Anomalous Hall effect in granular ferromagnetic metals and effects of weak localization

We theoretically investigate the anomalous Hall effect in a system of dense-packed ferromagnetic grains in the metallic regime. Using the formalism recently developed for the conventional Hall effect in granular metals, we calculate the residual anomalous Hall conductivity ${\sigma }_{xy}$ and resistivity ${\rho }_{xy}$ and weak localization corrections to them for both skew-scattering and side-jump mechanisms. We find that the scaling relation between ${\rho }_{xy}$ and the longitudinal resistivity ${\rho }_{xx}$ of the array does not hold, regardless of whether it is satisfied for the specific resistivities of the grain material or not. The weak localization corrections, however, are found to be in agreement with those for homogeneous metals. We discuss recent experimental data on the anomalous Hall effect in polycrystalline iron films in view of the obtained results.

Figure 1

Granular system and the classical picture of the Hall conductivity. The external Ohmic voltage $V_y$ is applied to the contacts in the $y$ direction. The resulting Ohmic current $I_y=G_T{V}_y$ running through the grain in the $y$ direction causes the Hall voltage drop $V_H=R_H{I}_y$ between its opposite banks in the $x$ direction. Since when calculating the Hall conductivity ${\sigma }_{xy}$ the total voltage drop per lattice period in the $x$ direction is assumed 0, the Hall voltage $V_H$ is applied with an opposite sign to the contacts in the $x$ direction (see bottom), causing the Hall current $I_x=G_T{V}_H=G_T^2{R}_H{V}_y$ [see Eq. (3.1)].

Figure 2

Diagrams for the residual anomalous Hall conductivity ${\sigma }_{xy}$ [Eqs. (3.1, 3.15)] (see Ref. 19 for details).

Figure 3

Diagrams for the Hall component ${⟨j_x{r}_y⟩}^{\text{ss}}$ of the current-coordinate correlation function (3.19) due to the skew-scattering mechanism.

Figure 4

Diagrams for Hall component ${⟨j_x{r}_y⟩}^{\text{sj}}$ of the current-coordinate correlation function (3.19) due to the side-jump mechanism. Diagrams (a) and (b) contain the relativistic corrections to the current [$\delta \hat{\mathbf{j}}$, Eq. (3.25)] and coordinate [$\delta \hat{\mathbf{r}}$, Eq. (3.26)] vertices, respectively.

Figure 5

Diagrams for the weak localization correction $\delta {⟨j_x{r}_y⟩}^{\text{ss}}$ to the skew-scattering Hall component ${⟨j_x{r}_y⟩}^{\text{ss}}$ [Eq. (3.24)] of the current-coordinate correlation function (3.19). The gray regions denote the Cooperons [Eq. (4.3)]. Each diagram depicted above stands for four diagrams: the ones not shown are obtained by reflecting the impurity lines through the diagram center and/or flipping the diagram upside down.

Figure 6

Diagrams for the weak localization correction $\delta {⟨j_x{r}_y⟩}^{\text{ss}}$ to the side-jump Hall component ${⟨j_x{r}_y⟩}^{\text{sj}}$ [Eq. (3.27)] of the current-coordinate correlation function (3.19). Diagrams (a) and (b) contain the relativistic corrections to the current [$\delta \hat{\mathbf{j}}$, Eq. (3.25)] and coordinate [$\delta \hat{\mathbf{r}}$, Eq. (3.26)] vertices, respectively. Each diagram in panel (a) stands for two diagrams: the one not shown is obtained by flipping the diagram upside down.