Local density of states at polygonal boundaries of $d$-wave superconductors

Besides the well-known existence of Andreev bound states, the zero-energy local density of states at the boundary of a $d$-wave superconductor strongly depends on the boundary geometry itself. In this work, we examine the influence of both a simple wedge-shaped boundary geometry and a more complicated polygonal or faceted boundary structure on the local density of states. For a wedge-shaped boundary geometry, we find oscillations of the zero-energy density of states in the corner of the wedge, depending on the opening angle of the wedge. Furthermore, we study the influence of a single Abrikosov vortex situated near a boundary, which is of either macroscopic or microscopic roughness.

Figure 1

(Color online) A $d_{x^2-y^2}$-wave superconductor with wedge-shaped boundary geometry. $\alpha$ denotes the opening angle of the wedge. The orientation between the $d$ wave and wedge is parametrized by the angle $\gamma$, which gives the rotation of the full-gap direction with respect to the bisecting line of the wedge. The origin $O$ represents the corner of the wedge.

Figure 2

(Color online) Zero-energy density of states of a $d_{x^2-y^2}$-wave superconductor for a wedge-shaped boundary with opening angle $\alpha =\pi ∕2$. The density of states is normalized to the density $N_0$ of the normal state. The effective scattering parameter is chosen to be $\delta =0.1{\Delta }_0$. The rotation of the $d$ wave with respect to the bisecting line is (a) $\gamma =-\pi ∕4$, (b) $\gamma =-\pi ∕8$, and (c) $\gamma =0$.

Figure 3

(Color online) Zero-energy density of states of a $d_{x^2-y^2}$-wave superconductor for a wedge-shaped boundary with opening angle $\alpha =\pi ∕4$. The density of states is normalized to the density $N_0$ of the normal state. The effective scattering parameter is chosen to be $\delta =0.1{\Delta }_0$. The rotation of the $d$ wave with respect to the bisecting line is (a) $\gamma =-\pi ∕8$, (b) $\gamma =0$, and (c) $\gamma =\pi ∕8$.

Figure 4

(Color online) Zero-energy density of states of a $d_{x^2-y^2}$-wave superconductor for a wedge-shaped boundary with opening angle $\alpha =\pi ∕3$. The density of states is normalized to the density $N_0$ of the normal state. The effective scattering parameter is chosen to be $\delta =0.1{\Delta }_0$. The rotation of the $d$ wave with respect to the bisecting line is (a) $\gamma =0$, (b) $\gamma =\pi ∕8$, and (c) $\gamma =\pi ∕4$.

Figure 5

(Color online) (a) Zero-energy density of states in the corner $O$ of a $d_{x^2-y^2}$-wave superconductor with wedge-shaped boundary geometry. $\alpha$ denotes the opening angle of the wedge, $\gamma$ the rotation of the maximum gap direction with respect to the bisecting line of the wedge. The global maximum in the middle of the picture corresponds to the well-known maximum of the Andreev bound states at a straight boundary line, with the nodal direction of the $d$ wave being perpendicular to the boundary. (b) Two examples for the relations (18, 19). The upper panel shows that for an opening angle of $\alpha =\pi ∕2$ an incoming trajectory always leaves the corner at the same angle. For an opening angle of $\alpha =\pi ∕3$, the directions of the incoming and outgoing trajectories are symmetric with respect to the bisecting line of the wedge (lower panel).

Figure 6

Oscillation of the zero-energy density of states in the corner $O$ of a $d_{x^2-y^2}$-wave superconductor with wedge-shaped boundary geometry. The nodal direction of the $d$ wave is fixed parallel to the bisecting line of the wedge, i.e., $\gamma =\pi ∕4$. Decreasing the opening angle $\alpha$ leads to oscillations of the density of states in the corner, with minima appearing at ${\alpha }_n^{+}=\pi ∕\left(2n\right)$ and maxima close to ${\alpha }_n^{-}=\pi ∕(2n-1)$. Both upper and lower envelope (dotted lines) are given analytically.

Figure 7

(Color online) Zero-energy density of states of a $d_{x^2-y^2}$-wave superconductor with wedge-shaped boundary geometries. A single Abrikosov vortex is pinned at a distance of $2\xi$ from each boundary line. All pictures presented here correspond to a specific wedge and orientation already shown earlier without a vortex. The same color palettes have been used for both the corresponding plot without vortex and the plot given here. (a) This picture refers to Fig. 2c. The vortex casts a shadowlike suppression on the Andreev bound states at each boundary. (b) This picture refers to Fig. 4a. Again, the Andreev bound states at the boundaries are locally suppressed. (c) This picture refers to Fig. 4c. The bound states in the corner induced by the boundary geometry are also affected but remain still considerable. Please note that (b) and (c) are given in a smaller scale than the corresponding pictures.

Figure 8

(Color online) Zero-energy density of states of a $d_{x^2-y^2}$-wave superconductor for a wedge-shaped boundary with opening angle $\alpha =3\pi ∕2$. A single Abrikosov vortex is situated at a vertical distance of $2\xi$ from the corner of the wedge. The horizontal distance of the vortex is (a) $2\xi$, (b) $\xi$, and (c) 0. In the top row, the nodal direction of the $d$ wave is perpendicular to both boundaries, which corresponds to $\gamma =0$. Below, the nodal direction is parallel to the bisecting line, i.e., $\gamma =\pi ∕4$.

Figure 9

(Color online) Zero-energy density of states of a $d_{x^2-y^2}$-wave superconductor in the vicinity of a surface. This corresponds to an opening angle of the wedge of $\alpha =\pi$. A single Abrikosov vortex is located at a distance of $2\xi$ from the boundary. In the top row, the boundary is smooth. In the lower row, the boundary is microscopically rough. (a) The nodal direction of the $d$ wave is perpendicular to the boundary, which corresponds to $\gamma =\pi ∕4$. (b) The rotation angle is $\gamma =\pi ∕6$. Compared to (a), the orientation changed by 15 deg.

Figure 10

The conformal mapping procedure used to obtain the phase distribution around an Abrikosov vortex in the vicinity of a boundary.

Figure 11

(Color online) Zero-energy density of states of a $d_{x^2-y^2}$-wave superconductor with a polygonal boundary geometry, which can be regarded as a boundary line with macroscopic roughness (faceting). (a) No vortex is present. (b) A single Abrikosov vortex is situated $2\xi$ away from the averaged boundary line.