1. 1 Superconductivity  pure metal metal with impurities 0.1 K Electrical resistance  is a material constant (isotopic shift of the critical temperature)

2. Superconductivity 2 The superconductivity was discovered in 1911 by Heike Kamerlingh Onnes at the Leiden University. At 4.2 K (-296°C), he observed a disappearance of resistivity in mercury. His experiments were made possible by the condensation of helium (1908). Heike Kamerlingh Onnes 1913 Nobel prize in physics

3. 3 Superconductivity Superconducting elements T [K] Al1.19 Cd0.56 Ga1.09 Hg4.00 In3.40 Ir0.14 La5.00 Mo0.92 Nb9.13 Os0.65 Pb7.19 Re1.70 T [K] Ru0.49 Sn3.72 Ta4.48 Tc8.22 Th1.37 Ti0.39 Tl2.39 U0.68 V5.30 Zn0.87 Zr0.55

4. 4 Isotopic Shift Material T [K]  Zn 0.87 0.45±0.05 Cd 0.56 0.32±0.07 Sn 3.72 0.47±0.02 Hg 4.00 0.50±0.03 Pb 7.19 0.49±0.02 Tl 2.39 0.61±0.10 Material T [K]  Ru 0.49 0.00±0.05 Os 0.65 0.15±0.05 Mo 0.92 0.33 Nb 3 Sn 18 0.08±0.02 Mo 3 Ir 0.33±0.03 Zr 0.55 0.00±0.05

5. 5 Superconductivity Superconductor in a magnetic field T HcHc normal state superconducting state TcTc Temperature dependence of the critical magnetic field Superconductor: Meissner effect Otherwise:   -10 -6

6. Meissner-Ochsenfeld effect 6

7. Magnetic levitation train 7

8. 8 Superconductor in a magnetic field External field: Inner field: Magnetization: Work per unit of volume (magnetization direction of a superconductor is opposite to the magnetic field direction) Energy of a superconductor within an magnetic field is higher than without an magnetic field This is caused by the “superconducting” electrons

9. 9 Transition between normal and superconducting state Thermodynamic consideration

10. 10 Superconductivity S.L. Bud’ko and P.C. Canfield: Temperature-dependent Hc2 anisotropy in MgB 2 as inferred from measurements on polycrystals, Phys. Rev. B 65 (2002) 212501.

11. Crystal structures of La 2-x Ba x CuO 4 and YBa 2 Cu 3 O 7-x 11 YBa 2 Cu 3 O 7-x Space group: Pmmm Lattice parameters: a = 3.856(2) Å b = 3.870(2) Å c = 11.666(3) Å a  b  c/3 La 2-x Ba x CuO 4 Space group: Bmab Lattice parameters: a = 5.33915(9) Å b = 5.35882(9) Å c = 13.2414(2) Å a  b a/  2 < c/3 < a

12. 12 Superconductivity normal state superconducting

13. 13 Theories of Superconductivity Super electrons : No scattering Entropy of the system is zero (the system is perfectly ordered) Large coherence length

14. 14 London Theory (Meissner Effect) Ohm:London: Maxwell: (static conditions) Meissner effect: Solution: B x

15. 15 Consequences of the London Theory

16. 16 Coherence Length The distance in which the width of the energy gap, in a spatial variable magnetic field, doesn’t change essentially. London:

17. 17 BCS Theory of Superconductivity J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106 (1957) 162. J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. 1. Interactions between electrons can cause a ground state, which is separated from the electronically excited states by an energy gap. However: there are also superconductors without an energy gap!

18. 18 BCS Theory of Superconductivity 2. The energy gap is caused by the interaction between electrons via lattice vibrations (phonons). One electron distort the crystal lattice, another electron “sees” this and assimilate his energy to this state in a way, which reduces the own energy. That’s how the interaction between electrons via lattice vibrations work.

19. 19 BCS Theory of Superconductivity 3. The BCS theory delivers the London penetration depth for the magnetic field and the coherence length. Thereby the Meissner effect is explained. London: Meissner: Coherence length: