SQUID-based superconducting microcalorimeter with in situ tunable gain Open Access

Cryogenic microcalorimeters such as superconducting transition-edge sensors (TESs)^1,2 or metallic magnetic calorimeters (MMCs)^3,4 have proven to be outstanding devices to measure the energy of x-ray photons with utmost precision. They rely on sensing the change in temperature of an x-ray absorber upon photon absorption using an extremely sensitive thermometer that is based on either a superconducting (TES) or a paramagnetic (MMC) sensor material. Thanks to their exceptional combination of an outstanding energy resolution and a quantum efficiency close to 100%, they are highly attractive for x-ray emission spectroscopy (XES), both at synchrotron light sources and in a laboratory environment.^5,6 They strongly relax the requirements on x-ray beam intensity as compared to state-of-the-art wavelength-dispersive x-ray spectrometers based on diffraction gratings or bent crystals.⁷ Moreover, they cover the entire tender x-ray range⁷ that is hardly accessible with both, grating and crystal spectrometers, and allow studying strongly diluted as well as radiation-sensitive samples that can, even at the most brilliant synchrotron light sources, only be investigated with greatest efforts.^5,6

Current cutting-edge TES- and MMC-based detectors achieve an energy resolution $Δ E FWHM$ of $0.72 eV$ for $1.5 keV$ photons⁸ and of $1.25 eV$ for $5.9 keV$ photons⁹ and provide a quantum efficiency close to 100% in this energy range. In terms of resolution, they come close to the resolution of wavelength-dispersive x-ray spectrometers while offering orders of magnitude higher efficiency. However, despite this great success, both detector types face some challenges that presently prevents these detectors from reaching an energy resolution in the range of $100 meV$ as required for investigating vibrations or d-d-excitations in soft x-ray spectroscopy or resonant inelastic x-ray scattering. It might hence be worthwhile to look for alternate detector concepts in addition to advancing the present leading technologies.

Here, we present an innovative SQUID-based microcalorimeter with in situ tunable gain that appears to be particularly suited for high-resolution soft x-ray spectroscopy. It is based on the strong temperature dependence of the magnetic penetration depth $λ ( T )$ of a superconducting material that is operated close to its critical temperature $T c$⁠. Figure 1 depicts a simplified schematic circuit diagram of our microcalorimeter, which we denote as λ-SQUID. It largely resembles a conventional low- $T c$ dc-SQUID that comprises two identical resistively shunted Josephson tunnel junctions with critical current $I c$ and normal state resistance R as well as a closed superconducting loop formed by two inductors with inductances $L λ$ and $L sfb$⁠, respectively. The Josephson junctions, the inductor $L sfb$ as well as the superconducting wiring are made from a superconducting material with critical temperature $T c ≫ T 0$ that is much greater than the operating temperature T₀ of the microcalorimeter. In contrast, the inductor $L λ$⁠, which we will refer to as λ-coil is made from a superconducting material with much lower critical temperature $T c λ ≳ T 0$ that exceeds only barely the device operating temperature T₀. It is worth mentioning that for a practical device, the operation temperature will be chosen according to the transition temperature of the λ-coil, and not vice versa. The λ-coil is coupled to the input coil with inductance $L in$ via the mutual inductance $M in = k L λ L in$⁠. Here, k denotes the magnetic coupling factor. Since the λ-coil is operated near its critical temperature, its magnetic penetration depth $λ ( T )$ depicts a strong temperature dependence, resulting in a temperature dependence of its inductance $L λ ( T ) = L λ , geo ( T ) + L λ , kin ( T )$⁠. Here, $L λ , geo ( T )$ and $L λ , kin ( T )$ represent the geometric and kinetic contributions to the total coil inductance. The magnetic penetration depth influences the geometric inductance $L λ , geo ( T )$ via the distribution of current density in the cross section of the inductor, which ultimately causes a temperature dependence of the mutual inductance $M in ( T )$⁠.

The device is biased either with a constant current $I b$⁠, with the resulting voltage drop $V SQ$ across the device being used as output signal, or with a constant voltage $V b$⁠, the current $I SQ$ running through the devices acting as output signal. In any case, the output signal depends on the total magnetic flux $Φ tot$ threading the SQUID loop. A current source supplying a constant current $I in$ is connected to the input coil. This current induces a magnetic flux $Φ ( T ) = M in ( T ) I in$ within the SQUID loop that is temperature dependent via the temperature dependence of the mutual inductance $M in ( T )$⁠.

By attaching a suitable x-ray absorber to the λ-coil, the absorption of an x-ray photon causes a temperature rise of this coil that is transduced into a change of magnetic flux within the SQUID loop. The latter can be sensed as a change of output signal. An additional coil with inductance $L fb$ is inductively coupled to the λ-SQUID via the inductor $L sfb$ and mutual inductance $M fb$⁠. This coil allows for additional control of the magnetic flux threading the SQUID loop by running an externally controlled current $I fb$ through it.

We fabricated a prototype device that is depicted in Fig. 2(a). It is based on the design of one of our custom-made dc-SQUIDs for MMC readout and features four superconducting loops that are connected in parallel [Fig. 2(b)]. Each loop is inductively coupled to a flux biasing as well as an input coil where the coil arrangement is engineered such that the mutual inductance between these coils is greatly reduced. The majority of the device including the actual input coil with inductance $L in$ and flux biasing coil with inductance $L fb$ is fabricated from niobium having a critical temperature $T c ≃ 8.9 K$⁠. The part of the SQUID loop that is coupled to the input coil acts as λ-coil and is made from aluminum with critical temperature $T c λ ≃ 1.2 K$⁠. We note that aluminum is an unfavorable sensor material for a real microcalorimeter (as the thermal noise at $≃ 1.2 K$ would prevent to reach an energy resolution in the range of $1 eV$ or below). However, this choice was triggered by our intention to use this device only for proving the suitability of our microcalorimeter concept and to estimate its sensitivity as well as the fact that an easy-to-deposit elemental superconductor with $T c$ in the range of $50 − 100 mK$ does not exist. We consequently postponed the development of a sophisticated deposition process for a superconducting material with suitable $T c$ and also omitted the particle absorber. A full working microcalorimeter comprising a low $T c$ material for the λ-coil as well as a suitable x-ray absorber will hence be the subject of future work.

We comprehensively characterized our prototype device in a ³He/⁴He dilution refrigerator that was tweaked to smoothly run up to temperatures of about $1.5 K$⁠. We used a resistive heater mounted at the mixing-chamber platform to control the operation temperature and bias and read out the detector using a direct-coupled high-speed dc-SQUID electronics.¹¹  Figure 3(a) shows the measured output voltage $V SQ$ across the device vs the applied input current $I in$ at two different operating temperatures T₀ close to the critical temperature $T c λ$ of the λ-coil. The periodic response of dc-SQUIDs is clearly visible. For small input currents, the tiny temperature-induced change of the mutual inductance hardly creates a measured voltage change $Δ V SQ$⁠. However, the larger the $I in$⁠, the larger was the $Δ V SQ$⁠. This directly resembles the prediction according to Eq. (1) and proves the in situ tunability of the gain factor. In Fig. 3(b), the gain coefficient $∂ V SQ / ∂ T$ is depicted vs both the input current $I in$ and the reduced operation temperature $t ̃ 0 = T 0 / T c λ$⁠, illustrating the large tuning range available with these two parameters.

To determine the temperature dependence of the mutual inductance $M in ( T )$ as well as the sensitivity coefficient $∂ V SQ / ∂ T$⁠, we performed two different types of measurements: For static measurements, we injected a linear current ramp into the input coil and monitored the output voltage $V SQ$⁠. The current change $Δ I in$ for a flux change of a single flux quantum $Φ 0$ then allows to determine the mutual inductance via $M in = Φ 0 / Δ I in$⁠. For flux ramp modulated^12,13 (FRM) measurements, we injected a constant input current $I in$ into the input coil and applied a periodic sawtooth-like current ramp with amplitude $I ramp$ and repetition rate $f ramp$ to the flux biasing coil. We chose the ramp amplitude $I ramp$ such that in each cycle an integer multiple of flux quanta was induced into the λ-SQUID. In this mode of operation, the output voltage $V SQ$ is modulated with a well-defined frequency. Any flux signal $Φ$ that is quasi-static with respect to the ramp repetition rate $f ramp$ is then transduced into a phase shift $Θ = 2 π Φ / Φ 0$ of the periodic output voltage $V SQ$⁠. The flux contribution $Φ ( T ) = M in ( T ) I in$ resulting from the constant current within the input coil changes with temperature and can be derived from the phase shift $Θ ( T )$ of the λ-SQUID response. The latter can be determined by demodulation of the SQUID output voltage. It is worth mentioning that a change in the SQUID output voltage can be caused by a shift of the working point, too, and may affect the shape and offset of the periodic output voltage. However, the FRM phase and thus demodulated flux signal remain unaffected. The FRM method hence allows to distinguish different contributions affecting the SQUID voltage.

We additionally performed an FRM measurement without applying an input current to check for parasitic inductance effects [see Fig. 4(a)]. Since $I in = 0$⁠, we cannot use the equation $Φ ( T ) = M in ( T ) I in$ to relate the signal flux $Φ$ to a change in mutual inductance $M in$⁠. In this scenario, no flux signal is induced via the input coil. Instead, the measured signal is caused by the temperature dependence of the inductance $L λ$ coupling to the bias current of the SQUID. The parasitic inductance effects are negligible as compared to the change of mutual inductance.

Figure 5 shows the predicted energy resolution(s) vs the total heat capacity $C tot$ of the detector for three different critical temperatures of the λ-coil. For these calculations, we assumed a conservative input current of $I in = 500 μ A$⁠, rise and decay times of $τ 0 = 1 μ s$ and $τ 1 = 1 ms$(Ref. 16) and $β = 1 / 2$ as well as an operating temperature of $T 0 = 0.95 T c λ$⁠. The contributions from SQUID noise (dotted) and thermal noise (dashed) scale differently with the heat capacity, leading to a crossover point below which the thermal noise is dominating. It is worth mentioning that the SQUID noise scales with total heat capacity as the heat capacity and hence the geometric size of the λ-coil must match the absorber heat capacity. As expected, a decrease in $T c λ$ (and thus T₀) leads to an overall improvement of the energy resolution as well as a shift of the crossover point toward higher heat capacities. To give a specific example, we want to assume an x-ray detector with a sensitive detection area of $250 × 250 μ m 2$ and a thickness d of 5, 8.6, and $50 μ m$ for the absorber materials Au, Bi, and Sn, respectively. Thus, the stopping power is roughly identical for all materials and exceeds 99.99% at photon energies up to $1 keV$⁠. We select gold, bismuth, and tin as absorber material as these materials are often used in the cryogenic microcalorimeter community.^9,17,18 Figure 5 shows that for a proper combination of absorber material and critical temperature of the sensor material, an energy resolution as low as $300 meV$ should be feasible. It should be mentioned that a Bi absorber operated close to 50 mK with a specific heat of $6.5 × 10 − 14 JK − 1$ would experience a temperature increase in $2.46 mK$ (or just below 5% of $T c λ$⁠) upon absorption of a 1 keV photon. It is thus expected to display some nonlinear behavior, which may potentially be predicted theoretically and compensated for. Moreover, it should be noted that increasing the input current $I in$ suppresses the SQUID noise contribution further [cf. Eq. (1)]. This allows, for example, to compensate for a potential readout noise degradation when using SQUID-based multiplexing techniques.¹⁹ 

In conclusion, we have presented an innovative concept for a SQUID-based superconducting microcalorimeter with an in situ tunable gain. It is based on the strong temperature dependence of the magnetic penetration depth of a superconductor close to its critical temperature $T c$ that affects the mutual inductance $M in$ between the SQUID loop and an input coil that is biased with a constant current. The latter can be easily tuned in situ. This allows, for example, for compensating a potential noise degradation when using cryogenic multiplexing techniques. We have designed, fabricated, and characterized a prototype device using aluminum as sensor material to study the temperature dependence of $M in$⁠. We find that there is no sign for any hysteresis effects that often spoil the performance of superconducting microcalorimeters. Using these data, we have made predictions of the achievable energy resolution. We found that the lower the total specific heat $C tot$ of the detector, the easier it is to suppress the λ-SQUID noise below the thermal noise floor. More specifically, we found that an energy resolution $O ( 300 meV )$ with a suitable combination of absorber and sensor material is possible.

We would like to thank A. Stassen for his support during device fabrication and greatly acknowledge fruitful discussions with G. Jülg. We acknowledge the financial support by the KIT Center of Elementary Particle and Astroparticle Physics (KCETA). Furthermore, C. Schuster acknowledges financial support by the Karlsruhe School of Elementary Particle and Astroparticle Physics: Science and Technology (KSETA).

The authors have no conflicts to disclose.

Constantin Schuster: Conceptualization (equal); Formal analysis (equal); Investigation (lead); Software (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Sebastian Kempf: Conceptualization (equal); Formal analysis (equal); Investigation (supporting); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.