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ISRN Condensed Matter Physics

Equivalent Single-Junction Model of Superconducting Quantum Interference Devices in the Presence of Time-Varying Fields

Roberto De Luca

Published Nov 30, 2011• 0 citations
Publisher

Abstract

The reduced dynamical model of a two-junction quantum interference device is generalized to the case of time-varying externally applied fluxes with a d. c. component and an oscillating addendum whose frequency is comparable with the inverse of the characteristic time for flux dynamics within the superconducting system. From the resulting effective single-junction model for null inductance of the superconducting loop, it can be seen that the critical current of the device shows a dependence on the frequency and amplitude of the oscillating part of the applied flux. It can therefore be argued that the latter quantities can be considered as control parameters in the voltage versus applied flux curves of superconducting quantum interference devices.

DOI
10.5402/2011/724384

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The reduced dynamical model of a two-junction quantum interference device is generalized to the case of time-varying externally applied fluxes with a d. c. component and an oscillating addendum whose frequency is comparable with the inverse of the characteristic time for flux dynamics within the superconducting system. From the resulting effective single-junction model for null inductance of the superconducting loop, it can be seen that the critical current of the device shows a dependence on the frequency and amplitude of the oscillating part of the applied flux. It can therefore be argued that the latter quantities can be considered as control parameters in the voltage versus applied flux curves of superconducting quantum interference devices.
We have fabricated and studied a promising kind of direct current superconducting quantum interference device (dc-SQUID) in which the magnitude and sign of the critical current of the individual Josephson junctions can be controlled by additional voltage probes connected to the junctions. We show that the amplitude of the voltage oscillations of the SQUID as a function of the applied magnetic field can be tuned and that the phase of the oscillations can be switched between 0 and π in the temperature range of 0.1–4.2 K using a suitable control voltage. This is equivalent to the external application of (n+1/2) flux quantum.
The dynamic properties of a superconducting quantum interference device (SQUID) containing arrays of Josephson junctions in a superconducting loop, and in particular the case of a four-junction SQUID, is analyzed theoretically via computer simulations. It is shown that phase locking of Josephson junctions determines the dynamic behavior of the SQUID. In the case of a stable phase-lock state hysteretic I-V curves as well as unusual voltage-flux dependencies appear. The influence of a small spread in the Josephson junction parameters upon the stability of the phase-lock state is investigated in parameter space.
By means of perturbation analysis, assuming small inductance values, the dynamical equations for the gauge-invariant superconducting phase differences in a dc superconducting quantum interference device (SQUID), containing junctions with different resistive and coupling parameters, are reduced to a single nonlinear differential equation. The resulting effective reduced potential shows that degenerate phase states exist for half-integer values of the applied flux number ${\ensuremath{\Psi}}_{\mathit{ex}}$ and for zero bias current. It can be also shown that, by small variations of the externally applied flux, this degeneracy can be removed and, for sufficiently low values of the SQUID parameter $\ensuremath{\beta}$, different magnetic states, characterized by opposite magnetic moments, can be reversibly realized.
The experimental results obtained with low noise dc superconducting quantum interference devices (dc-SQUIDs) are sometimes not well understood. In particular, when the critical current is high, very irregular voltage versus flux characteristics can be measured. On the other hand, when the critical current is low, the characteristics follow perfectly the resistively shunted junction model. We have studied the dynamics of a dc-SQUID in the presence of stray capacitance and shunting resistors imbalance. The equations describing a circuit model for this physical system are integrated numerically and the results are compared with voltage versus flux characteristics obtained from fabricated devices. We find that the theoretical analysis accounts very well for previously misunderstood experimental features of the dc-SQUID. This model can give more physical sense to the various regimes of behavior of the dc-SQUID giving more insight into the 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.
We study a 0-$\ensuremath{\pi}$ dc superconducting quantum interference device (SQUID) with asymmetric inductances and critical currents of the two Josephson junctions (JJs). By considering such a dc SQUID as a black box with two terminals, we calculate its effective current-phase relation ${I}_{s}(\ensuremath{\psi})$ and the Josephson energy $U(\ensuremath{\psi})$, where $\ensuremath{\psi}$ is the Josephson phase across the terminals. We show that there is a domain of parameters where the black box has the properties of a $\ensuremath{\varphi}$ JJ with degenerate ground state phases $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$. The $\ensuremath{\varphi}$ domain is rather large, so one can easily construct a $\ensuremath{\varphi}$ JJ experimentally. We derive the current phase relation and show that it can be tuned in situ by applying an external magnetic flux resulting in a continuous transition between the systems with static solutions $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi},\phantom{\rule{0.16em}{0ex}}\ensuremath{\psi}={\ensuremath{\varphi}}_{0} ({\ensuremath{\varphi}}_{0}\ensuremath{\ne}0,\ensuremath{\pi})$ and even $\ensuremath{\psi}={\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$. The dependence of ${\ensuremath{\varphi}}_{0}$ on applied magnetic flux is not $2\ensuremath{\pi}$ (one flux quantum) periodic.