Models of Ferromagnetism
Ion Ivan
Contents:
1. Models of ferromagnetism: Weiss and Heisenberg
2. Magnetic domains
Langevin Theory*
ignore the fact that magnetic moments can point only
along certain directions because of quantization
dS  2r 2 sin d
The probability of having angle between θ and θ+dθ at
temperature T is proportional to the fraction of shaded area
and the Boltzmann factor exp(B cos / k BT )

The average moment  z 
1

cos

exp(

B
cos

/
k
T
)
sin d
B
0
2

1
exp(

B
cos

/
k
T
)
sin d
B
0
2
z
1
x
 cothx   Lx  

x
3
n the number of magnetic moments per unit volume
* Magnetism
in condensed matter, Sthephen Blundell
1
 
 y e
y x
dy
1
1
e
y x
dy
x
B
k BT
, y  cos
1
n z
M
x
B

 
MS
n
3 3k BT

0  n   2
3k BT
Curie’s law
Weiss Theory of Ferromagnetism*
In 1907, Weiss developed a theory of effective fields
Magnetic moments in ferromagnetic material aligned in an
internal (Weiss) field:
Hw
HW = wM
w=Weiss or molecular field coefficient
H (applied)
*Fizica
Solidului, Ion Munteanu
M  n  L(x)
x
y
-average magnetization
  0 H ext  w  M 
kT
M
 L x ,
Ms
x
H ext
H ext
M
kT
RT
y

x


x

,
M S 0    w  M s
w  M s 0  w  M s 2
 MS
1 m ole N A  M S , k  R / N A
If Hext= 0
At T=Tc
y
M
 Lx ,
Ms
y
M
R T

 x,
2
M S 0  w  M s
Tc 
M/Ms
1
At Tc, spontaneous
magnetization
disappears and material
become
paramagnetic
0  w  M s 2
3 R
0
T/Tc
1
The Exchange Interaction
•Central for understanding magnetic interactions in solids
•Arises from Coulomb electrostatic interaction and
the Pauli exclusion principle
Coulomb repulsion
energy high
UC 
e2
4 0 r 2
Coulomb repulsion
energy lowered
The Exchange Interaction
Consider two electrons in an atom:
Hamiltonian:
H  H 1  H 2  H 12
2
Ze 2
H1  
1 
2me
4 0 r1
1
r12
er1
e- 2
r2
+
Ze
2
Ze 2
H2  
2 
2me
4 0 r2
H 12 
e
2
4 0 r12
2
2
2
j  2  2  2
x j y j z j
One orbital aproximation*
Because of the indistinguishability of electrons
 (1,2)   (2,1)
or
If the alectron are in different states
 (1,2) 2   (2,1)
 (1,2)   (2,1)
2
Pauli principle
this would conflict with the indistinguishability of
electrons because it is possible to know with
certainty that electron 1 si in state a and electron 2
is in state b
 (1,2)   a (1)  b (2)
1
 a (1) b (2)  a (2) b (1)
2
1
 a (1) b (2)  a (2) b (1)
 A (1,2) 
2
 S (1,2) 
If consider the spin of electron (1,2)   (1,2)   (1,2)
1
 a 1 b 2  a 2 b 1 1  1 2   2 1
2
2
 (1,2)
Total wave function must
be antisymmetrical
 1 2
 (1,2) 
1
 a 1 b 2  a 2 b 1
2
*Solid state electronics (Shyh Wang), Qunatum mechanics for chemists (David O. Hayward )
1
 1 2   2 1
2
 1 2
Singlet state S = 0
ms= 0
Triplet state S=1, ms= 1,0,-1
Using one electron approximation:
1
1 (r1 )2 (r2 )  2 (r1 )1 (r2 )
s (r1 , r2 ) 
2
1
1 (r1 )2 (r2 )  2 (r1 )1 (r2 )
A (r1 , r2 ) 
2
triplet
singlet
Using one electron approximation:
1
 1 (r1 ) 2 (r2 )  2 (r1 ) 1 (r2 )
s (r1 , r2 ) 
2
1
 1 (r1 ) 2 (r2 )  2 (r1 ) 1 (r2 )
A (r1 , r2 ) 
2
U12   * H 

singlet
triplet

1
*
*
*
*
3
3



(
r
)

(
r
)


(
r
)

(
r
)
(
H
)

(
r
)

(
r
)


(
r
)

(
r
)
d
r
d
r2 
1
1
2
2
2
1
1
2
3
1
1
2
2
2
1
1
2
1

2
K12  J12
2K12   1* (r1 ) 2* (r2 )H 12 1 (r1 ) 2 (r2 )d 3r1d 3r2   2* (r1 ) 1* (r2 )H 12 2 (r1 ) 1 (r2 )d 3r1d 3r2
Coulomb repulsion = 2K12
2J12 
*
*
3
3
*
*
3
3

(
r
)

(
r
)
H

(
r
)

(
r
)
d
r
d
r


(
r
)

(
r
)
H

(
r
)

(
r
)
d
r
d
r2
1
1
2
2
12
2
1
1
2
1
2
2
1
1
2
12
1
1
2
2
1


Exchange terms =2 J12
If J12 is positive
Lowest energy state is for triplet, with
e2
J12 
4 0
1
r1  r2
U12  K12  J12
*
*
3
3

(
r
)

(
r
)

(
r
)

(
r
)
d
r
d
r2


1
1
2
2
2
1
1
2
1

The energies of the parallel and antiparalel spin pairs differ by -2J12
The coupling energy
between spins of
neighboring atoms


 
exc  2J S1  S2  2J  S1  S2 cos
If J > 0,  exc is mininum if
 0
ferromagnetism
If J < 0,  exc is mininum if
  180
antiferomagnetism
Magnetic Domains*
Why do domains occur?
Competition between
Magnetostatic energy
Magnetostatic energy
Magnetostrictive energy
Magnetocrystalline energy
To minimise the total magnetic energy the magnetostatic energy
must be minimised. This can be achieved by decreasing the external
demagnetising field by dividing the material into domains
Magnetocrystalline energy
There is an energy difference associated with magnetisation
along the hard and easy axes which is given by the difference
in the areas under (M,H) curves.
This energy can be minimised by forming domains such that
their magnetisations point along the easy crystallographic
directions.
*http://www.msm.cam.ac.uk/doitpoms//tlplib/ferromagnetic/index.php
Magnetostrictive energy
Magnetostriction: when a ferromagnetic
material is magnetised it changes length
An increase in length along the direction of
magnetisation is positive magnetostriction (e.g.
in Fe), and a decrease in length is negative
magnetostriction (e.g. in Ni).
Domain walls*: The tranzition layer wich separates adjacent magnetic domains
The width of domain walls is controlled by
the balance of two energy contributions:
*Fundamentals
of magnetism, Mathias Getzlaff
Exchange energy
Anisotropy energy
Domain Wall Width


 
exc  2J S1  S2  2J  S1  S2 cos
When neighboring spins make small angles with each other
ex  2JS 2 1  cos    JS 2 2
If a is lattice constant, the exchange energy stored per unit area of tranzition region
 
NJS  
JS 2 2
N



, Na  The tickness of tranzition region
2
2
a
Na
2
2
In turning away from the easy axys the magnetization must increase its anisotropy
Energy per unit area: KNa, K is anisotropy constant.
The total energy per unit area tot
JS 2 2

 KNa
Na 2
The first term favors a large number N with spins involved in the domain wall whereas
the second term favors a small number. The energy minimum can be determined by
setting the first derivative to zero:
dtot
JS 2 2
 0   2 2  Ka
dN
a N
1/ 2
 JS 2 2 

Na  
Ka

