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Journal of Applied Physics

Shapiro steps in a dc quantum interference device with multiple identical junctions in each arm

Roberto De Luca, Francesco Romeo

Published Oct 1, 2005• 5 citations
Publisher

Abstract

Integer and half-integer Shapiro step amplitudes in the current-voltage characteristics of symmetric superconducting quantum interference devices (SQUIDS), containing more than one Josephson junction on each arm, are analytically calculated for small values of the parameter β and in the absence of noise. This analysis is made possible by the reduction of the dynamical equations of the multiple-junction SQUID to the dynamical equation of a single Josephson junction possessing a nonconventional current-phase relation.

DOI
10.1063/1.2071452

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Integer and half-integer Shapiro step amplitudes in the current-voltage characteristics of symmetric superconducting quantum interference devices (SQUIDS), containing more than one Josephson junction on each arm, are analytically calculated for small values of the parameter β and in the absence of noise. This analysis is made possible by the reduction of the dynamical equations of the multiple-junction SQUID to the dynamical equation of a single Josephson junction possessing a nonconventional current-phase relation.
A high-Tc dc superconducting quantum interference device (SQUID) with three Josephson junctions (JJs) in series in each of its arms has been fabricated. Its Shapiro steps were studied using microwave (rf) radiation of 10 GHz and weak magnetic fields. The appearance of giant Shapiro steps and of some of half-integer steps was observed. Separation between the adjacent Shapiro steps could be tuned by rf magnetic fields and small external dc magnetic fields. This phenomenon was analyzed by phase locking the JJs in the SQUID.
Abstract A superconducting quantum interference device (SQUID) comprising 0- and π-Josephson junctions (JJs), called π-SQUID, is studied by the resistively shunted junction model. The π-SQUID shows half-integer Shapiro-steps (SS) under microwave irradiation at the voltage V = ( ℏ / 2 e ) Ω ( n / 2 ) , with angular frequency Ω and half-integer n/2 in addition to integer n. We show that the π-SQUID can be a π-qubit with spontaneous loop currents by which the half-integer SS are induced. Making the 0- and π-JJs equivalent is a key for the half-integer SS and realizing the π-qubit.
An analysis expression for the stationary probability distribution of the symmetric superconducting quantum interference device with two Josephson junctions (dc SQUID) driven by thermal noise is derived from two-dimensional Fokker–Planck equations where the potential condition is satisfied. The analytical formulae for SQUID characteristics, e.g. the circulating current, the current–voltage relationship and the transfer function, are obtained by using the probability distribution. The ‘ripple’ phenomenon of the circulating current versus the applied flux, the effects of thermal fluctuations on current–voltage relationship and the optimum operating condition of dc SQUID system are represented.
We demonstrate a zero-crossing Shapiro step in an asymmetric two-junction superconducting quantum interference device (SQUID) driven by one sinusoidal RF signal. The operation is based on the imbalanced evolution of the junction phases due to the tilted interference diagrams. A test circuit was fabricated using Nb/AlO x /Nb junction technology. When we applied an RF signal of 10 GHz, we observed a Shapiro step crossing the zero-current axis at the voltage position of the first order. The polarity of the step voltage was switched by a DC control flux to the SQUID loop. The experimental current–voltage curves were quantitatively reproduced by simulation.
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 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.