Exploration of Altermagnetism in $\mathrm{RuO_{2}}$

TL;DR

RuO$_2$ is discussed as a canonical altermagnet, exhibiting zero net magnetization but momentum-space spin splitting due to spin-group symmetries. The review surveys the crystal and magnetic structure, the topological Dirac nodal lines, and the altermagnetic band splitting up to about $1.54$ eV predicted by DFT and observed in some spectroscopic probes. It then consolidates transport and optical signatures, including the crystal Hall effect with $σ_{xz} = σ_{xz}^{AHE} + σ_{xz}^{CHE}$, the spin-splitting torque, IASSE debates, TAMR, Kerr/Faraday responses, and SHG, highlighting symmetry control and interface effects. A central theme is that while bulk RuO$_2$ often shows no long-range magnetic order, thin films reveal magnetism strongly influenced by strain, stoichiometry, and interfaces, underscoring the need for bulk-sensitive probes and systematic control to ascertain intrinsic altermagnetism.

Abstract

The fundamental role of magnetic materials in modern science and technology has driven a rapid surge in research on unconventional magnetism in recent years. In particular, altermagnets, which simultaneously exhibit zero net magnetization in real space and anisotropic spin splitting in momentum space, have garnered significant interest for both fundamental physics and technological applications. Among these, $\mathrm{RuO_{2}}$ stands as the pioneering and most extensively studied altermagnet. While the intrinsic magnetic order of $\mathrm{RuO_{2}}$ is still a subject of active debate, numerous exotic phenomena characteristic of altermagnetism have been observed in $\mathrm{RuO_{2}}$ samples. In this review, we explore each facet of the altermagnetism through specific case studies in $\mathrm{RuO_{2}}$, systematically surveying its crystal and magnetic structures, electronic band properties, and transport phenomena. We critically assess the debate surrounding the intrinsic magnetism in $\mathrm{RuO_{2}}$, incorporating evidence from altermagnetic signatures in transport, as well as contrasting results from magnetic and spectroscopic measurements. Finally, possible future research directions in this field are discussed.

Figures (8)

• Figure 1: Timeline of research on the magnetism in $\mathrm{RuO_{2}}$. Key experimental and theoretical advances, highlighting reports of magnetic order and associated transport/spectroscopic signatures.
• Figure 2: Structural and magnetic characterization of $\mathrm{RuO_{2}}$. (a) Crystal structure of $\mathrm{RuO_{2}}$. Ru atoms (blue), O atoms (red), and magnetic order along the $[001]$ direction. The yellow arrows mark the local magnetic moment at the Ru atoms. (b) Cross-sectional scanning transmission electron microscopy images of $\mathrm{RuO_{2}}$ films grown on $\mathrm{TiO_{2}}$ substrate, confirming the similarity of their structure ref71. (c) Key spin-group symmetry element $[C_2||C_{4z}{\bm{t}}]$ of $\mathrm{RuO_{2}}$, which combines a two-fold spin-space rotation with a four-fold real-space rotation about the $z$-axis. $\mathrm{Ru_{A}}$ and $\mathrm{Ru_{B}}$ atoms are depicted in blue and yellow, respectively, and oxygen atoms are indicated in red. (d) Unpolarized neutron diffraction data taken at $295$ K. The results display enhanced intensity at odd-index reciprocal lattice points at room temperature, supporting the antiferromagnetic order and suggesting structural distortion in $\mathrm{RuO_{2}}$ref4. (e) Scattering intensities of resonant reflection (100) at different temperatures for (left) a bulk crystal and (right) a thin film $\mathrm{RuO_{2}}$. Both samples exhibit pronounced peaks near $H = 1$, persisting up to $400$ K, where $H$ denotes the reciprocal-space coordinate along the $(H,0,0)$ direction, and $H=1$ corresponding to a magnetic ordering wave vector $Q=(1,0,0)$, i.e., along the $[100]$ direction ref51. (f) Neutron diffraction of $\mathrm{RuO_{2}}$ single crystal. In the magnified region of the figure, the $(100)$ reflection exhibits negligible intensity, with no detectable magnetic scattering signal observed ref22. (g) Muon spin rotation and relaxation ($\mu$SR) in bulk $\mathrm{RuO_{2}}$. The curves were measured at different temperatures under zero (left) and weak (right) transverse magnetic fields ref22.
• Figure 3: Topological band structure and spin-splitting features in $\mathrm{RuO_{2}}$. (a) Dirac nodal lines (DNLs) in $\mathrm{RuO_{2}}$ref53. The calculated k-space trajectories of the three DNLs in the $\mathrm{RuO_{2}}$ Brillouin region (top). DNL3 shown in the DFT band structure model of $\mathrm{RuO_{2}}$ crosses the fourfold band along XR and a Dirac point (bottom). (b) Calculated band structure of $\mathrm{RuO_{2}}$, showing the crossing nodes and spin splitting ref20. (c) Energy bands of $\mathrm{RuO_{2}}$ near the Fermi level, without (red and blue) and with (black) relativistic SOC ref41. (d) Magnetic circular dichroism (top) and DFT calculations (bottom) of $\mathrm{RuO_{2}}$ spin polarization ref10. (e) 3D Fermi surfaces calculated for the paramagnetic and altermagnetic phases are compared with the measured Fermi surface ref10. The left parts are the calculated Fermi surface in 3D (top) and projected to 2D planes (bottom) in the paramagnetic phase of $\mathrm{RuO_{2}}$, the middle parts corresponds to the calculation of altermagnetic phase. The right parts are the Fermi surface obtained from topographic mapping experiments of $\mathrm{RuO_{2}}$ through the 3D Brillouin zone (top), and the photoelectron intensity at the plane Fermi energy (bottom).
• Figure 4: Comparison of ARPES and spin-resolved ARPES signatures in $\mathrm{RuO_{2}}$. (a) The existence of spin-polarized bands in $\mathrm{RuO_{2}}$. ARPES spectra along $M$-$\Gamma$-$M$ momentum cut (top) and corresponding calculated spin-resolved bands (bottom) ref28. (b) Calculated spin-polarized band structures along two orthogonal directions, $\Gamma_{1}$-$M$-$\Gamma_{2}$ and $\Gamma_{3}$-$M$-$\Gamma_{4}$ref28. (c)Schematic of the Brillouin-zone cross section (center) indicating the four measurement points used for extracting the spin-resolved energy distribution curves (EDCs). The surrounding panels show the spin-polarized EDCs at the left (EDC1), right (EDC2), lower (EDC3), and upper (EDC4) points, respectively. The differences between spin-up and spin-down intensities at these points are plotted in the adjacent upper and lower histograms ref28. (d) Comparison of ARPES spectra (top) and calculation (bottom) for the energy band structure of single crystal and thin film $\mathrm{RuO_{2}}$ref27. Panels i and iii correspond to the single crystal, while panels ii and iv correspond to the thin film. Neither sample exhibited the spin splitting predicted for the altermagnetic phase. (e) Spin-resolved EDCs measured at three selected momenta (1–3) and their corresponding spin polarization ($S_{y}$) spectra. The rightmost panel presents the spin-resolved ARPES intensity map, showing the momentum–energy distribution of the spin polarization ref27.
• Figure 5: Symmetry-tuned Hall responses in $\mathrm{RuO_{2}}$. (a) Calculated Hall conductivity ($\sigma_{xz}$) and the Hall conductivity components ($\sigma_{xz}^{\mathrm{AHE}},\sigma_{xz}^{\mathrm{CHE}}$) as functions of the sublattice magnetization (canting angle) ref40.(b) Calculated sign and magnitude dependence of $\sigma_{xz}$ on the altermagnetic strength $J$ref111. (c) Temperature-dependent Hall resistivity of $(110)$ film ref11. (d) Saturation of the Hall resistivity ref45. (e) Temperature-dependent Hall resistivity of ultrathin $\mathrm{RuO_{2}}$ films with thicknesses of $t = 1.7$ nm (top) and $t = 9.1$ nm (bottom) jeong2025metallicity.