Magnetism and Magnetic Materials
DTU (10313) – 10 ECTS
KU – 7.5 ECTS
Module 6
18/02/2011
Micromagnetism I
Reference material:
Blundell, section 6.7
Coey, chapter 7
These lecture notes
Mesoscale – nm-mm
Intended Learning Outcomes (ILO)
(for today’s module)
1.
2.
3.
4.
5.
6.
Explain why and how magnetic domains form
Estimate the domain wall width
Calculate demagnetizing fields in simple geometries
Describe superparamagnetism in simple terms
List Brown’s equation in micromagnetics
Explain how hysteresis arises in a simple Stoner-Wolfharth model
Flashback
M  M S BJ (y)
gJ m B J(B  M )
y
k BT
2
n
m
g m (J 1)M S
eff
TC  J B

3k B
3k B
Edge effects and consequences
This is a bit misleading
Dipoles
1  m  rr m 
H (r) 
3 5  3 

4   r
r 
dip
m0 m  r 
A(r) 
 3 
4   r 
Dipole field
Dipole vector potential
m1  rm 2  r
m0 m1 m 2

E
3


3

4   r
r5
Dipolar energy
EZ  m  B
 m B
Two interacting dipoles
Zeeman energy
Torque
m2
H12
H21
m1
Energy of magnetized bodies
dm  rdr
This is to avoid
double-counting
Each dipole (magnetic
moment) within a
magnetized body interacts
with each and every other.
The sum of all that is the
“self energy” of a
magnetized body.
dm2
dm1
r  r  r'r'  r  r'
1 m0 r  r'

dE 
3

drdr'
3
5
2 4   r  r'
r  r'

1 m0
E
2 4
Ed  
m0
2
r'
r'  r  r'r  r'


 drr   dr'r  r' 3  3
5
r  r'


 r  H d rdr
H d (r) 
dip
h
 (r  r')dr'
Recognize this?
It’s the dipole
field “density”.
The demagnetization field
The demagnetization field
m0
A(r) 
4

(r') (r  r')
dr'
3
r  r'
For spheres, ellipsoids, and a few other
shapes, the demag field is uniform
throughout the shape. In general, the
demag field is highly non-uniform.
B(r)    A(r)  m0 (r)   d (r)
=
B
+
M
H
Demag field for uniformly magnetized objects
m0 M 0
A(r) 
4

m  (r  r')
dr'
3
r  r'
m0 M 0
r  r'

m   D(r')
3 dr'
4
r  r'

m0 M 0
r 

m  D(r)  3 

4
r 
mk
A(k)  im0 M 0 D(k) 2
k
kmk
B(k)  m0 M 0 D(k)
k2
mk
 m0 M(k)  m0 M 0 D(k) 2 k
k
M0
m  k ikr
H d (r)   3  D(k) 2 ke dk
8
k
Introducing the characteristic function D(r), with
value 1 inside the object, and 0 outside, we
disentangle shape effects and get a convenient
expression for the demag field.
Representation of the demag field for
a uniformly magnetized tetrahedron
Demag energy and demag factors
H d (r)  
M0
8 3

D(k)
m  k ikr
ke dk   Nˆ (r)M
2
k
Demag field as a result of a tensor
operation on the magnetization
H (r)   Nˆ ij (r)M j
i
d
Nz
Ny
Nx
1
ˆ
N ij (r)   3
8
Ed  
m0
2

ki k j ikr
D(k) 2 e dk
k
   H d rdr 
m0
2
The demag tensor (a
function of position)
M i  Nˆ ij rdrM j
1
 m0 M 02V  N i mi2  K dV N x m x2  N y m y2  N z mz2 
2
ix,y,z
1
N i  Nˆ ii (r) 
V
 Nˆ
ii
(r)dr
Demag factors
The demag energy: a 2-form
involving the three demag
factors along main axes and
the magnetization direction
cosines
This is valid for any shape, provided its
magnetization is uniform.
Domain walls
Large dipolar
energy, no
exchange energy
Snaller dipolar
energy, some
exchange energy
Idem
Bloch walls: bulk,
thick objects
Neel walls: thin
films, thin objects
Cross-over
between dipolar
and domain wall
energies for a
sphere (idealized
model)
Wall width
E  2JS1  S2  2JS cos
2
dE     d   JS d 
2
2
2

 JS 2
a
 


 DW
E   0     DW   NJS 2  DW   JS 2
 
N
 N

NK
K
2
2



0





sin

d



 K  K sin 

K
 0
2a

2
 2  2 K 
2J
A
minJS
    S

aK
K
  a 2 
The strong commercial magnet NdFeB has
K=4.3e6 J/m3, and A=7.3e-12 J/m.
Estimate the domain wall width in this
material.
2
DW

K 
K
2
2JS 2
with A 
 a
The domain wall energy is
proportional to the area

 DW
  AK
Magnetocrystalline anisotropy
The crystal structure creates
anisotropy: some directions are
more responsive (“easier to
magnetize”) to applied fields
than others.
 K  K u sin 2   K u2 sin 4 
Uniaxial
 K  K1 m x2 m y2  m x2 mz2  m y2 mz2  K 2 m x2 m y2 mz2
u
M
Cubic
Consider a sphere of radius R magnetized along some easy axis u with anisotropy
constant Ku=4.53e5 J/m3 (value for Co). If the magnetization flips to –u, the energy
remains the same (up and down states are degenerate). But, in order to rotate from
+u to –u, the magnetization has to go through a high energy state, i.e. when M
points perpendicular to u. Suppose that the temperature is such that kBT is of the
same order of the energy barrier separating the degenerate states. What happens?
Stoner-Wolfharth
E  K dV (N x cos2  M  N y sin 2  M )  m0 M 0 H cos( M   H )
  sin 2  M  2h cos( M   H )
h
1
H
(N y  N x ) M 0

 0   M (h, H )
 M
y
M
The direction of M at
any given applied field
H
x
Single-domain hysteresis
is a consequence of
anisotropy (shape or
magnetocrystalline).
Brown’s equations
Ed  
m0 M 0
2
Kd
 3
8

 mr  H
d
rdr
mk  k
dk
2
k
2
EK  K u  mr  u 2 dr


E x  A  m x   m y   mz  d 3r
2
2
 A  m(r)   2 m(r) d 3r
2
The whole set of equations
provides a full description of the
energy landscape of a
micromagnetic system (such as the
one shown above) and drives its
evolution towards the ground state
of minimum energy
EZ  m0 M 0  mr  H app rdr
r  M 0mr
Sneak peek
Searching for ground states
Micromagnetic simulations
m
m
  m  Heff   m 
t
t
LLG equation
Magnetodynamics and evolution
Wrapping up
•Magnetic domains
•Bloch and Neel walls, and wall widths
•Dipolar/magnetostatic/demag energy
•Demagnetization fields and factors
•Stoner-Wolfharth hysteresis
•Magnetocrystalline anisotropy
•Brown’s equations
Please remember to:
•Install OOMMF on your laptop
•Familiarize a little bit with it
•Bring your laptop to class on Tuesday, February 22
Next lecture: Tuesday February 22, 13:15, KU (A9)
Micromagnetism II (MB)