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The resonance susceptibility of two-layer exchange-coupled ferromagnetic film with a combined uniaxial and cubic anisotropy in the layers
Journal of Magnetism and Magnetic Materials 419 (2016) 512–516
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
The resonance susceptibility of two-layer exchange-coupled ferromagnetic film with a combined uniaxial and cubic anisotropy in the layers N.V. Shul’ga n, R.A. Doroshenko Institute of Molecule and Crystal Physics, Academy of Sciences, 151, prospect Octobrya, Ufa 450075, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 9 April 2016 Received in revised form 23 June 2016 Accepted 23 June 2016 Available online 24 June 2016
A numerical investigation of the resonance dynamic susceptibility of ferromagnetic exchange-coupled two-layer films with a combined cubic and uniaxial magnetic anisotropy of the layers has been performed. It has been found that the presence of cubic anisotropy leads to the fact that much of the offdiagonal components of the dynamic susceptibility are nonzero. The change of the ferromagnetic resonance frequencies and dynamic susceptibility upon the magnetization along the [100], [010], and [011] directions have been calculated. The evolution of the profile of the dynamic susceptibility occurring during the magnetization has been described. The impact of changes in the distribution of equilibrium and dynamic components of the magnetization on the dependences of the components of the dynamic susceptibility and the ferromagnetic resonance frequency on the external magnetic fields has been discussed. & 2016 Elsevier B.V. All rights reserved.
Keywords: Ferromagnetic resonance (FMR) Two-layer film Dynamic susceptibility
1. Introduction Experimental and theoretical investigation in the field of magnonics causes an increasing interest in the study of multilayered structures [1,2]. This is due to the fact that structures with a periodic modulation of magnetic parameters give unprecedented control of spin waves. Multilayered yttrium-iron garnet (YIG) films are an interesting object of study largely due to their uniquely low magnetic damping [3]. The theory of ferromagnetic resonance for a film consisting of exchange-coupled ferromagnetic layers was developed in the phenomenological approach in the papers [4–12]. It has been described that two-layer film has two ferromagnetic resonance modes, between which there is an energy gap. These calculations are usually carried out in the ultrathin layers approximation or it is supposed that the external magnetic field is high enough for a film in the ground state to be magnetized uniformly [12]. The ferromagnetic resonance (FMR) remains the technique of choice to investigate the parameters of multilayered films [10,13]. Upon excitation of the resonance by uniform alternating magnetic field in two-layer film, two FMR lines, and a series of spin-wave resonance (SWR) lines between them were observed [14]. The SWR series is due to the reflection of spin waves on the boundary n
Correspondence to: IFMK, 151, prospect Oktyabrya, Ufa 450075, Russia. E-mail address: [email protected] (N.V. Shul’ga).
http://dx.doi.org/10.1016/j.jmmm.2016.06.054 0304-8853/& 2016 Elsevier B.V. All rights reserved.
of one of the layers. The number of SWR modes depends on the direction of the external magnetic field and the thickness of the layers of film [15]. The method suggested in [16] and based on the numerical calculation of the dynamic susceptibility makes it possible to study FMR frequencies in two-layer ferromagnetic films when the external magnetic field varies within wide limits. Since this method does not a priori postulate any shape of the coordinate dependence of variable components of magnetization, it become possible to study profiles of the FMR and SWR modes (the distribution of the dynamic susceptibility over the thickness). This method was used for the study of high-frequency properties a ferromagnetic film with the easy-plane and easy-axis anisotropy of the layers. It has been shown that at the nonzero parameter of the interlayer exchange interaction the dynamic components of the magnetization upon the ferromagnetic resonance are distributed through the film thickness inhomogeneously [17]. The evolution profiles of the of the FMR and SWR modes upon the changes in the strength [17] or in the direction of the external magnetic field for different thicknesses of the layers of film [18] has been described. Further, this method has been generalized to the case of ferromagnetic films with a combined cubic and uniaxial magnetic anisotropy of layers [19]. Further studies have shown that not only some features of dependences of the FMR frequencies on the field may indicate the changes taking place in the distribution of equilibrium and dynamic component of the magnetization. These changes can be judged by dependences of the integrated dynamic susceptibility
N.V. Shul’ga, R.A. Doroshenko / Journal of Magnetism and Magnetic Materials 419 (2016) 512–516
on the field. This topic is the subject of our article. We present the results of numerical studies of the high-frequency properties of a two-layer film whose parameters are close to a two-layer YIG film.
2. Model We consider two-layer exchange-coupled ferromagnetic film in an external magnetic field H. The geometry of the film is shown in Fig. 1. The layers of the film have uniaxial anisotropy of various signs and identical cubic anisotropy. The thicknesses of the layers di, i¼ 1, 2 are finite. The normal to the film coincides with the coordinate OX axis, and also with the axis [100] of the crystal and with the axis of uniaxial anisotropy. The direction of the magnetization M is characterized by a polar angle θ, which is counted off from the [001] axis of the crystal coinciding with OZ axis, and by an azimuthal angle ϕ, which is counted off from OX axis. The direction of the magnetic field H is characterized by a polar angle θh and by an azimuthal angle ϕh. It is assumed that the direction of the magnetization depends only on the coordinate x, and in the plane of the film the magnetization is distributed uniformly. The functional of the energy of the system includes the energy of cubic anisotropy, the energy of uniaxial magnetic anisotropy of the easy plane and easy axis types, the Zeeman energy, the energy of exchange interaction inside the layers and the energy of interlayer exchange interaction as follows: K1
2
W = ∑i = 1 ∫ dV ( Vi
−MiH +
Mi4
αi 2Mi2
{M
2 2 x, iMy, i
}
+ Mx2. iMz2. i + Mz2, iMy2, i −
Ku*, i Mi2
Mx2, i−
2
( ∂Mi/∂x) ) − ∫S M1JM2 M1M2dS.
where K1 is the first constant of cubic anisotropy, Mi are the saturation magnetizations of layers, Ku*, i = Ku, i − 2πMi2 are the efficient constants of magnetic anisotropy taking into account a demagnetizing influence of the surface of layers, αi are the constants of exchange interaction; and J is the constant of interlayer exchange interaction. The problem was solved numerically. The procedure of the solution is described here only briefly; it was considered in more detail in [16–19]. Each layer of the film is divided into uniformly magnetized flat cells. The magnetization of the cell with the number l is designated as ml. Thus, the energy functional is converted into the sum of the energies of these cells. Minimizing the energy by standard methods of multidimensional minimization [20], we obtain an equilibrium distribution of the magnetization in the two-layer film. The dynamics of the magnetization in each cell is described by a set of Landau–Lifshitz equations. By linearizing the obtained set of equations relative to small deviations of the vectors of magnetization of the cells from their equilibrium direction δml, we reduce the problem of obtaining the natural frequencies of the film to the solution of the generalized problem of eigenvalues for the thus-obtained set of linear equations. When calculating the dynamic susceptibility, a term of the form δmlh
Fig. 1. Coordinate system.
513
where h is the ac magnetic field, is added into the energy of the system. A relaxation term in the Gilbert form [21] is added to the set of Landau–Lifshitz equations. The solution of the set of equations obtained after the linearization yields the dynamic components of the magnetization δml. Then, the values of the components of the dynamic susceptibility of each cell are calculated: χi, k, l = δml, i / hk , (i, k)¼(x, y, z). Summing up them over all cells, we obtain the values of the components of the integrated dynamic susceptibility. Later in the article we consider its imaginary part . χ" i, k The calculation was carried out for the parameters of a twolayer film characteristic of an iron-garnet film M1 E30 G, M2 E70 G, JE 1 cm 1, αi E 107 erg/cm. The layer thicknesses are d1 E 0.5 10 4 cm and d2 E0.18 10 4 cm. The anisotropy para* E2 104 erg/cm3, * E 7 104 erg/cm3, meters are Ku,1 Ku,2 K1 E 2 104 erg/cm3.
3. Results and discussion Up to the saturation field, the ground state of the film is inhomogeneous. At low fields in case the sample with a combined cubic and uniaxial magnetic anisotropy of layers on the outer boundaries of the layers, the magnetization deviates toward one of the trigonal axes from the normal for the first layer, and from the film plane for the second layer. For the sample with a uniaxial magnetic anisotropy of layers we assumed that, without a loss of generality, it is possible to consider that the magnetization rotates in the plane (x, y), so that only the angle ϕ depends on the coordinate x. Let us consider the effect of changes in static and dynamic components of the magnetization on the behavior of the amplitude of the integrated dynamic susceptibility and frequency when the film is being magnetized by an external field. The field has been applied in one of three directions: along the [100], [011] or [010] axes. As these changes most influence the amplitude of the dynamic susceptibility of the lower FMR mode, later in this article we consider in more detail this mode. The amplitudes of the various components of the dynamic susceptibility vary considerably depending on the strength and direction of the external magnetic field. Let us consider the case when the external field is applied along the [100] axis. For the lower FMR branch of the sample with a combined cubic and uniaxial magnetic anisotropy all components are significant except χ″yx . Fig. 2 shows the dependences components of the integrated dynamic susceptibility of the lower FMR mode χ″ (Fig. 2a) and χ″yy (Fig. 2b), as well as the frequencies of zz lower and upper FMR modes (Fig. 2c) on the external magnetic field. Figs. 3 and 4 show the dependences of the susceptibility component χ″ of the lower and upper FMR modes on the cozz , l ordinate x for different values of the external magnetic field. At low fields, the lower FMR mode is localized in the first layer. It is distributed over a layer non-uniformly. The maximum of the distribution of the dynamic susceptibility is at the outer boundary of the first layer. With increasing of the field there is a shift of the localization of the lower FMR mode to the interlayer boundary. Then it is localized in the second layer. At the same time there is a change of localization of the upper FMR mode. It shifts from the second to the first layer. The displacement of the modes corresponds to the field H1 in throughout the range of variation of the Fig. 2. The component χ″ zz field is greater than other components. It is highest at low fields, and then, with the increase of the field, its amplitude decreases. When the field strength approaches the value Hst at which the
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N.V. Shul’ga, R.A. Doroshenko / Journal of Magnetism and Magnetic Materials 419 (2016) 512–516
Fig. 4. Dependences of the component of the dynamic susceptibility χ″ zz , l of the upper FMR mode on x for different values of the external magnetic field. The field is applied along the [100] axis.
Fig. 2. Dependences of the components of the integrated dynamic susceptibility of the lower FMR mode a) χ″ yy and c) the FMR frequencies on the external zz b) χ″ magnetic field. The field is applied along the [100] axis.
Fig. 3. Dependences of the component of the dynamic susceptibility χ″ zz , l of the lower FMR mode on x for different values of the external magnetic field. The field is applied along the [100] axis.
magnetic saturation of the sample is achieved, the amplitude of the component χ″ begins to increase and reaches a maximum at zz the field Hst. The amplitude of the components χ″yy at low fields is close to zero. When the field strength approaches to the value H1 the amplitude begins to increase and reaches a maximum near H1. Then there is its slow decline and a sharp jump to values close to zero after the saturation point. The components χ″ , yz , χ″ . χ″ xx χ″ xy xz behave similarly χ″yy . For upper FMR branch all components are significant except χ″ xz and χ″yx . The amplitudes of the components χ″ and χ″ exceeds all xx zz others. In the range of fields up to 500 Oe amplitude of the
component χ″ exceed χ″ . The maximum χ″ is at low fields and xx xx zz is located near the then its amplitude decreases. Maximum χ″ zz point of the closest approach of the FMR branches. For the sample with a uniaxial magnetic anisotropy of layers, for the lower mode only component χ″ is significant. The amplizz tudes of all other components are close to zero. The amplitude χ″ zz of the lower FMR branch reaches a maximum at the saturation point, and then rapidly decreases. The amplitude χ″ of the upper zz FMR branch has a maximum for lower fields, and then decreases. When the external field is applied along the [011] axis (Fig. 5), for the lower FMR mode all components are nonzero. The field in which there is the closest approach of the FMR branches is marked as H1. Near this field, there is a shift of the maximum of the dynamic susceptibility of the lower mode from the outer boundary of the first layer toward the interface between the layers (see Fig. 6). The displacement modes between layers is not observed. Its profile restores completely when the field strength exceeds the saturation field strength Hst. At this field, the minimum of the lower FMR branch is observed. The component χ″ (Fig. 5a) exceeds all zz other up to the point of the closest approach of the FMR branches. (Fig. 5b) exceeds all other. Its maximum is in Then component χ″ xx the field Hst, when its amplitude increases by several orders of magnitude. For upper FMR branch all components are significant except , yy , χ″yz , χ″ , χ″ remain significant and the χ″yx . The components χ″ xx χ″ zy zz component χ″ exceeds all other throughout the range of variation xx , , have the maximum at low of the field. The components χ″ χ″ χ″ xx yy zz fields, and then decrease. The components χ″yz , χ″ slowly increase. zy When the external field is applied along the [010] axis (Fig. 7), for the lower FMR branch only the components χ″ , xy , χ″yz , χ″ , χ″ xx χ″ zy zz are significant. In the field H1, the displacement of the lower FMR mode from the first layer into the second layer occurs (and, correspondingly, the reverse displacement of the upper mode). In this field, there is the closest approach of the FMR branches, and there are local extremes on the dependences of the susceptibility components. The maximum of the lower mode is on the interlayer boundary (see Fig. 8). There is the first minimum of the lower FMR branch in the field H2. In this field, the polar angle θ becomes identical in both of the layers and coincides with the angle θh. There is the reverse displacement of the FMR modes in the field
N.V. Shul’ga, R.A. Doroshenko / Journal of Magnetism and Magnetic Materials 419 (2016) 512–516
Fig. 5. Dependences of the components of the integrated dynamic susceptibility of the lower FMR mode a) χ″ xx and c) the FMR frequencies on the external zz b) χ″ magnetic field. The components χ″ xx are plotted on a logarithmic scale. The zz and χ″ field is applied along the [011] axis.
Fig. 6. Dependences of the component of the dynamic susceptibility χxx, l of the lower FMR mode on x for different values of the external magnetic field. The field is applied along the [011] axis.
H3. Though the lower mode is localized in the first layer, its maximum is on the interface between the layers, and the FMR signal is excited in the vicinity of the interlayer boundary, until the field strength approaches the saturation field Hst. Then the maximum is shifted to the outer boundary of the first layer and the amplitude of the component χ″ increases by several orders of zz , l magnitude. With increasing field amplitude χ″ decreases, and zz , l this component is distributed more uniformly over the layer. The amplitudes of the component χ″ exceed all others up to zz is higher than any other 1600 Oe (Fig. 7a). Then, the amplitude χ″ xx
515
Fig. 7. Dependences of the components of the integrated dynamic susceptibility of the lower FMR mode a) χ″ xx and c) the FMR frequencies on the external zz b) χ″ magnetic field. The components χ″ xx are plotted on a logarithmic scale. The zz and χ″ field is applied along the [010] axis.
Fig. 8. Dependences of the component of the dynamic susceptibility χxx, l of the lower FMR mode on x for different values of the external magnetic field. The field is applied along the [010] axis.
(Fig. 7b). The amplitude χ″ has a maximum in the field H1 and zz then it drops down to the deep minimum in the field Hst. The component χ″ from the minimum in the field H1 increases up to xx the sharp maximum in the field Hst. The component χ″ behaves xy . The components and behaves similarly to χ″ . similarly χ″ χ″ χ″ xx yz zy zz For upper FMR branch all components are significant except χ″yx and χ″ . The amplitudes of the component χ″ exceeds all others up xx zx is higher than any other. The to 200 Oe. Then, the amplitude χ″ zz has a maximum in the low field, and then it component χ″ xx
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N.V. Shul’ga, R.A. Doroshenko / Journal of Magnetism and Magnetic Materials 419 (2016) 512–516
decreases. The component χ″ has a maximum in the field H2, and zz then it decreases slowly. Let us consider the sample with a uniaxial magnetic anisotropy of layers. It is in an external magnetic field that lies in the film plane. For the lower FMR branch only the components χ″ , χ″ , xy xx χ″ zz are significant. The amplitude of the component χ″ exceeds all zz other up to the point of the closest approach of the FMR branches. exceeds all other. The Then the amplitude of the component χ″ xx beginning of changes in localization of modes corresponds to a local extreme in the dependence of the component χ″ . The closest xx approach of the FMR branches corresponds to the end of the localization of resonance modes. Minimum χ″ and maximum χ″ xx zz and χ″ occurs at the reaching saturation field. A significant inxy crease in the amplitude of the component χ″ is observed in this xx, l field similarly to that described above for the sample with a combined cubic and uniaxial magnetic anisotropy of layers. For the upper FMR branch all components are significant except χ″yx , χ″yy , χ″yz . The amplitude of the component χ″ is highest at zz low fields. Then it decreases until the field reaches the saturation field. The amplitude of the component χ″ slowly increases until xx the saturation point. The amplitudes of other components is less than χ″ and χ″ . These components have maximum values at low xx zz fields and then decrease slowly.
4. Conclusion The numerical study of the resonance frequency and the various dynamic susceptibility components for a two-layer ferromagnetic film having a combined cubic and uniaxial anisotropy was performed. It has been shown that the value of the amplitudes of the various components of the dynamic susceptibility varies considerably by changing the magnetic field. The presence of cubic anisotropy leads to the fact that the off-diagonal components are essential in addition to the diagonal component of the dynamic susceptibility. The evolution of the profile of the dynamic susceptibility occurring during the magnetization of the film has been investigated. In the lower fields the dynamic susceptibility distribution maximums are at the outer boundaries of the layers. With increasing of the field there is a shift of the localization of the lower FMR mode to the interlayer boundary. Depending on the direction of the magnetic field and the characteristics of the film, there are three variants of the profile evolution: displacement of
the modes between layers, double displacement of the modes between layers and a shift of the localization of the lower FMR mode to the interlayer boundary without displacement of the modes. At the saturation point a significant increase of the amplitude is typical for some of the dynamic susceptibility components. After magnetic saturation of the film the dynamic susceptibility distribution maximums are at the outer boundaries of the layers again. For the lower FMR branch the significant extremes in the dependences of integrated dynamic susceptibility components are observed at saturation points. There are lower extremes at low fields. Some of these extremes are caused by influence of cubic anisotropy. Other are observed at a shift of the localization of the lower FMR mode toward the interface between the layers. This shift is always accompanied by the closest approach of the FMR branches. A monotonic behavior is typical for the dependences of integrated dynamic susceptibility components of the upper FMR branches. Usually there is only one distinct extreme.
References [1] M. Krawczyk, D. Grundler, J. Phys. : Condens. Matter 26 (2014) 123202. [2] A.V. Chumak, V.I. Vasyuchka, A.A. Serga, B. Hillebrands, Nat. Phys. 11 (2015) 453. [3] A.A. Serga, A.V. Chumak, B. Hillebrands, J. Phys. D: Appl. Phys. 43 (2010) 264002. [4] B. Hillebrands, Phys. Rev. B 37 (1988) 9885. [5] A.V. Kobelev, Ya. G. Smorodinskii, Fiz. Tverd. Tela 31 (1989) 6. [6] A. Layadi, J.O. Artman, J. Magn. Magn. Mater. 92 (1990) 143. [7] Z. Zhang, L. Zhou, P.E. Wigen, K. Ounadjela, Phys. Rev. B 50 (1994) 6094. [8] N.G. Bebenin, A.V. Kobelev, A.P. Tankeyev, V.V. Ustinov, J. Magn. Magn. Mater. 165 (1997) 468. [9] J. Geshev, L.G. Pereira, J.E. Schmidt, Phys. B 320 (2002) 169. [10] J. Lindner, K. Baberschke, J. Phys. : Condens. Matter 15 (2003) R193. [11] A. Layadi, AIP Adv. 5 (2015) 057113. [12] A.L. Sukstanskii, G.I. Yampol’skaya, Phys. Solid State 42 (2000) 889. [13] K. Nesteruk, R. Zuberek, S. Piechota, M.W. Gutowski, H. Szymczak, Meas. Sci. Technol. 25 (2014) 075502. [14] A.M. Grishin, V.S. Dellalov, V.F. Shkar, E.I. Nikolaev, A.I. Linnik, Phys. Lett. A 140 (1989) 133. [15] N.K. Dan’shin, V.S. Dellalov, A.I. Linnik, V.F. Shkar’, Phys. Solid State 41 (1999) 1452. [16] N.V. Shulga, R.A. Doroshenko, Phys. Met. Metallogr. 113 (2012) 675. [17] N.V. Shulga, R.A. Doroshenko, Phys. Met. Metallogr. 114 (2013) 1063. [18] N.V. Shulga, R.A. Doroshenko, Phys. Met. Metallogr. 116 (2015) 150. [19] N.V. Shulga, R.A. Doroshenko, Phys. Met. Metallogr. 117 (2016) 124. [20] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Cambridge University, Cambridge 1992, p. 994. [21] A.G. Gurevich, G.A. Melkov, Magnetization Oscillations and Waves, CRC Press, Boca Raton, Florida 1996, p. 464.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
The resonance susceptibility of two-layer exchange-coupled ferromagnetic film with a combined uniaxial and cubic anisotropy in the layers N.V. Shul’ga n, R.A. Doroshenko Institute of Molecule and Crystal Physics, Academy of Sciences, 151, prospect Octobrya, Ufa 450075, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 9 April 2016 Received in revised form 23 June 2016 Accepted 23 June 2016 Available online 24 June 2016
A numerical investigation of the resonance dynamic susceptibility of ferromagnetic exchange-coupled two-layer films with a combined cubic and uniaxial magnetic anisotropy of the layers has been performed. It has been found that the presence of cubic anisotropy leads to the fact that much of the offdiagonal components of the dynamic susceptibility are nonzero. The change of the ferromagnetic resonance frequencies and dynamic susceptibility upon the magnetization along the [100], [010], and [011] directions have been calculated. The evolution of the profile of the dynamic susceptibility occurring during the magnetization has been described. The impact of changes in the distribution of equilibrium and dynamic components of the magnetization on the dependences of the components of the dynamic susceptibility and the ferromagnetic resonance frequency on the external magnetic fields has been discussed. & 2016 Elsevier B.V. All rights reserved.
Keywords: Ferromagnetic resonance (FMR) Two-layer film Dynamic susceptibility
1. Introduction Experimental and theoretical investigation in the field of magnonics causes an increasing interest in the study of multilayered structures [1,2]. This is due to the fact that structures with a periodic modulation of magnetic parameters give unprecedented control of spin waves. Multilayered yttrium-iron garnet (YIG) films are an interesting object of study largely due to their uniquely low magnetic damping [3]. The theory of ferromagnetic resonance for a film consisting of exchange-coupled ferromagnetic layers was developed in the phenomenological approach in the papers [4–12]. It has been described that two-layer film has two ferromagnetic resonance modes, between which there is an energy gap. These calculations are usually carried out in the ultrathin layers approximation or it is supposed that the external magnetic field is high enough for a film in the ground state to be magnetized uniformly [12]. The ferromagnetic resonance (FMR) remains the technique of choice to investigate the parameters of multilayered films [10,13]. Upon excitation of the resonance by uniform alternating magnetic field in two-layer film, two FMR lines, and a series of spin-wave resonance (SWR) lines between them were observed [14]. The SWR series is due to the reflection of spin waves on the boundary n
Correspondence to: IFMK, 151, prospect Oktyabrya, Ufa 450075, Russia. E-mail address: [email protected] (N.V. Shul’ga).
http://dx.doi.org/10.1016/j.jmmm.2016.06.054 0304-8853/& 2016 Elsevier B.V. All rights reserved.
of one of the layers. The number of SWR modes depends on the direction of the external magnetic field and the thickness of the layers of film [15]. The method suggested in [16] and based on the numerical calculation of the dynamic susceptibility makes it possible to study FMR frequencies in two-layer ferromagnetic films when the external magnetic field varies within wide limits. Since this method does not a priori postulate any shape of the coordinate dependence of variable components of magnetization, it become possible to study profiles of the FMR and SWR modes (the distribution of the dynamic susceptibility over the thickness). This method was used for the study of high-frequency properties a ferromagnetic film with the easy-plane and easy-axis anisotropy of the layers. It has been shown that at the nonzero parameter of the interlayer exchange interaction the dynamic components of the magnetization upon the ferromagnetic resonance are distributed through the film thickness inhomogeneously [17]. The evolution profiles of the of the FMR and SWR modes upon the changes in the strength [17] or in the direction of the external magnetic field for different thicknesses of the layers of film [18] has been described. Further, this method has been generalized to the case of ferromagnetic films with a combined cubic and uniaxial magnetic anisotropy of layers [19]. Further studies have shown that not only some features of dependences of the FMR frequencies on the field may indicate the changes taking place in the distribution of equilibrium and dynamic component of the magnetization. These changes can be judged by dependences of the integrated dynamic susceptibility
N.V. Shul’ga, R.A. Doroshenko / Journal of Magnetism and Magnetic Materials 419 (2016) 512–516
on the field. This topic is the subject of our article. We present the results of numerical studies of the high-frequency properties of a two-layer film whose parameters are close to a two-layer YIG film.
2. Model We consider two-layer exchange-coupled ferromagnetic film in an external magnetic field H. The geometry of the film is shown in Fig. 1. The layers of the film have uniaxial anisotropy of various signs and identical cubic anisotropy. The thicknesses of the layers di, i¼ 1, 2 are finite. The normal to the film coincides with the coordinate OX axis, and also with the axis [100] of the crystal and with the axis of uniaxial anisotropy. The direction of the magnetization M is characterized by a polar angle θ, which is counted off from the [001] axis of the crystal coinciding with OZ axis, and by an azimuthal angle ϕ, which is counted off from OX axis. The direction of the magnetic field H is characterized by a polar angle θh and by an azimuthal angle ϕh. It is assumed that the direction of the magnetization depends only on the coordinate x, and in the plane of the film the magnetization is distributed uniformly. The functional of the energy of the system includes the energy of cubic anisotropy, the energy of uniaxial magnetic anisotropy of the easy plane and easy axis types, the Zeeman energy, the energy of exchange interaction inside the layers and the energy of interlayer exchange interaction as follows: K1
2
W = ∑i = 1 ∫ dV ( Vi
−MiH +
Mi4
αi 2Mi2
{M
2 2 x, iMy, i
}
+ Mx2. iMz2. i + Mz2, iMy2, i −
Ku*, i Mi2
Mx2, i−
2
( ∂Mi/∂x) ) − ∫S M1JM2 M1M2dS.
where K1 is the first constant of cubic anisotropy, Mi are the saturation magnetizations of layers, Ku*, i = Ku, i − 2πMi2 are the efficient constants of magnetic anisotropy taking into account a demagnetizing influence of the surface of layers, αi are the constants of exchange interaction; and J is the constant of interlayer exchange interaction. The problem was solved numerically. The procedure of the solution is described here only briefly; it was considered in more detail in [16–19]. Each layer of the film is divided into uniformly magnetized flat cells. The magnetization of the cell with the number l is designated as ml. Thus, the energy functional is converted into the sum of the energies of these cells. Minimizing the energy by standard methods of multidimensional minimization [20], we obtain an equilibrium distribution of the magnetization in the two-layer film. The dynamics of the magnetization in each cell is described by a set of Landau–Lifshitz equations. By linearizing the obtained set of equations relative to small deviations of the vectors of magnetization of the cells from their equilibrium direction δml, we reduce the problem of obtaining the natural frequencies of the film to the solution of the generalized problem of eigenvalues for the thus-obtained set of linear equations. When calculating the dynamic susceptibility, a term of the form δmlh
Fig. 1. Coordinate system.
513
where h is the ac magnetic field, is added into the energy of the system. A relaxation term in the Gilbert form [21] is added to the set of Landau–Lifshitz equations. The solution of the set of equations obtained after the linearization yields the dynamic components of the magnetization δml. Then, the values of the components of the dynamic susceptibility of each cell are calculated: χi, k, l = δml, i / hk , (i, k)¼(x, y, z). Summing up them over all cells, we obtain the values of the components of the integrated dynamic susceptibility. Later in the article we consider its imaginary part . χ" i, k The calculation was carried out for the parameters of a twolayer film characteristic of an iron-garnet film M1 E30 G, M2 E70 G, JE 1 cm 1, αi E 107 erg/cm. The layer thicknesses are d1 E 0.5 10 4 cm and d2 E0.18 10 4 cm. The anisotropy para* E2 104 erg/cm3, * E 7 104 erg/cm3, meters are Ku,1 Ku,2 K1 E 2 104 erg/cm3.
3. Results and discussion Up to the saturation field, the ground state of the film is inhomogeneous. At low fields in case the sample with a combined cubic and uniaxial magnetic anisotropy of layers on the outer boundaries of the layers, the magnetization deviates toward one of the trigonal axes from the normal for the first layer, and from the film plane for the second layer. For the sample with a uniaxial magnetic anisotropy of layers we assumed that, without a loss of generality, it is possible to consider that the magnetization rotates in the plane (x, y), so that only the angle ϕ depends on the coordinate x. Let us consider the effect of changes in static and dynamic components of the magnetization on the behavior of the amplitude of the integrated dynamic susceptibility and frequency when the film is being magnetized by an external field. The field has been applied in one of three directions: along the [100], [011] or [010] axes. As these changes most influence the amplitude of the dynamic susceptibility of the lower FMR mode, later in this article we consider in more detail this mode. The amplitudes of the various components of the dynamic susceptibility vary considerably depending on the strength and direction of the external magnetic field. Let us consider the case when the external field is applied along the [100] axis. For the lower FMR branch of the sample with a combined cubic and uniaxial magnetic anisotropy all components are significant except χ″yx . Fig. 2 shows the dependences components of the integrated dynamic susceptibility of the lower FMR mode χ″ (Fig. 2a) and χ″yy (Fig. 2b), as well as the frequencies of zz lower and upper FMR modes (Fig. 2c) on the external magnetic field. Figs. 3 and 4 show the dependences of the susceptibility component χ″ of the lower and upper FMR modes on the cozz , l ordinate x for different values of the external magnetic field. At low fields, the lower FMR mode is localized in the first layer. It is distributed over a layer non-uniformly. The maximum of the distribution of the dynamic susceptibility is at the outer boundary of the first layer. With increasing of the field there is a shift of the localization of the lower FMR mode to the interlayer boundary. Then it is localized in the second layer. At the same time there is a change of localization of the upper FMR mode. It shifts from the second to the first layer. The displacement of the modes corresponds to the field H1 in throughout the range of variation of the Fig. 2. The component χ″ zz field is greater than other components. It is highest at low fields, and then, with the increase of the field, its amplitude decreases. When the field strength approaches the value Hst at which the
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Fig. 4. Dependences of the component of the dynamic susceptibility χ″ zz , l of the upper FMR mode on x for different values of the external magnetic field. The field is applied along the [100] axis.
Fig. 2. Dependences of the components of the integrated dynamic susceptibility of the lower FMR mode a) χ″ yy and c) the FMR frequencies on the external zz b) χ″ magnetic field. The field is applied along the [100] axis.
Fig. 3. Dependences of the component of the dynamic susceptibility χ″ zz , l of the lower FMR mode on x for different values of the external magnetic field. The field is applied along the [100] axis.
magnetic saturation of the sample is achieved, the amplitude of the component χ″ begins to increase and reaches a maximum at zz the field Hst. The amplitude of the components χ″yy at low fields is close to zero. When the field strength approaches to the value H1 the amplitude begins to increase and reaches a maximum near H1. Then there is its slow decline and a sharp jump to values close to zero after the saturation point. The components χ″ , yz , χ″ . χ″ xx χ″ xy xz behave similarly χ″yy . For upper FMR branch all components are significant except χ″ xz and χ″yx . The amplitudes of the components χ″ and χ″ exceeds all xx zz others. In the range of fields up to 500 Oe amplitude of the
component χ″ exceed χ″ . The maximum χ″ is at low fields and xx xx zz is located near the then its amplitude decreases. Maximum χ″ zz point of the closest approach of the FMR branches. For the sample with a uniaxial magnetic anisotropy of layers, for the lower mode only component χ″ is significant. The amplizz tudes of all other components are close to zero. The amplitude χ″ zz of the lower FMR branch reaches a maximum at the saturation point, and then rapidly decreases. The amplitude χ″ of the upper zz FMR branch has a maximum for lower fields, and then decreases. When the external field is applied along the [011] axis (Fig. 5), for the lower FMR mode all components are nonzero. The field in which there is the closest approach of the FMR branches is marked as H1. Near this field, there is a shift of the maximum of the dynamic susceptibility of the lower mode from the outer boundary of the first layer toward the interface between the layers (see Fig. 6). The displacement modes between layers is not observed. Its profile restores completely when the field strength exceeds the saturation field strength Hst. At this field, the minimum of the lower FMR branch is observed. The component χ″ (Fig. 5a) exceeds all zz other up to the point of the closest approach of the FMR branches. (Fig. 5b) exceeds all other. Its maximum is in Then component χ″ xx the field Hst, when its amplitude increases by several orders of magnitude. For upper FMR branch all components are significant except , yy , χ″yz , χ″ , χ″ remain significant and the χ″yx . The components χ″ xx χ″ zy zz component χ″ exceeds all other throughout the range of variation xx , , have the maximum at low of the field. The components χ″ χ″ χ″ xx yy zz fields, and then decrease. The components χ″yz , χ″ slowly increase. zy When the external field is applied along the [010] axis (Fig. 7), for the lower FMR branch only the components χ″ , xy , χ″yz , χ″ , χ″ xx χ″ zy zz are significant. In the field H1, the displacement of the lower FMR mode from the first layer into the second layer occurs (and, correspondingly, the reverse displacement of the upper mode). In this field, there is the closest approach of the FMR branches, and there are local extremes on the dependences of the susceptibility components. The maximum of the lower mode is on the interlayer boundary (see Fig. 8). There is the first minimum of the lower FMR branch in the field H2. In this field, the polar angle θ becomes identical in both of the layers and coincides with the angle θh. There is the reverse displacement of the FMR modes in the field
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Fig. 5. Dependences of the components of the integrated dynamic susceptibility of the lower FMR mode a) χ″ xx and c) the FMR frequencies on the external zz b) χ″ magnetic field. The components χ″ xx are plotted on a logarithmic scale. The zz and χ″ field is applied along the [011] axis.
Fig. 6. Dependences of the component of the dynamic susceptibility χxx, l of the lower FMR mode on x for different values of the external magnetic field. The field is applied along the [011] axis.
H3. Though the lower mode is localized in the first layer, its maximum is on the interface between the layers, and the FMR signal is excited in the vicinity of the interlayer boundary, until the field strength approaches the saturation field Hst. Then the maximum is shifted to the outer boundary of the first layer and the amplitude of the component χ″ increases by several orders of zz , l magnitude. With increasing field amplitude χ″ decreases, and zz , l this component is distributed more uniformly over the layer. The amplitudes of the component χ″ exceed all others up to zz is higher than any other 1600 Oe (Fig. 7a). Then, the amplitude χ″ xx
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Fig. 7. Dependences of the components of the integrated dynamic susceptibility of the lower FMR mode a) χ″ xx and c) the FMR frequencies on the external zz b) χ″ magnetic field. The components χ″ xx are plotted on a logarithmic scale. The zz and χ″ field is applied along the [010] axis.
Fig. 8. Dependences of the component of the dynamic susceptibility χxx, l of the lower FMR mode on x for different values of the external magnetic field. The field is applied along the [010] axis.
(Fig. 7b). The amplitude χ″ has a maximum in the field H1 and zz then it drops down to the deep minimum in the field Hst. The component χ″ from the minimum in the field H1 increases up to xx the sharp maximum in the field Hst. The component χ″ behaves xy . The components and behaves similarly to χ″ . similarly χ″ χ″ χ″ xx yz zy zz For upper FMR branch all components are significant except χ″yx and χ″ . The amplitudes of the component χ″ exceeds all others up xx zx is higher than any other. The to 200 Oe. Then, the amplitude χ″ zz has a maximum in the low field, and then it component χ″ xx
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decreases. The component χ″ has a maximum in the field H2, and zz then it decreases slowly. Let us consider the sample with a uniaxial magnetic anisotropy of layers. It is in an external magnetic field that lies in the film plane. For the lower FMR branch only the components χ″ , χ″ , xy xx χ″ zz are significant. The amplitude of the component χ″ exceeds all zz other up to the point of the closest approach of the FMR branches. exceeds all other. The Then the amplitude of the component χ″ xx beginning of changes in localization of modes corresponds to a local extreme in the dependence of the component χ″ . The closest xx approach of the FMR branches corresponds to the end of the localization of resonance modes. Minimum χ″ and maximum χ″ xx zz and χ″ occurs at the reaching saturation field. A significant inxy crease in the amplitude of the component χ″ is observed in this xx, l field similarly to that described above for the sample with a combined cubic and uniaxial magnetic anisotropy of layers. For the upper FMR branch all components are significant except χ″yx , χ″yy , χ″yz . The amplitude of the component χ″ is highest at zz low fields. Then it decreases until the field reaches the saturation field. The amplitude of the component χ″ slowly increases until xx the saturation point. The amplitudes of other components is less than χ″ and χ″ . These components have maximum values at low xx zz fields and then decrease slowly.
4. Conclusion The numerical study of the resonance frequency and the various dynamic susceptibility components for a two-layer ferromagnetic film having a combined cubic and uniaxial anisotropy was performed. It has been shown that the value of the amplitudes of the various components of the dynamic susceptibility varies considerably by changing the magnetic field. The presence of cubic anisotropy leads to the fact that the off-diagonal components are essential in addition to the diagonal component of the dynamic susceptibility. The evolution of the profile of the dynamic susceptibility occurring during the magnetization of the film has been investigated. In the lower fields the dynamic susceptibility distribution maximums are at the outer boundaries of the layers. With increasing of the field there is a shift of the localization of the lower FMR mode to the interlayer boundary. Depending on the direction of the magnetic field and the characteristics of the film, there are three variants of the profile evolution: displacement of
the modes between layers, double displacement of the modes between layers and a shift of the localization of the lower FMR mode to the interlayer boundary without displacement of the modes. At the saturation point a significant increase of the amplitude is typical for some of the dynamic susceptibility components. After magnetic saturation of the film the dynamic susceptibility distribution maximums are at the outer boundaries of the layers again. For the lower FMR branch the significant extremes in the dependences of integrated dynamic susceptibility components are observed at saturation points. There are lower extremes at low fields. Some of these extremes are caused by influence of cubic anisotropy. Other are observed at a shift of the localization of the lower FMR mode toward the interface between the layers. This shift is always accompanied by the closest approach of the FMR branches. A monotonic behavior is typical for the dependences of integrated dynamic susceptibility components of the upper FMR branches. Usually there is only one distinct extreme.
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