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Nonlinear Four-Magnon Scattering

Updated 1 September 2025
  • Nonlinear four-magnon scattering is a nonlinear interaction where two magnons produce two secondary magnons under energy and momentum conservation.
  • The process significantly reduces magnon attenuation lengths and broadens spectral peaks at high excitation levels, revealing a shift from linear to nonlinear dynamics.
  • Experimental μBLS techniques track spatial and spectral changes, providing key insights for accurately extracting damping parameters and optimizing magnonic device performance.

Nonlinear four-magnon scattering refers to a fundamental class of nonlinear interactions in magnetic systems whereby two magnons scatter and convert into two other magnons, subject to the conservation of energy and momentum. This process plays a central role in determining the behavior of magnons—quasiparticles representing collective excitations of the spin system—in regimes where excitation power, temperature, or geometry favor nonlinear dynamics. Four-magnon scattering processes govern magnon transport, relaxation, and spectral redistribution, and their experimental signatures and implications are directly accessible in contemporary magnonic materials through techniques such as microfocused Brillouin light scattering. The phenomenon establishes critical thresholds and effective limits on magnon propagation, nonlinear loss, device performance, and the interpretation of damping parameters.

1. Physical Mechanism of Four-Magnon Scattering

Four-magnon scattering is a quartic nonlinear process intrinsic to magnetic systems with exchange interactions. When a population of magnons is driven, typically by microwave (rf) excitation at a fixed frequency frff_{\mathrm{rf}}, low power ensures that spin dynamics persist in the linear regime. Beyond a critical excitation threshold, the amplitude of the driven (primary) magnon mode becomes high enough that magnon–magnon interactions—specifically, four-magnon interactions—dominate the nonlinear behavior.

The canonical four-magnon process involves two primary magnons excited at frff_{\mathrm{rf}} combining and scattering into two secondary magnons at frequencies f1=frfδff_1 = f_{\mathrm{rf}} - \delta f and f2=frf+δff_2 = f_{\mathrm{rf}} + \delta f. The process is governed by energy conservation,

f1+f2=2frff_1 + f_2 = 2f_{\mathrm{rf}}

and, for systems supporting a continuous dispersion, quasi-momentum conservation. This mechanism distinguishes four-magnon scattering from, for example, three-magnon splitting, which may be disallowed by the lack of available final states at certain frequencies.

Moreover, four-magnon scattering becomes efficient in systems with a broad, quasi-linear magnon dispersion, such as microstructured Co25_{25}Fe75_{75} waveguides. Under these conditions, secondary magnons with shifted frequencies ±δf\pm \delta f and corresponding wavevectors can always be found to conserve both energy and momentum (Hula et al., 2020).

2. Mathematical Formalism and Regimes

The theoretical representation of four-magnon scattering employs a quartic interaction Hamiltonian, usually constructed within the Holstein–Primakoff boson formalism. For the mode amplitudes aka_k, the relevant term in the Hamiltonian is:

H4=14Nk1,,k4δk1+k2,k3+k4W(k1,k2;k3,k4)ak1ak2ak3ak4H_4 = \frac{1}{4N} \sum_{k_1,\ldots,k_4} \delta_{k_1+k_2,\,k_3+k_4} W(k_1,k_2;k_3,k_4)\, a_{k_1}^\dagger a_{k_2}^\dagger a_{k_3} a_{k_4}

where W()W(\cdots) is the scattering vertex determined by exchange interactions. The lowest-order kinetic equation for the magnon population NkN_{k} contains a collision integral involving this H4H_4 vertex.

For spatially-extended systems, the nonlinear regime is characterized by a deviation from simple exponential decay:

I(x)=exp(2xLatt)+cI(x) = \exp\left(-\frac{2x}{L_\text{att}}\right) + c

where LattL_\text{att} is the magnon attenuation length. The presence of significant four-magnon scattering leads to a decrease of LattL_\text{att} due to redistribution of energy from the primary mode into secondary spectral components, notably visible as spectral broadening and as a reduced intensity of the directly excited magnon mode in experimental data (Hula et al., 2020). In the linear regime, LattL_\text{att} is uniquely determined by intrinsic Gilbert damping; whereas in the nonlinear regime, the addition of four-magnon interaction channels results in enhanced effective damping near the source.

3. Experimental Observations and Diagnostics

Microfocused Brillouin light scattering (μBLS) is employed to resolve the manifestation of four-magnon scattering in real devices. μBLS provides both spatial mapping—on the order of 350 nm resolution—and spectral sensitivity sufficient to track both direct and scattered magnon populations. The decrease in magnon propagation length at high excitation powers is a key signature of four-magnon scattering. Experiments in Co25_{25}Fe75_{75} waveguides demonstrate that for microwave powers above 15\sim15 dBm, the BLS signal at frff_{\mathrm{rf}} broadens and its spatial decay accelerates. Fittings of intensity decay close to the antenna yield apparent LattL_\text{att} values that are significantly shorter than those found at low power.

Honoring the distinction between the linear and nonlinear regimes is critical: spatially-resolved measurements reveal that LattL_\text{att} recovers to its linear value when exponential fits are performed over distances sufficiently far from the source (where four-magnon scattering is less active). Thus, nonlinear dissipation is spatially inhomogeneous and concentrated near the excitation region (Hula et al., 2020).

Table: Power-Dependent Phenomena in μBLS Measurements

Excitation Power Spectral Signature Measured LattL_\text{att} (near source) Physical Interpretation
Low Narrow peak at frff_{\mathrm{rf}} High Linear regime, only Gilbert damping
High Broadened spectrum Low Nonlinear regime, four-magnon loss

4. Implications for Damping Extraction and Device Characterization

Four-magnon scattering imposes strict limits on the reliable extraction of magnetic damping constants from propagation length measurements. Standard practice involves fitting the spatial decay of the magnon intensity to an exponential form to determine the intrinsic damping. If performed in or near the nonlinear regime, such analysis leads to an overestimation of damping, as the apparent losses now include additional, spatially-inhomogeneous four-magnon contributions. Thus, precise determination of intrinsic damping parameters in low-damping materials such as Co25_{25}Fe75_{75} requires maintenance of micro-magnon transport within the linear dynamic regime during μBLS experiments.

The necessity of power-dependent studies follows directly: only by establishing the threshold for the onset of nonlinear scattering and restricting fitting regimes to power levels well below this threshold can damping be correctly quantified (Hula et al., 2020).

Four-magnon scattering is a specific and distinct nonlinear process. Compared to three-magnon interactions—which may be forbidden by phase space constraints—four-magnon processes are often permitted by the structure of the magnon dispersion in planar waveguide systems. The redistribution of magnon energy in four-magnon scattering leads to a continuum of secondary modes rather than discrete sidebands, as detected in the width of BLS spectra. The underlying quartic nonlinearity is rooted in strong exchange interactions, in contrast to the dipolar mechanism that mediates three-magnon processes.

The broader impact of four-magnon scattering extends to various dynamical and thermal properties: it governs the efficiency of magnonic devices, sets bounds on spin-wave logic operations, and presents challenges to the interpretation of spin transport and damping. Four-magnon processes also play a role in limiting the coherence and transport of Bose–Einstein condensates of magnons and are central to nonlinear loss channels in magnon lasers and high-power spintronic circuits, as established by related work in quantized waveguides and multimode YIG systems.

6. Conclusion

Nonlinear four-magnon scattering sets a fundamental limit on magnon propagation in microstructured magnetic materials when operated at high excitation power. The transition from the linear to nonlinear regime is accompanied by a distinct redistribution of energy from the directly excited (primary) magnon mode to a continuum of secondary modes, a reduction in the apparent magnon attenuation length due to increased effective damping, and clear spectral signatures in μBLS experiments. Proper extraction of intrinsic damping values and understanding of device performance in low-damping materials such as Co25_{25}Fe75_{75} requires operation within the linear regime and careful consideration of the power-dependent onset of four-magnon processes. The mathematical framework developed to describe these phenomena, especially the energy-conservation condition f1+f2=2frff_1 + f_2 = 2f_{\mathrm{rf}} and the exponential intensity decay law, underpins the analysis and enables reliable comparison across materials and device architectures.

In summary, four-magnon scattering is a key nonlinear mechanism within magnon transport physics, with practical implications for device engineering, performance benchmarking, and the interpretation of high-power magnonic measurements.

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