detector-I pulses have
typical risetimes of (175$\pm$10) $\mu$s and decaytimes of (1000$\pm$40)
$\mu$s. Detector-II pulses have similar risetimes of (176$\pm$7) $\mu$s but
slightly faster decaytimes of (780$\pm$30) $\mu$s. See the text for the
definition of risetime and decaytime.

The analog voltage of the SQUID output goes through a 10 kHz anti-aliasing
filter before being acquired by an 18 bit analog-to-digital converter board
(NI PXI-6289) with a sampling frequency of 150 kHz. We use a custom C++ DAQ
software to continuously acquire data streams and save them to disk to process
offline. Offline processing is performed using Octopus [32], which is a modern
C++ and ROOT [33] based framework, to analyze continuous digitized waveforms.
We use a simple threshold-over-rms method to identify and trigger on thermal
pulses, and inject triggers at fixed intervals in order to produce a sample of
noise events. Octopus closely follows the implementation of frequency-based
“optimal filter” (OF) outlined in Ref. [34]. The OF maximizes the signal-to-
noise ratio for the estimation of the pulse amplitude, under the assumption
that the ideal pulse shape is known and the noise is stationary [35, 36]. The
ideal pulse template for the optimal filter is created by averaging a
selection of LED pulses of a certain amplitude. This is done after rejecting
events with a large baseline excursion, elevated noise levels, and requiring
that only one pulse exists in a 15 ms window after the measured triggered time
(Fig 4). On the other hand, the average noise power spectrum is created using
the noise events. The pulse amplitudes of the LED are then corrected for the
slow thermal drift of the detector by linearly regressing the known amplitude
versus the baseline values (cf. Ref. [37] ). Octopus also calculates basic
pulse shape parameters, on raw and OF-filtered pulses, which are used for data
processing and selection. We list a few of these parameters, which will be
heavily referenced in the rest of the paper.

  * •

Risetime: defined as the time it takes the leading edge of the pulse to go
from 10% to 90% of the maximum amplitude. Typically $\sim$175 $\mu$s for both
detectors.

  * •

Decaytime: defined as the time it takes for the falling edge of the pulse to
go from 90% to 30% of the maximum amplitude. Typical values of $\sim$1 ms and
$\sim$800 $\mu$s for Detector-I and Detector-II, respectively.

  * •

OFChisquare: statistic to measure the difference between the pulse and the
template pulse after OF. Typical values of $\chi^{2}/d.o.f\sim 1$.

  * •

OFDelay: time offset used to correct the pulse position relative to the
trigger time in the event window. Depends on the pulse amplitude and timing
jitter and has typical values of $\sim$10–20 $\mu$s.

##  IV Energy response and baseline resolution

We follow Cardani et al. [38, 39] to obtain an energy calibration based on the
Poisson statistics of the LED photons absorbed in our detectors. Each LED
trigger results in a certain number of photons ($N_{i}$) that deposit all of
their energy in the Si absorber. This gives rise to a Gaussian distribution
with a peak at the mean amplitude of $A_{i}$ and a width of $\sigma_{A_{i}}$
(Fig. 5). Assuming a baseline resolution of $\sigma_{0}$, we can write

$\sigma_{A_{i}}^{2}=\sigma_{0}^{2}+\left(R\cdot\sqrt{N_{i}}\right)^{2},$ (1)

where $R$ is the responsivity (dA/dN) of the detector per photon. Since
$A_{i}=R*N_{i}$, Eq. (1) reduces to

$\sigma_{A_{i}}^{2}=\sigma_{0}^{2}+R\cdot{A_{i}}.$ (2)

We obtain the responsivity $R$ by fitting $\sigma_{A}$ versus $A$ with the
functional form shown in Eq. (2). We then calculate the total number of
photons $N_{i}$ for each $A_{i}$. Since the mean energy of each photon is
known (2.1 eV for $\lambda$=600 nm), we get an absolute calibration for each
amplitude by multiplying the corresponding $N_{i}$ by the photon energy. We
treat the baseline noise as independent of amplitude and assume that the
Poisson fluctuation of the number of photons dominates the stochastic term
(second term in Eq. (1)). In Appendix B, we show that these are true for our
case.

Figure 5:  Amplitude histograms for LED excitations. We fit each peak with a Gaussian lineshape to obtain the centroid ($A_{i}$) and the width ($\sigma_{A_{i}}$). Figure 6:  $\sigma_{E}$ vs $E$ from LED excitation of Detector-I (without AR coating) and Detector-II (with AR coating) at 15 mK. The fitted function corresponds to Eq. (3) where $\sigma_{s}$ is assumed to be dominated by event-to-event fluctuations in the LED signal corresponding to Poisson statistics (see the text for details). The error bars are within the data markers, and we report the fit parameters in Table 2. The deviation between each data point and the fitted curves can be explained by the presence of telegraphic noise, which is not accounted for by Eq. (3). Table 2: Fit results from the fit of data to Eq. (3) for LED excitations. $\sigma_{n}$ and $\sigma_{s}$ are moderately anticorrelated. |  $\sigma_{n}$ (eV) |  $\sigma_{s}$ ($\sqrt{\text{eV}}$) | Correlation ($\sigma_{n}$,$\sigma_{s}$)
---|---|---|---
Det-I | 61$\pm$0.5 | 1.42$\pm$0.01 | -0.73
Det-II | 51$\pm$0.5 | 1.41$\pm$0.01 | -0.72

Once calibrated, $\sigma_{E}$ vs. $E$ is fitted again to

$\left(\frac{\sigma_{E}}{E}\right)^{2}=\left(\frac{\sigma_{s}}{\sqrt{E}}\right)^{2}+\left(\frac{\sigma_{n}}{E}\right)^{2},$
(3)

where $\sigma_{n}$ is the baseline noise of the detector (independent of the
amplitude) and $\sigma_{s}$ arises due to the stochastic nature of the primary
excitation signals.

We find that Detector-I has an $R_{n}\approx$ 480 m$\Omega$ and Detector-II
has an $R_{n}\approx$ 650 m$\Omega$. We operate Detector-I at $\sim$0.35
$R_{n}$ and Detector-II at $\sim$0.50 $R_{n}$. We could not operate both
detectors under similar bias conditions because we observed small
discontinuities and kinks for many bias points throughout the transition.
Similar observations of telegraph noise at certain bias voltages in TES and
instabilities have been previously reported in Refs. [40, 41, 42]. Bennet et
al. [43] show that discrete changes in the number of phase-slip lines could be
a possible explanation for the observed switching between discrete current
states in localized regions of bias. The complete characterization of the
transition, noise sources, and their mechanisms is currently under active
study and will be discussed in a separate paper. For the work shown here, it
was possible to bias the TESs in a narrow transition region, where the
current-voltage characteristics were smooth for small current excursions.

We can estimate the particle discrimination capabilities for $\alpha$/$\beta$
using the calibration curve shown in Fig. 6. CUPID needs a rejection factor
>99.9% to reduce the $\alpha$ background below $10^{-4}$
counts/(kg/keV/yr)[44, 45]. We can quantitatively show that our detectors
satisfy the CUPID requirements by using a classification metric for $\alpha$
and $\beta/\gamma$ distributions. We use the Bhattacharyya distance [46],
which is common for feature selection and is known to provide lower and upper
bounds of the classification error for normal distributions. The Bhattacharya
distance between two normal distributions can be written as

$d_{B}=\frac{1}{4}\cdot\frac{(\mu_{\beta/\gamma}(E)-\mu_{\alpha}(E))^{2}}{\sigma_{\beta/\gamma}^{2}(E)+\sigma_{alpha}^{2}(E)}\\\
+\frac{1}{2}\cdot{\text{ln}}\left(\frac{\sigma_{\beta/\gamma}^{2}(E)+\sigma_{alpha}^{2}(E)}{2\cdot\sigma_{\beta/\gamma}(E)\cdot\sigma_{alpha}(E)}\right)$
(4)

where $\mu$ and $\sigma$ are the average value and RMS of the $\beta/\gamma$
and $\alpha$ light distributions, respectively. Both $\mu$ and $\sigma$ depend
on the energy ($E$) of the incident particle in the LMO. Assuming a priori
equal probabilities for both distributions, the upper bound for the Bayes
classification error is given by [47, 48, 49]

$\varepsilon\leq\frac{1}{2}\cdot{\text{e}}^{-d_{B}}$ (5)

At $\sim$3 MeV we expect an average light energy of 1 keV for $\beta/\gamma$
particles. The quenching factor for $\alpha$ scintillation in LMO [13, 12] is
0.2 and therefore an average of 200 eV is expected on the light detectors from
$\alpha$’s. Using the respective $\sigma_{E}$’s (Fig 6) we obtain
$\varepsilon\leq$ 4.5$\times$10-8 and $\varepsilon\leq$ 3.4$\times$10-10 for
Detector-I and Detector-II, respectively, which demonstrates excellent
discrimination capabilities. The exponential dependence in Eq.(5) shows why
having a small baseline resolution and a higher light yield is necessary; the
classification error bound quickly increases to $\varepsilon\leq$
3.9$\times$10-4 if we increase the baseline noise ($\sigma_{n}$) to 100 eV.
Note that some of the previous publications [1, 44, 12] define the
discrimination power (DP) using only the first term of Eq. (4) with

${\text{DP}}=\frac{\left|\mu_{\beta/\gamma}(E)-\mu_{\alpha}(E)\right|}{\sqrt{\sigma_{\beta/\gamma}^{2}(E)+\sigma_{alpha}^{2}(E)}}$
(6)

and quote DP($E$=$Q_{\beta\beta}$)>3.1 as a requirement to reach an alpha-
induced background rate of < $1\times 10^{-4}$ counts/(kg/keV/yr). For
comparison, we cite a DP of 8.1 and 9.2 for Detector-I and Detector-II,
respectively.

Figure 7:  Energy distribution of noise triggers using LED calibration. We
measured a baseline resolution of (64$\pm$1) eV and (35$\pm$0.5) eV for
Detector-I and Detector-II, respectively. The noise distributions are fit to
Gaussian lineshapes to obtain the respective widths ($\sigma_{baseline}$).

We also calibrate the energy of the triggered noise events to obtain a
baseline noise resolution. We obtain a baseline noise resolution of (64$\pm$)1
eV and (35$\pm$0.5) eV at 15 mK for Detector-I and Detector-II, respectively
(see Fig. 7). The baseline resolution for Detector-I is consistent with
$\sigma_{n}$ obtained from the calibration fit (Table 2). However, Detector-II
has a smaller baseline resolution than its $\sigma_{n}$. The smaller
quiescent-state noise in Detector-II points to bias-dependent noise in its TES
sensor. The difference in baseline energy resolution ($\sigma_{n}$) between
Detector-I and Detector-II is due to the difference in their operating point
and the higher noise in Detector-II rather than being an effect of the AR
coating. We have verified that the AR coating does not adversely affect the
energy resolution. The fit results are presented in Table 2. Both detectors
meet the CUPID criterion for $\sigma_{n}$ < 100 eV .

##  V Timing resolution and pileup discrimination

In timing measurements, the pulse rise time is important because the timing
jitter depends on the slope-to-noise ratio rather than the signal-to-noise
ratio alone. The timing jitter ($\sigma_{t}$) distribution can be written as

${\sigma_{t}}=\frac{\sigma_{n}}{({dS}/{dt})_{S_{T}}}$ (7)

where $\sigma_{n}$ is the rms noise and the derivative of the signal
(${dS}/{dt}$) is calculated at the position of the trigger [50]. To measure
timing jitter, we use LED pulses to provide a start and stop signal for a
known time interval and measure the spread in the distribution of $\Delta
T=t_{start}-t_{stop}$. Fig. 8 shows the measured timing jitter as a function
of the pulse amplitudes for both detectors. The timing jitter is high for low-
energy pulses because noise affects the time at which a trigger is issued.
Nevertheless, we have measured a $\sigma_{t}$ < 20 $\mu s$ for all energies >
1 keV, which converges to $\sim$10 $\mu s$ above 3 keV. In the future, the
timing jitter for low-energy pulses can be further reduced using an optimal
triggering algorithm [51].

Figure 8:  Dependence of timing jitter $\sigma_{t}$ as a function of energy.
The errors are within the marker size. For pulses with lower energy, where the
SNR is poor, fluctuations in signal amplitude that cross the trigger threshold
translate into additional timing fluctuations. The jitter value flattens out
for high energy pulses because of the sampling rate (150 kS/s).

Timing jitter of $\sigma_{t}$ < 20 $\mu s$ is enough for the CUPID baseline
experiment, where we need to discriminate two $2\nu\beta\beta$ pileup events
that occur as close as $\sim$170 $\mu s$ from each other. To show the
discrimination power of the detector, we did a toy experiment in which we
produced pileup pulses by generating two LED pulses temporally close to each
other and comparing them to single-pulse signal events. We build a signal-like
event sample by carefully selecting single LED pulse events of $\sim$1 keV
energy. We characterize each event by five pulse-shape parameters and
represent each event with a point in this five-dimensional space. The five
pulse shape parameters are Risetime, Decaytime, OFChisq, OFDelay, and the sum
of the integral under the rising edge and the falling edge of the pulse
normalized to the total integral of the pulse from 10% of amplitude on the
rising edge to 30% of amplitude on the falling tail. The variables are
dependent on the pulse shape, which we expect to be modified in the presence
of pileup events. Identification of a pile-up event is then reduced to the
problem of outlier detection in a signal-like space. We use the Mahalanobis
distance ($d_{M}$) [52], a measure of the distance of a multidimensional point
($\vec{x}$) from a multivariate normal distribution that has a mean
$\vec{\mu}$ and the covariance matrix ${\Sigma}$. We then define the outlier
detection criterion by choosing a distance $k$ that satisfies

$d_{M}=\sqrt{{(\vec{x}-\vec{\mu})^{T}}{\Sigma^{-1}}{(\vec{x}-\vec{\mu})}}~{}>~{}k,$
(8)

where $k$ is chosen according to the number of variables used to build the
distance. The distance inherently considers the correlations between the
parameters and is always positive. We use Detector-I for the pileup experiment
since it has a larger baseline noise and higher timing jitter. We note that if
Detector-I is able to meet the CUPID criterion for timing, then Detector-II
should as well.

Figure 9:  Mahalanobis distance ($d_{M}$) distributions for single-pulse
events and double-pulse events (similar to pileup) as a function of the time
separation between double pulses. Figure 10:  ROC curves as a function of the
time separation between double pulses. The ROC curves are obtained using the
selection criteria of Eq. (8) where $k$ corresponds to the locus of a point
that is a certain distance from the false positive axis. For a large $k$ we
get an increase in the true positive rate, but also admit a higher number of
false positive events. We obtain a precision of >99% for pileup events that
are more than 160 $\mu$s apart.

The distance distribution for single-pulse events and pile-up-like double-
pulse events is shown in Fig. 9, while the Receiver Operating Characteristic
(ROC) curves as a function of temporal separation between pileup-like double
pulses are shown in Fig. 10. Each point in the ROC curve corresponds to a
predetermined distance cut $k$ which corresponds to a certain constant
distance from the False Positive axis shown in Fig. 10. An increase in $k$
increases the acceptance of single pulse events from its distribution (see
Fig. 9) but at the cost of letting in more pile-up like events. The precision
of our $d_{M}$ based selection criteria is defined as

$Precision~{}=~{}\frac{TruePositives}{TruePositives+FalsePositives}.$ (9)

Due to the fast rise time of our detectors, we can effectively discriminate
pileup events within $\sim$120 $\mu s$ with a signal efficiency of 96% and
accept only 1% of pileup events (precision $\sim$99%). For pileup events that
occur with a separation of at least $\sim$160 $\mu s$, we obtain a precision
of > 99%. We benchmark our pileup discrimination capabilities by calculating
an expected background rate using the data presented in Fig. 10. Chernyak et
al. [15] show that a background of $3.3\times 10^{-4}$ counts/(kg/keV/yr) is
expected for unresolved pileups below 1 ms in the region of interest of 100Mo.
We set the acceptance efficiency for a single pulse event at 98% and plot the
corresponding acceptance efficiency for pileup events as a function of the
separation time (Fig. 11). We use an exponential function to perform a
parametric fit of the data. The total acceptance rate between [0,1 ms] is the
area under the fitted curve. Since we do not have any measurements below 40
$\mu$s, we take a conservative approach and assume an acceptance rate of 100%
below 40 $\mu$s. Adding it to the area under the curve from 40 $\mu$s to 1 ms,
we get a total acceptance rate of 5.8% and hence a pileup background rate of
$1.9\times 10^{-5}$ counts/(kg/keV/yr). This is about three times lower than
the CUPID target of $5\times 10^{-5}$ counts/(kg/keV/yr). We caution the
reader that this exercise is merely a benchmark for the light detectors and
not the actual achievable background rate since we have performed our toy
experiment with equal amplitude pulses of 1 keV. In practice, pileup events
can have different amplitudes. Nevertheless, we have shown that our current
generation of detector is adequate to reach CUPID targets. We are working
towards improving our pileup discrimination capabilities by using machine
learning techniques that can achieve higher precision.

Figure 11:  Acceptance efficiency of pileup pulses (false positives) as a
function of the time separation between them. Data points correspond to 98%
signal efficiency for single pulse events (see Fig. 10). We use an exponential
for parametric fitting and calculate the area under the curve for the net
acceptance efficiency from 40 $\mu$s to 1 ms. In the absence of measurements,
we assume an acceptance efficiency of 100% below 40 $\mu$s.

##  VI Conclusion

We have demonstrated that a novel bilayer transition edge sensor-based light
detector is a promising technology for CUPID. The detectors can be operated
down to 10 mK and meet the energy resolution requirement of $\sigma_{n}$< 100
eV. The large surface area of the detector is crucial for detecting a low
light yield from a scintillating LMO crystal. Furthermore, we have shown that
adding a thin amorphous Si3N4 antireflective coating does not degrade detector
performance. In this work, we have also shown how we can tune the absorber-
sensor coupling and engineer the thermal conductances of the detector. More
importantly, we have achieved an excellent timing resolution and have
demonstrated that the current generation of detectors will suffice to meet the
CUPID requirements of an effective timing resolution of <170 $\mu s$. For our
next steps, we plan to produce more devices to understand the microscopic
physics of the TES and energy thermalization and the source of excess noise in
our TES. Work is underway to implement a frequency division multiplexing
readout scheme and to demonstrate its feasibility for the CUPID experiment.

###### Acknowledgements.

This work was supported by the US Department of Energy (DOE), Office of
Science under Contract No. DE-AC02-05CH11231 and DE-AC02-06CH11357, and by the
DOE Office of Science, Office of Nuclear Physics under Contract No. DE-
FG02-00ER41138. Superconducting thin films synthesis was supported by the DOE
Office of Science, Office of Basic Energy Sciences, Materials Sciences and
Engineering Division. Use of the Center for Nanoscale Materials, an Office of
Science user facility, was supported by the DOE Office of Science, Office of
Basic Energy Sciences under Contract No. DE-AC02-06CH11357.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Effect of Au pad on the thermal conductance of TESs

The addition of Au pads was a crucial step in the construction of our device.
Our initial tests of a wafer with a bare Ir-Pt TES had a small signal size
with a poor signal-to-noise ratio. This was because of the weak thermal
coupling between the sensor and the absorber, which was limited by the e-ph
coupling of the bilayer TES. In a separate experiment, we studied how thermal
coupling can be improved by using overlapping Au pads between the TES and Si
absorber. We instrumented a Si absorber with four bilayer TES sensors close to
the center of the Si wafer (Fig.12). Each TES has a 100 nm thick Ir film
deposited on top of a 50 nm thick Pt film, each with an area of 300 $\mu$m
$\times$ 300 $\mu$m. We then deposit Au pads on both sides of the TES to
increase the phonon coupling between the sensor and the absorber. The Au pads
have a thickness of 200 nm. We deposit Au of different surface areas for each
sensor (see Table 3) to systematically study the effect of Au on the coupling
between the absorber and the sensor.

Figure 12:  Detector with four TESs with different areas of Au that overlap
the sensors and the Si absorber. Additional copper clamps were added to the Si
wafer to increase the thermal coupling between the wafer and the copper bath
and make $T_{absorber}\approx T_{bath}$.

The typical procedure to measure the thermal conductance of a TES,
$G(T_{tes})$, involves measuring Joule power heating as a function of bath
temperature ($T_{b}$). We used the current-voltage (I-V) curves to obtain the
Joule power, for a particular resistance, at each bath temperature. The power
curve thus obtained can be fitted to the following:

$P_{e}(T)~{}=~{}k\cdot(T_{e,tes}^{n}-T_{b}^{n})$ (10)

where $P_{e}(T)$ and $T_{e,tes}$ are the power dissipated in the TES electron
system and its temperature, respectively.

Table 3:  Size of Au pad and $T_{c}$ for each of the four sensors shown in Fig. 12. Each sensor is flanked by Au film of the mentioned size on either side. See text for a detailed discussion on thermal conductance between TES and absorber, $G_{ea}$. In the absence of an Au layer, Sensor-A is susceptible to parasitic power, which explains the lower measured $T_{c}$. The Thevenin equivalent circuit parameters for Sensor-C TES readout were not precisely known and were assumed to be similar to the other circuits. This may explain the abnormally high $R_{n}$ measured for Sensor-C. Sensor | $R_{n}$ | Au pad size | $T_{c}$ | $G_{ea}(T_{c})$
---|---|---|---|---
| m$\Omega$ | $\mu m\times\mu m$ | mK | pW/K
A | 571 | No Au | 30 | 8 $\pm$ 1
B | 572 | 300 $\times$ 300 | 33 | 387 $\pm$ 9
C | 755 | 600 $\times$ 600 | 35 | 1788 $\pm$ 55
D | 495 | 900 $\times$ 900 | 35 | 1894 $\pm$ 40

The coefficient $k$ and the exponent $n$ define the thermal conductance
$G_{ea}$.

$G_{ea}(T_{e,tes})~{}=~{}n\cdot k\cdot T_{e,tes}^{n-1}.$ (11)

A simplified thermal model for our detector is shown in Fig. 13. We assume
that the absorber temperature ($T_{a}$) is close to the thermal bath
($T_{b}$). This assumption is valid only if the thermal conductance between
the absorber and the bath, $G_{ab}$, is much larger than that between the TES
and the absorber, $G_{ea}$, ensuring $T_{a}\approx T_{b}$. The assumption may
not hold for our setup where we are aiming for $G_{ab}$ << $G_{ea}$ to
increase the responsitivity for a quantum of energy deposited on the absorber,
nonetheless, Eq. (10) allows us to qualitatively study the effect of Au pads
on thermal conductance.

Figure 13:  Simplified thermal model of the detector. The TES system is shown
in the red dotted box. $T_{e}$ and $C_{e}$ are the temperature and heat
capacity of the electron system of the TES, respectively. The phononic
contribution to the heat capacity, $C_{p}$, of the TES films is extremely
small. $T_{p}\approx T_{a}$, the temperature of the Si absorber. $T_{b}$ is
the temperature of the thermal bath. Because we use superconducting Al wires,
we neglect any thermal connection between the electron system and the thermal
bath. $G_{ea}$ and $G_{ab}$ are thermal conductances (see the text for
details).

Eq. (10) can give an indication of the underlying physics that dominates the
conductance of the TESs. The heat flow from the substrate to the sensor is
determined by the electron-phonon coupling $G_{ep}$, as well as the Kapitza
boundary conductance between the metal film and the substrate $G_{K}$,

$\frac{1}{G_{ea}}=\frac{1}{G_{ep}}+\frac{1}{G_{K}}$ (12)

where $G_{ep}\propto V\cdot T^{5}$ and $G_{K}\propto A\cdot T^{3}$ with $V$
and $A$ being the volume and area of the TES, respectively. Since the
thickness of the deposited Au is the same for all sensors, $G_{ep}\propto
A\cdot T^{5}$ for the present setup. The value $n$ obtained from the fit to
the experimental data indicates the underlying thermal impedance mechanism.
All sensors in our device favor a fit with $n=5$. $G_{ea}$ also increases with
increasing Au pad area (see Table 3). Both observations suggest that our
devices’ thermal conductance is limited by the electron-phonon coupling and
not by the Kapitza impedance. We note that $G_{ea}(T_{c})$ does not increase
linearly with the Au pad area for very large sizes as expected. We based this
on the assumption that we have treated $T_{a}$ as a constant when the TES
power increases. The assumption is not valid for $G_{ab}$ < $G_{ea}$.

We also note that we have ignored the weak-link effects [53] due to which the
device’s critical current can change for $T_{b}$ < $T_{c}$. The weak link
effects are severe for very small TESs, but since our TESs are relatively
large, we assume that this will not significantly affect our approximation for
Eq. (10).

##  Appendix B Heater resolution vs LED resolution

This section qualitatively describes the efficacy of using Poisson statistics
of the LED pulses for calibration. In addition, we compare the resolution of
the LED pulses with that obtained by the Joule-heater excitation.
Unfortunately, since the heater is more susceptible to EMI/RF heating, getting
the TES to an appropriate bias point with high sensitivity was nearly
impossible. Therefore, we do not quote an absolute energy calibration and
$\sigma_{noise}$ in this section since we cannot precisely quantify the amount
of residual power and noise in the detector when the heater is connected. For
these reasons, we prefer using LED pulses over Joule-heater for detector
calibration.

Figure 14:  $\sigma_{A}$ vs. OF-Amplitude for heater and LED excitations.
$\sigma_{A}$ for heater excitations are nearly constant, while $\sigma_{A}$
for LED excitations can be fitted to the functional form described in Eq. 1.
The increase in $\sigma$ for LED excitations is due to Poisson statistics of
the number of photons from the LED.

Nevertheless, we can still compare the heater and LED excitations as long as
we keep the same baseline and temperature for both cases. The heater generates
phonon excitations $\mathcal{O}$(few meV) and, unlike LED photons, its
$\sigma_{A}$ should not have any amplitude dependence as long as the baseline
noise dominates the fluctuation of the number of phonons produced. Since each
LED photon has an average energy of 2.1 eV in our case, we expect Poisson
statistics to contribute significantly towards resolution for higher-energy
LED excitations. As shown in Fig.14, this is indeed the case. The heater
resolution is constant regardless of the excitation amplitude, while the LED
resolution shows an amplitude dependence that can be explained using Eq. (1).

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