Mutual Inductance Sensing SQUID: Cryogenic microcalorimeter based on mutual inductance readout of superconducting temperature sensors

Abstract

Superconducting microcalorimeters, such as superconducting transition-edge sensors and magnetic microcalorimeters, have emerged as state-of-the-art detectors for x-ray emission spectroscopy by combining near-unity quantum efficiency with excellent energy resolution. Despite these achievements, their resolving power has not yet reached the level required to rival modern wavelength-dispersive grating or crystal spectrometers. Here, we introduce a next-generation SQUID-based microcalorimeter concept that exploits the strong temperature dependence of the magnetic penetration depth of a superconductor operated close to its critical temperature. The resulting mutual-inductance-based readout enables in situ tunable signal amplification, while inherently avoiding hysteretic effects that commonly limit superconducting sensors. Experiments with prototype devices demonstrate robust and reproducible operation over a wide temperature range. Based on our measurements and modeling, we project that an energy resolution below 100meV should be achievable with an optimized absorber-sensor combination. This approach therefore represents a promising pathway towards next-generation cryogenic detectors for high-precision X-ray spectroscopy.

Figures (5)

• Figure 1: Schematic circuit diagram of a Mutual Inductance Sensing SQUID (MISS). The red-, black-, and purple-colored inductors are made from a superconducting material with critical temperature $T_\mathrm{c}$. The sensing layer (blue) is made from a superconducting material with significantly lower critical temperature, i.e., $T_\mathrm{c,sens} \ll T_\mathrm{c}$, and is in good thermal contact with a suitable X-ray absorber. The device is operated at temperature $T_0 \lesssim T_\mathrm{c,sens}$. A current source injects a constant current $I_\mathrm{in}$ into an input coil with inductance $L_\mathrm{in}$. Owing to the temperature-dependent diamagnetic response of the sensing layer, the mutual inductance $M_\mathrm{in}(\lambda(T))$ between the input coil and the inductively coupled SQUID loop segment with inductance $L_\mathrm{SQ,in}$ becomes temperature dependent. The green crosses indicate resistively shunted Josephson tunnel junctions.
• Figure 2: (a) Exploded view drawing, (b) simplified layout drawing, and (c) micrograph of the prototype Mutual Inductance Sensing SQUID (MISS). The sensing layer (silver in (a) and (c), blue in (b)), is made from Al ($T_\mathrm{c,Al} \simeq 1.2\,\mathrm{K}$), while the other superconducting device components are made from Nb ($T_\mathrm{c} \simeq 9.0\,\mathrm{K}$). In (b), the input and flux biasing coil as well as the SQUID loop are traced in yellow, purple and red, respectively.
• Figure 3: Measured mutual inductance $M_\mathrm{in}$ as a function of both temperature $T$ (bottom axis) and reduced temperature $\tilde{t} = T/T_\mathrm{c,Al}$ for both prototyoe MISS. For visibility purposes the MISSv2 data and fits have been horizontally shifted by $0.25\,\mathrm{K}$ and only a random sample of all data points for the original MISS are shown. For the original variant, we recorded three consecutive temperature cycles indicated by different symbols. The perfect overlap of these indicates the absence of hysteresis. The lines show fits of the measured data according to $M_\mathrm{fit,1}$ (dashed) and $M_\mathrm{fit,2}$ (dotted).
• Figure 4: a) Comparison of the change in mutual inductance as function of temperature (bottom axis) and reduced temperature $\tilde{t} = T/T_\mathrm{c,Al}$ for a $\lambda$-SQUID (green) and our MISS (red). The significantly larger change in mutual inductance of the MISS compared to the $\lambda$-SQUID is clearly visible. b) Comparison of the gain factor of the MISS (opaque with contour) and the $\lambda$-SQUID (colored), both calculated from the derivative of the fitted measurement data and assuming a $T_\mathrm{c}$ of $50\,\mathrm{mK}$ of the temperature-sensitive circuit element. The strong dependence on $I_\mathrm{in}$ and the operation temperature $T_0$ is visible as the gain factor changes over two orders of magnitude.
• Figure 5: Estimated achievable energy resolution as a function of total heat capacity for a MISS (solid lines) and a $\lambda$-SQUID (dashed lines). Red, green and blue represent critical temperatures of $100\,\mathrm{mK}$, $50\,\mathrm{mK}$, and $20\,\mathrm{mK}$ of the temperature-sensitive circuit element. Symbols represent the achievable energy when using a $250\,\upmu\mathrm{m} \times 250\,\upmu\mathrm{m}$ particle absorber made of gold, bismuth, or tin whose thickness was chosen to achieve a stopping power exceeding $99.99\,\%$ for soft X-ray photons photons. filled symbols refer to a MISS, open symbols to a $\lambda$-SQUID.