Intrinsic (non)-Gilbert damping in magnetic insulators calculated from a minimal model and \textit{ab initio} spin Hamiltonians

Abstract

We present an analytically solvable minimal model for the relaxation of low-frequency magnons in magnetic insulators arising from magnon-phonon and magnon-magnon interactions. The model establishes a direct connection between microscopic relaxation processes and Gilbert damping, and reveals how magnon decay evolves from bulk systems to the monolayer limit. We find that magnon-phonon coupling produces Gilbert damping of comparable magnitude in three- and two-dimensional magnets, with qualitative differences between flexural phonons in free-standing monolayers and three-dimensional phonons in substrate-supported layers. By contrast, non-Gilbert damping due to four-magnon scattering is strongly enhanced in two dimensions, where it becomes independent of spin-orbit coupling. To benchmark the model against real materials, we introduce a numerical approach for computing magnon damping from ab initio-derived spin Hamiltonians. We demonstrate that the central conclusions of the model remain valid for magnons in bulk YIG and in a monolayer of the van der Waals magnetic insulator CrSBr.

Figures (5)

• Figure 1: Magnon damping in the minimal toy model. (a) Temperature dependence of the damping due to the magnon-phonon interaction. (b) Temperature dependence of the damping due to the magnon-magnon interaction. (c) Magnon linewidth dependence on the energy compared with the Gilbert law (red solid lines) and with the non-Gilbert behavior derived from Eqs. (\ref{['mag2D']}) and (\ref{['mag2D']}), black dashed and dashed-dotted lines, respectively. (d) Dependence of the damping parameter $\alpha$ in zero magnetic field on the anisotropy parameter $\kappa$. (e,f) distribution of the magnons responsible for the magnon-magnon induced damping in 2D and 3D models, respectively.
• Figure 2: Magnon damping calculated from ab initio spin Hamiltonians. (a) Temperature-dependent damping in bulk YIG arising from magnon–magnon (m–m) interactions at $B=0.25\, {\rm T}$ and $B=1\, {\rm T}$, and from magnon–phonon (m–ph) interactions. Inset shows the positions of spin-up and spin-down Fe atoms in an 1/8 of YIG cubic cell. (b) Temperature-dependent damping in a CrSBr monolayer due to magnon–magnon (m–m) interactions at $B=0$ and $B=1\, {\rm T}$, and due to magnon–phonon (m–ph) interactions with both 2D and 3D phonons. The inset illustrates the monolayer crystal structure. (c) FMR linewidth $\delta\varepsilon$ in YIG, normalized to its value at $B=0.25\, {\rm T}$ The m–ph and m–m contributions are compared with the Gilbert damping and with Eq. (\ref{['mag3D']}), respectively. (d) FMR linewidth $\delta\varepsilon$ in a CrSBr monolayer, normalized to its value at $B=0$. The m–ph and m–m contributions are compared with the Gilbert damping and with Eq. (\ref{['mag2D']}), respectively. Results for both 2D and 3D phonons are shown, as indicated in the legend.
• Figure 3: (a) The processes responsible for damping. Color code: green arrow - ${\bf k}=0$ acoustic magnon; blue arrows—thermal magnons; red arrows—phonons. (b) Illustration of the separation of the dipole–dipole interaction into short-range (SR) and long-range (LR) contributions.
• Figure 4: (a) Low-energy magnon dispersion (black solid line) compared with the phonon dispersions obtained from ab initio calculations for a CrSBr monolayer (green solid line) and bulk CrSBr (blue dashed line). The black dashed line indicates the phonon energy that must be absorbed in the MC process. (b) Contributions of phonons with different wavevectors to the MC magnon–phonon damping in the case of two-dimensional phonons. (c) Corresponding contributions for three-dimensional phonons.
• Figure 5: (a) Comparison of the magnon dispersion of the ferromagnetic (FM) and antiferromagnetic (AFM) toy models with $\kappa = 10^{-2}$ and $10^{-3}$ as shown in legend. (b) Damping coefficient $\alpha$ at $T=50\,{\rm K}$ calculated for AFM toy model for different anisotropy parameters $\kappa$. The results are compared with power-law dependencies$\alpha \propto \kappa^\xi$.