Table Of Content21STINTERNATIONALSYMPOSIUMONSPACETERAHERTZTECHNOLOGY,OXFORD,23-25MARCH,2010.
Transition Edge Sensor Thermometry for On-chip
Materials Characterization
D. J. Goldie∗, D. M. Glowacka∗, K. Rostem∗† and S. Withington∗
∗Detector and Optical Physics Group
Cavendish Laboratory
University of Cambridge
Madingley Road
Cambridge
CB3 0HE, UK
1
1 Email: [email protected]
0 †Current Address
2
NASA-Goddard Space Flight Center
n
Greenbelt
a
J MD 20771 USA
1
3
]
ll Abstract—The next generation of ultra-low-noise cryogenic source at temperature Ts is given by the radiometer equation
a detectors for space science applications require continued ex-
s-h lpolworantoiiosneaonfdmgaotoedriaelnsercghyarsaecntseirtiisvtiitcysoaftculorwrentetmTprearnastiutiroens.ETdhgee σT2 = tmT∆s2f (1)
e Sensors (TESs) permitsmeasurements of thermal parameters of
m mesoscopicsystemswithunprecedentedprecision.Wedescribea wheretm isthemeasurementtimeand∆f isthemeasurement
radiometrictechniquefordifferentialmeasurementsofmaterials bandwidth.[2], [3] In practice the bandwidth is limited by the
t. characteristics at low temperatures (below about 3K). The source resistance and the input inductance of the SQUID to
a
technique relies on the very broadband thermal radiation that
m couples between impedance-matched resistors that terminate a a few 10’s of kHz. This gives σT ≃ 3mK for Ts = 500mK
with t = 1s. This is the precision that we found in
- Nb superconducting microstrip and the power exchanged is m
d measured using a TES. The capability of the TES to deliver practice.[4] TESs can be used to determine conductances by
n fast, time-resolved thermometry further expands the parameter measuring the power dissipated in the active region of the
o space: for example to investigate time-dependent heat capacity. device where electrothermal feedback (ETF) stabilizes the
c Thermal properties of isolated structures can be measured in TES at its transition temperature, T , as a function of the
[ c
geometries that eliminate the need for complicating additional
componentssuchastheelectricalwiresofthethermometeritself. temperature of the heat bath, Tb. These measurements are
1
Differentialmeasurementsalloweasymonitoringoftemperature widely reported particularly in the context of measurements
v
drifts in the cryogenic environment. The technique is rapid to of the thermal conductance of silicon nitride films, but a key
6
1 use and easily calibrated. Preliminary results will be discussed. limitation is that the technique measures thermal properties
9 averaged over a large temperature difference (i.e. between T
c
5 and T ) and the films under study must support additional
b
. I. INTRODUCTION
1 (generally superconducting) metalization to provide electrical
0 The problem of characterizing the thermal properties of connection. A problem arises if the thermal properties are
1 mesoscopic thin-film structures at low temperatures (below themselves a function of temperature. A technique for mea-
1
3K),inparticulartheirthermalconductancesandheatcapaci- suringconductancesorheatcapacitiesofmicron-scaleobjects
:
v tiesasafunctionoftemperature,remainsoneofthekeychal- rapidlyoverareasonabletemperaturerangewithoutadditional
Xi lengesfordesignersofultra-low-noisedetectors.Thisproblem overlying films certainly seems to be required. The proposed
is not confined exclusively to the detector community.[1] technique permits true differential measurements of conduc-
r
a These measurements require small, easily fabricated, easily tance (i.e with small temperature gradients), in a geometry
characterizedthermometers.Techniquesarealreadyusedsuch without complicating additional metalization. Measurements
as Johnson noise thermometry (JNT) using thin film resistors of heat capacities are also possible. The technique is easy to
as noise sources with dc-SQUID readout or measurements implement, simple to calibrate and rapid to use.
of thermal properties using Transition Edge Sensors (TESs) We recently demonstrated highly efficient coupling of very
butbothhavepracticallimitations.JNTcanperformmeasure- broadband thermal power between the impedance-matched
ments over a reasonable temperaturerange and is in principal terminationresistorsofasuperconductingmicrostriptransmis-
a primary thermometer but is in practice secondary because sion line.[5] The efficiency of a short microstrip (l 2mm
≃
of stray resistance in the inputcircuitto the SQUID thatmust was better than 97% for source temperatures up to 1.5K.
becalibrated.Theachievablemeasurementprecision,σ ,fora When the coupling efficiency is very high the power trans-
T
21STINTERNATIONALSYMPOSIUMONSPACETERAHERTZTECHNOLOGY,OXFORD,23-25MARCH,2010.
PTES δPh Ps whereν isthefrequency,Pν(Ti)=hνn(ν,Ti)andn(ν,Ti)is
the Bose-Einstein distribution, and we have assumed that the
Gm couplingisloss-less.IfthesourcetemperatureislowTs 3K
≤
C T C T allofthepoweriscontainedwell-withinthecut-offfrequency
TES c s s
of the microstrip andthe upperlimitof may be set to infinity.
G G Eq. 2 then has the solution
b sb
π2k2
P (T ,T )= b(T2 T2). (3)
m s c 6h s − c
T
b
The measurement conductance G (T ) = dP (T ,T )/dT
m s m s c s
Fig.1. Thermalcircuit ofasimplegeometry. is G (T ) = π2k2T /3h. G determines how changes in
m s b s m
source temperature affect power flow to the TES.
To calibrate the thermometry we need to know the low-
ferred along a superconducting microstrip between a source
frequency TES current-to-power responsivity, s (0). This is
I
andaTEScanbeusedasafast,accuratethermometer.[6]The
measured by applying slowly-varying power, δP , to the
h
practicaltemperaturesensitivityisdeterminedbytheTESlow-
known heater resistance on the TES island and measuring
frequency Noise Equivalent Power, NEP(0). The limiting
the change in detected current δI. The responsivity is then
temperature sensitivity is determined by the very-wide radio-
s (0)=δI/δP . As the source temperature is changed using
I h
frequency bandwidth of the microstrip coupling so that we
the source heater, the power incident on the TES changes as
substitute ∆f ∆ν in Eq. 1 and ∆ν = 34.5/T GHz/K is
s given by Eq. 3. Measuring the change in the current flowing
→
the equivalent rf-bandwidth of a blackbody source at temper-
throughtheTES,δI,determinesthechangeindetectedpower
ature T .
s δP =δI/s (0). Noting that δP =P(T ,T ) P(T ,T ),
m I m s c b c
−
and since ETF fixed the TES’s temperature at T , we can
c
II. TESTHERMOMETRY determine the source temperature to a good approximationas
We begin by reviewing the technique of TES thermometry δI 6h
T = +T2. (4)
for a simple geometry. In section II-B we describe the full s ssI(0)π2kb2 b
thermal circuit used in the measurements of conductances. In
The practical temperature measurement precision is deter-
sections II-C and II-D we describe how the conductances are
determined experimentally. mined by the low-frequency TES NEP(0) so that[6]
NEP2(0)
σ2 = . (5)
Ts 2t G2 (T )
A. A simple geometry m m s
Herewehaveintentionallyomittedthermalfluctuationsofthe
A thermal circuit for a simple geometry for TES thermom-
sourceislandwhichweconsiderasignalinthisgeometry.For
etry is shown in Fig. 1. A source of total heat capacity C
s NEP(0) = 2 10−17W/√Hz the achievable temperature
which could be a thermally isolated silicon nitride (SiNx) ×
precision is 30µK for t = 1s and T = 500mK. This is
island is connected to a TES also formed on a SiN is- m s
x two orders of magnitude better than JNT.
land by a conductance formed from a resistively-terminated
superconducting microstrip. Broadband power is transferred
B. The measured geometry
between the impedance-matched termination resistors of the
microstripwhichhasthermalconductanceGm.Overaportion InthefullgeometrytwosourceislandsS1,S2areconnected
ofitslengththemicrostripcrossestheSisubstrateso thatany bythesubjectundertesthereaconductance,G .Thesubject
12
phonon conductance is efficiently heat sunk to the bath. G may be more complicated.Each source island is connectedto
m
arises from conduction due to photons. The source and TES its own TES by a microstrip. Figure 2 shows the full thermal
areconnectedtoaheatbathattemperatureT byconductances circuit. We measure the quiescent TES currents to monitor
b
G and G respectively which include the contribution from and subtract small drifts in the substrate temperature or the
s sb
thephononconductanceofdielectricofthemicrostrip.Heaters electronics. The measurement precision of the temperature of
permit the source temperature to be varied and the TES to be either source, σ , with a measurementtime t is determined
T m
calibrated.Changesinthepowercouplingalongthemicrostrip by the low-frequencyTES NEP(0) so that
aterremmineaatsiuornerdesuissitnogr.tFhoerTloEwSswohuircchetiesminpecrloatsuerepsroTximit3yKto,aitlsl σ2 = NEP2(0) 1 + 1 + kbT12,2, (6)
s ≤ T1,2 2t G2 (T ) G2 (T ) C
of the power is contained within the pair-breaking threshold m (cid:18) m b m 1,2 (cid:19) 1,2
of the superconducting Nb, 2∆ /h 760GHz where h is and the temperaturemeasurementprecision for this differenc-
Nb
Planck’sconstantand∆ isthesuper≃conductingenergygap. ingapproachincludesacontributionfromthebathtemperature
Nb
The power transmitted between the source at temperature T measurement.We have also now explicitlyincludedthe effect
s
and the TES at its transition temperature T is given by of thermodynamic fluctuations in the temperatures of the
c
sources of heat capacity C since these fluctuations directly
2∆hNb affect the precision with w1h,i2ch the source temperaturecan be
P (T ,T )= [P (T ) P (T )]dν, (2)
m s c ν s − ν c determined.
Z0
21STINTERNATIONALSYMPOSIUMONSPACETERAHERTZTECHNOLOGY,OXFORD,23-25MARCH,2010.
PTES P0 P0+δP2 PTES P(T′,T)=G(T)δT.Ignoringfornowthesmallconductance
ofthemicrostrips,theinputpowersandresultanttemperatures
Gm G12 Gm are related by
C T C T C T C T
TES c 1 1 2 2 TES c P0 =P1b(T1,Tb) G12(T1,T2)(T2 T1) (8a)
− −
P =P (T ,T )+G (T ,T )(T T ) (8b)
0 2b 2 b 12 1 2 2 1
G G G G’ −
b 1b 2b b P =P (T′,T ) G (T′,T′)(T′ T′) (8c)
0 1b 1 b − 12 1 2 2− 1
P +δP =P (T′,T )+G (T′,T′)(T′ T′) (8d)
T 0 2 2b 2 b 12 1 2 2− 1
b
Subtracting 8a from 8c, 8b from 8d and using, for example,
Fig. 2. Thermalcircuit ofthe fullmeasurement. Thesubject under testis
shownhereasaconductance G12 butmaybemorecomplicated. T2′
′
P (T ,T ) P (T ,T ))= G (T)dT
2b 2 b − 2b 2 b 2b (9)
ZT2
=G (T )δT ,
2b 2 2
T ’
T2 2 where T = (T′ +T )/2 and the final equality follows since
u. 2 2 2
n a. tsettle tm tsettle tm tsettle tm δT2 is small, we find
nd I i1,2 Tb T1 T1’ aδnPd2T=2G=12((TT12+)(δTT′2+−δTT1+)+TG′)2/b4(.T2F)iδnTal2ly−tGhe1b(eTff1e)cδ(tT1o01f)
a 12 1 1 2 2
2 P + δ P conductance along the microstrip needs to be included. The
P1, 0 Tb P0 0 2 result is
G (T )=
12 12
Time in a.u.
δP G (T )+G (T ) δT + G (T )+G (T ) δT
2 2b 2 m 2 2 1b 1 m 1 1
Fig. 3. Schematic showing the time variation of the input powers (lower −
traces) andoutput currents (upper traces) foronecycle ofthe measurement. (cid:0) 2(δT(cid:1)2−δT1(cid:0)) (cid:1)
The sequence of measurement and settling times is indicated. The full . (11)
time-sequence represents a measurement frame. The measurement intervals
determine thetemperatures asshown.
D. Measurement of the conductances G , G
1b 2b
The conductance to the bath at a given temperature is
C. Measurement of the conductance G
12
measured in a two-step procedure where we measure the
We measure the interconnect conductance G in a three
12 quiescent TES current (P = 0) and the effect of applying
step process. A schematic of the measurementcycle is shown
equal power P to both islands and measuring the changes
0
in Fig. 3. In thefirst step the quiescentTEScurrentsaremea-
in T and T . The next temperature sample uses incremented
1 2
sured.ThendcpowerP isinputtobothsourceislandsraising
0 P . This measurement is rapid. With t = 0.82s, 500 data
0 m
the temperatures from the bath temperature T to T , T . We
b 1 2 pointsareacquiredinlessthan15minutes.Intheanalysiswe
find T = T and we quantify the effect of T = T later.
1 ∼ 2 1 6 2 use Eqs. 8a and 8b and assume T1 = T2. This introduces an
Finally power to S is stepped by an additionalsmall amount
2 inevitable error in the analysis and the magnitude is of order
δP raising the island temperatures to T′ and T′. Since both
2 1 2 ǫ = G (T )(T T )/P . Experimentally the error is
T′ T andT′ T aresmallthemeasurementisdifferential. | 12 12 2 − 1 | 1,2b
1− 1 2− 2 small 1%.Afifth-orderpolynomialisfittedtotheT1,2 P0
For the next temperature sample the process is repeated with ≪ −
data and the conductances G , G found by differentiation.
1b 2b
incremented P . The power steps are adjusted in software
0 Note that since the temperature difference T T may be
1,2 b
to give approximately constant increments and differences in −
large this measurement is temperature-averaged.
temperature across the range of temperatures measured. One
complete measurement cycle (i.e. P =0,P ,P +δP ), with
0 0 2 III. MEASURED DEVICE
a shortsettlingtime t betweeneachmeasurementstep to
settle
accommodate thermal response times, defines a ‘frame’ time. An optical image of the device described here is shown
The frame must be measured in a time less than the Allan in Fig. 4. The measured conductance labelled G12 is a long
time of the system. thin SiNx bar of dimensions 500 10 0.5µm3 formed
× ×
ThepowerflowacrossaconductanceGconnectingthermal by reactive ion and deep reactive ion etching of a nitride-
reservoirs at temperatures T, T′ can be written for notational coatedSiwafer.G12 carriesnoadditionalfilms.G12 connects
convenience as two larger 0.5µm-thick nitride source islands S1 and S2
T′ themselves isolated from the Si wafer by four supporting
P(T′,T)= G(T)dT =G(T′,T)(T′ T), (7) nitridelegs, two of length255µmtwo oflength358µmeach
−
ZT of width 15µm. One of the longer legs also carries a Nb
where the over-set line denotes averaging. If T′ T = δT microstripline.Themicrostripisterminatedbyanimpedance-
−
is small then the power flow can be linearized so that matched AuCu termination resistor at each end. Each island
21STINTERNATIONALSYMPOSIUMONSPACETERAHERTZTECHNOLOGY,OXFORD,23-25MARCH,2010.
10-5
10-6
τ
10-7 A
A
2
σ
10-8
τ-1
10-9 τ2
10-10
10-3 10-2 10-1 100 101 102
τ in s
Fig. 4. Optical image of the device under test. The central nitride bar
(labelled G12) of length 500µm, width 10µm connects the nitride source Fig. 5. Allan plot for the two SQUID systems with biased TESs and
tioslaTnEdsSsSi1n,tSh2e.uRpepseirsticvoerlnyertseromfinthaetedimmagicer.oTsthriepcloinnedsucrutanncferombettwheeeinslaenacdhs Tthbe=min2i5m9u±m0o.f5tmheKs.taTtihseticopσtA2im.um integration time, identified as τA, is at
nitrideislandandtheheatbathisformedfromfournitridelegs,twooflength
285µm,twoof355µmthatrundiagonally .
σ =0.54 mK
T
30.0 cy10
also supports a square AuCu resistor with Nb bias lines that n
e
u
can be used as a heater to modulate the temperature of the mK 20.0 Freq 5
issoluanrcdess.wThhicehmsiucpropsotrrtipTsErSusn.oRvoeurttihnegSoifwthaefemrtihceronsotrniptosninitrtihdies in 12 10.0 0 382 T i3n8 m4K 386
way ensures that the phonon conductance associated with the T 0.0
δ
microstrip dielectric is efficiently heat-sunk to the bath. The
TES islands also include AuCu resistors with Nb bias lines -10.0
thatallow the power-to-currentresponsivityofthe TESsto be
-20.0
measured. 0.0 0.2 0.4 0.6 0.8 1.0 1.2
AuCu resistors are 40nm thick, Nb bias lines and the T in K
ground plane of the microstrip are 250nm thick. The mi- Fig. 6. Difference in temperature between the two islands as a function
crostrip dielectric is sputtered Si0 and is 400nm thick. The of average temperature for equal applied powers. Note the improvement in
2
themeasurementprecisionasthetemperature increases.Theinsetshowsthe
TESsuseourstandardhighertemperatureMoCubilayerlayup
measuredtemperatureprecisionforrepeatedmeasurementsnear384mK.The
with40and30nmofMoandCurespectively.Thefabrication measuredprecisionσT =0.54mKcompareswellwiththecalculated value
route is identical with our usual process for MoCu TESs.[7] from Eq. 6 which gives σT = 0.47mK for the measurement time used
The TESs are voltage biased and read-out with SQUIDs. tm=0.82sandthemeasuredNEP(0)=1.2×10−16WHz−1/2.
The device is measured in a He-3 refrigerator with a base
temperature of 259 mK. The transition temperature of the
lines show the expected behaviour for white noise with a
TESs described here was Tc ∼ 485mK which is slightly slope τ−1 and for drift with a slope of τ2. This shows that
higherthanreportedinourearlierworkwiththesameMo-Cu
drift limits these measurements. Guided by the Allan plot,
bilayer lay-up. As a result the conductance to the bath of the
we chose a sample time t = 0.82s being 214 data points
TESs is increased and the measured TES Noise Equivalent m
sampled at 20kHz and t =100ms. For the conductance
Power is NEP(0)=1.2 10−16WHz−1/2. settle
× measurements with three power steps the total frame time is
2.8 s at each sample temperature.
IV. RESULTS AND DISCUSSION Applying equal power to both sources, G and G were
1b 2b
The Allan variance statistic, σ2(τ) where τ is the inte- determined. As discussed earlier the analysis introduces an
A
gration time, provides an exceptionally powerful diagnostic unavoidableerrorinthemeasurementofG andG depend-
1b 2b
of system stability and a plot of the variance as a function ing on T T . Figure 6 shows the difference in temperature
2 1
−
of τ identifies the optimum time for signal averaging.[8], δT =T T for equal powers applied to both islands as a
12 2 1
−
[9] In the Allan plot, a log-log plot of σ2 as a function functionofaverageislandtemperature.Thedifferenceissmall
A
of τ, underlying fluctuations with frequency-domain power andamaximumofabout5mKat500mK.SinceδT issmall
12
spectra varying as 1/fα show a τ(α−1) dependence. Hence wewillseethattheerrorisnegligible.Thetemperaturediffer-
white noise with α = 0 has a τ−1 characteristic. 1/f noise enceimpliesadifferenceinconductanceforthetwonotionally
shows no dependence on integration time and drift exhibits identical conductancesG , G of about 2% between 0.26
1b 2b
±
a dependence with 1 < α < 3. Figure 5 shows measured and 1.3 K. The temperature dependence of δT evident in
12
Allan variances for both SQUID systems with biased TESs Fig. 6 is also unexpected.It does notseem that the difference
andthebathtemperatureatT =259 0.5mK.Themeasured can be accounted for by experimental uncertainty (such as
b
±
characteristic indicates that white noise is reduced by time- in the calibration of the TES responsivities). One possibility
integration up to a maximum of order τ 5s. The dashed might be differences in the actual coupling efficiencies of the
A
≃
21STINTERNATIONALSYMPOSIUMONSPACETERAHERTZTECHNOLOGY,OXFORD,23-25MARCH,2010.
104 0.25
0.20
103
G in pW/K102 G , G T2 GTES G/G121b 00..1105 A/l ratio
1b 2b
101
0.05
G
12 T
100
0.1 1.0 0.5 1.0 1.5
T in K T in K
Fig.7. MeasuredconductancestothebathG1b,G2bandcalculatedsample Fig.8. Ratioofthemeasuredconductance G12 totheconductance tothe
conductance G12. Estimated errors for G12 are indicated. The dotted lines bathG1b.Thearrowindicatestheexpectedratioiftheconductances scaleas
indicate dependencies proportional to T2 and T. The single circle is the theareatolengthratioofthestructures.
measured GTES scaled by the A/l ratios to compare directly to G1b and
G2b.
the smaller contribution from the Nb wiring all of which
we assume are equal to that of SiN . The variation of the
x
microstrip lines. ratio is experimentally significant and may already illustrate
The inset of Fig. 6 shows a histogram of calculated tem- the difference between thermal properties averaged over a
peratures for repeated measurements of a source temperature temperature range and the true differential measurement.
near 384mK. The measured precision σ = 0.54mK com-
T
pares well with the value calculated from Eq. 6 which gives V. CONCLUSION
σt = 0.47mK for the measurement time used tm = 0.82s We have described our first true-differential measurements
and the measured NEP(0)=1.2 10−16WHz−1/2. of thermal conductances of a micron-scaled object at low
×
Figure 7 shows measured conductances G1b, G2b and G12 temperaturesusing microstrip-coupledTES thermometry.The
as a function of temperature. Representative error bars for temperatureprecisionis alreadysignificantly greater than that
G12 determined from the variance of repeated measurements achievable with JNT with the same measurement time. The
are indicated. We can now estimate the magnitude of the measurements are rapid and easily calibrated. The achieved
error in G1b and G2b. The maximum temperature difference precision already strongly suggests that the thermal transport
occurs near 500mK where δT12 5mK is greatest. We find characteristics of the nitride structure are not described by a
≃
ǫ 0.2%, but less than this over most of the temperature simple power-law across the temperaturerange 0.26 to 1.5K.
≃
range. This is considered acceptable. We also show the vari- The device under test can be fabricated without additional
ation with temperature if G = kTβ with k a constant. The metalization for wiring. It should be straight-forward to in-
dottedlinesshowdependenciesβ =1and2.Thereisastrong clude specific layers on the nitride test structure to measure
suggestion here that a simple power law does not account for particular thermal properties in a controlled manner. In the
theconductanceacrossthismeasurementrange.Atthehighest future we expectto be able to explore the temperaturedepen-
temperatures β > 2, at the lowest β < 2. A reduction of the dence of heat capacities of thin films such as SiO including
2
exponent may be expected at low temperatures if dominant possibly measurements of time dependent heat capacity. We
phononwavelengthsstarttobecomecomparabletothenitride will explore the effect on conductance of superposed layers:
thickness. The single dot in Fig. 7 is the measured GTES for for example, does thermal conductance in thin multilayers
one of the TESs obtained in the standard way by measuring really scale as total thickness? We also see the possibility of
the power plateau in the ETF region of the current-voltage exploring the engineering of the nitride to realise phononic
characteristicasafunctionofthebathtemperature.Thissingle structures,toachievereductionsoftheconductanceincompact
point represents approximately 2 hours of data taking. By nitride structures needed for the next generation of ultra-low-
contrasttheconductancestothebathG1b,G2bplottedinFig.7 noise detectors.
are found from 500 measurements of temperature acquired in
15minutes.A comparableacquistiontimemeasuresG12 with ACKNOWLEDGMENT
50 data points with additional signal averaging.
We would like to thank Michael Crane for assistance
Figure 8 shows the ratio of G12 to G1b as a function of with cleanroom processing, David Sawford for software and
temperature. The simplest model for power flow along, or
electrical engineering and Dennis Molloy for mechanical en-
the conductance of, a uniform bar would assume that the
gineering.
power scales as the ratio of cross-sectionalarea, A, to length,
l. From the dimensions of the nitride support legs for the
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12
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