Andreev Reflection Eigenvalue Density in Mesoscopic Conductors

The energy-dependent Andreev reflection eigenvalues determine the transport properties of normal-superconducting systems. We evaluate the eigenvalue density to get insight into the formation of resonant electron-hole transport channels. The circuit-theory-like method developed can be applied to any generic mesoscopic conductor or combinations thereof. We present the results for experimentally relevant cases of a diffusive wire and a double tunnel junction.

Figure 1

Left: schematic picture of a mesoscopic conductor connected to a normal (N) and a superconducting (S) reservoir. Right: the circuit theory representation of two mesoscopic conductors: (a) a diffusive wire in series with a tunnel barrier and (b) two tunnel barriers in series. The conductances of the respective circuit elements are shown.

Figure 2

Energy-dependent ARED of diffusive contacts. (a) ARED for a diffusive wire ideally connected to a normal and a superconducting reservoir. Energies are measured in units of the Thouless energy $E_c=\hslash D/L^2$ and the ARED is normalized to the curve for $E=0$. At energies around $E_c$, resonant electron-hole channels, $R\approx 1$, open up. (b) ARED with a contact resistance $G_T=G_N$ at the NS interface. For finite energies the ARED has an upper bound; there is a gap in the density for eigenvalues above a maximum eigenvalue $R_{\max }$. (c) ARED with a contact resistance $G_T=2G_N$ at the NS interface. For $E\lesssim 10E_c$ all eigenvalues contribute, whereas a gap opens up for higher energies.

Figure 3

Left panel: ARED for a double-barrier junction with $G_2=5G_1$ for different energies $E/E_c$. Right panel: limiting reflection eigenvalues $R_I$ (solid line) and $R_{II}>R_I$ (dashed line) as a function of energy for different conductance ratios $G_1/G_2$.