Curvature-tuned electronic properties of bilayer graphene in an effective four-dimensional spacetime

We show that in $AB$ stacked bilayer graphene low-energy excitations around the semimetallic points are described by massless, four-dimensional Dirac fermions. There is an effective reconstruction of the four-dimensional spacetime, including in particular the dimension perpendicular to the sheet, that arises dynamically from the physical graphene sheet and the interactions experienced by the carriers. The effective spacetime is the Eisenhart-Duval lift of the dynamics experienced by Galilei invariant Lévy-Leblond spin-$\frac{1}{2}$ particles near the Dirac points. We find that changing the intrinsic curvature of the bilayer sheet induces a change in the energy level of the electronic bands, switching from a conducting regime for negative curvature to an insulating one when curvature is positive. In particular, curving graphene bilayers allow opening or closing the energy gap between conduction and valence bands, a key effect for electronic devices. Thus, using curvature as a tunable parameter opens the way for the beginning of curvatronics in bilayer graphene.

Figure 1

The 4D massless Dirac cone $p_x^2+p_y^2+2p_u{p}_v$ is cut by the plane $p_v=m{v}_F$. The result is the nonrelativistic parabola $E=\frac{1}{2m}\left(p_x^2+p_y^2\right)$, where $E=-v_F{p}_u$. In our notation $p_Z=\frac{1}{\sqrt{2}}\left(p_u+p_v\right),p_T=\frac{v_F}{\sqrt{2}}\left(p_u-p_v\right)$.

Figure 2

Structures of bilayered graphene for different values of the 2D curvature. (a) $R<0$ corresponds to hyperbolic geometry, (b) $R=0$ to the flat graphene bilayer, (c) $R>0$ to spherical geometry.

Figure 3

Energy band gap between conduction and valence bands $E_R$ as a function of the (constant) curvature radius $l_R$. The arrows indicate the validity range of our approach.