Title: Surface pole modeling of field access magnetic bubble devices
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00098071/00001
 Material Information
Title: Surface pole modeling of field access magnetic bubble devices
Physical Description: vii, 147 leaves : ill. ; 28 cm.
Language: English
Creator: Lai, Fang-Shi Jordan, 1948-
Copyright Date: 1980
Subject: Magnetic bubble devices   ( lcsh )
Magnetostatics   ( lcsh )
Domain structure   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Statement of Responsibility: by Fang-shi Jordan Lai.
Thesis: Thesis (Ph. D.)--University of Florida, 1980.
Bibliography: Bibliography: leaves 143-146.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098071
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000100181
oclc - 07316086
notis - AAL5642


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I would like to express my gratitude and deep appreciation to the

chairman of my supervisory committee, Professor J. Kenneth Watson, for

his patient guidance, sincere encouragement and kind support throughout

the course of this research.

I am very grateful to Professor A. D. Sutherland, Professor H. J.

Monkhorst, Professor M. Zahn and Professor J. J. Hren for their invalu-

able suggestions for my dissertation, and for their participation on my

committee. I would like to thank Professors M. A. Uman and U. H. Kurzweg

for their participation on my committee.

Gratitude is also extended to Professor H. J. Yu and Professor S. Y.

Lee, supervisors of my master's thesis, for initiating my interests in

magnetic bubble devices. I also thank Mr. H. Akiba for his help in my

experimental work.

Financial support from the National Science Foundation is also




ACKNOWLEDGEMENTS ................ . . .. ii

ABSTRACT. ... . . . . . .... .. . . . . . v


I INTRODUCTION . . . . . . . . ... . .. 1

MAGNETIC DOMAINS . . . . . . . .. 7

2.1 Introduction . . . . .. . . .. .. 7
2.2 Review of Two-dimensional Discrete Fourier Series 8
2.3 Field Calculations of Arbitrary-shaped Magnetic
Domains . . . . . . . . . . . 13
2.4 Computer Results . . . . . . . 18
2.5 Bubble-bubble Interaction Problem . . . .. 24
2.6 Conclusion . . . . . . . . . . 33


3.1 Introduction . . . . . . . . . 34
3.2 Description of the Model . . . . . . . 35
3.3 Evaluation of Area Under Singularity . . . .. 41
3.4 Magnetostatic Energy of Permalloy Pattern . .. 44
3.5 Discussion . . . . . . . . . 45


4.1 Introduction . . . . . . . . . . 50
4.2 Rectangular Bar . . . . . . . . 51
4.3 Bubble Logic Circuit . . . . . . . 58
4.4 HI Permalloy Pattern . . . . . . ... 62
4.5 Chevron Permalloy Pattern . . . . . .. 64
4.6 Half Disk Permalloy Pattern . . . . .... 64
4.7 Conclusion . . . . . . . . ... 74


5.1 Introduction . . . . . . . . . 76
5.2 Limitations of Permalloy Pattern in Magnetic
Bubble Devices . . . . . . . 76
5.3 Preparation of Experiments . . . . . . 82

5.4 Bitter Solution Technique . . . . . . 84
5.5 Domain Wall Observation in the Permalloy Patterns . 85
5.6 Remanent States of I-bar Patterns . . . . 89
5.7 Phenomenological Model for Remanent States . . 94
5.8 Conclusion . . . . . . . . . . 112

VI CONCLUSIONS . . . .. . . . . . . . 114





DISTRIBUTIONS . . . . . . . . . . . 137

LIST OF REFERENCES . . . . . . . . . . . . 143

BIOGRAPHICAL SKETCH . . . . .. .. . . . . . . 147

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Fang-shi Jordan Lai

December 1980

Chairman: J. Kenneth Watson
Major Department: Electrical Engineering

The studies of this dissertation are concerned with three aspects of

permalloy patterns that are used as field access magnetic bubble devices:

the calculation of magnetostatic interaction with arbitrary-shaped mag-

netic domains, the development of a surface pole model to analyze arbi-

trary-shaped permalloy patterns, and a systematic study of their remanent

states by using a Bitter solution technique.

Magnetostatic interaction between the permalloy patterns and magnetic

domains is the basic step of the modeling problem. Calculations of field

distributions of arbitrary cylindrical-shaped magnetic domain arrays are

then developed as a primary step toward the completion of device modeling.

In the present method there is generated a scalar magnetostatic potential

#(r), in discrete Fourier series form, which is related to the specified

distribution of magnetic pole densities by Poisson's equation. The mag-

netic field strength is then derived from the gradient of the scalar

magnetostatic potential, using the orthogonal properties of Fourier

series to determine the unknown coefficients. A program has been prepared

by using the Fast Fourier Transform (FFT) algorithm and numerical results

are presented. Calculated results compare well with known results for a

specific lattice of magnetic bubble domains, a 24pm x 41.57um hexagonal


For the analysis of the permalloy patterns of field access magnetic

bubble devices, it is assumed in this dissertation that the magnetic poles

are distributed only on the surfaces. The whole model is derived by

assuming the permalloy has infinite susceptibility and each element is at

a constant magnetostatic potential. A usual (N+M) x (N+M) pattern matrix

can be obtained for N digitized subareas around permalloy edges and M

different permalloy patterns. In conjunction with two-dimensional dis-

crete Fourier series method, the potential well profile for arbitrary-

shaped permalloy patterns and magnetic domains can then be computed. The

numerical results are also discussed in comparison with results for

George-Hughes, Ishak-Della Torre and Simplified Fourier Series (SFS).

All the models have good agreement qualitatively, with some quantitative

discrepancies among their results.

Several typical propagation patterns are analyzed using the surface

pole model, such as the rectangular bar, HI, chevron and half disk patterns.

The versatility of this model is also demonstrated by calculating the

potential well profiles of a bubble logic circuit and a multiple chevron

pattern used to propagate a stripe domain.

Remanent states, which may limit the application of our above model,

have been studied in permalloy I-bars using a Bitter solution technique.

The permalloy I-bars were fabricated from vacuum deposited films, of

thicknesses ranging from 500 A to 4000 A. The width of the bars was

varied from Sum to 15um, and the length-to-width (aspect) ratio from 2

to 20. The results from the experiments showed that remanent states

tended to be found in thin, narrow-width and large aspect ratio permalloy

I-bars. A phenomenological model for the remanent state is developed

and is used as a basis for discussing the experimental results.


The rapid increases of bit density in magnetic bubble devices have

made them become possible competitors in the area of mass storage memory.

In engineering applications, the existence of a suitable model to study

the behavior of device operation becomes very important. This is the

major motivation from which the whole dissertation has been carried.

Typical field access magnetic bubble devices are fabricated from a mixed

rare earth garnet [52], permalloy propagation pattern, silicon dioxide

spacer and Gadolinium Gallium garnet substrate (see Figure 1). Infor-

mation in these devices is carried by the existence of, or absence of,

a magnetic bubble used to represent the binary code. A bias field,

usually from the permanent magnets, is used to sustain the cylindrical-

shaped magnetic bubble. A magnetic bubble can be collapsed at a certain

high bias field, which is called the bubble collapse field, or it can be

expanded into a stripe at some low bias field. In order for the bubble

device to work properly, there should exist some reasonable margins be-

tween these two fields [34,35]. These margins are called operating mar-

gins for magnetic bubble devices [48]. They are also used to evaluate

the device performances of magnetic bubble memory. Unfortunately, the

operating margin is not only affected by the bias field, but it is also

influenced oy the permalloy pattern design, the temperature variation,

the characteristics of magnetic garnet, the thickness of the silicon

dioxide, and the external rotating field. Since the magnetic garnet is

selected, the subsequent magnetic characteristics and temperature



x -


Fig. 1 Tne device structure of typical field access magnetic
bubble memory.The arrows are used to show the directions
of magnetization.

variation factor cannot be changed. The whole operating margin will,

then, largely depend on the permalloy pattern design [53,541.

Previous permalloy pattern designs have usually been a cut-and-try

process, relying almost exclusively on intuition and experience. Recently,

several different kinds of propagation patterns have been designed and

tested. Within just the framework of the field access device there are

TI [50], YI, Chevron, XI [48] and half disk propagation patterns [22], to

name just a few. To design a magnetic bubble memory, one will be faced

with a bewilderingly large number of different circuits. The cut-and-try

method will become the most tedious and time-consuming method to be used.

The problem, however, of the arbitrary-shaped permalloy pattern has

made the computation of the magnetization of this pattern extremely diffi-

cult. This is due to the non-uniformly distributed demagnetizing field

within the permalloy pattern. The complicated coupling between the perm-

alloy pattern and this non-uniformly distributed magnetic bubble magneto-

static field also increases the complexity of device modeling. Most

computations make a macroscopic assumption that the magnetization within

the permalloy pattern is a continuum, despite the well-known fact that

the permalloy pattern consists of a few magnetic domains [38]. Based on

this continuum assumption, different models have been developed to calculate

the continuum magnetization distribution in the permalloy pattern, as a.

response to the external applied field. George and Hughes [20,21] developed

a continuum model based on the assumption that the susceptibility of perm-

alloy pattern is infinite. In the entire mathematical modeling derivation,

the magnetization distribution M is the basic quantity used. The magneto-

static interaction energy can be obtained from this model. Ishak and Della

Torre [33] presented a continuum model by assuming that the susceptibility

of the permalloy pattern is finite and a function of an external

applied field. An iteration scheme is brought into their model to get a

converged magnetization distribution M in the permalloy pattern. Their

calculated results will be discussed in Chapter III in comparison with

our surface pole model.

There is only one known model based on a magnetic domain approach.

Khaiyer [36] suggested a two-dimensional domain model using closure con-

figuration. Only the straight wall moving is allowed in this model,

which makes it questionable if the non-uniformly distributed bubble mag-

netostatic field is considered. Almasi and Lin [4] proposed a new approach

to the modeling problem. In their elegant closed-form equation, the bubble

energy due to the permalloy pattern is considered, which is in direct con-

trast to the approaches used by the continuum and domain models. However,

the error in the energy value could be as high as 20%, and it is very

difficult to analyze the arbitrary-shaped permalloy patterns in which

a specific consideration should be made in advance [3].

This dissertation treats the bubble device modeling problem from a

perspective of pole distributions, rather than from a distribution of

magnetization M as in previous models. The present pole approach has

resulted in several advantages over the previously used magnetization

methods. The pole method avoids basic questions that concern the distri-

bution of M: whether a domain or a continuum distribution exists and

whether a unique distribution exists [301. Furthermore, the two dimen-

sional distributions are of a single scalar quantity (pole density) rather

than two components of a vector quantity (Mx,M ). The simplicity of the

physical model also leads to numerical simplifications by largely reduc-

ing the dimension of the matrix that must be solved. This leads to a

simple program implementation that makes it possible to address more com-

plicated problems.

The bubble-bubble interaction problem [9] is clearly understood.

The permalloy mediated bubble-bubble interaction, however, is not brought

into consideration in other models, except the one-dimensional simplified

Fourier series model [13]. It may become very important in future high

bit density bubble devices, where the small bubble will introduce high

magnetization. In such case, the bubble-bubble interaction cannot be

neglected. In most magnetic bubble devices, the magnetic bubble is not

propagated in exact cylindrical shape. The size fluctuation of the

bubble [21] will also affect the operating margin of the device. In

order to let our model easily treat the arbitrary-shaped magnetic domains,

which have not been considered in previous models, a discrete Fourier

series method is formulated in Chapter II.

In Chapter III, the mathematical background for a surface pole model

is derived. Permalloy-permalloy interaction [21] is also considered in

the derivations. Although it does not so strongly affect the total energy

formulation, neglected in pervious models, we include it to make the

analysis more complete. The comparison between the surface pole model and

other models is also discussed in this chapter.

Several propagation patterns are analyzed and discussed in Chapter IV

by using the newly developed surface pole model. Typical propagation

patterns, such as the rectangular pattern, HI pattern, Chevron pattern

and half disk patterns, are all displayed by their calculated potential

well profiles. The surface pole model is also demonstrated by analyzing

a bubble logic circuit and a multiple Chevron pattern used to propagate

a stripe domain, rather than a cylindrical-shaped magnetic bubble.

Large coercive forces are usually found in narrow width permalloy

patterns [2]. The device performance is rapidly decreased as the coercive

force is increased. The physical origin of this coercive force is not

yet clearly explained. The remanent state within the permalloy pattern,

however, is believed to be correlated with high coercivity. In Chapter V,

we discuss the results of observations on a series of different geometries

of permalloy I-bars. Three different classes of remanence are found from

our experiments, which used a Bitter solution technique for observing

domain walls. These experimental data are qualitatively discussed based

on a proposed zig-zag wall formation.

In summary, the original contributions claimed for this dissertation


(1) A new method for calculating the field distributions of period-

ically distributed arbitrary-shaped magnetic domain arrays

using a discrete Fourier series (DFS) technique.

(2) A new method for analyzing arbitrary-shaped permalloy patterns,

based on poles p rather than magnetization M.

(3) Implementation of the above two methods into an efficient com-

puter program which takes into account bubble-bubble and perm-

alloy-permalloy interactions. The computational algorithm takes

advantage of reduced matrix unknowns that result from the use

of a scalar pole distribution, rather than the components of

a vector M distribution.

(4) The versatility of the computer program is demonstrated by cal-

culating two new results: the analysis of a bubble logic cir-

cuit, and the multiple Chevron patterns used to propagate a

stripe domain.


2.1 Introduction

The determination of the bubble magnetostatic field is a basic step

in the modeling of field access magnetic bubble devices. The non-uniform

field distribution from the magnetic bubble leads to a complicated mag-

netostatic interaction with the permalloy pattern. Several methods [4,21,

33] used recently are based either on the continuous integration over an

isolated magnetic bubble, or else on a continuous Fourier series approach

for a bubble lattice. The basic domain configuration is cylindrical in

all the foregoing derivations.

This chapter deals with the computation of magnetostatic field dis-

tributions generated by arbitrary cylindrical-shaped magnetic domains,

using a two-dimensional discrete Fourier series method. We found that

using the discrete approach and the Fast Fourier Transform (FFT) algo-

rithm leads to a reduction of computer time in calculating the Fourier

coefficients. The method also eliminates the complicated integration

that arise due to arbitrary shapes of magnetic domains.

This chapter gives the method of analysis, and numerical results

that compare very well with known results for a hexagonal lattice of

magnetic bubbles with 24im x 41.57im periods [15]. A typical 32 x 32

points calculation needs seven seconds (Amdahl 470 computer system).

The sensitivity of the present calculation method to the number of mesh

points, used to digitize the magnetic domain arrays, is discussed. Only

around the symmetry point, where the sign of field changes, is the


accuracy observed to be affected. Comparisons between the isolated bubble

and the sparse bubble lattice arealso presented. Not too much discrepancy

is observed in the normalized radial field between these two methods.

However, the normalized z component magnetic field strengths are found to

have some discrepancies. The generality of the present method for arbi-

trary cylindrical-shaped magnetic domains is also demonstrated. Three-

dimensional plotting of normalized in-plane magnetic field strengths are

calculated for a hexagonal lattice of magnetic bubble, a honeycomb mag-

netic domain array and the stripe domain.

Bubble-bubble interaction is also discussed, due to its important

role in bubble logic design. The interaction energy is derived by

the two-dimensional discrete Fourier series method. The calculated volume

averaged demagnetizing factors for a magnetic bubble show a very good

comparison with the exact calculation.

2.2 Review of Two-dimensional Discrete Fourier Series

Let F(k,p) be a two-dimensional sequence which represents the sampled

data of a continuous two-dimensional signal F(x,y) in the xy plane [49].

In this sequence, k and p are integer numbers which can be varied from

-- to m, and |F(k,p)[ is the amplitude of the sampled data at coordinates

k and p. If we assume that the increment of two consecutive sampled data

is h along x direction and s along y direction, then F(k,p) can be shown


E(k,p) = F(x,y) (2-1)
x = kh, y = ps

For a linear time-invariant system, the basic principle of convolution

theorem is also valid here. If we assume that A(k,p) is a two-dimensional

discrete sequence and B(k,p) is another one, then the output of convolution

of these two sequences can be written as

C(k,p) = A(k,p) B(k,p)

= C I A(m,n) B(k-m,p-n)
m=-m n=-w

= B(m,n) A(k-m,p-n) (2-2)
m=-m n=-~

In this equation, k and p are varied from -- to .

Throughout this chapter, the two-dimensional discrete Fourier series

and discrete Fourier Transform will be used. Their characteristics of

convolution and properties of symmetry will be briefly introduced. Con-

sider that A(k,p) represents a periodic two-dimensional discrete sequence

with the periods M in x direction and N in y direction. This sequence can

then be represented by a two-dimensional Fourier series [49] as

M-1 N-1
A(k,p) = 1 M a(m,n) ej(2/M)mk ej(2 /)n (2-3)
m=O n=O

Actually, k and p are varied from negative infinity to positive infinity

in the kp space (xy plane in the continuous signal case). Ve know, how-

ever, that if the sequence is periodically distributed with the periods

M and N respectively, then A(k,p) = A(k+M,p+N) can be obtained. From

this fact, the domains for k and p are needed only within

0 k M 1
S p N 1

Similarly, the domains for m and n are defined within

S m (2-5)
The reason is that if we assume

j (2w/M) mk
ek (m) = e mk (2-6)

e (n) =e(2T/N) np

in the periods M and N, then eo (m) = eM (m), el (m) = eM+1 (m),

eo (n) = eN (n) and el (n) = eN+1 (n) etc. Consequently the set of
M x N complex exponentials in Equation (2-3) define all the distinct
complex exponentials in the entire inverse space domain. The multiplying
factor 1/MN, used here for convenience, actually has no important effect

on the nature of representation. A(m,n) are the discrete Fourier coef-

ficients. In order to get those coefficients, the following fact should
be used
1 for k=aM and p=bN
(a,b are integers)

n=1 elnp, (2-7)
r1 n e j(27/M) mk j(2T/N) np (2-7)
TM l L e e
m=O n=O
0 otherwise

Multiplying Equation (2-3) by e-j(2n/M)kr e-j(2e/N)pt and summing from
k=0 to M-l and p=0 to N-1, we can obtain the following expression

Ml ;I A(kp) -j(2T//M)kr e-j(21/N)pt
1 A(k,p) e e
k=O p=0

I N-1 1-1 1 N- ej(2Tr/M)k(m-r) ej(2T/N)p(n-t)
1 I I I I a(m,n) e
k=0 p=O m=0 n=0

Interchanging the summation m,n and k,p we can get the following form

1 N1 A(k,p) e-j(2I/M)kr e-j(2 /N)pt
k=O p=O

M- 1 N-1
S a(m,n)
m=O n=O

1 M N1 ej(2n/M)k(m-r) ,j(2/N)p(n-t)
k=O p=O

Using the Equation (2-7), we obtain

M-1 N-1
a(r,t) = A(k,p) e-j(2/M)kr ej(2I/N)pt
k=O p=O

The discrete Fourier coefficients a(m,n) can be obtained as

a(m,n) E A(k,p) e e(
k=O p=O



Thus the transformation pair for discrete Fourier series (DFS) can be

obtained by Equation (2-3) and Equation (2-11). If this periodically
distributed sequence has only non-zero terms during the finite duration

0 < k < M 1 and 0 < p < N 1 then the Equations (2-3) and (2-11)

can be modified to get the discrete Fourier Transformation (DFT) pair

as follows:

I m=O nO a(m,n) ej(2i/M)km ej(2Ir/)pn
M N m=0 n=0

A(k,p) =
L 0 otherwise

11 In A(k,p) e-j(2 /i)km e-j(2/N)pn
k=0 p=O
a(m,n) =
0 otherwise

< k < M-1

0 p < N-1


0 _m S M-1

0 < n < N-1


Implementing these equations with the Fast Fourier Transform algorithm

(FFT), we can reduce the computing time for coefficients substantially

from the usual method used. Throughout the entire dissertation, radix-2

decimation-in-frequency algorithm of FFT is applied [49]. The reason the

radix-2 algorithm is used is that their efficiency and simplicity can

make the program much more easily implemented. A non-radix-2 algorithm

can be used to increase the flexibility of this method to calculate the

field distribution at the expense of increasing the computer running

time [49].

The coefficients calculation between the DFS (Equations (2-3) and

(2-11)) and DFT (Equations (2-12) and (2-13)) is exactly the same when

those equations are implemented by the FFT algorithm, except that the

calculated coefficients for DFT are only a portion of the DFS coefficients.

However, convolution of those sequences (DFS and DFT) is quite different.

A circular convolution (which will not be discussed here, see [49]) should

be used in the DFS case.

The symmetry properties for the two-dimensional DFS can also be

applied to reduce the calculation time. In our case, all the desired sig-

nals are real numbers. Then the following properties exist as,

Re [a(m,n)] = Re [a(-m,-n)]

Im [a(m,n)] = Im [a(-m,-n)] (2-14)


0 m M 1

0 n N 1

2.3 Field Calculations of Arbitrary-shaped Magnetic Domains

In this section, the two-dimensional discrete Fourier series method

will be applied to calculate the magnetostatic field distributions for

arbitrary-shaped cylindrical magnetic domains. First of all, we assume

that the magnetic domains are uniformly magnetized along the z direction

as shown in Figure 2. This assumption is based on the fact that typical

magnetic material used for magnetic bubble devices should have a high

anisotropy constant parallel with the z direction and low magnetization.

With the period Ax in x direction and period y in y direction, as shown

in Figure 3, the periodically distributed magnetic surface poles are

positive in the z = -d plane and negative in the z = -(d+t) plane. The

distance is d between the calculation points and the upper surface of the

film, which has the thickness t. The magnetic film has magnetization MB,

which leads to the result that the magnetic domains have the magnetic

surface pole density 2MB on their upper and lower surfaces. The peri-

odically distributed magnetic surface pole density is digitalized by

M x N mesh points with the increment distance h in x direction and s

in y direction. The surface pole density can then be shown in two-

dimensional discrete Fourier series form by

1 M-1 N-I
Q(k,p) = M- N
m=O n=O

q(m,n) ej(27/M)km ej(2T/N)pn

where k,p are the coordinates of an arbitrary digitized point on the xy

plane (Figure 3). M and N are the total mesh points within one period

along the x and y directions, respectively. The relationship between

X and M is A = Mh; similarly, A = Ns in the y direction. The coef-
ficients of the discrete Fournier series in the inverse space domains
ficients of the discrete Fournier series in the inverse space domains



Fig. 2 Three-dimensional geometry of a magnetic domain structure,
where t is the thickness of magnetic film and d is the
distance between the calculation plane and the upper
surface of the magnetic film. Arrows show the direction
of magnetization.



Fig. 3 Definition of periods Ax,xy and digitized mesh points in
xy plane.

- ---

= I II i |

i T I 1




are q(m,n). Multiplying Equation (2-15) with a shape function, U(z), in

the z direction gives the whole magnetic pole density distribution, which

leads to the following form:

Q(F) = Q(k,p) U(z) (2-16)

where U(z) is defined as

S1 z = -d or z = -(d+t)

U(z) =
0 otherwise

According to Poisson's equation, the relationship between the magnetic

pole density and the scalar magnetostatic potential is given by the MKS


V2(7) = Q(r) (2-17)

Substituting Equations (2-15) and (2-16) into Equation (2-17) and using

the difference approximation to calculate the differential equation

v2(r) (see Appendix A for detailed derivation), we can compare the
respective Fourier coefficients between the two sides of Equation (2-17).

The scalar magnetostatic potential i(r) can then be derived as

iM-1 N-i 1 F ejkz(d+t) _ejkzd j kzz
) = q(m,n) e dkz
nm=0 n=0 [F(m,n) + kz

ej(2T/M)km ej(2ir/N)pn (2-18)


F(m,n) = [ 1 cos (2m/M)] [ cos (2n/N)
h s

and kz is the inverse space parameter of the Fourier transform in the z


Magnetic field strengths of x, y and z components can then be ob-

tained from the gradients of the scalar magnetostatic potential (r).

The results are

1 M-1 N-1
Hx(kp) = -1 1 E(m,n)
m=0 n=0

Sej(27/M)km ej(27/N)pn

M-1 N-1 l
H (k,p) = E(m,n)
m=0 n=o

ej(27/M)km j(2T/N)pn

Hz(k,p) -L

[-j sin (2mn/M)]


[- j sin (2nT/N)]

M1-I N-i
m Z T (m,n) E(m,n) ej(2/M)km ej(2 /N)pn
m=0 n=0




( 1 (mn e-T(mn)d ( e-T(mn)t) for m#0 or n#0
E(m,n) 2 T= mn

T(m,n) = F(mn)

E(o,o) = 7 q(o,o)

For any specified magnetic domain shape with periods \ and y one

can determine the discrete Fourier coefficients q(m,n) from Equation (2-15),

due to the known magnetic surface pole density 2MB within the magnetic

domains. All the coefficient terms can then be calculated by the inverse

Fourier series method, which uses the FFT algorithm.

2.4 Computer Results

This method is validated by coparision to the known results of

Druyvesteyn et al. [15]. Figure 4 shows the normalized z component of

magnetic field intensity, |hz l=hz(k,p)/MB|, of a hexagonal lattice

of cylindrical bubble with 24p1m x 41.57vm periods,and with ratio (bubble

radius/film thickness) of 2.4. These two calculations yield comparable

results for three different calculation distances above the upper surface

of the hexagonal lattice.

Figure 5 shows the sensitivity of the present method to the number

of mesh points calculated on the hexagonal lattice of magnetic bubbles.

The geometry is the same as shown in Figure 6 for d/t = 1. The normal-

ized magnetic field strength |h z varies along the x direction from the

middle of the period, shown in Figure 5. Figure 6 gives the y direction

variation. It is interesting to note that not too much discrepancy is

observed in the slow-varying position even if the mesh points are reduced

from 32 x 32 points to 8 x 16 points. The accuracy has been affected for

only the symmetry position,where the sign of field changes. We attribute

this phenomenon to the trunction of high order Fourier coefficient terms,

because there are fewer mesh points to digitize the magnetic domain array.

Coparison between the isolated bubble and sparse bubble

lattice with periods 10R x 10R is also shown in Figure 7 and Figure 8.

An isolated bubble scheme is usually used in previous models [4,13,21,33],

due to the fact that in typical field access magnetic bubble devices,the

distance between two consecutive bubbles is four times the bubble diameter.

This interaction effect, then, can be neglected. Figure 7 shows the nor-

malized radial field strength, h r[ = / Hx + FyH; / B I,as a function

of normalized distance, r/R. This figure shows that even though there

- r(pm)

Fig. 4 Calculated magnetic field, |hZ]=IHz/MBI, as a function of
in-plane position r; where r is the distance to the field
point from the mid-point of the period. The domain geometry
is shown in Fig. 6. Open and solid symbols are used to
indicate that they have different field directions.

o r (jm)

Fig. 5 Calculated magnetic field [hz[ due to the hexagonal lattice of
magnetic bubbles as a function of in-plane distance r along
the x direction. Open and solid symbols are used to indicate
that they have different field directions.

- r(rm)

Fig. 6 Calculated magnetic field ihzI as a function of in-plane
distance r along the y direction.The inset shows the hexagonal
geometry,with dimensions given in microns. Open and solid symbols
are used to indicate that they have different field directions.

Fig. 7 Calculated magnetic field strength,l h rI=VH4+/MB,
as a function of normalized distance r/R. R is the radius
of the magnetic bubble in sparse magnetic bubble lattice and
Druyvestyn's isolated bubble. t is the thickness of the
magnetic film.




1 --DFS
0 ODruyvesteyn's
-4 bubble D for R/t:0.05
2- / 0* change sign'

S1 2 3 4 5


Fig. 8 Calculated magnetic field strength, |hzI, as a function
of normalized distance r/R. Open and solid symbols are used
to indicate that they have different field directions.

exists some discrepancy between these two calculations beyond r/R=2,

isolated bubble calculation is a fairly feasible method to simulate the

actual periodically distributed bubble lattice in the radial fields.

There is a large discrepancy, however, in the normalized z field distri-

butions of these two methods, as shown in Figure 8. This phenomenon

probably results from the fact that the repulsive flux of neighboring

bubbles will force the sign of z component field strength to be changed

more quickly than that of an isolated bubble.

The generality of this method is demonstrated by calculating mag-

netic field strengths for various geometries, changing only the program

input data. Figure 9 shows three-dimensional plots for a hexagonal bubble

lattice in a radial field and z-component field respectively. The follow-

ing figures show the radial field and z-component field distributions

for the honeycomb magnetic domain array in Figure 10 and the stripe domain

in Figure 11. All these figures are calculated at the d/t = 1 plane above

the upper surface of magnetic film, and are plotted for their absolute

value of radial normalized field strength ]hrl and normalized z-component

field strength hz. The highest values correspond to unity, as all the

values are divided by the largest value.

2.5 Bubble-bubble Interaction Problem

Bubble-bubble interaction plays a very important role in some bubble

logic circuits. In Chapter IV, we will use a logic circuit to study the

permalloy mediated bubble-bubble interaction problem. Here, we present

the mathematical derivation for the bubble-bubble interaction by using

the two-dimensional discrete Fourier series method.

Assume that a periodically distributed magnetic bubble array is

called array A, as shown in Figure 12(a), with the periods x in x

.9 i '.




Fig. 9 Three-dimensional plots of (a) radial field !hrl
(b) z field hz ,for the hexagonal lattice of magnetic


*', ..- - ,,

" i ". '. ''
":', , ,,


,b/ X

S9.6 -

.224 ,
(a) u----f '

---- ^_^

Fig.10 Plots of (a) IhrI(b) hz of honeycomb domain array.
Dimensions are in microns.





a) I




" ' ,. '

I .


Fig. 11 Plots of (a) lhr[(b) hz of stripe domain array.Dimensions
are in microns.



> magnetic bubble


Fig. 12 (a) Periods and interaction distance are defined between
group A and group B magnetic bubble arrays.
(b) Coordinates are used in bubble-bubble interaction pro-

direction and A in y direction. In order to eliminate the bubble-bubble

interaction from another magnetic bubble in the same array A, x and X
x y
are set to a large number. The bubble-bubble interaction then comes only

from the bubbles between array A and array B, which is shown in Figure

12(a) by a dotted line. Array B magnetic bubble is also set to have the

periods A and y. The bubble-bubble interaction distance between these
x y
two bubbles is thus s in the y direction. The minimum interaction dis-
tance s equals 2R if the radius of the magnetic bubble is R. The distance

s can be set to zero, such that the calculated normalized number is the

bubble volume-averaged demagnetizing factor N, which will be discussed

later. The coordinate origin that we use here is set at the middle point

of the magnetic film, with a thickness t, as shown in Figure 12(b).

The z-component magnetic field strength, Hz, from the array A bubble
can then be derived by the previously shown method, and has the following

H- (7) = 1 N G(m,n) ej(2/M)km ej(2,/N)pn (2-22)
m=O n=O


q(m,n) e(-t/2) T(m,n) cosh [T(m,n)z] z f t/2
G(m,n) =
1 qm e-T(m,n)t z = t/2
2 q(m,n) e

T(m,n) and q(m,n) have the same definition as shown in Equation (2-21).
The interaction energy of these two interacted magnetic bubbles is then

E2b= 2 Mg Hz dv (2-23)

The factor 2 comes from the same definition shown before. MB is the

magnetization within the magnetic bubble domain and v is the volume of

the entire magnetic bubble. Substituting Equation (2-22) into Equation

(2-23), E2b can be obtained as follows:

2MB E M-1 N-I j(2/)km j(2T/N)pn
E2b = M R2 G n ei(mn)/N)pn Azi
i=l m=O n=O


where G.(m,n) has the same definition as Equation (2-22),except that the

z is replaced by an increment of Ai. The definition of Azi is shown as

t = C Azi (2-25)

The equivalent magnetic field strength can then be defined as

h = 2b (2-26)
zn 4 MB TR2t

By using the previous program, the bubble-bubble interaction energy can

then be calculated. If we set s=0, then hzn should be equal to the volume-

averaged demagnetizing factor. The calculated results are shown in Figure

13 in comparison with the exact calculation. The figure shows the com-

parable results between these two methods. Figure 14 shows the calculated

hz as a function of normalized separation distance s/R for five different

radii of the magnetic bubble. These data show that the interaction energy

is sharply increased when the separation distance s is less than four

times the radius of the magnetic bubble. The interaction energy can be

neglected as the distance s is larger than 8R.


0.8- - Exact Caiculation


U. 1 2


Fig. 13 Magnetic bubble volume-averaged demagnetizing factor N is
shown as a function of normalized magnetic film thickness,
t/D. D is the diameter of magnetic bubble. The data of exact
calculation are taken from Reference [4].



Rt =0.5
0.04- R =0.33

t = 0.25



0 2 4 6 8


Fig. 14 Normalized bubble-bubble interaction energy is shown as a
function of the normalized distance s/R.R is the radius of
magnetic bubble and t is the thickness of magnetic film.


2.6 Conclusion

A discrete Fourier series (DFS) method has been used to calculate

the fields of periodic domain arrays, with results that compare well with

their continuous counterpart. The discrete characteristics of DFS are

not only convenient for computing arbitrary-shaped domains, but they also

provide for reduced computer running time by program implementation using

the FFT algorithm [421.

A bubble-bubble interaction problem is also demonstrated by using

the DFS method. By comparing their results with exact calculation in

volume-averaged demagnetizing factors, the use of the DFS method is also

validated in the bubble-bubble interaction problem.


3.1 Introduction

Device models are very useful in engineering applications. They can

be used by device designers to predict the device performance numerically,

rather than dig into the cut-and-try experiments. Several papers [4,13,

21,33,43,45] that deal with this subject have been published. They all

consider the magnetization M as the basic quantity in their model formu-

lation. The two components, Mx and My in the two-dimensional modeling

have increased the complexity of their mathematical derivation.

In this chapter, the surface pole model will be derived. The basic

quantity of this model is the magnetic pole distribution, which is used

instead of the magnetization distribution. Thus the previous two unknown

components, Mx and My are replaced by the only one unknown component p.

This method reduces the complexity of mathematical formulation a great

deal [30].

In conjunction with the two-dimensional discrete Fourier series

technique, the permalloy mediated bubble-bubble interaction is considered

into the surface pole model. The permalloy-permalloy interaction is also

included [56]. All of these are major contributions to this newly devel-

oped surface pole model, and are not found in previous models.

The detailed mathematical analysis is shown in the following sections.

Its calculated results, which are compared with that of other models, are

also discussed. A good qualitative agreement is found by solving a 40 x

40 matrix in the surface pole model.

3.2 Description of the Model

By assuming the magnetization is continuously distributed within

the permalloy pattern, the continuum model views the magnetization as

the average of individual domain magnetization. In our model, a con-

tinuum scheme is applied throughout the entire chapter. Although the

domain configuration actually exists within the permalloy pattern, the

study of Huijer [27] shows that the continuum model is quite adequate to

describe the magnetization behavior of permalloy patterns.

In the present model, magnetostatic energy is the only energy term

considered, since it plays a dominant role in the behavior of field access

permalloy patterns. The mathematical form for this energy can be written


Ep p(F) 4(r) dv r p() 4d (r) dv (3-1)

where p(r)is the magnetic pole density within the permalloy pattern,

caused by the external field or bubble magnetostatic field, or both.

e (r) is the scalar magnetostatic potential caused by the external field.

{d(r) is the demagnetizing potential caused by the induced magnetic pole
density p(7). Both components of this equation are integrated over the

entire volume of the permalloy pattern.

In order to get the complicated distribution of demagnetizing poten-

tial Cd(r) more simply, the susceptibility of the permalloy is assumed

to be infinity. This leads to the results that the permalloy pattern

possesses only surface magnetic poles, and furthermore, that a constant

magnetostatic potential i, exists on the surface of each permalloy pattern.

The magnetostatic equipotential r can then be described in MKS unit of

1 r r(')
1 d is" + c~r) (3-2)

where ps(r") is the surface pole density (note: p = -V7. is used here,

rather than p = -Vuo' M)of permalloy pattern, and this equation is inte-

grated on the surface of this pattern. e( r) has the same definition of

the previous Equation (3-1). The demagnetizing potential Qd(r) is then

shown as:

1 ps(7")
: ) -d 4 I ds' (3-3)

We assume that the demagnetization field is large enough to suppress

the possible magnetization variation along the permalloy thickness. The

surface pole can then be observed around the edges of the permalloy

pattern only.

From some algebraical manipulations, the external field e (r) can

be written as

e () = He (r cos2 a + r sin2 a) + b(r) (3-4)

where a is the angle between the direction of the external field and that

of the x axis, as shown in Figure 15(a). H e is the absolute value of

the external field. Ob(r) is the non-uniformly distributed magnetostatic

potential, due to the external magnetic domains. Using the same period

definition of Chapter II, the permalloy pattern has Xx period in x di-

rection and A in y direction. The same increments h and s are also

applied to digitize the periodically distributed pemalloy patterns, as
shown in Figure 15(b).
If the surface pole density around the permalloy edges is assumed

constant in the sliced ith subarea, Ci, as shown in Figure 16, then
Equation (3-3) can be rewritten as [25]


r Exteral Field IHel


Definition of angle a between the directions of
external field and x axis.
Definition of permalloy pattern periods )x and y ,
and digitized increments h and s.

Fig. 15 (a)


Permalloy Pattern

* ^y

Fig. 16 Definition of sliced subareas around the permalloy edge.

I f- --

>^ I


Q d(r)= i Ci Pi (3-5)

Pi = 1-

L is the total subareas around the edge of a single permalloy element.

S' is the ith subarea of the permalloy edge. Subsequently, if there

exist M patterns within a x A period, and permalloy-permalloy inter-

action is considered only in the nearby eight periods around the desired

calculation period, as shown in Figure 17, then the mathematical form

can be modified from Equations (3-2) and (3-5) to include eight neighbor

periods and M permalloy patterns within one period shows:

-ej= Ci K p (3-6)
i=-l 1

N = Lp

K 1 pm.
ji 47 mO0 j

m imds
I^ r

tpis the constant magnetostatic potential of pth permalloy pattern within

the oth period. Qej is the averaged external field in the jth subarea of

permalloy edge. L are the total digitized subareas on the pth permalloy

pattern. S. is the ith subarea of the mth period, such that the oth

period is the desired calculation period. The r. is then the field dis-
tance within oth period and r. is the source distance between the oth

and mth periods.


Period 1 Patterns


1 2

3 ...........M

4 Period 0 5

6 7 8

Fig. 17 Period 0 is the desired calculation period,in which M
different permalloy patterns are shown. The nearest eight
periods are used to calculate the permalloy-permalloy

According to the principle of magnetic pole conservation, the total

magnetic pole of each permalloy pattern should be zero. There exist M

permalloy patterns within one period, so there should also exist the

following equations

ps (r) dsm = 0 m =1,2,3....,M (3-7)

where m is used to indicate that this equation belongs to mth permalloy

pattern. Equation (3-7) can then be rewritten as

SCm Q = 0 m = 1,2,3,...,M (3-8)
i=l 1 i

N = r L

Sim i {L
Qin= I
LO i {Lm}

Combined with Equation (3-6) and Equation -8), we can obtain an

(N+M) x (NxM) simultaneous equation. Solving this equation, the surface

magnetic pole density ps within one period and M constant magnetostatic

potentials of M permalloy patterns can be obtained [57]. This simulta-

neous equation is associated with the permalloy pattern only. Its results

can be stored and used for different magnetic bubble positions and dif-

ferent external field orientations.

3.3 Evaluation of Area Under Singularity

There are nine periods of permalloy patterns that have to be consid-

ered in the permalloy-permalloy interaction. In the oth period, the

argument of P?. approaches to infinity when i equals j, as shown in

Equation (3-6). If we expand P in the rectangular coordinate system,
then P can be rewritten as

J rt/2 [dzo dlo

S-t/2 [(x5 X$ )' + (yj Y )2 + (zj z o)2]


where t is the thickness of permalloy patterns and

dl = dx'2 + dy'2
dlio =dxio+ dyio

Integrating Equation (3-9) over the z domain from -t/2 to t/2, Equation

(3-9) can be rewritten as

Sn t/2 + R dl' (3-10)
-t/2 + R'2 + t2/4


Ri2 = (Xj X)2 + (yj y 0)

From this equation, it is very obvious that the argument of P approaches

to infinity when x = xo and yj = yo Fortunately, the logarithm func-

tion is a slow-varying function when it is approaching infinity. Exclusion
of the singular point in calculation of Equation (3-10) will not have too

much discrepancy in comparison with the exact calculated results.
Figure 18 shows the calculated results 4TPi0 of arbitrary chosen
subarea, which excludes the singular point, as a function of the number of

digitized points. In those calculations, the permalloy thickness is set
to 4000 A. It is very interesting to see from this figure that when the

digitized point is beyond 100 points, 4TTpiP starts to saturate at around

o -

2.0 x 10-6. In all the following calculations, 300 points is used to

digitize the subarea in which the singular point is found.

3.4 Magnetostatic Energy of Permalloy Pattern

The total magnetostatic energy of the permalloy patterns, that are

interacted with the external applied and magnetic bubble stray fields,
can be rewritten from Equation (3-1) as

E = ps) e(7) dv ps ) d(F) dv (3-11)

where ps(r) is the surface magnetic pole density in the permalloy
pattern. But the constant magnetostatic potential s on the permalloy
pattern is

c = e(r) + fd(F) (3-12)

Substituting Equation (3-12) into Equation (3-11), we have

E= p r) [ + e(7r)] dv (3-13)

Let us assume that the zero energy level is the energy of permalloy
pattern without introducing the magnetic bubble. The magnetostatic
energy of Equation (3-13) should then be subtracted by the energy which

is induced by the external applied field,

H (r cos2 a + r sin2 a)

Because this energy is independent of bubble positions and introduces a

constant shift on the calculated potential profile only, it can be com-

puted from Equation (3-13) by excluding the fb(r) term from te(r) shown
in Equation (3-4). Using Equation (2-18), the magnetic bubble

magnetostatic potential can be obtained. Combined with Equations (3-4),

(3-6), and (3-8), the surface pole distribution, which is introduced by

external field or bubble stray field, can be determined. Equation (3-13)

can be used to calculate the magnetostatic energy of the permalloy pattern.

The magnitude of hz, which is the potential well depth, is obtained by

normalizing E to the volume and magnetization of the magnetic bubble.


h = (3-14)
S (2MB) (R2h)

The direction of hz is parallel with that of the permalloy thickness [18].

The magnitude of hz will then locally modify the original bias field

acting on the magnetic bubble. Since the magnetic bubble tends to move

to a lower bias field, the profile of potential well depth will be very

useful in studying the propagation of the magnetic bubble under the perm-

alloy pattern.

3.5 Discussion

The calculated potential well profile for a rectangular bar has been

shown in Figure 19 for two different external field conditions, using the

surface pole model. The external field is applied along the longitudinal

direction of the permalloy bar. The potential well distribution of zero

external field results from the polarized permalloy bar. The polarization

effect mainly comes from the bubble's magnetostatic field. Symmetry dis-

tribution is found in this potential well profile, due to the movement of

the magnetic bubble under the permalloy pattern. Additional external

fields will break the symmetry situation and cause the magnetic bubble to

stay at one side only. The minimum potential well is located approxi-

mately 1.5pm inside of the permalloy edge. This fact is experimentally


- Surface Pole

-- George-Hughes

--- Ishak-Della Torre

-.%- SFS

Fig. 19 Calculated potential profiles by surface pole model are
compared with three other models. External applied fields
are 0 and 10 Oe respectively. Bar dimensions are 3x15x0.4 4m3.
Bubble parameters are D=6,m,h=3um and 4M B=200 G.


verified by Jones and Enoch [35]. In this figure, potential profiles

calculated by other models are also displayed. It seems that they are

all qualitatively consistent, but have some discrepancies among their

quantitative analysis.

In the surface pole model, the calculations are based on the assump-

tion that the permalloy has infinite susceptibility and no variation of

magnetization along the thickness direction. This leads to the conclu-

sion that only the surface poles can exist on the permalloy pattern.

In the Ishak-Della Torre model [33], the assumption is that the

permalloy susceptibility is finite and a function of the external applied

field. This assumption let them bring the iteration scheme into their

model. A carefully chosen under-relaxation factor should be decided on

at first, so that the iteration can converge. The Ishak-Della Torre

model also has a variation between the optimum under-relaxation factor

and the number of digitized mesh points. This makes the problem become

more complicated.

The George-Hughes model [20,21] assumes that the permalloy has in-

finite susceptibility. These should be only surface poles on the perm-

alloy pattern. From private communication with George, he agrees that

only surface poles can exist in the permalloy pattern, instead of surface

and volume poles as stated in his paper [20]. In the George-Hughes model,

however, volume poles are included in the calculations. The fundamental

quantity in their model is magnetization M within the permalloy pattern.

This brings about the following equation [21]:

H =- H = -7 7( F ) dv (3-15)
where i the external field and is the demagnetizing field. This
where H is the external field and Ho is the demagnetizing field. This

equation can then be expanded into two terms by vector calculus as

H = n ds' V M dv (3-16)
e r* Jv i -

where M n shows the surface poles and V. M represents the volume

poles. Based on the infinite susceptibility assumption, V-' M should

be zero. This should be set to zero explicitly in Equation (3-15), be-

cause in this equation, V1 M will not automatically become zero.

However, the George-Hughes model starts with this equation without setting

V' R = 0. Instead, they would introduce the volume poles in their cal-

culation. This situation leads to the conjecture that they overestimate

the value on the left side of Equation (3-15), which will introduce error

in their calculated results. Acutally, Ishak-Della Torre's model also

includes both volume and surface poles due to their non-constant finite

susceptibility. It is plausible that the discrepancy between Ishak-Della

Torre's and George-Hughes' results then could be caused from both George-

Hughes' overestimated results and the different assumed susceptibilities.

From the calculated results shown in Figure 19, it can be seen that the

discrepancy between Ishak-Della Torre's and George-Hughes' models is not

very much. This may lead to the conclusion that the assumed infinite

susceptibility may not affect the calculated results too much in this

permalloy sample. According to recent results by Grabau and Judy [24],

this conclusion can be true only when the product of the susceptibility

and the thickness-to-width ratio is larger than unity. If this is not

the case, the susceptibility will dominate the magnetization on the

permalloy bars [14,44].

The SFS (simplified Fourier series) model [131 has the deepest

potential well profile in Figure 19. This one-dimensional Fourier series

method may include overestimated magnetic poles in the model formulation.

They assume no variation of magnetization in the width and thickness

direction, but only in length direction. This assumption may be accurate

when the width of permalloy pattern is small, such that the demagnetiza-

tion field may cause no variations of magnetic moments between the center

and edges of the permalloy pattern. When the width of the permalloy

pattern is increased, as shown in our case where the width is 3vm, how-

ever, some degree of variation of magnetization along the width should

be expected. When this model takes the average along the width direction

and assumes no variation between the center and edge magnetic moments,

the magnetic poles density may be over-calculated. This effect may cause

the deepest potential well profile.

The cause of the discrepancy between the surface pole model and

other models is not clear. It is not known whether the two classes of

models result in the same total magnetic pole, or whether the difference

can be attributed to the pole distributions. In the latter connection,

an analogous calculation by Zahn [58] suggests that the magnetostatic

energy is greater for a volume distribution of poles than for a surface


Although George and Archer announced that their model is consistent

with experimental data [17], one would expect that some degree of quanti-

tative disagreement should exist, due to the simplified assumptions in all

the continuum models. Except for the magnetostatic energy, actually, the

wall energy, anisotropy energy, exchange energy, partial saturation, and

wall nucleation in permalloy patterns may account for the deviations of

the quantitative agreement between the theoretical and experimental data.

Excellent qualitative agreement is, however, still adequate for magnetic

bubble designers in engineering applications.


4.1 Introduction

The surface pole model developed in Chapter III is used to analyze

several permalloy patterns in this chapter. The potential well profile

for those patterns is computed and plotted as a function of bubble posi-

tions. The results are then used to characterize the individual perm-

alloy patterns.

The rectangular bar is the first permalloy pattern to be analyzed

because it is the basic building block for most existing patterns. The

three-dimensional plotting of the potential well profile is used to

demonstrate the bias depth changing as a function of applied fields.

The functional operation of the bubble logic circuit is the first one to

be verified by the numerical model. From the calculated result, we can

optimize the spacing of the structure in order to improve the logic func-

tion more efficiently. The multiple chevron pattern, with a propagating

stripe domain, is also calculated so that the potential well profile can

be plotted. The deeper potential well indicates that this permalloy

circuit has a slightly larger operation margin than that of a single

chevron pattern.

Half disk permalloy patterns are calculated for two different shapes

of asymmetric patterns and a symmetric pattern. Their computed results

have verified the experimental data that an asymmetric half disk pattern

operates better in gap crossing [8].

4.2 Rectangular Bars

The analysis of a rectangular bar in the dimension of 3 X 15 X 0.4um3

is shown in Figure 20. The rectangular bar is the basic building structure

for typical TI, YI, and XI patterns. It is, however, also the simplest

structure which can be used to demonstrate the results of numerical mod-


Three different external fields are applied along the longitudinal

direction of the bar. From their potential well profiles, it is easy to

see that the potential depth is not increased in proportion to the applied

field. The minimum potential well, where the center of the magnetic

bubble is located, is not at the edge of the bar. It is approximately

1.5im inside from the permalloy edge.

Figure 21 shows that the sequence of potential well depth, which is

plotted in three-dimensional diagram, is varied when the external field

rotates. The rectangular point is used to indicate the variation of the

potential well in accordance with the direction of the external field.

It is not used to point out the minimum potential well. Actually, from

Figure 21(c), the minimum potential well is still located at the position

near to the permalloy edge. It means that the magnetic bubble will not

move away from that position, even if the external field points perpen-

dicularly to the permalloy bar.

The interaction of permalloy mediated bubble-to-bubble and permalloy-

to-permalloy are studied in Figures 22 and 23. Figure 22 shows that when

the magnetic bubbles are located at position A and position A', respec-

tively, the potential well depth is decreased as the gap distance s is

decreased. In this figure, the bubble-bubble repulsive energy is not

taken into account. If we take the bubble-bubble repulsive energy


--1 2 .^-

Fig. 20 Calculated potential well profiles are shown as a function
of three different external fields. Bar dimensions are
3x15x0.4 um'. Bubble parameters are D=6im,h=3um and 4iMB=200 G.



w= 45


Fig. 21 Three-dimensional plots of potential well profiles at
different external field orientations. Bar dimensions are
3x15x0.4 um. External field strength is 10 Oe. Bubble pa-
rameters are D=6 um,h=3pm and 4M ,=200 G. The thickness of
Si02 is lum.

Fig. 21 Continued.

w= 90



(e i) "


Fig. 21 Continued.


m I-cu


c0 0


1n -

LA, a)
*C *I

0 C-
u m
3 *- Id

4. 0 --



0 O

0 *l 0

. .0 0.

-0 0 0
0 -- -

*- 0 s-

+C Q S


ao oz
"0 0Z

) CE


r- I-

-- C
M c)

..- S :




3 0
Or- n

o -o
(0 >C CO
5 *-


0 C

*r*- E

o 0 II
. a

r- C S-

C r- S-
0 .0 0

C..0 E
0 r- r
(0 .0
C3 0
0 -a
r- 0 .0



~o oz

calculation from Chapter II into the whole energy, the results are shown

in Figure 23. The potential well depth is decreased sharply when the

gap distance s is decreased. The potential well depth is almost flat

as the gap distance s is 0.5pm. This comes from the fact that the

permalloy pattern would be connected and become a long permalloy stripe.

In this case, the potential barrier would disappear and the magnetic

bubble would not move anymore. This is due to the fact that the gradient

of the potential well represents the force needed to push the magnetic

bubble. Double dipping in the s = 7.5im potential well is caused by a

small external field and large magnetization of a small magnetic bubble,

from which the permalloy pattern is polarized dominately by the magnetic


4.3 Bubble Logic Circuit

In this section, a simple magnetic bubble logic circuit is discussed.

In most of the existing bubble logic circuits [47], the bubble-bubble

interaction is used to act as a key function to select the desired output.

There are two possible tracks for the magnetic bubble to propagate, as

shown in Figure 24. When the magnetic bubble propagates from the right

side (not shown here) to the left side, the bubble may move through

Position 3 to Position 4, and then the upper track. It may also move

through Position 2 to Position 1, and then the lower track. Which way

it goes is entirely dependent on whether a magnetic bubble exists on the

Position P or not.

The potential well depths for these two cases are plotted in Figures

25 and 26, respectively. Figure 25 shows the situation when no bubble

exists in Position P. Assume that the magnetic bubble is located at

Position 0 originally, and that when the external field rotates from

10 Oe


--0- --3



Upper 3
Track -x-

Lower -r-

Fig. 24 The geometry of a bubble logic circuit. Bar dimesions are
shown in microns. Bubble parameters are D=6um,h=3um and
47rMB=200 G.

0./ .,,
































Q >

~a m/O



/ //

/ I,


Direction A to Direction C, the bubble does not move away from this

position. The only change that occurs is that the magnetic bubble be-

comes smaller, due to the shallow potential well depth. The magnetic

bubble moves through Position 3 to Position 4, however, until the exter-

nal field points to Direction D.

If a magnetic bubble exists at Position P, Figure 26 shows that it

will move through Position 2, from an original Position 0, to Position 1,

when the external field points to the direction D. If the magnetic bubble

in Position P represents the signal X and the bubble in Position 0 repre-

sents the signal Y, then the output from the lower track should be X Y,

AND logic function, and upper track should be X Y.

4.4 HI Permalloy Pattern

HI permalloy pattern is used in the major-minor loop [5,6,7,11] of

some magnetic bubble devices due to its simplicity in fabrication without

a gap in the long bar. Actually, its structure comes from the connection

of two T bars. This connection will decrease the potential well depth

and make it flat, however, as we have shown in Figure 22.

Figure 27 shows the calculated potential well profile for four

different external field orientations. In Field Orientation 2, the

potential well is almost flat. Because of this, the magnetic bubble

across the gap will be stripe-out rather than transfer. In Field Orien-

tation 3, the flat potential well will cause the bubble to stripe-out,

and to cover the space from Position B through Position C to Position D.

In order to avoid this, the magnitude of the external field should be

increased to create a sharp and definite potential well, as explained in

Figure 20. The minimum driving field for the magnetic bubble devices

fabricated by this pattern is therefore increased. This fact is



\ ;


/ piil _
If (-I 1

1 \N





E i
S- C=


C r-


0 0

4- yl





1- (

experimentally verified by comparing the minimum driving field require-

ments for HI, TI, and double-TI patterns [31]. The HI pattern always has

the highest minimum driving field.

4.5 Chevron Permalloy Pattern
The detector of magnetic bubble devices is usually in conjunction

with the bubble expander [5,9], which is built by the chevron permalloy

pattern to increase the signal-to-noise ratio. The magnetic domain

propagated under the multiple chevron pattern will be a stripe rather

than a cylindrical bubble shape. The potential well profiles caused by

the stripe domain under the triple chevron pattern, as well as by the

bubble domain under the single chevron pattern, are plotted in Figures

28 and 29, respectively. The potential well depth for the multiple

chevron type is much deeper than that of the single chevron pattern.

Because of this, the operating margin for the triple chevron pattern

should be slightly larger.

4.6 Half Disk Permalloy Pattern
The half disk propagation patterns represent the state-of-the-art

in bubble propagation patterns. It provides not only a large gap toler-

ance in its pattern generation, but also the lowest minimum device driving

field in all the patterns [39]. When the magnetic bubble starts to cross

the pattern gap, the half disk pattern provides two parallel poles between

the consecutive permalloy patterns that stretch the bubble into stripe.

The bubble then shrinks from its original position when the external field

rotates. This is in direct contrast to the typical TI pattern, which

introduce the orthogonal poles between the consecutive permalloy patterns.

A tight gap width between T and I patterns should be carefully






L \ /

\o C:,




0 0

i] p 1

) r4-


x ea

/ '* 0 / '4- I
/ *. /-. 0 C
/ ^ /r ^^ ^ Q
/ */ -.^ -">
/~ r-''^- 00-
/ ,0 .^' s
1 0 '* *'C T

lUL- -



o E

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/ 0

4 r- CL

1 o 00
/ 5- I*-

/ 0.

0)0 m -

N ^S^
N, -l-n

C n

designed in order to eliminate the possible failure of bubble crossing.

The stretching mechanism of bubble crossing in the half disk pattern

will slightly relieve this serious requirement. This makes the half disk

pattern a very suitable candidate for high density bubble devices, where

the gap problem of the TI pattern will increase the difficulty of device

processing. The lowest minimum driving field will also increase the ad-

vantage of the half disk pattern in temperature problems dealing with

high density devices.

Figure 30 shows the potential well profile for the asymmetric half

disk pattern at the six different field orientations to complete one cycle

of bubble propagating. The mark P in this figure indicates the minimum

potential well depth in that field orientation. Assuming that the mag-

netic bubble likes to seek the minimum bias position, then P should

point to the possible bubble location. From Figure 30(a), it can be seen

that the propagation of the magnetic bubble is not very smooth from

Position C2 through B2 to A2. The magnetic bubble lingers between Position

C2 and E2, and then moves abruptly to Position A2 when the field points

to Direction 3. As experimental data haveindicated, it is in this area

that the magnetic bubble can be easily collapsed [8]. There also exists

a small potential barrier between the gap, that is between Positions A2

and Fl. The propagation of the bubble from Position F1 to El is smoother,

as can be seen in Figure 30(b), than that of the previous period. If we

increase the width of the legs of the half disk pattern, the potential

barrier found in Figure 30 is suppressed. Because of this, it is expected

that there would be a smooth propagation of the magnetic bubble from

Position C2 to Position A2. This result is shown in Figure 31. A sym-

metric half disk is also used to calculate its potential profile, to be


1 EE


' r4


Lu\ -E

J., D

2 2
C 4 C o 4--- -r-


Ion c,
-S C)

-- 00

,+ 0
C4 -- I t )
C^ 0 *"^3 M T
r-, C> mM* LL.


E 0
5 o

7n -
- --- o

- A--


I-^ C
__ 7t

0 '

3- ,^- ----


- m:"L








4 -


r4- ,





C -

(U -
*i- Q
1< -


L, <


\ a

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- -' -__










5- 0


w -1
- U,
s- -I-


i- t/
? 0
0 S
I- 10
Q. C


c i


compared with its asymmetriccounterpart. From the result shown in Figure

32(a), it can be seen that there is a lingering problem between Position

C2 and B2, and a bad gap crossing potential profile due to the narrow

potential well in Position A2. This will cause the bubble to sometimes

collapse during gap crossing. These phenomena are experimentally veri-

fied [8], and prove that the asymmetric half disk pattern has better

operation in gap crossing.

4.7 Conclusion

The surface pole model, which is described in Chapter III, has been

used to analyze the rectangular bar, the bubble logic circuit, the HI

permalloy pattern, the chevron permalloy pattern, and the half disk perm-

alloy patterns. This model is capable of doing more than just analyze

the arbitrary-shaped permalloy patterns; it can also be used to calculate

the potential well depth profile by considering the bubble-bubble and

permalloy-permalloy interactions. Using the DFS technique developed in

Chapter II, we also can calculate the potential well profile of different

bubble shapes.

By analyzing the permalloy patterns, the surface pole model shows

the comparable qualitative agreement with the experimental observations.

In all the above calculations, a typical 40 X 40 matrix is solved to get

the unknown surface pole distributions and the constant magnetostatic

potentials of the permalloy patterns. Typical Amdahl 470 computer running

times are approximately one minute for a 40 X 40 pattern matrix generation,

and about twelve seconds for a complete analysis of twenty bubble positions

at one field orientation. The required memory for this model is close to

200K bytes.


For a 100 X 100 matrix, ten field orientations and ten bubble pos-

itions, a complete analysis for George-Hughes' model needs approximately

three minutes and 800K bytes in an IBM 370 system [21]. Ishak-Della

Torre's model needs eleven minutes and 110K words of memory in the

CDC-6400 system [32].


5.1 Introduction

The motivation of this study comes from the fact that the coercive

force of a permalloy pattern has been found to increase rapidly when the

width or thickness is decreased. This occurrence could degrade the de-

vice performance and may present a serious limitation of permalloy

technology for high bit density magnetic bubble devices. The physical

origin of this coercive force has not yet been explained. However, the

existence of the remanent state and its associated zig-zag wall config-

uration in various geometries of permalloy I-bars are studied in this

chapter in order to understand the nature of remanent states.

A Bitter solution technique is used to study the quasi-static mag-

netization response of I-bars due to external fields. Three different

classes of response have been identified from our prepared permalloy

samples, qualitatively discussed in Section 5.6. A phenomenological model

for the remanent state is afterward formulated to explain the experimental


5.2 Limitations of Permalloy Patterns
in Magnetic Bubble Devices

Magnetic bubble devices are progressing steadily to achieve higher

bit density. In 1967, bubble devices with a 75in diameter were reported.

Recently 2pm bubble diameters have been used, and 1 Mbit/cm2 experimental

prototypes have been announced [16]. This extremely rapid progress in ac-

hieving higher densities raises the question of whether it is still possible

to use permalloy patterns fabricated as field access bubble devices. It

is known that when the cell size of conventional field access bubble de-

vices goes down, the diameter of the magnetic bubble and the geometries

of permalloy patterns is reduced accordingly.

The one probable impacted limitation will be the lithographic reso-

lution problem. The natural characteristics of UV light in present con-

ventional exposure machines will limit the processing of permalloy patterns

to the 1 2vm minimum feature [9]. Further reduction in cell size seems

most likely when advanced techniques such as electron-beam and x-ray

lithography are used, or with significant advances in bubble device design

like the ones potentially offered by bubble lattice devices or contiguous

disk devices [6,12]. Kryder [39] has predicted that 1Im bubble can be

made in these devices (u 6 x 106 bits/cm2) based on the assumed drive

field, material, fabrication, and switching-current requirements. It

seems most likely that the contiguous disk and bubble lattice devices will

take advantages of the whole chip design, beyond the lpm bubble diameters.

It looks as if the l1m bubble diameter chip will be at the extremity of

these conventional* field access bubble devices. This conclusion, however,

is based on the assumption that a high permeability NiFe can be achieved,

and this does not seem to be the case in practical device fabrication.

A large coercive force is found in the narrow width permalloy patterns.

In some cases, this effect may cause undesirable magnetic behavior in the

performance of permalloy propagation patterns.

The relationship between the coercive force and the width of perm-

alloy stripes (stripe meaning the length of permalloy pattern is longer

than 100lm) has been studied by Kryder et al. [40] and Herd et.al. [26].

*"Conventional" is used here to distinguish between the typical field
access bubble devices and the field access bubble lattice.

As shown in Figure 33, the coercive force of permalloy stripe increases

sharply when the width is below 1Opm. All of these experiments use thick-

nesses of approximately 450 A and 300 A. The coercive force will decrease

when the thickness of the deposited permalloy film is increased [1], as

shown in Figure 34. From these two relationships, it becomes evident that

when the width of the permalloy pattern goes down, a thick film should be

used. Figure 35 shows the relationship between the coercive force and

the operating margin of the bubble device. It is interesting to see that

the permalloy pattern with the high coercive force not only increases the

minimum in-plane drive field for the bubble device, but also lowers the

operating margin very quickly. This high minimum drive field and low

operating margin could cause high power consumption in the magnetic

bubble devices, and thus unreliable operation.

The origin of high coercive force in narrow width permalloy patterns

is not yet explained clearly. A stable remanent state is usually found

in narrow width permalloy patterns. One possibility is that the remanent

states prevent the permalloy pattern from returning back to a demagnetized

state, and thus introduces the high coercive force. Without elimination

of the high coercive force in the permalloy pattern, the field access

bubble devices will become impractical, even if we can solve the litho-

graphic resolution problem. In the following sections, we formulate a

conceptual zig-zag wall model (remanent state model) and study its be-

havior with the function of thicknesses and aspect ratios of permalloy


Here, we will make a brief comment on our previous continuum modeling.

Susceptibility is assumed to be infinite in our surface pole model. This

means that zero coercive force is assumed on all the permalloy patterns.

0 A
a 4.7
S 0- o 2.7


40 0


0 8 3I 24 32 40

S W (pm

Fig. 33 Coercive forces are shown as a function of width of
permalloy stripes. These data are taken from.Reference
[40]. The thickness of permalloy film are close to 450 A.


1 2 4 668 2 4 681
101 102

2 4 68

-t (A)

Fig. 34 Coercive forces are shown as a function of the permalloy
thickness for three different deposition conditions. Data
are taken from Reference Wi].



[ii [ I I I I i 1

Substrate Temperati

I -- -196

Ii --- 50 C
.. 50o C
i --- 200


; /
,.i x, r a ?<

/ ^

^ V7 |






I -

ure _

nl I I


o 95-




55r NiFe 5400 A

0 20 40 60 80 100

Hxy (HOe)
H = 0.85

o H = 1.35 Oe

S Hc = 2.0

Fig. 35 Operating margin of magnetic bubble devices for three
different coercive forces of permalloy films. Data are
taken from Reference [2].

A small existing coercive force could cause some discrepancy on our model

predictions which is, however, still in the range of reasonable qualita-

tive expression. A high coercive force of the permalloy pattern will

overwhelm the applicability of our continuum model. In this case, however,

no conventional field access bubble devices will be of practical interest.

5.3 Preparation of Experiments

Several geometries of permalloy (80% Ni, 20% Fe) patterns are vacuum

evaporated on glass substrates, as shown in Table I. A standard IC wafer

cleaning procedure is applied to degrease the glass substrates (Appendix

B). Background pressure of approximately 10-7 Torr is used in our Varian

sputter-ion pumped vacuum system when the permalloy ingot in the ceramic-

coated tungsten basket starts to evaporate. The temperature of the sub-

strate holder is set at 2000C to enhance the adhesion between the eva-

porated permalloy film and the substrate. During the evaporation, a

small magnetic field is applied so that the direction of the anisotropy

constant is parallel to the length of the permalloy pattern. Thickness is

controlled by using a crystal monitor, calibrated by multiple beam inter-


Emulsion masks are processed on our camera system. Contact printing

is used to expose the glass substrate coated with a thin film of negative

Waycoat photoresist. Postbaking is strongly needed in order to prevent

the peeling of any photoresist from the following chemical etching. The

etching solution is slightly modified from the method used by Ma [44] and

Huijer [27] to decrease the etching rate (see Appendix B). The chemical

etching is the most critical step in the entire sample processing. A

reduction of the reaction rate will make it easier to control the etching

Table I. Geometries of permalloy

patterns used to observe the remanent

PARAM- W t Aspect Ratio
S '(width) (thickness)

A 5 2,3,
B 7.5 4000 4,5,
C 10 6,8,
D 15 10,20

A 5 2,3,
B 7.5 2000 4,5,
C 10 6,8,
D 15 10,20

A 5 2,3,
B 7.5 1000 4,5,
C 10 6,8,
D 15 10,20

A 5 2,3,
B 7.5 500 4,5,
C 10 6,8,
D 15 10,20

* There exist eight different aspect ratios of permalloy patterns
for each width.

of thin permalloy film. In order to compensate for the undercut caused

by chemical etching, slightly oversized permalloy patterns are made when

the emulsion mask is produced. Hot trichloroethylene solution is used

when removing the residue photoresist, because we find that typical

Waycoat photoresist striper J-100 will attack the surface of thin perm-

alloy film.

5.4 Bitter Solution Technique

In order to observe the magnetic effect of the permalloy patterns and

their remanent state, the Bitter solution technique is used. This solution

consists of fine ferro-magnetic colloidal particles with some kind of sol-

vent as its base [10]. The magnetites in this suspension are so small that

they perform Brownian motions. When the particles come close to the domain

walls, they are attracted by the stray field associated with them. The

resulting particle concentration above the walls can be observed by the


Several techniques were used in conjunction with the Bitter solution

in order to observe the wall patterns. Middelhoek [46] employed dark-field

microscopy to study the domain wall pattern. In that case, the light of

the microscope comes parallel to the surface of the observed sample, and

is reflected by the piled-up colloidal particles. The reflected light is

collected into the object lens and shows the white domain wall pattern in

the dark background. Special apparatus should be assembled to get the

parallel light needed for this dark-field microscopy. Khaiyerand O'Dell [37]

used the Nomarski interference contrast technique. This technique detects

the surface irregularities caused by the pile-up of colloidal particles

above the domain walls. It is hard to detect the Bloch wall with this

method, however, due to the smaller collection of colloidal particles

caused by the distributions of the Bloch wall stray field.

In our experiments, direct observation from the oil-immersion lens

is used to detect the slightly diluted Bitter solution pattern on the

permalloy surface. Light coming from the microscope is directed perpen-

dicularly on the permalloy surface. The piled-up colloidal particles

scatter the light, the dark domain walls show up against the bright back-

ground. A product of the Ferrofluidic Company, AO-1, in water base

is used throughout the experiments.

5.5 Domain Wall Observation in the Permalloy Patterns

Two basic domain structures were observed previously in permalloy

I-bars with thicknesses of approximately 3000 A [27]. These two struc-
0 0
tures also reappeared in our 4000 A and 2000 A thickness permalloy samples,

as shown in Figures 36(a) and 36(b), respectively. Two closure domains

are always created at both ends of these long permalloy bars, in order to

reduce the magnetostatic energy. From their wall movements, the plausible

domain magnetization can be shown in Figures 37(a) and 37(b), with respect

to the external field [28,29]. From the distributions of magnetization M

within these permalloy patterns, domain wall a should be the Bloch wall

and domain wall b the Neel wall. The slightly different apparent widths

of the Bitter solution pattern, comparing the domain walls a and b,

suggest that they should be two different types of walls. From the sche-

matic expression of Bloch and Neel walls, as shown in Figure 38, the Neel

wall should collect more magnetite in their width compared with that of

the Bloch wall. Another indication that domain wall a is a Bloch wall

and domain wall b is a Neel wall is shown in Figure 36(c). With this

picture, domain wall a is transferred from the Bloch wall to a crosstie

4000 A

2000 o

1000 A

500 t\

Fig. 36 Observed domain patterns at different thicknesses of
permalloy films.


Fig. 37 Two different domain configurations are observed at thick
permalloy films.

H field

""' / .,
S\ l -
S I I /

(a) Magnetic Domain

H field


Fig. 38 Schematic diagram for (a) Bloch domain wall (b) Neel domain

wall when the permalloy thickness is reduced from 4000 A to 1000 A.

Domain wall b is not found to have this same transition in the 1000 A

sample, however. It is known that the crosstie wall is the intermediate

state wall when the thickness of the permalloy pattern is reduced from

a certain thickness to a thinner one. The Bloch wall is observed in the

thick permalloy film, and. the heel wall prefers to stay in the thin

one. These observations are consistent with well-known properties of

domain walls [46].

5.6 Remanent States of I-bar Patterns

In this section, the Bitter solution pattern of the permalloy will

be observed at different external fields, so that the remanent state can

be investigated. The field is applied along the longer direction of the

permalloy I-bar. The effect of the field causes the domain wall to bulge

to one side. As it turns out, the domain with M parallel to the applied

field grows at the expense of the domain with M antiparallel. In all of

our experiments, three different cases were observed.

In the following sections, we will call the following phenomena

"Case B". This is when the permalloy bar is magnetized along the long

direction by the external field, from an originally demagnetized state to

partial saturation in the middle portion of the bar. The demagnetized

state will not come back again, even if the external field returns to

zero. If a small amount of reversed field is applied, however, the de-

magnetized state is then re-observed [29].

Figure 39 shows the sequence of responses to the magnetization

states to the different external fields under the Case B situation. In

these pictures, the sample is with a thickness of 2000 A, a width of 10nm,

and an aspect ratio of 6. The originally demagnetized state is found in

0 Oe

S Oe

10 Oe

3.6 Oe

Fig. 39 Magnetization sequence for permalloy pattern with a
thickness of 2000 A and L/W=6.

0 Oe

-3 Oe

-4.3 Oe

Fig. 39 Continued.

Figure 39(a). When a 10 Oe external field is applied, partial saturation

is observed within the middle portion of the permalloy bar, as shown in

Figure 39(c). As the field returns back from 10 Oe to 3.6 Oe, zig-zag

walls are created from both end sides. They are found completely within

the middle portion when the field returns back to zero, as shown in

Figure 39(e). Figure 39(f) shows that when a -3 Oe reverse field is

applied, the zig-zag walls then become turbid. All of the zig-zag walls

break down when a -4.3 Oe field is applied, and the demagnetized state

is re-observed again. It is interesting to note that in Figure 39(f),

the domain wall bulges upward even ff the applied field is -3 Oe in the

reverse direction. This shows that the net magnetization of this perm-

alloy pattern is still positive (assume it points to the right side of

this figure) when the external field is -3 Oe. In Figure 39(g), the net

magnetization is changed abruptly from a remanent positive state to a

negative one when the external field is -4.3 Oe.

Case A describes the permalloy patterns with zero remanent state

in their magnetization response. Zig-zag walls are temporarily found

during this magnetization cycle, but never exist when the external field

returns to zero. Figure 40 shows this magnetization sequence with the

sample fabricated by 2000 A thickness, 10m width and aspect ratio 4.

Figure 40(a) shows the demagnetized state with a diamond domain. When a

14 Oe external field is applied, the diamond domain is squeezed, as shown

in Figure 40(b). This also shows that the direction of magnetization N

within the diamond domain is nearly perpendicular to the longitudinal

direction of the bar. After this pattern is magnetized to saturation by

a 55 Oe external field, zig-zag walls appear as the field reduces to 8 Oe.

The demagnetized state is re-observed when the external field goes back

0 Oe

14 Oe

8 Oe

0 De

Fig. 40 Magnetization sequence for permalloy pattern with
a thickness of 2000 A and L/W=4.

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