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SUPERCONDUCTIVITY IN SYSTEMS WITH STRONG CORRELATIONS

SUPERCONDUCTIVITY IN SYSTEMS WITH STRONG CORRELATIONS. N.M.Plakida Joint Institute for Nuclear Research, Dubna, Russia. Dubna, 28.08.2006. Outline. ● Dense matter vs Strong correlations ● Weak and strong corelations ● Cuprate superconductors ● Effective p-d Hubbard model

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SUPERCONDUCTIVITY IN SYSTEMS WITH STRONG CORRELATIONS

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  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,jσti,jc+i,σcj,σ m 0 Strong corelations → unconventional metals, non-Fermi liquid Hubbard model H = ∑i,jσti,jc+i,σcj,σ + U ∑i ni ↑ ni ↓ where U > 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. W ws m -ws 0 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 ( 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. dx2-y2 px py Electronic structure of cuprates: CuO2 plain Undoped materials: one hole per CuO2unit cell. According to the band theory: a metal with half-filled band, but in experiment: antiferromagnetic insulators Cu3d dx2-y2orbital 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 pdσconduction band:half-filled antibonding 3d(x2 –y2) - 2p(x,y) band

  11. 2ed+Ud ed+ ep e2 ed D e1 dx2-y2 px py EFFECTIVE HUBBARD p-d MODEL 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), Xiσ2= ci –σn i σ Commutation relations for the Hubbard operators: [Xiαβ,Xj γμ ]+(−) =δi j δβγXiαμ Here anticommutator is for the Fermi-like operators as Xi0σ, Xiσ2, and commutator is for he Bose-like operators as Xiσ σ, Xi22, e.g. thespin operators: Siz = (1/2) (Xi++ –Xi –, – ), S i+ = Xi+–,Si– = Xi–+ , or thenumber 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 modelWe 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 exactself-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. cil Bi Bl ≈ tij Xj tml tij Gjm tml Xm 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. W ws m -ws 0 Spin-fluctuationpairing 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 ξ ― short-range AFM correlation length, ωs≈ J― cut-off spin-fluctuation energy.

  24. Estimate forTc 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: Δpd/ tpd= 2, ωs/ tpd= 0.1, ξ= 3, J = 0.4 teff, teff ≈ 0.14 tpd ≈ 0.2eV, 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. Unconventional d-wave pairing: ∆(kx, ky)~∆ (coskx - cosky) Fig. 2. ∆(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) 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 < 0 , e.g., for MgB2, d Tc / d P ≈ – 1.1 K/GPa, while for cuprates superconductors, d Tc / d P > 0 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 Superconductingd-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. 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 Superconductingd-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, 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 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 HUGEsuper-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

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