Nonlocal damping of spin waves in a magnetic insulator induced by normal, heavy, or altermagnetic metallic overlayer: A Schwinger-Keldysh field theory approach

Understanding spin wave (SW) damping and how to control it to the point of being able to amplify SW-mediated signals is one of the key requirements to bring envisaged magnonic technologies to fruition. Even widely used magnetic insulators with low magnetization damping in their bulk, such as yttrium iron garnet, exhibit a 100-fold increase in SW damping due to inevitable contact with metallic layers in magnonic circuits, as observed in very recent experiments [Bertelli et al., Adv. Quantum Technol. 4, 2100094 (2021)] mapping SW damping in a spatially resolved fashion. Here, we provide a microscopic and rigorous understanding of wave-vector-dependent SW damping using the extended Landau-Lifshitz-Gilbert equation with a nonlocal damping tensor instead of conventional local scalar Gilbert damping, as derived from Schwinger-Keldysh nonequilibrium quantum field theory. In this picture, the origin of nonlocal magnetization damping and thereby induced wave-vector-dependent SW damping is the interaction of localized magnetic moments of a magnetic insulator with conduction electrons from the three different types of metallic overlayers examined: normal, heavy, and altermagnetic. Due to the spin-split energy-momentum dispersion of conduction electrons in the latter two cases, the nonlocal damping is anisotropic in spin and space, and it can be dramatically reduced by changing the relative orientation of the two layers when compared to the usage of normal metal overlayer.