# Implications of Weak Link Effects on Thermal Characteristics of Transition-Edge Sensors

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## Abstract

Weak link behavior in transition-edge sensor (TES) microcalorimeters creates the need for a more careful characterization of a device’s thermal characteristics through its transition. This is particularly true for small TESs where a small change in the bias current results in large changes in effective transition temperature. To correctly interpret measurements, especially complex impedance, it is crucial to know the temperature-dependent thermal conductance, *G*(*T*), and heat capacity, *C*(*T*), at each point through the transition. We present data illustrating these effects and discuss how we overcome the challenges that are present in accurately determining *G* and *T* from I–V curves. We also show how these weak link effects vary with TES size. Additionally, we use this improved understanding of *G*(*T*) to determine that, for these TES microcalorimeters, Kaptiza boundary resistance dominates the *G* of devices with absorbers while the electron-phonon coupling also needs to be considered when determining *G* for devices without absorbers

## Keywords

Transition edge sensor (TES) Thermal conductance Weak link effects Kapitza resistance Electron-phonon coupling## 1 Introduction

*I*

_{ c }, with temperature,

*T*, depends on the device size,

*L*;

*I*

_{ c }(

*T*,

*L*)

*αe*

^{−L/ξ(T)}, where

*ξ*(

*T*) is the temperature-dependent coherence length [1] (Fig. 1e). Therefore, weak link effects become more prominent as the TES size is decreased, making device characterization more complex. Hence, a better understanding and method of TES characterization must take effect for continued detector improvements.

We have developed high fill-factor, high quantum efficiency, kilo-pixel arrays of TES microcalorimeters that provide high spectral resolution for x-ray astrophysics and solar physics applications. These devices [5, 6, 7, 8] are made of square Mo(45 nm)/Au(200 nm) bilayers (Fig. 1a) and have transition temperatures of ∼100 mK. Au or Au/Bi absorbers overhang the TES (Fig. 1b), supported through a small area attachment stem, to provide sufficient x-ray stopping power. Different applications have varying requirements, such as count rate and spatial scales. To meet these various specifications we are developing devices which have the same basic geometry as those developed for x-ray astrophysics applications [9], but with different TES sizes (to satisfy the needs of varying spatial scales), normal-metal features (to reduce noise at each size), absorber stem attachments (to maintain superior energy resolution at each size), and fabricated on the silicon substrate [10], instead of on silicon nitride membranes (speeding the device response to allow higher count rates). Here we focus solely on these ‘solid-substrate’ devices and compare characterization techniques for TESs varying in size, from square 12 μm Mo/Au bilayers to devices that are 20, 35, 50, and 140 μm square.

## 2 Determining *G* Considering Weak Link Effects

Our typical microcalorimeters that we have developed for x-ray astrophysics applications [9] use TESs that are 140 μm square and have thick absorbers, which dominate the heat capacity. In these devices, the physical properties such as *G*, *C*, and *T* _{ c }, are constant through the transition. While developing smaller TESs we have entered a new regime in which weak link effects are more evident, causing these devices to be highly current dependent where the operating temperature, and therefore *G*(*T*), change based on the bias current. For example, for a 12 μm square TES the operating temperature can vary by ∼20% throughout the superconducting transition while for a 50 μm square TES it varies by ∼10% and for a 140 μm square TES it only varies by ∼2.5%.

*G*, of a TES involves measuring the joule heating power as a function of bath temperature, which is achieved by recording how the sensor current varies with applied voltage (an I–V curve, see Fig. 1c) at various bath temperatures. Then, we perform a fit to the data of the form:

*K*and exponent

*n*, where

*T*is the TES temperature and

*T*

_{ b }is the bath temperature. The thermal conductance, at a given temperature, is then equal to:

*G*. We have found that improperly centering I–V curves introduces systematic errors up to 10%.

A second challenge in determining the thermal conductance of a TES is in dealing with weak-link effects. This typical procedure used to fit for *G* assumes the device’s critical current is constant with temperature for *T* _{ b }<*T* _{ c }. While this is a fair approximation for the 140 μm square TESs, it is no longer appropriate for small devices where weak-link behavior becomes stronger as the length scale, i.e. TES size, decreases. This highly current-dependent transition shape makes the determination of *G* challenging [11], as we no longer expect our data to be consistent with (1) globally. To accurately interpret measurements, it is crucial to calculate *G* at many discrete points through the TES transition by iterating the procedure described earlier in this section for many TES resistances.

*C*, by measuring the intrinsic pulse fall time,

*τ*, above the transition temperature, where there is minimal influence from electrothermal feedback due to the readout circuit, where

*C*(

*T*)=

*τ*(

*T*)

*G*(

*T*). Therefore we can verify that we determine the correct thermal conductance by comparing the calculated and expected heat capacity, which is calculated based on the measured absorber and TES volume and material properties. When fixing the exponent, the calculated heat capacity is consistent with expectation, for all but the smallest sized devices (Fig. 3). Thus, in the presence of weak-link effects, the best

*G*estimate is made by fixing the exponent, when the coupling physics is known, and only using the points measured at the lowest bath temperatures.

## 3 Kapitza vs. Electron-Phonon Coupling for Various Microcalorimeter Device Designs

To fully understand our devices, we want to know what physics dominates the thermal resistance for our solid substrate devices—electron-phonon coupling or Kapitza boundary resistance between the metal film and the substrate. Previously we conducted a limited study of 140 μm square devices, with *T* _{ c }’s between 50–120 mK, and found that the devices with absorbers were consistent with the cubic temperature dependence, or an exponent *n*=4, expected of a Kapitza resistance, with a boundary conductivity of 217*T* ^{3} W/m^{2} K, roughly 3 times lower than expected [9]. The limited data from devices with no absorbers suggested a significant electron-phonon contribution to the total thermal conductance [9].

Recently we have measured *G* of many more solid-substrate devices, including TESs with and without absorbers and with *T* _{ c }’s up to approximately 275 mK. The inclusion of this additional data allows us to further understand the contribution of the Kapitza resistance and electron-phonon coupling to the total thermal conductance. Since this data is from many differently sized devices, to directly compare them, *G* needed to be scaled to account for these size differences. Since, for Kapitza resistance, *G* is proportional to area, the total thermal conductance measured for each device was divided by area. The area used is the area of the Mo/Au bilayer plus the area of the stem extended past the bilayer minus the area under the bias leads. A separate analysis of test devices showed that the area of the bilayer under the bias leads does not contribute to the measured *G*.

*T*

^{3}W/m

^{2}K, closer to the theoretical value of ∼633

*T*

^{3}W/m

^{2}K [12]. Taking a closer look at the data, the 140 μm data alone does not go to very high temperatures so it is difficult to determine if different sized devices have different boundary conductivities or if the fit is skewed.

The measured *G* from devices without absorbers is lower than expected from a Kapitza resistance inferred from the devices with absorbers (Fig. 4). Currently we understand this difference to be due to the different volumes, roughly 100×, associated with devices with and without absorbers. Since the electron-phonon conductance is proportional to volume, instead of surface area, as is the case with Kapitza boundary resistance, the data indicates that the added volume of the absorber causes the electron-phonon conductance to become much higher than the Kapitza component, so that the latter then determines the total thermal conductance for devices with absorbers. However, without absorbers, the electron-phonon conductance is weak enough to impact the overall *G*, therefore lowering it. In order to obtain a good estimate of the electron-phonon coupling, and determine if superconducting effects change this coupling in the TES (depending on its SNS phase) [13], we need additional data points, especially at the higher temperatures.

## 4 Conclusions

The current dependence of the transition due to weak leak behavior, which is most pronounced in small devices, makes standard TES characterization more challenging, in particular the TES temperature, and therefore the thermal conductance, is not constant through the transition. To fully characterize a TES exhibiting weak link behavior, one must determine *G* at each point through the transition.

Fitting the power vs. temperature to calculate thermal conductance can be especially challenging for these small devices. In particular, for a good polynomial fit one must remove points at the highest temperatures, near *T* _{ c }. However, it can be difficult to determine which should be removed. We found that, especially for the smallest devices, the best technique for fitting the power vs. temperature curves was to fix the exponent.

When comparing thermal conductance data between many devices, with different transition temperatures and with and without absorbers, we see that devices with absorbers have Gs that are Kapitza dominated while devices without absorbers have Gs that are electron-phonon dominated.

## Notes

### Acknowledgements

This research was supported in part by an appointment to the NASA Postdoctoral Program at Goddard Space Flight Center (C.N. Bailey), administered by Oak Ridge Associated Universities through a contract with NASA.

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