Guided spin wave in monolayer CrSBr: Localization and spin-orbit coupling from dipolar field

TL;DR

This work develops a BdG framework for exchange-dipole spin waves in monolayer CrSBr, incorporating the long-range magnetostatic Green function to capture both static confinement and dynamic spin-orbit coupling of magnons. The static dipolar field acts as a confining potential that yields Damon-Eshbach edge modes, while the dynamic dipolar field introduces spin-orbit coupling and yields a Rashba-like term, producing nontrivial k-dependent chirality. Spin-wave spectra exhibit parity doublets due to inversion symmetry, with boundary conditions and edge effects playing key roles in mode localization. The results, validated by micromagnetic simulations, illuminate the nature of exchange-dipole magnons in 2D magnets and provide insight toward antiferromagnetic resonance phenomena in bulk CrSBr, with potential implications for 2D magnonic devices.

Abstract

Spin-wave spectrum of monolayer CrSBr waveguides was studied by numerically diagonalizing the Bogoliubov-de Gennes Hamiltonian derived from linearising the Landau-Lifshitz-Gilbert equation. In contrast to its short-range counterparts, the long-range dipolar field acts statically as a confining potential for spin wave, while the dynamic part couples the spin and orbit degrees of freedom, thus giving rise to spin-orbit coupling for spin wave. Due to the inversion symmetry of the Hamiltonian and the spinor structure of the wave function, spin-wave eigenstates form doublets with definite parity. Micromagnetic simulation tallies well with numerical calculation. Our study on spin-wave eigenstates in CrSBr waveguides sheds light on the nature of exchange-dipole spin wave in ferromagnetic slabs. We confirm particularly that the robustness of the Damon-Eshbach mode is not derived from topology, but rather from the static dipolar field. Moreover, a thorough knowledge on spin wave in monolayer CrSBr itself represents a step forward to understanding the more complicated antiferromagnetic resonance in bulk CrSBr.

Figures (20)

• Figure 1: Coordinate system used for spin wave in a monolayer of CrSBr. The rectangular slab is bounded by coordinate planes $x = \pm l$, $y = \pm w$, and $z = \pm t$. In the limit $l \rightarrow \infty$, the slab becomes an ideal, infinitely long, waveguide. An extended film has both $l \rightarrow \infty$ and $w$$\rightarrow \infty$.
• Figure 2: Static dipolar field for a wave guide when $\textbf{m} = \hat{y}$, Eq (\ref{['yd0']}). The jump at $y = w$ is exactly 1, as required by the continuity of the B field. $-d _0$ is symmetric with respect to $y$.
• Figure 3: $n ^ +$ spin-wave bands ($n = 0, 1, 2, 3$ from bottom to top) with field $\mu_0 H_a = 0.7$ T along $\hat{z}$ in a waveguide (solid lines) or a slab obtained from calculation (dashed lines) and micromagnetic simulation using OOMMF (symbols). Thick and thin dash-dotted lines correspond to Eq. (\ref{['wz']}) with $n_y \neq 0$ and $n_y = 0$, respectively. The error bar corresponds to the frequency resolution of the micromagnetic simulation.
• Figure 4: $n ^ +$ spin-wave bands with field $\mu_0 H_a = 0.2$ T along $\hat{y}$ in the notation of Fig. \ref{['zp']}. Thick and thin dash-dotted lines correspond to Eq. (\ref{['wy']}) with $n_y \neq 0$ and $n_y = 0$, respectively. The lowest band is a localized band, while the other bands correspond to $n = 0, 2, 3$ from bottom to top
• Figure 5: $n ^ +$ spin-wave bands with field $\mu_0 H_a = 0.6$ T along $\hat{x}$ in the notation of Fig. \ref{['zp']}. Thick and thin dash-dotted lines correspond to Eq. (\ref{['wx']}) with $n_y \neq 0$ and $n_y = 0$, respectively. Localized states with positive $x$-parity for the slab geometry are shown as triangles.