Reflection of surface spin waves from a layer of uniaxial ferromagnet

Statement of the problem

The great interest has the study of reflective properties of spin waves from multilayer ferromagnets. This is one of the modern ways of researches in the physics of multilayer structures. In this paper will be presented the basic formulas to obtain complex reflection coefficient of spin waves from the multilayer ferromagnetic. Also will be discussed the graphic dependence of reflection coefficient from material parameters, and frequency, external magnetic field.

Solving

Consider the system, which consist of three parts, collision planes are parallel to the plane yz. The first and the third (along the direction of the axis x) part is a homogeneous semi-infinite uniaxial ferromagnet, and among them is N- ferromagnetic layer with modulated exchange interaction constants a, uniaxial magnetic anisotropy b and saturation magnetization M₀. The layers are parallel to the plane yz and has a thickness a and b. The values of a, b і M₀ equal to values , , й , , in appropriate layers. Easy axis is parallel to the direction of the external constant homogeneous magnetic field and the axis z.

We need to get the equation for the magnetization dynamics of spin waves, which will be propagate over the surface of the material. To do this, write down the Landau-Livshitz excluding dissipative member:

Referring to the linear perturbation theory, the external constant magnetic field is represented as , where - the spatial energy density of magnet.

Based on the formalism of spin density distribution of magnetization represented as:

Through linearization of equations for the magnetization, we obtain the next solutions:

Using the condition fixing spins on the surface, we obtain the final equation for the magnetization dynamics:

, w – the frequency, – wave vector, .

- amplitude of the reflection of spin waves from multilayer structures with layers.

- amplitude of the reflection of spin waves from a semi-infinite multilayer structure.

Boundary conditions for non-ideal interaction between layers:

, .

Comparing the incident wave function, the reflected waves - , and the wave that passed through some period - , where and - the wave vectors of the incident (covered) and the reflected wave respectively, we arrive to the expressions for the amplitudes of reflection and passage of spin waves:

Discussion of results

We had got the expression for the complex reflection coefficient of spin waves from infinite uniaxial ferromagnet, and the reflection coefficient for multilayer ferromagnet having a finite number of layers.

Were built the graphical dependences of the spin-wave reflection from a multilayer ferromagnet. The result revealed a strong periodic dependence of the reflection coefficient on the frequency of the incident wave. Also are founded reflection coefficient dependence on the external magnetic field and a constant exchange interaction and uniaxial magnetic anisotropy, and changing of its makes it possible to obtain the required value of the coefficient of reflection.