Thermal fluctuations and longitudinal relaxation of single-domain magnetic particles at elevated temperatures

Abstract

We present numerical and analytical results for the swiching times of magnetic nanoparticles with uniaxial anisotropy at elevated temperatures, including the vicinity of Tc. The consideration is based in the Landau-Lifshitz-Bloch equation that includes the relaxation of the magnetization magnitude M . The resulting switching times are shorter than those following from the naive Landau-Lifshitz equation due to (i) additional barrier lowering because of the reduction of M at the barrier and (ii) critical divergence of the damping parameters.PACS numbers: 75.50. Tt, 75.40.Mg, 75.60.Nt, 75.50.Ss The theory of thermal fluctuations of small magnetic particles is one of the fundamental issues of modern micromagnetics. The conditions at which the particle becomes superparamagnetic define the thermal stability of the magnetized state and, therefore, is also valuable for technological application such as magnetic recording [1]. The basis of the theory has been introduced by Brown [2] who suggested to include thermal fluctuations into the Landau-Lifshitz (LL) dynamical equation as formal random fields whose properties are defined by the equilibrium solution of the correspondent Fokker-Planck (FP) equation. He also derived the Arrhenius-Néel formula to describe the relaxation rate in the axially symmetric case of single-domain particles which was lately generalized to the presence of external field [3,4]. Since the paper of Chantrell and Lyberatos [5], this Langevin-dynamics approach of Brown has been brought to numerical micromagnetics to model the thermal properties of an ensemble of interacting particles and, more generally, of ferromagnetic materials, interpreting the micromagnetic discretization elements as small particles. Generally speaking, the LL equation used in these simulations is a lowtemperature approximation only. Recently a generalized Landau-Lifshitz-Bloch (LLB) equation for a ferromagnet [6,8] was derived which is valid for all temperatures and includes the longitudinal relaxation. The deviations of the LLB dynamics from the LL dynamics should be pronounced at high temperatures, especially close to the Curie temperature T c . The validity of this approach has been confirmed by the measurements of the domain-wall mobility in crystals of Ba-and Sr-hexaferrites close to T c [9].Since the proposal of the heat-assisted magnetic recording (HAMR) [10] the problem of high-temperature thermal magnetization dynamics has become of large practical importance. The basic idea of HAMR is to write bits of information at elevated temperature (close to the Curie temperature, where the switching field is small) and store the information at room temperature. To achieve a significant areal density advantage, the use of high-anisotropy intermetallics such as Ll 0 FePt has been suggested [11]. Therefore, from both fundamental and applied points of view it is necessary to consider the micromagnetics of small particles (or magnetic grains) at elevated temperatures. The straightforward approach [12] uses the formalism of the sta...

Perspective: Ultrafast magnetism and THz spintronics

“…It is also possible to take the experimental temperature and extract the corresponding parameters directly from the reflectivity dynamics and get closest to the experimental temperature values. Historically, the first thermal model developed was the thermal macrospin model from Garanin,120 initially thought to describe temperature effect on spin systems. It is derived from an atomistic form of the Landau-Lifshitz-Gilbert equation of a spin ensemble in the presence of a thermal Langevin field.…”

Moving toward an atomistic reader model

“…The LLB equation does not conserve the magnitude of the magnetization, except for the special case , and may be more appropriate for applications in nanoparticle dynamics and in micromagnetic simulations with extremely fine discretizations. Garanin and Fesenko [15] have recently introduced thermal activation into the LLB equation, leading to a formalism capable of simulating thermally activated magnetization reversal and magnetization fluctuations. A detailed analysis of our dynamic data in terms of the LLB equation is beyond the scope of this paper and will be published elsewhere, but it is important to note that the LLB equation represents an intriguing alternative to the LLG equation, in that it does allow the longitudinal fluctuations along with any impact they might have on long-wavelength fluctuations.…”

Relaxation of Strongly Suppressed Modulus of Magnetization Following Ultrafast Demagnetization

“…Consideration of classical spins interacting with a thermal bath modelled by stochastic Langevin fields demonstrates that the Landau-LifshitzBloch equation valid at all temperatures [22][23][24]. Note that the relaxation term ˆr eff   H was first introduced by Bar'yakhtar [8] and gives the possibility to describe both the nonlinear, longitudinal relaxation of M, and the transversal relaxation of the magnetization.…”