Kinetic Inductance of Josephson Junction Arrays: Dynamic and Equilibrium Calculations

Abstract

We show analytically that the inverse kinetic inductance $L^{-1}$ of an overdamped junction array at low frequencies is proportional to the admittance of an inhomogeneous equivalent impedance network. The $i{j}^{th}$ bond in this equivalent network has an inverse inductance $J_{ij}\cos ({\theta }_i^0-{\theta }_j^0-A_{ij})$, where $J_{ij}$ is the Josephson coupling energy of the $i{j}^{th}$ bond, ${\theta }_i^0$ is the ground-state phase of the grain $i$, and $A_{ij}$ is the usual magnetic phase factor. We use this theorem to calculate $L^{-1}$ for square arrays as large as $180\times 180$. The calculated $L^{-1}$ is in very good agreement with the low-temperature limit of the helicity modulus $\gamma$ calculated by conventional equilibrium Monte Carlo techniques. However, the finite temperature structure of $\gamma$, as a function of magnetic field, is \underline{sharper} than the zero-temperature $L^{-1}$, which shows surprisingly weak structure. In triangular arrays, the equilibrium calculation of $\gamma$ yields a series of peaks at frustrations $f=\frac{1}{2}(1-1/N)$, where $N$ is an integer $\ge 2$, consistent with experiment.

14 pages + 6 postscript figures, 3.0 REVTeX