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We consider the polaron dynamics driven by Frohlich type, long wavelengthdominated electron-phonon interaction, for three different semi-metals: singleand bilayer graphene, and semi-Dirac, all grown on polar substrates such as,SiC or SiO_2. The problem of polaron has been studied by Feynman and others forordinary polar crystals. But the study of polaron formation in the context ofthe above-mentioned 2D semi-metals having non-scalar effective Hamiltonians isnovel. When SL and BL graphene are grown on polar substrates, their electronscan interact with the surface phonons of those polar substrates, as has beendiscussed by Fratini et al. That gives rise to the possibility of polaronformation in the context of SL and BL graphene, although they themselves arenon-polar. Semi-Dirac material, like SL and BL graphene, has been considered tobe grown on a polar substrate and the resulting polaron-dynamics has beeninvestigated. As was discovered by Pardo and Pickett, the interfaces of(TiO_2)_5/(VO_2)_3 heterostructure, in which semi-Dirac dispersion wasobserved, are non-polar. This justifies the treatment of semi-Dirac, for thepurpose of this paper, in the same footing as non-polar materials like SL andBL graphene. The electron self energy, or polaron energy, calculatedanalytically for BL graphene for small electron-momentum, is shown to varylinearly with the electron momentum. Despite the similarity between BL grapheneand ordinary polar crystals in the parabolic nature of the electronicband-structure in the absence of electron-phonon interaction, the linearenergy-momentum dispersion of BL graphene polarons stands in stark contrast tothe quadratic energy-momentum dispersion of the polarons produced in ordinarypolar crystals. In addition to the polaron energy, the decay rate(in theabsence of polaron formation), has been calculated for the above-mentionedmaterials.

Two Dimensional Polarons with Linearly Dispersing Self Energy and Other  Novel Features in the Study of Bi/Single Layer Graphene, and Semi-Dirac  Semi-metals on Polar Substrates