An elastic-shell-based theory for calculating the thermal conductance of graphene ribbons of arbitrary width w is presented. The analysis of vibrational modes of a continuum thin plate leads to a general equation for ballistic conductance sigma. At low temperature, it yields a power law sigma approximately T(beta), where the exponent beta varies with the ribbon width w from beta = 1 for a narrow ribbon (sigma approximately T, as a four-channel quantum wire) to beta = (3)/(2) (sigma approximately wT(3/2)) in the limit of wider graphene sheets. The ballistic results can be augmented by the phenomenological value of a phonon mean free path to account for scattering and agree well with the reported experimental observations.
In this section, we present a theory for Dirac cone splitting in graphene van der Waals heterostructures, where the substrate may induce chiral symmetry breaking. Our model is based on a tight-binding representation of single-layer graphene, subjected to an on-site potential representing the proximity of the substrate atoms. The fact that this local potential may differ at each site, provides a mechanism for chiral symmetry breaking and the corresponding splitting of the Dirac cone in graphene. From an effective tight-binding model for the graphene-heterostructure, a simple stoichiometric rule is obtained to predict the conditions for Dirac cone splitting in the energy spectrum.
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