Long range $p$-wave proximity effect into a disordered metal

We use quasiclassical methods of superconductivity to study the superconducting proximity effect from a topological $p$-wave superconductor into a disordered quasi-one-dimensional metallic wire. We demonstrate that the corresponding Eilenberger equations with disorder reduce to a closed nonlinear equation for the superconducting component of the matrix Green's function. Remarkably, this equation is formally equivalent to a classical mechanical system (i.e., Newton's equations), with the Green function corresponding to a coordinate of a fictitious particle and the coordinate along the wire corresponding to time. This mapping allows us to obtain exact solutions in the disordered nanowire in terms of elliptic functions. A surprising result that comes out of this solution is that the $p$-wave superconductivity proximity induced into the disordered metal remains long range, decaying as slowly as the conventional $s$-wave superconductivity. It is also shown that impurity scattering leads to the appearance of a zero-energy peak.

Figure 1

Potential landscape of a classical particle with motion describing the Green's function, for a normal metallic segment, in contact with a superconductor. Depending on the superconductor $(s$- or $p$-wave) potential is either $V_s$ or $V_p$. In the clean limit both converge to $V^{\mathrm{clean}}$.

Figure 2

Contour plot of the DOS of an infinite wire. There is moderate disorder $(\beta =1)$ in the normal segment $(x>0)$. The solid yellow marks the regions that are beyond the plot range (where $N/N_0>3.5)$. Notice the zero-energy peak in the normal segment.

Figure 3

DOS at the junction of infinite normal and superconducting segments. Three cases for the disorder in the normal segment are plotted: weak ($\beta =0.1$, blue), moderate ($\beta =1$, purple), and strong ($\beta =10$, red). Notice the suppression of DOS with the increase of disorder strength.

Figure 4

Components of $\widehat{g}$ $(g_1$: blue; $g_2$: purple; $g_3$: red) for a wire with infinite $p$-wave section and finite disordered section of length $L=5{\xi }_0$. Top panel: weak disorder $[\beta =1/\left(2{\tau }_{\mathrm{imp}}{\Delta }_0\right)=0.1]$. Middle panel: moderate disorder $(\beta =1)$. Bottom panel: strong disorder $(\beta =10)$. The Matsubara frequency is set to $\omega ={\Delta }_0/2$.

Figure 5

Weight of the zero-energy mode $\mathcal{M}\left(x\right)$ in a normal section with length $L=5{\xi }_0$ for three disorder strengths (blue: $\beta =0.1$; purple: $\beta =1$; red: $\beta =10$).