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Electron-phonon mediated heat flow in disordered graphene

We calculate the heat flux and electron-phonon thermal conductance in a disordered graphene sheet, going beyond a Fermi's Golden rule approach to fully account for the modification of the electron-phonon interaction by disorder. Using the Keldysh technique combined with standard impurity averaging methods in the regime $k_F l >> 1$ (where $k_F$ is the Fermi wavevector, l the mean free path), we consider both scalar potential (i.e. deformation potential) and vector potential couplings between electrons and phonons. We also consider the effects of electronic screening at the Thomas-Fermi level. We find that the temperature dependence of the heat flux and thermal conductance is sensitive to the presence of disorder and screening, and reflects the underlying chiral nature of electrons in graphene and the corresponding modification of their diffusive behaviour. In the case of weak screening, disorder enhances the low-temperature heat flux over the clean system (changing the associated power law from $T^4$ to $T^3$), and the deformation potential dominates. For strong screening, both the deformation potential and vector potential couplings make comparable contributions, and the low-temperature heat flux obeys a $T^5$ power law.