Optimized Designs for Very Low Temperature Massive Calorimeters

Matt Pyle,1, ∗ Enectali Figueroa-Feliciano,2 and Bernard Sadoulet1

1Physics Department, University of California Berkeley2Physics Department, Massachusetts Institute of Technology

(Dated: March 13, 2015)

The baseline energy-resolution performance for the current generation of large-mass, low-temperature calorimeters (utilizing TES and NTD sensor technologies) is > 2 orders of magnitudeworse than theoretical predictions. A detailed study of several calorimetric detectors suggests thata mismatch between the sensor and signal bandwidths is the primary reason for suppressed sensitiv-ity. With this understanding, we propose a detector design in which a thin-film Au pad is directlydeposited onto a massive absorber that is then thermally linked to a separately fabricated TES chipvia an Au wirebond, providing large electron-phonon coupling (i.e. high signal bandwidth), easeof fabrication, and cosmogenic background suppression. Interestingly, this design strategy is fullycompatible with the use of hygroscopic crystals (NaI) as absorbers. An 80-mm diameter Si lightdetector based upon these design principles, with potential use in both dark matter and neutri-noless double-beta decay, has an estimated baseline energy resolution of 0.35 eV, 20× better thancurrently achievable. A 1.75 kg ZnMoO4 large-mass calorimeter would have a 3.5 eV baseline res-olution, 1000× better than currently achieved with NTDs with an estimated position dependence∆EE

of 6×10−4, near or below the variations found in absorber thermalization in ZnMoO4 andTeO2. Such minimal position dependence is made possible by forcing the sensor bandwidth to bemuch smaller than the signal bandwidth. Further, intrinsic event timing resolution is estimatedto be ∼170 µs for 3 MeV recoils in the phonon detector, satisfying the event-rate requirements oflarge Qββ next-generation neutrinoless double-beta decay experiments. Quiescent bias power forboth of these designs is found to be significantly larger than parasitic power loads achieved in theSPICA/SAFARI infrared bolometers.

PACS numbers: 95.55.Vj, 95.35.+d, 29.40.Vj, 29.40.Wk

I. MOTIVATION

The success of experiments that use massive very low-temperature calorimeters (e.g. CRESST [1], CUORE [2],EDELWEISS [3] and CDMS [4]) in rare event searchesis quite natural since phonon vibrational modes with en-ergy above kbT freeze out and thus do not contributeto the crystal heat capacity, leading to T 3 heat capac-ity scaling for semi-conducting and insulating crystals.Consequently, detectors with a large active mass and ex-cellent energy resolution should be possible. To reiteratethe utility of this natural scaling law: for a given energydeposition, a giant 1-tonne Si crystal at 10 mK will havethe same temperature change as 1 g at 1 K .

At low temperature (∼100 mK), calorimetry based onTransition Edge Sensor (TES) technology is quite matureand has been used for energy measurement at virtuallyall energy scales from infrared-photon (∼100 meV) [5]to alpha detection (∼10 MeV) [6]. The measured energyresolution of well designed devices throughout this regimeroughly matches the theoretical expectation of

σ2E =

4kbT2c C

α

√√√√nF(Tc, Tb)ξ(I)

1− TnbTnc

(1)

where Tc is the superconducting transition temperature

∗ mpyle1@berkeley.edu

of the TES, α (TR∂R∂T ) is the unitless sensitivity parameter,

and the unitless terms within the square root all combineto usually be of order 2 [7].

Tc (mK)101 20 30 40 50

< pt [e

V]

10-2

10-1

100

101

102

103

104 Estimated Energy Resolution for Ideal CalorimetersCRESST Si/SOS LightCRESST 300g CaWO4CRESST 262g Al2O3Lucifer 330g ZnMoO4Edelweiss 400g Ge50g Si CNS (proposed)

FIG. 1. The naively estimated intrinsic energy resolution(Eq. 1) as a function of T for a variety of different massivecalorimeters versus their actual device performance (stars)[1, 3, 8–10]. The plotted 50 g Si coherent neutrino scatteringresolution (black circle) is simply a more detailed performanceestimate by [11], taking into account internal device thermalconductances and other non-ideal device characteristics, thatshows rough agreement with the simpler scaling law estimates.

In Fig. 1, we benchmark the measured performanceof the current generation of massive calorimeters oper-

arX

iv:1

503.

0120

0v2

[as

tro-

ph.I

M]

12

Mar

201

5

2

ating near 10 mK (stars) and a detailed proposal for a50 g Si coherent neutrino scattering detector [11] (blackcircle) against the theoretical scaling law of Eq. 1 assum-ing that the total heat capacity of the detector is 2×that of the absorber and that α = 20 (values of 10–500are common). Unfortunately we find that the energy-resolution performance of all current experiments is 2–3orders of magnitude worse than expected from Eq. 1.Consequently, either there are substantial experimentallimitations at very low temperature that are not takeninto account in Eq. 1, and [11], or the current generationof massive calorimeters could be significantly improved.

To gain insight into this energy resolution discrepancy,we will carefully study both the dynamics and noise ofthe original [8] and composite [1] CRESST phonon de-tectors as well as other calorimeters and then motivateand develop 6 design criteria that are applicable to verylow-temperature massive detectors:

1. The signal bandwidth (frequency scale at which en-ergy transfers from the absorber to the sensor) mustbe greater than or equal to sensor bandwidth,

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

3. The sensor bandwidth must be larger than the 1/fnoise threshold,

4. The sensor bandwidth must be large enough to sat-isfy event rate requirements, and

5. Microfabrication techniques should be used only onstandard thin wafers.

6. For applications that require minimal position de-pendence of energy estimators, we require that thatsignal bandwidth than the sensor bandwidth.

Finally, we present 2 prototype designs that simultane-ously achieve all of these design requirements: a single-photon-sensitive light detector and a 1.75 kg ZnMoO4

double-beta decay detector.

II. SIMULATING CRESST PHONONDETECTOR

Modeling of complex calorimeters with multiple cou-pled thermal degrees of freedom (DOF) like the CRESSTphonon detector has been a very active area of research,particularly within the x-ray and gamma-ray TES com-munities [15–18]. Thus, we need only apply maturestrategies, being careful to follow the conventions of [7].This is made even easier by the fact that CRESST has al-ready modeled the dynamics of their detectors [12, 19, 20]and even attempted to simulate noise [12]; so we willconcentrate in particular on developing analytical sim-plifications and physical intuition, neither of which hasyet been done.

CaWO4 absorberVa Absorber volume πx22x4cm3 [12]Ma Absorber mass 300 g [12]

CaAbsorber heatcapacity

ΓCaWO4VaT

3 132 pJK

@10mK

W TESAt Cross sectional area 7.5mmx200nm[12]lt Length 5.9 mm [12]Vt Volume Atlt 8.9x10−3mm3

Ct Heat capacity fscΓWVtT 22.7 pJK

@10mK

PtaPower flow fromTES to absorber

ΣepWVt(Tnt −Tna )

Gta

Thermal conduc-tance between TESand absorber

∂Pta∂Tt

142 pWK

@10mK

Gt intinternal TESconductance

LwfρW

Atlt/2

T 1.63 nWK

RnTES normalresistance

ρWltAt

300 mΩ [12]

TcTransitiontemperature

∼ 10 mK

Au wirebond: thermal link to bathTb Bath temperature 6 mK [13]ltb ∼ Wirebond length 2 cm [9]Atb Cross sectional area π x 12.52 µm2 [12]

PtbPower flow fromTES to bath

vfdeΓAtb6ltb

(T 2t − T 2

b

)19.2 pW

Gtb

Thermal conduc-tance from TES tobath

∂Ptb∂Tt

7.5 nW/K@10mK

Electronics properties

LInductance(SQUID+parasitic)

350 nH [14]

RlLoad resistor(shunt+ parasitic)

40 mΩ [14]

Tl Temperature of Rl 10mK [14]

SISQUIDSQUID CurrentNoise

1.2 pA/√

hz[14]

TABLE I. Estimated CRESST-II phonon-detector device pa-rameters using the material properties in Appendix A.

A. Dynamics

The original (non-composite) CRESST-II phonon de-tector is a large 300 g cylindrical CaWO4 crystal with aW TES (Tc ≈ 10 mK) and all of its required accessories(bonding pads, Al bias rails) fabricated directly upon itssurface, as shown in Fig. 2 [12]. Unfortunately, even aftersuch painstaking fabrication efforts, the thermal couplingbetween the TES and the large mass absorber, Gta, is still50× smaller than the thermal conductance between theTES and the bath via the direct electronic coupling of theAu wirebond, Gtb (cf. Table. I). This is because the Welectronic system (sensor) and the phonon system of theTES and absorber almost completely decouple at verylow temperatures since the thermal power flow scales asT 5.

From a modeling perspective, the primary consequence

3

Absorber

47

7.5mm

WAlAu

Thermallink

Heater

Thermometer5.9m

m

Cu-Kapton-CuHeat Sink

FIG. 2. Schematic of the TES and its connections on theoriginal CRESST-II phonon detector [12].

is that any realistic device simulation requires separatethermal DOF for the absorber, Ta, and the TES, Tt,in addition to modeling the current flowing through theTES It which will be measured through an inductivelycoupled SQUID. Thus in a way similar to CRESST [19],we will model the system as 3 coupled non-linear differ-ential equations

LdItdt

= Vb − It (Rl +Rt(Tt, It)) + δV

CtdTtdt

= I2tRt(Tt, It)− Ptb(Tt, Tb)− Pta(Tt, Ta) +Q+ δPt

CadTadt

= −Pab(Ta, Tb) + Pta(Tt, Ta) + δPa

(2)

where δV is a small voltage-bias excitation on top of theDC voltage-bias Vb that we will use to model Johnsonnoise as well as the small signal dynamics of the detec-tor. Likewise, δPt and δPa are power excitations into theTES and absorber respectively. This model can also beseen diagrammatically in Fig. 3. Variable definitions andestimated sizes can be found in Table I. One relativelyunique feature of the CRESST design is the capabilityto directly heat the TES electronic system through anadditional heater circuit (Q). When held constant, thisis effectively equivalent to being able to easily vary thebath temperature Tb on a detector by detector basis.

Taylor expanding to first order around the operatingpoint, [Ito, Tto, Tao] and then Fourier transforming, theseequations simplify to

jω + Rto(1+β)+Rl

LItoRtoαLTto

0

− ItoRto(2+β)Ct

jω +I2toRtoα

Tto−Gta−GtbCt

−Gta,aCt

0 −GtaCa jω +Gta,a+Gab

Ca

∆It(ω)

∆Tt(ω)∆Ta(ω)

=

δV (ω)L

δPt(ω)Ct

δPa(ω)Ca

(3)

which in the pertinent limit of L→ 0 can be simplified to only thermal DOF[jω + (L2D+1)Gta+Gtb

Ct

−Gta,aCt

−GtaCa

jω +Gta,a+Gab

Ca

] [∆Tt∆Ta

]=

[δPtCt

+ ItoRto(2+β)δVCt(Rl+Rto[1+β])

δPaCa

](4)

where

δIt =−ItoRtoα

(Rl +Rto [1 + β])TtoδTt+

1

Rl +Ro [1 + β]δV (5)

As is standard, β or ItoRto

∂Rt∂It

(Tto, Ito)|Tt characterizes theunwanted dependence of Rt on the current. Also notethat we have generalized Irwin’s loop gain parameter LIto

L2D = αRto −Rl

Rto(1 + β) +Rl

I2toRto

(Gtb +Gta)Tto(6)

to account for the 2 thermal DOF as well as the fact thatCRESST has purposely chosen to use a load resistor (Rl)that is of similar magnitude to Rto to suppress electro-thermal feedback. Finally, due to the possibility that Ttoand Tao could be macroscopically different, the thermalconductances ∂Pta

∂Ttand ∂Pta

∂Tacould have different values

and consequently we define

Gta,a =∂Pta∂Ta

=∂Pta∂Tt

(TaTt

)nep−1

= Gta

(TaTt

)nep−1

(7)

Inversion of the generalized impedance matrix, M2D,found in Eq. 4 leads to the current transfer functionsfor the thermal power response for direct heating into

4

Ct

Ca

Bath

Gtb

Gab

Gta

Absorber

TES

δPa

_+

RL

Vb

L

Rt

δPt QIt

FIG. 3. Simplified model of the CRESST phonon and lightdetectors with 2 thermal DOF as well as the voltage-bias andcurrent readout circuit

the TES

∂It∂Pt

=−ItoRto

Rl+Rto(1+β)

α

CtTtM−1

2D(1,1)(ω)

∼ −ItoRtoRl+Rso(1+β)

α

(L2D+1)

1

(Gtb+Gta)Tt

1

1 + jω/ωeff

∼ −1

Ito(Rto−Rl)L2D

L2D+1

1

1 + jω/ωeff

(8)

and into the absorber

∂It∂Pa

=−ItoRto

Rl+Rto(1+β)

α

CaTtM−1

2D(1,2)(ω)

∼ ∂It∂Pt

(Gta,a

Gta,a+Gab

)(1

1 + jω/ωta

) (9)

where

ωeff = (L2D+1)Gta+Gtb

Ct+

1

L2D+1

Gta,aGta+Gtb

GtaCa

+. . .

ωta=Gtb+Gta,a

Ca− 1

L2D+1

Gta,aGta +Gtb

GtaCa

+. . .

(10)

All of the approximations shown are valid only for thepertinent limiting case where Gta and Gab Gtb (cf.Table I).

Qualitatively, the last term of Eq. 9, which is an addi-tional pole that surpresses the bandwidth of ∂I

∂Parelative

to ∂I∂Pt

, is due to the fact that thermal energy must betransported across the tiny Gta before it is seen by theTES. Likewise, the middle term of Eq. 9 is a constantsuppression factor because a portion of δPa is shuntedthrough Gab. This suppression should be small for theCRESST phonon detector. However, for the CRESSTlight detector, the Au wirebond pad on the substrate has

an electron-phonon coupling that is larger than that ofthe W TES, and consequently this factor is significant[12]. Another detector with significant shunting is foundin CUORE where the vast majority of the absorber ther-mal signal flows through Gab [21].

Both the ∂I∂Pt

(Eq. 8) and ∂I∂Pa

(Eq. 9) transfer func-tions are important for understanding the signal responseof massive calorimeters, because some of the high-energyathermal phonons produced by a particle recoil in the ab-sorber may be collected and thermalized within the TES(δPt) before they thermalize within the absorber (δPa),since the electron-phonon coupling varies so significantlywith phonon energy and temperature. Consequently, atrue particle recoil within the absorber should be mod-eled within our 2 thermal DOF system as

∂It∂Eγ

=∂It∂Pt

∂Pt∂Eγ

+∂It∂Pa

∂Pa∂Eγ

=∂It∂Pt

(∂Pt∂Eγ

+Gta,a

Gta,a+Gab

1

1 + jω/ωta

∂Pa∂Eγ

)(11)

where the benefit of being able to write ∂I∂Pa

in terms of∂I∂Pt

plus additional factors is readily apparent.

B. Noise Estimation

With the dynamical response of the detector now mod-eled, we can estimate the magnitude of noise from ther-mal power fluctuations across Gtb, Gab, and Gta, theJohnson noise across Rt and Rl and finally the first stagesquid noise and compare their relative sizes by referencingthem to a thermal power signal flowing directly into theTES (δPt). This reference point was chosen purposely sothat intuition from simpler 1 DOF thermal systems couldbe used and to suppress differences due to CRESST’s useof a non-standard electronics readout scheme with a largeRl to suppress electro-thermal feedback.

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

SPtGtb(ω) = 4kbT2toGtbF(Tto, Tb,diffusive) (12)

where F is a noise suppression term with a value be-tween 1⁄2 and 1 to account for the fact that the powernoise across a thermal link between two different tem-peratures (a non-equilibrium situation) is less than thenaively derived equilibrium noise [22]. Now, estimationof the TFN across Gab is slightly more difficult since wemust refer it to the TES and thus

SPtGab =

(∂It∂Pa

/∂It∂Pt

)2

4kbT2aoGabF(Tao, Tb,diffusive)

∼(

Gta,aGta,a+Gab

)21

1 + ω2/ω2ta

4kbT2aoGabF

(13)

5

This noise is subdominant, even below the ωta pole sinceGab Gtb for both CRESST detectors (as well as forour new design).

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

SPtGta =

(∂It∂Pt− ∂It

∂Pa∂It∂Pt

)2

4kbT2toGtaF(Tto, Tao,ballistic)

∼(

1− Gta,aGta,a +Gab

1

1 + jω/ωta

)2

4kbT2toGtaF

(14)

The insensitivity to this noise below ωta is a direct rami-fication of energy conservation and is a general feature ofall “massless” internal thermal conductances. Of course,for the CRESST designs, this noise is again negligiblecompared to that from Gtb, simply due to their relativesizes.

TES Johnson noise is by far the most challenging tocalculate for 3 reasons. One must take into account theanti-correlation between δV and δPt [7, 22] as well as thenoise boost due to current sensitivity (non-zero β) [7].Finally, we must reference back to the TES input power.We do this in 2 steps. First, we calculate the Johnsonnoise referenced to current flowing through the TES (δIt)

SIt Rt =4kbTtoRto(1 + β)2

(Rl+Ro [1+β])2

(1 +

∂It∂Pt

Ito(Rto−Rl))2

∼ 4kbTtoRto(1 + β)2

(Rl+Ro [1+β])2

(1− L2D

L2D + 1

1

1 + jω/ωeff

)2

(15)

where we see that the anti-correlation leads to noise sup-pression at low frequencies for high L2D. Referencing toδPt, we obtain:

SPt Rt =4kbTtoRto(1 + β)2

(Rl+Ro [1+β])2

(1∂It∂Pt

+Ito(Rto−Rl)

)2

∼ 4kbT2to(Gta+Gtb)

(1+β)2

α2 I2toRto(Gta+Gtb)Tto

(1+

jωωeff

L2D+1

)2

(16)

Again, we find that as long as α is large and heating ofthe TES via external heating (Q) or a bath temperature(Tb) near Tc does not suppress I2

toRto too significantly,then the thermal fluctuations across Gtb dominate thenoise at low frequency. However, this is clearly not trueat high frequencies due to the zero at

ωeff

L2D+1 . In fact, it isthis TES Johnson noise term that sets the bandwidth foroptimal energy estimators of δPt, which we can estimate

by finding the frequency beyond which SPt Rt > SPtGtb .This frequency, which we will designate as the signal-to-noise bandwidth, is given by:

ωS/N δPt ∼Gta +Gtb

Ct

α

1 + β

√F GtbGtb +Gta

I2toRto

(Gta +Gtb)Tto(17)

10−2 100 102 104

10−4

10−3

10−2

10−1

frequency (hz)

σ (f

W/s

qrt(h

z))

CRESST 300g CaWO4: Athermal Power Noise ( δ Pt)

SquidRloadRtesG: TES−AbsorberG: TES−BathG: Absorber−Bath1/FtotalThermal SignalAthermal Signal

10−2 100 102 104

100

101

102

frequency (hz)

σ (p

A/s

qrt(h

z))

CRESST 300g CaWO4: Current Noise

SquidRloadRtesG: TES−AbsorberG: TES−BathG: Absorber−Bath1/FtotalThermal SignalAthermal Signal

FIG. 4. Simulated noise spectrum for CRESST phonon de-tector referenced to TES power δPt(top) and TES currentδIt (bottom). Dotted lines correspond to Taylor expansionsin the limit of L→0 and Gta & Gab Gtb and no 1/f noise.

For electronics designs with large electro-thermal feed-back, ωS/N δPt ∼ ωeff . However, for CRESST where L2D

has purposely been set to near 0 by choosing Rl ∼ Rto,ωS/N δPt ωeff . To reiterate, when referencing both thesignal and noise to current (Fig. 4 bottom), both thecurrent signal and the total current noise have a pole atωeff , and thus the signal-to-noise (which is equivalent torereferencing to TES power) remains flat all the way upto ωS/N δPt .

6

Unfortunately, neither the squid noise

SPtSQUID =1∂It∂Pt

SItSQUID (18)

nor the Johnson noise from the load resistor, Rl,

SPt Rl =4kbTtoRto(1 + β)2

(Rl+Ro [1+β])2

(1∂It∂Pt

+ItoRto(2+β)

)2

(19)can be trivially written in terms of Gtb. However, asshown in Fig. 4 they are subdominant to the combinationof Gtb TFN and Rt Johnson noise (this latter can also beseen through comparison of Eq. 19 to Eq. 15).

Finally, we would ideally like to have an estimate ofthe 1/f noise found in the current CRESST experimen-tal setup, particularly since there are some indicationsthat it could be an important contributor to the overallnoise in CRESST currently [12]. More importantly, wewould like to estimate the 1/f noise expected in a next-generation experimental setup to assess its effect on thedevice design. For the former, significant effort would berequired that is beyond the scope of this paper. For thelatter, we will use the noise performance of the SPIDERTES bolometers, which begin to be dominated by 1/fat 0.1 Hz, as guide to what could be achieved with ef-fort, since CMB experiments are quite sensitive to lowfrequency noise [23] and thus we will set a design goal ofωS/N δPt > 0.5 hz.

III. TES PHASE SEPARATION

Unfortunately, as first realized in the earliest daysof CRESST calorimeter development [19], the simple 2DOF block thermal model shown in Fig. 2 that we haveused for our CRESST-II phonon-detector resolution es-timates is not entirely accurate, since Gtb is also signif-icantly larger than the internal thermal conductance ofthe W TES, Gt int(cf. Table. I). In this situation, the dif-ference in temperature between the TES Tc and the effec-tive bath temperature is mostly internal to the TES, andthus there is a significant temperature gradient acrossthe sensor. In devices with large α, this thermal gra-dient is further exasperated by positive-feedback effectsof joule heating and the TES can separate into sharplydefined superconducting and normal regions, with only avery thin region within the transition that is sensitive totemperature fluctuations [24].

Although a computational simulation of phase separa-tion within the CRESST geometry is beyond the scopeof this paper (and of dubious value without a very care-ful matching to experimental data since the dynamicsdepend on the resistivity of the entire transition, ratherthan at a single operation point), we can qualitativelydiscuss the ramifications.

Most importantly, a phase separated TES has signif-icantly suppressed thermal sensitivity when biased. To

see this, we note that in the limit of α = Tρ∂ρ∂T → ∞

(i.e. the super conducting transition becoming infinitelysharp and infinitely sensitive to temperature variation),the DC response of the TES can be modeled analyticallyby tracking the fractional location of the superconduct-ing/normal transition in the TES, x, as a function of Vband Tb [25]. For the pertinent case where the electron-phonon coupling along the TES is negligible comparedto power flow through the connection to the bath at oneend of the TES, the thermal power flowing across thesuperconducting portion of the TES is constant and canbe matched to the joule heating in the normal portion ofthe TES:(

VbRl +Rnx

)2

Rnx =1

1− x

∫ Tc

Tb

dT Gt int(T ) (20)

Using this simplification, we obtain curves of Rt versusTb for a CRESST-like device for several values of Vb, asshown in Fig. 5.

7 7.5 8 8.5 9 9.5 1050

100

150

200

250

300

Tb[mK]

R t [mO

hm]

Simulated Voltage Bias Curves ( α = ∞ )

Vb=0.1µVVb=0.15µVVb=0.21µVVb=0.31µVVb=0.46µVVb=0.68µVVb=1µVVb=1.46µVVb=2.15µVVb=3.15µV

FIG. 5. Simulated Rt vs Tb curves for several values of Vbfor a phase-separated TES with an infinitely sharp transitioncurve (at Tc = 10 mK) that shows DC sensitivity suppressionwhen biased.

If not for phase separation, all of these curves would beinfinitely sharp step functions centered at Tb = Tc. Withseparation though, we observe significant degradation indevice sensitivity that worsens with increasing Vb. Basi-cally, a phase separated TES acts as if it has a very largeβ from a DC perspective. This exact behavior is seen inCRESST devices [9, 12].

Naively, one would think that this suppression in sig-nal amplitude would severely affect phonon power reso-lution at all frequency scales. However, this is not thecase. At low frequencies, thermal fluctuations across Gtbcompletely dominate the noise in a non-phase-separatedTES (Fig. 4 top), and thus both signal and noise are sup-pressed equally. Whereas at high frequencies, the total

7

noise is dominated by Johnson noise across the TES andthus the qualitative net effect is a suppression in ωS/N δPt .

Further, energy that is absorbed by the TES electronsystem in portions of the TES that are either fully nor-mal or superconducting must first diffuse to the tran-sition region before producing a measurable change incurrent, which adds an additional strongly expressed dif-fusive pole into all of the TES transfer functions. In aphase separated CDMS-II TES device for example, mea-sured ∂It

∂Vbhas 2 fall-time poles (plus the R/L pole) with

roughly similar weighting [26]. Beyond simply addingconfusion when trying to understand and model deviceperformance, the most important consequence is to againsuppress ωS/N δPt .

Finally, due to such large variations in the deriva-tive of the resistivity ( ∂ρ∂T ) across the sharp supercon-ducting/normal boundary, thermal power fluctuationswithin the TES directly couple to the total TES resis-tance Rt, and consequently the measured signal currentIt, giving us an additional noise source with fluctuationson all length scales [25]. Interestingly enough, in dis-cretized simulations of phase separated TES for CDMS-II [26], we have found that this additional noise source islargely counter balanced by a suppression in sensitivityto standard TFN noise across Gtb at frequencies belowthe longest diffusive pole, and thus Sp total primarily seesincreases at higher frequencies.

IV. BANDWIDTH MISMATCH BETWEENSIGNAL AND SENSOR

Before attempting to estimate the sensitivity ofCRESST detectors to recoils within the absorber thathave both thermal and athermal phonon components, itmakes sense to calculate the expected sensitivities of theCRESST detector to the two limiting cases of a Diracdelta thermal energy deposition into the absorber andthe TES. These estimates are shown in Table II.

σE : Eq. 1 1.0 eVσEa : 2D Simulated 34 eV (no 1/f) / 44 eV (1/f)σEt : 2D Simulated 0.5 eV (no 1/f) / 0.5 eV (1/f)σEγ Measured: “Julia” 420 eV [20])σEγ Measured: Composite 107 eV [1])

TABLE II. Simulated and measured CRESST phonon-detector energy resolutions. Note that the simulation doesnot include the effect of TES phase separation.

What is immediately striking is the significantly de-graded expected sensitivity for thermal energy depositionin the absorber (σEa) compared to that for direct absorp-tion in the TES (σEt). To understand the cause of thissuppression factor, we can write σEa in terms of σEt un-der the now well motivated assumption that the noise,when referenced to δPt, should be flat below ωS/N δPt(since it is dominated by the naturally flat Gtb):

limωtaωS/N δPt

σ2Ea =

1∫∞0

dω2π

4| ∂Pt∂Ea(ω)|2

SPt−total(ω)

∼ SPtGtb(ω = 0)(Gta,a

Gta,a+Gab

)2

ωta

∼ σ2Et

(Gta,a+GabGta,a

)2 ωS/N δPt

ωta

(21)

Basically, any signal whose bandwidth is smaller thanωS/N δPt will suboptimally use the sensor bandwidth,leading to poorer resolution as discussed in [27]. To seethis in another way, notice that both ωS/N δPt (Eq. 17)and Sp total (which is roughly SptGtb as in Eq.12) bothscale linearly with Gtb. Thus, as long as the bandwidth ofthe signal that is being measured is larger than ωS/N δPt ,the baseline energy sensitivity will be independent of thesize of Gtb; increasing Gtb increases the noise floor, butthis effect is balanced by an increase in sensor band-width. By contrast, when measuring signals with band-width below ωS/N δPt one has accepted a noise penaltybut the gained bandwidth is useless. In summary, be-cause ωta ωS/N δPt , the CRESST phonon detector hasrelatively poor sensitivity to phonon energy that is ther-malized within the absorber (δPa).

−20 0 20 40 60 80 100−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time [ms]

Puls

e A

mpl

itude

[V]

Event Pulse Shapes for CRESST Composite Detector

TES Event (δPt)

Absorber Event (δPa)

FIG. 6. Signal pulse shapes for γ-induced events in theCRESST composite detector “Rita” for events that interactin the TES chip (black) and in the absorber attached via anepoxy joint (blue). Clearly, absorber events have significantlysuppressed bandwidth and consequently suppressed resolu-tion, as in Eq. 21. Data taken from [20].

The athermal phonon signal that is thermalized in theTES bypasses the Gta restriction and therefore its band-width is not limited by ωta. However, it still takes timeto collect all of the ballistic athermal phonons rattlingaround in an absorber. In the SuperCDMS athermalphonon iZIP detector for example, this athermal-phononcollection bandwidth ωnt = 210 Hz while ωS/N δPt= 4 kHz

8

for the 90 mK W TES that CDMS has historicallyused[26]. Thus, they pay a resolution penalty of 5×due to the bandwidth mismatch between their athermalphonon signal and their sensor bandwidth, completelyanalogous to the thermalized sensitivity suppression.

Other examples of sensitivity suppression due to poorsignal /sensor bandwidth matching can be found in thenew CRESST composite detectors (Fig. 6) [20] as wellas the AMORE MMC based calorimeters [28]. In theCRESST composite detector, the athermal phonon sig-nal from an event in the absorber crystal must first betransmitted across an epoxy glue joint between the ab-sorber crystal and the substrate on which the TES isfabricated, yielding ωnt ∼7 hz (fall time ∼ 23 ms). Ifwe associate the fall-time pole of the phonon signals forevents hitting the TES chip substrate (∼ 50 Hz) withωeff and we further recognize that ωS/N δPt > ωeff forCRESST detectors (since they purposely bias to mini-mize electro-thermal feedback), then we can deduce thatthe new CRESST composite has a signal/sensor band-width suppression factor that is >10×. In fact, using our2D simulation, we estimate that this bandwidth-limitedenergy impulse into the TES, σEt(ωnt = 50 Hz), wouldhave 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 signalbandwidth must be larger than the sensor bandwidth,a design goal that, to our knowledge, has not been ac-complished by any very low-temperature large-mass de-tector but that is standardly implemented in 100 mKTES calorimeters for x-ray [29] and γ [18] applications.Please note that there is some subtlety in this designrule. CRESST detectors use pulse-shape differences be-tween absorber and TES events to suppress TES chipbackgrounds (Fig. 6); they have both an energy anda discrimination signal with different bandwidths. Thus,the optimal strategy is likely to choose ωta ∼ 1/2ωS/N δPtso that the device discrimination threshold is equal to itsenergy threshold.

Secondly, if we are able to design very low-temperaturecalorimeters that use the entire unsuppressed sensorbandwidth (design rule 1), we will become directly sensi-tive to TES phase separation. Consequently we impose asecond design constraint that Gtb Gt int. As an addedbenefit, we think device operation and analysis will besignificantly less complex, because we also naively ex-pect that phase-separated TESs are highly sensitivity toposition-dependent variations in Tc and other thin-filmproperties that are likely culprits for at least some ofthe non-linearities that have been found when biasingCRESST devices [12].

V. EXPECTED CRESST ENERGYRESOLUTION & ATHERMAL PHONON

COLLECTION EFFICIENCY

We have shown that the infinite bandwidth σEt in Ta-ble II significantly overestimates the athermal phononenergy sensitivity, since it does not account for the ather-mal phonon collection bandwidth, ωnt. Additionally, wemust account for the fact that only a fraction εnt of theathermal phonons are collected in the TES, and thus aparticle recoil in the absorber should be modeled as

dPtdEγ

=εnt

1 + jω/ωntdPadEγ

= 1− εnt(22)

For an upper bound on εnt, we note that optical pho-tons directly incident on a W TES have been measuredto deposit ∼ 42% of their total phonon energy into theW electronic system [30]. Another guide to the size ofεnt comes from the SuperCDMS iZIP athermal phonondetector, which measures a total athermal phonon col-lection that is 10–20% of the total deposited energy [26].Using a reasonable (but certainly not measured) value ofεnt = 20%, we estimate a σEγ = 15 eV. Although this is7× better than the best achieved resolution in a CRESSTcomposite detector (107 eV), it is also 30× worse than thenaive expectation of 0.5 eV, and thus we believe that sen-sor/signal bandwidth mismatch and non-unity athermalphonon collection efficiency are the dominant sensitivitydegradation mechanisms.

VI. PARASITIC POWER LOADING

−200 0 200 400 600 800 1000 1200 1400 1600 1800 2000−20

−10

0

10

20

30

40

50

60

70

time (us)

Ener

gy (k

eV)

Various Phonon Traces

GlitchSquare Wave GlitchNormal Pulse

FIG. 7. Environmental noise pulses (red and black) illustratethat SuperCDMS detectors are strongly susceptible to directTES heat by parasitic power noise.

Both design drivers so far discussed suggest that to be-come more sensitive an optimized very low-temperature

9

calorimeter should have lower Gtb than found in theCRESST phonon-detector design. One negative conse-quence of this change is greater sensitivity to parasiticheating. This is clearly a concern for SuperCDMS be-cause their devices produce TES heating signals due tocell phone usage in the laboratory (fortunately with avery distinct pulse shape from actual events, as shownin Fig. 7), and thus the old CDMS-II electronics do notadequately shield their TESs from high-frequency envi-ronmental noise. Although this is currently only an inter-esting nuisance, it highlights the parasitic power problemthat SuperCDMS faces. As they lower Gtb in their owndevices, this and other parasitic power sources heat thedetector to a greater and greater degree, eventually driv-ing the TES normal and rendering it inoperable. LowerGtb devices will require better environmental noise shield-ing.

Thus, it is reasonable to question if DC parasitic powernoise necessitates the use of large Gtb in CRESST phonondetectors, thereby breaking our first 2 design rules andlowering detector sensitivity. Further, if so, are all verylow-temperature calorimeters similarly limited? We donot believe this to be this case for 2 reasons. First,in the very same cryostat and with identical electron-ics, CRESST also operates light detectors that are Sior SOS wafers instrumented with the TES design shownin Fig. 8. Notice that for these detectors, the TES isnot directly connected to the bath through an Au wire-bond. Instead the coupling is through a fabricated Authin film impedance with an estimated Gtb= 21.1 pW/Kfor a Tc = 10mK which is 350× smaller than the Gtb ofthe Au wirebond used on their phonon detectors. Con-sequently, if DC parasitic power noise was even remotelyproblematic for the CRESST phonon detector, their lightdetector would be inoperable.

1450um

40um

1000

um

300um

500um 450um

100um

WAlAu

PhononCollectors

Heater

Thermometer

Cu-Kapton-Cu

HeatSink

ThermalLink

Absorber

FIG. 8. TES Sensor Geometry for CRESST-II light detector[12]

Of course, when trying to assess the validity of lower-ing Gtb during the optimization of TES-based light detec-tors (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 enormousemphasis on designing TES-based sensors for space-borninfrared spectrometry by SPICA/SAFARI, and they havebeen able to achieve parasitic power loads of ∼2 fW, 25×less than the estimated bias power, Pto, for the currentCRESST light detector and 104× less than their phonondetector [31]. Consequently, achieving parasitic powerloads that are much less than Pto for low Gtb devicesmay be difficult but should be possible.

VII. LIMITS ON SENSOR BANDWIDTH DUETO EVENT RATE

Unlike in 100 mK x-ray, γ and α calorimeters, theextremely low background and signal rates expected inunderground rare-event and exotic-decay searches havemeant that pileup (pulses from distinct interactions over-lapping due to high rates and finite bandwidth) rejectionand the ability to handle large event rates have not beenprimary design drivers.

In the future, however, this requirement will becomemuch more constraining for some applications. For ex-ample, the most daunting pileup requirements come fromreactor-sourced coherent neutrino scattering experimentsbecause the cosmic background is quite large due to min-imal rock overburden. As a first very rough estimate ofthe necessary start-time sensitivity, we note that the orig-inal CDMS shallow site experiment had a similar over-burden and used an anti-coincidence window of 25 µs foractivity in their plastic-scintillator muon veto [32]. Sincethe optimally estimated start-time resolution (assumingno pulse-shape dependence) is

σ2to =

1

E2

1∫∞0

dω2π

4ω2| dItdEγ(ω)|2

SI tot(ω)

(23)

which can be simplified to

σto ∼1

ωS/N δPt

σEE

(24)

when flat noise is assumed and ωS/N δPt is the lowest poleof the design, we estimate that a reactor-sourced coher-ent neutrino scattering experiment needs an ωS/N δPt of1.3 khz to have the requisite start-time resolution fornear-threshold events (E = 10σE). Unfortunately, sucha large ωS/N δPt is simply inconsistent with our otherdesign requirements for thermal calorimeters. Conse-quently, we believe that this application is better suitedto an athermal-only phonon detector design, as typifiedby a SuperCDMS detector.

Pileup rejection has also become one of the dominantdesign drivers for LUCIFER and other high-Qββ neutri-noless double-beta decay experiments because they ex-pect the background in their signal region to be dom-inated by un-vetoed pileup of lower energy 2 ν double-beta decay events for which pileup rejection is assumed to

10

be ineffectual below 5 ms [10]. Luckily, with such a largesignal, start-time resolutions of this order are achievable(see Sec. X) and thus, if required, this is a reasonabledesign requirement.

VIII. EASE OF FABRICATION &COSMOGENIC BACKGROUND SUPPRESSION:

SEPARATED TES CHIP

CUORE and EDELWEISS have continued to use NTDsensor technology, despite the many benefits offered byTES readout, including:

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

2. The SQUIDs used in TES readout have lower 1/fnoise than JFETs or HEMTs used for first stageamplification for NTDs;

3. Low-impedance sensors (TESs) are fundamentallyless sensitive to vibrationally induced capacitancechanges in readout compared to high-impedancesensors (NTDs); and

4. SQUIDs have significantly lower heat loads thanthe JFETs or HEMTs used in NTD readout, andcan therefore be placed significantly closer to thedetector, simplifying electronics and cryostat de-sign.

One reason for this is that both TES-based massive-detector groups (CRESST and CDMS) fabricated theirTESs directly upon the absorber surface, a feat thatrequired enormous fabrication process R&D since ev-ery facet of standard microprocessor fabrication (pho-tolithography, etching, thin film deposition) had to beretrofitted for thick and massive substrates. In fact, evenwith over a decade of R&D, fabrication yields were still asignificant resource drain until recently on CDMS. Fur-ther, the time and labor intensive nature of micropro-cessing fabrication means that absorbers spend a signifi-cant amount of time on the surface being cosmogenicallyactivated, certainly a disadvantage for low mass WIMPsearches, for example. This direct absorber fabrication isrequired since the W TES in CRESST has 2 distinct func-tions. First, it is a temperature sensor. This is the func-tionality that requires microprocessor fabrication tech-niques on multiple different thin film layers: Al for thesuperconducting bias rails, W for the TES, and Au for thethermal connection to bath. Second, the electron-phononcoupling within the W film acts as Gta. Because it is onlythis latter functionality that requires fabrication directlyupon the absorber, design goals of fabrication simplicityand minimum cosmogenic exposure require that the TESdoes not act as the thermal link between the sensor andthe absorber.

As illustrated in Fig. 9, we propose to deposit a large,single layer Au thin-film pad directly onto the large ab-sorber substrates that plays the role of Gta. This can

Au Pad on Absorber

Au Pad on Absorber

TES

Al

Gta

Gtb

Si TES Chip

Au

W

Absorber

FIG. 9. Optimized large-mass calorimeter sensor design.Only the large Au pad is directly fabricated on the large ab-sorber.

be done using only shadow mask techniques (albeit witha depostion machine modified for thick substrates) andconsequently fabrication should have very high yield andbe relatively hassle free since there is no photolithographyand etching. As an added benefit (in fact, perhaps themost important benefit), this permits use of any metalrather than being constrained to W; we choose Au whichhas an order of magnitude larger electron-phonon cou-pling than W for a given thermal capacitance.

The fabrication intensive TES can then be separatelyfabricated on standard thin substrates for fabricationease, where the material chosen is not necessarily identi-cal to that of the large absorber (Si, Ge, Al2O3, CaWO4).Further, each and every 100 mm wafer can produceover 20 devices. Thus, device fabrication throughputcould easily be 80× that of CDMS (in the standard Su-perCDMS fabrication procedure, 4 fully processed testwafers are produced for every detector). This physicalseparation also allows for testing of the TES sensor dieabove ground before connection to the absorber. This hassignificant advantages: the cost savings of sensor testingabove ground rather than in an underground laboratoryis substantial; and one can choose only the best sensorsto match with expensive absorbers, particularly useful inthe case of double-beta decay enriched crystals.

Thermal connection between the TES and the Gta ab-sorber pad is then accomplished via Au wire bonding,while the simpler mechanical connection can be accom-plished with epoxy in an arbitrary (and somewhat hap-hazard) manner; without any thermal-conductance re-quirements, very small-area single dot epoxy joints arepossible that do not mechanically stress the absorber [20]). Another possibility is that the TES chips are mechan-ically supported by the detector housing. Of course, theheat capacity of the Au wirebond between the TES andthe absorber (∼1 pJ/K at 10 mK) is entirely parasitic andis simply the price paid for the ease of fabrication. Mostimportantly, the thermal conductance of both the inter-nal pad, Gpad int, and that of the Au wirebond, Gbond int,must be much larger than Gtb so as to satisfy the band-width design rules discussed in Sec. IV.

11

It is interesting to reiterate how similar and yet howdistinct this design is from the CRESST composite de-tector [20]. On the one hand, this design is clearlyderivative. As with the CRESST composite detector, theTES is fabricated on a separate and much thinner sub-strate that drastically simplifies fabrication (and in theCRESST case, improves scintillation yield in the primaryabsorber). On the other hand, the difference is profound.In the CRESST design, energy transport between the 2systems is accomplished via phonon transport throughan epoxy joint that significantly suppresses both ather-mal and thermal signal bandwidths (making our band-width design objectives very difficult to achieve). Fur-ther, low-impedance phonon transport via epoxy cou-plings requires large cross-sectional areas that make thedetector much more prone to crack via differential ther-mal contraction among the epoxy, the absorber substrate,and the TES chip substrate. The CRESST composite de-tector mitigates this problem somewhat by using CaWO4

for both the TES substrate and the absorber. However,this limits possible detector materials. NaI, for exam-ple, is an attractive material to search for spin or or-bital angular momentum coupling dark matter [33]. Fur-thermore, scintillating NaI calorimeters would directlytest the anomalous annual modulation signal observed inDAMA [34]. Unfortunately though, NaI is very hygro-scopic and thus any complex photolithography is impos-sible.

In summary, this design should retain all the benefitsof TES performance and yet also have the fabricationsimplicity, cosmogenic benefits, and material freedom ofNTD based detectors.

This device design should also be critically comparedto the AMORE MMC calorimeter design [28]. Remov-ing the trivial difference of the use of an MMC insteadof a TES sensor element, all pertinent design rules thatwe have developed were followed with the single and veryimportant exception that their sensor bandwidth is muchlarger than their signal bandwidth (ωta). This choicesuppresses their zero energy sensitivity but even moreimportantly drastically increases their sensitivity to po-sition dependence (Secs. IX and X), a fact that severelylimits the viability of their current devices.

IX. POSITION DEPENDENCEREQUIREMENTS IN NEUTRINOLESS DOUBLE

BETA DECAY SEARCHES

Because of the approximately exponential shape of theWIMP nuclear-recoil energy spectrum, a slight positionor temporal systematic of <5% on the recoil energy es-timate in a CRESST phonon detector does not signifi-cantly affect their sensitivity. By contrast, a neutrinolessdouble-beta decay experiment like CUORE/LUCIFER[10] requires the maximum achievable energy resolutionat 3 MeV, and thus any variation in the pulse shape ormagnitude of the signal due to event location in the ab-

sorber must be minimized in addition to having excellentbaseline energy sensitivity. Specifically, a well designeddouble beta detector should be limited by systematics inthe absorber thermalization process, with ∆E

E measured

to be 5×10−4 in TeO2 [35]) and 2x10−3 in ZnMoO4 [10].

Array-compatible transition-edge sensor microcalorimeter !-ray detectorwith 42 eV energy resolution at 103 keV

B. L. Zink, J. N. Ullom, J. A. Beall, K. D. Irwin, W. B. Doriese, W. D. Duncan, L. Ferreira,G. C. Hilton, R. D. Horansky, C. D. Reintsema, and L. R. Valea!

National Institute of Standards and Technology, 325 Broadway MC 817.03, Boulder, Colorado 80305

!Received 11 March 2006; accepted 21 July 2006; published online 20 September 2006"

The authors describe a microcalorimeter !-ray detector with measured energy resolution of 42 eVfull width at half maximum for 103 keV photons. This detector consists of a thermally isolatedsuperconducting transition-edge thermometer and a superconducting bulk tin photon absorber. Theabsorber is attached with a technique compatible with producing arrays of high-resolution !-raydetectors. The results of a detailed characterization of the detector, which includes measurements ofthe complex impedance, detector noise, and time-domain pulse response, suggest that a deeperunderstanding and optimization of the thermal transport between the absorber and thermometercould significantly improve the energy resolution of future detectors. #DOI: 10.1063/1.2352712$

Low temperature microcalorimeters and microbolom-eters represent the state of the art in photon detection over awide range of wavelengths.1 For example, microcalorimetersbased on superconducting transition-edge sensors !TESs",which can measure the energy of x rays in the 6 keV regimeto within 2.4 eV, have potential uses for x-ray astronomyand x-ray microanalysis.2 For some time, researchers haverealized the potential of TES microcalorimeters for measur-ing hard x-ray and ! radiations, where a bulk absorber isrequired for sufficient absorption efficiency. Previous workhas demonstrated that attaching superconducting bulk tin ab-sorbers to TES microcalorimeters provides a potential routeto high-resolution !-ray detectors,3 but higher energy resolu-tion and the implementation of arrays of detectors are neces-sary for the most promising applications, which include pas-sive, nondestructive assay of nuclear materials such asplutonium isotopic mixtures4 and spent uranium fuelassemblies,5 and precise determination of the Lamb shift inheavy hydrogenlike atoms.6

In this letter we present experimental results obtainedwith a composite microcalorimeter in which the thermometeris an optimized, voltage-biased, Mo/Cu TES and photons areabsorbed in a superconducting bulk tin slab. The !-ray spec-tra show an energy resolution of 42 eV full width at halfmaximum !FWHM" at 103 keV, more than an order of mag-nitude better than typical high-resolution !-ray detectors. Wealso characterize the detector by comparing measurements ofthe TES complex impedance ZTES, current noise In, small-signal pulse response, and energy resolution to the predic-tions of thermal models. Our results suggest that a deeperunderstanding of the thermal transport in the device couldlead to further improvement in energy resolution.

Figure 1!a" is an optical micrograph of the compositemicrocalorimeter. The design of the Mo/Cu TES, which isthermally isolated from the bulk Si substrate with a silicon-nitride !Si–N" membrane, is described elsewhere.7 We pat-terned a 150 "m diameter, 20 "m tall post on the TES usinga negative photoresist and coated the top face with a thinlayer of glue. We then cut a 250 "m thick sheet of high-purity cold-rolled Sn into a 900–950 "m square, aligned the

TES and post to this absorber, and mated them to form acomposite microcalorimeter with estimated quantum effi-ciency of 25% for 100 keV photons. This technique allowsarrays of composite microcalorimeters to be assembled in asingle gluing step. Figure 1!b" shows a 16 pixel compositeTES !-ray detector array that we are currently testing using atime-division superconducting quantum interference device!SQUID" multiplexer. The energy spectrum of a 153Gd cali-bration source measured with the TES voltage biased suchthat the equilibrium resistance, R0=0.25Rn !normal state re-sistance Rn=8.3 m#", appears in Fig. 1!c".

The simplest thermal model of a composite microcalo-rimeter appears in Fig. 1!d". Ca represents the heat capacityof the absorber, which is linked to the TES heat capacityCTES via a thermal conductance Ga. Heat flow from the TESto the bath at Tb is largely through the Si–N membrane,

a"Electronic mail: bzink@boulder.nist.gov

FIG. 1. !Color online" !a" Side-view optical micrograph of the compositeTES microcalorimeter. !b" A 16 pixel composite TES array. !c" Spectrum ofoptimally filtered and drift-corrected pulse heights from 153Gd, with lines at97 and 103 keV. The inset graph shows the 103 keV peak, where the solidline is a least-squares Gaussian fit with $EFWHM=42 eV. A simple scalingpredicts that a similar detector with 4 mm2 collection area would have$EFWHM%84 eV. !d" Thermal model of the composite microcalorimeter.

APPLIED PHYSICS LETTERS 89, 124101 !2006"

0003-6951/2006/89"12!/124101/3/$23.00 89, 124101-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

169.229.32.36 On: Sat, 13 Sep 2014 04:43:47

FIG. 10. NIST γ calorimeter with σ= 21 eV at 103 keVusing a TES with Tc ∼ 100 mK. [18]

To guide us in developing design rules to minimize posi-tion dependence, let us look more carefully at the 100 mKTES γ calorimeter shown in Fig. 10 that achieved a∆EE <5×10−4[18]. First and foremost, the only ther-

mal and structural link between the absorber and thebath goes through the TES (Gab=0). To illustrate theimportance of this for suppressing position dependence,consider the design in [11] where the primary thermalpath from the TES to the bath is through the absorber(Gtb = 0, Gab 6= 0). In this case, athermal phonons pro-duced in the initial interaction will preferentially ther-malize in either the TES (Gtb) or in the metal pad ther-mally linked to the bath (Gab), depending on the event’slocation in the absorber and leading to significant un-wanted position dependence in δIt.

Unfortunately, Gab free designs are significantly moredifficult to achieve when the absorber weighs O(kg)rather than O(g), and thus this will be an aspirationalgoal rather than a requirement. We will solely requirethat Gab Gta, and that any Gab be diffusive ratherthan ballistic so that non-thermalized excitations wouldfind it difficult to escape through Gab.

Secondly, the size of the diffusive-phonon thermallink between the absorber and the TES in this γcalorimeter was chosen to be much slower than thethermalization/position-dependent time scale, ωnt ωta ωS/N δPt . This choice means that unwanted posi-tion information with frequencies around ωnt have 2 polesuppression over many decades in their current response.

Unfortunately, thermalization is very slow in the in-sulator and semiconductor absorbers used in massivecalorimeters compared to the superconductors and metalabsorbers used in γ and x-ray calorimeters. SuperCDMSiZIP detectors, for example, see variations in athermalphonon power absorbed by different channels on the samecrystal up to 300 µs after an event interaction. Such longposition-dependent thermalization times put downward

12

pressure on both ωta and ωS/N δPt when using this band-width design scheme. CUORE actually attempts to doexactly this in their NTD based calorimeter by using asmall dot of epoxy as Gta, with the hope of suppressingany non-thermalized position-dependent phonon signalfrom reaching their TES. Unfortunately, this choice incombination with poor electron-phonon coupling withinthe NTD means that Gab > Gta, and thus most of theabsorbed energy is not measured by the NTD but di-rectly shunted to the bath, suppressing baseline energysensitivity and increasing DC susceptibility to positiondependence [21].

In our proposed calorimeter design, the above band-width design scheme is very difficult to achieve be-cause the dominant phonon thermalization mechanism isthrough electron-phonon coupling within the metal padthat acts as Gta; ∂Pt

∂Eγ(Eq. 11) will always be non-zero.

Thus, we must be content with single-pole suppressionof position dependence. On the otherhand, since we nowcontrol the rate of athermal phonon thermalization (ωnt),we can maximize this single-pole suppression by mak-ing the pad as large as possible (i.e. set the design re-quirement that ωnt &ωta ωS/N δPt). As an aside, notethat suppression of position dependence via pole sup-pression is incompatible with the pulse-shape discrimina-tion between TES chip events and absorber events thatCRESST enjoys because it requires (ωta ∼ 1⁄2 ωS/N δPt).

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

dPtdEγ

= ε

dPadEγ

= 1− ε(25)

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

X. DESIGN SKETCH OF 1.75KG ZnMoO4

DETECTOR FOR NEUTRINOLESS DOUBLEBETA DECAY

In Table. III we flesh out the specifications and sim-ulated performance for a 1.75 kg ZnMoO4 device thatfollows all of the design rules so far discussed. Such amassive absorber (or the addition of parasitic heat ca-pacitance as was done in [6]) is required to keep Qββwithin the dynamic range of the TES. We recognize thatthe use of such large crystals increases the need for pileuprejection (i.e. timing requirements) [10].

The Au pad volume on the absorber was chosen to bequite large so that ωta ωS/N δPt . Notice that even withsuch a relatively large volume, the parasitic heat capac-ity of the pad is only ∼ 80% that of the ZnMoO4. The

ZnMoO4 absorberVa Volume π(40mm)2x80mm

TaoOperationaltemperature

9.33 mK

Ca Heat capacity ΓZnMoO4VaT3ao 191 pJ

K

Au thin film thermal link between TES and bathltb Length 900 µmAtb Cross sectional area 10 µm x 300 nm

Gtb

Thermal conduc-tance from TES tobath

vfde3

ΓAuTtoAtbltb

286 pWK

Au thermalization film on absorber (p1)Vp1 Volume 4π(9mm)2x300nm

Ct p1p1 component ofTES heat capacity

ΓAuVp1Tto 187 pJK

Gta

Thermal conduc-tance between TESand absorber

nepΣepVp1Tnep−1to 37 nW

K

Au wirebond between TES and absorberlw Length 1.0 cmAw Cross sectional area 4π(7.5µm)2

GwThermalconductance

vfde3γAuTto

Awlw

21.6 nWK

Ct wBond component ofTES heat capacity

4.6 pJK

W TESAt Cross sectional area 3.5 mmx150 nmlt Length 700 µm

TcTransitiontemperature

10 mK

∆T90−10 Transition width (T90%-T10%) 2 mK

TtoOperatingtemperature

9.33 mK

Ito Operating current 5.4 µA

CtWW component ofTES heat capacity

fscΓWVtTto 0.96 pJK

Gt intInternal TESconductance

LwfρW

Atlt/2

Tto 4.9 nWK

Rn Normal resistance ρWltAt

100 mΩ

Rto Operating point Rn/5 20 mΩ

α Thermal sensitivity TtoRto

∂R∂Tt

∣∣∣Ito

16.4

β Current sensitivity ItoRto

∂R∂It

∣∣∣Tto

0.05

Dynamical time constants

τL/RSquid inductor timeconstant

15 µs

τeff sensor fall time 365 ms

τtaAbsorber/TES cou-pling time constant

2.6 ms

Estimated resolution and saturation energy

σEEstimated baselineenergy resolution

3.54 eV

σtoEstimated timingresolution @ 3 MeV

170 µs

∆E/EPositiondependence

6x10−4

EsatAbsorber satura-tion energy

4.1 MeV

TABLE III. Optimized 1.75kg ZnMoO4 detector for next-generation double-beta decay experiments

13

thickness of the pad at 300 nm was a compromise. Onthe one hand we wanted to cover the the largest pos-sible surface area, so as to maximize ωnt. On the oth-erhand, we want the internal thermal conduction of thepad Gpad int Gta. Since the parasitic heat capacityof an individual Au wire bond is also relatively small(1.2 pJ/K) compared to the crystal, we used 4 and as-sumed an extra long 200 µm bond tail at both ends forgreater thermal conductance.

For the TES itself, we chose a W TES with anRn=100 mΩ that we would run low in the transition(20% Rn) to maximize the dynamic range. Further, wewould use Rl= 3 mΩ to be in the strong electro-thermallimit. This aspect ratio was chosen so that Gt int Gtband could potentially be further lowered. Since the heatcapacity of the TES is subdominant compared to boththe crystal and the Gta pad, the sole constraints on itssize are related to sensor performance related. Since β

increases with current density (which scales as V−1/2t ,

where Vt is the TES volume), we will choose the over-all TES size such that the current density is 1/2 thatfound in the current CRESST detectors. The use of low-resistivity Ir/Au or Mo/Au bilayers [19, 36, 37] is also cer-tainly possible and would suppress TES inhomogeneitiesbut with the potential for larger β.

Finally, Gtb, is a thin Au film impedance similar tothat used by CRESST in their original light detectors[12]. Its size is set so that ωS/N δPt is as small as allowedby 1/f noise constraints.

The simulated baseline energy resolution of 3.54 eV isclearly sufficient for double beta decay (Simulated NoisePSDs in Fig. 11). For a rough estimate of the positionsensitivity, we have simulated position-dependent pulseshapes for ε = 0, 5, and 10% (Eq. 25, Fig. 12), and usingthe standard optimum sensitivity estimator we estimatea ∆E

E of 6×10−4 which is just adequate for current pu-rity TeO2 and ZnMoO4. Further, SuperCDMS has beenable to suppress unwanted position dependence in theirphonon energy estimators by an additional factor of 2–4 using non-stationary optimum filters or multiple tem-plate optimum filters. Techniques such as these could beeven more viable for double-beta decay detectors sincethe residual dependence in SuperCDMS is due largely tovariations in Luke phonon production due to ionized e−

trapping [26].

This slight sensitivity to athermal phonons that sys-tematically biases our energy estimator does have onehidden silver lining: our devices should have sig-nificantly improved timing and pulse-shape rejectionfrom the largely decoupled NTDs currently used byCUORE/LUCIFER, a significant advantage. We findthat position-dependent systematics dominate our start-time sensitivity and limit our start-time resolution to 170µs (σto = 50 µs if noise limited).

10−2 100 102 104

10−4

10−3

10−2

10−1

frequency (hz)

σ (f

W/s

qrt(h

z))

1.74kg ZnMoO4 Detector (Proposed): Athermal Power Noise ( δ Pt)

SquidRloadRtesG: TES−AbsorberG: TES−BathG: Absorber−BathG: TES InternaltotalThermal SignalAthermal Signal

10−2 100 102 104

100

101

frequency (hz)

σ (p

A/s

qrt(h

z))

1.74kg ZnMoO4 Detector (Proposed): Current Noise

SquidRloadRtesG: TES−AbsorberG: TES−BathG: Absorber−Bath1/FtotalThermal SignalAthermal Signal

FIG. 11. Simulated noise referenced to TES power (top) andcurrent (bottom) for a 1.74kg ZnMoO4 detector.

10−2 10−1 100 101 102 103 104102

103

104

105

106

107

frequency [hz]

dIdE

[1/V

]

Position Dependence of Phonon Signals

Avg PulseSlow PulseFast Pulse

FIG. 12. Simulated pulse shapes for events from differentlocations in the absorber for a 1.74 kg ZnMoO4 detector.

XI. SINGLE PHOTON LIGHT DETECTOR

These very same design rules can also be applied to(hopefully) significantly improve the performance of verylow-temperature, large-area light detectors which are

14

used to distinguish electronic recoils from nuclear recoilsfor both dark matter and double-beta decay detectors.For CRESST, a single-photon sensitive detector wouldbe very beneficial because their low-mass WIMP sensi-tivity is suppressed by electron-recoil backgrounds leak-ing into the WIMP signal region [1], while for double-beta decay experiments, single-photon sensitivity couldpotentially be used for electron/nuclear-recoil discrimina-tion in non-scintillating crystals like TeO2 via Cherenkovlight collection [38].

The 30 mm × 30 mm Si chip used in the originalCRESST light detectors has a phonon heat capacity ofonly 220 fJ/K at 10 mK, and consequently the parasiticheat capacity of the Au wirebond between the Si TESchip and the large Si wafer (unfortunately) totally domi-nates the overall sensor heat capacity as outlined in TableIV. On the brightside, this allows us to increase the sizeof the Si chip to 80 mm diameter, thereby improving thescintillation collection efficiency without significant lossof energy sensitivity. The Au thin-film meander whichthermally connects the TES to the bath was chosen sothat τeff ∼ 50 ms, long enough to be minimally sensitiveto both the precise location of the scintillation absorptionin the Si and to stochastic fluctuations in the scintillationcreation/absorption time (O(1 ms) in many crystals), yetsmall enough to retain significant start-time informationthat could be vital for application of this technology todouble-beta decay with ZnMoO4 crystals[10]. Specifi-cally, the expected noise-limited optimal-filter start-timeresolution for a ∼ 1.5 keV scintillation signal accompa-nying a 3 MeV electron recoil in ZnMoO4 is found to be2.2 µs (assuming instantaneous photon release/capture).Clearly, this is a drastic underestimate for slowly scintil-lating crystals, but it suggests that pileup rejection withthe photon detector could be viable in fast scintillatingcrystals.

The expected energy-resolution performance of 0.36 eVis well below the single-photon quanta energy of ∼2 eVfor most scintillating crystals, and consequently this de-vice can be characterized as a highly efficient large-areasingle-photon detector. If additional sensitivity is re-quired, the TES could be directly fabricated onto theSi chip for improved performance. Of course, one wouldthen lose the fabrication rate benefit of being able tomake ∼ 20 devices per wafer. On the other hand, Siwafer processing is standard.

XII. CONCLUSIONS

The small heat capacitance of insulating and semicon-ducting crystals at very low temperatures suggests thateV-scale energy resolution is possible in massive, kilo-gram thermal detectors operating around 10 mK. Unfor-tunately, with achieved sensitivities 2–3 orders of mag-nitude worse than these expectations, the current gen-eration of detectors have yet to fully realize this po-tential. The dominant culprit for this discrepancy is

Si absorberVa Volume π(4 cm)2x525 µmMa Mass 6.0 g

TaoOperationtemperature

9.3 mK

Ca Heat capacity ΓSiVaT3ao 1.6 pJ

K

Au thin film thermal link between TES and bathltb Length 6.0 mmAtb Cross sectional area 5 µm x 300 nm

Gtb

Thermal conduc-tance from TES tobath

vfde3

ΓAuTtoAtbltb

23 pWK

Au thermalization film on absorber (p1)Vp1 volume π(3 mm)2x100 nm

Ct p1p1 component ofTES heat capacity

γAuVp1Tto 1.9 pJK

Gta

Thermal conduc-tance between TESand absorber

nepΣepVp1Tnep−1to 452 pW

K

Au wirebond between TES and absorberlw length 1.0 cmAw Cross sectional area π(7.5 µm)2

GwThermalconductance

vfde3

ΓAuTtoAwlw

5.4 nWK

Ct wBond component ofTES heat capacity

1.16 pJK

W TESAt Cross sectional area 1520 µm x 150 nmlt Length 300 µm

TcTransitiontemperature

10 mK

Ito Operating current 1.48 µA

TtoOperatingtemperature

9.3 mK

Gt intInternal TESconductance

LwfρW

Atlt/2

Tto 4.9 nWK

CtWW component ofTES heat capacity

fscΓWVtTto 0.18 pJK

Rn Normal resistance ρWltAt

100 mΩ

RtoOperatingresistance

Rn/4 20 mΩ

α Thermal sensitivity TtoRto

∂R∂Tt

∣∣∣Ito

16.4

β Current sensitivity ItoRto

∂R∂It

∣∣∣Tto

0.03

Dynamical time constants

τL/RSquid InductorTime Constant

15 µs

τeff Sensor Fall Time 56ms

τsa

Absorber/TESCoupling TimeConstant

2.6ms

Estimated Resolution

σEEstimated BaselineEnergy Resolution

0.36eV

σto

Estimated Tim-ing Resolution @1.5keV

2.2µs

TABLE IV. Optimized Light Detector

15

that electron-phonon coupling in both TESs and NTDsdrops rapidly with temperature, and thus the electronsystem within the sensor decouples from the phonon sys-tem within the absorber leading to the absorber phononsignal bandwidth being much smaller than the sensorbandwidth and/or shunting of the signal past the sen-sor through parasitic thermal conductance channels.

After a detailed study of current-generation detectors,we developed prototype designs for both an 80 mm di-ameter Si light detector and a 1.75 kg ZnMoO4 massivecalorimeter for which the sensor bandwidth is smallerthan the signal bandwidth yet still larger than an ex-pected 1/f noise threshold as well as the bandwidth re-quirements due to event rate. Consequently, we estimatedevice sensitivities of 0.4 eV and 3.4 eV that nearly matchthe expected scaling law performance. Further, we haveshown that these lower Gtb designs have achievable par-asitic power requirements via comparison to detectorsmade by SPICA/SAFARI for infrared spectrometry ap-plications.

Additionally, for fabrication simplicity and suppressionof cosmogenic backgrounds, both of these designs use asingle layer Au thin-film pad that is deposited directlyonto the absorber (using simple shadow mask techniques)for electron-phonon coupling rather than fabricating theentire TES onto the absorber. This pad is then thermally

connected via an Au wirebond to a separately fabricatedTES chip.

Finally, we have shown that position systematics on en-ergy estimators can still be suppressed to the level thatthey are subdominant compared to thermalization sys-tematics in both TeO2 and ZnMoO4 crystals. This isachieved despite purposely designing our detector to havelarge coupling between the electronic system within theTES and the phonon system in the absorber by requiringthat the sensor bandwidth be smaller than the athermalphonon collection bandwidth.

XIII. ACKNOWLEDGEMENTS

The authors would like to thank Ray Bunker, BlasCabrera, Raul Hennings-Yeomans, Kent Irwin, Alexan-dre Juillard, Yury Kolomensky, Nader Mirabolfathi,Wolfgang Rau, and Philippe Di Stefano for interestingand insightful discussions and comments.

Appendix A: Pertinent Material Properties

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16

Tungsten Properties

ΓWelectronic heat capacitycoefficient

107 Jm3K2 [39]

nepelectron-phonon thermalpower exponent

5 [40]

ΣepWelectron-phonon couplingcoefficient

0.32x109 Wm3K5 [40]

ρW W electrical resistivity 7.6x10−8Ωm [13]

fscsuperconductor/normalmetal heat capacity ratio

2.54

Gold Properties

ΓAuelectronic heat capacitycoefficient

66 Jm3K2 [41]

neAu free electron density 5.90x1028 1m3 [41]

vfAu fermi velocity 1.4x106ms

[41]

deAuelectron diffusion length(not annealed)

min(1µm,thickness)[42]

nepelectron-phonon thermalpower exponent

5

ΣepAuelectron-phonon couplingcoefficient

3.2x109 Wm3K5

Silicon Properties

ρnSi atomic number density 5.00x1028 1m3

TDSi Debye temperature 645K [43]

ΓSiphonon heat capacity coeffi-

cient: 12π4kbρnSi5T3DSi

0.6025 Jm3K4

CaWO4 Properties

ρn,CaWO4unit cell number density 1.267x1028 1

m3

TD,CaWO4Debye temperature 250K [44]

ΓCaWO4

phonon heat capacity coeffi-

cient:12π4kbρn,CaWO4

5T3D,CaWO4

2.6182 Jm3K4

ZnMoO4 Properties

ρn,ZnMoO4unit cell number density 1.267x1028 1

m3

TD,ZnMoO4Debye temperature 250K [44]

ΓZnMoO4

phonon heat capacity coeffi-

cient:12π4kbρn,ZnMoO4

5T3D,ZnMoO4

2.6182 Jm3K4

TABLE V. Pertinent Material Properties

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