Transport coefficients of Dirac ferromagnet: Effects of vertex corrections

As a strongly spin-orbit-coupled metallic model with ferromagnetism, we have considered an extended Stoner model to the relativistic regime, named Dirac ferromagnet in three dimensions. In a previous paper [J. Fujimoto and H. Kohno, Phys. Rev. B 90, 214418 (2014)], we studied the transport properties giving rise to the anisotropic magnetoresistance (AMR) and the anomalous Hall effect (AHE) with the impurity potential being taken into account only as the self-energy. The effects of the vertex corrections (VCs) to AMR and AHE are reported in this paper. AMR is found not to change quantitatively when the VCs are considered, although the transport lifetime is different from the one-electron lifetime and the charge current includes additional contributions from the correlation with spin currents. The side-jump and the skew-scattering contributions to AHE are also calculated. The skew-scattering contribution is dominant in the clean case as can be seen in the spin Hall effect in the nonmagnetic Dirac electron system.

Figure 1

The Feynman diagram of the velocity vertex. The filled (unfilled) triangle describes the full (bare) velocity vertex. The solid line represents the Matsubara Green's function. The dotted line and the cross symbol denote the impurity potential and concentration, respectively. Note that Eq. (2.7) is obtained from the equation given by this diagram after taking the analytic continuation $i{\epsilon }_m\to \epsilon +\omega +i0$ and $i{\epsilon }_n\to \epsilon -i0$ and taking the limit $\omega \to 0$.

Figure 2

The diagrammatic expressions of the diagonal conductivity and the two decomposed contributions from the bare bubble and the VCs. The lines and symbols are defined in the caption of Fig. 1. These diagrams are defined in the Matsubara formalism as well as Fig. 1, and Eq. (2.9) is obtained after rewriting the Matsubara summation of $i{\epsilon }_n$ using the contour integral, taking the analytic continuation $i{\epsilon }_m-i{\epsilon }_n\to \omega +i0$, and leaving the $\omega$-linear term which includes both retarded and advanced Green functions.

Figure 3

The diagrammatic expressions of the intrinsic, side-jump, and skew-scattering contributions to the AHE. The skew-scattering contribution can be given as the two Feynman diagrams for the case $n_{\mathrm{i}}\ll 1$ and $u{\sum }_{\eta }{\nu }_{0,0}^{\eta }\left(\mu \right)\ll 1$. Equations (2.11) and (2.13) are obtained through the procedures noted in the caption of Fig. 2.

Figure 4

The diagonal conductivities with the ladder-type VCs, ${\sigma }_A$ ($A=\perp ,\parallel$), for the typical three cases (i)–(iii) as functions of the chemical potential $\mu$. The inset of (i) describes the configuration of the directions in which the current flows and the vector of the “magnetization” and/or “spin”. They are normalized by ${\sigma }_0=e^2{m}^2{c}^3/{\hslash }^2\gamma$ with $\gamma =n_{\mathrm{i}}{u}^2{m}^{2}c/{\hslash }^3$. In the region of $\mu /m<0.6$, the chemical potential lies in the gap. In the regions of $0.6<\mu /m<1.4$ and $\mu /m>1.4$, the system has one and two Fermi surfaces, respectively. Those in the paramagnetic state ($M=S=0$) are also shown for comparison.

Figure 5

The $\mu$ dependencies of the the bare-bubble and the VC contributions to the diagonal conductivities for the typical three cases (i)–(iii). Note that the VC contributions are given by ${\sigma }_A^{}-{\sigma }_A^{\mathrm{b}}$. The additional contribution ${\sigma }_A^{\mathrm{add}}$, which is included by the VC contributions, is also shown for comparison. They are normalized by ${\sigma }_0$ given in the caption of Fig. 4.

Figure 6

The diagonal conductivities for $M=S=0$ with and without the ladder-type VCs. The VCs increase the conductivity, which can be understood as the fact that the forward scattering is dominant in the impurity scattering for the diagonal conductivity.

Figure 7

The chemical potential dependencies of the off-diagonal (Hall) conductivities for the three typical cases. The total Hall conductivities are no longer even/odd functions of $\mu$ because of the skew-scattering contribution ${\sigma }_{xy}^{\mathrm{sk}}$. Here, ${\sigma }_{xy}^{\mathrm{sk}}$ is shown for $n_{\mathrm{i}}u/m=0.5$.