First, I will discuss scanning tunneling
microscopy measurements on epitaxially grown graphene multilayers that
have revealed a splitting of the zeroth Landau level in certain regions
of space. One finds that a phenomenological theory that models the
multilayer system as a single graphene layer with a space-dependent
mass term in the Dirac equation fits the experimental data well. I will
show how an effective theory with such a mass term can be obtained
theoretically from a microscopically motivated tight-binding model.
Implications of this theory for transport in zero magnetic field will
be discussed.
Second, I will talk about the electron
dynamics in graphene bilayers with a commensurate interlayer rotation.
It has been shown by G. Mele that commensuration qualitatively modifies
the low-energy spectrum of graphene bilayers. That modification takes
two qualitatively different forms, depending on the sublattice exchange
parity of the bilayer. In each case the interlayer coupling causes a
splitting between the Dirac cones of the two graphene layers. The
cyclotron motion in the split cones undergoes interesting beating
phenomena, manifest in the energy spectrum as the “Dirac comb:” an
amplitude modulation of the Landau level spectrum that is discernible
at energies much larger than the typically small interlayer coherence
scale. This modulation is qualitatively different for the two
sublattice exchange parities of twisted graphene bilayers and its
period encodes the interlayer coherence scale. The Dirac comb thus
provides a powerful experimental probe of the magnitude of the
interlayer coherence splitting and the sublattice exchange parity of a
twisted graphene bilayer.