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Technical Physics

Equivalent parameters of a Josephson junction in a microwave SQUID structure

Orest G. Vendik, I. S. Danilov

Published Nov 1, 1999• 2 citations
Publisher

Abstract

The equivalent parameters of a Josephson junction in a microwave SQUID structure are calculated on the basis of relations obtained as a result of an analysis of the operation of an rf SQUID. This analysis is based on the sawtooth variation of the voltage on the resonator as a function of the constant flux bias. The quantitative characteristics permit regarding the Josephson junction as a linear impedance in the rf or microwave circuit, whose real and imaginary parts are controlled by the constant magnetic flux passing through the SQUID loop.

DOI
10.1134/1.1259524

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Journal of Applied Physics•May 1, 1982•3 citations
The equivalent parameters of a Josephson junction in a microwave SQUID structure are calculated on the basis of relations obtained as a result of an analysis of the operation of an rf SQUID. This analysis is based on the sawtooth variation of the voltage on the resonator as a function of the constant flux bias. The quantitative characteristics permit regarding the Josephson junction as a linear impedance in the rf or microwave circuit, whose real and imaginary parts are controlled by the constant magnetic flux passing through the SQUID loop.
The electrical equations of a resonant circuit fed with a radio-frequency current and inductively coupled to a weakly superconducting loop have been derived. The current-voltage characteristic of the rf superconducting quantum interference device can then be calculated and studied as a function of the junction resistance or of the resonance frequency, enabling its behavior to be better understood. As shown by Kurkijärvi, the large positive slope of the steps and the rounding of the characteristic result from thermal noise. However, in order to explain quantitatively the experimental results concerning, namely, point contact weak links which are generally oxidized or which have several contact points, it is useful to remark that the connection between the two superconductors constituting the Josephson junction may be made from n resistive paths. The mean thermal energy of the fluctuating flux is then close to nkbT instead of kbT if the inductances linking the contact points are large enough, that is, on the order of 10−11 H.
The dynamics of a small Josephson junction, driven either by a current source or by an external flux when the junction is the weak link element of a SQUID ring, were studied. Approximate analytical arguments and numerical simulations were used for the lumped circuit model of the junction. The various characteristic times of these devices were systematically obtained as functions of the parameters G and K of the junction and of the SQUID, respectively. The forms of the flux transitions and the times governing them are obtained for linear and periodic time variations of the external flux on the SQUID. A coupling of the SQUID to an LC circuit was treated. Limitations on the device parameters consistent with fast response, stability, and sensitivity were obtained both for the current driven junction and for the rf SQUID.
Some types of Josephson junctions such as long microbridges, S-N-S junctions, and junctions with a series palastic inductance are known to have nonsinusoidal current-phase relations which can be approximated by linear-periodic functions. In this paper the equivalent circuit equation of the resistively shunted junction model with linear-periodic current-phase relation is solved. The heights of zeroth and first constant-voltage steps in the I-V characteristics and the impedance at these steps are obtained analytically. This analysis shows that the impedance at the first step lies on a semicircle in the impedance plane with its center at the impedance of the zeroth step. The calculated impedance is compared with the experimental results by Claridge et al. The theory presented here can explain some features of experimental results which differ considerably from predictions based on a sinusoidal current-phase relation. Among these are the value of bias current corresponding to the center of the first step, and the appearance of ’’shoulders’’ in the imaginary part of the impedance at the first step.
Effects of magnetic fields and microwave radiation on I-V characteristics of YBa2Cu3Oy high-T, Josephson junctions, SQUIDS and Josephson arrays were reported in this experiment, with special emphasis focused on the effects of microwave radiation. In addition to the ordinary constant-voltage steps resulting from the ac Josephson effect, noninteger constant voltage steps are occasionally observed under microwave radiation. We attribute these noninteger constant-voltage steps to the flux coupled to junctions or SQUIDS. Resistive SQUIDS disclose V-Φmodulation without a biased current. The peak to peak value AV of a V-Φ curve is enhanced by microwave radiation.
The dc -impedance of an rf-biased R-SQUID is studied analytically for the nonhysteretic regime. A mathematical analysis of the intrinsic behavior of an R-SQUID, including the mutual interaction between the tank circuit and the SQUID loop, is presented. The results obtained give good agreement with some experimentally observed features. Although the main aim of the present paper was to find an analytic solution for the dc -impedance of an R-SQUID, the results given here can also be used to estimate the R-SQUID parameters and to optimize the performance of a R-SQUID noise thermometer. The theory presented here also provides a good basis for modelling the noise properties of a Josephson junction noise thermometer.
The microwave response of the dc quasiparticle tunneling current in Josephson tunnel junctions, where the Josephson current is suppressed by an external magnetic field, has been studied quantitatively in order to clarify its characteristics as a probe for the measurement of the junction capacitance. Extensive experiments for both small and long junctions are carried out for distinguishing between microwave behaviors of lumped and distributed constant junctions. It is shown that the observed voltage dependence of the dc quasiparticle tunneling current modified by an applied rf field is in good agreement with a theoretical result which takes into account the influence of the microwave circuit connected to the junction. The comparison between theory and experiment gives the magnitude of the internal rf field in the junction. Together with the applied rf field, this internal rf field leads to the junction rf impedance which is dominated by the junction capacitance in our experimental condition. In the case of lumped junctions, this experimental rf impedance is in reasonable agreement with the theoretical one with the junction capacitance estimated from the Fiske step of the distributed junction fabricated on the same substrate; the obtained ratio of the experimental impedance to the theoretical one is approximately 0.6–1.7. In the case of distributed junctions, however, experimental values of their characteristic impedances are approximately 0.2–0.3 of theoretical values calculated by assuming the one-dimensional junction model and taking account of the standing-wave effect in the junction.