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Journal of Applied Physics

Phase-locking and microwave impedance of Josephson junctions with linear-periodic current-phase relations

Norio Fujimaki, Yoji Okabe, Sogo Okamura

Published Jun 1, 1980• 3 citations

Abstract

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.

DOI: 10.1063/1.328055

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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.

Frequency locking effects in two interacting Josephson junctions are investigated. The junctions are described in the usual resistively-shunted-junction (RSJ) model and coupling between junctions is provided by a resonant circuit. The system can be used to simulate interactions between superconducting weak links in a resonant cavity or via a general impedance shunt. Various types of v-i curves in the voltage locking region are obtained in this calculation by using a first harmonic approximation in the high-frequency regime. Stability of order parameter phase oscillations for two identical junctions in the voltage-locked region is examined. We also show that dissimilar junctions can be synchronized in a resonant cavity using a common series bias current.

Calibrated measurements of both the real and imaginary parts of the microwave impedance of proximity coupled superconducting weak links have been made. The measurements were compared with analytic calculations based on several different models for the supercurrent Is and quasiparticle current. For dc bias currents I1 less than the critical current Ic, the experimental results are consistent with the Josephson relation Is=Ic sinφ, where φ is the quantum phase angle. However, for I1≳Ic, the Fourier component of Is at the Josephson frequency is found to be significantly less (∼0.65±0.1) than would be predicted from Is=Ic sinφ, and consistent with Tinkham’s analysis of a phase slip process. The experimental results indicate that the phase-dependent part of the quasiparticle current, if it exists at all, is no more than 10% of the total quasiparticle current. These results do not necessarily contradict previous measurements made on true tunnel junctions because of the fundamental differences between the two types of devices.

Previously published experimental results are used to deduce an equivalent circuit for this amplifier, based on the shunted junction model. It is found that the very small inductances of the junction shunt resistors play an important role in determining the shape of the current-voltage curves in the voltage-locked region. Once the circuit is determined using the shunted junction model a combination of an approximation method (the method of slowly varying amplitudes) and the shunted junction model is used to approximately maximize the power gain of the circuit. The maximum gain achieved in these simulations of the two-junction amplifier is 11.3. This gain occurred with a negative resistance imput impedance of −0.41R and a signal source impedance of 0.167R, where R is the junction shunt resistance. It is estimated that the bandwidth of this amplifier is about 1 GHz.

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.

Experimental I-V curves of microwave-driven Josephson tunnel junctions with resistive shunts are reported. The results are in very good agreement with numerical calculations using the resistively shunted junction model. In the low-frequency regime there are three distinct regions in the I-V curves and two different types of Shapiro steps. It is demonstrated that the overall shape of the I-V curves can be explained by using an adiabatic interpretation of the junction response. The two different types of Shapiro steps are related to a compensation effect between dc and rf bias currents.