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.