Transcript Document

Introduction to
Micromagnetic Simulation
Feng Xie
Ph.D. student
Major advisor: Dr. Richard B. Wells
Contents
Introduction to magnetic materials.
Ideas in micromagnetics.
Physical equations.
Field analysis.
ODE solver and coordinate selection.
Simulations for ideal cases.
Thermal effects
Summary
Magnetization

M
S
+p
 l

-p
N
Magnetic dipole
moment

m  pl

Magnetization
(magnetic dipole moment
per unit volume)

M

m
V
Magnetic Materials
Diamagnetic:
M
H
Most elements in the
periodic table, including
copper, silver, and gold.
Paramagnetic:
M
Include magnesium,
molybdenum, lithium, and
tantalum.
M
Ferromagnetic:
Iron, nickel, and cobalt.
Ferrimagnetic:
Ferrites.
CGS and SI Units
Quantity
Symbol
CGS
unit
SI unit
CGS value / SI
value
M
emu/cc
A•m-1
10-3
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)
Magnetization
Magnetic field
Hysteresis Loop
Scale Comparison
A magnetic force
microscopy (MFM)
image showing
Domain structure
Micromagnetic
explanation of
Domain structure
(Phenomenology)
Electron
Spins
(Quantum
theory)
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.
Micromagnetic
Assumptions

Magnitude: M  M s
H
M
The Landau-Lifshitz-Gilbert (LLG)
equation
Magnetic fields:
– Externally applied field
– Exchange field
– Demagnetizing field
– Anisotropy field
– Stochastic field or the stochastic LLG
equation
Physical Equations
The Landau-Lifshitz (LL) equation:

 

 
dM

  L M  H 
M  M H
2
dt
4M s
 

The Landau-Lifshitz-Gilbert (LLG) equation:





dM
 
dM 
  G M  H 
M 

dt
Ms 
dt 
where


4 L M s
 G   L 1   2 
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:

   L 
 
dM
  L M  H 
M  M H
dt
Ms
 

dMs2
0
dt
Field Analysis
• Applied field: DC + AC.
• Demagnetizing field: time consuming.
• Effective field:

E
H eff   
M
H eff , x  
E
E
E
, H eff , y  
, H eff , z  
M x
M y
M z
– Anisotropy field: uniaxial and cubic.
– Exchange field: quantum mechanic effect.
– Other fields.
DC Field Solution
DC field only + single grain
The solution is
H0
0
M
y
0
x
 sin cos L H 0 t   0 



M  M s  sin sin L H 0 t   0  


cos



where
tan

2
 tan
0
2
exp  L H 0 t
DC Field Simulations
Small Applied AC Field
• Small ac field + single grain
Hx = h cos(t)
Hy = h sin(t)
h << H0
H0

M
y

x
resonance
Demagnetizing Field
long distance
i
1
H dem 
Vi
where
N
 
vi j 1 S j


   


nS j  M r  r  r  2  3  N  
d r d r   Dij  M j
  3
r  r
j 1

1 
Dij   Tdxdydz
v vi
 
T r  

Sj
r  r nˆ  d 2 r 

3
r  r
• Consuming most of computation time.
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.
Fast Multipole Method
Source
Near Field
Middle Field
Far Field
Fast Fourier Transform
Fast Fourier Transform
– Convolution
N 
i

H dem   Dij  M j
j 1
– Symmetry in geometries
v_b(row)
v_a(row)
0, 3
0, 2
0, 1
0, 0
2, 1
2, 0
0, 3
3, 3
2, 3
3, 2
2, 2
1, 2
1, 1
1, 0
1, 3
0, 2
1, 3
0, 1
3, 1
0, 0
3, 0
u_a(column)
2, 1
1, 1
1, 0
3, 2
2, 2
1, 2
3, 3
2, 3
2, 0
3, 1
3, 0
u_b(column)
Anisotropy Field
Magnetocrystalline Anisotropy
H

M
Uniaxial Anisotropy:
Cubic anisotropy:
E=K0u+K1usin2+K2usin4+ E=K0c+K1c(cos21cos22
+cos22cos23+ cos23cos21)+
Exchange Field
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
H x ,i
1

M s ,i

jNN


2 Aij m j  mi 
2ij
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
Coordinates
|m|=1 
=?
m
m
z
m
z
z

y
m
z
z
y
y
+
x
x
y
x
z

x
z < 30
x
x
x < 30
ODE Solver
• Runge Kutta embedded 4th-5th method [2].
i 1


g ni  f  t n  ci hn , y n  hn  aij g nj 
j 1


i  1,, s
t n1  t n  hn
s
y q ,n 1  y n  hn  bq ,i g ni
i 1
• 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.201-222, September 1990.
Geometry
Top View
Side View


d
Layer 2
a
c-axis
dz

y
Layer 1
x
Default Simulation
Parameters

a
d

dz
0.5
1
0.1
1
0
heh
hev
-0.05 -0.05
Ms
K1


(emu/cc) (ergs/cc)
233
5105
[0,20 ] [0 ,360 ]
Various c-axis
distributions
Various exchange
constants
Various anisotropy
constants
Various thickness
Mixture of two anisotropy
constants
Stochastic LLG Equation
• Due to thermal fluctuation.
• Stochastic Landau-Lifshitz-Gilbert equation
 
   L 
  
dM     L M  H 
M  M  H dt
Ms


  



L
  L M  dh 
M  M  dh
M
s

Where h is a stochastic field with the property


 

E h t   0


 
 


2k BT
E hi t h j t  
 ij t  s    2 ij t  s 
M s v
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) 81-101.
Thermal effects
Domain Wall Simulation
(Side View)
Domain wall
Domain
Domain Wall Simulation
(Top View)
Bloch wall
Summary
Basic ideas in micromagnetic
simulation.
Algorithms in micromagnetic modeling.
Micromagnetic simulation results.