ANTIFERROMAGNETIC EXCHANGE AND SPIN-FLUCTUATION PAIRING IN CUPRATES
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1 ANTIFERROMAGNETIC EXCHANGE AND SPIN-FLUCTUATION PAIRING IN CUPRATES N.M.Plakida Joint Institute for Nuclear Research, Dubna, Russia CORPES, Dresden,
2 Publications and collaborators: N.M. Plakida, L. Anton, S. Adam, and Gh. Adam, Exchange and Spin-Fluctuation Mechanisms of Ssuperconductivity 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) S. Krivenko, A.Avella, F. Mancini, N.M. Plakida, SCBA within composite operator method for the Hubbard model Physica B, in press (2005)
3 Outline Two mechanisms of AFM pairing Effective p-d Hubbard model AFM exchange pairing in MFA Self-energy corrections in SCBA Results for T c and SC gaps T c (a) and isotope effect Comparison with t-j model
4 O3 Hg CuO 2 Ba T c max = 96 K Structure of Hg-1201 compound ( HgBa 2 CuO 4+δ ) After A.M. Balagurov et al. Tc as a function of doping (oxygen or fluorine) Abakumov et al. Phys.Rev.Lett. (1998)
5 WHY ARE COPPER OXIDES THE ONLY HIGH T c SUPERCONDUCTORS with T c > 100 K? Cu 2+ in 3d 9 state has the lowest 3d level in transition metals with strong Coulomb correlations Ud > pd = εp ε d. They are CHARGE-TRANSFER INSULATORS with HUGE super-exchange interaction J ~ 1500 K > AFM long range order with high T N = K Strong coupling of doped holes (electrons) with spins Pseudogap due to AFM short range order High-T c superconductivity?
6 EFFECTIVE HUBBARD p-d MODEL Model for CuO 2 layer: Cu-3d ( ε d ) and O-2p (ε p ) states = ε p ε d 2 t pd ~ 3 ev In terms of O-2p Wannier states 2ε d +U d ε d + ε p ε d ε 2 ε 1
7 Cell-cluster perturbation theory and Hubbard operators Exact diagonalization of the unit cell Hamiltonian H i (0) gives new eigenstates: E 1 = ε d -µ one hole d - like state: l σ > E 2 = 2 E 1 + two hole (p - d) singlet state: l > We introduce the Hubbard operators for these states: X αβ i =l iα > < iβ l with l α > = l 0 >, l σ >, l > Hubbard operators rigorously obey the constraint: X 00 i + X i + X i + X 22 i = 1 only one quantum state can be occupied at any site i. In terms of the projected Fermi operators: 0σ X i ci σ (1 n i σ), X σ2 i c i σ n i σ αβ Commutation relations: [X i, X γδ αδ γβ i ] ± = δ βγ X i ± δ δα X i
8 The two-subband effective Hubbard model reads: Kinematic interaction for the Hubbard operators:
9 Dyson equation for GF in the Hubbard model We introduce the (4x4) matrix Green Functions:
10 Equations of motion for the matrix GF are solved within the Mori-type projection technique: The Dyson equation reads: with the self-energy as the multi-particle GF:
11 Mean-Field approximation: zero order GF where frequency matrix: with frequency matrix of the normal state QP spectrum: Ω 2 (q) for UHD and Ω 1 (q) LHB matrix of anomalous correlation functions: e.g., SC gap for singlets (UHB)
12 Normal state MFA GF: one-hole Ω D (q) and two-hole Ω ψ (q) spectra Spectral weights: Hybridization: Dispersion in n.n. and n.n.n. approximation: where 1/ = 4 / n 2 Spin-correlation functions: < 0, > 0,
13 Spin-correlation functions gives a strong renormalization for spectra Normalization condition defines = at q = for a given AF correlation length the fitting parameter n = 1» a: n = 1.2 = a: = , = , = 0.202, = 0.03, n = 1.4 = = 0, no AF corelations
14 2-pole approximation for the effective Hubbard model: spectra and DOS n=1, ξ»a n=1.2, ξ = a n=1.4, χ s = 0
15 Self-energy corrections to the 2-pole approximation in the SCBA for the Hubbard model QMC LHB UHB E F Spectral density A(k,ω) U = 8 t, n = 0.75, T = 0.5 t E F Density of states A(ω) Krivenko et al. Physica B (2005)
16 Mean-Field approximation for the gap function Frequency matrix: where matrix of anomalous correlation functions anomalous correlation function SC gap for singlets in UHB PAIRING at ONE lattice site but in TWO subbands
17 Equation for the pair correlation Green function gives: For the singlet subband (UHB) : µ and E 2 E 1 : Gap function for the singlet subband in MFA : is equvalent to the MFA in the t-j model
18 ε 2 t 12 AFM exchange pairing µ W 0 ε 1 i j All electrons (holes) are paired in the conduction band. Estimate in WCA gives for T c ex :
19 Self-energy in the Hubbard model, where SCBA: B i t ij X j B j X m t lm t ij χ il G jm t lm Self-energy matrix:
20 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.
21 Spin-fluctuation pairing W ε 2 ε 1 i j µ ω s ω s 0 Estimate in WCA gives for T c sf :
22 Equation for the gap and T c in WCA The AFM static spin susceptibility Normalization condition: where ξ short-range AFM correlation length, ω s J cut-off spin-fluctuation energy.
23 Estimate for T c in the weak coupling approximation Effective spin-fluctuation pairing constant V s enhanced by exchange
24 T c (a) and pressure dependence For mercury compounds, Hg-12(n-1)n, experiments show dt c / da (K /Å), or d ln T c / d ln a 50 [ Lokshin et al. PRB 63 (2000) ] For exchange pairing Hg-1201F T c E F exp ( 1/ V ex ), V ex =J N(0), we get: d ln T c / d ln a = (d ln T c / d ln J) (d ln J / d ln a) 14 (1/ V ex ) 50, where V ex 0.3 and J t pd 4 ~ 1/a 14
25 For conventional, electron-phonon superconductors, d Tc / d P < 0, e.g., for MgB 2, 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 T N = 310K for La 2 CuO 4, T N 1.8 K [ G.Zhao et al., PRB 50 (1994) 4112 ] and α N = d lnt N /d lnm (d lnj / d lnm) 0.05 Isotope shift of T c : α c = d lnt c / d lnm = = (d lnt c / dln J) (d lnj/d lnm ) (1/ V ex ) α N 0.16
26 Equation for the gap and T c in WCA The AFM static spin susceptibility Normalization condition: where ξ short-range AFM correlation length, ω s J cut-off spin-fluctuation energy.
27 NUMERICAL RESULTS 0.13 sf + exch exch sf Parameters: pd / t pd = 2, ω s / t pd = 0.1, ξ = 3, J = 0.4 t eff, t eff 0.14 t pd 0.2 ev, t pd = 1.5 ev Fig.1. Tc ( in t eff units): (i)~spin-fluctuation pairing, (ii)~afm exchange pairing, (iii)~both contributions
28 Unconventional d-wave pairing: (k x, k y )~ (cosk x - cosk y ) FS Fig. 2. (k x, k y ) ( 0 < k x, k y < π) at optimal doping δ 0.13 Large Fermi surface (FS)
29 Comparison with the t-j model The Hamiltonian of the t-j model in X- operators reads: Interband hopping determines the exchange interaction: Matrix Green function for the X-operators: J ij = 4 (t ij ) 2 / where
30 Self-consistent system of equation in SCBA where the interaction is determined by spin- charge- fluctuations Spectral functions for the normal and anomalous GF:
31 Numerical solution of the linearized gap equation Interaction: Model spin susceptibility with parameters: AF cor.length ξ and ω s ~ J
32 Numerical results 1. Spectral functions A(k, ω) Fig.1. Spectral function for the t-j model in the symmemtry direction Γ(0,0) Μ(π,π) at doping: (a) δ = 0.1 (ξ=3), (b) δ = 0.4 (ξ=1).
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 < k x, k y < π) at doping δ = 0.1 (a) and δ = 0.4 (b).
35 4. Fermi surface and the gap function Φ(k x, k y ) Fig.4. Fermi surface (a) and the gap Φ(k x, k y ) (b) for the t-j model in the quarter of BZ (0 < k x, k y < π) at doping δ = 0.1.
36 CONCLUSIONS Superconducting d-wave pairing with high-t c mediated by the AFM superexchange and spinfluctuations 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 T c ~ E F W/2. T c (a) and oxygen isotope shift are explained. The results corresponds to numerical solution to the t-j model in (q, ω) space in strong coupling limit.
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