Stoner-Wohlfarth

This critical field called coercivity or switching field is also equal to If a field is appHed in between 0 and 90° the coercivity varies from maximum to zero. In the case of this special example the appHed field = H/. Based on the classical theory, Stoner-Wohlfarth (33)... 

This is the conclusion obtained by the most recent development of the Stoner-Wohlfarth theory (Edwards and Wohlfarth 1968). 

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). 

This critical field called coercivity ff. or switching field Ff., is also equal to FF. If a field is applied in between 0 and 90° the coercivity varies from maximum to zero. In the case of this special example the applied field Ha = Hs = Hc = Hk. Based on the classical theory, Stoner-Wohlfarth (33) considered the rotation unison for noninteracted, randomly oriented, elongated particles. The anisotropic axis can be due to the shape anisotropy (depending on the size and shape of the particle) or to the crystalline anisotropy. In the prolate ellipsoids b is the short axis and a the longest axis. The demagnetizing factors are IV (in the easy direction) and The demagnetizing fields can then be calculated by Hda = — Na Ms, and Hdb = — Nb Ms. The shape anisotropy field is Hd = (Na — Nb)Ms. Then the switching field Hs = Hd = (Na — Nb)Ms. 

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. 

Expanding this equation into powers into powers of and analyzing the stability of the local free-energy minimum at = 0 yields the Stoner-Wohlfarth coercivity... 

The Stoner-Wohlfarth approach works fairly well for very small particles, where VM = 0 is a good approximation. However, it has been known for decades that neither the Stoner-Wohlfarth theory nor the additional consideration of the curling mode account for the coercivity of real materials. For example, the coercivity of optimized permanent magnets is only 20-40% of the anisotropy field 2Kx nMs, and only a part of the discrepancy can be ascribed to the curling terms in Eq. (11). The reason is that real-structure imperfections make it impossible to consider the magnets as perfect ellipsoids of revolution. 

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. 

MODIFIED STONER-WOHLFARTH THEORY FOR HARD-MAGNETIC PARTICLE ARRAYS... 

The critical angle and the switching field was given by Stoner-Wohlfarth [8]... 

For the above mentioned FePt particles, the particle diameter is clearly smaller than the critical particle size given by Eq. (8) for coherent rotation. Furthermore the strength of the magnetostatic interaction field acting on nearest neighbor particles is only about 2% of the anisotropy field for a particle distance of 2 nm. Thus the Stoner-Wohlfarth theory can be applied. 

However, the classical Stoner-Wohlfarth theory is modified, in order to take into account the agglomeration of particles. 

The hysteresis loop of the classical Stoner-Wohlfarth theory follow from the subsequent minimization of the micromagnetic energy... 

In Eq. (10) the sum is over all crystallites in the particle ensemble. V, is volume of particle i, K is the uniaxial anisotropy constant, u, is the anisotropy direction of particle i, and J, is the magnetic polarization vector of the particle. In addition to the classical energy terms of the Stoner Wohlfarth theory we have to consider the exchange energy between the crystallites that agglomerate to a particle. We assume that in average three crystallites agglomerate as schematically shown in Fig. 2. 

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).

Dilute FePt C Cluster Film Stoner-Wohlfarth-Like Behavior... 

Understanding the magnetic properties of a collection of well-isolated clusters is of great interest for exploring FePt clusters as a media for EHDR beyond 1 Tera bit/in. Dilute FePt C cluster films were prepared by the multilayer method as described earlier. Thus Stoner-Wohlfarth-like behavior has been observed in a FePt C cluster film with FePt volume fraction of 5 %, [45], Figure 27 shows the zero-field-cooled (ZFC) and field-cooled (FC) magnetization curves for as-deposited and annealed FePt C cluster film with 5 vol. % FePt. 

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 ... 

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.

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... 

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.

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... 

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. 

---