Spin relaxation in metallic ferromagnets

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The Elliott theory of spin relaxation in metals and semiconductors is extended to metallic ferromagnets. Our treatment is based on the two-current model of Fert, Campbell, and Jaoul. The $d\ensuremath{\rightarrow}s$ electron-scattering process involved in spin relaxation is the inverse of the $s\ensuremath{\rightarrow}d$ process responsible for the anisotropic magnetoresistance (AMR). As a result, spin-relaxation rate $1/{\ensuremath{\tau}}_{\text{sr}}$ and AMR $\ensuremath{\Delta}\ensuremath{\rho}$ are given by similar formulas, and are in a constant ratio if scattering is by solute atoms. Our treatment applies to nickel- and cobalt-based alloys which do not have spin-up 3$d$ states at the Fermi level. This category includes many of the technologically important magnetic materials. And we show how to modify the theory to apply it to bcc iron-based alloys. We also treat the case of Permalloy ${\text{Ni}}_{80}{\text{Fe}}_{20}$ at finite temperature or in thin-film form, where several kinds of scatterers exist. Predicted values of $1/{\ensuremath{\tau}}_{\text{sr}}$ and $\ensuremath{\Delta}\ensuremath{\rho}$ are plotted versus resistivity of the sample. These predictions are compared to values of $1/{\ensuremath{\tau}}_{\text{sr}}$ and $\ensuremath{\Delta}\ensuremath{\rho}$ derived from ferromagnetic-resonance and AMR experiments in Permalloy.