The magnetic particle in a box: Analytic and micromagnetic analysis of probe-localized spin wave modes

The study of dynamic ferromagnetic phenomena at submicron lengthscales is important for the development of spintronic devices that exploit spin-torque,¹ such as spin-torque memory² and spin-torque oscillators.³ Understanding the behavior of magnetization dynamics in the presence of nonuniform fields is necessary to model confined dynamics such as in edge modes^4,5 and the coupling of magnetic dynamics to electromagnetic fields.⁶ The confinement of localized modes by the dipole field from a micron-sized particle combined with measurement by ferromagnetic resonance force microscopy (FMRFM) has shown promise as a non-invasive technique for the imaging of the internal fields of ferromagnetic films^7,8 as well as the study of size-dependent relaxation.⁹ The inhomogeneity of the precessing magnetization within the mode makes calculation of the resonance fields for these modes challenging, typically requiring time-consuming micromagnetic or numerical analysis. In this paper, we compare these two methods in addition to an analytic solution to the localized mode problem that facilitates computationally fast analysis in the regime, where exchange fields are negligible.

Micromagnetics have previously been used to model the complicated problem of in-plane localized modes,⁸ for which a simple solution is difficult to obtain due to the non-symmetric nature of the problem. Here, we apply micromagnetics to the modeling of localized modes in an out-of-plane geometry for which the modes are cylindrically symmetric and for which numerical solutions have previously been obtained.^7,9 We use an implementation of micromagnetics that calculates the eigenfrequencies of a system of exchange-coupled spins, taking into account both exchange and dipole fields. The result of micromagnetics in this case shows good agreement with the experimental data for the first four localized modes, as shown in Fig. 1, and the results are almost indistinguishable from the numerical solution published previously,⁹ confirming that micromagnetic calculations can accurately model the behavior of these localized modes.

It is illuminating to compare the mode profiles produced by micromagnetic simulations to the Bessel functions assumed for the numerical minimization technique. The resulting mode shapes are given in Fig. 2. From the figure, it is clear that the mode shapes found by micromagnetics are qualitatively similar to the Bessel function approximations used by the numerical technique, albeit with some modifications to the mode shapes. The main difference occurs at the edge of the mode. For the numerical minimization, the mode is defined by a radius outside which the mode is assumed to fall to zero, exhibiting a discontinuous derivative at the mode edge, while the mode shape from micromagnetics has a more gradual mode profile near the mode edge. This gradual mode profile is more physically permissible, as the abrupt cutoff of the Bessel mode would result in a divergent exchange energy in a continuous film. The computation time for the micromagnetic method, together with the accuracy of the Bessel function approximation suggests micromagnetics are unnecessary for calculating the resonance fields of localized modes in an out-of-plane geometry, but it is reassuring that the two techniques provide similar results for the resonance field of the localized modes.

Magnetization precession localized within an inhomogeneous field was suggested by Schlömann in 1964.⁶ He pointed out the similarity between the Landau-Lifshitz equation and the Schrödinger equation for a particle in a potential:

This simple problem was indeed experimentally verified by Kalinikos et al. in the 1980s¹⁰ using the dipole field from a magnet pole with a hole along its axis to localize precession modes. They derived an analytic solution for localized precession in a parabolic field well and sufficiently small wavevectors such that exchange fields are negligible, which is a reasonable assumption for localization radius larger than a micron. Here, we apply the solution found by Kalinikos to localized modes confined by the dipole field from a micron sized probe and find very good agreement with the experimental data for the observed discrete modes in thin film YIG.

The analytic solution¹⁰ is derived for a parabolic magnetic field well of the form $H(x)=H0(1+ax2)$⁠, where x is the lateral position in the film relative to the position of the probe. The boundary condition is

where x_n is defined as the position where k_x(x = x_n) = 0 and χ_n is the nth zero of the Bessel function J₀(x). Note that the boundary condition used in previous studies^7,9 describes a localized mode with a single wavevector k_n = χ_n∕R, where R is the mode radius. The two boundary conditions are equivalent if it assumed that k_n is an average wavevector that satisfies $knR=∫0xn kx(x)dx$⁠. However, the boundary condition involving the integral is more general, and physically it suggests that the mode varies its wavevector according to the local field H(x) and is injected at the position x_n with wavevector k = 0.

The resulting analysis¹⁰ yields the resonance field

where $4πMs$ is the effective saturation magnetization field of the film, t is the film thickness, and $Hext=ωγ−4πMs$ is the applied external field for the uniform mode. For our case, the total field at the center of the field well z = 0 including external applied field and the dipole field from the probe magnet of moment m_p at a distance z away is

and the parameter describing the parabolic field well from this probe magnet is found from a binomial expansion of the dipole field to be

The expected resonance fields H_n for the first four localized modes n = 1–4 are plotted in Fig. 3 as solid lines together with the experimental data. The experimental parameters are t = 25 nm, $4πMs=1608$ Oe, $mp=3.9×10−9$ emu, and $ω/2π=$ 4 GHz. This analytic solution shows very good agreement with the experimental data for n = 1, but begins to fail for the higher order modes. Nevertheless, the merit of the analytic solution is in its efficiency: it is at least 2 orders of magnitude faster to compute than the numerical minimization, and it is a good representation of the first localized mode resonance field. The discrepancy for higher order modes can be attributed to the fact that exchange fields are not negligible for these higher wavevector modes, and this is confirmed by the fact that similar results are produced when exchange fields are ignored for the numerical minimization technique.

In addition, the radius of the localized mode can be estimated by the analytic technique¹⁰ to be

This can be compared with the experimental resolution obtained by the analysis of the feature size in FMRFM images of a permalloy film in a previous study.⁷ This experimental resolution is shown together with the predicted radius obtained by analytic and numerical methods in Fig. 4. The numerical method overestimates the mode radius, while the analytic method underestimates it; however both methods at least qualitatively describe the experimentally obtained resolution.

In conclusion, we show that analytic and micromagnetic methods can be used to complement the established numerical minimization method to solve the resonance condition for probe localized modes. While the analytic method is the fastest, it is inaccurate at high k due the omission of exchange energy terms. Adding exchange energy to the calculation would be a useful extension to the technique, but attempts have so far only led to lengthy equations that in the end have to be solved numerically. Nevertheless, the current implementation of the analytic solution is very accurate for the first localized mode, indicating this method can be used to accurately model localized mode behavior for micron-sized modes, enabling localized mode experiments, and analysis to a wider audience without the need for complicated numerical analysis. On the other hand, micromagnetic simulations were found to accurately model the experimental data for all modes and provide insight into the true shapes of the modes; however, this is achieved at the cost of increased computational time.

This work was primarily supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award No. DE-FG02-03ER46054 (FMRFM measurement) and Award No. DE-SC0001304 (sample synthesis). This work was partially supported by the Center for Emergent Materials, an NSF-funded MRSEC under Award No. DMR-1420451 (structural characterization). This work was supported in part by Lake Shore Cryotronics, Inc. (magnetic characterization) and an allocation of computing time from the Ohio Supercomputer Center (micromagnetic simulations). We also acknowledge the technical support and assistance provided by the NanoSystems Laboratory at the Ohio State University.