An accurate and efficient 3-D micromagnetic simulation of metal evaporated tape

ELSEVIER Journal of Magnetism and Magnetic Materials 171 (1997) 190-208

~ Journal of anlnadgn etlsm magnetic materials

An accurate and efficient 3-D micromagnetic simulation of metal evaporated tape

M. Jones , J.J. M i l e s *

The Manchester School of Engineering, The Universi O, of Manchester, Manchester Ml3 9PL. UK

Received 16 January 1997; received in revised form 4 February 1997

Abstract

Metal evaporated tape (MET) has a complex column-like structure in which magnetic domains are arranged randomly. In order to accurately simulate the behaviour of MET it is important to capture these aspects of the material in a high-resolution 3-D micromagnetic model. The scale of this problem prohibits the use of traditional scalar computers and leads us to develop algorithms for a vector processor architecture. We demonstrate that despite the materials highly non-uniform structure, it is possible to develop fast vector algorithms for the computation of the magnetostatic interaction field. We do this by splitting the lield calculation into near and far components. The near field component is calculated exactly using an efficient vector algorithm, whereas the far field is calculated approximately using a novel fast Fourier transform (FFT) technique. Results are presented which demonstrate that, in practice, the algorithms require sub-O(N log(N)) computation time. In addition results of highly realistic simulation of hysteresis in MET are presented.

Keywords: Micromagnetics; Simulation; Metal evaporated tape; Vector processing; Fast Fourier transform

1. Introduction

A successful simulation of M E T must accurately represent its three dimensional and highly irregular microstructure. The curved column-like structure of the material undoubtedly influences its

*Corresponding author. Tcl.:+ 44 161 275 4501; fax: + 44 161 275 4501.

magnetic behaviour, yet micromagnet ic simulations to date have not involved this level of structural detail. The reason for this is that highly realistic models require large amounts of computat ion. The problems associated with high-resolution micromagnet ic modelling are well known. The most t ime-consuming part of simulation is the evaluation of the magnetostat ic interaction field, having an asymptot ic computa t ion time of O(N2). Al- though methods of reducing the time requirement without substantial loss of accuracy have been reported, the amoun t of computa t ion required for

0304-8853/'97/'$17.00 1997 Elsevier Science B.V. All rights reserved PII S0304- 88 53 (97)00048-6

M. Jones, j.d. Miles/Journal of Magnetism and Magnetic Materials 171 (1997) 190 208 191

a detailed model of MET prohibit the use of traditional scalar machines. Optimisation of these algorithms for massively parallel computer architectures is extremely difficult.

In this paper we present a new 3-D micromagnetic model which accurately represents the important aspects of MET structure. The problem of computing the magnetostatic interaction field are overcome by developing an efficient algorithm, suitable for use on vector computer architectures. The basic approach of the algorithm is similar to the popular particle particle/particle-mesh (PPPM) method. Here the interaction field calculation is split into two parts; the near field, arising from interactions between nearby magnetic elements; and the far field, which arises from all other interactions. Using this approach we develop a fast vector algorithm for the computation of the near field and a novel FFT-based far field algorithm. We demonstrate that in practice the algorithms require sub- O(N log(N)) asymptotic computation time. Results are presented which show hysteresis curves for a high resolution and realistic model of MET.

2. Related work

Several 3-D micromagnetic models have been reported previously. Tagawa et al. I l l use a 3-D cubic lattice with variable inter-granular spacing and grain orientation to investigate their effect on the mean interaction field in Ba-ferrite and metal particulate media. The grain centres are con- strained to lie on a regular 3-D cubic lattice. Zhu [2] extends a 2-D hexagonal lattice model to simulate a multi-layer thin film recording media. Again the magnetic centres lie on a regular grid. Victoria [3] models CoNi thin film samples formed by ob- lique evaporation. These consist of long columnar grains inclined at an angle to the plane of the film. The model similarly consists of columns of elliptical cross-section inclined at an angle and split into regular-sized segments. A pseudo-random structure was used by the same author, based on obser- vations made experimentally on real Ba-ferrite tape [4]. Here transverse and perpendicular structure is regular, whilst in the longitudinal axis, the start of each row of grains is chosen randomly.

In all of these models, the main computational problem is that of computing the magnetostatic interaction field. Since each of the N magnetic elements of a model interacts with every other this calculation has an O(N 2) complexity. This problem is part of a wider class of O(N 2) problems called N-body problems which arise in many of the physical sciences [5].

Several solutions exist to the N-body problem. When the bodies occupy a regular grid the interaction calculation can often be expressed as Fourier domain convolution. This can be computed in O(Nlog(N)) using FFT-based convolution [6]. When this is not the case several approximate methods may be considered. • Truncation: O(N). Here only the effect of near-

neighbour interactions is calculated. • Mean field approach: O(N). A refinement of

truncation in which the neglected interactions are approximated by the addition of a constant term which represents their mean value.

• Segmentation: O(N). Here the neglected material is segmented into regions which are assumed to be uniformly magnetised with the region average. In this way distant interactions are represented more accurately.

• Particle in cell methods: O(N log(N)). The irregular domain is covered with a regular mesh onto which the values at irregular points are interpolated. The calculation is performed only at mesh points using an F F T or other fast approach.

• Particle-particle/particle-mesh (PPPM) methods: O(N log(N)). The calculation is split into near and far parts. Here the interaction between near bodies is calculated exactly whereas that between distant bodies is calculated using the particle in cell method.

• Fast multipole method [7, 8]: O(N) to O(N log(N). Long-range forces or potentials between a particle and a cluster of particles are computed by means of a multipole expansion around the centre of the cluster. Short-range forces are computed exactly.

• Tree code methods [9]: O(N log(N)). The space containing the bodies undergoes hierarchical subdivision into a number of cubic spaces. If an adaptive tree code is used, this subdivision has

192 M. Jones, J.J. Miles'/Journal of Magnetism and Magnetic Materials" 171 (1997) 190 208

the property that densely populated regions are more heavily sampled. The tree representation of the hierarchy allows the average force/charge within each cube to be found in O(Nlog(N)) time. Interactions between distant bodies is calculated using the coarsest level of the tree whereas near-neighbour interactions use the finest level. This also has a complexity of O(N log(N)). Truncation is popular in early work on micro-

magnetic simulation [10-13]. However truncation introduces unacceptable error into the calculation. The mean field approach, although an improve- ment over truncation, is still inaccurate - especially if the grain size is large compared with the area of accurate calculation, or the scale of the magnetic structure is comparable to the system size.

Victoria adopts the segmentation approach using strips lying longitudinally [3, 4]. The field exerted by each of these on a particular grain is calculated using a line dipole. In a similar way, Nakatani et al. use rectangular segments [14].

A refinement to the segmentation method is to use a segment hierarchy. Miles and Middleton [15] employ a three-layer segment hierarchy. The bottom layer consists of individual elements, the second of subsegments and the third comprises complete segments. The magnetisation within each segment/subsegment is calculated prior to the field calculation. The interaction fields are calculated at each element by summing fields due to elements within a short distance, subsegments at intermedi- ate distances and entire segments for the remainder of the film. This scheme was optimised for a vector processor architecture.

Min and Zhu use the P P P M method to investigate grain-size dispersion in longitudinal thin-film media [16]. The multipole expansion has been utilised in several micromagnetic simulations. Blue and Sheinfein [17] demonstrate how a technique, similar to Greengard and Rokhlins, speeds up the magnetostatic calculation to O(Nlog(N)) whilst maintaining accuracy. Yuan and Bertram improve on this work by developing an O(N) algorithm for magnetostatic interaction [18]. They compare the algorithm to the generalised FFT method for regular grids of particles and conclude that, due to the book keeping overhead of the multipole method, the FFT is the fastest method in practice.

3. Simulation of M E T structure

The 3-D structure of MET is modelled by defining the cross-section of a set of close packed columns. The columns have constant cross-section, and consist of a number of elements formed by partitioning the thickness of the material into layers of equal thickness. The definition of a single column is shown in Fig. 1. Here the cross-section is defined in the X - Z plane. All columns have a common axis-shape which is formed by translating the start of each of the layers a distance in the Z-axis direction. The exact shape is defined using a function axis.

An important aspect of the material model is that the columns be arranged randomly in the X - Z plane. This is because, the use of regular structure introduces artificial regularities in the magnetisation states of the material [19]. However, when using FFT-based field calculation, the elements must be associated with a regular-grid. Although the randomly distributed magnetisation may be interpolated onto a regular-grid prior to field calculation [16], we seek to avoid this step by introducing a topology preserving randomness into a regular grid. This allows for a one-to-one correspondence to be defined between elements of the random structure and vertices of the regular grid. The far field can then be calculated using a regular-grid approximation; this is described in Section 5. Randomness is

/ / f

! / / J

Fig. 1. Some constants used in the hysteresis computation, Structure definition of a column of material.

M. Jones, J.J. Miles /Journal q[" Magnetism and Magnetic Materials 171 (1997) 190 208 193

introduced into column positions in the following way. 1. The cross-section of the columns is defined ini-

tially on a unit rectangular grid. The grid is periodic in both axes (X and Z).

2. Uniform spacing is added between the regular rectangles.

3. For each vertex of the grid two random numbers, rx and rz, are generated in the range [0, Random], where the parameter Random lies in the range [0, 1] and is used to control the amount of randomness. The vertex position is translated along the vector (rx, r=) t. All connect- ing lines are preserved.

4. After all the vertices have been translated each of the resulting distorted regions are tested to en- sure they have convex shape. Non-convex regions are adjusted to remove concavities.

5. Cubic splines are fitted within each of the irregular regions to represent the cross-sections of the columns.

6. Finally, the centre of mass of each region is found. The centres are the vertices of the irregular grid.

This process is illustrated in Fig. 2 for the case of a 15 × 15 set of columns with Random = 0.99 and spacing of 0.1 units. An example of the 3-D structure with four columns is shown in Fig. 3, in practice in the order of 104 columns would be used.

An interesting feature of this approach is that the packing density is invariant to changes in the Ran- dom parameter. This is because, the cubic splines always occupy the same percentage of the area of any irregular region. This is a useful property as it allows the effect of structural randomness to be investigated independently of changes in packing density.

Another useful feature of the method is that an upper limit on the number of different element shapes is simple to impose. This is desirable as each pair of different shapes interact in a different way. By reducing the number of such pairs the amount of work required to compute demagnetising factors is reduced (see Section 5).

The limit on element shapes is achieved using a simple modification to the algorithm above. In- stead of selecting rx and rz from a continuous range [0, Random] we randomly select rx from the set of

Fig. 2. An irregular structure defined upon a regular rectangular grid. The distorted grid (top) defines regions within which splines are fitted (bottom).

numbers {xi, i = 0 ... L} and rz from {zi, i = 0 .. L}. Here L is an integer parameter which is used to control the number of different shapes that result. Since each vertex of the irregular grid may be in one of L positions, there are only L 4 possible element shapes. For example when L = 2 there are only 16 possible element shapes.

194 M. Jones, J.J. Miles/Journal of Magnetism and Magnetic Materials' 171 (1997) 190 208

This is shown in Fig. 4. It is interesting to note that although the same 16 element shapes are repeated throughout the material, regularity has still been successfully removed.

Fig. 3. An example ofa 3-D structure model for a small sample of MET.

Fig. 4. Example of limited random structure. Here only 16 different element cross-sections are used throughout the material.

4. Simulation of material dynamics

The Landau Lifshitz equation represents the dy- namic behaviour of a magnetic particle under the influence of an applied field,

dM 7 ~7 dt - 1 +c~ ~MxHt-~ (1 +~)IM[ Mx(MxHt)'

(1)

where M is the particle magnetisation, H t the total field that it experiences, 7 is the gyromagnetic ratio, and c~ is the material dependent damping constant.

By representing a material as a collection of particles or 'elements', it is possible to simulate the magnetic behaviour of the material by solving a large set of ordinary differential equations (ODEs) with a given initial value. For examples of this approach see Refs. [15, 3, 14]. We perform this integration using a variable order Adams routine (NAG library d02chf).

The field Ht, in Eq. (1), is composed of fields arising from several sources,

Ht = Hext -k- H d + H e + Ha. (2)

Here Hcxt is the externally applied field, Hd is the demagnetising field (including self-demagnetisa- tion), Hc is the exchange field and Ha is a crystalline anisotropy field. Each of these are defined in Sec- tion 5.

5. Magnetostatic interaction field calculation

The magnetostatic interaction field at a particular element position is calculated as a sum over all other elements in the sample. This is a n O(N 2) calculation, which, for large-scale simulation is im- practical. To reduce this complexity the calculation is split into two parts; the near field, arising from the magnetisation of an element's nearest neighbours; and the far field, which arises from the

M. Jones, J.J. Miles"/Journal of Magnetism and Magnetic Materials" 171 (1997) 190 208 195

magnetisation of all other elements. A near neighbour region is defined such that an element lying at a is a neighbour of an element lying at b if,

Ix~--xb[<~N~ and l Y , - Y b I ~ N y

and lz,, - zbl <~ Nz, (3)

where a = (x~, y~, z~)' and b = (xb, Yb, zff. The field of an element at a arising from a near neighbour at b is defined using the near field tensor T,(a, b),

H(a) = T~(a, b)M(b), (4)

where a and b are irregular grid positions. The definition of the tensor for any pair of near neigh- bouts is

t12 t13~ T.(,,. b) = 4 E.

\t31 t32 t33/

where V. is the volume of element a and the elements ti~ are defined as

( f ( ~ (~ ~j(rb)(ri~r~) dZrb] d3r,, (6) '"=JJJLJJ Iru--rbl3 where the superscripts i and j denote the ith and j th components of a vector respectively, n(r) is the surface normal at a point on the surface of the element at b, r, represent position vectors of points in the volume of the elements at a, and rb are position vectors of points on the surface of the element at b. In this way the interaction between two elements is found by integrating over the surface of the element at b and the volume of the element at a. This assumes that magnetisation is constant within elements of the simulation. The tensors are pre-computed and stored for use in the simulation. The vector algorithm for computing the near field is described in Section 6.

The number of near-field tensors that are required when each of the N elements has M neighbours is in the order of N M tensors. For a large simulation, with N = l0 s elements and a near- neighbour region of 5 x 5 x 5 elements, this would require the computation in the order of 12.5 x 10 s tensors. This computational burden is greatly reduced if a limited set of element shapes are used. By exploiting the fact that, in this case, only a discrete

set of possible element shapes and positions are possible, the number of different tensors is much smaller than before. In the example above with L = 2 (16 shapes) only 244 000 demagnetising tensors need be calculated. Also, for a fixed number of material layers, this is an upper limit on the number required - the dimension of the simulation in the X and Z axes may be increased without introducing further near-field tensor computation.

The far-field interaction may be calculated using a far-field approximation which assumes that distant elements are dipoles. Further to this, each element position is approximated by its associated regular grid position. This allows the computat ion to be performed using a Fourier convolution approach without the usual interpolation step.

We will first describe the case when the columns of material are linear, and will then generalise this to the curved column case. Consider the field H(a) at a due to the magnetisation M(b) at b

H(a) = T(a - b)v(b)M(b), (7)

where a and b are position vectors (on the regular grid) of the elements, v(b) is the volume of the element at b, and the far-field tensor T is defined as

T ( r ) - 3 1 y2 4~ Irl 3 x y __ 1 y z , (8)

XZ y z Z 2 __ 1

where ~ = (x, y, z)'. Letting Mv(b)= v(b)M(b), the total field at a due to the elements in a region R is

H(a) = ~, T(a - b)Mv(b). beR

This convolution may be Fourier domain,

H = F - I [ F [ T ] x F[Mv]],

(9)

performed in the

(10)

where x represents matrix multiplication, and F [ ' ] the fast Fourier transform (FFT) operator.

All near neighbours are removed from the far- field computat ion by setting each element tij(r) of the kernel T(r) equal to zero for all values of r ~ [ - Nx . . . Nx, -- N y . . . Xy , - N z . . . g z ] .

Since the out-of-plane (Y-axis) boundary condition is not periodic, the convolution must be linear along this axis. This usually requires that the Y-axis

196 M. .Iones, J.J. Miles/Journal o['Magnetism and Magnetic Materials 171 (1997) 190 208

dimension be doubled and padded with zeros to avoid periodicity [181. We adopt a different scheme which avoids padding and also generalises the computation to the curved column case. This involves computing the field for each layer separately using layer specific kernel functions and 2-D convolution. Using F[ .1 to now represent the 2-D FFT operator, the field may be re-expressed as

g

H ; = ~ F ' [ F [ T I ] x F [ M ' v ] ] ( i = 1.. .Y), ¢11) /=1

where H; is the field for layer i, TI is the lth layer of the layer specific kernel T;, and M;, is the lth layer of the volume weighted magnetisation. This may be re-arranged to reduce the number of FFT operations,

H i = F - I I ~ / = I F [ T ' ] x F [ M ; " ] I ( i = I . . . Y ) . (12)

The layer specific kernels are defined as

Tl(x, z) = T((x, (I - i)~., z + axis(l) - axis(i))t), (13)

where ~ is the thickness of a material layer and axis(k) gives the Z-axis displacement of the kth layer. The layer specific kernels take account of the curvature of the columns by incorporating the relative Z-axis displacement between the layer in which the field is calculated, i, and where it arises, I.

The transformed kernel functions F [ T ] ] . . . F[T~] (l = 1 ... Y) are pre-computed and stored for use in the simulation.

As well as the interaction fields, described above, the total field also contains crystalline anisotropy and exchange terms. These, however, only involve nearest neighbour interaction, and since the computation is linear, it is possible to describe their effect as a 3 × 3 tensor operation which may be added to the appropriate near field demagnetising tensors. We describe each of these field calculations in turn.

5.1. Co,stalline anisotropy

Crystalline anisotropy is defined with reference to an easy axis vector e. The direction of the easy axis of each element is assigned randomly within user defined bounds. The near field tensor, of a par-

ticular element lying at a, representing self-demag- netisation is T,(a, a). This is updated in the following way:

2K t T,(a, a) : T,,(a, a) + ~ s e e , (14)

/2oMs

where K represents the anisotropy constant,/2o is the permeability of free space and Ms is the saturation magnetisation. This is repeated for each element.

5.2. Exchange

Exchange is treated as in Ref. [2] and incorpor- ated into the demagnetising tensors T,(a, b), for all pairs of elements which have a common surface (ones separated only by empty space). The tensors are updated in the following way:

2KC* T.ia, b) = T°Ia, bl + Z l , {15)

/20

where I is the 3 x 3 identity matrix and C* deter- mines the strength of the exchange coupling. For exchange along columns C* is calculated as

A C* - (16)

2Kd '

where A is the exchange constant and d~ is the along-column distance between element centres. For exchange between elements in the same layer but occupying neighbouring columns it is defined a s

AR C * - ( 1 7 )

2Kd 2'

where dp represents the in-plane distance between element centres, and R is a scaling factor with a value in the interval [0, l]. The scaling factor describes the extent to which the exchange field is reduced by the introduction of the spacing between columns.

5.3. Virtual jar field

Finally, a 'virtual' field contribution to each element is included in the computation. This is done to simulate a large bulk of magnetised material

M. &rues. J.d. Miles/Journal q[" Magnetism and Magnetie Materials 171 (1997) 190 208 197

× y

where

surrounding the element a long distance away. This is achieved by repeating the structure and magnetisation state of the material in blocks so that it forms an (2V_~ + 1)x (2V~ + 1) block array, where V~ and V~ define the X and Z dimensions of this array. The main computational block is the central block of the array. The field contribution from the blocks around the central one is included in the computation by modifying the kernel function T. This is done by modifying Eq. (8) so that

v, v: 3 1

i = - V, j = V:

xy XZ )

):2 ~ 3 ' z , (18)

(ix) i q = ( x , y , z ) ~ and r~i=-r+ 0 , (19)

jz

where X and Z are the physical dimensions of the repeated block. Thus, each of the blocks is defined to have the same magnetisation state as the main computational block. Since this only effects T no extra computation is required during the simulation.

6. Vectorisation of field computations

Since self-sorting vector algorithms are available for computing FFT and IFFT operations (for example see NAG Fortran library routine c06fut), the far-field computation is straightforward to vectorise efficiently. The near-field computation, sum- marised in Fig. 5 is more problematic. The aim here is to vectorise the large inner loop of this algorithm. To achieve this effectively, an indexing scheme is required such that within the inner loop;

(i) variables M, H and Tr are accessed contigu- ously, where T~(a, d) = T,,(a, a + d).

(ii) the periodic boundary condition be handled without conditional statements.

The following approach has been adopted. An n-dimensional index (qo . . . . . q,, 1) e {0. . .ao - 1, .... 0.. .a, , ~ - 1 } is mapped onto the address

For each layer (I=0..Y-I)

For each neighbour offset (d)

If 0<=(dy+l)<Y

For each grain position (a) in layer (1) ( bx= ax+ dxmod X bz= az+ dzmod Z by= ay+ dy

H(a) = H(a)+Tr(a,d)M(b ) }

Where Tr(a,d)=Tn(a,a+d)

Vectorise

Fig. 5. Ovcrview of the near-field algorithm.

space by

address(qo . . . . ,qn 1) = qo -~- ao(qt + al(q2 + ...

+ a,, 2q~. l). . .). (20)

Any discretized function may be represented as an array by defining a one-to-one mapping, I, from its domain to the n-dimensional index space. The inner loop of the algorithm traverses the domain of each of the functions in the order of in-layer rows. Constraint (i) requires that I be defined so that the image of the traversal (an index space traversal) is of the form (qo, ql) = (0, 0) . . . . . (ao - 1,0), (0, 1) . . . . . This then being mapped to a contiguous range of the address space by Eq. (20). In particular, the index for arrays M and H, which have identical domains, may be defined by the mapping,

l~da) = (a:,,a:,ar), a c R , (21)

where R is the material region discretized on the regular grid, and the index of Tr by the mapping,

IT(a, d) = (ax, a:, at, N~ + d~, Nz + dz, Ny + dy),

acR, deRu, (22)

where R.~. is the near-neighbour region. Constraint (ii) requires that no conditional tests

are performed within the inner loop, this is clearly broken by the algorithm since it uses the rood operator to impose the X- and Z-axis boundary conditions. Two main changes are necessary.

Firstly, to remove the problem of z-periodicity, arrays M and H are extended to allow parts to be duplicated. Fig. 6 shows the extended storage space

198 M. Jones, J.J. Miles/"Journal of Magnetism and Magnetic Materials 171 (1997) 190-208

X

Copy of B Region A T

Z

Low Mem

Region B

Copy of A

!

High Mem

Fig. 6. Copying regions of a layer to handle Z periodicity.

for a single layer of the magnetisation (or field) array. The convention used here is that elements ordered by scanning left to right and down are contiguous in memory. Due to z-periodicity, elements in region A have near neighbours in region B and so region B is copied to lower memory than region A to form the required adjacency. In the same way elements in region B have neighbours in region A and so region A is copied to a higher memory than region B. In addition, an extra empty row is inserted at the start and the end of the layer. These rows are explained later. The new indexing for the extended arrays M ' and H' is defined as

I'M(a) = (ax, az, ax), a ~ R' = {0... X - 1,

0 . . . Y - 1, - N= - 1 . . . Z + Nz}. (23)

Secondly, x-periodicity is overcome by mapping the elements of array Tr into a larger array T'r, where the indexing of T'r is defined by

I'T(a, at) = (ax, az, ay, N x + d~,

N z + d z + 1, N x + d y ) , a ~ R ,

d e R u = { - N x . . . N~, - N y . . . N~.,

- - 1 . . . + 1 } . ( 2 4 )

The new array T'~ is defined over a larger near- neighbour region R;4. The algorithm is modified so that d varies over this new region and the rood X

Nz!

(a)

X Enlarged ~ Neighbour R e g i o n ~

S Tensors ~ - - ~

(b)

Correct Positions

Fig. 7. The geometrical meaning of the modified tensor storage.

term is removed. The effects of the removal of the periodic boundary are compensated for by pre- sorting T'r so that

T;(I'r(a, d + B(a + d))) =

{ T r ( l r ( a , d ) ) f o r a e R , dER~. , (25)

f o r a ~ R , d ~ R ' N - RN,

where B is defined as

0 if0 ~< Vx < X,

B(v) = (0, 0, 1) t if vx < 0, (26)

(0,0, - 1)t i fv: ,~>X.

The geometrical meaning of this re-arrangement is shown in Fig. 7. In Fig. 7(a), the near-neighbour region of an element near the left-hand boundary of the material lies across the boundary. When the periodic boundary is not enforced, the region wraps around one-row lower in memory. Fig. 7(b) shows how this effect may be cancelled by extending the neighbour region to include an extra row at the start and end, and by shifting tensors in regions which undergo the wrapping effect. The extra rows in the M' and H' arrays shown in Fig. 6 are to prevent the algorithm from reading outside the new arrays when computing the field of elements on the first or last rows of the layer.

7. Evaluation of code performance

The asymptotic time requirement for the far field algorithm is O(N log(N)), and for the near field O(N). Thus the total asymptotic time requirement

M. Jones, J.J. Miles/Journal ( f Magnetism and Magnetic Materials 171 (1997) 190 208 199

is O(N log(N)). To measure the actual time requirement of the algorithm, the field was evaluated 20 times and the amount of CPU time spent in calcu- lating the near and far components was recorded. An average value of each is then used as a measure of the speed of the code. This procedure was repeated in a series of timing experiments where the number of elements N, the number of layers Y and the size of the neighbourhood region Nx, Ny, N~ are all varied to investigate how the time requirement is effected by scaling of the problem. The experiments are each described separately along with their results.

7.1. Experiment 1

A single layer of material was used with a fixed- neighbour region size of 3 x 1 x 3 elements. The number of elements in the material was varied so that the sample remained square. Fig. 8 shows the timing results compared to a complexity of O(N). The algorithm scales linearly with a slope smaller than unity. In this case, the vector speedup experi- enced as the size of the calculation increases ap-

pears to have more than compensated for the O(N log(N)) complexity. Fig. 9 shows the relative speed of the near- and far-field calculations. The calculation is dominated by the far field in this case.

7.2. Experiment 2

The effect of increasing the number of layers was investigated using layers with a dimension of 64 x 64 elements, and neighbour region size of 3 x 3 x 3 elements. The number of layers (Y) was varied from 3 to 8. The results are shown in Fig. 10. Here O(N) and O(Nlog(N) lines are plotted for comparison. It can be seen that the computat ion takes less than O(N log(N)) time for any number of layers tested. The actual time is closer to O(N) complexity. Fig. 11 shows the relative speed of the near- and far-field calculations. The far-field domi- nates the computat ion throughout.

7.3. Experiment 3

Experiment 2 was repeated with an increased neighbour region size of 5 x 5 x 5 elements. This is

0 . 3 - -

0.25

5 02

o) 0.15

g 0.t

> <

Timing results for single layer material

0.05

I I I I m

O0 1 2 3 4 5 6 7 Number of grains x 104

/ / / /

/ / / / /

/ / /

/ / . . ~ Test results

. / / -

, Y / / / j

/ /

L

Fig. 8. The average time for a full-field calculation plotted against the size of the simulation•

200 ,I/1. Jones, J.J. Miles ,," Journal q f Magnetism and Magnetic Materials 171 (1997) 190-208

0.2

0.18

0.16

r -

. 0.14'

~o.12

~ 0.1

0.08

o~

0.06 o)

0.04

0.02

Timing results for single layer material shewing near and far times seperately i i i i i i

Far field

Near field

0 L~ 0 1 2 3 4 5 6

Number of grains 7

x 1 0 4

Fig. 9. The relative cost of near and far parts of the field calculation for experiment 1.

0.5 Timing results for increasing number of layers (3x3x3 neighbours)

i i i i i i

0.45

0.4

"5 0.35 o

o ~

-~ 0.3

'E

o 0.25 a)

E

0.2

< 0.15

0.1

0.05

O(NIog(N))

I 1 I I i I I I I

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 Number of 64x64 layers

Fig. 10. The cost of the field computation with increasing numbers of 64 x 64 element layer,,

M. Jones, J,J. Miles/'Journal of Magnetism and Magnetic" Materials 171 (1997) 190 208

Timing results for P layer material showing near and far times seperately 0.11 --

0.1

0.09

~0 .08

-----007

o 0.06

~ 0.05

< 0.04!

0.03

0.02

i i i i

Near field

I. I I I

1 5 2 2.5 3 Number of grains

3.5 x 104


201

0.9

0.8

c 0.7 ._o

--~0.6 ( 9

-o ._~

0.5

5 I I )

E 0.4 Q

0 9 ,~ 0.3

0.2

0.1:

Timing results for increasing number of layers (5x5x5 neighbours) r i i

. . , ' "

I

3.5 4

O(Nlog(N)) . .

I I I I I I I

4.5 5 5.5 6 6.5 7 75. 8 Number of 64x64 layers

Fig. 12. The cost of field evaluation with a neighbourhood region of dimension 5 x 5 x 5 elements.

202 M. Jones, J.J. Miles'/Journal of Magnetism and Magnetic Materials 171 (1997) 190 208

0.3

0.25

g -~ 0.2

23 ..~

~= 0.15

E

g 0.1

<

0.05

Timing results for P layer material showing near and far times seperately

.4,-------

Far field

1 5 2 2.5 3 3.5 Number of grains x 10 4


as large as required in practice. As can be seen from Fig. 12, the algorithm still performed better than O(N log(N)), but the computation is dominated by the near field computation as shown in Fig. 13.

8. Simulation results

T h e c o m p u t a t i o n a l f r a m e w o r k w a s u s e d t o s i m u -

l a t e a s e c t i o n o f M E T w i t h a p h y s i c a l d i m e n s i o n o f

Fig. 14. Ttle cross-section of the material structure used in the simulation.

Table 1 Some constants used in the hysteresis computat ion

Parameter Value

Ms

Hext 2( ),

R V~ V=

5.2 x 105 A/m 2.3 x 105 A/m 1.0

- 2.2117 x 105 0.05 5 5

M. Jones, J.J. Miles ,; Journal q[ Magnetism and Magnetic" Materials 171 (1997) 190 208 203

x 10 s . . . . . .

i i :

] D U 1' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r- I o ] : : ~ i, i

.-~ o] - - 7 . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . r-i . . . . . . ~? . . . . . . . . . . . . . . . . . . . . . . . . . + . . . . . . . . 13b

I : [ :

-2 . . . . . . . . . . . . . . . i . . . . . . . : . . . . . . . . . . . . . . . . . . . . : . . . . .

t - 3 i

-45[ " j ~ i i i . . . . . . . . . . . . I . . . . . . . . . . ! . . . . . . . . ! . . . . . . . . -

- 2 . 5 - 2 - 1 . 5 -1 - 0 . 5 0 0 ,5 1 1.5 2 2.5 Field Z x 10 s

8 0 o o I . . . . . . . . . . . . . . . . . . . . . . . . . . .

! 7 0 0 0 ~ . . . . . ~ . . . . . . . . . :: . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . .

! ooo I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 0 0 4 . . . . . . . . . . . . . . ' . . . . . . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

°t i T

3o0o III ~' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i i |

2 0 0 0 ~ . i . . . . ! . . . . . . . . . . . . . . ) . . . . . . . . . . . . . . . . . . . ! . . . . . . . . . . . . . .

1 0 0 0 0 ~ : i -2 .5 - 2 - 1 .5 -1 - 0 . 5 0 0.5 1 1.5 2 2.5

Field Z x 10 s

Fig. 15. A simulated hysteresis loop (top) for a sample of MET. Asterisks are used to mark loop steps at which the magnetisat ion state of the material was investigated. The number of field calculations (bottom) for each field step,

204 M. Jones, J.J. Miles / Journal of Magnetism and Magnetic Materials 171 (1997) 190-208

960 x 150 x 960 nm. The structural simulation con- sisted of 10 layers of 64 x 64 elements arranged in straight columns, packed using the limited random approach described in Section 3. This gives a packing fraction of 0.85. Fig. 14 shows the cross-section of the columns used. The columns lean with at an angle of 50" to the vertical (Y-axis). Columns of material are separated using 0.75 nm spacing between units of the regular grid. The other constants used in the simulation are shown in Table 1. Here Ms is the saturation magnetisation for the material of the MET columns, the magnetisation saturation for the complete sample has a value of 4.4 x 105.

A simulated hysteresis loop for this structure is shown in Fig. 15 along with a profile showing the number of field evaluations used to perform each integration step. Over the 150 external field steps in the loop 57 000 field evaluations were required. The total computation time was 6 h 30 rain. This means that the field was evaluated 2.44 times per second on average. The material has a coercivity of 9.6 x 104 A/m and a remanence of 3.96 x 105 A/m. The magnetisation state of the material was sam-

pied at a series of 42 loop points. The hysteresis loop and sample points are shown in Fig. 15. In order to analyse these states, regions that had a negative Z-magnetisation were grouped. These we will call domains. The number of domains at each step is plotted in Fig. 16. Here, the vertical dashed line indicates the coercive point. It can be seen that as the loop is traversed, starting from a positive saturated magnetisation state, the number of domains increases to a maximum, which coincides with a point mid-way between coercive and remanent points. After this point, domain regions merge rapidly as the coercive point is approached.

The first layer of the material is displayed at points on the hysteresis loop shown in Fig. 17. The domains are clearly oriented along the Z-axis direction. This is the direction in which the columns lean.

The mean, minimum, and maximum domain sizes, at each point on the hysteresis loop, are plotted in Fig. 18. Again the vertical line in this figure represents the field strength at the coercive

350

300

250

o 200 a

150

z

1 O0

50

! ! ! , !

i i i ! i

i i il : i i i

......... ! ......... i ......... ii, . . . . . . . . . . . i . . . . . . . . ! . . . . . . . . i

i : il i i

i i !I i

i i i i i I

0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1.5 2 2.5

Field Z x 10 s

Fig. 16. The n u m b e r of d o m a i n s at each of the s a m p l e d hys teres is points .

M. Jones, J.J. Miles /Journal of Magnetism and Magnetic Materials" 171 (1997) 190-208

5 x l O S ! E p i I q

4 i i i/ / - " . . . . i . . . . . . . . ] " "

2 . . . . . . . . . . . . . . . . . . . . "

N 1 20 ~ 0 i

• / - 2 / - . -

/

- 3 . . . .

_ g

I i I I I I I [

-2 .5 - 2 -1.5 -1 -0.5 0.5 1 1.5 2 Field Z

2.5

x 10 5

Fig. 17. The hysteresis points for which magnetisat ion states are shown.

205

12

1 0 . . . . . . . . . . . . . . . . . . . . . . . .

E

c_

,_~ E

g

"5

! I

\

I

l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : \ : \ :

4 . . . . . . . . . . . . . . : . . . . . . . . i~ . . . . . . . } . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . .

o ~ !L i i i ~i i " 2 . . . . : . . . . . . ; . . . . . . ~L . . . . . X . . . . . . . . : ~ - ~ - ~ - . . . . . . . . . . i ~ " ~ , i . . . .

0 i i , i i i i -2 .5 - 2 -1 .5 -1 -0.5 0 0.5 1 1.5 2 2.5

Field Z x 10 s

Fig. 18. The mean of domain sizes. The dashed lines show the min imum and max imu m domain sizes.

206 M. Jones, .].J. Miles /Journal (?[' Magnetism and Magnetic' Materials 17l (1997) 190-208

10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 × ×

I 0 20 30 40 50 60 70 0 10 20 30 4 0 50 60 70 X X

70 0 10 20 30 40 50 60 70 X

0

10

N

4O

5O

o 10 20 30 40 50 60 ~ 0 70~ |0 20 30 40 50 60 70

× 3 8 ×

Fig. 19. The domains in the first layer of the material at the a series of eight points approaching coercivity.

M. Jones, J,J. Miles / Journal of Magnetism and Magnetic Materials 171 (1997) 190 208 207

point in the hysteresis loop. Near to the remanent point domain sizes remain approximately constant. The mean size of the domains can be seen to in- crease exponentially as the material undergoes re-

versal. The growth of magnetic domains is illustrated in Fig. 19.

The domains are further analysed to characterise their three-dimensional shape. This is accomplished

1 .6

= 1 .4

.gq 1.2

.>_

0 . 8

._c

0 . 6 . . . . . . . . .

0.4 . . . . . . . . . . :'

I 0.2 ............ i.

:1

i 0 :1

-0.2 - 1 . 5 - 1

. . . . . t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . .

i

. . . . . | . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

: i

. . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . .

; L i I L - 0 . 5 0 0 . 5 1 1 .5 2

F i e l d Z 2 . 5

X 10 s

1 . 8 P ~ 1

. . . . . . I . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . 1 .6

1 . 4 . . . . . .

1 . 2 . . . . . I . . . . . ! . . . . : . . . . . . ! . . . . ! . . . . . . . ! i "

. £

~0 .8

0.6

0.4

0.2

0 - 1 . 5 - 1 - 0 . 5 1 .5 0 . 5 2 2 . 5

F i e l d Z . ~ 5 X l i t

Fig. 20. The mean (top) and standard deviation (bottom) of domain orientation angle.

208 M. Jones', J.J. Miles/Journal of Magnetism and Magnetic Materials 171 (1997) 190-208

by finding the position covariance matrix for points within each domain. The eigenvalues of each covariance matrix correspond to the principle components of the shape of the corresponding region. The angle between the principle axis of a region (the eigenvector with largest eigenvalue) and the column axis is found for each region. The mean and s tandard deviation of the angle for each sample point is shown in Fig. 20. Note that here the sign of the angle has no meaning as the principle shape axis of a domain has an orientation but not a direction. It can be seen that domains have shapes which are elongated along the column axis. This charac- teristic persists up to the coercive point.

9. Conclusions

We have presented a comprehensive model for the micromagnet ic simulation of metal evaporated tape. A sophisticated structure model has been de- veloped which reflects both the isotropic distri- bution and curved axis of M E T columns, The structures randomness, curvature, lean angle, packing fraction and aspect ratio of the columns may be varied independently in the model. This provides the potential for future experiments on the effect of material structure on magnetic properties. At the same time the model structure allows the use of a novel fast field calculation algorithm. The algorithm is efficient, having an asymptot ic time requirement of O(N log(N)), allowing large structures to be simulated.

The simulation has been used to generate a hysteresis curve for a sample of MET. Analyses of magnetisat ion states at points along the hysteresis loop has shown that reversal occurs via domains that are elongated along the column axis, and that

merge along the direction of column lean as coercivity is approached.

Acknowledgements

The authors wish to acknowledge the suppor t of the E P S R C for this work. The computa t ions were performed on the Fujitsu VPX at Manchester Comput ing.

References

[1] 1. Tagawa, A. Takeo, Y. Nakamura, J. Appl. Phys. 75 (1) (1994) 6776.

[2] J.G. Zhu, IEEE Trans. Magn. 28 (5) (1992) 3267. [3] R.H. Victoria, J. Appl. Phys. 62 (1987) 4220. [4] R.H. Victoria, J.P. Peng, IEEE Trans. Magn. 25 (3) (1989)

2751. [5] L. Greengard, Science 265 (1994) 909. [6] M. Mansuripur, R. Giles, IEEE Trans. Magn. 24 (6) (1988)

2326. [7] L. Greengard, V. Rokhlin, J. Comp. Phys. 73 (1987) 325. [8] L. Greengard, V. Rokhlin, Comp. Phy. Commun. 48 (1988)

117. [9] J. Barnes, P. Hut, Nature 324 (1986) 446.

[10] G. Ronan, K. Matsuyama, E. Fujita, M. Ohbo, S. Kubota, S. Konishi, IEEE Trans, Magn. 21 (6) (1985) 2680.

[11] M. Mansuripur, J. Appl. Phys. 63 (12) (1988) 5809. [12] E. De La Torre, IEEE Trans. Magn. 21 (5) (1985) 1423. [13] G.F. Hughes, J. Appl. Phys. 54 (9) (1983) 5306. [14] Y. Nakatani, Y. Uesaka, N. Hayashi, Jpn. J. Appl. Phys. 28

(12) (1989) 2485. [15] J.J. Miles, B.K. Middleton, J. Magn. Magn. Mater. 95

(1991) 99. [16] T. Min, J.G. Zhu, J. Appl. Phys. 73 (10) (1993) 5548. [17] J.L. Blue, M.R. Scheinfein, IEEE Trans. Magn. 27 (6)

(1991) 4778. [18] S.W. Yuan, H.N. Bertram, IEEE Trans. Magn. 28 (5)

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Documents

An accurate and efficient 3-D micromagnetic simulation of metal evaporated tape