Quantum transport theory of anomalous electric, thermoelectric, and thermal Hall effects in ferromagnets

Abstract

In ferromagnets, the spontaneous magnetization bears the Hall effect through the relativistic spin-orbit interaction. Similar effects also occur in thermoelectric and thermal transport phenomena. Their mechanism, if it is of the intrinsic or extrinsic origin, has been controversial for many decades. We present a unified theory of these Hall transport phenomena in ferromagnetic metals with dilute impurities at the zero temperature, in terms of a fully quantum-mechanical transport theory for multiband systems with the self-consistent $T$-matrix approximation. This theory becomes exact with a single impurity and is appropriate for treating the dilute limit of the impurity concentration ${n}_{\mathrm{imp}}$. With the Fermi energy ${E}_{F}$ and the spin-orbit interaction energy ${E}_{\mathrm{SO}}$ being fixed $({E}_{F}>{E}_{\mathrm{SO}})$, three regimes and the associated two crossovers are found in the anomalous Hall conductivity ${\ensuremath{\sigma}}_{xy}$ as a function of ${n}_{\mathrm{imp}}$ that controls the longitudinal conductivity ${\ensuremath{\sigma}}_{xx}$. (i) In the superclean case with the relaxation rate $\ensuremath{\hbar}∕\ensuremath{\tau}\ensuremath{\lesssim}{u}_{\mathrm{imp}}{E}_{\mathrm{SO}}D$, the skew scattering arising from the vertex correction yields a dominant contribution that is inversely proportional to ${n}_{\mathrm{imp}}$, where ${u}_{\mathrm{imp}}$ is the impurity potential strength and $D$ is the density of states. With increasing $\ensuremath{\hbar}∕\ensuremath{\tau}$, this extrinsic skew-scattering contribution rapidly decays. (ii) In the moderately dirty regime ${u}_{\mathrm{imp}}{E}_{\mathrm{SO}}D\ensuremath{\lesssim}\ensuremath{\hbar}∕\ensuremath{\tau}\ensuremath{\lesssim}{E}_{F}$, ${\ensuremath{\sigma}}_{xy}$ becomes insensitive to the scattering strength because of the intrinsic dissipationless topological Berry-phase contribution. It is resonantly enhanced to the order of the quantization unit of conductance when an accidental degeneracy of band dispersions around the Fermi level is lifted by the spin-orbit interaction. Further increasing $\ensuremath{\hbar}∕\ensuremath{\tau}$, another crossover occurs to (iii) the scaling regime of ${\ensuremath{\sigma}}_{xy}\ensuremath{\propto}{\ensuremath{\sigma}}_{xx}^{\ensuremath{\varphi}}$ with $\ensuremath{\varphi}\ensuremath{\sim}1.6$, which has recently been verified by experiments on a wide class of ferromagnets. Similar behaviors also appear in the temperature-linear coefficient of the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$. The thermoelectric Hall conductivity ${\ensuremath{\alpha}}_{xy}$ strongly diverges in the clean limit when the Fermi level crosses edges of the avoided crossing, which may be observed by careful experiments. With increasing $\ensuremath{\hbar}∕\ensuremath{\tau}$, there occurs an interference between positive and negative contributions to ${\ensuremath{\alpha}}_{xy}$, which often leads to a sign change and obscures similar crossovers in the anomalous Nernst effect.