Domain walls in a chiral $d$-wave superconductor on the honeycomb lattice

We perform a fully self-consistent study of domain walls between different chiral domains in chiral $d_{x^2-y^2}\pm i{d}_{xy}$-wave superconductors with an underlying honeycomb lattice structure. We investigate domain walls along all possible armchair and zigzag directions and with a finite global phase shift across the domain wall, in addition to the change of chirality. For armchair domain walls we find the lowest domain wall energy at zero global phase shift, while the most favorable zigzag domain wall has a finite global phase shift dependent on the doping level. Below the van Hove singularity the armchair domain wall is most favorable, while at even higher doping the zigzag domain wall has the lowest energy. The domain wall causes a local suppression of the superconducting order parameter, with the superconducting recovery length following a universal curve for all domain walls. Moreover, we always find four subgap states crossing zero energy and well localized to the domain wall. However, the details of their energy spectrum vary notably, especially with the global phase shift across the domain wall.

Figure 1

Schematic diagram of the honeycomb lattice showing the atomic sites in sublattice A (filled circle) and sublattice B (circle) as well as the three nearest-neighbor bonds directions ${\mathbf{a}}_{\mathbf{1}},{\mathbf{a}}_{\mathbf{2}},{\mathbf{a}}_{\mathbf{3}}$. $c$ is the lattice constant and $l$ is the distance between sublattice sites in the horizontal direction. Dotted lines show all three ZZ (red) and AC (blue) domain wall directions.

Figure 2

Domain wall energy density $\varepsilon$ as a function of $\alpha$ for several $\mu$ and $J$ for sharp [(a), (c), and (e)] and self-consistent [(b), (d), and (f)] domain walls. Dotted horizontal line indicates the global minimum for the AC domain wall. For the sharp domain walls the bulk value of ${\mathrm{\Delta }}_b=\left|\mathbf{\Delta }\right|$ are the same as in the self-consistent calculations and given in each subfigure. Arrows indicate hard to see data points at $\alpha =\pi$.

Figure 3

Spatial profiles of the absolute values of the superconducting pairing $|\mathbf{\Delta }|$ for $\alpha =0,2\pi /3,\pi$ along ZZ [(a), (c), and (e)] and AC [(b), (d), and (f)] direction with a domain wall at $\tilde{x}=\tilde{y}\approx 150$ for the same $\mu$ and $J$ as in Figs. 2, 2 and 2. Thick lines indicate the configuration with lowest domain wall energy density.

Figure 4

Domain wall energy density minimum ${\varepsilon }_{\min }$ along ZZ (filled circle) and AC (cross) as a function of $\mu$ for $J=0.8,1.0,1.2$.

Figure 5

Spectrum (a) and spatial profile of the imaginary (dashed) and real (solid) parts of the $\mathrm{\Delta }$ character (b) for the AC DWI; the most stable domain wall at low doping levels. Low energy states within the gap in (a) are for self-consistent (black) and sharp domain walls (red), while the dashed horizontal line indicates the Fermi level. E and D denote edge and domain wall states, respectively, which are both doubly degenerate. Vertical line shows the $k$ point for which Fig. 7 is obtained. Here $\alpha =0,\mu =0.7,J=1.0$.

Figure 6

Spectrum (a) and spatial profile of the imaginary (dashed) and real (solid) parts of the $\mathrm{\Delta }$ character (b) for the ZZ DWII; the most stable domain wall at doping beyond the VHS. Low energy states within the gap in (a) are for self-consistent (black) and sharp domain walls (red), while the dashed horizontal line indicates the Fermi level. E and D denote doubly degenerate edge and singly degenerate domain wall states, respectively. Here $\alpha =\frac{2\pi }{3},\mu =1.1,J=0.8$.

Figure 7

Probability density of the four states closest to zero energy as a function of distance from the domain wall with E edge states [(a) and (b)] and D domain wall states [(c) and (d)] for the AC DWI at the $k_x$ value indicated by a vertical line in Fig. 5.

Figure 8

Half-width at half-maximum (HWHM) of domain wall at ${\alpha }_{\min }$ for ZZ (dotted) and AC (solid) domain walls as a function of ${\mathrm{\Delta }}_b$ for several values of $J$. Inset shows the relationship of ${\mathrm{\Delta }}_b$ with $\mu$.