Charge smoothening and band flattening due to Hartree corrections in twisted bilayer graphene

Doping twisted bilayer graphene away from charge neutrality leads to an enormous buildup of charge inhomogeneities within each moiré unit cell. Here, we show, using unbiased real-space self-consistent Hartree calculations on a relaxed lattice, that Coulomb interactions smoothen this charge imbalance by changing the occupation of earlier identified “ring” orbitals in the AB/BA region and “center” orbitals at the AA region. For hole doping, this implies an increase of the energy of the states at the $\mathrm{\Gamma }$ point, leading to a further flattening of the flat bands and a pinning of the Van Hove singularity at the Fermi level. The charge smoothening will affect the subtle competition between different possible correlated phases.

Figure 1

Left: A schematic representation of the moiré unit cell in twisted bilayer graphene, with indicated AA, AB, and BA regions (a larger twist angle shown). The unit cell itself is shown with a blue hexagon. The red dashed square indicates the region enlarged in the right figure. Right: The relative electron charge density $\delta n\left(\mathbf{r}\right)$ in the AB region of the unit cell at charge neutrality $\nu =0$ in layer 1 before Hartree corrections. Atoms that lie on top of atoms in the other layer have a charge deficiency (blue circles) and the remaining atoms have a charge excess (red circles). The size of the circles indicates the magnitude of $\delta n$, with a maximum of $\left|\delta n\right|\sim 0.0025$ at the AB point, here shown in the left bottom corner.

Figure 2

The relative electron charge density $\delta n\left(\mathbf{r}\right)$ as a function of distance from the AA center of the unit cell, for five different fillings ranging from charge neutrality ($\nu =0$) to the band insulator ($\nu =-4$). The top row shows the charge density without Hartree corrections, and the bottom row includes the self-consistent Hartree corrections. The blue dots represent all the atoms in the unit cell. The red line is a moving average, which shows the charge inhomogeneity between the AA center and the AB/BA regions of the unit cell at $\nu <0$. The Hartree correction significantly smoothens the initial charge inhomogeneity.

Figure 3

The electric potential ${\varphi }_i$ before (left) and after (right) the Hartree corrections as a function of the distance to the AA center, for various fillings. Note the approximate factor of 10 reduction in electric potential due to the Hartree corrections.

Figure 4

The local density of states measured at the AA region of the unit cell, in arbitrary units, for fillings $\nu =0$ through $\nu =-4$. At charge neutrality ($\nu =0$) the two Van Hove singularities are clearly separated. For partial fillings of the flat band, the Van Hove singularity remains close to the Fermi level. A similar effect that has been observed in Ref. [49].

Figure 5

The band structure after Hartree corrections are taken into account, for fillings from $\nu =0$ to $\nu =-4$. The thin gray lines represent the band structure prior to the Hartree corrections. The thick dashed line is the Fermi level at each given filling. Upon doping away from charge neutrality, the states at $\mathrm{\Gamma }$ are clearly pushed upwards, which is a feature of the charge smoothening. For the flat bands, this implies a reduction of the bandwidth.