Analytical results for giant Shapiro steps in Josephson arrays

Abstract

We study two-dimensional Josephson arrays driven by a combined dc plus ac current, and with an applied transverse magnetic field of f flux quanta per plaquette. We present ansatz solutions for sufficiently large frequencies, which are a generalization of the traveling wave solutions found by Marino and Halsey for the case of dc current driving. For f=1/2 and 1/3, we compute the widths of the first few Shapiro steps for both integer and fractional winding numbers. These expressions consist of products of Bessel functions of (iac/ωac), where iac and ωac are the amplitude and the frequency of the driving ac current, respectively, times a frequency-dependent factor for fractional steps. In the limit of large frequencies, we find that the fractional steps are suppressed, whereas the maximum integer step widths saturate to a frequency-independent value. We show that the suppression of the fractional steps is due to decrease of the vertical (i.e., perpendicular to the direction of flow of the injected current) supercurrent relative to the normal current, whereas the persistence of the integer steps is due to the existence of zero-frequency (though spatially varying) terms in the expansion for the gauge-invariant phase differences, for which the normal current vanishes. These results are in reasonable agreement with the numerical computations carried out by other groups.

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