Nonlocal transport in normal-metal/superconductor hybrid structures: Role of interference and interaction

We have measured local and nonlocal conductance of mesoscopic normal-metal/superconductor hybrid structures fabricated by e-beam lithography and shadow evaporation. The sample geometry consists of a superconducting aluminum bar with two normal-metal wires forming tunnel contacts to the aluminum at distances of the order of the superconducting coherence length. We observe subgap anomalies in both local and nonlocal conductance that quickly decay with magnetic field and temperature. For the nonlocal conductance both positive and negative signs are found as a function of bias conditions, indicating at a competition of crossed Andreev reflection and elastic cotunneling. Our data suggest that the signals are caused by a phase-coherent enhancement of transport rather than dynamical Coulomb blockade.

Figure 1

(Color online) (a) SEM image of the sample layout. (b) Closeup of the contact region, colorized for clarity. Two copper wires (A and B) are connected by overlap tunnel junctions to a weakly oxidized aluminum bar. The copper wires fork near the tunnel contact to allow four-probe characterization. As a by-product of shadow evaporation, additional unconnected parts of the sample, and a Cu/Al/oxide/Cu trilayer (striped), are formed. The contact configuration and measurement scheme is also indicated. The three metal layers (Cu2, Cu1, and Al) as created by shadow evaporation are indicated above the image.

Figure 2

(Color online) X-ray diffraction patterns of an unpatterned Al/oxide/Cu sandwich with substrate background subtracted. (a) $\theta \text{−}2\theta$ scan with expected positions and relative intensities of (111) and (200) reflections for both Al and Cu indicated by vertical bars. (b) Rocking curve with angle of inclination $\omega$ for the (111) reflections of Al and Cu.

Figure 3

(Color online) Local conductance $g_{\text{AA}}$ [(a) and (c)] and nonlocal conductance $g_{\text{AB}}$ [(b) and (d)] of sample I as a function of temperature $T$ at $V_{\text{A}}=0$ [(a) and (b)] and injector bias $V_{\text{A}}$ at $T=25\text{mK}$ [(c) and (d)]. The insets in (a) and (b) show the low-temperature region of both plots on an enlarged scale. The solid lines in (a) and (c) are fits to a standard BCS tunneling model.

Figure 4

(Color online) Local differential conductance $g_{\text{AA}}$ [(a) and (c)] and nonlocal differential conductance $g_{\text{AB}}$ [(b) and (d)] as a function of injector bias $V_{\text{A}}$ taken at different temperatures $T$ for sample I [(a) and (b)], and different magnetic fields $B$ for sample II [(c) and (d)].

Figure 5

(Color online) (a) $g_{\text{AB}}$ as a function of $V_{\text{A}}$ for different $V_{\text{B}}$. (b) Oscillatory part $g_{\text{osc}}$ of the nonlocal conductance $g_{\text{AB}}$ as a function of both $V_{\text{A}}$ and $V_{\text{B}}$. (c) $g_{\text{osc}}$ at $V_{\text{A}}=0$ (symbols) and local conductance $g_{\text{BB}}$ of contact B (line) as a function of $V_{\text{B}}$. (d) $g_{\text{shift}}$ (symbols) and derivative of the local conductance $d{g}_{\text{BB}}/d{V}_{\text{B}}$ as a function of $V_{\text{B}}$ (line). All data are taken at $B=0$ and $T=20\text{mK}$ from sample II. For an explanation of $g_{\text{osc}}$ and $g_{\text{shift}}$, see text.

Figure 6

(Color online) Normalized amplitudes of $g_{\text{osc}}$ and $g_{\text{shift}}$, together with the square of the amplitude of the subgap anomaly in the local conductance ${\left(g_{\text{AA}}\right)}^2$, as a function of (a) temperature $T$ and (b) applied magnetic field $B$. Data taken from sample I.

Figure 7

(Color online) (a) Local conductance $g_{\text{AA}}$ and (b) nonlocal conductance $g_{\text{AB}}$ of sample II in the normal state at $B=1.5\text{T}$ for different temperatures $T$. The solid line in panel (a) is a fit to the standard model of dynamical Coulomb blockade (Ref. 22).

Figure 8

(Color online) [(a) and (b)] Differential subgap conductance of sample II at $B=0$ and $T=20\text{mK}$ with regular background subtracted (symbols). The lines are three different fits described in detail in the text. (c) Measured nonlocal conductance amplitude $g_{\text{AB}}$ vs model prediction $g_{\text{AB}}=R_{\xi }{g}_{\text{AA}}{g}_{\text{BB}}\text{exp}\left(-d/\xi \right)$. See text for details.