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Parametrical and nonlinear magnetoelastic effects in magnetically ordered materials
introduction
The First Chapter presents a review of magnetoacoustics and defines its current state of the art and reveal its topical research problems. It starts from the description of the phenomenon of magnetoelastic waves in general and various parametric and nonlinear effects of these waves. Parametric instabilities in ferrites are discussed and different mechanisms that cause these instabilities under parallel and perpendicular electromagnetic pumping are analyzed. These instabilities were widely researched in the past and are currently used in various applications, notably in wave phase conjugation. Key features of phase conjugation are presented with a review of applications.
Magnetoelastic coupling is especially high in antiferromagnetic materials with easy plane anisotropy (AFEP), where the coupling introduces giant effective nonlinearity. The chapter continues with description of parametric and nonlinear processes in AFEP crystals that manifest themselves much brighter than in ferrites. Modern discoveries and trends in magnetoacoustics are discussed. Recently predicted nonlinear parametric instabilities in antiferromagnets are among the topical problems as they are anticipated to have unique features that differ them from regular parametric instabilities. Suggested explosive supercritical dynamics of these nonlinear processes is yet to be experimentally observed due to strong limitation mechanisms, but presents considerable scientific importance. In the end of the chapter as a result of presented state of the art analysis the scope of the present work is set and main goals are defined.
Magnetoelastic waves in ferromagnets and ferrites. Magnetoacoustic resonance
Condensed matter consists of subsystems of various physical natures. Excitations of these subsystems interact with each other in different ways that depend on the excitation amplitudes and coupling coefficients between the subsystems. The study of such linear and nonlinear interaction helps to understand better the materials. One of the examples of coupled systems is magnetically ordered materials. In such materials magnetic subsystem and elastic subsystem are interacting with each other via magnetoelastic coupling, causing the mixture of Chapter I. Parametrical and nonlinear magnetoelastic effects in magnetically ordered materials spin waves and elastic waves, so they no longer exist separately, but become coupled magnetoelastic waves.
The first works that interpreted magnetoelastic coupling as a reason of interaction between the elementary excitations like magnons (spin waves) and phonons (elastic waves) [1] were done by Akhiezer back in forties. Further work by Turov and Irhin that discussed coupled magnetoelastic waves as a separate phenomenon was published later in 1956 [2] with realization, that not only magnetoelastic coupling needs to be taken into account, but magnetoelastic waves themselves with their new properties present a fundamental interest in physics.
Active research of mangetoelastic waves started in 1958 with theoretical approach of coupled magnetoelastic waves in ferromagnets and ferroacoustic resonance by Akhiezer, Baryahtar and Peletminsky [4], Kittel’s work on spin waves and ultrasonic waves interaction [5], Spencer’s and LeCraw’s work on parametrical magnetoelastic resonance in yttrium iron garnet [3]. These works served as a foundation to the new important section of physics called magnetoacoustics and fueled up further research in this field. Yttrium iron garnet ferrite served as an important object for studies of magnetoelastic wave effects because the acoustic losses in this media were found to be an order of magnitude lower than in quartz [6]. A typical spectrum of coupled magnetoelastic waves in a ferrite is presented in Figure 1.1.
Dashed lines show “pure” spectra of spin waves and elastic waves . Elastic waves spectrum is determined as ωs = υsk where ωs is the elastic wave frequency, k is the wave vector and υk is the speed of sound.
The interaction that exists between spin waves and elastic waves makes the spectra coupled (solid lines) and causes a number of static and dynamic phenomena in magnets including ΔE-effect (elastic modules change due to magnetic field), spontaneous symmetry breaking, magnetoacoustic resonance (MAR) [8] and many others.
Figure 1.1. Spectrum of coupled magnetoelastic waves in ferrite. Magnon branch has two intersections with phonon branch υsk.
MAR appears as a rapid energy absorption in the conditions when frequencies and wave vectors of magnetic and elastic waves become identical, i.e. in the intersection points of spin and elastic spectra. These points correspond to the strongest magnetoelastic coupling in the system and are shown in Figure 1.1. In these points the spectra are actually being pushed away from each other. The frequencies of intersections can be easily obtained from the spectra equations above, there are two of them, in case when Θ = 0, and / . These frequencies generally appear in the hypersound band, generally lies in the frequencies region where elastic waves propagation is rather problematic, making MAR at of an interest. It is also important to notice that the term “MAR” is sometimes used also to describe the process of parametric excitation of acoustic oscillations with the help of lectromagnetic wave.
The coupling coefficient between magnetic and elastic subsystems in general theory of oscillations is determined by the energies relation:
where Wme is the energy of magnetoelastic interaction, Wm is the energy of magnetic subsystem and We is the energy of elastic subsystem. The value of coupling coefficient represents relative splitting of magnetic and elastic branches of the spectrum near the MAR point. The square of the coupling coefficient determines relative interaction of the magnetoelastic waves spectra as portion of energy that is being transmitted from one subsystem to another in the points far away from the MAR, including the important band of ultrasound waves, where , or in the magnets where MAR is not possible.
Magnetoelastic waves became an interesting object to study as it was relatively easy to excite them and vary their characteristics with different alternating elastic stress and magnetic field. They found important applications in different systems, including controllable delay lines [9,10], parametric amplifiers [11,12], bandwidth compression [13], etc. The easiest way to excite magnetoelastic waves is to use parametric pumping.
Parametric magnetoelastic instabilities in ferrites
Parametric excitation in general is a method of exciting and maintaining oscillations in a dynamic system (or coupled subsystems), in which excitation results from a periodic variation of some parameter in the subsystem. So magnetoelastic waves can be parametrically excited by variation of the parameters of magnetic or elastic subsystem. Nowadays parametric excitation in ferromagnets and antiferromagnets has become a very important tool that helps studying energy flows in nonequilibrium systems. The most common way to excite magnetoelastic waves is to apply an alternating magnetic field perpendicular or parallel to the bias magnetic field; however there are multiple mechanisms of parametric instabilities formation, that are discussed in this section.
Parametric instabilities excited by transverse pumping
First experimental observation of parametric resonance of magnetoelastic waves in ferrites under transverse pumping (when the alternating magnetic field h is applied perpendicular to the magnetization M) were made in the works of Damon [14], and Bloembergen and Wang [15] during ferromagnetic resonance experiments. They have found abnormal energy absorption that had a clear threshold. Suhl [16] explained this effect as parametric instability of uniform magnetization precession with regards to spin waves pair excitation with frequencies ω1 and ω2 and wave vectors and . He also formulated the condition of parametric resonance in continuous media:
where is the frequency of uniform precession and n = 1, 2, 3… The number n determines the order of the instability. Since , parametric instability creates pairs of waves with equal but opposite wave vectors. At first theoretical studies of the parametric processes were focused mostly on determinations of the processes thresholds and instability amplitude limitations. This process was called Suhl first order instability. In further works other mechanisms of magnetic, elastic and magnetoelastic instabilities that may be excited parametrically by means of magnetic pumping field h applied transverse to the bias field H were reported [3,17,18]. An important step was made when a summarizing theory of different transversely pumping mechanisms was published by Auld in his work [19].
In his work Auld actually classified known parametric instabilities, defining Suhl first-order instability [16], Suhl second order instability, reaction instability, magnetoacoustic resonance instability (MAR). A summary of these processes is presented in Figure.1.2.
There is spin-wave dispersion relation, is longitudinal elastic waves dispersion, and are two shear elastic waves. and show the upper and lower branches of the coupled shear magnetoelastic waves. and are the lower and upper branches of coupled longitudinal magnetoelastic waves. Possible instabilities are shown with the numbers and described in the figure.
The thresholds here will be shown in terms of critical pump precession angles . For Suhl first order instability mentioned earlier the critical pump precision threshold depends on the dipolar coupling strength and on the damping of the spin wave. When the elastic displacement vector angle with respect to external magnetic field 0 the threshold is infinite, so general form is:
Auld’s numerical calculations of thresholds showed that the MAR instability and reaction instability thresholds in YIG lie above Suhl instabilities [19]. The studies of the interaction of parametric coupling and travelling waves have shown that travelling waves can be parametrically amplified experimentally with transverse pumping field [11, 20-22]. The first parametric generation of magnetoelastic waves together with their amplification was experimentally reported by Damon and van de Vaart [23] at X-band frequencies and liquid helium temperatures, and later on at room temperatures and lower frequencies [24]. In further work on the phenomenon the net gain exceeded 55 dB at the temperature 1.5 K [25].
An important specificity of all the calculated parametric thresholds is that they are proportional to Qm 1/ 2 (the reaction instability varies at approximately same rate in the vicinity of its minimum value). The amplitude of the initial oscillations does not change the threshold. This is a typical behavior of the instabilities arising from those terms in the magnetocrystalline energy which are linear in the strain variations. Recently discovered parametric instabilities that arise from nonlinear strain variations have different threshold conditions that depend also on the initial oscillations amplitude. These instabilities are discussed in the section 1.8 of this chapter.
Table of contents :
Acknowledgement
Introduction
Chapter I. Parametric and Nonlinear Magnetoelastic Effects in Magnetically Ordered Materials
1.1 Chapter I introduction
1.2 Magnetoelastic waves in ferromagnets and ferrites. Magnetoacoustic resonance
1.3 Parametric magnetoelastic instabilities in ferrites
1.3.1 Parametric instabilities excited by transverse pumping
1.3.2 Parametric instabilities excited by parallel pumping
1.4 Magnetoelastic parametric wave phase conjugation
1.4.1 The phenomenon of wave phase conjugation
1.4.2 Principles of parametric wave phase conjugation in solids
1.4.3 Parametrically active media for wave phase conjugation.
1.4.4 Applications of phase conjugated ultrasonic beams
1.5 Magnetoelastic waves in antiferromagnets
1.5.1 Magnetoelastic coupling coefficient in antiferromagnets
1.5.2 The frozen lattice (magnetoelastic gap) effect in magnetic materials
1.5.3 Spectra of magnetoelastic excitations in easy plane antiferromagnets
1.5.4 Magnetoelastic coupling of critical resonator modes
1.6 Parametric excitation of magnetoelastic waves in easy plane antiferromagnets
1.7 Giant effective anharmonicity of easy plane antiferromagnets
1.8 Over-threshold nonlinearity of parametric magnetoelastic waves and excitations in antiferromagnets
1.8.1 Nonlinear effects due to the third order effective anharmonicity
1.8.2 Cubic nonlinearity in antiferromagnets
1.9 Multi-boson nonlinear effects in magnetic materials. Three waves coupling
1.10 Chapter I conclusion
Chapter II. Single Mode Three Quasi-Phonon Excitations and Supercritical Dynamics in Effective Anharmonicity Model
2.1 Chapter II introduction
2.2 Anharmonicity model of three quasi-phonon excitations
2.3 Three quasi-phonon interaction coefficient in perpendicular pumping geometry
2.4 Single mode three quasi-phonon excitations model
2.5 Threshold of the three quasi-phonon parametric instability
2.6 Explosive behavior of the parametric instability without higher order nonlinearity
2.7 Influence of cubic nonlinearity on the parametric instability
2.8 Compensation of the cubic nonlinearity in single mode three quasi-phonons excitations
2.9 Numerical simulations of the three quasi-phonon excitations using the anharmonicity model
2.10 Chapter II conclusion
Chapter III. Experimental Studies of Supercritical Explosive Dynamics of Single Mode Three Quasi-Phonon Instability
3.1 Chapter III introduction
3.2 Easy plane antferromagnetic resonators of α-Fe2O3 and FeBO3
3.3 Experimental setup for the studies of magnetoelastic properties
3.4 Magneto-elastic characteristics of α-Fe2O3 resonator
3.4.1 Spectrum of magneto-elastic coupled oscillations of α-Fe2O3
3.4.2 Dynamic properties of hematite contour shear mode
3.4.3 Nonlinear frequency shift of the magneto-elastic mode
3.4.4 Attenuation of contour shear mode in hematite resonator
3.5 Experimental technique of supercritical three quasi-phonon instability research
3.5.1 Geometry of the three quasi-phonon experiment
3.5.2 Resonator excitation for three quasi-phonon instability observation
3.5.3 Experimental setup for the explosive instability research
3.6 Supercritical single mode three quasi-phonon excitations in α-Fe2O3
3.7 Magnetoelastic characteristics of FeBO3 resonator
3.7.2 Dynamic properties of contour shear mode in FeBO3
3.7.3 Nonlinear frequency shift of the magneto-elastic mode
3.7.4 Attenuation of contour shear mode oscillations in iron borate resonator
3.7.5 Peculiarities of low temperature dynamics of FeBO3 resonator
3.8 Supercritical single mode three quasi-phonon excitations in FeBO3
3.9 Chapter III conclusion
Chapter IV. Strongly Nonlinear Model of Three Quasi- Phonon Excitations in AFEPs. Numerical Simulations
4.1 Chapter IV introduction
4.2 Strongly nonlinear model of three quasi-phonon excitations in a magnetoelastic AFEP resonator
4.3 Numerical simulations of single mode three quasi-phonon excitations in iron borate
4.3.1 Instability simulation for FeBO3 resonator using anharmonic approximation
4.3.2 Instability simulation for FeBO3 resonator using strongly nonlinear model
4.4 Numerical simulations of single mode three quasi phonon excitations in hematite
4.4.1 Instability simulation for α-Fe2O3 resonator using anharmonic approximation .
4.4.2 Instability simulation for α-Fe2O3 resonator using strong nonlinear model
4.5 Explosive instability gain dependence on the initial phase of pumping pulse
4.6 Chapter IV conclusion
Chapter V. Explosive Dynamics and Spatial Localization of Coupled Travelling Magnetoelastic Wave Triads
5.1 Chapter V introduction
5.2 Nonlinear three-wave parametric coupling in magnetoelastic system
5.3 Dynamic properties of experimental hematite crystal
5.4 Anharmonic model of three travelling waves coupling with parallel electromagnetic pumping
5.4.1 Three waves coupling under parallel electromagnetic pumping
5.4.2 Nonlinear phase shift of ultrasonic triads under parallel pumping
5.5 Numerical simulations program
5.6 Numerical simulations of three travelling waves coupling model with parallel pumping
5.6.1 Numerical solutions of parallel geometry model in subthreshold mode
5.6.2 Numerical solutions of parallel geometry model in supercritical mode
5.7 Anharmonic Model of Three Travelling Waves Coupling with Perpendicular Electromagnetic Pumping
5.7.1 Three Waves Coupling Under Perpendicular Electromagnetic Pumping
5.7.2 Nonlinear Phase Shift of Three Waves Coupling Under Perpendicular Pumping
5.8 Numerical Simulations of Three Travelling Waves Coupling Model with Parallel Pumping
5.8.1 Perpendicular geometry model numerical solutions in subthreshold mode
5.8.2 Perpendicular geometry model numerical solutions in supercritical mode
5.8.3 Nonlinear phase shift compensation
5.9 Chapter V Conclusion
Thesis Conclusion
Résumé étendu en Francais
References
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