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Theoretical predictions and recent experimental results suggest one canengineer spin Hall effect in graphene by enhancing the spin-orbit coupling inthe vicinity of an impurity. We use a Chebyshev expansion of the Kubo-Bastinformula to compute the spin conductivity tensor for a tight-binding model ofgraphene with randomly distributed impurities absorbed on top of carbon atoms.We model the impurity-induced spin-orbit coupling with a graphene-onlyHamiltonian that takes into account three differentcontributions~\cite{Gmitra2013} and show how the spin Hall and longitudinalconductivities depend on the strength of each spin-orbit coupling and theconcentration of impurities. Additionally, we calculate the real-spaceprojection of the density of states in the vicinity of the Dirac point forsingle and multiple impurities and correlate these results with theconductivity calculations.

Kubo-Bastin approach for the spin Hall conductivity of decorated graphene