Stationary states of superconducting interferometers with series junctions

A superconducting interferometer composed of two parallel arrays of series connected Josephson junctions (a simplified model of a high-${\mathit{T}}_{\mathit{c}}$ granular superconductor) is considered. The interferometer is supplied by dc current J from an external source and is linked by an externally applied magnetic flux ${\mathrm{\Phi }}_{\mathit{e}}$. We consider the associated problems of finding the stationary values of the current in function of ${\mathrm{\Phi }}_{\mathit{e}}$ and stationary values of flux in function of J. It is shown that both problems have identical solutions, which can be obtained by finding the extrema of a properly defined energy function G relative to the conditions given by the set of differential equations dJ=0, d${\mathrm{\Phi }}_{\mathit{e}}$=0, and one of the two conditions found to be complementary, either the fluxoid conservation relationship or the definition of the induced flux ${\mathrm{\Phi }}_{\mathit{i}}$ in terms of currents ${\mathit{J}}_1$ and ${\mathit{J}}_2$ through the arrays (J=${\mathit{J}}_1$+${\mathit{J}}_2$). The dc Josephson equations and the other of the two ‘‘magnetic’’ constraints are then derived as necessary conditions for the existence of energy extremum, i.e., it suffices that these relations are satisfied only locally at the extremum, a result which can be useful in the investigation of nonequilibrium processes. The stationary values, ${\mathit{G}}^{\mathrm{〈}\mathit{m}\mathrm{^}\mathrm{〉}}$, of G are found to depend only on ${\mathit{cphi}}_1$ and ${\mathit{cphi}}_2$(${\mathit{cphi}}_1$), the superconducting phase differences across the weakest junction in each array. Functions ${\mathit{G}}^{\mathrm{〈}\mathit{m}\mathrm{^}\mathrm{〉}}$ are labeled by different possible phase states 〈m^ 〉 of the system, generated by the numerable set of possible mappings of ${\mathit{cphi}}_1$ and ${\mathit{cphi}}_2$ into the phase differences across the other junctions of the system. By allowing the Josephson equations to be satisfied globally, and not only at the extremum, an analytical expression for the second order derivative of ${\mathit{G}}^{\mathrm{〈}\mathit{m}\mathrm{^}\mathrm{〉}}$ is obtained. Energy considerations confirm generally unstable and hysteretic behavior of systems containing series junctions.