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New roads opening in the field of Superconducting Materials after the discovery of MgB2 Sandro Massidda Physics Department University of Cagliari [email protected] http://www.dsf.unica.it/~sandro/ Outline •Most superconductors have been discovered by chance! •Can we do better? •Basic elements can be found in many SC and can serve as a guide in the search •Ingredients of conventional superconductivity: electrons and phonons. •The electron-phonon interaction in real materials. •Key concepts: Kohn anomaly, two-gap superconductivity, Fermi surface nesting, covalently bonded metals. •Applications to real materials: MgB2, CaSi2, intercalate graphite CaC6 , alkali under pressure Origin of “conventional” superconductivity: phonons produce an attraction among electrons (Cooper pairs) Lattice deformation Overscreening of e-e repulsion by the lattice Classical view of how a lattice deformation by a first electron attracts the second one First ingredient: Energy bands. Example of Cu Symbols are from experiments s bands nearly parabolic: free-electron d bands Narrow, filled k k (r l) k (r)ei kl Band dispersion from Bloch theorem carries the information on chemical bonding Similarity: bonding & anti-bonding molecular orbitals An interesting material: MgB2 Tc=39.5 K B planes Mg planes Isoelectronic to graphite, why so different? Energy bands of MgB2 3D p bands (strongly dispersed along G-A (kz)) 2D s bands (weakly dispersed along G-A) s bonding (px,py) p bonding & antibonding (pz orbitals) s k=(kx;ky;) (0,0,kz ) (kx;ky;p/c) sp2 E l e c t r o n i c p r o p e r t i e s o f MgB2 s Strong covalent s bonds p B B B 2-D s-bonding bands 3-D p bands Dispersion and bonding: p bands Mg - Mg + + + B B G k 0 A k 0,0,p / c s Graphite p MgB2 •Different dispersion along kz: 2D vs 3D s The presence of cations is crucial to get s holes. s holes are the origin of superconductivity Fermi surface of MgB2 B px and py ( s) B pz ( p) The FS is the iso-energy surface in k-space separating filled and empty states Second ingredient: Phonons Lattice deformation: Rls R0ls us (q)e iqRl 3Nat phonon branches at each wave vector q ' ss ' C (q) 2 det (q) 0 M s M s' Analogy with elementary mechanics: s atom cartesian component l lattice point k 2 M Force constants contain the response of the electrons to ionic displacement: fundamental ingredient First-principles calculations vs experiments Source of electron-electron attraction k+q k Virtual phonon q k’-q k’ BCS theory: superconducting gap k Vkk ' k' k ' 2 k ' k ' 2 Ek k 2 k 2 2 tanh k ' 2 k ' 2 2kBT excitation energies 2h D e 1 Exponential dependence on the coupling Tc 1.14 e 2 3.52 kBTc k ≈ 2p Coherence length 1 ELIASHBERG theory (1960): • attractive electron-phonon interaction: Eliashberg Spectral Function 2F() describes the coupling of phonons to electrons on the Fermi Surface 2 F ( ) 2 d Connection to normal state electrical resistivity : trT Pb and MgB2 Eliashberg functions Pb =1.62 Tc=7.2 K Low phonon frequencies MgB2 =0.87 Tc=39.5 K Large phonon frequencies Still, CaC6 has larger and similar but Tc=11.5 K !!! McMillan Equation Tc 1.2 1.04 e 1 * (10.62 ) represents the Coulomb repulsion and is normally fitted to experimental Tc N(EF ) I 2 M ph 2 N(EF) electronic density of states I e-ph interaction M nuclear mass ph average ph. frequency Exponential dependence Results of theoretical calculations for elemental superconductors: comparison with experiment Tc T=0 gap at EF M. Lüders et al. Phys. Rev. B 72, 24545 (2005) M. Marques et al. Phys. Rev. B 72, 24546 (2005) A. Floris et al, Phys. Rev. Lett. 94, 37004 (2005) G. Profeta et al, Phys. Rev. Lett. 96, 46003 (2006) Cagliari Berlin L’Aquila collaboration MgB2 superconductor, AlB2 no Phonon density of states Spectral function 2F() Comparable phonon DOS, very different 2F() 2 F( ) 2 d B1g Phonons in MgB2 E2g Anomalously low frequency E2g branch (B-B bond stretching) Large coupling of the E2g phonon mode with s hole pockets (band splitting) E2g=0.075 eV ≈ 1-2 eV !!! Electron doping destroys SC Phonon life-time MgB2 SC AlB2 not SC As soon as s holes disappear with e-doping, superconductivity disappears The width of Raman lines are proportional to the phonon inverse life-time. The difference between MgB2 and AlB2 indicates the different electron-phonon coupling in these two materials Kohn anomaly: LiBC, isoelettronic to MgB2 (Pickett) phonon frequency Stoichiometric compound is a semiconductor Strong renormalization of phonon frequencies Metallic upon doping Kohn anomaly High Tc predicted Unfortunately not found experimentally Kohn anomaly The electronic screening is discontinuous at 2kF (log singularity in the derivative of the response ) d q dq q2 k q > 2kF F For q>2kF it is not possible to create excitations at the small phonon energy q < 2kF FS For q<2kF the electronic screening renormalizes the phonon frequency A Kohn anomaly lowers the energy of E2g phonons in MgB2 2-dimensionality increases the effect Two band model for the electron phonon coupling (EPC) • stronger in s bands due to the coupling with E2g phonon mode • Experiments show the existence of two gaps: s and p. s p Fermi surface Two band model: experimental evidence R. S. Gonnelli, PRL 89, 247004 (2002) Specific heat: evidence of 2 gaps Two-gap structure associated with s and p bands Tunnelling experiments Two band superconductivity Tc depends on the largest eigenvalue of the inter- and intra- band coupling constants, nm and not on the average Impurities in two-gap superconductors have a pair-breaking effect as magnetic impurities in single-gap SC Unfortunately, the experimental situation is not so clear Parent structures to MgB2 CaGa2-xSix CaGa2 CaSi2 CaSi2 becomes Superconductor under pressure, Tc around 14 K Tc CaSi2: phase transitions and superconductivity Frozen-in B1g phonon: trigonal structure due to instability of p bands Trigonal MgB2 trT at high T CaSi2: instability of p bands; sp2 sp3 Large splitting at EF upon distortion DOS KSi2 CaSi2 Amplitude of trigonal distortion vs pressure and band filling Lowered frequencies in SC MgB2. CaBeSi? CaBeSi s bands at EF Intercalate graphite: CaC6 Tc=11.5 K The highest Tc among intercalated graphite compounds (normally Tc < 1 K) N. Emery et al. Phys. Rev Lett. 95, 087003 (2005) CaC6 Amount of Ca contribution Ca FS FS C p FS Phonons in CaC6: 21 modes Very high frequencies but also low frequency branches Orbital character CaC6: gap and orbital character k Gap k over the Fermi surface Superconductivity under pressure 29 elements superconducts under normal conditions 23 only under pressure: Lithium is the last discovered Tc(P) is a strongly material-dependent function* * C. Buzea and K. Robbie Supercond. Sci. Technol. 18 (2005) R1–R8 Aluminium under pressure…… 270 GPa Bonds get stiffer, frequencies higer …Al becomes a normal metal N(EF ) I 2 M 2ph Alkali metal under high pressure: many phase transitions Lithium is a superconductor under pressure CI16 42 hR1 … … 39 … fcc 7 0 9R Electron states of Li and K under pressure Charge on d states K 27 GPa Li 30 GPa Charge on p states Phonon dispersion in Li: softening and stiffening 26 GPa 26 GPa 0 GPa 0 GPa Why? Increasing the pressure a lattice instability driven by the Fermi surface nesting increases the electron-phonon coupling Pieces of Fermi surface connected by the same wave-vector q Phonon softening and lattice instability q q Imaginary frequency: instablility Orbital character at EF and superconductivity d character K Li p character Electron-Phonon Coupling Pressure Stiffer bonds (higher ’s) but higher coupling at low Theoretical predictions Summary • I presented an essential description of the properties and SC mechanisms in a few important materials • Each real material has plenty of interesting physics •SC needs material-adapted understanding where similar mechanisms can act in very different ways A15 Compounds Nb3Sn Tc=18 K it could be a Multigap SC Guritanu et.al. PRB 70 184526 (2004) Lattice distortions in Nb3Sn Free-energy of cubic and tetragonal V3Si c 1 a Nb3Sn Softening of elastic constant Softening of optical phonon mode Lattice distortions in A15 Band structure of Nb3Sn Large peak at EF Concepts in ELIASHBERG theory: • repulsive Coulomb interaction (Morel Anderson): Vel el FS 1 ln EF D The difference between electron (h/EF) and nuclear (1/D) time scales reduces the coulomb repulsion (retardation) Superconductivity results from the competition of opposite effects: Impurities in two-gap superconductors Irradiation by neutrons (Putti et al) Only in a C-doped sample the merging has been observed at 20 K (Gonnelli et coworkers) Electronic properties of Al-doped MgB2 Mg1-xAlxB2 x=0 x = 0.25 x = 0.33 x = 0.5 Electron-phonon spectral function 2F() Bands of CaSi2 in the ideal and distorted (full lines) structures Spectral function of Nb3Sn from tunnelling Many different results with many different values, ranging from =1.08 to 2.74! Non-magnetic impurities: Anderson theorem In the presence of disordered impurities the wave-vector k is not a conserved quantity: electrons cannot sneak anymore as Bloch suggested, if the potential is not periodic However, the impurity potential being static, V(r, t ), we still have stationary states: k n We can form Cooper pairs by time-reversal degenerate states k ,k n , * n Important physical conclusion: Tc does not change in a significant way due to the presence of impurities! Impurities: experiments Tc proportional to the low temperature resistivity, related to impurities induced by irradiation. Magnetic impurities: Gorkov-Abrikosov theory Magnetic impurities split the energy of states with spin and pair breaking effect Important physical conclusion: Tc is strongly depressed by the presence of magnetic impurities! d d Ni The presence of a static magnetic moment is incompatible with conventional superconductivity