02 - phenomenological theory of magnetism and classification of magnetic solitons

8/20/2019 02 - Phenomenological Theory of Magnetism and Classification of Magnetic Solitons

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2 P h e n o m e n o l o g i c a l T h e o r y o f M a g n e t i s m

a n d C l a s s i f ic a t io n o f M a g n e t i c S o l i to n s

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

t h e o r y o f f e r r o m a g n e t s a n d w e a k f e r r o m a g n e t s , th e p r o p e r t ie s o f t h e m a g -

n e t i c m a t e r i a l s w h i c h w e r e u s e d i n e x p e r i m e n t s , w i l l b e d i s c u s s e d i n t h i s

c h a p t e r . W e a ls o d e s c ri b e in b r i e f t h e s t r u c t u r e a n d p r o p e r t i e s o f m a g n e t i c

i n h o m o g e n e i t ie s , s o l i to n s w h o s e d y n a m i c s w i l l b e e x a m i n e d i n d e t a i l b e lo w .

2 .1 M a g n e t i z a t i o n . L a n d a u L i f s h i tz E q u a t i o n s

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

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

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

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

o n l y o n e f e r r o m a g n e t - E u O , e x h i b i t i n g t h i s b e h a v i o u r . H o w e v e r , o u r f u r t h e r

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

t h e r e f o r e , w e d is c u ss i n b r i e f t h e s t a t i c a n d d y n a m i c p r o p e r t i e s o f f e r r o m a g -

n e t s .

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

is d e t e r m i n e d b y g iv i n g a t e a c h p o i n t o f t h e m a g n e t t h e m a g n e t i z a t i o n t i m e -

d e p e n d e n t v e c t o r M = M ( r , t ). T h e f e r r o m a g n e t e n e r g y in t hi s a p p r o a c h

c a l le d , g en e r a ll y , m i c r o m a g n e t i s m , is w r i t t e n a s t h e m a g n e t i z a t i o n f u n c t i o n a l

21)

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

f u n c t i o n f M ) , M 2 = M 2 , d e t e r m i n e s t h e e q u i l i b r i u m l e n g t h o f t h e v e c t o r

I M ] a n d , a t lo w t e m p e r a t u r e s h a s a s h a r p m i n i m u m w h e n M ~- M 0 , M 0

i s 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 . T h e v a l u e o f M o ..~ 2 o s / a 3 , o b e i n g t h e

B o h r m a g n e t o n m o d u l u s , s - a t o m i c s pi n, a 3 - t h e u n i t c r y s t a l ce ll v o l u m e ,

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

a c c o u n t , i t is a s s u m e d t h a t t h e m a g n e t i z a t i o n M = M o r n , rn 2 = 1 , an d

t h e t e r m w i t h

f ( M )

is n e g l e c te d . T h e m a g n e t i z a t i o n h o m o g e n e i t y e n e r g y


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2 .1 M ag n e t i za t io n . L an d au -L i f s h i t z E q u a t i o n s 5

i s a s s o c i a t e d , i n i ti a ll y , w i t h t h e e x c h a n g e e n e r g y o f t h e f e r r o m a g n e t . T h u s ,

i t is i n v a r i a n t r e l a t i v e to h o m o g e n e o u s r o t a t i o n s o f t h e v e c t o r M r , t ) . T h e

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

e x c h a n g e i n t e g r a l

I , c~ ~ Ia2 /poM o,

w h e r e b y a i s t h e u s u a l l a t t i c e c o n s t a n t .

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

H e i s e n b e r g H a m i l t o n i a n i n p o w e rs o f t h e g r a d i e n ts o f M . F o r th e r h o m b i c

s y m m e t r y m a g n e t t h e t e n s o r c ~ k i s d i a g o n a l , a = d i a g

a x , a y , a z ) ,

t h e a x e s

x , y , z a r e c h o s e n a l o n g t h e c r y s t a l a x e s . T h e d i f f e r e n c e i n t h e c o n s t a n t s c ~

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

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

a x i s m a g n e t s , w i t h t h e a x i s a l o n g t h e z - a x i s , c ~ z h a s a v a l u e e q u a l t o O Z y

w h e r e a s i n c u b i c m a g n e t s t h e t e n s o r c ~ i k i s p r o p o r t i o n a l t o t h e u n i t t e n s o r ,

a i k = a ~ i k . T h i s s i m p l e e x p r e s s i o n i s g e n e r a l l y u s e d f o r t h e e a s y - a x i s a n d

r h o m b i c c r y s t a l s .

T h e s e c o n d t e r m i n 2 . 1 ) d e s c r i b e s t h e m a g n e t i c a n i s o t r o p y e n e r g y , w h i c h

a c c o u n t s f o r r e l a t i v i s t i c i n t e r a c t i o n s s p i n - o r b i t a l a n d d i p o l e - d i p o l e o n e s ) . I n

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

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

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

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

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

o n e u s u a l l y h a s t o d o t h e f o l l o w i n g : t h e a n i s o t r o p y e n e r g y h a s t o b e w r i t t e n a s

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

t o t r a n s f o r m a t i o n s o f t h e c r y s t a l s y m m e t r y , s o t h a t

o o

Wa = E E K(2 n) a /: M ~2 (2 .2 )

n l c ~ i

T h e n , o n l e a v i n g i n th i s s u m t h e f in i te v a lu e o f i n v a r i a n ts w i t h t h e m i n i m u m

n e c e s s a r y v a l u e o f t h e n u m b e r n t o c h o o s e t h e c o n s t a n t s K (2 ~) , c a l le d t h e

2 n - t h o r d e r a n i s o t r o p y c o n s ta n t s , o n e c a n fi nd th e m f r o m e x p e r i m e n t . F o r d e-

t a i l s o f t h i s p r o c e d u r e , s e e b e l o w f o r t h e d i s c u s s io n c o n c e r n i n g i ts a p p l i c a t i o n

t o s p e ci fi c m a g n e t i c m a t e r i a l s .

T h e t e r m - M H o d e s c ri b e s th e u s u a l Z e e m a n m a g n e t i z a t i o n e n e r g y i n

t h e e x t e r n a l m a g n e t i c f ie ld H 0 . A s f o r t h e l a s t t e r m i n (2 .1 ) it d e t e r m i n e s

t h e i n t e r a c t i o n o f m a g n e t i z a t i o n w i t h t h e s o -c a ll ed d e m a g n e t i z i n g f ie ld H ~

w h i c h i s g e n e r a t e d b y t h e m a g n e t i z a t i o n i t se l f a n d d e f i n e d b y t h e s o l u t io n o f

t h e e q u a t i o n s o f m a g n e t o s t a t i c s

d i v H m = - 4 7 r d i v M , r o t H m = 0 , (2 .3 )

w i t h a c c o u n t t a k e n o f t h e n a t u r a l b o u n d a r y c o n d i ti o n s, c o n t i n u i t y o f t a n g e n t

H , ~ a n d n o r m a l B = H m + 4 ~rM c o m p o n e n t s o n t h e b o u n d a r y o f t h e

m a g n e t i c m a t e r i a l . W i t h a l lo w a n c e fo r t h e c o n d i t i o n M 2 = M 2 = c o n s t i t

is c o n v e n i e n t t o i n t r o d u c e t h e a n g u l a r v a r i a b le s f o r t h e u n i t m a g n e t i z a t i o n

v e c t o r m = M / M o , m z = c o s 0 , m x + im y = s i n 0 e x p ( i ~ ) , a s a p o l a r a x i s z


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2.2 Magnetic Properties of Weak Ferromagnets 7

magnetization in the ground s tate is zero. A weak noncollinearity of sub-

lattices arises in weak ferromagnets due to Dzyaloshinskii-Moria exchange-

relativistic interaction. This gives rise to a nonvanishing spontaneous magne-

tizat ion of the weak ferromagnets. A spontaneous moment, in ferrites, appears

due to the inequivalence of sublattices, many ferrites such as ferrites-garnets

applied widely in solid devices - the electronics and computer technique) may

involve scores of magnetic sublattices. A sublattice structure of ferrites can

manifest itself in strong exchange fields of the order of exchange

H e ~ I / o )

or at high enough frequencies of the order of

g H e ) ,

and also near the com-

pensation point; see the review by

B a r ' y a k h t a r

and

I v a n o v

[2.6]. However, in

a good deal of the real cases, the dynamics of ferrite magnetization can be

described see

M a l o z e m o v

and

S l o n c h e v s k y

[2.7]) within the framework of the

equations 2.1 - 2.5) given above. Exclusion from these rules are the ferrites

near the point of compensation see [2.6]).

2 . 2 M a g n e t i c P r o p e r t i e s o f W e a k F e r r o m a g n e t s

The spin structures of weak ferromagnets WFM) are extremely diverse.

Among the large number of such magnets, it is possible to single out sev-

eral classes of magnetic materials, which are, in many of their properties,

similar; see the book by

T u w v

[2.8].

We discuss only two classes - the rhombic WFM such as orthoferrites

and easy-plane rhombohedron WFM. These are classes of WFM for which

the high-speed DW dynamics is studied in detail. We begin with the general

laws valid for all weak ferromagnets.

A two sublattice model will be used for the weak ferromagnet. In this

model the energy is determined by the functional of two magnetization den-

sities of the sublattices

M i ( r , t )

and

M 2 ( r , t ) ,

]MI[ = [M2[ = M0. It is

more convenient to introduce irreducible combinations of these sublattices:

the normalized magnetization vector rn and the antiferromagnetism vector l,

r n = ( M I + M 2 ) / 2 M o , I = ( M I - M 2 ) / 2 M o

2.7)

Since the sublattice magnetization lengths are constant

ml =0 , m 2+ I 2= 1 2.8)

The WFM energy represents the functional of m, and l can be written

as the integral of the energy density

w ( m ,

l),

1

w m , l ) = ~ V / ) 2 ~ - ~ ~ m 2 + ~ a l ) -~ W D -- h m 2 . 9 )

M

Here, 5 is the homogeneous exchange constant,

w ~ ( l )

is the magnetic

anisotropy energy,

Wa

can be wri tten as w2 + w4 + w6 + . . . , where w2,

w 4 , . . .


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8 2 . Ph en o m en o lo g i ca l T h e o ry o f M ag n e t i s m

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

c ~ (V l ) 2 d e s c r i b e s th e e n e r g y o f i n h o m o g e n e i t y o f t h e m a g n e t i z a t i o n s M 1 a n d

M 2 a n d h a s t h e s a m e s e n s e a s f o r t h e f e r r o m a g n e t . W h i l e w r i t in g d o w n ( 2 .9 ) ,

w e o m i t t h e t e r m

5rl 2

s ince , b ec au se o f m 2 + 12 = 1 , i t is re du ce d t o (~m 2 .

W e n e g l e c t t h e t e r m o f t h e f o r m a ~ ( V r n ) 2 a n d a l so t h e a n i s o t r o p y e n e r g y

d e p e n d e n c e o n m . T h i s i s s u b s t a n t i a t e d b y t h e s m a l l n e s s o f m , I rn [


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2.2 M agnetic Pro pertie s of W eak Ferromagnets 9

F in a l ly , t h e l a s t t e r m in th e en e r g y ( 2 .9 ) d e sc r ib es th e u su a l Zeem an en e r g y

in the f ie ld H = h M o .

A n ex am in a t io n o f t h e sp ec if ic c l a sse s o f wea k f e r r o m ag n e t s w i ll n o w b e

u n d e r t a k e n .

O r t h o f e r r i t e s . T h e g e n e ra l f or m u l a f or o r t h o fe r r it e s is R F e O 3 , w h e r e R is

a n i o n o f r a r e ~ a r t h e l em e n t s . T h e i ro n c a n b e s u b s t i t u t e d f o r c h r o m i u m ( or -

t h o c h r o m i t e s ) a n d r a r e - e a r t h e l e m e n ts fo r y t t r i u m ( t h e y t t r i u m o r t h o fe r r it e ) .

M o n o c r y s t a l s o f o r t h o f er r it e s a r e o b t a i n e d b y t h e s p o n t a n e o u s c r y s ta l li z a ti o n

m e th o d f r o m th e m e l t ag e [ 2 .1 1 ] , h y d r o th e r m a l sy n th es i s [ 2 .1 2 ], an d b an d

m e l t in g w i th o p t i ca l h ea t in g [2 .1 3], t h e l a t t e r p r o d u c in g th e l a r g es t an d o p -

t i ca l ly t r an sp a r en t c r y s t a l s . Be lo w, we g iv e th e o r th o f e r r i t e p r o p e r t i e s t o b e

u sed in th i s r ev i ew. Th e i r d e t a i l ed ex p e r im en ta l an d th eo r e t i ca l d a t a can b e

f o u n d i n t h e r e v i e w o f B e l o v an d K a d o m t s e v a [2 .1 4] an d in th e b o o k b y B e l o v

e t a l [ 2 1 5 ]

An o r th o f e r r i t e l a t t i c e i s a d e f o r m ed p e r o v sk i t e l a t t i c e . I t s sy m m et r y i s

d e s c r i b e d b y t h e s p a c e g r o u p D ~ . T h e u n i t c e ll h a s f o u r i ro n i o ns o c cu -

p y in g sy m m et r i c o c t a h ed r o n p o s i t i o n s ( 1 /2 , 0 , 0 ) , ( 1 /2 , 0 , 1 /2 ) , ( 0, 1 /2 , 1 /2 ) ,

( 0 , 1 / 2 , 0 ) , t h e i r s p i n s a r e d e n o t e d v i a S1 , 2 , 3 , 4 , r e sp ec t iv e ly . Th e

r a r e - e a r t h e l e m e n t i on s a r e o r d e r e d o n l y a t s u f f ic ie n tl y l ow t e m p e r a t u r e s

( T _< 1 0 K) ; t h e t em p e r a tu r e o f t h e i r o n io n m ag n e t i c o r d e r in g i s, o n th e

o th e r h an d , r e sp ec t iv e ly h ig h ( ab o u t 7 0 0 K) . T h i s m ak es i t p o ss ib le , in co n -

s id e r in g th e s t a t i c an d d y n am ic p r o p e r t i e s o f o r th o f e r r it e s , t o t ak e o n ly th o se

su b la t t i c e s a s so c ia t ed wi th m ag n e t i c i o n s in to acco u n t . Mo r eo v e r , t h e f i r s t

a n d t h i r d m a g n e t i c s u b l a t t ic e s c a n b e c o m b i n e d i n to o n e s u b l a tt i c e , a n d t h e

seco n d an d f o u r th - i n to an o th e r . Th u s , i n s t ea d o f f o u r su b la t t i c e s we can

c o n s i d er j u s t t w o . T h e i r m a g n e t i z a t i o n s a r e c o n n e c te d w i t h a t o m s p in s b y

t h e f o r m u l a e

m = ( 1 / 4 S ) [ ( S i + 2 ) + ( S s + S 4 ) ] ( M 1 -]- M 2 ) / 2 M o ,

l = ( 1 / 4 S ) [ ( S 1 J - S 3 ) - ( S 2 J - S 4 ) ]

= M 1 M 2 ) / 2 M o

A n a l y s i s o f t h e t r a n s f o r m a t i o n s o f v e c t o r s m a n d 1 r e v ea l s t h a t t h e c o m p o -

n e n t s m x a n d lz a r e t r an s f o r m ed in a s im i la r way ( b y th e r ep r e se n ta t io n F2 ) ;

m z a n d l x (b y t h e r e p r e s e n t a t i o n / 4 ) , t h e c o m p o n e n t m y g e t s t r a n s f o r m e d

b y th e r e p r e se n ta t io n / 3 , an d f in ally , t h e co m p o n e n t ly - b y t h e r e p r e se n t a -

t i o n F1 . Th u s , t h e an i so t r o p y en e r g y d en s i ty o f o r th o f e r r i te s u p to t h e f o u r th

o r d e r t e r m s in l~ can b e wr i t t en a s

1/~(0)/2 1~(0)/2

w 2 = ~ . 2 y + ~ - 3 z ,

4 4 ~ 2 2

W 4 ~ l l l x ] ~ 3 3 I z [ ~ t J l 3~ x~ z

1_~ 1212 1 ~ 121~ 1 4

2 ~ 1 2 ~ y - ~ , 2 s v ~ - ~ f 12 21 y

2.12)


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1 0 2 . Ph en o m en o lo g i ca l T h e o ry o f M ag n e t i s m

( t h e s u b s c r i p t

0

i s wr i t t en fo r /3~~ t33~ b ecau s e , f i r s t l y , t h e i r v a lu e i s

n o r m a l i z e d d u e t o t h e a c c o u n t t a k e n o f W D, s e e b e l o w , s ec o n d l y , i n d e s c r i b i n g

t h e D W , t h e c o n s t a n t s a r e f i tt e d in a d i ff e r e n t w a y ) .

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 ti o n e n e r g y c o n ta i n s t w o i n d e p e n d e n t

i n v a r i a n t s

2 . 1 3 )

F o r W F M m o m e n t o f o r t h o f e r r i t e s , o n e e as i ly a r r iv e s a t:

( 2 . 1 1 )

h e n c e , i t f o ll ow s t h a t t h e v e c t o r m is m a i n l y o r i e n t e d in t h e a c - t y p e p l a n e o f

t h e c r y s t a l . T h e a v a i l a b il i ty o f t h e W F M m o m e n t i s i m p o r t a n t f o r p o s s ib l e

e x p e r i m e n t a l s tu d i e s o f t h e D W d y n a m i c s in W F M .

I t w a s a s s u m e d f o r Wa a n d WD t h a t t h e c o o r d i n a t e a x e s x , y , z w e r e c h o s e n

a l o n g t h e c r y s t a l l l o g r a p h i c a x e s a , b, c , r e sp e c t i v e l y . T h u s t h e y - a x i s i s e v e n ,

a n d t h e x , z - a x e s a r e od d : t h e s y m m e t r y e l em e n t s a r e c al le d e v e n ( b y Tutor,

s e e [2 .8 ]) i f t h e i r a c t i o n d o e s n o t p e r m u t e s u b l a t t i c e s , a n d t h e y a r e c a l l e d

o d d i f t h e y d o s o. T h e o d d s y m m e t r y e l e m e n t e ff e ct o n m c o in c id e s w i t h t h e

a c t i o n o f t h e e v e n o n e , b u t , i n a d d i t i o n , i t c h a n g e s t h e s i g n o f t h e v e c t o r I .

F o r o r t h o f e r r i t e s a t h i g h t e m p e r a t u r e s , t h e v e c t o r 1 is o r i e n t e d a l o n g t h e

a - a x i s , a n d t h e v e c t o r m - a l o n g t h e c - a x i s , w h i c h is a c o n s e q u e n c e fr o m

e x p e r i m e n t a l o b s e r v at io n s (Wh i t e [2.16], see also [2.14,15]).

O n e o f t h e s p ec i fi c p r o p e r t i e s o f ra r e - - e a rt h o r t h o f e r r i t e s is t h e o c c u r e n c e o f

r e o r i e n t e d p h a s e t r a n s i t i o n s u n d e r t e m p e r a t u r e v a r i a t i o n . T h e r e o r i e n t a t i o n

o f t h e v e c t o r l , c a u se d b y t h e a n i s o t ro p y c o n s t a n t v a r i a t io n w i t h t e m p e r a t u r e

( t h is i s e x p l a i n e d o n t h e m i c r o s c o p i c l ev e l p r o c e e d i n g f r o m t h e f a c t t h a t w i t h

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

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

o r t h o f e r r i t e s , o c c u r s i n t h e (ac) p l a n e. T h i s a p p l ie s t o o r t h o f e r r i t e s o f t h u l i u m ,

s a m a r i u m , h o l m i u m , e t c . O n a c c o u n t o f ( 2 .1 1 ) , l r e o r i e n t s s i m u l t a n e o u s l y

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

l e n g t h c h a n g e s l i t t l e ( si n c e

d/de•


8/24

2 .2 M ag n e t i c P ro p e r t i e s o f W eak Fe r ro m ag n e t s 11

r e o r i e n t a t i o n t r a n s i t i o n i s m a n i f e s te d i n t h e d y n a m i c a l p r o p e r t ie s o f t h e D W ,

a d i s c u s s i o n o f s u c h t r a n s i t i o n s w i ll b e d e a l t w i t h i n d e t a i l .

U s i n g t h e f o r m u l a fo r m ( 2 .1 1 ), a n d t h e a n g u l a r v a r i a b l e s fo r t h e a n t i fe r -

r o m a g n e t i s m v e c t o r 1

lx = c o s 0 , ly = s i n 0 s i n ~ , lz = s i n 0 c o s ~

( i t is c o n v e n i e n t t o c h o o s e t h e p o l a r a x i s a l o n g t h e a - a x i s ) w e g e t:

W / M 2 = W o + 2 / 3 ( ~ ) s i n 2 0 + l b ( ~ ) s i n 4 0 , ( 2.1 4)

w h e r e W 0 is i n d e p e n d e n t o f t h e a n g le s ,

/ 3 (~ ) = /3 2 s i n 2 ~ + /3 1

C O S 2 ~ ,

b(~ ) =/31 1 - 2/312 s in 2 ~ - 2 /313 cos 2 ~ +/322 s in 4 ~ (2 .1 4 )

+ 2/323 s in 2 ~ cos 2 ~ +/333 cos 4 P

T h e e f f e c t iv e c o n s t a n ts / 3 1 a n d / 3 2, w h i c h w i ll f u r t h e r b e u s e d , a r e d e t e r m i n e d

b y

/32 = + - /3 11 +/312 ,

2x / 6 / 3~o)

/ 3 1 -- -- / 3 ~ 0 - - / 3 1 1 ~ - / 3 1 3

I n t h e e f f e c t i v e c o n s t a n t s , o n l y t h e t h e c o n s t a n t d ex i n u)D i s co n s id e red .

T h e e q u i l i b r i u m v a l u e s f o r t h e a n g le s 0 a n d ~ a r e f o u n d b y m i n i m i z i n g

t h e e n e r g y W . W e c o n s i d e r t w o p o s si b i li ti e s : ~ = 0 a n d ~ = ~ r/ 2. T h e v a l u e

= 0 c o r r e s p o n d s t o t h e r o t a t i o n o f t h e v e c t o r I i n t h e a c - p l a n e , w h i le

= 7c/2 c o r r e s p o n d s t o t h e r o t a t i o n i n t h e a b - p l a n e . B y a r o t a t i o n o f 1 i n

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

a p p l i e s t o a r e a l r o t a t i o n o f 1 i n t h e d o m a i n w a ll .

W h e n t h e v a l u e ~ i s f i x e d , m i n i m i z a t i o n o f ( 2 . 1 4 ) g i v e s t h e f o l l o w i n g

e q u i l i b r i u m v a l u e s fo r t h e a n g l e 0 a n d t h e e n e r g y W :

I. ~l l: 0 = O, ~r; W = M 2 W o ,

II . aS• 0 =

- 4 - I r / 2 ; W = M ~ W o +

(1 /2 ) / 3 (~ ) + (1 /4 )b (9 ~ ) ) .

T h e f ir s t o f t h e p h a s e s ~ l] i s s t a b l e o v e r t h e g i v e n t e m p e r a t u r e r a n g e ,

w h e r e / 3 ( ~ , T ) > 0 . T h e s e c o n d p h a s e ~P• i s s t a b l e f o r / 3 ( ~ , T ) < - b . I f b > 0 ,

t h e n a p a r t f r o m t h e i n d i c a te d p h a s es a t - b < / 3 < 0, t h e r e e x is ts t h e p h a s e

q s z , w h e r e s i n 2 0 = - / 3 / b , i .e . 0 ~ 0 , 7 r /2 . W e d o n o t d i s cu s s t h e p ro p e r t i e s o f

t h i s p h a s e , s i n c e a t t h e p r e s e n t t i m e , t h e d y n a m i c a l e x p e r i m e n t s a r e c a r r i e d

o u t o n l y i n t h e c o l li n e a r p h a s e s r a n d ~ •

T h e r e o r i e n t a t i o n f r o m t h e ~ t o t h e a -a x i s , w i t h d e c re a s in g t e m p e r a t u r e ,

c a n b e d e s c r i b e d a s s u m i n g t h a t /3 = 9 ~(0 , T ) i s a m o n o t o n o u s l y i n c r e a s i n g

t e m p e r a t u r e f u n c t i o n . I n t h i s c a s e w h e n T > T 1 , /3 (0 , T 1 ) = 0 t h e p h a s e qsil

w i th 1 [[ a i s s t ab l e , a n d w h e n T < T 2 , /3 (0 , T 2 ) = - b t h en t h e p h a s e ~ •


9/24

1 2 2 . Ph en o m en o l o g i ca l Th eo ry o f M ag n e t i s m

w i t h 1 ]] e i s s t ab l e . T h e an g u l a r p h as e i s r ea l i zed a t T2 < T < T1 , wh i ch i s

pos s ib le fo r b > 0 .

I f b < 0 , o n l y t h e p h a s e s ~511 a n d @ z p r o v e t o b e s t a b l e . T h e p h a s e t r a n -

s i ti o n o c c u r s as a f ir s t - o r d e r p h a s e t r a n s i t i o n a t t h e t e m p e r a t u r e s f o r w h i c h

WII =

W •

F o r ~ 0 = ~ r/ 2, t h e f u n c t i o n / ~ T ) - /3 7 c/2 , T ) is m o n o t o n o u s l y

i n c re a s in g w i t h t e m p e r a t u r e , f o r d y s p r o s i u m o r t h o f e rr i te . T h e b o u n d a r i e s o f

t h e s t a b i l i t y o f ~11 a n d ~ z p h a s e s a r e d e t e r m i n e d f r o m t h e s a m e c o n d i t i o n s

a s f o r t h e c a s e w h e n ~ = 0 . S i n c e b < 0 , t h e n T 1 < T 2 . T h e p h a s e t r a n s i t i o n

t e m p e r a t u r e T t is g o v e r n e d b y :

/ ~ T t ) + b / 2 = 0

T h e t e m p e r a t u r e

T t

s a t i s f i e s t h e i n e q u a l i t y

TI < T < T2

S i n c e i n t h e p h a s e @11 t h e m a g n e t i s m v e c t o r m = e z d / 5 , w h e r e b y r n is z e r o in

t h e p h a s e @ • t h is p h a s e t r a n s i ti o n i s a c c o m p a n i e d w i t h t h e m a g n e t i s m v e c t o r

j u m p A m z = d / 5 . T h e j u m p is d u e t o th e s p o n t a n e o u s m o m e n t a s s o c i a te d

w i t h t h e p r e s e n c e 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 te r a c t i o n .

I r o n B o r a t e . W e n o w di sc u ss th e p r o p e r t i e s o f i ro n b o r a t e F eB O 3 . T h i s m a g -

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

t h e o t h e r r e p r e s e n t a t i v e s o f t h i s c la ss b e i n g h e m a l i t e c ~ -F e 2 03 , c a r b o n a t e s

F e C O 3 , M n C O 3 , e t c . , se e [ 2.8 ]. T h e l a t t i c e F e B O 3 is o f a c a l c i t s t r u c t u r e , i t s

c ry s t a l c l a s s i s

D63d,

see

D i e h e

[2 .1 7]. T h e i r o n a t o m s p i ns f o r m t w o s u b l a t t i c e s

a n d a r e o r d e r e d a t a t e m p e r a t u r e o f 3 50 K .

F e B O 3 s u b l a t t i c e s p i n s l i e i n t h e b a s a l p l a n e . T o d e s c r i b e t h e a n i s o t r o p y

e n e r g y o f F e B O 3 , o n e c a n u s e t h e f o r m u l a e

9 , 1 4

w2 = ~12z , W = l z [ lx q- i /y) 3 q- l x - - i /y) 3] -k - ~ -z

Z 6 /~ 6 16

w6 = ] ~ [ /x i /y) 6 Ix - i /y) 6] -~- z

2 . 1 5 )

H e r e t h e z - a x i s is t a k e n a l o n g t h e t h i r d - o r d e r a x is , a n d t h e y - a x i s - a l o n g

t h e s e c o n d - o r d e r a xi s, l y in g in t h e b a s a l p l a n e . T h e s y m m e t r y p la n e s o f t h e

c lass D ~ h c o i n c id e w i t h t h e p l a n e s z x ) .

T h e f o u r t h - o r d e r a n i s o t r o p y r e s u lt s in a w e a k d e v i a t i o n o f t h e v e c t o r l

f r o m t h e b a s a l p l a n e w h e n 1 is n o t o r i e n t e d a lo n g t h e s e c o n d - o r d e r a x e s.

T h e d e v i a t io n a n g le ~ is a p p r o x i m a t e l y e q u a l t o - ~ 4 / / ~ ) c o s 3 ~ , w h e r e b y

t h e a n g l e ~ is c o u n t e d o f f f r o m t h e z - a x i s . I f w e d o n o t d is c u ss t h e d e v i a t i o n

e f f e ct s w e c a n e x c l u d e t h e c o m p o n e n t l z f r o m 2 .1 5 ) a n d w r i t e t h e e f f e c t iv e

a n i s o t r o p y e n e r g y i n a f o r m t y p i c a l o f h e x a g o n M m a g n e t i c m a t e r i a l s:

f l b

W a O , r = ~ COS2 0 + ~ s in 6 0 COS 6qo, b = /~6 -- 3 ~ 2 /2 ~ 2 . 1 5 ~ )


10/24

2.3 Do m ain W alls 13

S m a l l c o r r e c t i o n s t o f i, o f t h e o r d e r o f f i 2 / f l , an d t e rm s o f t h e t y p e 14, 16,

e t c ., c a n b e n e g l e c t e d in t h e a n a ly s is . W e n o t e t h a t a n i s o t r o p y in t h e i ro n

b o r a t e b a s a l p l a n e i s r a t h e r s m a l l .

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 e n e r g y c o n t a i n s se v e r a l i n v a r i a n t s ,

a m o n g w h i c h t w o o f t h e m o r e i m p o r t a n t a re :

W D = d e x m z l y - m v l x ) + d / 2 i ) m z [ Ix -

i /y) 3 -

l z +

i /y) 3] (2.16)

T h e f i rs t i n v a r ia n t d e t e r m i n e s t h e m a i n c o m p o n e n t o f t h e W F M m o -

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

a n t r e s u lt s in t h e a p p e a r e n c e o f t h e z - p r o j e c t i o n o f t h e w e a k m o m e n t

m z = d / 5 ) s i n 3 0 s i n 3 p . T h i s c o m p o n e n t i s s m a l l (n o m o r e t h a n 1 o f t h e

m a j o r c o n t r i b u t i o n to m a c c o u n t e d fo r b y d e • b u t i t is i m p o r t a n t f o r d e -

s c r i b i n g c e r t a i n F e B O 3 p r o p e r t i e s .

S i nc e t h e a n i s o t r o p y c o n s t a n t b is sm a l l, a n d t h e m a g n e t i z a t i o n r n o c c u rs ,

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

s u f f i c i en t l y w eak f i e ld (o f t h e o rd e r o f 1 00 O e) .

1 I I Y , m z b e i n g n o n z e r o , f r o m r o o m t e m p e r a t u r e s d o w n t o t h e g r o u n d

s t a t e o f i r on b o r a t e . E x p e r i m e n t s p e r f o r m e d b y D o r o s h e v e t a l. [2 .18] re-

v e a l t h a t l is s p i n - o r i e n t e d a t T -~ 7 K f r o m y t o x - a x e s ( w i t h d e c r e a s i n g

t e m p e r a t u r e ) . T h i s p h a s e t r a n s i t i o n t u r n s o u t t o b e t h e f i rs t o r d e r t ra n s i t io n .

2 . 3 D o m a i n W a l l s

W e n o w d i sc u s s t h e s t r u c t u r e a n d s t a t i c p r o p e r ti e s o f t h e m a i n m a g n e t i c

i n h o m o g e n e i t i e s r e a l i z e d i n m a g n e t i c m a t e r i a l s a n d o b s e r v e d i n e x p e r i m e n t s .

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

D W ) .

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

S i x t u s

a n d T o n k s [2.19], B l o c h [2.20], L a n d a u a n d L i f s h i t z [ 2 . 2 1 ] . T h e p h e n o m e n o n

o f a d o m a i n w a ll is a t t r i b u t e d t o t h e d i sc r e te d e g e n e r a c y o f t h e e n e r g y o f

t h e m a g n e t , i. e. t h e r e a r e s e v e ra l d i f fe r e n t d i r e c t i o n s o f m a g n e t i z a t i o n ( o r l )

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

w i t h a s i n g l e e a s y a x i s l = l e 3 t h e r e a r e t w o s u c h s t a t e s . I f i n o n e p a r t o f t h e

m a g n e t I - - - e 3 a n d i n t h e o t h e r I = - e 3 , t h e n t h e t r a n s i t i o n d o m a i n i n s i d e

o f w h i c h I r o t a t e s b y t h e a n g l e 7 r a 1 8 0 - d e g r e e d o m a i n w a l l ) s h o u l d e x i s t

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

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

t h a t t h e e q u i l i b r i u m d i r e c t i o n o f t h e v e c t o r l c o i n c i d e s w i t h t h e a - a x i s , a n d

w e c h o o s e e 3 I] a ; i . e . i n e q u i l i b r i u m t h e a n g l e 0 e q u a l s 0 o r 7 c . T h e a n a l y s i s

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

t h e m , a n a c - t y p e w a l l , c o r r e s p o n d s t o ~ - - 0 , i . e . t h e v e c t o r I r o t a t e s i n t h e

x z - p l a n e . A n o t h e r o n e , a n a b - t y p e w a l l , c o r r e s p o n d s t o a r o t a t i o n o f I i n t h e


11/24

1 4 2 . Ph en o m en o lo g i ca l T h e o ry o f M ag n e t i sm

x y - p l a n e ~ = 7 r/ 2) . T h e v a r i a t i o n o f t h e a n g l e 0 i n b o t h w a l ls i s d e t e r m i n e d

by:

d20

c ~ - f f - p s i n 0 c o s 0 = 0 , 2 .1 7)

w h e r e / 3 = 3 1 a n d / 3 = 3 2 f o r t h e a c - t y p e a n d a b - t y p e w a ll s, r e sp e c t iv e l y , s e e

E q . 2 . 1 4 ) .

T h e s o l u ti o n o f t h is e q u a t i o n , i n c o r p o r a t i n g t h e n e c e s s a ry b o u n d a r y c o n -

d i t i o n s 0 d : c c ) = 7r, 0 • e c ) = 0 ) , d e s c r i b e s b o t h t y p e s o f t h e D W a n d i s o f

t h e f o r m

t a n ~ = e x p , A i = , i = 1 , 2 2 .1 8 )

T h e s ig n s + o r - c o r r e s p o n d t o t w o p o s s ib l e d o m a i n w a ll s, t h e q u a n t i t y A i

d e n o t e s t h e w a l l t h i c k n e s s .

T h e s a m e d i s t r i b u t i o n c o r r e s p o n d s t o a 1 8 0 - d e g r e e d o m a i n w a l l i n t h e

f e r r o m a g n e t , f o r i t s d e s c r i p t i o n i t is n e c e s s a r y t o s u b s t i t u t e I f o r M / M o . T h e

i n d i c a t e d t y p e s o f w a l ls in t h e f e r r o m a g n e t c o r r e s p o n d t o t h e w e l l - k n o w n

B l o c h a n d N 6 e l w a ll s, t h e l a t t e r b e i n g e n e r g e t i c a ll y le ss a d v a n t a g e o u s a n d

u n s t a b l e i n m a s s iv e s a m p l e s . H o w e v e r , f o r o r t h o f e r r i t e s t h e v e c t o r r n d i s tr i -

b u t i o n i n b o t h t y p e s o f w a ll s d i ff e rs m o r e r a d i c a l ly t h is w a s f ir s t c o n s id e r e d

b y

B u l a e v s k y

a n d

Ginzburg

[2 .22], and

Farz td inov e t a l .

[2.2 3] ). Fo r t h e a c -

t y p e w a l l, t h e v e c t o r r n l ik e t h e v e c t o r 1 r o t a t e i n th e a c - p l a n e w i t h a n a l m o s t

c o n s t a n t l e n g t h , w h i c h fo l lo w s f r o m 2 .1 11 ). I n t h e a b - t y p e w a l l t h e v e c t o r r n

i s a l w a y s o r i e n t e d a l o n g t h e c - a x i s , a n d v a r i e s o n l y i n m a g n i t u d e , r n = 0 in

t h e c e n t r e o f th i s w a l l.

T h e p r e s e n c e o f t w o t y p e s o f D W w i t h t h e s a m e v a lu e s o f r n a n d I a t t h e

p o i n t s f a r f r o m a w a l l r a i s e s t h e q u e s t i o n o f t h e s t a b i l i t y o f t h e w a l ls . H o w e v e r ,

i t t u r n s o u t t h a t i n t h e a n a l y s is o f a D W a s a to p o l o g i c a l so l it o n , if t o t h e w a l l

t h e r e c o r r e s p o n d d if f e re n t v a l u e s o f r n a t ~ --4 + o c a n d ~ ~ - o o , t h e n t h e

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

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

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

I t i s i n t e r e s t i n g t o n o t e t h a t i f f o r a g i v e n w a l l t o ~ --* + c o a n d ~ - ~ - c ~

t h e r e c o r r e s p o n d t h e s a m e r n v a lu e s , a n d o n l y l v a lu e s a r e d if f e re n t , t h e n s u c h

a w a l l is n o t s t a b l e t o p o l o g i c a l l y a n d c a n b e e l i m i n a t e d . F o r t h i s i t su f fi c es t o

c r e a t e a r i n g d i s c li n a t i o n v e c t o r I d i s c o n t i n u i t y l in e ) a n d t h e n t o i n c r e a s e t h e

r a d i u s o f t h e r i n g s e e D z y a l o s h i n s k i i [2 .2 4]). T h e p o t e n t i a l b a r r i e r r e q u i r e d

t o o v e r c o m e is , f o r t h i s p r o c e s s , f i n it e .

T h e a n a l y s i s c a r ri e d o u t b y B a r y a k h t a r e t a l. [ 2.2 5] h a s s h o w n t h a t o n l y

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

w i t h i t t h e lo w e r v a l u e o f t h e c o n s t a n t ~ ) , i s s t a b le . T h e s e c o n d D W is

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

2.33)). G e n e r a l l y sp e a k in g , t h e c o n s t a n t s ~ i d e p e n d o n t e m p e r a t u r e i n a


12/24

2.3 Domain Walls 15

different way. Consequently, if the difference (f12 - ill) changes sign at some

temperature, one type of DW in the magnetic material should be transmuted

into a DW of the other type, on passing through this point. Such a trans-

formation has been found to occur in dysprosium orthoferrite DyFeO3 at

T = 150 K [2.26], walls of the ab-type have been found to occur at tempera-

tures below 150 K, whereas walls of the ac-type are found to occur above this

temperature. Notice that the reorientation of I in the DW is not the ordinary

spin reorientation, which occurs upon the reversal of the sign of one of the

constants fli (the smaller one), and not their difference. In DyFeO3 the spin

reorientation (the Moriya point) occurs, for example, at T = 40 K, i.e. at a

temperature significantly lower than the tempera ture at which the transfor-

mation of the ac-type walls into ab-type ones occurs. In the vicinity of the

ab-type wall transformation into the ac-type wall the dynamic properties of

DW have the unique peculiarities predicted in [2.27], but at present there are

no experiments to avail of in this temperature region.

It should be noted that in (2.17) or (2.18) the ~ axis direction against

the crystal axes should not be fixed, since the exchange interaction of the

type a(V/ ) 2 (see (2.1)) is independent of the direction of this axis. Generally

speaking, this formula is valid for the cubic crystals only, but describes the

exchange interaction in orthoferrites quite well. In experiments, the ~ axis

direction, i.e. the normal to the plane of DW, is fixed by the gradient of the

external magnetic field or demagnetizing field.

According to the various possible orientations, the walls of each type can

be divided into the following classes: quasi-Bloch walls (the vector m rotates

in the plane of the DW), quasi-N6el walls (the vector m is perpendicular

to the DW plane) and the so-called head- to-head walls . The last class

corresponds to a nonzero jump in the magnetization, i.e., a head-to-head

wall is charged, and produces a demagnetizing field at points far from itself.

Since the energy associated with the demagnetizing fields in orthoferrites

is small, the energies of these DW are close to one another. The difference

between these DW manifests itself especially strongly when allowance is made

for the magnetoelastic interactions (see Chaps. 4 and 5).

This classification of the DW and qualitative laws governing their struc-

ture holds true in the analysis of the DW in the easy-plane WFM of the

iron borate-type FeBO3. However, the quantitative differences are impor-

tant. This is connected with the essential difference in anisotropy constants

in the basal plane and outside the plane. Therefore, the wall with l rotation

in the basal plane of FeBO3 (in

f l / b ) U 2

see (2.15'), i.e. by several orders of

magnitude) is energetically more advantageous than the wall with l rotation

through a hard th ird -order axis. We discuss both types of the walls in more

detail in Chap. 3, but, only walls with a rotation in the basal plane are ob-

served in FeBO3. In the flee iron borate samples shaped as platelets parallel

to the basal plane both the N6el walls lying in the plane and the Bloch walls

dividing domains as platelets parallel to the basal plane are found to occur.


13/24

16 2 P h e n o m e n o l o g i c a l T h e o r y o f M a g n e t i s m

T o p r o d u c e s o l i t a r y N 6 el w a l l s (j u s t t h e s e w a ll s w e r e u s e d i n t h e d y n a m i c

e x p e r i m e n t s b y C he t k i n e t a l . [2 .2 8 ] a n d K i m a n d K h v a n [2 . 2 9 ] ) , o n e e x p l o i t s

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

p l a n e , f o r d e t a i l s s e e C h a p . 4 .

2 4 B l o e h L i n e s a n d P o i n t s

I n t h e p r e v i o u s s e c t io n , w e c o n s i d e re d t h e s i m p l e s t f o r m o f a D W , i .e . o n e -

d i m e n s i o n a l w a l ls w h e r e t h e m a g n e t i z a t i o n d e p e n d s o n o n e s p a c e v a r i a b le .

I t h a s, h o w e v e r , b e e n e x p e r i m e n t a l l y v e r if ie d t h a t t h e D W a r e f r e q u e n t l y

n o n - u n i d i m e n s i o n a l , i.e . i n v o lv e t h e m a g n e t i z a t i o n i n h o m o g e n e i t i e s in th e

w a l l p l a n e . I t h a s b e e n f o u n d t h a t s u c h in h o m o g e n e i t y c a n b e v ie w e d a s

b e i n g r e p r e s e n t a b l e b y a d y n a m i c a l o b j e c t , w h i c h c a n b e c h a r a c t e r i z e d b y a

t o p o l o g i c a l c h a r g e a n d m o v e u n d e r t h e a c t i o n o f t h e e x t e r n a l f ie ld s, e tc ; o r

s u e c i n t l y , i t c a n b e r e g a r d e d a s b e i n g a m a g n e t i c s o l i t o n .

T o g e t a n i n si g h t i n to t h e s t r u c t u r e o f m a g n e t i c i n h o m o g e n e i t ie s , w e s h a ll

p u r s u e a m o r e d e t a il e d e x a m i n a t i o n o f t h e s t r u c t u r e o f a D W i n t h e m a g n e t

o f r h o m b i c s y m m e t r y . F o r d ef in i te n e ss , w e t a k e u p a f e r r o m a g n e t w i t h t h e

f o l lo w i n g a n i s o t r o p y e n e r g y : u) a ( 1 / 2 ) ~ ( m 2 + m ~ ) + ( 1 / 2 )f l m 2~ . T h e v a r i -

a t i o n o f t h e a n g l e 0 i n t h e w a l l, i. e. , t h e c o m p o n e n t r n a l o n g t h e e a s y a x i s i s

d e t e r m i n e d b y t h e f o r m u l a ( 2 .1 8 ), w h e r e t h e s ig n =E d e t e r m i n e s t h e d i r e c t io n

o f m a t x = + o o , x i s t h e n o r m a l a x i s t o t h e w a l l.

T h e v a lu e s o f t h e t r a n s v e r s e c o m p o n e n t s ( r e x a n d m y ) a r e g o v e r n e d , w i t h

a c c o u n t t a k e n o f ( 2 . 1 8 ), by"

_ co s P0 s in 99o (2.19 )

m s c o s h ( z / A ) m y - c o s h ( x / A )

s o t h a t f o r r h o m b i c m a g n e t i c m a t e r i a l , t h e v a l u e 9 is m u l t i p l e o f 7 c/2 , ~ 0 =

7cn/2 ,

w h e r e n i s t h e i n t e g e r .

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

e y t h e n f o r a s ta b l e w a ll c o r r e s p o n d i n g t o m x = 0 , t h e r e a r e t w o v a lu e s o f

m y t o w h i c h t h e f o r m u l a e m y = c o s h ( x / A ) c o r r e s p o n d . T h u s , i f w e d e a l

o n l y w i t h a n e n e r g e t i c a ll y a d v a n t a g e o u s w a l l w i t h f ix e d v a l u e s o f m ~ ( + o o )

a n d r n z ( - e c ) , w e c a n in d i c a t e tw o t y p e s o f w a ll s t h a t d i ff e r i n m v a l u e in

t h e c e n t r e o f t h e w a l l ( t h e r e a r e a t o t a l o f f o u r t y p e s o f w a l ls w h e n t h e s ig n s

m ~ ( + e o ) a r e t a k e n i n to a c c o u n t ) . T h e t w o t y p e s o f w a l ls , c o r r e s p o n d i n g t o

~ 0 = 7 c/2 a n d 990 = - z c / 2 , a l s o d i f f e r i n t h e d i r e c t i o n o f r o t a t i o n o f t h e v e c t o r

r n w h e n m o v i n g f r o m x = - o o t o x = + e c .

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

i n g s e g m e n t s w i t h d i f f e r e n t v a l u e s ~ 0 , n a m e l y , 990 = 7 c/2 a n d 990 = - 7 r / 2 .

T h e s e w a l ls a r e o f t e n o b s e r v e d e x p e r i m e n t a l l y , e s p e c i a ll y in f i lm s o f b u b b l e -

m a t e r i a l s . L e t u s e x a m i n e t h e s t r u c t u r e o f s u c h a w a ll .


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2 .4 Bloch L ines and Poin t s 17

T h e e n e r g y p e r u n it a r e a o f a D W s e g m e n t c o r re s p o n d i n g to ~ = ~ / 2

i s t h e s a m e a s t h a t c o r r e s p o n d i n g t o - ~ / 2 , i .e . t h e w a l l s t a t e is t w o - f o l d

d e g e n e r a te . H o w e v e r , w h e n t h e t r a n s i t i o n f r o m o n e s e g m e n t t o t h e o t h e r

o c c u r s t h e a n g l e ~ s h o u l d g o t h r o u g h a ll i n t e r m e d i a t e v a l u es , w h i c h is e n -

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

b a s a l p l a n e ( / Y / 2 ) s i n 2 0 c o s 2 ~ . T h e r e is a ls o a n a d d i t i o n a l e x c h a n g e e n e rg y ,

a s i n 2 0 ( V ~ ) 2 / 2 , w h i c h a r i s es d u e t o t h e i n h o m o g e n e i t y a n g le ~ . T h u s , a s im -

i la r s it u a t i o n o c c u r s t o t h a t w h i c h o n e is c o n f r o n t e d w i t h i n s t u d y i n g a D W

i n a s i n g l e - a x i s f e r r o m a g n e t .

I n d e e d , i n b o t h c a s es w e c a n d i s ti n g u i s h b e t w e e n t w o e q u i v a l e n t s t a t e s

o f t h e s y s t e m : 0 = 0 a n d ~r f or t h e i n h o m o g e n e o u s m a g n e t i c m a t e r i a l , a n d

~ 0 = 7c/2 a n d - 7 r / 2 f o r t h e i n h o m o g e n e o u s w a ll . W h e n t h e t r a n s i t i o n f r o m

o n e s t a t e t o t h e o t h e r o c c u r s , a lo ss i n b o t h t h e e x c h a n g e e n e r g y a n d t h e

a n i s o t r o p y e n e r g y o c cu r s: ( 1 / 2 ) ~ s i n 2 0 fo r t h e i n h o m o g e n e o u s m a g n e t i c m a -

t e r i a l a n d ( 1 / 2 ) ~ s i n 2 0 c o s 2 ~ f o r t h e i n h o m o g e n e o u s w a ll . T h u s , a s i m i l a r

r e s u l t o f t h e a n a l y s i s fo r b o t h c a s es w i ll n o t b e s u r p ri s in g . I t t u r n s o u t t h a t

t h e d o m a i n w a l l s e g m e n t s w i t h d i ff er e nt ~ 0 a r e d iv i d e d b y t h e t r a n s i t i o n d o -

m a i n o f f i n it e t h i c k n e s s l o c a t e d i n t h e w a l l, se e F i g . 2 .1 . T h i s d o m a i n w h o s e

c e n t r e ( m a x i m u m d e v i a t i o n o f t h e ~ v a l u e f ro m - ~ r / 2 a n d 7 r /2 ) co i n ci d e s

w i t h s o m e l i n e i n t h e w a l l p l a n e i s k n o w n a s t h e B l o c h l i n e , w h i c h s u g g e s t s

i t s s i m i l a r i t y w i t h t h e B l o c h d o m a i n w a l l .

F i g . 2 . 1 T h e m a g n e t i z a t i o n d i s t r i b u t io n i n a d o m a i n w a l l w i t h a B l o c h l in e .

C a l c u l a t i n g t h e B l o c h l i n e s t r u c t u r e , e v e n w i t h o u t a l l o w a n c e f o r t h e d e -

m a g n e t i z i n g f i e l d t h e f i e l d / - / m i n t h e B l o c h l i n e i s n e e e s s a r e l y n o n z e r o S i n c e

i n i t d i v M ~ 0 ) , i s q u i t e a c o m p l i c a t e d p r o b l e m a n d c a n n o t b e a n a l y t i c a l l y

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

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


15/24

18 2 . Phenom enolog ica l Theory o f M agnet i sm

t h e D W : i t i s e i t h e r c l o s e d i n a r i n g o r a p p e a r s o n a c r y s t a l s u r f a c e t o g e t h e r

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

c a u s e d b y t h e t o p o lo g i c a l s t a b i li t y o f t h e s e i n h o m o g e n e i t i es o f m a g n e t i z a t i o n .

A t r e a t m e n t o f t o p o l o g i c a l st a b i li t y , i m p o r t a n t f o r c l a s si f ic a t io n a n d q u a l i t a -

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

b y ineev [2.30].

T h e B l o c h l i ne s in f e r r o m a g n e t f il m s a n d p l a t e l e t s a r e s p e c if ie d w i t h t h e i r

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

l a r t o t h e p l a t e l e t s u r f a c e a r e w e l l k n o w n , t h e y a r e t r a c e a b l e , f o r i n s t a n c e ,

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

f e r r i t e - g a r n e t s . I n p a r t i c u l a r , s e v e r a l s c o r e o f s u c h l i n e s c a n b e i n t h e D W o f

a c y l i n d r i c a l m a g n e t i c d o m a i n m a g n e t i c b u b b l e ) .

A b u b b l e w i t h v e r t i c a l B l o c h l i n e s i s c a l l e d a h a r d c y l i n d r i c a l d o m a i n .

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

p r o p e r t i e s , s e e [ 2 . 7 ] .

B l o c h l i n e d y n a m i c s w e r e e x p e r i m e n t a l l y i n v e s t i g a t e d b y m a n y a u t h o r s . I n

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

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

l in e s w a s i n v e s t i g a t e d i n y t t r i u m f e r r i t e - g a r n e t s , s ee [ 2.3 1]. T h e D W t h i c k n e s s

i n t h i s m a g n e t i c m a t e r i a l is a n o m a l o u s l y l a rg e m o r e t h a n 1 ~ m ) , w h i c h m a d e

i t p o s s ib l e t o d i r e c t l y o b s e r v e c e r t a i n B l o c h li ne s , t o d e t e c t t h e i r m o t i o n , t o

d e t e r m i n e t h e v i s co u s f r i c t i o n c o e ff ic i en t a n d t h e e f f ec t iv e m a s s .

T h e B l o c h l i n e s c a n a l s o b e o b s e r v e d a n d t h e i r d y n a m i c s c a n b e s t u d -

i e d f o r s t a n d a r d m u c h m o r e re f in e d ) D W , w h i c h a re o b s e rv e d in e p i t a x i a l

f il m s o f r a r e - e a r t h f e r r i te - g a r n e t s . C h a p t e r 9 d e a ls w i t h t h e a n a l y si s o f t h e s e

o b s e r v a t i o n s .

A s h a s a l r e a d y b e e n m e n t i o n e d , t h e c a l c u l a t i o n o f a B l o c h li n e s t r u c t u r e is

q u i t e a c o m p l i c a te d m a t h e m a t i c a l p r o b l e m w h o s e e x a c t s o lu t io n is u n k n o w n .

W e g iv e a n a p p r o x i m a t e s o l u t i o n w h i c h i s a s s u m e d t o b e t r u e f o r ~ t < < /3 . F o r

d e f i n i t en es s w e co n s i d e r a v e r t i c a l B l o ch li n e s i t u a t e d a l o n g t h e l i n e x = y = 0 .

In t h i s ea s e ~ = ~ y ) , 0 = 0 x , y ) ,

si n = tanh yv / ) ,

c o s 0 = t an h x / + c o s 2 2 . 2 0 )

T h e s e f o r m u l a e , w i t h t h e s u b s t i t u t i o n p ~ - - + 4 ~ , a r e u s e d t o d e s c r i b e t h e

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

i n t e r a c t i o n . I n t h i s c a s e , h o w e v e r, t h e y a r e o n l y q u a l i t a t i v e l y v a li d , s in c e t h e

e x p r e s s i o n f o r t h e d i p o l e e n e r g y , o f t h e f o r m 2 ~ r M e ~ ) 2 is , s t r i c t l y s p e a k i n g ,

o n l y c o r r e c t f o r a o n e - d i m e n s i o n a l d i s t r ib u t i o n o f m a g n e t i z a t i o n .

I t i s e v i d e n t t h a t f o r ~ > > f it t h e s c a l e o f v a r i a t i o n o f t h e a n g l e ~ i s m u c h

l a r g e r t h a n o f t h e a n g l e 0 , i . e . t h e e f f e c t i v e t h i c k n e s s o f a B l o c h l i n e A , e q u a l

t o x / J / Z i s s i g n i a c a n t l y l a r g e r t h a n t h e w a l l t h i c k n e s A = V / Z . T h e w a l l

t h i c k n e s s d e p e n d s o n ~ , A _ - - z 5 ~ ) , i . e . , A y ) a n d d e c r e a s e s n e a r t h e B l o c h

l ine


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2.4 Bloch Lines an d Poin ts 19

A V ) =

~/~ + 2 /3 /cosh2(y/A)

I n t h e c e n t r e o f t h e B l o c h l in e y = 0 )

A 0 ) - v ~

i .e . , t h e w a l l t h i c k n e s s is s m a l l e r t h a n b o t h , t h e B l o c h w a l l t h i c k n e s s A B =

a n d t h a t o f t h e N 6 e l w a l l

AN

v/O~/ /~ @/~t).

S o , t h e d o m a i n B l o c h ) w a l ls g e n e r a t e t h e B l o c h li ne s . T h i s h i e r a r c h y o f

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

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

e n e r g y . F o r i n s t a n c e , i n t h e c e n t r e o f t h e B l o c h li ne o f t h e f o r m 2 .2 0 ), t h e

m a g n e t i z a t i o n c a n t a k e t h e v a l u e s e x M o a n d - e z M 0 w h i ch c o r re s p o nd s t o

= 0 an d 7r fo r y = 0 , i. e ., i n t h e ce n t r e o f t h e l i n e .

I t f o l lo w s f r o m t h e B l o c h li n e o r i g in c o n s i d e r e d a b o v e , t h a t t w o B l o c h l i ne s

w i t h ~ y = 0 ) = 0 a n d 7v c a n b e j o i n e d a t s o m e p o i n t , th i s p o i n t i s c a l le d t h e

B l o c h p o i n t . U n l i k e t h e B l o c h l i n e , w h e r e t h e m a g n e t i z a t i o n i s e v e r y w h e r e

c o n t i n u o u s , s e e 2 . 2 0 ), t h e B l o c h p o i n t h a s n e c e s s a r i l y d i s c o n t i n u i t i e s in t h e

m a g n e t i z a t i o n f i e l d

M ( r ) .

I n o r d e r t o c o n v i n c e o n e s e l f o f t h i s , i t i s s u f f ic i e n t t o c o n s i d e r t h e b e h a v i o u r

o f t h e m a g n e t i z a t i o n f a r f ro m t h e B l o c h p o in t . I n m o v i n g aw a y f r o m t h e B l o c h

p o i n t a l o n g v a r i o u s d i r e c ti o n s ,

M ( r )

t a k e s o n d i f f e r e n t v a l u e s . F o r e x a m p l e ,

i f t h e D W p l a n e c o i n c i d e s w i t h y z - p l a n e , t h e B l o c h li ne s a r e p o s i t io n e d a l o n g

t h e z - a x i s , a n d t h e B l o c h p o i n t - a t t h e o r i gi n o f t h e c o o r d i n a t e s , t h e n y i e l d s

M --+ •

w he n y , z = 0 , x --+ i c e ,

M --+ -4-M0ey w h e n x , z = 0 , y --+ i c e , 2 .21)

M -~ =t=M0e~ w h en x , y = 0 , z -+ =t=ce

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

c a n fi n d a ll t h e r e m a i n i n g , in t e r m e d i a t e , w i t h r e s p e c t to 2 .2 1 ), m a g n e t i z a t i o n

v a l ue s . I n o t h e r w o r d s , i f w e e n c i r c le th e B l o c h p o i n t w i t h a s p h e r e o f r a d i u s

R > > A , t h e m a g n e t i z a t i o n o n t h i s s p h e r e t a k e s o n a l l p o s s i b l e v a l u e s .

I t i s q u i t e o b v i o u s t h a t i f t h e m a g n e t i z a t i o n f ie ld is c o n t i n u o u s e v e r y w h e r e ,

e x c e p t f o r t h e c o o r d i n a t e o r i g i n p o i n t, t h i s p r o p e r t y w i ll b e c o n s e r v e d f o r t h e

s p h e r e o f a n a r b i t r a r y r a d i us t h a t s u r r o u n d s t h e o r i g in o f t h e c o o r d in a t e s.

T h i s i s r i g o r o u s l y p r o v e d b y t h e a l g e b r a i c t o p o l o g y m e t h o d s , s e e [ 2.3 0]. I n

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


17/24


18/24

2 .5 E f f ec t iv e E q u a t i o n s o f t h e D y n a m i c s o f W F M M a g n e t i z a ti o n 2 1

i t

0 .5

f T ~

2 4 6 8 X

F i g . 2 . 2 T h e d e p e n d e n c e o f t h e m a g n e t i z a t i o n m o d u l u s o n t h e d i m e n s io n l es s co o r -

d i n a t e z , x = r

t o t h e s i z e o f t h e u n i t c e ll a , a = 1 2 .5 A . S i n c e in t h e u n i t c e l l o f t h e f e r r i t e -

g a r n e t t h e r e a r e s e v e ra l d o z e n s o f t h e i ro n m a g n e t i c i on s , o n e c a n a s s u m e

t h a t t h e p h e n o m e n o l o g i c a l de s c ri p t i o n o f t h e B l o c h p o i n t p e r f o r m e d a b o v e

is a ls o m e a n i n g f u l a t d i s ta n c e s r ~ a , a n d t h e f o r m u l a e o b t a i n e d a b o v e m a y

b e r e g a r d e d n o t o n l y a s e s t im a t e s . I t is i m p o r t a n t t h a t t h e a d d i t i n a l e n e r g y

b r o u g h t b y t h e B l o c h p o i n t i n to t h e d o m a i n w a ll is f in i te .

T h e B l o c h p o i n t , i n v i e w o f t h e f o re s ai d , c a n b e r e g a r d e d a s a r a t h e r e x o t i c

o b j e c t . N e v e r t h e l e s s , B l o c h l i n e s a n d B l o c h p o i n t s h a v e b e e n e x p e r i m e n t a l l y

p r o v e d t o o c c u r , t h e r e s u l ts b e i n g o f a r e li a b le n a t u r e , b y a n a l y s i n g t h e c r e-

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

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

B l o c h p o i n t a n d i ts f o r c e d m o t i o n i n t h e i r o n - y t t r i u m g a r n e t w a s f ir st c a rr i e d

o u t [2 .3 3]. T h e d y n a m i c s o f t h i s s o l it o n t u r n e d o u t t o b e p u r e l y d i s s i p a t iv e .

T h i s i s r a t h e r a n u n e x p e c t a b l e r es u lt , b e c a u s e th e i r o n - y t t r i u m g a r n e t h a s a n

e x t r e m e l y lo w m a g n e t i c r e l a x a t i o n c o n s t a n t . H o w e v e r , t h i s s tr o n g r e l a x a t i o n

o f B l o c h p o i n t s w a s e x p l a i n e d i n [2 .3 2] o n t h e b a s i s o f t h e e x c h a n g e r e l a x a t i o n

t h e o r y d e v e l o p e d b y V . B a r y a k h t a r , s e e b e lo w S e c t . 4.4 .

2 5 E f f e c t i v e E q u a t i o n s o f t h e D y n a m i c s

o f W F M M a g n e t i z a t i o n

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

W F M w h e n c o n t e m p l a t i n g t h e W F M s t a t i c p r o p e r t i e s . F o r i n s t a n c e t h e

s t r u c t u r e o f t h e s t a t i c D W i s m o r e d i v e r s e t h a n i n f e r r o m a g n e t s . H o w e v e r

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

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


19/24


20/24

2 .5 Effec tive Equat ions o f the Dynam ics o f W FM Mag net i za t ion 23

H av i n g u s ed t h e i n eq u a l i t i e s ~ < < 5 an d rn < ~ 1 t h e l a s t t e rm s i n 2 .27 )

w r i t t e n s c h e m a t i c al ly , c a n b e o m i t t e d . R e t a i n i n g t h e p r i n ci p le t e r m s o f

p / 5 ) ,

m , t h e d e s i r ed ex p re s s i o n fo r rn fo ll o w s f ro m 2 .2 7 ):

1 [ 2 h - D - l l , 2 h - D ) ) ] + ~ ~ x l 2 .28)

n = ~

T h e f i rs t t w o t e r m s i n t h i s f o r m u l a e x i s t i n th e s t a t i c c a s e a n d d e t e r -

m i n e t h e W F M m a g n e t i z a t i o n n o n c o l li n e a r it y o f s u b l a tt ic e s ) c a u s e d b y 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 s ee S e c t. 2 .2 ) a n d t h e i n f l u e n c e o f t h e e x -

t e r n a l m a g n e t i c f ie ld H =

h M o ,

r e s p ec t iv e l y . T h e l a s t t e r m is a s s o c i a te d w i t h

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

p r e c e ss i o n . T h i s t e r m l e a d s t o a n e s s e n t i a l d i ff e r e nc e b e t w e e n t h e d y n a m i c s

o f m a g n e t i z a t io n i n f e rr o m a g n e t s a n d i n W F M .

S u b s t i t u t i n g 2 .2 8 ) i n t o t h e e q u a t i o n fo r

O r e ~ O r )

o f t h e s y s t e m 2 .2 6 ) w e

o b t a i n t h e d e s ir e d d y n a m i c e q u a t i o n f or t h e v e c t o r l .

W e d o n o t e x p l i c i t ly w r i t e o u t t h i s e q u a t i o n , s i n c e i t c a n b e w r i t t e n a s

t h e E u l e r - L a g r a n g e e q u a t i o n

5 s

z • g = 0 2 .2 9 )

w h e r e t h e L a g r a n g i a n

s

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

L 1 , O l / O t , V / ) h a s t h e f o r m

M 0 2 _ 2 h )2

2.30)

F / g S M 0

I f w e n e g le c t a n i s o t r o p y a n d s e t h = 0 , D ~ = 0 , w e o b t a i n t h e L a g r a n g i a n

o f t h e L o r e n t z - i n v a r i a n t c h i r a l ~ - m o d e l , d i sc u s s ed a t l e n g t h i n m o d e r n f ie ld

t h e o r y . T h i s m o d e l is p r o v e n t o b e L o r e n t z - i n v a r i a n t . T h e p r e s e n c e o f t h e

t e r m w a l ) l e a d s t o t h e a n i s o t r o p i c g e n e r a l i z a t io n o f t h i s m o d e l , a n d t h e

t e r m s l i n e a r i n

O l / O t )

b r e a k t h e L o r e n t z - i n v a r ia n c e . I n d e s c ri b in g t h e W F M

c = g M o a S ) l / 2 / 2 ,

t h i s q u a n t i t y h a s t h e s e n s e o f t h e s p i n w a v e p h a s e v e l o c it y ,

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

i n t e r a c t i o n o n l y , h e n c e , i t i s m u c h l a r g e r t h a n t h e c h a r a c t e r i s t i c m a g n o n

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

i nt eg r al o f W F M J : c ~ - J a / l ~ ; a i s t h e l a t t i c e c o n s t a n t . C h a r a c t e r i s t i c v a l u e s

o f c a r e o f t h e o rd e r o f 1 0 3 m / s fo r J .-~ 1 0 K an d 1 0 4 m / s fo r I ~ 1 00 K .

I t is m o r e c o n v e n i e n t t o u s e t h e L a g r a n g i a n 2 .3 0) t h a n t h e i n i ti a l L a n d a u -

L i f s h it z e q u a t io n s . T h e m a i n a d v a n t a g e s a r e t h e s m a l l e r n u m b e r o f v a r ia b le s

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

d e s c r i p ti o n o f M 1 a n d M 2 ) a n d a ls o t h e i n v a r ia n c e w i t h r e s p e c t t o L o r e n t z

t r a n s f o r m a t i o n s i n s o m e s im p l e r v e r s io n s o f t h e m o d e l . T h e l a s t p o i n t w i ll b e


21/24

2 4 2 . Ph en o m en o l o g i ca l Th e o ry o f M ag n e t i s m

d i s c u s se d b e lo w , in d e s c r i b in g th e D W m o t i o n C h a p . 4 ) . I t s h o u l d b e n o t e d

t h a t t h e p r e s e n c e 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 d o e s n o t n e c e s s a r -

i ly re s u l t i n b r e a k i n g 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 d y n a m i c e q u a t i o n s f o r t h e

v e c t o r 1. I f t h e t e n s o r D i k i s a n t i s y m m e t r i c , D i k =

e i k j d j , d

i s t h e c o n s t a n t

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

D ~k, c a u s e d b y t h e e x c h a n g e - r e l a t i v i s t i c i n t e r a c t i o n ) , t h e t e r m w i t h D i k is

r e d u c e d t o t h e t o t a l d e r iv a t iv e i n t i m e

e ~ k j d j l x - ~ Ik = d d l )

i

i t is t a k e n i n t o a c c o u n t t h a t l 2 = 1 ). T h u s , t h e b r e a k i n g o f t h e L o r e n t z -

i n v a r i a n c e is d u e t o su f f i c i e n tl y w e a k e n o u g h r e l a t i v i s ti 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 i o n c o m p o n e n t s o n l y .

L e t u s e x a m i n e t h e s p in w a v e s p e c t r u m o f t h e l in e a r t h e o r y u s i ng t h e

e q u a t i o n s t h a t f o ll ow f r o m t h e L a g r a n g i a n 2 .3 0 ). W e c o n f in e o u r s e lv e s t o

t h e s i m p l e s t v e r s io n o f t h e m o d e l a s s u m i n g t h e t e r m s w i t h D i k a n d H t o b e

u n e s s e n t i a l f o r i n s t a n c e , D i k = e i k j d j , a n d H < < H a l l e ) l ~ 2 ) . I f w e a s s u m e

t h a t t h e v e c t o r 1 i n t h e g r o u n d s t a t e i s o r i e n t e d a l o n g s o m e x - a x i s , a n d

l i n e a ri z e t h e e q u a t i o n s n e a r t h i s e q u i l i b r iu m s t a t e , w e o b t a i n f o r

ly

a n d

l z

021z

Ot 2

c2 V2 / z + co ~ l z

= 0 ,

2 . 3 1 )

0 2 1 y c 2 V 2 1 y -4 - 0 2 2 1 y = 0

O t 2

w h e r e t h e n o t a t i o n s c o l , 0 ) 2 a r e i n t r o d u c e d :

0)1 = gMo,/7 /2, 0)2 = gMox/ /2 , ( 2 . a )

s ee Eq . 2 . 1 4 ) .

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

A F M ) o r W F M t h e r e c o r r e s p o n d t w o l i n e a rl y p o l a r i z e d w a v e s w i t h o s ci ll a-

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

0)2,2 ]9) ~- i k)21,2 q- C2/g2 , 2 .3 2 )

w h e r e c o k ) is t h e m a g n o n f r e q u e n c y w i t h t h e w a v e v e c t o r k , t h e q u a n t i t i e s

0)1 a n d 0 )2 r e p r e s e n t t h e m a g n o n a c t i v a t i o n . S i n c e t h e e q u a t i o n s 2 . 31 ) a r e

L o r e n t z - i n v a r i a n t t h e m a g n o n e n e r g y d e p e n d e n c e o n i ts m o m e n t u m is t h e

s a m e a s f o r t h e r e l a t i v i s t i c p a r t i c l e . I t is c l e a r t h a t w h e n c h >> c o l,2 t h e s p i n

w a v e s h a v e t h e l i n e a r d i s p e r s i o n l a w , 0)1 ,2 - - c k. T h e q u a n t i t y c h as , t h u s , t h e

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

s e e C h a p . 4 . I t c a n a l s o b e se e n , t h a t c c o i nc i de s w i t h t h e m i n i m u m w a v e

p h a s e v e l o c i t y w i t h t h e d i s p e r s io n la w 2 .3 2 ).

T h e E q s . 2 .3 2 ), a n d a l so t h e L a g r a n g i a n 2 .3 0 ) a r e o b t a i n e d in t h e l o n g -

w a v e l e n g t h a p p r o x i m a t i o n a n d h o l d t r u e o n l y w h e n A >> a , i .e . k


22/24

2 .5 E f fec ti v e E q u a t i o n s o f t h e Dy n am ics o f W FM M ag n e t i za ti o n 2 5

w h e r e a is t h e l a t t i c e c o n s t a n t . T h e c o n d i t i o n c k >> 031,2 ca n be re w ri t t e n as

k >> v /Y / c ~. S i n c e ~ >> a t h i s m e a n s t h a t t h e s p e c t r u m 2 .3 2 ) c a n b e

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

p a r t o f t h e s p ec t ru m : v /~ -/c~


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26 2 . Phenom enolog ica l Theo ry o f M agnet i sm

t h e f r e q u e n c y o f a n o r d i n a r y f e r r o m a g n e t i c r e s o n a n c e f o r H ~- 5 + 1 0 k O e .

H e r e a s o - c a l l e d m a g n e t o e l a s t i c g a p b e c o m e s e s s e n t i a l ( s e e t h e r e v i e w s b y

T u r o v a n d S h a v r o v, B a r y a k h t a r a n d T urov [2 .38] ) , wi th a l lowance fo r th i s

g a p w l r 0 a t H = 0 . T h e s e c o n d b r a n c h h a s t h e ( 2 .3 1 ~ ) - ty p e a c t i v a t i o n .

T h e s p i n w a v e v e l o c it y i s o f t h e o r d e r o f 15 k m / s . F o r t h e i r o n b o r a t e , t h e s p i n

w a v e v e l o c it ie s , w h e n k I [ c a n d k Z e , d i ff e r f r o m o n e a n o t h e r , a p p r o x i m a t e l y

b y 3 0 .

I n t h e p r e se n c e o f a D W t w o a d d i t io n a l b r a n c h e s o f m a g n o n s l o ca l iz e d

n e a r t h e w a l l, o c c u r i n t h e W F M ( se e , e . g. , Tsang e t a l . [2 .3 9]) . O n e o f t h em

h as a g ap l e s s d i s p e r s i o n l aw , w = c l k • I w h e r e k • is t h e w a v e v e c t o r l y i n g i n

t h e w a l l p la n e . T h i s w a v e i s d e s c r i b e d b y t h e b e n d i n g o s c i l la t io n s o f t h e D W .

T h e d i s p e r s i o n l aw o f t h e o t h e r l o c a l iz e d w a v e c a n b e r e p r e s e n t e d a s

~ 2 ( k • = ~ / w ~ - ~ + c 2 k 2 ( 2 . 3 3 )

I t i s a s s u m e d t h a t ~ 2 > / 3 1 , i . e . t h e r o t a t i o n i n t h e a c - p l a n e i s e n e r g e t i c a l l y

a d v a n t a g e o u s in t h e w a l l. A m u t u a l o sc i ll a ti o n o f M 1 a n d M 2 i n t h e D W ,

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

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

i n o r th o f e r r i t e s a m p l e s w i t h a d o m a i n s t r u c tu r e .

T h e L a g r a n g i a n ( 2 . 30 ) , t a k i n g a c c o u n t o f t h e c o n d i t i o n 12 = 1 , c a n b e u s e d

o n i n t r o d u c i n g t h e a n g u l a r v a ri a b le s f o r t h e v e c t o r 1. C h o o s i n g t h e e 3 - a x i s

a l o n g t h e e q u i l i b r i u m d i r e c t i o n l , w e w r i t e

13 = cos 0 , l l + if2 = s in 0 ex p( i~)

I n t e r m s o f 0 , ~ t h e L a g r a n g i a n d e n s i t y ( 2 .3 0 ) c a n b e w r i t t e n a s

{~22 [ (c9 0h 2 (c g ~ h 2 c~ sin2 0(Vqo)2]

L = M g [ \ O t J s in 2 0 \ 5 - / J / - ~ E ( V 0 ) 2 +

( 2 . 3 o ' )

- ~ o ( 0 , ~ ) + ( 0 , ~ ) ~ + ~ ( 0 , ~ ) ~ } ,

w h er e @ a( l) i s t h e e f f ec t i v e an i s o t ro p y en e rg y , @ ~ = w a + ( h l ) 2 / 2 5 . T h e l a s t

t e r m s l i n e a r i n

O0/Ot

a n d

O ~ / O t

a n d b r e a k i n g t h e L o r e n t z i n v a r ia n c e o f t h e

m o d e l , a r e d u e t o t h e p r e se n c e 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 io n a n d

t h e e x t e r n a l f ie ld . A s p ec if ic f o r m o f t h e f u n c t i o n s A I ( 0 , ~ ) a n d A2(O, ~ ) is

d e t e r m i n e d b y t h e f o r m o f th e t e n s o r D ~ k , s e e R e f . [ 2. 27 ]. W e n o t e o n l y t h a t

fo r t h e e f f ec t i v e b rea k i n g o f t h e L o ren t z i n v a r i an ce , a s u f f i c i en t ly s t ro n g f ie l d ,

H N ( ~ ) 1 / 2 M o ,

is n e c e ss a r y . S u c h f ie ld s i n e x p e r i m e n t o n D W d y n a m i c s a r e ,

g e n e r a l ly , n o t a p p l ie d , t h u s , w e a s su m e f u r t h e r t h a t H = 0 .

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

n a m i c s o f t h e n o r m a l i z e d m a g n e t i z a t i o n o f a f e r r o m a g n e t ( F M ) m . T o g o

o v e r t o t h e F M L a g r a n g i a n ( 2. 6) , i t is s u f fi c ie n t t o s u b s t i t u t e l f o r r n , t o

m a k e t h e l i m i t i n g t r a n s i t i o n c 2 --~ c ~ w h i c h e x c l u d e s t h e t e r m w i t h (O1/Ot) 2


24/24

2.5 Effective Equations of the Dynamics of WFM Magnetization 27

and also to choose A1, A2 such that A1 = 0, z~2 =

(1 /gMo) (cosO0 -

cos0),

00 =

c o n s t

We shall now return to an investigation of the dynamics of WFM mag-

netization. It seems to us, that the analysis on the basis of the Lagrangian

(2.30) is not only simpler than that based on a system of equations for m

and l, but it is also more adequate. The point is, that unlike a ferromag-

net, the applicability of the Landau-Lifshitz equations for AFM sublattice

magnetizations failed, until now, to be substantiated and some authors even

predicted this inevitability. However, the equations for l, following from the

Lagrangian 2.30),

can be substantiated by means of the symmetry analy-

sis, which was made by A n d r e e v and M a r c h e n k o [2.36] (see also the papers

by

Dzya losh insk i i

and

K u k h a r e n k o

[2.40], and

B a r y a k h t a r

[2.41], where the

principle of Onsager s symmetry of kinetic coefficients was used). The essence

of their arguments will be presented in what follows.

In accordance with the standard phenomenological approach, the La-

grangian describing the dynamics of some field is constructed using the in-

variants of the field components and their derivatives. The AFM is described

by the field of the vector l; because of Eq. (2.28) the magnet ization is ex-

pressed through l and should not be regarded as the dynamic variable. What

is the vector l, from the view-point of general symmetry of AFM withou t as-

sumption on sublattices? This can be explained having emphasized that if the

mean value of the spin densi ty (s} determining the magnet ization of the sys-

tem is zero the spin system of the magnetic material should be described by

the spin density moments - dipole, quadrupole, etc. In accord with the theory

of A n d r e e v and M a r c h e n k o [2.36] the antiferromagnet (or WFM) is described

by the spin density dipole moment which is determined by the vector I. The

vector I (unlike the magnetizat ion m) necessarily changes the sign under any

symmetry transformation, specifically under the transformations that rear-

range the magnetic atoms. Thus, for the AFM in the exchange approximation

there is no invariant of the type that determines the FM dynamics (linear in

O r n / O t

and cubic in the to tal power of m and

Ore~Or) .

The main dynamic

term in the AFM Lagrangian in the exchange approximation can be only

quadratic in O1/Ot, which agrees with the Lagrangian form (2.30). The terms

linear in O l / O t can arise due to the relativistic interactions only (and also the

invariant of the Dzyaloshinskii-Moriya-type interaction). To construct these

terms it is sufficient to take into account that under all symmetry transforma-

tions the vector (1 x O1/Ot ) transforms as the magnetization m. Hence there

arise the invariants with H and Dik. The coefficients of the dynamic terms

are determined from the condition that ( m n ) = - g ~ L /5 ( O ~ / O t ) , where ~ is

the angle of the rotation around the n-ax is, was coincident with the angular

moment projection onto this axis.

02 - phenomenological theory of magnetism and classification of magnetic solitons

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