Competition of topological and topologically trivial phases in patterned graphene based heterostructures

We explore the effect of mechanical strain on the electronic spectrum of patterned graphene based heterostructures. We focus on the competition of Kekulé-O type distortion favoring a trivial phase and commensurate Kane-Mele type spin-orbit coupling generating a topological phase. We derive a simple low-energy Dirac Hamiltonian incorporating the two gap promoting mechanisms and include terms corresponding to uniaxial strain. The derived effective model explains previous ab initio results through a simple physical picture. We show that while the trivial gap is sensitive to mechanical distortions, the topological gap stays resilient.

Figure 1

The phase diagram of the considered model based on Eq. (3) and the discussion below. Parameter set from the inside of the cones depicts a trivial gap, while outside of the cones the system is topological. The band gap is closed on the solid surfaces. Between the gray opaque shape and the cones the gap is located at zero momentum; everywhere else it is shifted away to a finite value.

Figure 2

Schematic structure of the unstrained model. The pale gray hexagons represent the unadulterated graphene substrate. Each pale gray bond corresponds to a hopping amplitude $t$. The red dashed lines denote the Kekulé-O type bond alteration with a characteristic strength of $\mathrm{\Delta }$. Each gray bond with a red dashed line on top corresponds to a hopping amplitude $t+\mathrm{\Delta }$. Brown lines connecting next-nearest neighbor sites symbolize the induced Kane-Mele type spin-orbit coupling. Each brown line corresponds to a spin dependent hopping amplitude $im$, in the direction of the arrows. For more details see the Appendix.

Figure 3

The low-energy spectrum of the considered tight-binding model (A21), denoted by solid orange lines, and the effective Dirac Hamiltonian (1), depicted with dashed black lines, for various parameters. Topological gaps are indicated by blue opaque coloring, while trivial gaps are highlighted with red. In all panels $m=0.04t$ and $\mathrm{\Delta }=0.08t$ unless indicated otherwise, for all cases $\theta =0$.

Figure 4

The magnitude of the gap obtained from the effective Dirac Hamiltonian is shown as the function of uniaxial strain for various model parameters. Red color indicates that the gap is trivial, while blue coloring shows that the gap is topological. The dashed line corresponds to $\mathrm{\Delta }=0$ and $m=0.04t$. The solid line shows the case when $\mathrm{\Delta }=0.04t$ and $m=0$. The dash-dotted line was obtained using $\mathrm{\Delta }=0.01t$ and $m=0.02t$ while in the case of the dotted curve it was $\mathrm{\Delta }=0.03t$ and $m=0.02t$.

Figure 5

Discretization of the Brillouin zone spanned by reciprocal lattice vectors ${\mathit{b}}_1$ and ${\mathit{b}}_2$. Each vertex denoted by a black dot and labeled by indecies $(n,m)$ is associated to a plaquette. The Berry flux, $F_{nm}^{\left(j\right)}$, is calculated for each band on each plaquette. White dots depict vertices whose plaquettes are located in the adjacent Brillouin zone.

Figure 6

Calculated Chern numbers and the distribution of the Berry flux over the Brillouin zone in the absence of strain for the three occupied bands for topological and trivial phases.