New roads opening in the field of Superconducting Materials after the discovery of MgB 2

1. 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/

2. 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 conventionalsuperconductivity: 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

3. 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

4. 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

5. An interesting material: MgB2 Tc=39.5 K B planes Mg planes Isoelectronic to graphite, why so different?

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

7. 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

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

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

10. 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

11. 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

12. First-principles calculations vs experiments

13. Source of electron-electron attraction k’-q k+q Virtual phonon k’ k

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

15. 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 :

16. Pb and MgB2 Eliashberg functions MgB2 Pb =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

17. 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

18. Results of theoretical calculations for elemental superconductors: comparison with experiment T=0 gap at EFD0 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

19. MgB2 superconductor, AlB2 no Phonon density of states Spectral function 2F() Comparable phonon DOS, very different2F()

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

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

22. 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

23. 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 Tcpredicted Unfortunately not found experimentally

24. Kohn anomaly The electronic screening is discontinuous at 2kF (log singularity in the derivative of the response ) q >2kF Forq>2kF it is not possible to create excitations at the small phonon energy For q<2kFthe 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

25.   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

26. Two-gap structure associated with  and  bands Tunnelling experiments

27. Two band superconductivity Tc depends on the largest eigenvalue of the inter- and intra- band coupling constants, nmand not on the average 

28. 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

29. Tc CaGa2-xSix Parent structures to MgB2 CaGa2  CaSi2 CaSi2 becomes Superconductor under pressure, Tc around 14 K

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

31. 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?

32. CaBeSi  bands at EF

33. 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)

34. CaC6  Amount of Ca contribution Ca FS FS C  FS

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

36. Orbital character CaC6: gap and orbital character Gap k over the Fermi surface

37. Superconductivity under pressure 29 elements superconducts under normal conditions 23 only under pressure: Lithium is the last discovered

38. Tc(P) is a strongly material-dependent function* * C. Buzea and K. Robbie Supercond. Sci. Technol. 18 (2005) R1–R8

39. 270 GPa Aluminium under pressure…… Bonds get stiffer, frequencies higer …Al becomes a normal metal

40. Alkali metal under high pressure: many phase transitions

41. Lithium is a superconductor under pressure CI16 42 hR1 39 … … … fcc 7 0 9R

42. Electron states of Li and K under pressure Charge on d states K 27 GPa Li Charge on p states 30 GPa

43. Phonon dispersion in Li: softening andstiffening 26 GPa 26 GPa 0 GPa 0 GPa

44. Phonon softening and lattice instability 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 q q Imaginary frequency: instablility

45. Orbital character at EF and superconductivity d character D K Li p character D

46. Electron-Phonon Coupling Pressure   Stiffer bonds (higher ’s) but higher coupling at low 

47. Theoretical predictions

48. 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

49. A15 Compounds Nb3Sn Tc=18 K it could be a Multigap SC Guritanu et.al. PRB 70 184526 (2004)

50. Free-energy of cubic and tetragonal Lattice distortions in Nb3Sn V3Si Nb3Sn Softening of elastic constant Softening of optical phonon mode