  We consider the polaron dynamics driven by Frohlich type, long wavelength
dominated electron-phonon interaction, for three different semi-metals: single
and 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 for
ordinary polar crystals. But the study of polaron formation in the context of
the above-mentioned 2D semi-metals having non-scalar effective Hamiltonians is
novel. When SL and BL graphene are grown on polar substrates, their electrons
can interact with the surface phonons of those polar substrates, as has been
discussed by Fratini et al. That gives rise to the possibility of polaron
formation in the context of SL and BL graphene, although they themselves are
non-polar. Semi-Dirac material, like SL and BL graphene, has been considered to
be grown on a polar substrate and the resulting polaron-dynamics has been
investigated. As was discovered by Pardo and Pickett, the interfaces of
(TiO_2)_5/(VO_2)_3 heterostructure, in which semi-Dirac dispersion was
observed, are non-polar. This justifies the treatment of semi-Dirac, for the
purpose of this paper, in the same footing as non-polar materials like SL and
BL graphene. The electron self energy, or polaron energy, calculated
analytically for BL graphene for small electron-momentum, is shown to vary
linearly with the electron momentum. Despite the similarity between BL graphene
and ordinary polar crystals in the parabolic nature of the electronic
band-structure in the absence of electron-phonon interaction, the linear
energy-momentum dispersion of BL graphene polarons stands in stark contrast to
the quadratic energy-momentum dispersion of the polarons produced in ordinary
polar crystals. In addition to the polaron energy, the decay rate(in the
absence of polaron formation), has been calculated for the above-mentioned
materials.
