High-Frequency and nonlinear dynamics of magnetic moment

Get Complete Project Material File(s) Now! »

High-Frequency and nonlinear dynamics of magnetic moment

In the first chapter we present general theoretical model and calculations on the second-order nonlinear dynamics and magnetostatic waves in thin ferromagnetic films that was used in the further research presented in the following chapters and could be also used as the base platform for other works in the area. The main results of the chapter are: the basic equations for HF and LF dynamics of magnetic system with a strong collinear magnetic order with an uniaxial anisotropy and magnetic relaxation in the substance in the approximation of a thin planar structure; the dispersion and main parameters of a magnetostatic wave propagating in a thin ferromagnetic plate.

General approach

General approach is useful as it introduces main concepts, system geometry and provides with several crucial theoretical results.

System geometry

The theory on low frequency (LF) elastic vibrations of the structure excited by LF electromagnetic field near SRT was already developed in ref. [28], [29]. Here this approach is extended for LF elastic vibrations excited by modulated HF electromagnetic field. The mechanism of such a process is explained by the combined contributions of two nonlinearities: the nonlinearity of magnetostriction and the nonlinearity of magnetic subsystem susceptibility.
The system geometry is presented in Fig. 1. We suppose that the thin film with magnetization M and thickness dm is placed on a thick nonmagnetic substrate with thickness d. The easy axis of the film is along the length of the sample corresponding to the x-axis and perpendicular to the external magnetic field H applied along the y-axis. The magnetization M has the direction defined by the competition between the external magnetic field and the anisotropy.
Fig. 1. Geometry of the system: e.a. – Easy axis, M – magnetization of the magnetic film, dm and d – thicknesses of the magnetic film and substrate respectively, β – parameter defining the position of the neutral line, H – external magnetic field.
Positions of the neutral lines defined by β are different for the static and dynamic resonance cases [40]. In the static case, the neutral line defined by the parameter β has the position minimizing the sum of elastic and magnetoelastic energies: β = 2/3. For the dynamic case, the vibration modes have an antisymmetrical distribution of deformations relatively to the mean section of the cantilever and therefore: β = 1/2.

Free energy of the system

Following the classical approach [41], [42] one can derive the nonlinear equations of motion for magnetization and elastic strains by retaining high-order terms in the free energy. Overall energy of the sample consists of three parts: elastic Fe, magnetoelastic Fme and magnetic Fm ones.
(1) We assume that the structure is elastically isotropic. Thus the elastic part of energy volume density can be written in the following form [40]:
(2) where uij are components of the strain tensor; C11 and C12 – the two elastic stiffness constants. In this expression, the difference between elastic modules of the substrate and the film was neglected.
Here E and are the integrated Young’s modulus and Poisson’s ratio of the material correspondingly. The magnetic part of energy volume density consists of the Zeeman energy, the anisotropy energy and the energy of demagnetization field.
The free energy of the system contains the magnetoelastic term that provides with the second order non-linearity of mechanical vibrations amplitude dependence on the magnetization. This causes the part of magnetoelastic demodulation phenomenon depending on high frequency magnetization precession amplitude.

READ  Nuclear Receptors and development of marine invertebrates

Magnetostatic approximation

This part of work is mostly based on the fundamental article of Damon and Eshbach [77]. It was carried out with the same geometry and assumptions. On the other hand, it provides with theoretical results derived in a quite different manner. It can be useful as an introduction to the calculations provided in the Chapter 4 and the Chapter 5. Moreover, here one can find the simplified expression for the MSSW dispersion equation that were not presented in the mentioned article.
It includes an alternating magnetization m that could be expressed from the Landau-Lifshits equation of magnetization vector motion through an alternating magnetic field h and a susceptibility
 tensor . The alternating field components can be expressed through potential according to (15). In  this way the components of m could be derived by the potential and then substituted directly to the obtained expressions (16). Thus we have the equation only for the potential . It is the Walker’s equation, which is defined by the susceptibility tensor .

Table of contents :

Introduction
1. High-Frequency and nonlinear dynamics of magnetic moment
1.1. General approach
1.1.1. System geometry
1.1.2. Free energy of the system
1.1.3. HF-dynamics of magnetic moment
1.1.4. LF-dynamics of magnetic moment
1.2. Magnetostatic approximation
1.2.1. Basics.
1.2.1.1. Full formulation of the problem
1.2.2. Frequency ranges for backward volume and surface magnetostatic waves
1.3. Conclusion
2. Ferromagnetic resonance and magnetoelastic demodulation in thin active films with an uniaxial anisotropy
2.1. Looking to the theory of phase transitions
2.2. Flexural bulk vibrations
2.3. Experimental setup.
2.3.1. Methods for hard axis setting
2.4. Microwave measurements
2.5. Optical measurements
2.6. Theory and experiment comparison
2.7. Conclusion
3. Nonlinear magnetoelectric effect in multiferroic nanostructure TbCo/FeCo-AlN in high frequency electromagnetic field
3.1. Experimental technique and results
3.2. Theory of nonlinear magnetoelectric conversion
3.3. Discussion
3.4. Conclusion
4. Propagation of surface magnetostatic waves in an one dimensional magnon crystal of variable thickness
4.1. Propagation of surface magnetostatic waves in a ferromagnetic film with variable thickness
4.2. Propagation of the surface magnetostatic wave in a ferromagnetic film with variable thickness and periodic magnetic inhomogeneity
4.3. Coarse WDM filters based on the phenomenon of MSW propagation in an one dimensional magnon crystal of variable thickness
4.4. Conclusion
5. Anomalous Doppler effect observed during propagation of magnetostatic waves in ferromagnetic films and ferrite–dielectric–metal (FDM) structures
5.1. Doppler effect observed during propagation of MSBVWs and MSSWs in a free ferromagnetic film
5.2. Doppler effect observed during propagation of MSSWs in a FDM structure
5.3. Anomalous Doppler effect on the equifrequency curves
5.4. Conclusion
General conclusion
Résumé étendu en Français
References

GET THE COMPLETE PROJECT

Related Posts