Interplay between curvature and in-plane magnetic field in bilayer graphene

For a two-dimensional electron gas (2DEG) in a uniform magnetic field, the effect of the in-plane component on the orbital motion of carriers is ignored because ``it can be gauged away.'' However, the effect of such a field on a massive quantum particle confined to a curved surface has been only recently explored [1]. We obtain the single-particle spectra for such a particle on a sphere, a cylinder, and a torus in the presence of a constant magnetic field. In addition to the geometric potential $V_G$ that arises due to the confinement on a curved surface, we find that in-plane field leads to energy shifts $\Delta E\propto V_G(R/l_B)^4$ where $R$ is the radius of curvature of the surface, and $l_B$ is the magnetic length for the in-plane field. With bilayer graphene as a model for massive quantum particle on a curved surface, we estimate the energy shift for a cylindrical geometry, and show that it is significant for typical experimental parameters. \\[4pt] [1] G. Ferrari and G. Cuoghi, Phys. Rev. Lett. {\bf 100}, 230403 (2008).