Topological Flat Bands in Graphene Super-Moiré Lattices

Moiré-pattern-based potential engineering has become an important way to explore exotic physics in a variety of two-dimensional condensed matter systems. While these potentials have induced correlated phenomena in almost all commonly studied 2D materials, monolayer graphene has remained an exception. We demonstrate theoretically that a single layer of graphene, when placed between two bulk boron nitride crystal substrates with the appropriate twist angles, can support a robust topological ultraflat band emerging as the second hole band. This is one of the simplest platforms to design and exploit topological flat bands.

Figure 1

Topological flat band in monolayer graphene with a super-moiré lattice potential. (a) Schematic of a double-aligned-moiré potential where the top and bottom layers have the same twist angle ($\theta =0.6°$). (b) The corresponding band structure does not have flat bands. (c) Schematic of a super-moiré lattice potential formed when monolayer graphene is encapsulated by slightly misaligned $h$-BN substrates (${\theta }_B=0°$ and ${\theta }_T=0.6°$). (d) The corresponding band structure has nonzero Chern numbers for the first two hole bands with the second hole band having a bandwidth $\lesssim 10\mathrm{meV}$.

Figure 2

Understanding the emergence of topological flat bands in monolayer graphene. (a) Schematic of the Brillouin zones and reciprocal lattice points of the super-moiré potential from the $k$ point of interest represented by the black dot at the center. The red, blue, and gray lines (dots) show the first-, second-, and super-moiré Brillouin zones (shells), respectively. (b) Bandwidth of the second hole band increases as a function of twist angle ${\theta }_T$ keeping ${\theta }_B=0°$ fixed. (c) Band structure of the $h\text{−}\mathrm{BN}/\mathrm{G}/h\text{−}\mathrm{BN}$ heterostructure including only the top moiré potential, i.e., coupling only to the blue dots in (a). (d) Band structure of the $h\text{−}\mathrm{BN}/\mathrm{G}/h\text{−}\mathrm{BN}$ heterostructure including only the bottom moiré potential (red dots). (e) Band structure of the $h\text{−}\mathrm{BN}/\mathrm{G}/h\text{−}\mathrm{BN}$ with the super-moiré potential (gray dots). Similar to our results in Fig. 1, the symmetry-based approach also results in an isolated and flat second hole band. Not shown: If we use only a single moiré potential with the parameters in Ref. [29], we reproduce their results, finding one mini-Dirac point in the valance band and three mini-Dirac points in the conduction band.

Figure 3

Effect of Hartree-Fock potentials on electronic spectrum of super-moiré $h\text{−}\mathrm{BN}/\mathrm{G}/h\text{−}\mathrm{BN}$ heterostructure. (a) Noninteracting band structure without Hartree and Fock corrections (dashed black line) and band structure for the self-consistent Hartree charge distribution correction due to filling the first hole band (solid red line). (b) Density of states without Hartree-Fock corrections (black line), with only Hartree corrections (red line), and with both Hartree and Fock corrections (blue line). The addition of Fock potential does not alter the energy spectrum of the system. (c),(d) Charge distribution of the $\mathrm{\Gamma }$ point wave function without and with the effect of Hartree potential, respectively.

Figure 4

Super-moiré topological flat bands in bilayer and trilayer graphene, with and without electric displacement field. (a) Low-energy bands of the $AB$ bilayer graphene exposed to super-moiré potentials without external displacement field, displaying various flat bands around charge neutrality. The first electron and hole bands are topological with positive $+1$ or negative $-1$ Chern numbers, respectively. (b) Low-energy bands of the $ABC$ trilayer graphene under super-moiré potential without an external displacement field, where the flat bands have Chern numbers greater than 1, which is different from the case of monolayer and bilayer. (c),(d) Energy bands of the $AB$ bilayer and $ABC$ trilayer graphene subjected to both super-moiré potential and external displacement field modeled by an electric potential energy difference of 40 meV. The indicated Chern numbers in (a) and (b) do not change in (c) and (d). The external displacement field adjusts the width of the low-energy bands, enhancing the diversity of possible experimental phase diagrams in the bilayer and trilayer graphene super-moiré systems.