Stoner-Wohlfarth model

If the material is ferromagnetic then the entropy, susceptibility and resistance at temperatures just above the Curie point are to be calculated in much the same way (i.e. in terms of spin fluctuations). A treatment of this problem starting from the Stoner-Wohlfarth model is due to Moriya and Kawabata (1973). 

Many ferromagnets are metals or metallic alloys with delocalized bands and require specialized models that explain the spontaneous magnetization below Tc or the paramagnetic susceptibility for T > Tc. The Stoner-Wohlfarth model,6 for example, explains these observed magnetic parameters of d metals as by a formation of excess spin density as a function of energy reduction due to electron spin correlation and dependent on the density of states at the Fermi level. However, a unified model that combines explanations for both electron spin correlations and electron transport properties as predicted by band theory is still lacking today. 

Figure 3 shows two hysteresis loops calculated with the modified Stoner-Wohlfarth model with high exchange interactions and low exchange interactions between the crystallites of a particle. Loop shape, remanence and coercive field are in good agreement with experimental results reported in [9],... 

Figure 3. Hysteresis loops for FePt particles calculated using the modified Stoner-Wohlfarth model for high exchange (A = 1.4 10"11 J/m) and low exchange (A = 0.4 10 11 J/m).

Figure 1. Angular variation of the normalized coercive field Hc(0)/Ha ( ) Stoner-Wohlfarth model and (o) 1/cos 6 dependence, approximately observed in usual hard magnets. The value of Hc(0) is arbitrary, //C(0)///A = 0.2 has been assumed. At large 6 values, when //c(0) > //sw(0), coherent rotation is favoured again. Inset definitions of the various angles involved in Eqs. 1-3.

Figure 16. Stoner-Wohlfarth predictions (a) schematic ellipsoidal cell used to calculate switching fields and the switching astroid in the Stoner-Wohlfarth model and (b) first quadrant of the switching asteroid, showing the switching boundary line.

Figure 17. Energy barriers and switching in the Stoner-Wohlfarth model (a) barrier between state 0 and state 1 when no field is applied, (b) reduced energy barrier under half-select conditions only easy axis fields applied, and (c) switching field distribution for fully selected bits and for half-selected bits. For the full lines, the separation from the center of the two distributions is about 10 o. In this situation half-select write errors do not occur. The dashed lines represent broadened switching field distributions. In this case the half-select and the full selected distributions start to overlap.

Let us consider the condition for thermal stability of the patterned-media with perpendicular anisotropy, based on a perpendicular M-H loop [28], The net M-H loop for a dot array is the statistic result of small Stoner-Wohlfarth model like square M-H loops of each magnetic dot. The thermally stable condition for a magnetic dot array is just that of the condition for a dot, which is the easiest to reverse among the whole dots. Using the beginning field of the reversal, namely, the nucleation field of the magnetic dot array, H, the condition is expressed as. 

A more complex magnetic behaviour is expected for RI compounds in which the second component is a 3d transition metal such as Mn, Fe, or Co. The magnetic behaviour of the transition metal component is now based on the magnetic polarization of the electronic d-bands. Consequently, in this section we summarize the theory of itinerant or band magnetism and its application to transport properties. We begin with the Stoner-Wohlfarth model and include a summary of recent works. 

As a result of the Stoner-Wohlfarth model, following the method of Edwards and Wohlfarth (and for higher terms de Chatel and de Boer, 1970) one obtains... 

Fig. 7 Schematic representation of the evolution of the magnetic properties of magnetic nanoparticles as a function of their volume and of the models suitable to describe them. The label (1) illustrates that the maximum magnetic field for which the linear response theory (Neel relaxation model) is valid decreases with increasing volume. The label (2) is the domain where incoherent reversal modes occur so Stoner-Wohlfarth model based theories are not valid anymore. The label (3) shows a plateau in the volume dependence of the coercive field. Reprinted with permission from Ref 41. Copyright 2011, American Institute of Physics.

The exponent m cannot be regarded as a fitting parameter but depends on the symmetry of the system. In most cases, m = 3/2 [16, 140, 158, 166, 167, 174, 175], but m = 2 for highly symmetric systems, such as aligned Stoner-Wohlfarth particles. In particular, the m = 3/2 law is realized for misaligned Stoner-Wohlfarth particles and for most domain-wall pinning mechanisms [5], Experimental values of m tend to vary between 1.5 to 2 [136, 158]. Linear laws, where m = 1, are sometimes used in simplified models, but so far it hasn t been possible to derive them from physically reasonable energy landscapes [5, 16, 176]. The same is true for dependences such as /H- l/H0 [177], where series expansion yields an m = 1 power law. 

The resistance to magnetization reversal indicates that an energy barrier separates the initial and the final magnetic states. This energy barrier is a consequence of magnetic anisotropy. This can be illustrated within the so-called Stoner-Wohlfarth (SW) model, in which reversal is assumed to occur by in-phase rotation of all moments (coherent rotation) [7], For HoPP, antiparallel to M, the energy may be expressed as ... 

Free-layer switching during the write process can be basically modeled in the Stoner-Wohlfarth coherent-rotation model (Ch. 4), which yields an astroid switching curve [71]. Figure 16(a) shows an ellipsoidal cell, with magnetization M and applied field H. In the Stoner-Wohlfarth approximation, the cell energy per unit volume is... 

Stancu A, Spinu L (1998) Temperature- and time-dependent Preisach model for a Stoner-Wohlfarth particle system. IEEE Trans Magnetics 34 3867-3875 Stein DL (1992) Spin glasses and biology. World Scientific, Singapore... 

The application of a pure itinerant model (such as the Stoner-Wohlfarth or the Fermi-liquid model) is limited in R-3d compounds. Such models, if they are valid, are confined to systems where R possesses no moment, e.g. Y, Lu, Zr. 

Based on the use of the Stoner-Wohlfarth theory for single-domain partieles, the Neel-Brown relaxation model, and equilibrium functions, Medahuoi et al. have composed a model that allows for a direct comparison of theoretical simulations and experimental results from hyperthermia experiments carried out in iron nanoparticles with particles sizes ranging from 5.5 to 28 nm." In the low field region, the optimum particle volume Vopi) can be calculated from the Neel-Brown model ... 

To maintain thermal stability, hence a condition EB/kBT= In (for) needs to be fulfilled. For z = 10 years storage, 109-10u Hz [28] and ignoring dispersions, i.e. assuming monodisperse particles, this becomes Es/kBT= 40-45. Reversal for isolated, well-decoupled grains to first order can be described by coherent rotation over EB. This simple model, as first discussed by Stoner and Wohlfarth in 1948 [29], considers only intrinsic anisotropy and external field (Zeeman) energy terms. For perpendicular geometry one obtains the following expression ... 

The first theoretical description of itinerant ferro- and paramagnetism in transition metals is due to Stoner and Wohlfarth (Wohlfarth, 1976). These authors use Fermi statistics to describe the conduction electrons in the d-band. They consider the case of a single conduction band in which the electrons interact via an intraatomic Coulomb interaction. This model is formally equivalent to the Hubbard model for correlations between electrons in a single conduction band. 

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