Half-integer Shapiro steps in single-plaquette Josephson-junction arrays in a magnetic field.

Abstract

We derive an equation for a single-plaquette array of Josephson junctions for an arbitrary number of flux quanta per unit cell f. We show that for f=1/2 this equation is equivalent to one derived for a 2\ifmmode\times\else\texttimes\fi{}2 Josephson-junction array in the f=1/2 ground state. We show that in the presence of an rf drive with frequency \ensuremath{\nu}, the system exhibits, for all nonzero values of f, integer and half-integer Shapiro steps at 〈V〉=nh\ensuremath{\nu}/4e, where n=1,2,3,. . . . In addition, very small subharmonic Shapiro steps at 〈V〉=nh\ensuremath{\nu}/2em, where m=1,2,3,. . . and not equal to n, are observed for all f not equal to zero. These particular steps, however, are found to be consistently smaller than the integer and half-integer steps concurrently present in the array I-V characteristics. Contrary to recent suggestions, we show that single-plaquette array behavior is not entirely consistent with that of large arrays of Josephson junctions and hence not responsible for the display of fractional giant Shapiro steps in these large arrays. Such behavior may, however, be responsible for the recent observation of half-integer steps in high-${\mathit{T}}_{\mathit{c}}$ grain-boundary junctions.