Magnetization dynamics

Contents
Rotation Physics
A magnetic moment m in the presence of a magnetic field H experiences a torque τ that attempts to bring the moment and field vectors into alignment. The classical expression for this alignment torque is given by
 ,
and shows that the torque is proportional to the strengths of the moment and field and to the angle of misalignment between them.
From classical mechanics, torque is defined as the time rate of change of angular momentum L or, stated mathematically,
 .
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 spinorbit 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 γ, such that
 .
The gyromagnetic ratio for a free electron has been experimentally determined as ^{[1]}. This value is very close to that used for Febased magnetic materials.
Taking the derivative of the gyromagnetic ratio with respect to time yields the relationship,
 .
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,
 .
Specifying that the applied magnetic field is in the z direction and separating the differential equation into its Cartesian components,
 ,
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 ^{[2]}.
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.
Atomiclevel dynamics involves interactions between magnetization, electrons, and phonons^{[3]}. 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 (electronspin relaxation)
 Lattice vibrations (spinphonon relaxation)
 Spin waves, magnons (spinspin relaxation)
 Impurities (spinelectron, spinphonon, or spinspin)
Damping results in a sort of magnetic field "viscosity," whereby the magnetic field H_{eff} under consideration is delayed by a finite time period δt. In a general sense, the differential equation governing precession can be rewritten to include this damping effect, such that,^{[4]}
 .
Taking the Taylor series expansion about t, while noting that , provides a linear approximation for the time delayed magnetic field,
 ,
when neglecting higher order terms. This approximation can then be substituted back into the differential equation to obtain
 ,
where
is called the dimensionless damping tensor. The damping tensor is often considered a phenomenological constant resulting from interactions that have not yet been fully characterized for general systems. For most applications, damping can be considered isotropic, meaning that the damping tensor is diagonal,
 ,
and can be written as a scalar, dimensionless damping constant,
 .
LandauLifshitzGilbert 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 LandauLifshitzGilbert equation,
 .
Since without damping is directed perpendicular to both the moment and the field, the damping term of the LandauLifshitzGilbert equation provides for a change in the moment towards the applied field. The LandauLifshitzGilbert equation can also be written in terms of torques,
 ,
where the damping torque is given by
 .
By way of the micromagnetic theory^{[5]}, the LandauLifshitzGilbert equation also applies to the mesoscopic and macroscopicscale magnetization M of a sample by simple substitution,
 .
References
 ^ "CODATA Value: electron gyromagnetic ratio," The NIST Reference on Constants, Units, and Uncertainty, <http://physics.nist.gov/cgibin/cuu/Value?eqgammaesearch_for=gyromagnetic+ratio+electron>
 ^ M. Getzlaff, Fundamentals of magnetism, Berlin: SpringerVerlag, 2008.
 ^ J. Stöhr and H. C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics, Berlin: SpringerVerlag, 2006.
 ^ M. L. Plumer, J. van Ek, and D. Weller (Eds.), The Physics of UltraHighDensity Magnetic Recording, Berlin: SpringerVerlag, 2001.
 ^ R. M. White, Quantum Theory of Magnetism: Magnetic Properties of Materials (3rd Ed.), Berlin: SpringerVerlag, 2007.
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
Magnetization — This article is about magnetization as it appears in Maxwell s equations of classical electrodynamics. For a microscopic description of how magnetic materials react to a magnetic field, see magnetism. For mathematical description of fields… … Wikipedia
Magnetization reversal — Magnetization reversal, or switching, represents the process that leads to a 180° reorientation of the magnetization vector with respect to its initial direction, from one stable orientation to the opposite one. Technologically, this is one of… … Wikipedia
Exchange Bias — Als Exchange Bias (EB) bezeichnet man eine unidirektionale Anisotropie (deshalb auch unidirectional exchange anisotropy genannt), die durch die Kopplung zwischen einem Ferro und einem Antiferromagneten entsteht. Der Exchange Bias bewirkt eine… … Deutsch Wikipedia
Micromagnetics — deals with the interactions between magnetic moments on sub micrometre length scales. These are governed by several competing energy terms. Dipolar energy is the energy which causes magnets to align north to south pole. Exchange energy will… … Wikipedia
Landau–Lifshitz–Gilbert equation — In physics, the Landau Lifshitz Gilbert equation , named for Lev Landau and Evgeny Lifshitz and T. L. Gilbert, is a name used for a differential equation describing the precessional motion of magnetization M in a solid. It is a modification by… … Wikipedia
Nanomagnet — A nanomagnet is a submicrometric system that presents spontaneous magnetic order (magnetization) at zero applied magnetic field (remanence). The small size of nanomagnets prevents the formation of magnetic domains (see single domain (magnetic)).… … Wikipedia
Micromagnetism — General= Micromagnetism deals with the interactions between magnetic moments on sub micrometre length scales. These are governed by several competing energy terms. Dipolar energy is the energy which causes magnets to align north to south pole.… … Wikipedia
Magnetic resonance imaging — MRI redirects here. For other meanings of MRI or Mri, see MRI (disambiguation). Magnetic resonance imaging Intervention Sagittal MR image of the knee ICD 10 PCS B?3?ZZZ … Wikipedia
Nuclear magnetic resonance spectroscopy of proteins — (usually abbreviated protein NMR) is a field of structural biology in which NMR spectroscopy is used to obtain information about the structure and dynamics of proteins. The field was pioneered by Richard R. Ernst and Kurt Wüthrich[1], among… … Wikipedia
Protein nuclear magnetic resonance spectroscopy — (usually abbreviated protein NMR) is a field of structural biology in which NMR spectroscopy is used to obtain information about the structure and dynamics of proteins. The field was pioneered by, among others, Kurt Wüthrich, who shared the Nobel … Wikipedia