  Despite many similarities between electronics and optics, the hopping of the
electron on a discrete atomic lattice gives rise to energy band nonparabolicity
and anisotropy. The crucial influences of this effect on material properties
and its incorporation into the continuum model have received widespread
attention in the past half century. Here we predict the existence of a
different effect due to the hopping of the electron across an atomically sharp
interface. For a general lattice, its influence on transport could be equally
important as the energy band nonparabolicity/anisotropy, but cannot be
incorporated into the continuum model. On the honeycomb lattice of graphene, it
leads to the breakdown of the conventional Klein tunneling -- one of the exotic
phenomena of relativistic particles -- and the onset of tilted Klein tunneling.
This works identifies a unique feature of the discrete atomic lattice for
transport, which is relevant for ballistic electronic devices at high carrier
densities.
