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Nonlinear modelling of superconducting devices and circuits

S. Withington, Philippa R Kennedy

Published Jun 25, 1990• 1 citation
OpenAlex

Abstract

Recent progress in the development of techniques for fabricating high-quality superconducting components, together with major advances in the theoretical understanding of quantum detection and mixing at millimetre and submillimetre wavelengths, have made the development of CAD tools for superconducting nonlinear circuits an important and exciting challenge. As a first step towards developing general purpose CAD techniques, the authors have devised algorithms, and written, in the C programming language, modular software for analysing the large-signal, small-signal and noise behaviour of quasiparticle tunnel-junction circuits. At present, they are using the software to study the complex behaviour of SIS mixers, however, the software can also be used for studying other modes of operation, and other tunnelling structures for which macroscopic quantum effects are important. >

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Recent progress in the development of techniques for fabricating high-quality superconducting components, together with major advances in the theoretical understanding of quantum detection and mixing at millimetre and submillimetre wavelengths, have made the development of CAD tools for superconducting nonlinear circuits an important and exciting challenge. As a first step towards developing general purpose CAD techniques, the authors have devised algorithms, and written, in the C programming language, modular software for analysing the large-signal, small-signal and noise behaviour of quasiparticle tunnel-junction circuits. At present, they are using the software to study the complex behaviour of SIS mixers, however, the software can also be used for studying other modes of operation, and other tunnelling structures for which macroscopic quantum effects are important. >
Over the last few years, considerable progress has been made in the development of techniques for fabricating high-quality superconducting circuits, and this success, together with major advances in the theoretical understanding of quantum detection and mixing at millimetre and submillimetre wavelengths, has made the production of CAD tools for superconducting nonlinear circuits an important new enterprise. A further motivation for modelling the behaviour of superconducting nonlinear circuits is that there is a range of circuits whose characteristics have not been studied in any detail. To model the behaviour of a quasiparticle mixer, it is first necessary to simulate the large-signal steady-state dynamics of the local-oscillator circuit. Once the large-signal operating point is known, it is then possible to perturb, either numerically or analytically, the underlying system of equations to gain information about the linear relationships between signal and noise variables. Clearly, if the simulator is to support advanced CAD facilities such as optimization, then it must be capable of calculating the large-signal behaviour of superconducting tunnel junctions efficiently.
Over the last few years, considerable progress have been made in the development of techniques for fabricating high-quality superconducting circuits, and this success, together with major advances in the theoretical understanding of quantum detection and mixing at millimeter and submillimeter wavelengths, has made the development of CAD techniques for superconducting nonlinear circuits an important new enterprise. For example, arrays of quasioptical mixers are now being manufactured, where the antennas, matching networks, filters and superconducting tunnel junctions are all fabricated by depositing niobium and a variety of oxides on a single quartz substrate. There are no adjustable tuning elements on these integrated circuits, and therefore, one must be able to predict their electrical behavior precisely. This requirement, together with a general interest in the generic behavior of devices such as direct detectors and harmonic mixers, has lead us to develop a range of CAD tools for simulating the large-signal, small-signal, and noise behavior of superconducting tunnel junction circuits.
To assist the development of superconducting millimetre and submillimetre-wave receivers, the authors have devised algorithms and written in the C programming language modular software for analysing the small-signal, large-signal and noise behaviour of SIS tunnel-junction circuits. Typical applications include the modelling of mixers, subharmonic mixers and direct detectors. To analyse the complex behaviour of a nonlinear SIS tunnel-junction circuit, it is first necessary to model the large-signal steady-state dynamics of the circuit; the underlying system of equations can then be perturbed, either numerically or analytically, to gain information about the linear relationships between signal and noise variables. A general-purpose simulator must therefore be capable of solving large systems of coupled nonlinear equations. >
The paper summarises preliminary work on the development of a general-purpose superconducting nonlinear-circuit simulator. After contrasting, from a CAD perspective, time and frequency domain descriptions of the macroscopic quantum behaviour of superconducting tunnel junctions, it is reported that the large-signal steady-state dynamics of a tunnel-junction circuit can be conmpletely decomposed by using the method of harmonic balance, and that the resulting system of coupled nonlinear algebraic equations can be solved through harmonic Newton. The principle is illustrated by briefly mentioning an analysis of inadvertent harmonic pumping in quasiparticle mixers.
Virtually all analyses of the superconductor-insulator-superconductor (SIS) quasiparticle mixer have employed the quantum theory of mixing in its three-frequency approximate form. This approximation is valid only in the limit of very large junction capacitance. For finite capacitance, these analyses may be seriously in error. To remedy this, a computer program has been developed which uses the quantum theory of mixing in its most general form, treating the large-signal properties of the mixer in the time domain. The terminating impedances at the harmonics of the local oscillator frequency and their sidebands can be arbitrary. Using this analysis, the effect of finite junction capacitance on one SIS mixer's performance is described. This gives an insight into the range of validity of the three-frequency model.
It has been shown both theoretically [1] and experimentally [2,3] that the very sharp nonlinearity in the conductance of superconducting tunnel junctions can be used to implement millimeter wavelength mixers having high gain and very low noise, leading to the possibility of receivers that achieve the quantum noise limit. The junctions involved contain a thin insulating barrier between two superconductors, and are thus known as superconductor-insulator-superconductor (SIS) junctions. They are physically identical to Josephson junctions, but in this application the conductance is via the tunneling of quasiparticles (unpaired electrons), and so the Josephson effect is not involved.