Institute of Solid State PhysicsTechnische Universität Graz

14. Magnetism

Nov 18, 2019

Diamagnetism

A free electron in a magnetic field will travel in a circle

Binduced

The magnetic created by the current loop is opposite the applied field.

Bapplied

Diamagnetism

Dissipationless currents are induced in a diamagnet that generate a field that opposes an applied magnetic field.

Current flow without dissipation is a quantum effect. There are no lower lying states to scatter into. This creates a current that generates a field that opposes the applied field.

= -1 superconductor (perfect diamagnet)

Diamagnetism is always present but is often overshadowed by some other magnetic effect.

~ -10-6 - 10-5 normal materials

Levitating diamagnets

Levitating pyrolytic carbonNOT: Lenz's law

dVdt

Levitating frogs

http://www.hfml.ru.nl/froglev.html

16 Tesla magnet at the Nijmegen High Field Magnet Laboratory

for water is -9.05 10-6

2000 Ig Nobel Prize for levitating a frog with a magnet

Andre Geim

Diamagnetism

A dissipationless current is induced by a magnetic field that opposes the applied field.

Diamagnetic susceptibilityCopper -9.810-6

Diamond -2.210-5

Gold -3.610-5

Lead -1.710-5

Nitrogen -5.010-9

Silicon -4.210-6

water -9.010-6

bismuth -1.610-4

M H

Most stable molecules have a closed shell configuration and are diamagnetic.

Paramagnetism

Materials that have a magnetic moment are paramagnetic.

An applied field aligns the magnetic moments in the material making the field in the material larger than the applied field.

The internal field is zero at zero applied field (random magnetic moments).

Paramagnetic susceptibilityAluminum 2.310-5

Calcium 1.910-5

Magnesium 1.210-5

Oxygen 2.110-6

Platinum 2.910-4

Tungsten 6.810-5

M H

Boltzmann factors

/

/

i B

i B

E k Ti

i

E k T

i

AeA

e

To take the average value of quantity A

Spin populations

1

2

exp( / )exp( / ) exp( / )

exp( / )exp( / ) exp( / )

B

B B

B

B B

N B k TN B k T B k TN B k TN B k T B k T

1 2( ) /exp( / ) exp( / )exp( / ) exp( / )

tanh

B B

B B

B

M N N VB k T B k TnB k T B k T

Bnk T

1NN

2NN

up

down

Paramagnetism, spin 1/2

2

tanhB B

B n B CBM nk T k T T

B

Bxk T

S

MM

for BB k T Curie law

Paramagnetism, spin 1/2

2

tanhB B

B n B CBM nk T k T T

B

Bxk T

S

MM

for BB k T Curie law

Curie law

Gd(C2H3SO4)39H2O

C is the Curie constant

for /BB k T M CB T

0B

dM CdB T

Z j J Bm g

In atomic physics, the possible values of the magnetic moment of an atom in the direction of the applied field can only take on certain values.

Atomic physics

, 1, 1,Jm J J J J

J L S

Lande g factor Bohr magneton

Total angular momentum

Orbital L + spin S angular momentum

Magnetic quantum number

Allowed values of the magnetic moment in the z direction

10 10 9 91 1 2 2

10 10 9 91 1 2 2

3 4 3 4 3 4 3 4

3 4 3 4 3 4 3 4

Cu d s Cu d s Cu d s Cu d s

Cu d s Cu d s Cu d s Cu d s

H H

Brillouin functions

( )/ /

( )/ /

1J B J J B B

J B J J B B

J JE m k T m g B k T

J JJ J

J J JE m k T m g B k T

J J

m e m edZm

Z dxe e

/J B Bx g B k T

sinh 2 12

sinh2

J

Jm x

J

xJZ e

x

Lande g factor

Bohr magneton

Average value of the magnetic quantum number

Brillouin functions

( )/ /

( )/ /

1J B J J B B

J B J J B B

J JE m k T m g B k T

J JJ J

J J JE m k T m g B k T

J J

m e m edZm

Z dxe e

/J B Bx g B k T

1 12 2

2 (J 1) 2

( ) (J )(J 1) 2

2 2 2

1 1

sinh 2 12

1 sinh2

J

Jm x Jx x x x x x

J

xJ x xJx x

x x xx

Z e e e e e e e

xJe e e e e

xe e e e

Lande g factor

Bohr magneton

Average value of the magnetic quantum number

Brillouin functions

1J B J J B

dZM ng m ngZ dx

2 1 2 1 1 1coth coth2 2 2 2

B BB

B B

g JB g JBJ JM ng JJ J k T J J k T

sinh 2 12

sinh2

J

Jm x

J

xJZ e

x

Brillouin function

Pauli paramagnetism

1 1 ( )2 21 1 ( )2 2

B F

B F

n n BD E

n n BD E

2 20

20

( )

( ) ( )

( )

B

B F B F

B F

M n n

M D E B D E HdM D EdH

Pauli paramagnetism is much smaller than the paramagnetism due to atomic moments and almost temperature independent because D(EF) doesn't change very much with temperature.

Paramagnetic contribution due to free electrons.

Electrons have an intrinsic magnetic moment B.

If EF is 1 eV, a field of B = 17000 T is needed to align all of the spins.

Hund's rules (f - shell)

The half filled shell and completely filled shell have zero total angular mom

Paramagnetism

2 1 2 1 1 1coth coth2 2 2 2

B BB

B B

g JB g JBJ JM Ng JJ J k T J J k T

Quantum Mechanics: The Key to Understanding Magnetism

John H. van Vleck

http://nobelprize.org/nobel_prizes/physics/laureates/1977/vleck-lecture.pdf

4fx5s25p6

Institute of Solid State PhysicsTechnische Universität Graz

Ferromagnetism

Below a critical temperature (called the Curie temperature) a magnetization spontaneously appears in a ferromagnet even in the absence of a magnetic field.

Iron, nickel, and cobalt are ferromagnetic.

Ferromagnetism overcomes the magnetic dipole-dipole interactions. It arises from the Coulomb interactions of the electrons. The energy that is gained when the spins align is called the exchange energy.

Schrödinger equation for two particles

is a solution to the noninteracting Hamiltonian, V1,2 = 0

1 2 1 1 2 2 1 2 2 1

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

1 1, ( ) ( ) ( ) ( )2 2

1 1, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2

A

S

r r r r r r

r r r r r r r r r r

1 2 1 1 2 2( , ) ( ) ( )r r r r

2

2 21 2 1 1 2 2 1,2 1 2( ) ( ) ( , )

2V r V r V r r E

m

Exchange (Austauschwechselwirking)

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

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

1= [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ]

A AH r r H r r r r H r r

r r H r r r r H r r

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

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

1= [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ]

S SH r r H r r r r H r r

r r H r r r r H r r

The difference in energy between the A and S is twice the exchange energy.

1 2 1 1 2 2 1 2 2 11, ( ) ( ) ( ) ( )2A r r r r r r

1 2 1 1 2 2 1 2 2 11, ( ) ( ) ( ) ( )2S r r r r r r

Exchange

The exchange energy can only be defined when you speak of multi-electron wavefunctions. It is the difference in energy between the symmetric solution and the antisymmetric solution. There is only a difference when the electron-electron term is included. Coulomb repulsion determines the exchange energy.

In ferromagnets, the antisymmetric state has a lower energy. Thus the state with parallel spins has lower energy.

In antiferromagnets, the symmetric state has a lower energy. Neighboring spins are antiparallel.

Ordered states have a lower entropy than free electrons.

Mean field theory (Molekularfeldtheorie)

BNM g SV

,1

MF iB

B J Sg

,,

i j i j B ii j i

H J S S g B S

Mean field approximation

,MF i i Bi

H S J S g B

sums over the neighbors of spin i

Heisenberg Hamiltonian

eliminate <S>

Exchange energy

Looks like a magnetic field BMF

magnetization

Mean field theory

z is the number of nearest neighbors

1 ( )2 B MF aE g B B

In mean field, the energy of the spins is

2 2MFB

VB zJMNg

We calculated the populations of the spins in the paramagnetism section

Spin populations

1

2

exp( / )exp( / ) exp( / )

exp( / )exp( / ) exp( / )

B

B B

B

B B

N B k TN B k T B k TN B k TN B k T B k T

1 2( )exp( / ) exp( / )exp( / ) exp( / )

tanh

B B

B B

B

M N NB k T B k TNB k T B k T

BNk T

Mean field theory

1 tanh2 2

B MF aB

B

g B BNM gV k T

For zero applied field

tanh cS

s

T MM MT M

and 2 4S B c

B

N zM g T JV k

Ms = saturation magnetization Tc = Curie temperature

Mean field theory

Experimental points for Ni.tanh mm

t

tanh cS

s

T MM MT M

Source: Kittel and 2 4S B c

B

N zM g T JV k

FerromagnetismMaterial Curie temp. (K)

Co 1388Fe 1043FeOFe2O3 858NiOFe2O3 858CuOFe2O3 728MgOFe2O3 713MnBi 630Ni 627MnSb 587MnOFe2O3 573Y3Fe5O12 560CrO2 386MnAs 318Gd 292Dy 88EuO 69 Electrical insulatorNd2Fe14B 353 Ms = 10 Ms(Fe) Sm2Co17 700 rare earth magnets

2

4

S B

cB

NM gV

zT Jk

Curie - Weiss law

1 tanh2 2

B MF aB

B

g B BNM gV k T

2 22 2

14 B a

B B

N VM g zJM BVk T Ng

Above Tc we can expand the hyperbolic tangent

2 2

4aB

B c

Bg NMVk T T

c

dM CdH T T

2 2MFB

VB zJMNg

Critical fluctuations near Tc

Solve for M

Curie Weiss Law

4cB

zT Jk

tanh( ) for 1x x x

Ferromagnets are paramagnetic above Tc

Ferromagnetic

Paramagnetic

Critical fluctuations near Tc.

Source: Kittel

Magnetic ordering

Ferromagnetism

Ferrimagnetism

Antiferromagnetism

Helimagnetism

All ordered magnetic states have excitations called magnons