Quantum Hall EffectFermi Surfaces

Magnetism

Quantum Hall Effect

Shubnikov-De Haas oscillations

xxx

x

y zxy

x

EjE Bj ne

Resistance standard25812.807557(18)

Quantum hall effect

y zxy

x

E Bj ne

Each Landau level can hold the same number of electrons. 0

2 20

zxy

hDB hne ve D ve

0 2c zm eBD

h

FE

If the Fermi energy is between Landau levels, the electron density n is an integer v times the degeneracy of the Landau level n = D0v

zc

eBm

0z

hDBe

Quantum hall effect

2xyh

ve

S. Koch, R. J. Haug, and K. v. Klitzing,Phys. Rev. B 47, 4048–4051 (1993)

Quantum Hall effect

Ibach & Lueth (modified)

On the plateaus, resistance goes to zero because there are no states to scatter into.

Edge states are responsible for the zero resistance in xx

B = 0

Fermi sphere in a magnetic field

B 0

Landau cylinders

Density of states 3d

convolution of andD(E)

E

3/ 2 1 12 2 -1 -3

2 2 1 10 2 2

/ 4 / 42( ) J m

8 / 4 / 4c cc

v c c

H E v g H E v gmD E

E v g E v g

2cm

D(E)

E0

c

Energy spectral density 3d

( 0) ( )FE

u T ED E dE

At T = 0

( ) ( ) ( )u E ED E f E

Fermi energy 3d

Periodic in 1/B

( )FE

n D E dE

Internal energy 3d

( )FE

u ED E dE

123/ 2

-31 12 22 2

0

22 J m

6

F

c

Ev

cc F F c

v

mu v E E v

12

12

3/ 2-3

2 2 10 2

2 J m

4

F

Fc

c

Ev Ec

v v c

m EdEuE v

At T = 0

Magnetization 3d

Periodic in 1/B

At finite temperatures this function would be smoother

de Haas - van Alphen oscillations

dumdB

Practically all properties are periodic in 1/B

Magnetization

dUMdH

Fermi sphere in a magnetic field

1

1 1 2

v v

eSB B

From the periodic of the oscillations, you can determine the cross sectional area S.

2FS k

2 2

12 2

Fc

kvm

2 2

12 2

v FeB kvm m

12

2

v

e SvB

12

1

2 1v

e SvB

Subtract right from left

Cross sectional area

Experimental determination of the Fermi surface

de Haas - van Alphen

Rhenium

SilverKittel

De Haas - van Alphen effect

The magnetic moment of gold oscillates periodically with 1/B

Kittel

http://www.phys.ufl.edu/fermisurface/periodic_table.html

Institute of Solid State PhysicsTechnische Universität Graz

Magnetism

diamagnetismparamagnetismferromagnetism (Fe, Ni, Co)ferrimagnetism (Magneteisenstein)antiferromagnetism

2 22 2 22 2

, 0 0 02 2 4 4 4A A B

i Ai A i A i j A Be A iA ij AB

Z e Z Z eeHm m r r r

Coulomb interactions cause ferromagnetism not magnetic interactions.

Institute of Solid State PhysicsTechnische Universität Graz

Magnetism

0B H M

M H

is the magnetic susceptibility

< 0 diamagnetic > 0 paramagnetic

magnetic induction field

magnetic intensity

magnetization

is typically small (10-5) so B 0H

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 NB k T B k TNB k T B k T

BNk T

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

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

check by synthetic division

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

2 2

2 2

x x

x xx x

e ee e e e

1xe

1 xe1 xe

0

J = 1/2

Brillouin function

Hund's rules from atomic physics

Electrons fill atomic orbitals following these rules:

1. Maximize the total spin S allowed by the exclusion principle

2. Maximize the orbital angular momentum L

3. J=|L-S| when the shell is less than half full, J=|L+S|when the shell is more than half full.

Hund calculated the energies of atomic states:

3 33 3 4 4 3 3

3 3 4 4 3 33 3

Ne p Ne pNe s Ne s Ar s Ar s Ne d Ne d

Ne s Ne s Ar s Ar s Ne d Ne dNe p Ne p

HH H H

He formulated the following rules:

H includes e-e interactions

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

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