Zero-energy bound state at the interface between an $s$-wave superconductor and a disordered normal metal with repulsive electron-electron interactions

In recent years, there has been a renewed interest in the proximity effect due to its role in the realization of topological superconductivity. Here, we study a superconductor–normal metal proximity system with repulsive electron-electron interactions in the normal layer. It is known that in the absence of disorder or normal reflection at the superconductor–normal metal interface, a zero-energy bound state forms and is localized to the interface [Fauchère et al., Phys. Rev. Lett. 82, 3336 (1999).]. Using the quasiclassical theory of superconductivity, we investigate the low-energy behavior of the density of states in the presence of finite disorder and an interfacial barrier. We find that as the mean free path is decreased, the zero-energy peak in the density of states is broadened and reduced. In the quasiballistic limit, the bound state eliminates the minigap pertinent to a noninteracting normal layer and a distinct peak is observed. When the mean free path becomes comparable to the normal layer width, the zero-energy peak is strongly suppressed and the minigap begins to develop. In the diffusive limit, the minigap is fully restored and all signatures of the bound state are eliminated. We find that an interfacial potential barrier does not change the functional form of the density of states peak but does shift this peak away from zero energy.

Figure 1

The pairing amplitude $F\left(x\right)$ (left curve) and pairing potential $\Delta \left(x\right)$ (right curve) profiles near the (ideal) interface between a superconductor and a normal metal with repulsive electron-electron interactions $\left({\lambda }_N>0\right)$.

Figure 2

Cross section of SN proximity geometry. Interface located at $x=0$ is taken to be perfectly transmitting throughout most of the paper, while vacuum–normal metal boundary at $x=d$ is assumed to be specularly reflecting. System is infinite in both the $y$ and $z$ directions.

Figure 3

Step model for $\Delta \left(x\right)$.

Figure 4

Local density of states as a function of energy in the ballistic limit and for perfectly transmitting SN interface [Eq. (17)] at various positions $x$, as shown in the legend. ${\Delta }_{N}d/v_F=0.3$.

Figure 5

Local density of states as a function of energy in the ballistic limit with a nonideal SN interface parametrized by reflection coefficient $R=0.05$. ${\Delta }_{N}d/v_F=0.4$; vertical dashed line corresponds to $E_0$, as given by Eq. (21).

Figure 6

Local density of states in the diffusive limit at the superconductor/normal metal interface $\left(x=0\right)$ plotted for various values of ${\Delta }_N$. The normal metal thickness is $d=\sqrt{D/2{\Delta }_S}$.

Figure 7

(a) Spatial profile of the pairing potential $\Delta \left(x\right)$ following from a self-consistent solution of Eqs. (4, 5, 6) with coupling constants ${\lambda }_S{N}_0=1$ and ${\lambda }_N{N}_0=-0.25$ and normal metal thickness $d=10{\xi }_S$. (b) Energy dependence of LDOS at fixed mean free path $\ell =100d$, shown for various values of $x$. (c) Energy dependence of LDOS at fixed position $x=0$, plotted for various values of mean free path $\ell$.

Figure 8

Energy dependence of LDOS at fixed position $x=0$ and mean free path $\ell =100d$, plotted at several different temperatures. LDOS follows from a self-consistent calculation of $\Delta \left(x\right)$ using Eq. (33); we choose ${\lambda }_S{N}_0=0.5$ and ${\lambda }_N{N}_0=-0.25$.

Figure 9

Local density of states at SN interface, calculated using exact (nonquasiclassical) BdG equation [Eq. (1)] with $d=5{\xi }_S$ and $\mu =100{\Delta }_S$. (a) Zoom of low-energy behavior. Both the numerical results (dots) and $1/E{ln}^{2}E$ functional form predicted by Eq. (16) (solid line) are plotted. Inset: Model form of pairing potential $\Delta \left(x\right)$. (b) Andreev structure of LDOS for energies $E\gtrsim v_F/d$.

Figure 10

Spatial dependence of the (unnormalized) wave function $u_E\left(x\right)$ corresponding to the minimal eigenvalue $E$ of the BdG equation (rapidly oscillating curve). Quasiclassical wave function [Eq. (3)] traces the envelope of the exact wave function (slowly varying curve). Parameters: $\mu =100{\Delta }_S$ and ${(1-k_{\perp }^2/k_F^2)}^{1/2}=v_x/v_F=0.3$. For $\Delta \left(x\right)$, we use the form shown in Fig. 7. Inset: Zoom of oscillations on the Fermi wavelength scale.