Trigonal Distortion of Valley Current in Bilayer Graphene

Bilayer Graphene (BLG) is extensively explored due to its remarkably electronic properties. Its band structure has two inequivalent but degenerate points K and K' at the corners of the Brillouin Zone, because their large separation there have been proposition to use a valley index, or pseudospin to generate valley dependent currents. In BLG these currents can be created by applying an step like bias, the topology of BLG in the two zones leads to the formation of a 1D chiral zero modes transverse to the applied bias. There are two such modes per spin per valley and the current have opposite direction in each valley. We explore how the trigonal warping of BLG affects these modes. Breaking the symmetry of the unit cell in BLG opens a gap, the shape of the Berry Curvature depends of the bias applied, for very low bias (few meV) the topological charge breaks in four well defined peaks, three having the same sign and magnitude and the fourth with opposite value, the overall charge remains unaltered, these fraction charges lead to a distortion of the chiral zero modes. We calculate the currents and show how the system evolves by tuning the effect of the trigonal warping.