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

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

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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

sandro.massidda@dsf.unica.it

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 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)

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

k

Band dispersion from Bloch theorem carries the information on chemical bonding

Similarity: bonding & anti-bonding molecular

orbitals

(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)

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

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

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

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

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

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 different2F()

Large coupling of the E2g phonon modewith s hole pockets (band splitting)

wE2g=0.075 eV

≈ 1-2 eV !!!

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 Tcpredicted

Unfortunately not found experimentally

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

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

Tc depends on the largest eigenvalue of the inter- and intra- band coupling constants, nmand 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

CaGa2-xSix

Parent structures to MgB2CaGa2 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)

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

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

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

Free-energy of cubic and tetragonal

Lattice distortions in Nb3SnV3Si

Nb3Sn

Softening of elastic constant

Softening of optical phonon mode

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)

Spectral function of Nb3Sn from tunnelling

Many different results with many different values, ranging from =1.08 to 2.74!

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

Non-magnetic impurities: Anderson theoremIn 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

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

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