Electronic structure of spontaneously strained graphene on hexagonal boron nitride

Hexagonal boron nitride substrates have been shown to dramatically improve the electric properties of graphene. Recently, it has been observed that when the two honeycomb crystals are close to perfect alignment, strong lattice distortions develop in graphene due to the moiré adhesion landscape. Simultaneously, a gap opens at the Dirac point. Here, we derive a simple low-energy electronic model for graphene aligned with the substrate, taking into account spontaneous strains at equilibrium and pseudogauge fields. We carry out a detailed characterization of the modified band structure, gap, local and global density of states, and band topology in terms of physical parameters. We show that the overall electronic structure is strongly modified by the spontaneous strains.

Figure 1

Comparison of the electronic structure for a model with different combinations of physical ingredients: unstrained graphene on $h\mathrm{BN}$ (a), strained graphene on $h\mathrm{BN}$ without pseudogauge potential (b), strained graphene electronically decoupled from $h\mathrm{BN}$ (c), and the complete model of a strained graphene on $h\mathrm{BN}$ including pseudogauge field (d). Represented are the low-energy band structure, the corresponding total DOS, and the LDOS in real space for four energies shown as dashed lines in the band structure and DOS. In the LDOS, the coloring corresponds to blue for zero and red for the maximum. We have included the value for the (nonperturbative) gap $\Delta$ at the primary Dirac point in each case.

Figure 2

Berry curvature and valley Chern numbers for the lowest subbands of models in Figs. 1 (top) and 1 (bottom). The four panels on the right depict, for each case, the Berry curvature and valley Chern number in the Brillouin zone (black hexagon, with center at the $\Gamma$ point) corresponding to the four subbands closest to zero (blue is positive, red is negative, gray is zero). The Berry curvature is also color coded in the band structure on the left. The valley Chern number ${\mathcal{C}}_s^{K_{\mathrm{gr}}}$ for each subband is also shown on the right side of the band structure.

Figure 3

Momenta contained in the low-energy Hamiltonian $\mathit{H}\left(\vec{r}\right)$ of Eq. (A18). According to their modulus, they are classified within the first star (red), second (green), or third (blue).