Introduction to Numerical Micromagnetism. Application to Mesoscopic - - PowerPoint PPT Presentation

Introduction to Numerical Micromagnetism. Application to Mesoscopic Magnetic Systems

Liliana Buda-Prejbeanu CEA / DRFMC / SPINTEC Grenoble, France Jean-Christophe Toussaint CNRS – Laboratoire Louis Néel Grenoble, France

Summer School Magnetism of Nanostructured Systems and Hybrid Structures Braşov 2003

Outline

Micromagnetics – theoretical background

• hypothesis & limits
• total free energy minimization (variational principle)
• static and dynamic equations

Micromagnetics – overview of the numerical implementation

• current state of the art
• finite difference approximation (fields & energies)
• errors & accuracy & validation

Application for mesoscopic ferromagnetic elements

• circular Co dots
• self-assembled epitaxial submicron Fe dots

References

1 Å 1 nm 10 nm 100 nm 1 µm 10 µm

Micromagnetism

10 nm 100 nm

mesoscopic scale

thin films sub-microns objects

Length Scale

L, l, e <1 µm ~ 4 Å Co (hcp)

magnetic device

1 Å

Atomic scale

individual spins

1 nm

nanoparticules clusters ultra-thin films

nanoscopic scale

Quantum Mechanics

1 µm 10 µm

macroscopic scale

micrometric

• bjects

bulk

Bulk

Experimental Scale

MFM, Lorentz microscopy,…

• spatial resolution limited (>20 nm)
• several possible configurations

Microscopic studies Local imaging

M r

Macroscopic studies Hysteresis curves

i

Mean values of the magnetization MOKE, SQUID,...

Vellekoop et al., JMMM. 190, 148 (1998).

Hypothesis

Classical theory of continuous ferromagnetic material

• smooth spatial variation of the magnetization vector

Continuous functions (space & time)

) , ( t r M r r

• magnetization

) , ( t r H r r

( ) [ ]

t r m E , r r

Thermal fluctuations neglected

) ( ), ( ), ( T K T M T A

u s ex

1 ) , ( ) , ( ) , ( = = t r m t r m M t r M

s

r r r r r r

Magnetization constant amplitude vector

≡

1963 - W. F. Brown Jr. 1907 P. Weiss / magnetic domains 1935 Landau-Lifshiz / domain walls Individual spins Continuous material

i

• fields
• energies
• J. F. Brown, Jr. : Micromagnetics, J. Wiley and Sons, New York (1963)

Total Free Energy (Gibb’s free energy)

Magneto-crystalline anisotropy

• the crystal symmetry axis
• easy direction

( )

[ ]

∫

⋅ −

V K

dV m u K

2 1 1

r r

( )

∫

∇

V ex

dV m A

2

r r

Magnetostatic interaction

• Maxwell’s equations
• magnetic charges distribution
• magnetic domains formation

( )

[ ]

∫

⋅ −

V dem s

dV m H m M r r r 2 1 µ

Zeeman coupling

• external applied field
• magnetization rotation

[ ]

∫

⋅ −

V app s

dV H m M µ r r

local interaction local interaction next neighbors long range interaction

Exchange interaction

• magnetic order (T<Tc ) (QM)
• parallels spins

Others contributions : surface coupling, ….

Micromagnetic equations m ( r )

magnetization distribution

Space of configurations

energy minima

( )

{ }

1 , = ∈ = m V r r m m r r r r r

total free energy functional

[ ] ( )dV

V m m E

∫

= r r ε magnetic stable state = minimum of the total free energy functional

[ ] [ ]

2

> = m E m E r r δ δ

1

2

= ⋅ = + → m m m m m m r r r r r r δ δ

⇐ variational principle

Micromagnetic equations – static equilibrium equations

n r

V

m r

S

m m r r r × = θ δ δ

( )

∫ ∫

⋅       ∂ ∂ × + ⋅ × − =

S ex V eff s

dS n m m A dV H m M µ E θ δ θ δ δ r r r r r r 2

( )

m H H u m u M µ K m M µ A H

D app K K s s ex eff

r r r r r r r r C 2 2

1

+ + + ⋅ + ∆ =

effective field

dV H m M µ E

eff V s

r r

∫

⋅ − = δ δ

• J. Miltat in Applied Magnetism (p. 221)
• A. Hubert, R. Schäfer : Magnetic Domains (p. 149)

i

[ ]( )

V r r H m

eff

∈ ∀ = × r r r r r

Brown’s equations

S r n m ∈ = ∂ ∂ r r ,

S r n m A n m A

ex ex

∈ ∂ ∂ = ∂ ∂ r r r ,

2 2 , 1 1 ,

Micromagnetic equations m ( r, t )

( )

{ }

1 , , , = ≥ ∈ = m t V r t r m m r r r r r

space & time dependence → energy Space of configurations

1 2

magnetization trajectory between two magnetic states

Micromagnetic equations - dynamics

Landau-Lifshitz-Gilbert Equation (LLG) relaxation H m precession m H

( )

( ) ( )

[ ]

eff eff

H µ m m H µ m t m r r r r r r

2

1 × × − × − = ∂ ∂ + αγ γ α

2 > =

e

m e g γ ) /( 10 105 . 1

5

As m g µ × × = = γ γ

γ = gyromagnetic ratio

2 ≅ g

g = Landé factor

. 1 001 . ÷ ≅ α

α = damping parameter

Magnetostatic Equations

Scalar potential formalism:

φ ∇ − = r r

D

H

( ) ( )

r r

m

r r ρ φ − = ∆

( ) ( )

r r

ext

r r φ φ =

int

[ ]( ) ( )

r r n

m ext

r r r r σ φ φ = − ∇ ⋅

int

S r ∈ r

3D

∞ → → r r r r , ) ( φ S r ∈ r

( ) ( )

∫∫ ∫

− ∇ − − ∇ − =

S m V m D

dS r r r G dV r r r G r H ' ' ) ' ( ' ' ) ' ( ) ( r r r r r r r r r r σ ρ

Green’s function formalism :

( ) ( ) ( ) ( ) ∫∫ ∫

− + − =

S m V m

dS r r G r dV r r G r r ' ' ' ' ' ' ) ( r r r r r r r σ ρ φ

r r G r r π 4 1 ) ( =

3

ℜ ∈ r r

3

4 ) ( r r r G r r r r π − = ∇

& i

• J. D. Jackson : Classical electrodynamics, New York (1962)

General Algorithm

Defining the problem

• geometry description
• material parameters
• initial conditions

(time & space)

( ) { }

appl s ex

H M A K t r m r r r , , , , ,

( )

( ) ( ) [ ]

eff eff

H µ m m H µ m t m r r r r r r

2

1 × × − × − = ∂ ∂ + αγ γ α

LLG time integration

• amplitude conservation

false

rr M H m

s eff

ε ≤ ×

max

r r

t→t+δt

Check equilibrium criteria

true

{ }

eq eq

E m , r

Stable state- numerical solution

= ∂ ∂ n m r

, M r r ⋅ ∇ − = ρ , M n r r ⋅ = σ

State characterization

• magnetic charges
• magnetostatic field
• fields & energy terms

k ex D H

ε ε ε ε ε + + + =

k ex D app eff

H H H H H r r r r r + + + =

analytical treatment

• macrospin approximation
• Bloch domain wall in bulk
• by linearisation
• near the saturation limit
• nucleation and switching of domains
• ferromagnetic resonance

numerical treatment

• powerful and efficient tools (if some rules are respected! )

static & dynamic equations non-linear non-local coupled partial differential 1 = ← m r

D

H r ←

2

, ∂ ∂ ←

Solution

Second order integro-differential equations

Current State of the Art

Finite element method (FEM / BEM) Finite difference approximation (FDA)

• regular mesh
• irregular meshes
• adaptive mesh refinement

Fredkin & Koehler Fidler & Schrefl Hertel & Kronmuller Ramstöck et al. …….

• restrictive geometry

W.F. Brown Jr (1965) Schabes et al., (1988) Berkov et al. Bertram et al. Donahue et al. Miltat et al. Nakatani et al. Toussaint et al. Scheinfein et al.

• J. -G. Zhu et al………….

Finite Difference Approximation (FDA)

( )

{ }

( )

{ }

V r r H H m V r r m m

eff eff

∈ = = ∈ = r r r r r r r r r 1 ,

{ }

{ }

N i H H m N i m m

i eff eff i i

.. 1 1 , .. 1

,

= = = = = r r r r r

Numerical Numerical discretisation discretisation

N – total number of mesh nodes

• infinite prisms

2D

: e.g. thin films

3D

• orthorhombic cells

: dots, wires, platelets, …

Finite Difference Aproximation (FDA) ( )

m M s

m

r r ⋅ ∇ − = ρ

( )

n m M s

m

r r ⋅ = σ

magnetic charges

( )

D H K K s s ex eff

H H u m u M µ K m M µ A H r r r r r r r + + ⋅ + ∆ =

1

2 2

[ ] ( )

[ ]

( ) ( )

[ ]

( ) ( )

[ ]

( ) ( ) ( )

[ ]

dV r m H r m M r H r m M µ r m u K r m A m E

V dem s app s K ex

∫

          ⋅ − ⋅ − − ⋅ − + ∇ = r r r r r r r r r r r r r r r r

2 1 2

2 1 1 µ

( ) ( ) ( )

x

h j i m j i m j i x m 2 , 1 , 1 , − − + ≅ ∂ ∂

The accuracy is dependent on the Taylor expansion order !

( ) ( ) ( ) ( )

2 2 2

, 1 , 2 , 1 ,

x

h j i m j i m j i m j i x m − + − + ≅ ∂ ∂

2D

• M. Labrune, J. Miltat, JMMM 151, 231 (1995).

i

Taylor expansion

( ) ( ) ( ) ( )

) ( , 2 1 , , , 1

3 2 2 2 x x x

h O h j i x m h j i x m j i m j i m + ∂ ∂ + ∂ ∂ + = +

( ) ( ) ( ) ( )

) ( , 2 1 , , , 1

3 2 2 2 x x x

h O h j i x m h j i x m j i m j i m + ∂ ∂ + ∂ ∂ − = −

Energies & Fields General Algorithm

Defining the problem

• geometry description
• material parameters
• initial conditions

(time & space)

( ) { }

appl s ex

H M A K t r m r r r , , , , ,

= ∂ ∂ n m r

, M r r ⋅ ∇ − = ρ , M n r r ⋅ = σ

State characterization

• magnetic charges
• magnetostatic field
• fields & energy terms

k ex D H

ε ε ε ε ε + + + =

k ex D app eff

H H H H H r r r r r + + + =

[ ] [ ]

m H m E

ex ex

r r r ,

Exchange

[ ] [ ]

m H m E

K K

r r , r

[ ] [ ]

m H m E

app H

r r ,

Magnetocrystalline anisotropy local terms direct evaluation

r

Zeeman

[ ] [ ]

m H m E

D D

r r , r

Magnetostatic long range interaction

• the stray field energy calculation involves amounts up to

a six-fold integration (90% of the computation time)

( )

m M s

m

r r ⋅ ∇ − = ρ

( )

n m M s

m

r r ⋅ = σ Stray field (HD) → constant magnetization cells

W.F. Brown Jr., A.E. LaBonte: JAP 36, 1380 (1965) A.E. LaBonte: J. Appl. Phys. 38, 3196 (1967) Schabes et al., JAP 64, 1347 (1988).

cst mijk = r

(ijk) (i`j`k`)

i

Volume charges Surface charges

( ) ( )

cell Ncells I I D I s S S m m D

V H m M dS dS r r r r µ E

∑ ∫∫∫∫

=

⋅ = − =

1 2

2 1 ' ' ' 8 r r r r r r µ σ σ π

( )

∑

= −

                    − =

N J r z y x r r zz zy zx yz yy yx xz xy xx s I D

J J I

m m m N N N N N N N N N M r H

1

) (

r r r

r r

• mean stray field upon the cell I

[N] = demagnetizing factor

Demagnetizing Factor (N)

[ ]

                    − = − =

z y x zz zy zx yz yy yx xz xy xx D

M M M N N N N N N N N N M N H r t r

1 N N N = + +

zz yy xx

Sphere Thin film ∞ Cylinder ∞

1 N =

⊥

N = 2 / 1 N =

⊥

N =

3 / 1 N =

N is a tensor with the trace :

! General: a uniformly magnetized ferromagnetic body gives rise to a non-uniform stray field. ! Exception : the ellipsoid

( )

m M s

m

r r ⋅ ∇ − = ρ

( )

n m M s

m

r r ⋅ = σ Stray field (HD) → volume + surface charges

(ijk)

σ ρ i

• K. Ramstöck et al., JMMM 135 , 97 (1994)

Volume charges Surface charges Stray field evaluated in the center of each cell.

( ) ( ) ∑∫∫ ∑ ∫

− ∇ − − ∇ − =

= Nsurf J S m I Nvol J V m I I D

J J

dS r r r G dV r r r G r H ' ) ( ' ' ) ( ) (

' ' 1 '

r r r r r r r r r r σ ρ

Constant and/or linear volume and/or surface charges Evaluation of a sum with a huge number of terms : N X N terms ! Computation time?

Fourier Transform implementation (TF)

( ) ( ) ( ) ( ) ∫∫ ∫

− ∇ − − ∇ − =

S m V m D

dS r r r G dV r r r G r H ' ' ' ' ' ' ) ( r r r r r r r r r r σ ρ

[ ] [ ] )

( ) ( ) ( r G r G r H

m m D

r r r r r r σ ρ ⊗ ∇ − ⊗ ∇ − =

[ ]

ℜ → +∞ ∞ − , : , g f

( )

∫

+∞ ∞ −

− = ⊗ ' ) ' ( ) ' ( ) ( dx x f x x g x g f

[ ] [ ] [ ]

g f g f TF TF TF ⋅ = ⊗

Theorem of the Convolution Product

[ ]

G ∇ TF

[ ]

m

ρ TF ] [ TF

m

σ

[ ] [ ] [ ] [ ]

[ ]

m TF G TF m TF G TF

D

H

σ ρ ⋅ ∇ + ⋅ ∇

−

=

r r

r

1

TF

) (r

m

r ρ

) (r

m

r σ

1 2 3 4

, ,

r

Discrete Fourier Transform & FFT

• Discrete functions (charges, interaction function,..)
• Discrete Fourier Transform → periodicity
• Zero padding technique allows to deal with non-periodic systems
• no. operations / iteration

N N N

2 2

log →

N=8 X 8 X 8 262144 → 4607

[ ]

G ∇ TF r

is evaluated once at the beginning of the simulation

FFT - T.W.Cooley et al, (1965); FFTW - M. Frigo et al, MIT, (1997)

regular mesh & double the size of the real system

i

Stray field (HD) – comparison i

• D. Berkov et al., IEEE Trans. Mag. 29(6) 2548 (1993)

M = cst

ρ=cst σ = c s t

ρ = cst σ = cst

Soft magnetic material : constant volume charges is the appropriated approximation!

• package GL_FFT by JC Toussaint (LLN)

General Algorithm

Defining the problem

• geometry description
• material parameters
• initial conditions

(time & space)

( ) { }

appl s ex

H M A K t r m r r r , , , , ,

( )

( ) ( ) [ ]

eff eff

H µ m m H µ m t m r r r r r r

2

1 × × − × − = ∂ ∂ + αγ γ α

LLG time integration

• amplitude conservation

= ∂ ∂ n m r

, M r r ⋅ ∇ − = ρ , M n r r ⋅ = σ

State characterization

• magnetic charges
• magnetostatic field
• fields & energy terms

k ex D H

ε ε ε ε ε + + + =

k ex D app eff

H H H H H r r r r r + + + =

LLG integration scheme & numerical stability

[ ]

H H m H H m H H H H m m r r r r r r r

2

) ( )) cos( 1 ( ) ( ) sin( ) cos( ) ( ) ( τ δτ τ δτ δτ τ δτ τ ⋅ − + × + = +

Toussaint et al., Proceedings of the 9th International Symposium Magnetic Anisotropy and Coercivity in Rare-Earth Transition Metal Alloys 2, 59 (1996)

) ( ) ( τ τ τ H m m r r r × − = ∂ ∂

Explicit scheme

2

1 α γ τ + = t µ

) ( ) ( ) ( ) ( τ τ α τ τ

eff eff

H m H H r r r r × + =

← normalized time Amplitude of the magnetization is implicitly conserved.

1 = m r

The field varies slowly in time.

) (t H r

Fast integration method

2

2 1         <

ex s l

h M µ t γ α δ von Neumann analysis → critical time step for stability

2 / 1 / =

ex

l h 1 . = α

m A M s / 10 8

5

× =

, ,

fs t 70

lim ≅

δ

i

General Algorithm

Defining the problem

• geometry description
• material parameters
• initial conditions

(time & space)

( ) { }

appl s ex

H M A K t r m r r r , , , , ,

( )

( ) ( ) [ ]

eff eff

H µ m m H µ m t m r r r r r r

2

1 × × − × − = ∂ ∂ + αγ γ α

LLG time integration

• amplitude conservation

false

rr M H m

s eff

ε ≤ ×

max

r r

t→t+δt

Check equilibrium criteria

true

{ }

eq eq

E m , r

Stable state- numerical solution

= ∂ ∂ n m r

, M r r ⋅ ∇ − = ρ , M n r r ⋅ = σ

State characterization

• magnetic charges
• magnetostatic field
• fields & energy terms

k ex D H

ε ε ε ε ε + + + =

k ex D app eff

H H H H H r r r r r + + + =

Mesh size & characteristics length scale ( ) ( )

2 2

sin       ∂ ∂ + = x A K

ex u

θ θ θ ε

Infinite Bloch wall y z

x Ku θ

x

              ∆ ± = exp arctan 2 ) ( x x θ

u ex

K A = ∆0

Bloch wall parameter Exchange length

2

2

s ex ex

M µ A l =

( ) ( ) ( )

2 2 2 2 2

sin 1 cos 2 1 , θ θ θ θ ε

ex ex s

A r x A M µ r +       ∂ ∂ + =

( )

             − ≅

ex

l r r exp arccos θ ‘vortex’ in a full disk

! Mesh size < ∆0 , lex

Ms(A/m) Aex(J/m) Ku(J/m3) ∆0 (nm) lex(nm) Co 1400 X 103 (1.5÷3) X 10-11 500 X 103 5.5 ÷ 7.7 3.4 ÷ 4.8 NiFe 800 X 103

• 1. X 10-11

1 X 103 100. 4.9

Validation Tests

err % 5 30

1 2

⇒ ° < −θ θ

° < − 30

1 2

θ θ

Néel variation Bloch variation

exchange magnetostatic (Néel) magnetostatic (Bloch) magnetocrystallin anisotropy

° < % 5 err

↓

Angular deviation between two adjacent cells smaller than 30° !

Numerical results & Analytical results

Radial position (lex)

Mesh effects

m A M s / 10 1400

3

× = m J A

ex

/ 10 4 . 1

11 −

× =

3 3

/ 10 500 m J Ku × =

nm 29 . 5

0 =

∆ nm lex 37 . 3 =

Mesh size (lex)

Energy density (erg/cm3)

Vortex state in a circular dot

Validation tests

Standard problems (NIST) Standard problem 1,2,3: static calculus Standard problem 4 : dynamic calculus

i

http://www.ctcms.nist.gov/~rdm/mumag.org.html

Simulation & experience MFM, Lorentz microscopy,…. Magnetization curves… Check points Check different equilibrium criteria… Check different field steps… Check different discretisations (doest it converge smoothly to limit value… Break the symmetry…

Free Open Source & Commercial software

OOMMF (Object Oriented MicroMagnetic Framework), M.Donahue & D. Porter

http://math.nist.gov/oommf/

SimulMag (PC Micromagnetic Simulator), developed by J. Oti

http://math.nist.gov/oommf/contrib/simulmag/

GDM2 (General Dynamic Micromagnetics), developed by B. Yang

http://physics.ucsd.edu/~drf/pub/

LLG Micromagnetics Simulator, developed by M. R. Scheinfein

http://llgmicro.home.mindspring.com/

MagFEM3D, developed by K. Ramstöck

http://www.ramstock.de/

MicroMagus, developed by D. V. Berkov, N. L. Gorn

http://www.micromagus.de/

Magsimus, Euxine Technologies

http://www.euxine.com/

Circular Co Disks Co(0001) NiFe 0.8µm Happ i

• J. Raabe et al. JAP 88, 4437 (2000)

i

• M. Demand et al. JAP. 87, 5111 (2000)

i

• T. Shinjo et al. Science 289, 930 (2000)

Magnetic Stable States

MFM Simulation

m A M s / 10 1400

3

× = m J A

ex

/ 10 4 . 1

11 −

× =

3 3

/ 10 500 m J Ku × = nm h h h

z y x

5 . 2 = = =

Material Parameters

i

Demand et al., JAP (2000), Prejbeanu Ph.D. thesis, Natali et al. PRL(2002)

Sizes & energies

Lift scan height 50nm φ= 200 nm h = 15 nm h = 5 nm

Internal vortex structure

↓

enhancement of the vortex

( )

⊥ ≠

u u

K K &

• J. Miltat in Applied Magnetism, R. Gerber et al Eds. Kluwer (1994)

i

Diameter effects (φ)

φ

Radial position (nm)

h=15 nm

Thickness effects (h)

h=10 nm h=5 nm

Demand et al., JAP. 87, 5111 (2000).

i

h=20 nm

Hehn et al., Science 272, 1782 (1996).

i

Diagramme des états fondamentaux Diagramme des états fondamentaux

Weak stripes

Thickness / lex

Ground state phase diagram

Diameter /lex

( )

⊥ ≠

u u

K K &

( )

=

u

K

Comparison disks & rings i

• S. Li et al. PRL (2001)

Self-assembled Epitaxial Submicron Fe Dots [001] [110] [1-10]

• P. O. Jubert, J.C. Toussaint
• O. Fruchart, C. Meyer

Laboratoire Louis Néel, Grenoble

Y Samson

CEA / DRFMC / SP2M / NM, Grenoble L~550nm, w~350nm, h~65nm

Hsat || [001] Fe/Mo(110) compact 3D dots

P.O. Jubert et al. PRB 64, 115419 (2001)

i

• 1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
• 1.0
• 0.5

0.0 0.5 1.0 // [001] // [1-10] // [110] Applied field (T) <M> / MS

T=300K

Diamond Landau

500 nm

Remanent Configurations (Happ=0)

Fe bulk parameters Ideal shape : hexaplot

m A M s / 10 1750

3

× = m J A

ex

/ 10 . 2

11 −

× = h = 60nm L = 600nm w = 300nm

3 4 1

/ 10 81 . 4 m J K × =

nm 7

0 ≅

∆ π nm lex 10 ≅ π

3

75 . 3 68 . 4 68 . nm × × 4

Mesh :

y x

Diamond State z = h/2 Landau state

• 1

+1

mz i

P.O. Jubert et al. EPL 63, 138 (2003)

Remanent Configurations

MFM measured MFM simulated simulated configuration

Landau state Diamond state Mp

• -
• -

MFM response ~ ∂zHz MFM tip ~ monopole Lift scan height 30nm

0.0 0.1 0.0 0.5 1.0

Applied field µ0Hx (T)

< mx > In plane hysteresis curve

+ 0.1°

Happ. [001]

z=h/2

• 1

+1

mz

Happ=0.09 T 0.08 T 0.06 T

0.00 T

In plane hysteresis curve & distortion

+ 0.1°

Happ. [001]

z=h/2

Applied field µ0Hx (T)

0.0 0.1 0.0 0.5 1.0

< mx >

• 1

+1

mz

Happ=0.09 T 0.08 T 0.0 T 0.06 T

Simulation & Experience

Landau states

0.0 0.1 0.2 0.3 0.0 0.5 1.0

Applied field µ0Hx (T)

< mx >

simulated

measured T=300K

• 1

+1

mz

Diamant state Simulation → remanent state is controlled by small external perturbation. MFM resolution is too low to distinguish between the two Landau states. Thermal fluctuation effect.

Selected bibliography - books

Basic book on micromagnetics

• W. F. Brown, Jr. : Micromagnetics

Intersience Publishers, J. Wiley and Sons, New York (1963)

i

• J. Miltat : Domains and domains walls in soft magnetic materials mostly

in Applied Magnetism, R. Gerber et al Eds. Kluwer (1994)

i

• A. Aharoni: Introduction to the theory of ferromagnetism

Clarendon Press, Oxford (1996)

i

• G. Bertotti: Hysteris in magnetism (for physicists,material scientists and engineerers)

Academic Press, (1998)

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• A. Hubert, R. Schäfer : Magnetic Domains

Springer (1998)

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Selected bibliography - articles

Articles on numerical aspects of micromagetics W.F. Brown Jr., A.E. LaBonte: J. Appl. Phys. 36, 1380 (1965) A.E. LaBonte: J. Appl. Phys. 38, 3196 (1967)

i

• J. Fidler, T. Schrefl : Journal of Physics D: Applied Physics, 33 R135-R156 (2000)

i

• D. Fredkin , T.R. Koehler: IEEE Trans. Mag. 26, 415 (1990)

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• R. Cowburn: J. Phys. D: Appl. Phys. 33, R1 (2000)

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Thank you! Mulţumesc! Merci!

Vortex configuration

Out of plane magnetization

m r

In-plane single domain state