Micromagnetic theory of non-uniform magnetization processes in magnetic recording particles

Journal of Magnetism North-Holland

Invited

and Magnetic

Materials

249

95 (1991) 249-288

review

Micromagnetic theory of non-uniform in magnetic recording particles Manfred

processes

E. Schabes

Honeywell, Received

magnetization

Inc., Test Instruments

Division,

4800 E. Dry Creek Rd., MS-108,

Littleton,

CO 80122, USA

4 June 1990

Recent micromagnetic results are discussed from a unified point of view. Emphasis is placed on a reorientation of the focus of micromagnetic theory towards a dynamical treatment of the magnetization processes in non-uniformly magnetized particles. The theory requires merely the knowledge of the micromagnetic energy functional and the prescription for the dynamical evolution equations. It is demonstrated that the dynamical aspects are especially important during irreversible switching. In the micromagnetic calculations of this paper the Landau-Lifshitz equations are taken as the evolution equations for irreversible switching. The distinction of reversible and irreversible processes leads to a rigorous definition of the notion of switching field. This yields a generalization of the classical nucleation field. In this context it is also necessary to carefully distinguish between equilibrium magnetization configurations and transient magnetization configurations. The latter magnetization states represent the non-linear dynamical generalizations of the classical nucleation modes. It is demonstrated from first principles that both the equilibrium states and the transient states can be described in simple topological terms for a wide range of magnetic particles as used in magnetic recording technology.

1. Introduction In the past few years the field of micromagnetits has experienced a renaissance of fresh research activity. The renewed interest is due to the rapid developments in magnetic data storage technology. High performance magnetic recording systems require spacings between magnetization transitions well below the 1 pm threshold [l] and area1 densities of 1 Gbit/in 2 have become possible [2]. As magnetics is entering the domain of nanostructure technology, a profound understanding of the associated magnetization processes is necessary. The fundamental structural units of most magnetic recording media are small particles or crystallites. This is obvious for particulate media, where fine magnetic pigments are dispersed in an organic binder, but applies also to most of the so-called thin film media. Although thin film recording media seem homogeneous and are often labeled as “continuous” thin films, examination of the mi0304-8853/91/$03.50

0 1991 - Elsevier Science Publishers

crostructure reveals grains with well defined grain boundaries. Hence, for both classes of recording media the fundamental micromagnetic processes occur at the level of the individual grains or particles, which are physically separated by either layers of organic polymers, as in the case of particulate media, or by phase segregation or growth boundaries, as they are formed during the deposition process of thin film media. In this paper we review recent results in the micromagnetic theory of non-uniform magnetization processes in magnetic recording particles. Rapid progress has been made in the modeling of magnetization phenomena due to the more widespread availability of supercomputers. In fact, certain issues of the magnetization processes can only be addressed by harnessing computational resources on a scale typical to the requirements of quantum chemistry or computational fluid dynamics. In the first sections of this paper a brief survey of magnetic recording particles is given. Subse-

B.V. (North-Holland)

250

M. E. Schabes / Mtcromagnetic

theor,, of non-uniform

quently. fundamentals of micromagnetic concepts are developed with emphasis on an ab initio treatment of the magnetization processes in non-uniformly magnetized particles. The focus on nonuniform magnetization configurations can, in fact, be considered as the dividing line between modern micromagnetics and the classical micromagnetic theory [3,4]. While classical micromagnetics typically refers to the uniformly magnetized state as the reference state. modern micromagnetics distinguishes between equilibrium and transient magnetization configurations, both of which are in general nonuniform magnetization states. The evolution of these magnetization configurations is treated as a dynamical process, and the distinction between reversible and irreversible processes plays a central role. Schabes and Bertram [5] recognized that such a reorientation of the focus of the micromagnetic point of view is necessary in order to achieve a unified analysis of the multitude of non-uniform magnetization processes in fine particles. In ref. [5] several tools and concepts were developed to this end. In particular, the classical field” was generalized to the term “nucleation notion of “switching field” by examining the irreversibility of magnetization processes. It was also recognized that the particle size and the magnetic material parameters are intimately related in the prediction of magnetization processes. This leads to a scaling relation between the particle size and the exchange constant [5]. The scaling relation simplifies the scenario of possible magnetization processes by linking particles with different material parameters and sizes. This is also useful in view of the general difficulty in measuring the value of the exchange constant. It was demonstrated from first principles [5], that the non-uniform magnetization configurations typical for magnetic recording particles can be described in simple topological terms. Investigation of transitions between the fundamental topologies leads to a generalization of the classical notion of “critical sizes” [5]. The size dependence of the remanence, of the switching field, of the hysteresis loops and of the angular dependence were presented [5]. In ref. [5] the irreversible switching mechanisms and the associated transient

magnetization

processes

states were computed by integration of the Landau-Lifshitz equations (see section 7.3). Schabes and Bertram continued the investigation of non-uniform micromagnetic processes in several papers [6-81 and included, in particular, the effects of non-uniform fields [7] and inhomogeneous particle morphology [8]. Micromagnetic studies by the author are also presented in refs. [9,10]. Much of the second part of the present paper will review results from refs. [5-91. Some of the material of the following sections will also include additional results, which have been obtained, while the author was with the Center for Magnetic Recording Research of the University of California, and which, in the years 1985-1989, the author presented to the public in quite a few seminar lectures and CMRR workshops, to which traditionally experts of the international micromagnetics community are invited; otherwise these additional results have been left unpublished [lo]. Recently the computations of ref. [5] have been extended by Lyberatos and Chantrell [II], who investigated the influence of thermal fluctuations on the magnetization processes of ref. [5], and by Nakatani, Uesaka, and Hayashi [12], who studied the effects of a finite damping as compared to the dynamics with infinite damping of the LandauLifshitz equations as had been used in ref. [5]. In both refs. [11,12] the topological results and the usefulness of the developed concepts of ref. [5] have been confirmed. Later Uesaka, Nakatani and Hayashi presented also calculations on elongated particles [13] and platelets [14]. Independently of the work of Schabes-Bertram, Fredkin and Koehler developed a numerical model for computing equilibrium magnetization configurations based on the finite element method [1520]. In particular, this made it possible to investigate non-uniform magnetization configurations in ellipsoidal particles. In the limit of small particle sizes this allows to establish the link of numerical micromagnetics to classical nucleation theory. So far, the computations of Fredkin and Koehler have been restricted to equilibrium states. Prior to ref. [5] numerical micromagnetic calculations had been carried out by Della Torre [21231. These calculations explored some of the methods for a numerical treatment of micromagnetics,

M. E. Schabes / Micromagnetic

theory of non-uniform

but generally did not attempt to focus on the dynamical issues of non-uniform magnetization processes [23]. Della Torre continued his work with Yan [24-281 and subsequently carried out a more comprehensive investigation. However, in refs. [22-281 it has been ignored that computation of transient states requires specification of the dynamical evolution equation of the magnetization. Merely presenting unconverged states of some iterative energy minimization scheme (possibly with acceleration methods for convergence in force) is generally meaningless, and remains unsatisfactory even for the case, where the unconverged states bear resemblance with dynamically computed results (see section 4.5). Although nucleation analysis is not within the main thrust of this paper, it should be pointed out, that a good understanding of nucleation calculations is prerequisite. The reader is referred to a comprehensive summary by Aharoni [29] and to a recent review by the same author [30], where several concepts of nucleation theory are clarified. The importance of nucleation analysis is also shown in Aharoni’s recent nucleation results [31,32], which proved to be very valuable in the micromagnetic theory of coated particles. In section 2 we characterize some of the most prominent magnetic properties of magnetic recording particles. Section 3 gives a brief discourse on the experimental status in fine particle research. Section 4 provides an overview of fundamental concepts of micromagnetic theory. In order to provide a condensed introduction for the reader, who is new to the field of micromagnetics, sections 4.1-4.4 include a discussion of the micromagnetic energy functionals and of several topics from classical micromagnetic theory. Sections 4.54.8 develop concepts necessary for a systematic treatment of non-uniform magnetization processes. In particular, the concept of switching field is rigorously formulated in terms of reversible and irreversible magnetization processes. The inherent dynamical aspect of the reversal process is emphasized by discussing the Landau-Lifshitz equations in section 4.6. Scaling laws are presented in section 4.7. In section 4.8 the topology of typical non-uniform magnetization configurations is characterized and the notion of helicity is introduced.

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251

Sections 5 and 6 contain brief discussions of analytical models and of certain types of micromagnetic models [33]. In section 7 various methods of numerical micromagnetics are discussed, but again, the importance of a dynamical treatment is emphasized. The following sections present firstprinciples calculations for cubic particles (section 8) elongated y-Fe,O, particles (section 9) cobalt modified y-Fe,O, particles (section lo), passivated and unpassivated iron particles (section ll), and the effects of inhomogeneous applied fields (section 13). Barium-ferrite particles are briefly discussed in section 12. Section 14 gives conclusions. Two short appendices further illustrate some of the first principles results by means of simplified “back-of-the-envelope” calculations.

2. Overview of magnetic recording particles In this section a summary of the properties of commonly used magnetic recording particles is given. No attempt is made for being exhaustive. For a comprehensive survey we refer to the work of Bate [34], whose compilation of data for magnetic particles still seems to be the most comprehensive one. It also contains a large number of useful references. An updated, yet less detailed review of fine particle data has been published by Koester [35]. Here it will suffice to briefly discuss some of the most striking magnetic properties of magnetic recording particles as a motivation for the subsequent topics. 2.1. Particle size Recent years of development have shown a steady trend for manufacturing particles with ever decreasing sizes. This is mainly due to the fact that media with smaller particles show improved signal to noise ratios and support higher bit densities [36]. Thus typical particle diameters have been reduced from several thousands of angstrom to several hundred angstrom. Azarian and Kuin [37], for example, produced y-Fe%O, particles with an average particle width of 580 A at an aspect ratio of about 6 : 1. For CrO, particles Azarian et al. measured an average par-

M.E. Schabes / Micromagnetic

252

theory

title width of 370 A. Metal particles may have even smaller particle diameters. Miyahara and Kawakami [38], e.g., manufactured iron particles with a width of about 200 A and aspect ratio of 6: 1. For a given ferromagnetic material the particle size determines to a large extent the uniformity of the magnetization within the particle. The larger the particle diameter, the more non-uniform magnetization processes can take place. However, as will be pointed out below, the influence of the particle size on the magnetic properties of the particle can only be understood in relation to the set of material parameters. A particle with strong exchange coupling will seem “magnetically smaller” (i.e. will have more uniform magnetization processes) than a similar particle with relatively less exchange interaction. 2.2. Material parameters The set of material parameters, which characterizes magnetic recording media, includes the saturation magnetization MS, the crystalline anisotropy constants K,, K,, . . . , the exchange constant A, and the magnetostriction constant L. The magnetoelastic properties of magnetic recording particles will not be discussed explicitly in this paper. It is noted, however, that the magnetostriction can be treated in a fashion similar to the magnetocrystalline anisotropy [39]. In the simplest case this merely leads to a renormalization of the uniaxial crystalline anisotropy constant. There is an extensive literature on measurements of magnetic material parameters. For a comprehensive overview we refer again to the above mentiontioned articles of Bate and Koester. Here it will suffice to provide a list of typical material parameters of common recording particles. This is done in table 1. Data are from ref. [35] and are given at room temperature. It turns out that the many possible combinations of particle sizes and material parameters of recording particles can be related to essentially two length scales [5-81, namely the exchange length

A,,, which is given by A,, = ~‘3,

to the wall width h, = 2m.

and

For many record-

of non-uniformmagnetiratron

processes

Table 1 Typical material parameters for several magnetic recording materials [35]. Also listed are estimates of the exchange length

I A,, = iA/Ms2

and of the wall width h, =,/m_ The values listed for X,, and X, correspond to an exchange constant of A = 1 X 10V6 erg/cm for the metals Fe, Co, Ni, and A = 1 x lo-’ erg/cm for the oxides Material

Fe co Ni CrO, Fe&), y-Fe@,

M,

K, (lo4

K, (lo4

a/M,

24m

(emu/ cm3)

erg/cm’)

erg/cm3)

(A)

(A)

1710 1430 483 480 480 350

45 430 - 4.5 22 -11 - 4.6

20 120 2.3 4 - 2.8 0

58 70 207 66 66 90

298 96 943 135 191 295

ing particles the value of X,, is the more relevant one, since the magnetization processes are usually dominated by the relative strength of the magnetostatic and exchange interaction. The most prominent exception to this rule are BaFe,,O,, platelets with a very high uniaxial crystalline anisotropy with easy direction perpendicuar to the plane of the platelets. Fig. 1 shows typical dimensions for X,, for a selection of magnetic materials. Since the values of the exchange constant A are generally not well

400

800 Maqnetization

1200

1600

2000

(emu/cm3)

Fig. 7. Exchange length h,, = /A/M: for typical material parameters of several magnetic materials. X,, is computed for three values of the exchange constant. (a) A = 1 X lO_‘erg/cm, (b) A=5XlO-‘erg/cm,(c) A=lxlO-6erg/cm.

M. E. Schabes / Micromagnetic theory of non-uniform magnetization processes

known, the results are presented for a range of A. For elongated particles X,, approximately indicates the particle radius, above which non-uniform magnetization processes become important during magnetization reversal. This statement can also be inferred from the nucleation theory of infinite cylinders, where X,, gives approximately the maximum radius for which uniform rotation is the nucleation mode. For these reasons it is useful to compare the particle size with the value of the exchange length. Iron particles, for instance, with large exchange constant and large magnetic moment density have an exchange length, which is comparable to that of y-Fe,O, particles with relatively smaller exchange and magnetostatic coupling. It is therefore expected that the magnetization processes in both particles will be similar (see section 4.8). Of course, iron particles will switch at a magnitude of the applied field, which is higher (as compared to the oxide particle) by approximately the ratio of the magnetization densities of the iron particles and the y-Fe,O, particles, respectively (since the crystalline anisotropies of the particles are relatively small in both cases).

2.3. Particle shape and morphology The particle shape has a profound effect on the magnitude of the switching fields. Most recording particles are elongated with aspect ratios anywhere from three to ten, although cubic particles of iron with potential applications as recording media have been reported [40]. The most important non-acicular recording particles are barium-ferrite particles [41]. Apart from the overall particle shape, the morphology of the boundary surfaces has to be considered. Sharp corners, or flat ends lead to large inhomogeneous demagnetizing fields, hence, the equilibrium magnetizations are non-uniform. This property distinguishes these particles from ellipsoidal particles, where uniform demagnetization fields and uniform magnetization configurations are obtained for sufficiently small particle size. Non-uniformity of the magnetization generally reduces the switching fields of the particle.

253

2.4. Homogeneous particles and inhomogeneous particle To distinguish particles, where the material parameters are approximately uniform throughout the particle volume, from those particles with a spatially varying composition, it is useful to separate the particles into the class of homogeneous particles and the class of inhomogeneous particles. Many oxide particles belong to the first class, whereas metal particles and surface modified oxide particles belong to the second class. Inhomogeneous particles with varying magnetic moment densities have in general non-uniform magnetization states even for ellipsoidal external shape, since div M # 0. An important class of inhomogeneous particles are those, where the magnetic properties of the outer particle shell differ from those of the particle core. This is the case for cobalt modified y-Fe,O, particles, where the magnetic crystalline anisotropy is enhanced in the shell region due to cobalt doping or cobalt adsorption [42]. Schabes and Bertram demonstrated [8] that layers of cobalt ferrite (CoFe,O,) as thin as one monlayer can lead to significant increases in the coercivity of y-Fe,O, particles, as has been suggested by Berkowitz et al. [43]. This remarkable finding will be discussed in section 10. Another class of particles, where the internal magnetic structure has a spatial variation, are iron particles. The inhomogeneity in iron particles arises from the fact, that the particles need to be passivated in order to be chemically stable. The outer shell of the passivated particles consists of various phases of iron oxide [38], which is grown during a controled passivation process. The oxide layer has a considerably lower magnetic moment density as compared to the metallic core of the particle (see section 11). Another important issue are imperfections. These can be surface defects (e.g. dendrites, edges, etc.) or volume defects (e.g. pores, grain boundaries, etc.). A recent study of the effects of pores on the magnetic properties of y-Fe,O, particles has been given, for example, by Wenbo [44]. Various issues on the subject of defects are discussed in refs. [29,45,46].

254

M. E. Schahes / Micromagnetic

theory

Defects generally cause non-uniformity of the magnetization close to the defect site, e.g., by generating additional magnetostatic sources on the walls of a void. In particular, this can lead to significant perturbations of the magnetization processes in the case, where the defect occurs in a region of the particle, which is nearly uniformly magnetized in the defect-free particle, e.g., in the central sections of an elongated particle. Flat ends and corners may be considered as a special class of defects. Similarly, non-uniform magnetization configurations may be the result of inhomogeneous applied fields, e.g., the field of a magnetic recording head, or the magnetostatic interaction field due to the other particles within the recording medium. The effects of inhomogeneous applied fields are discussed in section 13. 2.5. Thermal effects Temperature affects micromagnetic processes in two ways: (1) via the temperature dependence of the material parameters and (2) by the presence of thermal fluctuations. An explanation of the temperature dependence of the material parameters is beyond the scope of micromagnetic theory. Away from a phase transition, it is nevertheless possible to probe the temperature dependence of magnetization processes by micromagnetic methods. In this case temperature merely acts as a parametrization of the magnetization configurations. For fixed temperature, there are important thermal effects due to fluctuations. These effects become the more pronounced, the smaller the particle volume, since the height of the energy barriers, which separate the local minima of the energy surface in configuration space, diminishes with decreasing particle volume. If the particle size is small, thermal fluctuations have a high probability of overcoming the energy barriers. This leads to superparamagnetism [47]. Thermal fluctuations may be described by adding a random field to the non-fluctuating effective field [47] as obtained from the energy functional (see section 4.2). Statistical averaging of the Langevin equation yields the Fokker-Planck

of non-unrform

magnetization

processes

equation [47]. Recently Rode, Bertram and Fredkin have carried out a detailed study of the Fokker-Planck equation for a pair of interacting dipoles [48] and showed that the influence of thermal fluctuations on the magnetic relaxation time can be described by an effective activation volume which is generally larger than the physical volume and depends on the interaction strength. For long time scales the effect of the thermal fluctuations is usually referred to as magnetic viscosity [47,49,50]. An interesting modification of the classical NCel formula has been presented in ref. [51]. Recently Lyberatos and Chantrell presented micromagnetic calculations, which explicitly investigate thermal fluctuations in cubic particles [ll]. In particular, thermal effects on the flower and vortex states [5] and their switching field are computed. The lowering of the switching field and of the critical size for the onset of vortex formation is demonstrated explicitly. Furthermore, it is shown in ref. [ll] that the configurational anisotropy [5] of non-uniformly magnetized cubes reduces the superparamagnetic limit [ll].

3. Experimental

work

Experimental data on isolated magnetic particles are difficult to obtain because of the small particle sizes. Traditional magnetic measurement techniques usually determine bulk properties and require large magnetic signals. Data are therefore taken in many cases from (interacting) powders [52]. The interaction between particles masks to a great extent the single particle processes. To reduce the magnetostatic interaction between particles dilute suspensions have been used [53,54]. There are also some experiments on truly isolated particles [55]. Knowles’ method [56] requires the particles to be visible in the optical light microscope. Therefore the particle sizes are required to be rather large for this method. Recently arrays of well defined magnetic particles have been manufactured by means of electron beam lithography [57]. For permalloy particles a reduction of particle size from diameters

M. E. Schabes / Micromagnetic

theory of non-uniform

of several lo4 to about 1000 A results in an increase of the coercivity of almost three orders of magnitude. The greatest promise for studying magnetization processes in single particles lies in advanced methods of electron-microscopy, in particular, electron holography [58-601 and differential phase contrast Lorentz microscopy [61,62]. Other techniques, such as Mossbauer spectroscopy have been used extensively, e.g. in studying the details of the morphology of y-Fe,O, particles [43].

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255

measure the relative strength of the micromagnetic interaction mechanisms. The following will provide a brief description of these energy concepts. 4. I. Micromagnetic

energy

terms

The micromagnetic energy of a magnetic particle is written as the sum of several energy contributions

E,,, is the total free energy of the particle. is the Zeeman energy describing the interacEar? tion of the particle magnetization with an applied is the crystalline energy, E,, is the field. E,, exchange energy, Emag is the magnetostatic interaction energy. In general, there may be additional terms, e.g. the magnetoelastic energy contribution. However, in the following we will not consider magnetostriction explicitly (see section 2.2). Each of the terms of the above equation represents an energy functional. For completeness and to develop the formalism of the subsequent sections the explicit forms of each energy contribution will be given. The cgs system of units is used in the following formulae. Writing the magnetic moment density as M= M,a and the applied field as H~,pp(r;,pprthe Zeeman energy is given as

where 4. Micromagnetic processes

fundamentals

of magnetization

The systematic study of magnetization processes in magnetic particles requires a set of theoretical concepts, usually referred to as “micromagnetics ” [3,4]. Micromagnetic theory merges classical electrodynamics of continuous media and some branches of condensed matter physics. However, quantum mechanics - even though it provides the fundamental magnetic unit, i.e., the electronic spin _ is generally not considered explicitly. Instead, the magnetic sample is described by a set of macroscopic variables, (e.g., the directional cosines of the magnetization field within the sample), certain material parameters (e.g., the magnitude of the magnetic moment density at fixed temperature), and various phenomenological concepts. The latter aspect is clearly demonstrated, e.g., by examining the micromagnetic treatment of the crystalline anisotropy energy density. Micromagnetics uses an expansion of the crystalline anisotropy energy density, which satisfies the symmetry properties [63] of the underlying crystalline lattice and has the correct set of easy and hard directions. Micromagnetics per se is not concerned, e.g., with the question, of how the spinorbit interaction leads to the observed crystalline anisotropy [64]. Similar remarks apply to other micromagnetic energy contributions, in particular to the treatment of the exchange interaction. In either case, at the micromagnetic level quantum mechanics is assumed to have been accounted for by appropriate choice of continuous energy functionals and of the material parameters, which

Eapp= - Ms/ V

H.‘pp~+,p - a dx’.

(2)

where V indicates integration over the volume of the particle. The applied field and the magnetization can be spatially varying. The crystalline anisotropy is treated phenomenologically by writing E,,

=

/

u,,( CX)dx’, V

where u,,(a) is an expansion of the local crystalline energy density in terms of the directional cosines of the magnetization densny. For the uniaxial case with easy axis along the r-direction, e.g., u:,“(a) = - Ky’, where (Y= ((u, ,B, y) and K > 0. The case K < 0 yields easy plane anisotropy. For cubic crystals higher order terms need to be considered, e.g., u,,,(a) = K(&‘+ c?y’+ j3’y’).

M.E. Schabes / Micromagnetic

256

theory of non-uniform

A comprehensive tabulation of the various arrangements of easy axes and easy planes can be found, for instance, in ref. [65]. Switching processes involving higher order terms in the crystalline anisotropy may follow complicated hysteresis loops, even for the simple case of uniformly magnetized ellipsoids, as has been shown recently by Chang and Fredkin [66]. The exchange interaction is a quantum-mechanical effect. In the framework of micromagnetics it is treated in a semi-phenomenological fashion by deriving it from an energy function similar to a Heisenberg exchange Hamiltonian. Two neighboring spins S, and S, are exchange coupled with an energy given by v, = -2J,,s;s,,

(4)

exchange where J,, is the quantum-mechanical integral. For cubic crystals, for instance, it can be shown (see, e.g., ref. [67]) that the exchange energy is given by E,,

=

j-vA[(Va)2+ (VP>‘+ (ou)21~

where A is the exchange constant. In the tion of the above equation the spins are classically, i.e., their orientation is allowed in a continuous fashion. The magnetostatic interaction energy written as Em_=

-;

JV

H’*Mdx3,

derivatreated to vary may

be

(6)

where H’ is the demagnetizing field due to magnetostatic volume and surface charges. The magnetostatic volume charges are given by -div M and the magnetostatic surface charges are n. M, where n is the outward pointing surface normal. Writing H = HaPP + H’, H’ is required to satisfy the equations div( H + 4nM) curl(H)

= 0,

= 0, in V.

(7) (8)

The second equation is valid for the magnetostatic limit, where eddy currents are neglected. In this case it follows from the last equation, that H may

magnetization

be derived H’

= - v$.

tion inside

processes

from a scalar potential 9 such that Therefore $I satisfies a Poisson equathe particle volume, namely

I&$ = 4n div M, and Laplace’s &=Q,

(9)

equation (10)

in the space exterior of the particle. This set of partial differential equations has to be solved for appropriate boundary conditions, i.e. continuity of the potential at the particle boundaries and regularity at infinity. A detailed discussion of this classical topic of electrodynamics can be found, e.g., in refs. [3,4]. With the development of powerful finite element methods it has become feasible (with the aid of supercomputers) to solve the above set of partial differential equations numerically for arbitrary particle geometries, as has been demonstrated by Fredkin and Koehler [15-201. For certain geometries an alternative way of a numerical treatment of the magnetostatic interaction in three dimensions has been described by Schabes and Aharoni [68]. As previously in refs. [22,23], the particle is divided into an array of small interacting elements, the “micromagnetic within each element is array”. The magnetization of constant magnitude but may vary in its direction from element to element. The state of magnetization is described (down to a spatial resolution given by the size of the elements) by specifying the directional cosines of all elements. This procedure specifies uniquely the magnetostatic surface charges on the faces of the elements. Hence, the magnetostatic potential can be calculated by a two-fold integration over the surfaces of each element. Another two-fold integration of the potentials yields the magnetostatic interaction energy as a quadratic form in the directional cosines of the magnetization vectors. An analogous method was first applied in two dimensions by LaBonte [69]. Della Torre [22,23] used a similar discretization method with approximate dipolar terms for the magnetostatic interaction field in three dimensions. Schabes and Aharoni [68] showed that discretization with cubic

M. E. Schabes / Micromagnetic

elements allows for an exact analytic magnetostatic interaction integrals.

theory of non-uniform

form of the

4.2. The effective field The effective field is defined as the negative functional derivative of the total micromagnetic energy functional with respect to the magnetization. He,, = - 6E/6M.

01)

It is worthwhile to carry out the derivation explicitly in the case of the effective exchange field. The variation of the exchange functional is given by &,=2A

J

v[(va)*(Svol)+(v~).(6v~)

+(vv>W’~)] = 2A[ va -2A

/

dv

6a + VP W + VY 6Y],, Aa*6a

dV,

V

(12)

where 3V indicates the boundary of the particle. The surface terms of the second part of the above equation vanish, since va = 0 on the surface of the particle. Therefore the effective exchange field is given by He, = :Aa.

(13)

s

Discretization on a simple cubic lattice of discretization elements yields

Hex.;=

2AGa8,r Msd2

(14)

6

where the summation is taken over the six nearest neighbor elements and terms with zero torque have been neglected. The symbol d in eq. (14) is the lattice constant of the discretization grid. It is noted that the above expression differs from that given by Della Torre in refs. [22,23]. Also in refs. [5,6] the exchange energy is defined such that the quoted values of A should be read as only half their values, e.g. the value of 1 X 1O-6 erg/cm, as quoted in ref [5], corresponds to A =

magnetization

processes

257

0.5 x lop6 erg/cm, if A is used in its usual meaning as defined in the continuous exchange energy functional. The latter definition of A was used in the subsequent papers (refs. [7-lo]), and seems to have been used also in the later papers of Yang and Della Torre. In general, this is not a serious correction, since the exchange coupling in small particles is known only approximately and results can easily be resealed with respect to the particle size (see section 4.7). The computation of the discretized exchange field in three spatial dimensions on lattices other than simple cubic lattices generally requires appropriate averaging procedures. This has been reported by Fredkin and Koehler [19] and yields a correction to some of the initial results of finite element calculations. The effective field carries great significance since the expression M X Heff yields the torque onto the magnetization. Hence, in equilibrium, where the torque must vanish identically, magnetization states are distinguished by the condition that the effective field is parallel to the magnetization at every point of the particle. This condition can be used to design iterative methods for computing the equilibrium configurations. The fact that, in equilibrium, the effective field is parallel to the magnetization, can also be demonstrated by minimizing the micromagnetic energy subject to the constraint that the norm of the magnetization is constant. Of course, torques and magnetization configurations can also be derived without resorting to the effective field, if angular variables are used instead of the Cartesian components of the magnetization. In this case singularities require special treatment of the angular coordinates (see section 7.3). 4.3. Brown’s equations Brown derived a set of partial differential equations by computing the first variation of the total micromagnetic energy functional with regard to the magnetization. In equilibrium the requirement of vanishing first order variation of the free energy and the constraint of constant magnitude of the magnetization density (at constant temperature)

M.E. Schabes / Micromagneric

258

lead to the following ax

set of equations

au,

i

CllX

theory

2AAa-x+hlM”H aCx -2Aan-Ksn~an

=0

i

=0

inV,

(15)

onaV.

(16)

These equations are usually called Brown’s equations and have to be solved subject to appropriate boundary conditions (see ref. [4]). The expressions in the curly parentheses are (of course) proportional to the effective field. The complications arising from the non-linearities of Brown’s equations allowed for a rigorous analytical treatment of only but the simplest geometries of particle shapes. 4.4. Classical micromagnetic

concepts

In the context of Brown’s micromagnetic theory two uniquely defined quantities are introduced, namely the critical radius [70] and the nucleation field [71]. Both refer to the linear stability of the uniform magnetization state. For a recent discussion of these issues, see, e.g., Aharoni ]301. Magnetization reversal via uniform rotation was described by Stoner and Wohlfarth [72]. The Stoner-Wohlfarth model is applicable to sufficiently small ellipsoidal particles and approximately describes also the irreversible switching in sufficiently small particles of more complicated shapes [5]. Almost ten years after the classic paper by Stoner and Wohlfarth the curling mode was discovered independently by Brown [73,74] and Frei, Shtrikman and Treves [75]. A detailed study of the curling mode has been given by Aharoni v91.

The Stoner-Wohlfarth model of uniform rotation yields perfectly square loops for the case, here the field is applied parallel to the long axis of a prolate ellipsoid. The particle switches at a magnitude of the applied field, which (in the case of zero crystalline anisotropy) is given by H”“, = 4a(N,

- NZl)X,

where N, and N, are the demagnetization

(17) factors

of non-uniform magnetization processes along the long and short major axes of an ellipsoid of revolution (note: N, + 2N, = 1). For an infinite cylinder of radius R the curling nucleation field is given by H,,,

= 2 nl .OS&'S'

,

(18)

where S = R/ \IA/M,L, with A the exchange constant. The above formula is valid for the applied field parallel to the long cylinder axis. For the case of a finite angle between the applied field direction and the cylinder axis see ref. [76]. 4.5. Micromagnetic zation processes

theory of non-uniform

magneti-

For most particle shapes the classical definition of nucleation field is not practical, since - as was pointed out earlier - even very simple particle shapes yield non-uniform demagnetization fields. For these particles the uniform magnetization state is generally unstable as a remanent state or in a finite (homogeneous) applied field. Therefore it is useful to analyse the switching processes from a different point of view. Instead of focussing on the uniform magnetization state as a reference state, emphasis is placed on magnetization reversal as a dynamical process with the applied field as an external control parameter. It is noted that this control parameter can, in general, be a vector field (e.g. particle in the presence of the magnetic field of a recording head). Also other quantities, e.g., the particle size may be regarded as control parameters. In the simplest case of a particle in a uniform applied field the control parameter may be thought of as a single scalar quantity Harp, the magnitude of the applied magnetic field. Let the z-direction denote the direction of the applied field and let positive and negative values of Harp indicate that the applied field points into the positive and negative z-direction, respectively. Starting from some initial configuration M, corresponding to a large value of Happ,“, Happ is monotonically decreased. In general, the magnetization configurations will undergo initially a continuous and differentiable mapping M = M( Hap,,). In particular, if the control parameter takes on the

M.E. Schabes / Micromagnetic theory of non-uniform magnetization processes 1.5

0.5

,

,

,

,

,

,

,

-

I

I,

-----__-----

I

H,=H,

-1.0 -1.51

,

-

I I 0.0 _________.________’r----I 7 -0.5

,

I I

’

’ -400

’

’

-200 APPLIED

’

’ 0 FIELD

’

’ 200

’

’

400

”

Hopp (Oe)

Fig. 2. Typical hysteresis loop involving non-uniform magnetization configurations. Reversible branches, irreversible branches and the remanent magnetization are indicated. The switching field coincides with the coercive field for this particle. D/2X,, = 2.35, aspect ratio p = 3 : 1.

subsequence {H,, H,, Hz,. . . , H,,}, the magnetization configurations will yield the discrete mapping M,+M, -Mz ‘.. -IV,. Magnetization processes linking Mr and M,, are defined to be reversible, if and only if for any subsequence { H,,, H,, H,, . . . , H,, } and for the associated mapping Ma -+ Mr + M2 . . . -+ M, the subsequence { HO, H,, _ ,, Hn_2,. _ . , HO } yields the mapping M,, +Mn_, -+IW~_~ ... -+M,. If the magnetization processes are not reversible, the magnetization configuration is said to undergo irreversible processes on the interval [HO, H,,]. A value of the applied field is called a switching field H,, if the process M( H,) --* M( H, + c) is reversible for E > 0 and irreversible for c = 0, with Hi > H,. (An analogous statement holds for H, < H,, in this case we require reversibility for e -C 0.) The above definitions are illustrated in fig. 2, where a typical hysteresis loop for a particle with non-uniform magnetization configurations is plotted. Reversible and irreversible branches of the hysteresis loop are identified. The loop is non-square, since the non-uniformity of the magnetization configuration becomes larger as the applied field is reduced from an initially large value. of the As Harp is further reduced, a discontinuity magnetization and of the susceptibility occurs. This is at the switching field, where irreversible magnetization processes lead to the lower branch of the hysteresis loop.

259

It should be noted that each point on the reversible branches of the hysteresis loop corresponds to an equilibrium state (stable for given applied field). However, a calculation, which is restricted to computing equilibrium states, faces the dilemma, that at the switching field the initial and final state are not related in a continuous fashion, let alone, differentiability. This is unsatisfactory, since there is no real discontinuity, and the irreversible branch of the hysteresis loop is a sequence of transient states. The transient states are the non-linear dynamical generalizations of the classical nucleation modes. Stable equilibrium states are local minima of the energy surface in configuration space (i.e. the space of magnetization states) with positive curvature of the energy surface. Transient states are not extrema of the energy surface and are determined by the prescription of the irreversible dynamics. Therefore the unconverged results of an iterative energy minimization method are generally meaningless, especially, when various means for convergence acceleration have been used in the iteration. As the external control parameter (applied field) is varied, the energy surface undergoes a deformation. As long as the magnetization process is reversible, the deformation does not destroy the positive definiteness of the curvature matrix. In general, the particle state will be moving in configuration space. However, there is no independent motion of the magnetization configuration with regard to the energy surface: the particle state is “glued” to the topology of the hypersurface. (This and the fact, that the deformation of the hypersurface is a differentiable mapping, are the reason, why energy minimization methods are in many cases appropriate for computing the equilibrium magnetization configurations on the reversible branches of the hysteresis loop.) That picture changes dramatically at the onset of the irreversible branch of the hysteresis loop. The energy hypersurface is deformed in such a way that the curvature becomes zero in at least one direction. Any small perturbation of the magnetization configuration will set forward a motion of the particle magnetization configuration relative to the energy surface (which is fixed, if the applied field is kept stationary at H,). The irre-

260

M. E. Schabes / Micromagnetic

theory of non-uniform

versible switching process is therefore an intrinsically dynamical process. It is also noted, that in the case, where the applied field changes on time scales comparable to the time required for irreversible switching, it is necessary to consider the deformation of the energy surface, while the particle is moving relative to the hypersurface. The dynamical evolution of the magnetization configuration is particularly significant, if the particle is large. In this case the energy surface for H;lPP = H, may have a topology with several metastable states, and there may be several paths with negative curvature connecting the last state of the reversible branch of the hysteresis loop with local minima. The selection of such a metastable state depends on the detailed dynamics of the irreversible switching process. Recent studies on similar systems suggest that a state of “minimal stability” will be selected successively during the various stages of irreversible switching. This issue has become known as the “sand pile problem” [77,78], and has been investigated in domain processes in thin magnetic films [79]. For most modem recording particles and their ever decreasing sizes it seems that the energy surface is relatively simple, with essentially one local minimum as the possible target state for the equilibrium configuration after completion of the irreversible switching processes. In this case a less rigorous approach using merely energy minimization methods (e.g. steepest decent, conjugate gradients,. . . ) yields the correct result for the equilibrium states computed before and after irreversible switching, but still does not determine the transient magnetization configurations during irreversible switching. Iterative minimization algorithms sometimes mimic certain dynamical aspects of a dissipative system. In this case the dynamical transient states close to an equilibrium state may be estimated under certain circumstances by an unconverged iterative state. For large deviations from equilibrium this is generally not true. It is useful to further illustrate the above concepts for the case of the prolate Stoner-Wohlfarth ellipsoid. Let +._,r and (p, denote the direction of the applied field and the direction of the magnetization of the particle with respect to the long

magnetization

processes

particle axis. As has been pointed out already, for + app = 0, the loop is perfectly square and the reversible branches of the hysteresis loop are simply given by m=M/M,= +l. For the case ~&,r = 0 the reversible processes are degenerate: the state +,,, = 0 is stable (dE2/d&, > 0) for ( Happ ( < (H,,, ( and the magnetization process simply consists in the magnetization staying in either the up or down state. At Harp = H,,,, the curvature d E ‘/d&, becomes zero. The initial uniform state of magnetization is now unstable and switches to the other equilibrium. If this process is viewed in the one-dimensional configuration space, the particle is “rolling down” the slope of the energy surface to reach the state +,,, = 7~. For the case of the classical StonerWohlfarth model the switching process is the uniform rotation. It should be noted, however, that this is statement is not really a dynamical result, but rather a consequence of the assumption of one degree of freedom. If the ellipsoid were of a real magnetic material, the number of internal magnetic degrees of freedom is of course large, and in the classical micromagnetic continuum approximation infinite. In this case it is possible to show rigorously, that for sufficiently small size the onset of the switching process at the nucleation field is exactly by uniform rotation [29]. Uniform rotation and curling with finite amplitude in ellipsoids have been investigated by Fredkin and Koehler [15-191. particle has For %app# 0 the Stoner-Wohlfarth a magnetization loop which is non-square, since an increase of the magnitude of the applied field moves the location of the energy minimum such that c#+,,becomes finite. This process is a reversible magnetization rotation and the uniform magnetization state is stable for a finite +,. Stability is lost, when the applied field reaches the switching field and d E ‘/d& = 0.

4.6. Reversal dynamics For the computation of transient states the time evolution of the magnetization has to be specified. If thermal fluctuations are neglected, micromagnetics generally describes time evolution

M.E.

according

Schobes / Micromagnetic

to the Landau-Lifshitz

equations

theory of non-uniform

[80]

(19)

where y,, is the gyromagnetic ratio and X is the damping constant. The first term is the torque exerted onto the magnetization by the effective field. The second term describes in a phenomenological way dissipative processes. An alternative form of the dissipative term has been given by Gilbert as being proportional to M X ik?. A detailed discussion of the subtle distinctions between the two terms has been given by Mallinson [81]. The description of the origin of the dissipative mechanisms is beyond the scope of micromagnetic theory. Generally, dissipation is caused by eddy currents, magnon-phonon interaction and similar processes. For strongly coupled micromagnetic systems the angle between the effective field and and the magnetization is generally small (section 7.3) even during irreversible switching. The precessional cone has a small angle in this case and the dynamical evolution of the magnetization towards equilibrium is generally governed by the dissipative term (at least on the level of the micromagnetic description). In the limit, where the precessional term is neglected entirely, eqs. (19) are usually referred to as the infinitely damped Landau-Lifshitz equations.

4.7. Scaling laws

In ref. [5] Schabes and Bertram derived an important scaling relation between the particle size and the exchange interaction. The mathematical origin of this link is the fact, that - unlike the other terms of the micromagnetic energy functional - the exchange energy density is obtained by applying a differential operator to the magnetization field. In a uniform applied field the scaling relation between particle size and exchange constant is obtained by carrying out a global scale transformation of the energy functional [5]. This shows

magnetization

processes

261

that the switching field is invariant, if an increase of the exchange constant by a factor of h is accompanied by an increase in particle size D by a factor of JT; H,(D,

A) = H,(Jr;D,

AA).

(20)

The above scaling relation is important, since in many cases the value of the exchange constant is known only approximately. If crystalline anisotropy is neglected, the above scaling law can be extended to include also the magnetic moment density MS. In this case application of the scaling transformation shows that the relevant quantity, which determines the micromagnetic problem is the ratio

(21) Particles with the same value of p have the same micromagnetic properties, provided the value of the applied field is scaled proportionally to MS. In the subsequent sections we frequently will refer to the :fi = D/2A,, as a means of specifying particle sizes, since this allows application of micromagnetic results to a wider range of particles (e.g., compare magnetization processes in metallic iron particles with those in y-Fe,O, particles; see sections 2.2 and 11). This type of scaling has been used also previously in ref. [69]. If the applied field originates from fixed external sources of finite size (e.g. a recording head), the above scaling relations cannot be applied directly. In this case the applied field is generally non-uniform and a resealing of the external sources, e.g. of the length of the recording gap, is also required. As a concluding remark with regard to scaling relations, a curious property of magnetostatic interaction fields is remembered. If only the magnetostatic interaction within the particle were considered, the micromagnetic problem becomes entirely scale invariant, i.e. a magnetic sphere with radius 100 A is magnetostatically indistinguishable from a sphere of radius 1 km. This property stems from the fact that the magnetostatic interaction is long range.

262

M. E. Schabes / Micromagnetic

4.8. Classification helicity

of magnetization

theory of non-uniform

configurations

a*ds #P

ds. /+

P

processes

-

In ref. [5] it has been shown that the magnetization configurations of small magnetic particles can be described in simple topological terms. Two main topologies have been identified: the flower state and the vortex state [5]. Both magnetization configurations can be considered to be generalized curling states. Unlike the classical curling (as presented by Brown [73,74], Frei [75] and Aharoni [29]), the vortex configuration has a finite “curling amplitude” and is generally a three-dimensional topological object, The curling amplitude depends on the distance from the central particle axis (as in the classical curling mode). But furthermore, the curling may be “stronger” near the particle ends than halfway between the ends. Moreover, the sense of rotation at one particle end may be different from that at the other particle end. Depending on the shape of the transverse cross section of the particle, there may also exist an azimuthal dependence of the magnetization configuration. Since the azimuthal dependence of the magnetization is of minor concern for many cases, it is useful to introduce the line integral of the magnetization a = M/M along an oriented curve P [5]. The value of this integral normalized by the length of the path is called the helicity A of the magnetization configuration along path P, A=

magnetization

(22)

The helicity A characterizes both the sense of rotation and the extent to which the magnetization is tilted in various regions of the vortex (i.e. “the curling amplitude”). To describe a vortex state with long axis parallel to the z-axis the path P is chosen along the outer circumference of the particle and contained in a plane parallel to the xy-plane. The path is arbitrarily oriented counterclockwise. For fixed path P the helicity is a real valued function of the magnetization with values in the closed interval [ - 1, + 11. A counterclockwise rotation of the magnetization gives positive helicity while clock-

c> Fig. 3. Two basic topologies for equilibrium magnetization states small cubic particles. (a) The flower state; (b) projection of the magnetization onto the top plane; (c) the vortex state: (d) projection of the magnetization onto the top plane. (a) and (c) have been stretched slightly along the z-direction for clear separation of the planes (from ref. [5]).

wise rotation corresponds to negative helicity. For large aspect ratio the helicity of the vortex state may be of the same or opposite sign at the particle ends. In the first case the vortex state is called a + + vortex, in the second case it is called a + vortex. This will be discussed in section 9. For certain magnetization configurations the helicity vanishes identically. One such case is (trivially) the uniform state of magnetization. Another case with A = 0 occurs, when the projection of the magnetization onto the xy-plane has radial symmetry. This state is commonly encountered for small particle diameters and small magnitudes of the reverse field and has been called the “flower state” [5-lo] (fig. 3a, b). This state has also been discussed in refs. [I 1-I 31. The helicity can also be used to identify transitions in the topology of the equilibrium magnetization configurations, e.g. a transition from the flower state to a vortex state at certain magnitudes

M.E. Schabes / Micromagnefic

theory of non-uniform

of the applied field or at certain sizes. In general, during irreversible switching the helicity changes. The change affects the magnitude and possibly the sign of A. The helicity is (trivially) conserved for the switching from a positive flower state to a negative flower state. A change of sign is to be expected for the case, where the vortex state rotates “rigidly” about an axis perpendicular to the original vortex axis [5]. If the applied field is non-uniform, the equilibrium magnetization configurations may have significantly different helicities in the regions, where the applied field is high and those where the applied field is low. For strong field gradients (e.g. head field) this may even lead to a subdivision of the magnetization into a flower like region with zero helicity (at the particle end, which is submerged into a large record field) and into a vortex like region with finite helicity (at the other particle end). This aspect will be discussed in section 13.

5. Analytical treatment of non-uniform tion configurations

magnetiza-

In the following we list a few examples of analytical investigations of the reversal mechanism. In many cases this amounts to a Ritz parameterization, where an “ansatz” is made for the functional form of the magnetization configuration. This leads to a minimization problem with only few degrees of freedom and can easily be solved numerically. While insight into some aspects of the reversal process, e.g., of the energetits of certain particular magnetization configurations can be obtained this way, interpretation of the results has to be done cautiously, mainly for the reason, that a parameterization seriously biases the possible magnetization states. Calculations of the type of Ritz-parameterization have been presented, for instance, by Arrot, Heinrich and Aharoni [82], who described the magnetization reversal in a finite cylinder by using the magnetostatic surface charges on the top faces and on the sides of the cylinder as Ritz parameters. The charge density on the top of the cylinder is assumed uniform, while the charge density on the side surface varies linearly with displacement

magnetization

processes

263

parallel to the cylinder axis. Under these assumptions a finite amplitude curling pattern is computed. In the case, where the exchange interaction and the crystalline anisotropy are neglected, Van den Berg showed that the micromagnetic partial differential equations may be solved exactly [83]. This method has been extended by Bryant and Suhl, who also considered the effect of an applied field [84]. Recently Pu et al. [85] discussed both the problem of critical sizes and of the magnetization reversal in the context of bifurcation theory. While this approach is conceptually unifying, it did not proceed very far in terms of results for non-uniform magnetization states. In fact, it again focussed on the stability of magnetic matter in the uniform state. A similar statement applies also to the paper of Pinto [86], who put the StonerWohlfarth particle in the general framework of catastrophe theory. A step towards a general analysis of the stability of the non-uniform magnetization state has been undertaken by Hartmann [87], who derived a set of integro-differential equations for the stability of a general non-uniform state of magnetization. It was not yet possible to solve these equations except for cases, where the analytical solution can be found by traditional methods.

6. Micromagnetic

models

Until very recently the theoretical investigation of the reversal process was either restricted to linear nucleation theory or had to resort to ad hoc assumptions, if the non-linear switching mechanism was to be investigated. In the latter case the usefulness of the model critically depends on the intuition of its creator. A model of the reversal mechanism, that is quite remarkable in this respect was developed by Knowles [33], although some of his conclusions have later been criticized by Aharoni [30]. In the computation of the reversal process Knowles postulates that the switching process starts at the particle ends, and successively reverses the magnetization of adjacent spherical ele-

264

M. E. Schabes / Micromagnetic

theory of non-uniform

ments. This reversal mechanism is called the “flipping mode”. Dynamical calculations of the switching processes confirm Knowle’s premisse, namely that the irreversible switching process in elongated particles begins at the particle ends and proceeds towards the particle center. This will be discussed explicitly in section 9. A dynamical solution of the Landau-Lifshitz equations for a simple model further illustrates the flipping process (see appendix A). The sequential (flipping) character of irreversible processes has also important consequences for switching in an inhomogeneous applied fields, as shown in section 13 and appendix B.

7. “First principles” calculations The availability of supercomputers has made it possible to treat the reversal process without a priori assumptions as to the particular reversal modes. It has become customary to call such self-consistent calculations, which require only the prescription of the energy functional (from which the effective fields can be computed) and the definition of the dynamics for the time evolution of the magnetization (especially during irreversible switching) “first-principles” (see, for example, refs. [17,19,88]. The author concedes that such a point of view is valid, as long as it is kept in mind that the calculations are subject to the micromagnetic approximations. In particular, the treatment of the dissipative term in the Landau-Lifshitz equations is of some concern in this regard, since it is phenomenological. There is no obvious physical link between spin dynamics and micromagnetic dynamics and the validity of such a renormalization still needs to be shown. 7.1. Finite element calculations Fredkin and Koehler developed a model for computing magnetization states based on the finite element method [15-201. An advantage of using finite elements is the flexibility with regard to accomodating arbitrary particle shapes: Magnetization processes in a sphere can be computed with

magnetization

processes

the same ease as those in the more complicated shapes of physical particles. Fredkin and Koehler so far restricted the investigation to equilibrium magnetization configurations and do not explicitly consider an inherent underlying dynamical process. As has been pointed out above, in larger particles this may make it difficult to physically interpret the obtained hysteretic processes.

7.2. Iterative

methods

Iterative methods are suitable for computation of equilibrium states. These methods rely on some algorithm to successively improve the alignment between the magnetization and the effective field. In the following we will give a brief description of iterative procedures. Unless otherwise stated, a cubic discretization (see e.g. ref. [68]) is assumed, where (Y,= ( CY,,p,, -r,) is the magnetization of the i th element. Much of the following applies also to general discretization prescriptions. There are two types of iterative methods [89]. Gauss-Seidel iterations proceeds in the following way: First the effective field is computed at a given lattice site. Then the magnetization is adjusted in the direction of the local effective field. Another lattice site is visited, the field is recomputed and the magnetization is updated. This procedure continues, until all lattice sites are visited. This completes one single Gauss-Seidel sweep. The Jacobi method differs from the GaussSeidel method in the way the lattice sites are updated. Instead of using the updated magnetization from previous visits to other lattice sites within one sweep, the initial (old) magnetization configuration is used to compute the effective field at all lattice sites. The update of the magnetization is made after the fields at all lattice sites have been computed. While the rate of convergence is generally slower for Jacobi iterations, the overall computational time can be significantly shorter for Jacobi updates on massively parallel computers with a large number of processors. For a recent large scale micromagnetic calculation on such a computer see, e.g., ref. [90]. If the magnetization updates are small, the final result for the equilibrium state does not depend on

M. E. Schabes / Micromagnetic

theory o/non-uniform

which iteration procedure is used, nor on the particularities of the path for Gauss-Seidel updates and a clever choice of the path can significantly accelerate convergence. However, it must be emphasized, that the unconverged states are in general meaningless as physical magnetization configurations, since they are directly dependent on the path used, the update sequences, choice of relaxation parameters, etc. After convergence has been achieved it is useful to test the stability of the magnetization configuration by adding a small random perturbation to the magnetization [5]. The iteration is restarted with the perturbed state as the new initial configuration. If the state is unstable, the perturbation gets amplified instead of being reduced during the subsequent iterations. The perturbation mimics also the effects of thermal fluctuations. This has been investigated in detail by Lyberatos and Chantrell in ref. [ll]. 7.3. Dynamical

calculations

As has been pointed out, the reversal process is an essentially dynamical problem. Therefore it is necessary to solve a differential equation for the time evolution of the magnetization. In particular, if time evolution according to Landau-Lifshitz and Cartesian coordinates for the magnetization are used on a lattice with N elements, it is required to solve a coupled set of 3N non-linear ordinary differential equations. Transient states in small particles have been first computed dynamically in ref. [5] by Schabes and Bertram and later in refs. [6-lo]. The limit of inifinite damping (see section 4.6) was used, since the strong exchange interaction keeps the precessional cone small during switching [5]. The validity of the infinite damping approximation for strongly interacting micromagnetic systems has also been shown recently by Zhu and Bertram [91], and is further corroborated by the calculations of Nakatani, Uesaka and Hayashi [12]. The main topological aspects of the dynamical results of ref. [5] have been confirmed by the computations of ref. [12], where the Landau-Lifshitz equations with finite Gilbert damping were studied for a cubic particle. Similarly, the topological results

magnetization

processes

265

of section 9 for elongated particles are in good agreement with those of ref. [13]. There are various ways to carry out numerical integration of ordinary differential equations. The simplest methods are Runge-Kutta integration schemes of modest order (e.g. 4th order), as has been used e.g. by Victora [92] in calculations on barium ferrite. Schabes and Bertram [5-81 used Adams-Bashford numerical integration employing varying step size and extrapolation methods. A comprehensive description of the theory of multistep numerical integration has been given by Gear 1931. The appropriate choice of the numerical routine for integration is dictated not only by the system size, but also by the stiffness of the differential equations. Considering both the precessional and the damping motion turns the set of differential equations into a stiff set. Therefore explicit integration methods have to take small time steps in order to be stable. However, if the time steps become exceedingly small, accumulation of roundoff error occurs. This difficulty is overcome by the use of implicit methods. The application of these methods to micromagnetics has been investigated in ref. [12]. In the micromagnetic calculations presented in the sections 8-11 and 13 Adams-Bashford integration was used for computing the transient states. The initial conditions were in many cases an equilibrium state (obtained iteratively) in a field of magnitude just below the switching field. For large particle diameters of elongated particles starting conditions for the Adams solver were sometimes biased by adding a small helicity to the magnetization configuration to compare switching of the vortex states of even and odd helicity, respectively. It is noted that for small particle sizes and strong exchange coupling between neighboring elements the dynamical evolution of the magnetization becomes quite simple and predictable. Therefore, it may be expected that some of the dynamical aspects could be put into the form of cellular automata equations [94]. As the field of micromagnetics continues to mature, micromagnetic theory becomes essentially the discipline of studying from first principles the

M. E. Schabes / Micromagnetic

266

theory of non-uniform

magnetization dynamics of a dissipative non-linear dynamical system with careful mapping of initial conditions and phase space. The analysis will be similar in many ways to fluid dynamics, although micromagnetics is more complicated, since in the case, where the magnetization is represented by the field CY(X)= (a(x), p(x), y(x)), the underlying base manifold is non-flat (due to the constraint (Y*= 1). If, instead, angular variables are used, a covering atlas of coordinate charts [95] (i.e., a set of multiple coordinate systems) is required, since the azimuthal angle is not defined if the polar angle is 0 or 71 [96].

8. Cubic particles In this section we review calculations for particles of cubic shape [5,11,12] and apply the concepts developed in the earlier sections to describe the processes associated with non-uniform magnetization states. 8.1. Micromagnetic

energy

Using the results of section 14, the total micromagnetic energy density is written as

fl =-

Hw H,

A ’

'2=731

'3=

magnetization

tially two topologies, the flower state and the vortex state and have been described in section 4.8. Of course, if the particle size is larger (or the exchange coupling were smaller) much more complicated magnetization states are obtained, as the particles reach the multi-domain state. Examples of such magnetization states are of interest in paleomagnetism and were described recently by Williams and Dunlop [97]. By computing the remanent magnetization as a function of particle width a critical particle size can be identified, where the flower state becomes unstable in zero applied field and the vortex state is the remanent state. This critical size is a generalization of Brown’s notion of critical size [5]. The onset of the vortex state is in general a function of particle size and applied field. The exact value of the critical size is also susceptible to both thermal perturbations [ll] and to the details of the underlying dynamical evolution [12]. Fig. 4 shows the dependence of the magnetization loops for various particle sizes with the applied field parallel to one of the particle faces. For small particle diameter the loop is Stoner-Wohlfarth like and very square. For larger particle size the loops show the onset of the vortex as a marked decrease of the slope of the reversible branches. The squareness is decreased, if vortex formation occurs in a positive applied field. Along the decreasing branches of the hysteresis loop the helic-

M 2

Hk'

(24) o,,,,~ is the normalized magnetostatic energy density as derived in ref. [68]. The following calculations refer to a saturation magnetization M, = 370 emu/cm3 and uniaxial crystalline anisotropy of K = 1.8 x lo4 erg/cm3. The value of the exchange constant is A = 5 x lop7 erg/cm. Note that the definition of A differs from that of ref. [5] (see section 4.2).

v)

$

1.0 400

R 500

;

r" E 2 w

The equilibrium material parameters

configurations states in cubic particles with as specified above are essen-

I' __------'-____$-r"i $-Q---

-

-

0.5 -

0.0 -

-0.5

:

-

$ -l.O-

j;Fq5;

b: ij

;

1: I: 'I II ,;

: ; : ' ;

:I

: ii * :; I :t __-_&l_---;______-

,;

--

-1.51 -2.0

8.2. Magnetization

processes

-1.0 APPLIED

2.0

0.0

FIELD

H,,,;;Lk

Fig. 4. Size dependence of the hysteresis loop for cubic particles. Particle sizes are quoted for A = 5 x 10e7 erg/cm (from ref. [5]).

261

M.E. Schabes / Micromagnetic theory of non-uniform magnetization processes

ity of the vortex state increases. This is a reversible process. At the switching field irreversible transition to the reversed state occurs. The loop for 400 A is almost perfectly square and corresponds to flower to flower switching. At 500 and 520 A the irreversible switching is vortex to flower, at 520 A the irreversible switching is vortex to vortex. It is also noted that - whereas uniformly magnetized cubes have no shape anisotropy - nonuniformly magnetized cubic particles may have (in the uniaxial case) switching fields that are larger than the 2K/M,. This has been explained by Schabes and Bertram in terms of a configurational anisotropy associated with the magnetization state (rather than with the particle shape as in the case of uniformly magnetized particles) [5]. In ref. [5] two main switching mechanisms have been identified: (1) For small particle diameters irreversible switching proceeds via quasi-uniform rotation, which involves closing of the flower state in the initial phases of switching. (2) For larger particle sizes the switching of the vortex state takes place by rotating the vortex axis leading to an intermediate transverse vortex state. The main features of these processes, in particular the formation of a transverse vortex has also been confirmed in the detailed dynamical study of ref. [12], where a finite damping term was taken and time scales for transverse vortex formation have been computed. Thermal fluctuations, as reported in ref. [ll] do not change the switching mechanism, but generally lower the switching field. A transverse vortex state has also been reported by Fredkin and Koehler in ellipsoidal particles [19].

results for the undoped precursor particle will be presented. Some of this work has been published in ref. [6]. 9.1. Mathematical

The micromagnetic energy terms are the same as in the previous section, except for the crystalline anisotropy energy, which is of cubic symmetry [6]. Taking into account also the fact. that the preferred direction of crystalline growth is such that the [llO] direction coincides with the long particle axis, the total micromagnetic energy may be written as

with E, =

H;lPPMs d ’ A 9

particles

y-FeZOX particles have been used extensively for manufacturing magnetic media for large scale data storage and have been studied extensively. Improved magnetic properties have been achieved by doping y-Fe,O, particles with various transition elements, in particular cobalt. The effects of surface modification of y-Fe,O, particles due to Co doping or Co adsorption will be discussed in the next section. Here a summary of theoretical

K,d’ cZ=A’

M2d2 63 = \ A ’ (26)

In the above equation the energy is normalized with respect to the exchange interaction, rather than with respect to the anisotropy, since also the effect of small anisotropy is investigated. The second term of the energy function is obtained from the conventional fourth order term for the crystalline anisotropy %” = K,(a2p2

9. Elongated y-Fe,O,

model

+ ,‘y2

+ p2y2),

(27)

by carrying out first a rotation of 71/4 about the z-axis, which is chosen to be parallel to the long particle axis. This is followed by a rotation of 77/2 about the new y-axis. The following calculations were carried out out for a saturation magnetization MS = 350 emu/cm3, and cubic crystalline anisotropy of K, = -4.6 X lo4 erg/cm3. For comparison calculations were also carried out for uniaxial crystalline anisotropy K = 1K, (. The value of the exchange constant is less well known (see discussion of the value of the

268

zheo~v of non-uniform

M. E. Schabes / Muomugnetic

b I t t t t L t t t t + t , 6 4 I

t t

.

.

.

.

.

.

.

.

.

.

,

9

,

9

I

1 1

t

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I

magnetization

processes

t t t t t *

11111

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h

iiili

t t t I 11111 t f t I * et t . . . . . . . . b6, * t t t t t t t t t t t t t t t t

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4

width D to twice the value of A,,. Note also that the definition of the exchange constant A as used in ref. [6] differs by a factor of 2 from the one used here. It is easy to rescale the particle sizes or the value of A (see section 4.7). 9.2. Equilibrium magnetization

configurations

- re-

versible processes

Fig. 5. Transient states during irreversible switching of state with large exchange coupling. D/2h,, = 0.74, 1300 Oe. The magnetization is projected onto the longitudinal cross section. The sequence is arranged in increasing time.

a flower H,,,

=

central order of

exchange constant in section lo), and results are presented for the intermediate value of A = 5 x lo-’ erg/cm. In this case the exchange length A,, = 202 A. Particle diameters are referenced to the exchange length by giving the ratio of particle

The equilibrium magnetization configurations in elongated y-Fe,O, particles are generalized curling configurations and may be classified according to their topology in a similar fashion as has been done in section 4.8, i.e., as the flower state and as the vortex state. In the case of elongated particles the diminished exchange coupling between the particle ends warrants a closer examination of the reversible processes, which start at the particle ends and gradually (as the magnitude of the reverse field is increased) effect the magnetization farther away from the ends. These reversible processes are a prelude to the irreversible switching and have been called “vortex expansion” [6]. 9.3. Trunsient stutes - irreversible processes Irreversible processes propagate from the regions of non-uniform magnetization, as estab-

M. E. Schabes / Micromagnetic

theory of non-uniform

lished via reversible processes, from the particle ends towards the particle center. Depending on the particle diameter and aspect ratio, the dynamics of the irreversible processes can have a rich structure, which generally involves formation and propagation of transverse vortices, as has been described by Schabes and Bertram [7] in the particles of aspect ratio 6 : 1 in a non-uniform applied field. In the following we present results for typical topologies of the transient states obtained by numerical integration of the Landau-Lifshitz equations with infinite damping. The results are in good agreement with calculations using finite damping [13]. Ref. [27] shows similar configurations for the initial stages of the switching process, although the states of ref. [27] are merely unconverged results of an iterative energy minimization scheme. In this and the following sections the three-dimensional magnetization processes are further illustrated by projecting the magnetization onto cross sections, which are parallel to one of the long particle faces. The system of coordinates is such that the long particle axis is parallel to the z-axis, with x and y coordinates being the transverse directions. 9.3. I. Large exchange coupling - small particle size Fig. 5 shows the irreversible switching of a particle with D/2X,, = 0.74 and an aspect ratio 6 : 1 in an applied field of magnitude 1300 Oe. The magnetization is projected onto the central longitudinal cross section. The sequence of magnetization configurations is computed for uniaxial anisotropy and is arranged in order of increasing time. At the switching field the magnetization at the particle ends rotates. A large negative magnetization component is acquired, when the rotated magnetization encomprizes a volume of about cubic shape. Little shape anisotropy has to be overcome for complete reversal of this volume near the particle end (fig. 5.4). The quasi-uniform rotation of the magnetization at the particle ends generates a wall structure, which separates the reversed magnetization from the unreversed magnetization at the center of the

magnetization

processes

269

particle. The region of reversed magnetization expands by moving the wall towards the center of the particle (figs. 5.5-11). It is noted, that this process is similar to the flipping mechanism, which was postulated by Knowles [33]. The amount of lag between reversal of the particle center and reversal of the magnetization near the particle ends depends on the aspect ratio. Particles with large exchange coupling and small aspect ratio reverse by almost perfect uniform rotation. There is very little phase shift between the magnetization rotation at the particle center and the particle ends. Of course, this is related to the fact that the exchange length in these particles is not only comparable to the particle diameter but also to the particle length, while at a larger aspect ration the exchange interaction cannot span the entire particle length. 9.3.2. Small exchange coupling - large particle size If the particle size is increased or - equivalently _ the exchange coupling is decreased, the irreversible processes become more complicated than quasi non-uniform rotation. The following gives the reversal of a vortex state with odd helicity in a particle with D/2X,, = 1.56 and aspect ratio 3 : 1 in an applied field of magnitude 600 Oe. A sequence of transient states is depicted in fig. 6, which gives an overall three-dimensional view of magnetization configurations and in fig. 7. where the details of the irreversible switching are illustrated by projecting the magnetization onto the central longitudinal cross section. As fig. 7 shows, the irreversible switching mechanism is a superposition of (1) a curling motion in the transverse cross section (perpendicular to the long particle axis) and (2) a buckling motion in the longitudinal cross section. Figs. 7.1-2 reveal that in the wake of the longitudinal buckling there is formation of transverse vortices on the sides of the particle. This creates a continuous path of reversed magnetization connecting the bottom and the top of the particle. As a consequence, small vortices at the side of the particle are formed, which subsequently are pushed towards the particle boundary and disappear. The magnetization of the odd helicity vortex has inflection symmetry about the center of the

270

M.E. Schabes / Micromagnetic

b

theory of non-uniform

magnetization

processes

d

Fig. 6. Transient states during irreversible switching of a vortex state with odd helicity by integrating the Landau-Lifshitz equations in the limit of infinite damping. D/2X,, =1.56, aspect ratio p = 3: 1, H,,, = 600 Oe. (a)-(d) are arranged in order of increasing time.

Fig. 7. Transient states during irreversible applied field as in fig. 6. The magnetization

switching of a vortex state with odd helicity in a particle with material parameters and is projected onto the central longitudinal cross section. The sequence is arranged in order of increasing time.

M. E. Schabes / Micromagneiic theory

of non-uniformmagnetization

processes

Fig. 8. Transient states during irreversible switching of a vortex state with odd helicity in a particle with material applied field as in fig. 6. The magnetization is projected onto the front and rear faces of the particle, respctively. arranged in order of increasing time.

particle. For example, the magnetization vectors 06 the corners at the end of a body diagonal have the same directions. It is remarkable, that the transient magnetization configurations preserve this type of symmetry during irreversible switching. This is demonstrated in fig. 8, where the magnetization is projected onto a longitudinal front face and a longitudinal back surface, respectively. The wall, which separates the magnetization of opposite helicity, propagates in the direction of the applied field in the rear portion of the particle (labeled “back face” in fig. 8), while in the front portion of the particle (labeled “front face”) the wall propagates opposire to the applied field direction. If the aspect ratio is increased, in general, the complexity of the transient states grows and can involve multiple vortices. Again, the irreversible magnetization processes start at the particle ends and propagate towards the particle center. This is shown in fig. 9 for the reversal of a vortex state

271

parameters and The sequence is

with even helicity in a particle with D/2X,, = 2.35 and aspect ratio 6 : 1 in an applied field of magnitude 450 Oe. As shown in fig. 9, the initial formation of transverse vortices occurs at the sides of the particle as a small perturbation, which quickly grows in volume and moves towards the central particle axis (fig. 9.3). The transverse vortex at the top of the particle expands in the direction of the applied field, while the transverse vortex at the bottom of the particle expands in the direction opposite to the applied field. In either case, the magnetization acquires a large negative z-component on the downstream side of the vortex expansion. On the upstream side the magnetization carries out a longitudinal buckling motion, which leads to formation of a third transverse vortex (figs. 9.5-7). In the next phase of the irreversible switching the magnetization between the central transverse vortex and the adjacent vortices at the particle ends reverses. In this process the vortices seem to

272

M. E. Schuhes / Mmomagnetic

theory

of non-uniformmagnetization

processes

i

Fig. 9. Transient states during irreversible switching of a vortex state with even helicity. D/2h,, = 2.35, Happ = 450 Oe, aspect ratio p = 6: 1. The magnetization is projected onto the central longitudinal cross section. The sequence is arranged in order of increasing time. The magnetization is projected onto the central longitudinal cross section. The sequence is arranged in order of increasing time.

repel each other (fig. 9.8) and are pushed to the particle boundaries, where they shrink in volume, until they disappear (fig. 9.14). In view of the complexity of the irreversible processes of the type as depicted in fig. 9, it is instructive to discuss the evolution of the various energy terms. It is found that during the initial

stages of irreversible switching the decrease of the total energy is due to a reduction in the Zeeman energy, while the other energy terms may in fact increase. In particular, this applies to the exchange energy. Longitudinal buckling and/or formation of transverse vortices generally leads to a relative decrease of the magnetostatic contributions. In the

M. E. Schabes / Micromagnetic

theory of non-uniform

final stages of the irreversible switching, when transverse vortices are expelled, the exchange energy drops sharply: The magnetization reaches the reversed equilibrium state, where most of the volume of the particle is uniformly magnetized. The magnetostatic energy is generally larger after irreversible switching than before irreversible switching, in particular, if switching occurs from a vortex state to a flower state. This is due to the fact that the applied field after irreversible switching is pointing in the same direction as the average magnetization, which leads to a more uniform magnetization configuration and hence to larger magnetostatic surface charges at the particle ends. As a summarizing remark it is once more emphasized, that the understanding of the fact that the irreversible switching starts at the particle ends, relies on appreciating the underlying dynamics: demagnetizing fields at the particle ends enhance the reversal torque and consequently the rate of dissipation locally near the particle ends. A more difficult question is the selection of helicity type of the vortex states. Both the + + vortex (even helicity) and the + - vortex (odd helicity) are equilibrium states, but they are not entirely degenerate for magnetostatic reasons (e.g. the magnetization vectors at the end of a given edge are approximately perpendicular for the + + vortex, while they are approximately parallel for the + - vortex). The initial conditions seem to be important. If, for instance, a small helicity, which is of the same sign at either particle end, is added to the initial configuration, the magnetization configuration usually becomes a + + vortex (for appropriate particle size). In fact, this method was used in some cases to compare the reversal of the + + vortex with the + - vortex. Also the history of the applied field, such as prior application of a field at a large angle may be important. However, we did not attempt to investigate in detail the conditions which favor incipient formation of + + vortex over the + - vortex, since the overall reversal dynamics is similar for both type of vortices. The switching fields for the + + and the vortices are comparable, with the switching field for the + - vortex being somewhat higher. Moreover, it is expected that in realistic particles the helicity selection may be determined to a large

magnetization

processes

Fig. 10. Remanent mangetization of a particle with corner = 1.56, aspect ratio p = 3 : 1. The magnetization defect. D/2X,, is projected onto the transverse cross section. Integer labels indicate displacement from the bottom end of the particle, bottom = 1, top end = 15.

degree by variations of the local morphology at the particle ends, e.g., by small defects. This is demonstrated in fig. 10, where the equilibrium state of a particle, whose corner has been removed, is depicted. The magnetization is projected onto the transverse cross section of the particle. The labels indicate the displacement along the z-axis as measured from the bottom end of the particle. While the usual curling pattern is preserved at the intact particle end, the azimuthal symmetry is destroyed entirely by the defect at the other particle end. It is plausible to expect that a less severe defect may not entirely suppress the curling motion, but may lead to a preferred helicity selection at a given particle end. Small defects near the particle ends generally do not significantly perturb the irreversible switching mechanisms. This is due to the fact that particle ends are - strictly speaking - a special type of defect, anyhow. However, it is noted, that defects introduced near the particle center may lead to qualitatively different reversal mechanisms, because such a defect creates local magnetostatic interaction fields, which destabilize the magnetization adjacent to the defect. Hence, such a defect, if of sufficient size, is expected to generate ad-

M. E. Schabes / Micromagnetic

274

ditional non-uniform the particle center.

magnetization

processes

theory of non-uniform

magnetization

processes 1

near

I

%

9.4. Size dependence

With increasing particle size the exchange interaction becomes relatively weaker. Therefore the vortices at the ends of the particle occupy a larger fraction of the particle volume at a smaller value of the applied field. This larger non-uniformity of the magnetization leads to a decreased switching field. Moreover, the remanent magnetization along the z-direction is reduced. This is shown in fig. 11, where the average magnetization parallel to the long particle axis is plotted as a function of particle width. A pronounced reduction of the magnetization occurs, where the transition from the flower state to the vortex state takes place. This is completely analogous to the case discussed in section 8 for cubic particles and has similar consequences for the shape of the magnetization loops. Fig. 12 shows the switching field as a function of particle width for particles of aspect ratio 3 : 1 and 5 : 1, respectively. Particle sizes are quoted for A = 5 x lo-’ erg/cm, but are best related to the exchange length (indicated on the upper horizonthe switching field tal abscissa). For comparison,

I

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Fig.

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0.01

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2

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1200

(Angstrom)

Remanent magnetization as a function of particle width for aspect ratio 3 : 1. A = 5 X lo-’ erg/cm.

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A=5xlO-‘erg/cm

-

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600 -

O0

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400

800

Particle

Width

I 1200

(Angstrom)

Fig. 12. Size dependence of the switching field for aspect ratio 3: 1 (cubic and uniaxial anisotropy) and 5: 1. A = 5 x10-’ erg/cm. (uniaxial anisotropy). For comparison the StonerWohlfarth (SW) and the curling fields are shown.

of a uniaxial particle is included, where the easy axis is parallel to the long particle axis and the uniaxial anisotropy constant is equal to ( K, (. This demonstrates that the large angle of the easy cubic axis with respect to the long particle axis (= 30 “) reduces the magnitude of the switching field. In linear nucleation theory of uniformly magnetized particles, uniaxial crystalline anisotropy yields an additive constant equal to 2K/M, to the nucleation field. However, the switching field increases by less than 2K/M, in the case of nonuniform magnetization configurations. In fig. 12 the classical nucleation field for a Stoner-Wohlfarth particle is also plotted, although we are well aware, that the switching field of the parallelepipeds is not a nucleation field. The angular dependence of the switching field has been computed in ref. [6], where it has been shown that H, increases as a function of applied field angle. The rise in the switching field is most pronounced for large angles. This behavior is typical for the angular dependence of non-uniform magnetization configurations. It is also similar to the angular dependence of the classical curling nucleation field.

M. E. Schabes / Micromagnetic

10. Cobalt modified y-Fe,O,

theory of non-uniform

particles

Cobalt can be added to y-Fe,O, particles by either volume doping or surface adsorption. In both cases, the formation of a cobalt rich phase occurs in the iron oxide. In the case of surface adsorption it is believed that the outermost region of the particles contains cobalt ferrite (CoFe,O,). There is also evidence [43] that in the case of cobalt adsorption this outer region is restricted to a few atomic layers, possibly only one mono layer or less. This has been demonstrated recently also by Schabes and Bertram, who showed that extrapolation of first principles calculations via energy-volume scaling predicts that a layer of CoFe,O, as thin as 6 A yields an enhancement of the coercivity by approximately 300 Oe. This remarkable result will be discussed in section 10.2. The genera1 issue of surface modification is a large topic with many ramifications, since several physical mechanisms can be involved, e.g., changes in particle morphology, modification of the local particle chemistry, introduction of interface interactions,. . . , all of which may lead to a modification of the magnetic interactions within the particle. The situation is also complicated by the fact that the adsorption of materials, which are nonmagnetic in the bulk state may produce changes of the magnetic properties of the particle [98]. Two experimental results point to the key issues of a micromagnetic understanding of the reversal mechanisms in cobalt modified y-Fe,O, particles: (1) the enhancement of the coercivity of the modified particle over the precursor particle, (2) the fact that the enhancement of the coercivity saturates as a function of doping level. This effect has been demonstrated by Imaoka et al. [99] and will be referred to as the saturation effect. A rigorous treatment of all aspects of surface modification is a formidable task. However, it has been shown that a better understanding of the micromagnetic processes of cobalt modified particles may be obtained by discussing simplified situations [31,32,8], where surface modification is modeled by dividing the particle volume into a low anisotropy inner core region and a high anisotropy shell region. This approach will be pursued in the following. First, the linear nucleation theory

magnetization

processes

215

of cobalt modified y-Fe,O, particles will be briefly the non-linear switching discussed. Secondly, processes are presented. IO. 1. Linear nucleation theory The simplest way to gain insight into the mechanism of magnetization reversal of coated particles is by carrying out linear nucleation analysis. This has been done in detail by Aharoni for the the infinite cylinder [31,32]. Let R, and R, be the radii of the iron oxide core and of the entire particle, respectively; i.e., the thickness of the particle shell is given by showed that for a moDsllell = R, - R,. Aharoni ment density IV, = 425 emu/cm3, a uniaxial shell crystalline anisotropy Kshe,, = 2.9 x lo6 erg/cm3, and vanishing core anisotropy K,,,, = 0, the uniform magnetization state looses its stability via the curling mode for S, & 2 and via the buckling mode for S, 6 1, where S, = M,R,/fi. For the above ranges of S,, the nucleation mode is independent of R,. However, for intermediate core diameters 1 < S, < 2 the nucleation mode depends on both R, and R,. The assumption of uniaxial crystalline anisotropy for the shell region is probably reasonable (see e.g. ref. [42]), especially for thin coatings. The crystalline anisotropy of the core should be that of the precursor particle, namely, cubic. However, for the purpose of demonstrating the effects of shell thickness and doping levels of the particle shell it is sufficient to discuss the simpler case of uniaxial anisotropy. Using Aharoni’s method of ref. [31], it is easy to include also a finite uniaxial core anisotropy K,,,,. This shows that the crystalline anisotropy of the core raises the nucleation field by approximately 2K,,,,/M, and leads us to expect that the initial stages of irreversible switching will be processes involving mainly the core region rather than the shell region. This will be confirmed in the discussion of the non-linear results. There it will also become clear that these mechanisms together with a finite coupling between the shell and the core regions explain the saturation effect. This point is further underlined by noting that a finite nucleation field is obtained even in the asymptotic

216

M. E. Schabes / Micromagnetic

theory of non-uniform

case of infinite shell anisotropy, since there is only a finite core-shell coupling. Analogous statements will also derived for the switching field in the next section. It can also be shown that the nucleation field is approximately a function of the product of shell volume and shell anisotropy, i.e. H,,, = H,,,,( V&,, Kshe,,). This leads to a scaling relation between shell thickness and doping level and will be discussed further in the next section. 10.2. Non-uniform

magnetization

processes

In this section we present results for magnetization processes in cobalt modified y-Fe,O, particles, as obtained recently by Schabes and Bertram [8]. The following results refer to particles of aspect ratio 6: 1. The moment density MS = 350 emu/ cm3, the crystalline anisotropy of the core is K,,,,, = 4.6 x lo4 erg/cm3, the crystalline anisotropy of the shell is varied in the range 4.6 X lo4 erg/cm3 < Kshe,, I 1 X 10h erg/cm3. As in the nucleation calculations the crystalline anisotropies of both the shell and the core are taken uniaxial, although the core anisotropy may be more complicated. Two values for the ratio S = 0,,,/2h,, are considered, namely 1.56 and 2.35. D,,, is the total particle width and is 630 and 949 A, respectively, if a value of A = 5 x lo-’ erg/cm is used for the exchange constant. If Aharoni’s result [32] for the best fit of his nucleation calculation is used, namely, A = 0.9 x lo-’ erg/cm, the corresponding particle widths are 268 and 403 A, respectively. In the computations the particle shell is represented by the outermost layer of cubic elements. In most of the calculations the ratio of total particle width to core width is given by t = 5/3. For comparison a few results are also presented for t = 7/5. As indicated in the previous section, it is possible to estimate the effects of thinner coatings from the energy-volume scaling relation obtained in linear nucleation theory. The equilibrium states of the coated particle are similar to the precursor, i.e., they are generalized curling configurations such as the flower state and the vortex state. The remanence and the switching field are increased as the crystalline anisotropy of

magnetization

processes

the shell region becomes larger. Of course this is to be expected since the increased shell anisotropy leads to a more uniform magnetization, in particular at the ends of the particle, where the demagnetizing effects are most pronounced in the undoped particle. As has been pointed out in ref. [8], an increase of the shell anisotropy may also lead to a transition from the vortex state to the flower state, since the relative decrease of the total micromagnetic energy via magnetostaticly driven vortex formation is limited by the higher shell anisotropy. Moreover, at high values of the shell anisotropy the application of a reverse field may lead to a core reversed state, i.e. a state, where the magnetization of the shell remains unreversed (except at the particle ends), while the core magnetization irreversibly switches. In this case the resulting magnetization loops are distinguished by two switching fields, one for core reversal, and one for shell reversal. This is similar to loops of the two-component StonerWohlfarth model by Stavn and Morrish [lOO]. The magnitude of the jump of the average magnetization during core reversal depends on the ratio of core volume to shell volume and is small for thin coatings. It is also noted that for small total particle diameters typical of modern y-Fe,O, particles the core and the shell magnetization reverse at the same switching field. Of course, for given particle diameter, there is always a theoretical doping level of sufficient strength, to lead to the core reversed state. In ref. [8] the transient states during irreversible switching are obtained by numerical integration of the Landau-Lifshitz equations with infinite damping. For small enhancement of Kshs,, over K,,,, the transient states of the coated particles are similar to those of the uncoated particles (see section 9.3). For larger values of the shell anisotropy irreversible switching clearly starts in the core region. This is shown in fig. 13. The magnetization is projected onto longitudinal cross sections. The integer labels indicate the displacement from the longitudinal front face. Planes 1 and 5 are contained entirely in the high ansisotropy shell and are the front and the back face, respectively. Figs.

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Schabes / Micromagneric

processes

277

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irreversible switching of a cobalt modified y-Fe,O, particle D,,,/2h,, = 2.35. L&t = 0.20, erg/cm3, Happ = 2000 Oe, aspect ratio p = 6: 1. The magnetization is projected onto longitudinal cross sections, integer labels indicate the displacement from the front face, plane 1 and 5 are the front and back face, respectively, and are contained entirely in the high anisotropy shell. The sequence (a)-(d) is arranged in order of increasing time.

K,,,,, = 4.6 x lo4 erg/cm3,

during

Kshe,,= 1 x lo6

13a-d are arranged in order of increasing time. The irreversible processes start as a buckling of the core region (figs. 13a-b). Then portions of the core reverse. This involves formation of a transverse vortex in the core, although this is shown only in a coarse way at the resolution of fig. 13. In fig. 14 the formation of the transverse vortex is more obvious due to the increased resolution of a 7 x 7 transverse grid. The transverse core vortices subsequently propagate towards the particle center, leading to an intermediate core reversed

state. For large particle diameters, e.g., S = 2.35, this state is stable. For smaller particle size, S = 1.56, the core reversed state is not stable and shell reversal follows suit (for KShe,, I 1 X lo6 erg/cm3). This involves a flipping process of the surface magnetization starting at the particle ends. In fig. 15 the switching field is plotted as a function of doping level of the shell region. For small values of the enhancement of the shell anisotropy over the core anisotropy the rise in the switching field is quite pronounced and almost a

M. E. Schabes / Micromagnetic

278

theory of non-uniform

magnetization

processes

t I

I i : t 1 I I I t I I I t I I f t I I t I

Fig. 14. Transverse vortex formation during initial stages irreversible switching of a cobalt modified y-Fe,O, particle D,,,,/2X,, = 2.35. is DIhe,,/D,o, = 0.14, K,,,, = 4.6 X 10“ erg/cm’, Kahell = 1 X lo6 erg/cm3, H,,, = 1700 De, aspect ratio p = 6 : 1. The magnetization projected onto longitudinal cross sections, integer labels indicate the displacement from the front face. plane 1 and 7 are the front and back face, respectively, and are contained entirely in the high anisotropy shell.

linear function of Kshe,,. At a certain value of the doping level the enhancement of the switching field saturates. As indicated above, for large particle diameter and large doping level of the shell there may be two switching fields, one for core reversal and one I ,

’ EC=*.,5 P shell reversal

/’ /’ I’

-

0

5

K shell

(lo5

10

15

erdcm3)

Fig. 15. Switching field as a function of shell anisotropy for = 1.56 and S = 2.35, respectively. Dshe,, /D,,, = S ==&, /2L. 0.20, K,,,,, = 4.6 x lo4 erg/cm’, aspect ratio p = 6 : 1.

for shell reversal. This is shown by the dashed line in fig. 15 for S = 2.35. The existence of two separate switching fields is explained by the finite core-shell coupling. In ref. [8] it has been emphasized, that the core-shell coupling is the key element in understanding also the saturation effect of the switching field. The arguments are analogous to our discussion of the saturation effect of the nucleation field and are further underlined by noting that the switching field for the shell magnetization does not saturate, since it is approximately independent of the core-shell coupling in the limit of large shell anisotropy. This is quite in contrast to the switching field of the core magnetization, which is to a large extent determined by the core-shell coupling. For the smaller particle with S = 1.56 core and shell reversal occur at the same switching field for the specified range of values for the shell anisotropy. The switching field for the smaller particle saturates, since the particle diameter is sufficiently large such as to allow the initial irreversible processes to be dominated by the magnetic prop-

M. E. Schabes / Micromagnetic

theory of non-uniform

magnetization

processes

279

erties of the low anisotropy core. Once the nonuniform magnetization processes have proceeded to a certain stage, the shell magnetization is pulled along into the irreversible switching process. In ref. [8] the slopes of the approximately linear portions of the curves of fig. 15 have been compared with the increase of the nucleation field of a homogeneous particle with a volume averaged crystalline anisotropy given by 0 (28)

are the volumes of the core where K,,, and Shell and shell regions, respectively. This yields an increase of the nucleation field by 2K,,,/M,. If it is assumed, for the moment, that the crystalline anisotropy gives also approximately an additive term to the switching field, we obtain (29) For the particle diameters of fig. 15 this yields d HJdK,,,,,

and shell thicknessses

= 0.38 (kOe cm3/10s

erg).

(30)

Computation of the slope of the linear portion of the curve for S = 1.56 yields a value of 0.3 (kOe cm3/105 erg). The somewhat smaller increase is explained by the non-uniform magnetization processes in the core region. The increase of the switching field in absolute terms is significant, if the large crystalline anisotropy of cobalt ferrite is considered. If the approximate energy-volume scaling relation for the nucleation field (see section 10.1) is extended to the switching field, Schabes and Bertram [8] have shown, that a coating of cobalt ferrite as thin as 6 A yields an increase of the switching field of about 300 Oe as compared to the precursor particle. In the above estimate a value of K = 2.9 X lo6 erg/cm3 was used for the crystalline anisotropy of cobalt ferrite [loll and the exchange constant was taken to be A = 5 x 10 ~ ’ erg/cm. This remarkable result suggests that approximately one monolayer of cobalt ferrite is sufficient to explain the experimentally observed enhance-

I

1

I

30

60

90

19 (“) Fig. 16. Angular dependence of the switching field for S= D,,,/2X,, =1.56 and S = 2.35, respectively. D,hell/D,ot = 0.20, aspect ratio p = 6: 1. The dashed KC,,, = 4.6~ lo4 erg/cm3, line refers to the core reversal in the larger particle.

ment of the coercivity. This is in agreement with recent experimental evidence [43]. If the applied field is tilted at an appreciable angle with respect to the easy axis, the magnetization processes in the core are generally more uniform than compared to the case where the applied field is along the long particle axis [8]. In particular, the core carries out an almost uniform rotation during the initial stages of core reversal [S]. of the core For A,, = 45 O, e.g., the magnetization rotates first to an angle almost perpendicular to the easy axis. Then magnetization reversal proceeds from the particle ends by a process similar to the flipping mode. The uniformity of the magnetization of the transient states due to large shell anisotropy leads to an angular dependence which has some features of Stoner-Wohlfarth behavior. This is shown in fig. 16. For both particle sizes the switching field decreases slightly at about A,, = 45 O, and increases for larger angles. For very large angles the switching field exceeds the switching field obtained for the case, where the applied field is parallel to the long particle axis. This is due to the fact that for very large angles the irreversible switching of the core and shell magnetization approximately coincide. The switching field in this case is determined mainly by the crystalline anisotropy of the shell magnetization, which carries out an almost uniform rotation. Accordingly the switching field is quite large in magnitude.

280

M. E. Schahes / Micromagnetic

theory of non-uniform

11. Elongated iron particles In spite of the technological importance as high performance media there has been little theoretical work on the micromagnetics of iron particles until recently [9,10]. Iron particles must be passivated in order to be chemically stable. Unpassivated iron particles of the sizes useful for magnetic recording are pyrolithic. The passivation process creates a protective oxide layer on the surface of the particle [38]. The thickness of the oxide layer can be considerable (e.g. 10% of the particle radius). From a theoretical point of view the passivation layer turns the micromagnetic problem into an inhomogeneous problem, since the passivation layer has markedly different magnetic properties as compared to the particle core. Similar to the case of the cobalt modified y-Fe,O, particles, the magnetization processes are modeled by distinguishing a core region and a particle shell, which is represented by the outermost layer of cubic elements. However, the case of passivated iron particles is in many respects different from the case of cobalt modified y-Fe,O, particles. First, we note that the magnetic moment density of the iron core is very high (1710 emu/cm3) and yields a large shape anisotropy even for modest aspect ratio. In the calculations we neglect therefore the crystalline anisotropy. The magnetic moment density of the particle shell is greatly reduced compared to that of the core. The composition of the oxide layer is not known exactly. It may contain phases of both Fe,O, and Fe,O, as well as amorphous phases. For lack of a better value the magnetization density of the passivation layer is taken to be 480 emu/cm”, which corresponds to magnetite and should be a reasonable estimate. The composite morphology of the particle leads in this case to a non-local modification of the magnetostatic interaction energy density. This is qualitatively different from the case of cobaltmodification where the essential effect is a local change (confined to the shell region) of the crystalline anisotropy energy density. Of course, it is to be expected that also the exchange coupling within the particle shell and the particle core and the shell-core exchange coupling

magnetization

processes

all have their distinct values. However, this additional complication will not be investigated in the following and the exchange coupling is chosen to be A = 1 x lop6 erg/cm throughout the particle. A refinement of the results, which utilizes smaller values for the exchange constant within the particle shell and for the shell-core exchange coupling is left to future research. A similar remark applies also to the modeling of thinner oxide thicknesses. 11.1. Mathematical

model

In the present model the passivation layer leads to a local modification of the Zeeman energy in the shell region requiring the use of two coupling parameters with regard to the applied field. Hence the normalized Zeeman energy is written as

where

magnetiMshrll and MC,,,are the saturation zation of the shell and the core region, respectively. The other symbols have their usual meaning. The treatment of the magnetostatic interaction energy requires a more elaborate treatment, since ~ even though the modification of the magnetization density is confined to the passivation layer the magnetostatic interaction energy density is modified (non-locally) throughout the entire particle. Let Lhrll and CC,,, denote sums over the elements of the core and shell regions, respectively. The normalized total magnetostatic interaction energy is written as

with

shell

core

M. E. Schabes / Micromagneric theory of non-uniform magnetization processes

where z \s = c,, =

Mstst,e,,d2 A )

ecs=

M

Ms-she,,d ’ s-c0reA >

(34)

Mskred ’ ’

A

the quantity 9 is the normalized pair interaction energy density between cubic elements [681. 11.2. Micromagnetic properties unpassivated particles

of passivated

and

While the cobalt modification of y-Fe,O, particles leads to new topological features, such as the core reversed state, the passivation layer generally does not lead to significant differences in the magnetization configurations: Depending on the particle size and the applied field, the flower and the vortex state are the equilibrium configurations. The switching field is plotted in fig. 17 for both the homogeneous and passivated particle for aspects ratios in the range of 3 : 1 to 8 : 1, and the particle width in the range of 212 to 354 A. This choice is motivated by the typical particle sizes as taken from TEM pictures of iron particles [38]. The value of the particle width, as plotted in fig. 17, refers to the total particle width including the oxide layer, which is 20% of the total particle width. This is larger than usually obtained experi-

I

2500

t

F z

I

I

I

I

I

D=285 +...3----------+

I

1

lOOO-

: fn

50001

0

I

I

I

Particle

I

2000

1000 Length

I

I

I

3000

(Angstrom)

Fig. 17. Switching field for passivated and unpassivated iron particles as a function of particle length. The label D refers to the total particle width and is quoted in A.

281

mentally in iron particles. The modeling of thinner oxide layers requires finer discretization grids. It is seen from fig. 17 that the switching field is higher for the passivated particles. There are several reasons for the increased switching field. First, it is noted that the aspect ratio of the core of the passivated particle is 9.3, which is larger than the aspect ratio 6 of the (total) homogeneous particle. The effective magnetic aspect ratio is in this case larger than the geometrical aspect ratio of the particle boundary (which encomprises both the core and the shell), since the metallic particle core magnetostatically dominates the passivated particle. Moreover, the demagnetization effects are reduced at the ends of the passivated particle, since the discontinuity of the magnetization in going from the oxide to the adjacent vacuum is reduced by more than a factor of three as compared to the case of a metal-vacuum interface. Or, stated somewhat differently, a uniformly magnetized iron particle has a large magnetostatic surface charge density, whereas a passivated iron particle redistributes the magnetostatic charges into charges at the shell-core interface and charges at the shellvacuum interface. Hence, the resulting significant reduction of the magnetostatic charge density at the particle ends reduces the overall demagnetization effects. Furthermore, the torque exerted by the applied field onto the passivation layer is reduced by a as compared to the unpasfactor of Mshell/Mcore sivated particle. For the material parameters this is quite significant and results in a reduction of the torques by a factor of 0.28. Since the volume near the particle ends is the most crucial region for initiating irreversible magnetization processes, the reduction in torque carries special significance. Also the large magnetostatic interaction field with sources in the top or bottom section of the metallic core further stabilizes the magnetization of the oxide at the particle ends. It is emphasized, that a more detailed study of the passivation layer should include also a reduction of the exchange coupling within the particle shell. This is expected to lead to a smaller difference between the switching fields of the passivated and unpassivated particle, respectively. since

2x2

M. E. Schabes / Micromagnettc

theory

the magnetization of the particle shell will be less tightly coupled. Therefore the reduced torques from the applied field carry more weight and more readily lead to non-uniform magnetization within the particle shell, hence make the entire magnetization of the particle switch at a lower magnitude of the applied field.

12. Barium ferrite particles The micromagnetics of barium ferrite has attracted significant attention, since a large value of the coercivity makes barium ferrite suitable for high density magnetic recording. By appropriate doping the coercivity can be adjusted to be comparable to that of metal particles. Questions of corrosion resistance are less of an issue for barrium ferrite. However, the saturation magnetization of the particles is relatively low (= 340 emu/cm’) and similar to that of y-Fe,O, particles. Barium ferrite particles are small platelets with diameters of about 200 to 5000 A thick and ratios of thickness to diameter in the range of approximately 0.1 to 0.4 [35]. BaFe,,O,, is unusual also in many other respects. In particular, its large crystalline anisotropy along the hexagonal c-axis makes the direction perpendicular to the plane of the platelets the preferred direction for the magnetization. Therefore BaFe,,O,, can be used as a perpendicular recording medium. Uesaka, Nakatani and Hayashi [14] have shown from first principles by numerical integration of the Landau-Lifshitz-Gilbert equations, that the irreversible switching in thin barium ferrite platelets of thickness to diameter ratios of 0.1 and 0.2, respectively, starts in the particle center of the platelet. Irreversible switching begins (via vortex formation) at the particle circumference only, if the platelets are relatively thick (thickness to diameter ratios of 0.33). These results put in question the plausibility of which had been introduced “domain nucleation”, by Victora in ref. [92]. There, after it has been shown that the discrepancy between the large crystalline anisotropy field and the relatively lower observed coercivity cannot be explained by either

of non-umform

mugnetization

processes

thermal fluctuations, nor fluctuations in doping level, nor curling, a mechanism of “domain nucleation” is invoked, where a certain fraction of the particle magnetization near one of the corners of the hexagonal platelet is reversed “by hand”. This is far from being self-consistent and requires large amounts of energy. It is also not clear, why, if at all, such a domain should grow from the corner rather than from the center of the particle, where the demagnetizing fields are strongest in the case of a thin platelet. The proper choice of the exchange coupling constant in barium ferrite may be important for a micromagnetic understanding of the reversal processes. Uesaka et al. use a value of 1 x lo-’ erg/cm, whereas Victora’s value for the exchange coupling is 5 X lo-’ erg/cm. The results for the switching mechanism and for the switching field in ref. [14] seem more reasonable and could indicate that a low value of the exchange constant is indeed characteristic of barium ferrite. If Victora’s value of the exchange constant is scaled (see section 14.7) to that used in ref. [14], Victora’s particle has a diameter and thickness of 335 and 45 A, respectively, which would explain some of the difficulties of ref. [92]. As indicated in ref. [14], detailed micromagnetic investigations as to the effect of the exchange coupling in barium ferrite are currently being undertaken by Uesaka et al.

13. Inhomogeneous

applied fields

In the preceding sections one of the traditional assumptions of classical micromagnetics was maintained, namely, it was assumed that the particle is subjected to a uniform applied field. However, in many important situations the applied field is far from uniform. In particular, the field profiles of modern magnetic recording heads have large field gradients. An indication of this is obtained by noting that the gap length can be as small as 1200-3000 A in high density magnetic recording systems. This is of the same order of magnitude as the length of typical oxide or metal particles. It is therefore hardly possible to justify the classical micromagnetic assumption of a ho-

M. E. Schabes / Micromagnetic

theory of non-uniform

mogeneous applied field in these cases. Nevertheless, until recently [7], this aspect has been consistently ignored in all micromagnetic calculations. Schabes and Bertram showed [7] that the consequences of an inhomogeneous applied field can be described in a relatively simple fashion. In fact, many of the results obtained for homogenous applied fields can be generalized. This applies certainly to the class of fields, where the magnitude of the applied field is large at one particle end and decreases (monotonically) over the length of the particle. In many regards, this is one of the most important inhomogenous field profiles, and has been examined in detail in ref. [7], where without loss of generality the field profile was taken to be a Karlqvist head field. In the following we will review some of the results of ref. [7]. For the details of the geometrical arrangement of the particle with respect to the gap of the head we refer to ref. [7]. The inhomogeneity of the field applied to a particle may not only be due to head field gradients, but may also arise from the interaction field due to other particles in the medium. The results of ref. [7] can be used for some of the possible interaction mechanisms between particles, e.g., for the head-on arrangement of particles in a longitudinally oriented medium. Some effects of interaction fields between particles have been described in ref. [28]. A somewhat different situation arises, when two particles are perpendicular to each other with one particle end pointing towards the central section of the other particle. This case is similar to the presence of a defect in the particle morphology in the central section of the particle (see section 2.3). The calculations of a particle moving towards the gap at a low flying height show, that the magnetization processes for irreversible switching are initiated at the leading particle end. At the trailing particle end the magnetization may initially remain essentially in its remanent state, in particular, for the aspect ratio is large (e.g. 6 : 1) and the field gradients are high. This is explained by the fact that the particle length is quite larger than the exchange length, hence, the particle ends are only coupled by magnetostatic interaction,

magnetization

. ..a. . . . . . . . . _ .27 .*... -,..I

I,,.. ,.,.. . . .._,I ._.,,

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,,,.. , ,

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. , .*28..

..,, ..,I

, ,29

e-‘-f, ,e.q, /I J\.-,

* , ,d50

, ,.__

,,,_. II,_. . . ..,, ..,,

.

.

.21

283

processes

_.I,, ..,,, . _.. ,,\.. %,.._

I -15

_..,* I _ . .\ .u I I

> .

I,I.a , I._ . . . . --,,I

_. I I

.23 I

.24

/ ,a---. , ,-x., \ iI + t9 ,,-,A 1 -WV, ,

:::::,5 ,,._~,.__

{,CLL , ,c.t (,.,,, \ ,-,f r--r,

__.\. ,_.I, , , , , ,,... ,%..?

(,rr* ,,... , , . I,.,, *s*.+

,.~.. .-.. I I I >,..,

, 17

I .18

\, . . .

. . _ . .

. . .

, a , . _ .5 , . . . , .

_

*.

6,C.I 0

.

,,,

:::::a .,.,, *.. J ,.--, i-.x $ , . .w-f -..r,t

._..I I /. . . . . I I-\\ . . I. - ,2 ,,, ..I., h--Y/

. . . . .

--//

. .

,

15 t

+ +6

t t

Fig. 18. Equilibrium magnetization of a particle of aspect ratio 6: 1 with 16% of the volume submerged in a longitudinal applied field with magnitude 850 Oe. D/2X,, = 2.3. The magnetization is projected onto the transverse cross section. Integer labels indicate displacement from the bottom end of the particle, bottom 1, top end = 30.

which cannot significantly disturb the local equilibrium between exchange and magnetostatic interaction at the other particle end. For very large gradients this may lead to the presence of several topologies within the particle. This is shown in fig. 18, where the magnetization of a particle of aspect ratio 6 : 1 and D/2X,, = 2.3 is computed for the limiting case, where only 16% of the particle volume near the “leading particle end” is submerged in a longitudinal applied field. (The transverse components of the head field are neglected for the moment, since it turns out that for the particle the main features of the switching mechanism are determined by the longitudinal component.) In an applied field of 850 Oe the magnetization is reversed near the leading particle end. In fig. 18 the magnetization is projected onto transverse cross sections with the integer labels indicating the displacement form the leading par-

284

M. E. Schabes / Micromagneirc

theory

title end. Label 1 indicates the bottom plane, label 30 is the top plane of the particle. It is evident, that the reversed magnetization near the leading particle end is in a flower configuration, while the unreversed magnetization near the trailing particle end is in a. vortex configuration. The reversed flower configuration is separated from the unreversed magnetization by a “wall vortex”. This is a vortex, whose magnetization is almost entirely contained in the transverse cross section. Close to the central wall vortex the longitudinal magnetization component averaged over the transverse cross section is an odd function with respect to the plane of the central vortex, while the helicity is an even function about the central wall vortex. An increase of the applied field in the submerged region moves the position of the wall between reversed and unreversed magnetization towards the trailing particle end. The wall reaches its asymptotic position in an applied field of about 950 Oe. Even rather large field magnitudes do not move the wall across the half-length of the particle and therefore cannot switch the entire magnetization. This aspect has also been verified by carrying out dynamical calculations on partially submerged exchange coupled chain of spheres (see appendix B). It is also noted that the (linear) nucleation field of a partially submerged chain is generally larger than the nucleation field of the totally submerged chain. In the case of the partially submerged chain the nucleation field is obtained from a generalized Toeplitz determinant. This leads to a linear equation for the nucleation field for the case, where the first sphere is submerged in the applied field, and to a quadratic equation for the case, where the first and second spheres are submerged in the applied field. The enhancement of the nucleation field is due to the fact that the part of the chain, which is not coupled to the applied field, provides interaction fields, which stabilize the magnetization of the submerged part of the chain. It must be emphasized, that the notions “wall” or “domain” do not carry their conventional meaning in this context. They should be called “forced walls” or “forced domains”, since they are entirely dependent on the applied field. In

of non-uniformmagnetization

processes

particular, if the applied field is removed, the particle returns to its original remanent state. Again it is useful to consider the simple case of an exchange coupled chain of spheres, where it is easy to show that the interaction fields from the unreversed magnetization lead to the reversibility of forced domain formation. If a larger portion (e.g. 32%) of the particle of fig. 18 is submerged in an applied field the total particle magnetization reverses, since the wall extends significantly beyond the mid-plane of the particle. Hence the magnetization processes, which are initiated at the high field region of the particle propagate all the way towards the trailing particle end. In ref. [7] it is shown that typical recording particles generally do not contain fully developed forced 180 o walls even at rather low flying heights, where the gradients of the head field are most pronounced. Instead, certain magnetization configurations similar to the transient states (e.g. a transverse vortex structure near the leading end) in a uniform applied field become stabilized by the inhomogeneous applied field. In the case of low exchange coupling (or large particle size) irreversible switching of the entire magnetization occurs at a location, where at the leading end the average magnitude of the head field component parallel to the long particle axis is approximately equal to the uniform switching field. If the exchange coupling is high (or the particle size is small), irreversible switching occurs, when the volume average of the magnitude of the applied field is approximately equal to the uniform switching field. In perpendicular media, these results can be applied by simply rotating the long particle axis by 90 ’ and, in the case of low exchange coupling, considering the head field component parallel to the vertical column axis. In the case of columnar grains with large diameter and length comparable to the gap length the magnetization of the entire column switches irreversibly, when the magnitude of the magnetic component perpendicular to the medium surface is approximately equal to the uniform switching field. This result is also corroborated by experimental evidence [102]. The transient states during irreversible switch-

hf. E. Schabes / Micromagnetic theory of non-uniform magnerization processes

ing are similar to those obtained in a uniform field, except, that the irreversible processes are initiated only at one particle end (the high field end), instead of at both ends. The irreversible switching proceeds from the leading particle end to the trailing particle end, as has been shown explicitly in ref. [7].

14. Conclusions Micromagnetics has expanded its scope considerably and dynamical modelling of magnetization processes in non-uniformly magnetized particles has become possible from first principles. Careful consideration of reversibility and irreversibility and distinction between equilibrium and transient states are required. As the field of the micromagnetic theory of non-uniform magnetization configurations continues to mature, a more exhaustive investigation of the space of magnetization states is feasible. A systematic understanding of the dynamical features will be achieved by considering the magnetization field of a micromagnetic particle as a general dissipative dynamical system, which is to be investigated by the methods of modern non-linear science.

Acknowledgements The author wishes to express his gratitude to the Center for Magnetic Recording Research of the University of California, which provided support to the author in the years 1984-1989. In particular, we are grateful to Professor H. Neal Bertram and Professor John C. Mallinson.

Appendices The purpose of the following two appendices is to present an almost back-of-the-envelope type of model (or let’s say a model whose CPU requirements are very modest). Such a stripped down model can provide qualitative guidance for more complicated situations and should be consulted before a large scale calculation is undertaken on a

285

supercomputer. As additional remark it is noted that for small values of the exchange coupling within the spheres the curling mode is the most favorable nuleation mode [29]. As has been shown in section 9, the character of the non-linear reversal processes in particles with larger diameter is nevertheless sequential with vortex formation, at the particle ends and formation of transverse vortices, which propagate towards the center of the particle. It is illustrative to demonstrate these processes at a simpler level.

Appendix A. Magnetic chain in a uniform applied field Consider a chain of six spheres placed in an applied field parallel to the axis of the chain. The normalized micromagnetic energy density of the chain is given by %t = -C~l,aapp.,.ar-:Cat3-~7Ca,*a,+, I I +3

&,, i +.I

I (A.1)

cosines of the where a,, aapp, are the directional magnetization ‘and the applied field, respectively. e,, = Happ,,/Ho with H,, = 2K/M,, where K is the uniaxial anisotropy constant; cz = A/KD”, with A the exchange coupling between spheres (within each sphere the magnetization is taken uniform), and D the diameter of the spheres; ci = 7rMs2/12K; w,, are the pairwise dipole interaction energies of unit dipoles with the distance between dipoles measured in units of D. The fact that the chain consists of six spheres, enters in the computation of the magnetostatic interaction field, where for simplicity we take dipolar fields. For conceptual purposes of handling the exchange interaction we could equivalently consider a chain of cubes, the difference being an improved accuracy (on the order of 10%) of the magnetostatic interaction fields for nearest neighbors. This difference is not important for the following demonstration of the qualitative features of the irreversible switching mechanism in elongated particles.

M.E. Schabes / Micromagnetrc

286 l.O--

; q-+,-_

:

,,

1

’

‘\,_’ 0.5

I

r,=3.25 E2= 1 r,=0.6

-

E” 0.0 -

a

-0.5 . -1.0

-

!

I

I time

,’

0.5 /

,_

,

I

[a

u]

_-

the chain. The first and sixth sphere, the second and fifth, and finally the third and fourth sphere, flip in that sequence, with each pair rotating in the same direction, as can be seen from the plot of the transverse component. If the exchange coupling between the spheres is decreased, the essential character of the switching process is preserved, i.e., the switching starts at the ends of the particle and propagates towards the center of the particle. However, it is energetically favorable (magnetostatic interaction dominates) that adjacent spheres rotate in opposite directions (except for the two center spheres, which rotate in the same direction). This is similar in character to the fanning mode [103].

&2’1 r,=O 6

2.5

b

I

00

processes

El=325

0.0

-101

_

1.2

0.8

0.4

0.0

i

._._ I,~. I

magnetization

I

3.4

1.6

-

theory of non-uniform

I

I

0.4

0.8 time

[a

I

I

1.2

u]

Fig. 19. Time evolution of the magnetization of a chain of six spheres in a homogeneous applied field by integrating the Landau-Lifshitz equations with infinite damping. The magnetization components of all six spheres are plotted with integer labels indicating the number of the sphere. (a) longitudinal components; (b) transverse components.

Figs. 19a, b. show the irreversible switching by the flipping process. The transient states have been computed by solving the Landau-Lifshitz equations in the limit of infinite damping. The magnetization components of all six spheres are plotted and the integer labels attached to each curve indicate the sphere numbers, e.g., the label “1,6” refers to the magnetization of the first and the sixth spheres at the ends of the chain. The evolution of the longitudinal component is depicted in fig. 19a, the transverse component is shown in fig. 19b. During irreversible switching the spheres reverse in a sequence, which starts out at the ends of

0

0

3

6

9

time

[a

time

[a

3

12

u]

6

9 u]

Fig. 20. Time evolution of the magnetization of a chain of six spheres in an inhomogeneous applied field by integrating the Landau-Lifshitz equations with infinite damping. The longitudinal magnetization components of all six spheres are plotted with integer labels indicating the number of the sphere. (a) First sphere is submersed in the applied field; (b) first and second spheres are submersed in the applied field.

M. E. Schabes / Micromagneiic

theor)? of non-uniform

Appendix B. Magnetic chain in an inhomogeneous applied field

1161D.R. Fredkin [I71

The sequential character is particularly important in an inhomogeneous applied field, where it can lead to forced intra-particle domain states. The time evolution of the magnetization component parallel to the axis of a chain of six spheres is shown in fig. 20. Only part of the chain is coupled to the applied field. The spheres are labeled in the same way as in appendix A. Fig. 20a shows the case, where the first sphere is submerged in the applied field, fig. 20b shows the case, where the first two spheres are submerged in the applied field. It is noted that the remainder of the chain does not reverse even for large values of the field applied to the submerged part. This is due to the finiteness of the coupling between the spheres.

U81 u91 WI

WI PI ~231 ~241 ~251 1261 ~271

References [l] R. Wood, IEEE Spectrum [2]

[3] [4] [5] [6] [7] [8] [9] [lo] [II] [12] [13] [14] [15]

27 (1990) 32. C. Tsang, M. Chen. T. Yogi and K. Ju, presented at the 1990 Intermag Conference, Brighton, UK. IEEE Trans. Magn. MAG-26 (1990) in print. W.F. Brown Jr., Magnetostatic Principles in Ferromagnetism (North-Holland, Amsterdam, 1962). W.F. Brown Jr.. Micromagnetics (Wiley-Interscience. New York, 1963). M.E. Schabes and H. Neal Bertram, J. Appl. Phys. 64 (1988) 1347. M.E. Schabes and H. Neal Bertram. J. Appl. Phys. 64 (1988) 5832. M.E. Schabes and H. Neal Bertram. IEEE Trans. Magn. MAG-25 (1989) 3662. M.E. Schabes and H. Neal Bertram, J. Appl. Phys. 67 (1990) 5149. M.E. Schabes, submitted to the 35-th MMM Conf., San Diego (1990). M.E. Schabes, Ph.D. thesis in Physics, the University of California (1990) USA. A. Lyberatos and R.W. Chantrell. IEEE Trans. Magn. MAG-26 (1990) in print. Y. Nakatani. Y. Uesaka and N. Hayashi. Jpn. J. Appl. Phys. (1989) 2485. Y. Uesaka, Y. Nakatani and N. Hayashi. J. Appl. Phys. 67 (1990) 5143. Y. Uesaka, Y. Nakatani and N. Hayashi. J. Appl. Phys. 67 (1990) 5146. D.R. Fredkin and T.R. Koehler, IEEE Trans. Magn. MAG-23 (1987) 3385.

magnetization

P81 v91 [301 [311 ~321 [331 [341 1351

1361 [371 [381 1391 [401 [411 [421 [431 WI [451 [461 [471 [481

processes

287

and T.R. Koehler, IEEE Trans. Magn. MAG-24 (1988) 2362. D.R. Fredkin and T.R. Koehler, J. Appl. Phys. 63 (1988) 3179. D.R. Fredkin and T.R. Koehler. IEEE Trans. Magn. MAG-25 (1989) 3473. D.R. Fredkin and T.R. Koehler. J. Appl. Phys. 67 (1990) 5544. D.R. Fredkin and T.R. Koehler, presented at the 1990 Intermag Conf., Brighton, UK, IEEE Trans. Magn. MAG-26 (1990) in print. E. Della Torre, IEEE Trans. Magn. MAG-15 (1979) 1225. E. Della Torre, IEEE Trans. Magn. MAG-21 (1985) 1423. E. Della Torre. IEEE Trans. Magn. MAG-22 (1986) 484. Y.D. Yan and E. Della Torre, J. de Phys. 49 (1988) C8-1813. Y.D. Yan and E. Della Torre. IEEE Trans. Magn. MAG-24 (1988) 2368. Y.D. Yan and E. Della Torre. IEEE Trans. Magn. MAG-25 (1989) 3464. Y.D. Yan and E. Della Torre. J. Appl. Phys. 66 (1989) 320. Y.D. Yan and E. Della Torre. J. Appl. Phys. 67 (1990) 5370. A. Aharoni. Phys. Stat. Sol. 16 (1966) 3. A. Aharoni. IEEE Trans. Magn. MAG-22 (1986) 478. A. Aharoni, J. Appl. Phys. 62 (1987) 2576. A. Aharoni, J. Appl. Phys. 63 (1988) 4605. J.E. Knowles. J. Magn. Magn. Mat. 61 (1986) 121. G. Bate, in: Ferromagnetic Materials. vol. 2. ed. E.P. Wohlfarth (North-Holland. Amsterdam. 1980) p. 381. E. Koester and T.C. Arnoldussen, in: Magnetic Recording, vol. 1. eds. C.D. Mee and E.D. Daniel (McGraw-Hill. New York. 1987) p. 99. H.N. Bertram, IEEE Trans. Magn. MAG-22 (1986) 460. M.H. Azarian and P.N. Kuin. IEEE Trans. Magn. MAG-23 (1987) 192. T. Miyahara and K. Kawakami. IEEE Trans. Magn. MAG-23 (1987) 2877. A.H. Morrish. The Principles of Magnetism (Krieger. Malabar, 1983) p. 321. K. Bridger, J. Watts. M. Tadros. G. Xiao. S.H. Liou and C.L. Chien, J. Appl. Phys. 61 (1987) 3323. 0. Kubo, T. Ido. H. Yokoyama and Y. Koike, J. Appl. Phys. 57 (1985) 4280. A.E. Berkowitz, IEEE Trans. Magn. MAG-22 (1986) 466. A.E. Berkowitz, F.E. Parker, E. Hall and G. Podolsky, IEEE Trans. Magn. MAG-25 (1989) 3662. L. Wenbo, IEEE Trans. Magn. MAG-23 (1987) 89. W.F. Brown. Jr., Phys. Rev. 105 (1957) 1198. W.F. Brown. Jr., J. Appl. Phys. 33 (1962) 3022. W.F. Brown. Jr., Phys. Rev. 130 (1963) 1677. D. Rode. H.N. Bertram and D.R. Fredkin. IEEE Trans. Magn. MAG-23 (1987) 2244.

288

M.E. Schabes / Micromagnetic

theory of non-uniform

[49] A. Lyberatos, R.W. Chantrell and A. Hoare, IEEE Trans. Magn. MAG-26 (1990) 222. [50] S.B. Oseroff, D. Franks, V.M. Tobin and S. Schultz, IEEE Trans. Magn. MAG-23 (1987) 2871. [51] R. Victora, Phys. Rev. Lett. 63 (1989) 457. [52] W. Steck, J. de Phys. 46 (1985) C6-6. [53] W.C. Elmore, Phys. Rev. 54 (1938) 1092. [54] P.C. Scholten, IEEE Trans. Magn. MAG-11 (1975) 1400. [55] A.H. Morrish and S.P. Yu, Phys. Rev. 102 (1956) 670. [56] J.E. Knowles, IEEE Trans. Magn. MAG-1 (1980) 62. [57] J. Smyth, S. Schultz, D. Kern, H. Schmid and D. Yee, J. Appl. Phys. 63 (1988) 4237. [58] A. Tonomura. Rev. Mod. Phys. 59 (1987) 639. [59] S. Hasegawa, T. Kawasaki, J. Endo and A. Tonomura, J. Appl. Phys. 65 (1989) 2000. [60] M. Tsutsumi, F. Kugiya, S. Hasegawa and A. Tonomura, IEEE Trans. Magn. MAG-25 (1989) 3665. [61] J.N. Chapman, S. McVitie and I.R. McFadyen, Scan. Micr. Suppl. 1 (1987) 221. [62] S. McVitie and J.N. Chapman, IEEE Trans. Magn. MAG-24 (1988) 1778. [63] E. Ascher, Helv. Phys. Acta 39 (1966) 467. [64] L.D. Landau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon Press, Oxford. 1984) p. 138. [65] R.M. Bozorth, Phys. Rev. 50 (1936) 1076. [66] CR. Chang and D.R. Fredkin, J. Appl. Phys. 63 (1988) 3435. 1671 C. Kittel, Rev. Mod. Phys. 21 (1949) 541. [68] M.E. Schabes and A. Aharoni, IEEE Trans. Magn. MAG-23 (1987) 3882. 1691 A.E. LaBonte, J. Appl. Phys. 40 (1969) 2450. [70] W.F. Brown, Jr., J. Appl. Phys. 39 (1968) 993. [71] W.F. Brown, Jr., Phys. Rev. 124 (1961) 1479. [72] E.C. Stoner and E.P. Wohlfarth, Phil. Trans. Roy. Society (London) A240 (1948) 599. [73] W.F. Brown, Jr., Phys. Rev. 105 (1957) 1479. [74] W.F. Brown, Jr., Bull. Amer. Sot. (Ser. 2) 1 (1956) 323. (751 E.H. Frei, S. Shtrikman and D. Treves, Phys. Rev. 106 (1957) 446. [76] H.J. Richter, J. Appl. Phys. 65 (1989) 3597. [77] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381. [78] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. A 38 (1988) 364.

magnetization

processes

[79] X. Che and H. Suhl, Phys. Rev. Lett. 64 (1990) 1670. [80] L.D. Landau and E.M. Lifshitz, Phys. Z. Sowjet. 8 (1935) 153. [81] J.C. Mallinson, IEEE Trans. Magn. MAG-23 (1987) 2003. [82] A.S. Arrott, B. Heinrich and A. Aharoni, IEEE Trans. Magn. MAG-15 (1979) 1228. [83] H.A.M. van den Berg, J. AppI. Phys. 61 (1987) 4194. [84] P. Bryant and H. Suhl, Appl. Phys. Lett. 54 (1989) 78. [85] F. Pu, B. Li and B. Han, J. Magn. Magn. Mat. 59 (1986) 235. [86] M. Pinto, Phys. Rev. Lett. 59 (1987) 2798. [87] U. Hartmann, Phys. Rev. B37 (1988) 5299. [88] Proc. 34th Ann. Conf. on Magnetism and Magnetic Materials, The title of one session with several invited papers on micromagnetics was “Symposium: Computational Magnetics from First Principles”, see J. Appl. Phys. 67 (1990) p. xxiii. [89] D.M. Young and R.T. Gregory, A Survey of Numerical Mathematics, vol. II (Dover, New York, 1988) p. 1016. [90] M. Mansuripur and R. Giles, J. Appl. Phys. 67 (1990) 5555; see also Comput. Phys. 4 (1990) 291. 1911 J.G. Zhu and H.N. Bertram, J. Appl. Phys. 66 (1989) 1291. [92] R.H. Victora, J. Appl. Phys. 63 (1988) 3423. [93] C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice Hall, Englewood Cliffs, 1971). [94] S. Wolfram, Rev. Mod. Phys. 55 (1983) 601. [95] R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications (Addison-Wesley, London, 1983) p. 123. [96] Y. Nakatani and N. Hayashi, IEEE Trans. Magn. MAG24 (1988). [97] W. Williams and J. RoweII, Supercomput. Rev. 2 (1989) 43. [98] A.E. Berkowitz, J. Lahut, L. Levinson, I. Jacobs and D. Forester, AIP Conf. Proc. 29 (1976) 543. [99] Y. Imaoka, S. Umeki, Y. Kubota and Y. Tokuoka, IEEE Trans. Magn. MAG-14 (1978) 649. [loo] M.J. Stavn and A.H. Morrish, IEEE Trans. Magn. MAG-15 (1979) 1235. [loll M. Kishimoto, S. Kitaoka, H. Andoh, M. Amemiya and F. Hayama, IEEE Trans. Magn. MAG-17 (1981) 3029. [102] F. Jeffers, private communication. [103] IS. Jacobs and CF. Bean, Phys. Rev. 100 (1955) 1060.