Giant Shapiro Steps and Other Analytical Results in Josephson Arrays
Abstract
We study currentdriven twodimensional Josephson arrays with an applied transverse magnetic field of f flux quanta per plaquette. For DC current driving, there is a family of travelingwave solutions characterized by a constant phase shift delta. We present the analytical form of these solutions using a firstharmonic approximation, and compute their IV characteristics and dispersion relationship. The equation governing the long wavelength dynamics of this system is Burger's equation, which allows the formation of shocks (or fronts). States with delta in (2 pi f, pi) are diffusively stable, and relax through fronts which originate at the boundaries and propagate across the array. The group velocity measures the rate of change of the amplitude of the horizontal phase oscillations, whereas the diffusion coefficient measures the rate of change of the vertical amplitude. These solutions admit a straightforward generalization to the case where the system is driven by an additional AC current. For f = 1/2 and f = 1/3, we compute the widths of the first few Shapiro steps for both integer and fractional winding numbers. In the limit of large frequencies, we find that the fractional steps are suppressed, whereas the maximum integer step widths saturate to a frequencyindependent 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 zerofrequency (though spatially varying) terms in the expansion for the gaugeinvariant phase differences, for which the normal current vanishes.
 Publication:

Ph.D. Thesis
 Pub Date:
 1996
 Bibcode:
 1996PhDT........18M
 Keywords:

 TRAVELING WAVES;
 SUPERCONDUCTING ARRAYS;
 Physics: Condensed Matter