A1258150353BgsUC

1
SUPERCONDUCTIVITY IN SYSTEMS WITH STRONG
CORRELATIONS
N.M.Plakida Joint Institute for Nuclear
Research, Dubna, Russia
Dubna, 28.08.2006
2
Outline
? Dense matter vs Strong correlations ? Weak
and strong corelations ? Cuprate
superconductors ? Effective p-d Hubbard model
? AFM exchange and spin-fluctuation pairing ?
Results for Tc and SC gaps ? Tc (a) and
isotope effect
3
Dense matter ? strong correlations Weak
corelations ? conventional metals, Fermi
gas
W
H ?i,j s t i,j ci,s c j,s
m
0
Strong corelations ? unconventional
metals,
non-Fermi liquid

Hubbard model H ?i,j s t i,j ci,s c j,s U
?i ni ? ni ? where U gt W
U
4
Weak correlations
Strong correlations
1
2
3
2
Spin exchange
3
4
5
e1
Weak corelations ? conventional metals,
Fermi gas One-band metal
j
i
6
W
e2
m
0
t12
e1
j
i
Strong corelations ? two Hubbard subabnds
inter-subband hopping
7
e2
e1
j
i
and intra - subband hopping
8
O3 ? doping atom
0.12
0.24
Hg
CuO2
Ba
Tcmax 96 K
Structure of Hg-1201 compound ( HgBa2CuO4d
)
Tc as a function of doping (oxygen O2 or
fluorine F1 ) Abakumov et al. Phys.Rev.Lett.
(1998)
After A.M. Balagurov et al.
9
Electronic structure of cuprates CuO2 plain
Undoped materials one hole per CuO2 unit
cell. According to the band theory a metal with
half-filled band, but in experiment antiferromag
netic insulators
Cu3d dx2-y2 orbital and O2p (px , py)
orbitals Cu 2 (3d9) O 2 (2p6)
10
Electronic structure of cuprates Cu 2 (3d 9)
O 2 (2p 6)
Mattheis (1987)
Band structure calculations predict
a broad
pds conduction band half-filled
antibonding 3d(x2 y2) - 2p(x,y)
band
11
EFFECTIVE HUBBARD p-d MODEL
Model for CuO2 layer Cu-3d (ed) and O-2p
(ep) states, with ? ep - ed 3 eV,
Ud 8 eV, tpd 1.5 eV,
12
Cell-cluster perturbation theory

• Exact diagonalization of the unit cell
  Hamiltonian Hi(0) gives new eigenstates
• E1 ed - µ ? one hole d - like state
  l s gt
• E2 2 E1 ? ? two hole (p - d) singlet
  state l ?? gt

We introduce the Hubbard operators for these
states
Xiaß l ia gt lt iß l with l a gt l 0
gt, l s gt l ? gt, l ? gt, and l 2 gt l ??
gt Hubbard operators rigorously obey the
constraint Xi00 Xi?? Xi?? Xi
22 1 ? only one quantum state can be occupied
at any site i.
13
In terms of the projected Fermi operators Xi0s
ci s (1 n i s) , Xi s 2 ci s n i
s
Commutation relations for the Hubbard operators
Xiaß , Xj ?µ (-) di j d ß
? Xiaµ Here anticommutator is for the Fermi-like
operators as Xi0s, Xi s 2, and commutator is
for he Bose-like operators as Xis s, Xi 22,
e.g. the spin operators Siz
(1/2) (Xi Xi , ), S i Xi ,
Si Xi , or the number operator
Ni (Xi Xi ) 2 Xi22
These commutation relations result
in the kinematic interaction.
14
The two-subband effective Hubbard model reads
Kinematic interaction for the Hubbard
operators
15
Dyson equation for GF in the Hubbard model
We introduce the Green Functions
16
Equations of motion for the matrix GF are solved
within the Mori-type projection
technique
The Dyson equation reads
with the exact self-energy as the
multi-particle GF
17
Mean-Field approximation
Frequency matrix
where
anomalous correlation function SC gap for
singlets.
? PAIRING at ONE lattice site but in TWO subbands
18
Equation for the pair correlation Green
function gives
For the singlet subband µ ? and E2 E1
?
Gap function for the singlet subband in MFA
is equvalent to the MFA in the t-J model
19
AFM exchange pairing
W
e2
m
0
t12
e1
j
i
All electrons (holes) are paired in the
conduction band. Estimate in WCA gives for
Tcex
20
Self-energy in the Hubbard model
, where
SCBA
Self-energy matrix
21
Gap equation for the singlet (p-d)
subband
where the kernel of the integral equation in SCBA
defines pairing mediated by spin and charge
fluctuations.
22
Spin-fluctuation pairing
e2
e1
j
i
Estimate in WCA gives for Tcsf
23
Equation for the gap and Tc in WCA
The AFM static spin susceptibility
Normalization condition
where ? ? short-range AFM correlation
length, ?s J ? cut-off spin-fluctuation
energy.
24
Estimate for Tc in the weak coupling
approximation
We have
25
N.M. Plakida, L. Anton, S. Adam, and Gh. Adam,
JETP 97, 331 (2003). Exchange and
Spin-Fluctuation Mechanisms of Superconductivity
in Cuprates.
NUMERICAL RESULTS
Parameters ?pd / tpd 2, ?s / tpd
0.1, ? 3, J 0.4 teff, teff
0.14 tpd 0.2 eV, tpd 1.5 eV
? d0.13
Fig.1. Tc ( in teff units) (i)spin-fluctuation
pairing, (ii)AFM exchange pairing , (iii)both
contributions
26
Unconventional d-wave pairing ?(kx, ky) ?
(coskx - cosky)
Fig. 2. ?(kx, ky) ( 0 lt kx, ky lt p)
at optimal doping d 0.13
FS
Large Fermi surface (FS)
27
Tc (a) and pressure dependence
For mercury compounds, Hg-12(n-1)n, experiments
show dTc / da 1.3510 3 (K /Å), or d ln
Tc / d ln a 50 Lokshin et al. PRB 63
(2000) 64511
From Tc EF exp ( 1/ Vex
), Vex J N(0) , we get d ln Tc /
d ln a (d ln Tc / d ln J) (d ln J / d
ln a) 40 , where J tpd4 1/a14
28
Isotope shift 16 O ? 18 O
For conventional, electron-phonon
superconductors, d Tc / d P lt 0 , e.g., for
MgB2, d Tc / d P 1.1 K/GPa, while for
cuprates superconductors, d Tc / d P gt 0

• Isotope shift of TN 310K for La2CuO4 , ? TN
  -1.8 K
• G.Zhao et al., PRB 50 (1994) 4112
• and aN d lnTN /d lnM (d lnJ / d lnM)
  0.05
• Isotope shift of Tc ac d lnTc / d lnM
• (d lnTc / dln J) (d lnJ/d lnM ) (1/ Vex) aN
  0.16

29
CONCLUSIONS

• Superconducting d-wave pairing with high-Tc
• mediated by the AFM superexchange and
  spin-fluctuations is proved for the p-d Hubbard
  model.
• Retardation effects for AFM exchange are
  suppressed
• ?pd gtgt W , that results in pairing of all
  electrons (holes) with high Tc EF W/2
  .
• Tc(a) and oxygen isotope shift are explained.
• The results corresponds to numerical solution to
  the
• t-J model in (q, ?) space in the strong
  coupling limit.

30
Comparison with t-J model
In the limit of strong correlations, U gtgt t,
the Hubbard model can be reduced to the t-J
model by projecting out the upper band
Jij 4 t2 / U is the exchange energy for the
nearest neighbors, in cuprates J
0.13 eV
In the t-J model the Hubbard operators act only
in the subspace of one-electron states Xi0s
ci s (1 n i s)
31
We consider matrix (2x2) Green function (GF) in
terms of Nambu operators
The Dyson equation was solved in SCBA for the
normal G11 and anomalous G12 GF in the linear
approximation for Tc calculation
where interaction
The gap equation reads
32
Numerical results 1. Spectral functions A(k,
?)
Fig.1. Spectral function for the t-J model in the
symemtry direction G(0,0) ? ?(p,p) at doping
d 0.1 (a) and d 0.4 (b) .
33
2. Self-energy, Im S(k, ?)
Fig.2. Self-energy for the t-J model in the
symemtry direction G(0,0) ? ?(p,p) at doping
d 0.1 (a) and d 0.4 (b) .
34
3. Electron occupation numbers N(k) n(k)/2
Fig.3. Electron occupation numbers for the t-J
model in the quarter of BZ, (0 lt kx, ky lt
p) at doping d 0.1 (a) and d 0.4 (b) .
35
4. Fermi surface and the gap function F(kx, ky)
Fig.4. Fermi surface (a) and the gap F(kx, ky)
(b) for the t-J model in the quarter of BZ (0
lt kx, ky lt p) at doping d 0.1.
36
CONCLUSIONS

• Superconducting d-wave pairing with high-Tc
• mediated by the AFM superexchange and
  spin-fluctuations is proved for the p-d Hubbard
  model.
• Retardation effects for AFM exchange are
  suppressed
• ?pd gtgt W , that results in pairing of all
  electrons (holes) with high Tc EF W/2
  .
• Tc(a) and oxygen isotope shift are explained.
• The results corresponds to numerical solution to
  the
• t-J model in (q, ?) space in the strong
  coupling limit.

37
Electronic structure of cuprates strong electron
correlations Ud gt Wpd
At half-filling 3d(x2 y2) - 2p(x,y) band splits
into UHB and LHB Insulator with the
charge-transfer gap Ud gt ? ep - ed
38
Publications

• N.M. Plakida, L. Anton, S. Adam, and Gh. Adam,
• Exchange and Spin-Fluctuation Mechanisms of
  Superconductivity in Cuprates. JETP 97,
  331 (2003).
• N.M. Plakida , Antiferromagnetic exchange
  mechanism
• of superconductivity in cuprates. JETP
  Letters 74, 36 (2001)
• N.M. Plakida, V.S. Oudovenko,
• Electron spectrum and superconductivity in
  the t-J model at moderate doping.
• Phys. Rev. B 59, 11949 (1999)

39
Outline

• Effective p-d Hubbard model
• AFM exchange pairing in MFA
• Spin-fluctuation pairing
• Results for Tc and SC gaps
• Tc (a) and isotope effect
• Comparison with t-J model

40
WHY ARE COPPEROXIDES THE ONLY
HIGHTc SUPERCONDUCTORS with Tc gt 100 K?

• Cu 2 in 3d9 state has the lowest 3d level in
  transition metals
• with strong Coulomb correlations Ud gt?pd
  ep ed.
• They are CHARGE-TRANSFER INSULATORS
• with HUGE super-exchange interaction J
  1500 K gt
• AFM longrange order with high TN 300 500
  K
• Strong coupling of doped holes (electrons) with
  spins
• Pseudogap due to AFM short range order
• High-Tc superconductivity

41
EFFECTIVE HUBBARD p-d MODEL

2edUd
Model for CuO2 layer Cu-3d ( ed ) and O-2p
(ep ) states ? ep - ed 2 tpd In terms
of O-2p Wannier states
ed ep
e2
ed
D
e1