Geometrical enhancement of the proximity effect in quantum wires with extended superconducting tunnel contacts

We study Andreev reflection in a ballistic one-dimensional channel coupled in parallel to a superconductor via a tunnel barrier of finite length $L$. The dependence of the low-energy Andreev reflection probability $R_A$ on $L$ reveals the existence of a characteristic length scale ${\xi }_N$ beyond which $R_A\left(L\right)$ is enhanced up to unity despite the low interfacial transparency. The Andreev reflection enhancement is due to the strong mixing of particle and hole states that builds up in contacts exceeding the coherence length ${\xi }_N$, leading to a small energy gap (minigap) in the density of states of the normal system. The role of the geometry of such hybrid contacts is discussed in the context of the experimental observation of zero-bias Andreev anomalies in the resistance of extended carbon nanotube/superconductor junctions in field effect transistor setups.

Figure 1

Quasi-one-dimensional electron system (Q1DES) coupled to (a) semi-infinite and (b) finite length $L$ superconductor contacts. The interfacial barrier is depicted in black. The Q1DES width is chosen to be $W_N\approx {\lambda }_F∕2$, where ${\lambda }_F$ is the Fermi wavelength.

Figure 2

DOS $\nu \left(ϵ\right)$ of a ballistic Q1DES-superconductor system. The solid line corresponds to the semi-infinite geometry shown in Fig. 1a for which $\nu \left(ϵ\right)$ is calculated directly from the Green function of the system. The dashed line corresponds to the case of a finite (but relatively long $L∕{\xi }_S⪢1$) $S$ film [Fig. 1b] where we use the decimation technique.

Figure 3

(a) Decomposition of the DOS $\nu \left(ϵ\right)$ to that arising from the Q1DES (solid curve) and the superconductor (dashed curve). Both curves are scaled to compare with each other. (b) DOS in the proximity region for various interfacial transparencies. Inset: the minigap $E_g$ depends quadratically on the interfacial coupling ${\gamma }_{NS}$ at low transparencies, and starts deviating at higher values of ${\gamma }_{NS}$.

Figure 4

Andreev reflection probability for various $N∕S$ coupling constants as a function of the length of the proximity region (II) in Fig. 2(b). $R_A$ scales with the interfacial transparency and at short lengths is quadratic in $L$. For $L$ much larger than the proximity-induced coherence length ${\xi }_N\approx 5{\xi }_S$, the probability $R_A$ reaches unity as a manifestation of the reflectionless tunneling despite the low interfacial transparency (see inset for ${\gamma }_{NS}=0.2$).

Figure 5

(a) Andreev reflection coefficient for the geometry of Fig. 1a with ${\gamma }_{NS}=0.32$ and various values of the potential $U_0$ at (I)/(II) boundary. For $U_0=0$ the Andreev probability is exactly 1 for energies below $E_g$. (b) To fit the low- and high-energy peaks we use the formulas of the BTK model (Ref. 21) with parameters ${\Delta }_{\mathrm{BTK}}=E_g,$ $Z=0.278$, and ${\Delta }_{\mathrm{BTK}}=\Delta ,$ $Z=28$, respectively.

Figure 6

Andreev reflection $R_A$, normal transmission $T_N$, and normal reflection $R_N$ coefficients for the geometry of Fig. 1a with ${\gamma }_{NS}=0.32$ and various potential landscapes in region (II).

Figure 7

Differential conductance of the $N∕S$ hybrid in Fig. 5a at ultralow temperature $T⪡E_g∕k_B$.

Figure 8

Evolution of the differential resistance of the $N∕S$ hybrid in Fig. 5a, as a function of temperature from $k_{B}T\sim E_g<\Delta$ to $T⪡E_g$ for (a) vanishing and (b) finite potential step between regions (I) and (II).