Staggered Dzyaloshinskii-Moriya and canting angle in centrosymmetric altermagnetic and ferromagnetic phases: influence on the anomalous Hall effect and Weyl points

TL;DR

The paper develops a symmetry-preserving, Wannier-based method to compute the anomalous Hall conductivity ($AHC$) as a function of spin canting in ferromagnets and altermagnets, starting from a non-magnetic Hamiltonian that retains crystal symmetries and adding on-site spin splitting plus SOC with canting angles $( heta, \phi)$. Using SrRuO$_3$ as a model, it demonstrates how $AHC$ depends on spin-splitting $oldsymbol{\Delta}$ and canting, showing a sign change of $AHC$ achievable by canting even when collinear configurations yield near-zero $AHC$; it also links these transport changes to the evolution of Weyl points in the Brillouin zone. The work catalogs symmetry-allowed $AHC$ tensor components for the ferromagnetic and altermagnetic orders in space group 62 and shows that the central region of the electronic bandwidth is most sensitive to canting, with canting generally playing a secondary role except near zero $AHC$ in the collinear state. By combining DFT (with Hubbard $U$) and tight-binding with spin canting, the study connects microscopic magnetic structure to topological features, offering a versatile framework for exploring Berry-curvature-driven transport in magnetic oxides and related systems.

Abstract

We present a simple methodology to compute the anomalous Hall conductivity (AHC) as a function of the canting angles in ferromagnets and altermagnets, starting from a nonmagnetic Hamiltonian obtained from first-principles calculations that preserves the full symmetry of the crystal structure. Magnetism is introduced by including on-site spin splitting, spin-orbit coupling, and spin-canting angles. As a representative material, we study SrRuO$_3$, which supports spin canting and exhibits a sign change of the AHC. In the ferromagnetic phase, the low-energy AHC is found to be close to zero at the Fermi level, in agreement with experimental observations. We show that the dependence of the AHC on the relevant physical parameters is most pronounced in the central region of the electronic bandwidth. We determine the symmetry-allowed components of the AHC for different magnetic orders in the large family of transition-metal perovskite ABO$_3$ compounds with space group $62$, including the spontaneous in-plane anomalous Hall effect. Within density functional theory, we evaluate the range of spin-canting angles in SrRuO$_3$ and demonstrate that it is suppressed as electronic correlations increase. By analyzing the AHC as a function of the canting angle, we find that the collinear magnetic configurations contribute most to the AHC, while spin canting plays a secondary role in determining its magnitude in non-collinear ferromagnets and altermagnets. However, canting can become relevant and induce a sign change of the AHC when the collinear magnetic state exhibits an AHC close to zero. Finally, we investigate the locations of Weyl points in the Brillouin zone and their evolution as a function of the canting angle.

Figures (14)

• Figure 1: Comparison of the non-magnetic electronic band structures obtained from Wannier interpolation (blue dotted lines) and DFT calculations performed using the VASP code (red lines). The Wannier basis set was constructed using the $t_{2g}$ orbitals as trial wave functions. The Wannier-interpolated bands are in excellent agreement with the direct DFT results across the entire Brillouin zone.
• Figure 2: The altermagnetic phases of SrRuO$_3$. (a) A-type magnetic order, (b) C-type magnetic order, and (c) G-type magnetic order. The Néel vector is aligned along the $z$-axis in all three cases.
• Figure 3: A depiction of the three regions observed in the band structure in which the variation of AHC with spin-splitting has considerably different behaviour. The bright red (bright blue) region denotes the DOS of spin-up (down) electrons. The three regions of the AHC, at the bottom of the bands, around the Fermi level and above the Fermi level are represented in light red, light grey and light blue, respectively. In SrRuO$_3$, the t$_{2g}$ manifold is filled to two-thirds ($2/3$).
• Figure 4: AHC as a function of the spin splitting for the ferromagnetic phase with magnetization along the (a,b) $x$-direction with two non-zero components: $\sigma_{yz}$ and $\sigma_{zx}$ respectively. The first is the standard AHC, while the second is the spontaneous in-plane AHC. (c) $z$-direction where the only non-zero component is $\sigma_{xy}$.
• Figure 5: AHC as a function of the spin splitting for the A-type altermagnet with Néel vector along the (a) $x$-direction, where the only nonzero component is $\sigma_{xy}$. (b) $y$-direction where the only nonzero component is $\sigma_{xy}$. (c) $z$-direction where the only nonzero component is $\sigma_{zx}$. All magnetic configurations are constrained to be collinear and exhibit zero net magnetization.