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
