The superconducting transition in 4-D: temperature,current, resistance and heat capacity

M. A. Lindeman1*, K. A. Barger1, D. E. Brandl1, S. G. Crowder1,L. Rocks1, D. McCammon1, H. F. C. Hoevers2

1Department of Physics, University of Wisconsin, Madison, 53706, U. S. A.2SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA

Utrecht, The Netherlands

Received July 22, 2007, Accepted September 15, 2007We measured the superconducting phase transition of a transition-edgesensor (TES) provided by SRON. Using newly developed techniques, weobtain thousands of impedance measurements to find sensitivity

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α I, currentdependence

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βI , and heat capacity of the TES in the phase transition atvarious temperatures and bias currents. The resulting data illustrate theshape of the phase transition and probe the internal state of the device.PACS numbers: 74.25.Bt, 85.25.Am, 85.25.Oj.

1. INTRODUCTION

The resistance

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R of a transition edge sensor is a function of bothtemperature

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T and current

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I . A TES is a useful thermistor because of itslarge sensitivity to temperature, represented by the derivative

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α I = (R /T)(∂R /∂T) . However the current dependence, represented by

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βI = (R /I)(∂R /∂I), is significant and affects the performance of thesedevices. It is a parameter of the microcalorimeter dynamical model1 and of amodel non-equilibrium Johnson noise in a TES2. Despite the significance todetector optimization, measurements of large sections of the

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R(I,T) surfaceand its derivatives have not been previously published. This may be largelybecause it has been difficult to separate the effects of Joule heating on theTES resistance

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R (associated with

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α I) from more direct effects of biascurrent (associated with

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βI ). We present such data here, acquired usingnewly developed methods3.

The heat capacity

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C of a TES changes significantly as it goes fromsuperconducting to the normal metal state. Measurements of the way theheat capacity changes through the transition probe the internal state of theTES. We present measurements of the heat capacity as a function of current

* mlindeman@wisc.edu

M. A. Lindeman et al.

and temperature

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C(I,T) and contrast them with previous measurements ofheat capacity in the phase transition.

2. EXPERIMENTAL METHOD

We measured the superconducting phase transition of a transition edgesensor (TES) provided by Netherlands Institute for Space Research. Thedevice is a bilayer (12 nm of Ti / 53 nm of Au) suspended on a 1 µm SiNmembrane. It is rectangular TES (30 µm in the direction of current flow by160 µm wide), with no zebra stripes.

We measured a series of current vs. voltage (I-V) characteristics of thephase transition by sweeping the TES bias with a 0.01 Hz triangle wave andmeasuring the current in the TES using a Quantum Design dc SQUID, whileletting the bath temperature of the fridge drift slowly upward. We recordedthousands of measurements of TES current, bias voltage and bathtemperature. This data is plotted in Fig. 1a and 1b.

Data from temperatures above the phase transition was used toprecisely measure the thermal conductance

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G(T) from the TES through themembrane phonons to the bath. This conductance measurement isindependent of the I-V data in the phase transition. We used the thermalconductance to accurately compute the TES temperature

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T from the I-V andbath temperature, which is plotted on the x-axis of Fig. 1a. and 1b.

Simultaneous to the I-V measurements, we used 3 lock-in amplifiersand the equivalent circuit method4 to record the impedance of the TESimpedance at 3 frequencies. An example of these measurements and best fitof a TES calorimeter model5 are plotted in Fig. 1c. The three free parametersin the fit are

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α I,

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βI , and heat capacity

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C of the TES. A singularity occurs inthe impedance where the TES’s dynamic resistance is infinite. This is aproblem for fitting algorithms, which is avoided by fitting admittance

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(1/Z) .

3. RESULTS

The resistance vs. TES temperature is plotted in Fig.2a. for severalbias currents. The curves show a strong dependence of transition shape withbias current. A smoothing algorithm was used to plot a 3-D surface of thetransition in Fig 2b. Knowledge of the

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R(I,T) surface is useful fordesigning calorimeters and choosing the optimal bias of the calorimeter. Inthe past researchers have employed a tanh shaped transition that gets shiftedwith bias current as a first guess to the shape of the transition. The data showthat the transition both shifts and broadens with increasing current.Apparently, the bottom end of the phase transition shifts much more with

The superconducting transition in 4-D

Fig. 1. Current, voltage, temperature, and impedance measurements of aTES. We swept fridge temperature and TES bias while measuring current inthe TES, which is plotted in Fig. 1a. The voltage in Fig. 1b and TEStemperature is calculated from that data. For each of the data points in Fig1a. and 1b., we simultaneously measure the real and imaginary parts of theTES impedance at 3 frequencies using lock-in amplifiers and the equivalentcircuit technique. A TES model is fit to the impedance data as in Fig. 3c.,with

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α I,

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βI , and TES heat capacity

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C as free parameters.

bias current than the top end does. For each data point on the

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R(I,T)surface, we obtain a direct measurement of

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α I and

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βI from the impedancedata. This provides a rich multidimensional data set, which can be analyzeda number of useful ways. For example, we can plot all

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α I and

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βImeasurements for all resistances as in Fig. 3a and 3b. Most of the scatter in

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α I associated with it varying as a function of bias current. Or we can plot

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βIand

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α I vs. current for a particular resistance as shown in Fig. 3c and Fig. 3d.to see that

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α I varies greatly with current but

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βI does not.Following the analysis of Stephen Smith et al.6, we try to estimate

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βIgiven

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α I and the critical current cI using the following approximation

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α I = −βIdIdT R

TI

= −βIdIcdT

TI

(1)

The vertical trajectories of Fig. 1a trace the super current from

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−Ic to

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+IcWe measure

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dIc /dT to be -0.7 µA/mK. Our measurements are largelyconsistent with Smith’s results: In the middle of the transition,

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βI is roughly

M. A. Lindeman et al.

b.)

Fig. 2. Resistance as a function of TES current and temperature. We plot theresistance of contours of constant current in 2a. and the

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R(I,T) in 2b. Datafrom Fig. 1a and 1b is interpolated and smoothed to form these plots. Themeasurements show that the bottom end of the transition shifts more withbias current than the top end does.

10 times smaller than predicted by this analysis. The prediction is not exactbecause the directions of resistance contours, which are plotted in Fig. 3b,change with the resistance.

We also obtained independent measurements of

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α I and

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βI bycalculating gradients of the

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R(I,T) surface data. These gradientmeasurements were noisier than the impedance measurements but there wasgood agreement between the two types of

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α I and

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βI measurements. Thisverifies the accuracy of the impedance method and the accuracy of thethermal conductance

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G used to compute the TES temperature for the

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R(I,T) surface.We obtained a measurement of the TES’s heat capacity for every

point on the

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R(I,T) surface. We plot the heat capacity vs. resistance inFig. 4a. As expected, the measured values of the heat capacity in the phasetransition lie between the calculated normal state heat capacity of the TES,and the BCS heat capacity of the superconducting state. In Fig. 4b, we plotcontours of resistance on top of

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C(I,T) data. Interestingly, the heat capacityof the SRON TES lies near the BCS heat capacity throughout most of thephase transition. This differs from a measurement of heat capacity of a TESproduced by the NASA / Goddard Space Flight Center. In GSFC’s TES, theheat capacity varied more throughout the transition and was closer to thenormal state heat capacity.

The superconducting transition in 4-D

Fig. 3. Parameters

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α I and

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βI . All measurements of these parameters at allbias currents are plotted in a.) and b.). Much scatter results from thedependence o

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α I these on current. In c.) and d.), we cut the data by takingonly values near 28 mΩ and plotting them against TES current.

3. DISCUSSIONThe heat capacity result suggests the internal state is different between

the SRON TES and the GSFC TES. Perhaps the GSFC device is more proneto separate into superconducting and normal phases and thus a significantfraction of the TES is in the normal state. This might explain the differencesin excess noise between the two types of TESs. Perhaps phase separationcauses the observed excess electronic noise seen in GSFC TESs (withoutzebra stripes) through a percolation mechanism. The SRON TESs are moreprone to thermal fluctuation noise because they have higher resistance.Testing the phase separation / percolation hypothesis requires morecomparisons of heat capacity and excess noise between the different kinds ofTESs.

We also observe that the heat capacity is highly correlated with theresistance over a range of currents and temperature. Apparently, heatcapacity can be regarded as a function of TES resistance

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C = C(R) in thisTES.

M. A. Lindeman et al.

b.)

Fig. 4. (color on line) Heat capacity and resistance. The heat capacity vs.resistance profile of the TES in (a.) shows that the heat capacity of this TESlies close to the calculated BCS heat capacity throughout much of thetransition. We plot resistance contours on heat capacity in (b.). Heat capacityfollows the contours, showing a strong correlation between heat capacity andresistance. (Color on-line.)

REFERENCES

1 M. A. Lindeman, Ph.D. thesis (2000).2 K. D. Irwin KD, Nucl. Instr. Meth. Sec. A, 559, 718 (2006).3 M. A. Lindeman, K. A. Barger, D. E. Brandl, S. G. Crowder, L. Rocks, D.

McCammon, these proceedings.4 M. A. Lindeman, K. A. Barger, D. E. Brandl, S.G. Crowder, L. Rocks, andD. McCammon, Rev. Sci. Inst. 78, 043105 (2007).5 M A. Lindeman, S. Bandler, R. P. Brekosky, J. A. Chervenak, E. Figueroa-Feliciano, F. M. Finkbeiner, M. J. Li, and C. A. Kilbourne, Rev. Sci. Inst.75, 1283 (2004).6 S. J. Smith, S. R. Bandler, A. Brown, J. A. Chervenak, Fred M. Finkbeiner,N. Iyomoto, R. L. Kelley, C. A. Kilbourne, F. S. Porter, J. E. Sadleir, theseproceedings.