Effect of thermal phase fluctuations on the inductances of Josephson junctions, arrays of junctions, and superconducting films

We calculate the factor by which thermal phase fluctuations, as distinct from phase-slip fluctuations, increase the inductance $L_J$ of a resistively shunted Josephson junction (JJ) above its mean-field value $L_0.$ We find that quantum mechanics suppresses fluctuations when T drops below a temperature, $T_Q=ħ{/k}_B{\mathrm{GL}}_0,$ where G is the shunt conductance. Examination of the calculated sheet inductance $L_A{\mathrm{}\left(T\right)/L}_0\mathrm{}\left(T\right)\mathrm{}$ of arrays of JJ’s reveals that two-dimensional (2D) interconnections halve fluctuation effects, while reducing phase-slip effects by a much larger factor. Guided by these results, we calculate the sheet inductance $L_F{\mathrm{}\left(T\right)/L}_0\mathrm{}\left(T\right)\mathrm{}$ of 2D films by treating each plasma oscillation mode as an overdamped JJ. In disordered s-wave superconductors, quantum suppression is important for $L_F{\mathrm{}\left(0\right)/L}_F\mathrm{}\left(T\right)>0.14\mathrm{}$ (or ${T/T}_{C0\mathrm{}}<0.94).\mathrm{}$ In optimally doped YBCO and BSCCO quantum suppression is important for ${\lambda }^2\mathrm{}\left(0\right)/{\lambda }^2\mathrm{}\left(T\right)>0.25,$ where λ is the penetration depth.