  Electron spin relaxation in graphene on a substrate is investigated from the
fully microscopic kinetic spin Bloch equation approach. All the relevant
scatterings, such as the electron-impurity, electron--acoustic-phonon,
electron--optical-phonon, electron--remote-interfacial-phonon, as well as
electron-electron Coulomb scatterings, are explicitly included. Our study
concentrates on clean intrinsic graphene, where the spin-orbit coupling from
the adatoms can be neglected. We discuss the effect of the electron-electron
Coulomb interaction on spin relaxation under various conditions. It is shown
that the electron-electron Coulomb scattering plays an important role in spin
relaxation at high temperature. We also find a significant increase of the spin
relaxation time for high spin polarization even at room temperature, which is
due to the Coulomb Hartree-Fock contribution-induced effective longitudinal
magnetic field. It is also discovered that the spin relaxation time increases
with the in-plane electric field due to the hot-electron effect, which is
different from the non-monotonic behavior in semiconductors. Moreover, we show
that the electron-electron Coulomb scattering in graphene is not strong enough
to establish the steady-state hot-electron distribution in the literature and
an alternative approximate one is proposed based on our computation.
