Models of Ferromagnetism

Ion Ivan

Contents:

1. Models of ferromagnetism: Weiss and Heisenberg2. Magnetic domains

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