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Journal of Applied Physics

Multitone large-signal analysis of superconducting transition-edge sensors for astronomy

Karwan Rostem, S. Withington, D. J. Goldie

Published Aug 1, 2007• 3 citations
Publisher

Abstract

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.

DOI
10.1063/1.2763976

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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 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.
Multiplexed superconducting quantum interference device (SQUID) amplifiers have recently enabled the deployment of kilopixel arrays of superconducting transition-edge sensor (TES) detectors on a variety of receivers for astrophysics. Existing multiplexing techniques for TES arrays, however, have constraints due to aliasing of SQUID noise, the size of the required filtering elements, or the complexity of the room-temperature electronics that make it difficult to scale to much larger arrays. We have developed a Walsh code-division SQUID multiplexer that has the potential to enable the multiplexing of larger arrays or pixels with faster thermal response times. The multiplexer uses superconducting switches to modulate the polarity of coupling between N individual TES detectors and a single output SQUID channel. The polarities of the detector signals are switched in the pattern of an N × N Walsh matrix, so a frame composed of N orthogonal samples can be used to reconstruct the detector signals without degradation. We present an analysis of the circuit architecture and preliminary results.
When designing instruments for submillimeter-wave astronomy it is important to understand precisely how saturating superconducting tunnel-junction mixers behave. In this article, a procedure is described for calculating the behavior of tunnel-junction circuits when multitone high-level signals are applied. The procedure is based on a full quantum-mechanical description of photon-assisted tunneling, and uses Harmonic Newton to search for a set of voltages and currents that satisfy the circuit equations at every frequency present. A number of illustrative simulations are presented, and it is shown that Harmonic Newton can be used to model saturating superconductor–insulator–superconductor mixers in a robust and numerically efficient manner.
SQUIDs can be used as amplifiers for superconducting Transition Edge Sensor (TES) photon counting detectors. According to the physical process of photon counting, a TES signal simulator was developed to evaluate the performance of SQUID amplifier for TES detectors. A triangle simulation signal was determined to be a close match to that from a real TES photon counter. The dynamic range for the output voltage versus bias current was first characterized and then the current signals were incrementally coupled to circuit to determine the highest sensitivity of a series-array SQUID amplifier. The sensitivities are obtained by changing the triangle signal amplitude added to the TES signal simulator. The impedance changes can be deduced and further obtain the temperature changes of TES devices. When knowing the photon energy, TES design parameters can be known according to the thermal characteristics. In conclusion, the TES simulator can be well applied to guide the device design for effective readout.
We present a new software package, called QMix, for simulating the behavior of superconductor/insulator/ superconductor (SIS) mixers. The software uses a harmonic balance procedure to calculate the ac voltage across the SIS junction and multitone spectral domain analysis to calculate the quasiparticle tunneling currents. This approach has two major advantages over other simulation techniques: 1) it can include an arbitrary number of higher order harmonics, and 2) it can simulate the effect of multiple strong nonharmonic frequencies. This allows the QMix software to simulate a wide range of SIS mixer operation, including the effects of harmonics in the local-oscillator (LO) signal and gain saturation with respect to the RF signal power. In this article, we compare the simulated results from QMix to the measured performance of a 230-GHz SIS device, both to validate the software and to investigate the experimental data. To begin, we simulated the conversion gain of the mixer and we found excellent agreement with the experimental results. We were also able to recreate the broken photon step effect by adding higher and lower order harmonics to the LO signal. We then simulated a range of incident RF signal powers in order to calculate the gain saturation point. This is an important metric for SIS mixers because we require linear gain for reliable measurements and calibration. In the simulated results, we found both gain compression and gain expansion, which is consistent with other studies. Overall, these examples demonstrate that QMix is a powerful software package that will allow researchers to simulate the performance of SIS mixers, investigate experimental results, and optimize mixer operation. We have made all of the QMix software open-source and we invite others to contribute to the project.
In contemporary cosmic microwave background experiments, bolometric detectors are often background limited, and in this case the sensitivity of instruments can only be improved by increasing the number of background-limited detectors, and so contemporary TES receivers contain as many pixels as possible. Frequency-domain multiplexing (fMUX) is one strategy for reading out many detectors with one SQUID. For any readout system, it is important to carefully evaluate the thermal design of detector, in conjunction with the readout bandwidth, in order to ensure stable electro-thermal feedback (ETF). We demonstrate a novel technique for characterizing the thermal circuit of our detectors, using am AC-bias and the fMUX electronics. This technique is used to study the internal thermal coupling of a TES bolometer. We illustrate how the insights gathered by this technique have been instrumental in improving the stability of our multiplexed detectors for the south pole telescope (SPT).