Engineering spin-wave spectrum via the magnetization inertia tensor

Magnetic inertial dynamics has recently been predicted and experimentally demonstrated in two-sublattice ferromagnets such as CoFeB and NiFe permalloy. In this work, we investigate the spin-wave spectrum of such systems by incorporating the complete magnetic inertia tensor. By decomposing the tensor into symmetric and antisymmetric components, we identify isotropic, anisotropic, and chiral contributions to magnetic inertia. Within linear spin-wave theory, we find that the spectrum comprises two precessional and two inertial magnon bands. Remarkably, the upper precessional band intersects the lower inertial band within the Brillouin zone. Both cross-sublattice and chiral components of the inertia tensor act as effective control parameters for tuning these magnonic band structures. Furthermore, we show that the inertial spin-wave spectrum becomes nonreciprocal along propagation directions where the Dzyaloshinskii-Moriya interaction is finite. Strikingly, a similar nonreciprocity can also arise purely from chiral inertia, even in the absence of Dzyaloshinskii-Moriya interaction. Our findings establish magnetic inertia as a new pathway to engineer nonreciprocal magnon transport and ultrafast spintronic functionalities.

💡 Research Summary

In this paper the authors develop a comprehensive theory of spin‑wave dynamics in two‑sublattice ferromagnets (such as CoFeB and NiFe) that explicitly includes the full magnetic‑inertia tensor Δij. Starting from the inertial Landau‑Lifshitz‑Gilbert (iLLG) equation, they decompose Δij into three physically distinct parts: (i) an isotropic scalar inertia ηij, (ii) a symmetric traceless anisotropic component Sij, and (iii) an antisymmetric, axial‑vector component Cij, which they refer to as chiral inertia. The isotropic part reproduces the familiar scalar inertial relaxation time, while the symmetric anisotropic part merely renormalizes the isotropic term and is neglected for the rest of the analysis. The focus is therefore on the cross‑sublattice scalar inertia ηAB (and its reciprocal ηBA) and the cross‑sublattice chiral inertia Cab (and Cba).

Using linear spin‑wave theory, the authors expand the spin vectors around the uniform ground state (magnetization along the x‑axis) in terms of small angular deviations β1,β2, introduce circular variables β±, and perform a Fourier transform. The resulting equations of motion for the two sublattices can be written in a compact 2×2 matrix form that contains first‑order (precessional) terms, Gilbert damping, and second‑order (inertial) terms. Assuming a harmonic time dependence e−iωt, the eigenvalue problem reduces to a fourth‑order polynomial in ω for each circular polarization (+ and –):

 P± ω⁴ + Q± ω³ + R± ω² + S± ω + T± = 0

where the coefficients encode the inertia (ηAA, ηBB, ηAB, Cab), the exchange J(k), the Dzyaloshinskii‑Moriya interaction D(k), the effective fields ΩA,B, the gyromagnetic ratios γA,B, and the Gilbert damping αA,B. Because the polynomial is quartic, there are four distinct complex eigenfrequencies for each k‑vector: two correspond to the conventional precessional magnons (ωlp, ωup) and two to the inertial (nutational) magnons (ωln, ωun).

Numerical solutions reveal several key features:

1. Four‑band structure – The spectrum consists of two low‑frequency precessional branches and two high‑frequency inertial branches. The upper precessional branch (ωup) and the lower inertial branch (ωln) intersect inside the Brillouin zone, forming a band‑crossing point that can be tuned by the cross‑sublattice inertia η′ = ηAB = ηBA.

2. Control via cross‑sublattice inertia – Increasing η′ shifts the inertial branches upward in frequency and simultaneously reduces their effective damping, thereby enhancing group velocities. This provides a direct knob for engineering magnon dispersion and attenuation.

3. Chiral inertia as a non‑reciprocity source – The antisymmetric component Cab introduces a term proportional to iCab(k) in the dynamical matrix. Even when the DMI D is set to zero, a finite Cab produces a k‑dependent frequency shift that makes ω(k) ≠ ω(−k). Thus chiral inertia alone can generate non‑reciprocal spin‑wave propagation, a phenomenon previously attributed solely to DMI.

4. Interplay with DMI – When DMI is present, both the D term and the chiral inertia term contribute additively (or subtractively depending on polarization) to the non‑reciprocal splitting. The combined effect can be substantial, offering a route to amplify or suppress non‑reciprocity by adjusting Cab.

5. Effective parameters become k‑dependent – The presence of inertia modifies the effective gyromagnetic ratio γeff and effective damping αeff, making them functions of the wave vector. In particular, the inertial branches exhibit larger group velocities and lower αeff than the precessional branches, suggesting that high‑frequency nutational magnons could serve as low‑loss carriers for THz magnonics.

6. Experimental relevance – Reported inertial relaxation times in CoFeB (≈300 fs) and NiFe (≈1.6 ps) differ markedly from ab‑initio predictions (few femtoseconds). The authors argue that the tensorial nature of inertia, especially the cross‑sublattice and chiral components, can reconcile these discrepancies by effectively renormalizing the scalar η measured in pump‑probe experiments.

Overall, the paper establishes magnetic inertia—not merely as a scalar correction—but as a full tensorial quantity that can be decomposed into isotropic, anisotropic, and chiral parts. The cross‑sublattice scalar inertia and the chiral inertia act as independent control parameters for magnon band engineering, enabling (i) tunable band crossings between precessional and inertial modes, (ii) adjustable group velocities and damping, and (iii) non‑reciprocal magnon transport even in the absence of DMI. These findings open new avenues for designing low‑dissipation, direction‑dependent magnonic devices and for exploiting ultrafast nutational dynamics in THz spin‑tronic applications.