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Tutorial: The Physics of Superconductivity H. Kinder Technical University of München Onnes Meissner Landau Ginzburg Abrikosov Bardeen Cooper Schrieffer Bednorz Müller Outline • basics • normal state • superconducting state, overview • pair attraction • interplay of pairs • BCS Theory • zero resistance • Meissner state • mixed state • flux flow basics Quantum Mechanics for Engineers • Huygens 1691: light is a wave • Newton 1704: light consists of particles • Planck 1900: light-quanta E = h • Heisenberg 1925: there are no particles nor waves: both are manifestations of the same thing a square? a circle? a cylinder! basics QM Survival-Kit • Planck's formula: E h Energy Frequency h/2 • Heisenberg's uncertainty principle: x p h position momentum • De Broglie relation: p h / k wave length wave vector • Pauli principle: 2 Electrons (Fermions) have never the same state they differ in position, momentum, or spin normal state The Normal State • electrons in metals can move almost freely h h momentum is fixed position is uncertain: x L p x L sample dimension • 1 cm³ of Sn contains 5x1022 electrons: Pauli principle: all of their momenta must differ by p in 3 dimensions: 3 h N p3 max N p3 N h 3 L V pmax 3 N / V h 3 5 1022 /1cm3 21034 Js 2 1024 kg m / s v max p max / m e 2 1024 /10-30 m / s 2 106 m / s high speed! "Fermi velocity" normal state Momentum Space pz • states in momentum space py p average distance p p px L pz • ground state at T=0: states inside a sphere are occupied px to minimize total momentum "Fermi sphere" normal state Normal State at T > 0 • cross section of Fermi sphere: occupation probability f T=0 T1 T2 px pFermi Fermi momentum normal state Normal State Carrying a Current Fermi sphere is displaced by applied voltage: pz rigid displacement px I=0 f I>0 cross section: more electrons going right than left px pFermi SC state The Superconducting State, Overview • electrons in superconductors are bound to pairs "Cooper pairs" • all pairs have the same momentum P = p1+p2 no current: P = 0 or p1= -p2 • the total spin of each pair is zero opposite spins: • the binding energy of each pair is E binding 2 3.52 k BTc (BCS theory) • bound state orbitals depend on materials: L angular momentum L=0: s-wave L=2: d-wave metallic + MgB2 High Tc SC state The Superfluid Condensate • all pairs together form a classical wave (r, t) • the wave has amplitude and phase complex representation: ei N Pairs n Pairs 2 • the amplitude squared is the pair density V • frequently used terms: macroscopic wave function pair field pair amplitude superfluid order parameter gap parameter the condensate SC state Waves on a Ring ... n=1 n=2 n=3 • wave length must fit to the perimeter: n 2 r • wave length momentum current magnetic flux • i. e. the magnetic flux is quantized: = n0 h 0 2 1015 Tesla m2 pairs 2e n 0 SC state Josephson-Effect • dual beam-interference with electron pairs current current weak links B magnetic field B (10-5T) • Superconducting QUantum Interference Device, SQUID pair attraction How can two electrons form a pair? • isotope effect: 1 Tc Sn matomic atomic mass log(Tc) vibrations (=phonons) must play a role log(matomic) pair attraction Electron-Phonon Interaction • principle: Fröhlich 1950 electron at rest: moving electron: supersonic electron: e- e- e- screening overscreening anti-screening net charge; negative positive negative 1 Tc range of attraction speed of sound matomic isotope effect OK pair attraction Remarks on the Matress picture • demonstrates indirect interaction via another medium • however: suggests static attraction • isotope effect depends on mass: dynamic attraction matress should vibrate! pair attraction Effective Attraction in Cuprates • almost no isotope effect antiferromagnetism (AF) 300K and superconductivity (SC) closely related • phase diagram on hole doping: • neutron scattering: AF fluctuations persist in SC region • are these the matress?? no generally accepted understanding available yet interplay of pairs Pair Size h estimate from uncertainty principle: p x h x p p 2 2 2 1 (mv) p momentum p requires kinetic energy: E kin mv 2 2 2m 2m 2m available Energy: E binding 3.5 k B Tc for Ekin Ebinding : p 7 m k BTc h diameter: x 7 m k BTc with Tc 20 K: x 50 nm "coherence length 0" in reality: 0 1...10 nm HTS LTS interplay of pairs Overlap of Pairs N electron density in Sn was: 2 1022 cm 3 V in a volume of 0 : 3 103...105 pairs HTS LTS strong overlap! e– must fulfil the Pauli principle like in normal conductors the pairs are Fermions on the atomic scale interplay of pairs Synchronized Motion of Many Pairs this one breaks the ranks let all pairs go with same speed except for one maverick: • the maverick is crossing all other's ways not allowed by Pauli principle • maverick must evade to empty states with higher energy • this costs too much energy pair is broken up • conclusion: to mimimize energy, all pairs must march in lockstep! interplay of pairs Pairs Running in Lockstep • all pairs have their centers of gravity in the same momentum state "boson-like behavior", similar to photons in a coherent light wave • a nonzero momentum of the pairs corresponds to a transport current • why is the current frictionless? defect scattering would change the velocity, break the pair and cost energy elastic scattering is forbidden demonstration BCS theory BCS Theory • Bardeen, Cooper, and Schrieffer 1957 microscopic theory of superconductivity • BCS ground state (T = 0) in momentum space: occupation probability 1 2 p looks similar to NC at Tc 0 p -pF pF • big difference: Pair correlation: state p state p if p occupied, then also -p if p empty, then also -p BCS theory Quasiparticles • excited sates of the superconductor single electrons, broken pairs • anti-pair correlation: state p state p if p occupied, then -p empty if p empty, then -p occupied • minimum excitation energy = binding energy of pairs 2 3.52 k B Tc energy gap of the superconductor • quasiparticles exist only at finite temperatures BCS theory Superconductor at Finite Temperatures • a quasiparticle in state p: blocks 2 pair states p and -p: pair binding energy 2 is weakened more pairs are broken in thermal equilibrium broken pairs yield new quasiparticles • catastrophe occurs at some finite temperature: all pairs are broken up (T) critical Temperature Tc 0 T Tc zero resistance Supercurrents at T > 0 • normal state resistance: inelastic scattering 0 defects elastic scattering T • superconducting state: pair breaking phonons of Energy h 2 are abundant at TTc/2 inelastic scattering is not forbidden! dynamic equilibrium: pair breaking recombination can phonons stop the pairs? zero resistance Can Phonons Stop the Pairs? • on pair breaking, two quasiparticles are created: • the quasiparticles block two pair states • the blocked pair states move with the same speed as all other pairs • recombination can only occur with quasiparticles of the same speed • after recombination, the pair condensate goes on as before • the total momentum of the condensate is always conserved Meissner state Superconductor in Weak Magnetic Fields • in magnetic fields, the pairs don't fit together correctly binding energy will decrease • but they dont feel the field when they move! for physicists: the "kinetic momentum" can compensate the "field momentum" • consequence: a magnetic field sets the pairs in motion spontaneous supercurrents occur when sample is cooled in field 2 ns qp • "2nd London equation": jsuper Btotal mp the current is perpendicular to the field Meissner state Shielding • the supercurrents have a magnetic field of their own • Ampère's law: Bshielding 0 jsuper • one finds that the field is opposite to the external field • the total field falls off rapidly into the superconductor B SC Bext caracteristic length: Btotal 1 Bext "magnetic penetration depth" e jsuper x 100 nm surface Bshielding Meissner state Meissner Effect • is small, so macroscopic objects are virtually field-free magnetic fields are expelled from superconductors even when in-field-cooled • this holds only in weak fields when the supercurrents don't cost too much energy mixed state Superconductor in strong Magnetic Fields • in stronger fields, the condensate is no longer rigid supercurrents cost too much energy • how to reduce the currents? let the field come in! • simple behaviour. SC breaks down totally Type I superconductors • intelligent behavior: vortices Type II superconductors mixed state NbSe2 MgB2 LuNi2B2C mixed state Critical fields Type I: Type II: Bint Bint Bext Bext 0 Bc 0 Bc1 Bcth Bc2 fields up to 0.2 Tesla only can sustain much higher fields historically discovered first all technical SC are of Type II mixed state What makes the difference? interface energy between NC and SC in magnetic field: SC • > : more loss than gain NC positive interface energy Bext B(x) nsuper • > : more gain than loss negative interface energy x 0 spontanous creation of internal interfaces Eshielding saved material parameter / controls the behavior Ebinding lost "Ginzburg-Landau-Parameter" mixed state Vortices as "Interfaces" as many interfaces as possible: • disperse flux as finely as possible: • smallest possible flux in SC: 1 0 one flux quantum 0 SC SC vortex flux line Shielding current • div B 0 : vortices go throug from surface to surface, or they form rings flux flow Vortex motion magnetic field and transport current simultaneously: e. g. magnets, motors, transformers vortices Lorentz force vortices move at right angles with field and current; why? flux flow microscopic picture eddy field F Lorentz Force sample boundary force on a pair: F 2e vsuper B wants to push the pairs to the side Hall voltage forces the pairs to go straight but: counter force on vortices! motion of the vortex to the side! flux flow Resistance due to Flux Motion: • power consuption of one vortex: P1 F v vortex • N vortices: PN N F v vortex • energy conservation: V I transport V N F v vortex / I transport voltage drop! • conclusion: superconductor has resistance flux flow resistance flux flow Experimental Result ideal V low defect density higher defect density I Ic "technical" critical current • Ic depends on defect density • inhomogeneities are locking the vortices: "flux pinning" • v = 0 is enforced no work no voltage drop, no resisitance flux flow Pinning Mechanisms: segregation with small (or even NL) condensation energy is lost • segregation: vortex core can stay without cost in binding energy • to go on will cost again energy • i.e. segregation has a binding force "pinning - force" • particularly effective: defect sizee flux flow Jc tech as Function of Temperature and Field E j jc tech • decreases in magnetic field more vortices/pinning center • decreases with temperature thermal activation of vortices flux flow Pinning in external magnetic field Bi Hysterese! frozen-in flux ideal type II SC -Bc2 -Bc1 Bc1 Bc2 Ba virgin curve • Bc1 and Bc2 unchanged • pinning impedes entrance and exit of vortices • Bi is inhomogenous within the sample flux flow Field Distribution in the Sample: Bi B Ba Bc2 Ba grows Ba < Bc1 (Meissner) x Bc1 Bc2 Ba SL • surface: jump ideal magnetisation curve • inside: field gradient gradient of vortex density • vortices move only if their repulsion force is greater than the pinning force • gradient decreases with increasing field strengtn flux flow Bean Model: • density gradient shielding current y • macroscopic average over vortices: Bi 0 jAbschirm Ampère Bi • here: 0 jy x x • if B/ x small: j < jc vortices pinned • if B/ x larger: j > jc vortices are ripped away • move until everywhere j = jc "critical state" • remark: B measurement of dBi(x)/dx jc x