Kubo-Bastin approach for the spin Hall conductivity of decorated   graphene
  Theoretical predictions and recent experimental results suggest one can engineer spin Hall effect in graphene by enhancing the spin-orbit coupling in the vicinity of an impurity. We use a Chebyshev expansion of the Kubo-Bastin formula to compute the spin conductivity tensor for a tight-binding model of graphene with randomly distributed impurities absorbed on top of carbon atoms. We model the impurity-induced spin-orbit coupling with a graphene-only Hamiltonian that takes into account three different contributions~\cite{Gmitra2013} and show how the spin Hall and longitudinal conductivities depend on the strength of each spin-orbit coupling and the concentration of impurities. Additionally, we calculate the real-space projection of the density of states in the vicinity of the Dirac point for single and multiple impurities and correlate these results with the conductivity calculations. 
Jose H. Garcia and Tatiana G. Rappoport
cond-mat.mes-hall