Domain walls in superconductors: Andreev bound states and tunneling features

Domain walls can be formed in superconductors with a discrete degeneracy of the ground state, for instance, due to the breaking of time reversal symmetry. We study all cases where the formation of domain walls is possible in a tetragonal superconductor with the point group $D_{4h}$. We discuss both triplet and mixed singlet order parameters. It is found that in all cases the domain walls support subgap Andreev bound states, whose energies strongly depend on the direction of semiclassical propagation. The bound state contribution to the density of quasiparticle states exhibits peculiar features, which can be observed in tunneling experiments.

Figure 1

Schematic diagram showing the DW, the semiclassical trajectory of a Bogoliubov quasiparticle, and the order parameters ${\Delta }_{\pm }$ on both sides.

Figure 2

The variation of the ABS energy with the direction of the quasiparticle propagation for domains with $\mathit{\eta }\propto (1,\pm i)$ for different values of $\chi$.

Figure 3

The energy dependence of the ABS contribution to the DOS (in arbitrary units) for domains with $\mathit{\eta }\propto (1,\pm i)$ for different values of $\chi$.

Figure 4

The variation of the ABS energy with the direction of the quasiparticle propagation for domains with $\mathit{\eta }\propto (1,0)$ and $(0,1)$ for different values of $\chi$. Regions bounded by blue vertical dashed lines and not containing any bound states correspond to the directions of propagation for which the condition (9) is not satisfied.

Figure 5

The energy dependence of the ABS contribution to the DOS (in arbitrary units) for domains with $\mathit{\eta }\propto (1,0)$ and $(0,1)$ for different values of $\chi$. For $\chi =0$ the DOS has a delta-function singularity at $E=0$.

Figure 6

The variation of the ABS energy with the direction of quasiparticle propagation for domains with $\mathit{\eta }\propto (1,\pm 1)$ for different values of $\chi$. Regions bounded by blue vertical dashed lines and not containing any bound states correspond to the directions of propagation for which the condition (9) is not satisfied.

Figure 7

The energy dependence of the ABS contribution to the DOS (in arbitrary units) for domains with $\mathit{\eta }\propto (1,\pm 1)$ for different values of $\chi$. For $\chi =0$ the DOS has a delta-function singularity at $E=0$.

Figure 8

The variation of the ABS energy with the direction of quasiparticle propagation for the domains $d_{x^2-y^2}\pm i{d}_{xy}$ for different values of $\chi$ and $\rho$. Thin red line is for $\rho =0.2$ and thick blue line is for $\rho =0.8$. For $\chi =0$ the curves superpose on each other.

Figure 9

The energy dependence of the ABS contribution to the DOS (in arbitrary units) for the domains $d_{x^2-y^2}\pm i{d}_{xy}$ for different values of $\chi$ and $\rho$. Thin red line is for $\rho =0.2$ and thick blue line is for $\rho =0.8$. For $\chi =0$ they superpose on each other.

Figure 10

The variation of the ABS energy with the direction of quasiparticle propagation for the domains $d_{x^2-y^2}\pm is$ for different values of $\chi$ and $\rho$. Thin red line is for $\rho =0.2$ and thick blue line is for $\rho =0.8$.

Figure 11

The energy dependence of the ABS contribution to the DOS (in arbitrary units) for the domains $d_{x^2-y^2}\pm is$ for different values of $\chi$ and $\rho$. Thin red line is for $\rho =0.2$ and thick blue line is for $\rho =0.8$. For $\chi =\pi$ the DOS has delta-function singularities.

Figure 12

The variation of the ABS energy with the direction of quasiparticle propagation for the domains $d_{xy}\pm is$ for different values of $\chi$ and $\rho$. Thin red line is for $\rho =0.2$ and thick blue line is for $\rho =0.8$.

Figure 13

The energy dependence of the ABS contribution to the DOS (in arbitrary units) for the domains $d_{xy}\pm is$ for different values of $\chi$ and $\rho$. Thin red line is for $\rho =0.2$ and thick blue line is for $\rho =0.8$. For $\chi =\pi$ the DOS has delta-function singularities.