soliton pair dynamics in patterned ferromagnetic ellipses
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A U.S. Department of EnergyOffice of Science LaboratoryOperated by The University of Chicago
Argonne National Laboratory
Office of ScienceU.S. Department of Energy
Soliton pair dynamics in patterned
ferromagnetic ellipsesKristen Buchanan, Pierre Roy,* Frank Fradin,
Konstantin Guslienko, Marcos Grimsditch, Sam Bader, and Val Novosad
Magnetic Films GroupMaterials Science Division
AcknowledgementsL. Ocola, R. Divan, J. PearsonNSERC of Canada for a postdoctoral fellowshipArgonne - U.S. DOE Contract No. W-31-109-ENG-38Swedish Research Council (P. R.)
*Uppsala University, Sweden
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Magnetic Vortex State
Magnetic state Magnetic state ((magnetically-soft nanodots) depends on: ) depends on:
• Geometry: L and RGeometry: L and R
• Material: A and MsMaterial: A and Ms
Guslienko and Novosad, J. Appl. Phys. 96, 4451, 2004.
00
10
20
30
40
50
60
Do
t th
ickn
ess
L, (
nm
)
10 20 30 40 50 60
Dot Diameter 2R, (nm)
(Permalloy)
Polarization p = ± 1
Chirality c = ± 1
Vorticity (topological charge)
Vortex in a nanomagnet
• Flux closure state with central core
• Topological soliton
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Spin Excitations of a Magnetic Vortex
** Magnetostatic interactions dominate in sub-micron and micron-size dots **
High-frequency spin-waves, GHz range• Radial modes
• Azimuthal modes
Low-frequency eigenmodes,sub-GHz range• Translation (gyrotropic)
modes
Single vortex dynamics:
• Cylindrical
• Square/rectangular
• Elliptical
Vortex Pair Dynamics
in elliptic dots
Dynamic vortex interactions in:
• Tri-layer F/N/F dots
• Dense 2D dot arrays(theory/simulation)
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Vortex Dynamics: Translational Mode
Vortex core trajectory- Polarization dictates direction
Theory/simulations:Guslienko et al., J. Appl. Phys. 91, 8037,
2002
Experiment: Park et al., Phys. Rev. B 67, 020403 (R), 2003.Choe et al., Science 304, 420, 2004.Novosad et al., Phys Rev. B 72, 024455, 2005.
Shifted vortex core position
Energ
y
0.1 0.2 0.30.0
0.5
1.0
1.5
R 100 nm 150 nm 200 nm 250 nmEi
genf
requ
ency
(GH
z)
Dot aspect ratio =L/R
2 2 2 4 2 6 2 8 3 0 3 2
-0.05
0.00
0.05
0.10
M/M
s
Time (ns)0 108642
0 Oe100 Oe
Simulations of the vortex translational mode
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Elliptical Dots: Remanent State
• Magnetic force microscopy/micromagnetic simulations 2
m
1 m
40 nm Py
H
H
Static reversal of ellipses: Vavassori et al., Phys. Rev. B 69, 214404 (2004)
Mz
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Vortex Dynamics Experiment
Goal: Explore dynamic vortex interactions of vortex pairs confined in elliptical magnetic dots
Method: Microwave Reflection
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Single Vortex Dynamics for an Ellipse
100 200 300
0
20
R
eal I
mpe
danc
e D
eriv
ativ
e(a
rb. u
nits
)
(MHz)
0.05 0.10
100
200
300
(M
Hz)
Vertical Aspect Ratio, = 2L/(a+b)
experiment simulations theory
2b = 1 m
2a = 2 m
Thickness L= 40 nm
is Frequency
100 200 300
0
20
single pair
R
eal I
mpe
danc
e D
eriv
ativ
e(a
rb. u
nits
)
(MHz)
a/b ~ 2
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Experimental Mode Map: Vortex Pair
3 x 1.5 m2 ellipse, L = 40 nm
-50 0 50 1000
50
100
150
200
p1p
2 = +1
p1p
2 = -1
HH
(M
Hz)
H (Oe)
H // hrf
H // hrf
H hrf
H hrf
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Vortex Pair Modes
• Same frequency
• “Splitting”!Notation:
i = in-phaseo = out-of-phase
<Mx> = 0<My> = 0
<Mx> cos(t+)<My> sin(t+)
<Mx> = 0<My> sin(t+)
<Mx> cos(t+)<My> = 0
x
y
equilibrium
HH
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Micromagnetic Simulations – Single Vortex
0 5 10
-0.02
0.00
0.02
0.04
My/M
s
Time (ns)
simulation sine fit
-50 0 50
-50
0
50
y (n
m)
x (nm)
Py dotL= 40 nm2a = 1 m, 2b = 2 m
Ms = 700 emu/cm3
A = 1.3 erg/cmno anisotropyDamping = 0.008Gyromagnetic ratio: = 2.94 MHz/Oe
LLG, ScheinfeinOOMMF, NIST
134 MHz
Single translational mode frequency
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Dynamics of Interacting Solitons
red/blue represent My100 200 300
0
20
Rea
l Im
pede
nce
Der
ivat
ive
(arb
. uni
ts)
(MHz)
single pair
(o,o)
(o,i)
hr.f.
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Micromagnetic Simulations: Mode Map
-50 0 50 1000
100
200
300
x x
x x
x x
x x
x x
x x
H H
Freq
uenc
y, M
Hz
H, Oe
p1p
2 = +1
p1p
2 = -1
1.5 x 0.75 m2 ellipse, L = 40 nm
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Vortex Dynamics: Theory
Vortex coretrajectory
Shifted vortex core
Energ
y
Dampingt eff
HMM
1
M
rMH
W
eff
1) Landau-Lifshitz Gilbert equation
M(r): magnetization distributionW : energy Heff : effective magnetic field
0
X
XXG
W
dt
d
Applied to circular dots: Guslienko et al., J. Appl. Phys. 91, 8037, 2002
2) Representation in terms of core position X
Thiele et al., Phys. Rev. Lett, 30, 230, 1973
G : gyrovector
G : gyroconstant G=2MsL/L : dot thicknessMs : saturation magnetization
: gyromagnetic ratio
zG ˆGp))((),( tt XrMrM
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X1, p1
X2, p2
Vortex Pair Dynamics: Theory
0
,
1
2111
X
XXXG
W
dt
d 0
,
2
2122
X
XXXG
W
dt
d
21int21121 ,, XXXXXX WWWW
Equations of motion of the vortex cores: Gyrovectors: zG ˆjj Gp
21212
22
122
2121 2
1
2
1, YYXXYYXXW yxyx XX
Assume energy form:
yyxxG 1
2,1 yyxxG
12,1
Eigenfrequencies:
Prediction: 2121 True for simulations!
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Vortex Core Motion: Eigenvectors
Motion patterns match simulations!
1
1
1
1
xx
xx
i
i
2
2
1
1
xx
xx
i
i
1
1
1
1
xx
xx
i
i
2
2
1
1
xx
xx
i
i
1
2 1
2
2
2
1
1
Y
X
Y
X
iii YX ,X
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Conclusions
• First experimental data on magnetic vortex pair dynamics
• Core Polarizations:- Negligible static effect- Very important for dynamics
- Excitation direction- Mode map
- Theory/simulations agree on- Frequency product invariance- Core motion patterns
- Buchanan et al., Nature Physics (in press)
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Competing Energies
Exchange
Magnetostatic
Zeeman
Nanomagnetism
Competition between different energies at the
nanoscale will determine the fundamental properties of
nanomagnetsMagnetocrystalline
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Fabrication• Top Down: Lithography
1 m
Spin Coat Expose
Metallization Lift-off
http://chem.ch.huji.ac.il/~porath/NST2/Lecture%204/Lecture%204%20-%20e-Beam%20Lithography%202003.pdf
Develop
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Phase Diagram for Nanodots
Magnetic phase diagram for magnetically-soft nanodots
LL
2R2R
Magnetic state depends on: Magnetic state depends on: • Geometry: L and RGeometry: L and R• Material: A and MMaterial: A and Mss
Guslienko and Novosad, J. Appl. Phys. 96, 4451, 2004.
00
10
20
30
40
50
60
Do
t th
ickn
ess
L, (
nm
)
10 20 30 40 50 60
Dot Diameter 2R, (nm)
(Permalloy)
20
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Magnetic Vortex State
Vortex in a nanomagnet - nonlocalized solitonFlux closure state with central core
Polarization p = ± 1
Chirality c = ± 1
Vorticity q = 1
Outline• Vortex state – unique dynamic
excitations
• Vortex pair dynamics in elliptical dots
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X1, p1
X2, p2
Vortex Pair Dynamics: Theory
Dot energy for shifted vortices at positions Xj
0
,
1
2111
X
XXXG
W
dt
d 0
,
2
2122
X
XXXG
W
dt
d
21int21121 ,, XXXXXX WWWW
21212
22
122
2121 2
1
2
1, YYXXYYXXW yxyx XX
Equations of motion of the vortex cores
Gyrovectors: zG ˆjj Gp
Assume energy form:
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