Characterizing TES Power Noise for Future Single Optical-Phonon and Infrared-Photon Detectors

In this letter, we present the performance of a $100~\mu\mathrm{m}\times 400~\mu\mathrm{m} \times 40~\mathrm{nm}$ tungsten (W) Transition-Edge Sensor (TES) with a critical temperature of 40 mK. This device has a measured noise equivalent power (NEP) of $1.5\times 10^{-18}\ \mathrm{W}/\sqrt{\mathrm{Hz}}$, in a bandwidth of $2.6$ kHz, indicating a resolution for Dirac delta energy depositions of $40\pm 5~\mathrm{meV}$ (rms). The performance demonstrated by this device is a critical step towards developing a $\mathcal{O}(100)~\mathrm{meV}$ threshold athermal phonon detectors for low-mass dark matter searches.


I. INTRODUCTION
As dark matter (DM) direct detection experiments probe lower masses, there is an increasing demand for sensors with excellent energy sensitivity. Several athermal phonon sensitive detector designs have been proposed using superconductors 1 or novel polar crystals [2][3][4][5] as the detection medium. Additionally, experiments that use single infrared (IR) sensitive photonic sensors to read out low bandgap scintillators or multi-layer optical haloscopes for both axion and dark photon DM have also been proposed 6 .
Each of these designs would ultimately require sensitivity to single optical phonons or IR photons, corresponding to energy thresholds of O(100) meV 1-3,6 . Transition-Edge Sensor (TES) based detector concepts have been successfully applied in high-mass DM searches 7,8 , as well as IR and optical photon sensors 9 . The same concepts can also be used in these new applications, as the necessary energy sensitivities can theoretically be achieved 1,2 . Consequently, TESs are an excellent sensor candidate for these new light-mass dark matter applications.
As shown by Irwin and Hilton in Ref. 10, the fundamental limit of the energy resolution of a TES calorimeter is given by where α is the dimensionless temperature sensitivity, T c is the critical temperature, C is the heat capacity, and a) Electronic mail: cwfink@berkeley.edu. n = 5 is the thermal conduction power law exponent 11 . Noting that, for a metal in the low-temperature regime, the heat capacity scales with the volume of the TES For TES-based athermal phonon detectors 12 , the energy resolution is minimized when athermal phonons bounce in the crystal for times long compared to the characteristic time scale of the TES sensor 13,14 , as long as the surface athermal phonon down-conversion rate is negligible 15 . Since the sensor bandwidth scales at T 3 c , the volume of the TES also needs to scale as T 3 c in order to have the phonon collection bandwidth be less than or equal to the sensor bandwidth. In this limit, the total energy sensitivity will scale as σ 2 E ∝ T 6 c , suggesting that a low T c device is ideal for single optical-phonon sensitivity 14 .
In order to build a precise noise model of these athermal sensors, which incorporate a TES coupled to a phonon absorber, a variety of tungsten (W) TES chips were constructed with and without the absorber structures. In this letter, we present the TES noise as measured without the absorber structures for a single device with T c ≈ 40 mK.

II. EXPERIMENTAL SETUP AND DATA
A set of 4 W TES test devices was fabricated. The smallest of the TES structures was 25 µm × 100 µm × 40 nm, with each TES increasing in area by a factor of 4, keeping an aspect ratio of 1:4 and a fixed normal resistance (R N ). The TES mask design, as well as a photo of the experimental setup, can be seen in Fig. 1. Since environmental noise limited the smallest two chips from going through their superconducting (SC) transition, this study focused on the 100 µm × 400 µm × 40 nm TES.
The voltage-biased TES was studied at SLAC National Accelerator Laboratory in a dilution refrigerator at a base temperature (T B ) of 15 mK. The TES was mounted to a copper plate with GE varnish, which provided a robust thermal connection from the TES absorber substrate to the bath, which allowed the system to be modeled with a single thermal time constant. The current through the TES was measured with a DC SQUID array with a noise floor of ∼ 4 pA/ √ Hz and read out by an amplifier similar to the one in Ref. 16.
Multiple measures were put in place to mitigate electromagnetic interference (EMI). Pi-filters 17 with a cutoff frequency of ∼ 10 MHz were placed on all input and output lines to the refrigerator. Iron cable-chokes were placed around the signal readout cabling inside the vacuum interface connection, and the 4K and 1K cans were filled with broadband microwave-absorptive foam 18 to suppress radio frequency (RF) radiation onto TES structures. The outer vacuum chamber of the dilution refrigerator was surrounded by a high-permeability metal shield to suppress magnetic fields.
To characterize the TES, an IV sweep was taken by measuring TES quiescent current (I 0 ) as a function of bias current 19 (I bias ), with complex admittance data taken at each point in the IV curve, as done in Refs. 14 and 20. Data were also taken simultaneously with a second TES of dimension 200 µm × 800 µm × 40 nm on the same chip, operated at ∼ 40% R N in order to quantify the amount of remaining environmental noise that coupled coherently to both TES channels.

A. Parameter Estimation
From the IV sweep, both the DC offset from the SQUID and any systematic offset in the applied bias current are corrected for using the normal and SC state regions of the data. Using a shunt resistor (R sh ) of 5 ± 0.5 mΩ, we are able to calculate the quiescent bias power (Eq. 2), as well as the parasitic resistance (R p ), normal state resistance (R N ) and the TES resistance (R 0 ).
The critical temperature was found by measuring the TES resistance while increasing T B . The T c was de-  fined to be the temperature at which R 0 (T )| Tc = R N /2. The region of the IV curve where the TES is in electrothermal feedback is used to estimate the effective thermal conductance from the TES to the absorber substrate. This bias power has the form 10,14,20 where G T A is the thermal conductance of the TES to the substrate. These DC characteristics of the TES system can be seen Table I. For each point in transition, we do a least-squares fit of the complex admittance, using the standard small-signal current response of the TES 10 as defined in Eq. 3: In this fit, R 0 , R p , R sh 21 , β, and τ I are all free parameters. β is the dimensionless current sensitivity, τ I = τ /(1 − L ) is the constant current time constant, τ = G/C is the natural time constant, and L = I 2 0 R0α /GTc is the loop-gain. We put constraints on the fit by including the known prior values and covariance matrix for R 0 , R p , and R sh that were calculated from the IV sweep. Best fit values of β and the calculated sensor bandwidth τ − are shown in Fig. 2, while a typical fit of the complex admittance can be seen in Fig. 3.     10%RN . The shaded regions represent the 95% confidence intervals. A large difference can be seen between the measured NEP and the total noise model.

B. Noise Modeling
The normal-state noise is used to estimate the SQUID and amplifier noise, once the Johnson noise component of the TES at R n is subtracted out. The effective load resistance temperature 22 is estimated from the SC noise spectrum, resulting in T ≈ 37 mK. The theoretical and measured noise equivalent power (NEP) spectra of the TES in transition, as seen in Fig. 4, are calculated using the complex admittance fit parameters to estimate the power-to-current transfer function in Eq. 4: where Z(ω) is defined in Eq. 3. From the measured NEP, we estimate the energy resolution of a Dirac delta impulse of energy directly into the TES using an optimum filter (OF) 14,20,23 . We show this estimated energy resolution throughout the transition in the upper panel of Fig. 6. When the TES is operated at less than ∼ 20% R N , we calculate the resolution of the collected energy to be σ E = 40 ± 5 meV.

C. Environmental Noise Reduction
It is evident from Fig. 4 that the measured NEP is elevated from the theoretical expectation across the full frequency spectrum. We split the excess environmental noise into two categories. Excess noise that scales with the complex admittance and is present when the TES is biased in its normal or SC state, we call 'voltage-coupled', e.g. inductively coupled EMF. Noise that is only seen when the TES is in transition is referred to as 'powercoupled' noise. The excess voltage-coupled noise (S SC * ) can be modeled by scaling the SC power spectral density (PSD) by the complex admittance transfer function when the TES is in transition via Eq. 5. This modeled noise can then be subtracted from the transition state PSD in quadrature.
We expect environmental 'power-coupled' noise to be largely seen on both the 100 µm × 400 µm and 200 µm × 800 µm chips coherently, though we have seen evidence of power-coupled noise generated by the ethernet chip on our warm electronics to have significantly different couplings to different electronics channels. We can determine the correlated and uncorrelated components of the noise by using the cross spectral density (CSD), as described in Refs. 20 and 24. The scaled SC noise PSD and correlated part of the CSD are plotted with the measured PSD in Fig. 5 for a fixed R 0 . The two environmental noise sources can explain the peaks in the noise spectrum, but cannot explain the overall elevated noise level.
To investigate the possibility of the excess noise being explained by IR photons radiating onto the TES structure, we loosely model this system by multiplying the thermal fluctuation noise (TFN) by a scalar in order to make the total noise model match the measured NEP. This scale factor is shown in the lower panel of Fig. 6. The fact that this scale factor changes by an order of magnitude throughout the SC transition implies that this mechanism is not a dominant source of excess noise, as it should be independent of TES bias.
We can rule out the possibility of the excess noise being due to multiple thermal poles, as suggested in Refs. 25 and 26, as none of these models fit the data. This is also evident by noting the lack of additional poles in the complex admittance fits in Fig. 3.
We lastly consider the excess non-stationary noise, i.e. the dark rate of the detector. Eq. 4 was used to calibrate event amplitudes to units of energy. With a detector mass of 30.8 ng, we observe zero events with an 850 meV   threshold for 3040 sec. As a comparison, this is roughly twice the exposure as in Ref. 27 with a similar threshold.

IV. DISCUSSION
With an estimated energy resolution of 40 ± 5 meV (rms), this device has achieved comparable energy sensitivity to world leading optical/near IR TESs, but with a volume that is orders of magnitude larger, due to its low T c (Table II). It has immediate uses as a photon sensor in optical haloscope applications 6 . Fur- thermore, its large volume suggests that significant improvements in sensitivity can be made in short order; a 20 µm × 20 µm × 40 nm TES made from the same W film would be expected to have 4 meV (rms) sensitivity, provided that external environmental noise remains sub-dominant. For athermal phonon detector applications (Refs. 1-5), the expected resolution is also impacted by the athermal phonon collection efficiency, which is typically > 20% in modern designs 33 . Thus, small-volume crystal detectors (∼ 1 cm 3 ) should be able to achieve sub-eV triggered energy thresholds. Though such devices could not achieve the ultimate goal of single optical-phonon sensitivity, they could achieve the intermediate goal of sensitivity to single ionization excitations in semiconductors without E-field amplification mechanisms 8,34 , which have historically correlated with spurious dark counts. A decrease in TES volume and T c , along with concomitant improvements in environmental noise mitigation and the use of crystals with very low athermal phonon surface down-conversion, would additionally be necessary to achieve optical phonon sensitivity.