1. Technical University of München Tutorial: The Physics of Superconductivity H. Kinder Onnes Landau GinzburgAbrikosov BardeenCooperSchriefferBednorzMüller Meissner

2. Outline basics normal state superconducting state, overview pair attraction interplay of pairs BCS Theory zero resistance Meissner state mixed state flux flow

3. Quantum Mechanics for Engineers Newton 1704:light consists of particles Huygens 1691:light is a wave 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

4. QM Survival-Kit De Broglie relation: Heisenberg's uncertainty principle: Planck's formula: Pauli principle: Energy Frequency h/2  positionmomentum wave lengthwave vector they differ in position, momentum, or spin 2 Electrons (Fermions) have never the same state basics

5. The Normal State electrons in metals can move almost freely 1 cm³ of Sn contains 5x10 22 electrons: momentum is fixed position is uncertain:  x  L sample dimension Pauli principle: all of their momenta must differ by  p in 3 dimensions: high speed! "Fermi velocity" normal state

6. Momentum Space states in momentum space pzpz pxpx pp pp pypy ground state at T=0: states inside a sphere are occupied to minimize total momentum pzpz pxpx "Fermi sphere" average distance normal state

7. Normal State at T > 0 cross section of Fermi sphere: Fermi momentum occupation probability f pxpx p Fermi T1T1 T=0 T2T2 normal state

8. Normal State Carrying a Current Fermi sphere is displaced by applied voltage: rigid displacement f pxpx p Fermi I = 0 I > 0 pzpz pxpx more electrons going right than left cross section: normal state

9. The Superconducting State, Overview electrons in superconductors are bound to pairs "Cooper pairs" all pairs have the same momentum P = p 1 +p 2 bound state orbitals depend on materials: no current: P = 0 or p 1 = -p 2 opposite spins:  the total spin of each pair is zero L=0: s-wave metallic + MgB 2 L=2: d-wave High Tc the binding energy of each pair is (BCS theory) SC state

10. The Superfluid Condensate all pairs together form a classical wave the wave has amplitude and phase complex representation: the amplitude squared is the pair density frequently used terms: macroscopic wave function pair field pair amplitude superfluid order parameter gap parameter the condensate SC state

11. Waves on a Ring n=1n=2n=3... wave length must fit to the perimeter: wave length  momentum  current  magnetic flux i. e. the magnetic flux is quantized:  = n  0 pairs SC state

12. Josephson-Effect dual beam-interference with electron pairs weak links BB current magnetic field B (10 -5 T) current Superconducting QUantum Interference Device, SQUID SC state

13. isotope effect: How can two electrons form a pair? atomic mass Sn log(m atomic ) log(T c ) vibrations (=phonons) must play a role pair attraction

14. Electron-Phonon Interaction principle: Fröhlich 1950 e-e- electron at rest:moving electron: screeningoverscreening supersonic electron: anti-screening e-e- e-e- net charge; positivenegative isotope effect OK pair attraction

15. Remarks on the Matress picture demonstrates indirect interaction via another medium however: suggests static attraction matress should vibrate! isotope effect depends on mass: dynamic attraction pair attraction

16. Effective Attraction in Cuprates almost no isotope effect neutron scattering: AF fluctuations persist in SC region phase diagram on hole doping: are these the matress?? no generally accepted understanding available yet 300K antiferromagnetism (AF) and superconductivity (SC) closely related pair attraction

17. estimate from uncertainty principle: Pair Size momentum  p requires kinetic energy: available Energy:  diameter: with T c  20 K:  x  50 nm in reality:  0  1...10 nm HTSLTS for E kin  E binding : "coherence length  0 " interplay of pairs

18. electron density in Sn was: Overlap of Pairs  in a volume of : 10 3...10 5 pairs HTSLTS strong overlap!  e – must fulfil the Pauli principle like in normal conductors the pairs are Fermions on the atomic scale interplay of pairs

19. Synchronized Motion of Many Pairs let all pairs go with same speed except for one maverick: this one breaks the ranks the maverick is crossing all other's ways maverick must evade to empty states with higher energy this costs too much energy not allowed by Pauli principle  pair is broken up to mimimize energy, all pairs must march in lockstep! conclusion: interplay of pairs

20. 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 why is the current frictionless? a nonzero momentum of the pairs corresponds to a transport current demonstration defect scattering would change the velocity, break the pair and cost energy  elastic scattering is forbidden interplay of pairs

21. BCS Theory Bardeen, Cooper, and Schrieffer 1957 microscopic theory of superconductivity BCS ground state (T = 0) in momentum space: looks similar to NC at T c big difference: Pair correlation: if p occupied, then also -p 1 pFpF p occupation probability -p F if p empty, then also -p state BCS theory

22. Quasiparticles excited sates of the superconductor anti-pair correlation: single electrons, broken pairs minimum excitation energy = binding energy of pairs energy gap of the superconductor quasiparticles exist only at finite temperatures state if p occupied, then -p emptyif p empty, then -p occupied BCS theory

23. 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 catastrophe occurs at some finite temperature: all pairs are broken up  broken pairs yield new quasiparticles critical Temperature T c T  (T) 00 TcTc BCS theory

24. Supercurrents at T > 0 pair breaking phonons of Energy are abundant at T  T c /2 dynamic equilibrium: pair breaking recombination normal state resistance: inelastic scattering elastic scattering defects T  00 phonons superconducting state: inelastic scattering is not forbidden! can phonons stop the pairs? zero resistance

25. 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 zero resistance

26. Superconductor in Weak Magnetic Fields in magnetic fields, the pairs don't fit together correctly but they dont feel the field when they move! binding energy will decrease 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 nd London equation": the current is perpendicular to the field Meissner state

27. B B total x B ext SC B shielding surface Shielding the supercurrents have a magnetic field of their own Ampère's law: one finds that the field is opposite to the external field  the total field falls off rapidly into the superconductor caracteristic length: "magnetic penetration depth"  100 nm j super Meissner state

28. 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 Meissner state

29. Superconductor in strong Magnetic Fields in stronger fields, the condensate is no longer rigid how to reduce the currents? supercurrents cost too much energy let the field come in! simple behaviour. SC breaks down totally Type I superconductors intelligent behavior: vortices Type II superconductors NbSe 2 MgB 2 LuNi 2 B 2 C mixed state mixed state

30. Critical fields 0 B ext B int BcBc Type I:Type II: B c1 B c2 0 B ext B int B cth can sustain much higher fields all technical SC are of Type IIhistorically discovered first fields up to 0.2 Tesla only mixed state

31. What makes the difference? interface energy between NC and SC in magnetic field: x NC n super SC B ext B(x) 0  E binding lost E shielding saved  > : more loss than gain >  : more gain than loss spontanous creation of internal interfaces positive interface energy negative interface energy material parameter   /  controls the behavior "Ginzburg-Landau-Parameter" mixed state

32. 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0 vortex flux line vortices go throug from surface to surface, or they form rings SC Shielding current mixed state

33. Vortex motion e. g. magnets, motors, transformers magnetic field and transport current simultaneously: vortices Lorentz force flux flow vortices move at right angles with field and current; why?

34. microscopic picture force on a pair: Lorentz Force  Hall voltage forces the pairs to go straight but: counter force on vortices!  motion of the vortex to the side! flux flow eddy field sample boundary wants to push the pairs to the side

35. Resistance due to Flux Motion: power consuption of one vortex: N vortices: voltage drop! conclusion: superconductor has resistance flux flow resistance energy conservation: flux flow

36. Experimental Result I c depends on defect density "technical" critical current inhomogeneities are locking the vortices: "flux pinning" v = 0 is enforced  no work  no voltage drop, no resisitance flux flow V ideal low defect density higher defect density I IcIc

37. segregation with small  (or even NL) segregation: vortex core can stay without cost in binding energy Pinning Mechanisms:  condensation energy is lost "pinning - force" particularly effective: defect sizee   to go on will cost again energy i.e. segregation has a binding force flux flow

38. J c tech as Function of Temperature and Field decreases in magnetic fieldmore vortices/pinning center decreases with temperaturethermal activation of vortices E j j c tech flux flow

39. Pinning in external magnetic field pinning impedes entrance and exit of vortices B i is inhomogenous within the sample Hysterese! B c1 and B c2 unchanged frozen-in flux BiBi BaBa B c1 B c2 -B c1 -B c2 virgin curve ideal type II SC flux flow

40. Field Distribution in the Sample: surface: jump  ideal magnetisation curve B SL x B a < B c1 (Meissner) B a  B c2 B a grows BiBi BaBa B c1 B c2 inside: field gradient  gradient of vortex density gradient decreases with increasing field strengtn vortices move only if their repulsion force is greater than the pinning force flux flow

41. Bean Model: density gradient  shielding current  Ampère macroscopic average over vortices: here: if  B/  x small:j < j c  vortices pinned x y if  B/  x larger:j > j c  vortices are ripped away "critical state" remark: x B move until everywhere j = j c measurement of dB i (x)/dx  j c flux flow