Phonon-limited transport coefficients in extrinsic graphene

The effect of electron-phonon scattering processes over the thermoelectric properties of extrinsic graphene was studied. Electron-phonon interaction is formulated in the second quantization language, for chiral Dirac spinor fields and phonon Bose fields, within the deformation potential approximation. Electrical and thermal resistivity, as well as the thermopower, were calculated within the Bloch theory approximations. Analytical expressions for the different transport coefficients were obtained from a variational solution of the Boltzmann transport equation. The phonon-limited electrical resistivity $\rho_{e-ph}$ shows a linear in temperature dependence at high temperatures, and follows a $\rho_{e-ph}\sim T^{4}$ at low temperatures, in agreement with experiments. The phonon-limited thermal resistivity at low temperatures exhibits a $\sim T$ dependence and achieves a nearly constant value at high temperatures. The predicted Seebeck coefficient at very low temperature is $Q(T ) \sim \pi^{2} k_{B} T /(3e E_{F} )$, which shows a $n^{-1/2}$ dependence with the carrier density, in agreement with experiments.\\[4pt] [1] E. Mu\~noz, Journal of Physics: Condensed Matter {\bf{24}} (2012) 195302.\\[0pt] [2] E. Mu\~noz, J. Lu and B. I. Yakobson, Nano Letters {\bf{10}} (2010) 1652.