This book offers a modern, stimulating approach to the subject of nonlinear magnetization dynamics by discussing important aspects such as the Landau-Lifshitz-Gilbert (LLG) equation, analytical solutions, and the connection between the general topological and structural aspects of dynamics.
An advanced reference for the study and understanding of nonlinear magnetization dynamics, it addresses situations such as the understanding of spin dynamics in short time scales and device performance and reliability in magnetic recording. Topics covered include nonlinear magnetization dynamics and the Landau-Lifshitz-Gilbert equation, nonlinear dynamical systems, spin waves, ferromagnetic resonance and pulsed magnetization switching.
The book explains how to derive exact analytical solutions for the complete nonlinear problem and emphasises the connection between the general topological and structural aspects of nonlinear magnetization dynamics and the discretization schemes better suited to its numerical study. It is an exceptional research tool providing an advanced understanding of the study of magnetization dynamics in situations of fundamental and technological interest.
As data transfer rates increase within the magnetic recording industry, improvements in device performance and reliability crucially depend on the thorough understanding of nonlinear magnetization dynamics at a sub-nanoscale level. This book offers a modern, stimulating approach to the subject of nonlinear magnetization dynamics by discussing important aspects such as the Landau-Lifshitz-Gilbert (LLG) equation, analytical solutions, and the connection between the general topological and structural aspects of dynamics. An advanced reference for the study and understanding of nonlinear magnetization dynamics, it addresses situations such as the understanding of spin dynamics in short time scales and device performance and reliability in magnetic recording. Topics covered include nonlinear magnetization dynamics and the Landau-Lifshitz-Gilbert equation, nonlinear dynamical systems, spin waves, ferromagnetic resonance and pulsed magnetization switching. The book explains how to derive exact analytical solutions for the complete nonlinear problem and emphasises the connection between the general topological and structural aspects of nonlinear magnetization dynamics and the discretization schemes better suited to its numerical study. It is an exceptional research tool providing an advanced understanding of the study of magnetization dynamics in situations of fundamental and technological interest.
Front cover 1
Half title page 2
Title page 4
Copyright page 5
Dedication 6
Table of Contents 8
Preface 12
Chapter 1. Introduction 14
Chapter 2. Basic Equations for Magnetization Dynamics 34
Landau--Lifshitz equation 34
Landau--Lifshitz--Gilbert equation 40
Other equations for the description of magnetization dynamics 42
Landau--Lifshitz--Gilbert equation in normalized form 44
Chapter 3. Spatially Uniform Magnetization Dynamics 48
Spatially Uniform Solutions of LLG--Maxwell Equations 48
Structural Aspects of Spatially Uniform Magnetization Dynamics 54
Generalized Magnetization Dynamics 57
Analysis of Equilibrium Points of Magnetization Dynamics 64
Chapter 4. Precessional Magnetization Dynamics 70
Geometric Aspects of Precessional Dynamics 70
Analytical Study of Precessional Dynamics 76
Precessional Dynamics under Transverse Magnetic Field 84
Precessional Dynamics under Longitudinal Magnetic Field 91
Hamiltonian Structure of Precessional Dynamics 98
Chapter 5. Dissipative Magnetization Dynamics 104
Damping Switching in Uniaxial Media 104
Two-Time-Scale Formulation of LLG Dynamics and Averaging Technique 113
Magnetization Relaxation under Zero Applied Magnetic Field 118
Magnetization Relaxation under Applied Magnetic Fields 121
Self-Oscillations and Poincaré--Melnikov Theory 128
Chapter 6. Magnetization Switching 140
Physical Mechanisms of Precessional Switching 140
Critical Fields for Precessional Switching 144
Field-Pulse Duration for Precessional Switching 151
Switching under Nonrectangular Field Pulses (Inverse-Problem Approach) 157
Chapter 7. Magnetization Dynamics under Time-Harmonic Excitation 166
LLG Dynamics in the Presence of Rotational Invariance 166
Periodic Magnetization Modes 170
Quasi-Periodic Magnetization Modes 180
Bifurcation Diagrams 184
Nonlinear Ferromagnetic Resonance, Foldover, and Switching Phenomena 192
Magnetization Dynamics under Deviations from Rotational Symmetry 199
Chapter 8. Spin-Waves and Parametric Instabilities 206
Linearized LLG Equation 206
Spin-Wave Perturbations 215
Stability Analysis 223
Spin-Wave Instabilities and Instability Diagrams 230
Spin-Wave Perturbations for Ultra-Thin Films 239
Chapter 9. Spin-Transfer-Driven Magnetization Dynamics 246
Spin-Transfer Modification of the LLG Equation 246
Stationary States 252
Self-Oscillations 255
Phase Portraits and Bifurcations 259
Stability Diagrams 267
Systems with Uniaxial Symmetry 274
Chapter 10. Stochastic Magnetization Dynamics 284
Stochastic Landau--Lifshitz and Landau--Lifshitz--Gilbert Equations 284
Fokker--Planck Equation for Stochastic Magnetization Dynamics 296
Analysis of Magnetization Dynamics by using Stochastic Processes on Graphs 313
Stationary Distributions and Thermal Transitions 326
Stochastic Magnetization Dynamics in Uniaxial Systems 340
Autocorrelation Function and Power Spectral Density 347
Stochastic Magnetization Dynamics in Nonuniformly Magnetized Ferromagnets 358
Chapter 11. Numerical Techniques for Magnetization Dynamics Analysis 372
Mid-Point Finite-Difference Schemes 372
Mid-Point Finite-Difference Schemes for Stochastic Magnetization Dynamics 380
Numerical Techniques for Nonuniformly Magnetized Particles 386
Micromagnetic Simulations of Magnetization Reversal and Spin-Wave Excitation 398
Micromagnetic Simulations of Chaotic Dynamics 408
References 414
Index 460
Basic Equations for Magnetization Dynamics
Publisher Summary
This chapter deals with the origin of the Landau–Lifshitz (LL) equation, which is a dynamic constitutive relation that is compatible with micromagnetic constraints. The interactions with the thermal bath, which result in the physical phenomena of damping, are accounted for in the LL and Landau–Lifshitz–Gilbert (LLG) equations by introducing different damping terms. By using the appropriate linear combination of the LLG damping terms, the LL and LLG equations can be written in the mathematically equivalent form where the precessional term is the same as in the absence of the thermal bath. Equations for the free energy balance are also derived from the equations. Results show that the free energy is always a decreasing function of time when the external field is constant in time. The Bloch equation serves as an alternative to the LL and LLG equations in situations where the driving actions of applied magnetic fields are so strong that the magnetization magnitude is no longer preserved, at least during short transients before usual micromagnetic states have emerged. Micromagnetics is also reviewed.
2.1 Landau–Lifshitz equation
The Landau–Lifshitz equation for magnetization dynamics in ferromagnets can be construed as a dynamic constitutive relation that is compatible with micromagnetic constraints. To better understand the origin and nature of this equation, it is appropriate to start with a brief discussion of the micromagnetic description of ferromagnets subject to classical electromagnetic fields [10,79].
Micromagnetics is a continuum theory, which is highly nonlinear in nature and includes effects on rather different spatial scales such as short-range exchange forces and long-range magnetostatic effects. In micromagnetics, the state of the ferromagnet is described by the differentiable vector field (r,t) representing the local magnetization at every point inside the ferromagnet. When the temperature is well below the Curie temperature of the ferromagnet, the strong exchange interaction prevails over all other forces at the smallest spatial scale compatible with the continuum hypothesis. This fact is taken into account by imposing the following fundamental constraint:
M(r,t)|=Ms,(2.1)
which means that the magnitude of the local magnetization vector at each point inside the ferromagnet is equal to the spontaneous magnetization s at the given temperature . The direction of (r,t) is in general nonuniform, i.e., it varies from point to point. At equilibria, the spatial distribution of (r,t) results in extrema of an appropriate Gibbs–Landau free energy L(M(.);Ha). This free energy depends on the applied magnetic field a and the temperature . We omit the dependence of L and s on , since in the subsequent discussion the temperature will always be assumed to be uniform in space and constant in time.
The micromagnetic free energy L for a ferromagnet occupying the region is expressed as the following volume integral:
L(M(.);Ha)=∫Ω[AMs2((∇Mx)2+(∇My)2+(∇Mz)2)+fAN(M)−μ02M⋅HM−μ0M⋅Ha]dV.(2.2)
The first term inside the integral represents the exchange energy, which penalizes nonuniformities in the magnetization orientation. The constant is the so-called exchange stiffness constant; its value in ferromagnets is usually of the order of 10−11 J m−1. The second term AN(M) describes crystal anisotropy effects, while the two last terms represent magnetostatic energy and energy of interaction with the external magnetic field. The magnetostatic contribution is governed by the field M. This field is determined by solving the following magnetostatic Maxwell equations:
×HM=0,∇⋅HM=−∇⋅M,(2.3)
subject to the appropriate interface conditions at the ferromagnet surface. The applied field a is produced by external sources and, in subsequent discussion, it will be considered as a given vector function of space and time. The micromagnetic free energy may contain additional terms describing other energy contributions, for example magnetoelastic effects. These additional terms are beyond the scope of our discussion.
To find equilibrium magnetization states under given applied field a, the free energy variation GL with respect to arbitrary variations of the vector field (r) subject to the constraint (2.1) must first be determined. By using standard variational calculus, one obtains that GL corresponding to magnetization variation M(r) is given by the expression:
GL=−μ0[∫ΩHeff⋅δMdV−2Aμ0Ms2∮Σ∂M∂n⋅δMdS],(2.4)
where the second integral is over the surface of the ferromagnet, while /∂n represents the derivative with respect to the outward normal to . The effective field eff is defined as:
eff=Ha+HM+HAN+HEX,(2.5)
where AN and EX are the anisotropy field and the exchange field, respectively:
AN=−1μ0∂fAN∂M,HEX=2Aμ0Ms2∇2M.(2.6)
At equilibrium, GL=0 for any arbitrary variation M consistent with the constraint (2.1). Such a variation of will be of the form:
M=M×δv,(2.7)
where v is a small but otherwise arbitrary space-dependent vector. By substituting Eq. (2.7) into Eq. (2.4) and by taking into account that GL=0 for any arbitrary space-dependent variation v, one finds that at each point in the following equation is valid:
×Heff=0,(2.8)
whereas at each point on :
×∂M∂n=0i.e.∂M∂n=0.(2.9)
The above two forms of the boundary condition are equivalent because M/∂n is perpendicular to as a consequence of (2.1). Equation (2.8) is known as Brown’s equation; it expresses the fact that the local torque exerted on the magnetization by the effective field must be zero at equilibrium [133,134]. The boundary condition given by Eq. (2.9) is valid when no surface anisotropy is present. Surface anisotropy may give rise to pinning effects that substantially alter the response of the ferromagnet to external magnetic fields. In particular, spatially nonuniform magnetization modes may appear under spatially uniform driving fields in ellipsoidal ferromagnetic particles.
It is important to stress that Brown’s equation determines all possible magnetization equilibria regardless of their stability. However, according to the thermodynamic principle of free energy minimization, only L minima will correspond to stable equilibria and, thus, will be in principle physically observable. The information on the nature of equilibria can be obtained by computing the second variation of L and determining if it is positive under arbitrary variations of the vector field (r), subject to the constraint (2.1).
When ×Heff≠0, the system is not at equilibrium and will evolve in time according to some appropriate dynamic equation. The equation originally proposed by Landau and Lifshitz [429] is mostly used for the description of magnetization dynamics. This equation is based on the idea that in a ferromagnetic body the effective field eff will induce a precession of local magnetization (r,t) of the form:
M∂t=−γM×Heff,(2.10)
where >0 determines the precession rate. In the following, for simplicity we shall identify with the gyromagnetic ratio associated with the electron spin, which yields =2.2⋅105A−1ms−1. The dynamics described by Eq. (2.10) is such that the magnitude (length) of magnetization M| is conserved. Indeed, ⋅∂M/∂t=0 is an immediate consequence of Eq. (2.10). Thus, Eq. (2.10) is consistent with the fundamental...
Erscheint lt. Verlag | 20.4.2009 |
---|---|
Sprache | englisch |
Themenwelt | Schulbuch / Wörterbuch |
Informatik ► Office Programme ► Outlook | |
Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik | |
Naturwissenschaften ► Physik / Astronomie ► Festkörperphysik | |
Technik ► Elektrotechnik / Energietechnik | |
Technik ► Maschinenbau | |
ISBN-10 | 0-08-091379-2 / 0080913792 |
ISBN-13 | 978-0-08-091379-7 / 9780080913797 |
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