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.ithttp://www.dsf.unica.it/~sandro/

Outline

•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

•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

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

d bandsNarrow, filled

s bands nearly

parabolic: free-electron

Symbols are from experiments

Band dispersion from Bloch theorem carries the information on chemical

bonding Similarity: bonding &

anti-bonding molecularorbitals

k (r l) k (r)ei k l

k

An interesting material: MgB2

Tc=39.5 K

Isoelectronic to graphite, why so different?

B planes

Mg planes

sp2

Energy bands of MgB2

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

2D bands (weakly dispersed along -A)

k=(kx;ky;) (0,0,kz ) (kx;ky;/c)

s

bonding (px,py)

bonding & antibonding

(pz orbitals)

E l e c t r o n i c p r o p e r t i e s o f MgB2

2-D-bonding bands3-D bands

B

B

BStrong covalent bonds

A

Mg Mg

B

0,0, / ck

B

k 0

-+

++

Dispersion and bonding: bands

-

•Different dispersion along kz: 2D vs 3D

MgB2

Graphite

The presence of cations is crucial to get holes.

holes are the origin of superconductivity

Fermi surface of MgB2

B px and py ( )

B pz ( )

The FS is the iso-energy surface in k-space separating filled and empty states

Second ingredient: Phonons

s atom

cartesian component

l lattice point

Lattice deformation:

detCss ' '(q)

M sM s '

2 (q) 0

Rls R0ls us (q)eiqRl

3Nat phonon branches at each wave vector q

Force constants contain the response of the electrons to ionic displacement: fundamental ingredient

k

M 2Analogy with elementary

mechanics:

First-principles calculations vs experiments

Source of electron-electron attraction

Virtual phonon

qk k’

k+q k’-q

BCS theory: superconducting gap

k Vkk '

k '

2 k '2 k '

2k ' tanh

k '2 k '

2

2kBT

Ek k2 k

2 excitation energies

2hDe 1

k

≈2

Tc 1.14e 1

2kBTc

3.52

Coherence length

Exponential dependence on the coupling

ELIASHBERG theory (1960):

• attractive electron-phonon interaction:

22 ( )d

F

Eliashberg Spectral Function 2F() describes the coupling of phonons to electrons on the Fermi Surface

trTConnection to normal state electrical resistivity :

Pb and MgB2 Eliashberg functions

Pb MgB2

=1.62 Tc=7.2 K =0.87 Tc=39.5 K

Low phonon frequencies Large phonon frequenciesStill, CaC6 has larger and similar but Tc=11.5 K !!!

McMillan Equation

T

c1.2

e 1.04

1 * (10.62 )

represents the Coulomb repulsion and is normally fitted to experimental Tc

N(EF ) I 2

M ph2

N(EF) electronic density of statesI e-ph interaction M nuclear massph average ph. frequency

Exponential dependence

Results of theoretical calculations for elemental superconductors: comparison with experiment

TcT=0 gap at EF

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

Phonon density of states Spectral function 2F()

MgB2 superconductor, AlB2 no

2

2 F( )

d

Comparable phonon DOS, very different 2F()

Phonons in MgB2

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

E2g

B1g

Large coupling of the E2g phonon mode

with hole pockets (band splitting)

E2g=0.075 eV

≈ 1-2 eV !!!

Phonon life-time

As soon as holes disappear with e-doping, superconductivity disappearsThe 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

AlB2 not SCMgB2 SC

Electron doping destroys SC

Kohn anomaly: LiBC, isoelettronic to MgB2 (Pickett)

Stoichiometric compound is a semiconductor

Metallic upon dopingKohn anomalyHigh Tc predicted

Unfortunately not found experimentally

Strong renormalization of phonon frequencies

phon

on f

requ

ency

Kohn anomaly

The electronic screening is discontinuous at 2kF (log singularity in the derivative of the response )

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

2 Fq k

d q

dq

q > 2kF

FS

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)

• stronger in bands due to the

coupling with E2g phonon mode• Experiments show the existence of two gaps: and .

Two band model:experimental evidenceR. S. Gonnelli, PRL 89, 247004 (2002)

Fermi surface

Specific heat: evidence of 2 gaps

Two-gap structure associated with and bands

Tunnellingexperiments

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

CaGa2 CaSi2

CaSi2 becomes Superconductor under pressure, Tc around 14 K

CaGa2-xSixParent structures to MgB2

Tc

CaSi2: phase transitions and superconductivity

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

trT at high T

Trigonal MgB2

CaSi2: instability of bands; sp2 sp3

Amplitude of trigonal distortion vs pressure and band filling

CaSi2

KSi2

Lowered frequencies in SC MgB2. CaBeSi?

Large splitting at EF upon distortion

DOS

CaBeSi

bands at EF

N. Emery et al. Phys. Rev Lett. 95, 087003 (2005)

Intercalate graphite: CaC6 Tc=11.5 K

The highest Tc among intercalated graphite compounds (normally Tc < 1 K)

CaC6

Amount of Ca contribution

FS

Ca 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

Orbital character

k

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. RobbieSupercond. Sci. Technol. 18 (2005) R1–R8

Aluminium under pressure……

270 GPa

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

N (EF ) I 2

M ph2

Alkali metal under high pressure: many phase transitions

hR1

CI16

7

39

42

…

…

…

0 9R

fcc

Lithium is a superconductor under pressure

K

Li

Charge on p states

Charge on d states

27 GPa

30 GPa

Electron states of Li and K under pressure

Phonon dispersion in Li: softening and stiffening

0 GPa

26 GPa

0 GPa

26 GPa

Why?

Phonon softening and lattice instability

Increasing the pressure a lattice instability driven by the Fermi surface nesting increases the electron-phonon coupling

q

Pieces of Fermi surface connected by the same wave-vector q

q

Imaginary frequency: instablility

Orbital character at EF and superconductivity

Li

d character

p character

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 Kit could be a Multigap SC

Guritanu et.al. PRB 70 184526 (2004)

Lattice distortions in Nb3SnFree-energy of cubic and tetragonal

c

a 1

Softening of elastic constant

Softening of optical phonon mode

V3Si

Nb3Sn

Lattice distortions in A15

Band structure of Nb3Sn

Large peak at EF

Concepts in ELIASHBERG theory:

Superconductivity results from the competition

of opposite effects:

1 lnel el FS

F

D

VE

• repulsive Coulomb interaction (Morel Anderson):

The difference between electron (h/EF) and nuclear (1/D) time

scales reduces the coulomb repulsion (retardation)

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)

x = 0

x = 0.25

x = 0.33

x = 0.5

Mg1-xAlxB2

Electronic properties of Al-doped MgB2

Electron-phonon spectral function

2F()

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:

k n

We can form Cooper pairs by time-reversal degenerate states

k , k

n ,n*

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!

Ni

d

d The presence of a static magnetic moment is incompatible with conventional superconductivity