Chiral tunneling, tunneling times, and Hartman effect in bilayer graphene

Abstract

We investigate the chiral tunneling in a bilayer graphene $n\text{\ensuremath{-}}p\text{\ensuremath{-}}n$ junction in time aspect. The phase and dwell times are evaluated for various situations, including the effects of trigonal warping and band gap due to an external gate field. In the absence of band gap and for normal incidence, the chirality induces Klein effects; when the trigonal warping is excluded, the tunneling times are the same as the ordinary barrier tunneling, but including it leads to a perfect reflection and a new type of phase time is found. In the presence of band gap and/or for an oblique incidence, the Klein effects disappear and the tunneling times have peaks corresponding to resonant transmission maxima due to the Fabry-P\`erot-type interference of the oscillating waves allowed inside the barrier. The trigonal warping also leads to valley-polarized transmission for an oblique incidence in the absence or presence of band gap. As a result, the tunneling times and scattering angles of particles from one valley are different from the other valley. We observe that the Hartman effect exists only when the chirality selects pure evanescent waves as transmission channels.