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Electron spin diffusion and transport in graphene

We investigate the spin diffusion and transport in a graphene monolayer on SiO$_2$ substrate by means of the microscopic kinetic spin Bloch equation approach. The substrate causes a strong Rashba spin-orbit coupling field $\sim 0.15$ meV, which might be accounted for by the impurities initially present in the substrate or even the substrate-induced structure distortion. By surface chemical doping with Au atoms, this Rashba spin-orbit coupling is further strengthened as the adatoms can distort the graphene lattice from $sp^2$ to $sp^3$ bonding structure. By fitting the Au doping dependence of spin relaxation from Pi {\sl et al.} [Phys. Rev. Lett. {\bf 104}, 187201 (2010)], the Rashba spin-orbit coupling coefficient is found to increase approximately linearly from 0.15 to 0.23 meV with the increase of Au density. With this strong spin-orbit coupling, the spin diffusion or transport length is comparable with the experimental values. In the strong scattering limit (dominated by the electron-impurity scattering in our study), the spin diffusion is uniquely determined by the Rashba spin-orbit coupling strength and insensitive to the temperature, electron density as well as scattering. With the presence of an electric field along the spin injection direction, the spin transport length can be modulated by either the electric field or the electron density. (The remaining is omitted due to the limit of space)