Considering a Dual Exponential Model for Characterizing Dual−Tc Transition Edge Sensor Bolometer Prototypes for the CMB-S4 Experiment

We consider a dual-exponential characterization model for dual-Tc transition-edge sensor (TES) bolometer prototypes tailored to the CMB-S4 experiment. Each pixel integrates two superconducting transitions, a low-Tc ≈ 100 mK element for science observations and a high-Tc ≈ 500 mK element for elevated loading, wired in series to extend saturation power by nearly an order of magnitude without manual re-biasing. Devices were cooled in a dilution refrigerator at Tbath = 80mK and incremented at 10 mK and read out via a time-division-multiplexed SQUID chain, with a small square-wave voltage bias modulation applied to induce a step in bias current which results in a spike corresponding to the electrothermal feedback. Fitting the two-mode transient to Becker’s analytic model using Nelder–Mead and Powell algorithms yielded loop gains LI ∼ 1–20, intrinsic thermal time constants τ ∼ 20–80 ms, and for resistances ≥ 0.5 Rn found current sensitivities βI ≲ 0.5. Strong negative feedback compressed the closed-loop time constant to τeff ∼ 2 ms while the electrical time constant τel ∼ 10–50 µs remained an order of magnitude smaller, satisfying stability criteria although consistently underestimating τel. These quantities demonstrate promising results for further development of this dual-exponential modelling for large-scale CMB-S4 fabrication and integration into existing on-telescope work-cycles.