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Journal of Applied Physics

Analysis of the current-voltage characteristic of the rf SQUID

Philippe Lethuillier

Published Jun 1, 1993• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.353947

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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 measured the amplitude–frequency characteristics of radio frequency superconducting quantum interference devices (rf SQUIDs) over a temperature range between 65 and 79 K. Using the expressions derived from the recently developed rf SQUID theory, valid also at large thermal fluctuations, we determined from these data the basic parameters of high-transition-temperature superconductor (HTS) rf SQUIDs. These parameters were: (a) the high-frequency coupling coefficient between the rf SQUID and the tank circuit resonator, k, (b) the SQUIDs hysteretic parameter, β, (c) the critical current of the Josephson junction, Ic, (d) its normal resistance, Rn, and (e) its noise parameter, Γ. We found a good agreement with the values of β(Ic) and Rn determined directly after destructively opening the SQUID loop. In accordance with the theoretical predictions, our experimental results show that at large thermal fluctuation levels (T≅77 K), rf SQUIDs with large loop inductance operate in nonhysteretic mode up to β values exceeding 3. Furthermore, we have shown that the optimal energy sensitivity is attained in the nonhysteretic mode at a value of β distinctly higher than 1. A quantitative comparison of white noise predicted by the theory with that obtained from the experiment showed a reasonable agreement. We also discussed the contribution of the phase information to the SQUID’s signal and noise at optimum operation conditions, when a mixer was used as a signal detector.
Based on the hypothesis that the macroscopic superconducting shielding current on the surface of a porous granular superconductor is short lived due to the action of thermal noise, a dispersive-mode theory for a high-temperature radio-frequency superconducting quantum interference device (rf SQUID) is suggested. In this theory, the effect of the SQUID ring on the resonant tank circuit is an effective inductance Le{L}_{e}Le​, Le{L}_{e}Le​=k2{k}^{2}k2FLT{\mathrm{FL}}_{T}FLT​, where k is the coupling coefficient between the SQUID ring and the resonant tank circuit with inductance LT{L}_{T}LT​, and F is a periodic function of external flux ${\ensuremath{\Phi}}_{\mathrm{dc}}$ with a period of flux quantum ${\ensuremath{\Phi}}_{0}$. The phenomena that the weak-link-rf-SQUID flux-receiving area defined by the hole of the SQUID ring and the bulk-rf-SQUID flux-receiving area, which is determined by an annular surface region with a thickness of the effective penetration depth ${\ensuremath{\lambda}}_{e}$, can be interpreted. The theory is qualitatively consistent with available experimental results.
Partly resistive superconducting quantum interference devices (RSQUIDs) are used in noise thermometry in the liquid helium temperature range. Here we analyze one type of RSQUID that has not been thoroughly investigated until now: the direct current biased with two Josephson junctions and a resistive part connected in series with the junctions and closed by a superconducting ring. The resistive shunted model of the Josephson junctions is used for the analysis. Two cases of the RSQUID ring inductance were analyzed: (i) negligibly small and (ii) small but finite inductance. The expressions for the low frequency output signal, the current–voltage (I–V) characteristics, and the current flowing through the resistive part are obtained. In the dependence of the output frequency versus the dc current passed through the resistive part often an offset is observed experimentally. This phenomenon can be explained as an influence of an asymmetry in the critical currents of the junctions.
The output voltage of a dc superconducting quantum interference device (SQUID) containing a resistive section Rs oscillates at a frequency νq =Vs/ φ0 (φ0=flux quantum) when a current controlled voltage Vs is maintained across Rs. These oscillations have been observed from ∼10−1 to 1011 Hz, using Rs values ranging from ∼10−10 to ∼10−3 Ω in low inductance (∼10−10 H) point contact SQUID loops. At high frequencies, they manifest themselves in the dc current-voltage characteristics by inducing extra current steps.
Direct current-superconducting quantum interference devices (SQUIDs) characterized by asymmetric Josephson junctions have never been used for applications. In order to demonstrate their potential advantages, a throughout numerical analysis of different asymmetric configurations has been carried out. A damping resistance has been included in the SQUID circuit and the thermal noise associated with junction and damping resistances has been considered in the numerical model. Magnetic modulations and flux noise spectral densities have been computed as a function of many parameters (bias current, asymmetry, SQUID inductance, and damping resistance) and the performance of symmetric and asymmetric devices have been compared. The results show that, by properly optimizing the SQUID design, asymmetric SQUIDs are characterized by higher magnetic flux to voltage transfer coefficient and lower flux noise. As a result, asymmetric configurations can be very useful in all the applications where high sensitivity is required.
We have measured with high accuracy (100 ppm) and high precision (5--10 ppm) the impedance of a rf-driven resistive superconducting quantum interference device (SQUID) as a function of the amplitude and frequency of the radio-frequency (rf) bias, as a function of the current bias, and as a function of several other circuit parameters. We have developed the coupled differential equations for the resistive SQUID and for the rf tank circuit. The fit of the solutions of these equations to the data for nonhysteretic junctions is excellent. We were unable to develop solutions for the hysteretic case and thus simply present examples of typical data obtained in order to illustrate the differences with the nonhysteretic case. The relationship of this study to noise thermometry using resistive SQUID's is indicated.