We demonstrated that dipole-exchange spin waves traveling in geometrically
restricted magnetic thin films satisfy the same laws of reflection and
refraction as light waves. Moreover, we found for the first time novel wave
behaviors of dipole-exchange spin waves such as total reflection and negative
refraction. The total reflection in laterally inhomogeneous thin films composed
of two different magnetic materials is associated with the forbidden modes of
refracted dipole-exchange spin waves. The negative refraction occurs at a 90
degree domain-wall magnetic interface that is introduced by a cubic magnetic
anisotropy in the media, through the anisotropic dispersion of dipole-exchange
spin waves.Comment: 13 pages, 5 figure

Dae-Eun Jung

Dong-Soo Han

Jackson J. D.

Ki-Suk Lee

Sang-Koog Kim

Sangkook Choi

Youn-Seok Choi

Applied Physics Letters

English

arXiv

Crossref

Negative refraction of dipole-exchange spin waves through a magnetic twin interface in restricted geometry

A phenomenon of negative refraction of dipole-exchange spin waves (DESWs) was demonstrated by micromagnetic modeling, based on the fact that the DESWs' dispersion is anisotropic according to the relative orientation of the DESW propagation direction with respect to the orientation of local static magnetizations. Using this anisotropic dispersion behavior, the negative refraction of the DESWs was reproduced through a magnetic twin interface in a geometrically restricted medium of cubic in-plane anisotropy. This work verifies that the negative refraction of electromagnetic waves can also occur for DESWs propagating in restricted geometry through a specially designed magnetic interface.open12

Kim, Sang-Koog

Choi, Sangkook

Lee, Ki-Suk

Han, Dong-Soo

Jung, Dae-Eun

Choi, Youn-Seok

ScholarWorks@UNIST

Negative refraction of dipole-exchange spin waves through a magnetic twin interface inrestricted geometrySang-Koog Kim, Sangkook Choi, Ki-Suk Lee, Dong-Soo Han, Dae-Eun Jung, and Youn-Seok Choi  Citation: Applied Physics Letters 92, 212501 (2008); doi: 10.1063/1.2936294 View online: http://dx.doi.org/10.1063/1.2936294 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/92/21?ver=pdfcov Published by the AIP Publishing  Articles you may be interested in Interface boundary conditions for dynamic magnetization and spin wave dynamics in a ferromagnetic layer withthe interface Dzyaloshinskii-Moriya interaction J. Appl. Phys. 115, 233902 (2014); 10.1063/1.4883181  Micromagnetic simulations of spin-wave normal modes and the spin-transfer-torque driven magnetizationdynamics of a ferromagnetic cross J. Appl. 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Downloaded to IP:  114.70.7.203On: Mon, 27 Oct 2014 01:37:38Negative refraction of dipole-exchange spin waves through a magnetic twininterface in restricted geometrySang-Koog Kim,a Sangkook Choi, Ki-Suk Lee, Dong-Soo Han, Dae-Eun Jung, andYoun-Seok ChoiResearch Center for Spin Dynamics & Spin-Wave Devices, Seoul National University, and NanospinicsLaboratory, Department of Materials Science and Engineering, College of Engineering, Seoul NationalUniversity, Seoul 151-744, Republic of KoreaReceived 22 April 2008; accepted 3 May 2008; published online 27 May 2008A phenomenon of negative refraction of dipole-exchange spin waves DESWs was demonstratedby micromagnetic modeling, based on the fact that the DESWs’ dispersion is anisotropic accordingto the relative orientation of the DESW propagation direction with respect to the orientation of localstatic magnetizations. Using this anisotropic dispersion behavior, the negative refraction of theDESWs was reproduced through a magnetic twin interface in a geometrically restricted medium ofcubic in-plane anisotropy. This work verifies that the negative refraction of electromagnetic wavescan also occur for DESWs propagating in restricted geometry through a specially designed magneticinterface. © 2008 American Institute of Physics. DOI: 10.1063/1.2936294In recent years, negative refraction1,2 of electromagneticwaves ranging from microwave3,4 to infrared5 and even tovisible frequencies6 has become a hot research issue, owingto the facts that this phenomenon has not been achieved innaturally existing materials2 but has been realized with arti-ficially made metamaterials,3,5 photonic crystals,4 or utrathinplanar waveguides.6 The negative refraction of electromag-netic waves has become more interesting due to its practicalapplications to the realization of a superlens or superprism,either of which could be used to achieve subwavelength fo-cusing with resolution beyond the diffraction limit, or opticalnanodevices such as beam steerer and modulator.3,7Meanwhile, wavelike phenomena of dipole-exchangespin waves DESWs in a different physical system, e.g.,ordered magnetic materials of restricted geometry have beenexplored to pursue their fundamental understanding as wellas advanced technological applications.8 The DESWs are, inprinciple, low-lying collective excitations of magnetizationsMs in submicronsize magnetic elements, where both long-range dipolar and short-range exchange interactions betweenindividual Ms are primarily considered. The potential uses ofDESWs as signals in new-paradigm logic devices9,10 requirethe ability to steer the path of DESW rays propagating inmagnetic thin-film media without the loss of high-energyflux.11 Analogously to the case of electromagnetic waves,DESWs are also expected to be negatively refracted underpeculiar conditions, based on the anisotropic dispersion rela-tion that the magnitude of the wave vector k of the DESWs’propagation varies with its direction with respect to the ori-entation of the local static M. In fact, such anomalous refrac-tion of spin waves was investigated for only magnetostaticspin waves of wavelengths ranging from 10 m to 2 mmthrough a boundary between ferrite and ferrite-insulator-metal media,12 where exchange interactions were ignored.However, for the case of DESWs of wavelengths shorterthan 1 m and high frequencies larger than 10 GHz, travel-ing in magnetic waveguides of the nanometer length scale,such behaviors have not been studied. In this letter, we reporton an intriguing phenomenon of electromagnetic-wave-likenegative refraction of DESWs, as studied by numerical mi-cromagnetic simulations. In order to demonstrate the nega-tive refraction of DESWs propagating in nanoscale media,we have designed a specific model interface of magnetictwin structure in a medium of geometrically restricted thinfilm of cubic in-plane anisotropy.In the present modeling study, we chose a micromag-netic simulation approach13 based on the Landau–Lifshitz–Gilbert LLG equation,14 an approach that is one of the mostoptimized tools to elucidate electromagnetic-wave-like prop-erties of DESWs reflected at and refracted through a mag-netic interface between two dissimilar magnetic media. Anadvantage of this simulation approach is that one can obtainrealistic M dynamics of sufficient temporal 1 ps and spa-tial 1 nm resolutions, whereas experimental measure-ments are often impossible due to the relatively poor resolu-tions of the probing techniques. The governing LLGequation is given by M /t=−MHeff+ / MMM /t, with the phenomenological damping constant and the gyromagnetic ratio , where the M precesses underan effective torque MHeff attributed to Heff that is con-tributed from the exchange, magnetic dipolar, and magneticanisotropy interactions between the individual Ms and theZeeman interaction. Moreover, recent advanced computercapability allows for the micromagnetic simulation approachto provide quantitative information on the richelectromagnetic-wave-like properties of DESWs.Before going further, let us re-examine the basic law ofreflection and refraction with monochromatic electromag-netic waves of frequency f incident on an interface betweenoptically different media, not only in the real x-y plane butalso in the wavevector kx-ky plane, as illustrated in Figs.1a and 1b, respectively.15 When plane waves of wavevector k0 and group velocity vg0 are incident on an interface,some of them return to are reflected into medium 1 with kand vg, and the rest of them pass through the interface arerefracted and continue to propagate in medium 2 with kand vg, where vgkfmediumk. The optical rule governingaAuthor to whom correspondence should be addressed. Electronic mail:sangkoog@snu.ac.kr.APPLIED PHYSICS LETTERS 92, 212501 20080003-6951/2008/9221/212501/3/$23.00 © 2008 American Institute of Physics92, 212501-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:  114.70.7.203On: Mon, 27 Oct 2014 01:37:38relations between the incident, reflected, and refracted wavesis Snell’s law that satisfies the continuity of the tangentialcomponents of the k vector across the interface along the xaxis, such that kx0=kx=kx vertical dotted line in Fig. 1b.The wave propagation inside each medium is determined bythe dispersion relation of each medium. In Fig. 1b, theconstant frequency contours of the allowed modes in thedifferent media are plotted on the kx-ky plane, and are cir-cular in shape in the case of optically isotropic media. Eachpropagation direction is normal to the corresponding fre-quency dispersion contour surface at the condition kx0=kx=kx; thereby, the energy flow of electromagnetic wavespropagates along the allowed k vectors in the given isotropicmedia, as displayed in Fig. 1b; that is, the directions of kand vg for any incident, reflected, or refracted waves arealways parallel to each other.In contrast, in a certain medium of optical anisotropy, thefrequency contours of the allowed modes are anisotropicFig. 1c. Correspondingly, it is not necessary that vg beparallel to k according to the direction of k. In some cases,the direction of vg thick gray arrows can differ from that ofk thin gray arrows, as illustrated in Fig. 1c. Consequently,g,x and kx in an anisotropic medium can have either the sameor the opposite sign, according to the direction of k relativeto the optical axis of the medium, as shown in Fig. 1c,where g,xy=vg · eˆxy with the unit vector eˆxy along the xy axis. For the case of an interface between optically aniso-tropic dissimilar media, when g,x and kx have the oppositesign in medium 1 and the same sign in medium 2, the sign ofg,x should be opposite to that of g,x0 for the condition kx0=kx=kx, which is required for the negative refraction pos-sible, as visualized in Fig. 1d. Such a condition would bepossible either at an interface between a normal right-handedmaterial and a left-handed material of negative permittivityand permeability1,2 or at a special interface of twining struc-ture in uniaxial crystals, where the interface is a reflectionsymmetry plane.16Analogously to the case of electromagnetic waves as de-scribed above, DESWs are also expected to be negativelyrefracted under peculiar conditions, based on their aniso-tropic dispersion relation. In order to demonstrate the nega-tive refraction of DESWs, we designed a special model in-terface of magnetic twin structure because it could be readilyobtained from real samples such as cubic in-plane anisotropymaterial.17 The model system used for our numerical simu-lation was composed of 5- nm-thick Fe film of cubic mag-netic anisotropy.18 The cubic anisotropy allows the staticM orientation to saturate along the direction x ,y ,z= −1,1 ,0 in the Fe1 region y0 nm and along the direc-tion x ,y ,z= 1,1 ,0 in the Fe2 region y0 nm, as dis-played in Fig. 2a, resulting in a magnetic twin interfacewith a 90° domain wall along the x axis. The thorny shapeat the edges of such a finite-size model system was designedto align the local Ms near the edge boundaries along thedesired M orientations in both the Fe1 and Fe2 media, byminimizing the influence of dipolar interactions between themagnetic free poles at the edges.19 To excite monochromaticDESWs of f =25 GHz, incident on the magnetic twin inter-face with flat wavefronts parallel to the Fe1 /Fe2 interfacethe k vector is normal to the interface, we applied an os-cillating field Hosc=2501−cos 2	Ht Oe with a frequency	H=25 GHz only in the dash-dot boxed area in the Fe1 re-gion along the direction x ,y ,z= 1,1 ,0 perpendicular tothe static M direction x ,y ,z= −1,1 ,0. A snapshot imageof the resultant dynamic evolution of the Mz / M distributionfor the indicated area by the gray dotted box in Fig. 2a,taken at t=4.0 ns after applying the Hosc is shown in Fig.2b. This image clearly reveals that the DESW packets withthe wavefronts parallel to the Fe1 /Fe2 interface propagatedalong a direction 52° from the +x direction in the Fe1 and128° in the Fe2. This clearly exhibits not only that the direc-tion of vg differs from the k direction for both incident andrefracted DESW rays, but also that g,x and g,x0 have theopposite sign. This DESWs propagation behavior is similarto the negative refraction of electromagnetic waves.2FIG. 1. Color online Schematic illustrations of reflection and refraction ofelectromagnetic waves at an interface between optically different media, buteach is isotropic, in the real x-y plane and wave vector kx-ky plane, asshown in a and b, respectively. The purple and apricot circles on thekx-ky plane indicate the individual frequency contours of the allowedmodes in the same-color-coded media 1 and 2, respectively. The color-codedthin and thick arrows denote the k and vg of the electromagnetic wavespropagating in each medium, as indicated by the incident red, reflectedblue, and refracted green rays. c The frequency contour of the allowedmode in an anisotropic medium. The thin and thick gray arrows denoteexamples of the allowed k and vg in the medium, respectively. d Sche-matic diagram illustrating an example of negative refraction of electromag-netic waves through an interface between anisotropic media 1 and 2.FIG. 2. Color online a The model system employed, made of 5-nm-thickFe film, for the demonstration of DESWs’ negative refraction. b Plane-view, snapshot image of the spatial distribution of the local Mz / M, taken att=4 ns. The thick arrows indicate the directions of the group velocity vec-tors of the propagating DESWs in each medium.212501-2 Kim et al. Appl. Phys. Lett. 92, 212501 2008 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:  114.70.7.203On: Mon, 27 Oct 2014 01:37:38For further quantitative analysis, we made a fast Fouriertransform FFT of the dynamic evolution of the localMz / M distributions to obtain their power spectrum, asshown in Fig. 3a. On the kx-ky plane, there was only onestrong peak at kx ,ky= 0,0.07 nm−1. In the FFT power dis-tribution, the directions of vg for both the incident and re-fracted DESWs could be identified by overlapping the aniso-tropic frequency contours of f =25 GHz in both the Fe1purple and Fe2 apricot media.20 At the kx ,ky= 0,0.07 nm−1 peak, the directions of vg for the incidentand refracted DESW rays were determined to be 52° in theFe1 and 128° in the Fe2 media, respectively, from the +xdirection, as determined by the definition of vgkfFekFig. 3a. These angles are in quite good agreement withthe directions of the DESW packets represented by the imageof the Mz /Ms distribution, shown in Fig. 2b. Also, therange of the incidence angle of the k vector, where negativerefraction can occur for DESWs, were predicted from theconstant frequency contours of the DESWs shown in Fig.3b. By the definition of vgkfFek, the angles requiredfor g,x0=0 and g,x =0 were determined to be 116.1° and63.9° from the +x direction, respectively. Between these twoangles, the directions of g,x0 and g,x lie opposite to eachother, so that negative refraction of DESWs can take place inthe range of the direction of the k vector from 63.9° to116.1°. The 90° incidence angle of the k vector for thepresent simulation is just one of the probable angles for theoccurrence of DESWs’ negative refraction.In conclusion, we demonstrated, by micromagneticsimulations, that negative refraction of electromagneticwaves is also a possible phenomenon for exchange-dipolarspin waves based on their anisotropic dispersion behavior,that is, that the magnitude of the spin wave propagation wavevector varies according to its direction relative to the localMs. The demonstrated result provides a basic mechanism bywhich spin waves can be guided in waveguide media of re-stricted geometry.This work was supported by Creative Research Initia-tives ReC-SDSW of MOST/KOSEF.1V. G. Veselago, Sov. Phys. Usp. 10, 509 1968.2J. B. Pendry and D. R. Smith, Phys. Today 576, 37 2004.3R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 2001.4E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis,Nature London 423, 604 2003; P. V. Parimi, W. T. Lu, P. Vodo, and S.Sridhar, ibid. 426, 404 2003.5G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden,Science 312, 892 2006.6H. J. Lezec, J. A. Dionne, and H. A. Atwater, Science 316, 430 2007.7J. B. Pendry, Phys. Rev. Lett. 85, 3966 2000; H. Kosaka, T. Kawashima,A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys.Rev. B 58, R10096 1998; N. Engheta, Science 317, 1698 2007.8Spin Dynamics in Confined Magnetic Structures, Topics in AppliedPhysics, edited by B. Hillebrands and K. Ounadjela Springer, Berlin,2002, Vol. 83.9R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Lett. 93, 2572022004; S. Choi, K.-S. Lee, K. Yu. Guslienko, and S.-K. Kim, ibid. 98,087205 2007.10S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A.Katine, Nature London 437, 389 2005.11S. Choi, S.-K. Kim, V. E. Demidov, and S. O. Demokritov, Appl. Phys.Lett. 90, 083114 2007.12A. V. Vashkovskii and E. H. Lock, Phys. Usp. 47, 601 2004.13We used the OOMMF code. See http://math.nist.gov/oommf. In the presentmicromagnetic simulations, we used a constant cell size of 555 nm3 and a damping parameter of =0.01.14L. D. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 1935; T. L.Gilbert, IEEE Trans. Magn. 6, 3443 2004.15J. D. Jackson, Classical Electrodynamics Wiley, New York, 1975.16Y. Zhang, B. Fluegel, and A. Mascarenhas, Phys. Rev. Lett. 91, 1574042003.17H. P. Oepen and J. Kirschner, Phys. Rev. Lett. 62, 819 1989.18For Fe, we used the exchange stiffness A=2.510−11 J /m, the saturationmagnetization Ms=1.7106 A /m, and the cubic anisotropy constantK=4.6104 J /m3.19S. Choi, K.-S. Lee, and S.-K. Kim, Appl. Phys. Lett. 89, 062501 2006.20The frequency contours of DESWs correspond to the cases of the lowestquantization number along a film thickness. For details, see B. A.Kalinikos, in Dipole-Exchange Spin-Wave Spectrum of Magnetic Films,edited by M. G. Cottam, Linear and Nonlinear Spin Waves in MagneticFilms and Superlattices World Scientific, Singapore, 1994.FIG. 3. Color online a The FFT power-spectrum distribution of the localMz / M on the kx-ky plane, which are overlapped with the frequency con-tours of the allowed DESW mode of f =25 GHz in the indicated Fe1 and Fe2media. The wide arrows indicate the directions of the group velocities in thecorresponding media. b The same frequency contours in a are repre-sented in the polar coordinate system. The anticipated range of the incidenceangles of the k vector is indicated by the gray region, in which the DESWs’negative refraction takes place.212501-3 Kim et al. Appl. Phys. Lett. 92, 212501 2008 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:  114.70.7.203On: Mon, 27 Oct 2014 01:37:38

Total Reflection and Negative Refraction of Dipole-Exchange Spin Waves
  at Magnetic Interfaces: Micromagnetic Modeling Study

Total Reflection and Negative Refraction of Dipole-Exchange Spin Waves at Magnetic Interfaces: Micromagnetic Modeling Study

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