07 - Microscopic Theory of Relaxation of Domain Wall

8/20/2019 07 - Microscopic Theory of Relaxation of Domain Wall

1/17

7 M i c r o s c o p i c T h e o r y

o f R e l a x a t i o n o f D o m a i n W a l l

A s w a s r e m a r k e d a b o v e , a n i m p o r t a n t f a c t o r e f f e c t i n g t h e c h a r a c t e r o f t h e

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

M a g n e t i c r e l a x a t i o n i s d e t e r m i n e d b y t w o m a i n m e c h a n i s m s : i n t r i n s i c r e l a x -

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

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

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

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

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

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

o r t h o f e r r i t e s a n d a l s o t h e i m p u r i t y c e n t r e s o f F e + 2 , e t c . T h e i n d i c a t e d m e c h -

a n i s m s a r e o b s e r v e d e x p e r i m e n t a l l y s e e S e c t . 4 . 1 ) , t h e y a r e d i s t i n g u i s h a b l e

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

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

a n i s m c a u s e d b y C h e r e n k o v p h o n o n e m i s s i o n , c o n s i d e r e d i n C h a p . 5 , c a n b e

i m p o r t a n t , t o o .

7 1 G e n e r a l C o n s i d e r a t i o n s

T o d e s c r i b e t h e i n tr i n s ic r e l a x a t i o n p r o c e s se s tw o m a i n a p p r o a c h e s a r e u s e d:

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

q u a n t u m - m e c h a n i c a l c o n s i d e ra t io n o f t h e i n t e r a c ti o n b e t w e e n d i ff e re n t e x -

c i t a t i o n s o f t h e m a g n e t ( l in e a r a n d n o n l i n e a r o n e s ). S i n c e th e 6 0 s , th i s a p -

p r o a c h h a s b e e n r e g a r d e d a s t h e p r i n c ip a l o n e fo r l i n ea r a n d q u a s i - l i n e a r

ex c i t a t i o n s ( s ee e . g . [7 . 1 ] ) . In f ac t , n o a l t e rn a t i v e s h av e ev e r b een d i s cu s s ed .

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

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

henberg [7.2], Ba r yak htar e t a l. [7.3]. T h e m e t h o d s h a v e a c q u i r e d a n e l e g a n t

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

a l l m a g n e t s h a s b e e n w o r k e d o u t , s e e m o n o g r a p h [ 7 . 4 ] .

T h e m i c r o s c o p i c a p p r o a c h i s a d v a n t a g e o u s b e c a u s e i t m a k e s i t p o s si b le t o

f in d th e r e l a x a t i o n c h a r a c te r i s ti c d e p e n d e n c e o n t e m p e r a t u r e a n d p a r a m e t e r s

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

m e n t s . H o w e v e r , i n a p p l i c a t i o n t o n o n l i n e a r w a v e s , t h i s a p p r o a c h i s q u i t e


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7 .1 Ge nera l Cons idera t ions 97

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

v < < c . I n r e c e n t y e a r s i t h a s b e e n e l a b o r a t e d f o r t h e D W i n w e a k f e r r o m a g -

n e t s b y I v a n o v a n d S u k s t a n s k y [7.5], I v a n o v e t a l . [7.6].

T h e p h e n o m e n o l o g i c a l a p p r o a c h w a s f i rs t su g g e s t e d i n t h e c l as s ic a l p a p e r

b y L a n d a u a n d L i f s h i t z [7.7 ]. T h i s a p p r o a c h d o e s n o t c h a r a c t e r i z e t h e r e l ax -

a t i o n i n d e t a i l b u t m a k e s i t p o ss i bl e t o d e s c r i b e t h e g e n e r a l p i c t u r e o f t h e

r e l a x a t i o n o f n o n li n e a r e x c i ta t io n s . W i t h i n t h e f r a m e w o r k o f t h e p h e n o m e n o -

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

i n t r o d u c i n g a d d i t io n a l r e l a x a t io n t e r m s i n to t h e d y n a m i c e q u a t i o n s o f m o -

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

A c t u a l l y , t h e a p p r o a c h d i d n o t h a v e a l t e r n a t i v e s i n d e s c r i b i n g t h e r e l a x a t i o n

o f e s s e n t ia l l y n o n l i n e a r p e r t u r b a t i o n s , p r i m a r i ly , th e D W . H o w e v e r , m a n y a u -

t h o r s h a v e n o t e d i t s d r a w b a c k s , s e e

M a l o z e m o f f

a n d

S I o n c z e w s k i i

[7 .8] . The

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

m i n e d u s in g t h e D W m o b i l i t y a n d t h e f e r r o m a g n e t i c r e s o n a n c e li ne w i d t h ,

w h i c h d i f f e re d e ss e n t ia l l y f o r m a n y f e r r i t e s - g a r n e t s .

F r o m a t h e o r e t i c a l v i e w - p o i n t , t h e m a i n d r a w b a c k o f t h e r e l a x a ti o n t e r m

i n t h e f o r m o f L a n d a u a n d L i f s h it z ( o r t h e e q u i v a l e n t H i l b e r t f o r m ) c o n s i s ts

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

v ( k ) a t l a r g e k v a l u e s ( k > > l / A ) . S p e c if i ca l ly , u s i n g th e s e t e r m s g i v e s t h e

r e s u l t : V ~ A wk f o r t h e f e r r o m a g n e t a n d 7 ~ A cz0 f o r a w e a k f e r r o m a g n e t ,

2w h e r e a s , t h e c o r r e c t r e s u l t s a r e d i f f e r e n t: V e (

k2Wk

o( k 4 an d V e(

w k o( k 2,

r e s p e c t i v e l y , w h e n k --+ 0 s e e [7 .1 -4 ] . I n f a c t , i n d e s c r i b i n g t h e e x p e r i m e n t s

o n t h e p a r a m e t r i c e x c i t a t io n o f s h o r t - w a v e m a g n o n s , t h e p h e n o m e n o l o g i c a l

a p p r o a c h i s n o t e v e n d is c u s se d i n v i r t u e o f t h e a b o v e - m e n t i o n e d c o n s i d e r a -

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

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

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

[ 7.9 ,1 0] . I n t h e s e p a p e r s h e s u g g e s t e d a n e w f o r m o f t h e r e l a x a t i o n t e r m s

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

e x c h a n g e o r i g i n . F o r W F M w e h a v e d e s c r i b e d t h e m i n C h a p . 4 . I t h a s a l s o

b e e n s h o w n h o w t h e c r y s t a l s y m m e t r y a n d t h e h i e r a r c h y o f t h e d i ff e re n t in -

t e r a c t i o n s e f f e c t t h e s t r u c t u r e o f d i s s ip a t iv e t e r m s a n d t h e h i e r a r c h y o f t h e

c o r r e s p o n d i n g r e l a x a t i o n c o n s t a n t s . T h i s a p p r o a c h e x h i b i t s t h e r e g u l a r V ( k )

d e p e n d e n c e d u e t o a c c o u n t t a k e n o f t h e d i ss i p at iv e f u n c t i o n o f a n e x c h a n g e

o r ig i n . B u t i ts u s e r e q u i re s t h e d e t e r m i n a t i o n o f s e v e ra l r e l a x a t i o n c o n s t a n t s ,

f o r t h e W F M [7 .1 0] - t h e t h r e e o n e s A r, ~ , a n d s e w h i c h i s n o t a l w a y s p o s -

s ib le , e x p e r i m e n t a l l y . I n a d d i t i o n t o t h i s , f o r a n y p h e n o m e n o l o g i c a l t h e o r y

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

t i m e d i s p e r s i o n . T h i s is t y p i c a l b o t h f o r t h e i m p u r i t y a n d i n t ri n s ic r e l a x a t i o n .

F o r t h e i n t ri n s ic r e l a x a t i o n t h e f a c t t h a t t h e c o r r e s p o n d i n g m o d e l is c l os e t o

t h e c o m p l e t e l y i n t e g ra b l e o n e b y th e m e t h o d o f t h e i nv e rs e s c a tt e r i n g t h e o r y

p r o v e s t o b e a n e s s e n t i a l r e s t ri c t i o n o n t h e a p p l i c a b i l it y o f t h e p h e n o m e n o -

l o g i c a l t h e o r y , s e e R e f . [7 .5 ]. F i n a l ly , i t i s g e n e r a l l y o f i n t e r e s t t o c a l c u l a t e t h e


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98 7. Microscopic Theory of Relaxation of Domain Wall

DW mobility from first principles and to compare their absolute values and

temperature dependences with experiment.

This chapter deals with the results in this field. We hope the review of

theoret ical results in this field will be useful for a more adequate interpretion

of experiments on the dynamics of nonlinear excitations (magnetic solitons)

such as the DW.

7 . 2 I n t r i n s i c R e l a x a t i o n s

The problem of calculating the DW mobility in orthoferrites was raised in

the 70's by some authors (see [7.11,12]). But, actually, in these papers the

lifetime of magnons, with k = 0, was calculated. Using these data the relax-

ation constant A(T), used to calculate on the basis of (4.13), was calculated.

As it was noted in our previous review paper [7.13], a good agreement be-

tween the theoretical tempera ture dependence and the experimental one in

this approach seems to be accidental. (Below we shall present an additional

argumen tation in favour of this point of view).

We describe, schematically, the main concepts of the microscopic theory

of the intrinsic relaxation in magnets. The description of spin wave relax-

ation will proceed from an analysis of a gas of quasi-particles, incorporating

the magnon-magnon interaction. The relaxation is described as a decreasing

number of coherent magnons that form the spin wave with the given quasi-

momentum k due to the th ree- and four-magnon processes. When the DW

is retarded, we make use of the analysis of the magnon scat tering by the DW.

Let us specify the WFM. In the WFM Lagrangian (2.30') we separate

the terms quadratic in the components of the antiferromagnetism vector I.

A part o f the total Lagrangian of the WFM s containing only such terms

describes the so-called idealized WFM model whose specific properties were

mentioned in Ref. [7.5]. All the remaining terms in which the contribution of

anisotropy, non-squared in l (w4(/), w6(/)) is contained, and also the terms

with AI(0,

~) O0/Ot),

A2(0, qo)(0~/0t), will be regarded as a perturbation.

So, we write the Lagrangian as the sum of two terms:

s = s + A s ,

and take s in the form:

c~ 1 (7.1)

2 [(V0)2 + sin2 0(V~)2] - 2 (~1 s in2 0 sin 2 ~ + ~2 cos 2 0) ~ .

Here, the anisotropy energy is represented by w2 = (1/2)(Pll~ + ~2l~). We

then assume that /~2 > ~1, i.e. the wall with rotation 1 in zy-plane, (x-axis


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7.2 Int r ins ic Re laxat ions 99

i s a n e a s y a x i s ) i s s t a b l e . T h e p o l a r a x i s i s c h o s e n , h e r e , a l o n g t h e h a r d

y - a x i s . T h i s is m o r e c o n v e n ie n t w h e n t h e m a g n o n s a r e a n a l y z e d a t t h e D W

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

a n d i ro n b o r a t e w i t h a l lo w a n c e fo r t h e a n i s o t r o p y in t h e e a s y - p l a n e a n d , a ls o,

o t h e r W F M . T h u s , t h e m o d e l w i t h s = s is u n i v e rs a l e n o u g h . I t s h o u l d b e

n o t e d t h a t

A s

i s n o t s o u n i v e r s a l a n d d i f fe r s s i g n i f ican t l y , e .g . fo r t h e rh o m b i c

a n d r h o m b o h e dr o n W F M .

I t i s e a s y t o c o n v i n c e o n e s e lf t h a t t h e p r i n c i p a l d i ff e r e nc e o f t h e m o d e l s

w i t h s = s a n d w i t h

s = s + A s

i s p r e s en t i n a l mo s t a l l a s p ec t s o f

t h e p r o b l e m [ 7 .5 ]. T h u s , t h e s e m o d e l s w il l b e c o n s i d e r e d i n d i v i d u a l l y c a ll -

i n g t h e m i d e a li z e d ( fo r s = s a n d g e n e r a l iz e d ( w i t h a l lo w a n c e f o r

A s r e sp e c ti v el y . A n a l y z i n g th e i d ea li z ed m o d e l i s i m p o r t a n t b o t h f r o m t h e

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

A s

a re , g en e ra l l y , s m a l l (w 4 ,6 ~ w a an d

( d w o / g ~ o )

~ / ~ , w h e r e ~ 0 is t h e

m a g n o n f re q u e nc y ).

I n t e r m s o f t h e a n a l y s i s o f s m a l l o sc i ll a ti o n s o f t h e m a g n e t i z a t i o n a t t h e

b a c k g r o u n d o f t h e g r o u n d s t a t e ( h o m o g e n e o u s s t a t e w i t h 0 = 00, ~ = ~ 0 o r

i n h o m o g e n e o u s s t a t e t h a t i n c lu d e s t h e D W ) , w e w r i te t h e a n g u l a r v a ri a bl e s

fo r t h e v ec t o r l a s :

0 = e o + ~ ( r , t ) , ~ = ~ o + r , (7 .2 )

w h e r e 0o a n d ~ o c o r r e s p o n d t o t h e g r o u n d s t a t e ( in t h e p r e se n c e o f D W

0o = 0 o (~ ) , ~o = ~o (~ ) , ~ = x - v t ) . O n ex p a n d i n g t h e L a g ra n g i an s (7.1 )

i n p o w er s o f s m a l l v a l u es ~) an d r w e w r i t e i t a s:

s : s ~-s ~-s Jr s

w h er e s =- 0 i n v i r t u e o f t h e e q u a t i o n s o f m o t i o n , s ee [7 .5 ]. s i n v o lv es t h e

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

T h e t r a n s i t i o n t o t h e m a g n o n c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s c a n

b e p e r f o r m e d u s i n g t h e q u as i- -c la ss ic a l q u a n t i z a t i o n o f t h e W F M L a g r a n g i a n

( 2 .3 0 t) . T o t h i s e n d , i t is n e c e s s a r y t o i n t r o d u c e , f i r st ly , t h e c a n o n i c m o m e n t a

Po a n d P v c o r r e s p o n d i n g t o t h e f i e ld s o f a n g u l a r v a r i a b le s 0 a n d ~ ,

Po

= m-0--~, p~ = m sin 2 0

H e r e , c ~ M ~ / c 2 i s d e n o t e d b y m a n d a s i m p l e s t ( L o r e n t z - i n v a r i a n t ) v e r s i o n o f

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

A I ( O , ~ )

a n d A 2 ( 0, ~ ) a r e ta k e n i n t o a c c o u n t . T h e a b o v e m a g n i t u d e s a r e t h e n c a l le d

t h e f ie ld o p e r a t o r s w i t h t h e u s u a l c o m m u t a t i o n r e la t io n s :

[O ( r , t ) ,

p e ( r ' , t ) ] = i h S ( r -

r ' ) , [ ~ ( r , t ) , p ~ ( r t , t) ] - - - i h e ( r

- r ' ) ,

[ o = o o = ; o ] = [ o = o

a n d t h e W F M H a m i l t o n i a n is c o n s t r u c t e d a s t h e s a m e e x p a n s i o n in p o w e r s

Po

a n d O ; p ~ a n d ~ . I n a s i m p l e s t i d e a l iz e d m o d e l , t h e m a g n o n d y n a m i c s


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100 7 . Microscopic Th eor y of Re laxat ion of Dom ain W al l

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

H a m i l t o n i a n :

H = H 2 + H 4 + . . . .

H e r e H 2 i s a t w o - m a g n o n H a m i l t o n i a n i n t h e s t a n d a r d c a n o n i c a l f o r m :

{12

2 = d r ~ p 0 + P ~)

~

~ [r + ~ 2 V r + 1 + a)~)2 + A 2 V ~ ) 2 ]

7.3)

F o r c o n v e n i e n c e w e u se , h er e , t h e q u a n t i t y A - t h e s t a b l e D W t h i c k n e s s ,

w h i c h is a p a r a m e t e r w i t h t h e d i m e n s i o n s o f l e n g t h ; t h e m a g n i t u d e a = /3 2 -

/~ 1)/~ 1 d e t e r m i n e s t h e a n i s o t r o p y in t h e b a s a l p l a n e , a n d

w o = g M o v / - ~ / 2 .

I n t r o d u c i n g s ee , e .g ., [ 7. 5] ) t h e m a g n o n c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s

w i th t he m o m e n t u m

h k ,

v~= E i 2 m ~ k v a k + a + - k )e x p i k r )

E ~ / h A k + g + k ) e x p i k r )

r = 2 m ~ k V

~ ~. ~/-~Wk +

PO = 2 ._ .,1 V ~ , a k - a - k ) e x p i k r )

E i hm f 2 k A + e x p i k r )

~ = ~ - A - k )

w h e r e V is t h e m a g n e t v o lu m e ,

m a g n o n s i n t h e d i a g o n a l f or m :

H = ~ ~ k a ~a k h ~ kA~A k

k

7.4)

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

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

w k = ~ / w 2 + c 2 k 2 , ~ k = ~ / f 2 ~ c 2 k 2 , ~ = w ~ ( l + a ) ,

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

o s c i l l a t i o n s o f t h e a n g l e r ( o - m a g n o n s ) a n d a n g l e ~ ( 0 - m a g n o n s ) . T o t h e s e

t w o b r a n c h e s c o r r e s p o n d 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 p e r a t o r s a + , a k a n d

A ~ , A k , r e s p e c t i v e l y . T h e f o r m u l a e f o r m a g n o n f r e q u e n c i e s , o b t a i n e d u n d e r

t h e q u a s i - c l a s s i c a l q u a n t i z a t i o n , c o i n c i d e n a t u r a l l y w i t h f o r m u l a e ( 2 . 3 2 ) t h a t

f o l l o w f r o m t h e c l a s s i c a l e q u a t i o n s o f m o t i o n , w 1 ( k ) - - w k , w 2 ( k ) - - ~ k .


6/17

7.2 Intrinsic Relaxations 101

The difference in classification of the magnon branches in this and previous

chapters is due to the choice of the polar axis).

The Hami ltonian ,//2 , describes the ideal two-component gas of magnons

at th e background of the homogeneous ground state magnons at the DW

background will be discussed below). For an idealized model and homoge-

neous ground st ate 123 = 0, H3 = 0 and the magnon interactions are de-

scribed by the four-magnon Hamilto nian/ /4. The latter describes many pro-

cesses with participation of four 0-magnons, four ~-magnons and also two

O-magnons and two qo-magnons. We give a more compact formula for/-/4 in

terms of the field operators 0,

Po

and ~, p~:

f

4 = MO d r c-~m2p~v~ -

ar162 2

( 7 . 5 )

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

g e n e o u s i n t e r a c t i o n s a n d t h e l a s t t w o ( i n s q u a r e b r a c k e t s ) - t h e r e l a t i v i s t i c

i n t e r a c t i o n s , i n t h e g i v e n c a s e - t h e a n i s o t r o p y e n e r g y .

B e y o n d t h e i d e a l i z e d m o d e l , i . e . w h e n A / 2 i s t a k e n i n t o a c c o u n t , t h e t h r e e -

m a g n o n t e r m s / / 3 m a y a r i s e . T h e y a p p e a r d u e t o t h e t e r m s w i t h A I ( 0 , ~ ) a n d

A 2 ( 0 , ~ ) , a n d , a l s o , d u e t o t h e r h o m b o h e d r o n a n i s o t r o p y . T h e i r a m p l i t u d e s

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

a c c o u n t i n / 2 o a n d s i g n i f i c a n t i n H 4 .

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

t o n i a n : H = H 0 + H i n t , H i n t = / / 3 + H 4 - F u r t h e r a n a l y s i s o f t h e r e l a x a t i o n i s

d o n e i n a s t a n d a r d w a y u s i n g m a n y - b o d y t h e o r y m e t h o d s , w e l l e l a b o r a t e d ,

s e e [ 7 . 1 , 4 ] . L e t u s d i s c u s s w i t h a n i l l u s t r a t i v e p i c t u r e o f t h e r e l a x a t i o n .

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

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

a n i n t e r a c t i o n w i t h t h e r m a l m a g n o n s ( i n w h a t f o l l o w s w e p r e s e r v e P l a n c k s

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

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

w i t h a t h e r m a l m a g n o n , w i t h t h e f o r m a t i o n o f o n e m a g n o n ; a n d t h e d e c a y

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

scribe relaxation due to the coherent magnon scattering by a thermal one,

see Fig. 7.1

The contribution of the processes with participation of a greater number

of magnons has an additional small temperature multiplier:

T / T N T N

being

the NSel temperature. But the account taken of the four-magnon processes,

equally, with the t hree-magnon process, is necessary, since the t hree-magn on

Hamiltonian can be nonzero only when the small terms A/2 are taken into

account, hence, its contribution can be small and sometimes it is simply

equal to zero).


7/17

102 7 . Microscop ic Th eo ry o f Rela xa t ion o f Do main Wal l

I i

2 2 2

a) b) c)

F i g . 7 . l a - c T h r e e - m a g n o n ( a , b ) a n d f o u r - m a g n o n ( c ) pr o ce s se s t h a t c o n t r i b u te t o

th e s p in wav e r e l ax a ti o n . T h e s o li d li ne d en o te s a t h e rm a l m ag n o n an d t h e d as h ed

l in e d en o t e s a co h e ren t m ag n o n . 1 = k l , . . . , k l ,2 ,3 a re m ag n o n m o m e n ta

W e s h ow t h e r e su l ts o f t h e c a l c u la t io n o f t h e m a g n o n d a m p i n g d e c r e m e n t

o f t h e l o w e r b r a n c h , 7 ( k ) , w h i c h is d u e t o t h e c o n t r i b u t i o n o f / / 4 ( it is t h e

m a i n c o n t r i b u t o r i n r e a l m a g n e t s w h e n T > c 0, w h e r e e0 = h a;0 is t h e m a g n o n

a c t i v a t i o n e n e r g y ) . T h e e x p r e s s i o n f o r 7 ( k ) a t ~ r ~ 1 a n d T >> ~0 is w r i t t e n

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

= 2 7 t .3 Z ] 6 ~ l M o ) 2 k z ~ ) 2 T 3

~0 In T

a 0

2 T 3 m ] 7 . 6 )

c~ - - In - - + T 2

+ aJ--~ ~o e0 ~

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

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

i s o f a r e l a t i v i s t i c n a t u r e . I n t h e l o n g w a v e l e n g t h l i m i t , k --+ 0 , t h e d o m i n a n t

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

t i o n 7 e~; a t l a r ge k v a lu e s t h e m a i n c o n t r i b u t i o n i s d e t e r m i n e d b y t h e s u m o f

t w o e x c h a n g e t e r m s , 7 e a n d y~. T h e c o n t r i b u t i o n o f 7 r is sm a l l a t T >> e 0;

i t is w r i t t e n o u t f o r f u r th e r c o m p a r i so n w i t h t h e c o r r e s p o n d i n g p h e n o m e n o -

I og ic r e s u lt s . T h u s , i n t h e i d e a l iz e d W F M m o d e l , t h e m a i n c o n t r i b u t i o n t o

t h e a t t e n u a t i o n o f b o t h t h e l o n g a n d s h o r t w a v e l e ng t h m a g n o n s is g i ve n b y

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

B a r y a k h t a r e t a l. [7.3].

I t f o ll ow s f r o m t h e a n a l ys i s o f t h e e x p r e s s io n s f o r t h e d a m p i n g d e c r e m e n t

t h a t i t s c h a r a c t e r i s t i c t e m p e r a t u r e d e p e n d e n c e is ~ /e ~ T 3 , i .e . i t i s n o t s u c h

a s i n a f e r r o m a g n e t ( % e c

k 4 T 2 ,

see [7 .1]) or in calculat ion [7 .12].

S o m e c o n t r i b u t i o n s t o ~ , c a n e a s i l y b e c o m p a r e d w i t h t h e d i f f e r e n t t e r m s

i n t h e d i s s i p a t i v e f u n c t i o n s ( 4 . 5 ) , ( 4 . 6 ) .

I f w e c o m p a r e t h e r e s u lt s , i t b e c o m e s c le a r t h a t % c o r r e s p o n d s t o t h e

c o n s t a n t ) ' r , % a n d 3'~ t o t h e t w o t e r m s i n t h e e x c h a n g e d i s s i p a t i v e f u n c t i o n .

T h e c o m p a r i s o n o f t h e r e s u l ts f o r m a g n o n a t t e n u a t i o n y ie l ds


8/17

7.2 Intr ins ic Re laxa t ions 103

l n

1 - ~ ) 1 / 2 1 ~ ) 2

_ _ T i n T

7 . 7 )

w h e r e T . = / ~ I M ~ A 3 . T h i s e n e r g y p a r a m e t e r i s e q u a l, n u m e r ic a ll y , t o t h e D W

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

A 2 / 2 .

U s in g t h e u su a l v a lu e s i n W F M - t y p e o r t h o f e r -

r i te s ) f o r t h e D W e n e r g y, o 0 N 1 e r g / c m 2 a n d t h e D W t h i c k n e ss A ~ 1 0 - 6 c m ,

w e g e t t h a t T . c o r r e s p o n d s to t h e e n e r g y o f t h e o r d e r o f N 1 0 - * 2 e r g , o r

t h e t e m p e r a t u r e ~ 10 4 K . T h i s v a l u e is l a rg e , n o t o n l y a s c o m p a r e d t o ~ o

e0 ~ 1 0 + 2 0 K f o r o r t h o f e r r i te s ) , b u t a ls o t o t h e N 4 e l t e m p e r a t u r e v a l u e TN

T N is o f t h e o r d e r o f t h e e x c h a n g e i n t e g r a l I ~ 1 0a K f o r o r t h o f e r r i t e s ) , a n d

T . ~ 5 / /3 , ) 1 /2 T N ~ 1 0 I ) . F o r i r o n b o r a t e , t h e v a l u e o f T . is la r g e r t h a n f o r

o r t h o f e r r i t e s . I n e s s e n c e ,

T / T . )

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

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

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

a n a l y z e th i s p r o b l e m i t is n e c e ss a r y t o c o n s t r u c t t h e m a g n o n H a m i l t o n i a n a t

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

e x p l i c it l y o n t im e . A m o n g t h e m t h e r e m a y b e t h e t w o - m a g n o n o n e s , o f t h e

t y p e :

E

~ 12 ex p [ i k lx k2x)v t]a+la2, g r~ 2 x p [ i k l z

k 2 x v t ] a + A 2 ,

1,2

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

h i l a ti o n o p e r a t o r s o f ~ - o r 0 - m a g n o n s . T h e t h r e e - m a g n o n t e r m s a r e p r o -

p o r t i o n a l t o e x p [ i k l x - h2 x - k 3 x )v t ], w h e r e

k l

k2,

o r

k 3 a r e t h e m o m e n t a

o f m a g n o n s p a r t i c i p a t in g i n t h e p r o c es s. T h e m e t h o d o f c o n s t r u c t i n g t h e

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

s h a l l n o t c o n s i d e r i t h e r e , s ee t h e o r i g i n a l p a p e r [7 .5 ]. T h e c h a r a c t e r o f t h e

t i m e d e p e n d e n c e i s e a si ly u n d e r s t o o d b y a n a l o g y w i t h a s i m il ar d e p e n d e n c e

f o r t h e H a m i l t o n i a n o f p h o n o n e m i s s io n , s ee E q . 5 .7 ) . A s in t h e c a s e o f

p h o n o n e m i s s io n , t h e f o l lo w i n g h o ld s : 1 ) i n e l e m e n t a r y p r o c e ss e s , m o m e n t u m

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

w h i c h is a l o n g t h e x - a x i s ; 2 ) e q u a ll y , w i t h t h e m o m e n t u m t r a n s f e r q = q e x

i t t r a n s m i t s a n e n e r g y

q v

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

t h e F o u r i e r - c o m p o n e n t o f i n h o m o g e n ei ty , c a us e d b y t h e D W . T h u s , t h e w a ll

e f f e c t c a n b e d e s c r i b e d a s t h e a c t i o n o f a n e x t e r n a l f i e ld in v o l v in g a s e t o f

h a r m o n i c s o f t h e f o rm : e x p [ i h q x - v t) ]. I n th i s c a se , u n d e r t h e u s u a l d i a g r a m

r e p r e s e n t a t i o n o f t h e a c t io n o f t h e e x t e r n a l f ie ld t h e e l e m e n t a r y t w o - a n d

t h r e e - m a g n o n p r o c e s s e s a re s h o w n b y t h e g r a p h s i n F i g. 7 .2 .


9/17

104 7 . Microscop ic Th eor y o f Relax a t ion o f Do main W al l

1

q

2

q 3

b

Fig . 7 . 2 T w o - an d t h ree -m ag n o n p ro ces se s t h a t c o n t r i b u t e to t h e d o m a in wa l l

d am p in g . T h e s o li d li ne d en o t e s t h e t h e rm a l m ag n o n ; t h e ex t e rn a l f ie ld co r re s p o n d -

ing to the dom ain wal l i s deno ted by a dashe d l ine . 1 --- k l , . 9 k l,2 ,3 a re ma gnon

m o m e n ta , q is a Fo u r i e r co m p o n en t o f th e d o m a in wa l l f ie ld

C o m p a r i n g F i g s. 7 .1 a n d 7 .2 ( w i t h s u b s t i t u t i o n o f t h e l i n e d e s c ri b i n g a

c o h e r e n t m a g n o n f o r t h e l in e t h a t d e s c r ib e s t h e w a l l, m o r e e x a c t ly , i ts F o u r i e r

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

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

a n d o n e c a n f o r m u l a t e t h e f o l lo w i n g r ul e: t h e n - m a g n o n p r o c e s s e s i n w a l l

r e t a r d a t i o n c o r r e s p o n d to ( n + 1 ) - m a g n o n o n e s i n t h e m a g n o n r e l a x a ti o n .

T h i s c o r r e s p o n d e n c e c a n r e s u l t in a c o m p l e t e c o i n c id e n c e o f t h e r e s u l ts ( it

w i ll b e d i s c u s s e d b e l ow i n w h a t s e ns e t h e m a g n o n d a m p i n g d e c r e m e n t 7 k a n d

t h e w a l l v i s c o s it y co e f fi c ie n t r / c a n b e c o m p a r e d ) .

I t m a y h a p p e n t h a t t h e r e s u l t s o f t h e c a l c u l a t i o n o f 7 k a n d ~ a r e n o t

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

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

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

t h e f r e q u e n c y o f t h e e x t e r n a l f ie ld . E v i d e n t l y , fo r a s p in w a v e , t h e f r e q u e n c y

i s l a r g e e n o u g h ( a = cvk > ~ 0 , w 0 is t h e g a p i n t h e s p i n w a v e s p e c t r u m ) a n d

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

v - - * 0 . T h e s e c o n d r e a s o n i s m o r e r e f i n e d a n d i s a s s o c i a t e d w i t h t h e f a c t

t h a t t h e m a g n e t i c m o d e l s f o r W F M , i n t h e o n e - d i m e n s i o n a l c a se , a r e c lo s e

t o t h e S i n e - G o r d o n m o d e l, w h i ch is e x a c t l y i n te g r a b le b y t h e m e t h o d o f

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

is m a n i f e s t e d i n a d if f e re n t w a y i n t h e D W r e t a r d a t i o n , w h i c h i s d u e t o t h e

t w o - a n d t h r e e - m a g n o n p r o ce s se s an d , a ls o, m a k e s t h e m a g n e t i c i n te r a c ti o n s ,

e v e n t h e w e a k o n e s w h i c h b r e a k t h e e x a c t i n t e g r a b i l i t y o f t h e r e l e v a n t o n e- -

d i m e n s i o n a l m o d e l s , t h e d o m i n a n t o n e s. ( I n t h e c a l c u l a t i o n g i v en be l o w , th i s

c o r r e s p o n d s t o s e t t i n g A t ; e q u a l t o L ; o ) .

A f t e r t h e s e g e n e r a l r e m a r k s , w e g i v e a sc h e m e o f t h e s p ec i fi c c a l c u l a t i o n

o f t h e D W v i s c o s i t y 7. S i n c e t h e r e s u l t s f o r t h e i d e a l i z e d m o d e l w i t h 12 = 120

a n d t h e m o d e l w i t h a l l o w a n c e fo r A t ; a r e d if f e re n t , i n p r in c i p le , t h e y s h o u l d

b e c o n s i d e r e d i n di -c id u a ll y .


10/17

7.2 Int r ins ic Re laxat ions 105

T h e I d e a l i z e d M o d e l . T h e t w o - m a g n o n H a m i lt o n ia n d if fe rs a t s = s

f r o m E q . ( 7. 3) b y t h e t e r m i n s q u a r e b r a c k e t s . I t c a n b e t r a n s f o r m e d a s :

2

= - A 2 V 2 + 1 2 ~ = x - v t

c o s h 2 ( ~ / A ) '

T h e S c h r 5 d i n g e r o p e r a t o r L , w i t h r e f le c t io n l e ss p o t e n t i a l , h a s a w e l l - k n o w n

s e t o f e i g e n f u n c t i o n s . T h i s s e t i n vo l ve s t h e l o c a li z e d s t a t e :

1 ex p ( i ~ r ) , ( 7 .8 a)

= = v cosh /A)

w h e r e S is t h e D W a r e a , ~ = (0 , ~ y , ~ z ) w h i c h d e s c r ib e s t h e w a v e p r o p a g a t i n g

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

L fk = (1 + A 2 k 2 ) f k , f k = t a n h ( ~ / A ) - i k x A ex p( ik r ) , (7.8b)

+ k 2)v

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

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

m a g n o n s .

F o r a c o m p l e t e se t o f s t a t e s { f ~ , fk } w e c o m p a r e t h e m a g n o n c r e a t i o n a n d

an n i h i l a t i o n o p e ra t o r s an , a +'~, ak , f o r 9 - m ag n o n s an d A ~ , A +'~, A k , A + fo r

0 - m a g n o n s ( t h e f ie ld o p e r a t o r s a r e o b t a i n e d t h e n f r o m ( 7.4 ) b y s u b s t i t u t i o n

o f } - ]k ( ') f o r ~ ( - ) + ~ -]k( ') a n d a l s o o f t h e ex p o n e n t i a l e x p ( i k r ) / v / - ] fo r fn

o r f k , r e s p e c ti v e ly ) . I n t e r m s o f t h e o p e r a t o r s t h e t w o - m a g n o n H a m i l t o n i a n

a t t h e b a c k g r o u n d o f t h e D W t a k es , i n t h e c a se o f t h e i d e a li ze d m o d e l , t h e

d i a g o n a l f o r m :

T h e f re q u e n c ie s o f i n t r a d o r n a i n m a g n o n s , w k a n d / 2 k , a r e t h e s a m e a s in

t h e h o m o g e n e o u s c a se . T h e f r e q u e n c y w ~ =

l ~ {

d e s c r i b e s t h e b e n d i n g D W

o s c i ll a t io n s . T o t h e l o c a l iz e d s t a t e o f 0 - m a g n o n s c o r r e s p o n d s t h e f r e q u e n c y :

= 2 2k2, = - ,

a n d s p i n o s c i l l a t i o n s i n t h e D W , w h e n t h e l a t t e r i s n o t d i s p l a c e d . J u s t t h i s

m o d e d e s c r i b e s t h e l os s i n t h e D W s t a b i l i t y w h e n / ~ 1 > / 3 2 , s e e C h a p . 2 .

S o , t h e t w o - m a g n o n H a m i l t o n i a n o f t h e i d ea l iz e d m o d e l is d ia g o n a l a n d

g iv es n o D W r e la x a ti o n . T h e l a t te r c a n o n l y b e d u e to t h e t h r e e - m a g n o n

t e r m ( t h i s w a s f i r s t n o t e d b y A b y z o v a n d I v a n o v {7.1 4] fo r a f e r ro m ag n e t ) .

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

t e r m s , w e r e m a r k t h e f o ll o w in g , g i v e n b el ow .


11/17

106 7 . Microscopic Th eo ry of Re laxat ion of Do m ain W al l

T h e q u a d r a t i c H a m i l t o n i a n a t th e b a c k g r o u n d o f t h e i m m o b i l e w a l l c a n

b e d i a g o n a l i z e d f o r a n y m o d e l o f t h e n o n l i n e a r f ie ld , i n p a r t i c u l a r , f o r a n

a r b i t r a r y W F M . I t i s n e c e s s a ry to s o lv e, t h e n , a m o r e c o m p l i c a te d p r o b l e m

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

i n p r in c i p l e , p o ss ib l e. A s f o r d i a g o n a l i z in g t h e m a g n o n H a m i l t o n i a n a t t h e

b a c k g r o u n d o f t h e m o v i n g d o m a i n w a l l: t h i s i s p o s s ib l e fo r m o d e l s s u c h a s a n

i d e a l iz e d o n e . T h e t o t a l d i a g o n a l i z a t i o n , i .e ., t h e i n t r o d u c t i o n o f m a g n o n s a t

t h e b a c k g r o u n d , t h e m o v i n g w a l l n o t i n t e r a c t in g w i t h t h e m a g n o n s , is p o ss ib l e

f o r e x a c t l y i n t e g r a b l e S i n e - G o r d o n - t y p e m o d e l s o n l y . H 2 i s d i a g o n a t i z a b l e

b e c a u s e t h e m o d e l o f W F M , w i t h Z; = / 2 0 , is clo se t o t h e S i n e - G o r d o n m o d e l .

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

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

Ba r yakh tar e t a l .

[7 .15 ] , an d b y

Ogata

a n d

Wada

[7 .16] .

Zakharov

a n d

Schu lman

[7 .1 7 ] co n s i d e red a p o s s i b i l i t y o f

i n e l a s t i c p ro ces s e s i n ex ac t l y i n t eg rab l e mo d e l s .

W e c o m e b a c k t o a n a l y z i n g t h e D W r e t a r d a t i o n i n t h e i d ea l iz e d m o d e l .

S i n c e t h e t w o - m a g n o n H a m i l t o n i a n i s d i a g o n a l , t h e c o n t r i b u t i o n t o r e t a r d -

a t i o n c o m e s o n l y f r o m t h r e e - m a g n o n p r oc e ss e s d e sc r ib e d b y H ~ 3) . T h e d a m p -

i n g fo rce F3 =

- ( 1 / v ) ( d E / d t ) ,

w h e r e

( d E / d r )

is th e D W e n e r g y d i s s ip a t io n

r a t e , is d e t e r m i n e d b y t h e p r o b a b i l i t y o f t h e c o r r e s p o n d i n g p r o c e ss . A n a l y s i s

r e v e a l s p ro c e s s e s w i t h p a r t i c i p a t i o n o f b o t h o f t h e 0 a n d qo m a g n o n s , t h e

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

t h e co e f f i c i en t o f v i s co s i t y ~ g i v es t h e ex p re s s i o n [7 .5 ]:

r ] 3 - ~ l M g A 6 c

f l ( ~ ) + f 2 ( a ) ~00 + f 3 ( ~ ) In , (7 .9 )

w h e r e t h e f u n c t i o n s f l a n d f 3 a r e w e a k l y d e p e n d e n t o n or,

~5 .6 , a ~ 2 ~0 .9 , ~ ~ 2

f 1 = 1 0 - 2 1 4 . 0 , o r > > 1 f 3 ~ 1 0 - 3 [ 0 . 4 , ~ > > 1

a n d f 2 ~ 2 . 1 0 - 2 a t c r ~_ 2 , a n d d e c r e a s e s e x p o n e n t i a l l y w h e n a >> 1 . T h e

v a l u e o f (r = 2 .0 5 , c h o s e n fo r t h e c a l c u l a t io n s , c o r r e s p o n d s t o t h a t o f y t t r i u m

o r t h o fe r r i t e ; f o r t h e i ro n b o ra t e , cr >> 1 .

I t is e a si ly se e n t h a t t h e f o r m u l a h a s t h e s a m e p o w e r s o f t e m p e r a t u r e

a s t h a t f o r t h e m a g n o n d a m p i n g d e c r e m e n t ( 7 .6 ). T h u s , w e m a y h o p e t o

m a k e t h e r e s u lt s o f th e ~ ( k ) a n d r~ c a l c u la t io n , w i t h i n t h e f r a m e w o r k o f a

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

t h e a p p l i c a b i l i ty o f t h e d i s s ip a t i v e f u n c t i o n s ( 4 .5 ,6 ) f o r t h e i d e a l iz e d m o d e l .

T h e t e r m w i t h A~ g iv e s n o c o n t r i b u t io n t o t h e D W m o b i li ty , a n d t h e

m i c r o s c o p i c c a l c u l a t i o n o f th e f r i c ti o n a l f or c e a t n o n s m a l l v e l o c it ie s f a i le d

t o b e d o n e . T h u s , i t i s p o s s i b l e t o co mp are t h e co n s t an t s )~ r an d ) , e . I t i s

n a t u r a l t o a s s u m e t h a t t h e t e r m s i n 7 . 9 ) , w h i c h a r e p r o p o r t i o n a l t o T 2 , c a n

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

w i t h t h e c o n s t a n t h r . O n t h e c o n t r a r y , t h e t e r m s , p r o p o r t i o n a l t o T 3 a n d


12/17

7.2 Intrinsic Relaxations 107

T 3 In T should be compared with the exchange relaxation constant ;~e up to

a logarithmic multiplier (assuming inT/so ~- 3).

Using the above mentioned, we can get the data for the two relaxation

constants from two different microscopic formulae: for the DW retardation

and magnon attenuation. Having performed this analysis Ivanov and Suk-

stansky [7.5] obtained for ~r and ~e expressions that differ from one another

(up to (7.7)) by numerical multipliers only. This difference is small and does

not exceed 10 , which validates, to a certain degree of adequacy, the phe-

nomenological approach based on the dissipative function Q = Qr + Qe, given

in Chap. 4. The absence of an exact coincidence of temperatu re behavionr of

he in (7.7) and (7.9), and also the difference (even small, up to 10~0) in the

numerical coefficients, does not contradict the phenomenological approach

and can be associated with the fact that in the dissipative function Q for

each type of terms (relativistic and exchange), only invariants with minimum

possible powers of the components of the vector l are taken into account. In

particular, the relativistic term in Q for the rhombic AFM can involve the

invariant t 2 ~

2

r l z [ ( l • ) y ] , t h a t g i v e s n o c o n t r i b u t i o n t o t h e l i n e a r s p i n w a v e

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

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

e n o u g h . T h i s c o n f i r m s t h e a s s u m p t i o n m a d e b y V . G . B a r y a k h t a r i n [ 7 . 1 0 ]

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

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

T h u s , f o r t h e i d e a l i z e d W F M m o d e l t h e d a t a o n t h e m i c r o s c o p i c c a l c u -

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

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

B e c a u s e o f t h e a b o v e m e n t i o n e d , t h i s i s a n o n t r i v i a l f a c t . I n a n a l y z i n g t h e

g e n e r a l i z e d W F M m o d e l w e r e a l i z e t h a t t h i s a g r e e m e n t p r o v e s t o b e t h e

e x c e p t i o n r a t h e r t h a n t h e r u l e .

Th e Ge ne ra li ze d W F M Mo de l contains a nondiagonal addition to the

quadratic Hamiltonian H2, H2 = Ho AH. This addition is determined by

two factors: a part of anisotropy energy, nonquadratic in l~, results in the

term AHa, and a nonantisymmetric Dzyaloshinskii-Moriya interaction - to

the term AHD, and A H = AHa AHo. Let us discuss the contributions of

these terms in ~/(k) and 7.

Any account of A H violates, generally; the reflectionless of the poten-

tial in the two-magnon Hamiltonian. Thus, the contribution of two-magnon

processes such as 0 and ~-magnon scattering arises in the DW viscosity; for

all types of anisotropy which are relevant both for the rhombic and rhom-

bohedron WFM, this contribution is determined by the universal formula

[ 7 . 5 , 6 ] :

7 . 1 o )


13/17


where Aa is a numerical multiplier of the order of 10 -1 + 10 (for the contri-

bution w4 in iron borate Aa = 1.0), Ha and H~ are, respectively, the fields

of quadratic and nonquadratic anisotropy. We emphasize that these types

of processes, the temperature dependence of ~a e( T, and the quadratic de-

pendence on the constant of nonquadratic anisotropy are universal both for

ferromagnets [7.18,19] and for all weak ferromagnets, see [7.5,6].

According to the theory presented above, the contribution of two-magnon

processes to the DW viscosity should be compared to that of three-magnon

ones to the magnon damping decrement %(k). Here, there is no such univer-

sality: the terms w4(/) in the anisotropy energy give three-magnon processes

in the rhombohedron WFM, but do not give these processes in rhombic ones.

Here, one of the above indicated reasons of inadequacy of the phenomeno-

logical theory which is associated with the breaking of a hidden symme-

try of the idealized WFM model, determined by the similarity between this

model and the exactly integrable Sine-Gordon one, is manifest explicitly (see

[7.14,15,19]). It is clear that the contributions of A H a to r/a and 7(k) cannot

be described by any universal phenomenologic dissipative function taking no

account of this hidden symmetry (how this hidden symmetry should be

taken into account in the relaxation theory is not known so far).

For the contribution A H D to the dissipation of magnetic excitations, the

situation is different. This contribution is nonzero, but not for all DW, while

is meaningless for the ferromagnets, moreover, it is not nonzero for all DW in

weak ferromagnets. Among the walls concerned, this contribution is nonzero

only for the ac--wall in orthoferrite (i.e., just that wall, which is observed at

room temperature), but is zero for the ab-wall (see [7.5]). But AHD practi-

cally always gives a contribu tion to the magnon dissipation due to the three-

magnon processes.

For the viscosity coefficient ~]D, caused by AHD, one gets the formula:

(7.11)

where AD ~-- 10 -2, d is the component in the tensor

Dik

that induces the

breaking of the Lorentz-invariance (for the orthoferrite this is a constant at

the invariant

mxly + myIx),

see Chap. 2). The squared temperature depen-

dence is also universal for the given processes: when their contribution is

nonzero it is caused by the terms of the type 9~Ov~/cgtor v~O~/Ot, i.e., Po~,

Op~ in the Hamiltonian terms. When there is such a structure, then a quite

definite amplitude dependence on the frequencies of magnons, a~l and /21,

is realized, which just causes the temperature dependence ~?D. But the tem-

pera ture dependence of the corresponding contribution to y is different: for

orthoferrites ~'D O T, so that ~D ~ 0 only when cr > 3. Evidently, this data

cannot be made consistent at either choice of the dissipative function. In the

given case the inadequacy of the phenomenological description is associated,

apparently, with the availability of the strong time dispersion.


14/17

7.2 In tr ins ic Re laxat io ns 109

L e t u s p r e s e n t t h e n u m e r i c al e s t im a t e s o f a ll c o n t ri b u t io n s - t h e t h r e e -

m a g n o n 7 (3) ( 7 .9 ) a n d t w o - m a g n o n o n e s 7 (2) = Va + ~ D. W e f i rs t e s t i m a t e

@ 3 ). F o r y t t r i u m o r t h o f e r r i t e : e 0 ~ 1 5 K , A _~ 1 0 - 6 c m , c = 2 9 1 0 6 c m / s a n d

~ lM 0 2 _~ 1 0 6 e r g / c m 3. W i t h a l l o w a n c e t h e s e v a l u e s , i t t u r n s o u t t h a t i n t h e

t e m p e r a t u r e r a n g e T > 1 00 K t h e m a j o r c o n t r i b u t i o n t o 7 (3) c o m e s f r o m t h e

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

7 (3) = 2 . 6 . 1 0 - 4 ( T / 3 0 0 K ) 3 ( 7. 12 )

( h e r e a n d f u r t h e r t h e e s t i m a t e s o f ~ a r e in d i n . s / c m a ) . T h i s v a l u e, a t T N

3 0 0 K , i s m u c h s m a l l e r t h a n t h a t o b s e r v e d in e x p e r i m e n t s f o r t h e a c - w a l l o f

y t t r i u m o r t h o f e r r i t e . B e s id e s t h a t , i t h a s a d i f fe r e n t t e m p e r a t u r e d e p e n d e n c e :

~ e x p

X T 2 .

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

e x p e r i m e n t i s g i v e n b y t h e v a l u e o f t h e o r d e r o f ~ D . E q u a t i o n 7 . 1 2 ) s h o u l d

d e s c r i b e t h e r e t a r d a t i o n o f t h e o r t h o f e r r i t e a b - w a l l f o r w h i c h ~ D ---- b u t ,

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

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

a n d t h e c o n t r i b u t i o n o f @ 3 ) i s n e g l i g i b l y s m a l l a s c o m p a r e d t o t h a t o b s e r v e d

e x p e r i m e n t a l l y . H e n c e , i t i s n e c e s s a r y t o l o o k f o r o t h e r c o n t r i b u t i o n s t o r e -

l a x a t i o n .

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

g o o d a r g u m e n t i s o b t a i n e d w h e n @ 2 ) i s t a k e n i n t o a c c o u n t . F o r a n o r t h o f e r -

r i t e , w i t h a l l o w a n c e f o r t h e k n o w n p a r a m e t e r s a n d t h e e s t i m a t e d ~ 0 . 0 2 d e •

w e g e t :

~D -~ 1 0 - 3 ( T / 3 0 0 K ) 2 , ( 7 .1 3 )

w h i c h d e s c r i b e s w e l l t h e e x p e r i m e n t a l d a t a Tsang and White [7 .12]) , ob-

t a i n e d w i t h i n t h e t e m p e r a t u r e r a n g e 2 0 0 K < T < 4 0 0 K , ~e• -~ 1 . 4 - 1 0 - 3

a t T = 3 0 0 K , s ee C h ap . 4 .

T h e s m a l l e r m o b i l i t y v a l u e s ( l a r g e r t h a n 7 ) f o r t h e r a r e - e a r t h o r t h o f e r r i t e s

~ a > ~ f o r T i n , H x , D y c a n b e e x p l a i n e d b y t h e f a c t t h a t f o r t h e s e m a g n e t s

t h e v a l u e o f d/dex is la r g e r , a n d a ls o b y a d i r e c t c o n t r i b u t i o n o f t h e i m p u r i t y

r e l a x a t i o n ( se e b e lo w ) .

F o r i r o n b o r a t e , a s h a s b e e n m e n t i o n e d a b o v e , ~ D = 0 , a n d @ 3 ) i s n e g l i -

g i b l y s m a l l, @ 3) ~ 1 0 - 9 ( T / 3 0 0 K ) 3 . T h u s , th e D W r e t a r d a t i o n c a n b e d e t e r -

m i n e d b y ~ a . U s i n g t h e v a l u e s H~/H~ ~_ 0 .1 A _~ 1 0 - 5 c m ) o n e g e t s f o r t h e

v i s co s i t y co e f f i c i en t :

~a -~ 0 . 6 . 1 0 - 4 ( T / 3 0 0 K )

T h i s v a l u e is s o m e w h a t s m a l le r t h a n t h a t o b s e r v e d i n e x p e r i m e n t ( ac -

c o r d i n g t o t h e r e c e n t d a t a [ 7 .2 0 ] ~ x p -~ (1 .5 + 2 ) 9 1 0 - 4 d i n - s / c m 3 ) . S u c h a

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

m i n e d . T h i s is c o n n e c t e d w i t h t h e a n i s o t r o p y fi el ds - t h e h e x a g o n a l a n d t h e

r h o m b i c o n e g i v e n b y t h e p r e s s u r e ( t h e r h o m b i c o n e e f f e c ts t h e w a l l t h i c k -

n e s s ) .


15/17


Concluding, we note that the data of the microscopic theory for the DW

mobility in WFM, with allowance of the proper relaxation processes, describe

well the experiment for the qualitative samples containing no ion-relaxators.

For the W FM the situation appears to be more favourable than for the bubble

materials on the basis of the ferrites-garnets.

7 . 3 I m p u r i t y R e l a x a t i o n i n O r t h o f e r r i t e s

w i t h R a r e E a r t h I o n s

The increase in the magnon damping decrement when the rare-earth (R)

ions were added to ferrite-garnets has already been established in the 60 s.

At the present time the principal rules of this relaxation can be regarded

as decoded (see [7.8]). A considerable increase in the dissipation of magnetic

per turbat ion in the presence of rare -earth ions is associated with the existence

of two different mechanisms called the longitudinal (or slow) and transverse

(or fast) relaxation. This classification was primarily revealed in microscopic

theory.

The transverse relaxation mechanisms can be described on the basis of

phenomenological equations of the dynamics of magnetization of the R-

sublattice M with the standard relaxation term (in the form of Landau-

Lifshitz or Hilbert). In the microscopic approach, the transverse relaxation is

caused by the dynamic transitions

between the

R-ion levels under the action

of the exchange field of the iron (Fe) sublattice, the level broadening should

be really taken into account. The results of both approaches are consistent, in

particular, lead to the conclusion that there is a weak frequency dependence

on the spin waves damping (small time dispersion).

It has now been established (see [7.8]), that in the majority of ferrites with

R-ions the mechanism of longitudinal (or slow) relaxation is the basic one.

It has first been suggested by

Van Vleck

[7.21] how to describe the spin wave

damping. This mechanism is due to the modulation of R-ion levels under the

oscillations Fe-sublattice magnetization. The arrangement of populations of

these levels to instantaneous quasi~quilibrium ones is accompanied with the

transitions between them which result in dissipation. This approach can be

extended also to the nonlinear perturbations of the type of moving domain

walls. For ferrites-garnets, this is done in the

Teale

paper [7.22]. In what

follows, the theory was developed by Ivanov and Lyakhimets [7.23,24].

For orthoferrites with R-ions, similar calculations have not, so far, being

done (to our mind this is due to both greater complexity of the problem and

also the fact tha t rare-earth ferrites-garnets are widely used in technology).

The magnetic relaxation theory, including an analysis of the spin wave damp-

ing and DW retardation in orthoferrites with R-ions, has been constructed,

quite recently, by Ivanov and Lyakhimets see [7.24]. Not going into details,

we give the main results of this analysis.


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7 .3 Im pur i ty Relax a t ion in Or thofer r i tes wi th R are -Ea r th Ions 111

I t t u r n e d o u t t h a t t h e d i s s ip a t i v e c h a r a c t e r i s t ic s o f o r t h o f e r r i t e s , u n li k e

f e r r i te s - g a r n e t s , a r e e x t r e m e l y a n is o t ro p i c ev e n a t r o o m t e m p e r a t u r e . I n t h e

p h a s e w i t h / 0 [I a , m 0 [I c ax e s, t h e m a g n o n d a m p i n g d e c r e m e n t w i t h t h e

o s c i ll a ti o n / in t h e a b - a n d a c - p l a n e s ( a b - a n d a c - m a g n o n s ) is d e t e r m i n e d

b y d i f fe r e n t m e c h a n i s m s . F o r a b - m a g n o n s , o n l y t h e l o n g i t u d i n a l r e la x a t i o n

c o n t r i b u t i o n is i m p o r t a n t , f o r t h e a c - m a g n o n s - t h e t r a n s v e r s e o n e. C o r r e-

s p o n d i n g l y , f o r 5 a n d 7 a c, t h e d e p e n d e n c e o f 7 o n w i s e it h e r s i g n i f i c a n t o r

n eg l i g i b l y s ma l l , i. e ., t h e t i m e d i s p e r s i o n i s e i t h e r l a rg e o r s ma l l :

= ab( O ) r l / ( r l + o c 0 )

He re , F l l i s th e R - i on l eve l w id th ; gene ra l ly , FII ~ 1011s-1 .

Fo r l o w f r eq u en c i e s , w F ll A i s s w i t ch ed o fF ' b y t h e l aw : F f r N

1 / v

a t v >) F l lA. Th i s c ha rac te r i s t i c va lu e o f th e v e lo c i ty v0 a t /711 ~ 1011s -1 ,

A ~ 1 0 - 6 c m is o f t h e o r d e r o f 1 0 5 c m / s a n d m u c h sm a l l e r t h a n c . S w i t c h i n g

o f f ' o f c e r t a i n r e l a x a t i o n m e c h a n i s m ( t h e i n c r e a se i n t h e m o b i l i t y b y 5 + 7

t i m e s ) a t v > 2 . 1 0 5 c m / s w a s o b s e rv e d b y

K i m

a n d

K h w a n

i n y t t r i u m o r t h o -

f e r r i te [7 .2 5]. I t is i m p o r t a n t t o k n o w t h a t t h e f u n c t i o n

v ( H )

i s co n s i d e red

o n l y i n o n e o f t h e r a n g e s v ( ( v0 o r v 0


17/17


scores of kOe). The quantit ies/'li an d / ' • describe the relaxation rate of the

diagonal and nondiagonal components of the density matrix of R-ions, A(v)

is the DW thickness. Generally, r]• -~

( F l i F • i ~ -

0.1~ii, and the major

contribution comes from the longitudinal relaxation.

The estimation of the magnitude of this contribution for yttrium ortho-

ferrite with partial yttrium substitution for the rare-earth ions-relaxators,

yields formulae of the impurity relaxation contribution for the viscosity co-

efficient per unit DW area ~R:

r]a ~ 0.05y [din. s/ cm 3] ,

where y is the number of R-ions per unit cell. This value of U corresponds

to the mobility = 400/y [cm/s.Oe]. When estimating this, it was assumed

th at T = 300 K and the general parameter values were FII ~ 10ns -1, HR,e =

50 kOe. For X, the following expression was used: X =

y ~ / V o T , o

is the

Bohr magneton, V0 is volume of unit cell. This value agrees, on the whole,

with experimental data obtained by

R o s s o l

[7.26],

C h e t k i n e t a l.

[7.27].

The temperature dependence of the contribution from R-ions to the vis-

cosity coefficient U is determined by the temperature dependence of/-']] and

X (the remaining parameters in (7.19) exhibit a weaker temperature depen-

dence). For all orthoferrites, the quanti ty ~l increases with increasing tem-

perature. As for the magnitude of paramagnetic susceptibility of R-ions )ill,

the situation here is different. If for the majority of R-ions whose ground

state is magnetic,

X c< 1 / T ,

then for the europium ion Eu +a the quanti ty X is

determined by the contribution of high-lying levels and decreases with low-

ering temperature. In the relevant region of nitrogen and room temperatures

for Eu +3, the value of )l ~ (l /T ) e x p ( - w / T ) , w ~ 500K. This should result

in an essentially varying behaviour of v(T): an increase in ~(T) with lowering

temperature at

X c< 1 / T

and, correspondingly, a decrease in rl(T) for an ion

with nonmagnetic ground state. This behaviour of v(T) for EuFeOa and the

increase in ~(T) for orthoferrites Ho, Dy, etc. with lowering temperature was

observed by

R o s s o l

[7.26].

Concluding, we can sta te tha t for a WFM the existing theory can explain,

semi-quantitatively, the rules for DW relaxation using both the intrinsic and

impurity mechanisms.

07 - Microscopic Theory of Relaxation of Domain Wall

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