Spinwavetoolkit Reveals Analytical Spin-Wave Dynamics in Ferromagnetic Films and Bilayers

Researchers are increasingly focused on understanding and harnessing spin-wave dynamics for novel information technologies, but detailed theoretical modelling can be computationally expensive and time-consuming. Now, Jan Klíma from CEITEC BUT, Ondřej Wojewoda from MIT, and Jakub Krčma et al. have addressed this challenge with SpinWaveToolkit, a new open-source Python package for (semi-)analytical calculations in spin-wave physics. Combining established analytical models with a dynamic-matrix approach, the toolkit rapidly calculates key spin-wave properties and even simulates micro-focused Brillouin light scattering spectra, offering speed improvements of almost two orders of magnitude compared to conventional numerical methods. This advancement promises to accelerate research in magnonics by facilitating efficient experiment design, data interpretation, and parameter optimisation.

Modelling Spin Wave Dynamics with the SpinWaveToolkit Python Package is now straightforward and efficient

Scientists have developed SpinWaveToolkit (SWT), a new open-source Python package designed for the (semi-)analytical modelling of spin-wave dynamics in thin ferromagnetic films and exchange-coupled magnetic bilayers. This innovative work combines analytical models, rooted in Kalinikos-Slavin theory, with a semi-analytical dynamic-matrix approach, allowing for the precise calculation of crucial spin-wave characteristics.
These include dispersion relations, group velocities, decay lengths, mode profiles, and static equilibrium magnetization states, offering a comprehensive toolkit for researchers in the field. The research establishes a versatile framework for understanding and manipulating spin-wave behaviour at a fundamental level.

The team achieved a quantitative model of micro-focused Brillouin light scattering (BLS) within SWT, incorporating vectorial focusing, spin-wave Bloch functions, magneto-optical coupling, and Green-function propagation to accurately simulate experimentally measured BLS spectra. This capability bridges the gap between theoretical modelling and experimental observation, enabling direct comparison and validation of results.

By simulating BLS spectra, the package facilitates a deeper understanding of the underlying spin-wave dynamics and provides a powerful tool for interpreting experimental data. Experiments show that SWT has been rigorously validated against finite-element dynamic-matrix simulations performed with TetraX, across Damon-Eshbach, backward-volume, forward-volume, and oblique-field geometries.

This validation demonstrates excellent agreement with numerical simulations, while simultaneously reducing computation times by nearly two orders of magnitude. This significant speed-up allows for exploratory mapping of parameter spaces and facilitates the fitting of measured dispersion relations, offering a substantial advantage over traditional computational methods.

The study unveils a versatile and efficient framework for experiment design, interpretation, and parameter optimization in magnonics research. SWT’s ease of use and rapid calculation times enable not only systematic exploration of a wide range of parameters but also the precise fitting of experimental data, such as dispersion relations and Brillouin light scattering spectra. This breakthrough opens new avenues for the development of advanced magnonic devices and materials with tailored spin-wave properties, potentially impacting fields like data storage and information processing.

Spin-wave modelling using analytical theory, dynamic matrices and Brillouin light scattering simulation provides valuable insights into magnetic materials

Scientists developed SpinWaveToolkit (SWT), an open-source Python package for modelling spin-wave dynamics in thin ferromagnetic films and magnetic bilayers. The work combines analytical models derived from Kalinikos-Slavin theory with a semi-analytical dynamic-matrix approach to calculate dispersion relations, group velocities, decay lengths, mode profiles, and static equilibrium magnetization states.

Furthermore, the study implements a quantitative model of micro-focused Brillouin light scattering (BLS) incorporating vectorial focusing, spin-wave Bloch functions, magneto-coupling, and Green-function propagation to simulate experimentally measured BLS spectra. Researchers validated the package against finite-element dynamic-matrix simulations performed with TetraX for Damon-Eshbach, backward-volume, forward-volume, and oblique-field geometries.

This comparison demonstrated excellent agreement, with SWT reducing calculation times by nearly two orders of magnitude compared to the numerical simulations. The team engineered two models for infinite thin ferromagnetic films: a fully analytical model based on zeroth perturbation and a semi-analytical approach calculating eigenvalues of the interaction matrix.

The analytical model rapidly calculates dispersion relations for any in-plane and out-of-plane magnetization angle, while the numerical model currently supports arbitrarily in-plane and completely out-of-plane magnetization. To model interfacially-coupled magnetic bilayers, scientists implemented a two-macrospin model which semi-analytically solves micromagnetic energy equilibrium and then calculates the eigenvalues of the system matrix to retrieve the dispersion relation.

The study introduces a ‘Material’ class storing magnetic parameters in SI units, including saturation magnetization (Ms), gyromagnetic ratio (γ), exchange stiffness constant (Aex), Gilbert damping constant (α), and inhomogeneous broadening (∆H0). These material objects serve as direct inputs for the dispersion models.

For homogeneous thin films, the approach calculates the explicit expression for the spin-wave dispersion relation ω2 n = ωH + ωMl2 exk2 ωH + ωMl2 exk2 + ωMFn, where ωH = μ0γHext, ωM = μ0γMs, l2 ex = 2Aex M 2s μ0 −1, and k is the wavevector. This analytical implementation allows for totally pinned, unpinned, and partially pinned boundary conditions, and arbitrary propagation direction within the sample plane.

Spin-wave modelling and Brillouin light scattering simulation with SpinWaveToolkit

Scientists have developed SpinWaveToolkit (SWT), an open-source Python package for modelling spin-wave dynamics in thin ferromagnetic films and magnetic bilayers. The work combines analytical models, based on Kalinikos-Slavin theory, with a semi-analytical dynamic-matrix approach to calculate dispersion relations, group velocities, decay lengths, mode profiles, and static equilibrium magnetization states.

Additionally, SWT incorporates a quantitative model of micro-focused Brillouin light scattering (BLS) that accounts for vectorial focusing, spin-wave Bloch functions, magneto-coupling, and Green-function propagation to simulate experimentally measured BLS spectra. Experiments validating the package against finite-element dynamic-matrix simulations, performed with TetraX, demonstrate excellent agreement for Damon-Eshbach, backward-volume, forward-volume, and oblique-field geometries.

Crucially, SWT reduces calculation times by nearly two orders of magnitude compared to these numerical simulations. This speed increase allows for exploratory mapping of parameter space and facilitates the fitting of measured dispersion relations and related parameters. The analytical model within SWT calculates dispersion relations for any in-plane and out-of-plane angle of magnetization.

However, accuracy can decrease when modes approach each other in wavevector, frequency space, or in the region between dipolar and exchange dominated regimes. To address this, a numerical model calculates eigenvalues of the interaction matrix, currently limited to arbitrarily in-plane and completely out-of-plane magnetized layers.

For interfacially-coupled magnetic bilayers, such as synthetic antiferromagnets, SWT utilizes a two-macrospin model that semi-analytically solves micromagnetic energy equilibrium. The package introduces a ‘Material’ class storing magnetic parameters in SI units, including saturation magnetization (Ms), gyromagnetic ratio (γ), exchange stiffness constant (Aex), Gilbert damping constant (α), and inhomogeneous broadening (∆H0).

For example, the Nickel-Iron material definition sets Ms to 800e3 (A/m), Aex to 16e-12 (J/m), γ to 28.82e9π (rad.Hz/T), and α to 7e-3. The fully analytical implementation calculates spin-wave dispersion using the equation ω2 n = ωH + ωMl2 exk2 ωH + ωMl2 exk2 + ωMFn, where ωH = μ0γHext, ωM = μ0γMs, and l2 ex = 2Aex M 2s μ0 −1.

The MacrospinEquilibrium class numerically minimizes the free energy of a macrospin under an external field, with arbitrary demagnetizing and uniaxial anisotropy tensors. Total free energy density is calculated as εtot = −μ0Ms m · Hext + 1 2μ0M 2s m Ndm + 1 2μ0M 2s m Nunim, where m is the magnetization unit vector, Nd is the demagnetization tensor, and Nuni is the uniaxial anisotropy tensor. This allows for the calculation of dispersion relations, such as for the n = 0 mode, with increased efficiency.

Validating SpinWaveToolkit through Brillouin light scattering and dynamic matrix comparisons reveals strong agreement

Scientists have developed SpinWaveToolkit (SWT), an open-source Python package designed for modelling spin-wave dynamics in thin ferromagnetic films and exchange-coupled magnetic bilayers. SWT integrates analytical models, based on Kalinikos-Slavin theory, with a semi-analytical dynamic-matrix approach, allowing for calculations of dispersion relations, group velocities, decay lengths, mode profiles, and static equilibrium magnetization states.

Furthermore, the package incorporates a quantitative model of micro-focused Brillouin light scattering (BLS), simulating experimentally measured spectra by accounting for vectorial focusing, spin-wave Bloch functions, magneto-coupling, and Green-function propagation. The package’s validation against finite-element dynamic-matrix simulations using TetraX, across Damon-Eshbach, backward-volume, forward-volume, and oblique-field geometries, demonstrates excellent agreement with a reduction in calculation time of almost two orders of magnitude.

This speed, combined with ease of use, facilitates both exploratory parameter mapping and the fitting of measured dispersion relations, establishing SWT as a versatile framework for magnonics research, experiment design, interpretation, and parameter optimisation. The authors acknowledge a limitation in that the current models do not account for all possible material complexities or experimental conditions. Future work could focus on extending the package to incorporate additional physical effects and broadening its applicability to a wider range of magnonic systems, as suggested by the research team.