1. Beyond Zero Resistance – Phenomenology of Superconductivity Nicholas P. Breznay SASS Seminar – Happy 50 th ! SLAC April 29, 2009

2. Preview Motivation / Paradigm Shift Normal State behavior Hallmarks of Superconductivity –Zero resistance –Perfect diamagnetism –Magnetic flux quantization Phenomenology of SC –London Theory, Ginzburg-Landau Theory –Length scales: and  –Type I and II SC’s

3. Physics of Metals - Introduction Atoms form a periodic lattice Know (!) electronic states key for the behavior we are interested in Solve the Schro … … in a periodic potential K is a Bravais lattice vector Wikipedia

4. Physics of Metals – Bloch’s Theorem Bloch’s theorem tells us that eigenstates have the form … … where u(r) is a function with the periodicity of the lattice … Free particle Schro Wikipedia

5. Physics of Metals – Drude Model Model for electrons in a metal –Noninteracting, inertial gas –Scattering time  Apply Fermi-Dirac statistics damping term http://www.doitpoms.ac.uk/tlplib/semiconductors/images/fermiDirac.jpg

6. Physics of Metals – Magnetic Response Magnetism in media Larmor/Landau diamagnetism –Weak anti-// response Pauli paramagnetism –Moderate // response Typical  values – –  Cu ~ -1 x 10 -5 –  Al ~ +2 x 10 -5  minimal response to B fields –  r ~ 1  B =  0 H in SI linear response familiarly

7. Physics of Metals – Drude Model Comments Wrong! –Lattice, e-e, e-p, defects, –  ~ 10 -14 seconds  MFP ~ 1 nm Useful! –DC, AC electrical conductivity –Thermal transport Lorenz number  T –Heat capacity of solids Wikipedia Electronic contribution Lattice

8. Preview Motivation / Paradigm Shift Normal State behavior Hallmarks of Superconductivity –Zero resistance –Perfect diamagnetism –Magnetic flux quantization Phenomenology of SC –London Theory, Ginzburg-Landau Theory –Length scales: and  –Type I and II SC’s

9. Hallmark 1 – Zero Resistance Metallic R vs T –e-p scattering (lattice interactions) at high temperature –Impurities at low temperatures R Temperature Residual Resistance (impurities) Impure metal Electrical resistance R0R0 Lattice (phonon) interactions Pure metal T D /3

10. Hallmark 1 – Zero Resistance Superconducting R vs T R Temperature R0R0 Superconductor TcTc “Transition temperature”

11. Hallmark 1 – Zero Resistance Hard to measure “zero” directly Can try to look at an effect of the zero resistance Current flowing in a SC ring –Not thought experiment – standard configuration for high- field laboratory magnets (10- 20T) Nonzero resistance  changing current  changing magnetic field One such measurement  Superconductor Circulating supercurrent Magnetic (dipole) field From Ustinov “Superconductivity” Lectures (WS 2008-2009) I

12. Hallmark 1 – Zero Resistance Notes R = 0 only for DC AC response arises from kinetic inductance of superconducting electrons –Changing current  electric field Model: perfect resistor (normal electrons), inductor (SC electrons) in parallel Magnitude of “kinetic inductance”:  At 1 kHz, http://www.apph.tohoku.ac.jp/low-temp-lab/photo/FUJYO1.png

13. Hallmark 2 – Conductors in a Magnetic Field Normal metal Field off Apply field

14. Hallmark 2 – Conductors in a Magnetic Field Apply field Perfect (metallic) conductorSuperconductor Normal metal Cool Field off Apply field Apply field

15. Hallmark 2 – Meissner-Oschenfeld Effect Superconductor Cool Apply field B = 0  perfect diamagnetism:  M = -1 Field expulsion unexpected; not discovered for 20 years. B/  0 H -M H HcHc HcHc

16. Hallmark 3 – Flux Quantization Earth’s magnetic field ~ 500 mG, so in 1 cm 2 of B Earth there are ~ 2 million  0 ’s. first appearance of h in our description; quantum phenomenon Total flux (field*area) is integer multiple of  

17. Hallmark 3 – Flux Quantization Apply uniform field Measure flux

18. Aside – Cooper Pairing In the presence of a weak attractive interaction, the filled Fermi sphere is unstable to the formation of bound pairs electrons Can excite two electrons  above E f, obtain bound-state energy < 2E f due to attraction New minimum-energy state allows attractive interaction (e-p scattering) by smearing the FS The physics of superconductors Shmidt, Müller, Ustinov

19. Preview Motivation / Paradigm Shift Normal State behavior Hallmarks of Superconductivity –Zero resistance –Perfect diamagnetism –Magnetic flux quantization Phenomenology of SC –London Theory, Ginzburg-Landau Theory –Length scales: and  –Type I and II SC’s

20. SC Parameter Review g(H) H HcHc g normal state g sc state Magnetic field  energy density Extract free energy difference between normal and SC states with H c Know magnetic response important; use R = 0 + Maxwell’s equations … ?

21. London Theory – 1 Newton’s law (inertial response) for applied electric field Supercurrent density is We know B = 0 inside superconductors Faraday’s law Fritz & Heinz London, (1935)

22. London Theory – 2 London Equations Ampere’s law =0; Gauss’s law for electrostatics

23. Magnetic Penetration Depth - B(z) z Screening not immediate; characteristic decay length Typical ~ 50 nm m,e fixed –  uniquely specifies the superconducting electron density n s Sometimes called the “superfluid density” B0B0 SC

24. Ginzburg-Landau Theory - 1 First consider zero magnetic field Order parameter  Associate with cooper pair density: Expand f in powers of |  | 2  To make sense,  > 0,  (T) Free energy of superconducting state Free energy of normal state Need  > -Infinity; B > 0 Free energy of SC state ~ # of cooper pairs

25. Ginzburg-Landau Theory - 2 For  < 0, solve for minimum in f s -f n … http://commons.wikimedia.org/wiki/File:Pseudofunci%C3%B3n_de_onda_(teor%C3%ADa_Ginzburg-Landau).png

26. Know that f n -f s is the condensation energy: Ginzburg-Landau Theory - 3

27. Ginzburg-Landau Theory - 4 Momentum term in H: Now – include magnetic field Classically, know that to include magnetic fields …

28. Ginzburg-Landau Theory - 5 Free Energy Density

29. Ginzburg-Landau Theory - 6 Take  real, normalize Define Linearize in 

30. Superconducting coherence length -  x  (x) VacuumSC Superconductor  Characteristic length scale for SC wavefunction variation

31. London Theorymagnetic penetration depth Ginzburg-Landau Theorycoherence length   two kinds of superconductors! Pause

32. Surface Energy and “Type II” H(x) x   (x) H(x) x  (x) 

33. Surface Energy:  H(x)   (x) g magnetic (x) energy penalty for excluding B energy gain for being in SC state g sc (x) SC

34. Surface Energy:  H(x)   (x) g magnetic (x) energy penalty for excluding B energy gain for being in SC state net energy penalty at a surface / interface g net (x) g sc (x) SC

35. Surface Energy:  H(x)   (x) g magnetic (x) energy penalty for excluding B energy gain for being in SC state net energy gain at a surface / interface g net (x) g sc (x) SC

36. Type I Type II predicted in 1950s by Abrikosovelemental superconductors  nm  (nm) T c (K)H c2 (T) Al1600501.2.01 Pb83397.2.08 Sn230513.7.03  nm  (nm) T c (K)H c2 (T) Nb 3 Sn112001825 YBCO1.520092150 MgB 2 51853714

37. Type II Superconductors  H Normal state cores Superconducting region http://www.nd.edu/~vortex/research.html

38. London Theorymagnetic penetration depth Ginzburg-Landau Theorycoherence length   two kinds of superconductors The End