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Bistable magnetization profiles in thin magnetic films. Ramaz Khomeriki. Javakhishvili Tbilisi State University, GEORGIA. In Collaboration with. Jerome LEON, Miguel Manna, Université Montpellier 2, Montpellier, FRANCE. Reflected Power. Nonlinear Bistability in Pendula Chain. - PowerPoint PPT Presentation
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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
Beijing, June 9-12, 2008 2
Ref
lect
ed
Po
wer
Beijing, June 9-12, 2008 3
Nonlinear Bistability in Pendula Chain
Beijing, June 9-12, 2008 4
Nonlinear Standing Waves in Thin Magnetic FilmsNonlinear Standing Waves in Thin Magnetic Films
0 ,04 , HMHHMgdt
Md
Beijing, June 9-12, 2008 5
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
Beijing, June 9-12, 2008 6
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).
Beijing, June 9-12, 2008 7
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).
Beijing, June 9-12, 2008 8
..,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)
Beijing, June 9-12, 2008 9
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
Beijing, June 9-12, 2008 10
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
Beijing, June 9-12, 2008 11
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
Beijing, June 9-12, 2008 12
2
22
0
2
22
220
4
32
z
d
dHM
M
H
Beijing, June 9-12, 2008 13
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
Beijing, June 9-12, 2008 14
Thank You