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Journal of Applied Physics

Digital computer simulation of rf sputtering interferometer

R. H. Wang, G.J. Dick, James E. Mercereau

Published Mar 1, 1977• 1 citation
Publisher

Abstract

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.

DOI
10.1063/1.323726

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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.
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.
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.
This paper describes the I-V characteristics (as a function of applied magnetic field) and magnetic field response of a dc SQUID which incorporates two weak-link Josephson junctions in a superconducting ring. These characteristics are derived by digital computer simulation, using the Stewart-McCumber model. The effects of asymmetry (dissimilarity between the junctions) on SQUID critical current I c, normal resistance R and capacitance C are also simulated. With microwave excitation applied, the Josephson oscillations of the dc SQUID are found to be “frequency-locked” to the applied microwave signal even when critical currents I c of the two junctions differed, Subharmonic voltage steps are obtained for a symmetric dc SQUID in a magnetic field even when the current-phase relationship is purely sinusoidal.
The rf impedance of superconducting point contacts has been measured as a function of the quantum-mechanical phase difference $\ensuremath{\varphi}$ across the point contact. By representing the point contact as a parallel combination of an inductor $\mathfrak{L}(\ensuremath{\varphi})=(\frac{\ensuremath{\hbar}}{2e}){(\frac{\ensuremath{\partial}{I}_{p}}{\ensuremath{\partial}\ensuremath{\varphi}})}^{\ensuremath{-}1}$ and a resistor $R\ensuremath{\equiv}\frac{1}{G(\ensuremath{\varphi})}$ the current-phase relation ${I}_{p}(\ensuremath{\varphi})$ and the phase-dependent conductance $G(\ensuremath{\varphi})$ have been determined from measurements at 30 MHz on phase-biased niobium point contacts. For point contacts with sufficiently small critical current Ic{I}_{c}Ic​, the inductance was $\mathfrak{L}(\ensuremath{\varphi})=\frac{\ensuremath{\hbar}}{2e{I}_{c}cos\ensuremath{\varphi}}$ yielding the expected negative inductance branch for $\frac{\ensuremath{\pi}}{2}\ensuremath{\le}\ensuremath{\varphi}\ensuremath{\le}\ensuremath{\pi}$ and a sinusoidal current-phase relation. For larger critical currents there were departures from the sinusoidal form for the measured ${I}_{p}(\ensuremath{\varphi})$. There was a phase-dependent conductance that is an increasing function of $\ensuremath{\varphi}$ corresponding to a negative coefficient for the $cos\ensuremath{\varphi}$ term in the Josephson current.
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.
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.