Determination of magnetic anisotropy constants for bubble garnet epitaxial films using field orientation dependence in ferromagnetic resonances

Mat. Res. Bull., Vol. 16, pp. 957-966, 1981. Printed in the USA. 0025-5408/81/080957-1052.00/0 Copyright (c) 1981 Pergamon Press Ltd.

DETERMINATION OF MAGNETIC ANISOTROPY CONSTANTS FOR BUBBLE GARNET EPITAXIALFILMS USING FIELD ORIENTATION DEPENDENCE IN FERROMAGNETIC RESONANCES

Hiroshi Makino and Yasuharu Hidaka Basic Technology Research Laboratories, Nippon Electric Co., Ltd., l-l, Miyazaki 4-chome, Takatsu-ku, Kawasaki 213, Japan (Received May 27, 1981; Communicated by M. Nakahira)

ABSTRACT Cubic magnetoerystalline anisotropy constant K i as well as uniaxial anisotropy constant Ku and gyromagnetic ratio Y are precisely determined for (Ill) magnetic garnet epitaxial films from measurements on FMR field orientation dependence. Strict FMR conditions are derived from total free energy expressions, where the differences between magnetization direction and applied field direction are taken into consideration. By applying magnetic field in (~10) plane, FMR is measured to obtain the three best fitting parameters of K I , Ku and ¥. Present analysis is compared with Cronemeyer et al's analysis. Influence of sample misalignment on measurement accuracy is also presented. Similarly, for (ll0) garnet films with orthorhombic magnetic anisotropy, measurements are carried out for two crystallographic planes of (001) and (l~0), and the four best fitting parameters of KI , Ku , Aand ¥ are determined.

Introduction S e v e r a l t e c h n i q u e s are in c o m m o n use for the d e t e r m i n a t i o n of both u n i a x i a l

magnetic anisotropy constant (Ku) and cubic anisotropy constant ( K I ) for magnetic garnet films. Ferromagnetic resonance (FMR) (1,2), torquemeter (3), optical magnetometer (4-6) and magnetic hysteresisgraph (2) are example methods for such measurements. From the measurement accuracy viewpoint, FMR measurement is one of the most suitable methods. Especially, in contiguous disk (CD) high density bubble memory devices, KI is a very important material parameter to be measured accurately, because charged wall propagation mechanism is closely related to the KI value (7). Double layered structure is often used in CD devices, Using FMR, magnetic anisotropies for both drive-layer and bubble-layer garnets in such a composite structure are separately measurable. In addition, FMR is suitable for the characterization of anisotropy changes induced by ion-implantation, which is a key process in fabricating CD devices. Other important material parameters, such as damping parameter ( ~ ) and gyromagnetic ratio ( Y ), can be obtained simultaneously by FMR. 957

958

H. MAKINO, et al.

Vol. 16, No. 8

This paper describes an accurate method for determining magnetic anisotropy constants (KI and Ku) as well as Y, when 4~ Ms value is known. In order to obtain both K I and K. values for (Ill) films, FMR field orientation dependence must be measureG in addition to perpendicular and parallel resonances. Analytical equations to calculate FMR field orientation dependence for the (Ill) bubble garnet films were reported by Cronemeyer et al. (1) and the equation is applied for determinng KI and Ku values of garnet films for use in CD devices (2). However, Cronemeyer et al's paper does not show the exact solution to evaluate the discrepancy between applied field and magnetization directions. For X-band electron spin resonance, magnetization equilibrium direction does not coincide, in general, with the applied field direction, except for perpendicular and parallel resonances. In the present paper, the total free energy expression was employed to derive more strict FMR condition. The discrepancy between magnetization equilibrium direction and applied field direction are taken into consideration because the discrepancy greatly affects the KI value accuracy. For (lll) garnets, three material parameters, such as KI , Ku and y , were determined by the best fitting method. In addition, using the same technique, analytial equations were derived for the (ll0) oriented garnet films with orthorhombic magnetic anisotropies. In (ll0) films, measurements were carried out in two planes; i.e., magnetic fields were applied both within (001) and (ll0) planes. For (ll0) films, KI , K u, A and y values were determined by the best fitting method, where Ku is anisotropy constant between [U0] and [001], and A is the in-plane anisotropy constant. Resonance Conditions for (ill) Films Ferromagnetic resonance condition can be given by

-

l

E

Clll)

FIG. 1 Coordinate system for (lid film. H is applied in (ilO).

2E ,

C1)

where 0J is the angular frequency, y is gyromagnetic ratio, 0 and ~ are polar and azimuthal magnetization angles, respectively (8). DC magnetic field is applied in the (n0) plane for the (iii) films, as shown in Fig. 1. The total free energy of the system, Et ~., is the sum of Zeeman energy, 8~agnetization energy, uniaxial magnetic anisotropy energy and cubic magnetocrystalline anisotropy energy *, and is given by;

*Contribution from K2 was neglected in the present calculations.

Vol. 16, No. 8

GARNET EPITAXIAL FILMS

959

Etotal = - HMs(sine cos Csin $ + cos e cos 6) + 2~Ms2 cos2e + Kusin2e + Kl(41--sin4 e + ]1c o s

4

,/2. 3 e ÷. 2~-sm e cos e cos3¢ ).

(2)

In this equation, e and ¢ values at equlibrium must satisfy both 3 E/a0 =0 and E/3 ¢ =0. Equation 3 E/3 ¢ =0 always holds when qb=0. This means that Ms always lies in the (ll0) plane, in which H is applied. 3 E/3 e is given by; 3 EtotaI

. =-HMssin(B-~))

~'O

+

(Ku-2~Ms2)sin20 +KI{ - 7 s i n 4 0

_

1 ~sin2e

+~-(3sin20-4sin 4{)) } .

(3)

The second derivatives 3 2 Et,,,,~/3 e2,32 E,o,_,/3 ¢ 2 and substituted into Eq.(1). Then, the reg~'f/~hce conditio~i ~a~iven by;

01 +

{HresMs cos(

1

~22 +~sin 2 0(3

cos20

2

Etotal/~ S3 q5are

cos,

1

- 8 sin 2 0) }. (HresM s sin0 sinB -3 J 2 K 1 sin30 cos 0)] 1/2

(4)

In Eq. (4), H and 0 must satisfy 3 Etotal/) 0 =0 (Eq.(3)=0). At resonance, H in Eq.(3) corresponds to Hres. Especially, perpendicular and parallel resonance conditions have been determined and given by (9); 1 H//[lll] film normal, 0~

2K u

7 = a~_ + ( - i f S

2

4 K1

_ 4~ M s) - - 3 M

S

'

(5)

H//[112] in-plane,

2K K1 =H// • { H / / - ( ~ _ 4 ~ M s ) _ _ ~ _ ) . S

(6)

S

When M is known, K t , Ku, ~ and 0 are unknown parameters in Eq.(4). Experimental ~ata has been analyzed, making use of a computer program based on the following procedure. (i)

Putting an initial value for K 2 (for example, O e r g / c m 3 ), H L ( B--0 ° ) and H / / (6 --90 ° ) are used to calculate K u and ¥ values through Eqs. (5) and (6).

(ii)

H and 0 E~.e~) under values are c a l c u l a t e d for each 6 b e t w e e n 10a and 170 ° by solving the condition of Eq.(3)=0, without using any e x p e r i m e n t a l Hre s values for 6 =10 ° -170 °.

(iii) The coincidence between calculated and experimental data is judged by introducing total error sum for each field orientation (~ (error)2 =

If0

Z 6=0°

(Hrestalc -

960

H. MAKINO, et al.

Vol. 16, No. 8

H o e x p ) 2). By changing K 1 , Z (error) 2 is calculated. Since E (error) 2 is a p~F~ibolie function of K~ , as shown in Fig. 3(c), it is easy to find the best fitting combination of K1 , Ku and "f, which gives minimum E (error) 2 (iv)

In order to eliminate sample misalignment effect (i.e., setting error for 6 ), the input H s data are shifted backwards or forwards using the Lagrange interpolation formula.

FMR Field Orientation Dependence for Bubble Garnet Films In this section, application of the above mentioned analytical equations is described for evaluating magnetic anisotropy constants and gyromagnetic ratio. Figure 2 shows the measured plots for the resonance field orientation dependence, as well as calculated curves for K z =0, -3000, -5250, -6000 and -9000 erg/cm 3 . The material is (GdSmTmCa)3 (FeGe)5 O12 (4~ Ms=640 G) grown on (Ill) Gd3 Gas O12 substrate, which has been developed for use in drive layer material in double layered CD devices (10). In every theoretical calculation curves, y is chosen so that H at both 0° o . res . ( H / / [ l l l ] , film normal) and at 90 (H//[t12], in-plane) are in agreement w i t h expemmental values. In Fig. 2, angle discrepancy between DC magnetic field direction and equilibrium magnetization direction, 8 - (9, as a function of 6 is also shown, which is calculated by using best f i t t i n g material parameters of K u =28950 erg/cm 3 , K t =-5250 erg/cm 3 and c0 / y =4210 Oe. Figures 3(a) and 3(b) show the combination of K 1 , K u and ~ / y which satisfy both H . and H / / . Figure 3(c) is ~ (error) as a function of K1 • By determining the most sui'table parameters of K 1 , K u and ~( , E (error) 2 can be minimized down to several hundreds Oe 2 for summation of 18 points ( 0 ° ~ 170° by 10° step). Sample misalignment is another important f a c t o r which determines the accuracy for best fitting p a r a m e t e r s . A few degrees misalignment results in a different set of fitting p a r a m e t e r s . Figure 4 shows how the sample misalignment a f f e c t s the "best fitting" combinations for K 1, K u and ¥ . Two degree misalignment causes 10% error in K 1 • Therefore, measured Hre s set as a function of field orientation is required to be shifted to or from by a few de~rees to find best fitting p a r a m e t e r s which give absolutely minimum E ( e r r o r ) 2 . On the other hand it is not difficult to set samples such that the rotation axis coincides with [-110] (or H is exactly applied in ('110)). As shown in Fig. 2, angle discrepancy 6 - 0 is as large as 4 degrees for 6=40* and =140° for this sample. If the angle discrepancy is ignored and if the present best fitting p a r a m e t e r s (Ku=28950 erg/cm 3 , K I =-5250 e r g / c m ~ and ~/~f=4210 Oe) are directly substituted into C r o n e m e y e r et al's equation, 2K 2K 1 { H - ( - ~ - u - 4~ Ms) . a ( e ' ) - - ~ - - b ( e ' ) } S

$

2K u

x{ H - ( - ~ s

2K 1

- 4 ~ M s) ' c ( O ' ) - ~ d ( 0 ' )

where

a( 8')=-sin 2 0' ,

and

b( 8')=(3-16x-3y)/16, c( 8')=cos2 O' , d( 0 ')=-(x+3y)/4, x=(2 J2-sin 2 0' + cos 2 e ')/3, y=(4 J 2 s i n 4 8' - 7 cos 4 e')/9,

} ,

(7)

Vol. 16, No. 8

GARNET EPITAXIAL FILMS

GdSmTmCa)3 ( FeGe )5012 I

I

I

I

4TrMs= 6 4 0 I

f = 9.441 GHz

I

I

..-. K=(erg/cm3 ) ,' \

/,~, "l~---- 9 0 0 0

4500

-oooo

//1 /#

\ \~,(best \ \ 1 ~ fittin

961

and where O' is measured from in-plane [ll~.] ( 0 '= B+ 90°) (I), then calculated values are found to be always S hl}e~er than the experimental values, as open circles shown in Fig. 5. On the contrary, if Eq. (7) is used to find best fitting parameters by ignoring the angle discrepancy, one can get another set of parameters (Ku =29200 erg/cm3 , KI =-4270 erg/cm 3 and 0~/Y=4190 Oe). However, the crossed marks in Fig. 5 do not coincide with experimental plots perfectly, while the present analysis (solid dots) gave perfect coincidence.

v

i,/

\ \~;~- 525(

./ ,#

bJ

4000

( GdSrnTrnCo )3 ( FeGe)5012 47rMs=640G I

°

I

I

I

30000 U

29000

t 3500

Best fitting~'~ Ku = 28950 erg/cm 3 V

28000

[,,2] o I

I

I

I

o

30

60

90

[1~i]

Oli] [n.~)

I

I

120

150

looT]

l

180

[~i[]

4300 (b)

31× 4 2 o o

f

• ANGLE~ (DEGREE)

~'--'~"~

410(3. ~

f =9.441GH

5xlO 5

(c) z

~g z~

2

3

-2

~ a

t

-4

~

F--

~ H

L~ -...I

4

~ ~

2 0

50

60

90 120 150 180 ,, ANGLE B (DEGREE)

FIG. 2 Resonance field orientation dependence for (GdSmTmCa) 3 (FeGe) 5012 garnet film grown on (Ill) Gd 3 Gas O12. 4wM =640 G . S and f=9.441GHz. Experimental data are shown by filled circles, while theoretical values for K I =0, -3000, 5250,-6000 and-9000 erg/cm 3 are shown by curves. Angle discrepancy, - 0 , calculated by using the best fitting parameters, is also shown.

L-

~o

12/,

'

0

o

i

-2ooo -4000 -6ooo -sooo -,oooo --.- K I ( e r g / c m 3)

FIG. 3 Z (error) 2 , 0~/y function of K I .

and K u as a The best fitting

parameters are determined as the values which give minimum (error) 2 .

962

H. MAKINO,

~'32000

E

et

(GdSmTmCo)3(FeGe)5012 4~Msffi640G 4220 A I ~/X ffi4210 Oe 8 4210 ~TN.

I

I

I

I

|

I

f = 9.441GHz

® _-,_-

o

f = 9.441 GHz -7000

c. -sooo

KI= - ; 2 5 0 \

erg/cm3

I -3000

r~ d IJJ LL LU 4000 (,.3 Z
f 3500

6x I04

0

~

I

4500

:g 28000

Vol. 16, No.8

(GdSrnTrnCe)3(FeGe)5012 4rrMs=640G

4200 t

30000

E u

al.

I 30

I 60

! 90

I 120

[ 150

! 180

--,- ANGLE ,8 ( D E G R E E )

4 ~

j / m i n = Efor6°g~-2 (O

t -6 -4 -2 0 2 --=" MISALIGNMENT ANGLE ( DEGREE)

FIG. 4 Sample misalignment influence on K z accuracy, For each alignment, a combination of K1 , K and 0J/y , which gives minimum ~ ( e r r o r ) 2 , is obtained. One can find a combination which gives absolutely minimum Z (error) 2 , as depicted by arrows.

FIG. 5 FMR field orientation for ~ - measured data and present calculation by using Eq.(4) and best fitting parameters of K u =28950 erg/cm 3 , K z =-5250 erg/em 3 and c~/y =4210 Oe, ( ~ ) - - - o - - - calculation using Cronemeyer et al's equation, Eq.(7), and the present best fitting parameters (the same as (~)) and (~) - - - x - - calculation using Cronemeyer et al's equation, Eq.(7), by ignoring angle discrepany, which yield smallest ~ (error) 2 . However, agreement with experiments is not perfect, since angle discrepancy has been ignored. See text in detail.

Resonance Conditions for (ll0) Films

Using a similar analytical method, F M R can be also analyzed.

field orientation dependence for (110) films

(110) garnet films exhibit orthorhombie magnetic anisotropy in addition to cubic magnetoerystalline ani_sotropy. Therefore, resonance field orientation dependence should be measured for both 010) and (001) planes. As shown in Fig. 6, when D C magnetic field is applied in the (1i0) plane, which contains [110] (film normal) and [001] (in-plane), the total free energy is expressed by;

Vol. 16, No. 8

GARNET EPITAXIAL FILMS

963

Etota 1 = -HMs(sin 0 cos csin$ + cos0 cos ~) + 2 ~ Ms2COS20 + (Ku + Asin2 ¢ )sin20 Kl 7 +-~--{ c o s 4 0 ( 6 4 +

cos2¢

-

9

7 16 c o s 2 ¢ - 3 53

cos¢+~}

cos4¢)+cos20(~

+~cos2¢

+

cos4¢)

.

(8)

The definition of orthorhombic anisotropy constants, K u and A, is the same as in ref. (ll). Since 3E/3 ¢ =0 for ¢ =0, magnetization lies in the (ll0) plane. 3 E/30 can be given by; 3 Etota 1 30

- -HMs(COS0sin • - sin 0 cos B) + (Ku - 2 ~ Ms2) sin20

K1 . 3 + ~ - - t ~ sin 40 - sin 2 0).

(9)

The second derivatives of E 1 ( 32 E*,~*ol/302 ' 32E,,~,ol/~¢2 and 3 2 E . . . . / 303 ¢ ) are substituted into Eq.t~}~ind the resb~a~dce condition f(~r'~'he (ll0) films, when ~}/v(~0), is given by; I y iH//(gO) Mssin 0

[ {HresMs

cos(O- B)+2(Ku

-

2~M 2) cos20 + TK1 (3cos40 s

K1 5 e o s 4 0 -cos 20)}×{HresM s sin0 sin ~ + 2 A sin20 +-~-(

- 4eos20

+3)}3 1/2

(10)

In this equation, H and 0 must satisfy Eq.(9)=0. At resonance, H=Hres. Next, another resonance condition is derived, where H is applied in another orthogonal plane, which also contains film normal and in-plane directions. As shown in Fig. 7, when DC magnetic field is in the (001) plane, the total free energy expression is

OIo}

01o)

-c,Toj [ooi]

FIG. 6 Coordinate system for (ll0) film. H is applied in (I]0).

cooo i,,//1/~i~./

///J -" co@

07o]

FIG. 7 Coordinate system for (ll0) film. H is applied in (001).

964

H. MAKINO, et al.

Vol. 16, No. 8

Etota 1 = -HM s (sin 0 cos ¢ sin B + cos 0 cos 6) + 2 ~Ms2eOs20 + (Ku + A cos2qb ) sin20

+cos20 (~-

]eos2¢

+

cos4¢)+(-

eos2qb -

eos4¢

+ 6-4 )}"

(11)

In this ease again, since $E/8 ~ =0 for qb =0, magnetization lies in the same plane as the applied field. Derivative of Etota 1 is expressed as 3Et°tal=-HMs(e°s30 0 s i n 6 - s i n Ù e o s 6 ) + ( K u +

A-2

Ms2) sin20 - ~KI__s i n 4 0 .

(12)

and second derivatives are substituted into Eq.(1). Then, the resonance condition for (ll0) garnet films, when H//(001), is given by 0JI = I cos(0- 6)+2(Ku+A -27TMs2) eos2e -2Kleos4e} -YH//(001) Ms sine I { HresMs X{HresMssine sin~-2 Asin2e + ~IKl(3_2eos2e - e o s 4 e ) } ] 1/2.

03)

Especially, in the cases when DC magnetic fields are along [ll0], [Ii0] and [001], the resonance conditionsare (12); 1

H//[110] film normal 2K {(Hj_ +_~__u-4~rM

7 =

s

s

+

K1

2(K u + A ) )(H~_+ ~ - 4~TMs

s

H//[ll0] in-plane co [II0] 2(Ku + A) 2KI 7 = {(H// Ms +4~M s -~)(H/.

s

2K1 1/2 M ) } ,

(14)

s

[I]0]

2A KI I/2 Ms + ~ )}

(15)

and 3 H//[001] in-plane

.y

[001] 2Ku 2K1 [001] 2A 2K1 .{ (H//. . Ms + 4 ~Ms + - ~ s ) (H// +---~s+ -~s )} I/2

(16)

Using almost similar calculation procedure to that in the analysis for (Ill) films, the best fitting parameters can be obtained for Ki , Ku , Aand y , when magnetization M s is known. Examples of analysis for (110) garnet films with orthorhombie magnetic anisotropy are shown in Figs. 8 and 9. The materials are (EuLa)a (FeAIGa)5 O12 grown on (110) Nd3 Gas O12 and (YTmBi)3 (FeGa)5 O12 on (H0) Gd3 Gas O12 substrates, respectively (13,14). For (EuLa)3 (FeAIGa)5 O12/(II0)NdGG, the best fitting parameters yie.ld negative h and negative K I , while K u is positive. Easy axis is [110] because K u + ~- KI - 2 ~ M s 2 > 0 a n d K u + A -KI -2~Ms2 > 0. Medium axis is along[flfll] and the hard axis is [Ii0]. Due to relatively large I K i , as compared to Ku and A, a canting of the easy axis towards [111] (easy axis for e u b i e magnetoerystalline anisotropy) is observed (13), when growth temperature is raised. Fig. 9 is an example where A > O. [" K i [ is very small, hence, this material shows almost "true" orthorhombie magnetie anisotropy.

Vol. 16, No. 8

GARNET EPITAXIAL FILMS

( "I

i

i

(o) H/I

12000

EuLo)s(FeAIGo)5Out I I10) NdzGasOiz

i

m

(ITO)

"--'I0000 0

I

=r

i

f = 9 . 2 4 3 GH z

(b)

2000

j K I = Oerg/cm ~

-

r

(4TrMs = 3 8 2 G ) !

i

H# (001)

i

i

f = 9 . 2 4 3 GH z

8000

ooo

erg/cm K I == -20000

/t

6000

i

0000

A/

8000

965

6000

, / ~ . . ~ " ~

- 13600 "

Z 0

4000

4000

1

f--

2000

ffmmo]

o/I o

2000 [ml,]

[00,1

OH]

,I

,

I

,

~o

60

9o

~20

---'bANGLE ~

HO]

[i?0

I,

I

150

I 0

mso

[~00]

[riO]

I

I 90

, 30

(DEGREE)

, 60

[070] ,

,20

--,- A N G L E

OTO

I

~

,

I

,50

,80

(DEGREE)

FIG. 8

FMR field orientation dependence for (EuLa)3 (FeAIGa)5 O12 garnet films grown on (ll0) Nd 3 Gas Oz2. ( YTmBi )3 ( FeGa)30m i

I

I

I

'1

i

(llO)GdsGosOj2 I

12000

i

(4TrMs=I53.5G)

i

I

I

I

I

I

f2OOOl (o) H / / ( I T O )

0

•

f =9.454GHz

I (b) H//(O01)

f=9.454GHz

10000}

I0000

.J

8000 bJ L) Z

6000

Kl=Oerg/cm

Z 0 co ~J rr

erg/cm ~

8oool

.,~/,,

4000

3

",,,,'%~j-~ooo

t

,<-~"~,:,", K,=-,O000' ' '/ X\ 'C / /#

6ooo~

~,".vr'~-5°oo

;7/ ,/j

\'~', ~ ,~

,ooo j

, j - 700

~,,,j . o

,,//

2oooF

2000

,ioJ CH,.~ ,I o

30

60

(oo0 I

,

90

~2o

[Tr,j I,

~r~

~5o

m8o

ANGLE /3 (DEGREE)

FMR field o r i e n t a t i o n d e p e n d e n c e grown on 0]0) Gd 3 Gas Oz 2 •

0 io]

o~1 o

0oo]

[i~o~

(o]o)

, I ,

I

, l ,

30

60

90

~20

0]o ~50

I ~80

ANGLE ,/~ ( D E G R E E )

FIG. 9 for (YTmBi) 3 (FeGa) s O12

garnet

films

966

H. MAKINO, et al.

Vol. 16, No. 8

Conclusion Cubic magnetocrystalline anisotropy, KI , is one of the most important material parameters for contiguous disk bubble memory devices. An accurate method for determining KI as well as uniaxial anisotropy Ku was presented by analyzing asymmetric dependence of FMR H on field orientation B. In the present analysis, angle discrepancy between magneuzauon and applied field directions are strictly taken into consideration, and perfect coincidence was obtained between experimental and calculated Hres values. Present analysis was compared with Cronemeyeret al's equation. The importance of angle discrepancy was shown. Also, the influence of sample misalignment on measurement accuracy was presented. For (U0) garnet films with orthorhombic anisotropy, the four best fitting parameters of KI , Ku, A and ¥ are acculately determined, using similar techniques. Acknowledgment The authors wish to thank T. Furuoya, K. Matsumi, T. Hibiya, K. Suzuki, M. Mikami and H. Matsutera for many helpful discussions. Also, they would like to thank H. Honda and S. Nakamura for technical assistance. References I.

D.C. Cronemeyer,T.S. Plaskett and E. Klokholm, AIP Conf. Proc. 24, 586 (1975).

2.

A. Gangulee and R.J. Kobliska, J. Appl. Phys. 51, 3333 (1980).

3.

H. Uchishiba, T. Obokata and K. Asama, Japan. J. Appl. Phys. i_66,2291 (1977).

4.

P.W.Shumate, J. Appl. Phys. 44, 3323 (1973).

5.

T. Numata, Y. Ohbuchi and Y. Sakurai, Japan. J. Appl. Phys. 19, 289 (1980).

6.

J.-P. Krumme, P. Hansen and J. Haberkamp, Phys. Status Solidi (a) 12, 483 (1972).

7.

C.C.Shir and Y.S. Lin, J. Appl. Phys. 50, 4246 (1979)

8.

A.H.Morrish, "The Physical Principles of Magnetisms" (John Wiley and Sons, Inc. (1965)).

9.

M.D. Sturge, R.C. LeCraw, R.D. Pierce, S.J. Licht and L.K. Shick, Phys. Rev. B 7_, 1070 (1973).

I0.

H. Makino, Y. Hidaka, H. Matsutera and T. Hibiya, Abstracts of 4th Intq Conf. on Magnetic Bubbles, A-4 (1980).

II.

D.J. Breed, W.T. Stacy, A.B. Voermans, H. Logmans and A.M.J. van der Heijden, IEEE Trans Mag. MAG-13, I092 (1977).

12.

R.C. LeCraw, R.D. Pierce, S.L. Blank and R.Wolfe, IEEE Trans. Mag. MAG-13, 1092 (1977).

13.

H. Makino and Y. Hidaka, Proceedings of 2nd Annual Conf. on Magnetics in Japan, 22pA-5 (1978).

14.

S. Konishi, E. Engemann, J. Heidmann and T. Hibiya, Appl. Phys. Lett. 3_88, 467 (1981).