Realization of a vortex in the Kekule texture of molecular graphene at a Y junction where three domains meet

Following the recent realization of an artificial version of graphene in the electronic surface states of copper with judiciously placed carbon monoxide molecules inducing the honeycomb lattice symmetry [Gomes et al., Nature (London) 483, 306 (2012)], we demonstrate that these can be used to realize a vortex in a Kekule texture of the honeycomb lattice. The Kekule texture is mathematically analogous to a superconducting order parameter in the Bogoliubov–de Gennes equations, opening a spectral gap in the massless Dirac point spectrum of the graphene structure. The core of a vortex in the texture order parameter supports subgap states, which for this system are mathematical analogs of Majorana fermions in some superconducting states. The subgap states bind a fractional charge of $e/2$ to the vortex core, in effect delocalizing a single electron between two vortex cores, or between one vortex core and the system boundary. The Kekule texture as realized in the molecular graphene system realizes three different domain types, and we show that a Y junction between them realizes the coveted Kekule vortex.

Figure 1

Kekule pattern on the honeycomb lattice. The thick (red) links have stronger (or weaker) hopping strength on them. The unit cell is denoted by dashed (gray) lines.

Figure 2

The unit cell of the Kekule pattern. The six sites in the unit cell are denoted by $1,...,6$ (red). The unit cell is denoted by the dashed (gray) line. The (blue) Bravais lattice vectors are denoted by ${\mathbf{A}}_{1,2}$.

Figure 3

The three different Kekule domains, with a specific choice of unit cell. These three domains are characterized by an effective phase difference of $2\pi /3$.

Figure 4

Image of the Kekule texture Y junction, which we use in numerics. The graphene flake is a disk of radius $22.5$, using the convention mentioned in the text. The stronger hopping links are denoted by a thick (red) link between the (gray) points denoting the lattice sites. The boundaries between the three Kekule texture domains are denoted by (black) lines, indicating the Y-junction shape. The point where the three domain walls meet is a vortex core. An effective phase jumps by $2\pi /3$ at each domain wall.

Figure 5

The LDOS for $\lambda =1$ at different energies: $\epsilon =+1.5$ (a), $\epsilon =+0.5$ (b), $\epsilon =+0.1$ (c), $\epsilon =0$ (d), $\epsilon =-0.1$ (e), $\epsilon =-0.5$ (g), and $\epsilon =-1.5$ (g). The circles represent sites of the honeycomb lattice, taken here in the shape of a disk of radius $22.5$, using the conventions in the main text. The site coloring is such that dark (blue) points have a higher weight, and lighter (orange) points have lower weight. While LDOS scans above (a) and below (g) the gap show a rather uniform distribution of DOS, in the gap there clearly is some spatial structure. In particular, at $\epsilon =0$ we find a peak at the vortex core and at one spot on the disk edge (d), as expected.

Figure 6

Radial accumulated weight of the zero-mode wave function $D_1\left(R\right)$ (a) and of the LDOS at $E=0$ $D_2\left(R\right)$ (b). The probability density of the ${\psi }_{0-}$ wave function is depicted in (c) using the same convention as for the LDOS plots in Fig. 5.