University of Cagliari New roads opening in the field of Superconducting Materials after the discovery of MgB2 Sandro Massidda Physics Department University of Cagliari sandro.massidda@dsf.unica.it http://www.dsf.unica.it/~sandro/

Most superconductors have been discovered by chance! Can we do better? 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

Overscreening of e-e repulsion by the lattice Origin of “conventional” superconductivity: phonons produce an attraction among electrons (Cooper pairs) Lattice deformation Classical view of how a lattice deformation by a first electron attracts the second one Overscreening of e-e repulsion by the lattice

Similarity: bonding & anti-bonding molecular First ingredient: Energy bands. Example of Cu Symbols are from experiments s bands nearly parabolic: free-electron d bands Narrow, filled k 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?

sp2 s Energy bands of MgB2 (px,py) k=(kx;ky;) (0,0,kz ) (kx;ky;/c) 3D p bands (strongly dispersed along G-A (kz)) 2D s bands (weakly dispersed along G-A) p bonding & antibonding (pz orbitals) s s bonding (px,py) sp2 k=(kx;ky;) (0,0,kz ) (kx;ky;/c)

E l e c t r o n i c p r o p e r t i e s o f MgB2   Strong covalent  bonds B B B 2-D s-bonding bands 3-D p bands

Dispersion and bonding: p bands - + + Mg Mg B B - A G

The presence of cations is crucial to get  holes. MgB2 Different dispersion along kz: 2D vs 3D Graphite  The presence of cations is crucial to get  holes.  holes are the origin of superconductivity 

Fermi surface of MgB2 B px and py ( s) B pz ( ) The FS is the iso-energy surface in k-space separating filled and empty states

Second ingredient: Phonons Lattice deformation: 3Nat phonon branches at each wave vector q s atom  cartesian component l  lattice point Analogy with elementary mechanics: 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+q Virtual phonon k k’

BCS theory: superconducting gap Exponential dependence on the coupling  Coherence length k ≈ 2

attractive electron-phonon interaction: ELIASHBERG theory (1960): attractive electron-phonon interaction: Eliashberg Spectral Function a2F() describes the coupling of phonons to electrons on the Fermi Surface Connection to normal state electrical resistivity :

Pb and MgB2 Eliashberg functions =1.62 Tc=7.2 K =0.87 Tc=39.5 K Large phonon frequencies Still, CaC6 has larger  and similar  but Tc=11.5 K !!! Low phonon frequencies

Exponential dependence McMillan Equation  represents the Coulomb repulsion and is normally fitted to experimental Tc 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 T=0 gap at EF D0 Tc 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()

Anomalously low frequency E2g branch (B-B bond stretching) Phonons in MgB2 B1g E2g Anomalously low frequency E2g branch (B-B bond stretching)

Large coupling of the E2g phonon mode with s hole pockets (band splitting) wE2g=0.075 eV  ≈ 1-2 eV !!!

Electron doping destroys SC Phonon life-time MgB2 SC AlB2 not SC As soon as  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) Stoichiometric compound is a semiconductor Strong renormalization of phonon frequencies phonon frequency 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 ) q > 2kF For q>2kF it is not possible to create excitations at the small phonon energy For q<2kF the electronic screening renormalizes the phonon frequency q < 2kF FS 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  bands due to the coupling with E2g phonon mode Experiments show the existence of two gaps:  and .   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  and  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 Tc Parent structures to MgB2 CaGa2  CaSi2 CaSi2 becomes Superconductor under pressure, Tc around 14 K

CaSi2: phase transitions and superconductivity Frozen-in B1g phonon: trigonal structure due to instability of bands Trigonal MgB2

CaSi2: instability of  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  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)

 Amount of Ca contribution Ca FS FS C  FS

Phonons in CaC6: 21 modes Very high frequencies but also low frequency branches

CaC6: gap and orbital character 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

Alkali metal under high pressure: many phase transitions

Lithium is a superconductor under pressure CI16 42 hR1 39 … … … fcc 7 9R

Electron states of Li and K under pressure Charge on d states K 27 GPa Con l’aumentare della pressione gli stati d si portano in prossimità del livello di fermi, Da una parte delle bande con forte carttere d/p si avvicinano al livello di fermi, dall’altra, tutte le bande si ibridizzano maggiormente con gli stati d /p e ne assumono di più il carattere. Questo porta automaticamente a un aumento dell’accoppiemento elettrone fonone che è più forte per stati localizzati come quelli d e i legami p. I fononi inoltre sono influenzati dalla forma della superficie di fermi. Li Charge on p states 30 GPa

Phonon dispersion in Li: softening and stiffening 26 GPa 26 GPa 0 GPa 0 GPa

Why? q Pieces of Fermi surface connected by the same wave-vector q q 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 D K Li p character D

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 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): 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: l-m*

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

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: We can form Cooper pairs by time-reversal degenerate states 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  The presence of a static magnetic moment is incompatible with conventional superconductivity d  Ni