Disorder and temperature dependence of the anomalous Hall effect in thin ferromagnetic films: Microscopic model

We consider the anomalous Hall (AH) effect in thin disordered ferromagnetic films. Using a microscopic model of electrons in a random potential of identical impurities including spin-orbit coupling, we develop a general formulation for strong, finite range impurity scattering. Explicit calculations are done within a short range but strong impurity scattering to obtain AH conductivities for both the skew scattering and side-jump mechanisms. We also evaluate quantum corrections due to interactions and weak localization effects. We show that for arbitrary strength of the impurity scattering, the electron-electron interaction correction to the AH conductivity vanishes exactly due to general symmetry reasons. On the other hand, we find that our explicit evaluation of the weak localization corrections within the strong, short-range impurity scattering model can explain the experimentally observed logarithmic temperature dependences in disordered ferromagnetic Fe films.

Figure 1

Diagrams for side-jump contributions. Solid lines are impurity averaged Green’s functions. Shaded triangles with dashed lines represent impurity scattering amplitudes while the dotted line from a vertex denotes spin-orbit term in the velocity operator. The shaded vertex represents vertex corrections to the current density operator.

Figure 2

Diagrams for interaction corrections. Solid lines are impurity averaged Green’s functions, wavy lines denote screened Coulomb interactions, and dashed lines denote diffusion poles. There are two diagrams of type (a) and two of type (b).

Figure 3

Diagrams for weak localization corrections. Solid lines are impurity averaged Green’s functions and broken lines are impurity scattering amplitudes. Shaded cross is the Cooperon and shaded vertices are vertex corrections to the current density operator. There are two diagrams of type (b) and four diagrams of type (c).