Characterizing TES Power Noise for Future Single Optical-Phonon and Infrared-Photon Detectors
C.W. Fink, S.L. Watkins, T. Aramaki, P.L. Brink, S. Ganjam, B.A. Hines, M.E. Huber, N.A. Kurinsky, R. Mahapatra, N. Mirabolfathi, W.A. Page, R. Partridge, M. Platt, M. Pyle, B. Sadoulet, B. Serfass, S. Zuber
CCharacterizing TES Power Noise for Future Single Optical-Phonon andInfrared-Photon Detectors
C.W. Fink, a) S.L. Watkins, T. Aramaki, P.L. Brink, S. Ganjam, B.A. Hines, M.E. Huber,
3, 4
N.A. Kurinsky,
5, 6
R. Mahapatra, N. Mirabolfathi, W.A. Page, R. Partridge, M. Platt, M. Pyle, B. Sadoulet, B. Serfass, and S. Zuber Department of Physics, University of California, Berkeley, CA 94720, USA SLAC National Accelerator Laboratory/Kavli Institute for Particle Astrophysics and Cosmology, Menlo Park,CA 94025, USA Department of Physics, University of Colorado Denver, Denver, CO 80217, USA Department of Electrical Engineering, University of Colorado Denver, Denver, CO 80217, USA Fermi National Accelerator Laboratory, Batavia, IL 60510, USA Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA Department of Physics and Astronomy, and the Mitchell Institute for Fundamental Physics and Astronomy,Texas A&M University, College Station, TX 77843, USA (Dated: 19 June 2020)
In this letter, we present the performance of a 100 µ m × µ m ×
40 nm tungsten (W) Transition-Edge Sensor(TES) with a critical temperature of 40 mK. This device has a measured noise equivalent power (NEP) of1 . × -18 W / √ Hz , in a bandwidth of 2 . ± O (100) meV threshold athermal phonon detector for low-mass dark matter searches.Keywords: TES, Transition-Edge Sensor, Optical-Phonon, Infrared-Photon DetectorAs dark matter (DM) direct detection experimentsprobe lower masses, there is an increasing demand forsensors with excellent energy sensitivity. Several ather-mal phonon sensitive detector designs have been pro-posed using superconductors or novel polar crystals as the detection medium. Additionally, experiments thatuse single infrared (IR) sensitive photonic sensors to readout low–band gap scintillators or multi-layer optical halo-scopes for both axion and dark photon DM have also beenproposed .Each of these designs would ultimately require sen-sitivity to single optical-phonons or IR-photons, corre-sponding to energy thresholds of O (100) meV . Co-herent neutrino scattering experiments have made recentprogress using DM detector technology and are also in-terested in cryogenic detectors with very low thresholds .Transition-Edge Sensor (TES) based detector conceptshave been successfully applied in DM searches , as wellas IR and optical photon sensors . The same conceptscan also be used in these new applications, as the neces-sary energy sensitivities can theoretically be achieved .The energy resolution of a calorimeter can be esti-mated with an optimum filter (OF) from σ E = ε (cid:90) ∞ - ∞ dω π | p ( ω ) | S P ( ω ) , (1)where S P ( ω ) is the total power-referred noise spectrum, ε is the total phonon collection efficiency, and p ( ω ) ispower-referred pulse shape defined as p ( ω ) = 1 / (1 + jωτ ph ), with τ ph the athermal phonon collection time of a) Electronic mail: cwfi[email protected]. the detector. The resolution for a TES-based calorimeteris minimized when the noise is dominated by the intrinsicthermal fluctuation noise (TFN) between the TES andthe bath temperature ( T B ) . This noise can be writtenas S P ( ω ) ≈ k B T c GF ( T c , T B )(1 + ω τ − ) , (2)where k B is the Boltzmann constant, T c is the super-conducting (SC) critical temperature, G is the dominantthermal conductivity between the TES and the bath, and F ( T , T B ) is an O (1) scale factor accounting for the non-equilibrium nature of the thermal conductance. The ef-fective time constant in the strong electrothermal feed-back zero inductance limit (also neglecting effects of TESresistance and current sensitivity for simplicity) is definedas τ - ≈ C/ [ G (1+ α/n )], with α the dimensionless temper-ature sensitivity, C is heat capacity, and n is the thermalconduction power law exponent. Under this scenario, theintegral in Eq. (1) becomes σ E ≈ ε k B T c GF ( T c , T B )( τ ph + τ − ) . (3)If the energy of an incident particle is absorbed directlyby the TES, τ ph = 0, and ε = 1, the variance becomes σ E ≈ k B T c Cα (cid:114) n . (4)For a metal in the low-temperature regime, the heatcapacity scales with the volume of the TES (V TES ) as C ( T ) ∝ V TES T , suggesting σ E ∝ V TES T c . (5)For TES-based athermal phonon detectors, specificallyQuasiparticle-trap-assisted Electrothermal-feedback a r X i v : . [ phy s i c s . i n s - d e t ] J un Transition-edge sensors (QET) , the energy resolutionis minimized when athermal phonons bounce in thecrystal for times long compared to the characteristictime scale of the TES sensor (i.e. τ − < τ ph ) , aslong as the surface athermal phonon down-conversionrate is negligible . For low- T c W films, the thermalconductance is dominated by electron-phonon decou-pling, thus scaling as G ∝ V TES T n − c with n = 5, whichwas confirmed by measurement in the forthcominganalysis. In this case, the thermal conductance termis not cancelled from Eq. (3) and the baseline energyvariance of the detector will scale as σ E ∝ T c , suggestingthat a low- T c device is ideal for single optical-phononsensitivity.A set of 4 W TESs were fabricated on a 525 µ m thick1 cm × µ m × µ m ×
40 nm. Each subsequent TES increasedin area by a factor of four, keeping an aspect ratio of 1:4(width : length), which implies all the TESs have fixednormal resistance ( R N ). The TES mask design can beseen in left panel Fig. 1. Two sets of these 1 cm × T c = 40 mK, andthe other with TESs of T c = 68 mK. This letter focuseson the measurement and characterization of the low- T c µ m × µ m ×
40 nm TES (hereby referred to as theTES), but will also present characterization data fromthese other devices to elucidate scalings with T c and vol-ume.The voltage-biased TES was studied at the SLAC Na-tional Accelerator Laboratory in a dilution refrigerator ata base temperature of 15 mK. The Si chip was mountedto a copper plate with GE varnish. The current throughthe TES was measured with a custom DC Superconduct-ing Quantum Interference Device (SQUID) array systemwith a noise floor of ∼ / √ Hz, fabricated for the Su-perCDMS experiment, with a measured lower bound onthe bandwidth of greater than 250 kHz. The SQUID ar-ray was read out by an amplifier similar to the one inRef. 19.Multiple measures were put in place to mitigate elec-tromagnetic interference (EMI). Pi-filters with a cutofffrequency of 10 MHz were placed on all input and outputlines to the refrigerator. Ferrite cable-chokes were placedaround the signal readout cabling at 300 K, and the4K and 1K cans were filled with broadband microwave-absorptive foam to suppress radio frequency (RF) radia-tion onto TESs. The outer vacuum chamber of the dilu-tion refrigerator was surrounded by a high-permeabilitymetal shield to suppress magnetic fields. These measureswere the result of a systematic search of the system’s sus-ceptibility to environmental noise, and they lowered themeasured electrical noise by roughly an order of magni-tude. Despite these efforts, an unknown parasitic noisesource remained, which inhibited the smallest two low- T c TESs from going through their SC transition.To characterize the TES, IV sweeps were taken atvarious bath temperatures by measuring TES quiescentcurrent ( I ) as a function of bias current ( I Bias ), with 𝐺 " $ TES Absorber 𝐺 Bath 𝐺 " & TES 𝐼 %()* 𝑅 , 𝑅 *- 𝐿 𝑅 "/0
SQUIDreadout
FIG. 1. Left: TES design. The W is shown in red, whilethe blue represents Al bias rails. The Al connects to theleft and right sides of the TES. Middle: Thermal model forexperimental setup. For simplicity, only two TESs are shownin the model. Right: Electrical circuit. R sh is a shunt resistorwhich turns the current source ( I Bias ) into a voltage bias. Anyparasitic resistance on the shunt side of the bias circuit isabsorbed into the value used for R sh in this analysis. R p is theparasitic resistance on the TES side of the bias circuit. L isthe inductance in the TES line. R TES is the TES, which whenin transition takes on a value of R , and when its temperatureis above T c , it takes on a value of R N . complex admittance data taken at each point in the IV curve . Data were also taken simultaneously with thelargest low- T c TES (TES2) on the same Si chip, oper-ated at R ≈ R N , in order to attempt to quantifythe amount of remaining excess noise that coupled co-herently to both TES channels. From the IV sweep ateach temperature, both the DC offset from the SQUIDand any systematic offset in I Bias were corrected for usingthe normal and SC regions of the data. After this cor-rection, the parasitic resistance in the TES circuit ( R p ),the normal state resistance ( R N ), the TES resistance intransition ( R ), and the quiescent bias power ( P ) arecalculated (see Fig. 1 for circuit diagram).Since the Si chip contained multiple TESs, the ther-mal conductance between the Si substrate and the bath( G AB ) was measured by using one as a heater and oneas a thermometer. Knowledge of G AB allowed us to inferthe temperature of the Si substrate ( T A ) from a measure-ment of T B . See the middle panel of Fig. 1 for a thermaldiagram of the setup. Measuring P as a function of tem-perature from the IV sweeps, the thermal conductancebetween the TES and the Si substrate ( G T A ), T c , and n were fit to a power law . We measured that G AB wasroughly 3 orders of magnitude larger than G T A , meaningthat T A was effectively equal to T B and the system couldbe modeled as a single thermal conductance between theTES and the bath. The characteristics of the TES systemfrom the IV data are shown in Table I.For each point in transition, a maximum likelihood fitof the complex admittance was done, using the standardsmall-signal current response of the TES : Z ( ω ) ≡ R sh + R p + jωL + Z TES ( ω ) ,Z TES ( ω ) ≡ R (1 + β ) + R L − L β jω τ − L . (6)In this fit, L , R , R p , R sh , β , τ , and L are all freeparameters. L is the inductance in the TES bias circuit, TABLE I. Various calculated parameters of the TES. R (cid:3) or ‘R-square’ is the sheet resistance of the W film. R sh [mΩ] R p [mΩ] R N [mΩ] R (cid:3) [Ω] P [fW] G AB [ nJ / K ] G TA [ pJ / K ] T c [mK] T B [mK] T (cid:96) [mK]n5 . ± . . ± . ±
65 2 . ± .
26 31 ± . ± . . ± . ± ± ± . . . . . R /R N τ − [ µ s ] − − − β FIG. 2. Fitted values for β (purple dots) and effective elec-trothermal TES response time τ − (black crosses) as a functionof TES resistance. Frequency [Hz] | Z ( ω ) | [ Ω ] − π − π π π Φ ( Z ( ω )) [ r a d ] FIG. 3. A typical fit (cyan) of Eq. (6) to the complex admit-tance for the TES in the transition region, showing both themagnitude (black) and the phase (blue) for R ≈ R N . β is the dimensionless current sensitivity, τ is the naturalthermal time constant, and L is the loop-gain. We in-clude the estimates from the IV data of R , R p , and R sh as priors in the fit. Additionally, we include a prior on L ,measured from SC complex admittance data. The TESresponse times can also be measured from the complexadmittance data; defined as the rise and fall times of theTES response from a delta function impulse ( τ + and τ − ,respectively) . Best fit values of β and τ − are shown inFig. 2, while a typical fit of the complex admittance canbe seen in Fig. 3.The normal-state noise was used to estimate theSQUID and amplifier noise, once the Johnson noise com-ponent of the TES at R N was subtracted out. The ef-fective load resistance temperature was estimated fromthe SC noise spectrum, resulting in T (cid:96) ≈
37 mK, whichwas used to estimate the Johnson noise from R sh and R p . The TFN and TES Jonson noise components ofthe system were calculated as defined in the standardsmall-signal noise model , using the complex admittancefit parameters. The measured power spectral density Frequency [Hz] − − − − − − N EP [ W / √ H z ] ElectronicsTES LoadTFN TotalData
FIG. 4. Modeled noise components: TES Johnson noise (or-ange line), load resistor Johnson noise (red dashed), electron-ics noise (yellow dashed), thermal fluctuation noise (TFN)(purple alternating dashes and dots), and total modeled noise(purple dots), compared with the measured NEP (black line).The noise model and NEP are shown for R ≈ R N . Theshaded regions represent the 95% confidence intervals. (PSD) of the device in transition was converted into thenoise equivalent power (NEP) spectra with the power-to-current transfer function ∂I∂P ( ω ) = (cid:20) I (cid:18) − L (cid:19) (cid:18) jω τ − L (cid:19) Z ( ω ) (cid:21) -1 , (7)where Z ( ω ) is defined in Eq. (6). A comparison of thenoise model to the measured NEP for a typical operatingpoint in transition is shown in Fig. 4.From the measured NEP, the energy resolution of aDirac delta impulse of energy directly into the TES wasestimated using Eq. (1), with ε = 1 and τ ph = 0. Itcan be seen in the upper panel Fig. 5 that when theTES is operated at less than ∼ R N , the estimatedresolution of the collected energy is σ E = 40 ± . × -18 W / √ Hz for frequencies below 1 / (2 πτ − ), in abandwidth of 2 . T c TESs, using thesame analysis techniques, in Table II. The high- T c TESsalso observed a similar amount of excess noise. Despitethe elevated noise seen on both sets of TESs, the reso-lution scaling with volume and T c from Eq. (5) still ap-proximately holds.It is evident from Fig. 4 that the measured NEP iselevated from the theoretical expectation across the fullfrequency spectrum. We split the excess noise into twocategories. Excess noise that scales with the complex ad-mittance and is present when the TES is biased in its TABLE II. Energy resolution estimates for 68 mK T c TESscompared to the 40 mK T c TES described in this work. T c TES Dimensions σ E σ E a [mK] [ µ m × µ m × nm] [meV] [meV]Estimated Predicted from this workusing Eq. (5)40 100 × ×
40 40 ± × ×
40 44 ± ±
568 100 × ×
40 104 ±
10 89 ± a The resolution expected from a hypothetical device (with samephysical properties) by scaling the resolution of the low- T c TES( σ ) by Eq. (5), i.e. σ x = σ (cid:113) V x T c x / V T c normal or SC state, we call ‘voltage-coupled’, e.g. in-ductively coupled EMF. Noise that is only seen when theTES is in transition is referred to as ‘power-coupled’, e.g.IR photons radiating onto device. The excess voltage-coupled noise ( S SC ∗ ) can be modeled by scaling the SCpower spectral density (PSD) by the complex admit-tance transfer function when the TES is in transitionvia Eq. (8). This modeled noise can then be subtractedfrom the transition state PSD in quadrature. S SC ∗ ( ω ) = S SC ( ω ) (cid:12)(cid:12)(cid:12) [ Z ( ω )] R (cid:12)(cid:12)(cid:12) / (cid:12)(cid:12)(cid:12) [ Z ( ω )] R → (cid:12)(cid:12)(cid:12) (8)We expect ‘power-coupled’ noise from an environmen-tal origin to be largely seen on both the TES andTES2 coherently, though we have seen evidence of power-coupled noise generated by the Ethernet chip on ourwarm electronics to have significantly different couplingsto different electronics channels. We can determine thecorrelated and uncorrelated components of the noise byusing the cross spectral density (CSD) . The scaledSC noise PSD and correlated part of the CSD are plot-ted with the measured PSD in Fig. 6 for a fixed R .The two noise sources can explain the peaks in the noisespectrum, but cannot explain the overall elevated noiselevel.To investigate the hypothesis of the excess noise beingexplained by IR photons radiating onto the TES struc-ture, we loosely model this system by multiplying thethermal fluctuation noise by a scalar in order to make thetotal noise model match the measured NEP. This scalefactor is shown in the lower panel of Fig. 5. The factthat this scale factor changes by an order of magnitudethroughout the SC transition implies that this mecha-nism is not a dominant source of excess noise, as it shouldbe independent of TES bias.We can rule out the possibility of the excess noise be-ing due to multiple thermal poles , as none of thesemodels were able to explain the observed noise spectra.This is also evident by noting the lack of additional polesin the complex admittance in Fig. 3.The fact that the two smallest low- T c TESs (themost sensitive to parasitic power noise) were not ableto go through their SC transition, suggests that a non-negligible amount of the excess noise is environmental σ E [ m e V ] ( E s t i m a t e d ) . . . . . R /R N E x ce ss S T F N S c a l e F a c t o r FIG. 5. Top: Estimated energy resolution (from data)throughout the SC transition. Bottom: Scale factor needed toincrease S TFN to make the noise model match the measuredPSD. Frequency [Hz] − − − C u rr e n t N o i s e [ A / √ H z ] ModelData √ S SC ∗ Corrected Data Correlated
FIG. 6. Measured noise (black line), modeled ‘voltage-coupled’ noise (purple line), correlated noise (yellow dashed),measured noise with ‘voltage-coupled’ and correlated compo-nents subtracted (orange line), and theoretical noise model(purple dots) shown for R ≈ R N . The environmentalnoise model explains the peaks in the measured spectrum,but there is still discrepancy between the environmental noisecorrected data and the noise model. in origin. However, given the previous discussion, thisleaves open the possibility that some of this excess noiseis intrinsic to the TESs.With an estimated energy resolution of 40 ± T c (TableIII). It has immediate uses as a photon sensor in opticalhaloscope applications . Furthermore, its large volumesuggests that significant improvements in sensitivity canbe made in short order; a 20 µ m × µ m ×
40 nm TESmade from the same W film would be expected to have4 meV (rms) sensitivity, provided that we can reduceobserved excess noise and the volume scaling in Eq. (5)
TABLE III. Performance of state-of-the-art TES single pho-ton calorimeters/bolometers.TES T c TES Dimensions V TES σ E σ E √ V TES [mK] [ µ m × µ m × nm] [ µ m ] [meV] (cid:104) meV µ m (cid:105) W
125 25 × ×
35 21.88 120 25.7Ti
50 6 × . ×
56 0.13 47 128.2100 6 × . ×
56 0.13 47MoCu × ×
200 2000 295.4 6.6TiAu
106 10 × ×
90 90.0 48 16TiAu
90 50 × ×
81 202.5 ∼ b . b W (this) 40 100 × ×
40 1600.0 40 1 b The energy resolution in Ref. 32 is only an estimate from theNEP at a single frequency and the sensor bandwidth. holds.For athermal phonon detector applications , theexpected resolution is also impacted by the athermalphonon collection efficiency, which is typically > . Thus, small-volume crystal detec-tors ( ∼ ) should be able to achieve sub-eV trig-gered energy thresholds. Though such devices could notachieve the ultimate goal of single optical-phonon sensi-tivity, they could achieve the intermediate goal of sen-sitivity to single ionization excitations in semiconduc-tors without E-field amplification mechanisms , whichhave historically correlated with spurious dark counts.A decrease in TES volume and T c , along with concomi-tant improvements in environmental noise mitigation andthe use of crystals with very low athermal phonon sur-face down-conversion, would additionally be necessary toachieve optical phonon sensitivity.This work was supported by the U.S. Departmentof Energy under contract numbers KA-2401032, de-sc0018981, de-sc0017859, and DE-AC02-76SF00515, theNational Science Foundation under grant numbers PHY-1415388 and PHY-1809769, and Michael M. Garland.The main findings of this letter can be replicated fromthe presented data, but the full data that support thefindings of this study are available from the correspond-ing author upon reasonable request. Y. Hochberg, M. Pyle, Y. Zhao, and K. M. Zurek, J. High Energ.Phys. , 57 (2016). S. Knapen, T. Lin, M. Pyle, and K. M. Zurek, Phys. Lett. B , 386 (2018). S. Griffin, S. Knapen, T. Lin, and K. M. Zurek, Phys. Rev. D , 115034 (2018). N. Kurinsky, T. C. Yu, Y. Hochberg, and B. Cabrera, Phys.Rev. D , 123005 (2019). S. M. Griffin, K. Inzani, T. Trickle, Z. Zhang, and K. M. Zurek,Phys. Rev. D , 055004 (2020). M. Baryakhtar, J. Huang, and R. Lasenby, Phys. Rev. D ,035006 (2018). D. K. Papoulias, T. S. Kosmas, and Y. Kuno, Front. Phys. ,191 (2019). R. Agnese, Z. Ahmed, A. J. Anderson, S. Arrenberg, D. Balak-ishiyeva, R. Basu Thakur, D. A. Bauer, J. Billard, A. Borgland,D. Brandt, et al. , Phys. Rev. Lett. , 251301 (2013). R. Agnese, T. Aralis, T. Aramaki, I. J. Arnquist, E. Azadbakht, W. Baker, S. Banik, D. Barker, D. A. Bauer, T. Binder, et al. ,Phys. Rev. Lett. , 051301 (2018). A. H. Abdelhameed, G. Angloher, P. Bauer, A. Bento,E. Bertoldo, C. Bucci, L. Canonica, A. D’Addabbo, X. Defay,S. Di Lorenzo, et al. , Phys. Rev. D , 102002 (2019). S. W. Nam, A. Lita, D. Rosenberg, and A. J. Miller, in (Quebec City,Que., 2006) pp. 17–18. L. A. Zadeh and J. R. Ragazzini, Proc. IRE , 1223 (1952). M. Pyle,
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