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arXiv:1205.5693 · AIP Advances

Complex impedance, responsivity and noise of transition-edge sensors: Analytical solutions for two- and three-block thermal models

Ilari J. Maasilta

Published Oct 10, 2012• 45 citations
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Abstract

The responsivity and noise of a voltage-biased superconducting transition-edge sensor depends strongly on the details of its thermal model, and the simplest theory for TES response assumes a single heat capacity connected to the heat bath. Here, analytical results are derived and discussed for the complex impedance, the responsivity and the noise of a transition-edge sensor, when the thermal model is not simple but consists of either two or three connected heat capacities. The implications of the differences of the models are discussed, as well.

DOI
10.1063/1.4759111

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The responsivity and noise of a voltage-biased superconducting transition-edge sensor depends strongly on the details of its thermal model, and the simplest theory for TES response assumes a single heat capacity connected to the heat bath. Here, analytical results are derived and discussed for the complex impedance, the responsivity and the noise of a transition-edge sensor, when the thermal model is not simple but consists of either two or three connected heat capacities. The implications of the differences of the models are discussed, as well.
Complex impedance measurements are widely used to characterize superconducting transition edge sensors (TESs) in bolometers and microcalorimeters. Typically, models are fit to impedance data to find parameters which pertain to the performance of these detectors. After the parameters are determined, the models are then used to compute the response and noise of these devices. In this paper, we present general relationships between the measured impedance, the thermal response to power in the TES, and noise. We describe a method for measuring αIand βIof the superconducting phase transition, which does not require model fitting. We find bolometer response is determined from the impedance provided that the absorber is strongly coupled to the TES electron system. We also demonstrate how to calculate upper and lower limits on the noise directly from the impedance data without modeling. Additionally, the relations can be used to check the validity of the models and to understand what information can and cannot be obtained from measurements of impedance, response, and noise.
The current noise at the output of a microcalorimeter with a voltage biased superconducting transition edge thermometer is studied in detail. In addition to the two well-known noise sources: thermal fluctuation noise from the heat link to the bath and Johnson noise from the resistive thermometer, a third noise source strongly correlated with the steepness of the thermometer is required to fit the measured noise spectra. Thermal fluctuation noise, originating in the thermometer itself, fully explains the additional noise. A simple model provides quantitative agreement between the observed and calculated noise spectra for all bias points in the superconducting transition.
Impedance measurements of superconducting transition-edge sensors (TESs) are a powerful tool to obtain information about the TES thermal and electrical properties. We apply this technique to a 20 µm × 20 µm Ti/Au TES, suitable for application in the optical and near-infrared range, and extend the measurements up to 250 kHz in order to obtain a complete frequency response in the complex plane. From these measurements we obtain important thermal and electrical device parameters such as heat capacity C , thermal conductance G and effective thermal time constant τ eff that will be compared with the corresponding values obtained from noise measurements.
We have developed a model for the resistive transition in a transition edge sensor (TES) based on the model of a resistively and capacitively shunted junction, taking into account phase-slips of a superconducting system across the barriers of the tilted washing board potential. We obtained analytical expressions for the resistance of the TES, R(T, I), and its partial logarithmic derivatives αI and βI as functions of temperature and current. We have shown that all the major parameters describing the resistive state of the TES are determined by the dependence on temperature of the Josephson critical current, rather than by intrinsic properties of the S-N transition. The complex impedance of a pristine TES exhibits two-pole behaviour due to its own intrinsic reactance.
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.
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.