IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-23, NO. 5, SEPTEMBER 1987 3385

NUMERICAL MICROMAGNETICS BY THE FINITE ELEMENT METHOD

D. R. Fredkin Department of Physics

and Center for Magnetic Recording Research

University of California, San Diego La Iolla, CA 92093

and

T. R. Koehler IBM Almaden Research Laboratory

San lose, CA 95120

ABSTRACT

We use the finite element method to find the equilibrium magnetization of a continuum in which exchange and anisotropy energy occur. Preliminary results for a variety of two-dimensional systems, including irregularly shaped particles and systems of two particles, indicate that a11 of the essential physics is included cor- rectly. The mode of switching depends on the size of the system relative to the natural domain wall thickness parameter J(CMs/HK) in the expected way. Large particles switch via intermediate states containing closure patterns.

INTRODUCTION

We report preliminary results on the calculation of magnetization and magnetic field distributions in two-dimensional ferromagnetic systems of arbitrary shape. The calculations are micromagnetic, i.e., we adopt a continuum view of the medium and treat the magnetization as a field, but we otherwise proceed from first principles using the finite element method to perform the cal- culations. We decompose the spatial region into triangles and use linear interpolating functions within each element. There are dis- tinct advantages to the use of this computational technique: Regions and boundaries of arbitrary shape are easily accomodated using standard mesh generating techniques and existing preprocessors, and the treatment of exchange energy, that is the source of non-trivial non-linearity in the problem, is facilitated, as will be discussed later. The finite element method provides an efficient method for calcu- lation of the demagnetizing field. There is no difficulty in accomodating spatial variations of the material parameters.

Classical analytical workl-5 is limited to special geometries. In an ellipsoidal geometry the concept of nucleation field is well de- fined and can be determined by solution of a linear problem. In the general geometries considered herein, the magnetization is always non-uniform and must be determined by solution of a non-linear problem.

LaBonte6, Della Torre7, and Bertram and Schabesa have done calculations similar in spirit to ours. They eliminate the demagnetizing field by expressing it as a sum of the magnetostatic fields of their cubic elements. Their handling of exchange is some- what different from ours. Our use of arbitrarily shaped triangular elements permits more flexibility in the treatment of complex ge- ometries. Preliminary tests suggest that the method reported here is less subject to convergence problems.

THEORY Micromagnetics starts with an expression for the energy den-

sity:

1 u = - I B I ~ - M . B + - C I V M I +gm,.

1 2

ST 2 (1)

where we have used unixial crystalline anisotropy

am& = - K(M.&~ (2)

and have neglected magnetostriction in the present work. Eq. (2), with e as a unit vector parallel to the crystalline axis, specifies the crystaIline anisotropy energy; shape anisotropy is a consequence of the demagnetization field that is included in Eq. (1). The magnetic field B in Eq. (1) is given by B = V x A. We seek a local minimum of rhe total energy E = Su&x with respect to A and M, subject to the constraint that M = MSm with I m { = 1. Our exchange constant C differs by a factor of A@ from that used in Ref. 6 or used in the al- ternative form. A = C/2 in Ref. 2.

Except for the treatment of exchange, we follow the standard procedures used in the lowest order, two-dimensional finite element method. The region to be studied is subdivided into triangular ele- ments, using a fine mesh in the magnetic region (HA!) and a coarse mesh in the external region (BE). The applied magnetic field is re- presented by suitable boundary conditions on the external boundary of (H,$ There are two sets of unknowns in the problem. The values of the potential at the nodes constitute one set. The potential is in- terpolated linearly in each element, and the magnetic field is then determined and is constant in each element. The magnetization is taken to be constant in each element and these values are the other set of unknowns.

The preceeding does not provide a formal prescription for the quantity VM. In order to compute the exchange energy for a pair of adjaccnt elements, m, a’ in Fig. 1, we have adapted the following ansatz: The magnetization of each element is associated with the centroids c, c ’, because the value of a function at the centroid is the quantity which enters into the lowest order expression for the nu- merical integral of a function over a triangular region. These values of M are interpolated linearly in the quadrilateral lc2c’; VM is a constant in the quadrilateral and the third term of Eq. (1) can be evaluated. A portion of the total magnetization in each triangle be- comes coupled via the exchange interaction to the magnetization in each of the neighboring triangles and the three interactions exhaust the total magnetization.

a‘ 3’

Figure 1. Construction for determining the. exchange interaction

The resulting discretized expression for the total energy is between elements m and a’.

where Greek subscripts denote triangular elements, A is the area of an element, the summations run over the entire region or SP, only as appropriate and the evaluation of the exchange energy led to the appearance of the matrix

where i s the distance between the two non-common vertices of adjacent triangles. The vector potential enters into Eq. (4) through the relationship

0018-9464/87/0900-3385’$01~0001987 IEEE

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where ri, r j and rk are the coordinates of the vertices of triangle a and the cyclic sequence i + j - k is in the counter-clockwise sense. Substitution of Eq. (5) into Eq. (3) leads to the usual finite element formulation except that M appears explicitly instead of implicitly in the reluctivity. This is a more suitable formulation for a hysteretic medium.

There are several numerical options for finding a local extremum of the non-quadratic quantity E . Based on the observa- tion that only the second term on the right hand side of this Eq. (3) involves both A and m, we have adopted an iterative approach simi- lar to that used in Ref. 9. In one part of each cycle of the iteration, E is minimized with respect to the Ai with the current rn values as parameters; in the other part, E is minimized with respect to the m, with the current A values as parameters. This contrasts with a more commonly used approach in which an iteration cycle involves linearization of the entire expression for E about the current values of A and m, followed by a solution of the linear problem to obtain new values of the variables for the next cycle.

The result of minimizing E with respect to -4, is

Csj/j = j,,,. (6)

where Si. = [ ( T i - rk) . (rk - r j ) /AQ + ( r , - r,).(ri - ri)/AD]/16n when i, j are the common vertices and k , 1 the two othef vertices of two adjacent triangles a, p; Si.(j # i) = 0 otherwise and S;, = - XjSLj. In the source term in Eq. (6f,jA,,! = hfsZ,m,.(rk - r , ) / 2 , where the sum is over all triangles with the vertex i; it is the discretized expression for the magnetic current j,, = V X m 9. The set of equations for all i values of Eq. (6) form a matrix problem which can be solved by standard methods. However, for each new set of m values, we have chosen to solve for each Ai only once using the relationship Ai = [ - Zj+iSv4j + jM, i ] /Ai i , which may be obtained from Eq, (6). The Ai are updated as they are computed. An acceleration factor can be introduced into this expression and the resulting operation is equivalent to one step in the SOR method. With this approach, the elements of Scan be computed and discarded as they are used which saves a significant amount of storage.

The m minimization is equivalent to minimizing the set of equations

j

E, = - Msma.Ha - HKhf3(m,-eh)2/2 (7 1

which are in the form of the orientational energy of a Stoner- Wohlfarth particle in an applied field except that the field in triangle a, H a = B, + MsCXpXap/A, , has contributions from the magnetiza- tion in adjacent triangles because of the exchange interaction. These equations can also be solved sequentially, using the most recent magnetization information to compute the exchange field.

The solution of the entire set of equations used in each iter- ation is straightforward mathematically. Furthermore a natural convergence criterion emerges: As each equation in encountered, it requires very little additional numerical work to check how closely it is satisfied by the results of the previous iteration. When all of the equations are satisfied t o better than an assigned convergence crite- rion, the calculation is assumed to have converged. We rypically set the criterion to 10-9 of the mean energy considered in each stcp. After extensive numerical experimentation, we have occasionally observed meaningful changes in the final results for convergencc factors two or three orders of magnitude larger that this, but never for factors several orders of magnitude smaller.

This work is new and the numerical proccdure described above has proven to be fast enough so that little numerical experimentation has been performed. We have briefly tested solving the cntire set of A equations simultaneously as a matrix equation and found it to ’ne considerably slower than the method we use. It is obvious that the entire set of Eqs. (6) and (7) do not have to be solved in any order. One could randomly choose unsolved equarions in the sweep through an iteration cycle and we intend to implement this approach in the future. The equation set is also structured so that a temperature could be introduced and the statistical mechanics of the system in- vestigated with the Monte Carlo method.

RESULTS

We present some preliminary results for two-dimensional cal- culations on systems of one or two particles. Becausc the magnetiz- ation and the demagnetization field are confined to lie in a plane, the

calculations do not correspond to an easily achieved experimental situation. They could apply to a thin film sandwiched between superconducting layers. Our purpose is to demonstrate that we have incorporated correctly the essential physical elements that determine the magnetization and field distributions in ferromagnetic systems. In particular, the effects of shape anisotropy appear correctly from our calculations and switching by domain wall formation occurs only for sufficiently small values of the exchange energy relative to the total anisotropy energy.

All of the figures shown in the following were selected from results obtained by applying a strong enough horizontal field B, , directed toward the right, to saturate the sample and then reversing the field in steps until the sample was saturated in the opposite di- rection. In all cases, the crystalline anisotropy axis was horizontal, Af, = IOOemu/cc and H. = 5000e. In the figure captions, the units of B, are Gauss and C = 2nC is given, where C is in cgs units ap- propriate to Eq. (1) with length units in 1.1. The maximum width of all the particles shown here was 2p. The CPU requirements for a calculation with 40 reversal steps were typically 10 minutes on an 1BM 370/3081.

Octagon We calculated the switching behaviour for a regular octagon which, for us, was a convenient approximation to a circle, a system with no shape anisotropy. With constants corresponding to domain wall thickness smaller than the “particle” we found the intermediate state shown in Fig.2a. In this and in all other magnetization pattern plots, the center of the arrow is an the centroid of the finite element tri- angle and the arrows have constant length and point in the direction of the magnetization in the triangle. The system jumped directly from a state with uniform magnetization into this onc after a very small change in applied field. This behavior was a common feature of all of the calculations and one which we believe is real rather than a numerical artifact.

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(a) (b) Figure 2. (a) Magnetization pattern in an octagon at B, = -190 just

after nucleation of a structured state from one with uni- form magnetization and (b) just before the structured state vanished a t B, = -505. Here, C ’ = 0.01.

As the reversed applied magnet field is increased the closure struc- ture is swept out of the particle. Figure 2b shows the same particle just before its magnetization became essentially uniform.

Prolate octagon We flattened the octagon in the vertical direction to simulate an acicular particle. For the same constants that we used in Figs.2a,2b

Figure 3. Magnetization pattern in a prolate octagon a t B, = -290 with weak enough exchange, C ’ = 0.001, to support a complex pattern.

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we found that the uniform magnetization reversed without forma- tion of any intermediate structure and the hysteresis loop was per- fectly square. Reduction of the exchange constant by a factor of ten, however, resulted in the intermediate structure shown in Fig.3.

“Realistic” Real particles frequently have complex morpho1ogiesl0. Despite the limitations of our present two-dimensional calculation, we wanted to see how a particle with an irregular shape would behave. The shape of our sample particle can be seen in Fig.4. For strong ex- change the uniform magnetization reversed without formation of any intermediate structure and the hysteresis loop was perfectly

Figure 4. Magnetization pattern in a com lex shape at B, = -570 with intermediate exchange, C = 0.0025. Part of the particle has switched coherently, the rest will do so with a stronger reverse field.

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square. Reduction of the exchange constant by a factor of four re- sulted in the intermediate structure of Fig.4. For still weaker ex- change (or, equivalently, for a still larger particle) more complex intermediate states are found.

Interacting particles

Figure 5 shows two stages in the switching of a pair of rectan- gular “particles” in close proximity relative to their size. For this particular system and field sweep the upper particle switched with- out formation of intermediate closure structures of the kind seen in the lower particie in FigSb.

(a) ( b) Figure 5 . Magnetization pattern in closely spaced, interacting parti-

cles with C ‘ = 0:005. (a) At B, = -350, each particle is magnetized in an opposite direction. (b) With a stronger reverse field, B, = -350, the lower particle has formed a closure pattern.

For reduced exchange, intermediate closure structures develop in both particles. Figure 6 shows the pair of rectangles with the ex- change constant reduced by a factor of two.

In all of the examples shown, the formation of intermediate closure structures is accompanied by steps in the hysteresis loop. The details of these structures are sensitive to the exact constants applicable to the particles, their morphology and their relative posi- tions. The accompanying steps in the hysteresis loop would there- fore be averaged out in measurements on macroscopic systems. We leave it as open whether the hystersis loop of a single particle or of an isolated small cluster of two or three particles would exhibit the step structure we often find.

COMMENTS

The calculations were first principles calculations in the sense that a numerical method for solving the basic micromagnetics equations was devised, implemented and applied to a variety of sys- tems without intervention. The method is designed to mirror the physics of ferromagnetic systems, and the fact that features such as shape anisotropy and coherent rotation with strong exchange emerge indicate that the physics is modeled correctly. In addition, two re- sults of the calculations were not anticipated: In all cases, the system jumped abruptly between topologically different magnetization pat- terns with accompanying discontinuities in the hysteresis loops. In the case of the “realistic” particle, different sections rotated coherently at different field values. Since this result should not de- pend on the two-dimensional nature of the calculation, one could hope for its verification by experiments similar to those of Knowlesll.

It i s worth noting that our work here demonstrates the power of the finite element method to treat complicated geometries. One could not hope to treat our “realistic” particle analytically, and the interacting particles are too close for their interactions to be ap- proximated by a multipole expansion. These and other situations are easily handled using our numerical technique.

A new way of modeling micromagnetics was presented in this paper. The simplest test of the method was its application to a two- dimensionai system. Conceptually, the entire model is readily ex- tended to three-dimensions and we intend to do this with emphasis on the switching of single particles. We anticipate reformulating Eq. (1) in terms of H and a scalar potential cp for this extension. The exbected difficulties are the purely mechanical ones associated with mesh generation etc.

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