Electronic band structure and pinning of Fermi energy to Van Hove singularities in twisted bilayer graphene: A self-consistent approach

The emergence of flat bands in twisted bilayer graphene leads to an enhancement of interaction effects, and thus to insulating and superconducting phases at low temperatures, even though the exact mechanism is still widely debated. The position and splitting of the flat bands is also very sensitive to the residual interactions. Moreover, the low-energy bands of twisted graphene bilayers show a rich structure of singularities in the density of states, van Hove singularities, which can enhance further the role of interactions. We study the effect of the long-range interactions on the band structure and the van Hove singularities of the low-energy bands of twisted graphene bilayers. Reasonable values of the long-range electrostatic interaction lead to a band dispersion with a significant dependence on the filling. The change of the shape and position of the bands with electronic filling implies that the van Hove singularities remain close to the Fermi energy for a broad range of fillings. This result can be described as an effective pinning of the Fermi energy at the singularity. The sensitivity of the band structure to screening by the environment may open new ways of manipulating the system.

Figure 1

(a) Folding of the BZs of the twisted monolayers graphene. The BZ of layer 1 (red hexagon) is rotated by $-\theta /2$, while that of the layer 2 (blue hexagon) by $\theta /2$. The small black hexagons represent the mBZs forming the reciprocal moiré lattice. Inset: $K_{1,2}$ are the Dirac points of the twisted monolayers, which identify the corners of the mBZ. (b) mBZ. ${\mathit{G}}_{1,2}$ are the two reciprocal lattice vectors. The blue line shows the high symmetry circuit in the mBZ used to compute the bands shown in Fig. 3.

Figure 2

Values of $\delta {\rho }_G$ as a function of the filling, $n$, obtained for $\theta =1.{05}^{\circ }$.

Figure 3

Self-consistent bands of TBG in the presence of the Hartree potential, obtained for the twist angles $\theta =1.{12}^{\circ },1.{08}^{\circ },1.{05}^{\circ },0.{99}^{\circ }$ and fillings $n=-1$ (cyan), $n=0$ (red), $n=1$ (blue), and $n=2$ (green). The dotted lines show the bands corresponding to the opposite valleys, while the horizontal dashed lines represent the Fermi energies.

Figure 4

DOS for each of the results shown Fig. 3, color coded as in that figure. The vertical dashed lines represent the Fermi energy for each case.

Figure 5

Fermi surfaces (orange) and positions of the van Hove singularities (yellow) in the mBZ for twist angles $\theta =1.{08}^{\circ }$ and $1.{05}^{\circ }$ and filling fraction $n=1,2$.

Figure 6

Value of the DOS at the Fermi level as a function of the filling of the conduction band for the angles $\theta =1.{08}^{\circ }$ (a) and $\theta =1.{05}^{\circ }$ (b).

Figure 7

Band structure [panels (a1)–(a3)] and DOS [panels (b1)–(b3)] obtained at the magic angle $\theta =1.{08}^{\circ }$ and filling $n=-1,0,1$, as indicated in the legends. The gray area in the plots of the DOS represents the occupied fraction of the spectral weight.

Figure 8

Density of states obtained for a twist angle $\theta =1.{05}^{\circ }$, including the full Hartree-Fock interaction (solid lines) and only the Hartree part (dotted lines). The origin of the horizontal axis has been set at the Fermi level.

Figure 9

Bands and DOS for $V_0=15\mathrm{meV}$ (a) and $V_0=28\mathrm{meV}$ (b) at the angle $\theta =1.{08}^{\circ }$ and filling $n=-1$ (cyan), $n=0$ (red), $n=1$ (blue).

Figure 10

As in Fig. 9 but for $\theta =1.{05}^{\circ }$.

Figure 11

(a) DOS obtained for a twist angle $\theta =1.{05}^{\circ }$, including uniaxial strain $\varepsilon =0.6\%$ for a dielectric constant $\epsilon =4$. The zeros for each curve (denoted by the dashed lines) have been moved for each filling as indicated on the right-hand side. (b) Distorted mBZ in the presence of strain.

Figure 12

Value of the order parameter, $\delta {\rho }_G^{\left(m\right)}$, versus the number of steps in the iterative diagonalization, $m$, for the angle $\theta =1.{05}^{\circ }$ and filling $n=-1$ (cyan), $n=1$ (blue), and $n=2$ (green).

Figure 13

Bands (left) and density of states (right) of the model for twisted bilayer graphene with electron-hole symmetry and infinitely narrow bands at the neutrality point. The fillings considered are $n=-1,0,1,2$, as in Fig. 3. The Fermi velocity, $AB$ interlayer hopping, and relation between the Hartree potential and the charge density used are also as in Fig. 3. The twist angle is $\theta =1.{01}^{\circ }$, which is the first magic angle of the model.

Figure 14

Optical absorption, in arbitrary units, for the bands plotted in Fig. 3. The fillings are $n=-1$ (cyan), $n=0$ (red), $n=1$ (blue), and $n=2$ (green).