Emergence of flat bands in the quasicrystal limit of boron nitride twisted bilayers

We investigate the electronic structure and the optical absorption onset of close-to-${30}^{\circ }$ twisted hexagonal boron nitride bilayers. Our study is carried out with a purposely developed tight-binding model validated against density functional theory simulations. We demonstrate that approaching ${30}^{\circ }$ (quasicrystal limit), all bilayers sharing the same moiré supercell develop identical band structures, irrespective of their stacking sequence. This band structure features a bundle of flat bands lying slightly above the bottom conduction state which is responsible for an intense peak at the onset of the absorption spectrum. These results suggest the presence of strong, stable and stacking-independent excitons in boron nitride ${30}^{\circ }$-twisted bilayers. By carefully analyzing the electronic structure and its spatial distribution, we elucidate the origin of these states as moiré-induced $K$-valley scattering due to interlayer $\mathrm{B}\text{−}\mathrm{B}$ coupling. We take advantage of the physical transparency of the tight-binding parameters to derive a simple triangular model based on the B sublattice that accurately describes the emergence of the bundle. As our conclusions very general, we predict that a similar bundle should emerge in other close-to-${30}^{\circ }$ bilayers, such as transition metal dichalcogenides, shedding light on the unique potential of two-dimensional materials.

Figure 1

(a)–(c) Conduction bands of BN(1,3), BB(1,3), and BN(3,8) in TB (black solid) and DFT (red dashed). The TV of all structures have been aligned to 0.0 eV. In (c), tick bars on the canvas highlight notable energy intervals (see text). (d) Onset of IP absorption spectra of the same systems. The BN(3,8) spectra are computed only in the $\mathrm{\Gamma }$ point. All spectra have been broadened with a Lorentzian with variance 0.1 eV.

Figure 2

(a)–(c) Conduction bands of the BB(5,13), BB(4,11), and BB(11,30), respectively. (d)–(f) Same as (a)–(c) in the BN stacking. The TV of all structures have been aligned to 0.0 eV. (g) (h) IP absorption spectra of the BB(5,13) and BB(4,11) (red solid curves) and BN(5,13) and BN(4,11) (black dashed) obtained including all empty and occupied states (solid). Shaded and patterned areas are obtained by restricting the empty states between 4.34 and 4.48 eV. (i), (j) Blue circles: Radius proportional to the LDOS in the shallow conduction interval. Green circles: Radius proportional to ${\mathcal{C}}_{j_B}\left(\mathbf{d}\right)$ wherever it is higher than $\kappa$. See main text for details. Data are shown only for the upper layers of the (i) BB(3,8) and (j) BN(3,8) systems.

Figure 3

(a) TB conduction band of the monolayer in the (3,8) supercell (dashed red) and of the BB(3,8) with ${\gamma }^{BB}=0$ eV. (b), (c) The same bilayer with ${\gamma }^{BB}$ equal to 50% and 100% of the correct value. The top valence of all structures have been aligned to 0.0 eV. (d) Ball and stick model of the actual twisted bilayer. Dashed magenta line: the ${\gamma }^{BB}$ interlayer coupling. Red shaded area: the isolated monolayer. (e) Ball and stick model of the triangular lattice twisted bilayer made only of B sites. Dashed magenta line: the $\gamma$ interlayer coupling. (f), (g) Bands of the triangular model with $\gamma =0$ and 1.715 eV, respectively.