04 - Main Features of Stimulated Motion of Domain Walls

8/20/2019 04 - Main Features of Stimulated Motion of Domain Walls

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4 M a i n F e a t u r e s

o f S t i m u l a t e d M o t i o n o f D o m a i n W a l l s

Application of the methods described in the previous chapter allows the de-

termination of the dependence of the velocity, of the forced stimulated motion

of the domain wall in weak ferromagnets on an external field H. Superh igh-

speed photography has shown that the motion of the domain wall in ortho-

ferrites is not always uniform and it does not always remain rectilinear, viz.,

under certain conditions the shape of the wall becomes more complicated,

which can be described as a phenomenon of self-organization. This problem

will be discussed further in Chap. 8. For orthoferrites, the key features of

the dependence of the domain wall velocity on a magnetic field, when non-

one-dimensionali ty can be neglected, will be presented below. The following

details are of particular interest:

(a) the presence of a linear part in the dependence of

v H)

in low fields,

(b) abrupt anomalies of the shelf type at some chosen values of the velocity,

and

(c) sa turation of the velocity in high fields.

We will discuss these peculiarities in this chapter. The same analysis car-

ried out for iron borate demonstrated quite a different behavior. In iron borate

the stat ionary motion of the domain wall of the NSel type can occur only at

velocities less th an some definite velocity depending on the one-side pressure

compressing the specimen. At this velocity, which is less than the velocity of

transverse sound, the dynamic phase transition takes place and the domain

wall acts as the nucleus of the new phase. The peculiarities in the motion of

the domain walls in iron borate will be discussed in a separate section. An

elementary theoretical analysis of the experiments described above completes

this chapter.

4 1 M o b i l i t y o f a D o m a i n W a l l

In low magnetic fields the velocity of a DW is linearly linked with the field

through the mobility # as follows:

v p H


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48 4. Main Features of Stimulated Motion of Domain Walls

Such dependence was first obtained experimentally in Fe-Ni wires by

S i x t u s

and

Tonk s

[4.1]. This problem was considered theoretically by

L a n d a u

and

L i f s h i t z

[4.2].

Ro s so l

was the first who started the investigation of the

mobility of the orthoferrites DW stabilized by a gradient magnetic field [4.3].

He investigated the frequency dependencies of the DW shifts z(cz) in the

magnetic field H -- H0 exp iwt, and used the stroboscopic methods based on

the Faraday effect. The dependencies x(c~) were shown to have a relaxation

character. In the frequency range to 107 Hz, the inertia of domain wall in

orthoferrites is negligible. The mobility of some orthoferrites, over a wide

temperatu re range, was determined from the relaxation frequency wl at which

the amplitude of the domain wall shift decreases by v~, using relationship

X 21

- H0

Here, x0 is static shift of the domain wall in the field H0. The experiments

were made in thin orthoferrite platelets cut perpendicular to the e axis. The

independence of the measured domain wall mobility on the value grad H

was especially tested. Temperature dependencies (T) for the three various

YFeO3 specimens are presented in Fig. 4.1 [4.4]. Specimen A underwent care-

ful mechanical polishing and subsequent chemical smoothing. As a result, its

coercive force had a small value less than 0.10e. The specimen's mobility was

sharp ly increasing with decreasing temperature . Specimen B was prepared in

the same way as specimen A. Specimen C was mechanically polished much

deeper, resulting in a higher coercivity and larger inhomogeneities in the

specimen. After additional annealing in an oxygenic atmosphere, the coer-

civity of the specimen dropped to 0 .1 0e . In the temperatu re range of 340 to

180 K, the (T) dependencies are practically the same in all the specimens.

In specimen B, the mobility ceases to increase with a further decrease of

temperature, after which a small decrease in the mobility is observed. Spec-

imen C showed a more noticeable decrease in mobility. The DW mobility in

specimen A, particularly above 180 K, represents the t rue mobility of YFeO3.

The reasons for decreasing mobility of specimens B and C in the range

of low temperatures are not yet quite clear. It is most probably due to the

presence of the Fe 2+, Fe4+ ions and of the rare-earth ions in the lattice of

yt tr ium orthoferrites, as well as due to the crystal defects. Specimen A, which

is of higher quality, is likely to exhibit the DW relaxation caused by in ternal

processes inherent in this crystal, while the interaction with impurities and

defects is important in specimens B and C at low temperatures.

The temperature dependencies of mobility in several rare-earth ortho-

ferrites were investigated by

Ro s so l

[4.3]. These dependencies qualitatively

reproduce the curves for specimens B and C (Fig. 4.1). It should be particu-

larly noted, tha t the DW mobility in yttr ium orthoferrite becomes extremely

high at 77 K. Hu a n [4.5] was the first to notice tha t the temperature depen-

dence of the DW mobility for specimen A, in Rossol's work, was proportional

to T -2. He also was the first to at tribute this fact to the four-magnon re-


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4 1 Mobility of a Domain Wall 49

~ cm.s I Oe 1

10 ~

1 4

o sam ple A ~

sample C

10a

P ~

0 100 200 300 T~ K

Fig 4 1 Temperature dependencies of the domain wall mobility for three different

YFeOa samples [4 4]

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

d i d n o t t a k e i n t o a c c o u n t i t s s u b l a t t i c e s t r u c t u r e . A g o o d c o r r e l a t i o n o f t h e

e x p e r i m e n t a l t e m p e r a t u r e d e p e n d e n c i e s o f t h e D W m o b i l i t y o f Y F e O 3 w i t h

t h o s e c a l c u l a t e d a s w a s i n d i c a t e d i n [ 4 . 6 ] a p p e a r s t o b e a r a n d o m c o i n c i -

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

a r e d e s c r i b e d b e l o w .

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

S h u m e i t [ 4 . 7 ] w h o f o u n d t h a t t h e e x p e r i m e n t a l l y d e r i v e d r a t i o o f B l o c h a n d

N 6 e l D W m o b i l i t y B / N = 1 . 0 6 w h i c h i s c l o s e t o t h e t h e o r e t i c a l r e s u l t f o r

o r t h o f e r r i t e s .

R . L . W h i t e , T s a n g ,

a n d

R . M . W h i t e

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

o f t h e D W m o b i l i t y u s i n g t h e S i x t u s - T o n k s m e t h o d [ 4 . 8 ] . T h e m o b i l i t i e s o f

B l o c h a n d N 6 e l t y p e D W i n Y F e O 3 i n t h e t e m p e r a t u r e r a n g e 2 5 0 - 6 0 0 K

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

v ( H ) .

T h e s e d a t a

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

R o s s o l

[ 4 . 3 4 . 4 ] . T h e f o l l o w i n g v a l u e s o f

t h e m o b i l i t y o f B l o c h a n d N 6 e l w a l l s a t r o o m t e m p e r a t u r e w e r e o b t a i n e d b y

t h e a u t h o r s : 6 . 1 6 . 1 0 3 c m / s - O e ; 5 . 8 . 1 0 3 c m / s . O e . T h e s e v a l u e s a r e s o m e w h a t

h i g h e r t h a n t h o s e i n [ 4 . 7 ] b u t t h e r a t i o o f m o b i l i t i e s i s a g a i n e q u a l t o 1 . 0 6 .


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5 0 4 . M a i n F e a t u r e s o f S t i m u l a t e d M o t i o n o f D o m a i n W a l l s

4 2 M a g n e t o e l a s t i c A n o m a l i e s i n t h e D y n a m i c s

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

The investigation of the nonlinear dynamics of the DW in yttrium orthofer-

rite was carried out in the aforementioned work [4.8] using the Sixtus-Tonks

method. Figure 4.2 reproduced from this paper gives the experimental de-

pendencies of the velocities of the DW of Bloch N~el and the h ead- to -head

type on the magnetic field at room temperature. These dependencies are lin-

ear on the initial parts of the curves. The values of the domain wall mobilities

are determined from these linear parts.

k m

s 1 2 A - A X I S / p r - B - A X I S j l [ - i ~ . H E A D [

N E E L / / B L O C H ~ r i ~ ~ |

o

i d L I ~ I

o i b / 1 7 I

I I [ ] I

4 : y I f \ I

0 0 s 2 0 4 o s ~ 2 ~ 6 0 6 ~ 2

H/139 Oe H / 1 3 9 Oe H / 1 3 9 Oe

Fig. 4.2 Dependencies of the velocity of domain walls of different types in YFeOs

on the magnetic field obtained by the Sixtus-Tonks method [4.8]

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

n o n l i n e a r . F i g u r e 4 . 2 s h o w s t h a t t h e r e i s a r a t h e r w i d e r a n g e w h e r e t h e v e -

l o c i t y o f t h e D W i s c o n s t a n t , i . e . t h e s h e l f ' o f t h e d e p e n d e n c e o f v o n H

o f a l l t h e d o m a i n w a l l s u n d e r i n v e s t i g a t i o n . F o r t h e N ~ e l - t y p e o f D W , t h i s

s t a b i l i t y o c c u r s a t a v e l o c i t y o f 4 k m / s . T h e s h e l l s f o r t h e B l o c h - t y p e d o -

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

v H )

c u r v e , a t v e l o c i t i e s o f 4 a n d 8 k m / s . A s

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

d o m a i n w a l l s m o n o t o n i c a l l y i n c r e a s e r e a c h i n g 1 3 k m / s .

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

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

t h e r a n g e w h e r e t h e v e l o c i t y i s e q u a l t o 4 k i n / s , t h e a u t h o r s o b s e r v e d l a r g e

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

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

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

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

d e t a i l b e l o w .


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4.2 Magnetoelastic Anomalies in the Dynamics 51

Similar nonlinearities of the dependence of v on H for the DW in ortho-

ferrites were also observed by the method of collapse of the bubble domain,

the method of recording the DW transit time over the given distance, the

method of high speed single and double photography. The dependence of the

velocity of the DW motion on the magnetic field in a YFeO3 platelet, cut

perpendicular to the c axis, obtained by the method of the bubble collapse is

presented in Fig. 4.3. The bias fields were equal to 22.3 and 23.70 e. On the

dependence of v on H, the authors of [4.9] observed a very weak peculiarity at

the DW velocity of 4.8 km/s, more distinct peculiarities were observed at the

velocities 7 and 14 km/s. No saturation of the velocity of the DW motion in

the magnetic fields up to 370 Oe was observed. The maximum experimental ly

found value of the DW velocity equaled 25 km/s. All attempts to attribute

these peculiarities of the dependence of v on H in ytt rium orthoferrite at the

velocities 4 and 7 km/s to the Walker limiting velocity, taking into account

its orthorhombic anisotropy, were found to be incongruous. The studies have

shown that these values of the DW velocity coincide with the velocities of

the longitudinal and transverse sound in yttrium orthoferrite [4.8].

v k m / s

25

20 / o ~ ~ 1~

1.5 Y F e ~ 2 j ~

10

/

, / ~ o - Hb = 23 .7 0e

5 / 9 - Hb = 2 2 . 3 0 e

I I i i

100 200 300 H, Oe

Fig. 4.3 Dependence of the domain wall velocity in YFeO3 on the magnetic field,

obtained by the bubble collapse method for two values of the bias fields [4.9]

It should be noted that weak ferromagnet orthoferrites were found to be

the first magnetically ordered substances in which the velocity of the DW

motion had reached and exceeded the velocity of the sound. As mentioned

above, the interpretation of the peculiarities of the dependence of v on H,

at the values 4.1 and 7 km/s, is also supported by experimental results ob-

tained in the investigation of the dynamics of the intermediate-type DW in

TmFeOa [4.10].


6/17


7/17

4.3 Dynamics of Domain Walls in Iron Borate 53

sion for the width of the range over which the velocity is constant, AHt AH1

includes the coefficient of the sound attenuation. It depends substan tially on

the t empera ture and has only been, so far, determined for ErFeO3. The the-

ory predicted the existence of hysteresis in the dependence of v on H [4.6].

After the DW attains the supersonic velocity, its velocity should smoothly

decrease with decreasing H. In this case, the interval, where the DW veloc-

ity is constant, should not exist. The experiment, to be described at a later

stage, does not confirm this theoretical assumption. When the domain wall

moves at the supersonic velocity, decreasing the magnetic field results in a

sharp decrease in the domain wall velocity down to the velocity of sound. No

hysteresis is observed in the dependence of v on H. A possible interpretation

of this fact will be given in Chap. 6.

4 . 3 D y n a m i c s o f D o m a i n W a l l s i n I r o n B o r a t e

The investigation of the dynamics of domain walls in iron borate was carried

out at 290 K on platelets with their developed plane coinciding with the

basic one. The thickness of the platelets ranged from 20 to 100 m, while the

cross-sectional sizes were equal to several millimeters. A single 180 ~ domain

wall was formed with the help of on external one-side compressing pressure

applied to the plane of the specimen and to the gradient magnetic field. The

value of the pressure reached was 2.109 d in /cm2, and the value of the gradient

magnet ic field was varied up to 70 Oe/cm. The motion of the DW was caused

by the application of a pulsed magnetic field, with the pulse time rise equal

to 6 ns. The investigations were performed in [4.13] by the method of the

double-shot high speed photography [4.14].

The difference in rotations of the plane of polarization for two adjacent

domains with the platelet inclined around the horizontal axis did not ex-

ceed 1~ This was the major obstacle in the application of the method of

double-shot high-speed photography. An increase in the power of the laser

radiation, as compared to the case in investigating the dynamics of the DW

in orthoferrites, where the Faraday rotat ion is much higher, helped to resolve

this problem. The conventionally used superluminescence was replaced by

the generation of the dye laser pumped by a nitrogen TEA-TEA laser. The

duration of the light pulse was equal to 0.25 ns. The attainment of more

reliable recording of two sequential dynamic domain structures, both by the

method of double-shot photography (see Sect. 3.7) and in the DW contrast

was possible in the work described in [4.13]. Moreover, this technique made

it possible to simultaneously fix two or three half- tones and hence to inves-

tigate the dynamic domain structure and the profiles of the moving domain

wall in the real-time scale.

The analysis of the dependence of the DW velocity on the amplitude

of the driving magnetic field, has shown that the stationary motions are


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5 4 4 . M a i n F e a t u r e s o f S t i m u l a t e d M o t i o n o f Do m a i n W a l ls

p o s s i b l e o n l y a t v e l oc i ti e s l es s t h a n s o m e v e l o c i ty v l , d e p e n d i n g o n t h e o n e -

s id e e x t e r n a l c o m p r e s s i n g p re s s u re . A s t h e D W i n t h e m a g n e t i c f ie ld H 1

a c h i ev e s t h e v e l o c i t y v l , t h e D W b e c o m e s c o n s i d e ra b l y w i d e r t h a n a t t h e l o w

v e l o c it ie s a n d t r a n s f o r m s i n to a n e w d o m a i n . T h e d i r e c t io n o f m a g n e t i z a t i o n

i n t h e n e w d y n a m i c d o m a i n w a s r o t a t e d b y a n a n g le o f a b o u t 9 0 ~ w i t h r e s p e c t

t o t h e m a g n e t i z a t i o n , i n t h e i n i ti a ll y e x is t in g d o m a i n s . I n t h e p h o t o g r a p h , t h i s

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

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

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

t h e d y n a m i c d o m a i n o f 9 0 ~ n e i g h b o r h o o d . I n o t h e r w o r d s , a s t h e D W r e a c h e s

t h e v e l o c i t y v l , t h e d y n a m i c s p i n - r e o r i e n t a t i o n a l p h a s e t ra n s i t i o n t a k e s p l a c e

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

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

o n t h e e x t e r n a l p r e s s u r e c o m p r e s s i n g t h e s p e c i m e n a r e g i v e n in F i g . 4 .5 .

5

k m / s . 1 0 - 3 c m / s O e

_ ~ 0 ~ 0~0

e

2

200

150

100

p . 1 0 - 9 d i n / c m 2

F i g . 4 . 5 De p e n d e n c i e s o f t h e c r i t ic a l v e lo c i t y v l a t t h e b e g i n n i n g o f t h e d y n a m i c

o r i e n t a t i o n a l t r a n s i t i o n , i n wh i c h t h e 1 80 ~ DW d i s i n t e g r a te s , a n d i t s m o b i l i t y / x o n

t h e o n e - s i d e c o n t r a c t i n g p r e s s u r e i n a F e B Oa p l a t e [ 4 . 1 3 ]

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

2 . 1 0 5 c m / s - O e . T h e r e s u l t s o f t h e e x p e r i m e n t s l ea d t o a c o n cl u s io n a b o u t t h e

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

q u e s t i o n o f i t s e x i s t e n c e w a s t h e o r e t i c a l l y c o n s i d e r e d i n [ 4.1 5]. I t w a s s h o w n

t h a t t h e o n e - d i m e n s i o n a l d y n a m i c N ~e l D W b e c o m e s u n s t a b l e w h e n i t s v e -

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

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

v e l o c i t y t h a t t h e e f fe c ti v e c o n s t a n t o f a n i s o t r o p y c h a n g e s i t s s ig n . T h e o r i en -

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

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

p e n d s o n p r e s s u re . D e p e n d e n c i e s o f t h e v e l o c it y of s t a t i o n a r y D W m o t i o n i n


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10/17

56 4 . M ain Fea tu res o f S t im ula ted M ot ion o f Dom ain Wal ls

v , k m / s

20

o ~ . o . ~ . , ~ . o ~ e - ~

15

I

5

0 r i i

200 400 600 800 H , O e

F ig . 4 . 7 Dep en d en c e o f t h e d o m a in wa l l v e lo c it y i n a YFeOa p l a t e , cu t p e rp en -

d icu lar ly to the op t ica l ax i s , on the magnet ic f i e ld , ob ta ined by record ing the DW

tran s i t t im e over a g iven d is tance b e tween two l igh t spo ts [4.17]

F i g . 4 . 7 i n d i c a t e s a s u b s t a n t i a l d i f fe r e n c e . I n t h e m a g n e t i c f i e ld o f 6 0 0 O e ,

t h e D W v e l o c i ty a t t a i n s 2 0 k m / s a n d d o e s n o t c h a n g e w i t h a n i n c re a s e i n

t h e p u l s e d f ie ld t o 1 00 0 O e . S u b s e q u e n t e x p e r i m e n t s h a v e s h o w n t h a t t h e

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

t h e l im i t in g v e l o c i t y o f t h e D W i n o rt h o f er r it e s . T h u s , t h e m e t h o d d e s c r i b e d

i n [ 4 . 1 0 , 1 6 - 1 8 ] , a l l o w e d f o r t h e f i r s t t i m e t h e e x p e r i m e n t a l d e t e r m i n a t i o n o f

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

I n t h e a f o r e m e n t i o n e d w o rk , Chetkin a n d Campa [4 . 1 7 ] h av e i n d i ca t ed

t h a t t h e l i m i ti n g v e l o c i t y o f t h e 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 is e q u a l

t o t h e v e l o c i t y o f s p i n w a v e s o n t h e l i n ea r p a r t o f t h e i r d i s p e r si o n l a w , w h i c h

d e p e n d s o n l y o n t h e e x c h a n g e c o n s t a n ts a n d d o e s n o t d e p e n d o n t h e c o n s t a n t s

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

a s y m p t o t i c m a g n e t i z a t i o n o f t h e D W , R e f . [4.1 9]) a n d t h e c o n c l u s i o n s f r o m

a m o r e s t r i c t t h e o r y w h i c h le a d s to t h e L o r e n t z - i n v a r i a n e e o f t h e e q u a t i o n s

R efs . [4 .2 0 ,2 1 ], see C h a p . 2 ) . Us in g co n s t a n t s c~ an d 5 , w h ich w ere d e f i n ed

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

c = ~ g M 0 V ~ 4 .1 )

I t i s c o n v e n i e n t t o r e w r i t e t h i s f o r m u l a u s in g W l t h e g a p i n t h e s p e c t r u m o f

t h e l o w e r m a g n o n b r a n c h ) a n d t h e D W t h i c k n e s s A [4 .1 7,1 9].

C = C O l a 4 . 2 )

F o r m u l a c ~ ak TN h c a n b e u s e d t o e s t i m a t e t h e o r d e r o f m a g n i t u d e , h e r e

T N is t h e N 6 e l t e m p e r a t u r e a n d a i s t h e l a t t ic e c o n s t a n t .

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

t h e w av e v ec to r k : A -1


11/17

4.4 Lim iting Velocity 57

/j} S I

2 .1 0 s 1 1 1 ) 1 0 0 )

1013 ~ ( 1 1 0 )

0 I I I I I I r I t I I

1 9 2 (1 r / a )

F ig . 4.8 Spin-wave spe ctra of YFeO 3 calculated from exchange integrals inside the

Brillouin zone [4.22]

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

(see Fig . 4 .8 ) o r ca lcu la ted accord ing to fo rmula (4 .1 ) , agrees wel l wi th the

ex p e r im en ta l ly o b ta in ed v a lu e o f th e l im i tin g v e lo c i ty o f t h e d o m a in wa l l.

I n fa c t, a s su m i n g th a t 7 = 1 . 7 6 . 1 0 T O e - i s - 1 , H E = 6 M o 2 = 6 .4 - 10 Oe,

A = c~M3 2 = 4 .4 9 10 -7 e rg /cm , w here 11//0 i s m agn et iza t io n o f the i ron

s u b l a t t i c e , w e o b t a i n t h e v a l u e 2 . 1 0 6 c m / s f o r c , w h i c h is in g o o d a g r e e m e n t

w i t h t h e e x p e r i m e n t a l v a l u e .

Tw o im p o r t an t f ac to r s sh o u ld b e m en t io n ed . F i r s t ly , t h e v e lo c i ty o f sp in

wav es in o r th o fe r r i t e s d e p en d s wea k ly o n th e d i r ec t io n o f th e i r p ro p ag a t io n

( see F ig . 4 .8 ) . Th e ex p e r im en ta l an a ly s i s h a s sh o wn an i so t ro p y in th e v a l -

u es o f t h e D W l im i t in g v e lo c i t ie s . Th ese v e lo c it ie s , fo r B lo ch an d N6 e l wa l l s

in Tm FeO 3 , a r e p r ac t i ca l ly th e sam e . S eco n dly , t h e l im i tin g v e lo c i ty o f t h e

D W in c lu d es o n ly th e ex ch an g e co n s tan t s c~ an d 6 an d d o es n o t i n c lu d e th e

co n s tan t s o f an i so t ro p y ( in co n t r a s t w i th th e W alk e r l im i t in g v e lo c i ty fo r t h e

d o m a in wa l l s i n f e r ro m ag n e t s ) . Th e v a lu es o f t h e ex ch an g e co n s tan t s f o r v a r i-

o u s o r th o fe r r i t e s a r e c lo se to each o th e r an d , t h e r e fo r e , so sh o u ld b e th e v a lu es

o f th e l im i t in g v e lo c it i e s. M o reo v er , t h e ex ch an g e co n s tan t s o f o r th o fe r r i t e s

w e a k l y c ha n g e w h e n t h e t e m p e r a t u r e d e cr ea s es fr o m r o o m t e m p e r a t u r e d o w n

t o t e m p e r a t u r e o f l iq u id N 2 . T h e l im i t in g v e l o c i ty o f t h e i n t e r m e d i a t e - t y p e

D W i n T m F e O 3 a t 1 7 8 K a n d i n E u F e O 3 a t 7 7 K h a s t h e s a m e v a l u e a s

i n Y F e O 3 a t r o o m t e m p e r a t u r e s [ 4 . 6 ] . A l l t h e s e f a c t o r s a s w e l l a s t h e c o i n -

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

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

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

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

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

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

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

l i g h t p u l s e s o f 0 . 2 5 n s d u r a t i o n .


12/17

58 4 . M ain Fea tu res o f S t im ula ted M ot ion o f Do main Wal ls

v k m / s

15 F

r

s / -

I i

t

0 ~ J t r i r i p , , , , i q , , r i T , , , _ . 1 _

200 400 600 800 H Oe

Fig . 4 .9 D epend ence o f the d om ain wal l ve loc i ty in YFeO3 on th e m agne t ic f ie ld ,

o b t a in ed b y t h e d o u b l e h ig h s p eed p h o to g rap h y m e th o d ( . . . ) an d t h e t h eo re t i ca l

dep end enc e calcu lated from (4.3) for = 1 .3 .104 cm-s -1- Oe -1 an d c = 2 .106 cm /s

- - )

F i g u r e 4 . 9 g iv e s t h e d e p e n d e n c e o f v o n H i n t h e y t t r i u m o r t h o f e r r i t e

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

d i t i o n t o t h e p e c u l i a r it i e s m e n t i o n e d a b o v e a n d t h e l im i t i n g v e lo c i ty , t h e r e

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

y e t b e e n i n t e r p re t e d . T h e y a r e, p e r h a p s, d u e t o th e D W r e t a r d a t i o n c a u s e d

b y e x c i t a t i o n o f t h e D W o s c il la t io n s b y i rr e g u l a r it i e s in t h e c r y s t a l. T h e t h e -

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

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

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

e x p r e s s i o n :

v(H) = H

(4 .3 )

v /1 + , H / c ) 2

H e r e i s t h e D W m o b i l it y , c is t h e l i m i t i n g v e l o c it y , H i s t h e m a g n e t i c

f i el d [ 4. 20 ,2 1 ]. E x p r e s s i o n ( 4 .3 ) i s c o m m o n f o r t h e s y s t e m s d e s c r i b e d b y t h e

S i n e - G o r d o n e q u a t i o n w i t h d i s s i p a t i o n a n d a n e x t e r n a l f o r c e .

H a v i n g d e t e r m i n e d t h e m o b i li ty f r om t h e i ni ti a l p a r t o f t h e e x p e r i m e n t a l

c u r v e o f v ( H ) , w e c a n p l o t t h e c o m p l e t e c u rv e

v(H)

u s i n g ( 4 . 3 ) a n d c o m p a r e

i t w i t h e x p e r i m e n t . T h i s d e p e n d e n c e , f or ~ = 1 . 3 . 1 0 4 c m / s . O e a n d c =

2 . 1 06 c m / s , i s r e p r e s e n t e d i n F i g . 4 .9 b y t h e s o l id l in e . A s c a n b e s e e n ,

i t d e s c r i b e s t h e e n t i r e e x p e r i m e n t M c u r v e

v(H)

q u i t e we l l . T h u s , i t f o l l o ws

t h a t t h e d y n a m i c s o f t h e d o m a i n w a ll in o r t h o f e r r i t e s is q u a s ir e l a ti v i s ti c , i t ' s

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

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

o f t h e d o m a i n w a ll m o t i o n i n o r th o f e r r i t e s , a n d t h e t h e o r y o f f o r c e d m o t i o n ,

were g iv en i n R e f s . [4 . 6 , 4 . 2 0 , 4 . 2 1 ] an d a re d e s c r i b ed b e lo w.


13/17

4.5 Elementary Theoretical Analysis 59

4 5 E l e m e n t a r y T h e o r e t i c a l A n a l y si s

The problem of calculating the velocity of the DW steady-state motion, v,

due to the influence of the exte rnal magnetic field H, is based on the analy-

sis of two main factors associated with the driving force and analysis of the

dissipative force F v) . If the relaxation processes occurring in the system are

weak, the calculation of F v ) and, consequently, of the curve v H ) can be

based on the known solutions for the nondissipative medium. If the distribu-

tion of magneti zation within the wall is known, it is possible to calculate the

dependence of the retarding force affecting the DW on the velocity, in other

words, to find the function F v) . Setting this force equal to the external

force affecting the wall, we find the dependence of the domain wall veloci ty

on the external force. This approach is feasible due to the weakness of re-

laxation in the magnetic material and verified by the inequality g A H


14/17

60 4 . M ain Fea tu res o f S t imu la ted M ot ion o f Dom ain Wal ls

o f C h e r e n k o v r a d i a t i o n o f v a r i o u s q u a s ip a r t ic l e s ( e. g. , p h o n o n s ) . T h e c o n -

t r i b u t i o n o f t h e s e p r o c e s se s r e s u l t s in t h e f o r m a t i o n o f n a r r o w p e a k s i n t h e

c u r v e F ( v ) a t t h e v a l u e o f t h e w a l l v e l o c i t y c lo s e t o t h e p h a s e v e l o c i t y o f

q u as i p a r t i c l e s .

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

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

ap p ro ac h i s, f ir s tl y , t h e s i mp l e s t an d m o s t d e s c r i p t i v e an d , s eco n d l y , g i v es a

g o o d d e s c r i p ti o n o f t h e e x p e r i m e n t a l d e p e n d e n c e o f v o n H , i n o r t h o fe r r i te s

( se e F i g . 4 .9 ) . A m o r e c o n c i s e t h e o r y o f th e r e l a x a t i o n p r o ce s s es , b a s e d o n

mi c ro s co p i c t h eo ry , w i l l b e g i v en b e l o w , i n Ch ap . 7 .

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

f o r m :

Q = M o / 2 g / A r ( O l / O t ) 2 d r (4.5)

T h i s r e l a t io n r e s u lt s fr o m t h e r e l a x a t io n t e r m s i n t h e G i l b e r t o r L a n d a u -

L i f s h it z f o r m , u s e d in m o s t w o r k s o n t h e D W d y n a m i c s . A s L a n d a u a n d

L i f s h i t z n o t e d i n t h e i r c l a s s i c a l w o r k , t h i s t e r m d e s c r i b e s t h e r e l a x a t i o n p r o -

ces s e s o f r e l a t i v i s t i c n a t u re .

B a r y a k h t a r s h o w e d [4 .2 3] t h a t t h e e x c h a n g e c o n t r i b u t i o n t o t h e d i s si p a -

t i v e f u n c t i o n d i ff e rs f r o m (4 .5 ) b y i t s s t r u c t u r e . I n t h e c a s e o f w e a k f e r r o m a g -

n e t s , i t i n c l u d e s t w o c o n t r i b u t i o n s :

f { ) [V(Ol /Ot)] 2 + /Ve(O21 /Ot2)} dr

(4.6)

~

mo 2g

M o r e o v e r , a d e t a i l e d a n a l y s i s o f t h e r e l a t iv i s t ic r e l a x a t i o n h a s s h o w n t h a t

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

b e t a k e n i n t o a c c o u n t b y u s i n g t h e s u b s t i tu t i o n :

Ar fcq/N2~,~-~) A

Oli Olk

-+ ik - f f ( Ot

F o r t h e t h e o r e t i c a l a n a ly s i s o f t h e s t e a d y - s t a t e m o t i o n o f t h e D W , w e u s e

t h e 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 t h e o r y . I n t h i s v e r si o n o n l y t h e

e x c h a n g e - r e l a t iv i s t ic i n v a r ia n t o f th e e n e r g y o f t h e D z y a l o s h i n s k i i - M o r i y a in -

t e r a c t i o n i s t a k e n i n t o a c c o u n t , i . e . , t h e c o n d i t i o n A I ( O , qo) = A 2( 0 , g)) = 0 is

u s e d i n e q u a t i o n ( 2 .3 0 ) . T h e c o n s i d e r a t i o n o f t h e e f f ec t s o f t h e b r e a k i n g o f

t h e L o r e n t z - i n v a r i a n c e r e s u lt s in u n u s u a l e ff e c ts l ik e d y n a m i c r e c o n s t r u c t i o n

o f t h e d o m a i n w a l l s, r e 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 i n t h e w a l l (s ee

I v a n o v

et al.

[4.24],

G o m o n a i e t a l .

[4 .2 5] ). A c co rd i n g t o t h eo ry , i n t h e ca s e o f o r-

t h o fe r r i t e s , t h e s e e f f ec ts can b e o b s e rv e d o n l y a t r a t h e r h i g h v e l o c i ti e s , v ~ - c

( e x c e p t f or d y s p r o s i u m o r t h o f e r r i t e a t T = 1 50 K ) . N o e x p e r i m e n t s h a v e

y e t b een ca r r i ed o u t i n t h i s r eg i o n ; fo r t h i s r ea s o n , w e w i l l n o t d i s cu s s t h e s e

e f f e c ts h e r e i n a n d r e fe r t h e r e a d e r t o t h e o r i g in a l p u b l ic a t i o n s .


15/17

4 .5 E l e m e n t a r y T h e o r e t i c a l An a l y s i s 6 1

I n t h e L o r e n t z - i n v a r i a n t t h eo r y , th e t r a n s i t i o n t o t h e D W m o t i o n i s p e r -

f o r m e d b y c h a n g i n g x t o ( x - v t ) ( 1 - v 2 / c 2 ) - 1 / 2 i n t h e r e l e v a n t f o r m u l a e

( se e C h a p . 2 ) w h i c h d e s c r i b e s t h e d i s t r i b u t i o n o f I i n t h e s t a t i c w a l l. T h e

d i s t r i b u t i o n o f m a g n e t i z a t i o n is r e a d i ly d e t e r m i n e d f r o m f o r m u l a ( 2 .2 8 ).

T h i s p r o c e d u r e , f o r t h e a c - t y p e w a l l o f o r t h o f e r r i t e , gi v es ( w e re s t r ic t t h e

p r o c e d u r e b y t h e s i m p le s t a n t i s y m m e t r i c a l in v a r ia n t i n t h e D z y a l o s h in s k i i -

M o r i y a i n t e r a c t i o n ) y i e ld s :

l x = t a n h [ ~ / A l ( V ) ] , [y = 0 ,

2v/ (v)

= g M o c o s h

w h e r e ~ = x - v t .

1

l z c o s h [ { / A 1 ( v ) ] , ( 4 . 7 )

d

rnz

~

t a n h [ ~ / A l ( v ) ]

4 . s )

U n l i k e t h e s t a t i c c a se , th e m a g n e t i z a t i o n d e v i a t e s in th i s w a l l d u r i n g t h e

m o t i o n f r o m t h e w a l l p l a n e , t h a t is ,

m y ~ v / c ~ O .

I t i s n o t e w o r t h y t o m e n -

t i o n t h a t t h e r e l e v a n t c o m p o n e n t m is a n e v e n f u n c t i o n o f ~. T h e d e v i a t i o n o f

m f r o m t h e a c p l a n e e n t a i ls t h e d e v i a t i o n o f I f r o m th e s a m e p l a n e a t v ~ 0 .

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

r e l a t i o n m l = 0 . T h e v a l u e o f ly ~ 0 , w i t h a c c o u n t t a k e n o f b o t h i n v a r i a n t s

o f t h e D z y a l o s h i n s k i i - M o r i y a i n t e r a c t i o n ( se e [4 .2 5]).

I n t h e d o m a i n w a l l of t h e a b - t y p e , t h e m a g n e t i z a t i o n d u r in g t h e m o t i o n

r e m a i n s p a r a l l e l t o t h e c a x i s :

1

Ix = t a n h [ { / A 2 ( v ) ] , ly = c o s h [ ~ / A 2 ( v ) ] , lz = 0 , (4 .9 )

d 2 v / A 2 ( v )

(4 .10)

= g t a n h [ U & ( v ) ] + g eM 0 c o s h

I n t h e s e f o r m u l a e A ~ ( v ) = A i ( 1 - v 2 / c 2 ) 1 / 2 , i = 1 , 2 f o r wa l l s o f t h e a c - a n d

a b - t y p e s , r e s p e c t i v e l y , A ~ = ( a / / 3 i ) 1 / 2 , /31 a n d / 3 2 a r e e f f e c t iv e c o n s t a n t s o f

a n i s o t r o p y ( 2 . 1 4 ) .

U n l i k e t h e a c - w a l l , n e w c o m p o n e n t s o f v e c t o r s m a n d I d o n o t a p p e a r

a t v 0 i n t h e d o m a i n w a ll o f t h e a b - t y p e . H o w e v e r, t h e s y m m e t r y o f t h e

w a l l a t v 0 d e c r e a s e s , a s c o m p a r e d w i t h t h e c a se w h e n v = 0 , a n d t h e

s a m e d e c l i n e is o b s e r v e d a l s o f o r t h e a c - w a l l . T h e r e a s o n i s t h a t a t v = 0 t h e

f u n c t i o n m z ( { ) i s o d d a n d a t v 0 i t is n o t e v e n o r o d d . T h i s m e a n s t h a t a t

v = 0 , i t is p o s s i b l e t o i n t r o d u c e a g e o m e t r i c a l c e n t e r o f t h e w a l l ( t h a t i s, t h e

p o i n t a t w h i c h s i m u l t a n e o u s l y l x = O , m z = 0 a n d l y r e a c h es a m a x i m u m ) ,

w h i l e a t v 0 t h i s e l e m e n t o f s y m m e t r y is l os t.

T h u s , b o t h w a l ls i n o r t h o f e r r i te s e x h i b i t a r e d u c t i o n s y m m e t r y a t v 0 , a s

c o m p a r e d w i t h t h e c a s e o f v = 0 . T h i s f a c t, e st a b l i s h e d b y

B a r y a k h t a r e t al.

[4 .2 6] o n t h e b a s i s o f g e n e r a l m o d e l l e s s c o n s i d e r a t i o n s , i s o f g r e a t i m p o r t a n c e

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

i n d u c e d b y t h e v e lo c it y . I n t h e c a se o f o r t h o f e r r it e s a n d i ro n b o r a t e , t h e


16/17


17/17

4 .5 E l e m e n t a r y T h e o r e t i c a l A n a l y s i s 6 3

s h o u l d b e m e n t i o n e d , a g a in , t h a t i n d e r iv i n g t h e e f fe c t iv e e q u a t i o n a n d t h e

L a g r a n g i a n 2 .3 0 ), w e a s s u m e d t h a t a V 2 l ) > a o r

1 - v 2 / c 2 ) > a / A ) 2 = / 3 a 2 / c ~ ~ j 3 / 6 4 . 1 4 )

T h u s , t h e f o r m u l a e o f l o ng w a v e l e n g t h a p p r o x i m a t i o n 4 . 1 )- 4 . 1 3 ) , d e r i v e d

o n t h e b a s i s o f t h e L a g r a n g i a n 2 .3 0 ), c a n b e u s e d t o d e s c r i b e t h e m o v i n g d o -

m a i n w a l l e ls e w h e r e , e x c e p t f o r t h e n a r r o w r a n g e o f v e l o c i t ie s _~ / ~ /5 ) _~ 1 0 - 2

n e a r t h e w a l l l i m i t i n g v e l o c i t y c [4 .6 ]. T h e s o l u t i o n s d e s c r i b i n g t h e m o t i o n

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

2 . 2 6 ) , w i t h o u t t h e a p p r o x i m a t i o n ]rn ] < < ]/] o r a ~ 7 2 / ) < < 5 , a r e g i v e n i n

R e f . [4 .2 7]. I t s h o u l d b e n o t e d , h o w e v e r , t h a t t h e c o n d i t i o n o f t h e l o n g w a v e -

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

t h e m a g n e t 2 .1 ) o r 2 .9 ) . S t r i c t ly s p e a k in g t h e e n e r g y a ls o c o n t a i n s t h e c o m -

p o n e n t s o f t h e o r d e r a a 2 V 2 / ) 2 , a a 2 V / ) 4 , e t c , w h i c h c a n o n l y b e o m i t t e d

i n t h e l o n g - w a v e l e n g t h a p p r o x i m a t i o n . I n t h e f o r e m e n t i o n e d n a r r o w r a n g e o f

v e l o c it ie s , w h e n t h e D W t h ic k n e s s i s c o m p a r a b l e w i t h t h e l a t ti c e c o n s t a n t ,

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

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

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

s y s t e m o f a m a g n e t i c m a t e r ia l s e e B a r y a k h t a r e t al. [4.19]).

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