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When a crystal is subjected to a periodic potential, under certain circumstances it can adjust itself to follow the periodicity of the potential, resulting in a commensurate state. Of particular interest are topological defects between the two commensurate phases, such as solitons and domain walls. Here we report a commensurate-incommensurate transition for graphene on top of hexagonal boron nitride (hBN). Depending on the rotation angle between the lattices of the two crystals, graphene can either stretch to adapt to a slightly different hBN periodicity (for small angles, resulting in a commensurate state) or exhibit little adjustment (the incommensurate state). In the commensurate state, areas with matching lattice constants are separated by domain walls that accumulate the generated strain. Such soliton-like objects are not only of significant fundamental interest, but their presence could also explain recent experiments where electronic and optical properties of graphene-hBN heterostructures were observed to be considerably altered.

Laterally localized electronic states are identified on a single layer of graphene on ruthenium by low temperature scanning tunneling spectroscopy (STS). The individual states are separated by 3 nm and comprise regions of about 90 carbon atoms. This constitutes a highly regular quantum dot-array with molecular precision. It is evidenced by quantum well resonances (QWRs) with energies that relate to the corrugation of the graphene layer. The dI/dV conductance spectra are modeled by a layer height dependent potential-well with a delta-function potential that describes the barrier for electron penetration into graphene. The resulting QWRs are strongest and lowest in energy on the isolated 'hill' regions with a diameter of 2 nm, where the graphene is decoupled from the surface.