Design Considerations for the Optimization of $\lambda$-SQUIDs

Cryogenic microcalorimeters are key tools for high-resolution X-ray spectroscopy due to their excellent energy resolution and quantum efficiency close to 100%. Multiple types of microcalorimeters exist, some of which have already proven outstanding performance. Nevertheless, they cannot yet compete with cutting-edge grating or crystal spectrometers. For this reason, novel microcalorimeter concepts are continuously developed. One such concept is based on the strong temperature dependence of the magnetic penetration depth of a superconductor operated close to its transition temperature. This so-called $\lambda$-SQUID provides an in-situ tunable gain and promises to reach sub-eV energy resolution. Here, we present some design considerations with respect to the optimization of such a detector that are derived by analytic means. We particularly show that for this detector concept the heat capacity of the sensor should match the heat capacity of the absorber.