Nonreciprocal inertial spin-wave dynamics in twisted magnetic nanostrips

We develop a theoretical framework for inertial spin-wave dynamics in three-dimensional twisted soft-magnetic nanostrips, where curvature and torsion couple with magnetic inertia to generate terahertz (THz) magnetic oscillations. The resulting spin-wave spectra exhibit pronounced nonreciprocity due to effective symmetry breaking arising from geometric chirality and inertial effects. We show that this behavior is governed by a curvature-induced geometric (Berry) phase, which we analytically capture through compact expressions for dispersion relations and spectral linewidths in both nutational (THz) and precessional (GHz) regimes. Topological variations, including Möbius and helical geometries, impose distinct wavenumber quantization rules, elucidating the role of topology in spin-wave transport. These results position twisted magnetic strips as a viable platform for curvilinear THz magnonics and nonreciprocal spintronic applications.

💡 Research Summary

In this paper the authors develop a comprehensive theoretical framework for inertial spin‑wave dynamics in three‑dimensional twisted soft‑magnetic nanostrips. Starting from the inertial Landau‑Lifshitz‑Gilbert (iLLG) equation, they incorporate a magnetic inertia term ξ∂²m/∂t², which is known to generate nutation (THz‑frequency) modes in addition to the conventional precessional (GHz) modes. The nanostrip is modeled as a ruled surface rS(u,v)=ra(u)+v rr(u) characterized by a constant curvature κ and torsion τ. The local orthonormal basis (e₁,e₂,e₃) follows the strip’s geometry, with e₃≈the equilibrium magnetization direction (tangent to the axis). The torsion produces a continuous rotation of the transverse basis vectors, described by a twist angle θ(u)=∫τ du+C. This rotation is identified as a geometric (Pancharatnam‑Berry) phase that accumulates along the strip and directly enters the spin‑wave phase.

Linearizing the iLLG equation around the equilibrium state and assuming the strip is ultrathin (thickness ≈ exchange length) and much longer than its width, the authors reduce the problem to a one‑dimensional eigenvalue equation for the transverse components φ₁(u) and φ₂(u). By Fourier transforming along the axial coordinate u, they obtain a fourth‑order characteristic polynomial P(ω)=0 (Eq. 4). Its four complex roots correspond to two pairs of modes: the high‑frequency nutation branch (ωN) and the low‑frequency precessional branch (ωP). Closed‑form approximations for the real parts (dispersion relations) and imaginary parts (linewidths) are derived as Eqs. 5‑8. These expressions explicitly show how curvature κ, torsion τ, exchange length ℓex, inertia ξ, and Gilbert damping α influence the spectra.

Key physical insights emerge:

1. Curvature‑induced frequency shift – Both ωN and ωP acquire a blue‑shift proportional to κ², but curvature alone does not break reciprocity; the dispersion remains symmetric in k.

2. Torsion‑induced non‑reciprocity – The term ±2kτℓex² appears with opposite signs in the nutation and precessional branches, making ω(k)≠ω(−k). Consequently, group velocities and linewidths are direction‑dependent, providing a built‑in mechanism for non‑reciprocal spin‑wave propagation.

3. Inertial nutation at THz frequencies – For typical permalloy parameters (ξ≈4.5×10⁻²), the fundamental nutation mode lies around 600 GHz, well within the THz band, while the precessional Kittel mode stays in the GHz range. Inertia thus opens a new high‑frequency channel.

4. Linewidth asymmetry – The damping‑induced linewidths ΔωN and ΔωP also contain τ‑dependent terms, leading to a maximum (minimum) linewidth for nutation (precession) at k≈0 and opposite behavior at larger k.

5. Large‑k universal velocity – In the limit |k|≫π/ℓex, both branches converge to a universal group velocity v_g=±γMsℓex√ξ, independent of κ or τ, reflecting the dominance of the inertial term.

6. Topological quantization – For finite strips, boundary conditions enforce quantization of k. Closed Möbius strips (single‑twist topology) impose half‑integer quantization (k = (n+½)π/L), whereas open helical strips retain integer quantization (k = nπ/L). This demonstrates that the global topology of the surface directly controls the spin‑wave spectrum, analogous to optical Berry‑phase experiments in Möbius cavities.

The authors discuss experimental implications, suggesting that femtosecond laser pulses or high‑frequency microwave excitations could selectively drive the nutation mode, while the non‑reciprocal dispersion could be harnessed for unidirectional magnonic waveguides, isolators, or spin‑wave diodes. The theory contains no adjustable parameters; all quantities are expressed in terms of material constants (exchange stiffness, saturation magnetization, anisotropy, damping) and geometric descriptors (κ, τ, strip dimensions). Thus, the work provides a predictive, quantitative tool for designing three‑dimensional curvilinear magnonic devices operating at THz frequencies with built‑in non‑reciprocity, opening a pathway toward next‑generation curvilinear magnonics and topological spin‑tronic technologies.