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Beijing, June 9-12, 2008 1

In Collaboration with

Bistable magnetization profiles in thin magnetic films

Jerome LEON, Miguel Manna,Jerome LEON, Miguel Manna,

Université Montpellier 2, Montpellier, FRANCE

Ramaz Khomeriki

Javakhishvili Tbilisi State University, GEORGIA


Ref

lect

ed

Po

wer


Nonlinear Bistability in Pendula Chain


Nonlinear Standing Waves in Thin Magnetic FilmsNonlinear Standing Waves in Thin Magnetic Films

0 ,04 , HMHHMgdt

Md


0 ,04 , HMHHMgdt

Md

0 outside ;e inside e Solution Static 00 MMMHH zz

0000 e e Solution Dynamical HHHhMMMm zz

0 0 h(x,z,t); hh Φ hH y

11 0

,111

,11

222

2

2

2

yxzHzx

M

zxy

Hy

x

H

mm m, z

Φ

x

Φω

z

m

x

mω

x

Φm

z

Φm

dt

dm

z

Φm

dt

dm

MH2H

200M0 ,4 , gMgHHWhere


Linear Standing Wave Solution

pzkxk

pk

ω

ωiAem

,pzkxk

pk

ω

ωAem

dx , pzkxAeΦ

M

tiy

M

Htix

ti

coscos

coscos

2|| ifcossin

22

22

00

cos

2|| if2

yx

dxpti

m, m

pzeAeΦ

dx

Linear Dispersion Relation is Obtained

; kdk

p

kp

pω-ωωω MH 2tan;

22

220

2

zxx and HMH B d x 4of2at Condition Continuity

R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961).


Nonlinear Standing Waves

MHHyxz

yti

My

xti

M

Hx

ti

mmmm

cctztzekxω

ωkim

cctztzekxω

ωkm

cctztzekxΦ

20

222

22)3(222)1(0

22)3(222)1(

22)3(222)1(

22

..,,cos

..,,cos

..,,sin

0

0

0

0

||2

3

2

)1(2)3(2)3(2

)1(2)1(

0

2202)3(

0

)3(

0

2)3(

0

)1(

)3(0

)3(0

)1(

zkk

ikω

ωi

ω

ωii

t

iit

x

M

HHMHx

Hy

yx

R. Khomeriki, J. Leon, M. Manna, PRB, 74, 094414 (2006).


..,cos 0 ccztekxkxh tix

..,cos ccztekxkh pztix

p

p

; kdk

p

kp

pω-ωωω gMH 0;v2tan;

2

2

22

220

2

0||

4

3

2v 2

0

2202

2

2

M

HHg k

zzti

tz

Aezt

g

ti

vcosh,

tiezAzt ,cn,

0||

4

3

22

0

2202

2

2

20

M

HHMH kzkω

ω

ti

Comparison

p

A. K. Zvezdin and A. F. Popkov, Zh. Eksp. Teor. Fiz. 84, 606 (1983)


Linear Limit

pzkxk

pk

ω

ωiAem

,pzkxk

pk

ω

ωAem

dx , pzkxAeΦ

M

tiy

M

Htix

ti

coscos

coscos

2|| ifcossin

22

22

00

cos

2|| if2

yx

dxpti

m, m

pzeAeΦ

dx

Defining

2tan;22

220

2 kdk

p

kp

pω-ωωω MH

Linear Dispersion

Relation

d

kω -ωωMH

00

22

0

32

4

1 2

22

220

2

22

0

M

HHM

dz

d

ti


Boundary Value Problem

titixx ezt,zccehLthth

0 ..,0, 0

22

2200

2

22 32

;4

||M

H

MH ddω

ωzz

z

LzzbLzB 0 ;2/

22

1 22222

2

BBz

12

2 ,1 ;r ,2cn

2 I)

2

2222

2

B

BrBLzB

B


2

22

22

2

2 ,

2 ;r ,2sn

21 II)

B

Br

BrKLzB

B

2

22

22

2

2 ,

2

2 ;r ,2sn

1 III)

B

Br

BrLzB

B

K

tkxLzk

B

MH

cossin22

Bcos

0limit Linear

0

MHdk

022 2dxat Condition Continuity


2

22

0

2

22

220

4

32

z

d

dHM

M

H


Explicit Physical Solution

MHH

MHH

MHMH

yx

DB

Br

d

rrLzDmmmm

2

202

2

202

22

4

28 ,

2 ,

2B-2

,2snB(z) (z) K

MHH

MHH

MHMH

DmD

mr

D

m-

d

rrLzmm

2

202

222

20

4

28 ,

2 ,

21

4

,2sn(z) K


Thank You