Transcript Document
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 S 2
The spin Hamiltonian
A new set of orbitals
Superexchange
Crystal field splitting
J a* (r1 )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 Bmf
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
m J exp m J 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 mJ ngJ mB 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
J 1
BJ (y)
y
3J
2
gJ m B (J 1)M S nmeff
TC
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
Near TC (mean-field critical exponent)
M 1
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
1
, 0
T
Antiferromagnet
1
, TC
T TC
1
, TN
T TN
AFM with a strong magnetic field
E MBcos MBcos AM cos( ) cos2 cos2
2
2
MB
arccos
2
2AM
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)