  We investigate the atomic-scale tunneling characteristics of bilayer graphene
on silicon carbide using the scanning tunneling microscopy. The high-resolution
tunneling spectroscopy reveals an unexpected negative differential resistance
(NDR) at the Dirac energy, which spatially varies within the single unit cell
of bilayer graphene. The origin of NDR is explained by two near-gap van Hove
singularities emerging in the electronic spectrum of bilayer graphene under a
transverse electric field, which are strongly localized on two sublattices in
different layers. Furthermore, defects near the tunneling contact are found to
strongly impact on NDR through the electron interference. Our result provides
an atomic-level understanding of quantum tunneling in bilayer graphene, and
constitutes a useful step towards graphene-based tunneling devices.
