NASA Technical Reports Server (NTRS) 20090038699: Longitudinal Proximity Effects in Superconducting Transition-Edge Sensors

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Longitudinal Proximity Effects in Superconducting Transition-Edge Sensors
John E. Sadleir,1,2,∗ Stephen J. Smith,2 Simon R. Bandler,2 James A. Chervenak,2 and John R. Clem3
1Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801-3080
2NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 207701
3Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa, 50011–3160
(Dated: October14, 2009)
Wehavefoundexperimentallythatthecriticalcurrentofasquaresuperconductingtransition-edge
sensor(TES)dependsexponentiallyuponthesidelengthLandthesquareroot ofthetemperature
T. Asaconsequence,theeffectivetransitiontemperatureTc oftheTESiscurrent-dependentandat
fixedcurrentscalesas1/L2. WealsohavefoundthatthecriticalcurrentcanshowclearFraunhofer-
like oscillations in an applied magnetic field, similar to those found in Josephson junctions. The
observedbehaviorhasanaturaltheoreticalexplanation intermsoflongitudinalproximityeffectsif
theTESisregardedasaweaklinkbetweensuperconductingleads. Wehaveobservedtheproximity
effect in these devices overextraordinarily long lengths exceeding100 µm.
PACSnumbers: 74.25.-q,74.78.Bz,74.25.Op
A superconductor cooled through its transition tem- SincethesquarebilayersattheheartoftheTESarecon-
perature T while carrying a finite dc bias current nected at opposite ends to superconducting leads with
c
undergoes an abrupt decrease in electrical resistance transition temperatures well above the intrinsic transi-
from its normal-state value R to zero. Super- tiontemperatureofthe bilayers,superconductivity isin-
N
conducting transition-edge sensors (TESs) exploit this duced longitudinally into the bilayers via the proximity
sharp transition; these devices are highly sensitive re- effect. As we shallexplain later,many ofthe basic prop-
sistive thermometers used for precise thermal energy ertiesofourTESstructuresarewelldescribedbyregard-
measurements.1 TESmicrocalorimetershavebeendevel- ing them as SS′S or SN′S weak links.9–11
oped with measured energy resolutions in the X-ray and In this paper we report the properties of TESs based
gamma-raybandof∆E =1.8 0.2eVFWHMat6keV,2 onsquare(L L)electron-beam-depositedMo/Aubilay-
and∆E =22eVFWHMat97±keV,3 respectively—with ers consisting×of 55 nm Mo layers (T 0.9 K) to which
c
∼
thelatterresultatpresentthe largestreportedE/∆E of 210 nm of Au is added. The square side lengths L range
any non-dispersive photon spectrometer. TESs are suc- from 8 µm to 290 µm, and the normal-state resistance
cessfully used across much of the electromagnetic spec- per square is R =17.2 0.5 mΩ. The bilayers are con-
N
±
trum, measuring the energy of single-photon absorption nectedatoppositeendstoMo/Nbleadshavingmeasured
events from infrared to gamma-ray energies and photon superconducting transition temperatures of 3.5 and 7.1
fluxes out to the microwaverange.1 Despite these exper- K.12Furtherdetailsonthedevicefabricationprocesscan
imental successes, the dominant physics governing TESs be found in Ref. 13.
biased in the superconducting phase transition remains Ourmeasurementsaremadeinanadiabaticdemagne-
poorly understood.1 tizationrefrigerator(ADR)withmu-metalandNbenclo-
To achieve high energy resolution it is important to sures providing magnetic shielding for the TES devices
control both the TES’s Tc and its transition width ∆Tc. and SQUID electronics. The magnetic field normal to
Because the energy resolution of calorimeters improves the TES device plane is controlledby a superconducting
withdecreasingtemperature,they aretypicallydesigned coilwith the field value determined from the coilgeome-
to operate at temperatures around 0.1 K. For a TES, try and current. Measurements of the TES resistance R
this requires a superconductor with Tc in that range. are made by applying a sinusoidal current of frequency
While there exist a few suitable elemental superconduc- 5-10 Hz and amplitude I 50-250 nA, with zero dc
bias
∼
tors,thebestresultshavebeenachievedusingproximity- component, to the TES in parallel with a 0.2 mΩ shunt
coupled,superconductor/normal-metal(S/N)bilayers2,3, resistor(R ). The time-dependent TES currentis mea-
sh
for which Tc is tuned by selection of the thicknesses of sured with a SQUID feedback circuit with input coil in
the S and N layers.4 series with TES. When I is less than the TES crit-
bias
Therehavebeenavarietyofmodels4–8 usedtoexplain ical current I , R is zero, and all the ac current flows
c
the noise, T , and ∆T in TES bilayers, all assuming through the TES. However, when I > I and R > 0
c c bias c
spatiallyuniformdevices. Thoughsomehavebeenshown during part of the ac cycle, the TES current becomes
tobeconsistentwithcertainaspectsofparticulardevices, non-sinusoidal, and its maximum value I becomes less
they donotexplainmeasuredT and∆T inS/Nbilayer than I . The TES resistance R at the TES current I
c c bias
TESs generally. is then determined from R=R (I I)/I.
sh bias
−
In this paper we emphasize the importance of a phe- ThecriticalcurrentI ismeasured,withtheADRheld
c
nomenonthatsofarhasbeenneglectedinprevioustheo- atconstanttemperature, byramping the dc bias current
reticalstudiesofTESs: thelongitudinalproximityeffect. from zero and defining I as the TES current at the first
c
2
measured finite resistance (R 10 µΩ) across the TES.
∼
Record averaging is used at higher temperatures where
I becomes small.
c
The solid curves in Fig. 1(a) and (b) show measure-
ments of the critical current I over seven decades vs
c
temperature T. Note that although we find the in-
trinsic transition temperature of the Mo/Au bilayer is
T = 170.9 0.1 mK, at very low currents a zero-
cw
±
resistance state is measured up to much higher temper-
atures as the TES size is reduced, three times T for L
cw
= 8 µm. On the other hand, for the larger TES sizes
(L=130and290µm) the criticalcurrentI (T)decreases
c
rapidly with T near T . The observed I behavior as
cw c
functions ofbothT andthe lengthL providesstrongev-
idence that our TESs behave as weak-link devices. The
dotted curves in (a) and (b) show calculated values of
I using the Ginzburg-Landau theory described below.
c
In addition, at appropriately chosen temperatures, the
critical currents of these devices exhibit Fraunhofer-like
oscillationsasafunctionofanappliedmagneticfield,be-
haviorcharacteristicof Josephsonweaklinks.10,14–16 See
Fig. 1(c) for an example.
Because I depends upon T and L, the effective tran-
c FIG.1: (Coloronline)(a)andinset(b)Measured(solidlines
sition temperature T of the TES (the temperature at
c and markers) and theoretical (dotted curves) critical current
which an electrical resistance first appears, i.e., R
∼ Ic versus temperature T for square TESs with side lengths
10µΩ)is bothcurrent-dependentandlength-dependent. L ranging from 8 to 290 µm. The bold continuous segments
Figure 2(a) exhibits these effects. The points labeled at thelowest currentsare obtained byrecord averaging. The
T (I,L) are the effective transition temperatures at five intrinsic transition temperature of the Mo/Au bilayer weak
c
differentcurrentlevels(10 nAto 100µA) forthe datain links is Tcw = 170.9±0.1 mK (thin vertical lines). For T
Fig.1(a)and(b), showingthatT T foreachcurrent somewhatlargerthanTcw,Ic decaysapproximatelyexponen-
c cw
levelscalesapproximatelyas1/L2−(solidcurvefits)forL tiallywiththesquarerootofT−Tcw. Icalsodependsstrongly
upon L, which is particularly noticeable for the smaller de-
rangingfrom8to 290µm. ForeachL,T T depends
c cw
− vices. T and Lvaluesof theconstant current contours(hori-
upon the current.
zontaldashedlines)areplottedinFig.2(a). Inset(c)showsIc
Also shown in Fig. 2(a) are temperatures T
R=0.1RN vsappliedfieldfortheL=29µmdeviceshowingFraunhofer-
and TR=0.5RN for which the resistances are R = 0.1RN likeoscillations, similar tothoseseen in Josephson junctions,
and 0.5RN, respectively, from which we define ∆TR = providing further evidence that the TES exhibits weak-link
T T . We also define transition widths from behavior.
0.5RN − 0.1RN
the I measurements ∆T = T T and
c c1 c100nA c10nA
−
∆T = T T . In Fig. 2(b) we show that
c2 c1µA c10nA
−
these three measures of the transition width all vary ap- φ0 =h/2e.
proximately as 1/L2, shown by the dotted line. It also Near the center of the weak link, where f becomes
follows that T T scales linearly with the transition very small, the local penetration depth λ = λ /f be-
c cw r
−
width. comes very large. Moreover, in a thin film of thick-
Weak links in various SS′S or SN′S weak-link struc- ness d<λ, magnetic fields and currents spread out over
tureshavebeenstudiedexperimentallyandtheoretically the two-dimensionalscreening length (or Pearllength19)
bynumerousauthors. However,hereweuseasimplever- Λ = 2λ2/d = 2λ2r/df2. For all of our samples there is a
sionofGinzburg-Landau(GL) theory17,18 to explainthe range of temperatures T far enough above Tcw that at
resultsshowninFigs.1and2. This theory describesthe the center of the weak link Λ L and the current den-
≫
complex order parameter ψ(r), whose absolute square sity is j = xˆjx = xˆI/Wd, where I is the TES current.
ψ(r)2 is the superfluid density in the weak link. We The first and second GL equations given in Refs. 17 and
e|mplo|y the substitution ψ = ψ feiγ, where ψ is the 18 then depend only upon x and can be written as
r r
magnitudeoftheorderparameteratthereferencepoints
x = ±L/2 adjacent to the leads, f = |ψ|/ψr is the nor- f′′+ (t−1)f + κ2f3+ ˜j2 =0 (1)
malized order parameter, and γ is the phase. At the − ξ2 λ2 λ2f3
w r r
reference points, the local value of ψ2 is inversely pro-
r
portionalto the squareof the localpenetrationdepth λ and
r
via ψ2 = m/4µ e2λ2, and a characteristic reference cur-
r 0 r
rentdensityj canbedefinedviaj =φ /2πµ λ3,where ˜j =j /j = λ f2(γ′+2πA /φ ), (2)
r r 0 0 r x r − r x 0
3
f(x) df2
=x. (6)
Zf02 P(f2)
Thegauge-invariantphasedifferenceacrosstheweaklink
is20
L/2 2πA 2˜j 1 df2
′ x
φ= γ + dx= . (7)
−Z−L/2(cid:16) φ0 (cid:17) λr Zf02 f2P(f2)
TheintegralsinEqs.(5),(6),and(7)canbeevaluated
numerically as in Ref. 20 or in terms of elliptic integrals
as in Refs. 21 and 22. For given values of λ , κ, ξ , t,
r w
and L, the solutions of Eq. (5) reveal that ˜j is a single-
valued function of f , starting with the value ˜j = 0 at
0
f = 0, initially increasing linearly with f , rising to a
0 0
maximum value defined as ˜j , then returning to zero at
c
a larger value of f .
0
When t>1, the above equations reveal that ˜j(φ) is a
single-valued function of φ and has a functional depen-
dence close to˜j =˜j sinφ, similar to that ofa Josephson
c
junction. For 0 φ π, the reduced order parameter
f(˜j,x) at x = 0≤has≤its maximum value f = f(0,0)
00
when φ = 0, its minimum value 0 when φ = π, and
FIG. 2: (a) Measurements of the effective transition temper- a value between these two limits at the critical current
ature Tc at different currents and lengths. Markers Tc(I,L) when ˜j =˜jc and φ π/2.
≈
give the effective Tc from constant current contours of the When T > Tcw and L ξ(T) = ξw/√t 1, f 1
Ic(T,L) data in Fig. 1, with solid curves being 1/L2 fits for for a large fraction of the≫length L, and on−e may o≪mit
each current level. Markers T(R) give temperatures where thetermproportionaltof3 ontheright-handsideofEq.
R=0.1RN and 0.5RN. (b) Three different measures of the (1)toobtainthereducedorderparameterf(˜j,x). Inthe
transition width defined as differences between pairs of cor-
absence of a current, f(0,x) = f cosh(x/ξ)/cosh(L/2ξ)
r
responding points shown in (a), as labeled ∆TR, ∆Tc1, and near the center of the weak link, f = f(0,0) =
∆Tc2, showing 1/L2 scaling (dotted line). f /cosh(L/2ξ) 2f e−L/2ξatthecente0r0,andthegauge-
r r
≈
invariant phase difference across the weak link is φ = 0.
The parameter f , which is of the order of unity, would
where t = T/T is the reduced temperature, ξ(T) = r
cw beequaltounityifthelinearizedGLequationwerevalid
ξ /t 11/2 is the temperature-dependent coherence
w | − | overtheentirelengthLoftheweaklink;thesuppression
length, κ is the dimensionless Ginzburg-Landau param-
of f below unity occurs because the exact solution for
eter, A is the vector potential, and the primes denote r
x f(0,x)nearx L/2is stronglyinfluencedbythe term
derivatives with respect to x. (κ/λ )2f3 on t≈he±right-hand side of Eq. (1).
In this paper we are concerned chiefly with weak-link r
For nonzero current, f = f(˜j,0), the reduced order
behavior for which f(x) is an evenfunction of x and has 0
parameter at the center of the weak link, is suppressed
a minimum in the middle of the weak link, f(0) = f ,
where f′(0)=0. We can obtain an equation that dete0r- below f00, and the gauge-invariant phase difference φ
across the weak link obeys sin(φ/2) = f /f . The re-
mines how f depends upon L, t, and ˜j by multiplying 0 00
Eq. (1) by f0′, integrating the result, multiplying by f2, duced current is given by
and taking the square root, which yields the following
λ 4f2λ f f 2
equation, valid for 0 x L/2, ˜j = rf2 sinφ r r 0 1 0 e−L/ξ, (8)
≤ ≤ 2ξ 00 ≈ ξ f00 s − f00
df2/dx=P(f2), (3) (cid:16) (cid:17) (cid:16) (cid:17)
suchthatthereducedcriticalcurrentisgivenforT >T
cw
where and any L by the approximation
P(f2)= 2(f2 f2) κ2f4+ 2(t−1)+κ2f02 f2+ 2˜j2 , ˜jc =jc/jr =(λr/2ξ)f020 ≈(2fr2λr/ξ)e−L/ξ (9)
s − 0 hλ2r (cid:16) ξw2 λ2r (cid:17) λ2rf02i at the maximum of ˜j, where f0 = f00/√2 and φ = π/2.
(4) From Eq. (9), we may obtain the critical current as
suchthatf0 andf(x)canbe obtainedfromthe integrals Ic = jcLd = jr˜jcLd. Inferring Tcw = 170.9 mK from
the experimental data in Fig. 1(b) for L = 290 µm and
1 df2 L assuming κ = λ /ξ , we obtained ξ and λ by fitting
= and (5) r w w r
Zf02 P(f2) 2 the experimental Ic data for L = 8 µm at 250 mK and
4
375 mK. The dotted curves in Fig. 1(a) and (b) show I square root of the temperature T, (b) both the current-
c
calculatedusingξ =738nmandλ =79nminEq.(9) dependent effective transition temperature T and the
w r c
and Eq. (5), from which f was obtained.23 transition width scale as 1/L2, and (c) the TESs show
00
UnderconditionsforwhichEq.(9)isvalid,ifwedefine clearFraunhoferoscillationsasafunctionofappliedmag-
the effective transitiontemperature T (˜j) as the temper- netic field, characteristic of Josephsonweak links. It fol-
c
ature at which the first voltageappears along the length lows that the strength of superconducting order is not
of the TES when it carries a reduced current density uniformovertheTES.Ourfindingshaveimplicationson
˜j, we can determine T (˜j) or t (˜j) = T (˜j)/T by set- TES magnetic field sensitivity, which impacts required
c c c cw
ting ˜j = ˜j in Eq. (9) and solving for t (˜j), noting that limits on ambient magnetic field magnitude and fluctu-
c c
ξ =ξ /√t 1. The result is ations in TES applications. Proposed uses of the lon-
w
−
gitudinal proximity effect for TES applications include
(Tc−Tcw)/Tcw =(ξw2/L2)ln2(2fr2λr√tc−1/˜jξw). (10) (1) tuning the effective Tc of TES arrays by changing
L in mask design, which could compensate for bilayer
Since the dependence upon tc on the right-hand side is Tcw variability1 andincreaseyield,and(2)makingsmall
veryweak,becauseitappearswithintheargumentofthe TESs consisting of superconducting leads separated by
logarithm, Eq. (10) predicts that the current-dependent normalmetal,suchasAuwithT =0,avoidingtheuse
cw
transition temperature of the TES should scale very of S/N bilayers.
nearly as T T 1/L2 and that T should increase
c cw c
− ∝
as the square of the logarithm of the inverse TES cur- Our work at Goddard was partially funded under
rent. Similarreasoningleadstothe conclusionthatboth NASA’s Solar and Heliospheric Physics Supporting Re-
∆T1 and ∆T2 scale as 1/L2. Scaling of ∆TR can be un- searchandattheAmesLaboratorybytheDepartmentof
derstood using a simple model of the resistive transition Energy-BasicEnergySciencesunderContractNo. DE-
based on the assumption that R = (2xj/L)RN, where, AC02-07CH11358. We thank J.Beyer(PTBBerlin)and
for a given reduced current density ˜j, xj is the solution K.Irwin(NISTBoulder)forprovidingtheSQUIDSused
of ˜j =(λ /2ξ)f(0,x )2. in this work. We also thank F. Finkbeiner, R. Brekosky,
r j
We conclude that TESs behave as weak links. This andD.Kellyforessentialrolesindevicefabrication,and
conclusionisbasedonourexperimentalfindingsthat(a) C. Kilbourne, I. Robinson, F. S. Porter, R. Kelley, and
thecriticalcurrentatthefirstonsetofavoltagealongthe M. Eckart for useful discussion of these results and the
length depends exponentially upon the length L and the manuscript.
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