Wavefunction textures in twisted bilayer graphene from first principles

Abstract

Motivated by recent experiments probing the wavefunctions of magic-angle twisted bilayer graphene (tBLG), we perform large-scale first-principles calculations of tBLG with full atomic relaxation across a wide range of twist angles down to $0.99^\circ$. Focusing on the magic angle, we compute wavefunctions of the low energy bands, resolving atomic-scale details and moiré-scale patterns that form triangular, honeycomb, and Kagome lattices. By tuning the interlayer interactions, we illustrate the formation of the flat bands from isolated monolayers and the emergence of the band inversion and fragile topology at a sufficiently large interaction strength. We identify strong indicators of a new phase transition with increasing interlayer interaction strength, achievable with external pressure or a decrease in the twist angle. When this transition occurs, the upper and lower flat bands exchange their wavefunction character and symmetry eigenvalues, which may be correlated with the appearance of superconductivity with electron doping below the magic angle. Our study demonstrates the feasibility of using first-principles wavefunctions to help interpret experimental signatures of topological and correlated phases in tBLG.

Figures (4)

• Figure 1: Wavefunction of tBLG in the lower flat band at $\Gamma$ for a twist angle of $\theta=5.09^{\circ}$. (a) Modulus squared of the atomic-scale wavefunction $\vert\Psi\vert^2$, obtained from first-principles calculations. (b) Moiré-scale probability density $\vert \widetilde{\Psi} \vert^2 = \vert\Psi\vert^2 * \mathcal{N}$, obtained by a convolution with a Gaussian $\mathcal{N}$ with a standard deviation of $2.5$ Å, sketched in white above the $1$ nm scale bar in (a). The atomic positions are denoted by white and black dots in (a) and (b), respectively. The different stacking regions, AA, AB, BA and domain walls (DW), are labeled in (b).
• Figure 2: Emergence of the band inversion in magic-angle tBLG ($\theta = 1.08^{\circ}$). Each column shows a schematic atomic structure, the band structure, and wavefunction probability densities of the flat bands and dispersive bands for select values of the mean bilayer separation $z$, which effectively dictates the interlayer interaction strength. The band inversion is demonstrated by the energy of the AA$_z$ state and its precursors at larger $z$ (indicated by blue points in the band structure and blue borders around the probability densities) relative to the energy of the AA state(s) (indicated in red) in each column. DW states are indicated in green.
• Figure 3: Electronic band structure and wavefunctions at $\Gamma_{0\pm}$ of magic-angle tBLG as a function of $\Delta z = z-z_f$. The probability density of the projection of the wavefunction onto layer 1 and sublattice $A$, $\Psi^{A_1}_{\Gamma_{0\pm}}$, is shown. Cyan and purple borders (and the corresponding points on the band structures) denote wavefunction projections with different orientations.
• Figure 4: Transformation of the sublattice-projected microscopic wavefunctions of magic-angle tBLG under the $\mathcal{M}_y$ symmetry (shown in the center). (a) Layer and sublattice projections of the flat band microscopic wavefunctions $\Psi^{A_1}_{\Gamma_{0\pm}}$ and $\Psi^{B_2}_{\Gamma_{0\pm}}$ for $\Delta z = -0.075~\mathrm{\AA}$ with $\mathcal{M}_y$ applied to $\Psi^{A_1}_{\Gamma_{0\pm}}$. The inverted colors of the microscopic wavefunctions show that the upper flat band has a $\mathcal{M}_y$ eigenvalue of $\lambda_{\Gamma_{0+}}\! = -1$. (b) The same as (a), but for $\Delta z = -0.09$ Å. In this case, $\lambda_{\Gamma_{0-}}\! = -1$.