8/6/2019 Optimization of the Numerical Simulation of the Domain Walls Structure in Magnetically Ordered Media

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OPTIMIZATION OF THE NUMERICAL SIMULATION OF THE DOMAIN WALLS STRUCTURE IN MAGNETICALLY ORDERED MEDIA

Tanygin B.M., Tychko O.V.

Taras Shevchenko Kiev National University, Radiophysics Faculty, Glushkova 2, Kyiv, Ukraine, 03022.E-mail: RPDparadoxL@gmail.com

Introduction

An information about magnetization vector M spatial distribution (MD) in a

medium volume is basic at a theoretical investigation of magnetic states and

processes in a magnetically ordered media. One can obtain detailed informationabout domain wall (DW) structure using numerical methods. Some of them [1,2]have a high accuracy but are not always stable. Others [3] provide higher stability,

but they demand the considerable time of calculation. Therefore there is a necessity

of their optimization and construction of common approaches to use.Method optimization for numerical simulation of an equilibrium MD in DW

volume for a wide range of a thickness of a magnetically ordered sample is the aim

of this report. The results of the method optimization are shown by the example of2D numerical MD simulation in DW volume in a thin (001)-film with a n egative first

constant 1K of a magnetocrystalline cubic anisotropy (MCA) for various sample

thickness.

1. Variation problemLet's consider 2D-problem MD numerical simulation in cross-section of a thin

(100)-film. Let Oz- and Ox-axis is directed along film normal and crystallographic

[100] -direction. A free energy functional counting upon a unity of length along Oy -

axis look like: ( )[ ] += rr2dgG , where r is the Lagrange multiplier,

KmA gggg ++= is a volume energy density, ( ) ( )( )22 // zxAgA += is a anexchange 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, ( ) 2/122

1 pqqpK Kg = is a MCA

energy density, pq is the Kronecker symbol, .,,, zyxqp = .

Requirement of an equilibrium MD is 0=G or an absence of a magnetic

moment due to rotation ][ 0= effHM , where MH /geff = is an effectivefield: ( ) MeHH +++= MKMzxA ppmeff /2///2 312222 , where is anarbitrary real magnitude. The M relaxation to an equilibrium state is yielded by

means of its reorientation to an effective field direction [1-3] (M establishment).

An order of an establishment ofM orientationin finite elements (FE) can break

a problem symmetry. It is assumed to use the random numbers gen erator for finding

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

calculus.

2. Optimization of the demagnetization field calculation

Lets consider modeling of the demagnetization field of continuous totality of

JI uniformly magnetized 2D-FE [2,3]. The demagnetization fieldij

mH is

defined exactly or approximately with use of the operator N :( ) ( )

+

=

+

== =

=

ii

iik

jj

jjl

kl

I

k

J

l

klijm gjijlikgjijlik NNh ,,,,

,,,,

1 1

,where

Mij

m

ij

m /Hh = , g is a ratio of the FE geometry. The indexes denote FE numbers.

Components of the ( )gjimn ,,,, NN = operator look like:

=

zzzx

xzxx

NN

NNN . They

are determined by =xxN +1CKRx 2CK

Lx ; =zxN +1CK

Rz 2CK

Lz ; =xzN

= +3CKU

x 4CKDx ; =zzN +3CK

Uz 4CK

Dz , where ( ) [ ]( )iIninC = 111 ,

[ ]( ) [ ]( )ininC = 12 11 , ( ) [ ]( )jJmjmC = 113 , [ ]( ) [ ]( )jmjmC = 14 11 .

The K values with subscript (x andz) and superscript (R, L, Uand D) is equal

to x-and z components of a field created by the poles that are located on right,left, upper and under FE sides (here increase i andj counters defined as right ant up

direction) accordingly at unit kl projection on these sides. Quantity ji should

be incremented until the modeling result will not cease to depend on its value. In contrast to classical quadratic dependence [1,2] such algorithm allows to

make a modeling time of a linear function of a sample geometry.

The N operator has point and translation symmetry of FE lattice of partitions:

( ) ( )gnmKgmnK RzU

x /1,,,, = , ( ) ( )gnmKgmnKR

x

U

z /1,,,, = , ( ) =gmnKD

x ,,

( )gnmKLz /1,,= , ( ) ( )gnmKgmnKLx

Dz /1,,,, = ,

( )( )( )( )( )

( )

=

gmnK

gmnK

gmnK

gmnKL

c

czcxRc

czcxLc

Rc

,,1

,,

,,

,,

, ( )

( )( )

( )( )( )

+

=

gmnK

gmnK

gmnK

gmnK

Rc

czcxLc

czcx

R

c

Lc

,,1

,,

,,

,,

,

where xx zz= 1= , xz zx= 0= and c denotes x or z . At that [1]:

( ) ( )gmnNgmnN zzxx ,,,, = at 0|||| + mn . The last is agreed with similar

property of a dipoles static field in 2D space.Taking into account of symmetry of the demagnetization field operator allows to

speed up a numerical simulation of its com ponents in some times.

At a great distance FE creates a dipole field irrespective of its shape. Thereforeat a choice of FE shape it is necessary to proceed from comparison of a

demagnetization field at short distance (~ FE geometry) with a field of continuedMD. The measure of such comparison is the Lorentz local field. In 2D space it is

defined by a form-factor 2 : ( ) MB 2=Lm

. An analog of the Lorentz local field

is a demagnetization field of a solitary FE. For concerned FE the N operator has a

property [1]: ( ) ( ) 4,0,0,0,0 =+ gNgN zzxx . For squar e FE ( 1=g ) it would look

like ( ) ( ) 21,0,01,0,0 == zzxx NN i.e. register of form-factors of the continuous

totality of dipoles and square FE takes place. In 2D space the such regularity iscarried out for any shape FE (for example at FE totality of hexagonal shape).

However for square FE the effective field is formulated the most simple analytic

form.The choice of the square shape of 2D finite elements is sufficient for exact

calculation of a local part of an e nergy of dipole - dipole interaction.

3. Necessary conditionsIndependence of the simulated equilibrium MD from the FE shape and their

orientations is the important problem. Aharoni [4] first propose some necessary

conditions to prove conformity between the method discrete model andmicromagnetic approach. In describable case that conditions is given by:

( )=== rr2

321 dGGGG , where

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

( )[

++=2244

12 zxzxKG ( ) ] r22 / dA yy + ,

( )[ ++= 224413 yxyxKG ( ) ] r22 / dA zz + .4. Magnetization distribution symmetryA symmetry of initial MD determines a final simulated MD [2]. The last can be

equilibrium or metastable. Therefore problem of MD s ymmetry is important [2].

Let function ( )2/, fhzx fhzx and

( ) 22/, m =< fhzx , where 1m and 2m are the unit vectors alongmagnetization vector M1and M2in neighboringdomains volumes. If external field is

absent then ever isolated DW MD symmetry correspond to boundary conditionsymmetry. Thefirst reason of MD 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 Oy along 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:

=

1

2zSFS , 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

D

2

2,

2

1,

1

1S or

=

y

x

D

2

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 . In general case some other

transformations connect ever two magnetizations in DW volume, but the

examined groups describes MD symmetry agreed upon only boundary condition

symmetry. At the last case all reflection planes compared with M include a begin of

vector , reflection plane compared with R equals z2 or x2 transformation

coincides with Oxy plane or Oyz ( 0=x is a central point: ( ) ( )21 mm = )

respectively.

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

a group S of MD symmetry transformation in ever DW volume. At

( ) 012 =+ mm (i.e. at 180-DW) the group S looks like:

=1S

y

xz

x 2

2,

1

1

1

2,

1

1

2

1,

1

1,

where means an alternate 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 symmetryreduction 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 MD symmetry of all isolated DW with 2 =180 or 2180 respectively(tabl.1).

Table 1.

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

Bloch180-DW [5]

yy

xz

2

1,

2

2,

1

2,

1

1

1D-Brown and LaBonte

DW [1]

yy

x

x

z

2

1,

2

2,

2

2,1

1

Symmetrical LaBonte DW[5]

x

z

2

2,

1

1

Asymmetrical LaBonte

DW [2]

y2

1,

1

1

Asymmetrical Nel DW at

Hubert model [5]

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 [2] DW.

For more exact and fast modelling an initial MD should have symmetry

identical required.

5. ApplicationDescribed optimized method was used for modelling of equilibrium M

distribution in a volume of (001)-film of magnetically ordered medium with

62.4/2 12

=KMs at sample thickness 00 78.7346.2 fh , where

( ) 2/110 /KA= . Different random initial MD has an influence on time calculationbut not change final distribution.

- 12 .5 0 -6 .2 5 0 .0 0 6. 25 1 2. 50

0.0

0.2

0.4

0.6

0.8

1.0

z

x

x,

z

x/0, arb.units

-3 7. 50 - 18. 75 0. 00 18 .75 3 7. 50-0.2

0.0

0.2

0.4

0.6

0.8

1.0

x/0, arb.units

z

x

x,

z

a)

b)

-62 .5 0 - 31. 25 0. 00 3 1. 25 62 .5 0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

x/0, arb.units

z

x

x,z

-75.0 -37.5 0.0 37.5 75.0-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

x/0, arb.units

z

x

x,z

c)

d)

-112.4 -56.2 0.0 56.2 112.4

-0.6

-0.3

0.0

0.3

0.6

0.9

x/0, arb.units

z

x

x,

z

30 60 90 120 150

-0,8

-0,4

0,0

0,4

0,8

x/0, arb.units

y

y,

z

z

e) f)

Fig. 1. Equilibrium M distributionin a volume of (001)-film of magnetically ordered

medium with 62.4/2 12

=KMs at sample thickness fh : 046.2 (a);

038.7 (b); 030.12 (c); 07614 . (d); 014.22 (e); 078.73 (f) where

( ) 2/110 /KA= . Solid and dash lines correspond ( )4/, fhzx =M and( )4/, fhzx =M respectively.

With film thickness fh growth e in a range 0 fh 50 0 a transition from

Neel DW to Bloch DW takes place. It is accompanied by formation of domain

structure with 71-DW (with the 12 mmm = vector perpendicular to a surface

(001)-films) in volume initial (at fh = 0 ) 90-DW (with m vector parallel a

surface (001)-films).If MD has an oscillating nature DW width [5, 6] became large

or limitless in extreme case. We propose use a tangent to envelope of

dependences ( )xzyx ,, for determining the three width definitions zyx ,, .

MD symmetry of this DW is

=

y

DW

2

1,

1

1S . The equilibrium charge

distribution demonstrates this symmetry (Fig.2).

Fig 2. Equilibrium charge distribution in a volume of (001)-film of magnetically

ordered medium with 62.4/2 12

=KMs at sample thickness

076.14 =fh in shades of gray. Maximal charge densi ty 0.0728972

0 = .

The area dimensions along Ox and Oz axes are accordingly 00 8.1430 .

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 .units

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 3. Domain wall (90-DW) energy (a), energy density (b,c) and relation Am GG /

(d) depend on (001)-film width at material parameters =12/2 KMs 62.4 .

Here energy and energy density are measured in2

02M and 0

2M units.

Subscripts A, m and K mark exhange, demagnetization and an isotropy

energy components.Reference

[1]. W. F. Brown, A.E. LaBonte, J. Appl. Phys. 36, 1383, (1965).

[2]. A.E. LaBonte, J,Appl. Phys., 40, 2450 (1969).

[3]. L.I. Antonov, S.G. Osipov, M.M. Hapaev, PhMM, 55, 917 (1983).

[4]. A.Aharoni, J.Appl. Phys. 39, 861 (1968)[5]. Hubert A., Shafer R. Magnetic domains. The analysis of magnetic

microstructures. Berlin, Springer-Verlag, 1998.

[6]. B.A. Lilley. Phil.Mag., 41 (1950), 792.

z

ase, cite original work as: B.M. Tanygin, O.V. Tychko, Optimization of the numerical simulation of the domain walls structure in magneticallyered media, Abstracts of International Conference "Functional Materials" ICFM'2007, Ukraine, Crimea, Partenit (2007) 97.