We systematically study the analysis and simulation of ac-biased superconductor transition-edge sensor (TES) circuits. In these ac-biased circuits, the current and voltage of the TES experience large swings in both directions, and small-signal analysis around a dc steady state does not apply. To understand their electrical and thermal behavior, we rely on concepts and techniques from radio-frequency circuit simulation and introduce the periodic steady state and perform periodic ac analysis. We also construct TES device models based on a comprehensive two-fluid physical model and investigate the behavior of ac-biased TES circuits using advanced circuit simulators. By relating our findings to results for the analysis of dc-biased TES circuits, we give appropriate definitions for the current and temperature sensitivity of the TES in ac-biased circuits. Our work not only builds a rigorous foundation for theoretical analysis of ac-biased TES circuits but also introduces powerful simulation techniques valuable for their design and research.
We systematically study the analysis and simulation of ac-biased superconductor transition-edge sensor (TES) circuits. In these ac-biased circuits, the current and voltage of the TES experience large swings in both directions, and small-signal analysis around a dc steady state does not apply. To understand their electrical and thermal behavior, we rely on concepts and techniques from radio-frequency circuit simulation and introduce the periodic steady state and perform periodic ac analysis. We also construct TES device models based on a comprehensive two-fluid physical model and investigate the behavior of ac-biased TES circuits using advanced circuit simulators. By relating our findings to results for the analysis of dc-biased TES circuits, we give appropriate definitions for the current and temperature sensitivity of the TES in ac-biased circuits. Our work not only builds a rigorous foundation for theoretical analysis of ac-biased TES circuits but also introduces powerful simulation techniques valuable for their design and research.
It has long been recognized that weak-link effects play an important role in transition edge sensors (Sadleir et al., 2010 and Sadleir et al., 2011), especially when the intrinsic transition temperature of the leads is much higher than that of the device itself. The weak-link physics has also been identified as an important complication if the TES is read-out under MHz-range AC bias for multiplexing purposes (Gottardi et al., 2014). More recently, it has been proposed that the commonly observed “excess” Johnson noise could also be explained as a white Johnson noise component that is mixed down into the signal band from higher frequencies via Josephson oscillations on a non-linear I-V relation (Gottardi et al., 2021 and Wessels et al., 2021). In recent experiments with small scale Mo/Au (50μm length, 15μm wide) TES devices with Nb leads (fabricated at GSFC) with a small DC bias applied, we have been able to directly observe Josephson oscillations over a kHz to MHz frequency range. The spectrum of the oscillations is well described by a simple damped oscillator model, with its center frequency scaling in direct proportion to the voltage across the TES. The second harmonic is also observed at small bias. If nothing else, we show that the Josephson oscillations provide a straightforward way to determine the value of the shunt-resistor with accuracy of10−4or better, as well as the offset voltage of the bias circuit. This is complementary to our previous work demonstrating that Shapiro steps in the IV curve induced by AC magnetic fields can be used for shunt resistor calibration (Zhou et al., 2018). Further work will be needed to determine if this effect can be observed in larger TES or those with relatively low-Tccontacts, where Shapiro steps are too weak to observe. It remains to be seen whether additional insights on the mixed-down Johnson noise can be obtained from a physics-based model of the weak-link.
A simple transition-edge sensor (TES) is governed by two coupled, nonlinear equations, which when solved give the behavior of the device given the initial conditions. The two equations describe the full electrical and thermal characteristics of the device and the way in which the device interacts with the external electrical and thermal circuits. To date these coupled nonlinear equations have been solved analytically in the small-signal limit, by linearizing the equations about some operating point. The resulting coupled linear equations contain a wealth of interesting physics, and form the principal tool by which all detectors are currently designed and modeled. In this article we describe a numerical technique for solving the coupled nonlinear equations rigorously, without any assumptions about the magnitudes of the various signals and parameters that determine the behavior of a device. The technique is based on a harmonic balance algorithm that searches, in the frequency domain, for voltage, current, and temperature wave forms that are consistent with the solid-state physics of the device. Indeed, both the small and large signal limits can be simulated, and the full harmonic content of the system retrieved. In this article, we will outline the principal features of the algorithm, and show various results such as I−V curves, saturation when signal power is modulated, impedance response when the bias current is modulated and signal power kept constant, and pulsed signal power analysis that shows the dynamics of a device. Numerous extensions of this fundamental technique are now possible including multitone nonharmonic analysis, large-signal impedance calculation, modeling of complicated thermal circuits, noise models, and frequency-domain multiplexing. We believe that our algorithm will become the principle technique for analyzing the large-signal behavior of all TES detectors and on that basis we are developing computer aided design software.
We report the experimental evidence of the ac Josephson effect in a transition edge sensor (TES) operating in a frequency domain multiplexer and biased by ac voltage at MHz frequencies. The effect is observed by measuring the non-linear impedance of the sensor. The TES is treated as a weakly linked superconducting system and within the resistively shunted junction model framework. We provide a full theoretical explanation of the results by finding the analytic solution of the non-inertial Langevian equation of the system and calculating the non-linear response of the detector to a large ac bias current in the presence of noise.
Transition-edge sensor (TES) bolometers operate under strong electrothermal feedback, wherein power deposited on the bolometer is compensated by a corresponding change in electrical power dissipation. We present a comprehensive analysis of Johnson noise suppression that extends previous theoretical frameworks to the general case of AC-biased linear circuits with arbitrary parasitic impedances. Using a Thévenin-equivalent circuit formulation -- consisting of an ideal voltage source and complex series impedance external to the bolometer thermal island -- we derive analytical expressions for the noise-equivalent current in the presence of both bolometer and parasitic Johnson noise sources. Our analysis demonstrates that while the electrothermal feedback loop effectively suppresses Johnson noise originating from the bolometer resistance, it provides no suppression of Johnson noise from external series resistance. In the limit of high loop gain, the bolometer Johnson noise contribution vanishes while the parasitic contribution remains unsuppressed.
We present a detailed analysis of the fundamental noise sources in superconducting transition-edge sensors (TESs), ac voltage biased at MHz frequencies and treated as superconducting weak links. We have studied the noise in the resistive transition as a function of bath temperature of several detectors with different normal resistances and geometries. We show that the "excess" noise, typically observed in the TES electrical bandwidth, can be explained by the equilibrium Johnson noise of the quasiparticles generated within the weak link. The fluctuations at the Josephson frequency and higher harmonics contribute significantly to the measured voltage noise at the detector bandwidth through the nonlinear response of the weak link with a sinusoidal current-phase relation.
Understanding the weak thermal links in supporting legs of transition edge sensors (TES) is crucial for developing highly sensitive bolometers used in cosmic microwave background (CMB) detection. Conventional current-biased differential resistance measurements under varying temperature (R-T) used to characterizeRand critical temperature (Tc) of TES often encounter a phenomenon: the bias current will affect the transition temperature obtained duringTbdrop testing, as well as affect the obtained shape of the transition region. This behavior stems from the weak thermal links of TES under current bias, limitingR-Tmeasurements to only rough information about the TES transition regime sometimes. The detailed and accurate performance parameters typically relying on voltage-biased current-voltageI-Vmeasurements using using superconducting quantum interference devices (SQUIDs). In fact, the positive electro-thermal feedback induced by current bias contains valuable insights into the device's intrinsic properties. This work focuses on elaborating this underlying mechanism: by analyzing the current-induced suppression of measured transition temperature inR-Tmeasurements, we extract key parameters including thermal conductance (G), thermal exponent (n),Tc, and saturation power (Psat). This study provides intuitive calculations and visualizations of the electro-thermal behaviors induced by current bias in TES devices, and offers a simple and rapid complementary tool for electro-thermal parameter extraction. In batch TES fabrication, it can serve as a pre-screening tool to improve the efficiency of subsequent high-precision SQUID-basedI-Vcharacterization.