Semiclassical Klein Tunneling and Valley Hall Effect in Graphene
  We study the dynamics of semiclassical electrons in (gapped) graphene in two complementary limits, i.e. in the Klein tunneling and valley Hall effect regimes, by scattering wave packets off armchair step potentials and by exposing wave packets to a uniform electric field, respectively. Our numerical wave packet simulation goes beyond semiclassical analytical approximations and standard Klein tunneling treatments and allows to study intra- and intervalley scattering processes. We find distinct Klein tunneling characteristics for low and tall steps, which include unusual Berry curvature induced side shifts of the scattered wave packet trajectories. In the presence of a uniform field, our simulations capture the semiclassical valley Hall effect which manifests in the form of laterally shifted Bloch oscillations. Such anomalous trajectory corrections can be relevant for Klein tunneling experiments and electron optics devices. We present detailed simulation results. 
Christoph. M. Puetter, Satoru Konabe, Yasuhiro Tokura, Yasuhiro   Hatsugai, Kenji Shiraishi
cond-mat.mes-hall