Intrinsic Damping Phenomena from Quantum to Classical Magnets: An ab-initio Study of Gilbert Damping in Pt/Co Bilayer.*

A fully quantum mechanical description of the precessional damping is presented in the framework of the Keldysh Green function approach. We demonstrate that $\alpha_{GD}$ in the quantum case does not diverge in the ballistic regime due to the finite size of the total spin, $S$. In the limit of $S\rightarrow\infty$ we show that the formalism recovers the torque correlation expression for $\alpha_{GD}$ which we decompose into spin-pumping and spin-orbital torque correlation contributions. The formalism is generalized to take into account a self consistently determined dephasing mechanism which preserves the conservation laws and allows the investigation of the effect of disorder. We employed the formailsm to calculate the intrincic Gilbert damping constant of Pt/Co bilayer system. The dependence of $\alpha_{GD}$ on Pt thickness and disorder strength is calculated and the spin diffusion length of Pt and spin mixing conductance of the bilayer are determined and compared with experiments.