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Magnetic ordering
Ferromagnetism
Ferrimagnetism
Antiferromagnetism
Helimagnetism
All ordered magnetic states have excitations called magnons
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Ferrimagnets
Magnetite Fe3O4(Magnetstein)
Ferrites MO.Fe2O3
M = Fe, Zn, Cd, Ni, Cu, Co, Mg
Spinel crystal structure XY2O4
8 tetrahedral sites A (surrounded by 4 O) 5B
16 octahedral sites B (surrounded by 6 O) 9B
per unit cell
MgAl2O4Two sublattices A and B.
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Ferrimagnets
Magnetite Fe3O4
Ferrites MO.Fe2O3
M = Fe, Zn, Cd, Ni, Cu, Co, Mg
Exchange integrals JAA, JAB, and JBBare all negative
(antiparallel preferred)
|JAB| > |JAA|,|JBB|
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Mean field theory (Ferrimagnetism and Antiferromagnetism)
A B ANM g SV
, , ,1 1
MF A i AB B i AA AB B
B J S J Sg g
,,
i j i j B ii j i
H J S S g B S
Mean field approximation
Heisenberg Hamiltonian
Exchange energy
, , ,1 1
MF B i AB A i BB BB B
B J S J Sg g
B B BNM g SV
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Mean field theory
,1 ( )2A B MF A a
E g B B
The spins can take on two energies. These energies are different
on the A sites and B because the A spins see a different
environment as the B spins.
,1 ( )2B B MF B a
E g B B
,tanh MF A aAB
B BM N
k T
,tanh MF B aBB
B BM N
k T
Calculate the average magnetization with Boltzmann factors:
0 0, tanh AB B AA A aA s A
B
M M BM Mk T
0 0, tanh AB A BB B aB s B
B
M M BM Mk T
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Ferrimagnetism gauss = 10-4 T
oersted = 10-4/4x10-7 A/m
Kittel
D. Gignoux, magnetic properties of Metallic systems
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At low temperatures, below the Neel temperature TN, the spins
are aligned antiparallel and the macroscopic magnetization is
zero.
Spin ordering can be observed by neutron scattering.
At high temperature antiferromagnets become paramagnetic. The
macroscopic magnetization is zero and the spins are disordered in
zero field.
Antiferromagnetism
Negative exchange energy JAB < 0.
CT
Curie-Weiss temperature
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Antiferromagnetism
Average spontaneous magnetization is zero at all
temperatures.
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from Kittel
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Magnons
Magnons are excitations of the ordered ferromagnetic state
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Magnons
1 12 p p pJS S S
Energy of the Heisenberg term involving spin p
The magnetic moment of spin p is
p B pg S
1 12p p pB
J S Sg
This has the form -p.Bp where Bp is
1 12p p pB
JB S Sg
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Magnons
The rate of change of angular momentum is the torque
1 12p p pB
JB S Sg
p B pg S
1 12p p p p p p pdS B J S S S Sdt
If the amplitude of the deviations from perfect alignment along
the z-axis are small:
1 1
1 1
2 2
2 2
0
xp y y y
p p p
yp x x x
p p p
zp
dSJ S S S S
dtdS
J S S S Sdt
dSdt
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Magnons
1 1
1 1
2 2
2 2
0
xp y y y
p p p
yp x x x
p p p
zp
dSJ S S S S
dtdS
J S S S Sdt
dSdt
These are coupled linear differential equations. The solutions
have the form:
expx xp ky yp k
S ui kpa t
S u
( 1) ( 1)
( 1) ( 1)
2 2
2 2
x ikpa ik p a ikpa ik p a yk k
y ikpa ik p a ikpa ik p a xk k
i u e J S e e e u
i u e J S e e e u
Cancel a factor of eikpa.
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Magnons
4 1 cos( )0
4 1 cos( )i J S ka
J S ka i
These equations will have solutions when,
The dispersion relation is:
4 1 cos( )J S ka
2 2
2 2
x ika ika yk k
y ika ika xk k
i u J S e e u
i u J S e e u
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Magnon dispersion relation
4 1 cos( )JS ka
A phonon dispersion relationwould be linear at the origin
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Magnon density of states
4 1 cos( )JS ka
Mathematically this is the same problem as the tight binding
model for electrons on a one-dimensional chain.
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Ferromagnetic magnons - simple cubic
The dispersion relation in one dimension:
4 1 cos( )J S ka
The dispersion relation for a cubic lattice in three
dimensions:
2 cos( )J S z k
The magnon contribution to thermodynamic properties can be
calculated similar to the phonon contribution to the thermodynamic
properties.
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2 cos( )J S z k
Magnons
Dispersion relation is mathematically equivalent to tight
binding model for electrons.
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Long wavelength / low temperature limit
Dispersion relation: 2 22JSk a
The density of states: ( )D
Magnons are bosons:1
exp 1k
k
B
n
k T
5 / 2
0
( )
exp 1kB
D du T
k T
3 / 2vc T
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Magnons
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Neutron magnetic scattering
Neutrons can scatter inelastically from magnetic material and
create or annihilate magnons
n n magnonk k k G
2 2 2 2 2 2
2 2 2magnonn n crystal
k k Gm m m
0
k
k
G
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Antiferromagnet magnons
1
1
1
1
2 2
2 2
2 2
2 2
0 0
Axp By Ay By
p p p
Ayp Bx Ax Bx
p p p
Bxp Ay By Ay
p p p
Byp Ax Bx Ax
p p p
Az Bzp p
dSJ S S S S
dtdS
J S S S Sdt
dSJ S S S S
dtdS
J S S S Sdt
dS dSdt dt
exp 2
Ax Axp kAy Ayp kBx Bxp kBy Byp k
S uS u
i kpa tS uS u
a
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Antiferromagnet magnons
4 sin( )J S ka
RbMnF3
Brillouin zone boundary is at k = /2a
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Antiferromagnet magnons
4 sin( )J S ka
Mathematically equivalent to phonons in 1-d
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From: Solid State Theory, Harrison
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Student project
Make a table of magnon properties like the table of phonon
properties
1 student / column
Fe bccNi fccCo hcp
