Introduction

The calculation of the spin-wave dispersion f(k) is a frequently encountered task when dealing with magnetization dynamics. This JAVA app deals with tangentially magnetized thin films and typical structures made thereof. The following lines explain the assumptions made for the calculations and the different options.

The app relies on a theoretical description [1] which compromises both dipolar and exchange contributions. Note that the relevant equations were evaluated for the ideal case of totally unpinned spins, for the zeroth order mode (no quantization normal to the film). Pertubational corrections are excluded for the sake of simplicity (diagonal approximation). This approach is indeed good for thin films, it gets worse when the wavelength has the same order of magnitude as the thickness. In principle pertubation corrections gain importance, if eigenmodes - e.g. n=0 and n=2 - with same symmetry approach and intersect each other.

The first step is the choice of the material parameters, namely saturation magnetization Ms, exchange constant A, gyromagnetic ratio g, and Gilbert damping constant α. Two typical materials appear as predefined options - Ni80Fe20=Py (Permalloy) and yttrium iron garnet (YIG). For YIG the unit of the geometrical scales switches from nanometer to micrometer. The thickness of the film and the strength of the externally applied magnetic field H enter as a second set of adjustable parameters. Note that adjusting scrollbars is automatically updated in the graph, whereas clicking on checkboxes requires a subsequent click on the 'Calculate' button.

No confinement

With the standard options enabled, the app calculates the dispersion for an infinitly extended film, starting at k=0 going up to k_max. Below the main graph, a second graph shows the group velocity dw/dk. The wave vector and the external field span a certain angle. This angle and the magnitude of k_max can be adjusted by the scrollbars. In thicker films perpendicular standing spin waves (PSSWs) eventually become important. To judge the validity of the calculation, the first ten orders are calculated and shown if their frequency is inside a certain interval around the range given by the zeros-order mode. Left-clicking into the graph probes the spectrum (filled black circle). The chosen wave vector, frequency and group velocity df/dk are shown in the lower right panel, together with a schematic representation of H and k. The mode profile is also depicted.

Partial confinement

Confining the spin waves in one direction leads to the case of a waveguide. The wave vector is now quantized in the direction perpendicular to the side of the waveguide. The index of quantization can be adjusted, the first order solution is the standard option. The app distinguishes between two situations: either the external field is parallel or perpendicular to the side of the waveguide. In the first case the spin wave is a so-called Backward Volume mode (BWVM), whereas in the latter case one speaks of the Damon Eshbach (DE) modes. In the DE configuration the internal magnetic field H_int is reduced due to fictous magnetic charges at the boundaries of the waveguide. The local inhomogeneity of the internal field in principle demands more complex calculations of quantization integrals. Here, the values of H_int at the center is calculated relying on the theory of Joseph and Schlömann [2], in order to make a simple estimation. The app uses H_int(center) instead of the external field H to compute f(k), shown in the upper graph. When exciting the spin wave with standard antennas, the excitation efficiency depends on the width of the antenna (Fourier Transformation yields a sinc-like dependance). The green curve in the upper graph depicts this efficiency as a function of the wave vector. Higher order quantized waves have in  a second sinc-factor, which originates in the finite width of the waveguide (second order modes for example have a vanishing excitation efficiency). The lower graph again shows the group velocity.Left-clicking the upper graph again probes the spectrum. The scheme in the lower right panel now visualizes wave guide, antenna and the excited wave. In the BWVM configuration, the internal magnetic field is not reduced and the linear excitation with an antenna is related to the out-of-plane component. In this situation, the width of the antenna is less important. Note that damping leads to a finite propagation length.

Total confinement

Total lateral confinement results in a fully quantized spectrum. For this case, the app allows adjustment of the number of depicted eigenmodes, via the indicees m and n, and a variation of the lateral sizes, via l_z and l_y. The internal magnetic field is again reduced and estimated at the center for the calculations. The structure is treated as a rectangle. Note that this shape approximates elliptical or circular discs. Practical experience shows, that the eigenfrequencies of the latter do not differ strongly from the rectangular shape.
Left-clicking the graph selects the eigenmode which is nearest to the point clicked. The profile of the mode and frequency are shown in the lower right panel. Note that in very small samples, the confinement induced by the internal field leads to mode localization. Such edge modes generally have frequencies lower than the center modes, which are approximately described by the app. These edge modes eventually dominate the spectrum.

[1] Kalinikos & Slavin: J.Phys.C: Solid State Phys. 19 (1986) 7013-7033
[2] Joseph & Schlömann: J. Appl. Phys. 36 (1965) 1579-1593

Spin Wave App (Will open up in a seperate browser window)

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