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Physical review. B, Condensed matter

Dispersive-mode theory for the high-temperature rf superconducting quantum interference device

Chang-Xin Fan

Published Feb 1, 1990• 2 citations
Publisher

Abstract

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 ${L}_{e}$, ${L}_{e}$=${k}^{2}$${\mathrm{FL}}_{T}$, where k is the coupling coefficient between the SQUID ring and the resonant tank circuit with inductance ${L}_{T}$, 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.

DOI
10.1103/physrevb.41.2041

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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​={k}^{2}$${\mathrm{FL}}_{T}, 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 \ensuremathΦdc{\ensuremath{\Phi}}_{\mathrm{dc}}\ensuremathΦdc​ with a period of flux quantum \ensuremathΦ0{\ensuremath{\Phi}}_{0}\ensuremathΦ0​. 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 \ensuremathλe{\ensuremath{\lambda}}_{e}\ensuremathλe​, can be interpreted. The theory is qualitatively consistent with available experimental results.
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.
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.
A number of applications of high-temperature superconductor radio frequency superconducting quantum interference devices (rf SQUIDs) require a certain immunity of these sensors against electromagnetic interference (EMI). We have investigated effects of electromagnetic radiation in the high-frequency and ultrahigh-frequency range on various types of rf SQUIDs. It has been found that EMI of sufficient field strength reduces the voltage versus flux transfer function, and thus increases the flux noise of the SQUIDs. SQUIDs with a wire wound tank circuit coil have been found to be more sensitive to EMI than SQUIDs integrated into a superconducting microstrip resonator.
The operation of YBa2Cu3O7 rf superconducting quantum interference devices (SQUIDs) using a dielectric SrTiO3 resonator as tank circuit was tested. At 77 K, we attained for such resonators unloaded quality factors Q0 up to 7000 at resonant frequencies near 1 GHz, when measuring without electromagnetic shielding. For a washer SQUID with an area of 4×4 mm2 and an inductance of ≂100 pH, which was properly coupled to the resonator, we measured an approximately 60 μV peak-to-peak SQUID signal and a white flux noise spectral density of SΦ1/2=1×10−5Φ0/√Hz, corresponding to a field sensitivity of BN=105 fT/√Hz. With an additional single-layer flux transformer with a diameter of the pickup loop of 22 mm in flip-chip arrangement, the field sensitivity was improved to 30 fT/√Hz.
Experimental results on some radio-frequency superconducting quantum interference device (rf-SQUID) signal properties are presented. The quantum interferometer was made of ceramic YBa2Cu3O7−x and was due to a low critical current operated in the inductance or nonhysteretic mode. With bias current as reference, amplitude variation, and phase shift of the voltage over the tank circuit coupled to the SQUID were measured simultaneously. It is shown that there is qualitative agreement between calculations based on the resistivity shunted junction model and the data. Moreover, using phase detection, signal instabilities predicted for the rf-SQUID inductance mode were observed. These signal instabilities may be exploited to enhance the transfer coefficient for measured flux-to-output signal.
Conventional rf SQUIDs (superconducting quantum interference devices) consist of a superconducting loop interrupted by one weak link, and when read out via a tank circuit they produce output signals in the order of a few tens of microvolts. When the single weak link is replaced by a dc SQUID (a superconducting loop with two junctions) a so-called double quantization loop SQUID is formed. In this paper it is shown theoretically, and supported experimentally with thin-film devices of three different designs, that the general behaviour can be explained by a simplified picture of the fixed critical current of the single junction of the rf SQUID being replaced by an effective critical current of a dc SQUID loop that is varied by the applied magnetic flux (i.e., we have a rf SQUID with a variable β parameter). A more detailed analysis of the experimental data and the theory of this novel device reveals its interesting nonlinear character and shows the deviations from this simple picture. It is demonstrated experimentally that output signals in the order of ≥1 mV can be obtained with a suitable read-out mode for the tank circuit, a clear advantage when compared to conventional rf SQUIDs. With respect to dc SQUIDs, the coupling to integrated planar input coils will be unproblematic since no microwave resonances should occur in the structures.