Magnetic Inertial Dynamics in Ultrafast Spintronics

• Magnetic inertial dynamics is defined by the incorporation of acceleration terms into magnetization models, capturing ultrafast phenomena like nutation and switching.
• Generalized equations, derived from mesoscopic thermodynamics and microscopic relativistic treatments, predict THz resonance and novel spin-wave behaviors.
• Recent advances emphasize tensorial and chiral inertia effects, opening pathways for nonreciprocal magnonic devices and efficient ultrafast spintronic applications.

Magnetic inertial dynamics refers to the regime in magnetization dynamics where inertial (acceleration) effects, arising from the finite angular momentum of magnetic moments, play a key role on ultrafast timescales. Historically absent from standard models such as the Landau–Lifshitz–Gilbert (LLG) equation, inertial terms have been rigorously incorporated through both mesoscopic thermodynamics and microscopic relativistic treatments, leading to generalized dynamic equations. Their inclusion predicts phenomena—like nutation and novel ultrafast switching pathways—that are crucial in next-generation spintronic and magnonic devices as well as in the fundamental understanding of dynamic magnetism.

1. Theoretical Framework and Generalized Equations

In conventional magnetic dynamics, the magnetization vector $\mathbf{M}$ is governed by the LLG equation, which involves only first-order time derivatives: $$\frac{d \mathbf{M}}{dt} = \gamma \mathbf{M} \times \left( \mathbf{H} - \eta \frac{d \mathbf{M}}{dt} \right)$$ where $\gamma$ is the gyromagnetic ratio and $\eta$ the Gilbert damping parameter. This model does not include any explicit inertial (second derivative) contribution.

To resolve the absence of mechanical inertia, both phenomenological and microscopic approaches introduce a second time derivative of the magnetization, yielding the inertial LLG equation (ILLG): $$\frac{d\mathbf{M}}{dt} = \gamma \, \mathbf{M} \times \left[ \mathbf{H} - \eta \left( \frac{d\mathbf{M}}{dt} + \tau \frac{d^2\mathbf{M}}{dt^2} \right) \right]$$ The key new terms are:

• $\tau$: the angular momentum relaxation time, which sets the inertial timescale (typically femtoseconds),
• $\tau \frac{d^2\mathbf{M}}{dt^2}$: the acceleration term that introduces inertia,
• $\eta = I_1 / (\tau M_s^2)$, with $I_1$ a component of the inertia tensor.

Microscopically, when starting from the Dirac–Kohn–Sham Hamiltonian and applying the Foldy–Wouthuysen transformation, the inertial term naturally arises at order $1/c^4$ and is fundamentally a relativistic spin-orbit coupling effect (Mondal et al., 2017). The extended equation also connects the inertial damping tensor $\mathcal{I}$ to the real part of the inverse dynamic magnetic susceptibility tensor $\chi_m^{-1}$, thereby linking macroscopic dynamic behavior with microscopic electronic structure.

2. Nutation, Ultrafast Dynamics, and Resonance Phenomena

A core prediction of inertial magnetization dynamics is the occurrence of nutation—a rapid, damped oscillatory motion of the magnetization vector superimposed on conventional Larmor precession. Whereas precession is typically observed in the GHz range, nutation resonance appears in the THz regime.

Key formulas for resonance frequencies in a uniform ferromagnet (with static field $H$ and dimensionless damping $\alpha = \gamma \eta M_s$) are: $$\omega_\nu^\mathrm{(weak)} = \frac{1}{\alpha \tau} \sqrt{1 + \alpha \tau \gamma H}$$

$$\omega_\nu / (\gamma H) = \frac{\sqrt{1+\alpha\tau\gamma H}}{\alpha\tau\gamma H}$$

As $\tau$ or $H$ increases, $\omega_\nu$ approaches the THz regime, and the inertial contribution becomes experimentally significant. The scaling parameter $x = \alpha\tau\gamma H$ enables experimental data collapse onto universal curves for both precession and nutation frequencies (Olive et al., 2014).

Nutation’s experimental fingerprint is a second, higher-frequency resonance peak in ferromagnetic resonance (FMR) spectra. In materials with moderate to large damping ($\alpha$), inertial contributions also induce measurable shifts and broadening of the precession resonance, offering multiple routes for experimental detection.

3. Crossover: Inertial Versus Precessional Regimes

The angular momentum relaxation time $\tau$ delineates the crossover between inertial and conventional precessional dynamics:

• For $t \gg \tau$, inertial effects are negligible, and the standard overdamped, precessional LLG dynamics prevail.
• For $t \lesssim \tau$, the inertial term dominates, and the system exhibits memory of past acceleration—a haLLMark of inertial dynamics.

This crossover manifests as nutation in numerical simulations: after a rapid field change, rather than stopping instantaneously, the magnetization follows a path with looplike oscillations (nutation) before settling into precession.

4. Tensorial Inertia, Chirality, and Spin-Wave Effects

Recent advances reveal that the inertial parameter should be treated as a tensor ($\Delta_{ij}$), not just a scalar (Ghosh et al., 28 Aug 2024, Ghosh et al., 7 Sep 2025). This tensor decomposes into:

• Isotropic (scalar) inertia: shifts the nutation resonance down,
• Anisotropic (symmetric) inertia: introduces direction-dependent corrections,
• Chiral (antisymmetric) inertia: captured as $\varepsilon_{ijk} C_k$, where the vector $C$ yields handedness or directionality to inertial damping.

Implications include:

• Nutation resonance frequency redshifts with increased scalar and symmetric tensorial inertia,
• Chiral inertia leads to enhanced damping (broader linewidth) of the nutation mode,
• In multi-sublattice systems (e.g., two-sublattice ferromagnets), cross-sublattice and chiral inertial terms tune band crossings and enable nonreciprocal spin-wave propagation even in the absence of Dzyaloshinskii–Moriya interaction (DMI) (Mondal, 2021, Ghosh et al., 7 Sep 2025).

Linear spin-wave theory in these cases yields a quartic characteristic polynomial whose roots correspond to two precessional (Larmor-like) and two inertial (nutational) magnon bands. Experimental signatures include band crossing within the Brillouin zone and nonreciprocal dispersions in the presence of chiral inertia or DMI.

5. Statistical and Numerical Approaches

A statistical mechanical perspective extends the theory using a rotational Fokker–Planck equation derived in the enlarged $(\vec{e}, \vec{L})$ phase space, capturing both deterministic and stochastic inertial dynamics (Ciornei et al., 2010). The method of moments, via Taylor expansion of correlation functions, offers efficient analytic approximations for equilibrium properties and reveals sum rules (e.g., Gordon sum rule) imposed by the inertial term, ensuring physical high-frequency behavior (Titov et al., 2022).

For computational approaches, the additional acceleration term—raising the equation to second order in time—demands specialized algorithms. Structure-preserving integrators based on the implicit midpoint rule, sometimes employing auxiliary variables (generalized angular momentum), guarantee length and projection conservation as well as Lyapunov stability in the dissipative system (Ruggeri, 2021, d'Aquino et al., 2023). Multi-step and tangent plane schemes further ensure accurate capture of nutational and precessional dynamics, enabling large-scale simulation of ultrafast processes.

6. Applications: Ultrafast Switching, Magnonic Devices, and Nonreciprocity

Inertial magnetization dynamics enable new ultrafast switching protocols:

• The inertial term endows the magnetization with effective kinetic energy, facilitating deterministic switching across energy barriers within sub-picosecond timescales, outperforming thermally driven and purely precessional regimes (Ciornei et al., 2010, Fortunati et al., 22 Oct 2024).
• Analytical expressions for optimal field pulse duration and safe switching intervals can be derived using multiple time scale perturbation methods, yielding design criteria for ultrafast spintronic memory operation (Fortunati et al., 22 Oct 2024).
• The inertial regime enables resonant energy absorption at THz frequencies, opening new mechanisms for ultrafast information writing and reduction of energy costs (Neeraj et al., 2021).

Beyond single-domain systems, collective excitations in vortices, domain walls, and skyrmions display velocity-dependent effective masses and inertial reversal thresholds, explained by collective-coordinate models with deformation variables (Yoo et al., 2020, Wang et al., 4 Jul 2025).

Nonreciprocal magnonic devices exploit the combination of geometric chirality (torsion, Möbius topology), DMI, and chiral inertia to produce asymmetric spin-wave transport, tunable THz resonance modes, and engineered band gaps for magnonic logic and signal processing (d'Aquino et al., 7 Aug 2025, Ghosh et al., 7 Sep 2025).

7. Experimental Signatures and Open Directions

Experimental confirmation of inertial dynamics includes:

• Observation of nutation resonance peaks in THz FMR spectra for materials such as NiFe, CoFeB, and Co,
• Nonmonotonic field dependence of relaxation times in skyrmion-hosting compounds revealing double-peak structure at phase boundaries, consistent with inertial models (Saha et al., 28 Apr 2025),
• Pump-probe and spectroscopy methods targeting deviation of resonance peaks and line shapes as predicted by ILLG theory.

Open challenges remain in precise quantification and control of inertial parameters (e.g., sign and magnitude of the inertial relaxation time), disentangling inertial from other high-frequency effects (anisotropy, exchange, DMI), and realizing deterministic, low-loss switching pathways in antiferromagnetic and ferrimagnetic systems. Recent tensor generalizations and the interplay with topology, chirality, and geometry (nonorientable Möbius strips, curvilinear nanostructures) lay the foundation for architected ultrafast magnetism and magnonic technology at the THz frontier.