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

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 ${\sigma }_{xy}$ as a function of $n_{\mathrm{imp}}$ that controls the longitudinal conductivity ${\sigma }_{xx}$. (i) In the superclean case with the relaxation rate $\hslash ∕\tau \lesssim u_{\mathrm{imp}}{E}_{\mathrm{SO}}D$, 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 $\hslash ∕\tau$, this extrinsic skew-scattering contribution rapidly decays. (ii) In the moderately dirty regime $u_{\mathrm{imp}}{E}_{\mathrm{SO}}D\lesssim \hslash ∕\tau \lesssim E_F$, ${\sigma }_{xy}$ 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 $\hslash ∕\tau$, another crossover occurs to (iii) the scaling regime of ${\sigma }_{xy}\propto {\sigma }_{xx}^{\phi }$ with $\phi \sim 1.6$, 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 ${\kappa }_{xy}$. The thermoelectric Hall conductivity ${\alpha }_{xy}$ 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 $\hslash ∕\tau$, there occurs an interference between positive and negative contributions to ${\alpha }_{xy}$, which often leads to a sign change and obscures similar crossovers in the anomalous Nernst effect.

Figure 1

The simplest examples of (a) an accidental crossing of two band dispersions at a momentum ${\mathit{p}}_0$ and (b) an avoided crossing with a splitting of the dispersions by $2E_{\mathrm{SO}}$ at this momentum region.

Figure 2

(Color online) Electronic band dispersions of the present model given by ${\widehat{H}}_0$.

Figure 3

(Color online) Diagrammatic representation of the self-energy in the present self-consistent $T$-matrix approximation in the Keldysh space, which is composed of the infinite series of multiple Born scattering amplitudes. Here, underlined variables $\underset{̱}{\widehat{G}}$ and $\underset{̱}{\widehat{\Sigma }}$ are matrices in the Keldysh space. Note that the lesser component gives the integral equation for the vertex correction in the context of the Kubo formalism.

Figure 4

(Color online) The results for $D\left(\epsilon \right)$ and $D_z\left(\epsilon \right)$ obtained in the self-consistent $T$-matrix approximation for $v=3.59$, ${\Delta }_0=0.1$, and $2m{u}_{\mathrm{imp}}=0.2$.

Figure 5

(Color online) (a) The total anomalous Hall conductivity ${\sigma }_{xy}^{\mathrm{tot}}$, (b) the intrinsic contribution ${\sigma }_{xy}^{\mathrm{int}}$, and (c) the extrinsic contribution ${\sigma }_{xy}^{\mathrm{ext}}$ as functions of the Fermi energy $E_F$ and the Born scattering amplitude ${\gamma }_{\text{Born}}$ in an energy unit of $E_{\mathrm{res}}=1.0$. The parameters are chosen as $v=3.59$, ${\Delta }_0=0.1$, and $2m{u}_{\mathrm{imp}}=0.2$. Note the difference of the scales for ${\sigma }_{xy}$ in (a), (b), and (c).

Figure 6

(Color online) The total anomalous Hall conductivity ${\sigma }_{xy}^{\mathrm{tot}}$ as a function of $E_F$ and ${\gamma }_{\text{Born}}$ in an energy unit of $E_{\mathrm{res}}=1.0$. The parameters are chosen as $v=3.59$, ${\Delta }_0=0.1$, and $2m{u}_{\mathrm{imp}}=0.02$. Note the difference of the scale for ${\sigma }_{xy}$ compared with Fig. 5a.

Figure 7

(Color online) ${\sigma }_{xy}^{\mathrm{tot}}$, ${\sigma }_{xy}^{\mathrm{int}}$,${\sigma }_{xy}^{\mathrm{I}\mathrm{int}}$, ${\sigma }_{xy}^{\mathrm{II}\mathrm{int}}$, and ${\sigma }_{xy}^{\mathrm{ext}}$ at $E_F=0.9$ as functions of the $\hslash ∕\tau$ for the same parameter values as Fig. 5 except the impurity potential strength: (a) $2m{u}_{\mathrm{imp}}=0.02$, (b) 0.2, (c) 0.4, and (d) 0.6.

Figure 8

(Color online) ${\sigma }_{xy}^{\mathrm{tot}}$, ${\sigma }_{xy}^{\mathrm{int}}$, ${\sigma }_{xy}^{\mathrm{I}\mathrm{int}}$, ${\sigma }_{xy}^{\mathrm{II}\mathrm{int}}$, and ${\sigma }_{xy}^{\mathrm{ext}}$ as functions of the $\hslash ∕\tau$ for the same parameter values as Fig. 5 with $E_F=0.5$ and 1.5 for (a) and (b), respectively.

Figure 9

(Color online) Resistivity ${\rho }_{xx}\sim 1∕{\sigma }_{xx}$ for the same set of parameters as in Fig. 5, except for a variation of the impurity potential strength $2m{u}_{\mathrm{imp}}=0.02$, 0.2, 0.4, and 0.6. (a) and (b) are the linear and logarithmic plots. In (b), all the data points for $2m{u}_{\mathrm{imp}}=0.02$, 0.2, 0.4, and 0.6 are plotted with the same symbol for each value of $E_F$.

Figure 10

(Color online) Scaling plot of ${\sigma }_{xy}$ versus ${\sigma }_{xx}$ for the same sets of parameter values as in Fig. 8b, except $2m{u}_{\mathrm{imp}}$. The arrows show the extrinsic-intrinsic crossover scale of ${\sigma }_{xx}$ for $2m{u}_{\mathrm{imp}}=0.20$, 0.40, and 0.60. The horizontal solid line and the dotted curve represent the values given by the TKNN formula of Eq. (2) and the intrinsic contribution in the limit of $2m{u}_{\mathrm{imp}}\to 0$. The vertical dashed line gives the second crossover scale of ${\sigma }_{xx}(\sim \pi e^2∕h)$ to the dirty regime with the 1.6 power law.

Figure 11

(Color online) (a) The $T$-linear coefficients of the total anomalous thermoelectric Hall conductivity ${\alpha }_{xy}^{\mathrm{tot}}∕T$ and (b) the intrinsic contribution ${\alpha }_{xy}^{\mathrm{int}}∕T$ as functions of $E_F$ and ${\gamma }_{\text{Born}}$ in an energy unit of $E_{\mathrm{res}}=1.0$. The parameters are chosen as $v=3.59$, ${\Delta }_0=0.1$, and $2m{u}_{\mathrm{imp}}=0.2$, as in Fig. 5. Note the difference of the scales for ${\alpha }_{xy}$ in (a) and (b).

Figure 12

(Color online) Summary of experimental results on various ferromagnets, including transition metals, perovskite oxides, spinels, and magnetic semiconductors. The theoretical curve corresponding to Fig. 10 $(2m{u}_{\mathrm{imp}}=0.02)$ is also shown. The data are taken from Miyasato et al. (Ref. 20) for Gd, Fe, Ni, and Co films, Fe single crystals, $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$, ${\mathrm{La}}_{1-x}{\mathrm{Sr}}_x\mathrm{Co}{\mathrm{O}}_3$ ($x=0.20$ and 0.25), and ${\mathrm{Cu}}_{1-x}{\mathrm{Zn}}_x{\mathrm{Cr}}_2{\mathrm{Se}}_4$ ($x=0.0$, 0.2, 0.4, 0.5, 0.6, 0.8, and 0.9); from Lee et al. (Ref. 17) and Lyanda-Geller et al. (Ref. 55) for ${\mathrm{La}}_{1-x}{\left(\mathrm{Sr}\mathrm{Ca}\right)}_x\mathrm{Mn}{\mathrm{O}}_3$; from Iguchi et al. (Ref. 56) for ${\mathrm{Nd}}_2{\left(\mathrm{Mo}\mathrm{Nb}\right)}_2{\mathrm{O}}_7$; from Manyala et al. (Ref. 18) for ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$; from Lee et al. (Ref. 19) for MnSi; from Matsukura et al. (Ref. 59), Edmonds et al. (Ref. 60), Yuldashev et al. (Ref. 61), and Chiba et al. (Ref. 62) for ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$; from Ohno et al. (Ref. 63) and Oiwa et al. (Ref. 64) for ${\mathrm{In}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$; from Ueno et al. (Ref. 66), Cho et al. (Ref. 67), and Ramaneti et al. (Ref. 68) for anatase-$\mathrm{Co}\text{−}\mathrm{Ti}{\mathrm{O}}_2$; and from Toyosaki et al. (Ref. 69) and Higgins et al. (Ref. 70) for rutile-$\mathrm{Co}\text{−}\mathrm{Ti}{\mathrm{O}}_2$.