Nutational resonance modes in antiferromagnetic materials

Abstract

The Landau-Lifshitz-Gilbert (LLG) equation is well-established to describe the spin dynamics of magnetic materials. This first-order differential equation is based on the assumption that the spin angular momenta and corresponding magnetic moments are always parallel. While this assumption is largely unproblematic, both theoretical considerations and experimental results have indicated that the two may become separated on ultrafast timescales, giving rise to inertial dynamics along with a modified spin wave dispersion. Here, we apply linear spin wave theory to the inertial LLG equation to compute the eigenmodes of the altermagnetic materials SmErFeO₃ and α-Fe₂O₃. We find the largest influence of nutation on the magnetic resonances in the case of hematite, which exhibits both a sizeable shift of the resonance frequencies as compared to the inertia-free case and additional nutational resonances that are in a similar order of magnitude to the materials' higher-frequency precessional exchange modes. While the realistic magnitude of the inertial parameter remains an open question, we hope that our quantitative analysis provides the starting point for further experimental investigations.

Keywords: Antiferromagnetic Resonance; Nutation; Spintronics.

Fig. 1

Sketch of precessional (a) and nutational (b) eigenmodes of an antiferromagnet with inertia. A superposition of both modes (c) leads to a nutating precession of the sublattice magnetisation vectors . Because this is an antiferromagnet, there also exists another set of modes with the opposite sense of rotation.

Fig. 2

Properties of the precessional and nutational resonance modes in the toy model FM as a function of the inertial parameter : (a) resonance frequencies, (b) lifetimes, and (c) effective lifetimes. The black dotted lines indicate the leading-order terms.

Fig. 3

Properties of the precessional and nutational resonance modes for the toy model AFM as a function of the inertial parameter: (a) resonance frequencies, (b) lifetimes, and (c) effective lifetimes. The leading terms are indicated by the dashed lines for small  and the dotted lines for large . In (a), the two dotted lines are and for nutational and precessional modes, respectively. In (c), they are given by and . Because the AFM has two magnetic sublattices, each line represents two degenerate branches.

Fig. 4

Properties of precessional (blue, turquoise) and nutational (orange, red) resonance modes for the orthoferrite and hematite: (a, b) resonance frequencies, (c, d) lifetimes, and (e, f) effective lifetimes. In the orthoferrite, the higher-frequency modes (m3 to m8) come in pairs that are very close together but still distinct. In hematite, by contrast, all modes come in two exactly degenerate pairs. Again, leading terms are indicated by the dashed and dotted lines (for small and large , respectively).

Fig. 5

Dynamics of the sublattice magnetisation vectors of the nutational resonance modes (m5 to m8) for the orthoferrite (a–d) and hematite (e–h) model, sorted by frequency in ascending order. The net magnetisation vector is amplified by a factor of 250 in (a) and (b) and by a factor of two in the other plots (in ( e–f), the total magnetisation vanishes).

Fig. 6

Comparison of the precessional modes of the orthoferrite (EFO) and hematite (FO) to the resonance modes of the toy-model FM and AFM. In (a), for each resonance mode, the frequency normalised to its value in the inertia-free case is plotted versus the inertial parameter normalised to the period duration of that mode in the inertia-free case. Figures (b) and (c) show the same for the lifetimes and effective lifetimes.