Domain Wall in a Chiral $p$-Wave Superconductor: A Pathway for Electrical Current

Superconductors with $p_x\pm i{p}_y$ pairing symmetry are characterized by chiral edge states, but these are difficult to detect in equilibrium since the resulting magnetic field is screened by the Meissner effect. Nonequilibrium detection is hindered by the fact that the edge excitations are Majorana fermions, which cannot transport charge near the Fermi level. Here we show that the boundary between $p_x+i{p}_y$ and $p_x-i{p}_y$ domains forms a one-way channel for electrical charge. We derive a product rule for the domain wall conductance, which allows us to cancel the effect of a tunnel barrier between metal electrodes and the superconductor and provides a unique signature of topological superconductors in the chiral $p$-wave symmetry class.

Figure 1

Superconducting strip divided by a domain wall (dashed line, length $W$) into domains with $p_x\pm i{p}_y$ symmetry. The edge states ${\Psi }_L$, ${\Psi }_R$ of opposite chirality in the two domains are indicated by red arrows. These Majorana modes can carry heat current between contacts $N_L$ and $N_R$, but no electrical current. A normal-metal electrode $N_1$ at voltage $V_1$ injects charge into the domain wall, which is detected as an electrical current $I_2$ at the other end $N_2$. In an alternative measurement configuration (indicated in blue), contact $N_2$ measures a voltage $V_2$ without drawing a current.

Figure 2

Solid curves: probability distributions of the nonlocal conductance $G_{12}$ (in units of $e^2/h$) and nonlocal resistance $R_{12}$ (in units of $h/e^2$). These are results for a random distribution of the $4\times 4$ orthogonal scattering matrices $S_1$ and $S_2$. The dashed curve shows the narrowing effect on $P(G_{12})$ of a tunnel barrier in both contacts (tunnel probability $\tau =0.1$). In contrast, $P(R_{12})$ is not affected by a tunnel barrier.