1. Introduction to Micromagnetic Simulation
Feng Xie Ph.D. student
Major advisor: Dr. Richard B. Wells

2. Contents Introduction to magnetic materials. Ideas in micromagnetics.
Physical equations.
Field analysis.
ODE solver and coordinate selection.
Simulations for ideal cases.
Thermal effects
Summary

3. (magnetic dipole moment
Magnetization
Magnetic dipole
moment
Magnetization
(magnetic dipole moment
per unit volume) S N  +p -p l  

4. Magnetic Materials Diamagnetic: M H M
Most elements in the periodic table, including copper, silver, and gold.
Ferromagnetic:
Iron, nickel, and cobalt.
Ferrimagnetic:
Ferrites.
Paramagnetic: M
Include magnesium, molybdenum, lithium, and tantalum.

5. CGS and SI Units M H K m 0 Magnetization Magnetic field
Quantity
Symbol
CGS unit
SI unit
CGS value / SI value
Magnetization M
emu/cc
A•m-1
10-3
Magnetic field H Oe
A-turn•m-1
410-3
Anisotropy constant K
ergs/cc
J•m-3 10
Magnetic charge density m
unit pole/cm3
Wb•m-3
100/(4)
Permeability of vacuum 0 =1
H•m-1
107/(4)

6. Hysteresis Loop

7. Scale Comparison A magnetic force microscopy (MFM) image showing
Domain structure
Micromagnetic
explanation of
Domain structure
(Phenomenology)
Electron
Spins
(Quantum
theory)

8. Why Micromagnetics?
To provide magnetization pattern inside the material.
To explain some experimental results.
To simulate new materials.
To realize new properties of materials.
To provide material parameters to designers.

9. Micromagnetic Assumptions
Magnitude:
The Landau-Lifshitz-Gilbert (LLG) equation
Magnetic fields:
Externally applied field
Exchange field
Demagnetizing field
Anisotropy field
Stochastic field or the stochastic LLG equation

10. Physical Equations The Landau-Lifshitz (LL) equation:
The Landau-Lifshitz-Gilbert (LLG) equation:
where

11. Comments on Equations When <<1, LG=22.8 M (radHz/Oe).
From either the LL or the LLG equation:
In analysis, we prefer the form:

12. Field Analysis Applied field: DC + AC.
Demagnetizing field: time consuming.
Effective field:
Anisotropy field: uniaxial and cubic.
Exchange field: quantum mechanic effect.
Other fields.

13. DC Field Solution DC field only + single grain The solution is wherex y 0 0 H0
where

14. DC Field Simulations

15. Small Applied AC Field Small ac field + single grain H0 resonance
Hx = h cos(t)
Hy = h sin(t)
h << H0 x y   H0

16. Demagnetizing Field Consuming most of computation time. where
long distance
where
Consuming most of computation time.

17. Fast Algorithm
Two computational methods are in discussion: Fast Multipole Method (FMM) and Fast Fourier Transform (FFT).
FMM is good for very big sample size. It can be applied on either asymmetric or symmetric geometries.
FFT is good for small sample size. It can only applied on symmetric geometries.

18. Fast Multipole Method
Source
Near Field
Middle Field
Far Field

19. Fast Fourier Transform
Convolution
Symmetry in geometries
0, 0
1, 0
2, 0
3, 0
0, 1
1, 1
2, 1
0, 2
3, 1
1, 2
3, 3
2, 2
3, 2
2, 3
1, 3
0, 3
u_b(column)
v_b(row)
u_a(column)
0, 0
1, 0
2, 0
3, 0
0, 1
1, 1
2, 1
0, 2
3, 1
1, 2
3, 3
2, 2
3, 2
2, 3
1, 3
0, 3
v_a(row)

20. Anisotropy Field Uniaxial Anisotropy: E=K0u+K1usin2+K2usin4+
Magnetocrystalline Anisotropy H  M
Uniaxial Anisotropy:
E=K0u+K1usin2+K2usin4+
Cubic anisotropy:
E=K0c+K1c(cos21cos22
+cos22cos23+ cos23cos21)+

21. Exchange Field The effective exchange field is
It is mainly from electron spin coupling.
It is short-range so that we take into consideration only exchange energy between nearest-neighbor grains.
The effective exchange field is

22. Adjustable Parameters
Crystalline anisotropy
HCP or FCC
K1, K2, …
distribution of c_axis (how good is good)
Exchange constant A:
Different materials have different As.
Different parts may have different As (poly).
Sample size and shape.
Anisotropy .
Nonuniform Ms

23. Coordinates + y x z m x x z < 30 |m|=1  y x z m   y x z m
 = ?  y x z m z z +
x < 30

24. ODE Solver Runge Kutta embedded 4th-5th method [2].
Adaptive time step.
No need value of previous steps.
[2] J. R. Cash and A. H. Karp, “A variable order Runge-Kutta method for initial value problems
with rapidly varying right-hand sides,” ACM Transactions on Mathemathical Software, vol. 16,
no. 3, pp , September 1990.

25. Geometry x y a d
Top View
Layer 1
Side View
c-axis  dz
Layer 2 

26. Default Simulation Parameters a d  dz
heh
hev
0.5 1
0.1
-0.05
Ms (emu/cc)
K1 (ergs/cc)  
233
5105
[0,20 ]
[0 ,360 ]

27. Various c-axis distributions

28. Various exchange constants

29. Various anisotropy constants

30. Various thickness

31. Mixture of two anisotropy constants

32. Stochastic LLG Equation
Due to thermal fluctuation.
Stochastic Landau-Lifshitz-Gilbert equation
Where is a stochastic field with the property

33. SDE Solver
Stochastic LLG equation is a stochastic ODE with multidimensional Wiener process.
The strong order of Runge-Kutta methods cannot exceeds 1.5 [3].
Heun scheme is applied.
[3] K. Burrage and P.M. Burrage, “High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations,” Applied Numerical Mathematics 22 (1996)

34. Thermal effects

35. Domain Wall Simulation (Side View)

36. Domain Wall Simulation (Top View)
Bloch wall

37. Summary Basic ideas in micromagnetic simulation.
Algorithms in micromagnetic modeling.
Micromagnetic simulation results.

38. Question?