1 Introduction

Microcalorimeters based on superconducting transition-edge sensors (TESs) allow measurements of X-ray and \(\gamma\)-ray photons with energy resolution unmatched by commercial energy dispersive detectors. Spectrometers based on arrays with hundreds of TES microcalorimeters have been built at NIST and deployed at several national labs in the United States, Europe, and Japan [1,2,3]. These spectrometers are used for a wide variety of measurements, including X-ray emission spectroscopy of various types [4] and nondestructive analysis of nuclear materials [5, 6].

A TES microcalorimeter consists of two parts: (1) The TES itself is a superconducting thin film that acts as a sensitive thermometer due to its sharp change in resistance with temperature; (2) the absorber is a material that efficiently absorbs photons in the energy range of interest and is thermally coupled to the TES. Three key performance metrics are energy resolution \(\Delta E\), which determines the ability to resolve neighboring spectral lines, saturation energy \(E_{\textrm{sat}}\), which sets the maximum photon energy that can be accurately measured, and decay time \(\tau\) required for the microcalorimeter to recover after absorbing a photon, which limits the maximum count rate per pixel. These metrics can be tuned [7] by adjusting the TES equilibrium temperature T, as well as the heat capacities C and thermal conductances G in the schematic thermal circuit shown in Fig. 1a.

Fig. 1
figure 1

a Schematic thermal circuit for a TES and separate absorber connected to a bath at temperature \(T_\textrm{b}\). b Schematic cross section (not to scale) showing the Si wafer and SiN\(_x\) membrane supporting the TES and Sn absorber mounted on epoxy posts

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When \(\gamma\)-TES detectors have been tested at NIST and then mounted in another cryostat for deployment elsewhere, we have often observed degraded energy resolution compared to that measured in the NIST cryostat. This has consistently been accompanied by distorted current–voltage (IV) curves, as illustrated in Fig. 2b. The distortions vary among the individual pixels in an array. In some cases the critical current \(I_{\textrm{c}}\) at which the TES switches out of the fully superconducting state is simply reduced, as seen in the blue and orange curves. In other cases anomalous features appear at low voltage, as seen in the green and red curves.

Fig. 2
figure 2

a Sharp IV curves of four nominally identical \(\gamma\)-TES detectors in an environment with low stray B, showing large \(I_{\textrm{c}}\) and a sharp transition out of the superconducting branch at \(V_\textrm{TES} = 0\). b Distorted IV curves for the same four detectors with trapped flux, showing reduced \(I_{\textrm{c}}\) and, in some cases, anomalous features at small \(V_\textrm{TES}\). Voltage sweep starting at \(V = 0\) and \(T_\textrm{b}\) = 90 mK in both cases

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Among the potentially relevant differences between cryostats, such as wiring, thermal and magnetic shielding, filtering of electrical noise, and vibration, stray magnetic field B has long been the prime suspect for the cause of these IV distortions. This suspicion is supported by extensive work showing that X-ray TES detectors are strongly affected by B applied perpendicular to the TES plane [8]. However, prior to the results presented here, there was no direct empirical link between stray B and IV curves or \(\Delta E\).

2 Flux Trapping in \(\gamma\)-TES Detectors

In this section we describe the geometry of \(\gamma\)-TES detectors, consider how trapped flux may impact the TES itself, and estimate the increase in heat capacity from normal metal regions in the superconducting absorber.

Figure 1b shows a sketch of our typical \(\gamma\)-TES pixel [5]. Efficient absorption of photons with energy up to \(\approx\) 200 keV is provided by a bulk Sn absorber. The absorber and TES are supported by a silicon nitride membrane that determines \(G_1\), with the TES directly on the membrane and the absorber held above it by four epoxy posts that determine \(G_2\). Copper traces on the membrane (not shown) connect the epoxy posts to the TES, ensuring that heat from the absorber flows primarily to the TES and not directly to the bulk Si regions via the membrane.

If \(B \ne 0\) at the Sn when it becomes superconducting, persistent screening currents will arise to make \(B = 0\) for as much of the Sn as possible. Since Sn is a Type I superconductor, this will result in an intermediate state of macroscopic normal (N) and superconducting (S) domains, different from the mixed state of quantized flux vortices seen in Type II superconductors. For a thick plate of Type I superconductor with B applied normal to the plate, Lev Landau first showed that minimizing the sum of energies from domains, N/S interfaces, and surfaces predicts a structure of alternating N and S laminae oriented parallel to the applied field [9]. Most experiments, however, show more complicated structures that depend strongly on sample preparation and experimental details [10]

Whatever the actual configuration of trapped flux in the Sn absorber, we can expect it to have two effects of primary interest. The first effect is that the TES lying below the absorber will experience alternating high and low B regions that will reduce \(I_{\textrm{c}}\). Given the position and shape of the TES, the field from N domains oriented along z (see Fig. 1b) will have a stronger effect than that from domains parallel to the xy plane. The second effect is that \(C_2\) will be increased by the presence of the unpaired electrons in the N domains.

We can estimate the increased heat capacity from trapped flux as follows. For a plate of Type I superconductor cooled in an external field \(B_{\text {ext}}\), the flux trapped in the plate is determined by its demagnetizing factors \(({N_x}, {N_y}, {N_z})\) along directions (x, y, z). Using an analytical result for rectangular prisms [11] and the dimensions of our Sn absorber, \((L_x,L_y,L_z) = (1.45,1.45,0.50)\) mm, we find \(({N_x}, {N_y}, {N_z}) = (0.21,0.21,0.58)\). Since our absorber will trap about three times more flux from \(B_{\text {ext}}\) along z, we consider only this component of field in our heat capacity estimate, giving an overall trapped flux density \(B_{\text {trap}} = 0.58 B_z\). This flux is confined to the N domains of the Sn, where \(B \approx B_\textrm{c} = 31\) mT. Thus the fraction of Sn that is normal is given by \(F_\textrm{N} \approx B_{\text {trap}}/B_\textrm{c} = 0.58B_z/(31\) mT).

The specific heat capacities for Sn in both S and N states, \(c_\textrm{S}\) and \(c_\textrm{N}\), have been measured for \(T \approx (0.1\) to 1) K [12]. Our \(\gamma\)-TES pixels operate near \(T_\textrm{c} \approx 120\) mK, where the ratio \(c_\textrm{N}/c_\textrm{S} \approx 500\), so the fractional increase in \(C_2\) due to trapped flux in our Sn absorber is given by

$$\begin{aligned} \Delta C/C_2 = F_\textrm{N}(c_\textrm{N}/c_\textrm{S}) \approx 500 (B_{\text {trap}}/B_\textrm{c}) \end{aligned}$$
(1)

As an example, for a fairly large stray field of \(B_z = 10~\upmu \textrm{T}\), this estimate gives \(F_\textrm{N} = 2 \times 10^{-4}\) and \(\Delta C/C_2 = 0.1\). A static change of this amount is a modest effect, roughly comparable to the variation in C and G among pixels fabricated on the same wafer. However, since individual pixel gain is affected by \(C_2\), changes in trapped flux that occur during spectrum acquisition will degrade \(\Delta E\) unless the gain changes can be detected and accounted for in post-acquisition processing.

3 Experimental Results

The NIST adiabatic demagnetization refrigerator (ADR) used to test \(\gamma\)-TES pixels is also used to test X-ray TES pixels. When measuring the latter, holes in two layers of magnetic shielding at T = 3 K, each \(\approx\) 5 cm in diameter, are left uncovered to minimize absorption of photons with energy \(\lesssim\) 10 keV. In an oversight that proved serendipitous, the covers for these holes were not replaced during a subsequent cooldown of \(\gamma\)-TES pixels. The IV curves measured in this state showed distortions like those shown in Fig. 2b.

During the same cooldown, a cylindrical magnetic shield with an closed bottom and open top was placed around the ADR. This external shield was 1.14 times the diameter and 0.61 times the height of the ADR vacuum can, with a wall thickness of 1.5 mm. With the external shield in place during a magnetic cycle of the ADR, the distortions in the IV curves were greatly reduced. For each of the 27 pixels measured during this cooldown the effect of the shield was repeatable, i.e., after each cycle without the shield the distortions were obvious and after each cycle with the shield they were minimal or eliminated entirely. It is worth noting that the detector box reached \(T\approx 6\) K during each ADR cycle, above \(T_\textrm{c}\) of Sn but below \(T_\textrm{c}\) of Nb. Thus flux trapped in the Sn absorbers can readily change but any flux trapped in Nb components on the readout chips during the initial cooldown is expected to remain constant during ADR cycles.

Since a few pixels had IV distortions even with the external magnetic shield, we screened various components near the detector box for remnant magnetic field \(B_\textrm{rem}\) after opening the ADR. We placed these components near the bottom of the external shield, where the background was \(\lesssim 0.1~\upmu \textrm{T}\), and used a fluxgate magnetometer to measure the remnant field in different orientations. The small, single-axis magnetometer used for these measurements allowed components to be placed \(\approx 5\) mm from the active sensor area.

The remnant field for several components is given in Table 1. As expected, \(B_\textrm{rem}\) for brass screws is near the background level while for stainless screws it can be many times larger. While \(B_\textrm{rem}\) for stainless screws was oriented axially in most cases, there were exceptions. The results for coaxial cables and multipin connectors were less intuitive. The larger values for the SMP cables, which had Au-plated connectors, may be due an adhesion layer of Ni or other magnetic material under the Au. The nonmagnetic versions of the MDM-9 connectors were better overall, but \(B_\textrm{rem}\) near the sockets was still quite large.

Although we do not know the “safe” level for \(B_\textrm{rem}\), below which flux trapping is negligible in our detectors, we aimed to eliminate components with \(B_\textrm{rem}\) larger than about \(0.5~\upmu \textrm{T}\) located within a few cm of the detectors. Where replacement with a less magnetic component was not possible, we used annular or planar AC demagnetizers to reduce \(B_\textrm{rem}\). Moving components by hand through the peak field region and then far away, over a period of several seconds, induces many hysteresis cycles of gradually decreasing amplitude, which removes much of the magnetization that may have been induced previously by a strong static field. For some components the optimal orientation and type of demagnetizer was not intuitive and had to be discovered empirically. In particular, the annular demagnetizer was not always the best choice for coaxial cables and connectors, despite their axial symmetry.

Table 1 Remnant field in \(\upmu\)T measured \(\approx\) 5 mm from various components used near the detector box
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After eliminating the largest internal sources of stray B field, and installing the previously forgotten covers on the cold magnetic shields, we cooled down the same detector box with the same pixels. This allowed us to make a direct comparison between “Clean” and “Dirty” magnetic states. Figure 3 shows the histogram of \(I_\textrm{c}\) for the Dirty state with and without the external shield in place during an ADR cycle, and for the Clean state, where the external shield had no effect.

Figure 4 shows the histogram of \(\Delta E\) measured using 97 keV photons from a \(^{153}\)Gd source. The improvement is less dramatic than for \(I_\textrm{c}\) but still noticeable. For measurements lasting a day or more, we observed sudden changes in pulse size for this photon energy in some pixels, an effect known as “gain jumps". In the Dirty state, these jumps were more frequent without the external shield, which is consistent with the hypothesis that they involve changes in trapped flux, possibly from unusually large temperature excursions caused by background gamma rays absorbed in the TES itself.

The improvements in \(I_\textrm{c}\) and \(\Delta E\) from adding the external shield in the Dirty state indicate external sources of B had a significant effect on the detectors in this state, which is not surprising given the open holes in the shields at 3 K. As expected, in the Clean state with these holes covered, the external shield had no effect.

Fig. 3
figure 3

Histogram of \(I_\textrm{c}\) at \(T_\textrm{b} =\) 90 mK for three different magnetic environments, described in the legends. Bin width = 0.025 mA. Probability density is used because the number of detectors and/or measurements was not the same in all cases

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Fig. 4
figure 4

Histogram of \(\Delta E\) at \(T_\textrm{b} =\) 90 mK for three different magnetic environments, described in the legends. Legend shows mean ± standard error of the mean for each case. Bin width = 15 eV

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To further test the connection between stray B field and IV distortions, we made another comparison using similar \(\gamma\)-TES pixels mounted in a different detector package, known as a “microsnout” [13]. We used the same ADR as the comparison described above, after removing magnetic components and without holes in the internal magnetic shields. The microsnout has a protective Al sleeve whose lid is attached with four screws located a few mm from the corners of the TES array. Normally these are brass screws, which was the Clean state for this comparison. The Dirty state was created by using four stainless screws that had been intentionally magnetized along their axis. Fluxgate measurements of \(B_z\) at a distance of 16 mm from the plane of the lid ranged from (0.4 to 1.6) \(\upmu\)T over the area of the lid directly above the TES array. We expect \(B_z\) at the pixels was roughly an order of magnitude higher.

Figure 5 shows the histograms of \(I_\textrm{c}\) for the microsnout detectors with brass screws and magnetized screws in the sleeve lid. The suppression in \(I_\textrm{c}\) is even more dramatic than in Fig. 3, while the effect on \(\Delta E\) is again more modest. Apparently suppression of \(I_\textrm{c}\) in these detectors is evident before energy resolution is degraded significantly. This is convenient for confirming or ruling out effects due to stray B, since a measurement of \(I_\textrm{c}\) for every pixel takes much less time than a measurement of \(\Delta E\).

Fig. 5
figure 5

Histograms of \(I_\textrm{c}\) (a) and \(\Delta E\) (b) and (c), both at \(T_\textrm{b} =\) 90 mK, for a microsnout with magnetized vs. brass screws in the sleeve lid. For \(\Delta E\), legend shows mean ± standard error of the mean. Bin width = 0.025 mA for \(I_\textrm{c}\) and 15 eV for \(\Delta E\)

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4 Conclusion

Our results show that IV curve distortion is a sensitive probe of stray B field near \(\gamma\)-TES detectors, and that these distortions are correlated with degraded energy resolution. Screening components with a fluxgate magnetometer, and demagnetizing those that cannot be replaced, is now a routine step when building new cryostats and detector assemblies at NIST.