Quantum Hall Effect Fermi Surfaces Magnetism Quantum Hall Effect

Ex xx  jx E y Bz xy jx ne

Resistance standard Shubnikov-De Haas oscillations 25812.807557(18)  Quantum hall effect

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

E y Bz xy  jx ne EF Each Landau level can hold the same B hD h number of electrons. z 0 xy 22  ne ve D0 ve m eB D c z 0 2  h

hD eBz 0   Bz  c m e Quantum hall effect

h   xy ve2

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

Edge states are responsible for the zero resistance in xx

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

Ibach & Lueth (modified) Fermi sphere in a magnetic field

B = 0 B  0

Landau cylinders Density of states 3d

c D(E) D(E) mc convolution of and 2 

0 E E

  3/2 2m  HE  v11 g/4 HE  v g /4 DE( )c cc22 J-1 m -3 22  11 8  v0 Evg/4 Evg  /4 cc22

Energy spectral density 3d

At T = 0

uE() EDE ()()f E

EF u(0)() T ED E dE  Fermi energy 3d

EF nDEdE  () 

Periodic in 1/B Internal energy 3d

EF uEDEdE  () 

At T = 0

E vF 1 3/2 2 E 2m  c F EdE u  c J m-3 22   1 4  v0 1 Ev  c v 2 c 2 

EF 1 3/2 v2 2m  c uvEEvc 211 J m-3 22  cFFc22 6  v0  Magnetization 3d

du m  dB

Periodic in 1/B

At finite temperatures this function would be smoother

de Haas - van Alphen oscillations Practically all properties are periodic in 1/B

Magnetization dU M  dH Fermi sphere in a magnetic field

2 Cross sectional area Sk  F

 22k  v 1 F c 2 2m

eB  22k  v v 1 F mm2 2

2 eS1 2 eS1 v 2 v 1 2   Bv  Bv1 Subtract right from left 112 e S  BBvv1 

From the periodic of the oscillations, you can determine the cross sectional area S. Experimental determination of the Fermi surface

Rhenium

Kittel Silver de Haas - van Alphen 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 Physics Technische Universität Graz Magnetism

diamagnetism paramagnetism ferromagnetism (Fe, Ni, Co) ferrimagnetism (Magneteisenstein) antiferromagnetism

2222 2 22ZeAABe ZZe H  iA       iAiAijAB22mmeA,  444  000 r iAijAB r r

Coulomb interactions cause ferromagnetism not magnetic interactions. Institute of Solid State Physics Technische Universität Graz Magnetism

magnetic intensity   B 0 HM magnetic induction field   magnetization M   H  is the magnetic susceptibility  < 0 diamagnetic  > 0 paramagnetic

-5  is typically small (10 ) so B 0H Diamagnetism

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

Bapplied

Binduced

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

 ~ -10-6 -10-5 normal materials

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

NOT: Lenz's law Levitating pyrolytic carbon d V  dt Levitating frogs

 for water is -9.05  10-6

16 Tesla magnet at the Nijmegen High Field Magnet Laboratory http://www.hfml.ru.nl/froglev.html Andre Geim

2000 Ig Nobel Prize for levitating a frog with a magnet Diamagnetism

A dissipationless current is induced by a magnetic field that opposes the applied field.   M   H Diamagnetic susceptibility Copper -9.810-6 Diamond -2.210-5 Gold -3.610-5 Lead -1.710-5 Nitrogen -5.010-9 Silicon -4.210-6 water -9.010-6 bismuth -1.610-4

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).   M   H Paramagnetic susceptibility Aluminum 2.310-5 Calcium 1.910-5 Magnesium 1.210-5 Oxygen 2.110-6 Platinum 2.910-4 Tungsten 6.810-5 Boltzmann factors

To take the average value of quantity A

EkTiB/  Aei A  i eEkTiB/ i Spin populations

NBkTexp( / ) 1  B NBkTBkTexp( /BB ) exp( / ) NBkTexp( / ) 2  B NBkTBkTexp( /BB ) exp( / )

MNN()  12 exp(B /kT ) exp( B / kT )  N BB  exp(B /kTBB ) exp( B / kT )  B   N tanh  kTB Paramagnetism, spin 1/2

 B NBCB2 MNtanh  kTBB kT T Curie law for  BkT B

M

M S

 B x  kTB Paramagnetism, spin 1/2

 B NBCB2 MNtanh  kTBB kT T Curie law for  BkT B

M

M S

 B x  kTB Curie law

for  BkTMCBTB / dM C    C is the Curie constant dBB0 T

Gd(C2H3SO4)39H2O Atomic physics

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.

Total angular momentum J  LS Orbital L + spin S angular momentum

Magnetic quantum number

mJJJJJ  ,1,1, 

Allowed values of the magnetic moment in the z direction

Z  mgjJB Lande g factor Bohr magneton Brillouin functions

Average value of the magnetic quantum number

JJ Em()/JB kT mg JJBB BkT / meJJ me JJ 1 dZ mJ JJ  Em()/ kT mg BkT / Zdx eeJB JJBB JJ

Lande g factor xg JB BkT/ B Bohr magneton x J sinh 2J  1 mx 2 check by synthetic Ze J J x division sinh  2 Brillouin functions

J = 1/2 x x  x 22 sinh 2J  1 xxee J   mx 2 x x Ze J eeee22 J x x sinh  e 1 2  1 ex 1 ex 1 dZ 0 MNgm Ng JB J JBZdx  Brillouin function

21JJ 21gJB 1 1 gJB MNgJcothBB coth B  22J J kTBB 22 J J kT Hund's rules from atomic physics

Hund  calculated the  energies of atomic states:    HHHH Ne33 s Ne sNe33 p Ne p Ar 44 s Ar s Ne 3 d Ne 3 d Ne33 s Ne sNe33 p Ne p Ar 44 s Ar s Ne 3 d Ne 3 d H includes e-e interactions

He formulated the following rules: 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. Pauli paramagnetism

Paramagnetic contribution due to free electrons.

Electrons have an intrinsic magnetic moment B.

11 nn BDE()  22BF 11 nn BDE()  22BF

Mnn B () 22 M BFDE() B0 BF DE () H dM 2 DE() dH 0 BF

If EF is 1 eV, a field of B = 17000 T is needed to align all of the spins. 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. Hund's rules (f - shell)

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



21JJ 21gJB 1 1g JB MNgJcothBB coth B  22J J kTBB 22 J J kT Quantum Mechanics: The Key to Understanding Magnetism John H. van Vleck

4fx5s25p6

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