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Topics in Magnetism II. Models of Ferromagnetism Anne Reilly Department of Physics College of William and Mary
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Topics in Magnetism
II. Models of Ferromagnetism
Anne ReillyDepartment of Physics
College of William and Mary
After reviewing this lecture, you should be familiar with:
1. General source of ferromagnetism2. Curie temperature3. Models of ferromagnetism: Weiss, Heisenberg and Band
Material from this lecture is taken from Physics of Magnetism by Chikazumi
Estimating m ~ 10-29 Wb m and r ~ 1 Ǻ, UD~10-23 J (small, ~1.3K)
In ferromagnetic solids, atomic magnetic moments naturally align with each other.
However, strength of ferromagnetic fields not explained solely by dipole interactions!
N
S
321
r
mmU D
N
S
(see Chikazumi, Chp. 1)
In 1907, Weiss developed a theory of effective fields
Magnetic moments (spins*) in ferromagnetic material aligned in aninternal (Weiss) field:
Hw
H (applied)
Average total magnetization is:
0
0
sin)(
exp
sincos)(
exp
dkT
wMH
dkT
wMH
NMM
M
M
HW = wM
w=Weiss or molecular field coefficient
M = atomic magnetic dipole moment
*Orbital angular momentum gives negligible contribution to magnetization in solids (quenching)
Weiss Theory of Ferromagnetism
LNkT
wMHLN
dkT
wMH
dkT
wMH
NM MM
MM
M
M
)(
sin)(
exp
sincos)(
exp
0
0
Langevin functionConsider graphical solution:
M/Ms
1
0
T/Tc1
Tc is Curie temperature
At Tc, spontaneous magnetizationdisappears and material become paramagnetic
k
wNT eff
c 3
2M
(see Chikazumi, Chp. 6)
Weiss Theory of Ferromagnetism
k
wNT eff
c 3
2M
For Iron (Fe), Tc=1063 K (experiment), M=2.2B (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 KNickel (Ni), Tc= 631K
Heisenberg and Dirac showed later that ferromagnetism isa quantum mechanical effect that fundamentally arises from Coulomb (electric) interaction.
Weiss theory is a good phenomenological theory of magnetism,But does not explain source of large Weiss field.
•Central for understanding magnetic interactions in solids•Arises from Coulomb electrostatic interaction and the Pauli exclusion principle
Key: The Exchange Interaction
Coulomb repulsionenergy high
Coulomb repulsionenergy lowered
Jr
eUC
182
0
2
10~4
(105 K !)
The Exchange Interaction
Consider two electrons in an atom:
+
r1 r2
1 2
Ze
e- e-r12
120
2
20
2
2
2
10
2
1
2
1221
4
42
42
r
e
r
Ze
m
r
Ze
m
e
e
12
2
1
H
H
H
HHHH
Hamiltonian:
2
2
2
2
2
2
jjjj zyx
Using one electron approximation:
)()()()(2
1),(
)()()()(2
1),(
2112221121
2112221121
rrrrrr
rrrrrr
A
s
singlet
triplet
21, are normalized spatial one-electron wavefunctions
H
E
23
13
211222112*11
*22
*21
*1 )()()()()()()()()(
2
1rdrdrrrrrrrrE 321 HHH
We can write energy as:
23
222*22
3212
*1
13
121*21
3111
*1
)()()()(
)()()()(
rdrrrdrr
rdrrrdrr
22
11
HH
HH
23
13
21122*11
*2
23
13
22112*21
*1
)()()()(
)()()()(
rdrdrrrr
rdrdrrrr
12
12
H
H
23
13
22112*11
*2
23
13
21122*21
*1
)()()()(
)()()()(
rdrdrrrr
rdrdrrrr
12
12
H
H
Individual energies (ionization) = 2I1 + 2I2
Coulomb repulsion = 2K12
Exchange terms =2 J12
121221 JKIIE
We can write energy as:
Lowest energy state is for triplet, with 121221 JKIIE
Parallel alignment of spins lowers energy by:
2
31
32112
212
*21
*1
0
2
12 )()(1
)()(4
rdrdrrrr
rre
J
(if J12 is positive)
You can add spin wavefunctions explicitly into previous definitions:
)2()1(
)1()2()2()1(
)2()1(
)()()()(2
1),(
)1()2()2()1()()()()(2
1),(
2112221121
2112221121
rrrrrr
rrrrrr
A
s
(singlet)
(triplet)
1
0
0
1
Spin +1/2
Spin -1/2
You can add spin wavefunctions explicitly into previous defintions.
)2()1(
)1()2()2()1(
)2()1(
)()()()(2
1),(
)1()2()2()1()()()()(2
1),(
2112221121
2112221121
rrrrrr
rrrrrr
A
s
(singlet)
(triplet)
1
0
0
1
Spin +1/2
Spin -1/2
Heisenberg and Dirac showed that the 4 spin states above are eigenstatesof operator 21 SS
Heisenberg and Dirac showed that the 4 spin states above are Eigenstates of operator 21 SS
,
2, S
Hamiltonian of interaction can be written as (called exchangeenergy or Hamiltonian):
jiex SSJ 2H
(Pauli spin matrices)
Heisenberg Model
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
j
n
jjmBji SHSJSU
1 1
22
average molecular field due to rest of spins
Find, for a 3D bcc lattice: JkTc 446.2
For more on Ising model, see http://www.physics.cornell.edu/sss/ising/ising.htmlhttp://bartok.ucsc.edu/peter/java/ising/keep/ising.html
Heisenberg model does not completely explain ferromagnetism inmetals. A band model is needed.
Band (Stoner) Model
Assumes:
N
nIkEkE
N
nIkEkE
S
S
)()(
)()(
Is is Stoner parameter and describes energy reduction due to electron spin correlation
is density of up, down spins nn ,
Band (Stoner) Model
DefineN
nnR
(spin excess) RV
NM Bnote:
1
,/2/)(
~exp
)()(1
kTERIkEf
kfkfN
R
Fs
k
2/)(~
)(
2/)(~
)(
RIkEkE
RIkEkE
S
S
ThenN
nnIkEkE s
2
)()()(
~
Spin excess given by Fermi statistics:
Band (Stoner) Model
Let R be small, use Taylor expansion:
...)()(
~)(
24
1)(
)(~
)(1 3
3
3
RIkE
kf
NRI
kE
kf
NR s
ks
k
))~
((2
~)2(
~ 33 Fk
EEdkN
V
E
fdk
N
V
E
f
(at T=0)
...2
)('''!3
2)(')2/()2/(
3
x
xgxxgxxgxxg
with RIx s
f(E)
EEF
)(2 FEDV
D.O.S.: density of states at Fermi level
Band (Stoner) Model
)3(~ORIEDR sF
Density of states per atom per spin )(2
~FF ED
N
VED Let
Then
)3()~
1( OIEDR sF
Third order terms
When is R> 0?
0~
1 sF IED or 1~ sF IED
For Fe, Co, Ni this condition is true
Doesn’t work for rare earths, though
Stoner Condition for Ferromagnetism
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.
