  High mobility single and few-layer graphene sheets are in many ways
attractive as nanoelectronic circuit hosts but lack energy gaps, which are
essential to the operation of field-effect transistors. One of the methods used
to create gaps in the spectrum of graphene systems is to form long period moire
patterns by aligning the graphene and hexagonal boron nitride (h-BN) substrate
lattices. Here, we use planar tunneling devices with thin h-BN barriers to
obtain direct and accurate tunneling spectroscopy measurements of the energy
gaps in single- and bi-layer graphene-h-BN superlattice structures at charge
neutrality (first Dirac point) and at integer moire band occupancies (second
Dirac point, SDP) as a function of external electric and magnetic fields and
the interface twist angle. In single-layer graphene we find, in agreement with
previous work, that gaps are formed at neutrality and at the hole-doped SDP,
but not at the electron-doped SDP. Both primary and secondary gaps can be
determined accurately by extrapolating Landau fan patterns to zero magnetic
field and are as large as $\simeq$ 17 meV for devices in near perfect
alignment. For bilayer graphene, we find that gaps occur only at charge
neutrality where they can be modified by an external electric field. Tunneling
signatures of in-gap states around neutrality suggest the development of edge
modes related to topologically non-trivial valley projected bands due to the
combination of an external electric field and moire superlattice patterns.
