# Longitudinal Proximity Effects in Superconducting Transition-Edge Sensors

###### Abstract

We have found experimentally that the critical current of a square superconducting transition-edge sensor (TES) depends exponentially upon the side length and the square root of the temperature . As a consequence, the effective transition temperature of the TES is current-dependent and at fixed current scales as . We also have found that the critical current can show clear Fraunhofer-like oscillations in an applied magnetic field, similar to those found in Josephson junctions. The observed behavior has a natural theoretical explanation in terms of longitudinal proximity effects if the TES is regarded as a weak link between superconducting leads. We have observed the proximity effect in these devices over extraordinarily long lengths exceeding 100 m.

###### pacs:

74.25.-q,74.78.Bz,74.25.OpA superconductor cooled through its transition temperature while carrying a finite dc bias current undergoes an abrupt decrease in electrical resistance from its normal-state value to zero. Superconducting transition-edge sensors (TESs) exploit this sharp transition; these devices are highly sensitive resistive thermometers used for precise thermal energy measurements.IrwinHilton05 () TES microcalorimeters have been developed with measured energy resolutions in the X-ray and gamma-ray band of 0.2 eV FWHM at 6 keV, Bandler08 () and eV FWHM at 97 keV, Bacrania09 () respectively— with the latter result at present the largest reported of any non-dispersive photon spectrometer. TESs are successfully used across much of the electromagnetic spectrum, measuring the energy of single-photon absorption events from infrared to gamma-ray energies and photon fluxes out to the microwave range.IrwinHilton05 () Despite these experimental successes, the dominant physics governing TESs biased in the superconducting phase transition remains poorly understood.IrwinHilton05 ()

To achieve high energy resolution it is important to control both the TES’s and its transition width . Because the energy resolution of calorimeters improves with decreasing temperature, they are typically designed to operate at temperatures around 0.1 K. For a TES, this requires a superconductor with in that range. While there exist a few suitable elemental superconductors, the best results have been achieved using proximity-coupled, superconductor/normal-metal (S/N) bilayersBandler08 (); Bacrania09 (), for which is tuned by selection of the thicknesses of the S and N layers. Martinis00 ()

There have been a variety of modelsFraser04 (); Seidel04 (); Martinis00 (); Luukanen03 (); LindemanPerc06 () used to explain the noise, , and in TES bilayers, all assuming spatially uniform devices. Though some have been shown to be consistent with certain aspects of particular devices, they do not explain measured and in S/N bilayer TESs generally.

In this paper we emphasize the importance of a phenomenon that so far has been neglected in previous theoretical studies of TESs: the longitudinal proximity effect. Since the square bilayers at the heart of the TES are connected at opposite ends to superconducting leads with transition temperatures well above the intrinsic transition temperature of the bilayers, superconductivity is induced longitudinally into the bilayers via the proximity effect. As we shall explain later, many of the basic properties of our TES structures are well described by regarding them as SSS or SNS weak links.SS'S (); Clarke69 (); Likharev79 ()

In this paper we report the properties of TESs based on square () electron-beam-deposited Mo/Au bilayers consisting of 55 nm Mo layers ( 0.9 K) to which 210 nm of Au is added. The square side lengths range from 8 m to 290 m, and the normal-state resistance per square is 0.5 m. The bilayers are connected at opposite ends to Mo/Nb leads having measured superconducting transition temperatures of 3.5 and 7.1 K.LeadsTcComment () Further details on the device fabrication process can be found in Ref. Chervenak04, .

Our measurements are made in an adiabaticÊ demagnetization refrigerator (ADR) with mu-metal and Nb enclosures providingÊ magnetic shielding for the TES devices and SQUID electronics. The magnetic field normal to the TES device plane is controlled by a superconducting coil with the field value determined from the coil geometry and current. Measurements of the TES resistance are made byÊ applying a sinusoidal current of frequency 5-10 Hz and amplitude 50-250 nA, with zero dc component, to the TES in parallel with a 0.2 m shunt resistor (). The time-dependent TES current is measured with a SQUID feedback circuit with input coil in series with TES. When is less than the TES critical current , is zero, and all the ac current flows through the TES. However, when and during part of the ac cycle, the TES current becomes non-sinusoidal, and its maximum value becomes less than . The TES resistance at the TES current is then determined from .

The critical current is measured, with the ADR held at constant temperature, by ramping the dc bias current from zero and defining as the TES current at the first measured finite resistance () across the TES. Record averaging is used at higher temperatures where becomes small.

The solid curves in Fig. 1(a) and (b) show measurements of the critical current over seven decades vs temperature . Note that although we find the intrinsic transition temperature of the Mo/Au bilayer is 0.1 mK, at very low currents a zero-resistance state is measured up to much higher temperatures as the TES size is reduced, three times for = 8 m. On the other hand, for the larger TES sizes (=130 and 290 m) the critical current decreases rapidly with near . The observed behavior as functions of both and the length provides strong evidence that our TESs behave as weak-link devices. The dotted curves in (a) and (b) show calculated values of using the Ginzburg-Landau theory described below. In addition, at appropriately chosen temperatures, the critical currents of these devices exhibit Fraunhofer-like oscillations as a function of an applied magnetic field, behavior characteristic of Josephson weak links.Clarke69 (); Barone82 (); Miller84 (); Dobrosavljevic93 () See Fig. 1(c) for an example.

Because depends upon and , the effective transition temperature of the TES (the temperature at which an electrical resistance first appears, i.e., ) is both current-dependent and length-dependent. Figure 2(a) exhibits these effects. The points labeled are the effective transition temperatures at five different current levels (10 nA to 100 A) for the data in Fig. 1(a) and (b), showing that for each current level scales approximately as (solid curve fits) for ranging from 8 to 290 m. For each , depends upon the current.

Also shown in Fig. 2(a) are temperatures and for which the resistances are and , respectively, from which we define . We also define transition widths from the measurements and . In Fig. 2(b) we show that these three measures of the transition width all vary approximately as , shown by the dotted line. It also follows that scales linearly with the transition width.

Weak links in various SSS or SNS weak-link structures have been studied experimentally and theoretically by numerous authors. However, here we use a simple version of Ginzburg-Landau (GL) theorydeGennes (); StJames69 () to explain the results shown in Figs. 1 and 2. This theory describes the complex order parameter , whose absolute square is the superfluid density in the weak link. We employ the substitution where is the magnitude of the order parameter at the reference points adjacent to the leads, is the normalized order parameter, and is the phase. At the reference points, the local value of is inversely proportional to the square of the local penetration depth via and a characteristic reference current density can be defined via , where .

Near the center of the weak link, where becomes very small, the local penetration depth becomes very large. Moreover, in a thin film of thickness , magnetic fields and currents spread out over the two-dimensional screening length (or Pearl lengthPearl64 ()) . For all of our samples there is a range of temperatures far enough above that at the center of the weak link and the current density is , where is the TES current. The first and second GL equations given in Refs. deGennes, and StJames69, then depend only upon and can be written as

(1) |

and

(2) |

where is the reduced temperature, is the temperature-dependent coherence length, is the dimensionless Ginzburg-Landau parameter, is the vector potential, and the primes denote derivatives with respect to .

In this paper we are concerned chiefly with weak-link behavior for which is an even function of and has a minimum in the middle of the weak link, where . We can obtain an equation that determines how depends upon , , and by multiplying Eq. (1) by , integrating the result, multiplying by , and taking the square root, which yields the following equation, valid for ,

(3) |

where

(4) |

such that and can be obtained from the integrals

(5) |

(6) |

The gauge-invariant phase difference across the weak link isLikharev75b ()

(7) |

The integrals in Eqs. (5), (6), and (7) can be evaluated numerically as in Ref. Likharev75b, or in terms of elliptic integrals as in Refs. Mamaladze66, and Baratoff70, . For given values of , , , , and , the solutions of Eq. (5) reveal that is a single-valued function of , starting with the value at , initially increasing linearly with , rising to a maximum value defined as , then returning to zero at a larger value of .

When , the above equations reveal that is a single-valued function of and has a functional dependence close to , similar to that of a Josephson junction. For , the reduced order parameter at has its maximum value when , its minimum value 0 when , and a value between these two limits at the critical current when and .

When and , for a large fraction of the length , and one may omit the term proportional to on the right-hand side of Eq. (1) to obtain the reduced order parameter . In the absence of a current, near the center of the weak link, at the center, and the gauge-invariant phase difference across the weak link is . The parameter , which is of the order of unity, would be equal to unity if the linearized GL equation were valid over the entire length of the weak link; the suppression of below unity occurs because the exact solution for near is strongly influenced by the term on the right-hand side of Eq. (1).

For nonzero current, , the reduced order parameter at the center of the weak link, is suppressed below , and the gauge-invariant phase difference across the weak link obeys The reduced current is given by

(8) |

such that the reduced critical current is given for and any by the approximation

(9) |

at the maximum of , where and . From Eq. (9), we may obtain the critical current as . Inferring mK from the experimental data in Fig. 1(b) for = 290 m and assuming , we obtained and by fitting the experimental data for = 8 m at 250 mK and 375 mK. The dotted curves in Fig. 1(a) and (b) show calculated using = 738 nm and = 79 nm in Eq. (9) and Eq. (5), from which was obtained.Tmax ()

Under conditions for which Eq. (9) is valid, if we define the effective transition temperature as the temperature at which the first voltage appears along the length of the TES when it carries a reduced current density , we can determine or by setting in Eq. (9) and solving for , noting that . The result is

(10) |

Since the dependence upon on the right-hand side is very weak, because it appears within the argument of the logarithm, Eq. (10) predicts that the current-dependent transition temperature of the TES should scale very nearly as and that should increase as the square of the logarithm of the inverse TES current. Similar reasoning leads to the conclusion that both and scale as . Scaling of can be understood using a simple model of the resistive transition based on the assumption that , where, for a given reduced current density , is the solution of .

We conclude that TESs behave as weak links. This conclusion is based on our experimental findings that (a) the critical current at the first onset of a voltage along the length depends exponentially upon the length and the square root of the temperature , (b) both the current-dependent effective transition temperature and the transition width scale as , and (c) the TESs show clear Fraunhofer oscillations as a function of applied magnetic field, characteristic of Josephson weak links. It follows that the strength of superconducting order is not uniform over the TES. Our findings have implications on TES magnetic field sensitivity, which impacts required limits on ambient magnetic field magnitude and fluctuations in TES applications. Proposed uses of the longitudinal proximity effect for TES applications include (1) tuning the effective of TES arrays by changing in mask design, which could compensate for bilayer variabilityIrwinHilton05 () and increase yield, and (2) making small TESs consisting of superconducting leads separated by normal metal, such as Au with , avoiding the use of S/N bilayers.

Our work at Goddard was partially funded under NASA’s Solar and Heliospheric Physics Supporting Research and at the Ames Laboratory by the Department of Energy - Basic Energy Sciences under Contract No. DE-AC02-07CH11358. We thank J. Beyer (PTB Berlin) and K. Irwin (NIST Boulder) for providing the SQUIDS used in this work. We also thank F. Finkbeiner, R. Brekosky, and D. Kelly for essential roles in device fabrication, and C. Kilbourne, I. Robinson, F. S. Porter, R. Kelley, and M. Eckart for useful discussion of these results and the manuscript.

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