Electronic properties of curved few-layers graphene: a geometrical approach

We show the presence of non-relativistic L\'evy-Leblond fermions in flat three- and four-layers graphene with AB stacking, extending the results obtained in [Curvatronics2017] for bilayer graphene. When the layer is curved we obtain a set of equations for Galilean fermions that are a variation of those of L\'evy-Leblond with a well defined combination of pseudospin, and that admit L\'evy-Leblond spinors as solutions in an approriate limit. The local energy of such Galilean fermions is sensitive to the intrinsic curvature of the surface. We discuss the relationship between two-dimensional pseudospin, labelling layer degrees of freedom, and the different energy bands. For L\'evy-Leblond fermions an interpretation is given in terms of massless fermions in an effective 4D spacetime, and in this case the pseudospin is related to four dimensional chirality. A non-zero energy band gap between conduction and valence electronic bands is obtained for surfaces with positive curvature.