Pressure dependence of the magic twist angle in graphene superlattices

The recently demonstrated unconventional superconductivity [Cao et al., Nature (London) 556, 43 (2018)] in twisted bilayer graphene (tBLG) opens the possibility for interesting applications of two-dimensional layers that involve correlated electron states. Here we explore the possibility of modifying electronic correlations by the application of uniaxial pressure on the weakly interacting layers, which results in increased interlayer coupling and a modification of the magic angle value and associated density of states. Our findings are based on first-principles calculations that accurately describe the height-dependent interlayer coupling through the combined use of density functional theory and maximally localized Wannier functions. We obtain the relationship between twist angle and external pressure for the magic angle flat bands of tBLG. This may provide a convenient method to tune electron correlations by controlling the length scale of the superlattice.

Figure 1

Isosurfaces of the localized Wannier functions in bilayer graphene, with the colors indicating different signs. (Left) Side view: Vertical compression of the bilayer mainly causes the orbitals to overlap more, thus increasing interlayer coupling while leaving in-plane couplings mostly unaffected. (Right) Top view: The triangular shape and nodes (indicated by the sign change) introduce angular dependence effects in the interlayer coupling, neglected in empirical tight-binding models for bilayer graphene based on $p_z$ orbitals.

Figure 2

Calculated vertical external pressure as a function of interlayer distance between graphene layers (black crosses) with the the fit given in the text (red line). Inset: Compression dependence of the primary scaling parameters ${\lambda }_n$ of the interlayer coupling formula normalized by their values at $\epsilon =0$. Values for $n=0$, 3, and 6 are shown with crosses, circles, and triangles, respectively. The quadratic fit for ${\lambda }_0$ is given by the dashed line.

Figure 3

(a) Band structures for twisted bilayer graphene under compression $\epsilon$ from the ab initio tight-binding model. The flat-band regime is achieved at 5% compression for a twist angle of $1.{47}^{\circ }$, and at 10% compression for a twist angle of $2.{00}^{\circ }$. (b) The two coupled Dirac cones, shifted in momentum space due to the twist, and with interlayer coupling strength ${\lambda }_0$, are shown schematically. (c) Critical values of the compression parameter $\epsilon$ as a function of twist angle calculated by an ab initio $k·p$ model. The bandwidth of the eight bands closest to the Fermi level at the $\mathrm{\Gamma }$ point $\mathrm{\Delta }{E}_{\mathrm{\Gamma }}$ is shown in color with white representing the small bandwidth of the flat bands. The dashed red line gives the expected value of the compression to cause flat bands (see text for details).

Figure 4

Band structures of uncompressed twisted bilayer graphene with and without relaxation of the atoms. The black lines are bands for the unrelaxed system and the red lines for the relaxed system. The single particle gaps in the relaxed system are highlighted in pink.