Chiral magnetic excitations and domain textures of g-wave altermagnets

TL;DR

The paper investigates purely magnetic excitations in hexagonal $g$-wave altermagnets CrSb and MnTe, revealing a robust $g$-wave magnon splitting in CrSb that is insensitive to easy-axis anisotropy, while MnTe exhibits anisotropy-tuned, momentum-dependent magnon moments. It introduces a universal altermagnetic splitting metric $\lambda$ that combines magnon energy and magnetic moment and remains $g$-wave symmetric beyond the nonrelativistic limit. A continuum theory is developed showing that CrSb domain walls carry orientation-dependent net magnetization due to altermagnetic terms. Together, these findings illuminate how lattice symmetry and magnetic anisotropy shape spin-wave textures and magnetic domain textures, with potential implications for other AMs and spintronic applications.

Abstract

Altermagnets (AMs) constitute a novel class of spin-compensated materials in which opposite-spin sublattices are connected by a crystal rotation, causing their electronic iso-energy surfaces to be spin-split. While cubic and tetragonal crystal symmetries tend to produce AMs in which the splitting of electronic iso-energy surfaces has $d$-wave symmetry, hexagonal AMs, such as CrSb and MnTe, are $g$-wave AMs. Here we investigate the purely magnetic modes and spin-textures of $g$-wave AMs and show that they are drastically different for easy-axial (CrSb) and easy-planar (MnTe) materials. We show that in CrSb the splitting of the chiral magnon branches possesses $g$-wave symmetry, with each branch carrying a fixed momentum-independent magnetic moment. The altermagnetic splitting is not affected by the easy-axial anisotropy and is the same as that in the nonrelativistic limit. The magnon splitting of MnTe, however, does not strictly possess $g$-wave symmetry due to its easy-planar anisotropy. Instead, the magnetic moment of each branch becomes momentum-dependent, with a distribution that is of $g$-wave symmetry. To generalize the concept of the altermagnetic splitting beyond the nonrelativistic limit, we introduce alternative, directly observable splitting parameter which comprises both the magnon eigenenergy and its magnetic moment and possesses the $g$-wave symmetry in both easy-axial and easy-planar cases. The associated altermagnetic domain walls in easy-axial CrSb possess a net magnetization with an amplitude that depends on their orientation.

Figures (9)

• Figure 1: Schematic representation of the isolines of constant energy $\omega_\nu=\text{const}$ for MnTe (panel a) and CrSb (panel b) made for a constant $k_z>0$. Distribution of the amplitude of the magnon magnetic moment $\bm{\mu}$ along the isolines is shown by the color scheme. In both cases $\bm{\mu}||\bm{n}_0$ where $\bm{n}_0$ is the ground state Néel vector. For details of the magnon magnetic moment computation, see Section \ref{['sec:mmm']}. The green background hexagon indicates the size of the first Brillouin zone.
• Figure 2: Schematic crystal structure and exchange interactions in hexagonal CrSb and MnTe-type altermagnets. (a) the (001) plane of magnetic atoms (yellow discs: Cr or Mn) which form a triangular lattice with basis $\bm{e}_{1}$, $\bm{e}_{2}$. Thin lines connect next-nearest neighbors while thick lines border the hexagonal 2D Wigner-Seitz cells. Ferromagnetic Heisenberg exchange of strength $J_{\textsc{fm}}$ couples nearest neighbors in each layer. (b) cross-section of one of the vertical planes I, II or III. The antiferromagnetic exchange $J_{\textsc{afm}}$ couples nearest magnetic atoms in two neighboring layers at distance $c_0$. Altermagnetic Heisenberg bonds $\tilde{J}_{1(2)}$ are shown by blue (red) lines. The structure of the altermagnetic bonds is unique for all planes I, II, and III if the orientation of the cross-section (b) is consistent with the coloring of the unit cells (orange-green-magenta).
• Figure 3: Comparison of the magnon spectra \ref{['eq:disp-CSb']} for CrSb (top row) and \ref{['eq:disp-MnTe']} for MnTe (bottom row). The figures are built for the values of the parameters listed in Table \ref{['tab:params-discr']} and for $B=0$. Panels (a) and (f) show the evolution of two branches $\omega_{+}$ (orange) and $\omega_{-}$ (blue) within the 1st Brillouin zone (1.BZ) for $k_z=\text{const}$. The value of the splitting $\Delta\omega=\omega_{+}-\omega_{-}$ is shown by the color coding at the bottom. Surfaces of constant energy $\omega=\mathrm{const}$ are shown on panels (c) and (h) for CrSb and MnTe, respectively. Panels (b) and (g) aggregate panels (a,c) and (f,h), respectively, showing the surfaces of the constant energy for the constant $k_z$. On panels (b,d,g,i), the analytical dispersion (lines) is compared to the spectra extracted from the spin-lattice simulations (color density). Panel (k) shows the 1.BZ and the path $L'\Gamma L$ along which we computed the dispersions presented on panels (d,i). Panels (e) and (j) show precession of the neighboring magnetic moments of different sublattices for a spin wave with wave-vector $(k_xa_0, k_ya0, k_zc_0)=(0,\pi/\sqrt{3},\pi/4)$ for the case of CrSb and MnTe, respectively. The insets show the trajectories swept by the ends of the vectors $\bm{m}_\nu$.
• Figure 4: (a) Surfaces of constant magnon splitting $\Delta\omega^{\text{CrSb}}_{\bm{k}}=\pm8\gamma\delta\tilde{J}/\mu_s$ of the magnon branches of CrSb in the first BZ compared to (b) the hydrogen $g$-orbital with the corresponding quantum numbers $(n,l,m)$. Orange and blue color correspond to $\Delta\omega^{\text{CrSb}}_{\bm{k}}>0$ and $\Delta\omega^{\text{CrSb}}_{\bm{k}}<0$, respectively.
• Figure 5: The altermagnetically induced emergence of the chirality and magnetic moment in spin-waves for MnTe. The precession dynamics is reconstructed by means of Eq. \ref{['eq:dm-sw']} for the wave-vector $(k_xa_0, k_ya_0, k_zc_0)=(0,\pi/\sqrt{3},\pi/4)$. To illustrate the role of the altermagnetism, we introduce the scaling factor $\varsigma$ for the altermagnetic parameters $\varepsilon$ and $\delta\varepsilon$ of MnTe and show the precession dynamics for $(\varepsilon',\delta\varepsilon')=\varsigma(\varepsilon,\delta\varepsilon)$ with $\varsigma$ varying from 0 to 1. The easy-plane is shown by gray.