Zero-energy modes and valley asymmetry in the Hofstadter spectrum of bilayer graphene van der Waals heterostructures with hBN

We investigate the magnetic minibands of a heterostructure consisting of bilayer graphene (BLG) and hexagonal boron nitride (hBN) by numerically diagonalizing a two-band Hamiltonian that describes electrons in BLG in the presence of a moiré potential. Due to inversion-symmetry breaking characteristic for the moiré potential, the valley symmetry of the spectrum is broken, but despite this, the zero-energy Landau level in BLG survives, albeit with reduced degeneracy. In addition, we derive effective models for the low-energy features in the magnetic minibands and demonstrate the appearance of secondary Dirac points in the valence band, which we confirm by numerical simulations. Then, we analyze how single-particle gaps in the fractal energy spectrum produce a sequence of incompressible states observable under a variation of carrier density and magnetic field.

Figure 1

The $B=0$ minibands for the labeled superlattice perturbation parameters and $\theta =0$ (left panels), and the bandwidths of the magnetic minibands, shown separately for the $K^{\prime }$ and $K$ valleys, for the same superlattice parameters (central panels). A red line is used to indicate the unperturbed zero energy Landau level in the $K^{\prime }$ valley.

Figure 2

The full $(k_x,k_y)$ dispersion of magnetic minibands obtained by either diagonalizing effective Hamiltonians (8)-(9) (yellow), or the full numerical diagonalization of Hamiltonian (2) (turquoise).

Figure 3

The Landau level spectra of the valence band secondary Dirac point (sDP), the perturbation $u_{0,1,3}=\{0.032,-0.063,-0.055\}$, obtained from either Hamiltonian (12) for the ${\kappa }^{\prime }$ and $\kappa$ corners (red and blue dots) or Hamiltonian (2) (black dots). The gray solid lines are the Landau levels of the effective Hamiltonian, Eq. (13), using fitted parameters $E_0=-0.22vb,{\mathrm{\Delta }}_{\kappa }=-0.0057vb,\tilde{v}=0.38v,\overline{M}=-0.97ev/b$, and $\tilde{M}=-1.45ev/b$.

Figure 4

The evolution of energy gaps in the magnetic miniband spectrum as a function of density and magnetic flux, calculated for the same parameters as Fig. 1. Larger gaps are shown using a darker color, white represents a vanishing gap, and gray represents the particular gapless states associated with the zero-energy LL.