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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
[email protected]://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