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Journal of Applied Physics

Analog Computer Simulation of Weakly Connected Superconducting Rings

M. B. Simmonds, William H. Parker

Published Jan 1, 1971• 46 citations
Publisher

Abstract

A system comprising a weakly connected superconducting ring inductively coupled to a resonant L-C circuit has been studied using an analog computer. The weak link was characterized either by a current-phase relation of the form I = Ic sinφ or by ``supercurrent breakdown,'' and the results were compared. The effects of a normal resistance shunting the junction were investigated, as well as the effects of thermal current noise in the ring. Some criteria for achieving optimum coupling to the superconducting ring are discussed.

DOI
10.1063/1.1659605

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This paper reports the results of an experimental and phenomenological investigation of the static and dynamic behavior of weakly connected superconducting rings. The configuration is essentially a macroscopic superconducting ring of inductance $L$ incorporating a point contact as a weak link which determines the critical supercurrent in the ring, ${i}_{c}$. A phenomenological model for the stationary quantum states of the system as a function of an applied field is developed. The dynamic behavior is obtained directly from the time dependence of the applied fields. Experiments demonstrating both the stationary and the time-dependent magnetic behavior are described. The stationary behavior was obtained with a magnetometer incorporating a weakly connected ring as a sensor to measure the flux through the ring under test. The experimental results confirm the phenomenological model if the critical curr\`ent ${i}_{c}$ is greater than $\frac{{\ensuremath{\Phi}}_{0}}{2L(1+\ensuremath{\gamma})}$, where ${\ensuremath{\Phi}}_{0}=\frac{h}{2e}$ is the flux quantum and $\ensuremath{\gamma}$ is a material and geometric parameter which is usually small compared to unity. In the regime $L{i}_{c}>\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states are discrete, and the transitions between states are well defined and irreversible. If the critical current is not too large, the transitions generally occur only between adjacent states; that is, $\ensuremath{\Delta}k=\ifmmode\pm\else\textpm\fi{}1$. At large critical current, multiple quantum jumps are observed. On the other hand, if $L{i}_{c}<\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states merge into one another continuously and reversibly. In this case the magnetic behavior inthe neighborhood of the half-quantum points is related to the depairing or gapless regime in superconductivity. Measurements of the ac properties of the weakly connected ring at 30 MHz are interpreted directly in terms of the static properties under the influence of a time-varying applied field. In fact, no qualitative corrections to the theory are expected up to frequencies of the order of the superconducting energy gap.

Physical Review•May 10, 1967•310 citations
A system comprising a weakly connected superconducting ring inductively coupled to a resonant L-C circuit has been studied using an analog computer. The weak link was characterized either by a current-phase relation of the form I = Ic sinφ or by ``supercurrent breakdown,'' and the results were compared. The effects of a normal resistance shunting the junction were investigated, as well as the effects of thermal current noise in the ring. Some criteria for achieving optimum coupling to the superconducting ring are discussed.
A digital computer was used to predict the operational characteristics for two models of a superconducting interferometer inductively coupled to an rf tank circuit. The interferometer is modeled by a weak link shunted by a resistor and a superconducting inductor. For a Josephson model, the current iJ and the phase difference across the weak link φ are related by iJ=ic sinφ. We have also examined the consequences of the relationship iJ∝∇φ (1+cosφ) which would result from a modified Aslamazov-Larkin two-phase model, where the supercurrent depends on both the phase difference φ and the phase gradient ∇φ in the weak link. We find that the (1+cosφ) model with the proper boundary condition can reproduce essentially all aspects of the experimental result for interferometers with proximity effect bridges. In particular, the two-phase model is adequate in describing the temperature dependence of the rf response, whereas the Josephson model does not.
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.
We report here numerical calculations on the properties of superconducting weak links as mixers with external local oscillators (LO). Our calculations are based on the resistively shunted-junction model. We assume that the LO and the signal are ideal current sources of amplitudes ILO and IS and of frequencies ωLO and ωs, respectively. We calculate the IF voltage across the weak link as a function of the bias current, the LO current amplitude ILO, and the normalized signal frequency Ωs = ħωs/(2 e Ic R), where Ic is the critical current and R is the shunt resistance. If ωIF/ωLO and Is/Ic are small, we find a linear IF response. Using this method we find the frequency dependence of the IF response, estimate the conversion efficiency for the matched case in both the absence and the presence of noise, and calculate the required LO power. We have fabricated thin-film constrictions by using electron lithography. We have made constrictions of indium films with thicknesses of ~ 1 200 A with widths as narrow as 2 000 A and lengths as short as 1 000 A and have studied their I-V characteristics in the presence of RF radiation and measured their X-band mixing properties. We have measured an NEP of 10-13 W/Hz1/2 and a dynamic range of 23 dB (3 dB deviation from linearity) and estimate that, with optimum matching, these would become 3 × 10-18 W/Hz1/2 and 74 dB, respectively.
Weakly connected superconductors can function as very low power sensors by utilizing the coherence properties of superconductors. One configuration which has produced significant success incorporates a superconducting point contact and a small resistance element in a low-inductance superconducting circuit. When the critical supercurrent (ic) of the point contact is of the order of φ0/L, (φ0=h/2e, L=inductance of the ring), that circuit exhibits coherent quantum behavior. Used as a parametric amplifier, it has been used to measure ultralow voltages, limited by the thermal noise fluctuations in the resistance element R. The voltage sensitivity is approximated by 8kTR/φ0, where k is Boltzmann's constant and T the absolute temperature of the resistance R. We have measured 4×10−16 V across 1.7×10−10 Ω at 4.2°K with a signal-to-noise ratio approaching 10. The operation of this sensor follows from the Josephson oscillation of voltage-biased superconducting point contacts. A dc voltage V0 across the resistance of the circuit produces an oscillating current at the angular frequency ωJ=2πV0/φ0 and an oscillating voltage across the point contact at ωJ. Upon supplying an rf or microwave current at frequency ωD, the point contact mixes ωJ and ωD. Using a homodyne detector at ωD of bandwidth larger than ωJ, we can retrieve the oscillation at ωJ. Direct measurement of this frequency ωJ yields V0. Such superconducting circuits may be useful as cryogenic thermometers, as thermal radiation sensors, and in low-level spectroscopy.
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.
This paper reports the results of an experimental and phenomenological investigation of the static and dynamic behavior of weakly connected superconducting rings. The configuration is essentially a macroscopic superconducting ring of inductance LLL incorporating a point contact as a weak link which determines the critical supercurrent in the ring, ic{i}_{c}ic​. A phenomenological model for the stationary quantum states of the system as a function of an applied field is developed. The dynamic behavior is obtained directly from the time dependence of the applied fields. Experiments demonstrating both the stationary and the time-dependent magnetic behavior are described. The stationary behavior was obtained with a magnetometer incorporating a weakly connected ring as a sensor to measure the flux through the ring under test. The experimental results confirm the phenomenological model if the critical curr\`ent ic{i}_{c}ic​ is greater than $\frac{{\ensuremath{\Phi}}_{0}}{2L(1+\ensuremath{\gamma})}$, where ${\ensuremath{\Phi}}_{0}=\frac{h}{2e}$ is the flux quantum and $\ensuremath{\gamma}$ is a material and geometric parameter which is usually small compared to unity. In the regime $L{i}_{c}>\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states are discrete, and the transitions between states are well defined and irreversible. If the critical current is not too large, the transitions generally occur only between adjacent states; that is, $\ensuremath{\Delta}k=\ifmmode\pm\else\textpm\fi{}1$. At large critical current, multiple quantum jumps are observed. On the other hand, if $L{i}_{c}<\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states merge into one another continuously and reversibly. In this case the magnetic behavior inthe neighborhood of the half-quantum points is related to the depairing or gapless regime in superconductivity. Measurements of the ac properties of the weakly connected ring at 30 MHz are interpreted directly in terms of the static properties under the influence of a time-varying applied field. In fact, no qualitative corrections to the theory are expected up to frequencies of the order of the superconducting energy gap.