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10. Superconductivity

10. Superconductivity. Experimental Survey Occurrence of Superconductivity Destruction of Superconductivity by Magnetic Fields Meisner Effect Heat Capacity Energy Gap Microwave and Infrared Properties Isotope Effect Theoretical Survey

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10. Superconductivity

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  1. 10. Superconductivity • Experimental Survey • Occurrence of Superconductivity • Destruction of Superconductivity by Magnetic Fields • Meisner Effect • Heat Capacity • Energy Gap • Microwave and Infrared Properties • Isotope Effect • TheoreticalSurvey • Thermodynamics of the Superconducting Transition • London Equation • Coherence Length • BCS Theory of Superconductivity • BCS Ground State • Flux Quantization in a Superconducting Ring • Duration of Persistent Currents • Type II Superconductors • Vortex State • Elimination of Hc1 and Hc2 • Single Particle Tunneling • Josephson Superconductor Tunneling • DC Josephson Effect • AC Josephon Effect • Macroscopic Quantum Interference • High-Temperature Superconductors

  2. K.Onnes (1911) : ρ → 0 as T → TC

  3. Experimental Survey • ρ → 0 for T < TC . Persistent current in ring lasts > 1 yr. • NMR: supercurrent decay time > 105 yrs. • Meissner effect: superconductor = perfect diamagnet. Normal state SuperC state 4. BCS theory: Cooper pairs (k, –k ). See App. H & I.

  4. Occurrence of Superconductivity Occurrence: Metallic elements, alloys, intermetallic compounds, doped semiconductors, organic metals, … Range of TC: 90K for YBa2Cu3O7. .001K for Rh. Si: TC = 8.3K at P = 165 Kbar

  5. Destruction of Superconductivity by Magnetic Fields Magnetic field destroys superconductivity. in CGS units Magnetic impurities lower TC: 10–4 Fe destroys superC of Mo (TC= 0.92K ). 1% Gd lowers TC of La from 5.6K to 0.6K. Non-magnetic impurities do not affect TC .

  6. Meissner Effect B = 0 inside superC Normal state SuperC state For a long thin specimen with long axis // Ha, H is the same inside & outside the specimen (depolarizing field ~ 0) → Caution: A perfect conductor (ρ = 0) may not exhibit Meissner effect. Ohm’s law → → (B is frozen, not expelled.) Also, a perfect conductor cannot maintain a permanent eddy current screen → B penetrates ~1 cm/hr.

  7. Alloys / Transition metals with high ρ. Most elements ρ = 0 but B 0 in vortex state.

  8. HC2 ~ 41T for Nb3(Al0.7 Ge0.3). HC2 ~ 54T for PbMo6S8. Commercial superconducting magnets of ~1T are readily available.

  9. Heat Capacity → superC state is more ordered ΔS ~ 10–4kB per atom → only 10–4 e’s participate in transition. Al →

  10. → energy gap

  11. Energy Gap Comparison with optical & tunneling measurements → not For Ha= 0, n-s transition is 2nd order ( no latent heat, discontinuity in Ce , Eg →0 at TC ).

  12. Microwave and Infrared Properties EM waves are mostly reflected due to impedance mismatch at metal-vacuum boundary. They can penetrate about ~20A into the metal. Photons with ω < Eg are not absorbed → surface penetration is greater in superC than in normal state. For T << TC , ρs = 0 for ω < Eg . ρsρn for ω > Eg . (sharp threshold at Eg) For TTC , ρs 0 for all ω 0 ( screening of E incomplete due to finite inertia of e )

  13. Isotope Effect Isotope effect: → e-phonon interaction involved in superC. Original BCS: → Deviation from α = ½ can be caused by coulomb interaction between e’s. Absence of isotope effect due to band structure.

  14. Theoretical Survey • Thermodynamics of the Superconducting Transition • London Equation • Coherence Length • BCS Theory of Superconductivity • BCS Ground State • Flux Quantization in a Superconducting Ring • Duration of Persistent Currents • Type II Superconductors • Vortex State • Estimation of Hc1 and Hc2 • Single Particle Tunneling • Josephson Superconductor Tunneling • DC Josephson Effect • AC Josephon Effect • Macroscopic Quantum Interference • Thermodynamics Considerations • Phenomenological Models • Quantum Theory

  15. Thermodynamics of the Superconducting Type I superC: → no latent heat ( 2nd order transition) (continuous transition)

  16. London Equation London model: in London gauge: → → → London equation λL = London penetration length see flux quantization • Meissner effect not complete in thin enough films. • HC of thin films in parallel fields can be very high.

  17. Coherence Length Coherence lengthξ ~ distance over which nS remains relatively uniform. See Landau-Ginzburg theory, App I., for exact definition. Local properties = Average of non-local properties over regions ~ ξ . Minimum thickness of normal-superC interface ~ ξ . Spatial variation of ψ increases K.E. → High spatial variation of ψScan destroy superC. Let → for q << k

  18. → Critical modulation for destroying superC is Intrinsic coherence length: see Table 5 ξ in impure material is smaller than ξ0 . (built-in modulation) ξ & λ depend on normal state mfp l. see Tinkham, p.7 & 113. Pure sample: → Dirty sample: → ξ0 = 10 λL Type I Type II

  19. BCS Theory of Superconductivity BCS = Bardeen, Cooper, Schrieffer BCS wavefunction = Cooper pairs of electrons k and –k (s-wave pairing) • Features & accomplishments of BCS theory : • Attractive e-e interaction –U → Eg between ground & excited states. • Eg dictates HC , thermal & EM properties. • –U is due to effective e-ph-e interaction. • λ , ξ , London eq. (for slowly varying B ), Meissner effect, … • Quantization of magnetic flux involves unit of charge 2e. U D(εF) << 1 : θ = Debye temperature  Higher ρ → Higher TC (worse conductor → better superC)

  20. BCS Ground State T = 0 Cooper pair: 1-e occupancy with Teff = TC Super state: Cooper pair mixes e’s from below & above εF Normal state but due to –U. Cooper pair: ( k , –k ) → spin = 0 (boson) see App. H

  21. Flux Quantization in a Superconducting Ring Energy intensity for large number of photons: → Let ψ(r) be the super state wave function. Particle density n = ψ*ψ n = constant → Velocity operator: Particle flux: Electric current density: London eq. with

  22. Meissner effect: B = j = 0 inside superC → ψ measurable → ψ single-valued → Δ = 2 π s s Z  Flux quantization q = –2e → = fluxoid or fluxon Flux through ring : see Tinkham, p.121, for a derivation via Sommerfeld quantization rule Φext not quantized → Φsc must adjust

  23. Duration of Persistent Currents Thermal fluctuation : superC → normal : fluxoid escapes from ring Transition rate W = ( attempt freq ) ( Boltzmann factor for activation barrier ) Boltzmann factor for activation barrier = exp( −βΔF ) Free energy of barrier = ΔF = (minimum volume) (excess free energy density of normal state) minimum volume  Rξ2 . R = wire thickness excess free energy density of normal state = HC2 / 8 π. R = 10−4 cm, ξ = 10−4 cm, HC = 103 G, gives ΔF 10−7 erg. Note: estimate is good for T = 0 to 0.8 TC while ΔF → 0 as T → TC− β 10−15 erg at T = 10K → Attempt freq  Eg /  10−15 / 10−27  1012 s−1 Age of universe ~ 1018 s Exceptions: Near TC or in Type II materials.

  24. Type II Superconductors Electronic structure not much affected

  25. Ref: W.Buckel, “Superconductivity” Normal-Super Conductor Interface Lowering of energy due to field penetration Increase of energy due to destruction of Cooper pairs: Normal: Bulk superC: Interface energy at H = HC

  26. HC2 for Nb3Sn ~ 100kG. Thin films with H normal to surface Type I: Intermediate state Type II: Vortex state Fluxoid penetration reduces increase of energy due to flux repulsion.

  27. Vortex State Meissner effect starts breaking down when a normal core can be substained. Normal core radius is always ξ ; otherwise it’ll be bridged by surrounding ψS . Fluxoids well separated: fluxoid radius  λ = Field for nucleation of single fluxoid Closed-packed fluxoids: fluxoid radius ξ Type II: κ > 1 → λ > ξ → HC1 < HC2 Vortex state allowed. SuperC destroyed before fluxoid allowable → no vortex state. Type I: κ < 1 → λ < ξ → HC1 > HC2

  28. Flux lattice in NbSe2 at 1000 G & 0.2K. STM showing DOS at εF . EN – ES= Stabilization energy → Decrease of E for allowing H penetration: Total core energy wrt super-state Threshold for stable fluxoid: f = 0 at Ba= HC1. → →

  29. Single Particle Tunneling 2 metals Iand IIseparated by insulator C. Al Sn TC 1.14K 3.72K Glass + In contact + Al strip 1mm wide 2000A thick + Al2O3 20-30 A: S.P.T. 10A: J.T. + Sn strip Direct measurement of J.T. requires double junctions. See W.Buckel, “Superconductivity”, p.85.

  30. I, IIbothnormal: line 1

  31. I normal, IIsuper: T = 0, line 2 T  0, line 3 T0

  32. I , IIboth super: T  0

  33. Josephson Superconductor Tunneling • DC Josephson effect: • DC current when E = B = 0 • AC Josephson effect: • rf oscillation for DC V. • Macroscopic long-range quantum interference: • B across 2 junctions → interference effects on IS

  34. DC Josephson Effect T = transfer frequency →  Real parts:  → Imaginary parts:  → n1 n2 → DC current up to iC while V = 0.

  35. AC Josephson Effect V across junction:  Real parts:  →  Imaginary parts: Precision measure of e/ AC current with for V = 1 μV

  36. Macroscopic Quantum Interference Around closed loop enclosing flux Φ: For B = 0, For B 0, or zero offset due to background B periodicity = 39.5 mG Imax = 1 mA Junction area = 3 mm  0.5 mm periodicity = 16 mG Imax = 0.5 mA Prob 6

  37. High-Temperature Superconductors TC ceiling for intermetallic compounds = 23K.

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