# Superconducting detectors for rare event search experiments

Yong-Hamb Kim1,4,5, Sang-Jun Lee2, Byeongsu Yang3 1 Center for Underground
Physics, Institute for Basic Science (IBS), Daejeon 34047, Korea  2 SLAC
National Accelerator Laboratory, Menlo Park, CA 94025, USA  3 Department of
Physics and Astronomy, Seoul National University, Seoul, 08826, Korea  4
University of Science and Technology (UST), Daejeon 34113, Korea  5 Korea
Research Institute of Standards and Science (KRISS), Daejeon 34113, Korea
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Superconducting detectors have become an important tool in experimental
astroparticle physics, which seeks to provide a fundamental understanding of
the Universe. In particular, such detectors have demonstrated excellent
potential in two challenging research areas involving rare event search
experiments, namely, the direct detection of dark matter and the search for
neutrinoless double beta decay. Here, we review the superconducting detectors
that have been and are planned to be used in these two categories of
experiments. We first provide brief histories of the two research areas and
outline their significance and challenges in astroparticle physics. Then, we
present an extensive overview of various types of superconducting detectors
with a focus on sensor technologies and detector physics, which are based on
calorimetric measurements and heat flow in the detector components. Finally,
we introduce leading experiments and discuss their future prospects for the
detection of dark matter and the search for neutrinoless double beta decay
employing superconducting detectors.

††: Supercond. Sci. Technol.

###### Contents

  1. 1 Introduction
  2. 2 Astroparticle Physics Challenges
    1. 2.1 Dark matter
    2. 2.2 Neutrinos and neutrinoless double beta decay
  3. 3 Sensor Technologies
    1. 3.1 Motivation
    2. 3.2 Thermal calorimetric detection
    3. 3.3 Superconducting sensors
      1. 3.3.1 Transition edge sensors
      2. 3.3.2 Magnetic microcalorimeters
      3. 3.3.3 Kinetic Inductance Devices
      4. 3.3.4 Other technologies
    4. 3.4 Heat flow model
      1. 3.4.1 Acoustic mismatch model
      2. 3.4.2 Electron-phonon interactions in metal films
      3. 3.4.3 Electronic heat flow
    5. 3.5 Athermal phonon transfer processes
    6. 3.6 Phonon amplification by the Neganov-Trofimov-Luke effect
  4. 4 Application of superconducting detectors for rare event searches
    1. 4.1 DM detectors
      1. 4.1.1 CDMS
      2. 4.1.2 CRESST
      3. 4.1.3 Other low-temperature DM detectors
      4. 4.1.4 Nanometer-scale thermal calorimeters
    2. 4.2 Application to neutrinoless double beta decay search
      1. 4.2.1 AMoRE
      2. 4.2.2 Other low-temperature thermal calorimeters for $0\nu\beta\beta$ searches
      3. 4.2.3 Superconducting light detectors for $0\nu\beta\beta$ experiments
    3. 4.3 Noncalorimetric approaches for wave-like DM
      1. 4.3.1 Axion searches with microwave cavities
      2. 4.3.2 Direct detection with superconducting lumped element circuits
  5. 5 Conclusion

##  1 Introduction

During the last few decades, great progress has been achieved in low-
temperature detectors (LTDs) operating at sub-Kelvin temperatures, which have
played a major role in many scientific applications [1, 2, 3]. These LTDs
utilize novel properties of materials and technologies specialized for
operation at low temperatures to increase their detection sensitivity. In
particular, superconducting materials, circuits, and devices have become the
key components in many state-of-the-art LTDs. Throughout this review, we
define a superconducting detector as an LTD in which superconductivity plays a
critical role in detecting a signal. Moreover, we distinguish a detector from
a sensor, which we consider to refer only to the sensing element of a
detector. Thus, a typical superconducting detector can be said to consist of a
superconducting sensor and a target (or an absorber) for the particles or
interactions to be detected. It should be noted that although our definition
is not clear-cut in some situations, we view it as a plausible way to cover a
variety of techniques at play.

One of the most important advantages of superconducting detectors is their
detection sensitivity. Detection sensitivity is a rather complex (or loosely
defined) concept that encompasses the energy resolution, minimum detectable
energy, time resolution, detector volume/mass, detection efficiency, etc.
Unfortunately, there is no universal figure-of-merit expression for detection
sensitivity that would apply to every type of superconducting sensor/detector.
Instead, energy resolution is often considered the most critical
characteristic for sensors and detectors and is used to directly compare
different techniques. Superconducting sensors/detectors have demonstrated high
energy resolution far beyond the theoretical limit of conventional
semiconductor devices in any given energy region. Moreover, superconducting
detectors can also be made to have a high detection efficiency comparable to
that of semiconductor detectors. For this reason, superconducting detectors
have replaced semiconductor detectors in many applications, especially when a
stringent energy resolution requirement needs to be met.

Currently, superconducting sensors and detectors are being used over a wide
energy range. For example, in the detection of infrared (IR) and visible
photons, superconducting detectors demonstrate high speed and high efficiency
for the detection of single photons [4, 5]. High-performance single-photon
detectors of this kind have become an essential tool in quantum computing and
quantum communications. For the detection of X-rays in an energy range between
a few hundred eV and a few hundred keV, superconducting detectors have shown
much better energy resolution than semiconductor-based detectors while also
offering a high detection efficiency. Because of their unique combination of
high resolution and high efficiency, a few large-scale X-ray satellite
missions are being developed based on state-of-the-art superconducting
detectors [6, 7]. Superconducting detectors also provide superior performance
in gamma-ray and alpha radionuclide analysis [8, 9, 10]. In particular,
superconducting detectors equipped with metal 4$\pi$ absorbers enable decay
energy spectroscopy for alpha- and beta-emitting radionuclides, a new
spectroscopic method for the accurate measurement of spectral shapes and
activities [11, 12, 13]. This new method has also been adopted to study
important properties of neutrinos such as the rest mass through the end-point
measurement of beta-decay spectra [14, 15, 16].

Astroparticle physics experiments searching for rare events are another field
of application in which superconducting detectors play key roles. In this
field of study, unambiguous detections of rare events would lead to
groundbreaking discoveries of new particles or new physical processes. One
such example is the direct detection of dark matter (DM), which is being
pursued by a number of international projects. DM is a type of matter that is
known to exist in the Universe, as evidenced by many indirect measurements
[17, 18, 19, 20, 21, 22, 23]. Understanding the nature of DM is one of the
most mysterious but fundamental research topics in physics. Several leading DM
search projects have adopted superconducting detectors as their main detectors
[24, 25]. In particular, in experimental searches for particle-like DM,
superconducting detectors are primarily intended to measure heat signals
originating from the energy deposited in a target/absorber material when a DM
particle interacts with normal matter in the target. Superconducting detectors
enable the detection of heat signals with high resolution as well as a
significantly lowered minimum detectable energy (i.e., energy threshold)
compared with conventional detectors. To further increase the detector
sensitivity to DM-normal matter interactions, an additional detection channel
can be utilized, especially for a target consisting of a semiconductor or
scintillating material, from which charge or light signals, respectively, can
be measured together with the main heat signals. Such dual-channel detection
makes it possible to distinguish whether a detected signal comes from unwanted
background or indeed from the rare events being sought. Particle
identification (PID) of this kind is now an essential capability of rare event
search detectors.

Another example of the use of superconducting detectors in this field is the
search for neutrinoless double beta decay ($0\nu\beta\beta$). $0\nu\beta\beta$
is a hypothetical decay process that would occur if neutrinos have finite mass
and are their own antiparticles (i.e., Majorana particles) [26]. Its
experimental observation would not only reveal basic but unknown
characteristics of neutrinos but also provide a fundamental understanding of
the origin of the present matter-dominated Universe. A number of detection
technologies have been developed to probe the rare events of $0\nu\beta\beta$.
Superconducting detectors are one promising detection method because of their
energy resolution, efficiency and PID capability.

Here, we review superconducting detectors in the context of their
applicability for rare event search experiments in astroparticle physics. In
Section 2, we introduce the significance and challenges of two categories of
rare event experiments, namely, the direct detection of particle-like/wave-
like DM and the search for $0\nu\beta\beta$. In Section 3, we present the
basic detection principles of state-of-the-art superconducting sensors. This
sensor technology section includes a discussion on low-temperature heat
transfer processes as the essential mechanisms of particle detection in
superconducting detectors. In Section 4, we survey direct DM detection and
$0\nu\beta\beta$ search experiments based on superconducting detectors as well
as other types of LTDs. This application section includes new approaches based
on superconducting sensors coupled to low-temperature cavities and resonators
targeted at detecting wave-like DM. We also introduce newly developed
nanometer-scale superconducting detectors targeted at detecting ultralow-mass
particle-like DM.

##  2 Astroparticle Physics Challenges

As introduced in the previous section, the two main applications of
superconducting detectors addressed in this review are direct DM detection
experiments and $0\nu\beta\beta$ search experiments. These are two of the most
rewarding but challenging topics in contemporary astroparticle physics.
Moreover, they are the two main research areas that originally stimulated the
intensive development of then-new LTD technologies. Thus, it is timely to
describe the history and current status of these two topics before we review
the corresponding sensor technologies and their applications in detail in the
following sections.

First, we discuss DM, in particular, various DM candidates and strategies for
their direct detection. It is commonly believed that DM does, in fact, exist
based on astronomical observations. However, its existence has not been
confirmed through the direct detection of interactions between normal matter
and DM. Thus, the direct detection of DM is the first step in understanding
its properties. Then, we discuss neutrinos, in particular, the known and
unknown properties of neutrinos as well as the hypothetical $0\nu\beta\beta$
decay and its far-reaching implications in astroparticle physics.

These two areas of research are considered among the most straightforward and
revealing approaches for studying DM and neutrinos, respectively. However,
despite several decades of effort, no clear experimental demonstration has
been achieved, leading to calls for more in-depth investigations, possibly
involving entirely new detector techniques. Superconducting detectors and
related techniques are expected to play a major role in overcoming these
challenges by virtue of their excellent energy sensitivity, detection
efficiency, and PID capability.

Figure 1:  The proportions of normal matter, DM and dark energy in the
Universe as measured by the Planck spacecraft [21].

###  2.1 Dark matter

According to recent precise astronomical measurements of the cosmic microwave
background (CMB), normal (or ordinary) matter such as the protons, neutrons
and atoms composing stars and galaxies has been found to represent only 4.9%
of the entire contents of the Universe [21]. This observation also shows that
another type of matter called dark matter (DM) corresponds to approximately
26.2% of the mass of the Universe. DM does not appear to undergo
electromagnetic interactions but does interact via gravity. The most abundant
component, at 68.9%, is called dark energy, which is uniformly spread
throughout the Universe and is responsible for the repulsive force that is
accelerating the expansion of the Universe. The fractional contents of normal
matter, DM, and dark energy in the Universe are illustrated in Figure 1.

Although DM remains mysterious, our understanding of it has evolved with the
advancement of technology in physics and astronomy. According to a recent
review of the history of DM research [27], early scientific discussions on DM
began among astronomers in the 19th century as new astronomical observations
became available. For example, the uneven distributions of stars observed in
early astronomical photographs led to a question of whether the dark regions
are due to a lack of stars or the presence of light-absorbing matter in those
regions. One of the first quantitative estimations of the abundance of dark
objects/bodies was performed by Lord Kelvin, a well-known physicist in whose
honor the absolute temperature unit K was named. Kelvin estimated the mass of
the Milky Way galaxy from the velocity dispersion of the stars orbiting around
the galactic core, and argued that most of the stars must be dark bodies that
are too dim to be seen. Influenced by Kelvin’s estimation, Henri Poincaré, a
well-known mathematician and physicist, also performed an estimation of the
amount of DM (or matière obscure in French), although his estimation pointed
to much less DM than that of Kelvin [27]. Since then, many researchers have
invoked the possible existence of invisible matter to explain their
astronomical observations [28, 29, 30, 31, 32]. In particular, Fritz Zwicky
investigated Hubble’s observations of the motion of galaxies in the Coma
galaxy cluster and estimated based on the virial theorem that the mass of the
galaxy cluster as calculated from the observed velocity dispersion of the
galaxies was orders of magnitude greater than that expected from their visible
luminosity, which he ascribed to DM (or dunkle Materie in German) [30, 33,
34]. His work pointed to the existence of DM also at the galaxy cluster level.

Figure 2:  An example of the rotation curve measured from the spiral galaxy
Messier 33 [35]. The data points indicate observations of the speeds of stars
as a function of the distance from the galactic center. The dashed line is the
curve that is expected from the visible matter in the galaxy. The origin of
the radial distance represented on the x-axis is located at the center of the
galaxy (figure from [36]).

By the 1980s, the existence of DM in some form had become widely accepted.
This was mainly due to compelling observations of the velocity distributions
of stars, gas and dust in galaxies with respect to the distances from the
centers of these galaxies reported by Vera Rubin, Kent Ford and Ken Freeman
[17, 18, 37]. Figure 2 shows an example of the rotation curve of the stars in
a spiral galaxy. From the visible components, the rotation curve is expected
to slow down for stars in the outer spirals of the galaxy. However, the
observations show that stars far from the galactic center rotate much faster
than expected. This indicates that the amount of visible normal matter is not
sufficient to explain the measured rotation curve. Consequently, some
unobserved origin of gravitation (i.e., DM) must exist to hold the stars
tightly within the galaxy. Observational evidence from a number of other
sources has been reported that also indicates the existence of DM. Those
sources include gravitational lensing [19], the CMB [20, 21], the Bullet
Cluster [22], the total masses of galaxy clusters [23], and structure
formation in the early Universe [38].

Figure 3:  Three strong DM candidates, namely, axions, light DM and WIMPs, and
their expected mass ranges. When the mass of a DM particle is smaller
(greater) than $\sim 0.1$ eV, that particle shows wave-like (particle-like)
characteristics. Many other candidates that are not listed also exist; some of
them even have expected mass ranges outside of the mass range shown here [39].

As discussed above, scientists have found evidence for the existence of DM
mostly from its gravitational interactions with normal matter. However, almost
no observations are available to explain the nature or properties of DM. The
most pressing fundamental question about DM is “What is DM?” It is a rather
embarrassing question, but it accurately represents the current status of our
understanding about DM: many theoretical postulates exist, but none of them
has been proven to be the ultimate explanation. Following from this, “Can we
detect DM?” is another question of this kind. It is like looking for something
without knowing if it even exists. Thus, any experimental hint of the direct
interaction of DM with normal matter might be the first step towards answering
these questions. Such DM-normal matter interactions may involve the
generation, decay, or annihilation of DM.

Here, we briefly describe our present understanding of DM. Although the mass-
energy density of DM as a whole has been estimated with a relatively high
accuracy, its individual mass is a largely unknown parameter. There is a vast
possible range, between 10-22 eV and 1067 eV, set by various cosmological
constraints [40, 41, 42, 43, 44, 45]. Note that eV is used here as a unit of
mass that corresponds to a mass of eV$/c^{2}$, where $c$ is the speed of the
light. Figure 3 illustrates a partial region of the DM mass range that
includes the expected mass ranges of three strong DM candidates, namely,
axions, light DM, and weakly interacting massive particles (WIMPs). These
three candidates are further discussed below. Because of the wide range of
possible DM masses, very different properties of DM are expected depending on
its mass scale. For instance, if the mass were to fall in the range of
$10^{-22}\,\mathrm{eV}<m_{\mathrm{DM}}<0.1\,\mathrm{eV}$, the DM would behave
like waves with a de Broglie frequency $f$ determined by
$hf=m_{\mathrm{DM}}c^{2}$, where $h$ is the Planck constant and
$m_{\mathrm{DM}}$ is the DM rest mass. Thus, experiments targeting the
detection of DM in this mass range should be sensitive to signals originating
from the wave-like properties of the DM. Axions are a wave-like DM candidate.
On the other hand, for DM in the mass range of
$0.1\,\mathrm{eV}<m_{\mathrm{DM}}<$ $10^{19}\,\mathrm{GeV}$ ($\sim$the Planck
scale), its prominent properties would be those of a particle. Therefore, any
corresponding detection method should utilize its particle-like behavior.
WIMPs and light DM are particle-like DM candidates. There could also exist
intermediate-mass DM exhibiting the dual properties of waves and particles in
the intermediate mass region [46, 47, 48, 44, 45]. At a much higher mass scale
than the Planck scale, massive astrophysical compact halo objects (MACHOs) are
also considered DM candidates, including primordial black holes (PBHs), a
hypothetical species that formed soon after the Big Bang [39]. Below, we
further describe the three strong DM candidates, namely, WIMPs, light DM, and
axions.

  * •

WIMPs

As the evidence for the existence of DM has accumulated, many theoretical
models have been suggested to explain what the DM is. At the minimum, these
models should satisfy all constraints obtained from indirect observations of
DM. For example, the DM must be electrically neutral and stable against decay,
with a lifetime longer than the age of the Universe. Since none of the
Standard Model particles meets all necessary criteria to be the DM particle,
many hypothetical particles have been proposed based on various models. Among
them, WIMPs have been the most favored candidate.

On the one hand, WIMPs are favored in the context of the thermodynamic
evolution of the early Universe according to Big Bang cosmology. If they have
a mass of approximately 100 GeV and a cross section at the electroweak scale,
they well explain the cosmological abundance of DM. On the other hand, WIMPs
are also favored by a beyond-the-Standard-Model theory called supersymmetry
(SUSY). SUSY postulates that superpartners exist for all the Standard Model
particles (normal matter) and that they were created in great abundance in the
early Universe. Among the SUSY particles, the lightest one is expected to have
a mass in the range of 100 GeV to a few TeV and to be stable enough to have
persisted and constitute the present DM. In particular, their interactions
with normal matter are also at the electroweak scale [49, 50, 51, 52, 53, 54].
This accidental coincidence in mass and cross section between the two
hypotheses is known as the WIMP miracle. Without any better alternative, WIMPs
have naturally become the most preferred and the most sought-after DM
candidate.

Searching for WIMPs has been attempted in various ways. The Large Hadron
Collider (LHC), with its energy within the reach of the SUSY-predicted WIMP
mass, enabled a search for WIMPs that might have been created inside the
collider [55]. If WIMPs were indeed created by the colliding beams of protons,
the WIMPs would have left the detector undetected, resulting in a so-called
missing-momentum signal. However, no such signal has been detected [56].

A large number of underground experiments have also searched for signals
originating from the direct interaction of WIMPs with the detectors. Although
most of these experiments use distinct detection techniques, including
superconducting sensors, they share many common features. They all rely on the
particle-like nature of WIMPs, use ultrapure materials with low
radioactivities for the detectors and their supporting structures, and are
located in deep underground laboratories with heavy radiation shielding to
minimize any WIMP-mimicking background signals [25, 57, 58].

Figure 4 shows the DM limit curves for spin-independent (SI) interactions
between DM and normal matter (nucleon) obtained by a selected list of
underground experiments. Among all WIMP search experiments, including those in
this figure, the DArk MAtter/Large sodium Iodide Bulk for RAre processes
(DAMA/LIBRA) result is the only positive claim for WIMP detection at the
present time. Their claim comes from an annual modulation of the signal counts
in their WIMP signal window that has been observed by their NaI crystal
detectors for more than 13 annual cycles [59]. Their result is solid and
compatible with the WIMP masses predicted in the “traditional” WIMP DM
scenario. However, their claim has not been confirmed by any other experiment.
Early on, various LTD experiments, including those with superconducting
detectors, namely, Expérience pour Détecter Les WIMPs En Site Souterraine
(EDELWEISS) [60], the Cryogenic Dark Matter Search (CDMS) [61], and the
Cryogenic Rare Event Search with Superconducting Thermometers (CRESST) [62],
explored the DAMA/LIBRA region, but null results were found. Many other
experiments as well as the successors of these three LTD experiments have also
reported no sign of WIMP signals from various target materials. In particular,
experiments with liquid xenon targets, XENON1T [58], the Large Underground
Xenon (LUX) experiment [63], and the Particle and Astrophysical Xenon Detector
(PandaX)-II [64], showed detection sensitivities orders of magnitude higher
than the cross section predicted by the SUSY model for traditional WIMP
masses, but none of them detected a positive WIMP signal. In addition, another
experiment called COSINE-100 that uses the same type of NaI crystals as
DAMA/LIBRA also reported no WIMP signal in the DAMA/LIBRA region [65].

Due to the null results from the LHC and several underground experiments, the
DM model that was once thought to be the WIMP miracle has become highly
unlikely. However, this does not mean that WIMPs are no longer a DM candidate.
Rather, it means that the traditional WIMP predicted by a specific SUSY model
appears to have been ruled out, whereas other types of WIMPs based on other
SUSY models [66] as well as non-SUSY models [67] are still significant DM
candidates. As previously overlooked parameter space for DM is taken into
consideration, many experiments that were originally targeted towards the
traditional WIMP continue to improve their detection sensitivity in a wider
range of WIMP masses [39].

Figure 4:  DM direct detection experiments and their sensitivity limits for SI
interactions between DM and nucleons. The solid curves are from the
experiments that have resulted in the most stringent limits for certain mass
ranges [25, 68, 57, 69, 70, 58, 71]. The parameter space above these curves is
ruled out by the null results of the corresponding experiments. The two closed
regions represent the positive claim by DAMA/LIBRA. The gray region delineated
by the orange dashed curve marks the parameter space where inevitable
backgrounds from coherent neutrino scattering are expected to overwhelm the
DM-nucleon scattering signals for the cases of Ge targets [72]. The DM
sensitivity limits were extracted in March 2021 from Dark Matter Limit Plotter
[73].

  * •

Light DM

The absence of a significant signal strength for the traditional WIMP has
motivated new types of DM candidates based on several different theoretical
models [74, 75, 76, 77, 78]. There are several promising alternatives to the
traditional WIMP with lighter masses and particle-like characteristics. In
this review, we collectively call these alternatives light DM. Note that there
is no clear-cut distinction between light DM and WIMPs. For instance, a WIMP
with a lower mass than the originally expected range would be a light DM
candidate [74, 79].

An interesting light DM candidate is a sterile neutrino [80]. The sterile
neutrino is a hypothetical particle whose existence was hinted at by neutrino
oscillation experimental results unexpected from the three-flavor model of
ordinary neutrinos [81, 82]. Note that ordinary neutrinos are briefly
discussed in the following subsection. If sterile neutrinos exist with a very
weak interaction strength with normal matter and their mass is on the keV
scale, they are a viable DM candidate [83].

Direct detection of light DM requires a different approach from that of WIMPs.
When a DM particle of mass $m_{\mathrm{DM}}$ elastically scatters off of a
normal matter of mass $M$ at rest, the scattering causes the normal matter to
recoil with a kinetic energy $E$:

$E=\frac{\mu^{2}v^{2}}{M}(1-\cos\theta),$ (1)

where $\mu=m_{\mathrm{DM}}M/(m_{\mathrm{DM}}+M)$ is the reduced mass, $v$ is
the relative speed between the DM and the normal matter, and $\theta$ is the
scattering angle. In the case of light DM, where $m_{\mathrm{DM}}\ll M$, the
maximum recoil energy is approximated as $2m_{\mathrm{DM}}^{2}v^{2}/M$.
Obviously, this is much smaller than the recoil energy in the traditional WIMP
case, where $m_{\mathrm{DM}}$ and $M$ are of the same order. Hence, the energy
threshold of the detector should be significantly lowered in order to measure
such a small recoil energy.

On the other hand, the small mass of light DM has a positive effect; the
interaction rate of the DM-normal matter is proportional to the number density
of DM, which is inversely proportional to $m_{\mathrm{DM}}$ because the DM
abundance is rather fixed. It implies that a detector with a sufficiently low
threshold has a higher chance to probe the light DM as its mass is smaller.
This is where superconducting detectors can play a major role by virtue of
their excellent energy resolution and low energy thresholds. As seen from the
DM limit curves in Figure 4, the best limits in the low-mass region below a
few keV are set by the results from SuperCDMS with a Cryogenic PhotoDetector
(SuperCDMS CPD), CRESST-III, and CDMSlite. Also, more and more superconducting
detectors are being developed to probe even lower mass range.

Figure 5:  Experimental limits on the axion-photon coupling in the axion mass
range between 0.1 neV and 1 meV (or the frequency range between 24.18 kHz and
241.8 GHz). The shaded regions indicate the parameter space ruled out by the
null results of axion haloscopes [84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94,
95, 96, 97, 98, 99] and helioscopes [100]. The two diagonal lines with orange
shading represent the theoretical expectations from the KSVZ [101, 102] and
DFSZ [103, 104] models and their variants. This figure was plotted using Ref.
[105].

  * •

Axion and axion-like particles

Another strong candidate for the DM is axions. Axions are hypothetical
particles that were originally proposed to solve the so-called strong CP
problem in quantum chromodynamics (QCD). In particle physics, the charge (C)
and parity (P) symmetries determine whether the laws of physics remain the
same when a particle is interchanged with its antiparticle and when its
spatial coordinates are inverted to a mirror image, respectively. While the
combination of the two, i.e., CP symmetry is known to be violated in
electroweak interactions [106, 107, 108], no sign of CP violation has been
experimentally found in strong interactions, at least at an appreciable
strength, although CP is not an a priori conserved symmetry in QCD. This
puzzle is called the strong CP problem. In 1977, as a solution to the strong
CP problem, Roberto Peccei and Helen Quinn proposed a mechanism in which CP-
violating terms are suppressed by a new global symmetry [109, 110]. Based on
this solution, Frank Wilczek and Steven Weinberg separately proposed a new
type of particle [111, 112]. Since this particle could “clean up” the strong
CP problem, Wilczek named this particle the axion after a laundry detergent
[113].

If axions exist, they would not only solve the strong CP problem but also be a
promising DM candidate. However, the axion as proposed by Peccei, Quinn,
Wilczek, and Weinberg has been experimentally ruled out [114]. Nevertheless,
new theoretical mechanisms have emerged such that axions can have much weaker
coupling than the original version, giving rise to so-called invisible axions.
The two most viable models for the invisible axion are the Kim-Shifman-
Vainshtein-Zakharov (KSVZ) [101, 102] and Dine-Fischler-Srednicki-Zhitnitsky
(DFSZ) models [103, 104]. According to these two models, the axion coupling
parameters are proportional to the axion mass, as plotted in Figure 5. Since
the axion was conceived to have a very weak coupling, their mass is expected
to be very small, and thus, they are expected to exhibit wave-like behavior,
as indicated in Figure 3. Consequently, their direct detection is expected to
be extremely challenging and calls for a new approach.

In 1983, Pierre Sikivie pointed out that the axion proposed in these models
would be converted into detectable photons in a strong magnetic field [115].
Based on this idea, he also proposed novel methods of building axion
haloscopes and axion helioscopes to detect cosmological axions and solar
axions, respectively, in a strong magnetic field. Sikivie’s proposals
motivated a number of axion search experiments, as represented in Figure 5. To
date, however, no experiment has detected axions.

Axion DM is currently being sought using axion haloscopes, which attempt to
detect photons converted from axions inside the detector by a strong applied
magnetic field. Since the photon signal is expected to be extremely weak and
its frequency is unknown, it is critical to build a low-noise detector with
extreme sensitivity. In leading axion haloscope experiments, superconducting
sensors such as superconducting Quantum interference devices (SQUIDs) and
Josephson parametric amplifiers (JPAs) approaching the quantum limit are being
developed as key detector components [116, 117].

Experiments have also been performed to search for axion-like particles
(ALPs). As the name suggests, ALPs are similar to axions, but there is one
major difference. In contrast to axions, whose mass and coupling parameter are
proportional, the masses and coupling parameters of ALPs are independent of
each other. Many theoretical models [118] predict ALPs, and as such, several
experiments are aiming to detect them by scanning a wider parameter space than
is addressed in axion search experiments, particularly in the low-mass region,
as shown in Figure 5.

###  2.2 Neutrinos and neutrinoless double beta decay

The neutrino is a subatomic particle that is abundant in the Universe. Its
existence was first suggested by Pauli in 1932 to explain the apparent
violation of energy and momentum conservation observed in the beta decay of a
neutron. The theory of neutrinos was further developed by Fermi in 1934, and
the neutrino was first experimentally discovered by Cowan and Reines in 1956
[119]. It took such a long time for neutrinos to be directly detected because
they have very little interaction with matter via the so-called weak
interaction.

According to the Standard Model of particle physics, neutrinos come in three
different flavor states, namely, the electron neutrino ($\nu_{e}$), the muon
neutrino ($\nu_{\mu}$), and the tau neutrino ($\nu_{\tau}$), and they have
long been considered massless. However, since the late 1990s, a number of
experiments have shown compelling evidence that neutrinos convert from one
flavor to another, thus establishing that neutrinos have nonzero masses and
that each flavor is a different combination of three mass eigenstates
($m_{i}$, $i=1,2,3$) [120, 121, 122, 123]. These so-called neutrino
oscillation experiments have led to measurements of the differences in the
squares of the mass eigenvalues ($\Delta m^{2}_{ij}$), but their absolute mass
scale is still unknown.

$0\nu\beta\beta$ is a hypothetical decay process that can illuminate several
unknown key properties of neutrinos, including the absolute mass scale.
$0\nu\beta\beta$ can be thought of as a special case of double beta decay. In
a normal double beta decay ($2\nu\beta\beta$), two electrons ($2\beta^{-}$)
and two antineutrinos ($2\bar{\nu}_{e}$) are emitted, expressed as

$(Z,A)\rightarrow(Z+2,A)+2\beta^{-}+2\bar{\nu}_{e},$ (2)

where ($Z$,$A$) stands for a nucleus with atomic number $Z$ and mass number
$A$. This process, first proposed in 1935 [124], is now a well-established
second-order weak process and has been observed in many isotopes, although
with an extremely long half-life on the order of $T^{2\nu}_{1/2}\approx
10^{19-24}$ years [125]. In the meantime, double beta decay that is not
accompanied by the emission of two antineutrinos was independently proposed by
E. Majorana and G. Racah [126, 127] and further detailed by W.H. Furry [128],
expressed as

$(Z,A)\rightarrow(Z+2,A)+2\beta^{-}.$ (3)

This process can occur only if neutrinos are Majorana particles (i.e., a
neutrino’s antiparticle is identical to itself). Hence, the unambiguous
observation of $0\nu\beta\beta$ would confirm that neutrinos are Majorana and
not Dirac particles, thereby settling a major unknown about neutrinos [129].
Furthermore, its observation would lead to another profound consequence: it
would indicate that lepton number is not always conserved, as the lepton
numbers on either side of Eq. 3 differ by 2. The discovery of lepton number
violation would have far-reaching implications in cosmology as well as in
particle physics [130].

Experimentally, there is a large difference in the energy spectra of
$2\nu\beta\beta$ and $0\nu\beta\beta$. Since the energy carried by neutrinos
cannot be measured in a regular experimental setting and the recoil energy of
the daughter nucleus ($Z+2,A$) is negligible, only the energy spectrum of the
two electrons emitted from the decay is experimentally relevant. In the
$2\nu\beta\beta$ case, the energy carried by the two electrons forms a broad
spectrum with its end point equal to the Q-value of the decay, as shown in
Figure 6. On the other hand, since the total decay energy is split only
between the two emitted electrons in the $0\nu\beta\beta$ case, the
corresponding energy spectrum is simply a peak at the Q-value.

Figure 6: Energy spectra of the two electrons from the $2\nu\beta\beta$and
$0\nu\beta\beta$ processes. The $0\nu\beta\beta$ rate is highly exaggerated
for visibility. Backgrounds and statistical fluctuations are not included in
these spectra.

Without any further interpretation, clear detection of the $0\nu\beta\beta$
peak itself would already reveal two important conclusions: that neutrinos are
Majorana-type particles and that lepton number is not strictly conserved. Even
more valuable information could be obtained through a detailed quantitative
analysis of a detected $0\nu\beta\beta$ peak. In the case of the light
neutrino exchange model, which is a commonly accepted theoretical mechanism,
the decay rate $\Gamma_{0\nu}$ of the $0\nu\beta\beta$ process that would be
measured is expressed as

$\Gamma^{0\nu}=1/T^{0\nu}_{1/2}=G_{0\nu}\left|M_{0\nu}\right|^{2}m_{\beta\beta}^{2},$
(4)

where $T^{0\nu}_{1/2}$ is the $0\nu\beta\beta$ half-life of the element;
$G_{0\nu}$ and $M_{0\nu}$ are the phase space factor and nuclear matrix
element (NME), respectively, of the decay; and $m_{\beta\beta}$ is the
effective Majorana mass. Although the phase space factor $G_{0\nu}$ can be
rather accurately calculated in the framework of atomic physics [131], the NME
is subject to model-to-model variations by up to a factor of 3, thus
presenting a need for more accurate calculations [132]. Moreover, in addition
to the light neutrino exchange mechanism, there are also other theoretical
models that lead to $0\nu\beta\beta$ rates of the same order as that from the
light neutrino exchange model [133]. Consequently, due to the current
theoretical uncertainties, even a precisely measured $0\nu\beta\beta$ decay
rate would not translate into a well-defined $m_{\beta\beta}$ value or limit.
Thus, $0\nu\beta\beta$ should ultimately be measured using multiple isotopes.

During the last two decades, a number of experiments have been carried out to
search for the $0\nu\beta\beta$ process in various candidate isotopes.
Although different experiments have used different detection techniques, there
are several common strategies for increasing the detection sensitivity. First,
it is crucial to minimize any possible background events from the environment
as well as the detector itself. This means that any $0\nu\beta\beta$ search
experiment should be carried out in a deep underground laboratory where the
flux of cosmic muons is significantly suppressed. In addition, radiopure
materials must be used as the materials that constitute and surround the
detector. In particular, the amounts of impurities consisting of U and Th
compounds in the detector materials should be reduced to unprecedentedly low
levels. Second, the greater the exposure (i.e., the product of the amount of
the $\beta\beta$ isotope and the measurement time) is, the higher the chance
of detecting $0\nu\beta\beta$ becomes. Moreover, in the presence of a
background signal, a higher energy resolution would lower the count of
background events in the region of interest (ROI), thereby improving the
detection sensitivity, because the width of the ROI is determined by the
detector’s energy resolution. These strategies have motivated various
technological developments in terms of detection methods, active and passive
shielding, material purification and radioassays, laboratory
controls/maintenance, and online/offline analysis.

In several $0\nu\beta\beta$ search experiments, LTDs with crystal absorbers
have been chosen as the main detection technique, primarily due to their high
energy resolutions. It is also advantageous to use detectors of this type
because they provide great flexibility in the selection of the target crystal,
enabling an experiment to run with various crystals that contain different
$0\nu\beta\beta$ candidate isotopes. To date, 48Ca, 82Se, 100Mo, 116Cd, and
130Te have been tested using LTDs. These candidate isotopes are especially
suited for experiments using LTDs because of their relatively high Q-values,
which not only reduce environmental backgrounds in the ROI but also result in
a large signal size. Moreover, it is possible to realize event-by-event
detection with two different types of signals for PID using LTDs because the
phonon (heat) channel is almost always available in most materials. Active
$0\nu\beta\beta$ experiments based on low-temperature measurements are
reviewed in Section 4.2.

##  3 Sensor Technologies

In this section, we discuss the sensor technologies used in superconducting
sensors and detectors based on the low-temperature calorimetric measurement of
energy, also known as superconducting microcalorimeters. Although the focus is
on superconducting microcalorimeters, it should be noted that much of the
discussion herein also applies to any microcalorimeters in general.
Additionally, note that other types of superconducting sensors and detectors
are briefly discussed in Section 4.

###  3.1 Motivation

The development of superconducting microcalorimeters is motivated by the need
for energy sensitivity, i.e., a high energy resolution and a low energy
threshold, beyond the level that conventional semiconductor- or scintillator-
based detectors can achieve. Here, we present a brief comparison between
conventional detectors and superconducting microcalorimeters.

When a particle or radiation interacts with a target (absorber) material in a
detector, energy transfer occurs in the absorber. The details of the energy
transfer mechanism depend upon the type and energy scale of the radiation and
the material and geometry of the absorber. Hence, the absorber should be
properly chosen for a given type of radiation to ensure the efficient
conversion of the input energy into measurable physical quantities. The
microscopic interactions that occur during the energy transfer process,
although rather complex and often not fully understood, are critical for
achieving good detector performance.

The absorbers of conventional ionizing radiation detectors are made of gas,
liquid, or semiconductor materials in which the initial radiation creates
measurable quantities of charge (electrons and ions) or light. For instance,
in a semiconductor detector, an energy input creates electron-hole (e-h)
pairs, which are then collected and measured by means of an electric field
applied in the target volume. In general, the signal becomes larger as more
e-h pairs are created. Thus, the energy resolution of semiconductor detectors
is confined by the Fano limit, representing the statistical fluctuation in the
number of created e-h pairs [134]. The Fano-noise-limited energy resolution is
expressed as $E_{\mathrm{rms}}$ = $\sqrt{fwE}$. Here, $f$ is the material-
specific Fano factor, $w$ is the average energy needed to create an e-h pair,
which is about three times larger than the band gap of the semiconductor
because of energy loss in the creation of phonons; and $E$ is the input
energy. For Si-based detectors, where $w$ = 3.7 eV and $f$ = 0.115,
$E_{\mathrm{rms}}$ of 51 eV (or, equivalently, an $E_{\mathrm{FWHM}}$ = 120
eV) is achievable for $E$ = 6 keV, which corresponds to an energy resolving
power ($E/E_{\mathrm{FWHM}}$) of $\sim 50$. For higher-energy alpha particles,
Si-based detectors show higher energy resolving power of $\sim 650$
($E_{\mathrm{FWHM}}$ = 8.5 keV at 5.5 MeV) [135]. Similarly, the intrinsic
energy resolution limit of scintillator-based detectors also depends on the
number of created e-h pairs, which generate scintillation photons by
transferring their energy to the luminescent centers of the scintillator
materiel. However, in general they have poorer resolution than semiconductor-
based detectors due to a combination of low light yield and inefficient photon
collection.

Unlike these conventional detectors, low-temperature thermal detectors
(microcalorimeters) measure phonons created by the absorption of a particle or
radiation in an absorber made of a condensed matter that is typically
maintained at a temperature below 1 K. Although phonon measurement is more
efficient than other detection channels because the majority of the input
energy is converted into the phonon channel, achieving high-accuracy
measurements in this channel is often very challenging. This is because
phonons exist in every material, with a certain statistical frequency/energy
distribution, and thermal fluctuations between the constituents of the
detector can consequently overwhelm the phonon signal to be detected. However,
at low temperatures, phonon measurement can be the most sensitive detection
channel since the specific heat of most materials becomes very small,
resulting in a larger temperature change for a given energy input, and thermal
fluctuations are simultaneously greatly suppressed. Furthermore, at
sufficiently low temperatures, superconducting materials and superconducting
devices and electronics can be adopted to further improve the detector
sensitivity.

The fundamental limit of the energy resolution for such microcalorimeters is
primarily set by the thermodynamic energy fluctuation between the detector and
its thermal reservoir. In reality, however, noise from the readout electronics
and the sensor itself (e.g., Johnson noise in the case of a detector with a
resistive sensor) is often the primary factor limiting the energy resolution.
In recent years, significant improvements have been achieved in the
development of sensor materials and readout technologies, resulting in energy
resolutions that are better than those of conventional detectors by orders of
magnitude. For example, the highest energy resolutions achieved with
superconducting microcalorimeters are a 1.3–1.6 eV full width at half maximum
(FWHM) at 6 keV [136, 137, 138] and a 0.9–1.1 keV FWHM at 5.5 MeV [10, 9].
Furthermore, with a carefully chosen absorber and detection scheme, the high
resolution phonon measurement technique can be accompanied by measurements in
other detection channels, such as ionization and scintillation. Such
simultaneous measurement schemes have become the gold standard in rare event
search experiments using LTDs since they can be used to suppress background
events. In the following subsections, some of the most mature and powerful
superconducting sensors and their working principles will be discussed.

###  3.2 Thermal calorimetric detection

A typical microcalorimeter is composed of an absorber and a temperature
sensor, both of which have weak thermal links to a heat reservoir (also called
the heat bath). The initial interaction for the measurement of a particle or
radiation occurs in the absorber in the form of energy transfer. The
temperature sensor then reads out the temperature change in the detector
caused by the energy transfer in the absorber. Typically, there is a weak
thermal connection between the heat bath and the temperature sensor, while the
thermal connection between the absorber and the temperature sensor is much
stronger. The heat bath is maintained at a temperature below 1 K.

Figure 7 shows a thermal model diagram and a typical temperature signal from a
simple microcalorimeter. The absorber and the temperature sensor are connected
to each other with a thermal conductance $G_{\rm sensor-absorber}$. The
thermal conductance between the absorber and the heat bath is $G_{\rm bath}$.
Without any external energy input into the absorber or the temperature sensor,
they are in thermal equilibrium at temperature $T_{0}$. Any energy input
converted to thermal energy is eventually released into the heat bath.

The detection principle of a microcalorimeter is straightforward. Under the
assumptions that the heat transfer from the absorber to the sensor is much
more efficient than that from the heat link to the bath (i.e., $G_{\rm sensor-
absorber}$ $\gg$ $G_{\rm bath}$) and that the internal thermal conductivity of
the absorber is also much larger than $G_{\rm bath}$, the temperature change
$\Delta T$ in the sensor due to an energy deposition $E$ from a particle or
radiation as a function of time $t$ is

$\Delta T(t)=T-T_{0}=\frac{E}{C_{\rm absorber}+C_{\rm sensor}}e^{-t/\tau}$ (5)

for $t\geq 0$, with

$\tau=\frac{C_{\rm absorber}+C_{\rm sensor}}{G_{\rm bath}},$ (6)

where $C_{\rm absorber}$ and $C_{\rm sensor}$ are the heat capacities of the
absorber and the temperature sensor, respectively. In this calorimetric
detection scheme, $\Delta T$ corresponds to the signal pulse height, and the
time constant $\tau$ corresponds to the fall-time of the signal. This fall-
time constant can be engineered to meet specific experimental requirements on
the event-rate tolerance and to avoid excessive signal pile-up by fabricating
the thermal link with a desired $G_{\rm bath}$. However, $G_{\rm bath}$ should
be tuned carefully because it can also affect other critical detector
parameters such as energy resolution [139].

Figure 7: (a) A simplified thermal model of a typical microcalorimeter. An
input of energy ($E$) into the absorber causes a temperature increase, which
is measured by the temperature sensor. $T$ and $T_{0}$ denote the temperatures
of the sensor and the bath, respectively. (b) The expected temperature
response from an energy input. $C_{\rm tot}$ represents the sum of the heat
capacities of the two thermal components in the detector system, namely
$C_{\rm absorber}$ and $C_{\rm sensor}$.

Equation 5 implies that the signal size ($\Delta T$) can be maximized by
minimizing the heat capacities of the absorber and the sensor. One obvious
approach is to lower the temperature because the heat capacities of many
materials become very small at low temperatures. In addition, a material with
a small specific heat can be used as the absorber. For instance, a pure
dielectric crystal is a good candidate material because its specific heat can
be much smaller than that of a metallic or amorphous absorber at low
temperatures due to the $T^{3}$ dependence (the Debye law). Thus, using a
dielectric crystal as an absorber can make it possible to build a large-volume
or large-mass detector. However, it should be noted that a small specific heat
should not be the only criterion for choosing an absorber material for a rare
event search experiment. Additionally, other physical properties, such as the
long-term stability under low temperature and high vacuum, the internal
thermal conductance and the light yield (if a light signal is to be utilized)
should all be considered simultaneously.

Moreover, choosing the right temperature sensor among the various available
options is as important as the selection of the absorber material. In some
cases, the choice of the sensor can be made entirely independently of the
choice of the absorber, but more often, they are closely related. In the
following subsection, several different superconducting sensors will be
reviewed to help guide the selection of sensors for rare event search
experiments.

###  3.3 Superconducting sensors

Most thermometers used in everyday life and for industrial/scientific purposes
utilize the temperature dependence of a physical quantity. For instance,
mercury thermometers work because the volume of the liquid changes with
temperature. In a thermocouple gauge, the temperature dependence of the
contact potential of two dissimilar metals results in a voltage signal
indicative of a temperature. A platinum thermometer has a resistance that is
proportional to the temperature in its working temperature range.

Similarly, temperature sensors with extreme sensitivity can be developed by
measuring some temperature dependent property of the sensor material at low
temperatures ($<1$ K). In addition, superconducting circuits and electronics
can be used to maximize the sensitivity. As previously stated, in this paper,
we refer to an LTD that is made of a superconducting sensor material or
utilizes superconducting circuits/electronics as a superconducting detector.
Although the distinction is not clear-cut, a superconducting detector can be
either a thermal equilibrium detector (thermometer) or a nonequilibrium
quasiparticle detector. Quasiparticles are excitations (broken Cooper pairs)
in superconductors [140]. A thermometer-type detector measures the temperature
increase of the phonon system in a sensor material. The two most common types
of thermometers are resistive thermometers and magnetic thermometers. On the
other hand, a quasiparticle detector measures an excess of quasiparticles
created via energy absorption in a superconductor .

In the case of a resistive thermometer, in which the signal readout circuit
measures the change in the electrical resistance of the sensor material due to
a temperature change, the resistance change caused by the initial energy $E$
input into the absorber can be written as

$\Delta R=\left(\frac{\partial R}{\partial T}\right)\frac{E}{C_{\rm tot}},$
(7)

where $C_{\rm tot}$ is the sum of the heat capacities of the detector
components (primarily those of the sensor and the absorber). Here, the
temperature dependence of the resistance, $\partial R/\partial T$, is an
important parameter that determines the detector sensitivity.

In a similar way, a magnetic thermometer is another sensitive technology using
a sensor material with temperature-dependent magnetization (e.g., paramagnetic
Au:Er) and a superconducting circuit. A magnetic thermometer measures the
change in the magnetization of the sensor material, expressed as

$\Delta M=\left(\frac{\partial M}{\partial T}\right)\frac{E}{C_{\rm tot}}.$
(8)

Superconducting circuits and electronics enable accurate measurement of
$\Delta M$ for extreme detector sensitivity.

In this section, four different types of LTDs will be introduced, namely,
transition edge sensors (TESs), metallic magnetic calorimeters (MMCs); it is
also called as magnetic microcaloirmeters with the same acronym, kinetic
inductance devices (KIDs), and semiconductor thermistors. TESs and MMCs are
resistive and magnetic thermometers, respectively, and an KID is a
quasiparticle detector. Based on our definition, TESs, MMCs, and KIDs are all
superconducting sensors. Although, in general, thermistors are not
superconducting sensors, they are also discussed here because they are thermal
equilibrium detectors with a working principle similar to that of TESs or MMCs
and are popularly used in rare event search experiments.

####  3.3.1 Transition edge sensors

TESs are among the most sensitive temperature sensors utilizing the properties
of superconducting materials. A TES is a superconducting film operating at its
superconducting-normal transition temperature ($T_{\rm c}$). This
superconducting film often consists of a single material (an elemental
superconductor), such as tungsten (W). As an alternative, a bilayer consisting
of a superconductor and a noble metal, such as Mo/Au, Mo/Cu, or Ti/Au, is also
popularly chosen as the superconducting film. In the DM search experiments
CDMS and CRESST, W-TESs have been used, while bilayer TESs are often chosen
for high-resolution X-ray, gamma-ray, and alpha spectrometers. Using W or a
bilayer as the superconducting film makes it possible to tune $T_{\rm c}$. For
example, the $T_{\rm c}$ of a W film varies with its crystal structure
($\alpha$-, $\beta$-, or $\gamma$-phase [141]) and the environment in which
the film was deposited. The CRESST group uses $\alpha$-W with $T_{\rm c}\sim$
15 mK for their CaWO4 crystals, and CDMS uses mixed-phase W with a $T_{\rm c}$
of 50–100 mK in their detectors. In the case of a bilayer TES, its Tc can be
adjusted via the proximity effect by varying the thickness ratio between the
two layers [142].

The normal-state resistance of a TES is usually a few tens of m$\Omega$, with
a transition width of a few mK or less. Figure 8 shows a typical resistance
curve of a TES as a function of temperature near the superconducting-normal
phase transition “edge”. The transition width can be on the order of 0.1 mK
near a $T_{\rm c}$ of approximately 100 mK. The temperature dependence of the
resistance $(\partial R/\partial T)$ is very large at the transition, making
the TES a very sensitive thermometer.

TESs are, in general, operated in the voltage bias mode, such that a change in
the resistance of the TES causes a change in the current in the bias circuit.
This current change is measured with a SQUID. A simplified TES geometry and
the corresponding measurement circuit are illustrated in Figure 8. State-of-
the-art SQUID technology guarantees accurate and fast measurement of the
current changes with low noise. As shown in this figure, the TES film has a
weak thermal connection to a heat bath that is regulated to a temperature
below the $T_{\rm c}$ of the film ($T_{\rm bath}<T_{\rm c}$). With the
application of a bias voltage $V$ that is sufficiently high to break the
superconductivity of the TES film, the TES becomes resistive, and the
temperature of the TES is elevated from $T_{\rm bath}$ to within its
superconducting-normal transition region due to Joule heating: ${P}_{\rm
Joule}=V^{2}/R(T)$, where $R(T)$ is the resistance of the TES at $T$. With the
weak thermal link being the only thermal connection between the film and the
bath, the temperature of the TES is determined by the thermal differential
equation

$C\frac{dT}{dt}=-P_{\rm bath}+P_{\rm Joule}=-P_{\rm bath}+\frac{V^{2}}{R(T)},$
(9)

where $P_{\rm bath}$ is the power flowing from the TES to the heat bath, which
is a function of the TES temperature, the bath temperature, and the thermal
conductance between the TES and the bath.

Figure 8:  (a) Simplified electronic and thermal circuits of a TES. (b)
Typical $R$-$T$ characteristics of a TES near the superconducting transition.
The working point set by means of ETF is indicated by a black dot, and a
temperature increase due to an energy input is indicated by a red arrow.

Equation 9 has important implications. When a TES is at a temperature $T_{0}$
within its transition region under a constant bias voltage and there is no
external energy or power input, the TES is in thermal equilibrium, with
$P_{\rm bath}=P_{\rm Joule}=V^{2}/R(T)$. However, once the temperature of the
TES is increased (decreased) from $T_{0}$, the Joule heating decreases
(increases) because $R(T)$ has a positive slope in the transition region. This
effect is called negative electrothermal feedback (ETF). As a result, the
temperature of the TES is self-regulated at the quiescent temperature $T_{0}$.
The negative ETF mode has several advantages for particle detection with TESs.
Because they self-regulate their temperature within the transition region, a
large array of TESs in a detector can be stably operated even if they have
slightly different $T_{\rm c}$ values, and they also become less sensitive to
fluctuations in the bath temperature. Moreover, the recovery time of a TES is
much faster than the intrinsic time constant for a given energy input $E$
because the excess energy is effectively removed by a reduction in the Joule
heating, thereby reducing the dead time of the detector due to pulse pile-up.

Another advantage of using a TES is that the TES film can be directly
evaporated onto the surface of an absorber. Direct contact between the sensor
and the absorber enables efficient heat transfer between them, which results
in a much faster response time than those of other detectors such as
thermistors. The fast response time of TESs make them suitable for detecting
athermal phonons which can help increase the detector sensitivity and
facilitate the rejection of background signals.

TESs have been used as the temperature sensors in DM search projects such as
CDMS and CRESST, as will be discussed in Section 4.1. They have also been
extensively developed for other applications, such as X-ray and nuclear
spectroscopy. To the best of our knowledge, the best demonstrated FWHM energy
resolutions are 0.7 eV at 1.5 keV [143] and 1.3–1.6 eV for 5.9 keV X-rays
[136, 137], 65 eV for 208 keV gamma rays [144] and 1.0–1.1 keV for 5.3 MeV
alpha particles [9, 145]. These energy resolutions are improved by orders of
magnitudes compared to those of comparable commercial semiconductor detectors,
such as Si(Li) detectors and high purity Ge detectors for X-rays and gamma
rays and passive implanted planar Si detectors for alpha particles.

####  3.3.2 Magnetic microcalorimeters

MMCs are the most mature magnetic thermometers used in particle detection.
MMCs have become an important technology in various applications that require
high energy resolution over a wide energy range [139], and they have great
potential for use in rare event search experiments with crystal absorbers
[146, 147, 148]. The best achieved energy resolution of an MMC to date is a
1.6 eV FWHM for 6 keV x-rays [138], which is comparable to the best resolution
demonstrated by a TES. In an MMC setup designed for an energy range of 3–5
MeV, an FWHM of 0.86 keV has been obtained for the dominant Gaussian part of a
5.5 MeV alpha signal peak [10]. There is also another promising type of
magnetic thermometer called a magnetic penetration thermometer (MPT) [149],
but its maturity has not reached the level of MMCs, and it will not be
discussed here.

An MMC uses a paramagnetic material as the temperature sensor. This is because
the magnetization of a simple paramagnetic system is inversely proportional to
the temperature, following Curie’s law, and the temperature sensitivity can
consequently be very large at low temperatures. For many years, Au:Er, a
dilute magnetic alloy of gold doped with a small concentration of erbium, has
been the most popular choice of the paramagnetic sensor material for MMCs
because it maintains its paramagnetic properties at temperatures of tens of
mK, and the good thermal conductivity of the host metal Au guarantees fast
sensor thermalization at such low temperatures. Moreover, its thermal and
magnetic properties are well understood through mean field theory, which takes
into account the exchange interactions between the magnetic spins in a sensor
[150].

However, there is a drawback in using Au:Er as the sensor material of an MMC
because it shows excess heat capacity arising from nuclear quadrupole moments
of the gold nuclei under the electrical field gradient induced by the Er ions
[139, 150]. Since this excess heat capacity becomes the major contributor to
the total heat capacity at temperatures below $\sim$ 20 mK, Au:Er is not an
ideal sensor material for rare-event search experiments with large crystal
absorbers typically running at 10–20 mK. As a promising alternative, Ag:Er,
Er-doped silver [151, 138], has been quickly taking over the popularity of
Au:Er. A recent study where low temperature properties of MMCs based on two
types of sensor materials of Au:Er(1000 ppm) and Ag:Er(414 ppm) were measured
[152] demonstrated the Ag:Er MMC showed no noticeable excess heat capacity and
larger signal sizes than those from the Au:Er MMC.

MMCs utilize superconducting circuits and electronics to measure the changes
in magnetization of the sensor material. Typically, superconducting coils are
integrated on an MMC sensor chip to generate magnetic fields for magnetizing
the sensor material and to pick up magnetization changes in the sensor. The
signal picked up by the superconducting coils is read out by a dc-SQUID as a
voltage signal. Thus, the detection principle of an MMC can be expressed as
$E$ $\rightarrow$ $\Delta T$ $\rightarrow$ $\Delta M$ $\rightarrow$ $\Delta
V$, where $E$ is the input energy; $\Delta T$ and $\Delta M$ denote the change
in sensor’s temperature and magnetization, respectively; and $\Delta V$ is the
voltage signal of the SQUID.

Figure 9: (a) Simplified MMC setup with an absorber. (b) Typical $M$-$T$
characteristics of a Au:Er sensor in a magnetic field. A temperature increase
due to an input of energy into the absorber results in a change (reduction) in
the magnetization of the MMC sensor, as illustrated by the red arrow.

A simplified MMC setup is illustrated in Figure 9. Initially, MMCs for
particle detection were developed in a configuration in which a small piece of
Au:Er attached to an absorber is placed inside the superconducting loop of a
dc-SQUID. The SQUID loop itself was used as the pick-up coil of the MMC [139].
However, this early design imposed both fundamental and practical limitations.
First, it was very challenging to efficiently couple the sensor and the pick-
up coil in a reproducible way. It was also almost impossible to control the
thermal conductance between the sensor and the heat bath and build a detector
based on carefully worked-out thermal modeling. Furthermore, the magnetic
field applied to magnetize the sensor could easily affect the performance of
the SQUID. Most of these problems were overcome by lithographically
fabricating a whole MMC sensor chip. This state-of-the-art MMC design includes
an integrated superconducting pick-up/magnetization coil in the vicinity of
the sensor layer, maximizing the coupling between them and allowing the use of
ultralow-noise SQUIDs without their performance being affected by the magnetic
field used for the operation of the MMC. The thermal link between the sensor
and the heat bath is now controllable by on-chip thermalization pads.

MMCs have many advantages for use as sensors in rare event search experiments.
First, they have rather large heat capacities. Counterintuitively, this makes
an MMC suitable as the sensor for an absorber with a large heat capacity
because its performance will be less degraded by the added heat capacity of
the absorber. Second, their working temperature range is very wide, permitting
flexible optimization of the detector configuration depending on the detector
performance requirements [153, 148]. MMCs also exhibit excellent detector
linearity with a small deviation that can be reliably modeled.

The fast intrinsic thermalization of MMCs is another advantage for use in
particle detection. The intrinsic thermalization time is affected by the
interaction between conduction electrons and magnetic ions in the sensor
material. When the absorber is strongly connected to the sensor layer or the
sensor layer is used as its own absorber, an MMC can have an extremely fast
signal rise-time. An MMC with a gold absorber fabricated on top of the sensor
layer has demonstrated a rise-time shorter than 100 ns [154, 155].

In an application with a large crystal absorber, however, the rise-time of an
MMC is much slower (on the order of ms) because it is primarily determined by
the phonon generation mechanism and phonon dynamics in the crystal as well as
the thermal connection between the absorber and sensor rather than by the
intrinsic thermalization of the sensor [146]. Nevertheless, even with the slow
ms-scale overall rise-time with a scintillating crystal absorber, MMC signals
are sufficiently fast to distinguish the pulse shape difference between
electron- and alpha-induced events, which are considered signal and background
events, respectively, in $0\nu\beta\beta$ experiments [148]. Moreover, a rise-
time on this scale, which is approximately two orders of magnitude faster than
that of thermistor-based detectors, can help reduce the unresolved pile-up
(random coincidence) event rate, which can be one of the most significant
background sources for slow detectors in $0\nu\beta\beta$ experiments [156].

####  3.3.3 Kinetic Inductance Devices

A KID is a superconducting sensor with a very different detection mechanism
from those of the detectors discussed above. It is considered a nonequilibrium
detector and uses a change in kinetic inductance in a superconductor as its
main signal.

Superconductors enter the resistance-free state under a dc current (smaller
than a certain critical current) below a certain transition temperature.
However, they exhibit a nonzero resistance under an ac current due to a
complex surface impedance (or kinetic inductance) phenomenon. When a
superconducting strip, cooled below its $T_{\rm c}$, absorbs energy from a
particle or radiation, some of the Cooper pairs in the strip are broken,
generating quasiparticles. The higher the input energy is, the more
quasiparticles will be generated. These excess quasiparticles change the
kinetic inductance of a microwave resonance circuit, which can be measured as
a shift in the amplitude and phase of the resonance, as shown in Figure 10
[157]. This is the basic working principle of a KID. For the superconducting
strip materials in these detectors, Al, Al/Ti/Al, TiN, Ti/TiN, and PtSi are
often used [158].

Figure 10: Operation principle of a KID. (a) The quasiparticle density of
states ($N_{s}$) as a function of quansiparticle energy $E$. $\Delta$ is the
superconducting energy gap. A Cooper pair (denoted by C) is broken into two
quasiparticles by a photon of energy $h\nu$, affecting $N_{s}$. (b) A
schematic microwave resonant circuit of a KID. (c) Power vs. frequency ($f$)
and (d) phase vs. $f$ resonance profiles of the resonant circuit at 120 mK
(solid lines) and 260 mK (broken lines). Figure taken from [157].

The greatest advantage of KIDs is that they enable straightforward
multiplexing of many channels. A typical KID consists of many resonance
circuits with a high quality factor. By spreading the resonance frequency of
each resonance circuit over a large bandwidth (e.g., 2 MHz spacing over a 4
GHz bandwidth), it becomes possible to send a comb of frequency tones
corresponding to each resonator into the detector via a single feedline, and
its response can then be measured by room-temperature electronics. In
principle, more than a thousand resonators can be read out by a single
feedline. Because of their highly multiplexing capability and low noise, KIDs
have already been employed in many instruments to detect astronomical signals
ranging from the millimeter scale to the ultraviolet (UV) [159, 160].

In rare event search experiments, KIDs have not received as much attention as
in astronomical applications, mainly because when an KID is integrated as a
phonon detector (a thermal KID, or TKID), its energy resolution in the X-ray
regime is not sufficient to be competitive with other devices such as TESs
[161]. However, as the focus of DM searches moves towards lowering the
detection threshold, KIDs have arisen as an attractive solution. When a
coplanar waveguide feedline with many KID resonators is fabricated on a side
of an absorber wafer, the initial athermal phonons may break Cooper pairs in
the resonators near the event location. Each resonator would behaves like a
low- threshold phonon sensor on a target absorber helping position
reconstruction of the rare events [162] as in the R&D study of the BULky and
Low-threshold Kinetic Inductance Detectors (BULLKID) project [163]. Moreover,
in the Cryogenic wide-Area Light Detectors with Excellent Resolution (CALDER)
project [164], KIDs are also being developed as low-energy photon detectors
for a $0\nu\beta\beta$ experiment.

####  3.3.4 Other technologies

There are a few LTDs that do not use superconducting materials or
superconducting circuits but are popularly used in rare event search
experiments. One such detector is a doped semiconductor thermistor with a
strongly temperature-dependent resistance. Neutron-transmutation-doped (NTD)
Ge thermistors and ion-implanted Si thermistors are the two most common types
of thermistors used at low temperatures.

When a pure semiconductor such as Si or Ge is doped at a concentration
slightly below the critical doping for the metal-insulator transition, its
low-temperature resistance is governed by a process called Efros-Shklovskii
(ES) variable-range hopping [165] and can be modeled as

$R(T)=R_{0}\exp\left(\frac{T_{0}}{T}\right)^{1/2},$ (10)

with two characteristic constants, $R_{0}$ and $T_{0}$ [166]. The resistance
of a typical thermistor is 1–100 M$\Omega$ near 50 mK.

Figure 11: (a) Simplified circuit diagram of a thermistor setup with an
absorber. (b) Typical $R$-$T$ characteristics of a thermistor. An input of
energy into the absorber results in a change in the resistance of the attached
thermistor (represented by the red arrow).

Such thermistors are fairly easy to use because they can be operated with
conventional electronics such as junction field effect transistors (JFETs)
rather than requiring sophisticated superconducting electronics. They are
typically current biased, and the measured voltage across such a thermistor is
amplified by a JFET located at $\sim$100 K or at room temperature.

Of the two types of thermistors, NTD Ge thermistors are preferred in large-
scale experiments primarily because of their reproducibility and uniformity.
For example, the Cryogenic Underground Observatory for Rare Events (CUORE)
collaboration uses NTD Ge thermistors as the phonon sensors in their kilo-
pixel ton-scale detector. On the other hand, ion-implanted Si thermistors are
preferred in experiments where the individual detectors are small in size with
delicate electrical and mechanical structures because the dopant ions can be
implanted in a desired pattern through masking, and techniques for the
micromachining of silicon are well established. Thus, Si thermistors have been
a popular choice for spaceborne X-ray missions [166].

Both Ge- and Si- based thermistors have shown reasonable energy resolutions in
different energy regions. For instance, NTD Ge thermistors have achieved a
FWHM resolution of 3.1 eV at 5.9 keV X-rays [167], and Si thermistors have
achieved resolutions of 3.2 eV at 5.9 keV and 22 eV at 60 keV [168] (compared
to the best TES resolution of 1.3–1.6 eV and the best MMC resolution of 1.6 eV
at 6 keV and 10.2 eV at 42 keV [169]).

One drawback of thermistors is their rather slow response time, which is
caused by poor coupling between the conduction electrons and the phonons in
the thermistor. This slow response is unfavorable in experiments where the
rate of random coincidence events should be minimized as in $0\nu\beta\beta$
search experiments.

###  3.4 Heat flow model

To maximize the performance of a cryogenic detector, it is essential to
understand the heat flow mechanism between the thermal components of the
detector. This is especially true for rare event search detectors with large
dielectric absorbers because they involve one or more metal-dielectric, metal-
metal, and/or dielectric-dielectric interfaces. Another important heat flow
mechanism to be understood is the thermal coupling between the subsystems
(e.g., the phonon system and the electron system) of each detector component,
especially the sensor. In this subsection, three important heat flow
mechanisms based on the acoustic mismatch model, electron-phonon coupling, and
electronic thermal conduction will be discussed

####  3.4.1 Acoustic mismatch model

The heat transfer between two different media was studied theoretically and
experimentally in the 1960s–1980s [170]. In particular, the heat transfer
associated with thermal contact resistance, known as the Kapitza resistance
(or the inverse of the Kapitza conductance), was investigated in many
experiments involving liquid helium, various metals, and dielectrics. One of
the most successful models for the contact resistance was found to be the
acoustic mismatch model suggested by W. A. Little [171]. In this model, a
partial transmission of phonons across an interface between two media with
different densities and sound velocities is regarded as the origin of the
Kapitza resistance at low temperatures. In other words, the thermal resistance
between the two media is considered to be caused by the difference in their
acoustic impedances, i.e., the acoustic mismatch.

The essential result from the acoustic mismatch model is that the thermal
conductance $G_{\rm K}$ of the heat transfer is proportional to the contact
area $A$ of the interface and has a cubic temperature ($T$) dependence,
expressed as

$G_{\rm K}=\Lambda AT^{3},$ (11)

where $\Lambda$ is a proportionality constant that can be calculated based on
the model [170, 172]. A number of experiments involving interfaces between
various media show reasonable agreement with the predictions of this model
[170, 173].

####  3.4.2 Electron-phonon interactions in metal films

Studies of the interactions between electrons and phonons in metals have also
been a popular subject in modern physics. For instance, electron-phonon
interactions play a key role in explaining the mechanism of superconductivity,
which arises from indirect electron-electron interactions mediated by phonons.
Another significant implication of electron-phonon interactions is that they
change the effective mass of electrons in metals, offering explanations for
many characteristic properties of metals. Furthermore, electron-phonon
interactions are related to ultrasonic attenuation and the hot electron effect
in metals.

In the context of modeling the heat flow in metals at low temperatures, we
present a brief introduction to the hot electron effect originating from
electron-phonon relaxation processes in metals. The hot electron effect, a
temperature decoupling between the electrons and the lattice in a metal, is a
phenomenon that can be observed not only at low temperatures but also at much
higher temperatures. Many experiments using short laser pulses incident on
metal films have demonstrated the hot electron effect. For example, Schoenlein
et al. demonstrated that a high-intensity femtosecond laser pulse could induce
hot electrons at a temperature higher than 1000 K in gold, while the lattice
temperature remained at approximately 300 K [174]. P. B. Allen constructed a
theory for calculating the temperature relaxation of such hot electrons via
electron-phonon collisions governed by the Bloch-Boltzmann-Peierls formulas
[175]. The calculation assumes that the electrons first thermalize among
themselves as a result of heating from the short laser pulse and then
subsequently lose their energy to the lattice. An important conclusion of this
theory is that the heat transfer from the hot electrons to the lattice is
proportional to ($T_{\rm e}^{5}-T_{\rm l}^{5}$), where $T_{\rm e}$ and $T_{\rm
l}$ are the temperatures of the electrons and the lattice of the metal,
respectively. Allen claimed that this theory is also applicable at low
temperatures.

Accordingly, the heat transfer through the electron-phonon coupling in a metal
can be expressed as

$P_{\rm ep}=V\Sigma(T_{\rm e}^{5}-T_{\rm l}^{5}),$ (12)

where $V$ is the volume of the metal and $\Sigma$ is a material dependent
parameter that incorporates a coupling constant for the electron-phonon
interactions in the metal. According to Allen’s theory, $\Sigma$ is
approximately $2\times 10^{9}$ W/m3K5 for bulk gold based on a coupling
constant found from high temperature measurements [175]. For a small
temperature difference between $T_{e}$ and $T_{l}$, the corresponding thermal
conductance of the electron-phonon coupling can be approximated as

$G_{\rm ep}=5V\Sigma T_{e}^{4}.$ (13)

Based on the above discussion, the heat flow between a metal film and a
dielectric substrate primarily consists of two heat flow mechanisms connected
in series: the Kapitza resistance from the acoustic mismatch of the materials
and the impedance due to electron-phonon interactions. The conductance arising
from the series combination of these two thermal impedances is

$G=\left(\frac{1}{G_{\rm K}}+\frac{1}{G_{\rm ep}}\right)^{-1}.$ (14)

Since $G_{\rm K}$ is proportional to $AT^{3}$, and $G_{\rm ep}$ is
proportional to $VT^{4}$, $G_{\rm K}$ governs the conductance between the
metal film and the substrate at high temperatures. However, at sufficiently
low temperatures, $G$ is limited by $G_{\rm ep}$. When designing a detector
with a metal film, it is often useful to know at which temperature the two
conductances become equal, i.e., $G_{\rm K}\approx G_{\rm ep}$. For a metal
film with a thickness of $t=V/A$, the equal-conductance temperature $T$ in K
is approximately the inverse of $t$ in $\mu$m.

####  3.4.3 Electronic heat flow

The heat flow mechanism along a metal film or other metal structure depends on
the thermal conductivity of the conduction electrons. According to diffusion
theory for an electron gas, the electronic thermal conductivity $\kappa_{e}$
is equal to $\frac{1}{3}cv_{\mathrm{F}}\lambda$, where $c$ is the specific
heat, $v_{\mathrm{F}}$ is the Fermi velocity, and $\lambda$ is the mean free
path of the electrons. At sufficiently low temperatures, because the electron
mean free path is temperature independent, the thermal conductivity is
proportional to the specific heat of the metal. This results in the electronic
thermal conductivity being proportional to the temperature. Moreover, the
electronic thermal conductivity has a simple relation to the electrical
conductivity $\sigma$ for most noble metals at low temperatures, as follows:

$\frac{\kappa_{e}}{\sigma}=L_{0}T,$ (15)

where $L_{0}$ is the Lorentz number, which is equal to 2.45 $\times$ 10-8
W$\Omega$/K2. This relation is known as the Wiedemann-Franz law. It indicates
that the electronic thermal conductance of a metallic structure can be
estimated from its electrical resistance and geometry. In general, the
electrical resistivity $\rho$ of a metal decreases with decreasing temperature
until, at very low temperatures, it reaches a finite value due to the small
amounts of impurities and defects in the metal. The residual resistivity,
$\rho_{\mathrm{4K}}$, also depends on the microstructure of the metal such as
the grain size. When fabricating metal films for LTD applications, the
residual resistivity ratio (i.e., $\rho_{\mathrm{300K}}/\rho_{\mathrm{4K}}$)
is commonly used as an indicator of film quality and is often maximized to
achieve high thermal conductivity at low temperatures.

###  3.5 Athermal phonon transfer processes

In addition to the three thermal heat flow mechanisms discussed above,
athermal processes also need to be well understood to build LTDs with the
maximum achievable sensitivity, especially for detectors with dielectric
crystal absorbers. When an energy input is deposited locally in the electron
system of the absorber, initially, a cloud of energetic recoil electrons is
formed, which then loses the gained energy by producing high-energy phonons.
These phonons, having frequencies close to the Debye frequency of the absorber
material [176], are unstable even in a perfect crystal and quickly decay into
lower-energy phonons via anharmonic processes (downconversion). The lifetime
of a phonon with energy $E$ is proportional to $E^{-5}$ [177].

As the high-energy phonons lose their energy, their lifetime becomes longer,
and accordingly, their mean free path increases. For phonons with energies
corresponding to 30–50 K in a crystal such as silicon or sapphire, the mean
free path is on the order of several centimeters. Thus, for crystals of such
dimensions, which are a common choice in large-scale rare event experiments,
athermal phonons may travel ballistically around such a crystal until they
undergo further downconversion processes from inelastic scattering off of
defects or impurities (including isotope impurities) either at the surface or
within the crystal bulk. This downconversion can speed up when there is either
a normal metal or a superconducting feature present on the crystal surface, as
is the case in most cryogenic detectors, where an LTD sensor, typically
metallic, is in contact with the absorber either directly or indirectly via a
thin metal or superconducting film [178]. Such a film serves as a phonon
collector, through which efficient energy transfer from the crystal to the
sensor can occur. Thus, it is crucial to understand the anharmonic decay
process of athermal phonons to optimize the heat transfer and, hence, the
detector design.

One popular choice as a phonon collector for a crystal absorber consisting of
Si or CaWO4 is a superconducting Al film. A phonon with an energy more than
twice the superconducting gap of Al ($2\Delta_{\mathrm{Al}}\sim 340$ $\mu$eV)
may break a Cooper pair in the film, generating two quasiparticles. These
quasiparticles diffuse along the phonon collector film to the sensor attached
to the film. They then rapidly lose their energy to the electron system of the
sensor, raising the temperature of the sensor. Another popular phonon
collector is a gold film, especially when the sensor is an MMC. In this
configuration, athermal phonons will directly interact with electrons in the
gold film. Because of the good thermal conductance of the gold film itself and
that between the sensor and the film, the energy carried by the athermal
phonons is efficiently transferred to the sensor. Three different examples of
phonon collectors are depicted in Figure 12.

Figure 12: Simplified energy transfer processes in a phonon collector film
made of a superconductor (a and b) or a metal (c) fabricated on an absorber
crystal. The energy diagrams correspond to the superconductor phonon collector
films of (a) and (b). Note that $\Delta$ is the band gap of the superconductor
film, and $\Delta^{\prime}$ is a reduced gap originating from the proximity
effect in the vicinity of a TES film operating in its transition region. A
metal phonon collector such as that in (c) does not require Joule heating in
the energy transfer process to an MMC.

###  3.6 Phonon amplification by the Neganov-Trofimov-Luke effect

As discussed above, the sensitivity of a detector with a heat capacity $C$ to
an energy input $E_{0}$ is determined by the temperature increase $\Delta
T=E_{0}/C$. Thus, for a given detector configuration, it is nearly impossible
to increase the sensitivity of the detector. For a semiconductor absorber,
however, additional heat can be generated by applying an electric field,
resulting in an increased energy sensitivity. When an electric field is
applied to a semiconductor, e-h pairs generated by the absorption of a
particle or radiation drift along the electric field lines in the
semiconductor. This drift produces so-called Luke phonons in a process called
Neganov-Trofimov-Luke (NTL) phonon amplification. In the presence of the NTL
effect, the temperature increase due to an energy absorption of $E_{0}$ is
modified to

$\Delta T=(E_{0}+E_{\mathrm{NTL}})/C,$ (16)

where $E_{\mathrm{NTL}}$ is the amount of thermal energy converted from Luke
phonons produced via the NTL effect. Since $E_{\mathrm{NTL}}$ is generally
expected to increase with the number of created e-h pairs and the applied
electric field, it can be written as

$E_{\mathrm{NTL}}=\eta_{\mathrm{SE}}\eta_{\mathrm{NTL}}\cdot N\cdot e\cdot V,$
(17)

where $\eta_{\mathrm{SE}}$ and $\eta_{\mathrm{NTL}}$ are efficiency terms with
different physical origins, $N$ is the number of generated e-h pairs, $e$ is
the electron charge, and $V$ is the applied voltage across the electrodes. The
first efficiency term, $\eta_{\mathrm{SE}}$, known as the suppression
efficiency, depends on how well the electric field suppresses the premature
recombination of e-h pairs, which depends on the gap distance and the voltage
$V$ between the electrodes. The second term, $\eta_{\mathrm{NTL}}$, is the
efficiency associated with the creation of Luke phonons.

In the ideal case of perfect efficiencies $\eta_{\mathrm{SE}}$ and
$\eta_{\mathrm{NTL}}$, all the created e-h pairs are collected in each
electrode, generating a heat dissipation equal to $(e\cdot V)$ per pair. In
reality, however, the created e-h pairs may be recombined or trapped by
impurities inside the semiconductor or due to imperfect surface conditions,
causing a reduction in the generated heat. Thus, the amplification gain of the
NTL effect can be expressed as

$G_{\mathrm{NTL}}=\Delta T/\Delta T_{(V=0)}=1+\eta\frac{eV}{\epsilon},$ (18)

where $\eta$ is the product of $\eta_{\mathrm{SE}}$ and $\eta_{\mathrm{NTL}}$
and $\epsilon$ is the average energy needed to generate an e-h pair. With a
sufficiently large $V$, the gain becomes proportional to $V$. However, at much
higher voltages, the gain will saturate because the combined efficiency also
depends on $V$ and, in general, degrades at high voltages [179]. Since the
detector noise also tends to increase with increasing voltage, an optimal
voltage should be found to maximally utilize the NTL effect.

The ability to increase the detector sensitivity by means of the NTL effect
has been demonstrated in a number of applications based on various LTD
sensors, including TESs, MMCs and NTD Ge thermistors [180, 181, 182, 183].
These detectors showed reasonable amplification gains without significant
degradation in noise level, resulting in improved sensitivity. In particular,
sensitivity to single e-h pairs was recently demonstrated for visible photons
with an applied voltage of up to 160 V in a silicon crystal [182]. This
achievement may have a large impact, particularly in low-mass DM search
experiments.

In early-generation DM detectors based on calorimetric measurements with Si or
Ge absorbers, charge signals were measured together with heat signals [184,
185]. The primary reason for the simultaneous detection of charge and heat was
event discrimination between electron recoils and nuclear recoils. Since those
early-generation experiments were looking for heavy DM particles that would
interact more strongly with nucleons than with electrons, they used a small
bias voltage that could efficiently reject electron recoils. However, as the
focus of these experiments has shifted towards the detection of low-mass DM
particles, the main goal of detector design has become the lowering of the
energy threshold. This can be achieved by utilizing the NTL effect with a much
higher bias voltage, although as a trade-off, the detector becomes incapable
of event discrimination. Using single-charge-sensitive silicon detectors,
SuperCDMS has recently published significantly improved DM constraints in the
low-mass region [186].

Figure 13: Mass range of DM (top) and various detection methods with selected
ongoing projects (bottom). Particle-like DM is expected to deposit eV–MeV-
scale energy in the detector, resulting in ionization, scintillation, and/or
heat signals, as shown in the red boxes. While some experiments utilize a
single type of signal, others simultaneously measure two types of signals to
improve the detection sensitivity. Commonly used absorber materials are listed
in the black boxes for six different detection schemes. Ongoing DM search
projects and $0\nu\beta\beta$ projects (brown text) that utilize heat signals
are also listed next to the corresponding absorber materials. At the bottom
left, detection methods that utilize low-temperature microwave cavities and
superconducting lumped circuits are listed along with ongoing wave-like DM
search projects.

##  4 Application of superconducting detectors for rare event searches

In this section, we review how the superconducting sensor technologies
discussed in Section 3 can be used to develop a highly sensitive
superconducting detector for rare event search experiments by discussing
several real-life examples.

As discussed in Section 2, despite the abundance of compelling but indirect
evidence for the existence of DM, no direct detection has been achieved to
provide insight into the individual mass and coupling strength of DM. Thus, an
overwhelming number of theories have been proposed to explain the observed
indirect evidence and predict the properties of the DM. This situation has
resulted in a vast range of allowed masses for DM that span many orders of
magnitude, as shown in Figure 3 and in Figure 13. Since there is no single
detection method that can cover such a wide range, a detector should be
optimized for a rather narrow mass range. In Figure 13, we provide a list of
detection methods suitable for the direct detection of DM of different mass
ranges as well as for $0\nu\beta\beta$ search experiments that share common
detection schemes with the direct DM search experiments.

First, the detection of DM in the mass range above 1 eV relies on the
particle-like nature of DM. Such DM is expected to deposit energy in the range
of eV–MeV into the detector as a result of interactions between the DM and the
absorber material. Depending on the type of absorber material, this energy
deposition can result in ionization (charge), scintillation (light) and/or
heat (phonon) signals. The bottom right of Figure 13 shows six different
categories of target materials that allow single- or dual-channel
measurements. Dual-channel measurements have the advantage of enhanced
background rejection via particle identification, as explained below.

Superconducting detectors that utilize the heat (phonon) channel as their main
signal have excellent promise for use in particle-like DM searches because
such detectors have demonstrated superior energy resolution and a lower
detection threshold in the eV–MeV energy range. Moreover, the ubiquity of
phonon signals facilitates the development of a detector with an absorber that
allows dual-channel measurements.

In dual-channel detection methods (i.e., heat-ionization, heat-scintillation,
and ionization-scintillation), the ratio of the amplitudes of the two channels
differs depending upon the type of particle that primarily receives energy
upon interaction with a DM particle. For instance, in the heat-scintillation
case, an energetic ion that results from nuclear recoil due to DM-nucleon
scattering generates much less scintillation light than electrons or photons
(e/$\gamma$) of the same energy, as illustrated by Figure 14. This effect is
known as quenching of a scintillating material [187]. Similarly, in a
semiconductor, energetic ions generate fewer e-h pairs than electrons with the
same energy because they also lose their energy to phonons or other nuclei.

Dual-channel measurements have also been used in $0\nu\beta\beta$ search
experiments, where the main signal arises from two electrons with energy of a
few MeV produced by the decay. In these experiments, alpha-emitting
radionuclides inside and near the absorber crystals are crucial background
sources that can be efficiently removed by dual-channel detection.

The detection of DM candidates with masses below 1 eV is mostly based on the
wave-like nature of the DM. For instance, a tunable resonator composed of a
microwave cavity or a superconducting lumped circuit has been developed to
detect QCD axions, a strong DM candidate, through their conversion to photons
under a high magnetic field (Primakoff conversion) [115]. This type of
detection method, along with several ongoing projects, is illustrated at the
bottom left of Figure 13. Note that other promising technologies exist [39]
that are not listed here, some of which we briefly discuss later in the
section.

Figure 14:  The concept of particle identification in the dual-channel
detection of heat-scintillation or heat-ionization.

###  4.1 DM detectors

####  4.1.1 CDMS

The Cryogenic Dark Matter Search (CDMS) is an international collaboration for
direct detection of WIMPs that has pioneered many important superconducting
detector technologies. In particular, it is well known for its major
contributions to the development of TES and heat-ionization dual-channel
detectors based on quasiparticle diffusion in thin films and the NTL effect in
semiconductor absorbers. Most of these techniques were already discussed in
the previous section, but we will briefly revisit them when applicable.

The first phase of CDMS was called CDMS I (1998-2002), which was followed by
CDMS II (2003–2009) and then SuperCDMS Soudan (2011–2015). The current phase
is SuperCDMS SNOLAB, and a detector is being built in the deep underground
laboratory at SNOLAB.

With each phase of the experiment, the number and the total mass of the target
crystals increased. CDMS I included 6 detector modules composed of a Ge
crystal, with a total mass of 1 kg. The experiment was conducted at a shallow
underground laboratory (SUF, $\sim 10$ mwe (meter water equivalent)) and
achieved a total exposure of 30 kgd. CDMS II included 30 detector modules (4
kg of Ge in total) in a deeper underground laboratory of $\sim 2100$ mwe in
Soudan. With 400 kgd of exposure, CDMS II resulted in the then strongest limit
on the WIMP-nucleon spin-independent scattering cross section, nearly
rejecting the DAMA/LIBRA annual modulation region. SuperCDMS Soudan included
15 detector modules with a total Ge mass of 9 kg and was also conducted in
Soudan. The next-generation experiment SuperCDMS SNOLAB will be operated on
the order of 100 detector modules totaling approximately 200 kg. This latest
phase aims to probe not only the traditional WIMP mass region but also the
light DM region. For the latter, a fraction of the detectors are configured to
have an ultralow detection threshold.

With each successive phase, the CDMS detector design also continued to evolve.
However, the core measurement scheme remained the same: the detector consists
of multiple detector modules operating at cryogenic temperatures, each module
is equipped with a sizeable disk-shaped Ge or Si crystal as the target for DM-
normal matter interactions, and the temperature and ionization signals induced
by an energy input in the target crystal are simultaneously measured by a
temperature sensor and a field effect transistor (FET) amplifier,
respectively, as illustrated in Figure 13. The dual-channel measurement makes
it possible to distinguish nuclear recoil (NR) signals from electron recoil
signals, thus significantly improving the detector sensitivity to NR-like DM
signals, as previously discussed (see Figure 14).

The temperature sensors in the CDMS detectors, except for that in one early
generation called the Berkeley Large Ionization- and Phonon-mediated (BLIP)
detector, which is based on NTD Ge sensors, consist of thousands of tiny W
TESs spread over the surface of the target crystal in a hierarchical
arrangement, a unique feature of CDMS’s Z-sensitive ionization and phonon-
mediated (ZIP) detector. More specifically, CDMS I ZIP contained 4 independent
phonon channels, with each channel consisting of 37 cells and each cell
consisting of 12 TES elements coupled to Al “fins” or 12 quasiparticle-
assisted electrothermal feedback transition-edge sensors (QETs) as shown in
Figure 15 (a).

Figure 15: (a) Diagram of the phonon side of the CDMS I ZIP detector with a
100 g Si target crystal. Diagram of the phonon sensors for the 100 g Si ZIP
detector. The phonon sensor is divided into 4 channels labeled A to D. Each
channel consists of 37 cells, and each 5 mm $\times$ 5 mm cell consists of 12
TES elements connected in parallel. The TESs are coupled to Al quasiparticle-
collector fins. The figure is taken from Ref. [188]. (b) Diagram of a cell
consisting of 28 TES elements in the CDMS II ZIP detector. The figure is taken
and modified from Ref. [189].

In total, there were $4\times 444$ TES elements for each target crystal. The
TESs had a $T_{\rm c}$ of $\sim 100$ mK and were voltage-biased such that they
were in the extreme electrothermal feedback mode. This ensured stable
operation of such a large number of TES elements. The Al fins served as
quasiparticle collectors as well as the ground electrode of the ionization
detector and covered 82% of the top surface of the crystal. The phonon signal
was read out by SQUID arrays.

The CDMS II ZIP detector had a layout very similar to that of the CDMS I ZIP
detector but with an improved phonon collection efficiency. The QETs of the
CDMS I ZIP detector had a fill factor of nearly 100% near the QET coverage
area (82% of the top surface); because the average distance between generated
quasiparticles and the nearest TES was larger than the diffusion length of the
quasiparticles, some quasiparticles were unable to reach a TES within the time
constant of the TES. To solve this problem, the number of TESs per channel was
increased from 12 to 28, and the length of the Al fins was decreased in the
CDMS II ZIP detector as shown in Figure 15 (b). There were even more upgrades
in the SuperCDMS detectors, as described below.

The ionization channel of the CDMS detectors consists of charge-collection
electrodes deposited on both sides of the target crystal, which are used to
generate the electric field needed for the created e-h pairs to drift along
the field lines and to be collected in the electrodes [190]. To reduce the
loss of the ionization signal for surface events, a thin layer of amorphous
silicon was deposited between the electrode and the target crystal in every
detector except CDMS I ZIP [188]. In the CDMS I and II detectors, the
electrode on the top surface (the “phonon side”) served as the ground
electrode, and the electrode on the bottom surface (the “ionization side”) was
biased and connected to a current integrator. The generated e-h pairs under
electric fields created NTL phonons, which contributed to the total phonon
signal observed by the phonon sensor, as illustrated in Sec. 3.6. To prevent a
gradual drop in gain of the NTL phonon amplification due to charged trap
centers in the crystals, LEDs were used to routinely neutralize the crystals.
Moreover, to avoid creating too many NTL phonons, which would compromise the
detector’s PID capability, a relatively small bias voltage of a few V was used
in CDMS I and II.

In addition to exhibiting PID capability, the CDMS detectors have position
sensitivity, which can be utilized to further reject the background signal. As
described above, the CDMS I and II ZIP detectors have phonon sensors segmented
into four individual channels. By analyzing the pulse shape of a phonon signal
from the four channels (e.g., by comparing the arrival times), the lateral
position of the signal can be determined. Furthermore, the pulse-shape
difference between the bulk and surface events can be used to reject the
background signal occurring in the detector’s “dead layer” near the surface,
where the sensitivity of the ionization channel is compromised.

Now, we describe the SuperCDMS detectors. Based on the basic working principle
of the CDMS I and II ZIP detectors, SuperCDMS has involved significant
upgrades in its new interleaved Z-sensitive Ionization and Phonon sensors
(iZIP) detector. In this new detector, phonon sensors are patterned on both
sides of the crystal and interleaved with ionization electrode lines, as shown
in Figure 16.

Figure 16: (a) Diagram of an iZIP detector with a crystal 76 mm in diameter
and 25 mm thick used in SuperCDMS Soudan. (b) Photograph of an iZIP detector
with a crystal 100 mm in diameter and 33 mm thick and a copper housing to be
used in SuperCDMS SNOLAB. The figure was taken and modified from Ref. [191].

The phonon sensors on the top and bottom surfaces are separated into 4
channels in the SuperCDMS Soudan iZIP and 6 channels in the SuperCDMS SNOLAB
iZIP, covering the entire top and bottom surfaces of the crystal. The phonon
sensors also serve as the ground electrodes in both ZIPs and iZIPs. However,
unlike ZIP detectors, iZIP detectors have both positive (+2 V) and negative
(-2 V) bias electrodes. These changes were made to improve the sensitivity of
the iZIP detectors to surface events and to better extract the energy and
position information of each event.

There are several variants of the SuperCDMS Soudan and SNOLAB iZIP detectors,
such as CDMSlite and CDMS-HV. The main characteristic of these variants is
that they use a much higher bias voltage ($>50$ V) rather than a few V. This
higher voltage increases the gain of NTL amplification, thus improving the
energy resolution of the phonon channel and decreasing the energy threshold at
the expense of compromised PID capability. This was intended to enable probing
of the low mass region of the DM parameter space. SuperCDMS-CPD, a new
detector developed in collaboration with the Cryogenic PhotoDetector (CPD)
project [192], has taken a different approach. SuperCDMS-CPD uses a small
(approximately 10 g) Si crystal rather than massive Ge crystals. The use of
such a small crystal resulted in a substantially improved baseline energy
resolution of approximately 3.9 eV [68], which resulted in a promising limit
curve, as shown in Figure 4.

####  4.1.2 CRESST

The Cryogenic Rare Event Search with Superconducting Thermometers (CRESST) is
another multi-institution direct DM search experiment that uses TESs as
sensors of the DM-normal matter interaction. CRESST is known for its use of
scintillating crystals equipped with phonon-photon channel detection, as shown
in Figure 13. The collaboration recently completed the operation of CRESST-III
Phase 1 [193] and is in the preparation stage for CRESST-III Phase 2.

The first CRESST experiment (CRESST-I) used four sapphire (Al2O3) crystals
(262 g each) as the target material for the detection of DM-nucleon scattering
events [194]. Sapphire crystals were chosen because of their high Debye
temperature [195], which can make the specific heat of the detector very small
at cryogenic temperatures. To increase the resolution and sensitivity of the
detector, the operating temperature was decreased to as low as $\sim 15$ mK.
To match this low temperature, a TES composed of a W film (or a
superconducting phase transition (SPT) thermometer, as it was called within
the collaboration [196]) was developed. To achieve this very low
$T_{\mathrm{c}}$, an $\alpha$-phase W film was electron beam evaporated on
heated sapphire crystals in an ultrahigh vacuum environment [141]. For stable
operation of the detector at the $T_{\mathrm{c}}$ of the W film of $\sim 15$
mK, an active thermal feedback scheme in which the base temperature of the
cryostat was maintained at $\sim 6.5$ mK and an additional heater integrated
into the detector maintained the detector temperature at the $T_{\mathrm{c}}$
[197] was adopted, as shown in Figure 17.

[capbesideposition=right,top,capbesidewidth=4.5cm]figure[]

Figure 17: Schematic diagram of the active thermal feedback scheme. The Au
wire was used to make thermal contact between the heater and the tungsten
thermometer. The substrate was an Al2O3 crystal. The figure was taken from
Ref. [198]

The heater also served as a means to calibrate the detector.

The first DM limit result from the CRESST-I detector was released in 2002
[199]. Although these detectors had the highest sensitivity per unit mass
among cryogenic detectors in use at the time, this result did not have
sufficient sensitivity to rule out the “DAMA” region of WIMP’s annual
modulation. This limitation was mainly ascribed to the inability to
discriminate NR-like signals from gamma-dominated background events. The
collaboration implemented a PID solution to increase the detection sensitivity
by employing simultaneous detection of heat (phonon) and light (photon)
signals [200], which is a slightly different scheme from that of the CDMS,
whose PID mechanism is based on simultaneous detection of heat (phonon) and
ionization (charge) signals.

In certain scintillating crystals, electron- and photon-induced signals show a
different light yield than that due to heavy ions or neutrons, as shown in
Figure 14. Among several different scintillating crystals the CRESST
collaboration tested, CaWO4 showed the best performance at cryogenic
temperatures. Based on this finding, they made significant changes in their
detector design (CRESST-II): the sapphire crystals were replaced with CaWO4,
and light detectors were added, as shown in Figure 18.

Figure 18:  (left) Heat and light detector module. (right) Thermometer-heater
layout for the light detector. The figure was adopted from Ref. [201].

Note that the layout of the light detector is slightly different from that of
the phonon detector. As shown on the right side of the figure, Al phonon
collectors with large areas, which were meant to maximize the signal size for
a given energy deposition, were added.

The results of the commissioning phase of CRESST-II (completed in 2007) as
well as those of the subsequent CRESST-II phase 1 (2009–2011) showed improved
DM limits that excluded the DAMA region [201, 202, 203]. This result was
consistent with the null finding of other DM detectors, including the two LTD
experiments of CDMS and EDELWEISS. The design of the detector module was
updated in four different ways in CRESST-II phase 2 to clarify some events
with no or small amplitudes in the light channels that were unlikely from DM
scatterings or leakage events due to known backgrounds. One 300 g module of
CRESST-II phase 2 showed a 300 eV threshold, resulting in a large improvement
in the DM detection sensitivity [204, 205]. However, CRESST-II observed excess
events in the “acceptance” region in both experimental phases, which were
interpreted as background instead of positive WIMP signals.

The most recently completed CRESST-III phase 1 (2016–2018) focused on the low-
mass WIMP region. For this purpose, the mass of the CaWO4 crystals was reduced
to 24 g each (a factor of 10 reduction), significantly lowering the detector
threshold to below 50 eV [206]. Moreover, the new modules were designed to
employ crystal supporting rods composed of CaWO4, and each rod had a TES film,
as shown in Figure 19. This was to veto any events caused by the thermal
signal propagating to the target crystal through its holding structure, which
could explain the excess events in the acceptance region.

Figure 19:  Schematic diagram of the CRESST-III detector module. Note that the
detector holders are equipped with additional TESs as veto detectors to reject
events associated with the crystal holders. The figure was taken from Ref.
[193].

CRESST-III phase 1, with its lower threshold, resulted in an improvement in
the DM limit in the low-mass region ($\sim 500$ MeV/c2 and below) by an order
of magnitude. A new run of CRESST-III phase 2 with considerably increased
exposure is planned, which is expected to further extend the low-mass region
of the DM parameter space.

####  4.1.3 Other low-temperature DM detectors

Using TESs, CDMS and CRESST have shown remarkable progress and promising
results in direct DM detection covering the low-mass region of the DM
parameter space. Expérience pour DEtecter Les WIMPs En Site Souterrain
(EDELWEISS), another DM search project based on LTDs, utilizes NTD Ge
thermistors coupled to Ge target crystals as the phonon sensor. The Ge
crystals are also equipped with electrodes to measure the ionization signal.
This two-channel detection allows distinction of nuclear recoil events from
electron recoils in the target crystal.

EDELWEISS included three measurement phases, all of which were carried out at
the Modane Underground Laboratory in France. EDELWEISS-I, the first phase,
consisted of three detector modules composed of 320 g Ge crystals
(approximately 1 kg in total). EDELWEISS-I resulted in the first rejection of
DAMA’s positive claim of DM detection [207]. After this result, many other
experiments, including CDMS and CRESST, also ruled out DAMA’s claim. In
EDELWEISS-II, the collaboration operated 10 detector modules of 400 g Ge
crystals with a sensitivity of $4.4\times 10^{-32}$ cm2 for a DM mass of 85
GeV [208]. In EDELWEISS-III, 24 detector modules of 820–890 g Ge crystals were
operated to search for DM in the mass range of 1–20 GeV [209]. All three
phases yielded null results of the DM signal. Currently, as with CDMS and
CRESST, the EDELWEISS collaboration is shifting its focus to the detection of
low-mass DM [209].

EDELWEISS is acclaimed for the development of an efficient method to reject
surface events, which are one of major limiting factors of the sensitivity of
DM detection in the phonon channel. Surface events on a Si or Ge target
crystal may undergo incomplete charge collection, which can result in
misidentification of electron-induced surface events by nuclear recoil events
because, although due to a different mechanism, nuclear recoils also give rise
to a relatively small ionization signal for a given energy input. To solve
this issue, EDELWEISS designed a new detector with two sets of interdigitated
electrodes patterned on the surface of the Ge crystal [210, 211]. One set of
electrodes is used to collect ionization signals from bulk events, and the
other set is used to veto surface events. In the latest version of the
detector, called the fully interdigitated detector (FID), the electrodes cover
every surface of a crystal to improve the event discrimination efficiency. The
FID has demonstrated more than 99% efficiency [41]. Similar interdigitating
electrodes were also adopted in SuperCDMS’s iZIP detectors, as discussed in
the previous section.

####  4.1.4 Nanometer-scale thermal calorimeters

During recent decades, significant progress has been made in the development
and understanding of low-dimensional devices [212]. Many novel low-dimensional
devices are based on superconducting technologies and operate at sub-Kelvin
temperatures. These devices were motivated by various practical applications,
such as quantum computing and memory devices, but have also found their way
into astroparticle physics as detectors of rare event search experiments.
Here, we briefly introduce detectors based on low-dimensional devices (low-D
detectors).

In thermal calorimetric detection, a smaller heat capacity increases the
signal size, which is set by the temperature change for a given input energy,
as described in Eq. 5. One way to maximize the temperature increase is to make
a detector without an absorber and use the sensor itself as an absorber. The
heat capacity of such detectors can be further reduced by shrinking the
dimensions of the sensors. Thus, low-D nanometer-scale detectors have become a
promising choice for calorimetric detection of DM.

These nanometer-scale thermal detectors, however, are not always a viable
solution for the direct detection of DM, especially for WIMP-like DM particles
with a mass of approximately 100 GeV, because the small overall size of the
detector prevents it from being competitive in this mass region. On the other
hand, they can be an excellent choice for light DM because of their extremely
low detection threshold. Low-D detectors based on tunnel junctions are good
examples.

The Josephson-threshold calorimeter is a low-temperature thermal detector
based on a temperature-biased tunnel Josephson junction formed by different
superconductors [213]. Using the lower $T_{\rm c}$ junction element as the
sensor/absorber, these detectors are expected to be capable of photon number
resolution for photons from the mid-infrared region through ultraviolet region
(frequencies in the range of 30 THz to $9\times 10^{4}$ THz).

The graphene Josephson junction (GJJ) detector is another type of low-D
detector suitable for light DM. A GJJ uses a graphene film as the sensing
element [214, 215]. It utilizes the peculiar property of graphene in which the
electron density of states vanishes at the charge-neutrality point, which
leads to a highly suppressed electronic specific heat [216]. Recently, a DM
experiment was proposed based on GJJ detectors that can probe the DM parameter
space down to 0.1 keV [217].

The quantum capacitance detector is another low-temperature detector capable
of low-energy photon counting. When a photon is absorbed in a mesh absorber
composed of superconducting aluminum, the photon produces free electrons,
which tunnel into a single Cooper-pair box (SCB) island, a variable capacitor
embedded in a resonant circuit. A change in the SCB state due to electron
tunneling is recorded an RF signal. Recently, THz single photon detection was
demonstrated using this technique [218].

These small low-temperature devices are mostly in development and have not yet
been deployed in an active DM search experiment. However, as an increasing
number of DM search experiments report null results in the traditional WIMP-
like DM mass region and the focus shifts toward lower mass DM, these detectors
are expected to play a crucial role in next-generation DM search experiments.

###  4.2 Application to neutrinoless double beta decay search

As discussed in Section 2.2, the discovery of the $0\nu\beta\beta$ process
answers several fundamental questions in particle physics and astrophysics.
Because of the importance of this process, $0\nu\beta\beta$ searches have
become one of the most sought-after rare event experiments conducted in deep
underground laboratories. Among many detection technologies, thermal
calorimeters have played a major role in $0\nu\beta\beta$ searches during the
last two decades [129, 219, 220].

In thermal calorimetric detection, as discussed in Section 3, the detector
setup mostly consists of a temperature sensor and an absorber. For
$0\nu\beta\beta$ experiments based on thermal calorimetric detection, the
temperature sensor can be chosen from the high-resolution technologies
introduced in Section 3, while the absorber should be a dielectric crystal
containing double beta decaying isotopes.

The sensors should have a high resolution not only for the energy region of
$0\nu\beta\beta$ signals of a few MeV but also for a much wider energy range
to investigate background signals. For this reason, MMCs and NTD Ge
thermistors are suitable for the sensor of a $0\nu\beta\beta$ search
experiment because these sensors have a very wide practical dynamic range as
well as a well-characterized detector non-linearity, compared to other
superconducting sensors such as TESs and KIDs.

For absorber selections, scintillating crystals are preferred because unwanted
background signals can be discriminated through the simultaneous detection of
light (scintillation) and heat (phonon) signals, as shown in Figure 14.
Furthermore, these materials have better thermal properties than
noncrystalline materials.

To measure scintillation light for PID, a light detector composed of a thin Si
or Ge wafer can be employed, similarly to the CRESST setup for DM detection.
Note that the total energy of the scintillation photons absorbed in a light
detector is typically on the order of keV in $0\nu\beta\beta$ search
experiments that is orders of magnitude greater than in direct DM detection
experiments mainly because of the difference in the amount of energy absorbed
in the scintillating crystal. For such an energy scale, the light detector of
$0\nu\beta\beta$ search experiments has a large selection of sensor
technologies, including those with a relatively small dynamic range such as
TESs and KIDs.

Below, we discuss ongoing $0\nu\beta\beta$ projects in detail that have
adopted MMC and NTD Ge as both the heat and light sensors. We also discuss
future $0\nu\beta\beta$ projects that have chosen TESs and KIDs as the light
sensors as well as other related techniques used in light detectors.

####  4.2.1 AMoRE

The Advanced Mo-based Rare-event search Experiment (AMoRE) is a large-scale
international project to search for $0\nu\beta\beta$ of 100Mo in a deep
underground laboratory. Similar to the previously discussed DM detection
experiments, AMoRE employs superconducting detectors to probe extremely rare
$0\nu\beta\beta$ signals. The AMoRE detectors consist of heat and light
detectors to measure the heat and light signals, respectively, induced by
energy input in scintillating crystals, a detection scheme capable of PID
analogous to that of CRESST. However, unlike DM detectors, AMoRE detectors are
optimized to measure electron-induced events on the MeV energy scale.

AMoRE takes advantage of the high Q-value of $2\nu\beta\beta$ of 100Mo (3,034
keV). Because this Q-value is greater than the energy of most environmental
$\gamma$-rays, especially the 208Tl line of 2,615 keV, the background rate in
the region of interest (ROI) is significantly reduced. Nevertheless, another
source of background due to $\alpha$ decays inside or near the absorber
crystal may overwhelm the possible $0\nu\beta\beta$ signal unless it is
properly eliminated. This is why the AMoRE detectors require PID capability
for background rejection in addition to a high energy resolution.

AMoRE has involved tremendous effort in the development of heat and light
detectors as well as their target absorber crystals. The heat and light
detectors of AMoRE both employ high-resolution MMC sensors. For the absorber,
among several candidates, CaMoO4 and Li2MoO4 have been chosen based on their
high light yield and reasonable cost for mass production. In the first
cryogenic test of a small CaMoO4 with MMC, reasonable energy resolution in a
wide energy range was achieved, demonstrating the applicability of the
CaMoO4–MMC combination detector [147]. Soon, a detector with much larger
crystals of CaMoO4 and Li2MoO4 ($>200$ g) equipped with a separate light
detector was successfully built, which became the basis of the AMoRE detector
module [221, 222].

Figure 20: A simplified schematic (top left) and a photograph (bottom left) of
an AMoRE module as well as a photograph of the assembled AMoRE-I detector
tower with 18 modules (right). Each detector module includes one scintillating
crystal with two MMC channels for phonon and scintillation signals. The figure
was taken from [152]

The AMoRE detector consists of multiple modules, each consisting of a target
crystal of CaMoO4 or Li2MoO4 and a light detector enclosed by highly light-
reflecting material, as shown in Figure 20. In each module, two MMC sensors
are used, one for the heat (phonon) channel and the other for the light
channel. To increase the sensitivity of the heat channel, which is AMoRE’s
main signal, a gold film that serves as a phonon collector is evaporated on
one side of an absorber crystal. Then, an MMC sensor is thermally connected to
the phonon collector film via gold bonding wires. As discussed in the Section
3, the input energy in the crystal is transferred to the MMC sensor via
athermal and thermal processes of heat flow. The athermal phonons generated in
the crystal in the very early stage of energy absorption travel around the
crystal and have a significant probability of directly depositing their energy
onto the phonon collector before downconversion to thermal phonons. The heat
transferred to the phonon collector flows into the MMC sensor and eventually
to the heat bath. The light detector is built on the basis of an almost
identical heat flow mechanism, with two main differences: a Si or Ge wafer is
used as the absorber for lights from the main scintillating crystal, and input
energy, deposited by thousands of photons, is absorbed in a broad area than
being localized, as is the case for the heat detector.

The AMoRE detectors were carefully designed based on complete thermal modeling
that involves Kapitza conductance, electron-phonon interaction, and electronic
heat flow following the Wiedenmann-Franz law [223]. Thus, optimized AMoRE
detectors have shown an energy resolution of $\sim 15$ keV FWHM at 3 MeV.
Although 15 keV FWHM is considered a high resolution for a large detector, it
can be further improved by reducing the vibration of the cryostat [224] and by
reducing the position dependency [225]. In the current design, due to the
finite thermalization time of the large crystal, the signal size for a given
energy input varies depending on the position of the input, thus decreasing
the energy resolution. Employing an additional MMC sensor is expected to not
only solve the position dependency problem but also provide extra information
on the position of the event through the signal size difference between the
two sensors. This extra information can be used to distinguish localized
$0\nu\beta\beta$ events from delocalized high-energy $\gamma$-ray events
subject to multiple scattering inside the crystal. AMoRE detectors have also
demonstrated a fast rise-time of a few ms. This fast rise-time is realized by
the use of MMCs and gold phonon collectors and is much faster than that of a
crystal detector of similar dimensions equipped with an NTD Ge thermistor, a
popular choice in several $0\nu\beta\beta$ experiments. A fast detector is
very advantageous in cryogenic $0\nu\beta\beta$ experiments because it reduces
unresolved random pileups, which can be one of the most significant background
signals in searches for $0\nu\beta\beta$ signals of 100Mo in particular.

Recently, AMoRE finished its pilot stage run, which used 1.9 kg of
48depCa100MoO4 crystals, depleted in 48Ca and enriched in 100Mo, at Yangyang
underground laboratory (Y2L). 48Ca was depleted because 48Ca undergoes double
beta decay with a Q-value of 4,268 keV, higher than that of 100Mo, and it can
contribute to the background in the ROI that is indistinguishable from the
100Mo signal. The pilot run resulted in a $T_{1/2}^{0\nu}$ sensitivity of
9.5$\times$1022 years [225]. Presently, an upgrade version of AMoRE-I equipped
with 6 kg of 48depCa100MoO4 and Li2100MoO4 just started data collection.
Moreover, AMoRE-II has been fully funded, with a plan to build a 200 kg
detector in a new underground laboratory with a lower cosmic background to
achieve a limit sensitivity of $T_{1/2}^{0\nu}$ $>$ 5$\times$1026 years or,
equivalently, $m_{\beta\beta}$ $<$ (0.017–0.029) eV [226].

AMoRE’s MMC-based detection scheme has been adopted by another
$0\nu\beta\beta$ experiment: CAlcium fluoride for the study of Neutrinos and
Dark matters by Low Energy Spectrometer (CANDLES) [227, 228]. The CANDLES
experiment has been searching for $0\nu\beta\beta$ signals of 48Ca from CaF2
crystals using liquid scintillators as their main detector. As a strategic R&D
project, CANDLES recently carried out R&D experiments in collaboration with
AMoRE, successfully demonstrating its PID capability based on simultaneous
measurement of heat and light signals from a CaF2 crystal [227, 228].

####  4.2.2 Other low-temperature thermal calorimeters for $0\nu\beta\beta$
searches

Low-temperature thermal calorimetric detectors have long been studied for use
in $0\nu\beta\beta$ search experiments [229]. In particular, a series of
projects based in Italy have investigated the development of large-mass low-
temperature detectors for more than 30 years [230]. Preceded by Milano Double
Beta Decay (MiDBD) and CUORICINO, the Cryogenic Underground Observatory for
Rare Events (CUORE) project has utilized 800 kg TeO2 crystals as the
source/detector of 130Te $0\nu\beta\beta$ events and began collecting data in
2017. All three experiments were conducted in the Laboratori Nazionali del
Gran Sasso (LNGS) underground laboratory in Italy [230, 231, 232]. A new
project called the CUORE Upgrade with Particle Identification (CUPID) is under
development as a successor of the CUORE project but with an improved
background rejection capability [233].

The CUORE detectors use NTD Ge thermistors as temperature sensors of the TeO2
crystals. For each crystal, an NTD Ge thermistor is glued onto the crystal
surface. In this kind of setup, the heat flow between the crystal and the
sensor at low temperatures is governed by phonon transfer between interfaces
of difference media and interactions between phonons and electrons in the