1. SUPERCONDUCTIVITY IN SYSTEMS WITH STRONG CORRELATIONS
N.M.Plakida
Joint Institute for Nuclear Research,
Dubna, Russia
Dubna,

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 σ t i,j c+i,σ c j,σ m
Strong corelations → unconventional metals,
non-Fermi liquid
Hubbard model
H = ∑i,j σ t i,j c+i,σ c j,σ
+ U ∑i ni ↑ ni ↓
where U > W U

4. Weak correlations
Strong correlations 1 2 3 2
Spin exchange 3 4

5. One-band metal e1 Weak corelations → conventional metals, Fermi gas j

6. W e2 m
t12 e1 i j
Strong corelations → two Hubbard subabnds:
inter-subband hopping

7. -ws ws W m e2 e1 i j
and intra - subband hopping

8. Tc as a function of doping (oxygen O2– or fluorine F1– )
O  doping atom
0.12
0.24 Hg
CuO2 Ba
Tcmax = 96 K
Structure of Hg-1201
compound ( HgBa2CuO4+δ )
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 plainpx py
dx2-y2
Undoped materials:
one hole per CuO2 unit cell.
According to the band theory:
a metal with half-filled band,
but in experiment:
antiferromagnetic 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 pdσ conduction band: half-filled antibonding d(x2 –y2) - 2p(x,y) band

11. EFFECTIVE HUBBARD p-d MODELpx py
dx2-y2 ed
ed+ ep
2ed+Ud e1 e2 D
Model for CuO2 layer:
Cu-3d (εd) and O-2p (εp) states,
with Δ = εp − εd ≈ 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 = εd - μ → one hole d - like state: l σ >
E2 = 2 E1 + Δ → two hole (p - d) singlet state: l ↑↓ >
We introduce the Hubbard operators for these states:
Xiαβ = l iα > < iβ l with
l α > = l 0 >, l σ >= l ↑ >, l ↓ >, and l 2 >= l ↑↓ >
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:
Xi0σ = ci σ (1 – n i – σ) , Xi σ 2 = ci – σ n i σ
Commutation relations for the Hubbard operators:
[Xiαβ , Xj γμ ]+(−) = δi j δ β γ Xiαμ
Here anticommutator is for the Fermi-like operators as Xi0σ, Xi σ 2,
and commutator is for he Bose-like operators as Xiσ σ, Xi 22,
e.g. the spin operators:
Siz = (1/2) (Xi++ – Xi – , – ), S i+ = Xi+ –, Si– = Xi– + ,
or the number operator
Ni = (Xi Xi – –) 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 t12 e1 i je1 i j
All electrons (holes) are paired in the conduction band. Estimate in WCA gives for Tcex :

20. Self-energy in the Hubbard model
, where Xj Xm
tij
tml Bi Bl
Gjm
cil ≈
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
-ws ws W m e2 e1 i j
Estimate in WCA gives for Tcsf :

23. Equation for the gap and Tc in WCA
The AFM static spin susceptibility
Normalization condition:
where ξ ― short-range AFM correlation length,
ωs ≈ J ― cut-off spin-fluctuation energy.

24. Estimate for Tc in the weak coupling approximation
have:

25. ξ = 3, J = 0.4 teff, ↑ teff ≈ 0.14 tpd ≈ 0.2 eV, Parameters:
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:
Δpd / tpd = 2, ωs / tpd = 0.1,
ξ = 3, J = 0.4 teff,
teff ≈ 0.14 tpd ≈ 0.2 eV,
tpd = 1.5 eV ↑
δ=0.13
Fig.1. Tc ( in teff units):
(i)~spin-fluctuation pairing,
(ii)~AFM exchange pairing ,
(iii)~both contributions

26. ∆(kx, ky) ~∆ (coskx - cosky)
Unconventional d-wave pairing:
∆(kx, ky) ~∆ (coskx - cosky)
Fig ∆(kx, ky)
( 0 < kx, ky < π)
at optimal doping δ ≈ 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 ≈ – 1.35·10 3 (K /Å), or d ln Tc / d ln a ≈ – 50
[ Lokshin et al. PRB 63 (2000) ]
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. For conventional, electron-phonon superconductors,
d Tc / d P < 0 , e.g., for MgB2, d Tc / d P ≈ – 1.1 K/GPa,
while for cuprates superconductors, d Tc / d P > 0
Isotope shift: 16 O → 18 O
Isotope shift of TN = 310K for La2CuO4 , Δ TN ≈ −1.8 K
[ G.Zhao et al., PRB 50 (1994) 4112 ]
and αN = – d lnTN /d lnM ≈ – (d lnJ / d lnM) ≈ 0.05
Isotope shift of Tc : αc = – d lnTc / d lnM =
= – (d lnTc / dln J) (d lnJ/d lnM ) ≈ (1/ Vex) αN ≈ 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 >> 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, ω) space in the strong coupling limit.

30. Comparison with t-J model
In the limit of strong correlations, U >> 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: Xi0σ = ci σ (1 – n i – σ)

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. 1. Spectral functions A(k, ω)
Numerical results
1. Spectral functions A(k, ω)
Fig.1. Spectral function for the t-J model in the symemtry direction
Γ(0,0) → Μ(π,π) at doping δ = 0.1 (a) and δ = 0.4 (b) .

33. 2. Self-energy, Im Σ(k, ω)
Fig.2. Self-energy for the t-J model in the symemtry direction
Γ(0,0) → Μ(π,π) at doping δ = 0.1 (a) and δ = 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 < kx, ky < π) at doping δ = 0.1 (a) and δ = 0.4 (b) .

35. 4. Fermi surface and the gap function Φ(kx, ky)
Fig.4. Fermi surface (a) and the gap Φ(kx, ky) (b) for the t-J model in the quarter of BZ (0 < kx, ky < π) at doping δ = 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 >> 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, ω) space in the strong coupling limit.

37. Electronic structure of cuprates: strong electron correlations Ud > Wpd
At half-filling 3d(x2 –y2) - 2p(x,y) band splits into UHB and LHB Insulator with the charge-transfer gap: Ud > Δ = εp − εd

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, (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 COPPER–OXIDES THE ONLY HIGH–Tc SUPERCONDUCTORS with Tc > 100 K?
Cu 2+ in 3d9 state has the lowest 3d level in transition metals
with strong Coulomb correlations Ud >Δpd = εp – εd.
They are CHARGE-TRANSFER INSULATORS
with HUGE super-exchange interaction J ~ 1500 K —>
AFM long–range 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
2ed+Ud
Model for CuO2 layer:
Cu-3d ( εd ) and
O-2p (εp ) states
Δ = εp − εd ≈ 2 tpd
In terms of O-2p Wannier states
ed+ ep e2 ed D e1