Calculating the spin memory loss at Cu$|$metal interfaces from first principles

The role played by interfaces in metallic multilayers is not only to change the momenta of incident electrons; their symmetry lowering also results in an enhancement of the effects of spin-orbit coupling, in particular the flipping of the spins of conduction electrons. This leads to a significant reduction of a spin current through a metallic interface that is quantitatively characterized by a dimensionless parameter $\delta$ called the spin memory loss (SML) parameter, the interface counterpart of the spin-flip diffusion length for bulk metals. In this paper we use first-principles scattering calculations that include temperature-induced lattice and spin disorder to systematically study three parameters that govern spin transport through metallic interfaces of Cu with Pt, Pd, Py (permalloy) and Co: the interface resistance, spin polarization and the SML. The value of $\delta$ for a Cu$|$Pt interface is found to be comparable to what we recently reported for a Au$|$Pt interface [Gupta {\it et al.}, Phys. Rev. Lett. 124, 087702 (2020)]. For Cu$|$Py and Cu$|$Co interfaces, $\delta$ decreases monotonically with increasing temperature to become negligibly small at room temperature. The calculated results are in good agreement with currently available experimental values in the literature. Inserting a Cu layer between Pt and the Py or Co layers slightly increases the total spin current dissipation at these "compound" interfaces.