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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 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 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’ k-Q k´+Q Q k k´ The characteristic frequency of the phonon must then be the phonon frequency Q, Lecture 12 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 2 Q e2 V (Q ) 1 2 o (Q2 k 2s ) 2 Q e2 1 1 2 o (Q2 k 2s ) 2 Q 1 This shows that the phonon mediated interaction is of the same order of magnitude as the Coulomb interaction Clearly if <Q this (much simplified) potential is always negative. The maximum phonon frequency is defined by the Debye energy ħD =kBD, 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 ħD 2 E 2EF 2D exp N ( E ) V F Lecture 12 Superconductivity and Superfluidity The maximum (BCS) transition temperature N(EF)V is known as the electron-phonon coupling constant: ep N(EF )V / 2 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 1 1.75kBTc 2D exp ep which implies a maximum possible Tc of 25K ! Lecture 12 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 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 EF (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 Superconductivity and Superfluidity Superconducting transition temperature (K) Superconducting Materials 160 HgBa2Ca2Cu3O9 (under pressure) 140 HgBa2Ca2Cu3O9 120 TlBaCaCuO BiCaSrCuO 100 YBa2Cu3O7 Liquid Nitrogen temperature (77K) 80 60 (LaBa)CuO 40 20 Hg Pb Nb 1910 Lecture 12 NbC 1930 NbN Nb3Sn Nb3Ge V3Si 1950 1970 1990 Superconductivity and Superfluidity Superconducting compounds Perhaps the most widely used class of superconducting compounds are the A3B family which crystallise in the A-15 structure. B A 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 Six A15 compounds have transition temperatures over 17K Nb3Ge thin films held the record for the highest known Tc 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 Superconductivity and Superfluidity The A15 compounds A structural instability associated with soft phonon modes and a lattice distortion are believed to be responsible for the high transition temperatures Compound Tc B* V3Ga V3Si Nb3Sn Nb3Al Nb3Ga Nb3Sn 15.4K 17.1K 18.3K 18.9K 20.3K 23.0K 23T 23T 24T 33T 34T 38T B A Nb3Sn is the most widely exploited material for the construction of high field superconducting magnets for NMR, MRI etc Lecture 12 Superconductivity and Superfluidity The A15 compounds The materials properties that give the A15 compounds their relatively high Tcs give the compounds brittleness, which makes cable construction difficult: The so called Rutherford method is generally used Nb Nb3Sn Cu Cu Sn swaging Lecture 12 annealing Superconductivity and Superfluidity The Chevrel phase compounds The Chevrel phases were discovered in 1971 They are ternary molybdenum chalcogenides of the type MxMo6X8 M could be any one of a number of metals at rare earth (4f) elements and X is S, Se or Te The M atoms form a nearly cubic lattice in which the Mo6X8 uinits are inserted 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. Lecture 12 Superconductivity and Superfluidity The Chevrel phase compounds Some Chevrel compounds have relatively high transition temperatures, and very high critical fields Compound Tc B* SnMo6S8 PbMo6S8 LaMo6S8 PbMo6Se8 12K 15K 7K 3.6K 34T 60T 45T 3.8T Critical current densities as high as 3x105A.cm-2 have been observed at 4.2K Unfortunately the material is extremely brittle and making wires is problematic Lecture 12 Superconductivity and Superfluidity The nickel borocarbides The rare earth nickel borocarbides, discovered in 1994 have relatively high transition temperatures but also order magnetically at temperatures comparable to Tc Y Yb Lu Tm Er Ho Dy Tb Gd TN(K) Tc(K) 0 0 0 1.5 6.5 6 10 15 19.5 15 0 16 10.8 10.5 8.5 6.2 0 0 (g-1)2J(J+1) 0 (HF?) 0 1.17 2.55 4.5 7.08 10.5 15.5 …an ideal system for probing the interplay of superconductivity and magnetism Y, Lu, Tm, Er, Ho, Dy (Tb, Gd, Nd, Pr, Ce, Yb) Ni B C Superconductivity and Superfluidity Organic Superconductors The Bechgaard salts are nearly one dimensional conductors with very low carrier densities The electronic properties are extremely anisotropic Most of the class of compounds (TTMTSF)2-X, where X is an anion are only superconducting under pressure X ClO4 PF6 ReO4 Lecture 12 pc/kbar 0 9 9.5 CH3 Se Se CH3 CH3 Se Se CH3 TMTSF tetramethyltetraselenafulvane Tc 1.2K 1.2K 1.4K Superconductivity and Superfluidity Organic superconductors under pressure The systems are particularly interesting from a fundamental perspective Is the superconductivity “conventional”? Lecture 12 Superconductivity and Superfluidity Organic Superconductors The b-(BEDT-TTF)2X salts, where X is an anion such as I3, IBr2 or AuI2 are largely 2d organic superconductors X I3 bL I3 bH IBr2 Cu(NCS)2 Tc 1.2K 8.1K 2.5K 10K There is recent evidence that superconductivity in some of the BEDT compounds can only exist in high magnetic fields H H H H In this state the electron pairs may have finite momentum! Lecture 12 S S S S S S S S BEDT-TTF Bis-ethelenedithio-tetrathiafulvane Superconductivity and Superfluidity H H H H Organic superconductors Superconductivity and Superfluidity The Bucky balls Buckminsterfullerene contains 60 carbon atoms at the apices of a triacontaduohedron 7.1Å in diameter C60 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 A3C60 Compound Tc K3C60 K2 RbC60 Rb2KC60 Rb3C60 Cs3C60 19K 22K 25K 29K 47K Although the isotope effect is BCS-like in C60 there is some evidence that superconductivity might not be “conventional” Lecture 12 Superconductivity and Superfluidity