Unquenched orbital angular momentum as the origin of spin inertia

Abstract: The recent proposal and observation of spin inertia, and the consequent high-frequency spin nutation mode, have raised key questions for our understanding of magnetization dynamics, especially considering its high relevance for magnetic memories and ultrafast switching. Notwithstanding recent progress, a clear identification of spin inertia's physical origin thereby offering predictive power remains to be accomplished. Here, discussing general principles for identifying this physical origin, we examine unquenched orbital angular momentum (OAM) finding it to be a key candidate, despite its typically small value. Treating OAM and spin within a two-sublattice model, we derive the equivalent single-sublattice framework for magnetization dynamics making appropriate approximations. The latter naturally manifests the spin inertia term and parameter, which are otherwise introduced phenomenologically. The inertia parameter evaluated within our model is found to be in good agreement with its experimentally observed value in cobalt. We further delineate key experimental signatures that could verify or rule out the unquenched OAM as the origin of the observed high-frequency mode, and avoid a spurious optical mode in a two-sublattice ferromagnet from being identified as nutation. Our analysis offers a potential link between the recently emerged fields of orbitronics and spin inertia, thereby motivating investigations at their intersection.

• The paper demonstrates that unquenched orbital angular momentum, coupled via Russell-Saunders interaction, is the microscopic origin of spin inertia.
• It extends the traditional Landau-Lifshitz-Gilbert framework to include coupled spin and orbital dynamics, thereby explaining previously elusive high-frequency spin nutation modes.
• The effective model connects RS coupling strength and residual OAM magnitude to measurable spin inertia parameters, with predictions consistent with experimental data.

Unquenched Orbital Angular Momentum as the Physical Origin of Spin Inertia

Context and Motivation

Magnetization dynamics at ultrafast timescales are central to high-frequency applications such as magnetic memories and ultrafast reversible switching. The established Landau-Lifshitz-Gilbert (LLG) equation captures the precessional and damping dynamics of magnetization, but cannot explain recently observed high-frequency spin nutation modes and corresponding spin inertia phenomena. While earlier models commonly attributed spin inertia to phenomenological higher-order time derivatives or spin-orbit coupling, a robust microscopic origin for the experimentally measured spin inertia parameter (on the order of hundreds of femtoseconds) has remained unresolved, as ab initio calculations consistently underestimated its value. This work addresses the physical origin of spin inertia by linking it to unquenched orbital angular momentum (OAM) in solids.

Fundamental Principles and Modeling Framework

The central argument uses two symmetry- and dynamics-based constraints: (1) the spin inertia term is even under time-reversal, thus non-dissipative, and (2) any inertial (second order) term in the equations of motion for magnetization demands an additional coupled degree of freedom. Although OAM is typically quenched by the crystal field, a residual quantity always persists, as evidenced by deviations from the pure-spin $g$-factor in most magnetic materials. The authors propose that this residual OAM, coupled via Russell-Saunders (RS) spin-orbit interaction to spin, constitutes the relevant auxiliary degree of freedom, even though its amplitude is small relative to spin.

A two-sublattice model is constructed, with one sublattice representing spin and the other representing OAM, and crucially, the spin and OAM are coupled by an RS term parameterized by $\lambda$, which may be ferro- or antiferromagnetic. The LLG formalism is extended to coupled dynamics of the spin and OAM sublattices, and the collective excitations are analytically solved in the macrospin limit relevant for nanomagnets.

Key Results: Emergence of Spin Inertia and Nutation from Unquenched OAM

By systematically eliminating the OAM degree of freedom, the authors derive a closed effective equation for the spin magnetization. This equation naturally acquires a second time derivative, with a coefficient (the spin inertia parameter $\kappa$) fully determined by microscopic parameters: the RS coupling strength $\lambda$ and the relative magnitudes of spin and OAM. This result recovers the previously ad-hoc spin inertia term:

$$\dot{\mathbf{M}}_S = -\left|\gamma_S'\right| \mathbf{M}_S \times \mu_0 \mathbf{H}_{\mathrm{eff}} + \frac{\kappa}{|\mathbf{M}_S|} \mathbf{M}_S \times \ddot{\mathbf{M}}_S + \ldots$$

This analysis clarifies several crucial points:

• The spin inertia parameter $\kappa$ is nonuniversal and depends on RS coupling and OAM magnitude.
• The sign of $\kappa$, and thus the sense of nutational precession, encodes the sign (ferro- or antiferromagnetic) of RS coupling.
• For realistic materials, evaluating $\kappa$ with known $g$-factors, saturation magnetizations, and RS coupling constants yields values (e.g., $\sim 76$ fs for cobalt) quantitatively consistent with experimental measurements (2603.29421).
• The additional nutation resonance predicted by the model exhibits vanishing angular momentum and weak coupling to external fields, rationalizing its experimental elusiveness.
• A theoretical distinction is drawn between genuine spin nutation (OAM-based) and spurious high-frequency optical modes of multi-sublattice ferromagnets: the OAM-induced nutation mode yields an effective $\lambda$0-factor near unity (orbital value), while conventional optical modes yield $\lambda$1.

Numerical Comparisons and Experimental Consistency

Using material-specific parameters for Co ($\lambda$2, $\lambda$3, $\lambda$4 from atomic and crystal measurements), the predicted nutation frequencies and spin inertia constants are consistent with those observed in ultrafast experiments ($\lambda$5–$\lambda$6 s$\lambda$7, $\lambda$8–$\lambda$9 fs), addressing the persistent discrepancy between ab initio predictions and experiments (2603.29421). Notably, prior ab initio calculations neglected the explicit effect of OAM, supporting the present proposal.

Theoretical and Practical Implications

This analysis establishes unquenched OAM as a necessary and sufficient microscopic origin for the spin inertia phenomenon in magnetization dynamics. The following implications can be deduced:

• By quantifying spin inertia in terms of OAM content and RS coupling, the approach enables a priori predictions of high-frequency spin dynamic behavior across material classes.
• The result bridges two previously separate domains: the study of orbitronics (manipulation of OAM currents in solids) and ultrafast spin dynamics, suggesting tools from the former can be utilized to control inertial spin phenomena.
• Experimentally, the proposal can be tested by resolving the $\kappa$0-factor of the nutation mode with field-dependent spectroscopies and by material engineering to manipulate the OAM content.

Outlook and Future Directions

Microscopic understanding of spin inertia as arising from unquenched OAM sets the stage for targeted material searches and device engineering—especially exploiting recent advances in orbitronic control mechanisms to dynamically modulate OAM and, by extension, spin inertia. Given the potential to control the switching speed and spectral response of nanoscale magnets, this result is relevant both to fundamental magnetism and the design of next-generation spintronic and orbitronic devices.

It is anticipated that future work will integrate this OAM-driven spin inertia into advanced first-principles calculations, refine material-specific predictions of $\kappa$1, and experimentally validate the mode $\kappa$2-factor signature for nutation versus optical resonances.

Conclusion

Unquenched orbital angular momentum, though parametrically smaller than spin in most materials, is shown to be the physical source of spin inertia and the corresponding high-frequency nutation dynamics. By providing a concrete two-sublattice framework and comparison to experiment, the work establishes the relevance of OAM even in materials where its equilibrium value is minuscule and points to new directions for control of ultrafast spin phenomena by orbitronic means (2603.29421).