Thermal Fluctuations of a Single-Domain Particle

Abstract

A statistical ensemble of particles, with moment orientations (θ, φ), can be represented by a surface density W (θ, φ, t) of points on the unit sphere. The corresponding surface density J satisfies a continuity equation ∂W/∂t=−∇·J. With no thermal agitation, J=WṀ/Ms, where M is the vector magnetization (| M | = const = Ms); its rate of change Ṁ is assumed to be given by Gilbert's equation. To include thermal agitation, we may add to J a diffusion term −k′∇W; this gives directly the ``Fokker-Planck'' equation of a previous, more laborious calculation. When ∂/∂φ=0, the equation simplifies and can be replaced by a minimization problem, susceptible to approximate treatment. In the case of a free-energy function with deep minima at θ=0 and π, such treatment leads again to a result derived previously by a method adapted from Kramers and valid when v(Vmax−Vmin)/kT is at least several times unity (v=particle volume, Vmax and Vmin=maximum and minimum free energy per unit volume, k=Boltzmann's constant, T=Kelvin temperature). When the minima are not deep, a different treatment is necessary; this leads to a formula valid when v(Vmax−Vmin)/kT<<1.

Superparamagnetic relaxation times for mixed anisotropy and high energy barriers with intermediate to high damping: 2. Uniaxial axis in a 〈111〉 direction

“…If the uniaxial axis is in a h 111 i direction, n 0 = (1.3 -2.6) Â 10 10 Hz. Finally, the Brown [1963] expression for pure uniaxial anisotropy gives n 0 = (1.6-3.2) Â 10 10 Hz. Note that the predicted n 0 depends on the crystallographic orientation of the uniaxial axis, which is unknown.…”

“…The case K 0 1 = 0 (pure uniaxial) was treated by Brown [1963]. Assuming K 0 1 6 ¼ 0, let k = K u /K 0 1 .…”

Energy repartition in the nonequilibrium steady state

“…We show that gradients of external magnetic fields localize spin waves in the Wannier-Zeeman fashion, while weak many-body interactions (nonlinearities) lead to a mode-temperature renormalization. The LLMS equation encompasses all standard equations for classical spin dynamics, reducing to the (stochastic) Landau-Lifshitz-Gilbert (LLG) equation [12][13][14][15] and the Bloch equation [16] in respective limits. Our generic results should be widely applicable to describe the semiclassical dynamics of other thermodynamic systems such as Newtonian liquids, elastic solids, and Josephson junctions.…”

Thermal stability of magnetic states in submicron magnetic islands

“…A maximum along an MEP corresponds to a first order saddle point on the energy surface and gives an estimate of the activation energy within HTST. In adaptions of these rate theories to magnetic systems [6][7][8][9][10][11][12], the magnitude of the magnetic vectors is either assumed to be constant as orientation changes, or it is treated as a fast variable obtained from self-consistency calculations for fixed values of the slow variables specifying orientation [13]. The energy surface of a system of N magnetic moments is then a function of 2N degrees of freedom defining the orientation of the magnetic moments.…”

“…Brown [6,7] estimated the pre-exponential factor for remagnetization transitions in a single domain, uniaxial magnetic particle to be on the order of 10 9 -10 12 sec −1 . The size and shape of the particle as well as the materials properties will affect the value.…”