Frequency dependence of Shapiro steps in Josephson-junction arrays

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Abstract
We present an analytic model describing the dynamics of an N×N square array of overdamped Josephson junctions in a magnetic field producing 1/2 flux quanta per unit cell (f=1/2) biased with combined direct and alternating currents injected parallel to a row of junctions. In agreement with recent experiments and numerical simulations, we find Shapiro steps at voltages corresponding to integer multiples (giant steps) and half-integer multiples (fractional giant steps) of N times the corresponding single-junction Shapiro-step voltage. In the low-drive-frequency limit, our calculations show that each 2×2 group of cells in the array can be approximated by a single junction, leading to integer and half-integer Shapiro steps. At higher drive frequencies, a more complicated behavior of the 2×2 cell leads to a suppression of half-integer steps relative to integer steps. In a high-frequency expansion, we find the maximum width of a half-integer step to be inversely proportional to the ac drive frequency.