Coherent Description of the Intrinsic and Extrinsic Anomalous Hall Effect in Disordered Alloys on an Ab Initio Level

A coherent description of the anomalous Hall effect is presented that is applicable to pure and disordered systems. This is achieved by an implementation of the Kubo-Středa equation using the fully relativistic Korringa-Kohn-Rostoker method in combination with the coherent potential approximation. Applications to the pure ferromagnets Fe and Ni led to results in full accordance with previous work. For the alloy systems ${\mathrm{Fe}}_x{\mathrm{Pd}}_{1-x}$ and ${\mathrm{Ni}}_x{\mathrm{Pd}}_{1-x}$ very satisfying agreement with experiment could be achieved for the anomalous Hall conductivity. To interpret these results a detailed discussion of the skew and side-jump scattering processes is given.

Figure 1

The AHC of fcc ${\mathrm{Fe}}_x{\mathrm{Pd}}_{1-x}$ and fcc ${\mathrm{Ni}}_x{\mathrm{Pd}}_{1-x}$. The total AHC ${\sigma }_{xy}$ (full squares) has been calculated including the vertex corrections (${\sigma }_{xy}\equiv {\sigma }_{xy}^{\mathrm{VC}}$), while the intrinsic AHC ${\sigma }_{xy}^{\mathrm{intr}}$ (open squares) has been obtained by omitting them (${\sigma }_{xy}^{\mathrm{intr}}\equiv {\sigma }_{xy}^{\mathrm{no}\mathrm{VC}}$). In addition, experimental data [29] for ${\sigma }_{xy}$ (full circles) determined at $T=4.2\mathrm{K}$ are shown. Further, an estimation for the side-jump contribution ${\sigma }_{xy}^{\mathrm{sj}}$ to the extrinsic AHC of fcc ${\mathrm{Fe}}_x{\mathrm{Pd}}_{1-x}$ and fcc ${\mathrm{Ni}}_x{\mathrm{Pd}}_{1-x}$ calculated as the difference ${\sigma }_{xy}^{\mathrm{extr}}-{\sigma }_{xy}^{\mathrm{extr}}[\nabla ]$ is shown (open diamonds; see text).

Figure 2

The extrinsic AHC ${\sigma }_{xy}^{\mathrm{extr}}$ versus ${\sigma }_{xx}$ for fcc ${\mathrm{Fe}}_x{\mathrm{Pd}}_{1-x}$ and fcc ${\mathrm{Ni}}_x{\mathrm{Pd}}_{1-x}$. The straight lines represent extrapolations of the data for $x_{\mathrm{Pd}}\ge 0.9$ (Pd rich) and $x_{\mathrm{Ni}}\ge 0.9$ (Ni rich) to ${\sigma }_{xx}=0$.