1. “… at each new level of complexity, entirely new properties appear, and the understanding of this behavior requires research which I think is as fundamental in its nature as any other” Philip W. Anderson 1972 Si-crystal semiconductor MgB 2 superconductor 2 atoms Na x CoO 2 superconductor 3 atoms La 2-x Sr x CuO 4 superconductor 4 atoms DNA giant molecule Many atoms 1 atom From last lecture ….

2. Where could we find superfluidity? n p p He - 3 n p p n He - 4 1 millionth of a centimetre Helium Helium - 4 atoms are bosons  particles with integer spin. Helium - 3 atoms are fermions  particles with half integer spin.

3. Superfluids flow without resistance Normal fluid Superfluid 1938 Kapitza and Allen discover superfluidity in He-4

4. For T < 2.4Κ – gravity... If the bottle containing helium rotates for a while and then stops, helium will continue to rotate for ever – there is no internal friction (for as long as He is at T = -269 C or lower

5. 1938 Pyotr L. Kapitsa discovered the superfluidity of liquid Helium 4 Nobel Prize in 1978 1941-47 Lev Landau formulated the theory of quantum Bose liquid - 4He superfluid liquid. 1956-58 he further formulated the theory of quantum Fermi liquid. Nobel Prize in 1962 Early 1970s David M. Lee, Douglas D. Osheroff, and Robert C. Richardson discovered the superfluidity of liquid Helium 3. Nobel Prize in 1996 Anthony Leggett first formulated the theory of superfluidity in liquid 3He in 1965. Nobel Prize in 2003

6. Διάστημα: 3000 χιλιοστά απ ό το απ ό λυτο μηδέν (-273.15 C) 5 χιλιοστά από το απόλυτο μηδέν Χαμηλές θερμοκρασίες

7. LOW-T C Superconductors Lead (Pb) Mercury (Hg) Aluminum (Al) Gallium (Ga) Molybdenum (Mo) Zinc (Zn) Zirconium (Zr) Americium (Am) Cadmium (Cd) Ruthenium (Ru) Titanium (Ti) Uranium (U) Hafnium (Hf) Iridium (Ir) Beryllium (Be) Tungsten (W) Platinum (Pt)* Rhodium (Rh) 7.196 K 4.15 K 1.175 K 1.083 K 0.915 K 0.85 K 0.61 K 0.60 K 0.517 K 0.49 K 0.40 K 0.20 K 0.128 K 0.1125 K 0.023 K 0.0154 K 0.0019 K 0.000325 K

8. metals wood Conductors vs. Insulators plastics No free electrons to carry the current FREE ELECTRONS

9. The foam balls (containing small magnets) organise themselves based on the laws of minimum energy. This arrangement mimics the crystal lattice of a solid material.

10. What is Resistance? What is Resistance? ELECTRIC FIELD VOLTAGE DIFFERENCE IONS (+) ELECTRONS (-)

11. Electrical Resistance Thermal vibrations (phonons) of the ionic lattice Lattice defects Impurities RESISTANCE is caused by electrons colliding with: Cations Electrons

12. V I VsVs copper Liquid helium 4.2K (-269 ºC)

13. R T 77K 273K = 0ºC RoRo Impurities

14. V I VsVs Hg Liquid helium 4.2K (-269 ºC) Onnes (1911)

15. Low -T c Superconductivity Heike Kamerlingh Onnes (1911)

16. LOW-T C Superconductors Lead (Pb) Mercury (Hg) Aluminum (Al) Gallium (Ga) Molybdenum (Mo) Zinc (Zn) Zirconium (Zr) Americium (Am) Cadmium (Cd) Ruthenium (Ru) Titanium (Ti) Uranium (U) Hafnium (Hf) Iridium (Ir) Beryllium (Be) Tungsten (W) Platinum (Pt)* Rhodium (Rh) 7.196 K 4.15 K 1.175 K 1.083 K 0.915 K 0.85 K 0.61 K 0.60 K 0.517 K 0.49 K 0.40 K 0.20 K 0.128 K 0.1125 K 0.023 K 0.0154 K 0.0019 K 0.000325 K

17. BCS Theory John Bardeen Leon Cooper John Schrieffer (1957) No collisions Zero resistance

21. Meissner Effect 1933 – Walther Meissner and Robert Ochsenfeld T<Tc: external magnetic field is perfectly expelled from the interior of a superconductor

22. The energy gap and Bardeen- Cooper-Schrieffer theory The key point is the existence of energy gap between ground state and quasi-particle excitations of the system. 1. Existence of condensate. 2. Weak attractive electron- phonon interaction leads to the formation of bound pairs of electrons, occupying states with equal and opposite momentum and spin. 3. Pairs have spatial extension of order .

23. The electron-electron attraction of the Cooper pairs caused the electrons near the Fermi level to be redistributed above or below the Fermi level. Because the number of electrons remains constant, the energy densities increase around the Fermi level resulting in the formation of an energy gap.

36. Type I Type II

42. In summary … Characteristic lengths in SC London equation: The Pippard coherence length: Penetration depth is the characteristic length of the fall off of a magnetic field due to surface currents. Ginzburg-Landau parameter: for pure SC far from Tc temperature- dependent Ginzburg-Landau coherence length is approximately equal to Pippard coherence length Coherence length is a measure of the shortest distance over which superconductivity may be established The London equation shows that the magnetic field exponentially decays to zero inside a SC (Meissner effect)

43. Magnetic properties Dependences of critical fields on temperature. Phase boundaries between superconducting, mixed and normal states of type I and II SC.

44. Intermediate state (SC of type I) (Type I SC show a reversible 1st order phase transition with a latent heat when the applied field reached B c. At this particular field relatively thick Normal and SC domains running parallel to the field can coexist, in what is known as the intermediate state) Intermediate state of a mono-crystalline tin foil of 29  m thickness in perpendicular magnetic field (normal regions are dark) A distribution of superconducting and normal states in tin sphere (superconducting regions are shaded)

45. Mixed state (SC of type II) (In type II SC finely divided quantized flux vortices or flux lines enter the material over a range of applied fields below Bc, and remain stable over a range of applied fields, in what became known as the mixed state. If these flux lines are pinned by lattice defects or other agencies, type II SC can carry a large super-current: see development of useful high-field SC magnets.) Abrikosov: [1957] One vortex carries one quantum of the flux: Triangular lattice of vortex lines going out to the surface of SC Pb0.98In0.02 foil in perpendicular to the surface magnetic field Supercurrent Normal core Normal regions are approximately 300nm Closer packing of normal regions occurs at higher temperatures or higher external magnetic fields

46. Vortex characteristics Magnetic field of a vortex A quantum of magnetic flux is

47.  Normal core Vortex state of type II superconductors Type II Phase of GL pseudo-wave function changes by 2  when going around spatial lines where  is zero 0 1 ||||

48.  Normal core Vortex state of type II superconductors Type II In type-II SC field penetrates to the bulk of material in the form of vortices (or magnetic flux lines, or fluxons) Phase of GL pseudo-wave function changes for 2  when going around spatial lines where  is zero Each vortex represents magnetic flux quantum B/B eq 01 ||||

49. Critical current density Critical current is the maximum current SC materials can carry, above which they stop being SCs. If too much current is pushed through a SC, the latter will become normal, even though it may be below its Tc. The colder you keep the SC the higher the current it can carry. Three critical parameters Tc, Hc and Jc define the boundaries of the environment within which a SC can operate. Fig. demonstrates relationship between Tc, Hc and Jc (a critical surface). The highest values for Hc and Jc occur at 0K, while the highest value for Tc occurs when H and J are zero.

50. Josephson effect (see also hand-out) In 1962 Josephson predicted Cooper-pairs can tunnel through a weak link at zero voltage difference. Current in junction (called Josephson junction – Jj) is then equal to: Electrical current flows between two SC materials - even when they are separated by a non-SC or insulator. Electrons "tunnel" through this non-SC region, and SC current flows.

51. The Meissner-Ochsenfeld Effect Walter Meissner Robert Ochsenfeld (1933) T>T C T<T C Superconductor Magnet DIAMAGNETISM ?