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New roads opening in the field of Superconducting Materials after the discovery of MgB 2. Sandro Massidda Physics Department University of Cagliari Outline. Most superconductors have been discovered by chance!

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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

  • 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
origin of conventional superconductivity phonons produce an attraction among electrons cooper pairs
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


First ingredient: Energy bands. Example of Cu

Symbols are from experiments

s bands nearly parabolic: free-electron

d bands

Narrow, filled


Band dispersion from Bloch theorem carries the information on chemical bonding

Similarity: bonding & anti-bonding molecular



An interesting material: MgB2

Tc=39.5 K

B planes

Mg planes

Isoelectronic to graphite, why so different?


s bonding


p bonding &


(pz orbitals)


Energy bands of MgB2

3D p bands (strongly dispersed along G-A (kz))

2D s bands (weakly dispersed along G-A)


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 mgb 2
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




2-D s-bonding bands

3-D p bands



  • Different dispersion along kz: 2D vs 3D


The presence of cations is crucial to get  holes.

 holes are the origin of superconductivity

fermi surface of mgb 2
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
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

source of electron electron attraction
Source of electron-electron attraction



Virtual phonon



bcs theory superconducting gap
BCS theory: superconducting gap

Exponential dependence on the coupling 

Coherence length

k ≈2

eliashberg theory 1960
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 mgb 2 eliashberg functions
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


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 EFD0


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

mgb 2 superconductor alb 2 no
MgB2 superconductor, AlB2 no

Phonon density of states

Spectral function 2F()

Comparable phonon DOS, very different2F()

phonons in mgb 2



Phonons in MgB2

Anomalously low frequency E2g branch (B-B bond stretching)

large coupling of the e 2g phonon mode with s hole pockets band splitting
Large coupling of the E2g phonon modewith s hole pockets (band splitting)

wE2g=0.075 eV

 ≈ 1-2 eV !!!

phonon life time

Electron doping destroys SC

Phonon life-time


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 mgb 2 pickett
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


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


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

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 band superconductivity
Two band superconductivity

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

impurities in two gap superconductors
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 mgb 2



Parent structures to MgB2

CaGa2  CaSi2

CaSi2 becomes Superconductor under pressure, Tc around 14 K

casi 2 phase transitions and superconductivity
CaSi2: phase transitions and superconductivity

Frozen-in B1g phonon: trigonal structure due to instability of bands

Trigonal MgB2

casi 2 instability of bands sp 2 sp 3
CaSi2: instability of  bands; sp2  sp3

Large splitting at EF upon distortion




Amplitude of trigonal distortion vs pressure and band filling

Lowered frequencies in SC MgB2. CaBeSi?


 bands at EF

intercalate graphite cac 6 tc 11 5 k
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)

cac 6

 Amount of Ca contribution



C  FS

phonons in cac 6 21 modes
Phonons in CaC6: 21 modes

Very high frequencies but also low frequency branches


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


270 GPa

Aluminium under pressure……

Bonds get stiffer, frequencies higer …Al becomes a normal metal


Lithium is a superconductor under pressure





… … …






Electron states of Li and K under pressure

Charge on d states


27 GPa


Charge on p states

30 GPa


Phonon softening and

lattice instability


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



Imaginary frequency: instablility


Electron-Phonon Coupling

Pressure 

Stiffer bonds (higher ’s) but higher coupling at low 

  • 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
A15 Compounds

Nb3Sn Tc=18 K

it could be a Multigap SC

Guritanu PRB 70

184526 (2004)

lattice distortions in nb 3 sn

Free-energy of cubic and tetragonal

Lattice distortions in Nb3Sn



Softening of elastic constant

Softening of optical phonon mode

band structure of nb 3 sn
Band structure of Nb3Sn

Large peak at EF

concepts in eliashberg theory
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 superconductors54
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 mgb 2
Electronic properties of Al-doped MgB2


x = 0

x = 0.25

x = 0.33

x = 0.5

spectral function of nb 3 sn from tunnelling
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

However, the impurity potential being static, V(r, t ), we still have stationary states:

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

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
Impurities: experiments

Tc proportional to the low temperature resistivity, related to impurities induced by irradiation.

magnetic impurities gorkov abrikosov theory
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 