Microscopic origins of inertial magnetization dynamics

Technical Summary: Microscopic Origins of Inertial Magnetization Dynamics

Problem Statement
Recent ultrafast pump-probe, terahertz, and multi-tone spectroscopy experiments have revealed high-frequency components in ferromagnetic magnetization dynamics, specifically transient nutational oscillations occurring on picosecond and sub-picosecond timescales. While these observations have motivated the extension of the Landau-Lifshitz-Gilbert (LLG) equation to include an inertial term (the iLLG equation), the microscopic origin of this inertial behavior remains elusive. Existing theoretical justifications often rely on phenomenological arguments regarding the misalignment of magnetization and angular momentum vectors or classical treatments of the magnetization vector coupled to reservoirs. A critical gap exists in understanding the specific physical interactions that control the nutation time parameter ($\eta$) and explaining the significant variability in observed nutation frequencies and damping across different experimental setups and substrates.

Methodology
The authors derive magnetization dynamics from a microscopic magnon-phonon Hamiltonian using a non-Markovian quantum master equation (QME). The approach involves the following steps:

1. Microscopic Model: The spin system is described using the Holstein-Primakoff transformation within the spin-wave approximation, expressing magnetization components in terms of magnon operators. The system is treated as an open quantum system coupled to a phonon reservoir.
2. Interaction Mechanism: The authors analyze magnon-phonon coupling, identifying that for transverse magnetization dynamics (governed by $k=0$ magnon modes), the bilinear coupling via dipole-dipole interactions vanishes as momentum approaches zero. Instead, they establish that the primary mechanism is strain-modulated magnetic anisotropy, which couples transverse magnetization to optical phonon modes.
3. Non-Markovian Dynamics: A quantum master equation is employed to describe the magnon system's evolution, explicitly retaining memory effects. This allows the system's state at time $t$ to be influenced by its state at earlier times $s < t$, contrasting with Markovian approximations that assume instantaneous reservoir relaxation.
4. Green's Function Analysis: The phonon reservoir is analyzed via its retarded Green's function, decomposed into coherent quasiparticle poles (representing optical phonons) and an incoherent background.

Key Contributions and Results

• Microscopic Origin of Inertia: The paper demonstrates that inertial (nutational) dynamics arise directly from coherent interactions between the spin system and optical phonons in the lattice. The nutation frequency is identified with the optical phonon frequency ($\Omega_0$), typically on the order of a few terahertz.
• Role of Memory Effects: The study shows that allowing memory effects in the reservoir leads naturally to higher-order inertial terms in the average magnetization dynamics. The inertial character is inherited from the inertia of the optical lattice vibrations.
• Decomposition of Dynamics: The derived memory kernel separates the dynamics into two distinct parts:
  • Incoherent Background: At low frequencies, the incoherent part of the phonon Green's function leads to standard Gilbert-type damping (Ohmic dissipation).
  • Coherent Poles: The poles associated with optical phonons introduce high-frequency oscillations interpreted as nutation. The lifetime of these nutational oscillations is determined by the quasiparticle lifetime (damping) of the optical phonons.
• Explanation of Experimental Variability: The model provides a microscopic explanation for the variability in nutation timescales observed in experiments. Since the optical phonon frequency and lifetime are sensitive to substrate properties, interfaces, and phonon damping mechanisms, these environmental factors naturally account for differences in reported nutation parameters across different materials and setups.
• Comparison with iLLG: The authors derive a dynamical equation (Eq. 10) that is qualitatively similar to the phenomenological iLLG equation but differs quantitatively. Specifically, the amplitude of nutation in their microscopic model scales with the coupling strength squared ($|V_0|^2$) and the ratio of magnon to phonon frequencies, whereas the phenomenological iLLG amplitude is fixed by the inertial time $\eta$. The model predicts that the iLLG equation captures the high-frequency behavior only in specific limits and may overestimate amplitudes in the weak coupling regime.
• Longitudinal Dynamics: The formalism also predicts longitudinal magnetization dynamics (changes in magnetization magnitude) driven by the coupling to $k \neq 0$ magnon modes, providing a microscopic foundation for longitudinal relaxation models, particularly relevant near the Curie temperature.

Significance
The paper establishes magnon-phonon coupling, specifically via strain-modulated anisotropy and optical phonons, as the microscopic basis for inertial magnetization dynamics. By linking the nutation frequency directly to the optical phonon spectrum and the damping to phonon lifetimes, the work offers a verifiable physical mechanism for the inertial term. This understanding opens pathways for tailoring ultrafast spin dynamics and controlling magnetic states at terahertz frequencies by engineering substrate properties and phonon environments. Furthermore, the work clarifies the limitations of phenomenological iLLG models by showing that the inertial behavior is a specific consequence of coherent reservoir interactions rather than a generic property of angular momentum misalignment. The authors note that while the formalism can be extended to antiferromagnets, the strong hybridization between antiferromagnetic resonance frequencies and optical phonon modes requires further theoretical development.