Tarea IIIntroduccin al Magnetismo (I- 2015)Prof. Pedro LanderosFecha de entrega: da del certamen I1. Considere una pelcula ferromagntica delgada en presencia de un campo magntico DC (H0), como se muestra en la figura, donde es un vector unitario paralelo a la magnetizacin en equilibrio. Suponga que la normal al plano es un eje difcil de anisotropa.

Si el campo externo DC es lo suficientemente intenso como para que la magnetizacin sea uniforme, determine una ecuacin (trascendental) para calcularel ngulo M, que define la direccin de la magnetizacin en equilibrio. 1.a) Resuelva minimizando energa total.1.b) Resuelva a travs del campo magntico total (o efectivo), desde las soluciones estticas de la ecuacin de Landau-Lifshitz.1.c) Sirve la solucin obtenida en el caso que la normal al plano es un eje fcil de anisotropa? Explique.2. Considere un disco ferromagntico delgado magnetizado uniformemente en una direccin arbitraria, en un ngulo con el eje z (paralelo al eje de simetra) y con anisotropa cbica. Suponga tambin que la componente en el plano xy de se orienta en un ngulo con el eje x. a. Si se aplica un campo magntico en otra direccin arbitraria dada por los ngulos (), calcule la energa total del sistema. La energa dipolar exprsela entrminos del factor demagnetizante .b. Considere ahora dos cilindros delgados separados por una distancia d, medida a lo largo del eje z desde el centro del cilindro 1 al centro del cilindro 2. Calcule la energa total del sistema en el caso en que los cilindros tienen magnetizaciones arbitrarias dadas por los ngulos () y ().3. Considere una esfera de radio R uniformemente magnetizada en la direccin z. Resuelva las ecuaciones diferenciales para encontrar el potencial magnetosttico, el campo magnetosttico y la energa. Tambin resuelva el problema usando la solucin formal para el potencial magnetosttico. Use cualquiera de las siguientes expansiones:

Two of us have developed a general formalism withinwhich two magnon scattering and its influence may bestudied,14 for an ultrathin ferromagnet magnetized in plane.In the theory, we provide response functions that describe theresponse of the film to a finite wave vector applied field,including the influence of two magnon scattering, and fromthis we extracted expressions for the two magnon inducedlinewidth and frequency shift of the zero wave vector FMRmode. The general formalism applies to a variety of picturesof the defect structures. Shortly after the appearance of thetheory, Azevedo et al.15 reported data on the two magnoncontribution to both the linewidth and frequency shift of theFMR mode in Permalloy films. The very interesting analysisin this paper shows that these two effects are linked verynicely, in a fully quantitative manner, by the theory in Ref.14. The two magnon contribution to the linewidth is alsostrongly dependent on the wave vector of the spin wave ofinterest. That this is so has been demonstrated by comparingthe linewidth of the FMR mode with that of spin wavesexcited in BLS, on the same sample.16 Slebarski et al.16 de-veloped a theory of the wave vector dependence of the line-width, along with a description of two magnon damping inultrathin ferromagnets deposited on exchange biasing sub-strates. It is also the case that the picture set forth in Ref. 14provides an excellent account of the frequency dependenceof the FMR linewidth, on measurements which cover thevery broad frequency range from 2 to 80 GHz.17. The defectpicture introduced in Ref. 14 also leads to an excellent de-scription of the in-plane anisotropy found for the two mag-non linewidth in the data reported in Ref. 17.

Woltersdorf and Heinrich18 discussed the data on sampleswhere dislocation lines parallel to the film surfaces providethe activation mechanism for two magnon scattering in Fefilms grown on Pd 100 surfaces. The phenomenologicaltheory set forth in this paper provides a very good account ofthe data. The dislocation lines form a regular array to firstapproximation, with fourfold symmetry. Of course, disorderis necessary to initiate two magnon scattering. It would be ofinterest to construct a microscopic description of the matrixelement that couples the FMR spin wave to its short wave-length degenerate partners for this interesting physical situa-tion.

All of the papers above explore the case where the mag-netization of the film is in plane. An interesting question isthe effect on two magnon scattering of tipping the magneti-zation out of plane. In a brief note, McMichael et al.19 pre-sented data that demonstrated that as the magnetization istipped out of plane, the two magnon scattering shuts off.These authors presented numerical calculations appropriateto their sample, which agree nicely with the data. In a recentreview,13 it was noted that if one examines the dispersionrelation of dipole exchange spin waves for the case where themagnetization is tipped out of plane, spin waves degeneratewith the FMR mode and disappear when the magnetizationmakes the angle of 45 with the film plane. This is in goodagreement with the data presented in Ref. 19. Urban et al.20and Heinrich et al. also noted that the two magnon linewidthis not operative when the magnetization is perpendicular tothe film surfaces. Similar observations on a very differentsystem, a Heusler alloy grown on the InP 100 surface, are

reported in Ref. 21. These authors provide data that showthat the linewidth is less when the film is magnetized normalto its surface than when it is parallel.

There has been no systematic discussion of the influenceof tipping the magnetization out of plane on two magnonscattering in ultrathin ferromagnets, though as we have seenfrom the discussion above, various authors have appreciatedthat the mechanism is not operative when the magnetizationis tipped out of plane. In our view, it is important to have acomplete description of the phenomenon, including the outof plane geometry. In this paper, we extend the formalismdeveloped in Ref. 13 to this case. While our numerical workfocuses on the FMR mode of the film, our general theoreticaldevelopment provides the reader with complete responsefunctions for the film at finite wave vectors, including theinfluence of two magnon scatterings in their structure. Thus,if desired, with the response functions developed here, onecan explore two magnon effects in BLS where finite wavevector spin waves are excited. In Sec. II, we present thetheoretical development, and in Sec. III, we present numeri-cal calculations of the frequency and angle dependence ofthe two magnon damping rate for the FMR mode. While it is,indeed, the case that the effect is cut off when the magneti-zation makes an angle greater than 45 with the plane, theresults are striking in our view. Section IV is devoted toconcluding remarks.

II. THEORETICAL DISCUSSION

The geometry we consider is displayed in Fig. 1. We havea thin ferromagnetic film of thickness d. The film surfacesare parallel to the xz plane, and the y axis is normal to thefilm surfaces. This is the coordinate system used in Ref. 13.An external magnetic field of strength H0 is applied in the yzplane and, in our case, it makes the angle H with respect tothe z axis. The magnetization is canted out of plane to makean angle M with respect to the z axis. We shall also makeuse of the XYZ coordinate system indicated, where the Z axisis aligned with the canted magnetization, and the X axis co-incides with the x axis. We shall consider spin waves whosewave vector parallel to the film surfaces is k , and k is theangle between k and the z axis.

Z

Ms

H0

k||M

H

k||

dz

x,X

y Y

FIG. 1. Color online An illustration of the geometry used inthe present paper. The externally applied magnetic field H0 and themagnetization Ms lie in the yz plane and make the angles indicatedwith respect to the z axis. The xz plane coincides with the surface ofthe film. The film is taken to lie between y=d /2 and y=d /2 in thediscussions in the text. See also the text for the definition of otherquantities.

LANDEROS, ARIAS, AND MILLS PHYSICAL REVIEW B 77, 214405 2008

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donde es el ngulo entre los vectores y . son los Polinomios de Legendre, mientras que son los armnicos esfricos.

4. Considere un tubo con magnetizacin dada por:

,que representa una pared de dominio tipo vrtice. Invente un modelo para la magnetizacin de la pared de dominio y calcule las energas de intercambio, dipolar y anisotropa uniaxial (eje difcil en z). 5. Suponga un cilindro infinito con magnetizacin perpendicular al eje en la direccin x . Calcule su energa magnetosttica. Divida ahora la magnetizacin en dos dominios (ver figura) y calcule la energa magnetosttica (sin considerar la pared de dominio). De la misma manera divida ahora la magnetizacin en cuatro dominios y calcule su energa magnetosttica. Compare y discuta.

6. A partir del Hamiltoniano de Heisenberg derive la energa de intercambio en el continuo, para redes cubicas simples, bcc, fcc y hcp.