1. Superconductivity and Superfluidity The effects of lattice vibrations The localised deformations of the lattice caused by the electrons are subject to the same “spring constants” that cause coherent lattice vibrations, so their characteristic frequencies will be similar to the phonon frequencies in the lattice The Coulomb repulsion term, on the other hand, has a time scale defined by the plasma frequency and is therefore effectively instantaneous If an electron is scattered from state k to k’ by a phonon, conservation of momentum requires that the phonon momentum must be Q=k-k’ The characteristic frequency of the phonon must then be the phonon frequency  Q, k-Qk-Q k´+Q k´k´k Q The electrons can be seen as interacting by emitting and absorbing a “virtual phonon”, with a lifetime of  =2  /  determined by the uncertainty principle and conservation of energy Lecture 12

2. Superconductivity and Superfluidity The attractive potential It can be shown that such electron-ion interactions modify the screened Coulomb repulsion, leading to a potential of the form Clearly if  <  Q this (much simplified) potential is always negative. This shows that the phonon mediated interaction is of the same order of magnitude as the Coulomb interaction The maximum phonon frequency is defined by the Debye energy ħ  D =k B  D, where  D is the Debye temperature (~100-500K) The cut-off energy in Cooper’s attractive potential can therefore be identified with the phonon cut-off energy ħ  D Lecture 12

3. Superconductivity and Superfluidity The maximum (BCS) transition temperature N(E F )V is known as the electron-phonon coupling constant: ep can be estimated from band structure calculations and from estimates of the frequency dependent fourier transform of the interaction potential, ie V(Q,  ) evaluated at the Debye momentum. Typically ep ~ 0.33 For Al calculated ep ~ 0.23 measured ep ~ 0.175 For Nb calculated ep ~ 0. 35 measured ep ~ 0.32 In terms of the gap energy we can write which implies a maximum possible T c of 25K ! Lecture 12

4. Superconductivity and Superfluidity Bardeen Cooper Schreiffer Theory In principle we should now proceed to a full treatment of BCS Theory However, the extension of Cooper’s treatment of a single electron pair to an N-electron problem (involving second quantisation) is a little too detailed for this course Physical Review, 108, 1175 (1957) Lecture 12

5. Superconductivity and Superfluidity Bardeen Cooper Schreiffer Theory BCS theory requires: (a) low temperatures - to minimise the number of random (thermal) phonons (ie those associated with electron-ion interactions must dominate) (b) a large density of electron states just below E F (the electrons associated with these states are those that are energetically suited to form pairs) (c) strong electron phonon coupling BCS theory is an effective, all encompassing microscopic theory of superconductivity from which all of the experimentally observed results emerge naturally Ginzburg-Landau theory can be derived from BCS theory, and the phenomenological coefficients introduced by Ginzburg and Landau are related to quantities introduced in the microscopic theory Lecture 12

6. Superconductivity and Superfluidity Superconducting Materials 19101930195019701990 20 40 60 80 100 120 140 160 Superconducting transition temperature (K) Hg Pb Nb NbC NbN V 3 Si Nb 3 Sn Nb 3 Ge (LaBa)CuO YBa 2 Cu 3 O 7 BiCaSrCuO TlBaCaCuO HgBa 2 Ca 2 Cu 3 O 9 (under pressure) HgBa 2 Ca 2 Cu 3 O 9 (under pressure) Liquid Nitrogen temperature (77K) Lecture 12

7. Superconductivity and Superfluidity Superconducting compounds Perhaps the most widely used class of superconducting compounds are the A 3 B family which crystallise in the A-15 structure. The A-atoms are typically the transition metals V or Nb, whilst the B atoms are non-transition metals such as Sn, Al, Ga, Si, Ge B A Six A15 compounds have transition temperatures over 17K Nb 3 Ge thin films held the record for the highest known T c of 23K for a number of years up to 1986 This was thought to be close to the limit imposed by BCS theory Lecture 12

8. Superconductivity and Superfluidity The A15 compounds B A A structural instability associated with soft phonon modes and a lattice distortion are believed to be responsible for the high transition temperatures Compound T c B* V 3 Ga15.4K 23T V 3 Si17.1K 23T Nb 3 Sn18.3K 24T Nb 3 Al18.9K 33T Nb 3 Ga20.3K 34T Nb 3 Sn 23.0K 38T Nb 3 Sn is the most widely exploited material for the construction of high field superconducting magnets for NMR, MRI etc Lecture 12

9. Superconductivity and Superfluidity The A15 compounds The materials properties that give the A15 compounds their relatively high T c s give the compounds brittleness, which makes cable construction difficult: The so called Rutherford method is generally used Cu Nb Sn swaging annealing Nb 3 Sn Cu Lecture 12

10. Superconductivity and Superfluidity The Chevrel phase compounds The Chevrel phases were discovered in 1971 They are ternary molybdenum chalcogenides of the type M x Mo 6 X 8 M could be any one of a number of metals at rare earth (4f) elements and X is S, Se or Te Interestingly, these were the first class of superconductors in which magnetic order and superconductivity were found to coexist With M=Gd, Tb, Dy, Er the superconducting transition temperatures are between 1.5 and 2K, while the Neel temperatures are between 0.5 and 1K. The M atoms form a nearly cubic lattice in which the Mo 6 X 8 uinits are inserted Lecture 12

11. Superconductivity and Superfluidity The Chevrel phase compounds Some Chevrel compounds have relatively high transition temperatures, and very high critical fields Compound T c B* SnMo 6 S 8 12K34T PbMo 6 S 8 15K60T LaMo 6 S 8 7K45T PbMo 6 Se 8 3.6K3.8T Critical current densities as high as 3x10 5 A.cm -2 have been observed at 4.2K Unfortunately the material is extremely brittle and making wires is problematic Lecture 12

12. Superconductivity and Superfluidity The nickel borocarbides Y, Lu, Tm, Er, Ho, Dy Tb, Gd, Nd, Pr, Ce, Yb (Tb, Gd, Nd, Pr, Ce, Yb)NiCB T N (K)T c (K)(g-1) 2 J(J+1) Y 015 0 Yb 0 0 (HF?) Lu 0 16 0 Tm1.510.8 1.17 Er6.510.5 2.55 Ho 6 8.5 4.5 Dy10 6.2 7.08 Tb 15 0 10.5 Gd19.5 0 15.5 The rare earth nickel borocarbides, discovered in 1994 have relatively high transition temperatures but also order magnetically at temperatures comparable to T c …an ideal system for probing the interplay of superconductivity and magnetism

13. Superconductivity and Superfluidity Organic Superconductors The Bechgaard salts are nearly one dimensional conductors with very low carrier densities Se CH 3 TMTSF tetramethyltetraselenafulvane Most of the class of compounds (TTMTSF) 2 -X, where X is an anion are only superconducting under pressure The electronic properties are extremely anisotropic Xp c /kbar T c ClO 4 01.2K PF 6 91.2K ReO 4 9.51.4K Lecture 12

14. Superconductivity and Superfluidity Organic superconductors under pressure The systems are particularly interesting from a fundamental perspective Is the superconductivity “conventional”? Lecture 12

15. Superconductivity and Superfluidity Organic Superconductors The  -(BEDT-TTF) 2 X salts, where X is an anion such as I 3, IBr 2 or AuI 2 are largely 2d organic superconductors S S S S S S S S H H H H H H H H BEDT-TTF Bis-ethelenedithio-tetrathiafulvane X T c I 3  L 1.2K I 3  H 8.1K IBr 2 2.5K Cu(NCS) 2 10K There is recent evidence that superconductivity in some of the BEDT compounds can only exist in high magnetic fields In this state the electron pairs may have finite momentum! Lecture 12

16. Superconductivity and Superfluidity Organic superconductors

17. Superconductivity and Superfluidity The Bucky balls Buckminsterfullerene contains 60 carbon atoms at the apices of a triacontaduohedron 7.1Å in diameter C 60 itself is not a superconductor, but it can be doped with alkali metals (which form an fcc lattice with a lattice parameter of 10Å) giving A 3 C 60 Compound T c K 3 C 60 19K K 2 RbC 60 22K Rb 2 KC 60 25K Rb 3 C 60 29K Cs 3 C 60 47K Although the isotope effect is BCS-like in C 60 there is some evidence that superconductivity might not be “conventional” Lecture 12