Nonlocal correlations in a proximity-coupled normal-metal

We report evidence of large, nonlocal correlations between two spatially separated normal-metals in superconductor/normal-metal (SN) heterostructures, which manifest themselves as nonlocal voltage generated in response to a driving current. Unlike prior experiments in SN heterostructures, the nonlocal correlations are mediated not by a superconductor, but by a proximity-coupled normal-metal. The nonlocal correlations extend over relatively long length scales in comparison to the superconducting case. At very low temperatures, we find a reduction in the nonlocal voltage for small applied currents that cannot be explained by the quasiclassical theory of superconductivity, which we believe is a signature of new long-range quantum correlations in the system.

Figure 1

(a) False color scanning electron micrograph of the sample discussed in the text. Light areas are Au; the darker lines are Al. The numbers mark the contacts used for four-terminal differential resistance measurements. The size bar is 500 nm. (b) Local differential resistance as a function of the applied current $I_{\text{dc}}$. (c) Nonlocal resistance of the NSN configuration of sample shown in (a) as a function of $I_{\text{dc}}$. Inset: expanded view of the zero bias region. (d) Nonlocal resistance of the proximity-coupled normal-metal shown in (a) as a function of $I_{\text{dc}}$. Inset: expanded view of the zero bias region. All the measurements shown here are performed at 24 mK.

Figure 2

(a) False color scanning electron micrograph of the sample discussed in the text. Light areas are Au; the darker lines are Al. The numbers mark the contacts used for four-terminal differential resistance measurements. The size bar is 500 nm. (b) Local differential resistance $R_{19,28}$ as a function of the applied current $I_{\text{dc}}$. (c) Blue circles show the measured critical current, determined by the position of the peak in the differential resistance in (b), as a function of temperature $T$. The solid line shows a fit to the expected temperature dependence for a long SNS junction of length $L$.[20] Open red circles are the finite values of $I_{\text{dc}}$ at which the nonlocal differential resistance $R_{31,49}$ [Fig. 3a] has its minimum.

Figure 3

(a) Nonlocal resistances as a function of dc current bias for four nonlocal configurations, $R_{31,49}$, $R_{31,59}$, $R_{31,69}$, and $R_{31,79}$. (b) Data of (a), with the curves for $R_{31,49}$, $R_{31,59}$, $R_{31,69}$, and $R_{31,79}$ scaled so that their normalized peaks at $\pm 2.3$ $\mu$A match. (c) Nonlocal resistances $R_{31,79}$, $R_{41,79}$, and $R_{51,79}$, where the current injection terminal is changed, but the voltage contacts remain the same. (d) Data of (c), with the curves for $R_{31,79}$, $R_{41,79}$, and $R_{51,79}$ with both $x$ and $y$ axes scaled as described in the text.

Figure 4

(a) Schematic of the current separation model. The injected quasiparticle current $I_{\text{qp}}$ splits into two currents, $I_{\text{qp}1}$ and $I_{\text{qp}2}$, which go towards the two superconducting contacts $S_1$ and $S_2$, respectively. $I_{\text{qp}2}$ is compensated by a counterflowing supercurrent $I_s$ that flows from $S_2$ to $S_1$. (b) Results of the numerical simulations based on the quasiclassical theory of superconductivity for the nonlocal resistances for the geometry of (a) as a function of $I_{\text{qp}}$. The distance $L^{\prime }$ between the normal probe (on the normal-metal) and $S_1$ is $0.451L$, $0.578L$, $0.696L$, and $0.843L$, where $L$ is the length of normal-metal between $S_1$ and $S_2$. The position of $N_1$ is fixed at $0.196L$ from $S_1$. These values are chosen to match the sample geometry. (c) Calculated zero bias resistance as a function of temperature.

Figure 5

(a) Temperature dependence of $R_{\text{nl}}\left(0\right)$ for all four nonlocal resistance configurations. (b) Nonlocal differential resistance for the closest configuration at a number of different temperatures. (c) Temperature dependence of $-R_{\text{nl}}\left(0\right)$ for the configuration in (b) and temperature dependence of the local critical current $I_c$. (d) Data from (b) at 20 mK, 80 mK, and 160 mK, plotted as a function of the nonlocal voltage $V_{\text{nl}}$ obtained by numerical integration. Dotted lines are guides to the eyes for the points where 20 mK and 80 mK data deviate significantly from 160 mK data.