molecular field theory exchange interactionsoktyabr/nnse508/nnse508... · ferromagnetism –...

NNSE508 / NENG452 Lecture #15

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Lecture contents

• Ferromagnetism

– Molecular field theory

– Exchange interaction


2

• Spontaneous magnetization occurs in some

(Ferromagnetic) materials composed of

atoms with unfilled shells

• For some reason magnetic moments are

aligned even at relatively high temperature

• Hypothesis: magnetic order is due to strong

local magnetic field (Weiss effective field)

at the site of each dipole

with a constant g

• Consider a collection of N identical atoms

per unit volume, with total angular

momentum J, and use QM treatment of

atomic paramagnetism

Ferromagnetism – Molecular field theory

Values of Curie temperatures and spontaneous

magnetism (at 0 K in Gauss) for a few

ferromagnetic materials

From Burns, 1990

0loc aB B M g

0 0with

eff loc eff a J B a

B B B

B B M g J B My

k T k T k T

g g

sat JM M B y sat effM N


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• Now let’s find the spontaneous

magnetization (Ba = 0)

Solving equation against y :

• Depending on temperature spontaneous

magnetization can be either M=0 or finite

• At low y

We can find critical Curie temperature:

Ferromagnetism – Molecular field theory Solution of equation with Brillouin function

From Burns, 1990

0with

eff

B

My

k T

g sat JM M B y

0

BJ

eff sat

k Ty B y

M g

1( 1)

3J

JB y y

J

0

1

3

B C

eff sat

k T J

M J g

20 13

C J B

B

NT g J J

k

g

2

eff


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• At temperatures T > Tc , there is no

spontaneous magnetization, and we can find

temperature dependence of magnetization

(Ba > 0)

• Solving for M:

• Then susceptibility of ferromagnet in

paramagnetic region

• Some estimation for iron:

Ferromagnetism – Molecular field theory

From Burns, 1990

sat JM M B y

2

0

3

eff C

B

N TC

k

g

2

0

1

3 3

eff

sat a

B

NJM M y B M

J k T

g

0

with J B a

B

g J B My

k T

g

0

a

CM B

T C g

with Curie constant

C

C

T T

Reciprocal susceptibility vs. temperature for nickel

28 32; 1; 8.5 10 1.77 Jg J N m C K

1043 ; 588CT K g

61700 Gauss; 10 Gauss=100 M B T

Huge !


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• What is the reason for so high local

magnetic field ? – Exchange interaction

• Consider two electrons on two atoms. We

need to find the energy difference =

exchange integral Jex) :

• Their wavefunction is antisymmetric due to

Pauli exclusion principle

• Usually the interaction between the space

and spin parts is small, and the variables

can be separated :

• Antiparallel spins give antisymmetric spin

wavefunction, etc.

• We can construct wavefunctions for singlet

and triplet states with correct symmetry:

Ferromagnetism – Heisenberg exchange interaction

1 1 2 2 2 2 1 1, , , , , ,r s r s r s r s

Singlet Triplet

1 2ex exH J s s

1 2

3

4s s 1 2

1

4s s

1 1 2 2 1 2 1 2, , , , ,r s r s r r g s s

space spin

Wavefunction Singlet Triplet

Total Antisym. Antisym.

Spin part Antisym. Symmetric

Space part Symmetric Antisym.

1 2 1 2 2 1 1 2,S a b a bx x r r r r

1 2 1 2 2 1 1 2,T a b a bx x r r r r


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• The energy shift of singlet and triplet states can

be calculated from perturbation theory:

• The energy difference between the singlet and

triplet states

• At small a-b distance

• At large a-b distance

• BTW if the electrons are on the same atom, the

ion interaction change is zero, and

antiparallel spins are favorable = Hund’s rule

Ferromagnetism – Heisenberg exchange interaction-contd

1 1 2 2 1 2 2 1S S SE V V V V V

1 1 2 2 1 2 2 1T T TE V V V V V

2

1 2

0 2 1 12

1 1 1 1,

ab a b

eV r r

r r r r

1 2 1 2 2 14 4S T exc a b a bE E J V r r V r r

0 and singlet state is favorable excJ

2

1 2 2 1 1 2 2 1 1 2 2 1

0 2 12

1 1 18 4 4a b a b a b a b a b a b

a ab

er r r r r r r r r r r r

r r r

0 and triplet state is favorable excJ

0 excJ


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• Exchange interaction can be ferromagnetic

or antiferromagnetic depending on

interatomic distance

• Exchange interaction is electrostatic

(strong) in nature

• To correlate it with molecular field theory,

we can write:

• For Fe:

• Electrostatic interaction easily accounts for

this value

Ferromagnetism – Heisenberg exchange interaction contd.

Exchange integral vs. interatomic distance

rd – average radius of 4d electron

From Christman, 1988

ex J B loczJ s g B

,

ex ex i j J B i loc

i j i

H J s s g s B

Number of nearest neighbors

2

00 11J BJ B

ex

g ng MJ meV

zs z

g g


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n = number of (3d + 4s) electrons per atom

• For Ni (n=10) moment is 0.6B

Role of band structure

From Cullity, 2008

Density of states in 3d and 4s bands Dependence of the saturation magnetization on the

number n of (3d + 4s) electrons per atom

10.6H Bn


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• Energy is minimized by ordering spins into “domains”

• Net moment, M, would cause external field,

increase energy

• Magnetic domains cancel so that M = 0

• Natural ferromagnetism does not produce net

magnetic field

• To magnetize a ferromagnet, impose H

• Domain walls move to align M and H

• Defects impede domain wall motion

• Magnetization (Mr) retained when H removed

• Magnetic properties

Ms = saturation magnetization (All spins aligned with field)

Mr = remanent magnetization (Useful moment of permanent

magnet)

Hc = coercive force (Field required to “erase” moment)

Area inside curve = magnetic hysteresis (Governs energy

lost in magnetic cycle)

Ferromagnetic materials

Ferromagnetic material is always

locally saturated

Hysteresis loop of a ferromagnetic

material


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• High temperature:

- Spins disordered ⇒ paramagnetism

• Low Temperature (T < Tc)

- Spins align = ferromagnetism

• Elements: Fe, Ni, Co, Gd, Dy

• Alloys and compounds: AlNiCo, FeCrCo,

SmCo5, Fe14Nd2B

- Like spins alternate = antiferromagnetism

(RbMnF3)

- Unlike spins alternate = ferrimagnetism

• Compounds: Fe3O4 (lodestone, magnetite),

CrO3, SrFe2O3, other ferrites and garnets

Core magnetism – materials with

spontaneously ordered magnetic dipoles


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• Superexchange (transition metal oxides)

• Can be ferromagnetic or anfiferromagnetic

depending upon the energy of delocalization

of the p-electrons on M1 and M2

• Ordering temperature up to 900 K in ferrites

(NiFe2O4 - 863 K)

• Sign mostly negative, though ferromagnetics

are known: EuO (Tc=69K) or CrBr3 (Tc=37K)

• RKKY interaction (Ruderman-Kittel-Kasuya-

Yosida) - Indirect exchange over relatively large

distances trough spin of conduction electrons (4f

metals)

• Interaction oscillates with (kFR), Fermi

wavevector determines the wavelength of

oscillations

• The interaction is of the same order for all rare

earths, but ordering temperatures vary due to

magnetic moment: 19K for Nd, 289 K for Gd)

Other types of exchange interaction

Mn – O - Mn

RKKY interaction

kFR


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• Magnetic induction field is the same in a

Magnetic materils

From Goldberg, 2006


13 Physical constants

From Tremolet de Lacheisserie, 2005

molecular field theory exchange interactionsoktyabr/nnse508/nnse508... · ferromagnetism –...

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