Magnetization dynamics: Difference between revisions

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In physics, '''magnetization dynamics''' is the branch of [[solid-state physics]] that describes the evolution of the [[magnetization]] of a material.

==Rotation Physics==

[[Image:RotationPhysics.png|150px|right]]

A [[magnetic moment]] <math>m</math> in the presence of a [[magnetic field]] <math>H</math> experiences a [[torque]] <math>\tau</math> that attempts to bring the moment and field vectors into alignment. The classical expression for this alignment torque is given by

:<math>\boldsymbol{\tau}=\mu_0 \mathbf{m} \times \mathbf{H}</math>,

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Absent any other effects, this change in angular momentum would be realized through the dipole moment coming into rotation to align with the field.

== Precession ==

However, the effect of a torque applied to an [[electron]]'s magnetic moment must be considered in light of [[spin-orbit interaction]]. Because the magnetic moment of an electron is a consequence of its spin and orbit and the associated angular momenta, the magnetic moment of an electron is directly proportional to its angular momentum through the [[gyromagnetic ratio]] <math>\gamma</math>, such that

:<math>\mathbf{m}=-\gamma \mathbf{L}</math>.

The gyromagnetic ratio for a free electron has been experimentally determined as γ<mathsub>\gamma_e e</sub>&nbsp;= &nbsp;{{val|1.760 859 770760859644e11|(4411) \times 10^{11} \mathrm{|u=s^{-1} T^{-1}}<sup>−1</mathsup>⋅T⁻¹}}.<ref>"[http://physics.nist.gov/cgi-bin/cuu/Value?gammae CODATA Value: electron gyromagnetic ratio]," ''The NIST Reference on Constants, Units, and Uncertainty'', <[http://physics.nist.gov/cgi-bin/cuu/Value?eqgammae|search_for=gyromagnetic+ratio+electron http://physics.nist.gov/cgi-bin/cuu/Value?eqgammae|search_for=gyromagnetic+ratio+electron]></ref>. This value is very close to that used for Fe-based magnetic materials.

Taking the derivative of the gyromagnetic ratio with respect to time yields the relationship,

:<math>\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}=-\gamma \boldsymbol{\tau}</math>.

Thus, due to the relationship between an electron's magnetic moment and its angular momentum, any torque applied to the magnetic moment will give rise to a change in magnetic moment parallel to the torque.

Substituting the classical expression for torque on a magnetic dipole moment yields the [[differential equation]],

:<math>\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma\mu_0 \left(\mathbf{m} \times \mathbf{H}\right)</math>.

Specifying that the applied magnetic field is in the <math>z</math> direction and separating the differential equation into its Cartesian components,

:<math>\frac{\mathrm{d}m_x}{\mathrm{d}t}=-\gamma \mu_0 m_y H_z \qquad \frac{\mathrm{d}m_y}{\mathrm{d}t}=-\gamma \mu_0 m_x H_z \qquad \frac{\mathrm{d}m_z}{\mathrm{d}t}=0</math>,

it can be explicitly seen that the instantaneous change in magnetic moment occurs perpendicular to both the applied field and the direction of the moment, with no change in moment in the direction of the field .<ref name="getzlaff">M. Getzlaff, ''Fundamentals of magnetism'', Berlin: Springer-Verlag, 2008.</ref>.

== Damping ==

While the transfer of angular momentum on a magnetic moment from an applied magnetic field is shown to cause precession of the moment about the field axis, the rotation of the moment into alignment with the field occurs through damping processes.

Atomic-level dynamics involves interactions between magnetization, electrons, and phonons.<ref>J. Stöhr and H. C. Siegmann, ''Magnetism: From Fundamentals to Nanoscale Dynamics,'' Berlin: Springer-Verlag, 2006.</ref>. These interactions are transfers of energy generally termed relaxation. Magnetization damping can occur through energy transfer (relaxation) from an electron's spin to:

* Itinerant electrons (electron-spin relaxation)

* Lattice vibrations (spin-phonon relaxation)

* Spin waves, magnons (spin-spin relaxation)

* Impurities (spin-electron, spin-phonon, or spin-spin)

Damping results in a sort of magnetic field "viscosity," whereby the magnetic field <math>H_{eff}</math> under consideration is delayed by a finite time period <math>\delta{t}</math>. In a general sense, the differential equation governing precession can be rewritten to include this damping effect, such that,<ref>M. L. Plumer, J. van Ek, and D. Weller (Eds.), ''The Physics of Ultra-High-Density Magnetic Recording,'' Berlin: Springer-Verlag, 2001.</ref>

:<math>\frac{\mathrm{d}\mathbf{m}\left(t\right)}{\mathrm{d}t}=-\gamma\mu_0 \mathbf{m}\left(t\right) \times \mathbf{H_{eff}}\left(t-\delta t\right)</math>.

Taking the [[Taylor series]] expansion about ''t'', while noting that <math>\tfrac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}t}=\tfrac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}\mathbf{m}}\tfrac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}</math>, provides a [[linear approximation]] for the time delayed magnetic field,

:<math>\mathbf{H_{eff}}\left(t-\delta t\right)=\mathbf{H_{eff}}\left(t\right)-\delta t \frac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}\mathbf{m}}\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}+\dots</math>,

when neglecting higher order terms. This approximation can then be substituted back into the differential equation to obtain

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and can be written as a scalar, dimensionless damping constant,

:<math>\hat{\alpha}\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t} = \alpha \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}</math>.

[[Image:RotationPhysicsDamping.png|250px|right]]

== Landau-Lifshitz-Gilbert Equation ==

With these considerations, the differential equation governing the behavior of a magnetic moment in the presence of an applied magnetic field with damping can be written in the most familiar form of the [[Landau-Lifshitz-Gilbert equation]],

:<math>\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma \mu_0 \mathbf{m} \times \mathbf{H_{eff}} + \frac{\alpha}{m} \left( \mathbf{m} \times \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}\right)</math>.

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:<math>\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma \left( \boldsymbol{\tau} + \boldsymbol{\tau_{d}} \right)</math>,

where the damping torque is given by

:<math>\boldsymbol{\tau_{d}}=-\frac{\alpha}{\gamma m} \left( \mathbf{m} \times \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}\right)</math>.

By way of the [[Micromagnetism | micromagnetic theory]],<ref>R. M. White, ''Quantum Theory of Magnetism: Magnetic Properties of Materials'' (3rd Ed.), Berlin: Springer-Verlag, 2007.</ref>, the Landau-Lifshitz-Gilbert equation also applies to the [[Mesoscopic scale | mesoscopic]]- and macroscopic-scale [[magnetization]] <math>M</math> of a sample by simple substitution,

:<math>\frac{\mathrm{d}\mathbf{M}}{\mathrm{d}t}=-\gamma \mu_0 \mathbf{M} \times \mathbf{H_{eff}} + \frac{\alpha}{M} \left( \mathbf{M} \times \frac{\mathrm{d}\mathbf{M}}{\mathrm{d}t}\right)</math>.

== References ==

<references />

{{uncategorized|date=May 2009}}

[[Category:Magnetism]]