Introduction to Numerical Micromagnetism.

Application to Mesoscopic Magnetic Systems

Liliana Buda-PrejbeanuCEA / DRFMC / SPINTEC Grenoble, France

Jean-Christophe ToussaintCNRS – 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 filmssub-microns objects

Length Scale

L, l, e <1 µm

~ 4

Å

Co (hcp)

magneticdevice

1 Å

Atomic scale

individualspins

1 nm

nanoparticulesclustersultra-thin films

nanoscopic scale

Quantum Mechanics

1 µm 10 µm

macroscopicscale

micrometric objects

bulk

Bulk

Experimental Scale

MFM, Lorentz microscopy,…- spatial resolution limited (>20 nm)- several possible configurations

Microscopic studiesLocal imaging

Mr

Macroscopic studiesHysteresis curves

i

Mean values of the magnetizationMOKE, 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)

),( trM rr- magnetization

),( trH rr

( )[ ]trmE ,rr

Thermal fluctuations neglected

)(),(),( TKTMTA usex

1),(),(),(

=

=

trmtrmMtrM s

rr

rrrrMagnetization 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 dVmuK 21 1 rr

( )∫ ∇V

ex dVmA 2rr

Magnetostatic interaction- Maxwell’s equations - magnetic charges distribution- magnetic domains formation

( )[ ]∫ ⋅−V

dems dVmHmM rrr02

1 µ

Zeeman coupling- external applied field- magnetization rotation

[ ]∫ ⋅−V

apps dVHmMµrr

0

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

ener

gy

minima

( ){ }1, =∈= mVrrmm rrrrr

total free energy functional

[ ] ( )dVV

mmE ∫= rr ε

magnetic stable state = minimum of the total free energy functional

[ ][ ] 0

02 >

=mE

mEr

r

δδ

012

=⋅=

+→

mmm

mmm

rr

r

rrr

δ

δ⇐ variational principle

Micromagnetic equations – static equilibrium equations

nr

V

mrS

mm rrr ×= θδδ

( ) ∫∫ ⋅

∂∂

×+⋅×−=S

exV

effs dSnmmAdVHmMµE θδθδδ

rrrrrr 20

( ) mHHumuMµKm

MµAH DappKK

ss

exeff

rrrrrrrrC22

0

1

0

+++⋅+∆=

effective field

dVHmMµE effV

s

rr∫ ⋅−= δδ 0

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

[ ]( ) VrrHm eff ∈∀=×rrrrr 0

Brown’s equations

Srnm

∈=∂∂ rr

,0

Srn

mAnmA exex ∈

∂∂

=∂∂ r

rr,2

2,1

1,

Micromagnetic equations m ( r, t )

( ){ }1,0,, =≥∈= mtVrtrmm rrrrrspace & time dependence →

ener

gy

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

( ) ( ) ( )[ ]effeff HµmmHµmtm rrrrrr

0021 ××−×−=∂∂

+ αγγα

02

>=em

egγ )/(10105.1 5

00 Asmgµ ××== γγγ = gyromagnetic ratio

2≅gg = Landé factor

0.1001.0 ÷≅αα = damping parameter

Magnetostatic Equations

Scalar potential formalism: φ∇−=rr

DH

( ) ( )rr m

rr ρφ −=∆

( ) ( )rr ext

rr φφ =int

[ ]( ) ( )rrn mext

rrrr σφφ =−∇⋅ int

Sr ∈r

3D

∞→→ rr rr ,0)(φSr ∈r

( ) ( )∫∫∫ −∇−−∇−=S

mV

mD dSrrrGdVrrrGrH '')'('')'()( rrrrrrrrrrσρ

Green’s function formalism :

( ) ( ) ( ) ( )∫∫∫ −+−=S

mV

m dSrrGrdVrrGrr '''''')( rrrrrrr σρφ

rrG rr

π41)( =

3ℜ∈rr

34)(

rrrG r

rrr

π−=∇&

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

General Algorithm

Defining the problem-geometry description-material parameters-initial conditions(time & space)

( ){ } applsex HMAKtrmrrr ,,,,, 0

( ) ( ) ( )[ ]effeff HµmmHµmtm rrrrrr

0021 ××−×−=∂∂

+ αγγα LLG time integration- amplitude conservation

falserrMHm seff ε≤× maxrr

t→t+δt

Check equilibrium criteria

true

{ } eqeq Em ,r Stable state- numerical solution

0=∂∂ nmr,Mrr

⋅∇−=ρ ,Mnrr

⋅=σState characterization-magnetic charges-magnetostatic field-fields & energy termskexDH εεεεε +++=

kexDappeff HHHHHrrrrr

+++=

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 equationsnon-linearnon-local coupled partial differential

1=← mr

DHr

←2, ∂∂←

Solution

Second order integro-differential equations

Current State of the Art

Finite element method(FEM / BEM)

Finite differenceapproximation (FDA)

-regular mesh -irregular meshes-adaptive mesh refinementFredkin & KoehlerFidler & SchreflHertel & KronmullerRamstöck 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)

( ){ }( ){ }VrrHH

mVrrmm

effeff ∈=

=∈=rrrr

rrrrr 1, { }{ }NiHH

mNimm

ieffeff

ii

..1

1,..1

, ==

===rr

rrr

Numerical Numerical discretisationdiscretisation

N – total number of mesh nodes

-infinite prisms

2D

: e.g. thin films

3D

-orthorhombic cells

: dots, wires, platelets, …

Finite Difference Aproximation (FDA)

( )mM smrr

⋅∇−=ρ

( )nmM smrr⋅=σ

magneticcharges( ) DHKK

ss

exeff HHumu

MµKm

MµAH

rrrrrrr++⋅+∆=

0

1

0

22

[ ]( )[ ] ( )( )[ ]

( ) ( )[ ] ( ) ( )( )[ ] dVrmHrmMrHrmMµ

rmuKrmAmE

Vdemsapps

Kex

∫

⋅−⋅−

−⋅−+∇= rrrrrrrrr

rrrrrr

r

00

21

2

21

1

µ

( ) ( ) ( )xh

jimjimjixm

2,1,1, −−+

≅∂∂

The accuracy is dependent on the Taylor expansion order !

( ) ( ) ( ) ( )22

2 ,1,2,1,xh

jimjimjimjixm −+−+

≅∂∂

2D

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

Taylor expansion( ) ( ) ( ) ( ) )(,

21,,,1 32

2

2

xxx hOhjixmhji

xmjimjim +

∂∂

+∂∂

+=+

( ) ( ) ( ) ( ) )(,21,,,1 32

2

2

xxx hOhjixmhji

xmjimjim +

∂∂

+∂∂

−=−

Energies & FieldsGeneral Algorithm

Defining the problem-geometry description-material parameters-initial conditions(time & space)

( ){ } applsex HMAKtrmrrr ,,,,, 0

0=∂∂ nmr,Mrr

⋅∇−=ρ ,Mnrr

⋅=σState characterization-magnetic charges-magnetostatic field-fields & energy termskexDH εεεεε +++=

kexDappeff HHHHHrrrrr

+++=

[ ] [ ]mHmE exexrrr ,Exchange

[ ] [ ]mHmE KK

rr ,r

[ ] [ ]mHmE appH

rr ,

Magnetocrystallineanisotropy

local termsdirect evaluation

rZeeman

[ ] [ ]mHmE DD

rr ,r

Magnetostatic long range interaction

-the stray field energy calculation involves amounts up toa six-fold integration (90% of the computation time)

( )mM smrr

⋅∇−=ρ

( )nmM smrr⋅=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).

cstmijk =r (ijk)

(i`j`k`)i

Volume charges

Surface charges

( ) ( )cell

Ncells

IIDIs

S S

mmD VHmMdSdS

rrrrµE ∑∫∫∫∫

=

⋅=−

=1

20

0

21'

''

8

rrrr

rr

µσσπ

( )

∑=

−

−=

N

J

rz

y

x

rrzzzyzx

yzyyyx

xzxyxx

sID

JJI

mmm

NNNNNNNNN

MrH1

)(rrr

rr

-mean stray field upon the cell I

[N] = demagnetizing factor

Demagnetizing Factor (N)

[ ]

−=−=

z

y

x

zzzyzx

yzyyyx

xzxyxx

D

MMM

NNNNNNNNN

MNHrtr

1NNN =++ zzyyxx

Sphere Thin film ∞ Cylinder ∞

1N =⊥

0N =2/1N =⊥

0N =3/1N =

N is a tensor with the trace :

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

( )mM smrr

⋅∇−=ρ

( )nmM smrr⋅=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 SmI

Nvol

J VmIID

JJ

dSrrrGdVrrrGrH ')('')()( ''

1

' rrrrrrrrrrσρ

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

mV

mD dSrrrGdVrrrGrH '''''')( rrrrrrrrrrσρ

[ ] [ ] )()()( rGrGrH mmD

rrrrrrσρ ⊗∇−⊗∇−=

[ ] ℜ→+∞∞− ,:, gf ( ) ∫+∞

∞−

−=⊗ ')'()'()( dxxfxxgxgf

[ ] [ ] [ ]gfgf TFTFTF ⋅=⊗

Theorem of the Convolution Product

[ ]G∇TF

[ ]mρTF ][TF mσ

[ ] [ ] [ ] [ ][ ]mTFGTFmTFGTFDH σρ ⋅∇+⋅∇−= rrr

1TF

)(rm

rρ )(rm

rσ1

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 NNN 22 log→

N=8 X 8 X 8 262144 → 4607[ ]G∇TFr

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 systemi

Stray field (HD) – comparison

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

M = cst

ρ=cstσ=cstρ = 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)

( ){ } applsex HMAKtrmrrr ,,,,, 0

( ) ( ) ( )[ ]effeff HµmmHµmtm rrrrrr

0021 ××−×−=∂∂

+ αγγα LLG time integration- amplitude conservation

0=∂∂ nmr,Mrr

⋅∇−=ρ ,Mnrr

⋅=σState characterization-magnetic charges-magnetostatic field-fields & energy termskexDH εεεεε +++=

kexDappeff HHHHHrrrrr

+++=

LLG integration scheme & numerical stability

[ ] HHmHHmH

HHHmm

rrrrrrr

2

)())cos(1()()sin()cos()()( τδττδτδττδττ ⋅−+×+=+

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

)()( τττ

Hmm rrr

×−=∂∂

Explicit scheme2

0

1 αγτ

+=

tµ

)()()()( τταττ effeff HmHHrrrr

×+=

← normalized time

Amplitude of the magnetization is implicitly conserved. 1=mr

The field varies slowly in time.)(tHr

Fast integration method2

021

<

exs lh

Mµt

γαδvon Neumann analysis → critical time step for stability

2/1/ =exlh1.0=α mAM s /108 5×=, , fst 70lim ≅δ

i

General Algorithm

Defining the problem-geometry description-material parameters-initial conditions(time & space)

( ){ } applsex HMAKtrmrrr ,,,,, 0

( ) ( ) ( )[ ]effeff HµmmHµmtm rrrrrr

0021 ××−×−=∂∂

+ αγγα LLG time integration- amplitude conservation

falserrMHm seff ε≤× maxrr

t→t+δt

Check equilibrium criteria

true

{ } eqeq Em ,r Stable state- numerical solution

0=∂∂ nmr,Mrr

⋅∇−=ρ ,Mnrr

⋅=σState characterization-magnetic charges-magnetostatic field-fields & energy termskexDH εεεεε +++=

kexDappeff HHHHHrrrrr

+++=

Mesh size & characteristics length scale

( ) ( )2

2sin

∂∂

+=x

AK exu

θθθεInfinite Bloch wall

y

z

xKu

θ

x

∆

±=0

exparctan2)( xxθ

u

ex

KA

=∆0

Bloch wall parameter

Exchange length

20

2

s

exex Mµ

Al =

( ) ( ) ( )2

2

222

0 sin1cos21, θθθθε exexs A

rxAMµr +

∂∂

+=

( )

−≅

exlrr exparccosθ‘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%53012 ⇒°<−θθ

°<− 3012 θθ

Néel variation Bloch variation

exchangemagnetostatic (Néel)magnetostatic (Bloch)magnetocrystallin anisotropy°< %5err

↓Angular deviation between two

adjacent cells smaller than 30° !

Numerical results &

Analytical results

Radial position (lex)

Mesh effects

mAMs /101400 3×=

mJAex /104.1 11−×=

33 /10500 mJKu ×=

nm29.50 =∆

nmlex 37.3=

Mesh size (lex)

Ene

rgy

dens

ity (e

rg/c

m3)

Vortex statein a circular dot

Validation tests

Standard problems (NIST)

Standard problem 1,2,3: static calculus

Standard problem 4 : dynamic calculusi http://www.ctcms.nist.gov/~rdm/mumag.org.html

Simulation & experience

MFM, Lorentz microscopy,….Magnetization curves…

Check pointsCheck 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. Porterhttp://math.nist.gov/oommf/

SimulMag (PC Micromagnetic Simulator), developed by J. Otihttp://math.nist.gov/oommf/contrib/simulmag/

GDM2 (General Dynamic Micromagnetics), developed by B. Yanghttp://physics.ucsd.edu/~drf/pub/

LLG Micromagnetics Simulator, developed by M. R. Scheinfeinhttp://llgmicro.home.mindspring.com/

MagFEM3D, developed by K. Ramstöck http://www.ramstock.de/

MicroMagus, developed by D. V. Berkov, N. L. Gornhttp://www.micromagus.de/

Magsimus, Euxine Technologies http://www.euxine.com/

Circular Co Disks

Co(0001) NiFe

0.8µm

Hap

p

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

mAM s /101400 3×=

mJAex /104.1 11−×=33 /10500 mJKu ×=

nmhhh zyx 5.2===

Material Parameters

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

Sizes & energies

Lift scan height 50nmφ= 200 nmh = 15 nm

h = 5 nm

Internal vortex structure

↓enhancement of the vortex

( )⊥≠ uu KK &0

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 nmh=5 nm

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

h=20 nm

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

Diagramme des états fondamentauxDiagramme des états fondamentaux

Weak stripes

Thi

ckne

ss/ l

ex

Ground state phase diagram

Diameter /lex( )⊥≠ uu KK &0( )0=uK

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. ToussaintO. Fruchart, C. MeyerLaboratoire Louis Néel, Grenoble

Y SamsonCEA / 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=300KD

iam

ond

Lan

dau

500 nm

Remanent Configurations (Happ=0)

Fe bulk parametersIdeal shape : hexaplotmAM s /101750 3×=

mJAex /10.2 11−×=h = 60nm

L = 600nm

w = 300nm34

1 /1081.4 mJK ×=

nm70 ≅∆π

nmlex 10≅π375.368.468. nm××4Mesh :

y

x

Diamond Statez = h/2Landau state

-1

+1

mz

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

Remanent Configurations

MFMmeasured

MFMsimulated

simulatedconfiguration

Landau state Diamond state

Mp

- -- -

MFM response ~ ∂zHzMFM tip ~ monopole

Lift scan height 30nm

0.0 0.10.0

0.5

1.0

Applied field µ0Hx (T)

< m

x>

In plane hysteresis curve

+ 0.1°

Happ.

[001]

z=h/2

-1

+1

mz

Happ=0.09 T0.08 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

< m

x>

-1

+1

mz

Happ=0.09 T0.08 T

0.0 T

0.06 T

Simulation & Experience

Landau states

0.0 0.1 0.2 0.30.0

0.5

1.0

Applied field µ0Hx (T)

< m

x>

simulatedmeasured 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. : MicromagneticsIntersience Publishers, J. Wiley and Sons, New York (1963)

i

J. Miltat : Domains and domains walls in soft magnetic materials mostlyin Applied Magnetism, R. Gerber et al Eds. Kluwer (1994)

i

A. Aharoni: Introduction to the theory of ferromagnetismClarendon Press, Oxford (1996)

i

G. Bertotti: Hysteris in magnetism (for physicists,material scientists and engineerers)Academic Press, (1998)

i

A. Hubert, R. Schäfer : Magnetic DomainsSpringer (1998)

i

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) i

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

i

Thank you!Mulţumesc! Merci!

Vortex configuration

Out

of p

lane

mag

netiz

atio

n

mr

In-plane single domain state