^{1,a)}, S. Withington

^{1}and D. J. Goldie

^{1}

### 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 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.

I. INTRODUCTION

II. HARMONIC BALANCE ALGORITHM

III. NUMERICAL IMPLEMENTATION

A. Convergence

B. Constant signal power and bias current

C. Modulated signal power and bias current

D. Thermal circuits

E. Pulsed simulations

IV. CONCLUSIONS

### Key Topics

- Electrical resistivity
- 9.0
- Jacobians
- 8.0
- Electric currents
- 7.0
- Heat capacity
- 6.0
- Superconductivity
- 6.0

## Figures

A microstrip coupled TES structure fabricated by our group at Cavendish Laboratory for 100 GHz astronomy. The MoCu superconducting bilayer (large rectangle) is thermally isolated from the heat bath by narrow and thin legs. Power from the source is coupled to a microstrip, which runs down the lower left leg, and is terminated in a resistor on the island.

A microstrip coupled TES structure fabricated by our group at Cavendish Laboratory for 100 GHz astronomy. The MoCu superconducting bilayer (large rectangle) is thermally isolated from the heat bath by narrow and thin legs. Power from the source is coupled to a microstrip, which runs down the lower left leg, and is terminated in a resistor on the island.

A TES bias circuit with its Norton equivalent circuit. is the bias current, and are the bias and stray resistances respectively and is the sum of the SQUID input inductance and any other additional stray inductance. is the Norton current with being the equivalent admittance of the circuit.

A TES bias circuit with its Norton equivalent circuit. is the bias current, and are the bias and stray resistances respectively and is the sum of the SQUID input inductance and any other additional stray inductance. is the Norton current with being the equivalent admittance of the circuit.

The thermal circuit of the TES coupled to a heat bath at temperature . Power is fed into the device, which has heat capacitance . Power flows from the device to the heat bath through the weak thermal link .

The thermal circuit of the TES coupled to a heat bath at temperature . Power is fed into the device, which has heat capacitance . Power flows from the device to the heat bath through the weak thermal link .

The surface of a TES derived from Eq. (9). As more current passes through the TES, the sharpness of the transition edge falls.

The surface of a TES derived from Eq. (9). As more current passes through the TES, the sharpness of the transition edge falls.

Convergence of temperature for various guess temperatures. The TES has a solution near . For a at the critical temperature (solid line) convergence is reached in less than 20 iterations with the rest of the iterations being small adjustments to the small threshold defined for convergence. Even for a of 1 K (dashed line) the same solution is found. Thus, the presence of the sharp transition edge poses no problems for the algorithm.

Convergence of temperature for various guess temperatures. The TES has a solution near . For a at the critical temperature (solid line) convergence is reached in less than 20 iterations with the rest of the iterations being small adjustments to the small threshold defined for convergence. Even for a of 1 K (dashed line) the same solution is found. Thus, the presence of the sharp transition edge poses no problems for the algorithm.

Convergence of temperature to the same solution as in Fig. 5 with in all cases. Here, is changed from 0.1 (solid line) to 0.5 (the line with the sharpest fall to solution). The higher the value of , the quicker convergence is reached. The smaller the value of , the finer is the search, but less prone to errors. Note, the presence of the sharp transition poses no problem for the algorithm. We adopt a value in the range of as a compromise between speed and stability.

Convergence of temperature to the same solution as in Fig. 5 with in all cases. Here, is changed from 0.1 (solid line) to 0.5 (the line with the sharpest fall to solution). The higher the value of , the quicker convergence is reached. The smaller the value of , the finer is the search, but less prone to errors. Note, the presence of the sharp transition poses no problem for the algorithm. We adopt a value in the range of as a compromise between speed and stability.

The phases of the harmonic components of the current as a function of the phase of the signal power. Solid line represents the first harmonic , dashed line represents , dotted line represents and closely-dotted line represents . The phase of each harmonic wraps around in the correct order throughout the range of phase of the signal power. This demonstrates that the small and large signal behavior is predicted correctly by the algorithm.

The phases of the harmonic components of the current as a function of the phase of the signal power. Solid line represents the first harmonic , dashed line represents , dotted line represents and closely-dotted line represents . The phase of each harmonic wraps around in the correct order throughout the range of phase of the signal power. This demonstrates that the small and large signal behavior is predicted correctly by the algorithm.

A comparison of curves for the cases where the resistance of the superconducting bilayer is a function of temperature only, (solid line) and when the resistance is a function of both temperature and current, (dashed line). Signal power increases from zero and in steps of 1 pW for subsequent lines indicated by the direction of the arrow. The curves in the case of are less sharp, suggesting a decrease in the strength of negative electrothermal feedback.

A comparison of curves for the cases where the resistance of the superconducting bilayer is a function of temperature only, (solid line) and when the resistance is a function of both temperature and current, (dashed line). Signal power increases from zero and in steps of 1 pW for subsequent lines indicated by the direction of the arrow. The curves in the case of are less sharp, suggesting a decrease in the strength of negative electrothermal feedback.

The temperature of a TES as a function of increasing signal power. In the case where the resistance of the superconducting bilayer is a function of resistance only (solid line), the negative electrothermal feedback process keeps the temperature of the bilayer almost constant. However, when resistance is a function of temperature and current (dashed line), the temperature changes throughout, even from 0 to , suggesting a weakened electrothermal feedback process.

The temperature of a TES as a function of increasing signal power. In the case where the resistance of the superconducting bilayer is a function of resistance only (solid line), the negative electrothermal feedback process keeps the temperature of the bilayer almost constant. However, when resistance is a function of temperature and current (dashed line), the temperature changes throughout, even from 0 to , suggesting a weakened electrothermal feedback process.

The current through a TES as a function of time for signal powers modulated at 1 Hz. For a signal power of 0.2 pW p-p (solid line), the behavior is in the small signal limit. Signal power then increases in steps of 0.4 pW p-p up to 1.8 pW p-p (dash-dotted line). The device gradually saturates, and the current wave form gets clipped as the amplitude of the signal power increases.

The current through a TES as a function of time for signal powers modulated at 1 Hz. For a signal power of 0.2 pW p-p (solid line), the behavior is in the small signal limit. Signal power then increases in steps of 0.4 pW p-p up to 1.8 pW p-p (dash-dotted line). The device gradually saturates, and the current wave form gets clipped as the amplitude of the signal power increases.

The real and imaginary parts of the TES input impedance as a function of frequency assuming resistance is a function of temperature and current, . The impedance is calculated from 0.01 Hz to 10 kHz at various bias points. The small contour is the impedance calculated with the TES biased on the transition, with . The magnitude of the bias current increases by for each subsequently larger impedance contour, with the largest being for a bias current of . The shape of these contours resemble very closely those measured in experiments.^{12,13}

The real and imaginary parts of the TES input impedance as a function of frequency assuming resistance is a function of temperature and current, . The impedance is calculated from 0.01 Hz to 10 kHz at various bias points. The small contour is the impedance calculated with the TES biased on the transition, with . The magnitude of the bias current increases by for each subsequently larger impedance contour, with the largest being for a bias current of . The shape of these contours resemble very closely those measured in experiments.^{12,13}

A more complicated thermal circuit simulated using the algorithm. In this case an absorber at a temperature is linked to the TES with a thermal link .

A more complicated thermal circuit simulated using the algorithm. In this case an absorber at a temperature is linked to the TES with a thermal link .

The current response of a TES with the thermal circuit of Fig. 12 as a function of bias current at various frequencies. At 0.01 Hz (solid line) and are almost in phase. At 100 Hz (dashed line) and 1 kHz (dotted line), there is significant phase shift. At 10 kHz (closely-dotted horizontal line), the electrothermal time constants are far longer than the period of the applied bias current such that the device no longer responds to the modulations.

The current response of a TES with the thermal circuit of Fig. 12 as a function of bias current at various frequencies. At 0.01 Hz (solid line) and are almost in phase. At 100 Hz (dashed line) and 1 kHz (dotted line), there is significant phase shift. At 10 kHz (closely-dotted horizontal line), the electrothermal time constants are far longer than the period of the applied bias current such that the device no longer responds to the modulations.

(a) The current response of a TES current (solid line) as a function of time to pulses of signal power (dashed line). The pulse rate is 500 Hz. (b) The response current of the same TES subtracted from the equilibrium current response . The decay of the TES current back to the equilibrium state is a function of a single time constant in the case of the simple thermal model of Fig. 3.

(a) The current response of a TES current (solid line) as a function of time to pulses of signal power (dashed line). The pulse rate is 500 Hz. (b) The response current of the same TES subtracted from the equilibrium current response . The decay of the TES current back to the equilibrium state is a function of a single time constant in the case of the simple thermal model of Fig. 3.

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