Reflection of surface spin waves from the interface of uniaxial and biaxial ferromagnets in a planar magnetic field

The article investigates the process of reflection of surface spin waves passing through the interface of uniaxial and biaxial ferromagnets in a planar external magnetic field directed along the hard axis of ferromagnet. The problem is solved using the spin density formalism and the Landau-Lifshitz equations for the case of the absence of dissipation in the system. Geometrical optics formalism is used to describe the processes of refraction of surface spin waves propagating in the ferromagnetic medium with non uniform distribution of magnetic parameters. Quantum mechanical approach is used for calculation of the amplitudes of reflected and transmitted waves. It is shown that spin wave birefringence phenomenon appears at the interface of two uniform ferromagnetic components. Frequency and field dependencies of reflection coefficients for different branches of spin waves are obtained in the study. It is shown that it is possible to change the “optical” parameters of the system by only changing a magnitude of the external homogeneous magnetic field. It is also shown that reflection amplitude depends heavily on the angle of incidence.


INTRODUCTION
Recent advances in nanotechnology and nanoelectronics may enable one in employing high-frequency spin waves in various applications. In particular, the use of spin waves in practical applications is of interest since spin waves exhibit several peculiar characteristics. The wave approach is the most popular approach for describing behavior of propagating spin waves. This approach is successfully used, for instance, for determining various spectral characteristics of magnetic materials [1-4,].
Refraction index of bulk spin waves was calculated in paper [5]. Also the following array of information was discussed in this paper: usage of JWKB approximation for describing behavior of propagating bulk spin waves in uniaxial ferromagnets , behavior of bulk spin waves at the interface between two uniaxial ferromagnets with different magnetic parameters, influence of material parameters on reflection characteristics of bulk spin waves, calculation of parameters of devices that could use discussed phenomena.
Refraction and reflection of bulk spin waves in a biaxial ferromagnet was studied in [6]. The most interesting results were obtained for the case of propagation of surface spin waves in biaxial ferromagnets. The phenomenon of spin wave birefringence occurs in such systems. As demonstrated in [6] direction of magnetic field affects the birefringence phenomenon. E.g. birefringence occurs in planar magnetic field only for the case of surface spin waves.
The growing interest in spintronic devices [9,10] and dramatic progress in the field of nanotechnologies suggests the need of theoretical investigation of behavior of high frequency spin waves in the inhomogeneous media of different configurations. In particular, a series of experimental works of authors A. Serga, M. Kostylev, B. Hillebrands, A. Chumak [10][11][12][13][14] allowed to get the picture of possibilities of management, filtration of spin waves, and also creation of the modules for spin waves generation, switching, etc. in devices based on exchange spin waves [16,17].
This paper is devoted to the problem of reflection of spin waves in the structure consisting of two homogeneous media, one of which is an uniaxial ferromagnet, the second one is a biaxial ferromagnet. The structure is placed into magnetic field directed along the heavy axis of biaxial ferromagnet.

EXPERIMENT
Let's consider unbounded ferromagnet which consists of two half-infinite parts contacting along a plane xOz. The part situated in the area that corresponds to negative values of y possesses uniaxial anisotropy and has the value of saturation magnetization The heavy axis of biaxial ferromagnet and external magnetic field are directed along the axis Oy. Let's calculate refraction indices and reflection coefficients of spin waves in such structure.

1. Basic equations for surface spin waves in biaxial ferromagnet in a planar magnetic field
The energy density magnetic of the described configuration in the exchange (highfrequency) approximation under the assumption 2 j const  M has the following form [7,8]: where   x  -Heaviside step function; А -parameter, that characterizes the exchange interaction in the interface between the half-space at y = 0; We use the formalism of spin density [7], according to which the magnetization can be written as: where ( , ) j t  r -quasiclassical wave functions which play the role of the order parameter of spin density, r -radius vector in the Cartesian coordinates, σ -vector of Pauli matrices.
The principle of least action leads to the following equations for the Lagrangian j  in the case of absence of damping in the system [7]: Using linear perturbation theory and the fact that

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In uniaxial medium we obtain: in biaxial one: where .
On the surface z = 0 boundary conditions must be fulfilled [8]:

Refraction of spin waves
Basing on equations (8) and (9), we can find: If a spin wave wavelength  satisfies the condition of geometrical optics: where a -characteristic size of an inhomogeneity, then an analogue of classical Hamilton-Jacobi equation can be used: where k is wave number at the infinity from the side of incident wave. On the border between uniform uniaxial and biaxial ferromagnets we obtain refractive indices: (12) θ 1 is angle of incidence, θ 2 is angle of refraction.
Boundary conditions for ferromagnetic materials are determined by integrating the equations of motion of the magnetic moment in a small neighborhood of the interface and equating the result to zero with decreasing radius of the integration domain to zero.
So, we get the following system: Note that in the case of "ideal" exchange, that is the absence of defects at the interface (this corresponds to the values A ),and uniform saturation magnetization obtained expressions transform into standard exchange boundary conditions: 21 International Letters of Chemistry, Physics and Astronomy Vol. 28

3. Reflection of surface spin waves at the interface of two inhomogeneous biaxial media
Suppose, the incident wave function is given by   [15]. It is shown that it is possible to achieve the required ratio of intensities of reflected and transmitted waves by choosing appropriate set of parameters for the given spin wave frequency.
Furthermore, as follows from Fig. 2, the intensity of the reflection depends strongly on the value of the external uniform magnetic field, which enables one to change the intensity of the reflected wave in a wide range by only changing the value of the external magnetic field and keeping other parameters of the material constant.

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CONCLUSIONS
The JWKB approximation applied to the case of surface spin waves propagation in ferromagnets is used to show that spin wave birefringence phenomenon occurs at the interface between uniaxial and biaxial homogeneous ferromagnetic media. Each branch of refracted spin wave has its own shape of reflection intensity dependency. This fact enables one to establish required ratio of the "negative" branch reflection intensity to "positive" branch reflection intensity. Due to the fact that branches have different allowed zones one can completely get rid of one of the branches by changing magnetic parameters of the media. It's worth noting that it is possible to control reflection coefficient by changing value of the external magnetic field and keeping other parameters constant only if the exchange parameter is big enough. Value of the maximum transmission amplitude tends to zero as 0 A .
The results of the paper can be successfully used for the development of spin wave microelectronics devices (spin wave filters, spin wave analogues of optical devices, etc.). In particular, these calculations enable one to build a spin wave lens or mirror that has manageable focal distance and reflectiveness.