Topics in Magnetism
II. Models of Ferromagnetism
Anne Reilly
Department of Physics
College of William and Mary
After reviewing this lecture, you should be familiar with:
1. General source of ferromagnetism
2. Curie temperature
3. Models of ferromagnetism: Weiss, Heisenberg and Band
Material from this lecture is taken from Physics of Magnetism by Chikazumi
In ferromagnetic solids, atomic magnetic moments
naturally align with each other.
However, strength of ferromagnetic fields
not explained solely by dipole interactions!
 
m1  m2
U D 
3
r
N
S
N
S
Estimating m ~ 10-29 Wb m and r ~ 1 Ǻ, UD~10-23 J (small, ~1.3K)
(see Chikazumi, Chp. 1)
In 1907, Weiss developed a theory of effective fields
Magnetic moments (spins*) in ferromagnetic material aligned in an
internal (Weiss) field:
Hw
HW = wM
w=Weiss or molecular field coefficient
Average total magnetization is:
H (applied)

 M ( H  wM ) 
exp
0  kT  cos sin d
M  NM 
 M ( H  wM ) 
exp
0  kT  sin d
M = atomic magnetic dipole moment
*Orbital angular momentum gives negligible contribution to magnetization in solids (quenching)
Weiss Theory of Ferromagnetism

 M ( H  wM ) 
exp
0  kT  cos  sin d
 M ( H  wM ) 
M  NM 
 NML
  NML 
kT


 M ( H  wM ) 
exp
sin

d

0  kT 
Langevin function
Consider graphical solution:
Tc is Curie temperature
M/Ms
1
At Tc, spontaneous magnetization
disappears and material become
paramagnetic
0
T/Tc
(see Chikazumi, Chp. 6)
1
Tc 
2
NM eff
w
3k
Weiss Theory of Ferromagnetism
Tc 
2
NM eff
w
3k
For Iron (Fe), Tc=1063 K (experiment), M=2.2mB (experiment),
And N=8.54 x 1028m-3
Find w=3.9 x 108
And Hw=0.85 x 109 A/m (107 Oe)
Other materials:
Cobalt (Co), Tc=1404 K
Nickel (Ni), Tc= 631K
Weiss theory is a good phenomenological theory of magnetism,
But does not explain source of large Weiss field.
Heisenberg and Dirac showed later that ferromagnetism is
a quantum mechanical effect that fundamentally arises from
Coulomb (electric) interaction.
Key: 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
~ 10 18 J (105 K !)
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
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
1,2
singlet
triplet
are normalized spatial one-electron wavefunctions
We can write energy as:
H 
E



1
E   1* (r1 )2* (r2 )  2* (r1 )1* (r2 ) (H 1  H 2  H 3 )1 (r1 )2 (r2 )  2 (r1 )1 (r2 )d 3r1d 3r2 
2
  1* (r1 )H 11 (r1 )d 3r1   2* (r1 )H 12 (r1 )d 3r1
  1* (r2 )H 21 (r2 )d 3r2   2* (r2 )H 22 (r2 )d 3r2
Individual energies
(ionization) = 2I1 + 2I2
  1* (r1 )2* (r2 )H 121 (r1 )2 (r2 )d 3r1d 3r2 
Coulomb repulsion = 2K12
  2* (r1 )1* (r2 )H 122 (r1 )1 (r2 )d 3r1d 3r2
  1* (r1 )2* (r2 )H 122 (r1 )1 (r2 )d 3r1d 3r2
  2* (r1 )1* (r2 )H 121 (r1 )2 (r2 )d 3r1d 3r2
Exchange terms =2 J12
We can write energy as:
E  I1  I 2  K12  J12
Lowest energy state is for triplet, with E  I1  I 2  K12  J12
Parallel alignment of spins lowers energy by:
e2
1
3
3
J12 

(
r
)

(
r
)

(
r
)

(
r
)
d
r
d
r2
 2 1 1 2
1
2 
1

40
r1  r2
*
1
*
2
(if J12 is positive)
You can add spin wavefunctions explicitly into previous definitions:
(singlet)
1
1 (r1 )2 (r2 )  2 (r1 )1 (r2 ) (1)   (2)   (2)   (1)
s (r1 , r2 ) 
2
  (1)  (2)



1
1 (r1 )2 (r2 )  2 (r1 )1 (r2 )  (1)   (2)   (2)   (1)
A (r1 , r2 ) 
2
  (1)  (2)


 

(triplet)
1


          Spin +1/2
 0
 0

        
1
Spin -1/2
You can add spin wavefunctions explicitly into previous defintions.
(singlet)
1
1 (r1 )2 (r2 )  2 (r1 )1 (r2 ) (1)   (2)   (2)   (1)
s (r1 , r2 ) 
2
  (1)  (2)



1
1 (r1 )2 (r2 )  2 (r1 )1 (r2 )  (1)   (2)   (2)   (1)
A (r1 , r2 ) 
2
  (1)  (2)


 

1


          Spin +1/2
(triplet)
 0
 0

        
1
Spin -1/2
Heisenberg and Dirac showed that the 4 spin states above are eigenstates
of operator S1  S 2
Heisenberg Model
Heisenberg and Dirac showed that the 4 spin states above are
Eigenstates of operator S1  S 2
S,

 
2
,
(Pauli spin matrices)
Hamiltonian of interaction can be written as (called exchange
energy or Hamiltonian):
H ex  2 JSi  S j
J is the exchange parameter (integral)
Assume a lattice of spins that can take on values +1/2 and -1/2
(Ising model)
The energy considering only nearest-neighbor interactions:
n
n
j 1
j 1
U  2 JSi  S j  2m B H m  S j
average molecular field due
to rest of spins
Find, for a 3D bcc lattice:
kTc  2.446 J
For more on Ising model, see
http://www.physics.cornell.edu/sss/ising/ising.html
http://bartok.ucsc.edu/peter/java/ising/keep/ising.html
Band (Stoner) Model
Heisenberg model does not completely explain ferromagnetism in
metals. A band model is needed.
Assumes:
I S n
E (k )  E (k ) 
N
I S n
E (k )  E (k ) 
N
Is is Stoner parameter and describes energy
reduction due to electron spin correlation
n , n is density of up, down spins
Band (Stoner) Model
n  n
(spin excess)
Define R 
N
~
E
(
k
)

E
(
k
)

I
R
/
2
S

Then
~
E (k )  E (k )  I S R / 2
note: M  m B
I s (n  n )
~
E (k )  E (k ) 
2N
Spin excess given by Fermi statistics:
1
R
N
f  ,
f

(k )  f  (k )
k
 
~
 exp E (k )  I s R / 2  E F / kT
N
R
V

1
Band (Stoner) Model
Let R be small, use Taylor expansion:
2
 x 
g ( x  x / 2)  g ( x  x / 2)   g ' ( x)x  g ' ' ' ( x)   ...
3!
 2 
with x  I s R
1
R
N
f (k )
1
k E~(k ) ( I s R)  24 N
3
 f (k )
3
(
I
R
)
k E~(k )3 s  ...
3
V
V
~
 f 
 f 
k  E~   (2 )3 N  dk  E~   2 3 N  dk ( ( E  EF ))
(at T=0)
V
f(E)
  D( EF )
2
D.O.S.: density of states at Fermi level
EF
E
Band (Stoner) Model
V
~
Let DE F  
D( EF )
2N
Then
Density of states per atom per spin
~
R  DEF I s R  O(3)
Third order terms
~
( 3)
R(1  DEF I s )  O
When is R> 0?
~
1  DEF I s  0
or
~
DEF I s  1
Stoner Condition for Ferromagnetism
For Fe, Co, Ni this condition is true
Doesn’t work for rare earths, though
Heisenberg versus Band (itinerant or free electron) model
Both are extremes, but are needed in metals such as Fe,Ni,Co
Band theory correctly describes magnetization because it assumes
magnetic moment arises from mobile d-band electrons.
Band theory, however, does not account for temperature
dependence of magnetization: Heisenberg model is needed
(collective spin-spin interactions, e.g., spin waves)
To describe electron spin correlations and electron transport
properties (predicted by band theory) with a unified theory is still
an unsolved problem in solid state physics.