In this paper, we describe modeling of the effects of magnetic inhomogeneity on ferromagnetic resonance line width using eigen mode analyses of inhomogeneous thin magnetic films

Kunz, Andrew

McMichael, R. D.

Twisselmann, D. J.

English

epublications@Marquette

Marquette Universitye-Publications@MarquettePhysics Faculty Research and Publications Physics, Department of3-30-2003Eigen Modes and Ferromagnetic Resonance LineWidth of Inhomogeneous Thin FilmsR. D. McMichaelNational Institute of Standards & TechnologyAndrew KunzMarquette University, andrew.kunz@marquette.eduD. J. TwisselmannNational Institute of Standards & TechnologyAccepted version. Published as a part of the 2003 IEEE International Magnetics Conference(INTERMAG). DOI. © 2003 IEEE. Used with permission.Andrew Kunz was affiliated with Lawrence University at the time of publication. Marquette University e-Publications@Marquette  Physics Faculty Research and Publications/College of Arts and Sciences  This paper is NOT THE PUBLISHED VERSION; but the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in th citation below.  2003 IEEE International Magnetics Conference (INTERMAG), (March 30-April 3, 2003). DOI. This article is © Institute of Electrical and Electronic Engineers (IEEE) and permission has been granted for this version to appear in e-Publications@Marquette. Institute of Electrical and Electronic Engineers (IEEE) does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Institute of Electrical and Electronic Engineers (IEEE).   Eigen Modes and Ferromagnetic Resonance Line Width of Inhomogeneous Thin Films   R.D. McMichael National Institute of Standards and Technology, Gaithersburg. MD Andrew Kunz Lawrence University, Appleton, WI D.J. Twisselmann National Institute of Standards and Technology, Gaithersburg. MD  Abstract: In this paper, we describe modeling of the effects of magnetic inhomogeneity on ferromagnetic resonance line width using eigen mode analyses of inhomogeneous thin magnetic films.  As data rates approach and exceed 1 GHz in data storage and other magnetic devices. accurate measurements of magnetic damping are becoming increasingly important. Unfortunately, measurements of damping by ferromagnetic resonance are complicated by the presence of inhomogeneity. In this paper, We describe modeling of the effects of magnetic inhomogeneity on ferromagnetic resonance line width using cigenmode analyses of inhomogeneous thin magnetic films.  We start with a spin wave Hamiltonian identical to the one used in two-magnon calculations of inhomogeneous FMR line broadening [1]–[2][3],  𝐸𝐸 = � 𝜔𝜔(𝑘𝑘)𝑎𝑎𝑘𝑘∗𝑎𝑎𝑘𝑘𝑘𝑘 + � (𝐴𝐴𝑘𝑘,𝑘𝑘′𝑎𝑎𝑘𝑘∗𝑎𝑎𝑘𝑘′ + ℎ. 𝑐𝑐. )𝑘𝑘,𝑘𝑘′  (1)  Here 𝜔𝜔(𝑘𝑘) is the magnon dispersion relation that incorporates the magnetostatic and exchange interactions, 𝑎𝑎𝑘𝑘∗  and 𝑎𝑎𝑘𝑘 are the rnagnon creation and annihilation operators, and the Λ𝑘𝑘,𝑘𝑘′  terms arise from inhomogeneity. The inhomogeneity is modeled as an applied field 𝐻𝐻𝑝𝑝(𝒓𝒓) added to a uniform applied field, where 𝐻𝐻𝑝𝑝(𝒓𝒓) varies from grain to grain in Voronoi polygon “microstructures” with various typical grain sizes. In contrast to the two rnagnon model where 𝐴𝐴𝑘𝑘,𝑘𝑘′  is treated as a perturbation we diagonalize the Hamiltonian for a few thousand low 𝑘𝑘 spin wave modes to obtain the eigenmodes and eigenfrequencies of the perturbed film.[4] We then calculate the FMR spectrum as a superposition of the resonances due to each eigenmode.  We find that for small grain sizes, the eigenmode calculation is in good agreement with the two-magnon model, but for large grain sizes, the resonances become localized, and the spectrum is in good agreement with a local resonance model, The characteristic grain size 𝐷𝐷 for this transition is approximately the distance required for a spin wave to experience a phase shift of 𝜋𝜋 in a grain with perturbation field 𝐻𝐻𝑝𝑝 or,  ∂𝑘𝑘∂𝐻𝐻| 𝜔𝜔⟨𝐻𝐻𝑝𝑝⟩𝔪𝔪k𝐷𝐷 = 2(𝐻𝐻𝑝𝑝)m𝑠𝑠𝑀𝑀s𝑑𝑑 𝐷𝐷 = 𝜋𝜋(S. I) (2)  For example, in a 𝑑𝑑 = 3 nm thick film of Permalloy with perturbation fields on the order of 5 mT. the characteristic length scale is approximately 1 μm. Despite the fact that local resonance models ignore exchange and magnetostatic interactions, they have provided good descriptions of experimental data in some cases [5]–[6].  The nature of an inhomogeneously broadended FMR spectrum according to the eigenmode analysis is a superposition of resonances with a distribution of resonance frequencies.  This picture is very different from the two magnon model picture of a uniform precession mode with increased damping.   Figure 1. Peak-to-peak FMR line width for a 3 nm thick film of permalloy at 10 ghz calculated as a function of grain size for three values of the RMS perturbation field. 𝐻𝐻𝑝𝑝. Line width values from the two-magnon model and the local resonance model values are also plotted for each value of 𝐻𝐻𝑝𝑝. Low line width values at larger grain sizes occur when local resonance produces both the maximum and minimum FMR signal values.    Figure 2. Details of the eigenmode analysis at a grain size of 1400 nm and 𝐻𝐻𝑝𝑝 = 10.6.mT. a) FMR derivative signals (lines) calculated from eigenmode intensities and eigenfrequencies (solid spikes) and from grain area and. local applied field (dashed spikes). b) Half of the grain structure and c) the corresponding half of a gray scale map of the dynamic magnetization pattern corresponding to the eigenmode indicated by the arrow in a). Localization confines the dynamic magnetization mostly to two grains.  References 1. M. Sparks, R. Loudon, C Kittel, Phys. Rev., vol. 122, no. 79, 1961. 2. R. D. McMichael, M. D. Stiles, P. J. Chen, W. F. Egelhoff, J. Appl Phys., vol. 83, no. 7037, 1998. 3. R. Arias, D. L. Mills, Phys. Rev. B., vol. 60, no. 7395, 1999. 4. A. Kunz, R. D. McMichael, IEEE Trans. Mag. In press. 5. C. Chappell, K. Lc Dang, P. Beauvillain, H. Hurdequint, D. Renard, Phys. Rev. B. 34, vol. 3192, 1986. 6. S. Mizukanti, Y. Ando, T. Miyazaki, Jpn. J. Appl Phys., vol. 40, no. 580, 2001.   

Eigen Modes and Ferromagnetic Resonance Line Width of Inhomogeneous Thin Films

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Eigen Modes and Ferromagnetic Resonance Line Width of Inhomogeneous Thin Films

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