Superconductor-insulator transition in disordered Josephson-junction chains

We study the superconductor-insulator quantum phase transition in disordered Josephson-junction chains. To this end, we derive the field theory from the lattice model that describes a chain of superconducting islands with a capacitive coupling to the ground ($C_0$) as well as between the islands ($C_1$). We analyze the theory in the short-range ($C_1\ll C_0$) and in the long-range ($C_1\gg C_0$) limits. The transition to the insulating state is driven by the proliferation of quantum phase slips. The most important source of disorder originates from trapped charges in the substrate that suppress the coherence of phase slips, thus favoring superconducting correlations. Using the renormalization-group approach, we determine the phase diagram and evaluate the temperature dependence of the dc conductivity and system-size dependence of the resistance around the superconductor-insulator transition. These dependences have in general strongly nonmonotonic character, with several distinct regimes reflecting an intricate interplay of superconductivity and disorder.

Figure 1

Schematic depiction of a Josephson-junction chain. The superconducting grains are coupled via tunnel barriers that provide a capacitance $C_1$ and are connected to the ground via capacitances $C_0$. The superconducting phase of an island $i$ is denoted by ${\theta }_i$ and the corresponding number of Cooper pairs by ${\mathcal{N}}_i$.

Figure 2

Phase diagram for a JJ chain with short-range Coulomb interaction in the $\pi K_0–y$ plane. The chain is in the insulating phase (I) to the left of each transition line and in the superconducting phase (S) to the right of it. The black solid line (with stars) corresponds to the clean case ($D_{\xi }=D_Q=0$). The other two solid curves describe a chain without random stray charges ($D_Q=0$) but with random QPS fugacity: $D_{\xi }=0.1$ (red), $D_{\xi }=0.2$ (blue, open circles). The dashed curves include a small amount of stray charges ($D_Q={10}^{-12}$), with the same value of $D_{\xi }$ as on the solid line with the same color (same symbol).

Figure 3

(a) Temperature-dependent resistivity in the clean case for short-range Coulomb interaction. The numbers on the curves indicate the value of $\pi K_0$. The value of the fugacity is the same for every curve, $y=0.1$. Inset: Phase diagram in the $\pi K_0–y$ plane. The stars mark the position of the corresponding resistivity curve with the same color. To the left of the black line, the chain is in the insulating phase (I), while to the right it shows superconducting correlations (S). The dashed parts are qualitative extrapolations illustrating the flow towards the insulating (infinite resistivity) fixed point. (b) Dependence of the resistance with array length $N$ at $T=0$ with the same parameters as for the resistivity plot.

Figure 4

(a) Scaling of the resistivity with temperature in the presence of random offset charges for short-range Coulomb interaction. The numbers at the curves indicate the values of $\pi K_0$. The other parameters $D_Q={10}^{-3}$ and $y=6\times {10}^{-3}$ are the same for all curves. In the gray region random stray charges are weak (the boundary is at the temperature $T_Q$). The black curve ($\pi K_0=1.65$) corresponds to a superconducting system (although phase slips are relevant). The green and blue curve ($\pi K_0=1.5$ and 1.4) are in the insulating regime. The dashed blue and green lines, which correspond to the temperature range $T<T_{\mathrm{ps}}$, represent extrapolations to demonstrate the tendency at lowest temperatures (flow towards the Anderson insulator). Inset: Phase diagram in the $\pi K_0–y$ plane. The stars show the position of the parameters for the resistivity plots in the phase diagram. To the left of the black line, the system shows insulating behavior (I), while to the right it shows superconducting correlations (S). (b) Length-dependent resistance at $T=0$ for the same parameters as for the resistivity curves. The characteristic scales, which correspond to the temperature scales $T_Q$ and $T_{\mathrm{ps}}$ in the panel (a), are $N_Q$ (boundary of gray region) and $N_{\mathrm{ps}}$ (beginning of the dashed line, seen on the blue curve only).

Figure 5

Scaling of the temperature-dependent resistivity contributions originating from phase slips with homogeneous fugacity (a) and from phase slips with random fugacity (b). The numbers at the curves indicate the value of $\pi K_0$. The other parameters are the same for all curves: $y={10}^{-2},D_Q={10}^{-3}$, and $D_{\xi }={10}^{-4}$. The dashed lines at low temperatures are extrapolations illustrating the flow towards the insulating (infinite resistivity) fixed point. The inset in (a) shows the position of the systems in the phase diagram. Parameters that lie to the right of the black line show superconducting correlations (S), while those to the left are characterized by insulating behavior (I).

Figure 6

Temperature dependence of the resistivity (a) and length dependence of the zero-temperature resistance (b) of a clean JJ chain with screening length $\mathrm{\Lambda }=10$ and ultraviolet QPS amplitude $y=0.1$. The numbers at the curves indicate the value of $\pi K_0$ controlling the infrared scaling of the QPS amplitude. The dashed lines at low temperatures are extrapolations illustrating the flow towards the insulating (infinite resistivity) fixed point. The inset shows the position of each of the curves in the phase diagram of the system.

Figure 7

Temperature dependence of the resistivity (a) and length dependence of the zero-temperature resistance (b) of a JJ chain with screening length $\mathrm{\Lambda }=10$, weak random stray charges $D_Q={10}^{-3}$, and ultraviolet QPS amplitude $y=0.1$. The numbers at the curves indicate the value of $\pi K_0$ controlling the infrared scaling of the QPS amplitude. The dashed lines at low temperatures are extrapolations illustrating the flow towards the insulating (infinite resistivity) fixed point. The inset shows the position of each of the curves in the phase diagram of the system.

Figure 8

Attribution of lattice variables in the derivation of the sine-Gordon theory before (left) and after (right) the Poisson resummation over charges and of the Villain approximation.