Fermi sea term in the relativistic linear muffin-tin-orbital transport theory for random alloys

We present a formulation of the so-called Fermi sea contribution to the conductivity tensor of spin-polarized random alloys within the fully relativistic tight-binding linear muffin-tin-orbital (TB-LMTO) method and the coherent potential approximation (CPA). We show that the configuration averaging of this contribution leads to the CPA-vertex corrections that are solely due to the energy dependence of the average single-particle propagators. Moreover, we prove that this contribution is indispensable for the invariance of the anomalous Hall conductivities with respect to the particular LMTO representation used in numerical implementation. Ab initio calculations for cubic ferromagnetic $3d$ transition metals (Fe, Co, Ni) and their random binary alloys (Ni-Fe, Fe-Si) indicate that the Fermi sea term is small against the dominating Fermi surface term. However, for more complicated structures and systems, such as hexagonal cobalt and selected ordered and disordered Co-based Heusler alloys, the Fermi sea term plays a significant role in the quantitative theory of the anomalous Hall effect.

Figure 1

The calculated values of the total AHC (full diamonds) and its Fermi surface (open squares) and Fermi sea (open circles) contributions in random fcc Ni${}_{1-c}$Fe${}_c$ alloys as functions of Fe concentration. The values of the Fermi sea term are magnified by a factor of 10.

Figure 2

The calculated values of the total AHC (full diamonds) and its intrinsic (open triangles) and extrinsic (full triangles) parts in diluted bcc Fe${}_{1-c}$Si${}_c$ alloys as functions of Si concentration. The experimental AHC values [54] (open squares) and the calculated Fermi sea contributions (open circles) are displayed as well.