Effective spin-flip scattering in diffusive superconducting proximity systems with magnetic disorder

We revisit the problem of diffusive proximity systems involving superconductors and normal metals (or ferromagnets) with magnetic disorder. On the length scales much larger than its correlation length, the effect of sufficiently weak magnetic disorder may be incorporated as a local spin-flip term in the Usadel equations. We derive this spin-flip term in the general case of a three-dimensional disordered Zeeman-type field with an arbitrary correlation length. Three different regimes may be distinguished: pointlike impurities (the correlation length is shorter than the Fermi wavelength), medium-range disorder (the correlation length between the Fermi wavelength and the mean free path), and long-range disorder (the correlation length longer than the mean free path). We discuss the relations between these three regimes by using the three overlapping approaches: the Usadel equations, the nonlinear sigma model, and the diagrammatic expansion. The expressions for the spin-flip rate agree with the existing results obtained in less general situations.

Figure 1

Two types of spin-flip scattering: (a) local and (b) nonlocal. The diagrams show the way the magnetic scattering is included in the “cooperon” propagator (see Secs. 3, 5 for details). Thick solid lines represent the electron Green functions, thin dashed lines denote averaging over Gaussian disorder. The crosses and the open circles represent the potential and magnetic disorders, respectively.

Figure 2

Diagrammatic representations of the (a) local and (b) nonlocal contributions in the sigma-model calculation. Thick double lines correspond to the $Q$ matrix; closed loops of thin solid lines indicate the STr operation; open circles denote the “magnetic vertices” ${\widehat{\tau }}_3\hat{\mathit{\sigma }}$. Dashed lines are the magnetic disorder correlation functions $F_{ij}$; wavy lines are the propagators of the $W$ field. Panel (c) shows a potentially dangerous contribution to the propagator of the $W$ field. To avoid divergencies, a self-consistent “screening” of the $W$ propagator needs to be taken into account.