micromagnetic structure and symmetry of the 90 degree domain wall suppressed by the demagnetization...

8/7/2019 Micromagnetic Structure and Symmetry of the 90 degree Domain Wall Suppressed by the Demagnetization Field

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Preprint DRAFT 2011

Micromagnetic Structure and Symmetry of the 90 degree Domain

Wall Suppressed by the Demagnetization Field

B.M. Tanygina1, O.V. Tychkoa

aKyiv Taras Shevchenko National University, Radiophysics Faculty, Glushkov av.4g, Kyiv,

Ukraine, MSP 01601

1Corresponding author: B.M. Tanygin, 64 Vladimirskaya str., Taras Shevchenko Kyiv

National University, Radiophysics Faculty. MSP 01601, Kyiv, Ukraine.

E-mail: [email protected]

Phone: +380-68-394-05-52

Abstract.

A magnetization distribution in the yttrium-ferrite garnet (001) plate without

unidirectional anisotropy term is numerically considered for various sample thickness. The

smooth transition (from the to the domains) of the domain structure in case of

the plate thickness growth have been investigated. It was shown that growth of the plate

thickness leads origin of the periodical domain structure in the volume of the 90

domain wall between domains.

PACS: 75.60.Ch, 75.70.-i, 75.78.Cd

Please, cite original work as:B.M. Tanygin, O.V. Tychko,A magnetization distribution in a single domain wall volume,Proceeding of the II International Conference "Electronics and Applied Physics", Kyiv, Ukraine (2006) 40-41.

e-mail: [email protected]: http://sites.google.com/site/btanygin/


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Preprint DRAFT 2011

1. Introduction

It is well-known that problems of the applicability of the Bloch and Nel domain wall

(DW) model are usually solving via the micromagnetic simulation start from the work.

Usually changes of the micromagnetic structure in the DW volume or changes of the domain

structure (including domain magnetization reoritation) depend on the material sample

parameters are usually under consideration. Here we investigate both types of changes in the

same time.

An information about magnetization vector spatial distribution in a medium volume is

basic at a theoretical investigation of magnetic states and processes in a magnetically ordered

media. Reductive models that allow getting an analytical description of such distribution are

used for receiving this spatial distribution. Hence these models only can characterize a

domain structure and DW approximately, often qualitatively. One can obtain detailed

information about film media micromagnetic structure using numerical methods of

micromagnetic problem with initial and limit conditions.

2. Calculation technique

Let us consider 2D-problem numerical simulation in cross-section of a thin (100)-film.

Let Z and X axis are directed along film normal and crystallographic [100] direction

respectively. A free energy functional counting upon a unity of length along Yaxis look like:

( )[ ] += rr2

dgG , where r is the Lagrange multiplier, KmA gggg ++= is a volume

energy density, ( ) ( )22 // zxAgA += is a an exchange energy density, A is an

exchange constant, M/ M= , M is a saturation magnetization; ( ) 2/mmg HM = is a

demagnetization field energy density, mH is a demagnetization field,


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Preprint DRAFT 2011

( ) 2/1221 pqqpK Kg = is a magnetocrystalline anisotropy energy density, pq is the

Kronecker symbol, .,,, zyxqp =

Requirement of an equilibrium magnetization distribution is 0=G or an absence of a

mechanical moment due to rotation 0= effHM , where MH = /geff is an effective

field: ( ) MeHH +++= =

3

1

3

12222 /2///2

isiims

eff MKMzxA , where is an arbitrary

real magnitude, Q 4+= , )21 2/ sMKQ = . The M relaxation to an equilibrium state is

yielded by means of its reorientation to an effective field direction (M establishment).

An order of an establishment of M orientation in finite elements (FE) can break

problem symmetry. It is assumed to use the random numbers generator for finding of FE

counters i and j for which the establishment is yielded in the given step of calculus.

Let we introduce dimensionless values 0/xx = , 0/zz = ,2

/ sMgg = and

s

effeff

M/Hh = , where 10 /KA= . Then, expressions are given by:

( ) +

++=

= =

2/2/1//23

1

3

1

222222 m

i jijjizxQg h

//43

1

32222 ++

+=

=

m

iii

effzxQ heh ,

where smm

M/Hh = .

So, dimensionless energy of physical space [ ]= 0201 /,/ xx [ ]0201 /,/ zz is

2

0

2/ = sMGG . Let we break into identical rectangles: UU

I

i

J

jij

1 1= =

, where

[ ] 2/,2/2/,2/ zzzzxxxx jjiiij ++= . Now, energy is:

( ) ( )( ) ( ) ( )( ) ( ){

( ) ( )( ) ( ) ( )( )[ ] ( ) ( ) +

++

++=

= =+

+

2/2/12/11

2/112

3

1

3

1

222

11

2

1

2

11

2

1

mij

i jijjijjiijijjJji

ijiijijiIjiij

z

xQg

h


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Preprint DRAFT 2011

and effective field:

( ) ( ) ( ) ( )[ ] ( )( ++= + xQ iIjiijieffij 2/114 111h

( )( ) ( )( )[ ] ( ) 2/113

1

3

111 ++++ =

+mij

iiijJjijji z he ,

where demagnetization field mijh is

( ){ }( ) ( )zxddij kl

I

k

J

lij

mij

=

= =

//21 1

22 RRrRrRnrh ,

where ( )Rn is normal to the ij . The fieldm

ijh is given by the:

( )= =

=I

k

J

lkl

mij gjlik

1 1

,,Nh ,

where xzg = / . Let we determine ikn = and jlm = with integers

) xn xijkl = /err and ) zm zijkl = /err . Components of tensor ( )gmn ,,N are:

( ) ( ) ( )gmnKgmnKgmnN LxRxxx ,,,,,, += ;

( ) ( ) ( )gmnKgmnKgmnN LzRzzx ,,,,,, += ;

( ) ( ) ( )gmnKgmnKgmnN DxUxxz ,,,,,, += ;

( ) ( ) ( )gmnKgmnKgmnN DzUzzz ,,,,,, += ,

where values K with indexes x, z and R, L, U, D are numerically equal to the x, z th

components of the filed created by poles on the right, left, top and bottom sides of kl

respectively. The calculations with symmetry based optimization give the following:

( )

( ) ( )

( ) ( )

( )

( )

=

=

1,

0and1,

0and0,10

0and11

,,

mm,gn,K

mnn,m,gK

mn,m,gF,m,gF

mn,,m,gnFn,m,gF

gmnK

R

x

L

x

R

x

R

x

R

x

R

x

R

x


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Preprint DRAFT 2011

( )

( ) ( )

( ) ( )

( )

( )

=

+

=

1,

0and1,

0and0,10

0and11

,,

mm,gn,K

mnn,m,gK

mn,m,gF,m,gF

mn,,m,gnFn,m,gF

gmnK

Lx

R

x

R

x

R

x

R

x

R

x

L

x

( )

( ) ( )

( ) ( )

( )

( )

=+

=

1,

0and1,

0and0,10

0and11

,,

mm,gn,K

mnn,m,gK

mn,m,gF,m,gF

mn,,m,gnFn,m,gF

gmnK

R

z

L

z

R

z

R

z

R

z

R

z

R

z

( )

( ) ( )

( ) ( )( )

( )

=

+

=

1,

0and1,

0and0,10

0and11

,,

mm,gn,K

mnn,m,gK

mn,m,gF,m,gF

mn,,m,gnFn,m,gF

gmnK

L

z

R

z

RzRz

R

z

R

z

L

z

( ) ( )gnmKgmnK RzU

x /1,,,, =

( ) ( )gnmKgmnK RxU

z /1,,,, =

( ) ( )gnmKgmnK LzD

x /1,,,, =

( ) ( )gnmKgmnK LxD

z /1,,,, =

(n,m,g)f,g)(n,mf,g)(n,mf(n,m,g)F RxR

x

R

x

R

x 211 ++=

(n,m,g)f,g)(n,mf,g)(n,mf(n,m,g)F Rz

R

z

R

z

R

z 211 ++=

( ) ( ) ( )[ ] ( )( )

( )

==

=

=

=

0and0,0

0and0,log

0and0,log

0and0,2logarctan4

2

2

222222

mn

mn/gnn

mngmgm

mng/m+gnmgngm/nnmg

(n,m,g)fRx

( ) ( )[ ]( )[ ]

==

+=

0or/and0,2/sgn

0and0logarctan

22

222222

mn/gmmg

mn/g,m+gngnm(gm/n)mgn(n,m,g)f

R

z


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Preprint DRAFT 2011

Symmetry in these equations corresponds to the spatial symmetry . Also, there are

dipole symmetry in the 2D space ( ) ( )gnmgmn ,,,, NN = for 0mn , and expression the same

as the local Lorentz field ( ) ( ) 4,0,0,0,0 =+ gNgN zzxx .

Let us consider case when surfaces of the sample correspond to 01/zz = and 02/zz = .

Boundaries 01/xx = and 02/xx = are only the limit of the simulation area. They are not

real limits of the sample. The field of the surface charges on these limits should be removed:

( ) ( ) ( )[ ]{ }= =

+++=I

k

J

lklk

Lz

LxkI

Rz

Rx

mij KKKKgjlik

1 11

,, babaNh ,

where

00

01a ,

10

00b .

Independence of the simulated equilibrium magnetization distribution from the finite

element shape and their orientations is the important problem. Aharoni first proposed some

necessary conditions to prove conformity between the method discrete model and

micromagnetic approach. In describable case that conditions is given by:

( )=== rr2

321 dGGGG , where

( ) ++= 224411 zyzyKG ( ) ] r22 / dA xx + ,

( )[ ++= 224412 zxzxKG ( ) ] r22 / dA yy + ,

( )[ ++= 224413 yxyxKG ( ) ] r22 / dA zz + .

3. Magnetization distribution symmetry

A symmetry of initial magnetization distribution determines a final simulated magnetization

distribution. The last can be equilibrium or metastable. Therefore problem of magnetization

distribution symmetry is important. Here and hereinafter we talk about non-magnetic

symmetry classes. It means that magnetization is considered as polar vector as well.


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Preprint DRAFT 2011

Let function ( )2/, fhzx < specifies magnetization distribution in isolated DW

volume in film with thickness fh , where andXaxis is parallel to film surface. Than

the DW boundary conditions at infinity are ( ) 12/, m=> fhzx and

( ) 22/, m=< fhzx , where 1m and 2m are the unit vectors along magnetization vector

in neighboring domains volumes. If external field is absent then ever isolated DW

magnetization distribution symmetry correspond to boundary condition symmetry. The first

reason of magnetization distribution symmetry is a parallel orientation of film surface planes

(symmetry transformation group SFS ). The second reason is a mutual orientation of 1m , 2m

and direction Yalong which is a constant vector (group DS ). For describing groups ,SFS DS

lets define a

=

M

R

such as the transformation transferring vector from point r to other

position and changing its direction: ( ) ( )rr MR = . The group SFS consist of one

transformation that moving magnetization from one to other surface plane:

=12zSFS ,

where z2 is a reflection in plane perpendicular ze and 1 is a turn around the rotary onefold

axis. The group DS consist of transformation that relate among themselves 1m and/or

2m vectors:

=

y

x

x

D2

2,

2

1,

1

1S or

=

y

x

D2

2,

1

1S for 180-DW or non-180-DW

respectively, where x2 and y2 are reflections in planes those are perpendicular accordingly to

xe and ye .

All possible combinations of the transformations of a group DSF SS form a group S of

magnetization distribution symmetry transformation in ever DW volume. At ( ) 012 =+ mm

(i.e. at 180-DW) the group S looks like:


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Preprint DRAFT 2011

=1S

y

xz

x 2

2,

1

1

1

2,

1

1

2

1,

1

1,

where means an sequel fulfilment of symmetry operations:

=

21

21

2

2

1

1

MM

RR

M

R

M

R. After simplification it is obtained a multitude:

=

zz

x

yy

x

x

z

x

z

2

1,

2

2,

2

1,

2

2,

2

2,

2

1,

1

2,

1

11S , where zx 221 = and yxz 222 = . By

means of symmetry reduction it is possible to write group for non-180-DW):

=

yy

xz

2

1,

2

2,

1

2,

1

12S . Subgroups DWS of groups 1S or 2S describe magnetization

distribution symmetry of all isolated DW with 2 =180 or 2180 respectively (tabl.1).


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Preprint DRAFT 2011

Table 1. Symmetry (non-magnetic classes) of the 2D magnetic domain wall

DWS DW

zz

x

yy

x

x

z

x

z

2

1,

2

2,

2

1,

2

2,

2

2,

2

1,

1

2,

1

1

Classical

1D

Bloch

180-DW

yy

xz

2

1,

2

2,

12,

1

1

1D-Brown and LaBonte DW

yy

x

x

z

2

1,

2

2,

2

2,

1

1

Symmetrical LaBonte DW

x

z

2

2,

1

1

Asymmetrical LaBonte DW

y2

1,

1

1

Asymmetrical Nel DW at Hubert

model

An availability of all elements

yy

xz

2

1,

2

2,

1

2,

1

1in groups 1S and 2S is a necessary

condition of plane DW. An absence of all elements

zz

x

yy

x

2

1,

2

2,

2

1,

2

2,

1

1is sufficient

condition of asymmetric DW.

For more percise and fast modelling an initial magnetization distribution should have

symmetry identical as a final.


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Preprint DRAFT 2011

4. Results and discussion

Let we choose orts: ]100[||xe , ]010[||ye and ]001[||ze . The (001) plate width is fh and

modeling area is: 2/1 Lx = , 2/2 Lx = , 2/1 fhz = , 2/2 fhz = , where fhL 10= . The

direction of the will be determined by the polar and azimuth angle , which are counted

from the xe and ze respectively.

Let the 90 DW divides two domains with magnetizations ( )2/,4/ 11 == and

( )2/,4/ 22 == . The angular tolerance is710=tol for our simulation. The tolerance

for the comparison of the 1G , 2G and 3G in the above-mentioned necessary conditions is

given by the 1%. The magnetization distributions in the 90 DW at different widths fh for the

parameter 62.41 =Q (ferrite garnet at the temperature of the liquid Helium) are given at the

fig.1


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Preprint DRAFT 2011

-15 -10 -5 0 5 10 15

-80

-40

0

40

80

, , deg.

x/0

-40 -20 0 20 40

-100

-50

0

50

100

x/0

, , deg.

a) b)

-80 -40 0 40 80-150

-75

0

75

150

x/0

, , deg.

-80 -40 0 40 80-150

-75

0

75

150

x/0

, , deg.

c) d)

-150 -75 0 75 150

-200

-100

0

100

200

, , deg.

x/0

e)

Fig. 1. The equilibrium magnetization distributions in the volume of the 90 DW in the (001)

plate with the 62.4/2 12

=KMs for widths fh : a - 046.2 ; b- 038.7 ; c- 030.12 ; d-

07614 . ; e- 014.22 . Dashed, solid and chain-line are given for the 4/, fhzx = ,

2/, fhzx = and 4/, fhzx = respectively.


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Preprint DRAFT 2011

In case of the growth of the plate width from the 0 to the 20 0 transition from the Neel

DWs to Bloch DW is processing at the same time with the nucleations of the 71 DW based

domain structure with 12 = perpendicular to the (001) plate. This structure appears in

the volume of the initial DW.

The energy density of the DW is counted from the same value in the domain. In case of

the value 5.0/2 12


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Preprint DRAFT 2011

3 6 9 12 15

0

4

8

12

16

20

GK

Gm

GA

G

G,GA,G

m,G

Karb.units

hf/

0, arb.units

3 6 9 12 15

0.0

0.3

0.6

0.9

A

k

m

A,

m,

K, ar .un ts

hf/

0, arb.units

a) b)

3 6 9 12 150.9

1.2

1.5

1.8

2.1

, arb.units

hf/

0, arb.units

3 6 9 12 150.2

0.4

0.6

0.8

1.0

Gm/G

A, arb.units

hf/

0, arb.units

c) d)

Fig 2. The energies of the equilibrium 90 DW as function of the (001) plate width.

At the width of the (001) plate higher than 05.13 , the energy 0


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Preprint DRAFT 2011

If magnetization distribution has an oscillating nature then DW width (Lilley definition)

became large or limitless in extreme case.

The magnetization distribution symmetry of this DW is

=

y

DW21,

11S . The equilibrium

magnetic charge (magnetization divergence)distribution demonstrates on the fig. 3

Fig. 3. Example of the magnetization divergence distribution in the (001) plate cross-

section. Black corresponds to the north poles and white corresponds to the south poles.

Acknowledgements

I would like to express my sincere gratitude to my supervisor Prof. V.F. Kovalenko for

him outstanding guidance. Micromagnetic simulation has been performed on the cluster of the

Information and Computer Center of Kyiv National Taras Shevchenko University.

micromagnetic structure and symmetry of the 90 degree domain wall suppressed by the demagnetization...

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