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The Ginzburg-Landau and BCS Theories of Superconductivity

The Ginzburg-Landau and BCS Theories of Superconductivity. Daniel Wiegand Theoretisch-physikalisches Seminar über Probleme der Quantenmechanik , University of Heidelberg Jan 2011. What to expect …. h istorical overview. Tools I - the Landau Theory of Phase Transitions.

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The Ginzburg-Landau and BCS Theories of Superconductivity

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  1. The Ginzburg-Landau and BCS TheoriesofSuperconductivity Daniel Wiegand Theoretisch-physikalisches Seminar über Probleme der Quantenmechanik, University of Heidelberg Jan 2011

  2. Whattoexpect… historicaloverview Tools I - the Landau Theoryof Phase Transitions - the Ginzburg-Landau Theory (macroscopic) Tools II - secondQuantizationFormalism - the BCS Theory (microscopic)

  3. A briefHistoryofSuperconductivity

  4. Heike K. Onnes (1853-1926) 1911 thebeginnings Nobel Prize (1913) forhisinvestigations on thepropertiesof matter atlowtemperatures, […] all picturescourtesyof Emilio Segre Visual Archives

  5. Noelectricresistance Ideal diamagnetism ( Magneticfieldsareexpelledfromthebulkofthesuperconductor (Meissner effect) normal empirical phenomena superconducting T

  6. London theory charged (i.e. complexbosonic) fielddescribestheconducting quasi- particles, withthemacroscopicamplitude = outcome rot = -B = 1935 firstexplanationattempts H. and F. London (1953)

  7. Nobel Prize (1962) forhispioneeringtheoriesofcondensed matter […] 1937 the Landau theoryofphasetransitions Lev D. Landau (1908-1968)

  8. Alexei A. Abrikosov (*1928) Vitaly L. Ginzburg (1916-2009) 1950 the Ginzburg Landau theory (macroscopic) Nobel Prize (2003) forpioneeringcontributionstothetheoryofsuperconductorsandsuperfluids

  9. Leon Cooper (*1930) John Bardeen (1908 – 1991) John Schrieffer (*1931) 1957 the BCS theory (microscopic) Nobel Prize (1972) fortheirjointlydevelopedtheoryofsuperconductivity, usuallycalledthe BCS theory

  10. Nobel Prize (1987) fortheirimportantbreakthrough in thediscoveryofsuperconductivity in ceramicmaterials 1986 hotstuff Unit cellof Bi2Sr2Ca2Cu3O10

  11. The Landau TheoryofPhase Transitions

  12. introduction • Landau‘sassumptions • a phasetransitionisbetween a “symmetric“ phaseand a phaseofbrokensymmetries • bothphasesarecharacterizedby an order paramater • thefreeenergyistheminimumoffreeenergyfunction F, withrespecttotheorderparameter furthermore: F isanalyticalclosetoandthetransitionpointismarkedby a disconitunity

  13. theorder paramater T > T < nodipolemoment(p = 0) dipole moment(p =) The dipolemomentcharacterizesthephasetransitionas an orderparameter

  14. thefreeenergy F T > T = T < = For T ) + B(t) ) +

  15. The Ginzburg-Landau Theory

  16. theidea • At T = theconductorundergoes a phasetransition in a superconductingstate • Conductionparticledensity = istheorderparameter(same quasi particlesas in the London theory) • The freeenergyisgivenbythe integral overthefreeenergydensityofthesuperconductor vartiation Whichareknownasthe Ginzburg Landau equations

  17. = 0 0 = 0 = - Ginzburg-Landau coherencelength = “descriptionofthespatialmodulationoftheorderparamter“

  18. 𝘹 = - = = Ginzburg-Landau penetrationdepth = “descriptionofthedecayofpenetratingmagneticfields“

  19. Ginzburg-Landau coherencelength Ginzburg-Landau penetrationdepth = = Ginzburg-Landau parameter

  20. outlook Abrikosov andGorkov i) Fluxthrough a superconducting ring is quantized (fluxlines) Abrikosovvortices ii) The Ginzburg Landau theoryisthemacrsoscopiclimitofthe BCS theory (Gorkov 1959) itis not limited to a temperaturescloseto comparisonyields = 2 = 2e

  21. Fock Space, Quasi Particlesand all theRest

  22. manyparticles the Fock Space/occupationnumberbasis ℂ ⊕ H ⊕ S(H ⊗ H) ⊕ S(H⊗ H ⊗ H) ⊕… thebasisvectorsbeeingspannedby = all operators on F canbeexpressedthrough c and creationoperator = = destructionoperator

  23. Fermi systems groundstate/Fermi vaccum excitedstatestate In theparticle-hole picture

  24. The BCS Theory

  25. theidea metal = cubiclatticeof positive ions + “smeared out“ electrons i) an electrondeformsthelatticethroughcoulombinteraction, (creating a phonon) – with < ii) anotherelectronisattrackteddue tothenow positive vicinityofthefirstelectron (absorbingthephonon) iii) bothelectrons form a quasi particle, a Cooper pair, oflower energythentheinitialsingleeletronenergy = Eigenstates = Cooper pair operators =

  26. BCS Hamiltonian + + = = = ) =

  27. BCS Hamiltonian (Fröhlich 1952) + + + … = = = = = )

  28. findingtheBCS groundstate in general = = 2N(0

  29. excitedstates/Bogoliubov-Valatin-transf. = = = = + smallterms

  30. comparisonwithexperiment (T)

  31. tosumup… Ginzburg-Landau The minimizationofthefreeenergygaveus the Meißner effect (penetrationdepth!) thespatialexpansionof a quasi particle (coherencelength!) BCS The variationofphonon-electron-Hamiltoniangaveus thetransitiontemperature the minimal energyΔ(T) neededtoexcite a cooper pair, andthusthereasonforthevanishingresistance

  32. “… so superconductivityis a real grab-bag, filledwithgoodiesfor all kindsofphysicists.“ R. Mattuck ThankYouForYour Patience

  33. MattuckA Guide to Feynman Diagrams in theMany-Body Problem Fetter/WaleckaQuantum TheoryofMany-Particle Systems Toledano/ToledanoThe Landau Theoryof Phase Transitions Landau/LifschitzCourse on theoreticalphysics V/IX EschrigTheoryofsuperconductivity Literature L.N. Cooper Bound electron pairs in a degenerate Fermi gas (physical review, 104/4, 11/15/1956) J. Bardeen, L.N. Cooper, J.R. Schrieffer Theory of superconductivity (physical review, 108/5, 07/08/1957)

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