Citation
Surface pole modeling of field access magnetic bubble devices

Material Information

Title:
Surface pole modeling of field access magnetic bubble devices
Creator:
Lai, Fang-Shi Jordan, 1948- ( Dissertant )
Watson, J. Kenneth ( Thesis advisor )
Sutherland, Alan D. ( Reviewer )
Monkhorst, Hendrik J. ( Reviewer )
Uman, Martin A. ( Reviewer )
Kurzweg, Ulrich H. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1980
Language:
English
Physical Description:
vii, 147 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Domain walls ( jstor )
Fourier series ( jstor )
Magnetic domains ( jstor )
Magnetic fields ( jstor )
Magnetic polarity ( jstor )
Magnetism ( jstor )
Magnetization ( jstor )
Magnetostatic fields ( jstor )
Magnets ( jstor )
Permalloys ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Domain structure ( lcsh )
Electrical Engineering thesis Ph. D
Magnetic bubble devices ( lcsh )
Magnetostatics ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
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 magnetic domains, the development of a surface pole model to analyze arbitrary- 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 4(r), in discrete Fourier series form, which is related to the specified distribution of magnetic pole densities by Poisson's equation. The magnetic 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 24um x 41.57um hexagonal array. 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 (M+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 discrete 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 of thicknesses ranging from 500 A to 4000 A. The width of the bars was varied from 5pm 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.
Thesis:
Thesis (Ph. D.)--University of Florida, 1980.
Bibliography:
Includes bibliographic references (leaves 143-146).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Fang-shi Jordan Lai.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
023441068 ( AlephBibNum )
07316086 ( OCLC )
AAL5642 ( NOTIS )

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SURFACE POLE MODELING OF FIELD ACCESS
MAGNETIC BUBBLE DEVICES














BY

FANG-SHI JORDAN LAI


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE ':i. :':"LE''' .
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA













ACKNOWLEDGMENTS

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

appreciated.













TABLE OF CONTENTS

Page

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

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

CHAPTER

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

II MAGNETOSTATIC FIELD DISTRIBUTIONS OF ARBITRARY-SHAPED
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

III MODELING OF PERMALLOY PATTERNS . . . . . .. 34

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

IV ANALYSIS OF PERMALLOY PATTERNS . . . . . .. 50

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

V STUDY OF REMANENT STATES IN PERMALLOY PATTERNS . .. 76

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

APPENDICES

A DETAILED DERIVATIONS OF CHAPTER II . . . .... .116

B EXPERIMENTAL PROCEDURE TO FABRICATE PERMALLOY PATTERNS .117

C COMPUTER PROGRAM (PL/I) TO CALCULATE POTENTIAL WELL . .120

D COMPUTER PROGRAM (PL/I) TO PLOT THREE-DIMENSIONAL
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


SURFACE POLE MODELING OF FIELD ACCESS
MAGNETIC BUBBLE DEVICES

By

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

array.

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.













CHAPTER I
INTRODUCTION

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















Z BI

rotating



x -


garnet


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

include:

(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.













CHAPTER II
MAGNETOSTATIC FIELD DISTRIBUTIONS
OF ARBITRARY-SHAPED MAGNETIC DOMAINS

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

7







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

as


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
(2-4)
S p N 1


Similarly, the domains for m and n are defined within







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

j (2w/M) mk
ek (m) = e mk (2-6)
(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
(2-8)







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
(2-9)


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


(2-10)


(2-11)


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:


M-I N-I
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

(2-12)

0 _m S M-1

0 < n < N-1

(2-13)







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)

where

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


(2-15)































T


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.



























A-U


~)


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


- ---


= I II i |


i T I 1


origin
k,p)



b


i







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

equation

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)

where

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

direction.

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)]


(2-19)


[- 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


(2-20)



(2-21)


where


( 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.


























A




d
2



6-





13
1 --DFS
0 ODruyvesteyn's
data
-4 bubble D for R/t:0.05
2- / 0* change sign'


S1 2 3 4 5


R

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
x
















.9 i '.


-.

-~-~
I.


(b)


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


..




*', ..- - ,,


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


--





















,b/ X

S9.6 -



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
























(b)
---- ^_^


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


.?



Y

..x
I.


-~----






















-----I-*
24-
a) I


41.57



.1_



II


" ' ,. '

I .
I






(b)


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


N?"












I B


> magnetic bubble

o


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-
blem.







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
form:

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

where


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

(2-24)

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

L
t = C Azi (2-25)
i=l

The equivalent magnetic field strength can then be defined as

E2b
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 DFS


0.8- - Exact Caiculation
N K


0.S











U. 1 2
02-6

t,
D


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].
















0.06

hR



Rt =0.5
0.04- R =0.33

t = 0.25

-0.2


0.02







0 2 4 6 8



5R

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.




33


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.













CHAPTER III
MODELING OF PERMALLOY PATTERNS

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

as:

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)
Ir-<







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]















'Permailoy
Pattern


r Exteral Field IHel

Xa)
la)


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)

(b)





















Permalloy Pattern






* ^y


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


I f- --


>^ I


SC1







L
Q d(r)= i Ci Pi (3-5)
1=1

where
ds'
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:

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

where
M
N = Lp
p=l

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-
J
tance within oth period and r. is the source distance between the oth

and mth periods.














2


Permalloy
Period 1 Patterns


___3A







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
interaction.







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

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

where
M
N = r L
m=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,
13
then P can be rewritten as

ji
J rt/2 [dzo dlo

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

(3-9)

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

where

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
31










































































o -
a.







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.

Thus

E
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


















(Cm)


- 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.


MODELS








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

distribution.

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.














CHAPTER IV
ANALYSIS OF PERMALLOY PATTERNS

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-

eling.

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































(Oe)


(j2m)
--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.























(a)
Y



OX




w= 45
E











(b)

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
















w=1800







(e i) "




(e)


Fig. 21 Continued.











*C




m I-cu

0
E
Ss-L





c0 0
















CC
a)
r-








1n -











LA, a)
s-U
-CM
*C *I



















0 C-
4CI
u m
3 *- Id










4. 0 --


0-0















cu
5--
sc





0 O



0 *l 0





. .0 0.

-0 0 0
0 -- -




*- 0 s-

UCQ
+C Q S




U-3


ao oz
"0 0Z
















) CE





+--



CL
0
r- I-









-- C
M c)





CO
..- S :
C

4-.-
o-c

Om

U*


3 0
Or- n
U II




o -o
(0 >C CO
5 *-

OWE

0 C


*r*- E

o 0 II
. a

r- C S-


C r- S-
0 .0 0

C..0 E
I10
0 r- r
(0 .0
C3 0
0 -a
r- 0 .0




(J


U-


COOZ
~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

bubble.


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




0OD
(um)



--0- --3


75
T P

6

Upper 3
Track -x-



Lower -r-
Track
30









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./ .,,


7




U/


0-













4-,
0
4-



-0



Cu
rO



CT
0,














-r

sr-
r-0







0
La









CJ
0










0,












L-'.
S-
C










fl



Q.








i-0





















I

r-
3
C







0.


to
O


o


cr-



r-
.0


C-
0

.Q



r*

C
-o

r-


C



C,
a












r
a









Q >
0
S-a


~a m/O

a



S//



/ //


/ I,

1'








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



















O



0








CN J
\ ;
0






I





/ piil _
If (-I 1


1 \N


I
r


I
t

r


E






E i
4-'
E II
S- C=



oa



















C r-
S11
a)


SE
(U



0 0



















L2
4-
QL..
4- yl
cv


01

10



C.






CL
S II
'-

U-
r
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



















cu


0


0








S-
0





W E
S-



























L \ /

\o C:,









.-1
E

0















-0
U


0 0








i] p 1

) r4-


&-









x ea






-17r
/ '* 0 / '4- I
/ *. /-. 0 C
/ ^ /r ^^ ^ Q
/ */ -.^ -">
/~ r-''^- 00-
/ ,0 .^' s
1 0 '* *'C T


lUL- -






66





E


<(0
0
o E



4J (U




oo




















s -
a C
















O (.
0.


2a




U
1 v,





0
O-




r
oo .

UC
/ 0



4 r- CL


1 o 00
/ 5- I*-



/ 0.











Sr=
0)0 m -








m
\NOr
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








68


1 EE
















0



' r4
ai-
























-we-
-CU



Lu\ -E








J., D





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



U)
Scu









Ion c,
-S C)











-- 00






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






69


E 0
5 o
















7n -
- --- o




- A--
















I I


CC
I-^ C
__ 7t
C-,

0 '
















3- ,^- ----






















E




- m:"L
ICN)L


-)


.)

E
C





>,










S-.
0


4--
+-












C
S-






4 -
C








C[















r4- ,











c.
-5C



















e-
C
I-
4-C










0
U,
LO






LC-
'4-









C -
Cs-











(U -
*i- Q
1< -

S-








L, <












0
0

\ a






























,- _













00

- -' -__

'U-


C
1)
0
























e-,
0



E
S-





01







C





r.



U-






C)
4'









5- 0
U,



4-
U1O
1m












w -1
- U,
s- -I-













wEC
0

4-
VI
tULC
i- t/
? 0
0 S
I- 10
Q. C
















LL-











































































c i

0,







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.




75


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].














CHAPTER V
STUDY OF REMANENT STATES IN PERMALLOY PATTERNS

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

data.

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

I-bars.

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
Oe
S 0- o 2.7

1.7

40 0



20-



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.






A


1 2 4 668 2 4 681
101 102


2 4 68
103


0
-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.


[ii [ I I I I i 1





Substrate Temperati

I -- -196

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

I I









; /
,.i x, r a ?<

/ ^

^ V7 |


2,0


1.21


0.8


0.4-


I


I -


ure _


nl I I




81











o 95-

-5r




75t-



65



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-

ferometry.

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
states


patterns used to observe the remanent


ETERS
PARAM- W t Aspect Ratio
S '(width) (thickness)
SAMPLES Jm A L/W

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

microscope.

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

















0
4000 A












0
2000 o











0
1000 A












50
500 t\


Fig. 36 Observed domain patterns at different thicknesses of
permalloy films.























(a)


Fig. 37 Two different domain configurations are observed at thick
permalloy films.













H field

""' / .,
S\ l -
S I I /








(a) Magnetic Domain
Wall




H field









(b)


Fig. 38 Schematic diagram for (a) Bloch domain wall (b) Neel domain
wall.







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.




Full Text

PAGE 1

SURFACE POLE MODELING CF FIELD ACCES3 MAGNETIC BUBBLE DEVICES BY FANG-SHI JORDAN LAI DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1 930

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ACKNOWLEDGMENTS 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 invaluable 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 appreciated.

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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii ABSTRACT v CHAPTER I INTRODUCTION 1 II MAGNETOSTATIC FIELD DISTRIBUTIONS OF ARBITRARY-SHAPED 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 III MODELING OF PERMALLOY PATTERNS 34 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 IV ANALYSIS OF PERMALLOY PATTERNS 50 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 V STUDY OF REMANENT STATES IN PERMALLOY PATTERNS 76 5.1 Introduction 76 5.2 Limitations of Permalloy Pattern in Magnetic Bubble Devices 76 5.3 Preparation of Experiments 82 1 T 1

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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 APPENDICES A DETAILED DERIVATIONS OF CHAPTER II 116 B EXPERIMENTAL PROCEDURE TO FABRICATE PERMALLOY PATTERNS . .117 C COMPUTER PROGRAM (PL/I) TO CALCULATE POTENTIAL WELL . . . .120 D COMPUTER PROGRAM (PL/I) TO PLOT THREE-DIMENSIONAL DISTRIBUTIONS 137 LIST OF REFERENCES I 43 BIOGRAPHICAL SKETCH I 47

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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 SURFACE POLE MODELING OF FIELD ACCESS MAGNETIC BUBBLE DEVICES By 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 magnetic domains, the development of a surface pole model to analyze arbitrary-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 4(r), in discrete Fourier series form, which is related to the specified distribution of magnetic pole densities by Poisson's equation. The magnetic 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

PAGE 6

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 24um x 41.57um hexagonal array. 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 (M+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 discrete Fourier series method, the potential well profile for arbitraryshaped 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 o thicknesses ranging from 500 A to 4000 A. The width of the bars was varied from 5pm to 15um, and the length-to-width (aspect) ratio from 2 to 20. The results from the experiments showed that remanent states vi

PAGE 7

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.

PAGE 8

CHAPTER I INTRODUCTION 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 yery 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). Information 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 cylindricalshaped 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 between these two fields [34,35]. These margins are called operating margins 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 Dy 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 1

PAGE 9

Z B permalloy pattern magnetic garnet magnetic bubble Fig. 1 Tne device structure of typical field access magnetic bubole memory. The arrows are used to show tne directions of magnetization.

PAGE 10

variation factor cannot be changed. The whole operating margin will, then, largely depend on the permalloy pattern design [53,54], 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 difficult. This is due to the non-uniformly distributed demagnetizing field within the permalloy pattern. The complicated coupling between the permalloy pattern and this non-uniformly distributed magnetic bubble magnetostatic 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 permalloy pattern is infinite. In the entire mathematical modeling derivation, the magnetization distribution M is the basic quantity used. The magnetostatic interaction energy can be obtained from this model. Ishak and Delia Torre [33] presented a continuum model by assuming that the susceptibility of the permalloy pattern is finite and a function of an external

PAGE 11

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 configuration. Only the straight wall moving is allowed in this model, which makes it questionable if the non-uniformly distributed bubble magnetostatic field is considered. AlmasiandLin [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 contrast 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 distribution of M: whether a domain or a continuum distribution exists and whether a unique distribution exists [30]. Furthermore, the two dimensional distributions are of a single scalar quantity (pole density) rather than two components of a vector quantity (M X ,M ). The simplicity of the physical model also leads to numerical simplifications by largely reducing the dimension of the matrix that must be solved. This leads to a simple program implementation that makes it possible to address more comolicated problems.

PAGE 12

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 ^ery 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

PAGE 13

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 include: (1) A new method for calculating the field distributions of periodically 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 computer program which takes into account bubble-bubble and permalloy-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 calculating two new results: the analysis of a bubble logic circuit, and the multiple Chevron patterns used to propagate a stripe domain.

PAGE 14

CHAPTER II MAGNETOSTATIC FIELD DISTRIBUTIONS OF ARBITRARY-SHAPED MAGNETIC DOMAINS 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 magnetostatic 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 distributions 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) algorithm leads to a reduction of computer time in calculating the Fourier coefficients. The method also eliminates the complicated integrations that arise due to arbitrary shapes of magnetic domains. This chapter gives the method of analysis, and numerical results that compare yery well with known results for a hexagonal lattice of magnetic bubbles with 24um x 41.57ym 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 7

PAGE 15

accuracy observed to be affected. Comparisons between the isolated bubble and the sparse bubble lattice are also 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 arbitrary cylindrical-shaped magnetic domains is also demonstrated. Threedimensional plotting of normalized in-plane magnetic field strengths are calculated for a hexagonal lattice of magnetic bubble, a honeycomb magnetic 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 ^ery 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 «, 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 as F(k,p) = F(x,y) I (2-1) I 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

PAGE 16

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) CO CO = I I A(m,n) B(k-m,p-n) m =_oo n =-°° = I I B(m,n) A(k-m,p-n) (2-2) m =_oo 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. Consider 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 A(k.p) ^ T T a (m ,n) e^ M >™ k j^<» fM , im m=0 n=0 {C 6 ' Actually, k and p are varied from negative infinity to positive infinity in the kp space (xy plane in the continuous signal case). We know, however, that if the sequence is periodically distributed with the periods M and M 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 OiUH-1 (2-4) OipiN-1 Similarly, the domains for m and n are defined within

PAGE 17

10 < m < M 1 < n < N 1 The reason is that if we assume e k (m) = e^ 2 */"" mk e („) = e J '< 2 * /N > "V P (2-5) (2-6) in the periods M and N, then e Q (m) = e^ (m), e-j (m) = e M+1 (m) , e (n) = e., (n) and e, (n) = e M+ , (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 coefficients. In order to get those coefficients, the following fact should be used M-l "1-1 I L m=0 n=0 e J(2VM) mk e J(2VN) np 1 , for k=aM and p=bN (a,b are integers) (2-7) "0 , otherwise Multiplying Equation (2-3) by e JV ' ' e JV ' ,v and summing from k=0 to M-l and p=0 to N-l, we can obtain the following expression Y Y A (k.p) e" j ^ /M)kr e -J(2*/N)pt k=0 p=0 M-l N-l M-l N-l L L a(m,n, k=0 p=0 m=0 n=0 e J(2VM)k(m-r) e J(2^/N)p(n-t) (2-8)

PAGE 18

11 Interchanging the summation m,n and k,p we can get the following form Y U l A(k.p) e-^ 2ir/]A ^ e -J(2Vn)pt k=0 p=0 'V ''f a(m , n) 1 "j 1 "j 1 e J(2,/H)k( m -r) e ](2,/N)p(n-t) m=0 n=0 MN k=0 p=0 Using the Equation (2-7), we obtain M-l N-l a(r,t) = I I A(k,p) e* k=0 p=0 j(2ir/M)kr -j(2ir/N)pt (2-9] (2-10) The discrete Fourier coefficients a(m,n) can be obtained as M-l M-l a(m,n) = I I A(k,p) e j(2ir/M)km -j(2ir/N)pn :2-n) k=0 p=0 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 < k < M 1 and < p < N 1 , then the Equations (2-3) and (2-11) can be modified to get the discrete Fourier Transformation (DFT) pair as follows: M-l N-l A(k,p) = Mi I I a(m,n; J(2TT/M)km J(2ir/N)pn , m=0 n=0 , otherwise M-l N-l f L k=0 p=0 I A(k,p) e-^ Z ^km e -J(2i-/N)pn atm.n. < k < M-l < p < N-l (2-12) < m < M-l < n < N-l (2-13) , otherwise

PAGE 19

12 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 signals are real numbers. Then the following properties exist as, Re [a(m,n)] = R e [a(-m,-n)] Ira [a(m,n)] = Im [a(-m,-n)] (2-14) where < m < M 1 < n < N 1

PAGE 20

13 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 X in x direction and period X 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 M„, which leads to the result that the magnetic domains have the magnetic surface pole density 2M R on their upper and lower surfaces. The periodically 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 twodimensional discrete Fourier series form by Q(k> p,. ± 1 V V „(..„) Jl*/*)* .Jta'/'OP" (2-15) ' '''' m=0 n=0 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, X = Ms in the y direction. The coefx x y ficients of the discrete Fournier series in the inverse space domains

PAGE 21

14 magnetic magnetic domains film ,ongin 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.

PAGE 22

15 -origin magnetic domains Fig. 3 Definition of periods ^ X ,A and digitized mesh points in xy plane.

PAGE 23

16 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(r) = Q(k,p) U(z] where U(z) is defined as (2-16) r 1 , z = -d or z = -(d+t) U(z) = * , otherwise According to Poisson's equation, the relationship between the magnetic pole density and the scalar magnetostatic potential is given by the MKS equation V 2 *(r) = Q(r) (2-i7; Substituting Equations (2-15) and (2-16) into Equation (2-17) and using the difference approximation to calculate the differential equation V 2 !})(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 4>(r) can then be derived as r° (f) = 1 M-l N-l I I m=0 n=0 2tt q(m,n) e jk z (d + t) , e jk z d jRzZ dk[F(m,n) + k z ] e J(2TT/M)km e J(2TT/M)pn :2-18) where (m,n] -4 T l cos (2mir/M)] + -| T 1 cos (2mr/N)] h s

PAGE 24

17 and k is the inverse space parameter of the Fourier transform in the z direction. Magnetic field strengths of x, y and z components can then be obtained from the gradients of the scalar magnetostatic potential (r). The results are , M-l M-l i M k >p> = m I I i E(m ' n) [ j sin ( 2mTT / M )3 x ,1N m=0 n=0 h e J(2TT/M)km e J(277/N)pn (2-19] , M-l N-l -, H v^ k 'P) = W I I 1 E(m ' n) [ " j s1n ^ 2nTr/N )^ ' y m=0 n=0 . e j(27T/M)km e J(27r/N)pn (2-20) H z (k,p) -^ V V T (»,„) E(m,n) e^ M ) km e^^" (2-21) m=0 n=0 where E(m , n) « 1 afei e "T(m,n)d ( _ e -T(m,n)t } f ^ Qr ^ 2 T(m,nJ T(m,n) = ^F(m,n) E(o,o) = | q(o,o) For any specified magnetic domain shape with periods I and > one can determine the discrete Fourier coefficients q(m,n) from Equation (2-15), due to the known magnetic surface pole density 21% within the magnetic domains. All the coefficient terms can then be calculated by the inverse Fourier series method, which uses the FFT algorithm.

PAGE 25

l; 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, |h | = |h (k,p)/M„| ,. of a hexagonal lattice of cylindrical bubble with 24ym x 41.57ym periods, and with ratio (bubble radius/film thickness) of 2.4. These two calculations yield coparable 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 5 for d/t = 1. The normalized magnetic field strength |h | 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 1 OR x 1 OR 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 normalized radial field strength, jh I = I/IT 2 + FF / M Las a function r x y o of normalized distance, r/R. This figure shows that even though there

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19 • O Druyvesteyn's data • change sign «f i i i t i i i i ' ' 2 4 6 8 10 12 + r(jjm) Fig. 4 Calculated magnetic field, h ? |=|H z /M ? |, 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.

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20 o

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21 Fig. 6 Calculated magnetic field |h 2 | 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.

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22 to 1 64!MlO 18 64r2 21 8610 s 10 218644 2bubble TO 1 3 Fig. 7 Calculated magnetic field strength , | h r I =v4ix +H y/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.

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23 10 bubble • ODruyvesteyn's data for R/ f = o.05 • change sign >r Fig. 8 Calculated magnetic field strength, |h z |, as a function of normalized distance r/R. Open and solid symbols are used to indicate that they have different field directions.

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24 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 distributions 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 magnetic 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 following 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 [ h | and normalized z-component field strength h z . 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 ^/ery 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 ^ in x

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25 W&Z (a) *?;;, \^;;'.V ':-f/££" 1 ;"•;' ' ' '^^^'f; 1 ;..--^^,' 'i -^ 'x; : W XI li.V (b) Fig. 9 Tnree-dimensional plots of (a) radial field !h r | (b) z field h z ,for the hexagonal lattice of magnetic bubbles.

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26 JM$ (a) ,iS^£ >*.'.". ; : :V-;r 'J'"-^?*^ '•-u w Fig. 10 Plots of (a) !hr|(b) hz of honeycomb domain array. Dimensions are in microns.

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27 (a) r k— 24 — H? (b) Fig. 11 Plots of (a) |h r |(b) h z of stripe domain array. Dimensions are in microns.

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23 * Y / B magnetic bubble T i i magnetic bubble •+X (b) 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 problem.

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2S direction and A in y direction. In order to eliminate the bubble-bubble interaction from another magnetic bubble in the same array A, ^ and A 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 A The bubble-bubble interaction distance between these two bubbles is thus s in the y direction. The minimum interaction distance 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, H z , from the array A bubble can then be derived by the previously shown method, and has the following form: H 2 (7) =W L Y V G(m,n) e^ M ^ ^(^Jpn (2 . 22) m m=0 n=0 where G(m,n; qtm ,n) e ( " t/2) T(m ' n) cosh [T(m,n)z] , Izl t t/2 1 , v -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 hb Z h H z dv (2 " 23)

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30 The factor 2 comes from the same definition shown before. M R 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), E 2b can be obtained as follows: E = ^i 2 S V N y ] G (m n) e J(2ir/H)km j(2rr/N)pn E 2b MN ^ ^ l Q G i( m,n) (2-24) where G.(m,n) has the same definition as Equation (2-22) , except that the z is replaced by an increment of Az.. The definition of Azis shown as L t = I Az. (2-25) i = l ^ The equivalent magnetic field strength can then be defined as h = ^ (2-26) zn 4 M 2 B 7iR 2 t 3y using the previous program, the bubble-bubble interaction energy can then be calculated. If we set s=0, then h should be equal to the volumeaveraged demagnetizing factor. The calculated results are shown in Figure 13 in comparison with the exact calculation. The figure shows the comparable results between these two methods. Figure 14 shows the calculated h 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.

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31 ® DfS Exact Calcuiation -> t 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].

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32 0.08 r 0.06 magnetic bubble "zn 0.04 0.02 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.

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33 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 [42]. 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.

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CHAPTER III MODELING OF PERMALLOY PATTERNS 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 formulation. The two components, M and M , 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, M v and M , are replaced by the only one unknown component p. x y 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 developed 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. 34

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35 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 continuum 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 mathmatical form for this energy can be written as: f E p = I p{r) 4> e (r) dv j p(r) d (r) dv (3-1 where p(^)is the magnetic pole density within the permalloy pattern, caused by the external field or bubble magnetostatic field, or both. 6 (r) is the scalar magnetostatic potential caused by the external field. cj>j(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 potential i>A~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 , exists on the surface of each permalloy pattern. The magnetostatic equipotential cj> can then be described in MKS unit of 6 = JL -§ ds+ $» (3-2) t-ii 177 77^ I -

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36 where P_(r*) is the surface pole density (note: p = -V-M is used here, rather than p = -Vu *M) of permalloy pattern, and this equation is integrated on the surface of this pattern. cf> (r) has the same definition of the previous Equation (3-1). The demagnetizing potential $.(r) is then shown as:

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37 Permalloy Pattern External Field |Hel -*X (a) Fig. 15 [a) Definition of angle a between tne directions of external field and x axis. [b) Definition of permalloy pattern periods X and X and digitized increments h and s. y

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38 Permalloy Pattern Fig. 16 Definition of sliced subareas around the permalloy edge.

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39 d (D i I C f P, (3-5) where p i ds i r r. 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 x X period, and permalloy-permalloy interx y 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 j, c i K ji *p (3 6) where M N" I, L d p=l 1 3 K ii = 4tt ^ P ii J1 w m=0 J1 f d S : p m = ™ J ' j im 1 ^p is the constant magnetostatic potential of pth permalloy pattern within the oth period. . is the averaged external field in the jth subarea of permalloy edge. L are the total digitized subareas on the pth permalloy pattern. Sis the ith subarea of the mth period, such that the oth period is the desired calculation period. The ris then the field distance within oth period and r. is the source distance between the oth r im and mth periods.

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40 Period 1 Fig. 17 Period is the desired calculation period, in which M different permalloy patterns are shown. The nearest eight periods are used to calculate the permalloy-permalloy interaction.

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41 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 p™ (r) ds m = 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 N I q Q im =0 m = 1,2,3 M (3-8) i=l n m where M N = 7 L m=l Q= • S i m i e V '° • i i < L m> Combined with Equation (3-6) and Equation (3-8), we can obtain an (N+M) x (NxM) simultaneous equation. Solving this equation, the surface magnetic pole density p within one period and M constant magnetostatic potentials of M permalloy patterns can be obtained [57]. This simultaneous equation is associated with the permalloy pattern only. Its results can be stored and used for different magnetic bubble positions and different external field orientations. 3.3 Evaluation of Area Under Singularity There are nine periods of permalloy patterns that have to be considered in the permalloy-permalloy interaction. In the oth period, the

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42 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 ji ~ 4tt t/2 dzr dir TO 10 t/2 [(x. x: ) 2 + (y. y: ) 2 + (z, zl ) 2 f 2 (3-9) where t is the thickness of permalloy patterns and dK = dx^ 2 + dyr 2 10 10 •'io Integrating Equation (3-9) over the z domain from -t/2 to t/2, Equation (3-9) can be rewritten as P° -JJi 4tt In t/2 + / r; 2 + t 2 /4 •t/2 + r: 2 + t 2 /4 dl (3-10; where From this equation, it is very obvious that the argument of P • • approaches to infinity when x. = x" and y. = y' . Fortunately, the logarithm function 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 4irp9. 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 o to 4000 A. It is yery interesting to see from this figure that when the digitized point is beyond 100 points, 4ttP°. starts to saturate at around

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43 Z a.

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44 2.0 x 10" . 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 = P (r) e (r) dv 2 P s (r) d (r) dv (3-11) where p (r) is the surface magnetic pole density in the permalloy pattern. But the constant magnetostatic potential on the permalloy pattern is > e (r) + * d (r) Substituting Equation (3-12) into Equation (3-11), we have (3-12) \ P s (r) [4> + e (r)] 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 cos 2 a + r sin 2 a) e v Because this energy is independent of bubble positions and introduces a constant shift on the calculated potential profile only, it can be computed from Equation (3-13) by excluding the b (F) term from
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45 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 h , which is the potential well depth, is obtained by normalizing E to the volume and magnetization of the magnetic bubble. Thus E n h = ^ (3-14) Z (2M B ) (irR 2 h) The direction of h is parallel with that of the permalloy thickness [18]. The magnitude of h 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 permalloy 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 distribution 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 approximately 1.5um inside of the permalloy edge. This fact is experimentally

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46 -15— *| 2 H — 0"") — — Surface Pole George-Hughes Ishak -Delia Torre -•— SFS MODELS Fig. 19 Calculated potential profiles by surface pole model are compared with three other models. External applied fields are and 10 Oe respectively. Bar dimensions are 3><15x0.4 urn 3 Bubble parameters are D=6um,h=3um and 4ttI v L=200 G.

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47 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 assumption that the permalloy has infinite susceptibility and no variation of magnetization along the thickness direction. This leads to the conclusion 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 infinite susceptibility. These should be only surface poles on the permalloy 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 Q = -V M • V(|r r^l" 1 ) dv' (3-15) J where IT is the external field and H is the demagnetizing field. This

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48 equation can then be expanded into two terms by vector calculus as M • n r r ds' V'M dv" (3-16) where M • n shows the surface poles and VM 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), because in this equation, V"M will not automatically become zero. However, the George-Hughes model starts with this equation without setting V' • M = 0. Instead, they would introduce the volume poles in their calculation. 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 GeorgeHughes' 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 [13] has the deepest potential well profile in Figure 19. This one-dimensional Fourier series

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49 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 demagnetization 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 3ym, however, 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 distribution. Although George and Archer announced that their model is consistent with experimental data [17], one would expect that some degree of quantitative 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.

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CHAPTER IV ANALYSIS OF PERMALLOY PATTERNS 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 positions. The results are then used to characterize the individual permalloy 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 function 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]. 50

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51 4.2 Rectangular Bars The analysis of a rectangular bar in the dimension of 3 X 15 X 0.4um 3 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 model ing. 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 1.5um 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 perpendicularly to the permalloy bar. The interaction of permalloy mediated bubble-to-bubble and permalloyto-permalloy are studied in Figures 22 and 23. Figure 22 shows that when the magnetic bubbles are located at position A and position A', respectively, 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

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52 15— » Fig. 20 Calculated potential well profiles are shown as a function of three different external fields. Bar dimensions are 3x15x0.4 urn 3 . Bubble parameters are D=6ym,h=3um and 4uM =200 G.

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53 w = q 11 iii III 1 1 ffMfi II i'»''f ill 1 1 II I i I I i i I I ! ! i 1 1 i 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 parameters are D=6 um,h=3ym and 4ttM =200 G. The thickness of Si0 o is lum. B

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54 !c) w=135 (d) Fig. 21 Continued.

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55 w =180° W4J I '! II, 1 ' :'!l ! tf I ; ' I I ! I ! ,: iP" (e) Fig. 21 Continued.

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56

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57 S_ i— 0) E-rcu e

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58 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 0.5ym. 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 = 7.5um 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 bubble. 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 tnese 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 originally, and that when the external field rotates from

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59 10 Oe (jjm) U pper Track Lower Track 15 -I 3 T 1 lo* *1 30 Fig. 24 The geometry of a bubble logic circuit. Bar dimesions are shown in microns. Bubble parameters are D=6um,h=3um and 4uM =200 G.

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60 o— \ \ \ a

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61 4->

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62 Direction A to Direction C, the bubble does not move away from this position. The only change that occurs is that the magnetic bubble becomes smaller, due to the shallow potential well depth. The magnetic bubble moves through Position 3 to Position 4, however, until the external 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 represents 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 Orientation 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

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63
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64 experimentally verified by comparing the minimum driving field requirements 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 tolerance 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

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65 cs H *• c 0) .c c Ci £ 2 o \ ^ u— ta— I cot \ V<_ \ /

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66 E

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67 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 advantage 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 magnetic 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 have indicated, 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 Fl 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 symmetric half disk is also used to calculate its potential profile, to be

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68 £ CNl Vas<,_

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69 h-SH 0) in «£3

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70 E 3> r

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71

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72 O X j\tr a— va<

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73

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74 compared with its asymmetric counterpart. 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 verified [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 permalloy 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.

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75 For a 100 X 100 matrix, ten field orientations and ten bubble positions, 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 11 OK words of memory in the CDC-6400 system [32].

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CHAPTER V STUDY OF REMANENT STATES IN PERMALLOY PATTERNS 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 device 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 configuration 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 magnetization 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 data. 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 75ym diameter were reported. Recently 2ijti bubble diameters have been used, and 1 Mbit/ cm 2 experimental prototypes have been announced [16], This extremely rapid progress in achieving higher densities raises the question of whether it is still possible 76

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77 to use permalloy patterns fabricated as field access bubble devices. It is known that when the cell size of conventional field access bubble devices 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 resolution problem. The natural characteristics of UV light in present conventional exposure machines will limit the processing of permalloy patterns to the 1 2ym 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 lum bubble can be made 1n these devices (^ 6 x 10 6 bits/cm 2 ) 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 lum bubble diameters. It looks as if the lum 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 permalloy stripes (stripe meaning the length of permalloy pattern is longer than lOOum) 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.

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78 As shown in Figure 33, the coercive force of permalloy stripe increases sharply when the width is below lOym. All of these experiments use thicko o 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 lithographic resolution problem. In the following sections, we formulate a conceptual zig-zag wall model (remanent state model) and study its behavior with the function of thicknesses and aspect ratios of permalloy I-bars. 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.

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79 W t/J m 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

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80 2.4 O 2.0i.S 1.2 0.8 0.4 TTT l I T t — r—r Substrate Temperature _ -196' 50° C -•— • 20 0^ » y N y \ n 5 * 4 6 8 10 1 4 6 8 1 10 2 4 6 10 3 rH *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 fl] .

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81 v O -* H xy (Oe) • H c = 0.85 O H c = 1.35 Oe O H c = 2.0 Fig. 35 Operating margin of magnetic bubble devices for three different coercive forces of permalloy films. Data are taken from Reference [2] .

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82 A small existing coercive force could cause some discrepancy on our model predictions which is, however, still in the range of reasonable qualitative 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~ Torr is used in our Varian sputterion pumped vacuum system when the permalloy ingot in the ceramiccoated tungsten basket starts to evaporate. The temperature of the substrate holder is set at 200°C to enhance the adhesion between the evaporated 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 interferometry. 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

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83 Table I. Geometries of permalloy patterns used to observe the remanent states

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84 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 permalloy 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 solvent 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 microscope. 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 ' Del 1 [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

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85 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 perpendicularly on the permalloy surface. The piled-up colloidal particles scatter the light, the dark domain walls show up against the bright background. A product of the Ferrofluidic Company, AO-1 , in water base is used throughout the experments. 5.5 Domain Wall Observation in the Permalloy Patterns Two basic domain structures were observed previously in permalloy o I-bars with thicknesses of approximately 3000 A [27]. These two struco o 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 schematic 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

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36 inno a :o o 500 A Fig. 36 Observed domain patterns at different thicknesses of permalloy films.

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87 (a! (b] Fig. 37 Two different domain configurations are observed at thick permalloy films.

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n \\\ !;/ /,t H field Magnetic Domain Wall H field ** — © taii (b) Fig. 38 Schematic diagram for (a) Bloch domain wall (b) Meel domain wall .

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89 wall when the permalloy thickness is reduced from 4000 A to 1000 A. o 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 Neel 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 !l . 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 demagnetized 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 o these pictures, the sample is with a thickness of 2000 A, a width of lOum, and an aspect ratio of 6. The originally demagnetized state is found in

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90 Oe 3 Oe 10 Oe 3.* Oe Fig. 39 Magnetization sequence for permalloy pattern with a thickness of 2000 A and L/W=6.

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91 Oe -3 Oe -4.3 Oe Fig. 39 Continued.

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92 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 permalloy 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 o sample fabricated by 2000 A thickness, lOum 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 M 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

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93 Oe 14 Oe » 8 Oe Oe Fig. 40 Magnetization sequence for permalloy pattern with a thickness of 2000 A and L/W=4.

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94 to zero. This situation can be seen in Figure 40(d), which shows the diamond domain relocated at a different position in this pattern. Case C describes a different situation in which no demagnetized state o is found. This case is observed in almost all 500 A thickness, 5um width and large aspect ratio samples. Figure 41 shows this magnetization o sequence with the sample processed by 500 A thickness, lOum width and aspect ratio 8. Even when no external field is applied, 41(a) shows this remanent state when only a few walls can be observed. After this sample is magnetized by a 55 Oe external field, zig-zag walls are found in a reversed external field -11 Oe situation. The crosstie wall is observed in the left side of this sample, as shown in Figure 41(b). Another remanent state is observed in Figure 41(c), when the field reaches -15 Oe. 5.7 Phenomenological Model for Remanent States It is assumed that all the remanent states are associated with the existence of zig-zag walls, except when the permalloy pattern is saturated. In the latter case, the saturation domain configuration persists at its remanent state. The existence of zig-zag walls is studied in this section by looking at the results of our experimental data. o Figure 42 shows the experimental data of a typical array of 4000 A samples. This array of 32 different bar geometries was etched from a single 0.4ym permalloy film. It is interesting to note that the chance of observing the zig-zag walls is increased when the aspect ratio is increased or the bar width is decreased. This trend is also observed by the experimental data shown in Figure 43. Few of the examples of Case A, however, are found in this thin permalloy bar family. The observed relationship between the existence of the zig-zag walls and thickness is summarized in Figure 44. As can be seen, the decreasing of the permalloy

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o r, Oe fa) VLJI 11 Oe (b) Fig. 15 Oe (c) 41 Magnetization sequence for permalloy pattern with a thickness of 500 h and L/W=6.

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_ 159 CASE • A a B O C 96 4000 A NiFe 5 10 • 7.5 8 10 20 —• Aspect Ratio W Fig. 42 Experimental observations of remanet classes for 4000 A permalloy I-bars.

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97 CASE 9 A a 3 o c 1000 A NiFe i5r 10 7.5 cin G O i t b 8 10 Aspect Ratio { V) Fig. 43 Experimental observations of remanent classes for 1000 A permalloy I-bars. To

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c< 10 la CASE » A B C 98 10 ym Width NiFe 0.5 a 10 7o Aspect Ratio ( /• W Fig. 44 Observed remanent classes in the lOum width permalloy films. These classes are plotted with the functions of aspect ratios and film thicknesses.

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99 thickness will increase the chance of the zig-zag walls in the permalloy pattern to be observed. Before these data can be explained, we will discuss magnetization reversal. Assume that the middle portion of the permalloy pattern has been entirely saturated. The Bloch wall will then disappear for the. length 1 , as shown in Figure 45(a). If the external field is reduced to zero, the total magnetostatic energy will be large, due to the appearance of magnetic poles on curvatured domain walls. This demagnetization field provided by the magnetic poles is antiparallel to the magnetization M. This field can then cause the magnetization configuration to be changed, in order to reduce the energy of the entire system. It is suggested by Prutton that introduction of walls would require more energy than magnetization rotation [51], Magnetization rotation, however, would cause the magnetic poles to be generated on permalloy edges. In order to reduce the magnetostatic energy that comes from these magnetic poles, a magnetization configuration like the one found in Figure 45(b) could be arranged. Due to the very strong demagnetization field on the edges of the permalloy, closure domains can be formed to eliminate all the magnetic poles found here. Figure 45(c) shows a possible magnetization configuration on the remanent state. Figure 45(a) shows the final remanent state associated with zig-zag walls, and Figure 46(b) shows the detailed zig-zag structures of Figure 46(a). In order to simplify our model formulation, we assume that the magnetization M between domain a and domain b forms Neel wall p with the angle 8= p, as defined by the inset shown in Figure 47. It is assumed that there are no magnetic poles generated in domain walls p and k. If we assume 1. = 1 , the total zig-zag wall energy with length 1 can be

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100 [a) h > * / \ / \ / R — r — (b) h

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101 Fig. 46 (a) Zig-zag wall observed in permalloy pattern, (b) Magnified zig-zag wall structure.

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102 re §1 y 4 _*Z A —

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103 represented by E Nfc| = 4 l s t sec 3 a N (p) + (2N + 1) 1„ t a N (k) (5-1) where t is the thickness of the permalloy bar, N is the total number of zig-zag walls within length 1 , o\.(p) is the surface wall energy density for the Neel wall with an angle p and o\.(k) with an angle k. We can then approximate the wall energy in the (L 1 ) portion of the permalloy bar by E BW ML-l s )ta b (5-2) where a, is the surface wall energy density for the Bloch wall, and L is the total length of the permalloy pattern. The total wall energy of the entire permalloy bar, then, will be E W = E MW + E BW < 5 3 > From the assumed condition of zero magnetic poles on domain walls p and k, we can derive the following restrictions as p = tt/2 3 (5-4) k = 7T 23 (5-5) k < tt/2 (5-6) From Equation (5-5), we can get 3 > ir/4 (5-7) We approximate the Neel wall surface energy densities o..(p) and a f ,(k) by [46] a N (p) = a N (1 cos p) 2 (5-8)

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104 and ff N (k) = ^ (l cos k) 2 (5-9) where o„ is the 180° Neel wall surface energy density. The total wall energy of the permalloy bar then can be shown as E TW = (L " V t % + 4 ] s t sec B (1 " sin 6 ^ a N + (2N + 1) l w t (1 cos 2B) 2 a N (5-10) Except for the wall energy, the applied energy is zero in the remanent state case and the anisotropy energy of the permalloy bar should be very small in the polycrystalTinefilm [51]. They are all neglected in our following formulation. The magnetostatic stray field energy for the configuration shown in Figure 47 is not calculated exactly because of the difficulty in obtaining a closed form expression for it. However, an approximation is made by considering the magnetization distribution in the permalloy bar as < M > % M $ sin k l s /L = 1,,/L • sin 2s • ^ s (5-11) < M x > % l s /L • cos 23 • M s (5-12) where M s is the saturation magnetization of permalloy films, and the factor 1 /L is added to make the averaged < M > and < M > be zero s 3 x y when the zig-zag wall length 1 becomes zero. The x component of the magnetostatic energy is calculated by approximating the permalloy bar as an inscribed ellipsoid. The demagnetization factor and demagnetizing field are represented respectively by

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105 N * t/L (5-13) H , * cos 28 M (5-14) UX i 2 S The total x component of magnetostatic energy is then estimated as u w t 2 I 2 ^^~ cos 2 28 M 2 (5-15) L 2 -Mx ~ z UWi ^ P "s where w is the width of the permalloy pattern. The polarity of the < M > component is changed periodically by the distance of one half of 1 /N, shov/n in Figure 47. This makes the calculation of magnetostatic energy extremely difficult. However, if we approximate this energy by assuming w » t, it can be estimated as [40] ^o '•c t2 E My ^-2-j-S sin 2 23 M\ (5-16) The total energy of the permalloy pattern can then be written as the sum of Equations (5-10), (5-15) and (5-16) E y = (L l s ) t a fa + 4 l s t sec 8(1 sin 8) 2 a N I 2 t 2 + (2N + 1) 1 t (1 cos 2B) 2 a N + -^— M 2 (sin 2 23 + \ cos 2 28) (5-17) Due to experimental observations, we found that 1 is decreased when N is decreased. The length of 1 /N is also found to be approximately a function of the width of the permalloy pattern, when the zig-zag form becomes stable. We, therefore, approximate the length of 1 /N as 1, -| ^ bW (5-18) where b equals cot 3 in Figure 47, when we make the assumption that

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106 l k = l w -w/2 If we set N=l as a critical value for the existence of zig-zag walls in the following energy calculations, then Equation (5-17) can be rewritten as E y = wt 2 „ . 3 /, ooU. (R b) a fa + 4b sec 3(1sin 3) 2 o„ + j (1 cos 28) M^ MMsin 2 23 + C0s2 2B (5-19) R "s v °'" H R where R is the aspect ratio of the permalloy pattern, R = L/w. The calculated results of Equation (5-19) are plotted in equi-energy lines shown from Figure 48 through Figure 52. From Figures 48 to 50, these calculated equi-energy lines are plotted by varying the 3 angle in comparison with our experimental data shown in Figure 42. It is interesting to note that if all the zig-zag walls found in Cases B and C were only one zig-zag (that means N=l), then the transition energy for transferring between the states A and B could be close to 3x10" erg at 3= 50°. This result is shown in Figure 48. When the angle of 3 is increased to 60°, it -h the transition energy is decreased from 3x10 erg to 1x10 erg, shown in Figure 49. It decreases to 5x10" erg when the angle of 3 is 70°, shown in Figure 50. If experimental data were under our assumed N=l condition (N were observed to vary from one to six in actual cases), a large 3 anqle could let the ziq-zag walls form become more unstable. This is because remanent Case A could be easily obtained at lower transition enerqy, Fiqure 51 shows the calculated results in comparison with experimental data o of 1000 A permalloy samples, shown in Fiaure 43. These results are calculated by assuming 3 = 50°. The surface energy densities of Bloch and Neel walls are changed, due to the thin permalloy film [46,51], If the transition energy assumed is still 3x10" erg, as shown in Figure 48, o could expect that all the permalloy samples with 1000 A thickness would

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107 A • B C o 4000 A NiFe E 1 15

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108 A • B ' D C O 4000 A NiFe Fig. 49 Calculated equi-energy lines of permalloy film with 6=60 . o,,o,. and 4ttM are the same as those shown in Fig. 48. b N s

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10.9 A • B C O 4000 A NiFe Fig. 50 Calculated equi-energy lines of permalloy film with 3=70 . a. ,a,, and 4ttM are the same as those shown in Fig. 48.

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no Fig. 51 Calculated equi -energy lines of permalloy films with 3=50 , a b =3.8 erg/cm?a N =7.8 erg/cm 2 and 4irM =10* G.

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Ill A • B a C o 10 jim Width NiFe 1 2 3 4 5 6 7 8 9 10 L /w Fig. 52 Calculated equi-energy lines of permalloy films with 6=50 , a =4 erg/cm 2 a =9 erg/cm 2 and 4ttM =10"* G. b 3 N s

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112 have remanent states. The equi-energy lines which are calculated as a function of permalloy thickness are also plotted in Figure 52. These results are calculated by assuming the Bloch and Nee! surface wall energies are constant. These may cause the transition energy to be low at only 1x10" erg here. Remanent states are rarely found in small aspect ratio, thick and wide permalloy patterns. The suggested reason is that in small aspect ratio samples, the magnetostatic energy could be large, which would let the remanent states become very unstable. According to Equation (5-19), the magnetostatic energy varies as the inverse square of the aspect ratio. For wide and thick samples, Equation (5-19) suggests that both the wall and the magnetostatic energies are increased when the thickness and width are increased. This could cause the remanent state to become very unstable and, therefore, let the remanent class transition happen. 5.8 Conclusion A series of permalloy I-bars has been prepared and used to observe the remanent states by using the Bitter solution technique. The nature of remanent states in permalloy elements is studied to explore a limitation of their applications in field access magnetic bubble devices. Actually, remanent states could prohibit domain wall movement and introduce the large coercive force of wall moving. Thus it degrades device performance largely and destroys the applicability of our surface pole model . The studies of this experiment show that narrow-width, large aspect ratio and thin permalloy films tend to have the remanent states. A proposed phenomenological model explains these tendencies as being below an

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113 energy threshold. The suggested reasons are also qualitatively discussed. From the experimental observations, it may suggest that in the future high bit density devices, suitable permalloy patterns would have small aspect ratio and slightly large thickness to avoid increasing coercive force.

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CHAPTER VI CONCLUSIONS The major contribution of this research is the modeling of bubble devices, considering the magnetic surface pole distribution as the basic quantity of derivation. The pole-based model has the advantage that the number of unknowns in the pattern matrix have been cut in half. To make our numerical modeling more complete, bubble-bubble and permalloy-permalloy interactions have all been considered. The computer running time, however, is still within a reasonable range when compared with other models. In order that this model can analyze the arbitrary-shaped magnetic domains, a two-dimensional discrete Fourier series method is applied. This makes the analysis of magnetostatic field distributions of magnetic domains simpler as well as more flexible. In conjunction with the surface pole derivation, this model is not only used to analyze the arbitraryshaped permalloy patterns, but also to analyze the arbitrary-shaped magnetic domains propagated under the permalloy patterns. Several typical permalloy patterns, such as the rectangular bar, the HI pattern, the chevron pattern and the half disk pattern have been studied and discussed. The new calculated results of a bubble logic circuit and a multiple chevron pattern used to propagate a stripe domain were also analyzed to show the generality of the model. The research has also studied the occurrence of magnetic remanent states in small permalloy elements. The study consists of experimental observations of classes of remanence of permalloy elements, as a function of element thickness, width and aspect ratio (L/w). A phenomenological 114

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115 model is proposed, and is used to give a plausible energy interpretation to the observations. This research work has brought up various other topics that are promising for further investigations. These are: (1) The two-dimensional discrete Fourier series method can be extended into the three-dimensional case. It can be used to calculate the field distributions of arbitrary-shaped permanent magnet arrays. In conjunction with our developed sectioned spheroid model [41,56], the calculation of field distributions of arbitrary-shaped permanent magnet array will be yery interesting to compare with that of the finite elements method. (2) The surface pole model can be easily extended into a threedimensional model in order to explore which basic quantitymagnetization M or magnetic pole distribution— could be more accurate. The three-dimensional case can also be used to analyze the permalloy patterns which could be affected by a bias field. (3) The surface pole model can be revised to study the bubble size fluctuation problem by implementing an algorithm. The upper and lower bounds of the operating margin can be calculated from this. (4) The periodic nature of our surface pole model can also be extended to analyze the magnetoresi stance responses of periodically built magnetic bubble detectors. (5) The study of remanent states should be yery interesting to extend to explore different processing procedures or permalloy pattern structures [2], in order to decrease the coercive force.

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APPENDIX A DETAILED DERIVATION OF CHAPTER II Due to the magnetic film current-free media, the scalar magnetostatic potential field can be assumed as _ r i M-l N-l *(**) = M I I A(m,n; im m=0 n=0 dk. J(2ir/M)km r J(2tt/N) pn (A-l) where A (m,n) are the Fourier series coefficients in the inverse space domain. Using the difference approximation [34] and Equation (A-l), V 2 ("r) can be written as r i M-l N-l V 2 *(r) = mmI I A(m,n) " u m=0 n=0 2 [2 cos (2m7r/M) 2] h + 1 2 [2 cos (2mr/N) 2] k 2 I e jk z* dk. e J(2TT/M)km e J(2Tr/N)pn (A-2) From Equation (2-17), Q(r) can be written as [23] r 1 M-l N-l Q(r) = Wr I I q(m,n; m m=0 n=0 g. [e JM . e Jkz(d + t) ] • e jkzZ dk z / e j(27T/M)km e j(27T/N)pn . (A-3) Replacing the A(m,n) coefficients in Equation (A-l), Equation (2-18) can then be obtained. 116

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APPENDIX B EXPERIMENTAL PROCEDURE TO FABRICATE PERMALLOY PATTERNS A. Glass Clean Step 1. Agitate acoustically in Trichloroethylene (TCE) for five minutes. 2. Scratch with cotton tip in TCE solution. 3. Rinse with acetone (ACE) solution. 4. Rinse with methanol solution. 5. Rinse in DI water. 6. Dry with N„ gas. B. Evaporation Step 1. Load permalloy ingot in ceramic-coated tungsten crucible. 2. Set substrate holder at 200° C. 3. Pump down background pressure to around 10 Torr. o 4. Typical 4000 A evaporation needs around 12 ^18 minutes. C. Mask Making Step 1. Cut rubylith with desired pattern. 2. Make premask by camera system in 20:1 ratio. 3. Develop the glass film for approximately five minutes. 4. Fixing the glass film takes about fifteen minutes. 5. The final mask is processed in the repeater with a 10:1 ratio. D. Pattern Printing Step 1. Clean the evaporated glass using the same procedure as in Step A. 2. Coat the glass with photoresist in spinner with 5000 rpm for twenty seconds. 117

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118 3. Prebake the sample in a 65°C oven for twenty minutes. 4. Print the sample in UV printing machine for twenty seconds. 5. Develop the exposed sample for twenty seconds by using Waycoat developer. 6. Rinse the developed sample for twenty seconds by using n-Butyl Acetate. 7. Postbake in a 100°C oven for twenty minutes. E. Etching Step 1. The etching solution consists of HC1 , HNCU, and DI water in the ratio of 1:1:3. 2. Increase the temperature of the etching solution up to 50°C. 3. Dip the sample in the etching solution at a suitable etching time, by the range of 10-60 seconds for film thicknesses varied from 500 A to 4000 A.

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APPENDIX C COMPUTER PROGRAM (PL/I) TO CALCULATE POTENTIAL WELL

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LIST OF REFERENCES 1 K. Y. Ahn, "Magnetic Properties Observed During Vacuum of Permalloy Films," J. Appl . Phys ., Vol. 37, PP 1481-1482, 1966. 2 K. Y. Ahn and A. M. Tuxford, "Control of Coercivity in NiFe Films for Single Level Masking Bubble Devices," IEEE Trans . Mag . , Vol. MAG-15, PP 997-999, 1979. 3 G. S. Almasi, "Gap Tolerant Half Disk Bubble Device Margins," IEEE Trans . Mag ., Vol. MAG-14, PP 40-45, 1978. 4 G. S. Almasi and Y. S. Lin, "An Analytical Design Theory for Field Access Bubble Domain Devices," IEEE Trans . Mag ., Vol. MAG-12, PP 160-202. 1976. 5 J. L. Archer, L. Tocci, P. K. George and T. T. Chen, "Magnetic Bubble Domain Devices," IEEE Trans . Mag ., Vol. MAG-8, PP 695-700, 1972. 6 A. H. Bobeck, P. I. Bonyhard and J. E. Geusic, "Magnetic Bubble-An Emerging New Memory Technology," Proc . IEEE . , PP 1176-1195, 1975. 7 A. H. Bobeck and H. E. D. Scovil, "Magnetic Bubble," Scientific American , PP 78-90, 1971. 8 P. I. Bonyhard and J. L. Smith, "68K Bit Capacity 16um Period Magnetic Bubble Memory Chip Design with 2um Minimum Features," IEEE Trans . Mag ., Vol. MAG-12, PP614-617, 1976. 9 H. Chang, "Magnetic Bubble Technology: An Integrated Circuit Magnetics for Digital Storage and Processing," IEEE Press , 1975. 10 S. Chikazume, Physics of Magnetism , John Wiley, Inc., New York, New York, 1967. 11 M. S. Cohen and H. Chang, "The Frontiers of Magnetic Bubble Technology," Proc . IEEE , PP 1196-1206, 1975. 12 F. A. DeJone and W. F. Druyvesteyn, "Bubble Lattices and Their Defects," AIP Conf . Proc , PP 130-134, 1972. 13 D. B. Dove, J. K. Watson, E. Huijer and H. R. Ma, "A Simplified Fourier Series Method for the Calculation of Magnetostatic Interactions in Bubble Circuits," AIP Conf. Proc, PP 44-45, 1975. 143

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144 14 D. B. Dove, J. K. Watson, H. R. Ma, and E. Huijer, "Permeability Effects on the Magnetization of Thin Permalloy I Bars," J. Appl . Phys ., Vol. 47, PP 2237-2238, 1976. 15 W. F. Druyvesteyn, D. L. A. Tjaden, and J. W. F. Dorleijn, "Calculation of the Stray Field of a Magnetic Bubble, with Application to Some Bubble Problems," Philips Res . Repts ., PP 7-27, 1972. 16 R. E. Fontana, Jr., D. C. Bullock, and S. K. Singh, "Characteristics of a 1 Mbit/cm 2 Magnetic Bubble Memory," Paper 33-1, INTERMAG Conf ., 1980. 17 P. K. George and T. L. Archer, "Experimental Confirmation of Field Access Device Modeling," AIP Conf . Proc , No. 24, PP 116-121, 1975. 18 P. K. George and T. L. Archer, "Magnetization Distributions, Magnetostatic Energy Barriers and Drive Fields for Permalloy Bars," J_. Appl . Phys ., PP 444-448, 1973. 19 P. K. George and T. T. Chen, "Magnetostatic Potential Wells and Drive Fields in Field Access Bubble Domain Drive Circuits," Appl . Phys . Let . , PP 263-264, 1972. 20 P. K. George and A. J. Hughes, "Bubble Domain Field Access Device Modeling, Part I," IEEE Trans . Mag ., Vol. MAG-12, PP 137-147, 1976. 21 P. K. George and A. J. Hughes, "Bubble Domain Field Access Drive Modeling, Part II," IEEE Trans . Mag ., Vol. MAG-12, PP 148-159, 1976. 22 J. S. Gergis, P. K. George, and T. Kobayashi , "Gap Tolerant Bubble Propagation Circuit," IEEE Trans . Mag ., Vol. MAG-12, PP 651-653, 1976. 23 Y. U. Geronimus and M. Y. Tseythlin, Table of Integrals , Series and Products , Academic Press, New York, New York, 1965. 24 J. J. Grabau and J. H. Judy, "A Demagnetization Eigenfunction Analysis of Linear Transverse Magnetization in Long, Thin Stripe," IEEE Trans . Mag ., Vol. MAG-15, PP 1488-1490, 1979. 25 R. F. Harrington, Field Computation by Moment Methods , MacMillan, Inc. New York, New York, 1968"! (See also D. M. Fye and E. L. Grinberg, "Magnetic Field Controller Analysis Using the Method of Moments," IEEE Trans . Mag . , to be published). 26 S. R. Herd, K. Y. Ahn and S. M. Kane, "Magnetization Reversal in Narrow Stripes of NiFe Thin Films," IEEE Trans . Mag . , Vol. MAG-15, PP 1824-1826, 1979. 27 E. Huijer, Comparison of Continuum and Domain Models for Magnetization Process in Permalloy Thin Film Segments , Ph.D. Dissertation, University of Florida, 1977.

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145 28 E. Huijer, D. B. Dove and J. K. Watson, "Small Region Magneto-optic Measurements of Permalloy I Bars," 3M Conf . , 1977. 29 E. Huijer and J. K. Watson, "Hysteretic Properties of Permalloy I Bars," 3M Conf ., Paper 7E-10, 1978. 30 E. Huijer, J. K. Watson, and D. B. Dove, "Magnetostatic Effects in I-Bars, a Unifying Overview of Domain and Continuum Results," IEEE Trans . Mag ., Vol. MAG-16, PP 120-126, 1980. 31 S. Igarashi, K. Igarashi, A. Hirano, S. Orihara, and K. Yamagishi, "Design of a 3ym Bubble 80 Kbit Memory Chip," AIP Conf . Proc , No. 24, PP 48-50, 1975. 32 W. Ishak, Modeling and Optimization of Bubble Memory Field Access Propagation Circuits , Ph.D. Dissertation, McMaster University, Canada, 1978. 33 W. Ishak and E. Delia Torre, "Modeling of Field Access Bubble Devices, 1 IEEE Trans . Mag ., Vol. MAG14, PP 1042-1053, 1978. 34 E. Isaacson and H. B. Keller, Analysis of Numerical Methods , John Wiley, Inc., New York, New York, 1966. 35 M. E. Jones and R. D. Enoch, "An Experimental Investigation of Potential Wells and Drive Fields in Bubble Domain Circuits," IEEE Trans . Mag ., Vol. MAG-10, PP 832-835, 1974. 36 I. Khaiyer, "Models for I and T Pattern Permalloy Bars Based Upon Bitter Pattern Observation of Domain Wall Movements," IEEE Trans . Mag ., Vol. MAG-11, PP 114-118, 1975. 37 I. Khaiyer and T. H. O'Dell, "Domain Wall Observation of Permalloy Overlay Bars by Interference Contrast Technique," AIP Conf . Proc . , No. 24, PP 37-38, 1975. 38 G. S. Krinchik and 0. M. Benidze, "Magneto-optic Investigation of Magnetic Structures under Micro-resolution Conditions," Soviet Phys . JETP , PP 1081-1087, 1975. 39 M. H. Kryder, "Magnetic Bubble Device Scaling and Density Limits," IEEE Trans. Mag., Vol. MAG-15, PP 1009-1016, 1979. 40 M. H. Kryder, K. Y. Ahn, N. J. Mazzeo, S. Schwarzl and S. M. Kane, "Magnetic Properties and Domain Structures in Narrow NiFe Stripes, 1 IEEE Trans . Mag ., Vol. MAG-16, PP 99-104, 1980. 41 F. S. Lai and J. K. Watson, "A Sectioned Spheroid Model for Cylindrical Permanent Magnets," IEEE Trans . Mag . , Vol. MAG-16, PP 473476, 1980.

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146 42 F. S. Lai and J. K. Watson, "Field Distributions of Arbitrary-shaped Magnetic Domains Calculated by Two-dimensional Discrete Fourier Series," INTERMAG Conf ., Paper 26-6, 1980. 43 Y. S. Lin, "Analysis of Permalloy Circuits for Bubble Domain Propagation," IEEE Trans . Mag . , Vol. MAG-8, PP 375-377, 1972. 44 H. R. Ma, Magnetostatic Fields in Thin Film Permalloy Bar Arrays , Ph.D. Dissertation, University of Florida, 1976. 45 H. Matsutera and Y. Hidaka, "Bubble Propagation Pattern Analysis," 3M Conf . , Paper 6B-9, 1978. 46 S. Middelhoek, Ferromagnetic Domains in Thin NiFe Films , Ph.D. Dissertation, University of Amsterdam, Holland, 1961. 47 R. C. Minnick, P. T. Bailey, R. M. Sanford, and W. L. Semon, "Magnetic Bubble Logic," WESCON Proc . , Session 8, Paper 2, 1973. 48 T. H. O'Dell, Magnetic Bubble , John Wiley, Inc., New York, New York, 1974. 49 A. V. Oppenheim and R. W. Schafer, Digital Signal Processing , McGraw-Hill, New York, Mew York, 1975. 50 A. J. Pennerski, "Propagation of Cylindrical Magnetic Domains in Orthoferrites," IEEE Trans . Mag., Vol. MAG-5, PP 554-577, 1969. 51 M. Prutton, Thin Ferromagnetic Films , Butterworths, London, England, 1964. 52 L. K. Snick, J. W. Nielsen, A. H. Bobeck, A. J. Kurtsig, P. C. Michaelis and J. P. Reekstin, "Liquid Phase Epitaxial Growth of Uniaxial Garnet Films, Circuit Deposition and Bubble Propaaation," Appl . Phys . Lett ., PP 89-91, 1971. 53 A. A. Thiele, "Device Implications of the Theory of Cylindrical Magnetic Domains," B.S.T. J. , PP 727-775, 1971. 54 A. A. Thiele, "The Theory of Cylindrical Magnetic Domains," B.S.T. J_. , PP 3287-3335, 1969. 55 J. K. Watson, Applications of Magnetism , John Wiley, Inc., New York, New York, 1980. 56 J. K. Watson, H. R. Ma, D. B. Dove, and E. Huijer, "Proximity and Interaction Effects in Arrays of I-bars," IEEE Trans . Mag . , Vol. MAG-12, PP 669-671, 1976. 57 D. M. Yang and R. T. Gregory, A Survey of Numerical Mathematics , Addison Wesley, New York, New York, 1973. 58 M. Zahn, Electromagnetic Field Theory : A Problem Solving Approach , John Wiley, Inc., New York, New York, 1979.

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BIOGRAPHICAL SKETCH Fang-shi Jordan Lai was born in Taiwan, China, on September 19, 1948, He received the B. S. degree in electrical engineering from National Cheng Kung University, Tainan, Taiwan, in 1971. After graduation, he served as a Technical Officer to maintain the communications equipment in the Chinese Air Born Division from 1971 to 1973. From 1973 to 1975, he was a technical staff member of the Chinese Government Radio Administration, Taipei, Taiwan. He received the M. S. degree from National Taiwan University, Taipei, Taiwan, in 1977. His master's research was involved in the gapless structure implementation for magnetic bubble devices. From 1977, he has been working towards the Ph.D. degree in the area of magnetism at the University of Florida, Gainesville, U. S. A. 147

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. / A cfl\ J7i Kenneth Watson, Chairman Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. '^X*~ \j ~r d^C\ Alan D. Sutherland Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Hendrik^U Monkhorst Associate Professor of Physics

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. a c Martin A. Uman Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. //W Ul rich H. Kurzweg Professor of Engineering Sciences This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as Dartial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1980 Dean, Graduate School

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UNIVERSITY OF FLORIDA 3 1262 08666 223 5


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