Interaction of massless Dirac electrons with acoustic phonons in graphene at low temperatures

Interaction of massless electrons with the acoustic phonons is studied in two-dimensional (2D) graphene at low temperatures by calculating phonon drag thermopower Sᵍ and hot-electron energy-loss rate $F(T)$. Sᵍ and $F(T)$ are studied as a function of temperature T and electron concentration nₛ. For very low temperatures Sᵍ~T³ and F(T)~T⁴ in contrast to Sᵍ~T⁴ and F(T)~T⁵ of unscreened deformation-potential coupling in usual 2D systems. We find that Sᵍ is related to the phonon limited mobility μₚ by Sᵍ{{μ}}ₚ={v}ₛ{Λ}{T}^{{-}1}$ (${v}ₛ$ is the phonon velocity and ${Λ}$ is the phonon mean-free path) validating Herring's law for linear dispersion of electrons in graphene. In the low-temperature limit ${S}ᵍ$, $F(T){~}{n}ₛ^{{-}1/2}$. For comparison diffusion thermopower ${S}ᵈ$ is calculated and ${S}ᵈ{~}T$, ${n}ₛ^{{-}1/2}$. Our results are compared with those in the usual 2D systems.