Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Quasi-stable vortex magnetization structuresin nanowires with perpendicular anisotropy
Kristof M. LebeckiUniversitat Konstanz, Konstanz, Germany
Michael J. DonahueNIST, Gaithersburg, Maryland
13-May-2009
1
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
MFM of cobalt wires
Experimental wire result and conjectured explanation:†
Wire radius: 50 nm
†Y. Henry, K. Ounadjela, et al., Eur. Phys. J. B 20, 35 (2001).
2
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
200 nm Co film with perpendicular anisotropy
3
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Analytic theory‡
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Assumem(x,y,z) = m(z)
‡G. Bergmann, J.G. Lu, et al., Phys. Rev. B 77, 054415 (2008).
4
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Analytic theory‡
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Assumem(x,y,z) = m(z)
u[M||z ]^k2/u00
aex/u00
(10-2) smin θmin umin/u00 u[M||x ]^
000.0830.0830.1250.125
0.681.360.681.360.681.36
2.31.752.11.62.11.5
0.70.31.00.81.00.9
0.333 880.341 370.378 830.396 90.397 040.417 71
0.340.340.4250.4250.470.47
0.50.50.50.50.50.5
‡G. Bergmann, J.G. Lu, et al., Phys. Rev. B 77, 054415 (2008).
5
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Model schematic
z
y
x (easy axis)Model Parameters
Ms: 138 kA/mK1: 200 - 500 kJ/m3
K2: 0 - 150 kJ/m3
A: 13 - 52 pJ/mRadius: 30 - 200 nm
infiniteextent (z)
6
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Discretization error for sinusoidal state
0 . 5 1 2 3 4 50 . 3 3 3 4
0 . 3 3 3 6
0 . 3 3 3 8
S i m u l a t i o n T h e o r y , R e f . 1 T h e o r y , i m p r o v e dNo
rmaliz
ed ene
rgy de
nsity
C e l l s i z e ( n m )
7
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Analytic theory‡
��
�
�
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��������� ���������
�
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Assumem(x,y,z) = m(z)
‡G. Bergmann, J.G. Lu, et al., Phys. Rev. B 77, 054415 (2008).
8
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Micromagnetic simulations
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Assumem(x,y,z) = m(z)
z-vortex(non-periodic)
y-vortices(periodic)
9
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
z-vortex state, radius dependence
y
z
x (easy axis)
Radius: 30.4 nm Radius: 200 nm
10
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
y-vortex states, radius dependence
Radius: 30.4 nm
z
y
x (easy axis)
Radius: 200 nm
11
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Thin film, micromagnetic simulation
D.G. Porter and M.J. Donahue, HMM 2001.12
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Energy density for y-vortex state
0 1 0 0 2 0 00 . 1
0 . 2
0 . 3
S i m u l a t i o n : y - v o r t i c e s s t a t e S i m u l a i o n : z - v o r t e x s t a t e T h e o r y : u n i f o r m ( 0 0 1 ) s t a t e T h e o r y : s i n u s o i d a l s t a t e
Norm
alized
energy
densi
ty
N a n o w i r e r a d i u s , r ( n m )
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Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Multiple metastable y-vortex states
Radius: 40 nm.
Simulation length: 168 nm
Simulation length: 372 nm
Simulation length: 512 nm
14
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Relative energy densities (40 nm radius)
7
2 pairs 3 pairs
15
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Effects of simulation window size
16
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Infinite Plate*
Infinite Rod
α = (p-tz)/2p
∗H. Kronmuller and M. Fahnle, Micromagnetism and theMicrostructure of Ferromagnetic Solids (Cambridge, 2003).
17
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Variation of tz with r
r (nm) tz (nm)
30 4640 5250 5164 5680 54
100 62128 68200 70
In this range,
α = (p − tz)/2p ∈ [0.25, 0.4]
18
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Periodicity formula
p(r) = 2
√8r√
AK1
f (α)4µ0M2s /π
3 + 2α2K1
here
α = 0.25 =⇒ f (α) ≈ 0.5259
α = 0.4 =⇒ f (α) ≈ 0.130887
19
Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
y-vortex period
0 1 0 0 2 0 00
2 0 0
4 0 0
S i m u l a t i o n r e s u l t s S t r i p e d o m a i n t h e o r y , �= 1 / 4 S t r i p e d o m a i n t h e o r y , �= 0 . 4
Period
, p (nm
)
N a n o w i r e r a d i u s , r ( n m )
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Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Summary table
Material constants Sinusoidal state Vortex-like states
K1 K2 A ssin θsin usin uzvort uyvort p(MJ/m3) (MJ/m3) (pJ/m) (nm)
0.41 0.0 26 2.24 0.6849 0.3344 0.2268 0.247 1680.41 0.0 52 1.75 0.3212 0.3422 0.2832 0.307 2200.41 0.1 26 2.09 0.9555 0.3788 0.2463 0.263 1680.41 0.1 52 1.57 0.8499 0.3972 0.3094 0.322 2160.41 0.15 26 2.09 1.0251 0.3972 0.2551 0.270 1760.41 0.15 52 1.53 0.9452 0.4179 0.3213 0.323 2160.20 0.03 13 - - - 0.1274 0.148 1440.50 0.0 13 2.73 1.0409 0.3687 0.2062 0.221 150
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Quasi-stable vortexmagnetizationstructures in
nanowires withperpendicularanisotropy
K.M. Lebecki,M.J. Donahue
Summary
I Wide range of material constants and wire radiiconsidered.
I Lowest non-saturated energy in z-vortex and periodicy-vortices states.
I y-vortex periodicity in rough agreement with experiment
I Z-vortex and periodic y-vortices have comparableenergy.
I y-vortex period described using simple quasi-stripedomain theory.
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