Quasi-stable vortex magnetization structures in nanowires ...MDonahue/talks/hmm2009-cowires.pdf ·...

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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

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).

200 nm Co film with perpendicular anisotropy

Analytic theory‡

��

Assumem(x,y,z) = m(z)

‡G. Bergmann, J.G. Lu, et al., Phys. Rev. B 77, 054415 (2008).

Analytic theory‡

��

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

Model schematic

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)

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

C e l l s i z e ( n m )

Analytic theory‡

��

Micromagnetic simulations

��

z-vortex(non-periodic)

y-vortices(periodic)

z-vortex state, radius dependence

x (easy axis)

Radius: 30.4 nm Radius: 200 nm

y-vortex states, radius dependence

Radius: 30.4 nm

x (easy axis)

Radius: 200 nm

Thin film, micromagnetic simulation

D.G. Porter and M.J. Donahue, HMM 2001.12

Energy density for y-vortex state

0 1 0 0 2 0 00 . 1

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

alized

energy

N a n o w i r e r a d i u s , r ( n m )

Multiple metastable y-vortex states

Radius: 40 nm.

Simulation length: 168 nm

Relative energy densities (40 nm radius)

2 pairs 3 pairs

Effects of simulation window size

Infinite Plate*

Infinite Rod

α = (p-tz)/2p

∗H. Kronmuller and M. Fahnle, Micromagnetism and theMicrostructure of Ferromagnetic Solids (Cambridge, 2003).

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]

Periodicity formula

p(r) = 2

√8r√

f (α)4µ0M2s /π

3 + 2α2K1

α = 0.25 =⇒ f (α) ≈ 0.5259

α = 0.4 =⇒ f (α) ≈ 0.130887

y-vortex period

0 1 0 0 2 0 00

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 )

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

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.

Quasi-stable vortex magnetization structures in nanowires ...MDonahue/talks/hmm2009-cowires.pdf ·...

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