Two-magnon scattering in the framework of the Lippmann-Schwinger equation

Technical Summary: Two-Magnon Scattering in the Framework of the Lippmann-Schwinger Equation

Problem Statement
Controlling magnetic losses is critical for the development of efficient information-processing technologies. While magnetic losses are typically described by the Gilbert damping parameter within the Landau–Lifshitz–Gilbert framework, experimentally measured ferromagnetic resonance (FMR) linewidths often contain extrinsic contributions beyond intrinsic damping. These include sample inhomogeneities, multi-magnon scattering processes (specifically two-, three-, and four-magnon scattering), and field-drag effects. Previous studies have largely separated these linewidth contributions phenomenologically without thoroughly exploring the underlying physical mechanisms governing the in-plane angular dependence of the linewidth. There is a need for a framework that links crystallography, the effective field on the sample scale, and two-magnon scattering to accurately identify dominant relaxation processes.

Methodology
The authors introduce a theoretical framework based on the Lippmann–Schwinger scattering equation and the Born approximation to model spin-wave scattering from weak defect potentials.

• Theoretical Model: Defects are treated as localized perturbations of the magnetic effective field that do not significantly alter the spin-wave dispersion. The scattered-wave amplitude is expressed as the product of the Fourier image of the defect's effective field contribution and the density of spin-wave states (Bloch function) in reciprocal space.
• Scattering Mechanism: The model posits that two-magnon scattering conserves frequency, requiring scattered magnon states to lie on the FMR isofrequency contour. The scattering efficiency is determined by the overlap between the perturbation of the effective field and degenerate spin-wave states.
• Experimental Validation: The framework was applied to yttrium iron garnet (YIG) films grown on gadolinium gallium garnet (GGG) substrates with (111) and (110) orientations. Samples (70 nm on GGG(111) and 24 nm on GGG(110)) were characterized using X-ray diffraction and vibrating sample magnetometry. FMR measurements were conducted in a flip-chip geometry, rotating the sample to determine in-plane anisotropy.
• Data Analysis: The FMR linewidth was analyzed by fitting the data to a model summing three contributions: inhomogeneous broadening, Gilbert damping, and two-magnon scattering strength ($\Gamma$).

Key Results

• Theoretical Insight: The analysis reveals that scattering is not solely determined by defect density but by the spatial symmetry of the defect potential and its reciprocal-space overlap with spin-wave states. For example, an elongated rectangular defect aligned with the external magnetic field suppresses scattering, whereas alignment perpendicular to the field allows scattering into backward volume modes, increasing the linewidth.
• Experimental Observations:
  • Intrinsic Damping: The extracted Gilbert damping parameter ($\alpha$) was found to be essentially identical for both orientations ($\approx 8\text{--}9 \times 10^{-4}$), indicating that intrinsic damping is not the source of the observed differences.
  • Anisotropy: The YIG/GGG(110) sample exhibited enhanced inhomogeneous broadening near $\phi = 90^\circ$, attributed to its reduced symmetry, whereas the YIG/GGG(111) sample showed nearly isotropic broadening.
  • Two-Magnon Scattering: The two-magnon scattering strength ($\Gamma$) displayed pronounced angular anisotropy in both samples. The maxima in scattering strength followed the high-symmetry crystallographic directions of the respective substrates.
• Correlation: The results establish a direct link between the perturbed effective field (likely arising from anisotropic surface morphology, such as elongated steps or line defects) and the observed linewidth anisotropies. The scattering is strongest when the dominant perturbation is elongated perpendicular to the magnetic field direction at which the maximum occurs.

Significance and Claims
The paper claims to provide a direct link between crystallography, the effective field on the sample scale, and two-magnon scattering. By utilizing the Lippmann–Schwinger framework, the authors demonstrate that the strength of two-magnon scattering can be controlled by the orientation between the magnetic field and the defect landscape. This approach offers a pathway to identify, suppress, or engineer extrinsic linewidth contributions. While demonstrated here for the uniform FMR mode, the framework is presented as general and applicable to propagating spin waves. This understanding is identified as essential for minimizing losses in magnonic devices and for designing geometries where defect-induced scattering is either avoided or deliberately exploited. The authors note that the specific origin of the effective field perturbations in the studied films remains to be fully determined but suggest that the relationship between perturbation geometry and scattering can be quantitatively characterized through further engineering of topographic features.