Klein tunneling and valley-polarized scattering in bilayer graphene with trigonal warping

We study the effect of trigonal warping on the Klein tunneling and scattering of electrons in bipolar junctions of bilayer graphene. We derive average velocities of electrons and obtain closed forms of transmission and reflection amplitudes. For normal incidence the amplitude and the average velocity in transmission zone are obtained to be zero, from which we show that the Klein effect of perfect reflection can appear in a thin potential barrier as well as in a potential step. The trigonal warping also causes variation in curvatures of isoenergy curves and breaks the valley degeneracy. This brings about valley-polarized scattering of electrons in the bipolar junctions, which can be used to produce a valley-polarized electron beam with bilayer graphene.