The onset of additional high-frequency resonance in ferromagnets due to inertia of magnetization has been shown first theoretically [1] and then experimentally [2,3]. The existence of sub-THz resonant frequency in ferromagnetic materials and ability to excite the magnetization nutation coupled to its precession significantly broadens the potential range of applications of ferromagnets in ultrafast magnetism. While several theoretical studies have been mostly addressing the dependence of the nutation resonance frequency on the inertial relaxation time [4,5], the understanding of the influence of the materials parameters on the overall magnetic susceptibility behavior including the precession and nutation magnetic resonances is still missing. We present an analytical study of the linearized high-frequency susceptibility tensor using the Polder-Ansatz for the ferromagnetic systems exhibiting the inertia of magnetization and resolve the effect of Gilbert damping constant and inertial time as well as applied magnetic field on the intensity and linewidth of nutation resonance. We show that the inertia in general leads to the reduction of the ferromagnetic resonance frequency for both aligned and non-aligned modes, and illustrate it with examples of single-crystalline thin films with cubic and uniaxial anisotropy. For an out-of-plane applied magnetic field the frequency dependence of resonance field becomes non-linear, in contrast to conventionally used Kittel formula. We also find that the nutational resonance frequency increases with the magnetic anisotropy and field. We find that a larger inertial relaxation parameter leads to an increase of nutation amplitude and to the decrease of nutation resonance linewidth, thus, favoring its experimental detection. In contrast, the nutation resonance in ferromagnetic materials with high Gilbert damping is expected to be broad and of low amplitude. The relative intensity of the nutation peak in susceptibility as compared to precession is boosted significantly by the increase of inertial term which further confirms that the materials with higher inertial relaxation parameter might be the best candidates for investigating the novel phenomena of inertial spin dynamics. Our study provides a quantitative pathway to identify effects of inertia in broadband microwave spectroscopy. We conclude that inertia needs to be taken into account for an accurate evaluation of magnetic parameters such as magnetic anisotropy and g-factor, and for an interpretation of spin dynamics experiments, especially at high magnetic fields (> few T). Our findings might contribute to a new concept of high speed information processing with THz frequencies which has recently moved into the focus of spintronics.

[1] J.-E. Wegrowe, M.-C. Ciornei, Am. J. Phys. 80, 607 (2012)

[2] K. Neeraj et al., Nature Phys. 17, 245 (2021)

[3] V. Unikandanunni et al., PRL 129, 237201 (2022)

[4] M. Cherkasskii, M. Farle, A. Semisalova, Phys. Rev. B 102, 184432 (2020)

[5] M. Cherkasskii, I. Barsukov, R. Mondal, M. Farle, A. Semisalova, Phys. Rev. B 106, 054428 (2022)