08 - non one-dimensional dynamics of domain walls in week ferromagnets

8/20/2019 08 - Non One-dimensional Dynamics of Domain Walls in Week Ferromagnets

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8 . N o n o n e D i m e n s i o n a l D y n a m i c s

o f D o m a i n W a ll s in W e a k F e r r o m a g n e t s

8 .1 N o n o n e D i m e n s i o n a l i t y o f S u p er s o n ic D y n a m i c s

o f D o m a i n W a l ls i n O r t h o f er r it e s

The investigations of the dynamics of the domain wall in y ttr ium orthofer-

rite presented above have shown that there exist large fluctuations in times

of the domain wall transit over a given distance in the region of supersonic

velocities of DW motion. The range of magnetic fields where these fluctua-

tions are most noticeable is marked by two vertical lines in Fig. 4.7. A similar

situation was registered by the method of Sixtus and Tonks for the domain

wall of the head-to-h ead type. It is natural to associate this instability with

the unsteady character of the domain wall supersonic motion. In this re-

spect it was interesting to register the shape of the dynamic domain wall

and then to est imate its velocity particularly in the supersonic range. Using

the method of single-shot high speed photography in Ref. [8.1] the authors

have shown th at an initial ly rectilinear domain wall statically stabilized by

the gradient magnetic field as presented above does not change its shape

in dynamics up to the velocity of transverse sound. Upon further increase

of the pulsed magnetic field moving the domain wall the latter exhibits the

semi-spherical formations - leading parts of a characteristic size of several

hundred micrometers which move faster than the rectilinear parts of the do-

main wall in the same field. This distort ion in the shape of the domain wall

moving at supersonic velocity is perhaps one of the reasons for the fluctu-

ations of transit time of the orthoferrite domain wall over a given distance

which was obtained by registering two light spots with the help of the mag-

netooptical method and previously by the method of Sixtus and Tonks for

the head-to-head type of domain wall. The rectilinear parts on the domain

wall decrease as the leading parts increase. A light pulse dura tion of 8 ns was

used in the above-mentioned experiment of a single-shot high speed photo-

graph of the moving domain wall [8.1]. During this time the domain wall

moving at supersonic velocity passes over large distances and therefore it

was interesting to substantia lly reduce the dura tion of the light pulses. This

was atta ined by exci tation of superluminescence of a dye by a N2-1aser with

transverse discharge [8.2]. In Ref. [8.3] the dynamics of the domain wall in

orthoferrites was investigated with the use of red light pulses with a duration


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114 8. Non-one-Dimensional Dynamics of Domain Walls

of abo ut 1 ns. Th e dye oxazine wa s u sed as a source of superluminescence.

Th e appea rance of deformations on the orthoferrite do ma in wall occurring

on passing to supersonic velocity ma de it necessary to use the met ho d of

doubl e-sho t high speed phot ograph y, wh ich allows one to register two posi-

tions of the mo vi ng do ma in structure in the process of one passage of the

do ma in wall along the specimen, as described above.

Figure 8.1 presents so me of these double dy na mi c structures in a YF eO 3

platelet, of thickness 120 #m , in a pulse d ma gn et ic fields up to 2 k O e at 290 K.

T h e interval be tw ee n tw o light pulses wa s equal to 15 ns. In ma gn et ic fields

up to 170 Oe, the initially rectilinear dom ain wall, stabilized by t he gradient

ma gn et ic field of 300 O e/ cm , does not c han ge its initial sha pe (Fig. 8.1 a).

In high fields the shape of a mo vin g do ma in wall changes. Th e do ma in wall

sh ap e be co me s curved, the velocities of its various parts b e co m e different, as

can be det erm ine d fro m the double- shot photogr aphs. Th e rectilinear parts

of the d om ai n wall continue to mo ve at the velocity close to the transverse

velocity of so un d (Fig. 8.2 a, c). Th e leading parts m o v e conside rably faster.

Fig. 8.1a-e Double high speed photographs of the dynamic domain wall in YFeO3

cut perpendicularly to the optical axis at a temperature of 290 K, in magnetic fields

of: a) H = 127 Oe, b) H = 185 Oe, c) H = 195 Oe, d) H = 1200 Oe and e)

H = 1950 Oe [8.3]

Figure 8.1 b, c present p hotog raphs which describe another typ e of un-

stead y supersonic motion of the doma in wall. The photogra phs show th at

various pa rts of the domain wall in the same magnetic field move at different

supersonic velocities. This situation is observed when the domain wall veloc-

ity only marginally exceeds the velocity of transverse sound, and causes the

fluctuations of measured intervals of time in which the domain wall passes


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8.1 Non-on e-Dimen sional Supersonic Dynamics 115

the d i s tance be tween the two l igh t spo ts . Success ive pho tographs in F ig . 8 .1 ,

p ro duc ed in d i f f e ren t pu lsed m agne t ic f ie lds, a llow the de te rm ina t ion o f the

va lues o f the dom ain wal l ve loc i ty .

F ig u r e 8 .2 s h ow s s ev e ra l p i c tu r e s o f d y n am ic d o m a in s t r u c tu r e s o b t a in e d

b y m ean s o f t h e m e th o d o f d o u b l e h ig h s p eed p h o to g r a p h y i n t h e s am e Y F eO 3

pla te le t a t 110 K. In th i s case the mobi l i ty o f the dom ain wal l was equa l to 2 .

104 cm /s -O e , i .e . it was twice as la rge than a t 290 K. Th e in te rva l be tween the

l igh t pulses was equa l to 5 ns. The ampl i tud es o f non -on e-d im ens ion a l i ty on

the domain wal l inc rease wi th a r i s e in i t s mobi l i ty . The t r ans i t ion f rom one-

d im en s io n a l m o t io n t o t h e n o n - o n e - d im en s io n a l s u p e r s o n i c m o t io n b eco m es

m o r e a b r u p t w i th i n c rea s in g m o b il it y . F o r a m o b i li t y o f 5 . 1 0 3 c m / s . O e ,

t h e n o n - o n e - d im en s io n a l i t i e s a r e s m a ll an d b eco m e v e r y d i s t in c t w h en t h e

m o b i l i t y i s eq u a l t o 2 . 1 0 4 cm / s . O e .

Fig

8.2 a- e Double high speed photographs of a dynamic domain wall in an YFeO3

plate cu t pe rpendicularly to the optical axis at a temp eratu re of 110 K in magnetic

fields of differe nt stren gths: a,b ,e) H = 380 Oe, e) H = 160 Oe, d) H =

750 Oe [8 3]

Th e pos i t ions o f the l ead ing par t s on the do main wal l, pa r t i cu la r ly near

the t r ans i t ion to superson ic ve loc i t i es , change f rom case to case and , f ina l ly ,

the en t i r e dom ain wal l moves a t th e sam e superson ic ve loc i ty , which can be

me asure d f rom F ig. 8 .2. The rad i i o f the curv a tu re o f the dom ain wal l, which

can b e m eas u r ed w i th t h e u s e o f F ig . 8 .2 , g rea t l y d ep en d o n t h e m ag n e t i c

f ie ld in which the domain wal l i s moving.

F ig u r e 8 .3 g iv es t h e d ep en d en ce o f t h e cu r v a tu r e r ad iu s o f t h e d o m a in

w a l l p e r p en d i cu l a r t o t h e s u r face o f t h e Y F eO 3 p l a t e le t , cu t p e r p en d i cu l a r l y

to t he op t ica l ax i s , on the m agne t ic f ie ld . Increas ing the m agne t ic f ie ld causes


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1 16 8 . N o n , m e-D i m en s i o n a l Dy n am i cs o f Do m ai n W alls

R , m

700

500

300

o

100 - -- - ~~ ~ ~ 1 7 6

f

500 1500 2500 H , O e

F i g . 8 . 3 Dep en d en ce o f th e cu rv a t u re r ad i u s o f a s u p e r s on i c d o m a i n wa l l o n t h e

driv ing magnet ic f ie ld [8 .3]

t h e r a d i u s t o d e c r e a s e , f ir st s h a rp l y , fr o m 7 0 0 t o 3 00 m , w i t h t h e s u b s e q u e n t

r a n g e o f m o n o t o n i c c h a n g e it s v a l u e d r o p s t o 1 2 0 m .

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

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

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

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

v e l o c it y . T h i s i s o b s e r v a b le i n a s e ri es o f p h o t o g r a p h s o f t h e d o m a i n w a l l

m o v i n g i n a m a g n e t i c f i e ld o f 6 00 O e a t v a r i o u s m o m e n t s i n t i m e , p r e s e n t e d

i n [8.3 ]. T h e v i s i b le w i d t h o f t h e d o m a i n w a l l m o v i n g a t a v e l o c i t y o f 4 k m / s

is sm a l l. I n c r e a s e o f t h e v e l o c i t y l e ad s t o a n a b r u p t w i d e n i n g o f t h e w a ll . T h i s

w i d t h r e a c h e s 5 0 - 6 0 m i n 2 - 3 n s a f te r p a s s in g t h r o u g h t h e s o u n d b a r r i e r

(see F ig . 8 .4 a ) .

T h e d o m a i n w a ll b lu r r i n g , d u e t o i t s m o t i o n a t a v e l o c i t y o f 10 - 15 k m / s

w i t h i n t h e d u r a t i o n o f t h e l i g h t p u l s e, is e q u a l t o 10 - 1 5 m . T h i s b l u r r i n g

c a n b e s u b s t a n t i a l l y r e d u c e d b y u s i n g s h o r t e r l i g h t p u ls e s. W h e n l ig h t p u ls e s

w i t h a d u r a t i o n o f 0 .2 5 n s a r e u s e d , t h e g e n e r a l p i c t u r e o f t h e v i si b le w i d e n i n g

o f t h e d o m a i n w a l l r e m a i n s t h e s a m e , b u t t h e m a x i m u m w i d e n in g d e c r e a se s

b y 10 - 1 5 m .

I n a f e w n a n o s e c o n d s a f t e r th e m a x i m u m w i d e n i ng , t h e v i s i bl e w i d t h o f t h e

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

o f b l u r r i n g d u r i n g t h e l ig h t p u ls e s. T h i s w i d e n i n g o f t h e d o m a i n w a l l re s u l ts ,

p e r h a p s , f r o m i t s in c l i n a t io n t o t h e s p e c i m e n ' s s u r fa c e , c a u s e d b y t h e t r a n s i -

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

o f s o u n d , t h e d o m a i n w a l l i n o r t h o f e r r i t e s ce a se s t o b e o n e - d i m e n s i o n a l a n d

b e c o m e s a t h r e e - d i m e n s i o n a l o b j e c t . A l l t h e a b o v e - m e n t i o n e d w a s i n re f er -

e n c e t o t h e d o m a i n w a ll , w h ic h , a t v e l o c i ti e s b e lo w t h a t o f s o u n d , i s p e r p e n -

d i c u l a r t o t h e s u r f a c e o f t h e o r t h o f e r r i t e p l a t e l e t a n d l ie s i n a p l a n e p e r p e n -

d i c u l a r t o t h e a a x i s . A s i n e - s h a p e d d o m a i n w a l l , i n c l i n e d t o t h e s p e c i m e n ' s

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


5/18

8 .1 No n -o n e -D im en s io n a l Su p e r s o n ic Dy n am ics 1 17

A ~ / t i n

100 _-'\

6O

o o o , , .o 9 - ' 8 " ~ 1

f

I

0 10 20 30 t , n s

F ig . 8 . 4 a , b T im e d ep en d en c e o f t h e v i si b le t h ick n es s o f a DW in YFeO3 a t H =

1000 Oe fo r dom ain wall s: (a ) pe rpend icu lar and (b ) inc l ined to the sam ple ' s su r face

before motion [8 .4]

p e n d i c u l a r t o t h e a a x is , in o r t h o f e r r i t e p l a t e l e t s p e r p e n d i c u l a r t o t h e o p t i c a l

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

i t s t r a n s i t i o n t o s u p e r s o n i c v e l o c i t i e s [8 .4 ]. H o w e v e r , d i s t o r t i o n s o f t h e i n i t ia l

s h a p e a r e s u b s t a n t i a l l y s m a l le r t h a n f o r t h e i n i t ia l l y p l a n e D W , w i t h t h e s a m e

m o b i l it y o f t h e d o m a i n w a l l. T h e p l ac e s w h e r e n e w n o n u n i f o rm i t i e s a p p e a r

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

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

s u p e r s o n ic m o t i o n t h a n p l a n e on es . P e c u li a r w e d g e - s h a p e d p o i n ts a p p e a r o n

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

a t p l a c e s, w h i c h c o r r e s p o n d t o t h e m o s t r e t a r d e d p a r t s o n t h e d o m a i n w a ll s,

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

f o r t h e i n i t i a l l y p l a n e d o m a i n w a l l .

I f t h e g r a d i e n t o f t h e m a g n e t i c f i el d i n c re a s es , r e a c h i n g 2 5 00 O e / c m , t h e

s i n e - s h a p e d d o m a i n w a l l b ec o m e s p l a n e a n d i n cl in e d to t h e s p e c im e n s u rf a c e

a t a n a n g l e o f 4 5 - 4 7 ~ T h e d e p e n d e n c e o n t i m e o f t h e v i si b le w i d t h o f t h e

m o v i n g d o m a i n w a l l o f t h i s t y p e i n a m a g n e t i c f ie l d o f 1 0 0 0 O e i s g i v e n i n

F i g . 8 .4 b . I t i s s e e n t h a t t h e v i si b le w i d t h o f t h e d o m a i n w a l l g r e a t l y d e c r e a s e s

w i t h t i m e a n d a p p r o a c h e s 10 m , i .e ., t h e v a l u e w h i c h , a s m e n t i o n e d a b o v e , is

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

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

o f t h e d o m a i n w a l l le a v es t h e c p l a n e , w h e r e i t i n i ti a l ly w a s l o c a t e d a n d

b e c o m e s p e r p e n d i c u l a r t o t h e s p e c im e n ' s s ur fa c e . T h e t i m e d u r i n g w h i c h th e

r o t a t i o n o f t h e d o m a i n w a l l t a k e s p l a ce i s e q u a l t o 1 0 n s . T h i s s u b s t a n t i a l l y

e x c e e d s t h e t i m e a c c e l e r a t i o n o f i ts l e a d i n g p a r t s u p t o t h e s u p e r s o n i c v e lo c i ty ,

a n d , i s cl os e t o t h e r e l a x a t i o n t i m e o f s u p e r s o n i c n o n - o n e - d i m e n s i o n a l i t i e s

o n t h e i n i ti a l ly p l a n e d o m a i n w a ll .


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118 8. Non one Dimensional Dynamics of Domain Walls

8 2 K i n k o n t h e D o m a i n W a l l i n O r t h o f e r r i t e s

Un de r conditions of high mobility, a sharp flexure - kink prop agat es along the

do ma in wall in ytt riu m orthoferrite mo vi ng at a velocity of transverse sound.

Th e velocity of the kink can reach abou t 20 km/s . As described above, in

passing t hr oug h the velocity of sound, semicircular leading sections app ear on

the do ma in wall. Un de r conditions of high mobility, attaining 2 9 104 cm /s . Oe

at Ii0 K, the velocity of these sections of the do ma in wall can reach i0 -

15 km/ s. If the am pli tud e of the driving ma gn eti c field is increased further,

in the conditions of the exp eri men t described in Ref. [8.5], up to 12 0- 14 0 Oe

at ii0 K, then the character of the do ma in wall mot ion sharply changes, as

can by seen from Fig. 8.5.

Fig. 8.5 Ph otogr aphs of two successive positions of a kink on an YF eO 3 do ma in

wall in 5 ns at ii0 K. The time interval between neighboring pictures is 3 ns [8.5]

Th e left part of the dom ai n wall has not passed the soun d barrier yet,

an d rem ain s rectilinear, mo vi ng at the velocity of transverse so und. Th e su-

personic velocity of the right part, whi ch has move d mu ch forward, sharply

decreases to the velocity of sound. Thu s, a kink m ov in g fro m right to left ap-

pears on t he do ma in wall, wit h its left an d right sides mo vi ng at the velocity

of transverse sound. Figure 8.5 sh ows successive positions of the kink after

5 ns. Th e interval be tw ee n two successive do ma in wall positions on the pho-

tog rap hs is 3 ns. Th e shar peni ng of the kink, as it mo ve s fro m right to left,

is clearly visible in the figure. Th e velocity of the kink, u, along the do ma in

wall mov in g at a velocity of v -- 4.1 km /s is equal to 19.5 km/ s. Me as ur ed

values of the velocities u, v and the limiting velocity c are linked th ro ug h the

relation:

u 2 + v 2 = c 2 8 .1 )


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8 .2 Kink on the Dom ain W al l in Or thofer r i t es 119

T h i s r e l a t io n f ol lo w s f r o m e x p e r i m e n t a n d c a n b e o b t a i n e d i n a l i n ea r a p p r o x -

i m a t i o n f r o m t h e t h e o r y [8 .6 ]. T h e s p a t i a l d e r i v a t iv e s o f t h e d o m a i n w a l l s h i f t

a t t h e b e g i n n i n g a n d e n d o f t h e k i n k h a v e d i s c o n ti n u it ie s a n d t h e k i n k i ts e l f is

a s e c t i o n o f a s t r i c t l y r e c t i l in e a r d o m a i n w a l l i n c l in e d a t a n a n g l e o f 4 5 ~ T h e

a m p l i t u d e o f t h e k i n k d e c r ea s e s a s g r a d H , s t ab i li z in g t h e d o m a i n w a l l in -

c r e a se . E x i s t e n c e o f t h e k i n k i s c l o se l y a s s o c i a t e d w i t h t h e g r a d i e n t m a g n e t i c

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

s e c t i o n o f t h e d o m a i n w a l l f a ll s i n t o t h e l ow e r m a g n e t i c f ie ld . A s a r e s u l t,

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

s o u n d . T h e m a g n e t i c f i el d i n w h i c h a k i n k i s f o r m e d , a n d t h e a n g l e o f i t s i n cl i-

n a t i o n a r e d e t e r m i n e d b y t h e v e l o c it y o f t h e d o m a i n w a ll su p e r s o n ic m o t i o n

a t w h i ch r e l a t i o n 8 .1 ) is s a t i sf i ed .

T h e a p p e a r a n c e o f t h e k i n k i n d ic a t e s t h a t t h e r e i s n o h y s te r e si s in t h e

d e p e n d e n c e o f v o n H u p o n p a s s in g t h r o u g h t h e v e l o c i ty o f s o u n d . T h i s

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

i s su cces s i v e l y a f f ec t ed b y t w o p u l s e s o f t h e f i e ld o f o p p o s i t e p o l a r i t i e s H 1

a n d / / 2 . F i g u r e 8 .6 sh o w s t h e g r a p h s o f v a g a i n s t H o f t h e d o m a i n w a ll i n

y t t r i u m o r t h o f e r r i t e in a n i n c r e a s in g m a g n e t i c f i e l d / / 1 F i g . 8 .6 a ) , a n d i n a

d ec reas i n g f i e ld H 1 + / / 2 F i g . 8 .6 b ) [8.7 ].

v k m / s

o o

e~ l~~ e

10 ~

OQ

e 9

5 oo , , ~ ,o~,,O~OoOo o

o . ~ b

0 I I I ~ t t

4 0 80 H , 0 e

F i g . 8 .6 a ,b D ep en d en ce o f t h e d o ma i n w a l l v e lo c it y i n Y FeO 3 o n t h e mag n e t i c

f ield . a) Increa sing H o) and b) decreasing H -)

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

v e l o c i t y , i n t h e r a n g e o f A H t = 6 0 O e , r e m a i n e d c o n s t a n t a n d w a s e q u a l t o

t h e v e l o c i t y o f s o u n d . I n t h e r a n g e I 0 - 1 5 O e , t h e v e l o c i t y s h a r p l y i n c r e a s e d

t o 1 4 k m / s . T h e n , a n o p p o s i t e l y p o l e d f i e l d , w i t h i n c r e a s i n g a m p l i t u d e , w a s

a p p l i e d ; w h e r e b y t h e b e h a v i o r o f t h e g r a p h o f v H 1 § a n d t h a t o f v H )

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

h y s t e r e s i s i n t h e v i c i n i t y o f t h e v e l o c i t y o f s o u n d . W i t h a n i n c r e a s e i n H 2 ,

t h e d o m a i n w a l l v e l o c i t y h a d t o f o l l o w t h e d e p e n d e n c e 4 . 3 ) v H 1 + / / 2 ) .

T h e a b i l i t y t o e x p e r i m e n t a l l y o b s e r v e t h e k i n k m e a n s t h a t t h e t r a n s i t i o n t o

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

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


8/18

1 20 8 . No n -o n e -D im en s io n a l Dy n am ics o f Do m ain W alls

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

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

H d u r i n g a ti m e t c rl i n o r d e r t h a t i ts s h a p e s h o u l d s h a r p l y c h a n g e a n d l e a d i n g

s e m i c i r c u l a r f o r m a t i o n s a p p e a r o n i t [ 8 . 7 ] .

T h e d e p e n d e n c e o f t h i s t i m e o n H i s p r e s e n t e d i n F ig . 8 . 7 a . A s H i n -

c r e a se s , t h e t i m e o f t h e d o m a i n w a l l m o t i o n a t t h e v e l o c i t y c l os e t o t h e

v e l o c i t y o f s o u n d r e q u i r e d t o i n i t ia t e t h e d e v e l o p m e n t o f n o n u n i f o r m i t i e s , d e -

c r e a s e s . I n t h e e x p e r i m e n t , t h e m a x i m u m t i m e t c r l w a s a b o u t 4 0 n s . A t t h e

m o m e n t o f t i m e tc r2 , t h e e n t i r e d o m a i n w a l l pa s s es t o t h e s u p e r s o n i c v e l o c i t y

o f m o t i o n . A t t cr l < t < tc r~ , t h e c o e x i s t e n c e o f p l a n e a n d c u r v e d p a r t s o f t h e

d o m a i n w a l l i s o b s e r v e d . T h e i n t e r v a l: t C rl - tc r2 i s t h e e f f e c t iv e t i m e f o r t h e

d o m a i n w a l l tr a n s i t io n t o t h e s u p e r s o n ic m o t io n . T h e e x p e r i m e n t s h ow s t h a t

w i t h h i g h m o b i l i ti e s , e s p e c i a l l y i n f ie ld s c l os e t o t h e b e g i n n i n g o f t h e t r a n -

s i t i o n t o s u p e r s o n i c v e l o c i ty , t h e v a l u e s o f t c rl e x h i b i t e d l a r g e f l u c t u a t i o n s .

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

o r t h o f e r r i t e s is u n s t e a d y . T h e d e p e n d e n c e o f t~ rl a n d t c r2 o n H a r e p r e s e n t e d

in F ig . 8 .7 .

t~ IlS

6O

4O

20

0

o \

: 0 ~ 0 0 0

i ~

5 0 0 1 0 0 0

H O e

Fig . 8 . 7 Dep en d en ce o f t h e t r an s i t i o n t im e t o s u p e r s o n ic m o t io n o n t h e m a g n e t i c

f ield o f: a ) l ead ing par t s o f D W a nd b ) the whole dom ain wall [8 .7 ]

C o m p a r i s o n o f t h e s e d e p e n d e n c i e s s h ow s t h a t i n t h e e x p e r i m e n t a l c o n d i -

t i o n s t h e m a x i m u m v a l ue o f t h e t r a n s i t io n t i m e f or t h e d o m a i n w a l l t o s t a r t

m o v i n g a t t h e s u p e r s o n i c v e l o c i t y is e q u a l t o 3 0 n s a n d d e c r e a s e s w i t h a n i n -

c r e a s e o f H . T h e s e r e s u l t s w e r e v er if i ed b y t h e a b o v e d e s c r i b e d e x p e r i m e n t s

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

w a s i n v e s t i g a t e d .


9/18

8 . 3 I r r e v er s ib i li ty o f N e a r s o n i c D W D y n a m i c s

8 . 3 I r r e v e r s i b i l i t y o f N e a r s o n i c D y n a m i c s

o f D o m a i n W a l l s i n O r t h o f e r r i t e s

121

T h e n e w e ff ec t r e s u l ts f r o m a c h a n g e i n t h e d i r e c t io n o f t h e d o m a i n w a l l

m o t i o n t o t h e o p p o s i t e o n e [8.7]. A s s u m i n g t h a t t h e d o m a i n w a ll , w i t h m o t i o n

e f fe c t e d b y t h e p u l s e d m a g n e t i c f ie ld H 1 a t t h e v e l o c i ty o f t r a n s v e r s e s o u n d ,

s t a r t s t o e x p e r i e n c e t h e a c t i o n o f t h e p u l s e d m a g n e t i c f ie ld /- /2 i n t h e d i r e c ti o n

o p p o s i t e t o H 1 ; t h e n t h i s w o u ld c a u s e t h e d o m a i n w a l l t o s t o p , a f te r w h i c h

i t w o u l d c o n t i n u e t o m o v e i n t h e o p p o s i n g d i re c t io n .

F i g u r e 8 .8 , a a n d b , p r e s e n t s t h e d e p e n d e n c e o f v o n H 1 a n d v o n H i + H 2 ) ,

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

c o n s t a n t ,

AHt

= 7 0 O e , is o b s e r v e d o n t h e

v H1)

c u rv e . I f t h e d o m a i n w a l l

m o v e s a t t h i s v e l o c i t y f o r a s u ff ic i e n tl y lo n g t i m e , v H1 + H 2 ) s u b s t a n t i a l l y

d i ff er s f r o m v H 1 ) . T h u s ,

v H1 + H2)

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

t i a l ly n a r r o w e r t h a n t h a t f o r m o t i o n i n t h e i n it ia l d ir e c ti o n . T h u s , v H1 +H 2 )

e x h i b i t s a p e c u l i a r i t y a t v = S 1, w h i c h d o e s n o t o c c u r f o r m o t i o n i n t h e i n i t i a l

d i r e ct i o n . I n t h i s c a s e, th e t i m e d u r i n g w h i c h t h e d o m a i n w a ll m o v e s a t t h e

v e l o c i t y o f s o u n d i n t h e i n it ia l d i r e c ti o n p l a y s a n i m p o r t a n t r ol e. I t h a s b e e n

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

AHt i n t h e v H1 + H2) c u r v e b e c o m e s n a r r o w e r .

v k m / s

2 0 o O e o e e o e e o e e

o e e o e O O o o o o o e o m e o ~ e ~ 9

1 5

o o 0 O ~ *

,,,s,o%~ o' '

lo b :

, ~ o

o9

O

0 ( l i t I (

i ( f l r l 1 1 ( I ,

1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 1 0 00 H , O e

F i g . 8 . 8 D e p e n d e n c e o f t h e D W v e l o c it y i n Y F e O 3 o n a ) d ir e ct a n d b ) o p p o s i t e

m a g n e t i c f ie l d s a ft e r D W m o t i o n i n a d i r e c t f ie l d a t s o n i c v e l o c i t y [ 8.7 ]

T h e d e p e n d e n c e o f t cr o n t e m p e r a t u r e i s g i v e n i n F i g . 8 .9 . A t T = 1 00 K ,

Zcr = 1 0 u s ; a t T = 2 6 5 K , tc r = 1 0 0 n s . T h e s h o r t e n i n g o f t h e i n t e r v a l A H t

i n t h e r e v e r s e m o t i o n o f t h e d o m a i n w a l l is fo l lo w e d b y a s u b s t a n t i a l d e c r e a s e

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

v e l o c i t y o f s o u n d i n t h e fi el d H 1 § T h i s f a c t d e s c r i b e d a b o v e c a n r e s u l t

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

c r y s t a l l a t t i c e , c a u s e d b y t h e m o t i o n o f t h e d o m a i n w M 1.


10/18

1 22 8 . No n -o n e -D i m en s i o n a l Dy n am i cs o f o m a i n Walls

t ~ n s

1

5

o

~

i o j

j o

0 i f I i

100 200 T K

F i g . 8 . 9 T e m p e r a t u r e d e pe n d e n c e o f

t h e t i m e

o f DW m o t i o n i n a d i r ec t m ag n e t i c

f ield at a sonic veloci ty wh ich

i s n e c e s s a r y

fo r no t iceab le changes in son ic and

superson ic dyn am ics a t the bac k mot ion o f a DW [8.7 ]

A n a t t e m p t t o e x p e r i m e n t a l l y o b s e rv e t h is k i n d o f d e f o r m a t i o n a n d t h e

g e n e r a t i o n o f s o u n d d u r i n g s u p e r so n i c tr a n s i t io n w a s m a d e in e x p e r i m e n t s o n

l i g h t r e f l e c t i o n f l o m t h e m o v i n g d o m a i n w a l l . A s d e s c r i b e d a b o v e i n C h a p . 3 ,

t h e i d e a o f t h i s e x p e r i m e n t c o n s is t s in t h e i n v e s t i g a ti o n o f t h e f r e q u e n c y s h if t

o f v i s ib l e l ig h t r e f l e c t e d f r o m t h e d o m a i n w a l l d u e t o t h e D o p p l e r e f f e c t.

T h e l ig h t r e f l e c te d f r o m t h e m o v i n g d o m a i n w a l l c h a n g e s it s p o l a r i z a t io n ,

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

t h e d o m a i n w a l l, d o e s n o t c h a n g e i t s p o l a r i z a t i o n . A t 2 K ,

Demokritov et

al.

[ 8.8 ] s u c c e e d e d i n o b s e r v i n g t h e d e p a r t u r e o f t h e r e g i o n o f d e f o r m a t i o n ,

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

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

t o 1 0 -7 s .

8 4 D i s s i p a t i v e S t r u c t u r e s i n S u p e r s o n i c M o t i o n

o f D o m a i n W a l l s i n O r t h o f e r r i t e s

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

I n t h e p r o c e ss o f t r a n s it i o n t o s u p e rs o n ic m o t i o n , so m e n o n - o n e - d i m e n s i o n a l

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

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

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

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

i m p o r t a n t t h a t t h is e f f ec t o c c u r s i n a h o m o g e n e o u s m e d i u m a n d i n a u n if o r m

( a l o n g t h e D W ) e x t e r n a l m a g n e t i c f ie ld i n t h e D W p l a n e , w h i c h a l lo w s o n e

t o a s s u m e t h a t t h e s y s t e m is s e l f- o r g a n iz i n g . U s u al ly , t h e f o r m a t i o n o f t h e

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

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

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

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

Hz

w h e r e z i s t h e d i r e c t i o n o f t h e


11/18

8.4 Dissipative Structures in Supersonic DW Motion 123

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

i s o l a t e d D W w i t h t h e h e l p o f e x t e r n a l m a g n e t s a s i s s h o w n i n F i g . 3 . 9 b . T h i s

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

d e m a g n e t i z i n g f i e l d s o f t h e n e i g h b o r i n g d o m a i n s . H a g e d o r n s h o w e d [ 8 . 9 ] t h a t

i f t h e e a s y a x i s o f m a g n e t i z a t i o n i s p e r p e n d i c u l a r t o t h e s p e c i m e n ' s p l a n e ,

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

s o m e c r i t i c a l v a l u e . F o r Y F e O 3 , t h i s v a l u e i s e q u a l t o I 0 0 0 O e / c m . A s g r a d H

d e c r e a s e s b e l o w t h e c r i t i c a l v a l u e , t h e D W c u r v e s p r o d u c e t h e s t r u c t u r e w i t h

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

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

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

r e g u l a r s t r u c t u r e s o n t h e D W .

I t h a s b e e n s h o w n a b o v e , t h a t w h e n p a s s i n g t h r o u g h t h e s o u n d b a r r i e r ,

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

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

d i s t i n c t , a s t h e v a l u e o f A H t / S t i n c r e a s e s [ 8 . 7 ] . H e r e , i s t h e D W m o b i l -

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

t h e v e l o c i t y o f s o u n d , S t , o n t h e c u r v e

v ( H ) .

T h e o r e t i c a l c a l c u l a t i o n s o f t h e

v a l u e o f A H t a r e g i v e n i n C h a p s . 5 a n d 6 . I n R e f . [ 8 . 1 0 ] i t w a s e x p e r i m e n t a l l y

f o u n d t h a t n o n - o n e - d i m e n s i o n a l i t i e s o n t h e D W u p o n t r a n s i t i o n t o s u p e r -

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

a l o n g t h e D W a r e s t r i c t l y r e g u l a r . F i g u r e 8 . 1 0 r e p r e s e n t s s e v e r a l d o u b l e - s h o t

h i g h s p e e d p h o t o g r a p h s o f d y n a m i c d o m a i n s t r u c t u r e s i n a Y F e O 3 p l a t e l e t

a t 2 9 0 K . T h e D W m o b i l i t y w a s e q u a l t o 1 0 4 c m / s . O e . 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 r e g i o n p a s s e d b y t h e D W d u r i n g t h e t i m e i n t e r v a l b e t w e e n

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

e a c h l i g h t p u l s e i s e q u a l 0 . 2 5 n s .

F i g u r e 8 . 1 0 a , s h o w s h o w w e a k r e g u l a r d i s t o r t i o n s , o f o b v i o u s l y n o n -

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

t h e s e d i s t o r t i o n s i n c r e a s e a n d g r a d u a l l y o c c u p y t h e e n t i r e D W . T h e i n t e r s e c -

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

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

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

D W .

Wedge-shaped points where the spatial derivative of the DW shift rup-

tures appear at the intersection of two neighboring non-one-dimensional

sections. Figures 8.10 b c represent the dynamic regular structures which ap-

pear on the DW in y tt rium orthoferrite in magnetic fields of 150 and 1000 Oe.

The time delays between the light pulses are 15 and 2 ns respectively. The

photographs given here show tha t the period of the dynamic structure on the

DW does not change with time. The dependence period of the structure on

the magnetic field is given in Fig. 8.11.

In the experiment the maximum period was equal to 1200pm. As the

field H in which the trans ition to supersonic motion took place increased


12/18

124 8. Non-one-Dimensional Dynamics of Domain Walls

Fig . 8.10a-e Periodic structures on a supersonic DW of yttr ium orthoferrite:

(a) the appearance of a structure, At = 2 ns, (b) the structures in the magnetic

fields of H~ = 150 Oe, At = 15 ns (c) and Hz = 1000 Oe, At = 2 ns [8.10]

A, m

1200

800

400

O O

O O

J I J

400 800 1200 H, Oe

Fig. 8.11 Dependence of the period of the structures on a supersonic DW of YFeOa

in a magnetic field at room temperature [8.10]

the period, first, sharply decreased, and then gradually reached a value of

250 m. The amplitud e of the structur e increased until the rectilinear sections

of the DW remained. The amplitude of the structure was maximum at the

mom ent of collapse of the rectilinear sections of the DW.

Figure 8.12 demonstrat es the evolution of the struc ture wi th time. For a

period of 1200t~m, the t ime of evolution was equal to go ns, after that, the

ampl itu de remained c onsta nt during the entire time of observation. This time

is substantially longer than the time of magnetic relaxation. For this reason,

one can speak of the stati onary character of the regular structure in magnetic

fields which are close to the minimum magnet ic field of the tran sit ion to


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8 .5 R e l a x a t i o n o f t h e No n - o n e - D i m e n s i o n a l i t i e s 1 25

h / h ~

oo-7 .. /

0.5 o ~ - . _ 0 ~ . 9

0 ~ I I I I I I I I I

20 40 60 80 t , n s

F i g . 8 . 1 2 D e p e n d e n c e o f t h e r e l a ti v e a m p l i t u d e o f a s t r u c t u r e o n a s u p e rs o n i c D W

of YFeO 3 w i th d i f f e r en t pe r iod s )~ on t im e . ;~ = 260 t im ( - - - ) , )~ = 50 0# m (o o o ) ,

= 12 00 tim ( |174 [8.10]

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

s t r u c t u r e s d e c r e a se s . A f t e r t h e a m p l i t u d e a t t a i n s i t ' s m a x i m u m , t h e p r o c e s s

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

t h e r a t io o f t h e m a x i m u m a m p l i t u d e o f t h e s t r u c t u r e t o i ts p e r io d w a s e q u a l

t o 0 . 2 .

I t s h o u l d b e n o t e d t h a t t h e b o u n d a r y c o n d i ti o n s a t t h e D W e n d s in t h e c o il

p r o d u c i n g t h e m a g n e t i c f ie ld h a v e n o e ff ec t o n t h e p e r i o d o f t h e s t r u c t u r e s.

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

s t r i c t l y p a r a l l e l t o t h e i n i ti a l s t a t i c p o s i t i o n o f t h e r e c t i l i n e a r D W [8 .1 0].

8 . 5 R e l a x a t i o n o f t h e N o n o n e D i m e n s i o n a l i t i e s

o n t h e M o v i n g D o m a i n W a l l i n Y t t r i u m O r t h o f e r r i t e s

N o n - o n e - d i m e n s i o n a l i t i e s o n t h e d y n a m i c D W in a n u n if o rm m a g n e t i c fie ld

a p p e a r o n l y a t s u p e r s o n i c v e l o c it ie s . U s i n g a n o n u n i f o r m l o c a l f ie ld , it is

p o s s i b le t o p r o d u c e d i s t o r ti o n s o n t h e D W a t a n y v e lo c i t y o f i ts m o t i o n e v e n

i f i t i s l e ss t h a n t h e v e l o c i t y o f s o u n d [8 .1 1]. A s t h e d i r e c t i o n s o f t h e l o c a l a n d

t h e u n i f o r m m a g n e t i c f ie ld s c o in c id e , a le a d i n g s e c ti o n a p p e a r s o n t h e D W .

F i g u r e 8 .1 3 ( 1 ) g i v e s t h e d e p e n d e n c e o f t h e t i m e o f r e l a x a t i o n o f a d is -

t u r b a n c e o n t h e r e c t i l i n e a r D W o n i ts v e l o c i t y [ 8.1 1]. T h i s d e p e n d e n c e i s n o t

r e g u l a r . I n t h e r e g io n s w e r e t h e D W v e l o c it ie s a r e c o n s t a n t , a n d c l o se to t h e

v e l o c it i es o f t r a n s v e r s e a n d l o n g i t u d in a l s o u n d , t h e r e l a x a t i o n t i m e e x h i b i t s

s h a r p m a x i m a . A s t h e D W v e l o c i t y i n c re a s e s f u r th e r , a n d a l m o s t r ea c h e s t h e

l i m i t in g v e lo c i ty , t h e r e l a x a t i o n t i m e o f t h e n o n - o n e - d i m e n s i o n a l i t i e s s t a r t s

i n c r e a s i n g a g a i n . T h i s is a c o n s e q u e n c e o f t h e q u a s i r e l a t i v i s t i e n a t u r e o f t h e

D W d y n a m i c s in w e a k f e r ro m a g n e t s a n d c o r r e sp o n d s t o t h e r e s u lt s o f t h e -

o r e ti c a l e s t im a t i o n s o f t h e r e l a x a ti o n o f t h e D W d i s tu r b a n c e s w i t h s m a l l

a m p l i t u d e [ 8 . 6 ] .


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126 8. Non-one-Dimensional Dynam ics of Dom ain W alls

r , n s , ~ 3 ' 3

100 / ~ ~ ~ ~ ~ 7 ~ ~ ~ ~ ,/ f

50

0 I I

5 10 15

v} k m / s

F ig . 8.1 3 Dependence o f the relaxation time of disturbances on a dynamic D W of

YFeO 3 on its velocity. Experiment for goin g ahead (1) an d retardin g (2) distur-

bances. Calculations for two different values of distu rba nce dim ensions along th e

DW: n = 600/zm (3), L = 800/zm (3') [8.11]

A s im i l a r d ep en d en ce a l so o ccu r s in th e ca se fo r l o ca l d i s tu rb an ces wh ich

o p p o se th e u n i fo rm f i e ld d r iv in g th e d o m a in wa l l ; t h i s d ep en d en ce i s d e -

p ic t ed in F ig . 8 .1 3 (2 ) . T h e an a ly s is o f t h e r e l ax a t io n o f s ing le n o n - o n e -

d i m e n s i o n al it i es o n t h e D W c a n b e c a r r ie d o u t o n t h e b a s is o f t h e e q u a t io n :

019 + p _ V • 1 7 7 = 2 M s H + f ( p ) (8.2)

O t r

wh ere q i s t h e co o rd in a te o f t h e cen te r o f t h e wa l l, p = m O q / O t - rag1 i s the

m o m en tu m d en s i ty , m = m 0 [1 + (V• 2 - ( i t/ c ) 2 ] - 1 / 2 i s the " re la t iv is t ic"

m a s s o f t h e D W , f ( p ) i s t h e r e t a rd in g fo r ce d u e to d i s s ip a t io n in th e e l a s t i c

s u b s y s t e m o f t h e c r y s ta l . I n t h e f i rs t o rd e r o f n o n l in e a r p e r t u r b a t i o n t h e o r y

a b o u t t h e s m a l l p a r a m e t e r e = c r / L = 1 0 -2 , wh e re c i s t h e D W l im i tin g

v e lo c ity , r i s t h e t im e o f m ag n e t i c r e l ax a t io n , L is t h e l en g th o f n o n - o n e -

d imensiona l i ty , equa t ion (8 .2 ) i s equ iva len t to the Burgers equa t ion fo r ~ =

- Oq/Oz)

O ~ Og~ 1 0 2 ~

- - + e - - - 8 .3 )

O r O x R O x 2

w here t / = t / r and x ~ = x / c w are d imension less var iab les .

1 r O f

R - + - - - -

1 - v 2 / c 2 r n c O v

i s t h e an a lo g o f th e Re in o ld s n u m b er , v i s t h e D W v e lo ci ty . As i s k n o wn , b y

m e an s o f th e n o n l in ea r su b s t i tu t io n o f th e v a r i ab le s ~ = - 2 c ( O ~ / c g x ) / R ~

(8 .3 ) i s reduced to the l inear d i f fus ion equat ion :

~ _ 1

0 2 ~

O V R O x ~2


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8 .6 The Nonl inear M agnetoe las t ic W ave in I ron Bo ra te 127

T h u s i t i s p o s s ib l e t o g e t t h e g en e ra l s o lu t i o n fo r q [8 . 1 1 ] :

@ 2 7 ~ ( _ _ _ ~ R h l/ 2

q = v t + - - ~ in t ~4~r rtc2 j

o o

f R

e x p = ~

F o r t h e p a r a b o l i c i n it ia l d i s t u r b a n c e o f t h e D W

A0[1 - (2x /L) 2] 1 /2 , L < x < L

= 2 - - 2

L L

q ( 0 , x < - - ~ o r x > ~

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

l a x a t i o n t i m e w e r e m a d e w i t h t h e f ol lo w i ng v a lu e s o f t h e D W p a r a m e -

t e r : m 0 = 5 . 1 0 - 1 2 g / c m 2 , p = 1 0 4 c m / s - O e , A 0 =

L / 5 ,

AH1 = 40 Oe ,

A H t = 3 0 O e . T h e c u r v e s 3 a n d 3 i n F i g . 8 . 1 3 s h o w t h e t h e o r e t i c a l d e p e n -

d e n c e o f t h e r e l a x a t i o n t i m e o f t h e a m p l i tu d e o f t h e n o n l in e a r d i s t u r b a n c e o n

t h e D W v e l o ci ty , c a l c u l a t e d f o r L = 6 0 0 p m a n d 8 00 p m , r e s p e c ti v e ly . T h e y

q u a l i t a ti v e l y d e s c ri b e t h e e x p e r i m e n t a l r e su l ts . T h e m a x i m a o f t h e r e l a x a t io n

t i m e c o r r e s p o n d t o t h e m a x i m a R . A t v > S t a n d v > S 1, t h e r e e x i s t s m a l l

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

8 . 6 T h e N o n l i n e a r M a g n e t o e l a s t i c W a v e i n I r o n B o r a t e

M a g n e t i c a n i s o t r o p y i n t h e b a s ic p l a n e o f i r o n b o r a t e is n o t l ar g e . A n i n t e n s e

a c o u s t i c w a v e , p r o p a g a t i n g i n th e b a s i c p l a n e o f i r o n b o r a t e , c a n c a u s e a

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

p h a s e t r a n s i t i o n . T h e r e s u l t s o f t h e i n v e s t ig a t i o n s o f t h e i n t e r a c t i o n o f t h e

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

i r o n b o r a t e p l a t e l e t a r e p r e s e n t e d i n F i g . 8 . 1 4 [ 8 . 1 2 ] .

O b s e r v a t i o n s w e r e m a d e w i t h t h e h e lp o f t h e F a r a d a y e f fe c t i n t h e p l a t e l e t

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

t ic a l a x e s s o t h a t t h e v e r t ic a l a n d h o r i z o n ta l c o m p o n e n t s o f m a g n e t i z a t i o n

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

s p e e d p h o t o g r a p h y , w i t h t h e p u l s e o f l ig h t f r o m a n o x a z i n d y e la s e r, o p e r -

a t i n g a t a w a v e l e n g th o f 5 30 nm , p u m p e d b y a n i tr o g e n T E A - T E A l as e r,

w a s u s e d . T h e d u r a t i o n o f t h e l ig h t p u ls e w a s e q u a l to 0 . 25 n s. T h e s p e c i m e n

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

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

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

g e n e r a t e d w i t h t h e h e lp o f a p i e z o t r a n s d u c e r w i t h a f r e q u e n c y o f 3 .5 M H z

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


16/18

128 8 . N o n o n e D i m e n s i o n a l D y n am i cs o f D o m a i n Wal l s

Fig. 8.14 Nonlinear magnetoelastic w ave with transverse corrugation mov in g in

thin Fe B O3 plates an d obser ved with t he help of the Far ada y effect [8.12]

core, wh ic h serves as an acoustic delay. In the ab sen ce of a puls ed vol tage

on the piezotransducer, the Far ada y rotation across the spec imen was ho-

mogeneo us. This homoge neity did not change u p to a deformation of about

10 -6, created by the acoustic wa ve in the specimen. At an amp lit ude of de-

formation of about 3.1 0 -6, a magnetoelastic wav e may be observed. In this

case, the spe cim en of iron borate w as divided into a nu mb er of alternating

bright and dark ban ds of period of 500 m and with diffuse, weak ly man -

ifested walls directed pe rpen dicu lar to the wav e vector [8.13]. Th e velocity

of the magnetoel astic wa ve wa s equal to 1.8 • 0.2 km /s [8.12]. Th e rota-

tion of the plane of polarization wa s less tha n 0.5 ~ . As the voltage on the

piezotran sducer increased, the amp litu de of defo rmat ion in the acoustic wa ve

reaching 10 -5 , the hom oge nei ty inside the aforementione d propagating ban ds

disa ppea red [8.13]. An additional do ma in structure, with a chan ging ma gn e-

tization direction, app ear ed in every but on e of these movi ng bands. Th e

walls bet wee n the dyn ami c dom ain s bec ame mor e distinct. The structures

exhibi ted a characteristic distortion in the sh ap e of the regular plan e front,

resemb ling the structures on the dy na mi c DW in orthoferrites resulting fro m

supe rsoni c instability (Fig. 8.10). Th e structure, with transverse corrugation,

mo ve d as one piece at a velocity of 1.8 4- 0.2 km/s . Wi th in the accur acy of

the experiment, the velocity did not depe nd on the mag net ic field, the am-

plitude of sou nd or the specimen's thickness, wh en the latter varied from 30


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8 .6 Th e Nonl inear M agnetoe las t ic W ave in I ron Bo ra te 129

t o 7 5 m . I f t h e d i v e r g e n c e o f t h e a c o u s t i c w a v e is n o t l a r g e, t h e s t r u c t u r e s

a r e a l m o s t r e g u l a r [ 8 . 1 3 ] . W h e n t h e a c o u s t i c w a v e d i v e r g e s , t h e s t r u c t m ' e d i -

v e r g e s t o o ( F ig . 8 .1 4 ). I f t h e s o u n d a m p l i t u d e p u m p i n g i n t h e s p e c i m e n is

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

c e r t a i n i n c r e a s e in t h e i m a g e c o n t r a s t i s o b s e r v e d a n d t h e f r o n t d o m a i n w a l ls

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

f ie ld , H , i n t h e d i r e c t i o n o f t h e w a v e p r o p a g a t i o n r e s u l ts i n a n i n c r e a s e o f

t h e t r a n s v e r s e s iz e s o f t h e d a r k r e g io n s a n d a d e c r e a s e i n t h e s iz e s o f b r i g h t

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

o p p o s i t e d i r e c t i o n r e s u l ts i n t h e o p p o s i t e e f f e ct . I n t h e s e c a se s , th e p e r i o d o f

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

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

T h e d e p e n d e n c e o f t h e p e r i o d o f t h e s t r u c t u r e o n H • is p r e s e n t e d in

F i g . 8 .1 5 . T h e p e r i o d s m o o t h l y d e c r e a s e s a s t h e f ie ld g r ow s , i ts m i n i m u m

m e a s u r e d v a l u e b e i n g e q u a l t o 8 0 m . I n t h e d a r k r e g io n s o f t h e d y n a m i c

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

v e c t o r , w h i l e i n t h e b r i g h t r e g i o n s , i t is c lo s e t o t h e o p p o s i t e d i r e c t i o n o f t h i s

v e c t o r . T h e n e i g h b o r h o o d o f t h e d o m a i n s i s n o t s t r i c t l y 1 80 ~ A n i n c r e a s e

o f t h e m a g n e t i c f i el d a b o v e 4 0 O e , d i r e c t e d e i th e r p e r p e n d i c u l a r t o t h e w a v e

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

s t r u c t u r e i n t h e s p e c i m e n , t h a t is, i n t h e u n i f o r m m a g n e t i z a t i o n o f t h e c r y s t a l.

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

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

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

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

f l u x c lo s e s b e h i n d t h e s t r u c t u r e . I t i s p o s s ib l e t h a t i n t h e c l o s u re s o m e o t h e r

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

o t h e r b y t h e B l o c h w a l ls p a r t i c i p a t e . I t i s a l so p o s s ib l e t h a t t h e r e i s n o c l o s in g

o f t h e f lu x b e h i n d t h e n o n - e q u i l i b r i u m d y n a m i c s t r u c tu r e .

T h e p h y s i c s o f t h e d e s c r i b e d p h e n o m e n o n m a y b y c o n s i d e r e d a s f ol lo w s .

T h e v a l u e o f t h e a n i s o t r o p y c o n s t a n t i n t h e b a s ic F e B O 3 p l a n e is s m a l l a n d

is m a i n l y d e t e r m i n e d b y t h e e x t e r n a l p r es s u re . T h e a c t io n o f t h is o n e - s i d e d

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

s t r u c t u r e i n t h e s p e c i m e n w i t h 1 80 ~ d o m a i n w a ll s w h i c h i s p a r a ll e l t o o n e o f

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

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

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

l o c a l c o m p r e s s i o n i n t h e w a v e , t h e a n i s o t r o p y i n c r e as e s; w h e r e a s t h e d e f o r m a -

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

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

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

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

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

c u r . A s a r e s u l t, t h e m a g n e t i c m o m e n t s i n t h e r e g i o n o f t h e w a v e s tr e t c h i n g


18/18

1 30 8 . No n -o n e -D im en s io n a l Dy n am ics o f Do m ain W alls

)~, m

200

100

t

I I

10 20

H Oe

Fig . 8 . 1 5 Dep en d en ce o f t h e c o r ru g a t i o n p e r i o d o f a n o n li n ea r m ag n e to e l a s t ic wav e

in a th in p la te o f FeBO3 on the mag net ic f ie ld perp end icu lar to th e wave vec to r [8 .12]

w i ll b e c o m e p a r a l le l t o t h e d i r e c t i o n o f i ts p r o p a g a t i o n , a n d i n th e r e g i o n

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

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

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

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

f r o n t w h i c h d i s t i n c t l y e x h i b i t s t h e s a m e s p ec if ic p o in t s a s o n t h e s u p e r s o n i c

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

I t w o u l d b e a ls o w o r t h w h i l e t o d i sc u ss t h e l ow v e l o c i t y o f t h e m a g n e t o o p -

t i c a l v is u a l i z e d n o n l i n e a r m a g n e t o e l a s t i c w a v e , w h i c h is 5 t i m e s s m a l l e r t h a n

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

a n d , a s sh o w n b y o u r m e a s u r e m e n t s , a t l e a st is 3 t i m e s s m a l l e r t h a n t h e r e -

l o c i t y o f t h i s s o u n d i n a t h i n i r o n b o r a t e p l a t e l e t . I t i s e q u a l t o 6 .1 k m / s a t a

f r e q u e n c y o f 3 .5 M H z . I t is p o s si b le t h a t t h e n o n l i n e a r m a g n e t o e l a s t i c w a v e

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

s t re s s e d , t h a t t h e d e p e n d e n c e o f t h e v e l o c i t y o f t h e 1 80 ~ F e B O 3 D W o n t h e

a m p l i t u d e o f t h e d r i v i n g m a g n e t i c f ie ld , g i v e n a b o v e i n F i g . 4 .6 , is c o n s t a n t

o v e r a c e r t a i n r e g i o n , a t w h i c h v = 1 .9 k m / s .

08 - non one-dimensional dynamics of domain walls in week ferromagnets

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