Micromagnetism• Equations• Discretization FEM/Euler• Newton Algorithm• Results

Micromagnetism - Philippe Corboz, 29.10.2002 – p.1/15

EquationsLandau-Livshits-Gilbert equation:

dM

dt= −γM × Heff −

γα

Ms

M × (M × Heff) .

Effective magnetic field:

Heff = −∂E(m)

∂m

Micromagnetism - Philippe Corboz, 29.10.2002 – p.2/15

EquationsGibbs free energy:

E(M) =

∫

Ω

A

M2s

|gradM|2 + Φ(M/Ms) − 2µ0He · M dx + 1

2µ0

∫

R3

|gradψ|2 dx .

Potential equation:

µ0∆ψ = div M inR3

M satisfies Neumann boundary conditions

Micromagnetism - Philippe Corboz, 29.10.2002 – p.3/15

Discretization: TimeTimestepping:

δmn+ 1

2 = −mn+ 1

2 × hn+ 1

2

eff − αmn+ 1

2 × (mn+ 1

2 × hn+ 1

2

eff )

hn+ 1

2

eff := η∆mn+ 1

2 +Qd (d · mn+ 1

2 ) − gradψn+ 1

2 + he(tn+ 1

2)

with

dm

dt(t

n+ 1

2

) ≈ δmn+ 1

2 :=mn+1 − mn

τ, m(t

n+ 1

2

) ≈ mn+ 1

2 :=mn+1 + mn

2

Micromagnetism - Philippe Corboz, 29.10.2002 – p.4/15

Discretization: Spatial

Micromagnetism - Philippe Corboz, 29.10.2002 – p.5/15

Discretization: Weak formWeak form of LLG and potential equation:

1

1 + α2

∫

Ω

(δmn+ 1

2 − αmn+ 1

2 × δmn+ 1

2 ) · v dx =

=

∫

Ω

η grad(mn+ 1

2 × v) · gradmn+ 1

2 −Q(mn+ 1

2 × d)(d · mn+ 1

2 )v+

+ (mn+ 1

2 × gradψ) · v − (mn+ 1

2 × he) · v dx

∫

D

gradψ · gradϕ dx =

∫

D

mn+ 1

2 · gradϕ dx .

for all v ∈ (H1(Ω))3, and ϕ ∈ H10 (D)

Micromagnetism - Philippe Corboz, 29.10.2002 – p.6/15

DiscretizationLLG → |m| is invariant in timeNot the case for our discrete equations.Need to introduce a reduced integration for coupling term:

∫

Ω

η grad(mn+ 1

2 × v) · gradmn+ 1

2 dx

Micromagnetism - Philippe Corboz, 29.10.2002 – p.7/15

DiscretizationLLG → |m| is invariant in timeNot the case for our discrete equations.Need to introduce a reduced integration for coupling term:

∫

Ω

η grad(mn+ 1

2 × v) · gradmn+ 1

2 dx

↓↓↓

∫

Ω

η grad(I(mn+ 1

2 × v)) · gradmn+ 1

2 dx

Micromagnetism - Philippe Corboz, 29.10.2002 – p.7/15

Newton-Algorithm

fl(m) =kA1(ml − ml + α(ml+2ml+1 − ml+1ml+2))

+ η(A3l+2ml+1 − A3

l+1ml+2) − Al+1ml+2 + Al+2ml+1 = 0

Start with m = m

Do while norm(f(m)) > tol

• mit+1 = mit − (Df)−1f(mit)

• if it > itmax, break and error

Micromagnetism - Philippe Corboz, 29.10.2002 – p.8/15

Results: ExamplesStrong coupling vs weak coupling

Micromagnetism - Philippe Corboz, 29.10.2002 – p.9/15

Results: Energy contributions

0 10 20 30 40 50 60 70 80 90 100−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

time

ener

gy

exchange energyanisotropy energyexternal fieldmagnetic energytotal energy

Micromagnetism - Philippe Corboz, 29.10.2002 – p.10/15

Results: Different mesh widths

0 20 40 60 80 100 1200.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

time

ener

gy

4*ho2*ho3/2*hoho

Micromagnetism - Philippe Corboz, 29.10.2002 – p.11/15

Results: Different timesteps

0 1 2 3 4 5 6 7 8 9 100.28

0.3

0.32

0.34

0.36

0.38

0.4

time

ener

gy

0.10.050.01

Micromagnetism - Philippe Corboz, 29.10.2002 – p.12/15

Results: Varying Size of D

0 10 20 30 40 50 60 70 80 90 1000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

time

ener

gy

D=4D=8

Micromagnetism - Philippe Corboz, 29.10.2002 – p.13/15

Results: Convergence

0 100 200 300 400 500 6001.5

2

2.5

3

timestep

# ite

ratio

ns

Micromagnetism - Philippe Corboz, 29.10.2002 – p.14/15

Results: Movie example

Dynamical evolution of the magnetization

Micromagnetism - Philippe Corboz, 29.10.2002 – p.15/15