1. University of Cagliari
New roads opening in the field of Superconducting Materials after the discovery of MgB2
Sandro Massidda
Physics Department
University of Cagliari

2. 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

3. 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

4. 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

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

6. 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)

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. 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 

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. 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 :

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

17. 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

18. 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, (2005)
M. Marques et al. Phys. Rev. B 72, (2005)
A. Floris et al, Phys. Rev. Lett. 94, (2005)
G. Profeta et al, Phys. Rev. Lett. 96, (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. Anomalously low frequency E2g branch (B-B bond stretching)
Phonons in MgB2
B1g
E2g
Anomalously low frequency E2g branch (B-B bond stretching)

21. Large coupling of the E2g phonon mode with 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 Tc predicted
Unfortunately not found experimentally

24. 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

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, (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, nm and 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. Parent structures to MgB2
CaGa2-xSix Tc
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, (2005)

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

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

36. 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. Aluminium under pressure……
270 GPa
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 9R

42. 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

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

44. 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

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
(2004)

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

51. Lattice distortions in A15

52. Band structure of Nb3Sn
Large peak at EF

53. 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*

54. 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)

55. Electronic properties of Al-doped MgB2
Mg1-xAlxB2
x = 0
x = 0.25
x = 0.33
x = 0.5

56. Electron-phonon spectral function

57. Bands of CaSi2 in the ideal and distorted (full lines) structures

58. Spectral function of Nb3Sn from tunnelling
Many different results with many different  values, ranging from  =1.08 to 2.74!

59. 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!

60. Impurities: experiments
Tc proportional to the low temperature resistivity, related to impurities induced by irradiation.

61. 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