Magnetism and Magnetic Materials
DTU (10313) – 10 ECTS
KU – 7.5 ECTS
Module 5
Sub-atomic – pm-nm
Mesoscale – nm-mm
15/02/2011
Magnetic order
Macro – mm-mm
Intended Learning Outcomes (ILO)
(for today’s module)
1.
2.
3.
4.
List the various forms of magnetic order in magnetic materials
Calculate the room-T magnetization of a given ferromagnet
Relate exchange interactions with the ”molecular field” in Weiss models
Explain the peak in magnetic susceptibility at the Neel temperature in antiferromagnets
Flashback
Hˆ spin  2JS1  S2
The spin Hamiltonian

A new set of orbitals
Superexchange
Crystal field splitting
J
*
*

(r
)

 a 1 b (r2 )Hˆ a (r2 )b (r1 )dr1dr2
The exchange integral
Ferromagnetism
Hˆ  2 Jij Si  S j  gm B  Si  B
i j
B mf  
i
2
gm B
J S
ij
j
j
We define an effective field
acting upon each spin due to
exchange interactions
Hˆ  gm B  Si  B  B mf 
i
Bmf  M
In the Weiss model for
ferromagnetism, exchange
interactions are responsible for
the huge “molecular field” that
keeps moments aligned.
The Hamiltonian now looks just like the
paramagnetic Hamiltonian, except
there’s a field even with no applied field
We relate the molecular field with the
“order parameter”, i.e. the magnetization
Review Brillouin paramagnetism
J
 mJ exp mJ x
mJ 
m J  J
J
 exp m x
J=1/2
gJ m B B
, x
k BT
J=5
J
m J  J
M  ngJ m B m J  ng J m B JBJ (y)  M S BJ (y)
2J 1  1
 y 
2J 1
g m JB
BJ (y) 
coth
y coth , y  J B
 2J  2J
2J 
2J
k BT
meff  gJ m B J(J 1)
ms  gJ m B J
J=3/2
3 S(S 1)  L(L 1)
gJ  
2
2J(J 1)
The spontaneous magnetization
By solving numerically the
two equations, we determine
the spontaneous magnetization
(in zero applied field) at a
given temperature
M  M S BJ (y)
gJ m B J(B  M )
y
k BT
T>TC
T=TC
T<TC
BJ (y) 
J 1
y
3J
2
gJ m B (J 1)M S nmeff
T 

 C
3k B
3k B

Re-estimate the effective
molecular field Bmf=MS if TC
is 1000 K and J=S=1/2.
The temperature dependence M(T)
The case of Nickel (S=1/2)
M  (TC  T )1/2
M 1
Near TC (mean-field critical exponent)
Low T (as required by thermodynamics)
Estimate the room-T M/Ms of
Fe (J=S=3/2, Tc=1043 K)
Ferromagnet and applied field
T>TC
T=TC
T<TC
Increasing B
M  B1/ 3 T  TC
Origin of the molecular field
If we assume that exchange
interactions are effective over z
nearest-neighbours, we find:
When L is involved (e.g. 4f ions), only a part of S
contributes to the spin Hamiltonian: de Gennes factor
2zJ
 2 2
ng m B
So that we reveal the
proportionality between Tc and
the exchange constant
2zJ (J 1)
TC 
J
3k B
This is valid when L is
quenched (3d ions) and,
therefore, J=S
J
L
L+2S=J+S
S
(gJ-1)J
2zJ(gJ 1)2

ngJ2m 2B
2zJ (J 1)gJ 1
TC 
J
3k B
2
Antiferromagnetism
B    M  M  
B    M  M  
Neglect those for now (but they are
important for a realistic theory)
 g J m B J  M m 
M   M S BJ 

k BT


M  M  M
M  M  M
Staggered magnetization
(order parameter)

2
gJ m B (J 1)  M S n  meff
TN 

3k B
3k B
The magnetic susceptibilities
Paramagnet
Ferromagnet
Antiferromagnet
1

,   TC
T  TC
1
 ,  0
T

1

,   TN
T  TN

AFM with a strong magnetic field
E  MB cos  MBcos   AM 2 cos(   ) 

 MB 
  arccos

2
2AM   

cos2   cos2  

2
Types of antiferromagnetic order
Simple cubic
BCC
Ferrimagnetism and helical order
E  2NS 2 (J1 cos  J2 sin 2 )
J1  4J2 cos sin   0

J1
cos  
4J2


Ferrimagnets: important technologically
for their non-metallic nature and flexible
magnetic response
Sneak peek
B
M
B  m0 (M Hd )
M
 Shape effects and magnetic domains
Wrapping up
•Ferromagnetism
•Spontaneous magnetization
•Ferromagnetic-to-Paramagnetic transition at Tc
•Antiferromagnetism
•Susceptibilities and Curie-Weiss laws
•Ferrimagnetism
•Helical order
Next lecture: Friday February 18, 8:15, KU room 411D
Micromagnetics I (MB)