Topological Properties of Electrons in Honeycomb Lattice with Detuned Hopping Energy

Abstract

Honeycomb lattice can support electronic states exhibiting Dirac energy dispersion, with graphene as the icon. We propose to derive nontrivial topology by grouping six neighboring sites of honeycomb lattice into hexagons and enhancing the inter-hexagon hopping energies over the intra-hexagon ones. We reveal that this manipulation opens a gap in the energy dispersion and drives the system into a topological state. The nontrivial topology is characterized by the index associated with a pseudo time-reversal symmetry emerging from the C6 symmetry of the hopping texture, where the angular momentum of orbitals accommodated on the hexagonal "artificial atoms" behaves as the pseudospin. The size of topological gap is proportional to the hopping-energy difference, which can be larger than typical spin-orbit couplings by orders of magnitude and potentially renders topological electronic transports available at high temperatures.

Figure 1. Hopping texture in honeycomb lattice and emergent orbitals.

(a) Honeycomb lattice with hopping energies between NN sites: t₀ inside hexagons as denoted by the green bonds and t₁ between hexagons by red ones. The red dashed hexagon is the primitive cell of triangular lattice with lattice vectors , and lattice constant . Numbers in circle index atomic sites within a hexagon. (b) Emergent orbitals in the hexagonal artificial atom.

Figure 2. Band inversion and topological phase transition.

(a,b) Current densities in the pseudospin-up channel (p₊ or d₊) and pseudospin-down channel (p₋ or d₋) respectively. Band dispersions for the system given in Fig. 1: (c) t₁ = 0.9t₀ (Inset: Brillouin zone of the triangular lattice), (d) t₁ = t₀ and (e) t₁ = 1.1t₀. Blue and red are for |p_±〉 and |d_±〉 orbitals respectively, and rainbow for hybridization between them. The on-site energy is taken ε₀ = 0.

Figure 3. Topological edge states.

(a) Band dispersion of a ribbon system of 36 hexagons with t₁ = 1.1t₀ cladded from both sides by 10 hexagons with t₁ = 0.9t₀. (b) Real-space distribution of the in-gap states associated with the red solid dispersion curves in (a). (c,d) Real-space distributions of current densities in pseudospin-up and -down channels at the momenta indicated by the red and green dots 1 and 2 in (a) within the rhombic area sketched by dashed line in (b); the excess currents in pseudospin-up and -down channels are indicated by red and green arrows in (b).

Figure 4. Conductances of the topological phase.

(a) Schematic configuration of a six-terminal Hall bar where a topological sample (light blue region) with t₁ = 1.1t₀ is embedded in a trivial environment (gray region) with t₁ = 0.9t₀. The size of topological scattering region is 240a₀ × 120a₀, and the width of each semi-infinite lead is 40a₀. The injected current flows along the edges of topological sample as indicated by the red parts between electrodes. (b) Longitudinal and Hall conductances of the Hall bar as a function of energy of incident electrons. The on-site energy is taken ε₀ = 0. A rhombic topological sample is taken for ease of calculation.

Figure 5. Hopping-energy and sample-size dependence of longitudinal conductances.

Longitudinal conductance G_xx of the topological sample given in Fig. 4(a) as a function of the energy of injected electrons: (a) for several typical values of inter-hexagon hopping integrals [1.05t₀, 0.95t₀], [1.1t₀, 0.9t₀] and [1.2t₀, 0.8t₀], where the first (second) inside bracket is for the topological (trivial) region; (b) with several typical system sizes , where the width of the electrodes is fixed at 40a₀ and the inter-hexagon hopping integral is fixed at t₁ = 1.1t₀ and t₁ = 0.9t₀ for the topological and trivial regions respectively.

Figure 6. Edge states and conductances of the topological phase in the presence of SOC.

(a) Dispersion relations and (b) longitudinal conductances of the topological system same as that given in Fig. 4(a) except that finite SOC is included.

Figure 7. Schematics for hopping textures with C₆ symmetry.

Molecular graphene realized by decorating the Cu [111] surface with a triangular lattice of CO molecules: (a) t₁ > t₀ generating topological state, (b) t₁ < t₀ for trivial state. Gray balls are CO molecules decorated by STM techniques, and red thick bonds are shorter than green thin ones which generates the hopping textures.