09 - Dynamics of Bloch Lines and Their Clusters

7/26/2019 09 - Dynamics of Bloch Lines and Their Clusters

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9 D y n a m i c s o f B l o c h L i n e s

a n d T h e i r C l u s t e r s

Bloch lines are another example of topological magnetic solitons observed

in real magnets. Some properties of Bloch lines were described in Chap. 2.

Usually, vertical Bloch lines VBL) can exist inside the domain wall in thin

magnet ic films. VBL are perpendicular to the film plane. If the nonuniformi-

ties in the direction of the normal to the film are neglected, then the usual

domain wall appears to be a linear soliton rather than a plane one as is the

case in an infinite magnet), and VBL represents a point topological soliton

a kink) on this line.

The investigation of the dynamics of these kinks and their collisions com-

prises an interesting task for physics and may result in a broadening of the

ordinary understanding of multisoliton interactions in quasi-one-dimensional

systems, which are represented by the domain walls of a ferromagnet with

strong uniaxial anisotropy. Moreover, the investigations of the VBL dynamics

are also of interest in connection with the creation of devices of a superdense

magnetic memory proposed by

onishi

[9.1]. In these devices, information

is stored in pairs of lines. Such systems of memory on VBL, including the

devices for recording, t ransmitting and readout of information, are being de-

veloped in a number of laboratories worldwide.

The dynamics of VBL have been studied in less detail than the dynamics

of domain walls. However, in recent years certain results, both experimental

and theoretical, have been obtained in this field. We would like to present

these results in our review without claiming a comprehensive coverage of this

fast developing area of the physics of magnetic solitons. This chapter will

begin with the general theory of the VBL motion and a discussion of the

origin of the gyroscopic force; since the latter determines the peculiarities of

the VBL dynamics which are important for the registration of VBL. Further

theoretical sections of the chapter will be correlated with experiment and

calculations will follow the relevant experimental results.


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132 9 . Dy nam ics o f Bloch L ines and Th ei r Clus te rs

9 . 1 G y r o s c o p i c D y n a m i c s o f V B L

W e c o n f i n e o u r s e lv e s t o t h e a n a l y si s o f t h e a b o v e m e n t i o n e d c a s e w h e n w e c a n

n e g l ec t t h e d e p e n d e n c e o f m a g n e t i z a t i o n o n t h e c o o r d i n a t e z , p e r p e n d i c u l a r

t o t h e f i l m p l a n e ( w h i c h i s t h e c a s e i n t h e e x p e r i m e n t ) . I n t h e t h e o r e t i c a l

d e s c r i p ti o n o f t h is g e o m e t r y , t h e V B L a r e c h a r a c t e r iz e d b y a t w o - d i m e n s i o n a l

d i s t r i b u t i o n o f m a g n e t i z a t i o n , M = M x , y ) , w h e r e x , y a r e t h e c o o r d i n a t e s

i n t h e f il m p l a n e . F o r a s p e c i f ic d e s c r i p t i o n o f t h e f u n c t i o n M x , y ) , w e c a n

u s e t h e a p p r o x i m a t e f o r m u l a ( 2. 20 ) , a c c o r d i n g t o w h i c h ( f o r t h e V B L s i t u a t e d

a t t h e p o i n t x = 0 , y = 0 o n t h e d o m a i n w a l l p a r a l l e l t o t h e y a x i s ) :

m y = t a n h y / A ) s i n O , m x [ 1 /c o s h y /A ) ]

s i n 0 ,

m z

= co s 0 ,

9 . 1 )

an d co s 0 = t a n h [ z / A y ) ] . H e r e , A a n d A a r e t h e t h ic k n e s s o f t h e d o m a i n

w a l l a n d t h e V B L , r e s p e c t i v e l y .

T h e c h a n g e in t h e a n g l e ~ i n t r a c i n g c o m p l e t e l y a l o n g t h e d o m a i n w a l l

c e n t e r ( i. e. , a l o n g t h e l in e x = 0 ) , is e q u a l t o 1 8 0 ~ T h i s s - V B L r e s e m b l e s a

~ r - k in k o r a 1 8 0 ~ d o m a i n w a ll . B l o c h l in e s, w i t h t h e m a g n e t i z a t i o n r o t a t i o n

o f 36 0 ~ (a l s o c a l le d 2 ~ r - V B L ) , c a n e x i s t i n a d d i t i o n t o ~ r - V B L . T h e 2 7 r - V B L

a r e c a u s e d b y t h e m a g n e t i c f i e ld a p p l i e d i n t h e p l a n e o f t h e f il m a n d r e m o v i n g

t h e e q u i v a le n c e o f t h e m a g n e t i z a t i o n d i re c t io n s + e y a n d - e y .

O n e m o r e i m p o r t a n t a n a l o g y f or V B L c a n b e i n di c a te d , i n a d d i ti o n t o t h e

a n a l o g y w i t h k in k s . I t is s e e n f r o m f o r m u l a ( 9 .1 ) t h a t w h e n t h e m a g n e t i z a t i o n

v e c t o r m t r a c e s r o u n d t h e V B L a l o n g t h e c l o se d l o o p i n t h e f i lm p l a n e a t a

s u f fi c ie n t d i s t a n c e f i 'o m t h e V B L c e n t e r , t h e m a g n e t i z a t i o n v e c t o r r n s w e e p s

i n a n e a s y p l a n e z y a t t h e a n g l e 3 60 ~ I t r e s e m b l e s t h e b e h a v i o r o f t h e s u-

p e r f lu i d c o n d e n s a t e p h a s e i n t r a c i n g r o u n d t h e v o r t e x c e n t e r i n t h e s u p e r f l u i d

l iq u id . D e s p i t e t h e f a c t t h a t t h i s p r o p e r t y m a y a p p e a r t o b e o n l y a f o r m a l

o n e , t h e V B L e x h i b i t s c e r t a i n p r o p e r t i e s o f t h e m a g n e t i c v o r t e x , w h i c h is

m a i n l y m a n i f e s t e d i n t h e p r e s e n c e o f a s p e c if ic g y r o sc o p i c f o r c e a c t i n g o n t h e

m o v i n g V B L a n d l a rg e l y d e t e r m i n i n g i ts d y n a m i c s .

G y r o s c o p i c p r o p e r t i e s o f t w o - d i m e n s i o n a l m a g n e t i c s o l it o n s w e r e f i rs t o b -

s e r v e d in t h e e x p e r i m e n t s o n t h e d y n a m i c s o f m a g n e t i c b u b b l e s i n t h i n m a g -

n e t i c f i lm s . T h e s e p r o p e r t i e s a r e d i s p l a y e d i n s i m i l a ri t y o f m o t i o n o f t h e m a g -

n e t i c b u b b l e t o t h a t o f a c h a r g e d p a r t i c l e in a m a g n e t i c f ie ld ( o r t o a v o r t e x

i n a l i q u id i n t h e p r e s e n c e o f v i s co u s f r ic t i o n u n d e r t h e a c t i o n o f t h e M a g n u s

f o r c e ) i n a r e a l e x p e r i m e n t ; i t r e s u lt s i n t h e m o t i o n o f a m a g n e t i c b u b b l e i n a

d i r e c t i o n f o r m i n g s o m e a n g l e w i t h t h e a p p l i e d f or c e. T h e t h e o r y o f t h e m o t i o n

w a s d e v e l o p e d b y S lonczewsk i [9 .2 ] and Thiele [9 .3 ] , see the monograph [9 .4 ] .

T o d e s c r i b e t h e g y r o s c o p i c f o r ce fo r t h e c a s e o f a r b i t r a r y t w o - d i m e n s i o n a l

n o n u n i f o r m i ti e s o f m a g n e t i z a t i o n ( V B L , m a g n e t i c b u b b le s ), w e u s e a m e t h o d

( s ee I v a n o v a n d S tephanov i ch [9 .5 ]) a n a lo g o u s t o o n e o f t h o s e i n R e f s . [9 .2 -4 ] .

T h e p r e s e n c e o f t h e g y r o s c o p i c f o r c e i s d u e t o t h e e x i s t e n c e o f a t e r m

l i n e a r w i t h r e s p e c t t o t h e t i m e d e r i v a t iv e o f t h e a n g l e qo i n t h e L a g r a n g i a n

o f t h e f e r r o m a g n e t ( 2 .6 ) . T o o b t a i n t h e r e s u l t s f o r a p o s s i b l y w i d e r c la s s o f


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9 . 1 G y r o s c o p i c D y n a m i c s o f V B L 1 3 3

m a g n e t s , w e w il l p r o c e e d f r o m t h e g e n e r a l L a g r a n g i a n o f ( 2. 3 ff ) o b t a i n e d i n

C h a p . 2 f or a n a r b i t r a r y w e a k fe r ro m a g n e t . T h i s L a g r a n g i a n c o n t a in s b o t h

t h e q u a d r a t i c a n d l i n ea r c o m p o n e n t s w i t h r e s p e c t t o O 0 / O t a n d O ~ / O t . T h e

L a g r a n g i a n d e n s i t y c o r r e s p o n d i n g t o t h e l i n e a r t e r m s i s :

M J [z l o , ) oo /o t) + o ,

T o o b t a i n a m o d i f i c a t i o n f o r t h e c a s e o f a f e r r o m a g n e t i n ( 2 . 3 f f ) , i t i s s u f f i c i e n t

t o o m i t t h e c o m p o n e n t s w i t h ( 0 0 / 0 t ) 2 a n d O ~ / O t ) 2 ( f o r m a l l y t o a l l o w c 2 t o

a p p r o a c h o o ) a s w e l l a s t o s e t A 1 - --- , A 2 -- - - i / g M o ) l - c o s 0 ) .

A c c o r d i n g t o t h e g e n e r a l r u l e , t h e d e n s i t y o f t h e m a g n e t i z a t i o n f i e l d m o -

m e n t u m

p is d e t e r m i n e d b y th e f o r m u l a p = - ( O L / O O ) V O - ( O L / O @ ) V ~ ,

w h e r e L i s t h e L a g r a n g i a n d e n s i ty . T h e s o l i t o n m o m e n t u m 0 =

O ( r - v t ) ,

= ~ o ( r - v t ) f o r t h e m o s t g e n e r a l L a g r a n g i a n o f (2 . 3f f) c a n b e w r i t t e n i n

t h e f o r m :

f c~ [ V O ( v V O ) + s in 2 0 V 9 9 ( v V ~ ) ]

= m o d 3 x { ~

- [ A l 0 , ) v 0 + A 2 0 , }

9 . 2 )

T h i s e x p r e s s i o n i s s u b s t a n t i a l l y s i m p l i f i e d f o r a f e r r o m a g n e t f o r w h i c h

c 2 - - + 0 A I = 0 ; i n t h i s c a s e t h e s u b i n t e g r a l e x p r e s s i o n t a k e s t h e w e l l k n o w n

f o r m I / g M o ) l - c o s 0 ) V ~ . H e n c e , t h e d o m a i n w a l l m o m e n t u m i s p r o p o r -

t i o n a l t o t h e a n g l e o f t h e m a g n e t i z a t i o n d e v i a t i o n f r o m t h e w a l l s u r f a c e .

N o w w e c o m e b a c k t o t h e g e n e r a l p r o b l e m . W e t a k e o n l y t h e c a s e s o f s m a l l

v e l o c i t i e s o f t h e s o l i t o n m o t i o n , i . e . , w e c a l c u l a t e t h e d e p e n d e n c e o f p o n v i n

a l i n e a r a p p r o x i m a t i o n w i t h r e s p e c t t o v . T o m e a s u r e t h e v o r t e x m o m e n t u m

i n t h i s a p p r o x i m a t i o n , i t i s n e c e s s a r y t o d e t e r m i n e t h e s t r u c t u r e o f t h e m o v i n g

s o l i t o n i n t h e l i n e a r a p p r o x i m a t i o n w i t h r e s p e c t t o v . T h i s m e a n s t h a t o n e

c a n w r i t e O (r I ) a n d q o ( rl ), w h e r e r ~ = r - v Q i n t h e f o r m ( 9 . 3 ):

O ( r ) = O o ( r ) + v O ( r ) , ~ ( r ) = ~ o ( r ) +

v ( p ( r ) ,

(9 .3 )

w h e r e O o ( r ) a n d ~ 0 ( r ) d e s c r i b e t h e s o l u t i o n a t v = 0 . W e s h a l l d e a l w i t h t h e

e q u a t i o n s l in e a r iz e d w i t h r e s p e c t t o v a n d ~ . E v e n i n t h i s c a se , t h e p r o b l e m

is v e r y c o m p l i c a t e d a n d c a n n o t a l w a y s b e s ol v ed . H a v i n g t a k e n i n to a c c o u n t

t h e a b o v e - s a i d , w e w i ll d i sc u s s t h e s o l i to n m o m e n t u m .

T h e f i rs t t e r m i n (9 .2 ) is p r o p o r t i o n a l t o v a n d h e n c e t h e u n p e r t u r b e d

s o l u t i o n : 0 = 0 0, ~ = ~ 0 c a n b e u s e d . A f e w s i m p l e t r a n s f o r m a t i o n s r e d u c e

t h i s s o l u t i o n t o t h e f o r m : ( E / c 2 ) v , w h e r e E i s t h e e n e r g y o f t h e s t a t i c s o l i to n .

T h i s d e t e r m i n e s t h e t r iv i a l ( L o r e n t z - i n v a r i a n t ) c o n t r i b u t i o n t o t h e s o l i to n

e f fe c ti v e m a s s . I t i s m o r e c o m p l i c a t e d t o d e a l w i t h t h e s e c o n d t e r m . T a k i n g

i n t o a c c o u n t ( 9. 3) , w h i c h c o n t a i n s b o t h a l in e a r c o m p o n e n t w i t h r e s p e c t t o v ,

a n d a p a r t i n d e p e n d e n t o f v . T h e c o m p o n e n t l in e a r w i t h r e s p e c t to t h e s o l i to n

v e l o c i t y d e t e r m i n e s t h e a d d i t i o n o f m t o t h e s o l i to n e f fe c ti v e m a s s A m , w h i c h

c a n b e r e p r e s e n t e d i n th e f o r m : m = E / c 2 + A m . I t s h o u ld b e n o t e d t h a t t h e


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1 34 9 . Dy n a m i c s o f B l o c h L i n es a n d T h e i r C l u s t e r s

m a g n i t u d e o f A m c a n n o t b e so s m a ll as c o m p a r e d w i t h m . F e r r o m a g n e t s

c a n s e r v e a s a g o o d e x a m p l e , s in c e , f o r m a l l y in t h i s c a se ; c 2 - + c o , E / c 2 ~ O ,

a n d t h e e n t i re e f fe c ti v e m a s s i s d e t e r m i n e d o n l y b y t h e m a g n i t u d e o f A m .

T h e t e r m , P 0 , is o f f u n d a m e n t a l i m p o r t a n c e . I t r e m a i n s f in i te a s v - ~ 0 .

I t i s t h i s t e r m w h i c h d e t e r m i n e s t h e g y r o s c o p i c e ff ec ts i n t h e d y n a m i c s o f

n o n - o n e - d i m e n s i o n a l s o li to n s . T o c a l c u la t e th i s v a l u e it is e n o u g h t o u s e

t h e s t a t i o n a r y s o l u t i o n 0 = 0 o, ~ = ~ 0 . F o r P 0 , w e c a n r e a d i l y o b t a i n t h e

e x p r e s s i o n :

P0 = - M 0 2 f { A I 0 , ~ )V 0 0 + A 0 0 , ~ 0 ) V ~ 0 } d 3 x ,

( 9 . 3 )

w h i c h is c a lc u l a t e d w i t h t h e u s e o f t h e s t a t i o n a r y s o lu t io n : 0 = O o r ) , ~ =

Y X, w h e r e r , X a r e t h e p o l a r c o o r d i n a t e s i n x y - p l a n e . F i n a l ly , t h e f o ll o w i n g

f o r m u l a is d e ri v e d f or t h e v o r t e x m o m e n t u m :

p = p o + m v

I t c a n b e d e m o n s t r a t e d s e e [9 .5 ]), t h a t t h i s is t h e v a l u e w h i c h p l a y s t h e

r o le o f t h e d y n a m i c m o m e n t u m , t h a t is , i n t h e p r e s e n c e o f t h e e x t e r n a l f or ce ,

F 0 , a c t i n g o n t h e v o r t e x , t h e e q u a t i o n o f i ts m o t i o n h a s t h e f o rm :

d p / d t : F o r m . d v / d t : F - d p o / d t 9 .4)

I t b e c o m e s c l ea r f ro m t h e s e c o n d f o r m o f t h i s e q u a t i o n , w i t h t h e u s e o f

t h e s o l i t o n a c c e l e r a t i o n d v / d t , t h a t t h e v a l u e o f d p o / d t c a n h a v e t h e s e n s e

o f s o m e f o rc e a c t i n g o n a s o l it o n . L e t u s f in d th i s f o r c e. A s s u m i n g t h a t t h e

m a g n e t i z a t i o n i n s o l it o n s d e p e n d s o n t h e c o o r d i n a t e s a n d t i m e in t h e c o m b i -

n a t i o n : r - v t , r : x , y , 0 ) , w e g e t :

= h M o 2 f D O , ~ ) v x V ~ V 0 0 ) ) d x d y , 9 .5)

g

w h e r e h is t h e f il m t h i c k n e s s a n d t h e d e f i n it io n : D 0 ,

~ ) = O A 1 / O ~ - O A 2 / 0 0 ,

i s u s e d . N o t e t h a t t h e v a l u e o f

D O , ~ )

d o e s n o t c h a n g e w i t h a d d i t i o n o f t h e

t o t a l t i m e d e r i v a t i v e o f t h e a r b i t r a r y f u n c t io n o f 0 a n d ~ t o t h e L a g r a n g i a n ) .

T h e v a l u e o f F g c a n b e c o m p a r e d w i t h t h e g y r o t r o p i c fo r ce . I t e x h i b i ts

p r o p e r t i e s s i m i l a r t o t h o s e o f k n o w n g y r o t r o p i c f o rc e s; li k e t h e L o r e n t z f o r c e

a n d t h e M a g n u s f o r c e. F i r s tl y , F ~ i s p r o p o r t i o n a l t o t h e v e l o c i t y v . S e c o n d l y ,

i t i s p e r p e n d i c u l a r t o t h e v e l o c i t y v , v F g = 0 ; i . e. , i t i s a f o r c e w h i c h d o e s

n o t p e r f o r m a n y w o r k . I t f ol lo w s f r o m t h e s t r u c t u r e o f F g , t h a t t h e g y r o f o r c e

m a y d if fe r f r o m z e r o o n l y i n t h e c a s e o f a t w o - d i m e n s i o n a l d i s t r ib u t i o n s o f

t h e m a g n e t i z a t i o n , m , o r t h e v e c t o r , I , i n a s o l it o n .

S i n c e 8 0 = O o x , y ) a n d ~ 0 = ~ 0 x , y ) , t h e v e c t o r [~700 x V ~ 0 ] i s p e r -

p e n d i c u l a r t o t h e f il m p la n e . H e n c e , t h e f o r m u l a fo r t h e g y r o f o r c e c a n b e

r e w r i t t e n i n t h e f o r m :

Fg G[vx~] , ( 9 . 6 )


5/37

9 .1 Gy roscop ic Dyn amics o f VB L 135

w h e r e ~ i s a u n i t v e c t o r a l o n g t h e n o r m a l t o t h e f i l m , t h e c o n s t a n t G c a n b e

c a l c u l a t e d i n e a c h p a r t i c u l a r c a se . T h e f o r m u l a f o r t h e g y r o f o r c e i n t h e f o r m

o f ( 9. 6) is s im i l a r t o t h e f o r m u l a f o r t h e L o r e n t z f o r c e o f a c h a r g e d p a r t i c l e

e m o v i n g in a m a g n e t i c f ie ld H = H ~ , i f w e s u b s t i t u t e G f o r

e l l ~ c , c

b e i n g

t h e v e l o c i t y o f l i g ht .

N o w , w e c a l c u l a t e t h e v a l u e o f t h e c o n s t a n t , G , f o r v a r i o u s t w o - d i m e n s i o n -

a l s o li to n s w h i c h a r e i m p o r t a n t f o r t h e p r o b l e m o f t h e V B L d y n a m i c s . L e t

u s b e g i n w i t h a n a n a l y s i s o f F g f o r t h e m a g n e t i c b u b b l e . T h e d o m a i n r a d i u s ,

R , i s u s u a l l y g r e a t e r t h a n t h e d o m a i n w a l l t h i c k n e s s . C o n s e q u e n t l y , w e c a n ,

a p p r o x i m a t e l y , c o n s i d e r t h e a n g le , 8 , t o b e d e t e r m i n e d b y t h e f o r m u l a : c os 8 -~

t a n h [ ( r - R ) / A ] ; h e r e w e u s e t h e p o l a r c o o r d i n a t e s r a n d X ; r = 0 d e t e r m i n e s

t h e c e n t e r o f t h e m a g n e t i c b u b b l e . I t i s c l e a r f r o m t h e g e n e r a l f o r m u l a f o r F g

t h a t i f 0 = 8 ( r ) , t h e n o n l y t h e d e p e n d e n c e o f ~ o n X is e s s en t ia l . A s s u m i n g

t h a t 9 = ~ ( X ) , w e g e t t h e f o ll o w in g f o r m u l a fo r t h e c o n s t a n t o f t h e g y r o f o r c e

G :

G = h M o 2 f D 8 , ~ ) d 8 d~

(9 .7 )

F o r a s o l i t a r y V B L i n t h e p l a n e d o m a i n w a ll , t h e b a s i c c o o r d i n a t e d e p e n d e n c e

o f t h e a n g l e 8 i s c o n n e c t e d w i t h t h e r o t a t i o n o f m o r l i n t h e w a l l a n d o n

t h e d e p e n d e n c e o f t h e a n g l e g ~ w i t h t h e r o t a t i o n o f t h e s e v e c t o r s i n t h e a r e a

i n s i d e t h e w a l l . I n t h i s c a s e t h e f u n c t i o n s o f 8 a n d q o 0 c a n b e a p p r o x i m a t e d b y

f u n c t i o n s w h i c h d e p e n d o n l y o n o n e c o o r d i n a t e . W e c h o o s e t h e x a x i s a l o n g

t h e d i r e c t i o n n o r m a l t o t h e w a l l, s o t h a t 80 = Oo x) , ~ 0 - - ~ 0 (Y) . In t h i s c a s e ,

f o r m u l a ( 9 . 7 ), f o r t h e c o n s t a n t o f t h e g y r o f o r c e G , a ls o c a n b e o b t a i n e d .

N o w w e d is cu s s t h e e f fe c t o f t h e g y r o f or c e o n t h e m o t i o n o f t w o - d i m e n s i o n -

a l m a g n e t i c s o l i t o n s . F i r s t , w e c o n s i d e r f e r r i t e s - g a r n e t s . F o r t h e s e m a t e r i a l s

D O , ~ ) = 1 / g M o ) s i n

8 0 ; w h i c h d o e s n o t d e p e n d o n t h e a n g l e p . H e n c e

G = h M o / g ) 2 A ~ , (9 .8)

w h e r e A ~ is t h e c h a n g e i n t h e a n g l e ~ i n t r a c i n g c o m p l e t e l y r o u n d t h e V B L .

I n m a g n e t i c b u b b l e s , t h e m a g n e t i z a t i o n a t e a c h p o i n t o f t h e d o m a i n w a l l

c e n t e r i s p a r a l le l t o t h e w a l l p l a n e d u e t o t h e e n e r g y o f d e m a g n e t i z i n g f ie ld s.

T h e r e f o r e , i n t r a c i n g r o u n d t h e b u b b l e i n t h e d o m a i n w a ll , t h e a n g l e ~ o b t a i n s

a n i n c r e m e n t e q u a l t o 2 ~r . H e n c e , f o r t h i s b u b b l e G = 4 ~ h M o / g . I f t h e r e

e x i s ts s o m e n u m b e r o f 7 r -V B L i n t h e d o m a i n w a l l of t h e b u b b l e a n d e a c h

B l o c h l i n e g i v es a n a d d i t i o n a l i n c r e m e n t o f t h e a n g l e e q u a l t o 7f, t h e n G =

27r(2 +

S ) h M o / g ) ,

w h e r e S is t h e a l g e b ra i c s u m o f t h e V B L s ig n s w h i c h

is c a l c u l a t e d w i t h a n a c c o u n t t a k e n o f t h e s i g n o f t h e a n g l e in c r e m e n t , A ~ ,

c a u s e d b y t h i s V B L .

I f G ~ 0 , t h e n t h e m o t i o n o f t h e m a g n e t i c b u b b l e d u e t o th e a c t io n o f

t h e e x t e r n a l f o rc e , F , ( u su a l ly , t h e m a g n e t i c fi el d g r a d i e n t V H ~ ) p r o c e e d s a t

a n a n g l e c~ t o t h e f o rc e . T h e s t a t i o n a r y m o t i o n is p r o v i d e d b y t h e c o n d i t io n :

- ~ v + F g + F e = 0 , w h e r e ~ i s t h e b u b b l e v i s c o s i t y c o e f f ic i e n t. S i n c e

Fg_l_v,

w e


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g e t f r o m t h i s e q u a t i o n t h a t t h e v a l u e o f c~ d o e s n o t d e p e n d o n t h e e x t e r n a l

f o r c e

t a n s = + 2 ) , 9 . 9 )

t h a t is , t h e a n g l e o f d e f l e c ti o n c a n t a k e a n u m b e r o f d i s c r e t e v a lu e s . C o n d i -

t i o n ( 9 . 9 ) , c a l l e d t h e g o l d e n r u l e i n t h e t h e o r y o f m a g n e t i c b u b b l e s , a ll o w s

o n e to d e t e r m i n e t h e n u m b e r o f V B L i n th e d o m a i n w a l l o f t h e b u b b l e . T h e

n u m b e r o f V B L c a n a m o u n t t o s e v e ra l fa c t o r s o f t e n . I n f a c t, t h e f i r st i n fo r -

m a t i o n a b o u t t h e e x i s te n c e of V B L i n t h e d o m a i n w a lls o f f e r r i t e - g a r n e t s w a s

o b t a i n e d i n t h e a n a l y s is o f t h e m a g n e t i c b u b b l e s d e f l e c ti o n f r o m t h e d i r e c t i o n

o f g r a d H , s e e [9 .4 ].

F o r a s o l i t a r y V B L i n t h e r e c t i l i n e a r d o m a i n w a l l, t h e v a l u e o f G =

27chMo/9,

fo r ~ r -VB L ; an d i s

47rhMo/g,

f o r 2 7 c- V B L . W i t h i n t h i s g e o m e -

t r y , t h e g y r o f o r c e l e a ds t o a m u t u a l e f fe c t o f t h e m o v i n g d o m a i n w a l l a n d

V B L l o c a t e d i n th i s w a ll . W e m a y s a y t h a t t h e g y r o f o r c e is a p p l ie d t o t h e

V B L . I f w e a s s u m e t h a t t h e n o r m a l t o t h e w a l l is p a ra l l e l t o t h e x a x i s, t h e n

t h e g y r o f o r c e is d i r e c t e d a l o n g t h e y a x is a n d s h i ft s t h e V B L a l o n g t h e w a l l

w h e n t h e w a ll m o v e s (v]lx). I f t h e m a g n e t i c f ie l d is a p p l i e d a l o n g t h e y a x i s ,

i .e ., i n s u c h a w a y t h a t t h e V B L h a s t o m o v e a lo n g t h e w a l l, t h e n v l ] y a n d

Fgt[X. T h e p r o b l e m o f t h e V B L m o t i o n c a u s e d b y th e m o t i o n o f t h e w a l l w a s

f i r s t s o l v e d b y S lonczewsk i [ 9 . 2 ] , b y a s s u m i n g t h a t t h e w a l l r e m a i n s r e c t i l i n -

e a r. I f t h e D W v e l o c it y is e q ua l t o v , t h e n t h e v e l o c i t y o f t h e s t e a d y - s t a t e

m o t i o n o f V B L d u e t o t h e g y r o f or c e is d e t e r m i n e d b y t h e c o n d it io n :

G v = ? } B L u (9 .10)

w h e r e ~ BL is t h e c o e f f ic i e n t o f t h e V B L v i s c o u s f r i c t io n . F o l l o w i n g f o r -

m u l a ( 4 .5 ) f o r t h e d i s s ip a t i v e f u n c t i o n o f a f e r r o m a g n e t , a n d u s i n g f o r m u l a

( 9. 1) f o r t h e d i s t r i b u t i o n o f m a g n e t i z a t i o n i n a V B L , w e c a n r e a d i l y g e t:

?]BL ---~ 4 . ~ ( M o / g ) ( A / A ) 9 I t t h e n f ol lo w s t h a t t h e r a t i o o f t h e V B L v e l o c i t y t o

t h e w a l l v e l o c i t y is d e t e r m i n e d o n l y b y t h e r e l a x a t i o n c o n s t a n t , A , a n d t h e

r a t i o o f t h e w a l l w i d t h t o t h e V B L i s:

u / v = 7 r A / 2 A A (9 .11)

A c c o r d i n g t o t h i s f o r m u l a , t h e r a t i o ( u / v ) c a n b e q u i t e l a r g e , e v e n f o r

s m a l l d a m p i n g . H o w e v e r , t h e e x p e r i m e n t i s u s u a l l y n o t d e s c r i b e d in t e r m s o f

s u c h a s im p l e f o r m u l a . T h e e x p e r i m e n t a l v a l u e o f ( u / v ) f or th e s a m e m a t e r i a l

a p p e a r s t o d e p e n d o n t h e w a ll v e l o c i t y v ( f o r m o r e d e t a i ls s e e t h e n e x t s ec -

t i o n o f t h i s c h a p t e r ) . T h e r e a s o n is t h a t t h e D W f l e x u re a t t h e p l a c e w h e r e

t h e V B L i s s i t u a t e d s h o u l d b e t a k e n i n t o a c c o u n t w h e n t h e D W m o t i o n i s

a n a l y z e d , g i k i f o r o v a n d S o n i n [9 .6 ] an d Zvezd in a n d P o p k o v [ 9. 7] c a r r i e d o u t

t h e a n a l y s is o f t h i s p r o b l e m , w i t h t h e D W f l e x u re b e i n g t a k e n i n t o a c c o u n t .

F o r a c l u s t e r c o n t a i n i n g n V B L ' s o f t h e s a m e s i gn w h i c h c o r r e s p o n d s t o

th e i n c rem en t o f t h e an g l e qv b y 2 7 rn , t h e v a lu e o f G = 2~r n h M o / g c a n b e

q u i t e l a r g e . O n t h e b a s i s o f t h i s , Chetkin et al . [ 9 .2 4 ] p r o p o s e d a n d r e a l i z e d


7/37

9 .1 Gyro scop ic Dy nam ics o f VB L 137

a m e t h o d f o r t h e o b s e r v a t i o n o f c l us t er s o f V B L o n a m o v i n g d o m a i n w a l l

b y a n a l y z i n g t h e w a l l f le x u r e. T h e a d v a n t a g e o f t h i s m e t h o d , w h i c h w il l b e

c o n s i d e r e d i n m o r e d e t a i l s b e l o w , c o n s i s t s i n t h e f a c t t h a t i t a l l o w s o n e t o

r e c o r d th e V B L d y n a m i c s in m a g n e t s o f b u b b l e - t y p e m a t e r ia l s w i t h t h i n

d o m a i n w a ll s.

C o n c l u d i n g , w e w i l l di sc u s s t h e q u e s t i o n o f t h e p r e s e n c e o f a g y r o f o r c e in

o t h e r m a g n e t s , a n d i n w e a k f e r r o m a g n e t s W F M ) f i rs t o f a ll. T h e g y r o f o r c e

is n a t u r a l l y a b s e n t i n t h e L o r e n t z - i n v a r i a n t m o d e l o f W F M . I t c a n e x i st a t

Z~l ~

0

or z~ ~

0 a n d c a n b e a s so c i a te d e i th e r w i th t h e n o n - a n t i s y m m e t r i c

D z y a l o s h i n s k i i - M o r i y a i n t e r a c t io n o r w i t h t h e p r e s e nc e o f a s t r o n g m a g n e t i c

f i e l d , i . e . , t h e f a c t o r s w h i c h l e a d t o

D O , ~ ) = O A 1 / O ~ - O A 2 / O 0 ) ~ O .

T h e g e n e r a l f o r m u l a e 9 .6 ) o r 9 .7 ) c a n b e u s e d f o r a n a r b i t r a r y m a g n e t b y

c h a n g i n g o n l y t h e f o r m o f t h e f u n c t io n

D O , ~ )

i n t h e f o r m u l a f o r G . H o w e v e r ,

t h e r e s u l t s o f t h e a n a l y s is o f G f o r W F M a r e n o t s o s i m p l e a n d u n i v e r sa l , a s

i n t h e c a s e o f a f e r r o m a g n e t .

L e t u s b e g i n w i t h a n a n a l y s i s o f t h e e f f e c t o f a s t r o n g m a g n e t i c f i e ld .

D u e t o 2 .3 0 ) t h e c o r r e s p o n d i n g t e r m i n t h e L a g r a n g i a n is p r o p o r t i o n a l t o

H l x O 1 /O t ).

A f t e r s i m p l e t r a n s f o r m a t i o n s , w e g e t t h e f o r m u l a :

D O , ~ ) = 81 H l ) / g S M o 2 ,

w h e r e l is t h e p r o j e c t i o n o f 1 o n t h e p l a n e p e r p e n d i c u l a r t o t h e e a s y a x i s

o f t h e W F M . A s p r e v i o u sl y , w e c h o o s e t h e a x is c a l o n g th e e a s y W F M a x is ,

a n d t h e

a c

p l a n e a s a p l a n e o f t h e t w i s t i n t h e w a ll . I n s e r t in g t h e a n g u l a r

v a r i ab l e s 13 = co s 0 , 11 + i /2 = l e x p i ~ , I = s i n 0 , we g e t a fo rm u l a fo r G o f

t h e f o rm :

G = - 2 7 { H 3 s i n O c o s O d O

y v

+ / s i n 2 O d O [ H 1 / c o s q o d q D + H 2 f s i n g ) d q o ] }

9 .12)

T h e i n t e g r a t i o n l i m i t s i n 9 .1 2 ) a r e s e t , d e p e n d i n g o n t h e m a g n e t i c s o l it o n

b e i n g c o n s i d e r e d . I n t h e c a s e o f a d o m a i n w a l l w i t h a V B L , b o t h t h e a n g l e

0 a n d t h e a n g l e ~ c h a n g e f r o m z e r o t o ~r o r f r o m 7r t o z e r o ) , d e p e n d i n g o n

t h e s ig n o f t h e w a l l o r t h e V B L . H e n c e , t h e c o n t r i b u t i o n t o G i n t h e c a s e o f

a V B L is m a d e o n l y b y t h e c o m p o n e n t / - / 2 , i .e ., o n ly t h e c o m p o n e n t o f t h e

m a g n e t i c f i e ld p e r p e n d i c u l a r t o t h e p l a n e o f I tw i s t s i n t h e w a l l

M e l i k h o v

a n d

P e r e k h o d

[9.8]). O t h e r c o m p o n e n t s o f H a r e n o t i m p o r t a n t a n d f o r 7 r - V B L

a = 1 6 7 c H u ) / g ~ ,

9 .13)

h e r e , L , is t h e u n i t v e c t o r w h o s e d i r e c t i o n c o i n c id e s w i t h t h e n o r m a l t o t h e

p l a n e , w h e r e t h e v e c t o r l r o t a t e s i n t h e d o m a i n w a l l fa r f r o m t h e B l o c h li ne .

A c c o r d i n g t o 9 . 1 2) , G = 0 f o r a 2 7 c -V B L .

I n t h e c a s e o f a m a g n e t i c b u b b l e , i n w h i c h ~ c h a n g e s f r o m 0 t o 2 7r , t h e

v a l u e o f G , d e t e r m i n e d b y t h e m a g n e t i c f ie ld a c c o r d i n g t o 9 .1 2 ) , a p p e a r s t o

b e e q u a l t o z e r o a t a n y o r i e n t a t i o n o f t h e f ie ld .


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138 9 . Dy nam ics o f Bloch Lines and Th ei r Clus te rs

T h e c o n s i d e r a t i o n o f t h e e f fe c t o f t h e D z y a l o s h i n s k i i - M o r i y a i n t e r a c t i o n

( t o b e m o r e e x a c t , o f i t s n o n - a n t i s y m m e t r i c p a r t w h i c h g iv e s r i se t o v i o l a t i o n

o f t h e L o r e n t z - i n v a r i a n c e o f t h e v e c t o r I d y n a m i c s ) w a s m a d e f or rh o m b i c a n d

u n i a x ia l w e a k f e r r o m a g n e t s

O k s y u k

[ 9 . 9 ] ) . I t h a s b e e n f o u n d t h a t a l t h o u g h

t h e r e is a g r e a t v a r i e t y o f f u n c t i o n s

D O , ~ )

w h i c h a p p e a r i n t h i s c a s e , t h e

v a l u e o f t h e c o n s t a n t o f t h e g y r o s c o p i c f o r c e G is e q u a l t o z e r o b o t h f o r a

m a g n e t i c b u b b l e a n d a ~ r - V B L .

T h u s , t h e m a n i f e s t a t i o n o f g y ro s c o pi c e ff e ct s i n t h e d y n a m i c s o f t w o -

d i m e n s i o n a l m a g n e t i c s o li to n s in w e a k f e r r o m a g n e t s is e x t r e m e l y li m i t e d a n d

is n o t s o u n i v e r s a l a s i n t h e c a s e o f f e r r o m a g n e t s . I t is q u i t e p o s s i b le t h a t

i t r e s u l t s f r o m t h e f a c t t h a t t h e i n t e g r a l f o r f e r r o m a g n e t s w h i c h i s i n c l u d e d

i n t h e f o r m u l a f or v h a s a f u n d a m e n t a l m a t h e m a t i c a l m e a n i n g , v i z ., it is

p r o p o r t i o n a l t o t h e d e g r e e o f m a p p i n g o f t h e ( x, y ) p l a n e o n t o t h e s p h e r e

m 2 = 1 ; i n t h e c a s e o f w e a k f e r r o m a g n e t s t h i s r e l a t i o n i s a b s e n t .

9 2 E x p e r i m e n t a l M e t h o d s

o f R e c o r d i n g t h e B l o c h L i n e s

I t w a s n e c e s s a r y t o d e v e l o p m e t h o d s o f r e c o r d i n g t h e V B L i n g a r n e t f i lm s in

o r d e r t o i n v e st ig a t e t h e d y n a m i c s a n d c r e a t e t h e m e m o r y o n V B L . A t f i rs t it

w a s b e l i e v e d t h a t is a v e r y c o m p l i c a t e d e x p e r i m e n t a l p r o b l e m a n d , f o r t h i s

r e a so n , t h e m a j o r e f fo r ts w e r e c o n c e n t r a t e d o n m a t h e m a t i c a l m o d e l l in g a n d

n u m e r i c a l c a l c u l a ti o n o f t h e d y n a m i c s o f V B L . T h e m a t t e r is t h a t t h e c h a r -

a c t e r i s t i c s i z es o f V B L A B L ~-- 1 0 - 5 c m a n d t h e w i d t h o f d o m a i n w a l l s ( D W )

A _~ 1(} 5 + 1 0 - 6 c m , a r e m u c h s m a l l e r t h a n t h e w a v e l e n g t h o f l ig h t . F o r

t h i s r e a s o n i t is d if f ic u lt t o u s e o r d i n a r y m a g n e t o o p t i c a l m e t h o d s f o r r e c o r d -

i n g V B L . I t i s n o t e w o r t h y to i n d i c a t e a l a r g e se r ie s o f w o r k s b y Niki tenko,

Dedoukh et al .

( s ee [ 9.1 0] a n d r e f e r e n c e s t h e r e i n ) . U s i n g o r d i n a r y m a g n e -

t o o p t i c a l m e t h o d s , t h e a u t h o r s i n v e s ti g a te d th e s u b d o m a i n s i n a n o m a l y w i d e

( a b o u t 1 m ) d o m a i n w a l ls o f t h e p l a t e s o f y t t r i u m f e r r it e g a r n e t . H o w e v e r ,

i t w a s i m p o s si b le t o a p p l y t h e r e s u lt s o b t a i n e d a n d t h e m e t h o d s d e v e l o p e d

t o b u b b l e m a t e r i a l s , w h i c h a r e o f g r e a t i n t e r e s t f o r p r a c t i c a l a p p l i c a t i o n . T h e

i n v e s t i g a t i o n o f V B L i n th e s e m a t e r i a l s w a s f o u n d t o b e a t a s k f a r f r o m b e -

i n g h o p e le s s . T h e f i rs t e x p e r i m e n t a l o b s e r v a t i o n o f B l o c h l in e s w a s r e p o r t e d

i n [9.11]. C a r r y i n g o u t t h e i r e x p e r i m e n t s o f t h e d o m a i n w a l l in I n v a r w i t h

t h e u s e o f f e r ro f l u id , t h e a u t h o r s f o u n d t h a t t h e d o m a i n w a l l c o n t a i n s r e g i o n s

w i t h o p p o s i t e l y d i r e c t e d tw i s t s o f t h e m a g n e t i z a t i o n v e c t o r .

B y u s i n g a t r a n s m i s s i o n L o r e n t z e le c t r o n m i c ro s c o p y , a n i m a g e o f a V B L

i n t h e b u b b l e d o m a i n s o f t h i n f il m s o f c o b a l t w e r e o b t a i n e d [9 .1 2]. H o w e v e r ,

t h is m e t h o d r e q u ir e s t h a t f e r r o m a g n e t i c f ilm s w i t h o u t s u b s t r a t e s c a n b e p o -

s i t i o n e d i n t h e v a c u u m .

M a g n e t o o p t i c a l e f f e c ts w e r e f o u n d t o b e u s e f u l f o r r e c o r d i n g V B L . I f t h e

m a g n e t i z a t i o n o f t h e D W w a s p a r a ll e l t o t h e s p e c i m e n s u r f a c e, e i t h e r t h e


9/37

9.2 Methods of Recording the VBL 139

equatorial or meridianal Kerr effect was employed [9.13]. If the DW mag-

netization was perpendicular to the specimen surface, the Faraday effect for

optically transparent ferromagnets or the polar Kerr effect for nontransparent

ferromagnets were used. The presence of VBL in the garnet films is detected

by the spectra of oscillation of the DW [9.14], ballistic overshot [9.4], and

the DW sweep in the high frequency gradient magnetic field [9.15]. In this

case the mobility of the DW regions, containing VBL's decreases. According

to [9.16], it is also possible to register VBL by subjecting the DW to the

effect of the sequence of high frequency pulses. Then the DW sections which

do not contain VBL oscillate, remaining parallel to themselves and producing

distinct images. The sections of the DW which contain VBL produce diffused

images since they move along the elliptic trajectories due to the action of the

gyroscopic force.

The Bloch line in the domain wall in iron whisker monocrystals was re-

corded using a magnetic force microscope. Anti-paral lel magnetized segments

on the DW are clearly distinguished by opposite contrast which indicates the

transition from the forces of attraction to the forces of repulsion, acting at a

spherical probe. The adjacent DW segments are inclined to the direction of

the easy magnetization in the film plane at an angle of about 3~ [9.17].

Th i a v i l le A r n a u d e t a l.

[9.18] proposed the use of light diffraction to

record the static VBL in ferrite garnet films with perpendicular anisotropy.

Since the intensity of light, which is diffracted on the VBL, is small, it was

proposed to observe the diffraction by the dark field method in which the

straight light beam does not reach the microscope objective lens. For this

purpose an eccentric diaphragm was set before the condenser lens. The im-

age in the microscope was produced only by the beams diffracted from the

DW and VBL. A modified scanning laser microscope manufactured by Zeiss

was used. The linearly polarized beam from the He-Ne laser operating at a

wavelength of 0.63 #m could be deflected by two rotat ing mirrors and focused

on the specimen in a spot with a diameter of 1.3 #m. The diffracted light

was recorded with a photomultiplier. The signal from the photomultiplier

through the frame memory was fed onto a TV screen. A modified scheme of

the optical apparatus for observation of VBL was elaborated by

Th e i l e

and

E n g e m a n n

and is presented in Fig. 9.1 [9.19].

This modification is based on the main idea of

Thiavi l le et a l .

[9.18] de-

scribed above. A microprism is added to provide the inclined incidence of

light from a mercury arc lamp on the specimen. The VBL were studied in

one-s ided epitaxial Bi-containing ferrite--garnet films, having a large Faraday

rotation. A typical picture produced with the use of the dark-field method

and based on the diffraction of light on a DW with a VBL is given in Fig. 9.2.

The stripe domains in the photo appear to be dark, the domain walls

directed perpendicular to the plane of incidence of light are grey. Small, more

bright and less bright regions, marked by + , are well observed on the DW.


10/37

140 9. Dynamics of Bloch Lines and Their Clusters

ob jec t ive lens

d i f fr a r ed l ig h t t d i r ec t b earn

~] garn6t~--h~ple

rop r i sm

2::~ condensor lens

1 ~ ~ ha lf s top

Fig. 9.1 Experimental set-up for VBL observation in a polarizing light microscope

using the dark field diffraction [9.19]

In a pulsed magnetic field acting along the DW, these regions shift in the

opposite directions (Fig. 9.2).

This becomes clear if we assume that, for example, brighter DW regions

contain a solitary 7r-Bloch line with positive cr+-charges and less bright re-

gions contain ~r lines with negative a_-charges. The magnetization in the

central part of the DW converges to a VBL with the ~r_-charges and di-

verges from a VBL with the cr+-charges. For this reason these lines shift in

the opposite directions in the magnetic field acing along the DW. The motion

of a VBL along the DW also occurs in the pulsed magnetic field perpendicular

to the film due to the action of the gyroscopic force. The polarization of the

light diffracted on the DW and the VBL is perpendicular to the polarization

of the incident light.

For distinct observation of the diffraction picture it is necessary to use

ferrite garnet films with Bi in which, as noted above, the Faraday ro tation is

large. This rotation is important to produce the VBL images. The nonzero

projection of magnet ization on the wave vector of the light exists in the center

of the VBL. The role of the Faraday rotation in the formation of the DW

contrast in the diffraction picture is not quite clear since the magnetization

in the DW and the wave vector of the light are perpendicular to each other.

Until recently it has been believed that brighter regions on the DW contain

Bloch lines with r charge and less bright regions contain Bloeh lines with r

charge. However, the change in the angle of the incident light falling on the

ferrite-garnet film performed in Ref. [9.20] shows th at brighter regions on the

DW can become less bright and vice versa. In other words, the determinat ion

of th e signs of the cr chm ges of VBL described above was no t found to have an


11/37

9.2 Method s of Recording the VBL 141

Fig . 9.2 a,b Init ial position of three VBLs - - (a) dark and white spots, marked

by + an d (b) VB L positions after apply ing an im pla ne field pulse [9.19]

abso lu te m eaning . Perhaps , the fo rmat ion of the V BL image in the d i f f r ac t ion

picture can resul t f rom the inf luence of a wal l micro defo rma t ion [9 .21].

Hav ing record ed the ini t ia l pos i t ions of several sol i tary VB L of the DW ,

and us ing the above ment ioned method, i t i s poss ible to observe a shi f t of

these VB L in the pu l sed magnet i c f ie ld ac t ing a long the DW. An exam ples o f

these shi f ts are given in Fig. 9 .2 a ,b which show a num ber of VBL (a) pr ior an d

(b) a f t e r the i r m ot ion in the pu l sed magnet i c f ie ld wi th the sharp f ron t, the

f l a t peak and long fa l l t ime . Cons ider ing tha t VBL moves un i formly dur ing

the f la t peak of the ma gnet ic f ie ld pulse and havin g recorded two pos i t ions

of VBL, before and af ter thei r shi f t , the authors of [9 .19] could measure the

depen dence of an average ve loc i ty o f the VBL on the magn et i c f ie ld .


12/37


9 .3 R e g i s t r a t i o n o f t h e D y n a m i c s o f V B L

b y t h e M e t h o d o f H i g h S p e e d P h o t o g r a p h y

Mat s uyama

a n d

Kon i sh i

[9 . 2 2 ] an d

Niki forov

a n d

Son in

[ 9.6 ] p r e d i c t e d t h e

p r e s e n c e o f a D W f l e x u r e n e a r a m o v i n g v e r t ic a l B l o c h li n e d u e t o t h e a c t i o n

o f a g y r o s c o p i c f o r c e . H o w e v e r , i n t h e c a s e o f a s o l i t a r y 1 r - B l o c h l i n e t h e

a m p l i t u d e o f t h e f l ex u r e s h o u l d b e le ss t h a n 0 .5 m a n d i t is r a t h e r d i ff i cu l t

t o o b s e r v e i t w i t h t h e u s e o f m a g n e t o o p t i c a l t e c h n i q u e s .

T h e a p p e a r a n c e o f g r e a t e r D W f le x u r es f o r t h e c l u s t e r s c o n s is t in g o f m a n y

V B L w a s m o r e l ik e ly t o b e e x p e c t e d . F i r s t l y t h e s e e x p e r i m e n t s w e r e c a r r i e d

o u t b y L ian a n d umphrey [9 .2 3]. T h e y o b s e r v e d t h e b u l g e s - f l e x u r e s o n t h e

m o v i n g D W w i t h t h e c l u s t e r s o f V B L . T h e s e f l e x ur e s w e r e la r g e i n a m p l i t u d e

a n d r a t h e r e x t e n d e d i n sp a ce . T h e a u t h o r s u s e d th e m e t h o d o f h i g h - s p e e d

p h o t o g r a p h y w i t h a n o n e - t i m e i l l u m i n a ti o n o f t h e f e r r i t e - g a r n e t f ilm s b y t h e

p u l s e d l i g ht . T h e a u t h o r s o f t h i s w o r k d i d n o t m e a s u r e t h e v e l o c i t y o f V B L

c l u st e rs a n d d i d n o t e s t i m a t e t h e a m o u n t o f V B L i n th e c lu s te r . H o w e v e r , t h e y

n o t e d t h a t l a r g e b u l g e s m o v e f a s t e r t h a n s m a l l o n e s . T h i s h o l d s f o r s e v e r a l

s o l i t a r y n o n l i n e a r w a v e s b u t t h i s is n o t t r u e f o r t h e c a s e o f t h e V B L , w h e n

t h e s i t u a t i o n i s o p p o s i t e . A s w i l l b e s h o w n b e l o w , s m a l l V B L c l u s t e r s m o v e

f a s t e r t h a n l a r g e on e s . B u t i n t h e c a s e o f v e r y l a r g e c lu s t e r s t h e s i t u a t i o n

a g a i n c h a n g e s f o r t h e o p p o s i t e o n e . T h i s f a c t a n d t h e r e a s o n s f or t h i s c h a n g e

w i l l b e d i s c u s s e d b e l o w .

T h e m e t h o d o f d o u b l e h i g h - s p e e d p h o t o g r a p h y i s u s ef ul fo r e x p e r i m e n -

t a l l y d e t e c t i n g a n d m e a s u r i n g t h e v e l o c i t y o f t h e m o v i n g V B L a n d i ts c l u st e rs .

T h i s m e t h o d a ll ow s o n e t o r e c o r d t w o s e q u e n t ia l p o s i t io n s o f t h e D W f l e x u re

n e a r t h e m o v i n g V B L a n d t o m e a s u r e t h e v e l oc i ti e s o f V B L a n d D W i n t h e

r e a l t i m e s c a l e . T h e s e e x p e r i m e n t s w e r e f i r s t p e r f o r m e d i n R e f . [ 9 . 2 4 ] . T h e

s i n g l e r e c t i l i n e a r D W i n a ( B i L a T m ) 3 ( F e G a ) 5 0 1 2 g a r n e t s a m p l e w a s s t a b i -

l iz e d b y a m a g n e t i c f ie ld p e r p e n d i c u l a r t o t h e s a m p l e s u r f a ce w i t h a g r a d i e n t

o f 1 50 0 O e / c m in th e d i r e c ti o n p e r p e n d i c u la r t o t h e D W . A s in g le D W b e i n g

s t r a i g h t i n s t a ti c s w a s m o v e d b y t h e p u l s e d m a g n e t i c f i e ld p e r p e n d i c u l a r t o

t h e f il m s ur f a c e . T h e m a g n e t i c p a r a m e t e r s o f t h e f il m w il l b e g i v e n b e lo w .

U s i ng t h e F a r a d a y e f f ec t a n d t w o - f o l d h i g h - s p e e d p h o t o g r a p h y , a c c o r d i n g t o

t h e p r o c e d u r e d e s c r i b e d a b o v e i n C h a p . 3 , i t i s p o s s i b l e t o d e t e c t t w o p o s i -

t i o n s o f t h e m o v i n g D W w i t h t h e V B L c l u s t e rs in t h e r e a l t i m e s c a le . S u c h

a p h o t o g r a p h i s g i v e n i n F i g . 9 . 3 . T h e d a r k b a n d i n F i g . 9 . 3 a r e p r e s e n t s t h e

r e g i o n w h i c h t h e D W p a s se s d u r i n g a t i m e d e l a y b e t w e e n t w o l i g h t pu l se s .

I t w a s e q u a l t o 0 . 4 s . T h e l i g h t p u ls e s o f 8 n s d u r a t i o n w e r e p r o d u c e d b y

t w o n i t r o g e n l a se r s p u m p i n g t h e s u p e r l u m i n e s c e n c e o f t h e d y e o xa z i n e. T h e

r e q u i r e d t i m e i n t e r v a l b e t w e e n t h e l ig h t pu l se s w a s o b t a i n e d w i t h t h e h e l p o f

a n e l e c t r o n i c d e l a y l in e . I n F i g . 9 .3 a t h e D W m o v e s d o w n w a r d s .

T h e p h o t o g r a p h s c l e a r ly s ho w a n o n - o n e - d i m e n s i o n a l a s y m m e t r i c f o r m a -

t i o n s , s o l i t a r y f le x u r a l w a v e m o v i n g f r o m r i g h t t o l e f t a n d l a g gi n g b e h i n d t h e

D W . T h e a m p l i t u d e o f a s o l i t a r y w a v e w a s e q u a l t o 5 t~ m. T h e m a x i m u m v e -


13/37

9.3 Regis t rat ion of the Dyna mics of VB L 143

Fig . 9 .3 a ,b Double h igh speed photograph of a so l it a ry fl exure wave accompanying

a m oving vert ical Bloch l ine in a dom ain w al l of the garnet f ilm in the contrast :

a) of the dom ains and b) of the dom ain wal l [9 .24]

l o c i t y o f t h e D W f le x u re a c c o m p a n y i n g th e V B L c l u s te r w a s e q u a l t o 5 7 m / s .

A t w o - f o l d p i c t u r e o f a s o l it a r y w a v e o f t h e D W f le x u re w h ic h a c c o m p a n i e s

t h e V B L c l u s te r i n t h e c o n d i t i o n o f t h e D W c o n t r a s t w a s a l so r e p r o d u c e d

i n F i g . 9 .3 b . The change i n t he d i r ec t i on o f t he D W ve l oc i t y r e su l t ed i n a

c h a n g e i n t h e d i r e c t i o n o f t h e V B L c l u s te r m o t i o n . T h u s , t h e V B L c l u s te r

m o ved u nde r t h e i n f luence o f t he gy roscop i c fo rce .

Th e m o t i on o f a so l i t a ry 7~-Bloch l ine unde r t he ac t i on o f t he m a g-

ne t i c f i e l d pu l s e d i r ec t ed a l ong t he DW was i nves t i ga t ed by Ro n a n e t al

i n Re f. [9.25]. Th e pos i t i on o f a VB L i n t he spec i m en wi t h a d i m ens i on l e s s

d a m p i n g p a r a m e t e r a = 0 .1 1 w a s r e g is t e r e d a ls o a s a D W f l ex u r e o f a n a s y m -

m e t r i c s h a p e . N o p h o t o g r a p h s o f t h e D W f le x u re a re g i v e n i n t h i s r e f er e n c e.

A n a s y m m e t r i c a l l y s h a p e d f le x u re w a s p r e s e n t e d i n t h e f o r m o f a d r a w i ng .

T h e d e p e n d e n c e o f t h e V B L v e l o c i t y o n t h e i n - p l a n e m a g n e t i c f ie ld w a s o b -

t a i n e d . T h e s a t u r a t i o n v e l o c i ty o f t h e V B L w a s e q ua l to 3 6 m / s , w h i c h is

subs t an t i a l l y l e s s t han i t s t heo re t i ca l va l ue .


14/37

144 9 . Dy nam ics o f Bloch Lines and Th ei r Clus te rs

A m o r e d e t a il e d s t u d y o f t h e V B L c l u st e r d y n a m i c s w a s p e r f o r m e d i n

R e f. [9.26]. T h e e p i t a x ia l g a r n e t - f e r r i t e f ilm ( B i L a T m ) 3 ( F e G a ) 5 0 1 2 w i t h

a t h i ck n e s s o f 7 # m h a d a s t r ip e d o m a i n w i d t h e q u a l t o 4 7 p r o , w h e r e b y

4 ~ M = 1 00 G s , t h e q u a l i t y - f a c t o r Q = 4 5, 7 = 1.7"10 7 s - l O e - 1 - T h e s o l i t a r y

D W w a s s t a b i l iz e d b y t h e g r a d i e n t m a g n e t i c f ie ld , a s d e s c r i b e d a b o v e . T h e

o p t i c a l r e s o l u t i o n o f t h e s y s t e m w a s n o t w o r s e t h a n 0 .5 # m . T h e o b j e c t i v e

l en s o f t h e m i c r o s c o p e w a s p l a c e d n e a r t h e s u r f a c e o f t h e g a r n e t - f e r r i t e f il m

u n d e r o b s e r v a t i o n , a s r e q u i re d f o r s u c h w o r k w i t h a l a r g e m a g n i f i c a ti o n . T h e

c o i l , p r o d u c i n g t h e p u l s e d m a g n e t i c f i e l d , a n d t h e f l a t m a g n e t s , p r o d u c i n g

t h e g r a d i e n t f ie ld , w e r e l o c a t e d j u s t u n d e r t h e s u b s t r a t e o n w h i c h t h e f il m

u n d e r o b s e r v a t i o n w a s g r o w n . T h i s r e s u l t e d i n t h e p r e s e n c e o f a s m a l l i n -

p l a n e c o m p o n e n t o f t h e m a g n e t i c f ie ld i n t h e s p e c i m e n , d i r e c t e d a l o n g t h e

D W . T h e m a g n e t i c f ie ld d i r e c t e d a l o n g t h e D W c o u l d al so b e p r o d u c e d b y a

s p e c ia l c oi l. T h u s , o n l y th e 2 ~ N - B l o c h l in e s c o u l d e x is t in t h e D W .

F i g u r e 9 .4 s h ow s t h e d e p e n d e n c e o f t h e v e l o c i t y o f t h e D W m o t i o n o n

t h e a m p l i t u d e o f a p u ls e d m a g n e t i c f ie ld H z p e r p e n d i c u l a r t o t h e s p e c i m e n ' s

s u r fa c e . U s i n g t h is d e p e n d e n c e , i t w a s p o s si b le t o d e t e r m i n e t h e D W m o b i l it y ,

w h i c h w a s f o u n d t o b e e q u a l t o 1 4 0 c m . s - 1 9 O e - 1 , a n d t h e d i m e n s i o n le s s

d a m p i n g p a r a m e t e r w a s f o u n d t o b e : c~ = 0 .3 8 .

v~

3

2

1

S

o

~

i I I I ~

0 2 0 4 0 6 0 H O e

Fi g . 9 . 4 Dep en d en ce o f t h e d o m a i n wa l l v e l o c it y i n t h e g a rn e t f ilm s o n t h e m ag n e t i c

field [9.26]

T h e n , u s i n g t h i s v a l u e , i t w a s p o s s i b l e t o f i n d t h e W a l k e r c r i t i c a l f i e l d

Hcr

2 ~ r a M = 2 0 O e . T h e c u r v e o f v H) d i s t i n c t l y e x h i b i t e d a p e a k v e l o c it y ,

t h e v a l u e o f w h i c h w a s f o u n d t o b e e q u a l t o 3 0 m / s , w h i c h i s v e r y c l os e t o

t h e W a l k e r ve lo c it y. T h e a v e ra g e v e l o c i ty o f t h e D W a b r u p t l y d r o p p e d a s

t h e m a g n e t i c f i el d w a s f u r t h e r i n c r e as e d . T h e i n i ti a l s h a p e o f t h e m o v i n g

D W c h a n g e d a n d t h e s t r u c t u r e s s h o w n i n Fi g. 9 .5 a p p e a r e d . T h e s e s t r u c t u r e s

r e s u l t e d b e c a u s e a t H >

Hcr

t h e V B L a r o s e a l on g th e e n t i r e D W a n d g r o u p e d

i n c l u st e r s, l o c a t e d a t d i s t i n c t p o i n t s o f t h e m o v i n g D W .


15/37

9.3 Reg istration of the Dyna mics of VB L 145

Fig. 9.5 Dynamic structure on a moving domain wall in a magnetic f ield greater

than Her [9.26]

Th e DW ins tab i l i ty in the r eg ion wi th nega t ive d i f f eren t ia l mob i l i ty on the

d ep en d en ce o f t h e D W v e lo c i ty o n H is v e r y im p o r t an t f o r t h e f o r m a t io n o f

these s t ruc tu res . In mag ne t ic f ie lds lower tha n cr t h e s p o n t an eo u s g en e r a t i o n

of a VB L i s no t o bserved in a moving DW. Und er these cond i t ions, the o n ly

s o u r ce o f a V B L in t h e D W w as t h e cu r r en t l o o p o r a s o l it a r y co n d u c to r

pho to l i thograph ica l ly depos i ted on the spec imen or on a spec ia l subs t ra te .

W h en cu r r en t p u l se s o f co n s t an t a m p l i t u d e an d d u r a t i o n b e tw een 2 0 an d

100 ns passed th rough th i s loop , i t was poss ib le to p roduce unwind ing pa i r s

o f 2~rN c lus te r s o f Bloch l ines on the D W, wi th a twis t in the az im utha l

ang le in the DW in the oppos i te d i rec t ion , i . e . , w i th topo log ica l charges o f

oppos i te s igns . These VBL c lus te r s moved in oppos i te d i rec t ions dur ing the

DW mo t ion and i t was possib le to inves t iga te the d ynam ics o f each o f these

c lu s te r s w i th t h e u s e o f tw o - f o ld h ig h - s p eed p h o to g r ap h y .

F igure 9 .6 p resen ts a s er ies o f these pho tograp hs p roduc ed in the dom ain

co n t r a s t . E ach p h o to g r ap h d i s t i n c tl y s h ow s tw o p os i ti o n s o f t h e m o t io n f r o m

u p to d o w n D W , r eco r d ed w i th d e l ay t im e A t i n t h e p r o ces s o f o n e D W p as-

sage a long the spec imen . Th e f i r st upper pos i t ion o f the D W cor respond s to

th e l i g h t - d a r k t r an s i t i o n i n F ig . 9 .6 . T h e s eco n d p o si t io n r ep r e sen t s t h e d a r k -

l igh t t r ans i t ion . Th e da rk ban d represen ts the spec imen s r eg ion in which the

DW passes dur ing the t ime in te rva l be tween two l igh t pu lses . The exper i -

me nta l me tho d o f r ecord ing th i s band i s descr ibed above in Chap . 3 . Th e


16/37

146 9. Dyna mics of Bloch Lines and The ir Clusters

Fig. 9. 6a -d Double high speed photographs of soli tary f lexure waves accompanying

moving vertic al Bloch lines in a g arnet film in the contra st o f doma ins [9.26]

p h o to g r ap h s d i s t i n c t ly s h ow th e n o n - o n e - d im en s io n a l f o rm a t io n s s o l i t a ry

w av es o f t h e D W f l ex u r e o f an a s y m m et r i c a l s h ap e l agg ing b eh in d t h e D W

and moving a long i t f rom righ t to l ef t.

A ch an g e o f t h e d i r ec ti o n o f t h e D W m o t io n ch ang es t h e d i r ec t io n o f

prop aga t ion o f the so l i t a ry wave a long the DW. T hus a so l it a ry wave o f the

D W f l ex u r e acco m p an ie s t h e V B L c lu st e r m o v in g d u e t o t h e ac t i o n o f t h e

gyroscopic force.

At la rge values of the dimensionless dam ping pa ram ete r c~ -~ 0 .4 the

so l i t a ry wave has an asym metr ic a l shape . T he f ron t o f the so l i t a ry wave

is muc h shar per th an the t r a i l ing par t . T he so l i t a ry wave fron t inc l ina t ion

increases as the w ave amp l i tude and th e D W ve loc i ty increase . The ve loc i ty

of the DW is mea sured f rom the d i s tance be tween the two pos i t ions o f i t s


17/37

9.3 Registrat ion of the Dynamics of VB L 147

rec t i l in ea r h o r i zo n ta l p a r t s an d th e d e lay t im e o f th e l i g h t p u lse s . Th e v e lo c i ty

o f t h e V B L c l u s t e r is d e t e rm i n e d f r o m t h e d i s t a n c e b e t w e e n t h e m a x i m a o f t h e

s o l i t a ry w a v e p r o p a g a t i n g a l on g t h e D W . A g r o w t h o f t h e s p a t i a l d e r iv a t i v e

a t t h e so l i t a ry wav e f ro n t i s ex p l i c i t l y o b se rv ed o n in c r eas in g th e am p l i tu d e

o f th e so l i t a ry wav e . Th u s , i t is p o ss ib l e to d e te rm in e th e p o s i t io n a n d th e

v e lo c i ty o f t h e V B L c lu s t e r , q u i t e accu ra t e ly . I t s s h ap e ch an g es su b s ta n t i a l ly

u p o n f u r t h e r i n c re a s e o f t h e s o l i ta r y w a v e a m p l it u d e . A r e g io n w i t h a n a l m o s t

v e r t i ca l t an g e n t ap p e a r s o n th e f ro n t o f t h e so l i t a ry wav e . Th i s r eg io n b eco m es

m o re d i s t in c t w i th an in c r ease in th e D W v e lo c i ty (F ig . 9.6 d ). I n F ig . 9 .6 a - c ,

t h e D W v e l o c i t y i s e q u a l t o 2 0 m / s , a n d t h e p e a k v e l o c i t y is e q u al t o 3 0 m / s .

In F ig . 9 .6 d , t h e DW v e lo c i ty v i s eq u a l t o 3 0 m /s , t h e p eak v e lo c i ty i s

5 5 m / s .

A s t h e a m p l i t u d e o f t h e s o l i t a r y w a v e i n c r e a s e s , t h e t o t a l l e n g t h o f t h e

D W f l e x u r e d e c r e a s e s . T h e s h a p e o f t h e s o l i t a r y w a v e , i n t h i s c a s e , r e s e m b l e s

t h e s h a p e o f t h e s h o c k w a v e . T h e r a t i o o f t h e V B L c l u s t e r v e l o c i t y t o t h e

D W v e l o c i t y d e c r e a s e s , w h i c h i s d u e t o t h e i n v o l v e m e n t o f a l a r g e p a r t o f t h e

D W i n t h e m o t i o n o f t h e V B L c l u s t e r . I n i t i a l l y m e n t i o n e d b y

i k i f o r o v

a n d

S o n i n

[ 9 . 6 ] , t h i s p h e n o m e n o n w a s b e l i e v e d t o g i v e r i s e t o s t r o n g n o n l i n e a r

d a m p i n g i n t h e m o t i o n o f a V B L . N o n l i n e a r d y n a m i c s o f ~ - B l o c h l i n e w a s

s t u d i e d t h e o r e t i c a l l y b y

Z v e z d i n

a n d

P o p k o v

[ 9 . 7 ] . A s m a l l c h a n g e i n t h e

d y n a m i c s h a p e o f s o l i t a r y w a v e s i n t h e s e c o n d p o s i t i o n ( l o w e r p o s i t i o n ) o f

t h e D W , i n F i g . 9 . 6 , i s c a u s e d b y t h e e x i s t e n c e o f a g r a d i e n t m a g n e t i c f i e l d

s t a b i l i z i n g t h e D W i n t h e s p e c i m e n a n d l e a d i n g t o a s m a l l d e c r e a s e o f t h e

D W v e l o c i t y . T h e s h a p e s o f s o l i t a r y w a v e s a c c o m p a n y i n g t h e V B L c l u s t e r s a t

t h e D W v e l o c i t i e s f r o m I 0 t o 2 0 m / s w e r e o f s i m i l a r c h a r a c t e r . T h e s o l i t a r y

w a v e s i n v e s t i g a t e d i n R e f . [ 9 . 2 6 ] a r e s t a t i o n a r y a t a m p l i t u d e s u p t o 1 0 m ,

a n d a t t h e g i v e n v e l o c i t y o f t h e D W t h e i r v e l o c i t i e s a r e c o n s t a n t .

F i g u r e 9 . 7 p r e s e n t s t h e e x p e r i m e n t a l r e s u l t s o f t h e d e p e n d e n c e o f t h e V B L

c l u s t e r s v e l o c i t y o n t h e a m p l i t u d e o f s o l i t a r y w a v e s a c c o m p a n y i n g t h e m f o r

D W v e l o c i t i e s o f I I , 1 5 a n d 2 0 m / s . T h e m i n i m u m a m p l i t u d e o f t h e s o l i t a r y

w a v e o b s e r v e d i n t h e e x p e r i m e n t [ 9 . 2 6 ] , a t a D W v e l o c i t y o f 2 0 m / s , w a s

e q u a l t o 0 . 8 m . I n t h i s c a s e , t h e V B L c l u s t e r m o v e d a t a v e l o c i t y o f 8 0 m / s .

T h i s v e l o c i t y i s l e s s t h a n t h e l i m i t i n g v e l o c i t y o f a 7 ~ - V B L d e t e r m i n e d b y

t h e e x p r e s s i o n S - - ~ ' ( 8 ~ A ) 1 / 2 . T h e r a t i o o f t h e m a x i m u m v e l o c i t y o f t h e

V B L c l u s t e r u t o t h e D W v e l o c i t y v i s i n t h e v i c i n i t y o f 4 f o r a l l v , a s s e e n

i n F i g . 9 . 7 . I t i s s u b s t a n t i a l l y l e s s t h a n t h e r a t i o o f t h e s e v e l o c i t i e s f o r t h e

s ing le 7c- l ine ob ta ined by lonczewski [9.2] and repres en te d a bov e (9.11). For

th e sp ec im en u n d e r o b se rv a t io n , t h i s r a t io sh o u ld b e c lo se to 2 5 . Su ch a b ig

d i f fe r en ce i s c au sed b y th e co n t r ib u t io n to d am p in g o f th e l a rg e D W f l exu re

a c c o m p a n y i n g t h e m o v i n g V B L c l u s t e r . A s t h e D W v e l o c i t y d e c r e a s e s t o

3 m /s , t h e so l i t a ry wav es o f t h e D W f lex u re a r e s ti ll d i s t in c t ly seen . As th e

a m p l i t u d e o f t h e s o l i ta r y w a v e s a c c o m p a n y i n g t h e V B L c l u s te r i nc re a se s ,

th e v e lo c i ty o f t h e l a t t e r d ec r ease s an d , a s seen in F ig . 9 .7 , ap p ro ach es th e

D W v e lo c it y . P e r h a p s , e x p e r i m e n t a l l y o b s e r v e d m i n i m u m a m p l i tu d e s o f t h e


18/37


u m / s

1 2

80

40

o

o

9 X~o( 0 0

0

Oo 9 ~z

0 0

0 9 XooX 9 9

t

5 10 q, m

Fig . 9 . 7 Dep en d en c i e s o f t h e VB L c lu s t e r v elo ci ti e s o n t h e am p l i t u d e o f s o l it a ry

deflec t ion waves at d ifferent values of the dom ain wal l veloci t ies : v = 11 . ) , 17 x ) ,

a n d 2 0 o ) m / s [ 9 . 2 6 ]

s o l i t a r y w a v e s o n t h e D W c o r r e s p o n d t o t h e m i n i m a l l y p o s si b le c l u s t e r o f

V B L c o n s i s t i n g o f o n e 2 7 r - B l o c h l in e in t h e p r e s e n c e o f t h e m a g n e t i c f ie l d

a l on g t h e D W . T h e f in a l a n sw e r t o t h e q u e s t io n a b o u t t h e n u m b e r o f B l o c h

l in e s i n t h e c l u s te r , a c c o m p a n i e d b y t h e e x p e r i m e n t a l l y d e t e r m i n e d s o l i t a r y

w a v e o f t h e D W f l ex u r e , c a n b e g i v e n, o n l y a f t e r a c o m p a r i s o n w i t h t h e

t h e o r e t i c a l r e s u lt s .

T h e d y n a m i c s o f t h e B l o c h li ne i n th e d o m a i n w a l l o f t h e f e r r i t e - g a r n e t

w i t h p e r p e n d i c u l a r a n i s o t r o p y a t o n l y o n e sm a l l v e lo c i ty o f t h e D W w a s n u -

m e r i c a l l y a n a l y z e d b y

Nakatani

a n d

Hayashi

n R e f . [9 .2 7]. U s i n g t h e L a n d a u -

L i f sh i tz e q u a t i o n o f t h e m a g n e t ic m o m e n t m o t i o n , w i t h t h e p e r io d i c b o u n d a r y

c o n d i t i o n s a l o n g t h e D W a n d p e r p e n d i c u l a r t o i t, t h e d y n a m i c p r o fi le o f t h e

s o l i t a r y w a v e o f t h e f l e x u re o f t h e D W a c c o m p a n y i n g t h e s in g le 7 r - B l oc h li n e

w a s o b t a i n e d b y n u m e r i c a l m e t h o d s . T h e s e p r o f i l e s f o r f e r r i t e - g a r n e t w i t h

c~ = 0 .1 4 , 4 ~ M = 1 79 G a u s s , K = 9 8 83 .6 e r g / c m 3, A = 1 . 3 . 1 0 - T e r g / c m ,

i n w h i c h t h e d o m a i n w a ll m o v e s d u e t o t h e a c t i o n o f t h e m a g n e t i c f ie ld

Hz

= 1 0 e p e r p e n d i c u l a r t o t h e f il m s u r f a c e , a r e g iv e n i n F i g . 9. 8.

T h e f ig u r es a t e a c h p r o fi le i n d i c a t e t h e t i m e t i n n a n o s e c o n d s a f t e r t h e

b e g i n n i n g o f t h e p u l s e o f t h e m a g n e t i c f ie ld . T h e m o t i o n o f t h e D W a n d

V B L b e c o m e s s t a t i o n a r y a f t e r a t i m e i n t e r v a l t , e q u a l t o 1 5 u s . B e c a u s e o f

b o t h t h e s m a l l D W v e l o c i t y a n d n o t v e r y l a r g e v a lu e o f c~, i t is p o s si b le t h a t

t h e s h a p e o f t h e d o m a i n w a l l a c c o m p a n y i n g t h e m o v i n g V B L is o n l y s li g h t ly

a s y m m e t r ic a l . T h e v e l oc it ie s o f t h e D W v a n d V B L u c a n b e m e a s u r e d o n

t h e b a s i s o f t h e d a t a i n F i g . 9 .8 . I n t h i s c a se , i t w a s f o u n d t h a t t h e r a t i o

u v = 3 0 .1 i s v e r y c l o se t o t h e t h e o r e t i c a l v a l u e c a l c u l a t e d w i t h t h e h e l p

o f 9 .1 1 ). U n f o r t u n a t e l y , t h e V B L d y n a m i c s a t h i g h D W v e l o c i t ie s a n d l o n g

t im e i n t e rv a l s t was n o t co n s id e red i n R e f . [9 . 2 7 ] .


19/37

9.3 Registration of the Dynamics of VBL 149

-0.5

20

16

12

8

4

0

-5 0 5 y / A v B L

x A

Fig. 9.8 Dynamic profiles of domain wall deflection waves accompanying a moving

solitary 7r-VBL calculated from the Landau-Lifshitz equation for different moments

of time [9.27]

The dynamic profiles of the solitary waves of the DW flexure accompa-

nying the VBL clusters containing 2, 10, etc. numbers of Bloch lines, were

calculated by Zvezdin and Popkov in Ref. [9.26] from the Slonczewski equa-

tions, by assuming a linear dependence of the azimuthal angle of twist, ~,

in the cluster on the coordinate. In general, they qualitatively correspond to

the experimental profiles given in Fig. 9.6, but the steepness of the leading

fronts is much less than in the experiment. The theoretical dependencies of

the VBL clusters velocity on the amplitude of the solitary waves of the DW

flexure were calculated in Ref. [9.26]. These dependencies qualitatively cor-

relate with the experimental ones represented in Fig. 9.7 but go much higher

than the latter. This is likely to be due to the fact that the steepness of the

solitary wave leading front obtained in the experiment is substant ially larger

than in the calculations performed in Ref. [9.26]. In fact, from the equality

of the energy dissipated by the VBL cluster and the energy obtained by this

cluster due to the action of the gyroscopic force, Zvezdin and Popkov found

a relationship between the DW velocity v and the velocity of the cluster u,

in the following general form see Ref. [9.26]):

9 . 1 4 )

Here, A is the DW width; ~ x) and

q x)

should be determined from the

Slonczewski equations. It is seen from 9.14) th at with increasing

Oq/Ox,

the

velocity of the VBL cluster, 5~, at the given velocity of the DW decreases.

Using the experimental data, it is possible to determine, directly, the function

q x) and then to calculate the function ~ x), using the Slonczewski equations

and, thus, to find the topological charge of the cluster from the profile of

q x),

recorded by high-speed photography. In the experiments described above, at

the given velocity of the DW, smaller VBL clusters, accompanied by solitary

waves of the DW flexure of a smaller amplitude moved faster than large

clusters. The calculations, made by Zvezdin and Popkov, have shown that as

the number of Bloch lines in the cluster increases, starting from the value


20/37

150 9 . Dyn amics o f Bloch L ines and Th ei r Clus te rs

o f a b o u t 5 0 - 6 0 , i ts v e l o c it y , a t t h e g i v e n v e l o c i t y o f t h e D W , i n c r e a s e s . I t

is q u i t e p o s si b le t h a t t h e c o n c l u s io n m a d e b y

L i a n

a n d

H u m p h r e y

[9.23] of

l a r g e f l e x u r e s o n t h e D W a c c o m p a n y i n g t h e b i g V B L c l u s t e r s , m o v e f a s t e r

t h a n t h e s m a l l o n es , r e su l t s f r o m th i s f a c t i. e. , t h e s e a u t h o r s d e a l t w i t h v e r y

l a r g e V B L c l u s t e r s .

F i g u r e 9 .9 s h ow s t h e e x p e r i m e n t a l d e p e n d e n c i e s o f t h e v e l o c i t y o f m i n i m a l

V B L c l u s te r s u o b s e r v e d i n R e f. [ 9.2 6] o n t h e D W v e l o c i t y v . T h e s e c l u s te r s

w e r e o b s e r v e d o n t h e m o v i n g D W l o c a t e d i n a s m a l l i n - p l a n e m a g n e t i c f i e l d

d i r e c t e d a l o n g t h e D W , a n d s e e m e d t o c o n t a i n t w o p a i r s o f B l o c h l in e s.

u m / s

120

8

Z /

z

400f////~>/ /

27c

/ / ' /

/ /

/ 4vF

/ 2 / / G - -

/ / o . / /

s 2 0 3 0 v ,

m ' / s

F ig . 9 . 9 E x p e r im en ta l d ep en d en ce o f t h e s m a l l e st V B L c lu s te r v e lo c it y o n t h e

do m ain wal l veloc i ty (o o o) an d s im ilar depe ndenc ies for 2~, 47r, 6~ a nd 8~r V BL ,

ca lcu la ted f rom Slonczewsk i equa t ions a t the in -p lane magnet ic f i e ld a long the

d o m a in wa l l Hx - - 0 ( s o l i d l i n e ) , and H~ = 64 Oe ( d a s h e d l i n e )

T h e e x p e r i m e n t a l f u n c t i o n

u ( v ) ,

f ro m R ef . [9 . 2 6 ] , p re s en t ed i n F ig . 9 . 9 ,

is s u b s t a n t i a l l y n o n l i n e a r . A t a D W v e l o c i t y o f 2 0 m / s , a c h i e v e d i n t h i s e x -

p e r i m e n t , t h e v e l o c i t y o f t h e o b s e r v e d V B L c l u s t e r w a s e q u a l t o 8 0 m / s . A s

v d e c r e a s e s , t h e d e r i v a t i v e d u / d v i n c r e a s e s a n d r e a c h e s 1 0 a t v = 3 m / s . A t

l ow e r v el o c it i es o f t h e D W , w h e n t h e a b o v e d e r i v a t iv e is s u p p o s e d t o g r o w

f u r t h e r , t h e a m p l i t u d e o f t h e s o l i ta r y w a v e o f t h e D W f le x u r e a c c o m p a n y i n g

t h e V B L c l u s t e r b e c o m e s v e r y s m a l l a n d h a r d l y o b s e rv a b l e . F i g u r e 9 .9 a ls o

s h o w s t h e c a l c u l a t e d f u n c t i o n s u ( v ) f o r t h e V B L c l u s te r s c o n s i s ti n g o f 2

4vr, 6 ~ , 8 ~ B l o c h l in e s. T h e s e d e p e n d e n c i e s w e r e d e r i v e d f r o m t h e n u m e r i c a l

s o l u t i o n o f t h e S l o n c ze w s k i e q u a t i o n s f o r

q ( x - a t )

a n d

~ ( x - u t ) .

T h e c a l -

c u l a t i o n s w e r e m a d e f o r t h e f e r r i t e - g a r n e t f i lm i n a n a b s e n c e o f a n i n - p l a n e

m a g n e t i c f ie ld a n d i n t h e p r e s e n c e o f a n i n - p l a n e m a g n e t i c f ie ld o f 64 O e

d i r e c t e d a l o n g t h e D W . T h e s e c o n d c a l c u l a t e d f u n c t i o n s f a l l b e l o w t h e f i r s t


21/37

9 .3 Reg is t ra t ion o f the Dyn am ics o f VB L 151

o n es . T h e d i s si p a ti o n o f e n e r g y b y th e V B L c l u st e r in t h e i n - p l a n e m a g n e t i c

f ie ld , d i r e c t e d a l o n g t h e D W , i n c re a s e s. T h e p r o fi le s o f t h e s o l i t a r y w a v e s o f

t h e D W f l ex u r e i n t h e i n - p l a n e m a g n e t i c f ie ld d i r e c t e d a lo n g t h e D W a r e

l es s e x t e n d e d i n s p ac e . T h e e x t e n s i o n o f t h e l e a d i n g f r o n t d e c r e a s e s b y s e v-

e r a l ti m e s . T h e s m a l l e s t c l u s te r o b s e r v e d in t h e e x p e r i m e n t [9 .26 ] p r o b a b l y

c o n s i s t e d o f 4~r o r 6 ~ r - B l o c h l in e s , s in c e t h e e x p e r i m e n t a l f u n c t i o n u v) is

c l o s e t o t h e c a l c u l a t e d o n e f o r 47r a n d 6 7 r - B l o c h li n es , a s s e e n f r o m F i g . 9 .9 .

T h e f u n c t i o n s u v) o b t a i n e d f r o m t h e S l o n cz e w s k i e q u a t i o n s f o r al l t h e c lu s -

t e r s , a s s u m i n g a l i n e a r d e p e n d e n c e q o(x ), g o c o n s i d e r a b l y a b o v e o f t h e s i m i l a r

e x p e r i m e n t a l f u n c t i o n s [9.26]. T h e a m p l i t u d e s o f s o l i t a r y w a v e s o f t h e D W

f l e x u r e a c c o m p a n y i n g t h e V B L c l u s t e rs c o n s i s t in g o f 2 % 4~r a n d 6~r B l o c h

l in e s a t a D W v e l o c i t y o f 2 0 m / s , w e r e c a l c u l a t e d f r o m t h e S l o n c ze w s k i e q u a -

t i o n s a n d w e r e f o u n d t o b e e q u a l t o 0 .4 8, 0. 96 , a n d 1 . 4 4 m , r e s p e c ti v e ly . T h e

c a l c u l a t i o n s w e r e p e r f o r m e d f o r t h e f o ll o w in g p a r a m e t e r s o f t h e f e r r it e - -g a r n e t

f il m : 4 ~ rM = 1 0 0 G , a = 0 .4 , 7 = 1 . 8 . 1 0 T O e - i s - 1 , ( r = 2 . 1 0 - 7 e r g / c m 2,

Q = 4 5 , g r a d H z = 2 00 0 O e / c m . T h e s e p a r a m e t e r s a r e c lo se t o t h e p a r a m -

e t e r s o f t h e i n v e s ti g a te d s p e c i m e n s. T h e e x p e r i m e n t a l l y ob s e r v e d a m p l i t u d e

o f a s o l it a r y w a v e o f t h e D W f le x u re a c c o m p a n y i n g t h e m i n i m a l V B L c lu s-

t e r a t t h e D W v e l o c i ty o f 2 0 m / s w a s e q u a l t o 0 .8 m . T h e c o r r e l a t io n o f

t h i s v a l u e w i t h t h e r e p r e s e n t e d a b o v e c a l c u l a t e d o n e s g iv e s a v a l u e o f 47r f o r

t h e t o p o l o gi c a l c h a rg e o f t h e V B L c lu s te r s. T h e q u e s t io n o f d e t e r m i n i n g t h e

t o p o lo g i c a l c h a rg e o f t h e V B L c l u st e r f r o m t h e e x p e r i m e n t a l o b s e r v a t io n o f

t h e p r o fi le o f a s o l i t a r y w a v e o f t h e D W f l e x u re a c c o m p a n y i n g t h i s c l u s t e r

w i ll b e f u r t h e r c o n s i d e r e d in t h e n e x t p a r a g r a p h i n c o n n e c t i o n w i t h t h e d i s -

c u s s io n o f t h e e x p e r i m e n t s o n t h e c o ll is io n a n d s o l i to n - l i k e b e h a v i o r o f t h e

V B L c l u s t e r s .

F i g u r e 9 .1 0 s ho w s t h e d e p e n d e n c e o f t h e a v e r a g e v e l o c i t y o f t h e m o t i o n o f

a s o l i t a r y ~ r - B lo c h l in e o n a n i n - p l a n e m a g n e t i c f ie ld d i r e c t e d a l o n g t h e D W .

T h i s d e p e n d e n c e w a s o b t a i n e d b y Theile a n d Engemann [9.1 9] . Us in g t h e d a rk

f ie ld m e t h o d o f l i g h t d i f fr a c t io n , t h e y r e c o r d e d t h e p o s i t io n s o f t h e s e l in e s o n

t h e D W b e f o r e a n d a f t e r s h i ft in g . T h e a u t h o r s o f R e f . [9 .19] a ls o s u p p o s e d t h a t

t h e B l o c h l i ne s m o v e u n i f o r m l y d u r i n g t h e f l at to p o f t h e d r i v i n g m a g n e t i c

f i e ld p u l s e . I t i s s e e n t h a t t h e V B L v e l o c i ti e s , t h u s o b t a i n e d , a r e n o n l i n e a r l y

c o n n e c t e d w i t h t h e a m p l i t u d e o f t h e m a g n e t i c f ie ld a c t in g a l o n g t h e D W . T h e

m a x i m u m a c h ie v e d v e l o c it y w a s e q u a l t o 8 0 m / s . T h e s c a t t e r i n g o f p o i n ts o n

t h e f u n c t i o n u v) i s r a t h e r l a r g e . I n g e n e r a l , t h i s f u n c t i o n i s s i m i l a r t o o n e s

g iv en ab o v e i n F ig . 9 . 9 .

F r o m F i g . 9 .1 0 i t is c l e a r ly se e n t h a t a l a r g e c o e r c iv e f o r c e r e a c h e s 6 0 e .

M o r e r e c e n t s t u d i e s c a r r i e d o u t w i t h t h e s a m e m e t h o d g a v e a w i d e r a n g e o f

t h e c o e r c i v e f o r c e f r o m 2 t o 8 0 e , f o r v a r i o u s 7 r -B l o c h l in e s. T h e m a x i m u m

v e l o c i t i e s o f 7 r- a n d 4 7 r - B l o c h l in e s , o b s e r v e d u p t o n o w , e x p e r i m e n t a l l y , w e r e

e q u a l t o 8 0 m / s , w h i c h i s c o n s i d e r a b l y le ss t h a n t h a t c a l c u l a t e d f r o m t h e

ex p re s s io n fo r t h e l im i t i n g v e lo c i t y , S , o f a s in g l e 7 c -l in e .


22/37

152 9 . Dyn amics o f Bloch L ines and Th ei r Clus te rs

100

80

60

40

20

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F i g . 9 . 1 0 Dep en d en ce o f VB L v e lo c it y o n an i n -p l an e f ie ld p u l s e am p l i t u d e o b -

tain ed by th e m et ho d of l ight d iffract ion in a dark f ield [9.19]

O f c e r t a i n i n t e r e s t a r e t h e e x p e r i m e n t s w h e n a s u f f i c i e n t l y h i g h c o n s t a n t

i n - p l a n e , m a g n e t i c f i e ld i s d i r e c t e d a l o n g t h e D W . A s a r e s u lt , i t is p o s s i b l e

t o s u b s t a n t i a l l y i n c re a se t h e p e a k v e l o c it y o f t h e D W a n d e x p a n d t h e r a n g e

o f t h e D W v e l o c it i es a t w h i c h n o g e n e r a t i o n o f t h e a d d i t i o n a l V B L o n t h e

d y n a m i c D W o c c u rs , a n d t o i n v e s ti g a te t h e d y n a m i c s o f t h e V B L c l u st e rs d u e

t o t h e a c t i o n o f t h e g y r o s c o p i c f o rc e a t h i g h D W v e l oc i ti e s. T h e e x p e r i m e n t a l

d e p e n d e n c i e s o f t h e D W v e l o c i t y o n t h e m a g n e t i c f i el d f o r f e r r i t e - g a r n e t ,

w i t h = 3 0 0 c m / s . O e , 4 7rM s = 1 00 G , Q = 4 5 , a t t w o v a l u e s o f t h e i n - p l a n e

m a g n e t i c f ie ld d i r e c t e d a l o n g t h e D W , H ~ = 4 0 a n d 6 0 O e , a r e si m i l a r t o t h e

d e p e n d e n c i e s g i v e n i n F i g . 9 .9 ( M s is t h e s a t u r a t i o n m a g n e t i z a t i o n ) . I n t h i s

c a s e , t h e p e a k v e l o c i t i e s a r e e q u a l t o 7 5 a n d 1 0 0 m / s , r e s p e c t i v e l y .

F i g u r e 9 .1 1 r e p r e s e n t s t h e d e p e n d e n c i e s o f t h e v e l o c i t y o f t h e V B L c l u s te r s

m o t i o n o n t h e v e l o c i t y o f t h e D W a t t h e t w o , a b o v e - g iv e n , v a lu e s o f t h e i n -

p l a n e m a g n e t i c f ie ld d i r e c t e d a l o n g t h e D W . T h e s e d e p e n d e n c i e s a r e s t r o n g l y

n o n l i n e a r . T h e m a x i m u m v e l o c i t i e s d o n o t e x c e e d 8 5 m / s .

A s t h e i n - p l a n e f ie ld g r ow s , r e a c h in g t h e m a x i m u m v e l o c i t y b e c o m e s in -

c r e a s i n g l y s l ow . A g r e a t i n c r e a s e o f t h e p e a k v e l o c i t y d o e s n o t r e s u l t i n a

s i gn i f ic a n t in c r e a s e in t h e m a x i m u m v e l o c i t y o f t h e V B L c l u st e r. W i t h a n

i n c r e a s e o f t h e i n - p l a n e f ie ld a n d p e a k v e l o c it y , a p p r o a c h t o t h e m a x i m u m

v e l o c i ty b e c o m e s m o r e a n d m o r e s m o o t h . T h e v e l o c it y o f t h e V B L d u s t e r

a p p r o a c h e s t h e p e a k v e l o c i ty o f t h e D W . T h u s , t h e g r o w t h o f t h e i n - p l a n e

f ie ld d o e s n o t p r o v i d e a s u b s t a n t i a l i n c r e a s e in t h e v e l o c i t y o f t h e V B L c lu s -

t e r a n d a t t a i n m e n t o f t h e l i m i ti n g v e l o c i t y o f t h e 1 r -B l o c h l in e . T h e r e a s o n

s e e m s t o h a v e s o m e t h i n g t o d o w i t h t h e s p a t i a l c o m p r e s s io n o f t h e p r of il e ,

q x - u t ) , o f a s o l i t a r y w a v e o f t h e D W f l ex u r e a c c o m p a n y i n g t h e V B L c lu s -

t e r , i n v o l v in g a l a rg e s e c t io n o f t h e D W i n t h e m o t i o n , w i t h a s u b s t a n t i a l

i n c r e m e n t i n t h e d e r i v a t iv e 0 r A s a r e s u l t , i n r e l a t i o n ( 9 . 1 4 ) b e t w e e n t h e

v e lo c it ie s o f t h e D W a n d t h e V B L c lu s te r , t h e t e r m p r o p o r t i o n a l t o c ~ r 2


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9 .4 S o l i t o n

Documents

09 - Dynamics of Bloch Lines and Their Clusters