The relative interfacial thermal contraction as a possible origin of the low-energy excess in cryogenic calorimeters

This paper proposes that the low-energy excess observed in cryogenic calorimeters arises from surface dislocations nucleated by thermal contraction mismatches between the absorber and underlying SiO$_2$ layers, offering a solid-state explanation and suggesting detector design modifications to test and mitigate this background.

The Gist

Imagine you are trying to build the most sensitive microphone in the world to hear a single whisper from a ghost (in this case, a dark matter particle). This microphone is a cryogenic calorimeter—a super-cold crystal detector. It's so sensitive that it can detect the tiniest bit of energy.

However, there's a problem. Instead of hearing just the ghost, the microphone is picking up a lot of static noise at the very bottom of the energy scale. Scientists call this the "Low-Energy Excess" (LEE). It's like a rising hum that gets louder as you look at lower and lower energies, and nobody knows what's making it.

This paper proposes a new theory for what causes that hum. Here is the explanation in simple terms:

1. The "Shrinking Suit" Problem

Think of the detector as a sandwich. The bottom layer is a heavy crystal (the absorber), and the top layer is a very thin, glass-like coating (amorphous SiO2) sitting right under the sensor.

When you cool this sandwich down from room temperature to near absolute zero (colder than outer space), everything shrinks. But different materials shrink at different rates.

• The Analogy: Imagine a wool sweater (the crystal) and a tight plastic wrap (the glass coating) stuck together. If you put them in a freezer, the wool shrinks a lot, but the plastic shrinks very little. Because they are stuck together, the wool tries to pull away, but the plastic holds it back. This creates a lot of tension or stress at the seam where they meet.

2. The "Snap" That Makes Noise

The authors suggest that this tension gets so strong that the materials actually "slip" or break at the microscopic level.

• The Analogy: Imagine a rubber band stretched too tight. Eventually, it snaps. When it snaps, it releases a tiny "pop" of energy.
• In the detector, this "snap" is called a dislocation. It's a tiny defect in the crystal structure that forms because the two layers are fighting over how much to shrink. When these defects form or relax, they release a tiny burst of energy (phonons) that the sensor picks up. This burst looks exactly like a particle hit, creating the "Low-Energy Excess" noise.

3. Why the "Double Microphone" Didn't Fix It

Scientists tried to fix this by building detectors with two sensors (Double-TES) on the same crystal. The idea was:

• If a particle hits the crystal, it will trigger both sensors at the same time.
• If the noise comes from the surface (the seam), it should only trigger one sensor, so they can ignore it.

The Paper's Twist: The authors explain why this trick might not work for this specific type of noise.

• The Analogy: Imagine the two sensors are on opposite sides of a room, and the "snap" happens right in the middle. If the room is made of a material that bounces sound waves perfectly, the "pop" from the snap might bounce off the first sensor, travel through the room, and hit the second sensor too.
• Because the crystal and the sensor have different "sound speeds" (phonon dispersion), the high-energy "pop" from the stress might bounce around inside the crystal and trigger both sensors. This makes the surface noise look like a real particle event, fooling the double-sensor system.

4. The Proposed Solutions

The authors suggest building new detectors to test their theory and stop the noise:

• Match the Shrinkage: Use materials that shrink at the exact same rate. They suggest using a specific type of crystal orientation and a tungsten sensor that "fits" perfectly, so no tension builds up.
• Match the Sound: Use materials that transmit sound waves better, so the "pop" doesn't bounce around and trigger both sensors. This would help the double-sensor system tell the difference between a real particle and a stress-induced "pop."

Summary

The paper argues that the mysterious "Low-Energy Excess" noise isn't caused by ghosts or unknown particles, but by the detector itself getting stressed as it cools down. The different layers are shrinking at different speeds, causing microscopic "snaps" that look like signals. By matching the materials better, we might be able to silence this noise and finally hear the real signals we are looking for.

Technical Summary

Technical Summary: Relative Interfacial Thermal Contraction as a Possible Origin of the Low Energy Excess in Cryogenic Calorimeters

Problem Statement
Low-threshold cryogenic calorimeters are pivotal for rare-event searches, such as dark matter detection, due to their exceptional energy resolution. However, their sensitivity is currently limited by a "low-energy excess" (LEE), an unexpected rise in the event rate at low energies attributed to an unknown background. While various hypotheses have been proposed, existing data presents a complex picture: the LEE does not scale with absorber mass in CRESST-III but does show mass scaling in TESSERACT; it can be "recharged" by warming the cryostat and decays rapidly upon re-cooling; and it exhibits pulse shapes identical to bulk absorber events. Crucially, neutron calibration data suggests radiation damage is not the primary cause, and double-TES (Transition-Edge Sensor) modules designed to reject surface backgrounds have failed to eliminate the LEE, as the excess persists in the coincident-event band.

Methodology and Theoretical Framework
This work proposes that the LEE originates from the relative thermal contraction coefficient (TEC) mismatch between the crystal absorber and the amorphous SiO$_2$ layer situated beneath the TESs (specifically in CRESST detectors). The authors employ a multi-faceted approach:

1. Elastic Energy Modeling: The authors formulate a simple elastic model based on the theory of elasticity for crystals. They calculate the elastic energy density ($u$) generated by the strain ($\epsilon$) resulting from cooling a CaWO$_4$ absorber from 60 K to 10 K. Using TEC data for CaWO$_4$ (along $a$ and $c$ axes) and SiO$_2$, they estimate the total elastic energy released during cooldown.
2. Dislocation Nucleation Analysis: Drawing on materials science literature regarding epitaxial thin films and bulk inclusions, the paper posits that the shear stress induced by TEC mismatch can exceed a threshold displacement energy (TDE), leading to surface dislocation nucleation. The relaxation of these dislocations is hypothesized to generate the observed phonon bursts (LEE).
3. Phonon Propagation and Double-TES Analysis: The authors analyze why double-TES modules fail to reject these events. They apply the concept of phonon dispersion curve mismatch. If the absorber supports higher-frequency phonon modes than the TES or phonon collector, high-energy phonons generated at the interface are reflected rather than transmitted. These reflected phonons propagate through the crystal and can trigger both TESs, mimicking a coincident bulk event and evading surface-rejection cuts.

Key Results

• Energy Scale Consistency: The calculated elastic energy released during a cooldown from 60 K to 10 K for a CaWO$_4$ crystal is approximately 2.8 MeV (for $c$-axis orientation) and 0.3 MeV (for $a$-axis orientation). This is comparable in order of magnitude to the integrated energy of the LEE spectrum observed in CRESST-III warm-up tests (~20 MeV), suggesting that thermoelastic stress is a viable energy source for the LEE.
• Orientation Dependence: The model explains the variability in LEE "recharging" between different CRESST modules. Modules with crystals oriented along the $c$-axis show different thermal contraction behaviors compared to those along the $a$-axis, potentially explaining why some modules exhibit recharged LEE after 30 K warm-ups while others do not.
• Failure of Surface Rejection: The study demonstrates that the standard surface-rejection logic of double-TES modules is compromised by interfacial thermal boundary resistance and phonon dispersion mismatches. High-energy phonons generated at the interface due to dislocation relaxation are reflected back into the absorber, allowing them to be detected by both sensors as coincident events.
• Mass Scaling: The authors suggest that while surface dislocations explain the non-mass-scaling component (CRESST-III), bulk dislocation nucleation around inclusions (due to TEC mismatch) could explain the mass-scaling behavior observed by TESSERACT.

Proposed Solutions and Tests
To validate this hypothesis and mitigate the LEE, the authors propose three specific detector designs:

1. Double $\alpha$-SiO$_{2,\parallel}$/W-TES: Utilizing crystalline SiO$_2$ oriented parallel to the plane, which has a TEC similar to Tungsten (W) below 100 K, eliminating the amorphous SiO$_2$ layer and minimizing TEC mismatch.
2. Double CaWO$_{4,a}$/W-TES: Using CaWO$_4$ crystals oriented along the $a$-axis (which matches W's TEC) equipped with W-only TESs, avoiding other film layers.
3. Double GaAs-TES: While GaAs still faces TEC mismatch, its phonon dispersion curves better match potential phonon collectors. This design aims to improve the transmission of interface-generated phonons, potentially shifting events from the coincident band to single-TES bands, thereby restoring surface-rejection power.

Significance and Claims
The paper claims to provide a unified physical mechanism that accounts for the majority of LEE observations, including the recharging behavior, the pulse shape similarity to bulk events, and the failure of double-TES surface rejection. The authors explicitly state that their hypothesis fulfills all observations except the specific mass-scaling rule observed by TESSERACT below 30 eV, which they attribute to a potential bulk dislocation component rather than surface effects.

The work does not claim to have definitively proven the origin of the LEE but rather motivates further investigation through specific experimental tests. It highlights that the current interpretation of double-TES data may be flawed due to phonon reflection at interfaces and suggests that detector designs focusing on TEC matching and phonon dispersion compatibility are necessary to either eliminate the LEE or conclusively test this hypothesis. The authors emphasize that dedicated atomistic simulations (molecular dynamics and density-functional theory) are ongoing to bridge the macroscopic elastic model with microscopic phonon generation mechanisms.