Vortex noise and fluctuation conductivity in Josephson-junction arrays

Abstract

We study the vortex number noise ${S}_{v}(\ensuremath{\omega})$ and fluctuation conductivity ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ in two-dimensional Josephson-junction arrays at three different applied magnetic fields, corresponding to zero, one-half, and $\frac{1}{24}$ of a flux quantum per plaquette $(f=0$, $\frac{1}{2}$ and $\frac{1}{24}$). ${S}_{v}$ and ${\ensuremath{\sigma}}_{1}$ are obtained by numerically solving the equations for the coupled overdamped resistively-shunted-junction model with Langevin noise to simulate the effects of temperature. In all three cases, we find that ${S}_{v}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}3/2}$ at high frequencies $\ensuremath{\omega}$ and flattens out to become frequency independent at low $\ensuremath{\omega}$, indicative of vortex diffusion, while ${\ensuremath{\sigma}}_{1}\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{-}2}$ at sufficiently high $\ensuremath{\omega}$ and $\ensuremath{\sim}{\ensuremath{\omega}}^{0}$ at low frequencies. Both quantities show clear evidence of critical slowing down and a simplified scaling behavior near the normal-to-superconducting transitions at $f=0$ and $f=$$\frac{1}{2}$, indicating that the vortex diffusion coefficient is approaching zero and the charge-carrier relaxation time is diverging at these temperatures. At $f=$$\frac{1}{24}$, there is no clear phase transition; instead, the vortex diffusion coefficient diminishes continuously as the temperature is lowered towards zero. The critical slowing down of ${S}_{v}(\ensuremath{\omega})$, but not its frequency dependence, is in agreement with recent experiments on the flux noise ${S}_{\ensuremath{\Phi}}(\ensuremath{\omega})$ in Josephson-junction arrays, which show a $1/\ensuremath{\omega}$ frequency dependence. We speculate about some possible reasons for the absence of a $1/\ensuremath{\omega}$ frequency regime.