Extrinsic anomalous Hall effect in altermagnets

TL;DR

This work analyzes extrinsic versus intrinsic anomalous Hall conductivity in altermagnets, focusing on two-band effective models that capture the spin-splitting protected by spin-group symmetries. Using the Kubo-Středa formalism with short-range disorder, it shows that in class A altermagnets the extrinsic AHC is comparable to the intrinsic AHC in the large exchange-splitting limit, while in class B the extrinsic contribution is negligible. The nonanalytic SOC dependence of the intrinsic AHC, arising from lifting spin degeneracy along nodal planes, underpins the strong coupling between magnetization, SOC, and disorder in class A. The results highlight the importance of including extrinsic terms when comparing theory with experiments in altermagnets and offer a framework to distinguish material classes via AHC responses to SOC, DM interactions, and magnetization.

Abstract

We find the extrinsic anomalous Hall conductivity (AHC) to be comparable to the intrinsic one in roughly half of the altermagnetic spin Laue groups in the limit of large exchange splitting. In materials with a finite Dzyaloshinskii-Moriya type interaction, the extrinsic contribution is essential even in the clean limit. In other altermagnets it is mostly negligible. This peculiar behavior is linked to the nonanalytic dependence of the intrinsic AHC on spin-orbit coupling. Both originate from the lifting of the spin degeneracy along the nodal planes as the weak spin-orbit coupling breaks the nonrelativistic spin symmetry.

Figures (6)

• Figure 1: (a) The representative ${}^1m_z {}^2m_x {}^2m_y$ of class A. (b) The representative ${}^24/{}^1m{}^2m_y{}^1m_d$ of class B. The horizontal mirror symmetry, $m_{\hat{z}}$ is common to the two systems. The symmorphic mirror symmetry operations, $m_{\hat{X}}$ and $m_{\hat{Y}}$, where $\hat{X} =( \hat{x} + \hat{y} )/\sqrt{2}$, $\hat{Y} =( \hat{x} - \hat{y} )/\sqrt{2}$ are present in the class B representative only. These symmetries ensure the SOC vanishes along the symmorphic mirror plane intersections. In particular, while in class B the spectrum at $\Gamma$-point remains spin degenerate at finite SOC, in the class A this degeneracy is lifted.
• Figure 2: (a) The parametrization of the frame rotation by a three Euler angles, $\psi_E,\theta_E,\varphi_E$. The full rotation is the rotation around $\hat{z}$ by $\varphi_E$ followed by rotation around $\hat{x}$ by $\theta_E$ and by $\psi_E$ around $\hat{z}$. (b) A rotation that aligns the $z'$ axis with the Néel vector $\mathbf{N}$ defined by $\theta_E = \theta_N$, $\psi_E = \phi_N + \pi/2$, and we choose $\varphi_E = -\pi/2$, where $\theta_N$ and $\phi_N$ are polar and azimuthal angles of $\mathbf{N}$.
• Figure 3: The Berry curvature $\Omega^{(-)}_{yz}(\mathbf{k})$ for the band $E_{\mathbf{k}}^-$ shown at the outer Fermi surface, $E_{\mathbf{k}}^-=E_F$ in the two-band limit $1/|\mathbf{N}|=0$. The other two components of the $\hat{\Omega}$ tensor vanish for $\mathbf{N}\parallel \hat{y}$. The Berry curvature of the second inner band $E_{\mathbf{k}}^+$, $\hat{\Omega}^{(+)}(\mathbf{k}) = -\hat{\Omega}^{(-)}(\mathbf{k})$ is not shown. Panels (a) and (b) show $\Omega^{(-)}_{yz}(\mathbf{k})$ at the the larger of the two Fermi surfaces, $E_{\mathbf{k}}^-=E_F$ for the models representing class A and class B $d$-wave altermagnets, respectively as introduced in Sec. \ref{['sec:Model']}. The parameters, $E_0=E_F/3$, $t_A = 0.01 E_F$, $\lambda=\lambda_z = 0.24 t_A$ are the same for both panels, and are in the large altermagnetic splitting limit, $t_A\gg\mathrm{max}\{\lambda,\lambda_z\}$. Panels (a) and (b) are qualitatively similar.
• Figure 4: AHC $\hat{\sigma}$ of a class A representative as a function of the SOC, $\lambda_z$ in the two-band limit $1/|\mathbf{N}|=0$, $\bar{\lambda} = 0.09\lambda_z$, $\Delta_\mathrm{A}/E_F = 0.02$, and $k_F = 0.3$. The magnetization is set to zero. The thin (red), dot-dashed (blue), and solid (black) curves show the intrinsic $\hat{\sigma}_{in}$, extrinsic $\hat{\sigma}_{ex}$, and total $\hat{\sigma}$ contributions, respectively. The separate $\hat{\sigma}_{in}$ and $\hat{\sigma}_{ex}$ contributions are obtained numerically from Eqs. \ref{['eq:sigma_Berry_3']} and \ref{['sigma^I_b1']}, respectively. Dashed straight (yellow) lines passing through the origin are plotted based on Eqs. \ref{['h_expand6']} and \ref{['Extr7']} valid at small $\lambda_z$. Dashed horizontal (green) lines validate Eqs. \ref{['h_expand7']} and \ref{['h_expand9']} at large $\lambda_z$.
• Figure 5: AHC $\hat{\sigma}$ of a class B representative as a function of the SOC, $\lambda_z$ in the two-band limit $1/|\mathbf{N}|=0$ for $\lambda = 0.1\lambda_z$, $\Delta_\mathrm{A}/E_F = 10^{-4}$. The magnetization is set to zero. The solid (black) curve is $\hat{\sigma}$ obtained numerically from Eq. \ref{['eq:sigma_Berry_3']}. Dashed straight (yellow) lines passing through the origin show the asymptotic scaling Eq. \ref{['h_expand3a']} as small $\lambda_z$. Dashed (green) line indicates the logarithmic scaling, Eq. \ref{['h_expand8']} at large $\lambda_z$ with properly adjusted additive constant.