Crystallographic features of magnetostatic waves spectra in ferrite films

Influence of crystallographic magnetic anisotropy on dispersion laws of magnetostatic spin waves is discussed. Classification of waves spectra in single crystal films of ferrites is given. The scale of magnetic anisotropy influence on spectra of volume and surface types of waves is investigated. The film model with cubic magnetic anisotropy is constructed and expressions for components of a tensor of the effective demagnetization anisotropy factors are deduced. Results of spectra calculation of magnetostatic waves in a film of yttrium iron garnet are given.


Introduction
One of the perspective directions of functional electronics is development of devices, in which transformation of signals is carried out by means of magnetostatic spin waves (MSWs). The greatest interest is represented by MSWs extending in films of ferrite [1,2]. Their phase speed is some orders less than velocity of light and variations of fields in space than in time are more essential to them. The theory of such waves can be constructed on the basis of the equation of the movement of magnetization and Maxwell's equations taken in magnetostatic approach.
Important feature of films is their perfect crystal structure and as a result films possess anisotropy of magnetic properties. Crystallographic magnetic anisotropy belongs to the factors defining dispersive characteristics of MSWs. Radiation effects and changes of temperature lead to changes of anisotropy parameters and it influences on dispersion waves laws. To the present time many aspects of such an influence are studied. Private results relating to the films of the cubic ferrites are together with common approaches to a description of the MSWs spectra in the films with arbitrary magnetic anisotropy type were obtained [3 -8]. The orientation of the investigations just to such ferrites is caused by wide use in the spin-wave electronics of the yttrium iron garnet films (YIG, Y 3 Fe 5 O 12 ), which are grown up on the single crystal gadolinium gallium garnet (Gd 3 Ga 5 O 12 ) substrates. In the present work the model, in which crystallographic orientation of a cubic ferrite film is the varied parameter, is built and investigated. Thus the considered earlier separate orientations become only special cases of the developed model.

General ratios
The studied model is presented in figure 1. The ferrite film thickness of d and the unlimited sizes in the plane is magnetized before saturation by an external constant magnetic field. The vector of magnetization M 0 forms an angle θ with the film plane. Two systems of coordinates are entered into consideration. In xyz system the y axis is parallel to the film plane and the z axis is parallel to the M 0 . In ξηζ system the ξ axis is directed along a normal to a film n and the η axis along a wave vector of MSW k. The φ angle defines a direction of the η axis regarding the xz plane. The dispersive equations (DEs) of the magnetostatic waves are deduced by joint integration of the linearized equation of the magnetization movement without an exchange and losses and Maxwell's equations taken in magnetostatic approach and with the corresponding electrodynamic boundary conditions. As a result the DEs can be presented in the form of the equations expressed through the components of the magnetic permeability tensor of the film material taken in ξηζ system at a reverse inequality. In the xyz coordinates system the tensor of magnetic permeability has a view is the gyromagnetic ratio; H e is the external magnetizing field strength; a ij N are the tensor components of effective demagnetizing anisotropy factors [9]; f is the wave frequency. In passing to the ξηζ coordinate system, the components of tensor  are transformed as follows The analysis of the spectra of the main types MSWs is based on the equations (1), (2).

The perpendicularly magnetized layer when
, and it is also possible to combine y and η axes and therefore equality 2    is carried out.
The equation (4) describes modes of direct volume MSW. ez H is a projection of the vector of the external magnetizing field strength to a magnetization vector.  The equation (5) describes modes of the backward volume waves.

Tangentially magnetized layer when
The equation (6) corresponds to bulk wave and (7) Just surface MSW represents the greatest practical interest.
In experiment orientation of the external magnetizing field but not orientation of magnetization is usually controlled. In the anisotropic and uniformly magnetized film at . It is physically clear that this difference of subjects is less than intensity of the magnetizing field in comparison with the magnetic anisotropy effective fields is more. The detailed consideration of a given question is a separate task and goes out beyond the frames of carried out investigation of dispersion laws. It is possible to note however that on the formulas given above can be calculated angular derivative of frequency and at 0 . So the corrections of dispersion laws connected with consideration of small deflections of the angles from the considered values will have the second order of smallness.
The example of the spectrum calculated on formulas (6) and (7)    ) the large part of MSW spectrum with wave vectors perpendicular to the magnetization will occupy by a surface type wave. Besides spectra of the volume waves arise in the considered conditions only due to crystallographic magnetic anisotropy and in isotropic model these spectra aren't present.

Anisotropy of the surface wave spectrum in a cubic ferrite film
In connection with the considered task we will note that YIG films with crystallographic orientation of {111} type are most widely applied now in spin-wave devices. At the same time some characteristics of devices can be improved when using films of other orientations. In particular the thermostability of SMSW spectra in the YIG films with orientations of {110} and {100} type is higher than in the {111} films.
We shall build a model, allowing executing of calculations for the films oriented along by any of the crystallographic planes passing across an axis of <110> type. Then, for example, all three mentioned orientations {111}, {110}, {100} become special cases of such model. Geometrical aspects of placed task are represented in figure 3.