Topics in Magnetism II. Models of Ferromagnetism Anne Reilly Department of Physics College of William and Mary.

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.