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) mc convolution of and 2 0 E E 3/2 11 2m HE cc v22 g/4 HE v g /4 DE( )c J-1 m -3 22 11 8 v0 Evg/4 Evg /4 cc22 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 vF 1 3/2 2 E 2m c F EdE u c J m-3 22 1 4 v0 1 Ev c v 2 c 2 EF 1 3/2 v2 2m c uvEEvc 211 J m-3 22 cFFc22 6 v0 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 Bv1 Subtract right from left 112 e S BBvv1 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, 444000 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 -9.810-6 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 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 2.310-5 Calcium 1.910-5 Magnesium 1.210-5 Oxygen 2.110-6 Platinum 2.910-4 Tungsten 6.810-5 Boltzmann factors To take the average value of quantity A EkTiB/ Aei A i eEkTiB/ 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 BkTMCBTB / dM C C is the Curie constant dBB0 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 ex 1 ex 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: HHHH Ne33 s Ne sNe33 p Ne p Ar 44 s Ar s Ne 3 d Ne 3 d Ne33 s Ne sNe33 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 21gJB 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.