STRESS-RELATED PHENOMENA AT GROOVING EDGES IN HETEROEPITAXIAL GARNET FILMS

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STRESS-RELATED PHENOMENA AT GROOVING EDGES IN HETEROEPITAXIAL GARNET FILMS

D. Klein, J. Engemann

To cite this version:

D. Klein, J. Engemann. STRESS-RELATED PHENOMENA AT GROOVING EDGES IN HET-

EROEPITAXIAL GARNET FILMS. Journal de Physique Colloques, 1985, 46 (C6), pp.C6-131-C6-

135. �10.1051/jphyscol:1985623�. �jpa-00224870�

Colloque C6, suppl6ment au n09, Tome 46, septembre 1985 page C6-131

S T R E S S - R E L A T E D PHENOMENA A T GROOVING EDGES I N H E T E R O E P I T A X I A L GARNET

F

I LMS

D. Klein and J. Engemann

Uiziv. o f WuppertaZ, Dept. EZectricaZ Engineering, 5600 WuppertaZ I , F.R. G.

Rksum6 - La s t r u c t u r e d e s p a r o i s e n t r e domaines dans d e s couches d e g r e n a t o r i e n t k e s s e l o n ( 1 1 1 )

,

avec d e s a n i s o t r o p i e s magnet i q u e s e t d e s champs locaux a s s o c i e s aux changements brusques dans 1 ' Q p a i s s e u r d e l a couche, e s t ktudige thkoriquement en u t i l i s a n t l a mCthode d e RITZ. La s t r u c t u r e i n t e r n e d e s p r o i s e n t r e domaines obtenues d a n s l e v o i s i n a g e immgdiat d e s bords d e s i l - l o n s e s t d6formi.e. S i l e p r o f i l d ' g p a i s s e u r d ' u n e couche d e g r e n a t e s t u t i l i s k pour C t a b l i r un p o t e n t i e l B une dimension en vue d e l a propagation d e s l i g n e s d e Bloch v e r t i c a l e s dans une mkmoire 2 l i g n e s - a e Bloch, l e s f o r c e s n $ c e s s a i r e s au dkplacement d e s p a r o i s y s e r o n t un peu p l u s f o r t e s que pour l e s p a r o i s qui s e t r o u v e n t l o i n d e t e l s changements brutaux d ' k p a i s s e u r .

A b s t r a c t

-

The domain wall s t r u c t u r e s i n ( 1 1 1 ) - o r i e n t e d g a r n e t f i l m s with l o c a l magnetic a n i s o t r o p i e s and f i e l d s a t a b r u p t changes i n f i l m t h i c k n e s s a r e i n v e s t i g a t e d t h e o r e t i c a l l y u s i n g t h e R I T Z s method. I t i s found

t h a t t h e i n t e r n a l s t r u c t u r e of domain w a l l s i n t h e immediate v i c i n i t y of grooving edges i s d i s t o r t e d . S i n c e a topographic shaping of a g a r n e t . f i l m is used f o r s e t t i n g a one-dimensional p o t e n t i a l f o r v e r t i c a l Bloch l i n e propagation i n t h e Bloch l i n e memory, t h e a c t u a l f o r c e s r e q u i r e d t o move t h e w a l l w i l l be somewhat higher a s compared t o w a l l s f a r away of such a b r u p t changes i n f i l m t h i c k n e s s .

I

-

INTRODUCTION

Observations of magnetic domains a t ion-milled s t e p s i n g a r n e t f i l m s /1/ i n d i c a t e d t h a t a b r u p t changes i n f i l m t h i c k n e s s have a s t r o n g i n f l u e n c e on t h e wall i n t h e v i c i n i t y of t h e edge. These phenomena can be used f o r l o c a l l y s t a b i l i z i n g domains and a l s o f o r s e t t i n g a one-dimensional p o t e n t i a l f o r v e r t i c a l Bloch l i n e propagation /2/

i n t h e Bloch l i n e memory / 3 / . I n t h i s paper we r e p o r t on t h e i n t e r a c t i o n of domain w a l l s with l o c a l i z e d s t r e s s f i e l d s and p r e s e n t a model which p r e d i c t s t h e t w i s t a n g l e s and t h e e n e r g i e s of w a l l s a t grooving edges i n h e t e r o e p i t a x i a l /4/ g a r n e t f i l m s .

I1

-

MAGNETIC ANISOTROPY

The a n a l y s i s of t h e i n t e r a c t i o n of domain w a l l s with etched s t r u c t u r e s

i n g a r n e t f i l m s assumes t h e i d e a l i - .+.

zed s t r u c t u r e a s shown i n f i g . 1, and t t i e r e f o r e s i m p l i f i e s t h e calcu-

l a t i o n t o t h e two-dimensional case. hz

A s f u r t h e r s i m p l i f i c a t i o n s we as- h,

sume, t h a t t h e nonmagnetic s u b s t r a - I I

t e and t h e magnetic f i l m a r e c u b i c

+., - ,--

1 ^{I I}

and have t h e e l a s t i c c o n s t a n t s of - -

-

- - - - - ^***,_,--- + + + + +

1

+ Y I G /5/. I n s p h e r i c a l p o l a r coordi-

n a t e s t h e a n i s o t r o p y may be w r i t t e n Fig. 1

-

Schematic drawing of a g a r n e t

a s f i l m with t h i c k n e s s e s hl and h2, which

c o n t a i n s a planak domain wall a t x=x _W

.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985623

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PHYSIQUE

I11

-

WALL ENERGY PER UNIT WALL LENGTH

0.0

E = - K cos 2

8 +

AEa

a U (1 j

where

yl

is the uniaxial anisotropy parameter and A,,E the induced stress anisotropy /6/. The analysis of stress in the magnetic film was carried out with a finite element program /7/

confined to the elastic cross-aniso- _CJ tropy of (111)-oriented garnets /8/.

It is straightforward to show that eqn. +

-.'

(1) can be rewritten as a quadratic N form of the direction cosines of the magnetization and a symmetrical tensor with the eigenvalues K 5 K 2 Z K

.

^These

eigenvalues may be use& direct?y

,

to obtain the local anisotropy in the vi- cinity of etched microstructures in a strained garnet film. Fig. 2 shows t3e

variation of the normalized (to 2xMS)

-1.0

The wall energy per unit wall length for planar domain walls in garnet films with localized stress and magnetic fields can be expressed as

-

-

-

0

4

4 - ^[fi

c 0 s 8 - ~ - hn cos (P -h for x < O ^{2 f i} -hl for x > O 2

uniaxial anisotropy q and stress in-

4.5 5.0 5.5 6.0 0 . 5

duced inplane anisotropy q. with z for x =-I. 14 Pm, where the z-a+&raged value

< q > as a function of x has a minimum. W

9 'lip

The increase of q to the film surface Fig.22

-

Dependence of the normalized (to at is accompanied by the increase 2 TIM uniaxial anisotropy q and stress _S of q.

.

The local minimum of q . at induced inplane anisotropy q. on position z/h2&P:14 to

chang:.

of the through film thickness r forlg =-1.14 pm.

W direction of the stress induced inplane The parameters, which are used in this

anisotropy. calculation are listed in table I.

where

2

q'. =

-

_{+ 9 + qscos} ² _{( P + q. sin} ²

( 9 - 9 .

)

1P =P

hncos (P

8-oo= arc sin

-

9'

(see ref. 9 and 10 for similar expressions of the local wall energy density in orthorhombic bubble materials). The azimuthal angles (P (direction of magnetization in the wall center) and (P. (direction of the local inplane anisotropy in the' (111)-plane) are measured #om the x axis. A, h and q are the exchange stiffness, the normalized (to 4 K M ) m gnetic field due tonmagnet% surface and volume charges, and the normalized (to $TI??) local shape anisotropy, respectively. The local shape anisotropy may be obtained gy calculating the demagnetizing field of the stepped surface of Fig. 1 and h =h (q,x,z) by a modification of the stray field, given by Holz and Hubert /11/.

~8

f?nd the azimuthal angle (P and the wall energy per unit length e eqn. (2) is integrated over the film thickness, using third-order finite-dyfference formulae with error estimates /12/, and minimized with respect to the twist angle (RITZs method).

Although the choice of trial functions for any variational calculation is to some extent arbitrary, we choose the orthonormalized ansatz

N

(p = cl

+ fl

Ccncos(n-1)~z'

n=2 ⁽⁵⁾

which satifies the exchange boundary condition at the film surfaces. The reduced coordinate z' is normalized to h for x < O and to h for x>O. Clearly we cannot

1 .

expect the exact solution with $his ansatz for a flnlte numbgr N of the variational parameters, so our numerical results have an accuracy of 10

,

which can be improved by the increasing of N. However, the significant advantage of the finite series solution according to eqn. (5) is to find the starting values by a real fast Fourier transform of the contour of the minimum magnetostatic energy (A=O). In the second step we calculate the wall structure by a numerical minimization of the wall energy per unit wall length.

V

-

NUMERICAL RESULTS

As an example, fig. 3 shows the structures of the twisted domain walls at

x

=-1.14 pm (solid line, referring to fig. 2) and x = 00 (dashed line). The average vayue of twist angle

<p>

is determined by the bglance between the local demagnetization energy and the local magnetoelastic energy due to the abrupt change of the film thickness at x=O and approaches ll/2 in the limit 1x1 -& 0 0 . In general ew decreases/

increases at positions in the garnet film, where the induced strain produces compression/tension. In the case with neglibible magnetoelastic interactions,<y>

is lower/higher than K/2 for x <O/x> 0.

Fig. 3 - Plot of the azimuthal angle of the magnetization at the center of a domain wall at x =-1.14 pm (solid line) and x =iW (dashed line) as a function of the position in the garnet film with thickness h2. W

The calculated wall energy per unit wall length e versus position x (normalized to Ah=h -h ) in the garnet film plotted in fig. 4 sgows a significant variation in the

2 1

immediate vicinity of the edge and has a local minimum at x/Ahz-2.

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Fig. 4

-

Wall energy per u n i t w a l l l e n g t h ew a s a f u n c t i o n of t h e p o s i t i o n x (normalized t o Ah=h -h ) f o r t h e para- meters f i s i e d t a b l e I .

V I

-

DISCUSSION AND CONCLUSIONS

According t o f i g . 4 , one e x p e c t s t h a t t h e s p i n arrangement of domain w a l l s i n t h e immediate v i c i n i t y of t h e edge cannot f o l l o w t h e a b r u p t changes i n e i n s i d e t h e wall. Therefore t h e changes of wall e n e r g i e s have t o be averaged o v e r t h e wall width. W

A d e t a i l e d a n a l y s i s of t h e d a t a with d i f f e r e n t changes of f i l m t h i c k n e s s e s , with higher s t r e s s e s i n f i l m s due t o t h e mismatch i n l a t t i c e s p a c i n g , and of wall s t r u c - t u r e s u s i n g t h e s t e a d y s t a t e approximation i s n o t undertaken i n t h e p r e s e n t paper.

More complete d a t a a r e r e q u i r e d before a conclusion should be drawn. However, we o u t l i n e t h e two conclusions from t h e foregoing:

1) I n t h e v i c i n i t y of a b r u p t changes of f i l m t h i c k n e s s l o c a l v a r i a t i o n s of t h e magnetic a n i s o t r o p y and magnetic f-ields d i s t o r t t h e i n t e r n a l s t r u c t u r e of domain w a l l s .

2) Since a topographic shaping of a g a r n e t f i l m i s used f o r s e t t i n g a one-dimensional p o t e n t i a l f o r v e r t i c a l Bloch l i n e propagation i n t h e Bloch l i n e memory, t h e a c t u a l f o r c e s r e q u i r e d t o move t h e w a l l w i l l be somewhat h i g h e r a s compared t o w a l l s f a r away of such a b r u p t changes i n f i l m t h i c k n e s s .

TABLE I. Parameters used in the calculations.

h2= 4.27pm ~TcM, = 195 G

h , = .9xh2 K, = 8230 erglcm3

A,,, = - 3 . 0 4 ~ 1 0 - ~ ~ = 2 . 6 3 ~ 1 0 - ~ e r g l c m A,,, = - 1 . 4 4 ~ 1 0 - ~ u,=-lx10~ dyn/cm2

ACKNOWLEDGMENTS

The a u t h o r s wish t o thank D . Krumbholz f o r h e l p f u l d i s c u s s i o n s and S t i f t u n g Volkswagen- werk f o r t h e f i n a n c i a l s u p p o r t .

REFERENCES

/1/ K l e i n , D. and Engemann, J . , J . Magn. Magn. Mat. 45, (1984) 399.

/2/ K l e i n , D. and Engemann, J . , J . Appl. Phys. 57, (1985) 4071.

/3/ Konishi S , , IEEE Trans. Magn., MAG-19, (1983) 1838.

/5/ Haussiihl, S., 2 . Naturforsch. 31a (1976) 390.

/6/ Chikazumi, S., Physics of Magnetism (Wiley, New York 1964) Part 3 , Chap. 8 /7/ Bathe, K. J., Applications Using ADINA, Report AVL 82448-6, Dept. Mech. Eng.,

M.I.T. (1977)

/8/ Engemann, J., 2. Res. Rep., Contract No. 83128, (VW-Foundation, 1985).

/9/ Krumbholz, D., Heidmann, J., Engemann, J. and Kosinski, R. A . , IEEE Trans. Magn.

20 (1984) 1138.

/lo/ Engemann J., 6. Res. Rep., Contract No. 423-7291-NT 2576, (German Ministry of Research and Technology BIVIFT, 1984).

/11/ Holz, A. and Hubert, A., 2. angew. Phys. 26 (1969) 145.

/12/ Gill, P. E. and Miller, G. F., Comp. J. 15 (1972) 80.