1 Introduction

Planned for launch in the early 2030s onboard the Athena X-ray observatory [1], the X-ray Integral Field Unit [2] will operate an array of more than 3000 Transition Edge Sensor (TES) microcalorimeters providing an unprecedented energy resolution of 2.5 eV (FWHM at 7 keV) over a field of view of 5’. At its orbit around the L2 Lagrangian point, the X-IFU will be subject to a continuous flux of high-energy cosmic rays (CR). The most energetic of these charged particles will go through the instrument structure, creating secondary particles in their passing. Primary and secondary particles will regularly deposit energy in the X-IFU pixels themselves and in the detector Si wafer frame. The former will create false events and generate the so-called instrument non-X-ray background [3]. Most of it will be rejected by a cryogenic anticoincidence detector (cryoAC) located less than 1 mm away from the main detector [4]. Energy depositions in the wafer will induce heat waves across the wafer, which may perturb the pixels’ signal and eventually the energy measurement of X-ray photons. This effect was critical for the Planck/HFI instrument [5], and this paper aims at estimating the impact of these CR-induced thermal fluctuations on the X-IFU energy resolution.

2 Study Inputs

2.1 Energy Deposition in the X-IFU Wafer

Fig. 1
figure 1

Left Schematic of the muntin structure suspending the TES array on a SiN membrane. A coating (typically gold) covers the bottom of each muntin as well as its side up to 80 % (limited by the deposition angle). Figure adapted with permission from [7]. Right Spectrum of CR-induced energy depositions in the Si frame and muntin structure (color figure online)

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The X-IFU pixels are comprised of a Mo/Au TES and an overhanging Bi/Au absorber providing the required X-ray stopping power. The TESs themselves will be suspended on a Si wafer by a thin SiN membrane to create the appropriate weak thermal link to a \(\sim \) 50-mK thermal bath. This suspension is created by a back-etch up to the SiN membrane, forming a muntin structure as shown in Fig. 1 (left). The expected energy deposition in both the Si wafer main frame and the muntin structure was characterized with an 83-s-long Geant4 simulation performed under solar minimum conditions (maximum GCR flux, [3]). Figure 1 (right) shows the resulting spectra. With its surface of \(\sim 63\,\mathrm{cm}^2\), the Si frame sees a high count rate of 183 cts/s with an average deposited energy of \(\sim 450\) keV. In contrast, the smaller muntin structure (\(\sim 0.9\,\mathrm{cm}^2\) top surface) receives only 7 cts/s with a mean impact energy of \(\sim 250\) keV. For simplicity, all individual energy depositions simulated by Geant4 from a single primary CR event, including that of the associated secondary particles, were summed together to create a single event. The impact of this hypothesis compared to resolving all depositions was studied in [6] and shown to moderately overestimate the final impact on the energy resolution.

2.2 TES Wafer Thermal Response

Fig. 2
figure 2

The two studied heatsinking designs. For simplicity, the real geometries were approximated with centrally symmetric equivalents. Left The baseline configuration with 1000 wirebonds around the perimeter of the cryoAC (wb1 location) and a gold back coating of 3.5 \(\upmu \)m thickness. Right Optimized configuration with an additional Pd coating to dampen the thermal response near the TES array and a split in the Au coating on the frame to limit the conductivity toward the periphery. See more details in [7] (color figure online)

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The X-IFU wafer thermal response was modeled and characterized in [7] (see also [6] for a similar simulation effort) for two heatsinking designs as a function of the CR impact position for a pixel at the center and at the edge of the TES array (see Fig. 2): The first configuration corresponds to the currently baselined heatsinking approach with gold wirebonds connecting the wafer gold back coating (3.5-\(\,\upmu \)m gold layer deposited on the back side of the wafer—see also Fig. 1) to the focal plane 50-mK structure around the perimeter of the cryoAC (wb1 location). The second design has been optimized by adding on top of the gold coating a palladium layer close to the TES array to dampen the frame response. The wafer backside coating was also removed between the wb1 location and an outer second wirebonding ring in order to cut the lateral thermal conductivity between the outer frame and the TES array.

Figure 3 (left) shows a comparison of the simulated wafer thermal response in these two configurations. As expected from the high specific heat capacity of palladium, the optimized design shows smaller and slower temperature profiles for events close to the TES array, while the split in the backside coating efficiently cuts the events in the outer frame. Figure 3 (right) in turn shows the response (simulated for the baseline design) to events in the muntin structure located directly below the TES detectors. In this case, we see very sharp temperature profiles, rising up to a significant fraction of a Kelvin, but decreasing below the \(\upmu \)K level in \(\sim 1\) ms.

Fig. 3
figure 3

Simulated temperature pulses measured at the center of the array for CR impacts at varying distances (500 keV). Left Simulation of impacts in the Si frame for both studied heatsinking configurations. Right Simulation of impacts in the muntin structure below the pixels (color figure online)

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3 Effect on the X-IFU Energy Measurement

3.1 TES Response to Thermal Fluctuations

Fig. 4
figure 4

Left Response of a TES located at the center of the array to a frame CR impact at 11 mm distance (edge of the TES array) in the baseline heatsinking configuration (resulting temperature profile shown in red), compared to an X-ray pulse. Middle and right Energy offset measured on a 7-keV X-ray event recorded in the central pixel from a CR event at varying time offset and distance for frame (middle) and muntin (right) hits. All profiles are shown for the baseline heatsinking design and for the average energy deposition (450 keV for the frame and 250 keV for the muntins) (color figure online)

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The response of the TES detectors to thermal bath fluctuations was simulated using the xifusim X-IFU readout chain simulator [8]. To do so, we modeled the wafer response as a change of the bath temperature of the pixels, assuming that the detector reaction does not influence the propagation of the heat wave. This constitutes a worst case as the TES response can only absorb part of the wave’s energy. Figure 4 (left) compares the response of an X-IFU pixel to a CR impact to that of a normal X-ray pulse. We see that the TES reacts much more slowly to the former than to the latter (\(\sim \) 5 times longer 90 % rise time). This is due to the fact that a bath temperature fluctuation has to go through the weak SiN thermal link to reach the TES when an X-ray photon directly thermalizes in the detector.

We further characterized the effect of thermal pulses on the energy measurement by simulating the energy error induced on a 7-keV X-ray pulse as a function of the time separation between the CR and photon hits (see Fig. 4, middle and right). The characteristic shape of this time dependence is due to the shape of the optimal filter used and is similar to what is seen, e.g., for cross-talk effects [9]. We see that significant energy errors can only occur within a few ms time coincidence. This will be quite common for the numerous frame hits, but with only 7 muntin hits per second (see Fig. 1, right), the likelihood of such an overlap is very limited.

3.2 Average Energy Resolution Degradation

Fig. 5
figure 5

Example of bath temperature timelines for the baseline wafer configuration. Left For CR hits in the Si frame. Right For CR hits in the muntin structure (color figure online)

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Fig. 6
figure 6

Histogram of the energy errors induced by cosmic ray hits in 3 simulation scenarios (orange). For comparison, a realization of the 2.5-eV FWHM instrument Gaussian broadening is shown in blue. The quoted FWHM values were measured from the 88th and 12th percentiles of the distribution. The rarest, largest energy errors (some eVs) are not plotted for visibility purposes (4, 1 and 5 events for the left, middle and right histograms respectively). Left Frame hits for a side (worst case) pixel and the baseline heatsinking design. Middle The same CR hits and victim pixel but with the optimized design. Right Muntin hits for the central pixel (worst case) (color figure online)

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To estimate the overall effect on the X-IFU energy resolution of multiple CR impacts, it is necessary to simulate the TES response to not only single temperature profiles but also to a complete bath temperature timeline. Such a timeline can be generated as follows: For each CR impact, its arrival time is drawn randomly from a Poisson distribution and its deposition energy selected from the relevant Geant4 CR event list. The location of the energy deposition is then selected uniformly over the wafer, assuming a flat spatial distribution. We then interpolate the induced temperature fluctuation using the pre-calculated temperature profiles obtained at different locations on the wafer (see Sect. 2.2). Finally, all resulting wafer responses are rescaled with energy and added together as the thermal model was found to be linear (see [6] and [7]). Figure 5 shows examples of such timelines for CR hits both in the Si frame and in the muntin structure. We can see that fluctuations as high as a few tens of \(\upmu \)K are already quite common in 10-s-long simulations. We further note that a pixel on the edge of the array will see larger fluctuations from the frame than one in the center of the array and inversely for hits in the muntins.

We thus simulated for all configurations a series of 10 000 such timelines and modeled their impact on a concomitant 7-keV X-ray event. Figure 6 (left) shows the distribution of the measured energy reconstruction errors for frame hits in the baseline heatsinking configuration and for a pixel on the edge of the array (constitutes a worst case compared to the central one). We see that the distribution is largely non-Gaussian, but features a small inner broadening of 0.06-eV FWHM, well within the corresponding 0.2-eV allocation in the X-IFU energy resolution budget. Errors larger than this 0.2-eV limit only represent 1.2 % of the distribution. When comparing against a realization of the 2.5-eV instrument resolution Gaussian broadening, we see that these larger offsets will induce negligible non-Gaussianities. Figure 6 (middle) gives the same distribution for the optimized configuration, showing that it could offer a factor of two improvements in resolution degradation. Finally, we see from Fig. 6 (right) that hits in the muntin structure lead to an even narrower broadening due to the low associated CR count rate.

4 Triggering Capability

Fig. 7
figure 7

Comparison between the 5 \(\sigma \) trigger level and the energy at which a CR deposition can induce different energy offsets. Left Comparison made for hits in the frame and the baseline heatsinking design. The trigger pixel is assumed to be on the edge of the array. Right Comparison made for hits in the muntin structure for a pixel and its trigger located at the center of the array (color figure online)

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As seen in the previous section, the most energetic CR energy depositions may lead to energy errors in the few eV range when coincident with the reconstructed X-ray event. Ideally, we would want such large errors to be flagged out from the scientific data. Due to the much slower rise of the pixel response to a bath temperature change than to an X-ray (see Fig. 4, left), the standard X-IFU trigger mechanism, however, cannot efficiently detect even the largest signals. We found through simulation that the optimal signal-to-noise ratio for thermal events is actually obtained for a pixel timeline filtered with a \(\sim \) 1-ms averaging window (for comparison, the standard trigger operates with a 12.8-\(\upmu \)s window).

To quantify this trigger’s efficiency, we compared its 5 \(\sigma \) threshold level with the energy deposition required to induce various energy offsets on coincident X-rays. As seen in Fig. 7 (left), for the baseline heatsinking design, a trigger on an edge pixel could allow removing all energy offsets larger than \(\sim \)1 eV for central pixels and \(\sim 2.5\) eV for its neighboring pixels. Assuming a 2-ms exclusion window applied to the full array when an event is detected, this would be achieved with an almost negligible dead time penalty (\(< 0.1\) %). This solution is therefore attractive, but may not be strictly needed as shown in the previous section. A better solution may be to instead download a down-sampled timeline of a selection of pixels to more generally monitor any low-frequency environmental variations. This will be investigated in subsequent studies. This could allow partially correcting for the largest muntin hits, for which a simple trigger appears to be mostly ineffective (see Fig. 7, right).

5 Conclusion

In this paper, we investigated through simulations how the thermal fluctuations induced by cosmic ray hits on the X-IFU TES array Si wafer could affect the instrument energy resolution. We showed that with the baseline heatsinking design, this effect should remain well below its corresponding allocation in the instrument resolution budget and introduce negligible non-Gaussianities in the detector’s response. An optimized heatsinking configuration can further reduce this effect by a factor of two. We also showed that a dedicated trigger mechanism operating on the edge pixels of the array could detect and veto some of the most energetic frame events. Of course, this purely model-based study now warrants a dedicated experimental verification, which shall be conducted in the current phase B of the X-IFU project. This may for instance include the measurement of the TESs response to individual energy depositions at known locations in the wafer using radioactive alpha sources and/or heaters and a comparison with the corresponding model. A first attempt at such a comparison was made with an Hitomi/SXS [10] wafer and showed encouraging results (see more details in [7]).