SURFACE POLE MODELING OF FIELD ACCESS
MAGNETIC BUBBLE DEVICES
BY
FANGSHI 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 ARBITRARYSHAPED
MAGNETIC DOMAINS . . . . . . . .. 7
2.1 Introduction . . . . .. . . .. .. 7
2.2 Review of Twodimensional Discrete Fourier Series 8
2.3 Field Calculations of Arbitraryshaped Magnetic
Domains . . . . . . . . . . . 13
2.4 Computer Results . . . . . . . 18
2.5 Bubblebubble 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 Ibar 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 THREEDIMENSIONAL
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
Fangshi 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 arbitraryshaped mag
netic domains, the development of a surface pole model to analyze arbi
traryshaped 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 cylindricalshaped 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 twodimensional 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
GeorgeHughes, IshakDella 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 Ibars using a Bitter solution technique.
The permalloy Ibars 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 lengthtowidth (aspect) ratio from 2
to 20. The results from the experiments showed that remanent states
tended to be found in thin, narrowwidth and large aspect ratio permalloy
Ibars. 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 cutandtry
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 cutandtry
method will become the most tedious and timeconsuming method to be used.
The problem, however, of the arbitraryshaped permalloy pattern has
made the computation of the magnetization of this pattern extremely diffi
cult. This is due to the nonuniformly distributed demagnetizing field
within the permalloy pattern. The complicated coupling between the perm
alloy pattern and this nonuniformly 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 wellknown 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 twodimensional domain model using closure con
figuration. Only the straight wall moving is allowed in this model,
which makes it questionable if the nonuniformly distributed bubble mag
netostatic field is considered. Almasi and Lin [4] proposed a new approach
to the modeling problem. In their elegant closedform 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 arbitraryshaped 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 bubblebubble interaction problem [9] is clearly understood.
The permalloy mediated bubblebubble interaction, however, is not brought
into consideration in other models, except the onedimensional 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 bubblebubble 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 arbitraryshaped 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. Permalloypermalloy 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 cylindricalshaped 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 Ibars. 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 zigzag 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 arbitraryshaped magnetic domain arrays
using a discrete Fourier series (DFS) technique.
(2) A new method for analyzing arbitraryshaped 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 bubblebubble and perm
alloypermalloy 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 ARBITRARYSHAPED 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 nonuniform
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 cylindricalshaped magnetic domains,
using a twodimensional 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 cylindricalshaped magnetic domains is also demonstrated. Three
dimensional plotting of normalized inplane magnetic field strengths are
calculated for a hexagonal lattice of magnetic bubble, a honeycomb mag
netic domain array and the stripe domain.
Bubblebubble interaction is also discussed, due to its important
role in bubble logic design. The interaction energy is derived by
the twodimensional 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 Twodimensional Discrete Fourier Series
Let F(k,p) be a twodimensional sequence which represents the sampled
data of a continuous twodimensional 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) (21)
x = kh, y = ps
For a linear timeinvariant system, the basic principle of convolution
theorem is also valid here. If we assume that A(k,p) is a twodimensional
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(km,pn)
m=m n=w
= B(m,n) A(km,pn) (22)
m=m n=~
In this equation, k and p are varied from  to .
Throughout this chapter, the twodimensional 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 twodimensional discrete sequence
with the periods M in x direction and N in y direction. This sequence can
then be represented by a twodimensional Fourier series [49] as
M1 N1
A(k,p) = 1 M a(m,n) ej(2/M)mk ej(2 /)n (23)
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
(24)
S p N 1
Similarly, the domains for m and n are defined within
S m
(25)
SOnn
The reason is that if we assume
j (2w/M) mk
ek (m) = e mk (26)
(26)
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 (23) 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, (27)
r1 n e j(27/M) mk j(2T/N) np (27)
TM l L e e
m=O n=O
0 otherwise
Multiplying Equation (23) by ej(2n/M)kr ej(2e/N)pt and summing from
k=0 to Ml and p=0 to N1, we can obtain the following expression
Ml ;I A(kp) j(2T//M)kr ej(21/N)pt
1 A(k,p) e e
k=O p=0
I N1 11 1 N ej(2Tr/M)k(mr) ej(2T/N)p(nt)
1 I I I I a(m,n) e
k=0 p=O m=0 n=0
(28)
Interchanging the summation m,n and k,p we can get the following form
1 N1 A(k,p) ej(2I/M)kr ej(2 /N)pt
k=O p=O
M 1 N1
S a(m,n)
m=O n=O
1 M N1 ej(2n/M)k(mr) ,j(2/N)p(nt)
k=O p=O
(29)
Using the Equation (27), we obtain
M1 N1
a(r,t) = A(k,p) ej(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
(210)
(211)
Thus the transformation pair for discrete Fourier series (DFS) can be
obtained by Equation (23) and Equation (211). If this periodically
distributed sequence has only nonzero terms during the finite duration
0 < k < M 1 and 0 < p < N 1 then the Equations (23) and (211)
can be modified to get the discrete Fourier Transformation (DFT) pair
as follows:
MI NI
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) ej(2 /i)km ej(2/N)pn
k=0 p=O
a(m,n) =
0 otherwise
< k < M1
0 p < N1
(212)
0 _m S M1
0 < n < N1
(213)
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, radix2
decimationinfrequency algorithm of FFT is applied [49]. The reason the
radix2 algorithm is used is that their efficiency and simplicity can
make the program much more easily implemented. A nonradix2 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 (23) and
(211)) and DFT (Equations (212) and (213)) 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 twodimensional 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)] (214)
where
0 m M 1
0 n N 1
2.3 Field Calculations of Arbitraryshaped Magnetic Domains
In this section, the twodimensional discrete Fourier series method
will be applied to calculate the magnetostatic field distributions for
arbitraryshaped 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 M1 NI
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
(215)
T
Fig. 2 Threedimensional 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.
AU
~)
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 (215) 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) (216)
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) (217)
Substituting Equations (215) and (216) into Equation (217) 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 (217).
The scalar magnetostatic potential i(r) can then be derived as
iM1 Ni 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 (218)
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 M1 N1
Hx(kp) = 1 1 E(m,n)
m=0 n=0
Sej(27/M)km ej(27/N)pn
M1 N1 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)]
(219)
[ j sin (2nT/N)]
M1I Ni
m Z T (m,n) E(m,n) ej(2/M)km ej(2 /N)pn
m=0 n=0
(220)
(221)
where
( 1 (mn eT(mn)d ( eT(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 (215),
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 slowvarying 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
inplane position r; where r is the distance to the field
point from the midpoint 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 inplane 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 inplane
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 threedimensional plots for a hexagonal bubble
lattice in a radial field and zcomponent field respectively. The follow
ing figures show the radial field and zcomponent 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 zcomponent
field strength hz. The highest values correspond to unity, as all the
values are divided by the largest value.
2.5 Bubblebubble Interaction Problem
Bubblebubble 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 bubblebubble interaction problem. Here, we present
the mathematical derivation for the bubblebubble interaction by using
the twodimensional 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 Threedimensional 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) uf '
(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 bubblebubble interaction pro
blem.
direction and A in y direction. In order to eliminate the bubblebubble
interaction from another magnetic bubble in the same array A, x and X
x y
are set to a large number. The bubblebubble 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 bubblebubble 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 volumeaveraged 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 zcomponent 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 (222)
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 eT(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 (221).
The interaction energy of these two interacted magnetic bubbles is then
E2b= 2 Mg Hz dv (223)
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 (222) into Equation
(223), E2b can be obtained as follows:
2MB E M1 NI j(2/)km j(2T/N)pn
E2b = M R2 G n ei(mn)/N)pn Azi
i=l m=O n=O
(224)
where G.(m,n) has the same definition as Equation (222),except that the
z is replaced by an increment of Ai. The definition of Azi is shown as
L
t = C Azi (225)
i=l
The equivalent magnetic field strength can then be defined as
E2b
h = 2b (226)
zn 4 MB TR2t
By using the previous program, the bubblebubble 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
026
t,
D
Fig. 13 Magnetic bubble volumeaveraged 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 bubblebubble 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 arbitraryshaped domains, but they also
provide for reduced computer running time by program implementation using
the FFT algorithm [421.
A bubblebubble interaction problem is also demonstrated by using
the DFS method. By comparing their results with exact calculation in
volumeaveraged demagnetizing factors, the use of the DFS method is also
validated in the bubblebubble 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 cutandtry 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 twodimensional 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 twodimensional discrete Fourier series
technique, the permalloy mediated bubblebubble interaction is considered
into the surface pole model. The permalloypermalloy 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 (31)
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) (32)
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 (31). The demagnetizing potential Qd(r) is then
shown as:
1 ps(7")
: ) d 4 I ds' (33)
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) (34)
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 nonuniformly 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 (33) 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 (35)
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 permalloypermalloy 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 (32) and (35) to include eight neighbor
periods and M permalloy patterns within one period shows:
N
ej= Ci K p (36)
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 permalloypermalloy
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 (37)
where m is used to indicate that this equation belongs to mth permalloy
pattern. Equation (37) can then be rewritten as
N
SCm Q = 0 m = 1,2,3,...,M (38)
i=l 1 i
where
M
N = r L
m=l
Sim i {L
Qin= I
LO i {Lm}
Combined with Equation (36) 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 permalloypermalloy interaction. In the oth period, the
argument of P?. approaches to infinity when i equals j, as shown in
Equation (36). If we expand P in the rectangular coordinate system,
13
then P can be rewritten as
ji
J rt/2 [dzo dlo
St/2 [(x5 X$ )' + (yj Y )2 + (zj z o)2]
(39)
where t is the thickness of permalloy patterns and
dl = dx'2 + dy'2
dlio =dxio+ dyio
Integrating Equation (39) over the z domain from t/2 to t/2, Equation
(39) can be rewritten as
Sn t/2 + R dl' (310)
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 slowvarying function when it is approaching infinity. Exclusion
of the singular point in calculation of Equation (310) 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 106. 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 (31) as
(
E = ps) e(7) dv ps ) d(F) dv (311)
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) (312)
Substituting Equation (312) into Equation (311), we have
E= p r) [ + e(7r)] dv (313)
Let us assume that the zero energy level is the energy of permalloy
pattern without introducing the magnetic bubble. The magnetostatic
energy of Equation (313) 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 (313) by excluding the fb(r) term from te(r) shown
in Equation (34). Using Equation (218), the magnetic bubble
magnetostatic potential can be obtained. Combined with Equations (34),
(36), and (38), the surface pole distribution, which is introduced by
external field or bubble stray field, can be determined. Equation (313)
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 = (314)
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
 GeorgeHughes
 IshakDella 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 IshakDella 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 underrelaxation factor should be decided on
at first, so that the iteration can converge. The IshakDella Torre
model also has a variation between the optimum underrelaxation factor
and the number of digitized mesh points. This makes the problem become
more complicated.
The GeorgeHughes 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 GeorgeHughes 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 (315)
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 (316)
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 (315), be
cause in this equation, V1 M will not automatically become zero.
However, the GeorgeHughes 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 (315), which will introduce error
in their calculated results. Acutally, IshakDella Torre's model also
includes both volume and surface poles due to their nonconstant finite
susceptibility. It is plausible that the discrepancy between IshakDella
Torre's and GeorgeHughes' 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 IshakDella Torre's and GeorgeHughes' 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 thicknesstowidth 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 onedimensional 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 overcalculated. 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
threedimensional 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 threedimensional 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 bubbletobubble and permalloy
topermalloy 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 bubblebubble repulsive energy is not
taken into account. If we take the bubblebubble 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 Threedimensional 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 Icu
0
E
SsL
c0 0
CC
a)
r
1n 
LA, a)
sU
CM
*C *I
0 C
4CI
u m
3 * Id
4. 0 
00
cu
5
sc
0 O
0 *l 0
. .0 0.
0 0 0
0  
* 0 s
UCQ
+C Q S
U3
ao oz
"0 0Z
) CE
+
CL
0
r I
 C
M c)
CO
.. S :
C
4.
oc
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 bubblebubble
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
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rO
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0,
r
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r0
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C
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3
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o
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a
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a
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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 majorminor 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 stripeout rather than transfer. In Field Orien
tation 3, the flat potential well will cause the bubble to stripeout,
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
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E
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S C=
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experimentally verified by comparing the minimum driving field require
ments for HI, TI, and doubleTI 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 signaltonoise 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 stateoftheart
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
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/ *. /. 0 C
/ ^
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lUL 
66
E
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a C
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0.
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UC
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/ 5 I*
/ 0.
Sr=
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m
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N ^S^
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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
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0
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ai
we
CU
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C 4 C o 4 r
U)
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r, C> mM* LL.
69
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+
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C
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c.
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Cs
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0
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00
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1)
0
e,
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E
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01
C
r.
U
C)
4'
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4
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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 arbitraryshaped permalloy patterns; it can also be used to calculate
the potential well depth profile by considering the bubblebubble and
permalloypermalloy 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 GeorgeHughes' model needs approximately
three minutes and 800K bytes in an IBM 370 system [21]. IshakDella
Torre's model needs eleven minutes and 110K words of memory in the
CDC6400 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 zigzag wall config
uration in various geometries of permalloy Ibars are studied in this
chapter in order to understand the nature of remanent states.
A Bitter solution technique is used to study the quasistatic mag
netization response of Ibars 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 electronbeam and xray
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 switchingcurrent 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 inplane 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 zigzag wall model (remanent state model) and study its be
havior with the function of thicknesses and aspect ratios of permalloy
Ibars.
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 107 Torr is used in our Varian
sputterion 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 J100 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 ferromagnetic 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 darkfield
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 piledup 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 darkfield microscopy. Khaiyerand O'Dell [37]
used the Nomarski interference contrast technique. This technique detects
the surface irregularities caused by the pileup 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 oilimmersion 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 piledup colloidal particles
scatter the light, the dark domain walls show up against the bright back
ground. A product of the Ferrofluidic Company, AO1, 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
Ibars 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 wellknown properties of
domain walls [46].
5.6 Remanent States of Ibar 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 Ibar. 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 reobserved [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, zigzag
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 zigzag walls then become turbid. All of the zigzag walls
break down when a 4.3 Oe field is applied, and the demagnetized state
is reobserved 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. Zigzag 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, zigzag walls appear as the field reduces to 8 Oe.
The demagnetized state is reobserved 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.
