the plane bloch domain wall structure in (110) plate of magnetically ordered medium with...

8/6/2019 THE PLANE BLOCH DOMAIN WALL STRUCTURE IN (110) PLATE OF MAGNETICALLY ORDERED MEDIUM WITH NEGATIV

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THE PLANE BLOCH DOMAIN WALL STRUCTURE IN (110)

PLATE OF MAGNETICALLY ORDERED MEDIUM WITH

NEGATIVE CUBIC MAGNETOCRYSTALLINE ANISOTROPY

Bogdan. anygin,lexandr V. ychko

1. The plane Bloch domain wall (DW) definition.

zO~

nW

M

M1M2

DW plane

normal

Domain

DW

Domain

~

=(M^nW)=const

M=(M2 M1)nW

Coordinate system


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2. Unrestricted crystal

2.1. The plain Bloch DW variation problem

( ) ( )[ ] 02/1/,,,

22

1

2=

+= =

dxdydzKjAEzyxji

ijjii

where ( ) ( ) ( )]001[^cos,]010[^cos,]100[^cos MMM === zyx ;A and K1is an exchange constant and magnetocrystalline anisotropy first

constant respectively , ij is the Kronecker symbol.

Plane Bloch DW surface energy density:

[ ] ( ) ( )[ ]{ }

~~,~~,

~~sin2

2

1

~

~

2/1

1

2deeA AA = ,

Magnetization distribution in DW volume:

[ ] ( ) ( )[ ]{ }

~~,~~

,~

/~

sin~~

0

2/1

1

2deeAz AA =

where, Ae is ( )222222

1 xzzyyxK ++ as a function of~

and ~

.

2.2. The DW type definition in unrestricted crystal

Traditionally DW typein unrestricted crystal means a scalar value

2(2-DW). Here 2is an angle between M1 andM2

An angle determines the DW plane equilibrium orientation (itassigns by an angle ) and all equilibrium parameters (particularly

equilibrium value of ( ) ).

At K1


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3. Restricted crystal: (110)-plate of cubic crystal withK1k3

;

;

010

100

001

1

21

TT

T

T

=

=


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4. Magnetization turn trajectories classification

The trajectories at large DW area The trajectories at small DW area

A multitude of all possible M turn trajectories in DW volume is

determined by all possible combinations of different (short (S-), long (L-)

and middle (M-) paths accordingly at ~ and ~ = , where12

~~~ = ) paths with various (right (R-) and left (L-) rotations)

rotations.

A common definition of rotation direction is:

RC 0 forL- rotation of magnetization vector,

where

( ) ( ){ } { }( ) ( )[ ]( ) ++= 1sgnsgnsgn1sgn 1 BnAMMMBMB WRC ;[ ] ( )( ) ( ) ( ) ( )[ ] cossgn1cos2/cossgn2/cossgn1 2 +++= MMPMA 1 ,

( )( ) [ ] ( ) ( )[ ] cossgn12sin/cossgncossgn1 2 ++= MMPB 1 ,

P is an arbitrary unit vector with 0=mP , ( )( )[ ]BA ]110[]110[sgn= .A kind of path is determined by parameter:

0pC forL- path of magnetization vector turn,

where

[ ]{ } ( )cossgnsgn 21 RWRp CCC == mmn

nW

M2

M1

[110]

Left turn

Long path

Right turn

Short pathnW

M2

M1

[110]

Right turn

Long path

Left turn

Short path


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5. Table. Energy density and structure of the equilibrium plane Bloch

DW in (110)-platesInter.

angle

DW type Trajectory type DW width (Lilley definition) and

energy (equilibrium state)

, 2, , , Rotation Path , ,/ 0 arb.units.

0S ,arb.units.

0

71 90 90 R-,L- S- 54.783 3.650 0.738

L- 54.385 13.028 1.899

109 90 90 R-,L- S- 81.519 10.143 0.994

L- - - -

60

71 60 45 R- S- - - -

L- 86.389 16.502 1.500

L- S- 74.248 4.163 0.497

L- 45.275 15.987 2.107109 45 60 R- S- 42.482 12.083 1.509

L- 82.427 12.026 2.508

L- S- 61.095 7.614 1.183

L- - - -

90

71 0 90 R-,L- S - 90.000 4.264 0.461

L- 42.813 20.792 2.263

109 90 0 R-,L- S- 90.000 9.710 0.981

L - 90.000 2 2.290

Arb. 180 Arb.90 R-,L- M- 82.788 8.135 2.004

35.264 M- 90.000 1.829

where ||/ 10 KA= , || 10 KA= .

Conclusion

1.There is three ( ( )1,1,3/1 =I , ( )1,1,3/1 =II , ( )3/1,1=III )-DW in (110)- sample with equilibrium parameters identical to acase of the unrestricted crystal.

2.The minimum -DW (2180) energy density is at =90, RL*-

turn and S- path. The minimum -DW (2=180) energy density is at=35.264.

3.Degeneration removal between R-and L-rotationofm in a volume of

some -DW takes place for 71-DW and 109-DW at interfacialangle value =60.

the plane bloch domain wall structure in (110) plate of magnetically ordered medium with...

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