Micromagnetics: Basic Principles principle only determine nucleation fields for the system, and do not predict necessarily correctly the state of the system after magnetization reversal. A small class of materials exhibit the important Consequently, a lot of work has gone into the property of long range magnetic order. Funda- development of dynamic approaches which use simu- mentally, this arises because of the so-called ‘‘exchange lations based on the Landau–Lifshitz equation of energy’’whichcan,undercertaincircumstances,leadto motion. This is probably the technique in most parallel alignment of neighboring atomic spins. Given common use today. the link between the atomic spin and magnetic The other area in which considerable development moment, the spin order gives rise to the observed long has taken place during the 1990s is that of the range magnetic order. The origin of the long range calculation of magnetostatic fields, which because of magnetic order is a fundamental scientific problem its complexity forms the largest part of most micro- which has been studied extensively, both theoretically magnetic calculations. A number of techniques are and experimentally and is now understood in some available and it is the intention in this review to outline depth. However, within there is a major each technique and give consideration to the cir- problem when relating fundamental atomic properties cumstances under which each one is most applicable. to magnetization structures and magnetization reversal mechanisms in bulk materials. The problem was described vividly by Aharoni 1. Energy Terms (1996) in his discussion of magnetostatic effects. The essential idea is that the material properties are 1.1 Exchange Energy determined by the exchange energy, which in principle Exchange energy forms an important part of the is effective on a length scale of subnanometers, and the covalent bond of many solids and is also responsible magnetostatic energy, which has contribution from for ferromagnetic coupling. The exchange energy is the boundaries of the material itself and which given by introduce a different length scale entirely. The chal- lenge of micromagnetics is to develop a formalism in lk : Eexch 2JS" S# (1) which the macroscopic properties of a material can be simulated including the best approximations to the where J is referred to as the exchange integral. S" and fundamental atomic behavior of the material. S# are the atomic spins. Clearly for ferromagnetic In brief, the classical approach to micromagnetics ordering J must be positive. This so-called direct replaces the spin by classical vector field which initially exchange coupling is somewhat idealized and ap- allows the determination of magnetostatic fields within plicable to only a few materials rigorously. the system. In addition to this a different approach to A number of other models exist, including itinerant the exchange interaction much be formulated which electron ferromagnetism and indirect exchange can replace the quantum mechanical exchange in- interaction or Ruderman–Kittel–Kasuya–Yoshida teraction with a formalism appropriate for the limit of (RKKY) interaction. Generally speaking, however, continuous material. This, coupled with an energy Eqn. (1) is the form usually taken for the exchange minimization approach, forms the basis of classical interaction with the value of J dependent on the micromagnetics which will be outlined in a later detailed atomic properties of the material. section. The history of micromagnetics starts with a 1935 paper of Landau and Lifshitz on the structure of a wall 1.2 Anisotropy between two antiparallel domains, and several papers by Brown around 1940. A detailed treatment of The term anisotropy refers to the fact that the micromagnetism is given by Brown in his 1963 book properties of a magnetic material are dependent on the (Brown 1963). directions in which they are measured. Anisotropy For many years micromagnetics was limited to the makes an important contribution to hysteresis in use of standard energy minimization approaches to magnetic materials and is therefore of considerable determine domain structures and classical nucleation practical importance. The anisotropy has a number of theory to determine magnetization reversal mechan- possible origins. isms in systems with ideal geometry. Arguably, the current interest in micromagnetics arises from the availability, from about the mid-1980s onward, of (a) Crystal or magnetocrystalline anisotropy. This large-scale computing power which enabled the study is the only contribution intrinsic to the material. It of more realistic problems which were more amenable has its origins at the atomic level. First, in materials to comparison with experimental data. with a large anisotropy there is a strong coupling One important realization during this period was between the spin and orbital angular momenta the fact that energy minimization approaches in within an atom. In addition, the atomic orbitals are

1 Micromagnetics: Basic Principles

H

H NNNN H S d N Hd S N

SSSS (a) (b)

Figure 1 Sample with an anisotropic shape with a magnetic field applied in two perpendicular directions: (a) parallel to the short axis; here the ‘‘free poles’’ are separated by a relatively short distance, leading to a large Hd, (b) parallel to the long axis; poles separated by a smaller distance, which leads to a small value of Hd. generally nonspherical. where M is the magnetization and Nb and Na are Because of their shape the orbits prefer to lie in ‘‘demagnetizing factors’’ in the short axis and long certain crystallographic directions. The spin–orbit axis directions. N and N depend on the geometry so l b a l #\ coupling then assures a preferred direction for the that Keff 0 for a sphere and Keff M 2 for a magnetization—called the easy direction. To rotate needle-like sample. the magnetization away from the easy direction costs energy— and anisotropy energy. As might be expected the anisotropy energy depends on the lattice structure. (c) Stress anisotropy. This arises from the change (i) Uniaxial anisotropy occurs in hexagonal crystals in atomic structures as a material is deformed. It is such as cobalt. related to the phenomenon of ‘‘magnetostriction’ which is important for sensor applications. E l KVsin#θjhigher terms (2) Here θ is the angle between the easy direction and the (d) Magnetostatic effects and domain formation. magnetization, K is the anisotropy constant, and V the Magnetostatic fields are a natural consequence aris- volume of the sample. The higher order terms are ing from any magnetization distribution. The magnet- small and usually neglected. This has one ‘‘easy axis’’ ostatic fields are fundamental to the micromagnetic with two energy minima, separated by energy maxima. problem and, importantly, introduce another length This energy barrier leads to hysteresis. scale. Magnetostatic effects give rise to magnetization (ii) Cubic anisotropy; for example, iron, nickel structures on a length scale orders of magnitude greater than atomic spacings. Consequently it is \ l j α#α#jα#α#jα#α# E V K! K" ( " # # $ " $) (3) impossible to deal with both magnetostatic and Here α gives the direction cosine, i.e., the cosine of the exchange effects rigorously in micromagnetic calcu- angle between the magnetization direction and the lations at least with the current computer facilities. crystal axis. The formulation of the magnetostatic interaction field problem will be given in detail later but for the moment it is simply necessary to introduce the fact (b) Shape anisotropy. Consider a uniformly mag- that because of magnetostatic fields a magnetized netized body. By analogy with dielectric materials we body has a magnetostatic self energy given by can refer to the magnetization (magnetic polariz- E l "N M# (6) ation) creating fictitious ‘‘free poles’’ at the surface. mag # d These lead to a ‘‘demagnetizing field’’ Hd which acts where Nd is a factor depending on the shape of the in opposition to H. Figure 1 shows a sample with an sample. anisotropic shape with a magnetic field applied in two This is, of course, related to the statement (Aharoni perpendicular directions. The energy increases as Hd 1996) that the energy of a system depends on the increases. For an ellipsoid of revolution it can be boundary of the material. Clearly the energy given by shown that Eqn. (6) can be reduced if the magnetization M is reduced. As a result of this the material has a tendency lk #θ E Keff Vsin (4) to break into domains in which the magnetization i.e., the same form as for uniaxial anisotropy. θ is the remains at the spontaneous magnetization appropriate angle between the long axis of the sample and the for a given material at a given temperature with the magnetization direction. Also, domains oriented in such a way as to minimize the overall magnetization and hence the magnetostatic l " k # Keff #(Nb Na)M (5) self energy. 2 Micromagnetics: Basic Principles

However, creating a boundary between two do- the summation being carried out over nearest neigh- φ mains also requires energy and the actual domain bors only. The ij represents the angle between two structure depends on a minimization of the total neighboring spins i and j. It should be noted that in exchange and magnetostatic self-energy. This in a Eqn. (8) the energy of the state in which all spins are sense was the first application of micromagnetics. aligned has been subtracted and used as a reference It is interesting to note that as the size of a system state. This is legitimate as long as it is done con- Q φ Q $ Q k Q reduces the magnetostatic self energy reduces and at sistently. For small angles, ij mi mj , a first- some point it becomes energetically unfavorable to order expansion in a Taylor series is form a domain structure since the lowering of the Q k Q l Q :] Q magnetostatic energy is not sufficient to compensate mi mj (si )mj (9) for the energy necessary to create the . Below this size a sample is referred to as where s is a position vector joining lattice points i and and here the magnetization processes are dominated j. by the rotation of the magnetic moment against the Substituting, Eqn. (9) into Eqn. (10) gives anisotropy energy barrier. l # Q :] Q# Eexch JS (si )mj (10) i si 2. Foundations of Micromagnetics where the second summation is over nearest neighbors. It is not possible to neglect any of the three major Changing the first summation to an integral over the energy terms, exchange, anisotropy, and magneto- whole body, the result is that for cubic crystals static. The detailed magnetic behavior of a given l l ] # material depends on the detailed balance between Eexch & We dV; We A( M) (11) these energy terms. It is not even possible to add the V magnetostatic and anisotropy terms as a perturbation to the exchange energy term and use a quantum where mechanical solution of this problem. Currently the (]M)# l (]M )#j(]M )#j(]M )# (12) only realistic approach is to ignore the atomic nature x y z of matter, to neglect quantum effects, and to use The material constant A l JS#\a for a simple cubic classical in a continuum description of a lattice with lattice constant a. This represents a major magnetic material. Essentially, we assume the mag- step in the formulation of micromagnetism: we have netization to be a continuous vector field M(r), with r related the fundamental atomic properties to the the position vector. Thus we write spatial derivatives of the magnetization in the con- tinuum approximation. The atomic properties are M(r) l M m(r); m : m l 1 (7) s included via the exchange integral J which in micro- magnetic terms is essentially a phenomenological where Ms is the saturation magnetization of the material. The basic micromagnetic approach is to constant which can be determined from experimental formulate the energy in terms of the continuous data. magnetization vector field and to minimize this energy in order to determine static magnetization structures. Classical nucleation theory can be used to study modes 2.3 Magnetostatic Fields of magnetization reversal and nucleation fields. The Here we consider only the magnetostatic field arising energy terms are formulated as follows. from the magnetization distribution itself and not any externally applied field which is trivial to add to the overall energy. The magnetostatic or demagnetizing 2.1 Anisotropy Energy field Hd is governed by Eqns. (13) and (14) This remains a local calculation and is straightforward ]i l to carry out using the expressions for either cubic or Hd 0 (13) ] j π l uniaxial symmetry given earlier. (Hd 4 M) 0 (14) Eqns. (13) and (14) are written in cgs units, i.e., B l j π 2.2 Exchange Energy H 4 M. Since the curl of Hd is 0, the demagnetizing field can be derived from a scalar potential, The exchange energy is essentially short ranged and lk]φ involves a summation over the nearest neighbors. Hd (15) Assuming a slowly spatially varying magnetization the exchange energy can be written Substitution of Eqn. (15) into Eqn. (14) yields l #φ# ]#φ lk π]: Eexch JS ij (8) 4 M (16) 3 Micromagnetics: Basic Principles which because of its analogy with Poisson’s equation 2.4 Brown’s Equations in electrostatics leads to the definition of a volume magnetic charge density given by The final step in the formulation of classical micro- ρ lk π]: magnetics is to minimize the total energy which can be 4 M (17) written as follows Thus we can solve for the magnetostatic field by E l solving for the potential using Eqn. (16) subject to tot boundary conditions which determine the continuity A E G C ] #j k : jHd of the normal component of B and of the tangential & A( M) Eanis Msm Ha dV (25) V B F 2 H D component of Hd. : k l where H is the externally applied field. Here, E is n (Bext Bint) 0 (18) a anis i k l the anisotropy energy density. n (Hd;ext Hd;int) 0 (19) The approach uses standard variational principles where n is a unit vector pointing outward from the but the derivation is somewhat protracted. Essentially, surface. setting the first variation of the total energy to zero In terms of the scalar potential the equivalent leads to two equations. The first is a surface equation conditions are A C cm cm φ l φ 2A mi l 0  l 0 (26) ext int (20) c c B n D n cφ cφ k lk4πM : n (21) :c \c l : l c ) c ) since m m n 0 by virtue of m m 1. The n ext n int second is a volume equation, Thus again the surface of a bulk magnetized body is A C i 2A]# j j j l determining the overall response of a phenomenon m m Hd Ha HK 0 (27) having its origins at the atomic level. Finally we can B Ms D write expressions for the potential and demagnetizing field as follows: where the anisotropy field HK is defined as c ]:M(rh) M(rh) : n lk 1 Eanis φ(r) lk dVhj dSh (22) HK (28) & Q k h Q & Q k h Q M cm V r r S r r s (rkrh)]:M(rh) Eqn. (27) can be written H lk& dVh d Q rkrh Q$ i l V m Heff 0 (29) (rkrh)M(rh) : n where the effective field H is j& dSh (23) eff Q rkrh Q$ S 2A H l ]#mjH jH jH (30) The integrals in Eqn. (23) can be interpreted as fields eff M d a K arising from volume and surface change densities ρ l s k4π]:M and σ l 4πM(r) : n, respectively. Eqn. (27) states that the equilibrium solution is found Although Eqns. (22) and (23) represent elegant by making the magnetization lie parallel to the local closed form solutions for the potential and de- field. Eqns. (26)–(30) are referred to as Brown’s magnetizing field, they are not the best form for equations and form the basis of the classical micro- numerical computation. Essentially, the total energy magnetic approach for the solution of stationary of the system involves an integral over the volume as problems. follows:

l 1 : 3. Analytical Solutions Emag & Hd MdV (24) 2 V 3.1 Stoner–Wohlfarth Theory which because of Eqn. (23) involves a six-fold in- Stoner–Wohlfarth (SW) theory is the simplest micro- tegration. In terms of the numerical problem this magnetic model since it is assumed that all spins involves a scaling with N#, where N is the number of remain collinear thus removing the exchange term. elements into which the body is discretized. This of Consider a particle with uniaxial anisotropy and easy course leads to rapid degradation of computational axis at some angle ψ to the applied field. The speed with system size and in practice an alternative magnetization of the particle has an angle θ to the easy θkψ solution must be found for the calculation of Hd. The axis and to the applied field. The magnetic techniques involved will be described in detail later. moment of the particle is MV, where V is the particle

4 Micromagnetics: Basic Principles

M

cba

H

d

Figure 2 Hysteresis loop for a SW particle with the field applied parallel to the easy axis. volume. Assuming that the atomic magnetic moments over the maximum energy, E . Thus there is an max l k always remain parallel (coherent rotation) the energy energy barrier to rotation, of value Eb Emax Emin.It is the sum of the anisotropy and field energies, i.e., is straightforward to show that

E G E G E l KVsin#θkMVHcos(θkψ) (31) M#H# MH E l KV 1k kMVH k(kMVH) b 4K# 2K The orientation of the magnetic moment is de- F H F H termined by minimizing the energy E, that is we set E # # G \ θ l l jM H dE d 0. This gives KV 1 # (35) F 4K H dE l 2KVsinθcosθjMVHsin(θkψ) l 0 (32) Therefore, the energy barrier becomes zero when θ \ l lk \ lk d (MH 2K) 1, that is, when H 2K M HK. At this point, the coerciŠe force, the magnetization In general there are four solutions of Eqn. (32), two rotates into the other minimum, i.e., into the field minima and two maxima (which separate the minima). direction (points c to d). Thus HK is the maximum We can illustrate the behavior using the special case of coercive force for a particle. For a particle oriented at ψ l 0, that is, the easy axis is aligned with the field, some angle ψ the behavior consists of a mixture of Then Eqn. (32) becomes reversible and irreversible rotation. ψ The coercive force at some angle is less than HK. 2KVsinθcosθjMVHsinθ l 0 (33) For a system of particles with randomly orientated θ l θ l π easy axes an average over all orientations gives a Eqn. (33) has a solution at sin 0, i.e., 0or . hysteresis loop that has a remanence of half the Further solutions occur at 2KVcosθ lkMVH,so saturation magnetization and a coercivity of 0.479HK. that The SW model is still extensively used, especially in materials where the long-range interaction effects are cosθ lkMH\2K (34) more important than intrinsic grain properties. Gen- which are maxima. erally, agreement with experiment is obtained for a These results can be used to illustrate the reversal low ‘‘effective value’’ of HK which takes some account process in the following way using Fig. 2. A large of the nonuniform magnetization processes as is positive field is first applied to the sample, taking it to discussed later in relation to thin film simulations. point (a) on the hysteresis loop. When the field is reduced to zero the energy maximum stops the system 3.2 The Nucleation Problem from becoming demagnetized. The magnetization still lies along the easy direction, i.e., in this case parallel to Consider a ferromagnetic body in a magnetic field the applied field. This gives a remanent magnetization large enough to cause saturation. The field is slowly (point b). In negative fields the magnetization would reduced to zero and then increased in a negative sense. prefer to reverse, in order to be parallel to the field At some point the original state becomes unstable and direction. In order to do this it would have to rotate the magnetization makes a transition to a new energy

5 Micromagnetics: Basic Principles minimum. The field at which the state becomes If the energy surface has a complex topology it may unstable is the ‘‘nucleation field.’’ This can be de- contain a number of energy minima accessible during termined analytically for a few simple geometries. In a the magnetization reversal process. Energy minimiz- sense, SW is the simplest nucleation theory. The ation is a very poor technique for predicting the limitations of nucleation theory can be seen by correct minimum. considering the final state after transition. Magnetization reversal is intrinsically a dynamic In a complex system there may be more than one phenomenon and in order to predict magnetization accessible minimum, rather than the one state of SW states correctly after reversal we should in principle theory. Consequently, nucleation theory cannot pre- take account of the dynamic behavior of the system as dict the magnetization curve of anything but the far as is possible. Victora (1987) first used a dynamic simplest materials and in this sense its predictions are approach in studies of longitudinal thin films. The more mathematically than physically interesting. Nu- approach is based on the Landau–Lifshitz (L–L) merical micromagnetics with its novel techniques for equation of motion of an individual spin, which has determining the magnetic state at any point of the the form hysteresis loop is where predictions become accessible αγ to experimental verification. dM l γ i k ! i i !M H M M H (36) dt Ms γ α 4. Numerical Micromagnetics Here, ! is the gyromagnetic ratio and is the damping constant. In Eqn. (36) the first term leads to gyro- 4.1 Calculation of Stationary States: Energy magnetic procession. In the absence of damping this Minimization Versus Dynamic Approach will be eternal and would not lead to the equilibrium The first numerical micromagnetic approaches fol- state in which the magnetization is parallel to the local lowing the resurgence of interest in micromagnetic field. calculations in the mid-1980s were based essentially on The second term represents damping. This ensures energy minimization. Della Torre (1985, 1986) uses that the system eventually reaches the equilibrium this approach in some calculations on the behavior of position. It should be noted that other forms of the elongated particles. The approach works reasonably dynamic equation are possible, including the Gilbert well but is prone to numerical instabilities. form. However, in the limit of small damping, these A more sophisticated minimization technique was forms are equivalent although the Gilbert equation is developed by Hughes (1983) in studies of longitudinal probably more physically correct. thin film media consisting of strongly coupled spheri- Also, in the limit of large damping, in which case the cal grains (the strong coupling arising from exchange second term of Eqn. (36) dominates, the Landau– interactions between grains). In these materials strong Lifshitz equation is equivalent to a steepest decent cooperative reversal is important and a straightfor- energy minimization approach. The L–L equation is ward energy minimization technique was problem- the most widely used dynamic approach currently. atical during the magnetization reversal process. Eqn. (36) is easy to solve numerically for a single A number of sophisticated energy minimization spin. However, the micromagnetic problem is repre- approaches are possible using standard numerical sented by a set of strongly coupled equations of techniques. For example a number of workers have motions which is a somewhat more difficult numerical used conjugate gradient techniques. However, the problem. The types of system considered can for basic problem arises from the nature of the micro- convenience be considered as of two types. magnetic problem itself. Essentially, it is relatively (i) Weakly coupled systems. Typical examples here easy to track a well-defined energy minimum as it might be a set of grains with uniform magnetization evolves with the magnetic field. This is usually the coupled by magnetostatic and perhaps weak exchange case, for example, in the decrease from magnetic interactions. For this type of system a simple nu- saturation towards the remanent state where no merical technique such as the Runge–Kutta approach irreversible behavior occurs. The energy minimum in with adaptive step size will probably suffice. which the system resides is essentially a local energy (ii) Strongly coupled systems. These examples tend minimum on a very complex energy surface. The to have strong exchange coupling, for example, in the magnetization reversal process can be seen as the case of studies of nonuniform magnetization processes disappearance of the local energy minimum due to in a single grain or a set of strongly exchange coupled the action of a sufficiently large field applied in a grains. In terms of the differential equations the system sense opposite to the original ‘‘positive’’ saturation in this case is referred to as ‘‘stiff.’’ Although Runga– direction. Kutta will still give a solution, the step size tends to The magnetization reversal process takes place become very small and the computational times when the minimum in the positive field direction involved, increasingly long. It is more usual under vanishes and the system makes a transition into an these circumstances to use an alternative technique, energy minimum closer to the negative field direction. for example, predictor–corrector algorithms.

6 Micromagnetics: Basic Principles

The numerical technique essentially entails the mining the effects of exchange coupling in the determination of the local field H at each point in the materials. system followed by a determination of the small For example, standard micromagnetic calculations magnetization changes using Eqn. (36). The whole with an ordered system lead to an enhancement of the system is updated simultaneously at each time step remanent magnetization due to intergranular ex- after which the local field is recomputed and this change coupling but also predict a decreased coercive procedure is carried out until the system evolves into force due to the effects of cooperative magnetization an equilibrium state. Equilibrium can be determined reversal. In a system with a random microstructure, using a number of criteria: however, the effects of the cooperative reversal are (i) Magnetization changes at a given step below a decreased and although the exchange coupling leads certain minimum value, and to an enhanced remanence the reduction in the scale of (ii) Comparison of the direction of the local mag- the cooperative reversal can also lead to a small netization and the local field. Essentially the con- enhancement of the coercive force. Consequently, it is vergence is determined dependent on the maximum important to stress that the microstructure of a value of MiH for the system. material needs to be modeled as realistically as The technique used depends to some extent on the possible. system studied. In a sense this is the easiest part of In the case of particular materials produced by the numerical micromagnetics. A number of numerical solidification of a fluid precursor, for example, by the techniques are available for the solution of Eqn. (36) polymerization of a ferrofluid or the production of and it is not too difficult to create algorithms based on particulate recording medium from a magnetic dis- the dynamic approach which are robust in terms of persion, it is known that magnetostatic interactions finding stationary states and also reliable in finding are important because they give rise to flux closure and new stationery states after the nucleation of a mag- consequent modifications to the microstructure. netization reversal event. Vos et al. (1993) have demonstrated the importance Two problems remain however. The first is the of microstructure by comparing the chaining effects determination of the local field H which is a rather which occur in standard particulate media with the difficult and specific problem on which considerable behavior of barium ferrite. In the latter case the effort has been expended. However, a further problem particles consist of small platelets which tend to form is in the microstructure of the material itself and the long stacks under the influence of magnetostatic approximations which need to be made in order to interactions. The calculations of Vos et al. indicate make the problem amenable to a numerical solution very different behavior for the two types of micro- whilst retaining some physical realism. The following structure. However, the actual microstructures used section considers some aspects of the problem of were created using an ad hoc procedure. microstructure in micromagnetics. Chantrell et al. (1996) have demonstrated the importance of the correct simulation of microstructure in particulate systems, and have developed a soph- 4.2 Microstructural Simulations isticated molecular dynamic approach Coverdale et al. 2001 to the prediction of microstructures. Numerical micromagnetics involves a discretization of space. The behavior of the spin associated with each small volume element is described by Eqn. (36). (b) Granular thin metallic films. Although these Because of the exchange and magnetostatic inter- are similar to particulate materials in that they con- actions this leads to a finite set of coupled equations of sist of well-defined grains, these materials are gener- motion of the form of Eqn. (36) which have to be ally considered as a separate case, essentially because solved numerically. The spatial discretization is an of their different applications and also because they important part of the problem. consist, to a first approximation at least, as a single layer of grains. In these systems, clearly magnetostatic interactions (a) Particulate systems. Nanostructured particu- between the metallic grains will be very strong and late systems are essentially of two types. The first is there is also the possibility of some exchange coupling so-called granular magnetic solid, essentially a hetero- across grain boundaries. Generally these materials are genous alloy consisting of magnetic particles in a non- alloys, for example, of chromium. The intention is that magnetic matrix. This is often produced by sputter- the chromium segregates to grain boundaries giving ing, evaporation, or laser ablation. These systems rise to good magnetic isolation. have a microstructure consisting of relatively ran- The first models of these materials were produced domly distributed grains and can be modeled by plac- from the mid–1980s onwards and in many ways ing grains at random into the computational cell. remain largely unchanged. The models consist of The spatial disorder in this system has been shown spherical grains which are assumed to rotate co- (El Hilo et al. 1994) to be an important factor in deter- herently and can therefore be treated as a single spin,

7 Micromagnetics: Basic Principles situated on a regular hexagonal lattice. The reason for where this uniformity is that it simplifies the magnetostatic field calculation because fast Fourier transform tech- m 3(m : r)r H lk jj j niques can be used (as described in the following ij r$ r& section). Many simulations still use this rather idealized is the interaction field at element i due to element j. m model, even though it has been shown that it dras- i and mj are the magnetic moments of each element and tically overestimates the cooperative reversal. Miles r is the spatial separation. and Middleton (1990) were the first to develop a Clearly the determination of fields via Eqn. (37) is simulation based on an irregular grain structure with a an N# problem. In order to make the calculation distribution of particle volumes. This and later work feasible for large systems the usual approximation is to (Walmsley et al. 1996) demonstrate conclusively that divide the problem into a direct summation over some the size of magnetic features in the materials is region V around a given site i and to use a continuum critically affected by the microstructure and, as one approximation to the field outside this region. For a might expect, is significantly smaller in materials with spherical cut off the result is of the form (in cgs units) some disorder. The other problem with the assumed microstructure M is the intrinsic assumption of coherent reversal, which l  j4 k Hi Hij NdM (38) means that the intrinsic coercivity (assumed to be due j ? V 3 to coherent rotations) is probably overestimated. The parameters hint and C* represent the magneto- The first term is a direct summation within a region V static and exchange coupling strengths with respect to surrounding i. Outside this region the material is the intrinsic coercivity and are empirical parameters treated as a uniformly magnetized continuous material which essentially hide the micromagnetic deficiencies of magnetization M (in cgs units). The problem of of the model. At attempt to produce a realistic determining the field at the center of the spherical cut microstructure is described by Schrefl and Fidler out than reduces to solutions of (1992). The microstructure is produced via a voronoi construction. Essentially this entails, for a two-dimen- (rkrh) : M(rh)n# H l & dS (39) sional system at least, distributing points at random in d Q rkrh Q$ space and bisecting the line between points to produce cV grain boundaries. This produces an irregular grain over surfaces cV representing the inside surface of the structure and has been used in granular three- cut out region and the outside of the magnetized body. dimensional systems (Schrefl and Fidler 1992) and Eqn. (39) follows directly from Eqn. (23) given the longitudinal thin films (Tako et al. 1996). assumption of a uniform magnetization, ]:M l 0. These models represent the state of the art in terms The second and third terms in Eqn. (38) are the of microstructural simulations. It is desirable to carry ‘‘Lorentz’’ and ‘‘demagnetizing’’ fields arising from out a discretization of the systems at the subgrain the internal and external surfaces respectively, with Nd level, since this allows the correct simulation of a shape-dependent demagnetizing factor. These fac- nonuniform magnetization processes. The uniform tors arise directly from analytical solution of Eqn. mesh is no longer appropriate. Finite element methods (39). are necessary, and are outlined in Micromagnetics: Finite Element Approach. 5.2 Hierarchical Calculation 5. Magnetostatic Field Calculations The hierarchical approach is a more sophisticated Here we consider relatively common and easily applied calculation which improves on the Bethe–Peierls– approaches. More sophisticated techniques are nece- Weiss approximation. Essentially, a dipole sum is ssary for calculations using finite elements, and these again carried out within a central area around the will be outlined in Micromagnetics: Finite Element point at which the field is to be calculated. In the Approach. simplest case the remaining material is represented by ‘‘cells’’ of similar size, taken as having a total moment obtained by adding all the moments in the cell. This is 5.1 Dipole Sum—the Bethe–Peierls–Weiss then used to calculate the field due to the cell at the Approximation central point and a summation of these contributions This is by far the easiest approach to adopt. We start is taken. This approach was used by Miles and with the interaction field written as Middleton (1991). It is possible to improve the accuracy by adding higher order (multipole) terms. l  Such fast multipole methods are often used in elec- Hi Hij (37) j  i trostatic calculations.

8 Micromagnetics: Basic Principles

The beauty of the hierarchical calculation is that it Brown Jr. W F 1963 Micromagnetics. Interscience, New York can be applied to systems with disordered micro- Chantrell R W, Coverdale G N, El-Hilo M, O’Grady K 1996 structures which is not easily the case for methods such Modelling of interaction effects in fine particle systems. J. as the fast Fourier transform, a description of which Magn. Magn. Mater. 157/158, 250–5 follows. Coverdale G N, Chantrell K W, Veitch K J 2001 J. Appl. Phys. in press Della Torre E 1985 Fine particle micromagnetics. IEEE Trans. 5.3 Fast Fourier Transform Techniques Magn. 21, 1423–25 Della Torre E 1986 Magnetization calculation of fine particles. This was developed by Mansuripur and Giles IEEE Trans. Magn. 22, 484–9 (Mansuripur and Giles 1988, Giles et al. 1990) specifi- El-Hilo M, O’Grady K, Chantrell K W 1994 The effect of cally for simulation of magneto-optic recording media interactions on GMR in granular solids. J. Appl. Phys. 76, and has since become widely used. Its advantage is 6811–13 that the use of FFTs give rise to a scaling with NlnN Giles K C, Kotiuga P K, Humphrey F B 1990 Three-dimen- # sional micromagnetic simulations on the connection machine. rather than N . The disadvantage is the need for a J. Appl. Phys. 67, 5821–3 lattice in order to carry out the discrete Fourier Hughes G F 1983 Magnetization reversal in cobalt–phos- transform. The problem can be formulated in a general phorous films. J. Appl. Phys. 54, 5306–13 way using the following three-dimensional represen- Mansuripur M, Giles K 1988 Demagnetizing field computation tation formulas: for dynamic simulation of the magnetization reversal process. IEEE Trans. Magn. 24, 2326–8 l  i(k : x) M(x) mpqr e pqr (40) Miles J J, Middleton B K 1990 The role of microstructure in pqr micromagnetic models of longitudinal thin film magnetic media. IEEE Trans. Magn. 26, 2137–39 φ l ψ i(kpqr : x) (x) pqre (41) Miles J J, Middleton B K 1991 A hierarchical model of longi- pqr tudinal thin film recording media. J. Magn. Magn. Mater. 95, l l π \ l 99–108 Here, kpqr (kp, kq, kr) with kp 2 p Lz, kq π \ l π \ Schrefl T, Fidler J 1992 Numerical simulation of magnetization 2 q Ly, kr 2 r Lz. By substitution into Eqn. (16) and comparing coefficients it is straightforward to reversal in hard magnetic materials using a finite element method. J. Magn. Magn. Mater. 111, 105–14 show that Tako K M, Wongsam M, Chantrell K W 1996 Micromagnetics 4πi of polycrystalline two-dimensional platelets. J. Appl. Phys. 79, φ lk m : k (42) 5767–9 pqr Q Q# pqr pqr kpqr Victora R 1987 Quantitative theory for hysteretic phenomena in CoNi magnetic thin films. Phys. ReŠ. Lett. 58, 1788 Determination of k]φ then gives the field value. Vos M J, Brott R L, Zhu J G, Carlson L W 1993 Computed hysteresis behaviour and interaction effects in spheroidal See also: Micromagnetics: Finite Element Approach; particle assemblies. IEEE Trans. Magn. 29, 3652–7 ; Coercivity Mechanisms; Mag- Walmsley N S, Hart A, Parker D A, Chantrell R W, Miles J J netic Hysteresis 1996 A simulation of the role of physical microstructure on feature sizes in exchange coupled longitudinal thin films. J. Magn. Magn. Mater. 155, 28–30 Bibliography Aharoni A 1996 Introduction to the Theory of Ferromagnetism. J. Fidler, R. W. Chantrell, T. Schrefl, Clarendon Press, Oxford and M. A. Wongsam

Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials: Science and Technology ISBN: 0-08-0431526 pp. 5642–5651

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