Anatomy of inertial magnons in ferromagnets

We analyze dispersion relations of magnons in ferromagnetic nanostructures with uniaxial anisotropy taking into account inertial terms, i.e. magnetic nutation. Inertial effects are parametrized by damping-independent parameter $\beta$, which allows for an unambiguous discrimination of inertial effects from Gilbert damping parameter $\alpha$. The analysis of magnon dispersion relation shows its two branches are modified by the inertial effect, albeit in different ways. The upper nutation branch starts at $\omega =1\mathrm{/}\beta$, the lower branch coincides with FMR in the long-wavelength limit and deviates from the zero-inertia parabolic dependence $\simeq {\omega }_{FMR}+D{k}^2$ of the exchange magnon. Taking a realistic experimental geometry of magnetic thin films, nanowires and nanodiscs, magnon eigenfrequencies, eigenvectors and $Q$-factors are found to depend on the shape anisotropy. The possibility of phase-matched magneto-elastic excitation of nutation magnons is discussed and the condition was found to depend on $\beta$, exchange stiffness $D$ and the acoustic velocity.