Optimized Designs for Very Low Temperature Massive Calorimeters

Optimized Designs for Very Low Temperature Massive Calorimeters

Matt Pyle mpyle1@berkeley.edu Physics Department, University of California Berkeley    Enectali Figueroa-Feliciano Physics Department, Massachusetts Institute of Technology    Bernard Sadoulet Physics Department, University of California Berkeley
July 10, 2019
Abstract

The baseline energy-resolution performance for the current generation of large-mass, low-temperature calorimeters (utilizing TES and NTD sensor technologies) is orders of magnitude worse than theoretical predictions. A detailed study of several calorimetric detectors suggests that a mismatch between the sensor and signal bandwidths is the primary reason for suppressed sensitivity. With this understanding, we propose a detector design in which a thin-film Au pad is directly deposited onto a massive absorber that is then thermally linked to a separately fabricated TES chip via an Au wirebond, providing large electron-phonon coupling (i.e. high signal bandwidth), ease of fabrication, and cosmogenic background suppression. Interestingly, this design strategy is fully compatible with the use of hygroscopic crystals (NaI) as absorbers. An 80-mm diameter Si light detector based upon these design principles, with potential use in both dark matter and neutrinoless double-beta decay, has an estimated baseline energy resolution of 0.35 eV, 20 better than currently achievable. A 1.75 kg ZnMoO large-mass calorimeter would have a 3.5 eV baseline resolution, 1000 better than currently achieved with NTDs with an estimated position dependence of 610, near or below the variations found in absorber thermalization in ZnMoO  and TeO. Such minimal position dependence is made possible by forcing the sensor bandwidth to be much smaller than the signal bandwidth. Further, intrinsic event timing resolution is estimated to be 170 s for 3 MeV recoils in the phonon detector, satisfying the event-rate requirements of large next-generation neutrinoless double-beta decay experiments. Quiescent bias power for both of these designs is found to be significantly larger than parasitic power loads achieved in the SPICA/SAFARI infrared bolometers.

pacs:
95.55.Vj, 95.35.+d, 29.40.Vj, 29.40.Wk
preprint: APS/123-QED

I Motivation

The success of experiments that use massive very low-temperature calorimeters (e.g. CRESST Angloher et al. (2014), CUORE Arnaboldi and et al (2008), EDELWEISS Armengaud and et al (EDELWEISS-II) (2011) and CDMS Ahmed and et al (CDMS II) (2010)) in rare event searches is quite natural since phonon vibrational modes with energy above T freeze out and thus do not contribute to the crystal heat capacity, leading to heat capacity scaling for semi-conducting and insulating crystals. Consequently, detectors with a large active mass and excellent energy resolution should be possible. To reiterate the utility of this natural scaling law: for a given energy deposition, a giant 1-tonne Si crystal at 10 mK will have the same temperature change as 1 g at 1 K .

At low temperature (100 mK), calorimetry based on Transition Edge Sensor (TES) technology is quite mature and has been used for energy measurement at virtually all energy scales from infrared-photon (100 meV) Lita, Miller, and Nam (2008) to alpha detection (10 MeV) Horansky and et al (2008). The measured energy resolution of well designed devices throughout this regime roughly matches the theoretical expectation of

(1)

where is the superconducting transition temperature of the TES, () is the unitless sensitivity parameter, and the unitless terms within the square root all combine to usually be of order 2 Irwin and Hilton (2005).

Figure 1: The naively estimated intrinsic energy resolution (Eq. 1) as a function of for a variety of different massive calorimeters versus their actual device performance (stars) Angloher and et al (2009); Angloher et al. (2014); Sisti and et al (2001); CUO (2012); Armengaud and et al (EDELWEISS-II) (2011). The plotted 50 g Si coherent neutrino scattering resolution (black circle) is simply a more detailed performance estimate by Formaggio and Figueroa-Feliciano (2012), taking into account internal device thermal conductances and other non-ideal device characteristics, that shows rough agreement with the simpler scaling law estimates.

In Fig. 1, we benchmark the measured performance of the current generation of massive calorimeters operating near 10 mK (stars) and a detailed proposal for a 50 g Si coherent neutrino scattering detector Formaggio and Figueroa-Feliciano (2012) (black circle) against the theoretical scaling law of Eq. 1 assuming that the total heat capacity of the detector is 2 that of the absorber and that (values of 10–500 are common). Unfortunately we find that the energy-resolution performance of all current experiments is 2–3 orders of magnitude worse than expected from Eq. 1. Consequently, either there are substantial experimental limitations at very low temperature that are not taken into account in Eq. 1, and Formaggio and Figueroa-Feliciano (2012), or the current generation of massive calorimeters could be significantly improved.

To gain insight into this energy resolution discrepancy, we will carefully study both the dynamics and noise of the original Angloher and et al (2009) and composite Angloher et al. (2014) CRESST phonon detectors as well as other calorimeters and then motivate and develop 6 design criteria that are applicable to very low-temperature massive detectors:

  1. The signal bandwidth (frequency scale at which energy transfers from the absorber to the sensor) must be greater than or equal to sensor bandwidth,

  2. The transition edge sensor (TES) must not phase separate,

  3. The sensor bandwidth must be larger than the noise threshold,

  4. The sensor bandwidth must be large enough to satisfy event rate requirements, and

  5. Microfabrication techniques should be used only on standard thin wafers.

  6. For applications that require minimal position dependence of energy estimators, we require that that signal bandwidth than the sensor bandwidth.

Finally, we present 2 prototype designs that simultaneously achieve all of these design requirements: a single-photon-sensitive light detector and a 1.75 kg ZnMoO double-beta decay detector.

Ii Simulating CRESST Phonon Detector

CaWO absorber
Absorber volume x2x4cm Pantic (2008)
Absorber mass 300 g Pantic (2008)
Absorber heat capacity 132 @10mK
W TES
Cross sectional area 7.5mmx200nmPantic (2008)
Length 5.9 mm Pantic (2008)
Volume 8.9x10mm
Heat capacity 22.7 @10mK
Power flow from TES to absorber
Thermal conductance between TES and absorber 142 @10mK
internal TES conductance 1.63
TES normal resistance 300 Pantic (2008)
Transition temperature 10 mK
Au wirebond: thermal link to bath
Bath temperature 6 mK Angloher and et al (2005)
Wirebond length 2 cm Sisti and et al (2001)
Cross sectional area x 12.5m Pantic (2008)
Power flow from TES to bath 19.2 pW
Thermal conductance from TES to bath 7.5 nW/K @10mK
Electronics properties
Inductance (SQUID+ parasitic) 350 nH Henry et al. (2007)
Load resistor (shunt+ parasitic) 40 m Henry et al. (2007)
Temperature of 10mK Henry et al. (2007)
SQUID Current Noise 1.2 pA/Henry et al. (2007)
Table 1: Estimated CRESST-II phonon-detector device parameters using the material properties in Appendix A.
Figure 2: Schematic of the TES and its connections on the original CRESST-II phonon detector Pantic (2008).

Modeling of complex calorimeters with multiple coupled thermal degrees of freedom (DOF) like the CRESST phonon detector has been a very active area of research, particularly within the x-ray and gamma-ray TES communities Figueroa-Feliciano (2001, 2006); Smith et al. (2009); Zink et al. (2006). Thus, we need only apply mature strategies, being careful to follow the conventions of Irwin and Hilton (2005). This is made even easier by the fact that CRESST has already modeled the dynamics of their detectors F.Probst and et al (1995); Kiefer et al. (2009); Pantic (2008) and even attempted to simulate noise Pantic (2008); so we will concentrate in particular on developing analytical simplifications and physical intuition, neither of which has yet been done.

ii.1 Dynamics

The original (non-composite) CRESST-II phonon detector is a large 300 g cylindrical CaWO crystal with a W TES (T 10 mK) and all of its required accessories (bonding pads, Al bias rails) fabricated directly upon its surface, as shown in Fig. 2 Pantic (2008). Unfortunately, even after such painstaking fabrication efforts, the thermal coupling between the TES and the large mass absorber, , is still 50 smaller than the thermal conductance between the TES and the bath via the direct electronic coupling of the Au wirebond, (cf. Table. 1). This is because the W electronic system (sensor) and the phonon system of the TES and absorber almost completely decouple at very low temperatures since the thermal power flow scales as .

From a modeling perspective, the primary consequence is that any realistic device simulation requires separate thermal DOF for the absorber, , and the TES, , in addition to modeling the current flowing through the TES which will be measured through an inductively coupled SQUID. Thus in a way similar to CRESST F.Probst and et al (1995), we will model the system as 3 coupled non-linear differential equations

(2)

where is a small voltage-bias excitation on top of the DC voltage-bias that we will use to model Johnson noise as well as the small signal dynamics of the detector. Likewise, and are power excitations into the TES and absorber respectively. This model can also be seen diagrammatically in Fig. 3. Variable definitions and estimated sizes can be found in Table 1. One relatively unique feature of the CRESST design is the capability to directly heat the TES electronic system through an additional heater circuit (). When held constant, this is effectively equivalent to being able to easily vary the bath temperature on a detector by detector basis.

Figure 3: Simplified model of the CRESST phonon and light detectors with 2 thermal DOF as well as the voltage-bias and current readout circuit

Taylor expanding to first order around the operating point, and then Fourier transforming, these equations simplify to

(3)

which in the pertinent limit of can be simplified to only thermal DOF

(4)

where

(5)

As is standard, or characterizes the unwanted dependence of on the current. Also note that we have generalized Irwin’s loop gain parameter to

(6)

to account for the 2 thermal DOF as well as the fact that CRESST has purposely chosen to use a load resistor () that is of similar magnitude to to suppress electro-thermal feedback. Finally, due to the possibility that and could be macroscopically different, the thermal conductances and could have different values and consequently we define

(7)

Inversion of the generalized impedance matrix, , found in Eq. 4 leads to the current transfer functions for the thermal power response for direct heating into the TES

(8)

and into the absorber

(9)

where

(10)

All of the approximations shown are valid only for the pertinent limiting case where and (cf. Table 1).

Qualitatively, the last term of Eq. 9, which is an additional pole that surpresses the bandwidth of relative to , is due to the fact that thermal energy must be transported across the tiny before it is seen by the TES. Likewise, the middle term of Eq. 9 is a constant suppression factor because a portion of is shunted through . This suppression should be small for the CRESST phonon detector. However, for the CRESST light detector, the Au wirebond pad on the substrate has an electron-phonon coupling that is larger than that of the W TES, and consequently this factor is significant Pantic (2008). Another detector with significant shunting is found in CUORE where the vast majority of the absorber thermal signal flows through Alessandrello et al. (1992).

Both the (Eq. 8) and (Eq. 9) transfer functions are important for understanding the signal response of massive calorimeters, because some of the high-energy athermal phonons produced by a particle recoil in the absorber may be collected and thermalized within the TES () before they thermalize within the absorber (), since the electron-phonon coupling varies so significantly with phonon energy and temperature. Consequently, a true particle recoil within the absorber should be modeled within our 2 thermal DOF system as

(11)

where the benefit of being able to write in terms of plus additional factors is readily apparent.

ii.2 Noise Estimation

With the dynamical response of the detector now modeled, we can estimate the magnitude of noise from thermal power fluctuations across , , and , the Johnson noise across and and finally the first stage squid noise and compare their relative sizes by referencing them to a thermal power signal flowing directly into the TES (). This reference point was chosen purposely so that intuition from simpler 1 DOF thermal systems could be used and to suppress differences due to CRESST’s use of a non-standard electronics readout scheme with a large to suppress electro-thermal feedback.

With this choice of reference, thermal fluctuation noise (TFN) across is flat and can be estimated as

(12)

where is a noise suppression term with a value between \sfrac12 and 1 to account for the fact that the power noise across a thermal link between two different temperatures (a non-equilibrium situation) is less than the naively derived equilibrium noise McCammon (2005). Now, estimation of the TFN across is slightly more difficult since we must refer it to the TES and thus

(13)

This noise is subdominant, even below the pole since for both CRESST detectors (as well as for our new design).

Next, we estimate our sensitivity to thermal fluctuations between the electronic and phonon systems (i.e. across ). We must be cognizant that these fluctuations are anti-correlated to conserve energy; if thermal power randomly flows into the phonon system, it must be flowing out of the electronic system and vice-versa. Thus, our sensitivity is

(14)

The insensitivity to this noise below is a direct ramification of energy conservation and is a general feature of all “massless” internal thermal conductances. Of course, for the CRESST designs, this noise is again negligible compared to that from , simply due to their relative sizes.

TES Johnson noise is by far the most challenging to calculate for 3 reasons. One must take into account the anti-correlation between and Irwin and Hilton (2005); McCammon (2005) as well as the noise boost due to current sensitivity (non-zero ) Irwin and Hilton (2005). Finally, we must reference back to the TES input power. We do this in 2 steps. First, we calculate the Johnson noise referenced to current flowing through the TES ()

(15)

where we see that the anti-correlation leads to noise suppression at low frequencies for high . Referencing to , we obtain:

(16)

Again, we find that as long as is large and heating of the TES via external heating () or a bath temperature () near does not suppress too significantly, then the thermal fluctuations across dominate the noise at low frequency. However, this is clearly not true at high frequencies due to the zero at . In fact, it is this TES Johnson noise term that sets the bandwidth for optimal energy estimators of , which we can estimate by finding the frequency beyond which . This frequency, which we will designate as the signal-to-noise bandwidth, is given by:

(17)
Figure 4: Simulated noise spectrum for CRESST phonon detector referenced to TES power (top) and TES current (bottom). Dotted lines correspond to Taylor expansions in the limit of 0 and & and no noise.

For electronics designs with large electro-thermal feedback, . However, for CRESST where has purposely been set to near 0 by choosing , . To reiterate, when referencing both the signal and noise to current (Fig. 4 bottom), both the current signal and the total current noise have a pole at , and thus the signal-to-noise (which is equivalent to rereferencing to TES power) remains flat all the way up to .

Unfortunately, neither the squid noise

(18)

nor the Johnson noise from the load resistor, R,

(19)

can be trivially written in terms of . However, as shown in Fig. 4 they are subdominant to the combination of TFN and Johnson noise (this latter can also be seen through comparison of Eq. 19 to Eq. 15).

Finally, we would ideally like to have an estimate of the noise found in the current CRESST experimental setup, particularly since there are some indications that it could be an important contributor to the overall noise in CRESST currently Pantic (2008). More importantly, we would like to estimate the 1/ noise expected in a next-generation experimental setup to assess its effect on the device design. For the former, significant effort would be required that is beyond the scope of this paper. For the latter, we will use the noise performance of the SPIDER TES bolometers, which begin to be dominated by at 0.1 Hz, as guide to what could be achieved with effort, since CMB experiments are quite sensitive to low frequency noise Bonetti et al. (2009) and thus we will set a design goal of 0.5 hz.

Iii TES Phase Separation

Unfortunately, as first realized in the earliest days of CRESST calorimeter development F.Probst and et al (1995), the simple 2 DOF block thermal model shown in Fig. 2 that we have used for our CRESST-II phonon-detector resolution estimates is not entirely accurate, since is also significantly larger than the internal thermal conductance of the W TES, (cf. Table. 1). In this situation, the difference in temperature between the TES and the effective bath temperature is mostly internal to the TES, and thus there is a significant temperature gradient across the sensor. In devices with large , this thermal gradient is further exasperated by positive-feedback effects of joule heating and the TES can separate into sharply defined superconducting and normal regions, with only a very thin region within the transition that is sensitive to temperature fluctuations Cabrera (2000).

Although a computational simulation of phase separation within the CRESST geometry is beyond the scope of this paper (and of dubious value without a very careful matching to experimental data since the dynamics depend on the resistivity of the entire transition, rather than at a single operation point), we can qualitatively discuss the ramifications.

Most importantly, a phase separated TES has significantly suppressed thermal sensitivity when biased. To see this, we note that in the limit of (i.e. the super conducting transition becoming infinitely sharp and infinitely sensitive to temperature variation), the DC response of the TES can be modeled analytically by tracking the fractional location of the superconducting/normal transition in the TES, , as a function of and Luukanen and et al (2003). For the pertinent case where the electron-phonon coupling along the TES is negligible compared to power flow through the connection to the bath at one end of the TES, the thermal power flowing across the superconducting portion of the TES is constant and can be matched to the joule heating in the normal portion of the TES:

(20)

Using this simplification, we obtain curves of versus for a CRESST-like device for several values of , as shown in Fig. 5.

Figure 5: Simulated vs curves for several values of for a phase-separated TES with an infinitely sharp transition curve (at  mK) that shows DC sensitivity suppression when biased.

If not for phase separation, all of these curves would be infinitely sharp step functions centered at . With separation though, we observe significant degradation in device sensitivity that worsens with increasing . Basically, a phase separated TES acts as if it has a very large from a DC perspective. This exact behavior is seen in CRESST devices Sisti and et al (2001); Pantic (2008).

Naively, one would think that this suppression in signal amplitude would severely affect phonon power resolution at all frequency scales. However, this is not the case. At low frequencies, thermal fluctuations across completely dominate the noise in a non-phase-separated TES (Fig. 4 top), and thus both signal and noise are suppressed equally. Whereas at high frequencies, the total noise is dominated by Johnson noise across the TES and thus the qualitative net effect is a suppression in .

Further, energy that is absorbed by the TES electron system in portions of the TES that are either fully normal or superconducting must first diffuse to the transition region before producing a measurable change in current, which adds an additional strongly expressed diffusive pole into all of the TES transfer functions. In a phase separated CDMS-II TES device for example, measured has 2 fall-time poles (plus the pole) with roughly similar weighting Pyle (2012). Beyond simply adding confusion when trying to understand and model device performance, the most important consequence is to again suppress .

Finally, due to such large variations in the derivative of the resistivity () across the sharp superconducting/normal boundary, thermal power fluctuations within the TES directly couple to the total TES resistance , and consequently the measured signal current , giving us an additional noise source with fluctuations on all length scales Luukanen and et al (2003). Interestingly enough, in discretized simulations of phase separated TES for CDMS-II Pyle (2012), we have found that this additional noise source is largely counter balanced by a suppression in sensitivity to standard TFN noise across at frequencies below the longest diffusive pole, and thus primarily sees increases at higher frequencies.

Iv Bandwidth Mismatch between Signal and Sensor

Before attempting to estimate the sensitivity of CRESST detectors to recoils within the absorber that have both thermal and athermal phonon components, it makes sense to calculate the expected sensitivities of the CRESST detector to the two limiting cases of a Dirac delta thermal energy deposition into the absorber and the TES. These estimates are shown in Table 2.

: Eq. 1 1.0 eV
: 2D Simulated 34 eV (no 1/f) / 44 eV (1/f)
: 2D Simulated 0.5 eV (no 1/f) / 0.5 eV (1/f)
Measured: “Julia” 420 eV Kiefer et al. (2009))
Measured: Composite 107 eV Angloher et al. (2014))
Table 2: Simulated and measured CRESST phonon-detector energy resolutions. Note that the simulation does not include the effect of TES phase separation.

What is immediately striking is the significantly degraded expected sensitivity for thermal energy deposition in the absorber () compared to that for direct absorption in the TES (). To understand the cause of this suppression factor, we can write in terms of under the now well motivated assumption that the noise, when referenced to , should be flat below (since it is dominated by the naturally flat ):

(21)

Basically, any signal whose bandwidth is smaller than will suboptimally use the sensor bandwidth, leading to poorer resolution as discussed in Sadoulet (1996). To see this in another way, notice that both (Eq. 17) and (which is roughly as in Eq.12) both scale linearly with . Thus, as long as the bandwidth of the signal that is being measured is larger than , the baseline energy sensitivity will be independent of the size of ; increasing increases the noise floor, but this effect is balanced by an increase in sensor bandwidth. By contrast, when measuring signals with bandwidth below one has accepted a noise penalty but the gained bandwidth is useless. In summary, because , the CRESST phonon detector has relatively poor sensitivity to phonon energy that is thermalized within the absorber ().

Figure 6: Signal pulse shapes for -induced events in the CRESST composite detector “Rita” for events that interact in the TES chip (black) and in the absorber attached via an epoxy joint (blue). Clearly, absorber events have significantly suppressed bandwidth and consequently suppressed resolution, as in Eq. 21. Data taken from Kiefer et al. (2009).

The athermal phonon signal that is thermalized in the TES bypasses the restriction and therefore its bandwidth is not limited by . However, it still takes time to collect all of the ballistic athermal phonons rattling around in an absorber. In the SuperCDMS athermal phonon iZIP detector for example, this athermal-phonon collection bandwidth = 210 Hz while = 4 kHz for the 90 mK W TES that CDMS has historically usedPyle (2012). Thus, they pay a resolution penalty of 5 due to the bandwidth mismatch between their athermal phonon signal and their sensor bandwidth, completely analogous to the thermalized sensitivity suppression.

Other examples of sensitivity suppression due to poor signal /sensor bandwidth matching can be found in the new CRESST composite detectors (Fig. 6) Kiefer et al. (2009) as well as the AMORE MMC based calorimeters Kim et al. (2014). In the CRESST composite detector, the athermal phonon signal from an event in the absorber crystal must first be transmitted across an epoxy glue joint between the absorber crystal and the substrate on which the TES is fabricated, yielding 7 hz (fall time 23 ms). If we associate the fall-time pole of the phonon signals for events hitting the TES chip substrate ( 50 Hz) with and we further recognize that for CRESST detectors (since they purposely bias to minimize electro-thermal feedback), then we can deduce that the new CRESST composite has a signal/sensor bandwidth suppression factor that is 10. In fact, using our 2D simulation, we estimate that this bandwidth-limited energy impulse into the TES, , would have a resolution of 5 eV, 10 worse than the Dirac-Delta sensitivity.

In summary, our first and most important very low-temperature detector design objective is that the signal bandwidth must be larger than the sensor bandwidth, a design goal that, to our knowledge, has not been accomplished by any very low-temperature large-mass detector but that is standardly implemented in 100 mK TES calorimeters for x-ray Saab et al. (2006) and Zink et al. (2006) applications. Please note that there is some subtlety in this design rule. CRESST detectors use pulse-shape differences between absorber and TES events to suppress TES chip backgrounds (Fig. 6); they have both an energy and a discrimination signal with different bandwidths. Thus, the optimal strategy is likely to choose so that the device discrimination threshold is equal to its energy threshold.

Secondly, if we are able to design very low-temperature calorimeters that use the entire unsuppressed sensor bandwidth (design rule 1), we will become directly sensitive to TES phase separation. Consequently we impose a second design constraint that . As an added benefit, we think device operation and analysis will be significantly less complex, because we also naively expect that phase-separated TESs are highly sensitivity to position-dependent variations in and other thin-film properties that are likely culprits for at least some of the non-linearities that have been found when biasing CRESST devices Pantic (2008).

V Expected CRESST Energy Resolution & Athermal Phonon Collection Efficiency

We have shown that the infinite bandwidth in Table 2 significantly overestimates the athermal phonon energy sensitivity, since it does not account for the athermal phonon collection bandwidth, . Additionally, we must account for the fact that only a fraction of the athermal phonons are collected in the TES, and thus a particle recoil in the absorber should be modeled as

(22)

For an upper bound on , we note that optical photons directly incident on a W TES have been measured to deposit 42% of their total phonon energy into the W electronic system Burney et al. (2006). Another guide to the size of comes from the SuperCDMS iZIP athermal phonon detector, which measures a total athermal phonon collection that is 10–20% of the total deposited energy Pyle (2012). Using a reasonable (but certainly not measured) value of , we estimate a = 15 eV. Although this is 7 better than the best achieved resolution in a CRESST composite detector (107 eV), it is also 30 worse than the naive expectation of 0.5 eV, and thus we believe that sensor/signal bandwidth mismatch and non-unity athermal phonon collection efficiency are the dominant sensitivity degradation mechanisms.

Vi Parasitic Power Loading

Figure 7: Environmental noise pulses (red and black) illustrate that SuperCDMS detectors are strongly susceptible to direct TES heat by parasitic power noise.

Both design drivers so far discussed suggest that to become more sensitive an optimized very low-temperature calorimeter should have lower than found in the CRESST phonon-detector design. One negative consequence of this change is greater sensitivity to parasitic heating. This is clearly a concern for SuperCDMS because their devices produce TES heating signals due to cell phone usage in the laboratory (fortunately with a very distinct pulse shape from actual events, as shown in Fig. 7), and thus the old CDMS-II electronics do not adequately shield their TESs from high-frequency environmental noise. Although this is currently only an interesting nuisance, it highlights the parasitic power problem that SuperCDMS faces. As they lower G in their own devices, this and other parasitic power sources heat the detector to a greater and greater degree, eventually driving the TES normal and rendering it inoperable. Lower devices will require better environmental noise shielding.

Thus, it is reasonable to question if DC parasitic power noise necessitates the use of large in CRESST phonon detectors, thereby breaking our first 2 design rules and lowering detector sensitivity. Further, if so, are all very low-temperature calorimeters similarly limited? We do not believe this to be this case for 2 reasons. First, in the very same cryostat and with identical electronics, CRESST also operates light detectors that are Si or SOS wafers instrumented with the TES design shown in Fig. 8. Notice that for these detectors, the TES is not directly connected to the bath through an Au wirebond. Instead the coupling is through a fabricated Au thin film impedance with an estimated = 21.1 pW/K for a = 10mK which is 350 smaller than the of the Au wirebond used on their phonon detectors. Consequently, if DC parasitic power noise was even remotely problematic for the CRESST phonon detector, their light detector would be inoperable.

Figure 8: TES Sensor Geometry for CRESST-II light detector Pantic (2008)

Of course, when trying to assess the validity of lowering during the optimization of TES-based light detectors (something we also discuss due to applicability for both dark matter and double-beta decay experiments), the above explicitly pertinent observation is insufficient. Fortunately, however, there has recently been enormous emphasis on designing TES-based sensors for space-born infrared spectrometry by SPICA/SAFARI, and they have been able to achieve parasitic power loads of 2 fW, 25 less than the estimated bias power, , for the current CRESST light detector and 10 less than their phonon detector Goldie et al. (2011). Consequently, achieving parasitic power loads that are much less than for low G devices may be difficult but should be possible.

Vii Limits on Sensor Bandwidth due to Event Rate

Unlike in 100 mK x-ray, and calorimeters, the extremely low background and signal rates expected in underground rare-event and exotic-decay searches have meant that pileup (pulses from distinct interactions overlapping due to high rates and finite bandwidth) rejection and the ability to handle large event rates have not been primary design drivers.

In the future, however, this requirement will become much more constraining for some applications. For example, the most daunting pileup requirements come from reactor-sourced coherent neutrino scattering experiments because the cosmic background is quite large due to minimal rock overburden. As a first very rough estimate of the necessary start-time sensitivity, we note that the original CDMS shallow site experiment had a similar overburden and used an anti-coincidence window of 25 s for activity in their plastic-scintillator muon veto Abrams et al. (2002). Since the optimally estimated start-time resolution (assuming no pulse-shape dependence) is

(23)

which can be simplified to

(24)

when flat noise is assumed and is the lowest pole of the design, we estimate that a reactor-sourced coherent neutrino scattering experiment needs an of 1.3 khz to have the requisite start-time resolution for near-threshold events (). Unfortunately, such a large is simply inconsistent with our other design requirements for thermal calorimeters. Consequently, we believe that this application is better suited to an athermal-only phonon detector design, as typified by a SuperCDMS detector.

Pileup rejection has also become one of the dominant design drivers for LUCIFER and other high-Q neutrinoless double-beta decay experiments because they expect the background in their signal region to be dominated by un-vetoed pileup of lower energy 2 double-beta decay events for which pileup rejection is assumed to be ineffectual below 5 ms CUO (2012). Luckily, with such a large signal, start-time resolutions of this order are achievable (see Sec. X) and thus, if required, this is a reasonable design requirement.

Viii Ease of Fabrication & Cosmogenic Background Suppression: Separated TES Chip

CUORE and EDELWEISS have continued to use NTD sensor technology, despite the many benefits offered by TES readout, including:

  1. A TES is fundamentally more sensitive than a NTD (larger );

  2. The SQUIDs used in TES readout have lower noise than JFETs or HEMTs used for first stage amplification for NTDs;

  3. Low-impedance sensors (TESs) are fundamentally less sensitive to vibrationally induced capacitance changes in readout compared to high-impedance sensors (NTDs); and

  4. SQUIDs have significantly lower heat loads than the JFETs or HEMTs used in NTD readout, and can therefore be placed significantly closer to the detector, simplifying electronics and cryostat design.

One reason for this is that both TES-based massive-detector groups (CRESST and CDMS) fabricated their TESs directly upon the absorber surface, a feat that required enormous fabrication process R&D since every facet of standard microprocessor fabrication (photolithography, etching, thin film deposition) had to be retrofitted for thick and massive substrates. In fact, even with over a decade of R&D, fabrication yields were still a significant resource drain until recently on CDMS. Further, the time and labor intensive nature of microprocessing fabrication means that absorbers spend a significant amount of time on the surface being cosmogenically activated, certainly a disadvantage for low mass WIMP searches, for example. This direct absorber fabrication is required since the W TES in CRESST has 2 distinct functions. First, it is a temperature sensor. This is the functionality that requires microprocessor fabrication techniques on multiple different thin film layers: Al for the superconducting bias rails, W for the TES, and Au for the thermal connection to bath. Second, the electron-phonon coupling within the W film acts as . Because it is only this latter functionality that requires fabrication directly upon the absorber, design goals of fabrication simplicity and minimum cosmogenic exposure require that the TES does not act as the thermal link between the sensor and the absorber.

Figure 9: Optimized large-mass calorimeter sensor design. Only the large Au pad is directly fabricated on the large absorber.

As illustrated in Fig. 9, we propose to deposit a large, single layer Au thin-film pad directly onto the large absorber substrates that plays the role of . This can be done using only shadow mask techniques (albeit with a depostion machine modified for thick substrates) and consequently fabrication should have very high yield and be relatively hassle free since there is no photolithography and etching. As an added benefit (in fact, perhaps the most important benefit), this permits use of any metal rather than being constrained to W; we choose Au which has an order of magnitude larger electron-phonon coupling than W for a given thermal capacitance.

The fabrication intensive TES can then be separately fabricated on standard thin substrates for fabrication ease, where the material chosen is not necessarily identical to that of the large absorber (Si, Ge, AlO, CaWO). Further, each and every 100 mm wafer can produce over 20 devices. Thus, device fabrication throughput could easily be 80 that of CDMS (in the standard SuperCDMS fabrication procedure, 4 fully processed test wafers are produced for every detector). This physical separation also allows for testing of the TES sensor die above ground before connection to the absorber. This has significant advantages: the cost savings of sensor testing above ground rather than in an underground laboratory is substantial; and one can choose only the best sensors to match with expensive absorbers, particularly useful in the case of double-beta decay enriched crystals.

Thermal connection between the TES and the G absorber pad is then accomplished via Au wire bonding, while the simpler mechanical connection can be accomplished with epoxy in an arbitrary (and somewhat haphazard) manner; without any thermal-conductance requirements, very small-area single dot epoxy joints are possible that do not mechanically stress the absorber Kiefer et al. (2009) ). Another possibility is that the TES chips are mechanically supported by the detector housing. Of course, the heat capacity of the Au wirebond between the TES and the absorber (1 pJ/K at 10 mK) is entirely parasitic and is simply the price paid for the ease of fabrication. Most importantly, the thermal conductance of both the internal pad, , and that of the Au wirebond, , must be much larger than so as to satisfy the bandwidth design rules discussed in Sec. IV.

It is interesting to reiterate how similar and yet how distinct this design is from the CRESST composite detector Kiefer et al. (2009). On the one hand, this design is clearly derivative. As with the CRESST composite detector, the TES is fabricated on a separate and much thinner substrate that drastically simplifies fabrication (and in the CRESST case, improves scintillation yield in the primary absorber). On the other hand, the difference is profound. In the CRESST design, energy transport between the 2 systems is accomplished via phonon transport through an epoxy joint that significantly suppresses both athermal and thermal signal bandwidths (making our bandwidth design objectives very difficult to achieve). Further, low-impedance phonon transport via epoxy couplings requires large cross-sectional areas that make the detector much more prone to crack via differential thermal contraction among the epoxy, the absorber substrate, and the TES chip substrate. The CRESST composite detector mitigates this problem somewhat by using CaWO for both the TES substrate and the absorber. However, this limits possible detector materials. NaI, for example, is an attractive material to search for spin or orbital angular momentum coupling dark matter Anand, Fitzpatrick, and Haxton (2014). Furthermore, scintillating NaI calorimeters would directly test the anomalous annual modulation signal observed in DAMA Nadeau et al. (2014). Unfortunately though, NaI is very hygroscopic and thus any complex photolithography is impossible.

In summary, this design should retain all the benefits of TES performance and yet also have the fabrication simplicity, cosmogenic benefits, and material freedom of NTD based detectors.

This device design should also be critically compared to the AMORE MMC calorimeter design Kim et al. (2014). Removing the trivial difference of the use of an MMC instead of a TES sensor element, all pertinent design rules that we have developed were followed with the single and very important exception that their sensor bandwidth is much larger than their signal bandwidth (). This choice suppresses their zero energy sensitivity but even more importantly drastically increases their sensitivity to position dependence (Secs. IX and X), a fact that severely limits the viability of their current devices.

Ix Position Dependence Requirements in Neutrinoless Double Beta Decay Searches

Because of the approximately exponential shape of the WIMP nuclear-recoil energy spectrum, a slight position or temporal systematic of 5% on the recoil energy estimate in a CRESST phonon detector does not significantly affect their sensitivity. By contrast, a neutrinoless double-beta decay experiment like CUORE/LUCIFER CUO (2012) requires the maximum achievable energy resolution at 3 MeV, and thus any variation in the pulse shape or magnitude of the signal due to event location in the absorber must be minimized in addition to having excellent baseline energy sensitivity. Specifically, a well designed double beta detector should be limited by systematics in the absorber thermalization process, with measured to be 510 in TeO Bellini, Biassoni, and et al (2010)) and 2x10 in ZnMoO CUO (2012).

Figure 10: NIST calorimeter with = 21 eV at 103 keV using a TES with T 100 mK. Zink et al. (2006)

To guide us in developing design rules to minimize position dependence, let us look more carefully at the 100 mK TES calorimeter shown in Fig. 10 that achieved a 510Zink et al. (2006). First and foremost, the only thermal and structural link between the absorber and the bath goes through the TES (=0). To illustrate the importance of this for suppressing position dependence, consider the design in Formaggio and Figueroa-Feliciano (2012) where the primary thermal path from the TES to the bath is through the absorber (, ). In this case, athermal phonons produced in the initial interaction will preferentially thermalize in either the TES () or in the metal pad thermally linked to the bath (), depending on the event’s location in the absorber and leading to significant unwanted position dependence in .

Unfortunately, free designs are significantly more difficult to achieve when the absorber weighs rather than , and thus this will be an aspirational goal rather than a requirement. We will solely require that , and that any be diffusive rather than ballistic so that non-thermalized excitations would find it difficult to escape through .

Secondly, the size of the diffusive-phonon thermal link between the absorber and the TES in this calorimeter was chosen to be much slower than the thermalization/position-dependent time scale, . This choice means that unwanted position information with frequencies around have 2 pole suppression over many decades in their current response.

Unfortunately, thermalization is very slow in the insulator and semiconductor absorbers used in massive calorimeters compared to the superconductors and metal absorbers used in and x-ray calorimeters. SuperCDMS iZIP detectors, for example, see variations in athermal phonon power absorbed by different channels on the same crystal up to 300 s after an event interaction. Such long position-dependent thermalization times put downward pressure on both and when using this bandwidth design scheme. CUORE actually attempts to do exactly this in their NTD based calorimeter by using a small dot of epoxy as , with the hope of suppressing any non-thermalized position-dependent phonon signal from reaching their TES. Unfortunately, this choice in combination with poor electron-phonon coupling within the NTD means that , and thus most of the absorbed energy is not measured by the NTD but directly shunted to the bath, suppressing baseline energy sensitivity and increasing DC susceptibility to position dependence Alessandrello et al. (1992).

In our proposed calorimeter design, the above bandwidth design scheme is very difficult to achieve because the dominant phonon thermalization mechanism is through electron-phonon coupling within the metal pad that acts as ; (Eq. 11) will always be non-zero. Thus, we must be content with single-pole suppression of position dependence. On the otherhand, since we now control the rate of athermal phonon thermalization (), we can maximize this single-pole suppression by making the pad as large as possible (i.e. set the design requirement that ). As an aside, note that suppression of position dependence via pole suppression is incompatible with the pulse-shape discrimination between TES chip events and absorber events that CRESST enjoys because it requires ( \sfrac12 ).

As a rough but conservative estimate of the position-dependent systematics in both the energy and start-time estimators, we assume that the thermal power signals into the TES and absorber can be modeled as

(25)

where we vary from 0–10%, using SuperCDMS position dependence as a rough guide (we have pushed to maximally accentuate the position dependencies).

X Design Sketch of 1.75kg ZnMoO Detector for Neutrinoless Double Beta Decay

In Table. 3 we flesh out the specifications and simulated performance for a 1.75 kg ZnMoO device that follows all of the design rules so far discussed. Such a massive absorber (or the addition of parasitic heat capacitance as was done in Horansky and et al (2008)) is required to keep within the dynamic range of the TES. We recognize that the use of such large crystals increases the need for pileup rejection (i.e. timing requirements) CUO (2012).

ZnMoO absorber
Volume (40mm)x80mm
Operational temperature 9.33 mK
Heat capacity 191
Au thin film thermal link between TES and bath
Length 900 m
Cross sectional area 10 m x 300 nm
Thermal conductance from TES to bath 286
Au thermalization film on absorber (p1)
Volume 4(9mm)x300nm
p1 component of TES heat capacity 187
Thermal conductance between TES and absorber 37
Au wirebond between TES and absorber
Length 1.0 cm
Cross sectional area 4(7.5m)
Thermal conductance 21.6
Bond component of TES heat capacity 4.6
W TES
Cross sectional area 3.5 mmx150 nm
Length 700 m
Transition temperature 10 mK
Transition width (T-T) 2 mK
Operating temperature 9.33 mK
Operating current 5.4 A
W component of TES heat capacity 0.96
Internal TES conductance 4.9
Normal resistance 100
Operating point R/5 20
Thermal sensitivity 16.4
Current sensitivity 0.05
Dynamical time constants
Squid inductor time constant 15 s
sensor fall time 365 ms
Absorber/TES coupling time constant 2.6 ms
Estimated resolution and saturation energy
Estimated baseline energy resolution 3.54 eV
Estimated timing resolution @ 3 MeV 170 s
Position dependence 6x10
Absorber saturation energy 4.1 MeV
Table 3: Optimized 1.75kg ZnMoO detector for next-generation double-beta decay experiments

The Au pad volume on the absorber was chosen to be quite large so that . Notice that even with such a relatively large volume, the parasitic heat capacity of the pad is only 80% that of the ZnMoO. The thickness of the pad at 300 nm was a compromise. On the one hand we wanted to cover the the largest possible surface area, so as to maximize . On the otherhand, we want the internal thermal conduction of the pad . Since the parasitic heat capacity of an individual Au wire bond is also relatively small (1.2 pJ/K) compared to the crystal, we used 4 and assumed an extra long 200 m bond tail at both ends for greater thermal conductance.

For the TES itself, we chose a W TES with an =100 m that we would run low in the transition (20% R) to maximize the dynamic range. Further, we would use R= 3 m to be in the strong electro-thermal limit. This aspect ratio was chosen so that and could potentially be further lowered. Since the heat capacity of the TES is subdominant compared to both the crystal and the pad, the sole constraints on its size are related to sensor performance related. Since increases with current density (which scales as V, where V is the TES volume), we will choose the overall TES size such that the current density is 1/2 that found in the current CRESST detectors. The use of low-resistivity Ir/Au or Mo/Au bilayers F.Probst and et al (1995); Damayanthi et al. (2006); Smith et al. (2014) is also certainly possible and would suppress TES inhomogeneities but with the potential for larger .

Finally, , is a thin Au film impedance similar to that used by CRESST in their original light detectors Pantic (2008). Its size is set so that is as small as allowed by 1/f noise constraints.

Figure 11: Simulated noise referenced to TES power (top) and current (bottom) for a 1.74kg ZnMoO detector.

The simulated baseline energy resolution of 3.54 eV is clearly sufficient for double beta decay (Simulated Noise PSDs in Fig. 11). For a rough estimate of the position sensitivity, we have simulated position-dependent pulse shapes for = 0, 5, and 10% (Eq. 25, Fig. 12), and using the standard optimum sensitivity estimator we estimate a of 610 which is just adequate for current purity TeO and ZnMoO. Further, SuperCDMS has been able to suppress unwanted position dependence in their phonon energy estimators by an additional factor of 2–4 using non-stationary optimum filters or multiple template optimum filters. Techniques such as these could be even more viable for double-beta decay detectors since the residual dependence in SuperCDMS is due largely to variations in Luke phonon production due to ionized e trapping Pyle (2012).

Figure 12: Simulated pulse shapes for events from different locations in the absorber for a 1.74 kg ZnMoO