# A 4-by-4 high impedance NbSi TES array for high resolution X-ray spectro-imaging

Benjamin Criton, Stefanos Marnieros, Jean-Luc Sauvageot, Xavier de la Broïse, Laurent Bergé

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Benjamin Criton, Stefanos Marnieros, Jean-Luc Sauvageot, Xavier de la Broïse, Laurent Bergé. A 4-by-4 high impedance NbSi TES array for high resolution X-ray spectro-imaging. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2025, 1075, pp.170369. 10.1016/j.nima.2025.170369. hal-04995233

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Submitted on 10 Jun 2025

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Full Length Article

# A 4-by-4 high impedance NbSi TES array for high resolution X-ray spectro-imaging

Benjamin Criton<sup>a, ID, *, 1</sup>, Stefanos Marnieros<sup>b, ID, 1</sup>, Jean-Luc Sauvageot<sup>a, 1</sup>, Xavier de la Broïse<sup>a, ID, 1</sup>, Laurent Bergé<sup>b</sup>

<figure><img src="image_3.png" /></figure>

ARTICLE INFO

Keywords:

Nbium–Silicon

Transition-Edge-Sensors

SOI

Si membranes

DRIE

Micro-calorimeter array

Si heat conduction

ABSTRACT

We conduct an R&D project on high impedance TES detectors aiming to minimise the focal plane power consumption while having high spectral and spatial resolutions. In this article we review the development of 4-by-4 Nbium–Silicon (NbSi) Transition-Edge-Sensor (TES) arrays. The 16 membrane pixels are suspended by thin (5 µm thick) Silicon bridges that ensure electrical wiring and a weak thermal link to the heat sink. We fabricate these arrays on Silicon On Insulator (SOI) wafers with different pixel-to-substrate bridge lengths, widths, orientations and numbers to experimentally determine the Silicon heat conduction at 100 mK. We also design different NbSi TES meanders to measure the electron–phonon decoupling effect, the various heat capacities (Si, NbSi, absorber) and the slope of the superconducting transitions. These chips will help to better understand and control the physical parameters for future optimised developments. This constitutes, in addition with our 50 mK CMOS multiplexing ASIC and our 4 K HEMT-SiGe amplifier, a scaled-down prefiguration of an X-ray cryogenic detector chain for astrophysics.

1. Introduction

This research focuses on high impedance (~2 MΩ at normal state) NbSi TESs for X-ray astrophysics as a possible path to increase the pixel number of the future X-ray space telescopes. To achieve high spatial resolution of wide astronomical objects such as the intra-cluster medium, it is necessary to increase the size of the array as aimed in the future Athena telescope, embedding around 3000 240 µm wide pixels [1]. High impedance detectors can be read-out by low-noise, low-frequency HEMT transistors [2] at 4 K, in the idea of easing the transfer to the hotter cryo-cooler stages of as much as possible of the power load of the focal plane and 300 mK stage, which are major impediment for increasing the number of pixels, and in particular the pre-amplification function classically performed by SQUID. Furthermore, we use a classical CMOS technology to multiplex the 16 pixels at 50 mK [3]. The use of HEMTs and CMOS technologies greatly simplifies the readout of our high impedance TESs compared to SQUID multiplexing/readout. The 50 mK focal plane is constituted of the TESs array and the CMOS multiplexer, and the amplification is placed on the 4 K stage. This present paper focuses on the development of the 4-by-4 NbSi TES arrays as a first step towards much larger arrays. We

base this work on simulations and successful measurements of X-ray pulses achieved previously in our lab on single pixels suspended by Al/Au wire bondings [4,5]. Furthermore, data on low temperature heat conduction of Silicon, as well as heat capacity of superconducting NbSi and Tantalum absorbers, are scarce, if not absent. Thus, we develop our set of arrays to be able to measure these aforementioned parameters. The on-chip heater, normally used for the active electro-thermal feedback, is here also used for static and transient measurements to determine the thermal parameters of our chips. The active feedback temperature stabilisation consists in permanently dissipating a very low thermal power (depending on the heat conduction to the sink) on the heater and transiently reducing it when an X-ray photon is absorbed, so as to always keep the temperature of the pixel as constant as possible. This feedback signal is generated from the thermometer readout signal, amplified and applied at one terminal of the on-chip heating resistor [4–6]. It allows the TES thermometer to be current biased at a defined power, independent of the thermal link to the sink.

This article reviews the TES functioning under active feedback as well as system modelling and optimisations to then focus on the 4 × 4 array design and fabrication steps and finally show (using simulations)

<footnote><sup>a</sup> Université Paris-Saclay, CEA, IRFU, Gif-sur-Yvette, 91191, France</footnote>

<footnote><sup>b</sup> Université Paris-Sud, CNRS/IN2P3, IJCLab, Bât. 108 Orsay, 91405, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: benjamin.criton@cea.fr (B. Criton), stefanos.marnieros@ijclab.in2p3.fr (S. Marnieros), jean-luc.sauvageot@cea.fr (J.-L. Sauvageot), xavier.de-la-broise@cea.fr (X. de la Broïse), laurent.berge@ijclab.in2p3.fr (L. Bergé).</footnote>

<footnote n="1">¹ These authors contributed equally to this work.</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Transient modelling of a pixel and its readout electronics (see text). $R_{ch}$ and $R_{th}$ are the electrical resistances of the heater and the thermometer, $R_{pol}$ is the biasing resistance of the thermometer, $R_i$ and $R_z$ form a voltage divider to adjust the electro-thermal feedback gain, $V_{cr}$ and $V_{pol}$ are the heater and thermometer constant (filtered) biasing voltages. $C_{flex}$ is the parasitic capacitance of the electrical links between the pixel and the electronics.</figcaption></figure>

<caption>**Table 1**</caption>

Pixel physical parameters at the bias point where the TES resistance is around 400 kΩ and the sensitivity $\beta$ is near its maximum. We see the strong decoupling of the temperature for the heater due to a large voltage bias whereas the decoupling between the TES electrons and the membrane phonons is very small (even though the thermal link is a factor of two stronger in the heater due to its higher electronic temperature).

<table><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Heat sink temperature (T<sub>S</sub>)</td><td>35 mK</td></tr><tr><td>Si phonon temperature (T<sub>p</sub>)</td><td>90.7 mK</td></tr><tr><td>NbSi Thermometer electronic temperature (T<sub>T</sub>)</td><td>92.24 mK</td></tr><tr><td>NbSi Heater electronic temperature (T<sub>R</sub>)</td><td>189.7 mK</td></tr><tr><td>Heat sink to Si heat resistance (R<sub>p</sub>)</td><td>1 GK/W</td></tr><tr><td>NbSi e-ph decoupling constant</td><td>600 W K<sup>-5</sup> cm<sup>-3</sup></td></tr><tr><td>NbSi Thermometer decoupling resistance (R<sub>T</sub>)</td><td>3.87 &times; 10<sup>9</sup> K/W</td></tr><tr><td>NbSi Heater decoupling resistance (R<sub>H</sub>)</td><td>1.79 &times; 10<sup>9</sup> K/W</td></tr><tr><td>NbSi Thermometer volume</td><td>1.2 &times; 10<sup>-9</sup> cm<sup>3</sup></td></tr><tr><td>NbSi Heater volume</td><td>1.4 &times; 10<sup>-10</sup> cm<sup>3</sup></td></tr><tr><td>Absorber Volume</td><td>1.25 &times; 10<sup>-6</sup> cm<sup>3</sup></td></tr><tr><td>Si membrane volume</td><td>1.25 &times; 10<sup>-6</sup> cm<sup>3</sup></td></tr><tr><td>NbSi Thermometer heat capacity (C<sub>T</sub>)</td><td>1.51 &times; 10<sup>-14</sup> J/K</td></tr><tr><td>NbSi Heater heat capacity (C<sub>R</sub>)</td><td>2.39 &times; 10<sup>-15</sup> J/K</td></tr><tr><td>Absorber heat capacity (C<sub>A</sub>)</td><td>1.29 &times; 10<sup>-14</sup> J/K</td></tr><tr><td>Si membrane heat capacity (C<sub>P</sub>)</td><td>7.05 &times; 10<sup>-15</sup> J/K</td></tr><tr><td>V<sub>pol</sub></td><td>10 mV</td></tr><tr><td>V<sub>cr</sub></td><td>-17 mV</td></tr><tr><td>R<sub>pol</sub></td><td>10 MΩ</td></tr><tr><td>R<sub>ch</sub></td><td>3 MΩ</td></tr><tr><td>Thermometer biasing current (i<sub>th</sub>)</td><td>0.96 nA</td></tr><tr><td>TES slope β<sub>0</sub> = dR/dT</td><td>151 MΩ/K</td></tr><tr><td>Amplifier gain (A<sub>i</sub>)</td><td>-60</td></tr><tr><td>TES resistance at bias point (R<sub>th</sub>)</td><td>427 kΩ</td></tr></tbody></table>

that we can experimentally retrieve the thermal conductivities and heat capacities of the pixels.

## 2. TES modelling and optimisation

### 2.1. Large signal model and linearisation

Our X-ray microcalorimeters are modelled with 5 different bodies, each characterised by a different temperature: the X-ray absorber (at temperature $T_A$), the electrons of the superconducting NbSi TES at $T_T$, the electrons of the normal metal NbSi heater at $T_R$, the heat sink (Si wafer thermally connected to the cryostat) at $T_S$ and the phonons of the

Silicon membrane at $T_p$. The four first bodies are linked to the phonon bath of the Si membrane of the pixel by a thermal conductance $R_A$, $R_T$, $R_R$ and $R_P$, each of them having their respective noise spectrum presented in Table 2. The heater, thermometer, Si membrane and absorber also have their respective heat capacity $C_R$, $C_T$, $C_P$ and $C_A$. For the NbSi alloys, we still do not have heat capacity data in the superconducting transition and thus use the law determined for NbSi Anderson insulator derived in [7]. For electron-phonon (el-ph) decoupling, we use the law determined in [8]. We assume that the Kapitza resistance between the Si and NbSi phonons (this Kapitza implicitly included in $R_T$ as it is in series) is negligible [8,9]. Thus, we do not expect Thermal Fluctuation Noise (TFN) from the conduction between Si and NbSi phonons as the Kaptiza is assumed to be low for such long meanders. We however included the electron-phonon decoupling noise between these two temperatures. In Mo/Au TES, the large Debye temperature difference between both elements (Kapitza) and a proximity effect with the Nb leads could explain an internal TES TFN as seen in [10]. For the specific heat of both the Tantalum absorber and Si membrane we use the Debye model with their respective Debye temperatures. With this brief description, we can establish the transient model of our system, presented in Fig. 1. It includes a thermal part (yellow, left) and an electrical part (blue, right) interacting with each other through the heat sources ($P_{iR}$ and $P_{iT}$), introducing in the thermal part the Joule power resulting from the current flow in the heater ($I_{ch}$) and the thermometer ($I_{th}$), while the electrical resistance of the thermometer ($R_{th}$) in the electrical part depends on the temperature of the thermometer electrons ($T_T$). The X-ray heat deposition is modelled using a pulse power source $P_v$ on the absorber and its energy is given by $E = P\delta t$, with $\delta t$ the duration of the pulse. The amplifier of gain $A_l$ amplifies the voltage across the thermometer ($V_{th} = I_{th}R_{th}(T_T)$) to provide the output signal $A_lV_{th}$. This signal is used after a selectable attenuation ($R_1/R_2$ resistor bridge) to drive the feedback voltage on the heater resistance ($R_{ch}$), and thereby the electro-thermal feedback.

This model is used to perform transient electro-thermal simulations under LTSpice (as presented in Fig. 3). We also establish from this a biasing algorithm based on analytical expressions that computes the bias point (i.e. the voltages, currents and temperatures in the model without photon absorbed) for a given resistance of the TES in the transition. This gives also the values of all the parameters of the model (thermal capacities and resistances, etc.) as shown in Table 1 for a typical bias. Finally, the model is linearised at the chosen bias point, resulting in the block diagram of Fig. 2, which allows to
<figure><img src="image_5.png" /><figcaption>Fig. 2. Linearised block schematic of the transient model of Fig. 1. The input is the $dP_e$ Dirac and the output is the $dV_I$ impulse response $\beta_0$ is the slope $dR_{th}/dT_T$ at the bias point.</figcaption></figure>

Source: Figure taken from [5].

<figure><img src="image_6.png" /><figcaption>Fig. 3. Evolution of temperatures (phononic: $T_{ph}$, electronic: $T_e$), output voltage ($V_e$) and heater Joule power ($P_{Jh}$) with (1/K = 0.17) and without (1/K = 0.005) active electro-thermal feedback with a 5.9 keV photon absorbed at t = 1 ms. The heater Joule power is proportional to the square of the output voltage and its integral gives a compensated energy of 5.57 keV. This simulation is conducted on a 500 × 500 × 5 µm membrane pixel with a 2 MΩ TES (at normal state) biased at 420 kΩ where the slope is 150 MΩ/K. The expected resolution calculated with this particular pixel is 7.24 eV.</figcaption></figure>

compute the impulse response and the noise spectrum (Fig. 4) using the noise expression in Table 2. In low resistivity, voltage-biased TES research, linearisation is also performed but the analysis mostly lies on the complex impedance of the TES to evaluate its behaviour and response (time constants) [11]. A matrix formalism is also used to solve the linearised expressions [12]. In our approach, we use the Bode formalism as developed in [13] since we explicitly have a multi-loop system (feedback+TES self heating). The Bode plots show phase and gain margins for stability, static gain and time constants. It is however more difficult to extract from the Bode plot the multiple time constants

<caption>Table 2</caption>

Noise sources included in the small signal model. All thermal links present a white TFN, Johnson noise of the three main resistors are taken. The amplifier noise model is fitted from experimental data on our 4 K amplifiers.

<table><thead><tr><th>Noise source</th><th>Noise expression</th></tr></thead><tbody><tr><td>Thermometer Johnson noise</td><td>e<sub>R<sub>th</sub></sub> = √(4k<sub>B</sub>R<sub>th</sub>T<sub>T</sub>) [V/√Hz]</td></tr><tr><td>Heater Johnson noise</td><td>e<sub>R<sub>th</sub></sub> = √(4k<sub>B</sub>R<sub>th</sub>T<sub>R</sub>) [V/√Hz]</td></tr><tr><td>Biasing resistor Johnson noise</td><td>e<sub>R<sub>pol</sub></sub> = <sup>R<sub>th</sub></sup>/<sub>R<sub>th</sub>+R<sub>th</sub></sub>√(4k<sub>B</sub>R<sub>pol</sub>T<sub>S</sub>) [V/√Hz]</td></tr><tr><td>Thermometer el-ph decoupling noise</td><td>e<sub>R<sub>j</sub></sub> = √(<sup>2k<sub>B</sub></sup>/<sub>R<sub>j</sub></sub> <sup>T<sub>j</sub>+T<sub>e</sub></sup>/<sub>T<sub>j</sub></sub>) [W/√Hz] [14]</td></tr><tr><td>Heater el-ph decoupling noise</td><td>e<sub>R<sub>k</sub></sub> = √(<sup>2k<sub>B</sub></sup>/<sub>R<sub>k</sub></sub> <sup>T<sub>k</sub>+T<sub>e</sub></sup>/<sub>T<sub>k</sub></sub>) [W/√Hz] [14]</td></tr><tr><td>TFN Si to heat sink</td><td>e<sub>R<sub>p</sub></sub> = √(<sup>4k<sub>B</sub>T<sub>p</sub><sup>2</sup></sup>/<sub>R<sub>p</sub></sub>) √(<sup>T<sub>p</sub></sup>/<sub>T<sub>p</sub></sub>) <sup>T<sub>p</sub>/(k<sub>B</sub>T<sub>p</sub>)</sup> <sup>dT</sup>/<sub>dT</sub> [W/√Hz] [15]</td></tr><tr><td>TFN Si to absorber</td><td>e<sub>R<sub>a</sub></sub> = √(<sup>4k<sub>B</sub>T<sub>a</sub><sup>2</sup></sup>/<sub>R<sub>a</sub></sub>) √(<sup>T<sub>a</sub></sup>/<sub>T<sub>a</sub></sub>) <sup>T<sub>a</sub>/(k<sub>B</sub>T<sub>a</sub>)</sup> <sup>dT</sup>/<sub>dT</sub> [W/√Hz] [15]</td></tr><tr><td>Amplifier noise model</td><td>e<sub>n</sub> = <sup>1×10<sup>-9</sup></sup>/<sub>τ</sub> <sup>1+τp</sup>/<sub>p</sub> [V/√Hz]</td></tr></tbody></table>

of the model. I. J. Maasilta in [11] shows that the more complex the thermal system is, the higher the noise and the lower the responsivity. This can be explained because the bodies are “simple-passive” heat capacity blocks. In our case, we have the usual phonon heat block and the absorber block forming the classic two-blocks model. But to this we add two other heat capacity blocks: the electrons of the heater and of the thermometer. These added block have heat capacity and thermal conductances varying with temperature. Thus, the analysis of the impact of the number of block in the system on the noise is less straight-forward. Optimisation of each physical parameter has to be addressed and the expected measurements on the matrices presented here will give us tendencies to which we will build our simulations. The small signal model also permit the computation of the energy resolution for different parameter values (see Figs. 5 and A.2). The pixel used for the following simulations, is a square Si membrane of 500 µm width and 5 µm thickness on which are deposited the heater and the thermometer (resp. ~2 mm and ~10 mm length, volumes given in Table 1). The Absorber is either glued using Stycast 1266 or attach using Indium bump. The thermal link to the heat sink is set to 1 nW/K.

In many of the fabricated matrices, the thermal link from the membrane to the heat sink is controlled using different number of Si bridges on each pixel. The thermal link in the simulations of Fig.
<caption>**Table 3**</caption>

Power consumption summary at the 50 mK, per pixel, of our TES development, put in perspective with the Athena X-IFU FDM/TDM options. We can still reduce the power consumption at 50 mK by placing the biasing resistor at 300 mK with a spectral resolution degradation of only 0.7%, leading to 55 pW at 50 mK, still 4 times higher than the Athena estimates due to the current multiplexing dissipation. Also, the shunt resistor can be placed at 300 mK without significant addition of noise (~2% degradation of the resolution in eV).

<table><thead><tr><th>Parameter</th><th>Pixel of Table 1</th><th>Pixel of Table 1, normalised G</th><th>Athena</th></tr></thead><tbody><tr><td>TES thermal biasing</td><td>57 pW for 1 nW/K</td><td>4.6 pW for 0.08 nW/K</td><td>2.5 pW for 0.08 nW/K [17]</td></tr><tr><td>Shunt resistor</td><td>10 pW</td><td>10 pW</td><td>2.5 pW</td></tr><tr><td>Multiplexing</td><td>50 pW (CMOS) [3]</td><td>50 pW (CMOS)</td><td>8 pW (SQUID) [18]</td></tr><tr><td>Total</td><td>120 pW</td><td>65 pW</td><td>13 pW</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Spectra of the different noise sources identified in our model presented in **Fig. 2**. Noises are output referred (gain of 60 on the amplifier). The impulse response of the system is added in black dots. The spectral resolution of this pixel is 2.1 eV and the thermal link to the heat sink is set to 100 pW/K. Johnson noises and thermometer el-ph decoupling noises are predominant. At low and high frequency, amplifier noise is the largest.</figcaption></figure>

3 is set to 1 GK/W ($G = 1$ nW/K conductance). This link is much stronger than for the Athena TESs: ~80 pW/K [16,17], explaining the strong dissipation in the heater exhibited in the simulation of **Fig. 3**: 57 pW (ten times higher than for Athena) to keep the membrane at the transition temperature (90 mK) of the NbSi TES. The reason for this stronger link only lies in the heritage of former tests and measures on wire-bonding suspended pixels and for higher resolution pixels, we set this thermal link to 100 pW/K. For the Athena pixels described in [16], the dissipation in the TES can be estimated as follows: $P_{TES} = (T_c - T_{bath}) \times G_{bath} \approx (88 \times 10^{-3} - 55 \times 10^{-3}) \times 80 \times 10^{-12} \approx 2.6$ pW to which has to be added the dissipation in the shunt, roughly of the same amount (voltage bias), leading to 5 pW per pixel. In our case, by decreasing the thermal conductance to the sink by a factor of 12.5 (thus reaching the Athena numbers [1,16]), we will dissipate only 4.6 pW per pixel and we will also improve spectral resolution as shown in **Fig. 5**. To this, we also need to add the dissipation in the biasing resistor providing a constant current to the TES: 10 MΩ under 0.96 nA, leading to 15 pW per pixel in total for our solution, a factor of three higher than the dissipation of the Athena X-IFU pixels, but these biasing resistors can be placed to higher temperature (typically 300 mK) without significant resolution degradation. **Table 3** shows the different power dissipations for our design in perspective with the Athena proposal (with FDM or TDM) that, however, fulfils other design choices: smaller pixels, longer signal duration, smaller energy dynamics. The main difference in dissipation comes from the multiplexing, far more dissipative in our current set-up. It must be reduced in the future, in particular by using a finer CMOS technology (here 0.35 µm) and less parasitic capacitances.

## 2.2. NbSi high impedance TES functioning

Before describing the fabrication process of our NbSi arrays, we present shortly the functioning of the current-biased high impedance TESs. To reduce the strong and inherent decoupling of the electronic and phononic temperatures in the NbSi TES alloy, it has been decided

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Evolution of the spectral resolution with the heat resistance between cold bath (wafer) and pixel (suspended membrane).</figcaption></figure>

to bias the superconducting thermometer with a very small constant current. However, the temperature coefficient of the TES being positive, this current biasing is inherently unstable. To cope with this situation, we add on the pixel membrane a purely resistive NbSi heater (made from a different proportion of co-evaporated Nb and Si). The Joule power dissipated in the heater is controlled by the readout electronics: the voltage applied to the heater is proportional to the strongly and negatively amplified thermometer's voltage (i.e. proportional to the temperature for small variations). Thus, when the temperature of the detector rises, the power dissipated in the heater falls (because the Joule dissipation in the heater is always much higher than that in the thermometer to thermally set the pixel to the transition temperature), tending to keep the temperature constant on the membrane: an active electro-thermal feedback is implemented [4]. Using a block-diagram
<figure><img src="image_9.png" /><figcaption>Fig. 6. Comparison of the time responses of a suspended pixel with either a 1 GK/W thermal link or a 10 GK/W thermal link. We see the rise in amplitude caused by the weaker link conjugated with the lengthening of the pulse duration.</figcaption></figure>

formalism, we can derive a forward chain/loop gain and a feedback chain gain: see Appendix.

Fig. 3 gives the transient evolution of different parameters of the detector (electronic temperature of the TES, phononic temperature, output voltage and heater Joule power dissipation) with two different feedback gains: $1/K = 0.005$ and $1/K = 0.17$ (see Appendix for calculation). We see that with a high thermal feedback (gain of $1/K = 0.17$), the temperature swing of the electrons of the NbSi TES is shorter and somehow smaller compared to no feedback ($1/K = 0.005$) due to the much higher drop of the heater power and thus the faster cooling of the detector. This has the advantage of limiting the variations of the electronic capacity of the TES $C_T$ and the decoupling resistance $R_T$. The oscillation observed with feedback is due to the delay between readout and the actual temperature feedback on the heater, largely dominated by the parasitic capacitance between the detector and the 4 K electronics. For both simulations, the biasing point is the same, i.e. the thermometer resistance is maintained at 400 kΩ and its Joule dissipation is below 1 pW (same biasing current). Without feedback, the output amplitude is higher but will saturate more easily for higher photon energy, and will take hundreds of ms to return to the operating point due to self heating of the unstable current biasing. Low feedback also makes stability and ease of thermal biasing more difficult. Small signal simulations have confirmed that even though the thermal signal is reduced by the active feedback, the signal to noise ratio is not affected (see Fig. A.2), and even reinforced thanks to the attenuation of the electron-phonon decoupling, allowing to obtain resolutions around 2 eV. Noise spectra for the (simulated) 2 eV resolution pixel are given in Fig. 4 as well as its impulse response (black dots). The power dissipation of this optimised pixel is 2 pW (1.6 pW in the heater) due to a sink conductance of only 100 pW/K (similar to Athena data). There are multiple trades-off between choices of physical parameters values. Indeed, as the system is in closed-loop, each parameter has a complex impact on the noise and on the sensitivity. For instance, increasing the biasing current of the thermometer will reduce the impact of the amplifier noise as the signal will be increased. The feedback from the latter will maintain the TES in its bias point. However, a higher bias current results in a higher dissipation on the electrons system ($T_T$), thus a higher decoupling. This decoupling is noisy and as it is one of the most dominant, it will degrade in turn the resolution. We thus expect an optimum in bias current in the TES. The HEMT noise must be the lowest possible and a lower than 1 nV/√Hz is achievable [2].

## 3. NbSi array design and fabrication

### 3.1. Array design

We design 16-pixel arrays on SOI wafers with a 5 µm top Si layer. Each pixel comprises a high normal state resistance NbSi TES thermometer and a resistive NbSi heater [5,6] as shown in Fig. 8, where

the heater is on the right and the thermometer meander on the left, occupying nearly the entire membrane surface. Pixels are suspended by different bridge configurations to study the thermal conductivity of Silicon around the transition temperature. Furthermore, we want to see the effect on heat conduction of the bridges' directions with respect to the wafer crystal orientation, as anisotropy could occur [19]. Moreover, different heat conduction modelling such as the diffusive model ($\kappa_{dif}$), the Casimir model ($\kappa_{Cas}$) or the Ziman model (specular reflection) lead to different heat conduction values. The Ziman model, introducing a $(1+p)/(1-p)$ coefficient is not studied here since $p$ is an experimental parameter. For a bridge of width $w = 10$ µm, length $l = 100$ µm and thickness $e = 5$ µm and a phonon mean free path taken to be $\lambda = 1.12\sqrt{e\bar{w}}$ [20] and assuming no temperature gradient (100 mK), we obtain in the case of a diffusive heat conduction following Eq. (1) a thermal resistance of 250 GK/W, whereas the Casimir model described by Eq. (2) gives a thermal resistance of 150 GK/W.

$$ \kappa_{dif} = \frac{1}{3} C \nu \lambda = 3 \frac{\rho_{Si}}{M_{Si}} \mathcal{N}_{A} k_{B} \frac{4\pi^4}{15} \frac{T^3}{\Theta_D^3} \nu \lambda \quad [\text{W/K/m}] \quad (1) $$

$$ \kappa_{Cas} = \frac{2\pi^2 k_B^4}{15\hbar^3} \frac{1}{\nu^2} \lambda T^3 \quad [\text{W/K/m}], \quad (2) $$

with $\nu$ the speed of sound in Si (taken at 5000 m/s) and $\Theta_D$ the Debye temperature of Si. The choice between these different models lies with the geometrical aspect-ratio of the samples namely the comparison of the length with respect to the phonon mean free path [20]. We thus develop a wide variety of parallelepipedic bridges for the 16 pixels of a chip. We fabricate pixels with bridges respectively of 60 µm or 120 µm length, 10 or 30 µm width and a number of bridges per pixel ranging from 4 to 32. In all cases, the bridge thickness is 5 µm (the thickness of the top layer of the SOI). Three bridge orientations are studied: 45° and perpendicular-parallel to the wafer primary flat. Applying Eq. (1) to our set of bridges, we obtain thermal resistances ranging from 0.91 GK/W for the strongest thermal link to 76G K/W for the weakest link framing perfectly the values used in our pixel simulations. The strong effect on spectral resolution of the thermal link between heat sink and pixel can be seen in Fig. 5: a strong thermal link short circuits the heat flux from the absorber to the heat sink directly, reducing the heat rise on the decoupled thermometer. Spectral resolution is strongly dependent on the strength of the link to the heat sink for large conductances since the energy is shunted to the sink instead of being accumulated in the heat capacity of the pixel: $dT = E/C$. For small heat conductances, the energy resolution tends to converge towards a constant value as the energy is not shunted any more but the TFN (inherent to all calorimeters whether voltage or current biased) becomes higher. A competition between signal amplitude rise and increasing TFN begins. It is foreseen that for very weak thermal links, the amplitude of the signal does not increase but the noise does and thus the resolution begins to be degraded. Moreover, this link is an important parameter when multiplexing pixels. Indeed, the shorter the pulse duration the lesser sampling points we will have to reconstruct the signal (see Fig. 6). However, a very low thermal link, resulting in long pulse durations, causes increased number of pile-ups and imposes to reduce the dissipation in the heater and thermometer to keep the thermometer at working thermal bias point and stabilised, the latter causing a reduction in the input energy dynamics since less heat reserve is available to reduce the heat dissipation in the heater when an X-ray is absorbed. It is also important to have a sufficient amount of samples to generate the active feedback to the heater. Indeed, as the TESs are current-biased, thermal stability is only fulfilled by the active heaters. When multiplexing, the feedback is not continuous any more and is instead maintained constant whilst the pixel is not read-out and updated only when the pixel is read [3]. All in all, there is a trade-off between high sensitivity and linearity on a large energy range, and the variety of bridge sizes in our test arrays will enable to find the best heat link.
<figure><img src="image_10.png" /><figcaption>Fig. 7. Fabrication steps, pixel cut view: (1) Gold hybridisation pad lift-off. (2) Lift-off NbSi heater (resistive NbSi) and Al heater contact+wiring. (3) Lift-off NbSi TES and Al TES contact+wiring. (4) Selective Al etching. (5) Protective SiO₂ lift-off on NbSi TES and heater. (6) Front-face Si RIE until buried SiO₂. (7) Back-face Si DRIE stopping at SiO₂ + SiO₂ etching. Yellow: Au or Al pad for absorber attachment; red: resistive NbSi heater; green: superconducting NbSi TES; blue: Al wiring and pads; orange: SiO₂ protective layer; grey: Si; black: SiO₂ BOX.</figcaption></figure>

## 3.2. Array fabrication

The fabrication process of our 16-pixel chips can be broken down into seven steps shown in Fig. 7. First, we deposit the Au or Al hybridisation pad. The second step is the lift-off of the NbSi heater and of the Al heater wiring without breaking vacuum to ensure optimal contact between Al and NbSi. Then, we proceed to the lift-off of the thermometer, superconducting NbSi and its Al wiring in the same vacuum. The fourth step is the etching of Al on the NbSi TES and heater meander, using a chemistry with excellent selectivity of Al over NbSi. The Al wiring and pads are protected from the etching using a photoresist. A SiO₂ layer is deposited on the pixels to protect them from the etching phases and air oxidation of the NbSi. The gold hybridisation pad is not covered by the SiO₂ layer. The sixth step consist in the etching of the top Si layer to create the pixel patterns (the central membrane and the bridges). The etching process stops on the SiO₂ Buried Oxide Layer (BOX) after etching the 5 µm of Silicon. The final step is the back face DRIE and etching of the BOX. This phase is challenging if we want to obtain vertical walls, no over-etching and no black Si. Yet, since all pixels have the same aspect ratio, the process can be optimised for the specific geometry and succeeded with high yields: only one broken bridge on an entire wafer of 25 chips. The transition temperature of our NbSi TES being quite sensitive to annealing and the disorder of the amorphous alloy is a key parameter [21], it is important to master this final DRIE step that can generate heat and unwanted front face etching. The central Au or Al pad visible in Fig. 8 is deposited for future absorber hybridisation tests using Indium bump. Tests on glueing Tantalum absorbers are mechanically successful to date. However, for the first static measurements on our chips we will not glue any absorbers as we want to evaluate beforehand some conduction properties of the membranes. In our experimental apparatus, the chip is glued onto a gold plated Cu surface of a “daughter” PCB using a silver-filled epoxy (Epo-tek E4110 passing NASA STM E595) increasing thermal homogeneity of the Silicon substrate (heat sink) and ensuring

a strong thermal link to the cryostat cold plate. We use Al bondings to prevent any heat flow from the 4 K “hot” electronics towards the pixels via the wiring through the “mother” 50 mK PCB. Chips (16-pixel arrays) are diced from the SOI 3" wafer using DRIE (same last step of array fabrication) at a square format of side 1 cm.

Regarding the NbSi alloy, there are three parameters used to adjust the transition temperature ($T_c$). Two of them are fabrication parameters: the thickness of the NbSi meander and the ratio of Nb in the amorphous $Nb_xSi_{1-x}$ alloy [21–23]. The third parameter to tune the transition is annealing [24,25] which changes the conductivity of the NbSi 2D sheet film due probably to topological short range ordering. Annealing below 200 °C will cause a lowering of the transition temperature and an increase of the normal-state resistance. A good control of the $T_c$ is achieved at IJCLab and different experiments have been developed using NbSi TESs with $T_c$ ranging from 50 mK to 480 mK [26,27]. In our developments, the thickness of the NbSi TES meander is 30 nm. This is achieved with a precision of 1 nm using quartz crystals and INFICON controller. No morphological granularity nor inhomogeneity have been reported with Atomic Force Microscopy and Rutherford BackScattering methods [21,28]. We designed pixels with either 5 µm or 10 µm meander widths, and lengths reaching roughly 10 mm. The resistive NbSi heater meander is deposited using another fabrication step (first step in Fig. 7) since its Nb proportion in NbSi does not lead to a superconducting material but a purely resistive one (see [21]), and it has a sheet resistance roughly a factor of 10 higher than the normal resistance of the superconducting TESs, so the heater meander can be 10x shorter than the TES at equal resistance.

# 4. Determination of the thermal parameters of pixels using simulations

Before designing our pixels, we ensure that it is possible to retrieve the different important physical parameters we want to measure: thermal conduction to the heat sink, thermal electron–phonon decoupling
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Top left: A 32-bridges pixel of one of the fabricated arrays presented in the bottom image. The central pad in the middle of the pixel is a Au under bump metallisation for Ta absorbers. Top right: layout of the pixel presented on the left image. The red NbSi meander is the TES thermometer whilst the short blue meander is the resistive NbSi on-chip heater. The bridges are 60 µm long, 30 µ wide and 5 µm thick. The pixel membrane is a square of side 500 µm. Bottom: a 4-by-4 array of NbSi TESs, the chip is 1 × 1 cm and presents different pixel configurations.</figcaption></figure>

of the TES, and total heat capacity. The first measurement to conduct will be the determination of the superconducting transition at very low thermometer biasing current so that we can assume no added Joule power on the pixel and a very low electron-phonon decoupling leading to equating electronic and phononic temperatures. Then, we can determine the heat conduction between the pixel and the heat sink (i.e. the Si substrate) using the on-chip heater. We will achieve this measurement by increasing the Joule power dissipated in the heater while maintaining the same temperature on the substrate $T_S$, thus creating a temperature gradient between the pixel and the substrate. With the same low current bias (to equalise $T_T$ and $T_P$), we will measure the phononic temperature of the pixel and deduce the heat conduction to the heat sink, knowing the Joule power on the heater. The simulation result presented in Fig. 9 shows that it is possible to retrieve the heat conduction in the sensitive part of the superconducting transition (a few mK). Furthermore, this measurement has already been successfully achieved on wire bonding suspended pixels. The value obtained is only for a very short temperature range as the transition is very sharp but it remains pertinent as it is our working temperature. In fact, it is also possible to obtain a wider temperature range lower than the transition temperature by maintaining the thermometer at a constant temperature (using the heater dissipation) and varying the heat sink temperature (cryostat) below the temperature of the TES.

The next simulation presented in Fig. 10 shows that we are also able to determine the electron-phonon decoupling heat conduction (incorporating a constant Kapitza resistance but assumed to be negligible in a first approximation). To obtain this physical parameter, we sweep the biasing current of the thermometer (by sweeping the voltage on its 10 MΩ load resistor) leading to an increase in decoupling. With the help of the table R($T_T$) measured at very low biasing current, when $T_T = T_P = T_S$, we deduce the corresponding electronic temperature $T_T$ by measuring the resistance (which depends only on the electronic temperatures). The phononic temperature $T_{ph}$ is calculated knowing the Joule power $P_{th}$ through the thermometer and the thermal conductance between the pixel and the heat sink (determined with the previous measurement). It is thus a direct calculation : $R_T = (T_T - T_P)/P_{th}$ [K/W].

Finally, Fig. 11 shows how the averaged frequency response of a set of heater-injected pulses [5] can provide a way to determine the heat capacity of the device. For that, we need to assume that the system's thermal response is well approximated by a low pass filter (with one or two poles). A simulation without absorber gives the thermal time constants of the bare pixel (Si, NbSi meanders and superconducting Al wiring) using a two poles fitting of the frequency response. Then, with the knowledge of the time constant and the previously determined pixel to heat-sink thermal conductance, we can
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Simulation of the determination of Si thermal resistance value. The value is valid only in the sensitive part of the transition, i.e. between 90 mK and 200 mK. The heating power is the power applied on the on-chip resistive NbSi meander.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Simulation of the determination of the thermal electron-phonon decoupling value. The oscillations are caused by numerical approximations.</figcaption></figure>

determine the heat capacity of the bare pixel. By difference, it is possible to determine the heat capacity of different absorbers and attachment methods. For instance, we determine in simulation a pixel heat capacity of 60.9 fJ/K without any absorber (the input theoretical value was 56.6 fJ/K) and an absorber heat capacity of 30.5 fJ/K for an input capacity of 31.8 fJ/K. This represents an error of 4% on the determination of the absorber's heat capacity, which is acceptable. Experimentally, this will be a quite straightforward measurement : we compute the Fourier transform of the mean pulse from a set of heater-injected pulses and fit it with a two poles transfer function. However, the thermometer biasing current must be kept low in order to prevent any thermometer passive positive electro-thermal feedback, and the heater feedback must be disconnected. This procedure will enable measurements of different types of absorbers, the Tantalum heat capacity, the effect of thin layers of Al superconducting traps on quasi-particles life time [29,30], and of different thermal links (epoxy or In bumps) to the Si membrane. For epoxy attachment of the absorber, we foresee using Stycast 1266 as used by the Goddard Space Flight Center for the XRS instrument of Suzaku spacecraft [31]. Evaporated Bi-Au absorbers could also be studied as they exhibit excellent spectral resolution performances [1,32] and a better filling factor compared to hybridised absorbers. However, their fabrication steps might cause an unwanted annealing of the superconducting NbSi, and they might present a higher heat capacity than the Ta option. The Tantalum absorber will be diced in square of side 500 µm (same size as the membrane of the pixels). Its thickness is 12.5 µm and it is annealed in high vacuum to extract dissolved Hydrogen [33,34]. A thin layer of Al is deposited without breaking vacuum to avoid Hydrogen migration

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Evolution of the frequency response of a pulse parametrised by absorber volume (i.e. heat capacity). A second order low pass filter allows the determination of the cut-off frequencies.</figcaption></figure>

and to create the quasi-particle traps. Indium bumping of the absorber to the membrane is the baseline option and has been successfully tested in our labs on 8-by-8 matrices [35].

## 5. Conclusion

In this article we have presented the design and fabrication of 4-by-4 high impedance NbSi TES arrays. Each pixel embeds the high impedance NbSi superconducting thermometer and the resistive NbSi heater. We have designed pixels with different thermal link configurations in order to evaluate the Silicon heat conduction and eventually produce arrays with optimised thermal links for high spectral resolution. We show that in our simulation environment it is possible to retrieve the key physical parameters of our pixels : heat sink to pixel thermal resistance, electron-phonon decoupling and pixel/absorber heat capacity, essentials for further optimisations. We have determined with a first low temperature measurement a normal state resistance of 2.9 MΩ at 200 mK and experimental measurements down to 30 mK will resume as soon as the dilution refrigerator will be repaired.

### CRediT authorship contribution statement

**Benjamin Criton:** Writing – review & editing, Writing – original draft, Investigation, Formal analysis, Conceptualization.
**Stefanos Marnieros:** Investigation, Conceptualization.
**Jean-Luc Sauvageot:** Investigation, Conceptualization.
**Xavier de la Broïse:** Investigation, Formal analysis, Conceptualization.
**Laurent Bergé:** Investigation.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

We want to greatly thanks the IJC Lab team for the work they are accomplishing to fabricate and provide us with the detectors samples, detectors matrices and absorbers. This work was partially funded by Labex FOCUS (ANR-11-LABX-0013) and Labex P2IO (ANR-10-LABX-0038), both managed by the Agence Nationale de la Recherche (France).
<figure><img src="image_15.png" /><figcaption>Fig. A.1. Forward chain gain as a function of bias current in the TES. The gain is quite constant as long as the current $i_{th}$ is below 1 nA (i.e. $1 - i_{th}^2 \beta_0 R_T \approx 1$). Above this value, the TES begins to be unstable, and biasing without feedback is impossible. The region where the forward gain is negative is not explored experimentally and translates an inherently unstable region: the TES self heating is too strong and the thermometer becomes a constant resistor.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. A.2. Spectral resolution as a function of feedback amplifier gain (1/K) in the previous equations. The resolution does not vary much with feedback gain which means that the feedback does not add noise to the system and that the reduction in the temperature swing does not degrade the resolution.</figcaption></figure>

Appendix. Loop gain calculations

The input of the forward chain is the phononic temperature $T_P$ and its output the amplified voltage at the amplifier's output $V_I$. We have:

$$dV_I = A_I dV_{th} = A_I i_{th} dR_{th} = A_I i_{th} \beta_0 dT_T, \qquad (A.1)$$

since at the bias point and assuming a constant bias current $i_{th}$, we have by linearisation $R(T_{T_0}) = R_0 + dR/T_{T_0} dT_T$. Furthermore, we have:

$$dT_T = dT_P + dP_{JT}R_T = dT_P + dR_{th}i_{th}^2 R_T. \qquad (A.2)$$

Thus, by injecting Eq. (A.2) in (A.1) and factorising, we obtain:

$$dV_I = A_I \frac{i_{th} \beta_0}{1 - i_{th}^2 \beta_0 R_T} dT_P \qquad (A.3)$$

The forward loop gain is thus $(A_I i_{th} \beta_0)/(1 - i_{th}^2 \beta_0 R_T)$, and using the data of Table 1, we obtain a gain $dV_I/dT_P$ of -18.85 V/K ($A_I$ is negative) and the voltage decreases at the amplifier output when the temperature rises (in agreement with Fig. 3). We see that the stronger the coupling between electron and phonons the lower the open loop gain. However, as long as we keep the thermal link much smaller than $1/i_{th}^2 \beta_0$ (i.e. $1 - i_{th}^2 \beta_0 R_T \approx 1$), the reduction in loop gain is not even seen. For $i_{th}^2 \beta_0 R_T \ge 0$, we have an unstable thermometer (the solution is to reduce the current bias). Fig. A.1 shows the evolution of the forward chain gain (Eq. (A.3)) as a function of bias current in the TES.

The feedback chain/loop has for input the output voltage of the amplifier (output of the forward chain) and for output the phononic temperature $T_P$. Indeed, the feedback chain has for role to modify the phononic temperature with respect to the TES electronic temperature. The gain is determined using a Taylor expansion to the 2nd order. We note $V_H$ the voltage at the amplifier side of the heater resistor, $V_{B_0}$ the voltage at the bottom of the heater (this is always a constant voltage $V_{cr}$) and K the amplifier gain for the feedback loop (fixed by resistors $R_1$ and $R_2$ of Fig. 1). We have $V_H = V_{H_0} + A_I/K dV_{th}$ and $V_B = V_{B_0}$. Thus, the power dissipated in the heater is:

$$P_{JR} = \frac{(V_H - V_B)^2}{R_{ch}} = \frac{1}{R_{ch}} \left[ V_{H_0}^2 + \left( \frac{A_I}{K} dV_{th} \right)^2 + \frac{2A_I V_{H_0}}{K} dV_{th} + V_{B_0}^2 - 2(V_{H_0} + \frac{A_I}{K} dV_{th})V_{B_0} \right]. \qquad (A.4)$$

By taking only the derivative part of this equation, we have

$$dP_{JR} = \frac{1}{R_{ch}} \left[ \left( \frac{A_I}{K} dV_{th} \right)^2 + \frac{2A_I}{K} (V_{H_0} - V_{B_0}) dV_{th} \right]. \qquad (A.5)$$

Yet, $dT_P = dP_{JR}R_P$ when we shut-down the TES forward chain. We obtain finally the relation between the voltage of the TES and the phononic temperature change from the heater:

$$dT_P = \frac{R_P}{R_{ch}} \left[ \left( \frac{A_I}{K} dV_{th} \right)^2 + \frac{2A_I}{K} (V_{H_0} - V_{B_0}) dV_{th} \right]. \qquad (A.6)$$

If we take only the first order Taylor expand, the feedback gain is $(2A_I/K)(V_{H_0} - V_{B_0})$. For $K = 21$, $V_{B_0} = V_{cr} = 17$ mV, $A_I = -60$, $R_P = 1.8$ GK/W, $R_{ch} = 3$ MΩ, and assuming $V_{H_0} \approx 0 \ll V_{B_0}$ V, we obtain a feedback loop gain $dT_P/dV_{th}$ of -58 K/V, or $dT_P/dV_I$ of 0.97 K/V. This is a low value, compared to the forward loop gain, since in this numerical application, the feedback gain K of the amplifier is small. The amplified signal $V_I$ is divided by a factor of 21 from the resistor bridge $R_1/R_2$ before being applied to the heater. We see that we have a very good control of this feedback gain by adjusting the value of K and/or $A_I$.

The values of both the forward and feedback chain gains have been validated using LTSpice simulations of the electro-thermal model presented in Fig. 1.

We thus have a system with a current biased TES of a given gain determined above and stabilised with a feedback signal from the TES signal that reduces the heat dissipation on the pixel when the TES temperature rises. The feedback gain, also determined above, is adjustable using different resistor values for $R_1$ and $R_2$. The feedback gain is positive with respect to the amplifier output voltage $V_I$ meaning that when $V_I$ rises, the temperature of the phonons also rises. However, as the forward chain gain is negative, the output voltage of the amplifier decreases when the phononic temperature rises. Thus, when the phononic temperature rises the amplifier voltage decreases, thus the heater power decreases so that the temperature decreases. Thermal stabilisation is achieved.

Data availability

Data will be made available on request.

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