Designer quantum spin Hall phase transition in molecular graphene

Graphene was the first material predicted to be a time-reversal-invariant topological insulator [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)]; however, the insulating gap is immeasurably small owing to the weakness of spin-orbit interactions in graphene. A recent experiment [K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C. Manoharan, Nature (London) 483, 306 (2012)] demonstrated that designer honeycomb lattices with graphenelike “Dirac” band structures can be engineered by depositing a regular array of carbon monoxide atoms on a metallic substrate. Here, we argue that by growing such designer lattices on metals or semiconductors with strong spin-orbit interactions, one can realize an analog of graphene with strong intrinsic spin-orbit coupling, and hence a highly controllable two-dimensional topological insulator. We estimate the range of substrate parameters for which the topological phase is achievable, and consider the experimental feasibility of some candidate substrates.

Figure 1

Schematic illustration of proposed experimental setup, involving a triangular array of CO molecules (black circles) placed on a metallic substrate. The surface electrons of the metal are repelled by the CO molecules, and therefore occupy the sites of the dual lattice (red squares), which form a honeycomb lattice.

Figure 2

Band structure of “molecular graphene” on a periodic (honeycomb-lattice) substrate, e.g., the surface of Au. (a) Reciprocal lattice structure, showing the substrate reciprocal lattice (gray solid hexagons) as well as the first Brillouin zone (BZ) of the reciprocal “molecular graphene” superlattice (dashed shaded hexagons) near the $\Gamma$ point (red dot) of each substrate unit cell. Solid and dashed arrows denote reciprocal lattice vectors of the substrate (${\mathbf{Q}}_i$) and of the superlattice (${\mathbf{G}}_i$), respectively. The two inequivalent Dirac cones are represented using triangles. (b) “Dirac” band structure for states at corners of the first superlattice BZ. The six corner momenta form two inequivalent groups (blue and red, respectively ${\Lambda }_i,{\overline{\Lambda }}_i$); (c) states at the three momenta marked (e.g.) in blue hybridized via the lattice potential to form two low-lying states and one higher-energy state. All three states are coupled via the intraband SOC. (d) Schematic one-dimensional cut through the band structure, showing the band gaps due to the substrate lattice potential ($\sim$$W_1$) and superlattice potential ($\sim$$V_G$), as well as the multiple superlattice BZs that fit within the first substrate BZ. We are primarily interested in states at the edges of the first superlattice BZ (shaded ellipse), which are connected by the interband spin-orbit coupling (SOC) to the corresponding momenta in the upper band.

Figure 3

Phase diagram of “molecular graphene” as a function of the spin-orbit coupling (SOC) parameter $\gamma$, introduced in the main text, and the work function $W_0$ of the substrate (which can be tuned to some extent via gating). The quantum spin Hall insulator is achieved in the region shaded blue.