Collective Cooper-Pair Transport in the Insulating State of Josephson-Junction Arrays

We investigate collective Cooper-pair transport of one- and two-dimensional Josephson-junction arrays. We derive an analytical expression for the current-voltage characteristic revealing thermally activated conductivity at small voltages and threshold voltage depinning. The activation energy and the related depinning voltage represent a dynamic Coulomb barrier for collective charge transfer over the whole system and scale with the system size. We show that both quantities are nonmonotonic functions of the magnetic field. We propose that formation of the dynamic Coulomb barrier and its size scaling are consequences of the mutual Josephson phase synchronization across the system. We apply the results for interpretation of experimental data in disordered films near the superconductor-insulator transition.

Figure 1

Sketch of the considered array geometries. An external current $I$ is injected from the left through the electrode having the superconducting phase ${\chi }_L$ and extracted through the right electrode with the phase ${\chi }_R$. Upper panel: One-dimensional array of $N$ superconducting islands (squares) connected by two Josephson junctions (crosses) to neighbors corresponding to experimental system of 10. Lower panel: Two-dimensional $M\times N$ Josephson-junction array.

Figure 2

(a) Activation energy $T_0\equiv {\Delta }_c/k_B$ plotted as a function of logarithm of the sample length (squares are the data from 4). (b) The experimental data from 3: activation energy (circles, left axis) and voltage threshold (squares, right axis) as functions of the magnetic field $B$. The lines are according to Eq. (14): $E_J^{2\mathrm{D}}(B)$, with values $\alpha E_J/E_c=0.8$ and $A_{\mathrm{loop}}=1.4\times {10}^{-3}\mu {\mathrm{m}}^2$ fits $T_0$ (solid line); $V_T(B)$ is the 1D quantity and is fitted by $E_J^{1\mathrm{D}}(B)$ with the same $A_{\mathrm{loop}}$ and $\tilde{\alpha }{E}_J/E_c=0.96$ (dashed line) reflecting slightly different geometric factors.