Atomistic substrate relaxation effects in the band gaps of graphene on hexagonal boron nitride

Abstract

We assess the impact of atomistic substrate lattice relaxation schemes in the primary band gap at charge neutrality and the secondary valence band gap of graphene on hexagonal boron nitride (G/h-BN) as a function of twist angle. For zero twist angle, the primary gap decreases from $\sim 30$~meV in fully relaxed suspended G/h-BN bilayers, to $\sim 9$~meV when the remote h-BN substrate layer is kept rigid, and down to $\sim 3$~meV in completely rigid structures. In the presence of relaxations, the primary gap shows a maximum near $\sim 0.6^{\circ}$ coinciding with energetic stabilization due to alignment between the moiré pattern and the graphene lattice vectors, while the secondary valence band gap drops from $\sim 12$~meV down to zero beyond twist angles of $\sim 1^{\circ}$. A small but finite primary gap on the order of $\sim 1$~meV, with a mass sign favoring electronic occupation of carbon atop boron, persists across twist angles from $0^{\circ}$ to $30^{\circ}$ for all sliding configurations, and switches sign for twist angles between $30^{\circ}$ and $60^{\circ}$.

Figures (15)

• Figure 1: (Color online) Number of atoms in the commensurate supercells constructed using the four indices defined in Eq. \ref{['twistangle']}. A lattice tolerance of $\pm 0.01$ Å is allowed with respect to the equilibrium h-BN lattice constant $a_{h-BN}^0 = 2.504$ Å, increasing the density of accessible twist angles via smaller commensurate cells. The graphene lattice constant is held fixed at $a_G^0 = 2.4602$ Å. The bottom panels show the elastic energy correction per atom, calculated using Eq. (\ref{['equlibriumDistance']}), when the graphene or h-BN lattice constants are varied from their equilibrium values using the same force fields employed in the energy minimization. Total energies are shifted by $-7.394$ eV/atom for graphene and $-6.690$ eV/atom for h-BN to set the energy minimum to zero.
• Figure 2: (color online) Top view of the aligned G/h-BN moiré pattern generated with indices $i = 55, j = 0, i^\prime = 54, j^\prime = 0$. In the AA stacking region, the carbon atoms are on top of boron and nitrogen atoms, respectively. In the AB region, the carbon atom is on top of the nitrogen atom and the center of the graphene hexagon is on top of the boron atom. In the BA region, the role of B and N is reversed with respect to AB. The moiré length is $L_M = 135.311$ Å for zero angle and decreases for increasing twist angles following Eq. (\ref{['eq:MoireLength2']}).
• Figure 3: (color online) Matrix elements $H_{ii}({\bm d}({\bm r}_{<kl>}))$ (top panels) and $H_{ij}({\bm d}({\bm r}_{<kl>}))$ (bottom panels) from Eq. (\ref{['eq:Onsite']}) and Eq. (\ref{['eq:HAB']}) respectively. $C_1$ and $C_2$ are the two sublattices from the graphene layer and $B$ and $N$ are the atoms from the h-BN layer. We illustrate both the real and imaginary parts of the Hamiltonian. The site energies are plotted relative to their spatial averages given in Table \ref{['Table:OnsiteABC']} shifted to zero.
• Figure 4: (top panels) ${\bm d}$-dependent tunneling map for an interlayer distance of $3.35$ Å, focusing here on the real part of the $C_1-B$ interaction. The left panel is obtained from DFT, the middle is obtained by fitting the $p$ and $q$ parameters from the TB model in Eq. (\ref{['STCtunneling']}) while the right panel is the difference between the DFT and TB panels. The bottom panel illustrates this fitting for different interactions at different interlayer distances, setting $\exp{\left[{(r_z -v)}/{w}\right]}$ in Eq (\ref{['STCtunneling']}).
• Figure 5: Electronic band structure for the rigid configuration (dashed lines) and the suspended configuration (solid lines). The increase in primary gap and decrease in secondary gap due to relaxation effects is clearly visible, where the primary gap goes from $3.51$ meV to $20.3$ meV and the secondary gap from $16.41$ meV to $14.86$ meV. The inset shows the Brillouin Zone for G/h-BN with a red hexagon indicating the $\Gamma-M-K-\Gamma$ along which the band structures are calculated. Here we set to zero the Fermi level inside the gap by shifting the bands by $-0.73$ eV.