models of ferromagnetism
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Models of Ferromagnetism. Ion Ivan. Contents:. Models of ferromagnetism: Weiss and Heisenberg Magnetic domains. ignore the fact that magnetic moments can point only along certain directions because of quantization. Langevin Theory *. - PowerPoint PPT PresentationTRANSCRIPT
Langevin Theory*
drdS sin2 2
1
1
1
1
0
0
sin21
)/cosexp(
sin21
)/cosexp(cos
dye
dyey
dTkB
dTkB
xy
xy
B
B
z
ignore the fact that magnetic moments can point only along certain directions because of quantization
The probability of having angle between θ and θ+dθ at temperature T is proportional to the fraction of shaded area and the Boltzmann factor )/cosexp( TkB B
The average moment
cos, yTk
Bx
B
3
1coth
xxL
xxz
n the number of magnetic moments per unit volume Tk
Bx
n
n
M
M
B
z
S 33
Tk
n
B3
20
Curie’s law* Magnetism in condensed matter, Sthephen Blundell
In 1907, Weiss developed a theory of effective fields
Magnetic moments in ferromagnetic material aligned in aninternal (Weiss) field:
Hw
H (applied)
HW = wM
w=Weiss or molecular field coefficient
*Fizica Solidului, Ion Munteanu
Weiss Theory of Ferromagnetism*
)(xLnM -average magnetization
kT
MwHx ext
0
ASA
S
ext
ss
ext
sS
s
NRkMNmole
M
Hx
Mw
RT
Mw
Hx
Mw
kT
M
My
xLM
My
/,1
,
,
200
If Hext= 0
,
,
20
xMw
TR
M
My
xLM
My
sS
s
x
At T=TcR
MwT s
c
3
20
1
01
M/Ms
T/Tc
At Tc, spontaneous magnetizationdisappears and material become paramagnetic
•Central for understanding magnetic interactions in solids•Arises from Coulomb electrostatic interaction and the Pauli exclusion principle
The Exchange Interaction
Coulomb repulsionenergy high
Coulomb repulsionenergy lowered
20
2
4 r
eUC
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
Pauli principle
One orbital aproximation*
Because of the indistinguishability of electrons
)1,2()2,1()1,2()2,1( or
22 )1,2()2,1(
If the alectron are in different states )2()1()2,1( ba this would conflict with the indistinguishability of electrons because it is possible to know with certainty that electron 1 si in state a and electron 2 is in state b
)1()2()2()1(2
1)2,1( babaS
)1()2()2()1(2
1)2,1( babaA
If consider the spin of electron )2,1()2,1()2,1(
)2,1(
Total wave function must be antisymmetrical
)2,1(
12212
11221
2
1 baba
12212
1baba
21
12212
1
21
*Solid state electronics (Shyh Wang), Qunatum mechanics for chemists (David O. Hayward )
Singlet state S = 0 ms= 0 Triplet state S=1, ms= 1,0,-1
Using one electron approximation:
)()()()(2
1),(
)()()()(2
1),(
2112221121
2112221121
rrrrrr
rrrrrr
A
s
singlet
triplet
Using one electron approximation:
)()()()(2
1),(
)()()()(2
1),(
2112221121
2112221121
rrrrrr
rrrrrr
A
s
singlet
triplet
23
13
211222112*11
*22
*21
*1
*12 )()()()()()()()()(
2
1rdrdrrrrrrrrHU 3H 1212 JK
23
13
21122*11
*22
31
322112
*21
*1 )()()()()()()()( rdrdrrrrrdrdrrrr 1212 HH122K
23
13
22112*11
*22
31
321122
*21
*1 )()()()()()()()( rdrdrrrrrdrdrrrr 1212 HH122J
Coulomb repulsion = 2K12
Exchange terms =2 J12
If J12 is positive Lowest energy state is for triplet, with 121212 JKU
2
31
32112
212
*21
*1
0
2
12 )()(1
)()(4
rdrdrrrr
rre
J
The energies of the parallel and antiparalel spin pairs differ by -2J12
cos22 2121 SSJSSJexc The coupling energy
between spins of neighboring atoms
If J > 0, exc is mininum if 0
If J < 0, exc is mininum if 180
ferromagnetism
antiferomagnetism
Magnetic Domains*
Why do domains occur?
Magnetostatic energyMagnetostrictive energyMagnetocrystalline energy
Competition between
Magnetostatic energy To minimise the total magnetic energy the magnetostatic energy must be minimised. This can be achieved by decreasing the external demagnetising field by dividing the material into domains
Magnetocrystalline energyThere is an energy difference associated with magnetisation along the hard and easy axes which is given by the difference in the areas under (M,H) curves.
This energy can be minimised by forming domains such that their magnetisations point along the easy crystallographic directions.
*http://www.msm.cam.ac.uk/doitpoms//tlplib/ferromagnetic/index.php
Magnetostrictive energyMagnetostriction: when a ferromagnetic material is magnetised it changes length
An increase in length along the direction of magnetisation is positive magnetostriction (e.g. in Fe), and a decrease in length is negative magnetostriction (e.g. in Ni).
Domain walls*: The tranzition layer wich separates adjacent magnetic domains
The width of domain walls is controlled by the balance of two energy contributions:
Exchange energy
Anisotropy energy *Fundamentals of magnetism, Mathias Getzlaff
cos22 2121 SSJSSJexc
When neighboring spins make small angles with each other
222 cos12 JSJSex
If a is lattice constant, the exchange energy stored per unit area of tranzition region
NaNa
JS
aN
NJS,
2
22
2
22
In turning away from the easy axys the magnetization must increase its anisotropy Energy per unit area: KNa, K is anisotropy constant.
The total energy per unit area KNaNa
JStot
2
22
The tickness of tranzition region
The first term favors a large number N with spins involved in the domain wall whereas the second term favors a small number. The energy minimum can be determined by setting the first derivative to zero:
KaNa
JS
dN
d tot 22
22
0 2/122
Ka
JSNa
Domain Wall Width