High-frequency susceptibility tensor in the inertial regime: Prediction for subterahertz magnonics in ferromagnets

Inertial spin dynamics alters the resonant spin wave excitation resulting in the modification of the resonance field and relaxation in microwave spectroscopy of ferromagnets described by the conventional Landau-Lifshitz-Gilbert (LLG) equation. Here we present a complete theoretical framework which quantitatively predicts the dynamic response of the magnetization for ferromagnetic systems exhibiting inertia. We compute the tensor of the high-frequency magnetic susceptibility analytically using the linearization of the inertial LLG equation, with fewer approximations than in the inertial Smit-Beljers approach. Our solution yields the precession (ferromagnetic resonance) and nutation resonance frequency, relaxation and intensity for an arbitrary free energy density, allowing straightforward quantitative analysis of inertial effects in magnetic resonances. Using the typical parameters of permalloy as an example we calculate the impact of materials parameters including Gilbert damping and inertial relaxation time on the conventional magnetic resonant response as well as the amplitude, linewidth, and frequency of the typically subterahertz nutation resonance.