Evaluating Gilbert damping in magnetic insulators from first-principles

Magnetic damping poses a significant impact on the performance of various magnetic and spintronic devices, making it a longstanding focus of research. The strength of magnetic damping is usually quantified by the Gilbert damping constants in the Landau-Lifshitz-Gilbert equation. Here we propose a first-principles-based approach to evaluate the damping constant contributed by spin-lattice coupling in magnetic insulators. The approach involves effective Hamiltonian models and spin-lattice dynamics simulations. As a case study, we applied our method to Y₃Fe₅O₁₂, MnFe₂O₄, and Cr₂O₃. Their damping constants were calculated to be 0.8×10^-4, 0.2×10^-4, 2.2×10^-4, respectively at a low temperature. The results for Y₃Fe₅O₁₂ and Cr₂O₃ are very close to the experimental results, while the large discrepancy in MnFe₂O₄ can be attributed to the inhomogeneity and small band gap in real samples. The stronger damping observed in Cr₂O₃, compared to Y₃Fe₅O₁₂, essentially results from its stronger spin-lattice coupling. In addition, we confirmed a proportional relationship between damping constants and the temperature difference of subsystems, which had been reported in previous studies. These successful applications suggest that our approach can be a viable candidate for estimating the Gilbert damping constant in magnetic insulators.