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Journal of Applied Physics

Analysis of threshold curves for superconducting interferometers

Rick Peterson, C.A. Hamilton

Published Dec 1, 1979• 72 citations
Publisher

Abstract

Threshold curves for multijunction superconducting interferometers have been calculated previously, showing general agreement with observed features, especially in symmetric cases. We here add some more details to the analysis, paying particular attention to the effects of asymmetries in coupling, inductance, or critical currents. Feed-loop inductance and flux quantization in the feed loop can be important. A changing lobe pattern over many periods, asymmetries within a period, shifting patterns between runs spanning a warm-up, and sudden changes in pattern because of noise in the environment are all quantitatively explainable on the basis of this model. By use of a single ’’calibration curve’’, the inductance for symmetric two- or three-junction interferometers can be obtained immediately.

DOI
10.1063/1.325954

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arXiv:0707.4395•Physical Review B•Jul 30, 2007•5 citations
Threshold curves for multijunction superconducting interferometers have been calculated previously, showing general agreement with observed features, especially in symmetric cases. We here add some more details to the analysis, paying particular attention to the effects of asymmetries in coupling, inductance, or critical currents. Feed-loop inductance and flux quantization in the feed loop can be important. A changing lobe pattern over many periods, asymmetries within a period, shifting patterns between runs spanning a warm-up, and sudden changes in pattern because of noise in the environment are all quantitatively explainable on the basis of this model. By use of a single ’’calibration curve’’, the inductance for symmetric two- or three-junction interferometers can be obtained immediately.
Thin-film Josephson-junction quantum-interference devices require only a small amount of magnetic flux to switch them between the voltage state and the superconducting state. They are sensitive to any spurious magnetic flux present at the device. The magnetic flux may be trapped in the superconducting ground plane, particularly in ground plane holes. This paper analyzes the effects of the trapped magnetic flux on the threshold curves of three-junction Josephson interferometers. The threshold curves of the interferometer are calculated automatically by using a circuit optimization program. The automatic procedure of calculating the threshold curves of an interferometer can be applied to any Josephson-junction interferometer device. The threshold curves of both symmetric and asymmetric interferometers are calculated. We find that the threshold curves of a symmetric three-junction interferometer changes periodically as a function of the effective coupled trapped magnetic flux with a period of Φ0. The threshold curves are shifted and severely distorted in shape by the presence of the coupled trapped magnetic flux. The degeneracy of the threshold curves of a symmetric interferometer makes the unique quantitative determination of the trapped magnetic flux difficult. However, the degeneracy is removed in an asymmetric interferometer. The presence of the unbalanced trapped magnetic flux greatly shifts the entire threshold curve of an asymmetric interferometer.
A technique for the analysis of complex superconducting circuits containing one or more current biased inductively coupled Josephson junction interferometers is described. The equations of motion for the system are shown to separate into a set of non-linear junction equations relating the voltage drops across the junctions to the currents passing through the junctions, and a set of linear equations expressing the quantization of the fluxoid around the superconducting loops and the corresponding linear circuit equations for the normal loops. In many cases, the linear equations can be integrated, resulting in an effective elimination of the normal circuit loops from the problem. The circuit is thus reduced to that of an equivalent set of coupled SQUIDs whose device parameters are a function of the original circuit parameters and the inductive coupling strengths. The dc SQUID tuned linear amplifier is analyzed to display the method. As a result of this analysis, the optimal noise temperature of a tuned SQUID amplifier is derived as a function of the current and voltage noise spectral densities, Svand Sj, and forward transfer function, Vφ. The results obtained demonstrate that the behavior of the SQUID cannot be isolated from that of the rest of the circuit in the manner assumed in previous analyses. The existence of a quantum noise limit on the noise temperature of any phase preserving linear amplifier is used to infer that the noise figures must satisfy the constraint[(S_{v}/2LV_{\phi}^{2})(LS_{j}/2)-S_{vj}^{2}]^{1/2}>\hbarThe behavior of several simple circuits containing a pair of inductively coupled current biased interferometers is discussed to display the extension of the method to circuits containing more than one interferometer.
We have demonstrated high-Tc interferometers using YBaCuO trilayer junctions, which are two-junction-type interferometers with a direct current injection. In order to realize a small loop inductance value without a ground plane, the interferometers were designed to have loop inductance with a vertical stacked structure by overlapping junction bridges on base electrodes. The interferometers showed clear threshold characteristics in the control current dependence of the gate current at temperatures below 70 K. The temperature dependence of the loop inductance showed good agreement with a calculation result based on a strip-line model. The sheet inductance was estimated to be 0.85 pH/□ at 4.2 K from the threshold characteristics of the interferometers by changing the bridge length between the two junctions, which was in good agreement with the design value.
Planar interferometers with two Nb-NbOx-Pbln functions have been fabricated. Because of the peculiar configuration adopted, both a symmetric and an asymmetric current bias are possible on the same sample. The loop area is on a plane perpendicular to the cross section of the functions, and, therefore, the flux coupled to them is negligible. ThreshoM curves have been measured for different values of the characteristic parameter fl = LI1/qbo, where 1i and L are the maximum Josephson current and the inductance of the interferometer and C~o is the flux quantum. The dependence of the current modulation depth upon fl has been also investigated. A comparison of the experimental data with the theory shows quantitative agreement.
The theory of self-induced resonances in asymmetric two-junction interferometer device is presented. In real devices, it is impossible to have an ideal interferometer free of imperfections. Thus, we extended previous theoretical approaches, introducing a model that contains several asymmetries: Josephson current ϵϵϵ, capacitances $\ensuremath{\chi}$, and dissipation $\ensuremath{\rho}$ presented in an equivalent circuit. Moreover, nonconventional symmetry of the order parameter in high temperature superconducting quantum interference devices forced us to include phase asymmetries. Therefore, the model has been extended to the case of $\ensuremath{\pi}$-shift interferometers, where a phase shift is present in one of the junctions.