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IEEE Transactions on Applied Superconductivity

Direct Observation of Josephson Oscillations in DC-Biased Transition-Edge Sensors

Felix T. Jaeckel, Conjeepuram V. Ambarish, Ayshea Banes, Haiyue Duane, Natalie Lesnjak, Miriam M. Marino, D. McCammon, Sophia Nowak, Avirup Roy, Shenggao Xu, J.-Q. Yan, J. S. Adams, S. R. Bandler, James A. Chervenak, Renata Cumbee, Fred Michael Finkbeiner, Joshua D. Fuhrman, Samuel V. Hull, Richard L. Kelley, Caroline Anne Kilbourne, Frederick Scott Porter, Kazuhiro Sakai, S. J. Smith, Nicholas A. Wakeham, Edward J. Wassell

Published Jan 24, 2025• 0 citations
Publisher

Abstract

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 (50$\mu$m length, 15$\mu$m 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 of$\mathbf {10^{-4}}$or 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-$\mathbf {T_{c}}$contacts, 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.

DOI
10.1109/tasc.2025.3533468

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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 (50μ\muμm length, 15μ\muμm 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 of10−4\mathbf {10^{-4}}10−4or 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-Tc\mathbf {T_{c}}Tc​contacts, 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.
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.
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.
It has been recently demonstrated that superconducting transition edge-sensors behave as weak-links due to longitudinally induced superconductivity from the leads with higher Tc. In this work we study the implication of this behaviour for transition-edge sensors (TES)-based bolometers and microcalorimeter under ac bias. The TESs are read-out at frequencies between 1 and 5 MHz by a frequency domain multiplexer based on a linearised two-stage SQUID amplifier and high-Q lithographically made superconducting LC resonators. In particular, we focus on SRON TiAu TES bolometers with a measured dark noise equivalent power of 3.2 \\times 10 -19 W/\\sqrt{Hz} developed for the short wavelength band for the instrument SAFARI on the SPICA telescope.
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.
Measurements have been made of the low-frequency spectral density of the voltage noise in current-biased resistively shunted Josephson tunnel junctions under conditions in which the noise mixed-down from frequencies near the Josephson frequency (${\ensuremath{\nu}}_{J}$) to the measurement frequency ($\ensuremath{\ll}{\ensuremath{\nu}}_{J}$) is in the regime $h{\ensuremath{\nu}}_{J}>{k}_{B}T$. In this limit, quantum corrections to the mixed-down noise are important. The spectral densities measured on junctions with current-voltage characteristics close to the Stewart-McCumber model were in excellent agreement with the predicted values, with no fitted parameters. The mixed-down noise for a wide range of bias voltages was used to infer the spectral density of the current noise in the shunt resistor at frequency $\ensuremath{\nu}$. With no fitted parameters, this spectral density at frequencies up to 500 GHz was in excellent agreement with the prediction $(\frac{2h\ensuremath{\nu}}{R})coth(\frac{h\ensuremath{\nu}}{2{k}_{B}T})$. The presence of the zero-point term, $\frac{2h\ensuremath{\nu}}{R}$, at frequencies $h\ensuremath{\nu}>{k}_{B}T$ was clearly demonstrated. The current-voltage characteristics of a junction with ${\ensuremath{\beta}}_{L}\ensuremath{\equiv}\frac{2\ensuremath{\pi}{L}_{s}{I}_{0}}{{\ensuremath{\Phi}}_{0}}\ensuremath{\sim}1$ and ${\ensuremath{\beta}}_{C}\ensuremath{\equiv}\frac{2\ensuremath{\pi}{I}_{0}{R}^{2}C}{{\ensuremath{\Phi}}_{0}}\ensuremath{\ll}1$, where I0{I}_{0}I0​ is the critical current, CCC is the junction capacitance, and Ls{L}_{s}Ls​ is the shunt inductance, showed structure at voltages where the Josephson frequency was near a subharmonic of the LsC{L}_{s}CLs​C resonant frequency. The additional nonlinearity of the $I\ensuremath{-}V$ characteristic caused mixing down of noise near higher harmonics of the Josephson frequency, thereby greatly enhancing the voltage noise. The measured noise was in good agreement with that predicted by computer simulations.
Transition-edge sensors (TESs) are two-dimensional superconducting films utilized as highly sensitive detectors of energy or power. These detectors are voltage biased in the superconducting-normal transition where the film resistance is both finite and a strong function of temperature. Unfortunately, the amount of electrical noise observed in TESs exceeds the predictions of existing noise theories. We describe a possible mechanism for the unexplained excess noise, which we term "mixed-down noise." The source is Johnson noise, which is mixed down to low frequencies by Josephson oscillations in devices with a nonlinear current-voltage relationship. We derive an expression for the power spectral density of this noise and show that its predictions agree with measured data.