Kinetic Inductance of Josephson Junction Arrays: Dynamic and Equilibrium Calculations

TL;DR

This paper analytically relates the inverse kinetic inductance of overdamped Josephson junction arrays to an equivalent impedance network, enabling large-scale calculations that match Monte Carlo results and reveal temperature-dependent structural differences.

Contribution

It introduces an analytical method to compute inverse kinetic inductance in Josephson arrays using an equivalent impedance network, matching Monte Carlo results and exploring temperature effects.

Findings

Inverse kinetic inductance proportional to equivalent impedance network admittance.

Good agreement between analytical calculations and Monte Carlo simulations.

Temperature affects the structure of the helicity modulus, with finite temperature showing sharper features.

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 $ij^{th}$ 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 $ij^{th}$ 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 $\geq 2$, consistent with experiment.