2362 IEEE TRANSACTIONS ON MAGNETICS, VOL. 24, NO. 6, NOVEMBER 1988

NUMERICAL MICROMAGNETICS OF SMALL PARTICLES

D. R. Fredkin* Department of Physics

and Center for Magnetic Recording Research

IJniversity of California, San Diego La Jolla, CA 92093

and

T. R. Koehler IBM Almaden Research Laboratory

San Jose. C A 95120

ABSTRACT

l’he results of applying our micromagnetic code to a homoge- neous spherical particle which is large enough to support inhomo- geneous magnetization arc given. For smaller particles, magnetization reversal is by coherent rotation. Larger particles in- itially exhibit curling as the applied magnetic field is reduced from a saturating value. Then one of two new behaviours is observed: For weak crystalline anisotropy, the axis of the curling state rotates and bends, and the magnetization reversal process is reversible, or nearly so. For strong crystalline anisotropy, a sudden discontin- uous transition occurs to a vortex state with axis perpendicular to the anisotropy axis, and the vortex moves across the particle as re- versal of the applied magnetic field continued. The formation and disappearance of the vortex are irreversible but all other aspects of the process are reversible.

INTRODCJCTION

We have started to investigate the magnetization distributions that occur in three-dimensional ferromagnetic systems when they are subjected to a slowly varying external magnetic field, a s in the measurement of a hysteresis loop. l‘he calculations are micromag- netic, i. e., we adopt a continuum view of the medium and treat the magnetization as a field, but we otherwise proceed from first prin- ciples. We have already described two-dimensional micromagnetic calculations using the methods of the present paper’, and we have described some of the refinements necessary for three-dimensional finite element calculations*. In this paper, we shall discuss some of the phenomena which we have observed when we subject a homogenous sphere to a numerical hysteresis loop.

We chose the sphere for our first three-dimensional calcu- lations because there are some rigorous theoretical results3 against which we could compare our calculations. We do find that, for a suitably large sphere, the state of uniform magneti~ation becomes unstable and there is a transition to a ‘‘curling” magnetization state, in agreement with theory. Prom that point onward in the hysteresis loop, nonlinearities take over and a variety of hitherto unexpected phenomena occur: rotation of the curling axis, displacement of thc curling axis from the center of the sphere, deviation of the curling axis from linearity, and, for suficiently large uniaxial crystalline anisotropy, an abrupt transition to a “vortex” state with axis pcr- pendicular to the conimon direction of the applied magnetic field and the anisotropy axis. We shall describe the variation of the vortex state, once formed, as the applied magnetic field is varied.

We emphasize that our methods arc in no way adapted or specialized to the spherical geometry, and the methods and com- puter codes work equally well for arhitrary geometries. including systems of several particles. We are reporting results only for a

* Supported in part by a grant from the National Science Fovnda- tion

sphere at the present time because of a number of reasons: the wealth of new phenomena which we observed for the sphere still requires additional investigation. the time consuming character of the calculations, and space for the present article. Other geometries will be discussed in the future.

MATHEMATICAL AND NUMERICAL METHODS

Our micromagnetic analysis starts with the free energy density

u = - (HI2 - M-H + $ C(VM12 + 8aoni,y, 8n

with H = - VQ. We seek a stationary point of I / = Su&x as a functional of I$ and M subject to the constraint that IMI = M,, where M , is the known saturation magnetization. We have used uniaxial crystalline anisotropy

’The numerical procedure used to find the stationary point of U was based on the finite element method. The magnetic region, and a much larger surrounding region of free space, is divided into tetrahedra. The potential I$ is interpolated linearly and the magnetization M is taken to be constant in each tetrahedron. The details of the finite clement calculation of 4 and the calculation of M, including the efPectivc shape anisotropy of each tetrahedron4, are fully described in Ref. 2.

Figure 1. Details of the triangularization on the surface of thc sphere from a viewpoint along the anisotropy axis. The hemispheres above and below the plane of the page are shown a t the left and right respectively.

The specific system reported on here is a spherical magnetic particle of radius a. The surrounding region is a sphere of radius 5a. Each sphere has 92 nodes on its boundary. ‘The triangulari7ation of the surface of either sphere is shown in Fig. I from a perspective in which the anisotropy axis is pcrpcndicular to the plane of the page. The pattern was obtained by starting with the surface structure of a soccer ball, which is shown by the solid lines in the figure, and adding points at the center of each of the five and six sided panel?. The added points were moved radially out- ward so that the lay on the surface of the sphere. l h e dotted lines

001 8-9464/88/ 1 100-2362%01 .00O 1988 IEEE

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a finite amount. Under this condition, the state of the magnetization is determined by the nonlinear micromagnetic equations derived by variation of(I) .

The direction of magnetization will he indicated in all magnetization pattern plots by elongated rectangular pyramids centered in an element and oriented along the magnetization or tha t element. If the entire base of a pyramid 'can be seen, the magnetization is directed into the page. If only the edges of the base can be seen, the indicated direction is out of the page. Several aspects o f a pyramid for magnetization directed into and out of the page are shown in Fig. 3. In the magnetization pattern plots, an indication of the absolute spatial orientation will be given by in- cluding Cartesian axes for a right handed system with the z-axis drawn with a heavier line.

connect the new points to the old. The surface pattern is antisym- metric with respect to reflection in the plane of the page, as shown .by the two views in the figure. Points were added to the interior regions so that they were uniformly distributed, but without any particular pattern. This was done to avoid biasing the sample by imposing a numerical symmetry. The points in the outer region were denser in the vicinity of the inner sphere. There wcre 381 points in the interior of the smaller sphere and 1192 in the shcll between the spheres. A Del-aunay tesselation5 of the system, using a computer program developed for our numerical micromagnetic work, generated 2639 elements inside the inner sphere and 7.534 in- side the shell.

We subjected the magnetic particle to a numerical hysteresis loop in which a uniform field was applied to the inner sphere by fixing the values of 4 a t the nodes of the outer sphere a t appropri- ate values. First, a saturating magnetic field was applied parallel to the anisotropy axis. The magnetic field was then stepped through a reversal process. We will emphasize the qualitative features of the magnetization field, which will be prcsentcd as plots of the magnetization patterns which evolve during the stepping process. However, quantities such as average magnetization and individual energy contributions were also recorded numerically and are avail- able. The range of material parameters for the systems treated to date are shown in Fig. 2 where the c a m treated are indicated by crosses. This figure is rather sparse because, as will be discussed later, the computer runs take a very long time. For the same reason some of our results have been obtained using a rather coarse field step.

X

X

X

0.0 3.0

Figure 2. Material parameter values in dimensionless units for which calculations have been performed. The values for which curling is expected theoretically are those in the re- gion below the dotted line.

RESULTS

Our choice of material parameters was designed to concentrate in the area where the particle is just large enough to support a nonuniformly magnetized state. 'This boundary was determined numerically to be at C/a2=0.06. Only coherent rotation is seen for larger values of C/a2, and no pictorial results for that. case will be given since the magneti~ation is uniform and directed either along or opposed to the applied field direction.

For smaller values of C/a2, the first phenomenon observed during the reversal is curling. The amplitude of the curling state, measured by the maximum angle between the magnetization and the anisotropy axis, increases continuously from 7ero a s the mag- netic field is reduced through the nucleation value. We did not compare the detailed features of the curling state with theoretical predictions. In fact, we do not know of any rigorous calculations of the curling state for applied field below the nucleation value by

Figure 3.Various views of the pyramid used to indicate the magnetization direction in the magnetbation pattern plots. The direction is in the plane of the page for the four outer pyramids. The direction rotates 22.5" hetween ad- jacent pyramids. The rotation is out of and into the page for the top and bottom rows respectively.

Pig. 4 shows three aspects of the curling pattern in the IfK/bf3 = 2.5 case just hefore the transition to the vortex state. Fig. 4a shows a view along the anisotropy direction of a slice of the sphere through the origin, oriented perpendicular to the viewing direction. To avoid clutter, all figures show only a slice of the sample perpendicular to the viewing direction. The maximum curl- ing angle is a t the edge of the sphere and is about 4.5 degrees. Fig. 4b shows a transverse view of a centered slab. Ilere, thc tilt is in- ward at the right hand side, zero at the Center and outward at the left hand side. A transverse view of a slab displaced 0 . 7 5 ~ from the origin is shown in Fig. 4c. Some variation in tilt can also he oh- served in this figure.

As the reversal progressed, two quite dimerent and totally un- anticipated behaviors wcre seen, depending on the value of I IK/M, .

The large I IK/Ms hchavior was seen only for I IK/Mq = 2.5. 'There, curling started a t a particular field value and increased with decreasing field up to a point a t which the curling configuration became unstable and a sudden transition to a vortex like state was made. The transition occurred completely in one field step no matter how small the step size. The smallest tried was 0.005M,. The resulting vortex state was transverse: its axis was perpendicular to the anisotropy axis (which, for convenience, we take as the z -di- rection). The vortex formed somewhat off center. Its axis was lo- cated in the x - y plane, but did not go through the origin. As the reversal continued, the vortex remained in the x - y plane hut moved, passing through the origin a t an applied field of zero and continuing to move until it eventually passed out the side. The transverse vortex has also been scen in numerical studies of rcctan- gular particles6.

Fig. 4 showed the curling state just before the transition to the vortex state. The appearance of a centered slab of the transverse vortex from a point along the vortex axis just arter nucleation is shown in Fig. Sa. All of the magnetization vectors at the perimeter are within a few degrees of being in the, plane of the page, but they are not all inclined at the same angle to'that plane or, consequently, to the vortex axis. This is typical for an off center vortex. The displacement of the vortex axis from the center is clearly seen in Pig. Sb, which shows a view down the z-axis, transvcrse to the

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I

A AA

A

I

Figure 5. Transverse vortex inimcdiately after formation, (a) view down the vortex axis (transverse to the applied field di- rection) o f a centcrcd slab showing slight off center posi- tion of core, (h) view transverse to the vortex axis of a centcred slah showing the extreme variation in orientation of the magneti7ation field, and (c) same viewpoint as (a), but the slab is 0 . 8 ~ from the conter of the sphere.

Figure 4. ( 'r~rling magncti7ation field for fairly large curling angle in the l fK/MT = 2 . 5 case just hcforc nucleation of a transverse vortex, (a) view down the applied field direction of a CCII-

tcrcd slah, (h) vicw transverse to the applied field direction of a centcred slah. and (c) same viewpoint as (h), hut the skth is 0 . 7 5 ~ from the centcr of the sphere.

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vortex axis, of a centered slab. One should note in this picture that the vortex axis is not quite straight. Supporting evidence for the latter observation is shown in Fig. 5c, which has the same viewpoint as Fig. 5a, but the slab is 0.8~ from the center of the sphere. The apparent center of the pattern in Fig. 5c is somewhat closer to the origin than that of Fig. 5a.

The motion of the vortex axis during reversal was described above. The appearance of the magnetization pattern during this process is illustrated in Fig. 6 for two values of applied field. A view a t zero applied field along the z-direction of a centered slab is shown in Fig. 6a, and a transverse view, along the vortex axis, a t the same field is shown in Fig. 6b. Both pictures show that the vortex is quite well centered. The net magnetization of the sphere is very nearly zero for this configuration. A comparison of the Cartesian axes between Figs. 6a and 5b shows that the vortex axis rotated as the applied field was reduced to zero.

As noted above, when the vertex axis passes through the ori- gin, it becomes straight and the magnetization in the equatorial perimeter is in a plane perpendicular to the axis. Upon additional field reversal, the axis again migrates further from the origin and the distortions seen in Fig. 5 reappear and eventually become enhanced. The first phase of this process is illustrated in Fig. 6c, which is a view similar t o that of' Fig. 5b of' the magnetization pattern a t an applied field of I?K/M,r = - 0.5. This is slightly greater in magnitude than the positive field at which the transverse vortex formed.

'The appearance of the vortex just before it disappears is shown in Fig. 7, where the energy balance considerations that make the transverse vortex a favorable state under certain conditions can be visuali7,ed. The applied field here is H K / M , = - 1.1 and the viewpoints of Figs. 7a-7c are analogous to those of Fig. Sa-5c. There is obviously considerable flux closure in Fig. 5a and partial in Fig. 7a. In both, there is substantial alignment along the anisotropy axis, which is energetically favorable and explains why this state is favored for high HK/MS. The alignment is easily visu- ali7,ed in Pig. 7 b the magneti7,ation of the entire right hand side of' the slab is well aligned with the anisotropy axis (which is perpen- dicular to the page in this view). As the magnitude of the applied field increases, and the vortex axis moves from the center to the edge of the sphere, the anisotropy energy decreases and the mag- netic moment increases. but the topology remains the same, pro- viding a mechanism for the smooth reversal of the net magnetization. This process is reversible until the vortex leaves the sphere, a point which has been tested numerically. A view trans- verse to the vortex axis reveals considerable bending in the axis, as exhibited in Fig. 7b. Further evidence of the bending is seen in Fig. 7c, which is from the same viewpoint as in Fig. 7a but the slab is displaced upward by 0 . 7 ~ . The vortex is just leaving the sphere at this elevation, and the axis is not seen at slightly higher elevations. The vortex pattern is quite distorted and it is not possible to prcscnt complete details in a few pictures.

The hysteresis loop for the process described above is shown by the solid line in Fig. 8 . , where the average magnetization of the entire sphere normalized by the saturation magnetization, M,, is plotted as a function of applied field. The lower portion of the hysteretic part and its continuation through the origin is the trans- verse vortex state. As noted above, it was determined numerically that the magnetic behavior is completely reversible in this region and reversal of the direction of tnagnetic field simply reverses the motion of the vortex. The discontinuity as the transverse vortex exits may be less than indicated in rig. 8. The field step size used this region was 0.02S/M,, which was five times that used in explor- ing the suddenness of the formation of the vortex. Its onset is the discontinuity in the upper branch of the loop. In the same branch, the onset ofcurling is where M, becomes noticeably less than unity. The onset is probably smoother than indicated; a field step size or 0.12S/M, was used here. Samples with the four lowest H K / M Z values shown in Fig. 1 exhibited similar behavior to the extent that the reversal process also started with curling. Subsequently, however, the all evolved into what were basically curling states, but with the curling axis tilted away from the anisotropy axis. These field con-

'

Figure 6. Motion of transverse vortex as applied field changes. (a) view transverse to the vortex axis of a centered slab at zero applied field showing the centered position of the core, (b) view along the vortex axis of a centered slab at zero applied field also showing the centered position of the core, and (c) view transverse to the vortex axis of a cen- tered slab at an applicd Geld of IIK/Mr = -0.5 showing thc continued motion of the vortex axis with further field re- versal.

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m 1

0

0

0

Figure 7.Transverse vortex ahout to leave the sphere: (a) view down the vortex axis of a ccntercd slab showing pro- nounced olTccntcr position of core, (b) view transverse to the vortex axis of a ccntercd slab showing curvature of the axis, and (c) came viewpoint as (a), but the slah is dis- placed 0 . 7 ~ from the ccnter of thc sphere out of the plane of the page.

figurations resemble those already illustrated except for two fea- tures: first, the axis tilt, and second, the tilted axis develops a slight "S" shape with the end regions of the axis aligned more parallel to the anisotropy axis than the midpoint region. Because of their general three dimensional orientation, these states are difficult to prcsent visually in a meaningful way with only a few pictures, and none will be shown here. The essence is that they are tilted, curling states.

In general as the applied field decreased toward zero for thcse states, the curling angle increased and the tilt increased until, a t zero applied field, the magnetization field had evolved into that of the transverse vortex. Visually, the above process appeared to be completely smooth and its mirror image eventually seemed to lead to a complcte and smooth reversal. There were some systematic differences in the runs, as will be discussed in the following.

The hysteresis loops for the two extreme values of the low I I K cases are shown in Fig. 8. As i n the large II, case, the onset of curling is where M, becomes somewhat less than unity. The IJ,/M,= I case is drawn with a dashed line. The discontinuity at an applied field of about 1.25M, signifies a sudden transition from a curled state with axis along the anisotropy axis to one with the axis tilted about 20". With a coarse step run, there appeared to hc hysteresis associated with this transition. Finer step runs in both applied field direction show no hysteresis. As the field is decreased in magnitude below this transition, the axis tilts as dcscribed above.

The I IK/Ms = 0 case is shown by the dotted line in Pig 8 . Here, the onset of the tilted state is very gradual. It can be readily re- cognized at an applied field of about I IK/Ms = 1 and probably starts earlier. Thc change of slope near the origin is associated with a rapid change in the axis tilt. I t is difficult to discern in the drawing, but the line does not go completely through the origin. The data shown was obtained with an applied field step s i x of O.OOS/M, near the origin. Ijarlicr runs with a coarser step size exhibited hysteresis loops there. This run, which is a few points from completion, ap- pears destined to go through the origin. The apparent discontinuity near the origin occurs where the finer step run was started and in- dicates that the run should cover a larger region. It is also possible that, especially with no intrinsic anisotropy, small irregularities in the shape anisotropy of the elements could add somc randomness to the numcrical results. One would expect that such effects would he diminished hecausc there is a healing length associated with the system due to the exchange interaction.

I .o

- I .o

Figure 8 . I Iysteresis loops for l/K/Ms = 0, 1 and 2.5 arc shown hy the dotted, dashcd and solid lincs respcctivcly. All had C = 0.02/4n. In the minor loop, the lowcr I M,I values arc from the transverse vortcx state.

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DISCUSSION

We have studied the micromagnetic behavior of a few small spherical particles of direrent size and different material parame- ters. We have found the expected behavior in one case: coherent rotation in particles that are small enough to be strictly single do- main, and unanticipated behavior in other cases involving slightly larger particles. The unexpected results were an irreversible transi- tion from a curling state with no visually detectable axis tilt to a transverse vortex state for large IJ,/M,. and, for lower values of H,/M,, a behavior whose details vary from case to case, but can generally be described as rotation by curling accompanied by a tilt- ing of the curling axis. The latter appears to be fully reversible with the smallest field step sizes used, but did not appear so for larger step sizes. However, all of the details have not yet been resolved.

The original motivation in starting our three dimensional work with a study of the sphere was to be able to compare the numerical results with analytical theory. In practice this has proved to be difficult for various reasons. The analytical theory considers when curling should start, but cannot rigorously predict what happens after it does start, which is our major result. This may be beyond the range of analysis. The precise value of the field at which curling or coherent rotation start is obscured in the calculation by a nu- merical effect. In the scalar potential formulation of the finite ele- ment method, the transverse component of I 1 is conserved exactly, but the normal component of D is not. This seems to produce a slippage of the applied field between the boundary and the magnetic region. Specifically, if the magnetization in the magnetic sphere is set and held uniform and aligned along the z-axis and only phi is varied, the resulting calculated internal field in the sphere should be I f z = - 4RM,/3, I!,, = [ I , = 0 In fact, it comes out to be quite uniform, but too small in magnitude by an amount that depends upon details of the tesselation (about 80/0 for the system used in the calculation reported in this paper). It changes correctly from this value when a uniform field is applied via the boundary conditions. Our explorations of this phenomenon indicate that its net effect is such that the applied field value a t which an event takes place is somewhat off, but the nature of the event is correct. In a sample with coarser mesh, the field slippage is larger and the onset of curl- ing and the transition to the transverse vortex are displaced in ap- plied field value by a corresponding amount. We are currently working on a promising method for treating this problem, but the work is in the early stages. We also intend to implement and use second order elements, which should refine the results numerically.

The treated cases, as summarized in Fig. I , are rather sparse and only a partial exploration of these has been made with smaller step sizes in applied field. The sole reason for this lack of more detailed results is that the computational requirements for the runs are large. The calculations were mostly all done on an IDM 370/3081. On this machine, the proper unit of time for the runs is the CPU week and the nurnber of these units for a typical run is several to many. A few runs have recently been made using an IDM 370/3090 and this decreased the CPlJ time by a factor of 2.5. The computational time requirements increase considerably when the magnetization field pattern is about to change character. For ex- ample, the number of iteration steps to convergence increases by at least an order of magnitude when curling is about to start, hut before there is obvious visual evidence of curling. This behavior is well known in computational simulations of phase transitions.

The numerical study of the micromagnetics of a simple ferromagnetic sphere has yielded several interesting and unantic- ipated results. Much remains to hc done on this system. The cases I l , > 2.SM, and 2.SM, > 11, > M , should be explored for the C value used here and smaller values of (I should be tried. The same techniques can also hc immediately applied to many other geon- etries.

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REFERENCES

D. R. Fredkin and T. R. Koehler, MAG23, 3385 (1987). D. R. Fredkin and T. R. Koehler, J. ( I 988).

IEEE Trans. Magn.

Appl. Phys. 63, 3179 ,- --, S. Shtrikman and D. Treves, in Magnetism, Ed. G. T. Rad0 and 11. Suhl, Vol. 111, (Academic Press, New York, 1963). D. R. Fredkin and T. R. Koehler, to be published. The siginifcance of the Delaunay tesselation in this context is discussed in “.rhree-Dirnensional Finite Element Mesh Gener- ation Using Delaunay Tesselation”, D. N. Shenton and Z. .I. Cendes, IEEE Trans. Magn. MAG-21, 2535 (1985). Iiowever, we do not use the algorithm described in this paper. Manfred E. Schabes and 11. Neal Bertram, to be published.