A theory of superconductivity in cuprates

A microscopic theory of superconducting pairing mediated by antiferromagnetic (AFM) exchange and spin-fluctuations is developed within the effective p − d Hubbard model for the CuO₂ plane. It is proved that retardation effects for AFM exchange interaction are unimportant and result in pairing of all...

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Datum:	2005
1. Verfasser:	Plakida, N.M.
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Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2005
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spelling	irk-123456789-1210592017-06-14T03:04:57Z A theory of superconductivity in cuprates Plakida, N.M. A microscopic theory of superconducting pairing mediated by antiferromagnetic (AFM) exchange and spin-fluctuations is developed within the effective p − d Hubbard model for the CuO₂ plane. It is proved that retardation effects for AFM exchange interaction are unimportant and result in pairing of all electrons in the conduction band and high Tc proportional to the Fermi energy. The spin-fluctuations caused by the kinematic interaction give an additional contribution to the d-wave pairing. Tc dependence on the hole concentration and lattice constants (or pressure) is studied. Small oxygen isotope shift of Tc is explained. The data are compared with the results for the t − J model. Розроблено мікроскопічну теорію надпровідного спарювання через антиферомагнітний (АФМ) обмін та спін-флуктуації в рамках ефективної p − d моделі Хаббарда для площини CuO₂. Доведено, що запізнюючі ефекти для АФМ обмінної взаємодії є неважливими і приводять до спарювання всіх електронів у зоні провідності та високої Tc, пропорційної до енергії Фермі. Спін-флуктуації, спричинені кінематичною взаємодією, дають додатковий внесок у спарювання d-типу. Досліджується залежність Tc від концентрації дірок та сталої гратки (чи тиску). Пояснено малий зсув Tc від ізотопів кисню. Отримані дані порівнюються з результатами для t − J моделі. 2005 Article A theory of superconductivity in cuprates / N.M. Plakida // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 845–858. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 74.20.-z, 74.20.Mn, 74.72.-h DOI:10.5488/CMP.8.4.845 http://dspace.nbuv.gov.ua/handle/123456789/121059 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A microscopic theory of superconducting pairing mediated by antiferromagnetic (AFM) exchange and spin-fluctuations is developed within the effective p − d Hubbard model for the CuO₂ plane. It is proved that retardation effects for AFM exchange interaction are unimportant and result in pairing of all electrons in the conduction band and high Tc proportional to the Fermi energy. The spin-fluctuations caused by the kinematic interaction give an additional contribution to the d-wave pairing. Tc dependence on the hole concentration and lattice constants (or pressure) is studied. Small oxygen isotope shift of Tc is explained. The data are compared with the results for the t − J model.
format	Article
author	Plakida, N.M.
spellingShingle	Plakida, N.M. A theory of superconductivity in cuprates Condensed Matter Physics
author_facet	Plakida, N.M.
author_sort	Plakida, N.M.
title	A theory of superconductivity in cuprates
title_short	A theory of superconductivity in cuprates
title_full	A theory of superconductivity in cuprates
title_fullStr	A theory of superconductivity in cuprates
title_full_unstemmed	A theory of superconductivity in cuprates
title_sort	theory of superconductivity in cuprates
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2005
url	http://dspace.nbuv.gov.ua/handle/123456789/121059
citation_txt	A theory of superconductivity in cuprates / N.M. Plakida // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 845–858. — Бібліогр.: 19 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT plakidanm atheoryofsuperconductivityincuprates AT plakidanm theoryofsuperconductivityincuprates
first_indexed	2025-07-08T19:07:18Z
last_indexed	2025-07-08T19:07:18Z
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fulltext	Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 845–858 A theory of superconductivity in cuprates N.M.Plakida Joint Institute for Nuclear Research, 141980 Dubna, Russia Received July 11, 2005 A microscopic theory of superconducting pairing mediated by antiferromag- netic (AFM) exchange and spin-fluctuations is developed within the effec- tive p − d Hubbard model for the CuO2 plane. It is proved that retardation effects for AFM exchange interaction are unimportant and result in pairing of all electrons in the conduction band and high Tc proportional to the Fermi energy. The spin-fluctuations caused by the kinematic interaction give an additional contribution to the d-wave pairing. Tc dependence on the hole concentration and lattice constants (or pressure) is studied. Small oxygen isotope shift of Tc is explained. The data are compared with the results for the t − J model. Key words: high-temperature superconductivity, strong electron correlations, Hubbard model, antiferromagnetic exchange interaction, spin-fluctuations PACS: 74.20.-z, 74.20.Mn, 74.72.-h 1. Introduction A unique property of cuprates is their belonging to charge-transfer insulator with a small splitting energy between 3d copper and 2p oxygen levels and large Coulomb correlations in 3d copper states. These result in a huge antiferromagnetic (AFM) superexchange interaction of the order of J ' 1500 K which brings about a long- range AFM order in the undoped regime and causes strong AFM dynamical spin fluctuations in the superconducting state. The AFM spin fluctuations can also be responsible for anomalous normal state properties of cuprates (see, e.g. [1]) and for the superconducting pairing as proposed by Anderson [2]. In in a number of studies of the reduced one-band t-J model (see, e.g. [3–6]) it was demonstrated that the instantaneous AFM exchange interaction mediates the d-wave pairing with a high Tc. However, to prove the AFM pairing mechanism one has to consider the original two-band p− d model for CuO2 layer [7] without reducing the interband hopping to the effective exchange interaction in one subband of the t-J model. In this paper we describe a microscopic theory of superconductivity within the c© N.M.Plakida 845 N.M.Plakida effective p − d Hubbard model [8–10]. By applying the Mori-type projection tech- nique to the matrix Green function in terms of the Hubbard operators, the Dyson equation is derived [11]. It is proved that in the mean-field approximation (MFA) the d-wave superconducting pairing mediated by the interband exchange interacti- on occurs similar to the t-J model. The self-energy is calculated in the non-crossing approximation (or the self-consistent Born approximation) which gives an addition- al contribution to the d-wave pairing mediated by spin-fluctuations caused by the kinematic interaction in the intraband hopping. The results of numerical solution of the gap equation are presented for the superconducting Tc as a function of hole concentration and the superconducting gap as a function of the wave-vector [11]. Two remarkable features for cuprate superconductors which distinguish them from the conventional ones, i.e., the increase of Tc with pressure and small oxygen isotope shift of Tc, are explained [6] as well. These results for the two-band p− d model are compared with calculations for the t-J model [5]. 2. Effective Hubbard model 2.1. Dyson equation We consider the original two-band p− d model for the CuO2 layer [7] where two bonding oxygen orbitals px and py and the copper 3dx2−y2 orbital are taken into account as shown in figure 1. By applying the sell-cluster perturbation theory [8–10] Figure 1. Effective two-band p − d model for CuO2 layer [7]. we can reduce it to the effective two-band Hubbard model with the lower Hubbard subband occupied by one-hole Cu d-like states and the upper Hubbard subband occupied by two-hole p − d singlet states H = ε1 ∑ i,σ Xσσ i + ε2 ∑ i X22 i + ∑ i6=j,σ { t11ij Xσ0 i X0σ j + t22ij X2σ i Xσ2 j + 2σt12ij (X2σ̄ i X0σ j + H.c.) } , (1) where Xnm i = \|in〉〈im\| are the Hubbard operators for the four states n,m = \|0〉, \|σ〉, \|2〉 = \| ↑↓〉, σ = ±1/2 = (↑, ↓) , σ̄ = −σ. Here ε1 = εd−µ and ε2 = 2ε1 +∆ 846 A theory of superconductivity in cuprates where µ is the chemical potential and ∆ = εp − εd is the charge transfer ener- gy (see [8]). The superscript 2 and 1 refers to the singlet and one-hole subbands, respectively. The hopping integrals are given by tαβ ij = Kαβ 2tνij where t is the p − d hybridization parameter and νij are estimated as: ν1 = νj j±ax/y ' −0.14, ν2 = νj j±ax±ay ' −0.02. The coefficients Kαβ < 1 , and for the singlet subband, e.g., we have teff ' K222tν1 ' 0.14t and the bandwidth W = 8teff . If we take the standard parameters, ∆ = 2t ' 3 eV we get for the ratio ∆/W ' 2 which shows that the Hubbard model (1) corresponds to the strong correlation limit. The chemical potential µ depends on the average electron occupation number n = 〈Ni〉 = ∑ σ 〈Xσσ i 〉 + 2〈X22 i 〉, (2) where the number operator is Ni = ∑ σ Xσσ i + 2X22 i . The Hubbard operators en- tering (1) obey the completeness relation X00 i + X↑↑ i + X↓↓ i + X22 i = 1 (3) which rigorously preserves the constraint of no double occupancy of any quantum state \|in〉 at each lattice site i. To discuss the superconducting pairing within the model Hamiltonian (1), we introduce the four-component Nambu operators X̂iσ and X̂† iσ and define the 4 × 4 matrix Green function (GF) [12] G̃ijσ(t − t′) = 〈〈X̂iσ(t) \|X̂† jσ(t′)〉〉, G̃ijσ(ω) = ( Ĝijσ(ω) F̂ijσ(ω) F̂ † ijσ(ω) − Ĝjiσ̄(−ω) ) , (4) where X̂† iσ = (X2σ i X σ̄0 i X σ̄2 i X0σ i ) and Ĝijσ and F̂ijσ are normal and anomalous 2 × 2 matrix components, respectively. By applying the projection technique for equation of motion method for GF (4), we derive the Dyson equation in (q, ω)- representation [11]: ( G̃σ(q, ω) )−1 = ( G̃0 σ(q, ω) )−1 − Σ̃σ(q, ω), G̃0 σ(q, ω) = ( ωτ̃0 − Ẽσ(q) )−1 χ̃, (5) where τ̃0 is the 4× 4 unity matrix and χ̃ = 〈{X̂iσ, X̂ † iσ}〉 . The zero-order GF within the generalized mean field approximation (MFA) is defined by the frequency matrix which in the site representation reads Ẽijσ = Ãijσχ̃ −1, Ãijσ = 〈{[X̂iσ, H], X̂† jσ}〉. (6) The self-energy operator in the Dyson equation (5) in the projection technique method is defined by a proper part (having no single zero-order GF) of the many- particle GF in the form Σ̃σ(q, ω) = χ̃−1〈〈Ẑ(ir) σ \| Ẑ(ir)† σ 〉〉(prop) q,ω χ̃−1. (7) 847 N.M.Plakida Here the irreducible Ẑ-operator is given by the equation: Ẑ (ir) σ = [X̂iσ, H]− ∑ l ẼilσX̂lσ which follows from the orthogonality condition: 〈{Ẑ (ir) σ , X̂† jσ}〉 = 0. The equati- ons (5)–(7) provide an exact representation for the GF (4). However, to calculate it one has to use approximations for the self-energy matrix (7) which describes the finite lifetime effects (i.e., the effects of inelastic scattering of electrons on spin and charge fluctuations). 2.2. Mean-field approximation In the MFA the electronic spectrum and superconducting pairing are described by the zero-order GF in (5). By applying the commutation relations to the Hubbard operators we get for the frequency matrix (6): Ãijσ = ( ω̂ijσ ∆̂ijσ ∆̂∗ jiσ − ω̂jiσ̄ ) . (8) The normal component ω̂ijσ defines quasiparticle spectra Ω1,2(q) for two Hubbard subbands of the model in the normal state which have been studied in detail in [8]. As an example, in figure 2 and figure 3 the dispersion Ω1,2(q) (solid lines) and the density of states (DOS) are shown for the undoped case, n = 1, and for the overdoped case, n = 1.4, respectively. For n = 1 an insulating state is observed with the Fermi level (dotted line) being between the subbands with a dispersion defined by the next nearest neighbour hopping, while for n = 1.4 the Fermi level is in the singlet subband with a dispersion defined by the nearest neighbour hopping. �� ! "� �#�$%&� "� Figure 2. Hole dispersion curves along the symmetry directions (left panel) and the corresponding DOS (right panel) for the half-filled case, n = 1, for the pa- rameters ∆ = 2t ' 3 eV in the model (1) [8]. The anomalous component ∆̂ijσ defines the gap functions for the singlet and one-hole subbands, respectively, (i 6= j): ∆22 ijσ = −2σt12ij 〈X 02 i Nj〉, ∆11 ijσ = −2σt12ij 〈(2 − Nj)X 02 i 〉. (9) Using the definitions of the Fermi annihilation operators: ciσ = X0σ i + 2σX σ̄2 i , we can write the anomalous average in (9) as 〈ci↓ci↑Nj〉 = 〈X0↓ i X↓2 i Nj〉 = 〈X02 i Nj〉 848 A theory of superconductivity in cuprates '( ') * +,+-./0123 0454 3 0* 5 3 045 3 0454 3 063 +7 8 9 : ;: ;9 <= 8 >?@ ABC>DEFEGDHGI@ <J<KLM>GI@ Figure 3. The same as in figure 2 for the overdoped case, n = 1.4 [8]. since other products of the Hubbard operators vanish according to the multiplica- tion rule for the Hubbard operators: Xαγ i Xλβ i = δγ,λX αβ i . Therefore the anomalous correlation functions describe the pairing at one lattice site but in different Hubbard subbands. The same anomalous correlation functions were obtained in MFA for the ori- ginal Hubbard model in [13–15]. To calculate the anomalous correlation functi- on 〈ci↓ci↑Nj〉 in [13,15] the Roth procedure was applied based on a decoupling of the operators on the same lattice site in the time-dependent correlation function: 〈ci↓(t)\|ci↑(t ′)Nj(t ′)〉 . However, the decoupling of the Hubbard operators on the same lattice site is not unique (as has been really observed in [13,15]) and turns out to be unreliable. To escape uncontrollable decoupling, in [14] kinematical restrictions imposed on the correlation functions for the Hubbard operators were used which, however, have not produced a unique solution for superconducting equations either. In our approach we perform a direct calculation of the correlation function 〈X02 i Nj〉 without any decoupling by writing the equation of motion for the cor- responding commutator GF Lij(t − t′) = 〈〈X02 i (t) \| Nj(t ′)〉〉 as follows: (ω − ε2) Lij(ω) ' 2δij〈X 02 i 〉 + ∑ m6=i,σ 2σ t12im { 〈〈X0σ̄ i X0σ m \|Nj〉〉ω−〈〈Xσ2 i X σ̄2 m \|Nj〉〉ω } , (10) where we neglected the intraband hopping \|tαα im \| � ε2 ' ∆ . After applying the spec- tral theorem and neglecting exponentially small terms of the order of exp(−∆/T ) � 1 , we obtain the following representation for the correlation function at sites i 6= j for the singlet subband in the case of hole doping [11]: 〈X02 i Nj〉 = − 1 ∆ ∑ m6=i,σ 2σt12im〈X σ2 i X σ̄2 m Nj〉 ' − 4t12ij ∆ 2σ 〈Xσ2 i X σ̄2 j 〉. (11) The last equation is obtained in the two-site approximation, m = j, usually applied for the t-J model. The identity for the Hubbard operators, X σ̄2 j Nj = 2X σ̄2 j was used as well. This finally permits us to write the gap function in (9) in the case of 849 N.M.Plakida the hole doping as follows: ∆22 ijσ = −2σ t12ij 〈X 02 i Nj〉 = Jij〈X σ2 i X σ̄2 j 〉. (12) This result is similar to the exchange interaction contribution to the pairing in the t- J model with an exchange energy Jij = 4 (t12ij )2/∆. In the case of electron doping, an analogous calculation for the anomalous correlation function of the one-hole subband 〈(2 − Nj)X 02 i 〉 gives ∆11 ijσ = Jij 〈X 0σ̄ i X0σ j 〉 for the gap function. Therefore, we may conclude that the anomalous contributions to the zero-order GF (5) are just the conventional anomalous pairs of quasi-particles. Their pairing in MFA is mediated by the exchange interaction which has been studied in the t- J model (see, e.g., [3,5]) and there are no new “composite operator excitations” (“cexons”) proposed in [15]. 2.3. Self-energy The self-energy matrix (7) can be written in the form Σ̃ijσ(ω) = χ̃−1 ( M̂ijσ(ω) Φ̂ijσ(ω) Φ̂† ijσ(ω) − M̂ijσ̄(−ω) ) χ̃−1 , (13) where the 2× 2 matrices M̂ and Φ̂ denote the normal and anomalous contributions to the self-energy, respectively. The self-energy (13) is calculated below in the non-crossing (NCA) or the self- consistent Born approximation (SCBA). In SCBA, the propagation of the Fermi- like and Bose-like excitations in the many-particle GF in (13) are assumed to be independent of each other as shown schematically in figure 4. This approximation Figure 4. Self-consistent Born approximation for the self-energy (14). is given by the decoupling of the corresponding operators in the time-dependent correlation functions for different lattice sites (i 6= j, l 6= m) as follows 〈Bi(t)Xj(t)Bl(t ′)Xm(t′)〉 ' 〈Xj(t)Xm(t′)〉〈Bi(t)Bl(t ′)〉. (14) Using the spectral representation for these correlation functions we get a closed system of equations for the GF (4) and the self-energy components (13) [11]. Below we explicitly write down only the anomalous part of the self-energy for the singlet band which is relevant in the further discussion: Φ22 σ (q, ω) = 1 N ∑ k \|t(k)\|2 +∞∫ −∞ +∞∫ −∞ dω1dω2 ω − ω1 − ω2 1 2 ( tanh ω1 2T + coth ω2 2T ) × χ′′ s (q − k, ω2) { −(1/π)Im[K2 22F 22 σ (k, ω1) − K2 21F 11 σ (k, ω1)] } . (15) 850 A theory of superconductivity in cuprates The kinematic interaction for the nearest and the second neighbors is given by t(k) = t1(k) + t2(k) = 8t [ν1γ(k) + ν2γ ′(k)] , where γ(k) = (1/2)(cos kx + cos ky) and γ′(k) = cos kx cos ky . The pairing interaction is mediated by spin-fluctuations defined by the susceptibility χ′′ s (q, ω) = −(1/π)Im〈〈Sq \| S−q〉〉ω+iδ which comes from the bosonic correlation functions 〈Bi(t)Bl(t ′)〉 in (14). For the hole doped case, at frequencies \|ω, ω1\| � ωs � W close to the Fermi surface (FS) ( ωs 6 J is a characteristic spin-fluctuation energy) we can use the weak coupling approximation (WCA) to calculate the first term in the self-energy (15). The contribution from the second term F 11 σ (k, ω1) is rather small since the one-hole band lies below the FS at the energy of the order of ∆ � W . Neglecting it and taking into account the contribution from the exchange interaction in MFA (12), we arrive at the following equation for the superconducting gap in the singlet subband: Φ22(q) = 1 N ∑ k [ J(k − q) − K2 22 λ(k,q − k) ] Φ22(k) 2E2(k) tanh E2(k) 2T , (16) where λ(k,q − k) = \|t(k)\|2χs(q − k, ω = 0) > 0 . The quasiparticle energy in the singlet band is given by E2(k) = [Ω2(k)2+Φ22(k)2] , where Ω2(k) is the quasiparticle energy in the normal state as shown in figure 3. Similar considerations hold true for an electron doped system, n 6 1 when the chemical potential lies in the one- hole band, µ ' 0. In that case, the WCA equation for the gap Φ11(q) is quite similar to (16). 3. Numerical results and discussion To solve the gap equation (16) we used the following model for the static spin- fluctuation susceptibility: χs(q, 0) = χ0 1 + ξ2[1 + γ(q)] , (17) where ξ is the AFM correlation length and the constant χ0 = 3(2−n)/(2πωsC1) with C1 = (1/N) ∑ q{1 + ξ2[1 + γ(q)]}−1 is defined from the normalization condition: (1/N) ∑ i〈SiSi〉 = (3/4)(1 − \|1 − n\|). Let us first estimate the superconducting transition temperature Tc by solving the gap equation (16) for a model d-wave gap function Φ22(q) = ϕd (cos qx − cos qy) ≡ ϕd η(q) in the standard logarithmic approximation in the limit of weak coupling. Integrating both sides of (16) over q multiplied by η(q) results in the following equation for Tc: 1 = 1 N ∑ k [ J η(k)2 + λs (4γ(k))2η(k)2 ] 1 2Ω2(k) tanh Ω2(k) 2Tc , (18) where λs ' t2eff/ωs . For the exchange interaction mediated by the interband hopping with large energy transfer ∆ � W the retardation effects are negligible which results in the coupling of all electrons in a broad energy shell of the order of the bandwidth W and high Tc [6]: Tc ' √ µ(W − µ)exp(−1/λex), (19) 851 N.M.Plakida where λex ' J N(δ) is an effective coupling constant for the exchange interaction J and the average density N(δ) of electronic states for doping δ. By taking into account both contributions we can write the following estimation for Tc: Tc ' ωs exp(− 1 λ̃sf ), λ̃sf = λsf + λex 1 − λex ln(µ/ωs) , (20) where λsf ' λs N(EF) is the coupling constant for the spin-fluctuation pairing. By taking µ = W/2 ' 0.35 eV, ωs ' J ' 0.13 eV and λsf ' λex = 0.2 for estimation we get λ̃sf ' 0.2 + 0.25 = 0.45 and Tc ' 160 K, while only the spin-fluctuation pairing gives T 0 c ' ωs exp(−1/λsf) ' 10 K. The results of numerical solution of the gap equation (16) are shown in figure 5 for the superconducting transition temperature Tc(δ) [11]. The following parameters are used: ξ = 3, J = 0.4teff , ωs = 0.15 eV and teff = K222tν1 ' 0.2 eV. The maximum 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 T c δ=n-1 (i) (ii) (iii) Figure 5. Left panel: superconducting Tc(δ) (in units of teff ' 0.2 eV) for (i) spin- fluctuation interaction (solid line), (ii) exchange interaction (dashed line), (iii) for the both contributions (dotted line). Right panel: wave-vector dependence of the gap function Φ22(k) over the first quadrant of the BZ at optimum doping (δ = 0.13). The circles plot the Fermi surface. The +/− denote gap signs inside the octants [11]. Tc ∼ 280 K (dotted line) is achieved for the chemical potential µ = EF ' W/2 at the optimal doping δopt ' 0.12. The spin-fluctuation interaction produces much lower Tc (solid line) since it couples the holes in a narrow energy shell, ωs � EF , near the Fermi surface (FS). This interaction is rather weak at the FS close to the AF zone boundary along the lines \|kx\| + \|ky\| = π where the main contribution coming from the nearest neighbor hopping vanishes: t1(k) ∝ γ(kx, \|ky\| = π − \|kx\|) = 0. We can confirm the AFM pairing mechanism by considering the Tc dependence on pressure or lattice constants. While in electron-phonon superconductors, Tc decreas- es under pressure, in cuprates, Tc increases with compression of the in-plane lattice constant a. In particular, in mercury superconductors dTc/da ' −1.35×103 K/Å [16] 852 A theory of superconductivity in cuprates and for Hg–1201 compound we get d lnTc/d ln a ' −50. From (19) we get an esti- mate: d ln Tc d ln a ' d ln Tc d ln J d ln J d ln a ' − 14 λ ' −47 (21) which is quite close to the experimentally observed one. Here we use λ = JN(δ) ' 0.3 and take into account that for the exchange interaction we can use an estimate J(a) ∝ t4pd where tpd ∝ 1/(a)7/2 for the p − d hybridization [17]. Concerning an oxygen isotope effect in cuprates, on substituting the 18O oxygen for 16O, we can also estimate it from (19). By using the experimentally observed isotope shift for the Néel temperature in La2CuO4 [18]: αN = −(d ln TN/d ln M) ' (d ln J/d ln M) ' 0.05 we obtain αc = − d ln Tc d ln M = − d ln Tc d ln J d ln TN d ln M ' αN λ ' 0.16 (22) for λ ' 0.3 which is close to experiments: αc = −d ln Tc/d ln M 6 0.1 . 4. Comparison with the t -J model Now we compare the results for the original two-band p − d model for CuO2 layer (1) with the calculations for the t-J in [5]. In that paper, a full self-consistent numerical solution for the normal and anomalous GF in the Dyson equation was performed in the strong-coupling limit allowing for the quasiparticle renormaization and finite life-time effects caused by the self-energy operators which were neglected in the above calculations for the Hubbard model. In the limit of strong correlations the interband hopping in the model (1) can be excluded by perturbation theory which results in the effective t-J model Ht−J = − ∑ i6=j,σ tijX σ0 i X0σ j − µ ∑ iσ Xσσ i + 1 4 ∑ i6=j,σ Jij ( Xσσ̄ i X σ̄σ j − Xσσ i X σ̄σ̄ j ) , (23) where only the lower Hubbard subband is considered with the hopping energy tij = −t11ij . The exclusion of the interband hopping results in the instantaneous exchange interaction Jij = 4 (t12ij )2/∆. The superconducting pairing within the model (23) can be studied by considering the matrix GF for the lower Hubbard subband in terms of the Nambu operators: Ψiσ and Ψ+ iσ = (Xσ0 i X0σ̄ i ): Ĝij,σ(t − t′) = 〈〈Ψiσ(t)\|Ψ+ jσ(t′)〉〉, Ĝijσ(ω) = Q ( G11 ijσ(ω) G12 ijσ(ω) G21 ijσ(ω) G22 ijσ(ω) ) . (24) Here we introduced the Hubbard factor Q = 1 − n/2 depending on the average number of electrons n = ∑ σ〈X σσ i 〉. By applying the projection technique as described above we get the Dyson equa- tion which can be written in the Eliashberg notation as Ĝσ(k, ω) = Q ωZσ(k, ω)τ̂0 + (Eσ(k) + ξσ(k, ω) − µ̃)τ̂3 + Φσ(k, ω)τ̂1 (ωZσ(k, ω))2 − (Eσ(k) + ξσ(k, ω) − µ̃)2− \| Φσ(k, ω) \|2 , (25) 853 N.M.Plakida where τ̂i are the Pauli matrices. The quasiparticle energy Eσ(k) in the normal state and the renormalized chemical potential µ̃ = µ − δµ are calculated in the MFA as discussed above (for details see [5]). The frequency-dependent functions ω(1 − Zσ(k, ω)) = 1 2 [Σ11 σ (k, ω) + Σ22 σ (k, ω)], ξσ(k, ω) = 1 2 [Σ11 σ (k, ω) − Σ22 σ (k, ω)] are defined by the normal components of the self-energy Σ22 σ (k, ω) = −Σ11 σ̄ (k,−ω) . The gap function is specified by the equation: Φσ(k, ω) = ∆σ(k) + Σ12 σ (k, ω) , ∆σ(k) = 1 NQ ∑ q J(k − q)〈X0σ̄ −qX 0σ q 〉 . (26) The self-energy is calculated in SCBA (14) as in the Hubbard model: Σ11(12) σ (k, ω) = 1 N ∑ q g2(q,k − q) +∞∫ −∞ +∞∫ −∞ dzdΩ ω − z − Ω 1 2 ( tanh z 2T + coth Ω 2T ) × A11(12) σ (q, z) [ −(1/π)ImD±(k − q, Ω + iδ) ] , (27) where the interaction g(q,k − q) = t(q)− 1/2 · J(k − q) and the spectral densities are defined by the corresponding GF: A11(12) σ (q, z) = − 1 π ImG11(12) σ (q, z + iδ). (28) The electron-electron interaction is caused by the spin-charge fluctuations defined by the boson-like commutator GF: D±(q, Ω) = 〈〈S(q) \| S(−q)〉〉Ω ± 1/4〈〈n(q) \| n(−q)〉〉Ω as in the Hubbard model. Figure 6. Spectral density A11 σ (q, ω) (28) (left panel) and ImΣ11 σ (k, ω + iδ) (27) (right panel) along the symmetry direction M(π, π) → Γ(00) at hole concentra- tion δ = 0.1. Energy ω is measured in units of t, with J = 0.4 t [5]. As we see, the equation for the self-energy (27) is similar to (15) obtained for the Hubbard model if we disregard in the latter the small contribution from the second 854 A theory of superconductivity in cuprates subband ∝ F 11 σ (k, ω1). However, contrary to the gap equation (16) in the WCA for the Hubbard model, for the t-J model in the equation (26) the frequency-dependent self-energy contribution Σ12 σ (k, ω) (27) is taken into account. Moreover, in [5] for the t-J model a full self-consistent solution for the normal GF G11 σ (k, ω) in (25) and for the corresponding self-energy Σ11 σ (k, ω) (27) was performed. The results of the calculations are shown in figure 6 in the left panel for the spectral density A11 σ (q, ω) (28) and in the right panel for the ImΣ11 σ (k, ω + iδ) along the symmetry direction M(π, π) → Γ(00) in the BZ. These results for the hole concentration δ = 0.1 and the AFM correlation length ξ = 3 in the model spin susceptibility (17) demonstrate quasiparticle-like peaks only in the vicinity of the Fermi level and anomalous behavior for the self-energy ImΣ11 σ (k, ω+iδ) ∝ ω close to the Fermi level. The occupation number N(k) = (1/Q)〈Xσσ k 〉 shown in the left panel of figure 7, reveals only a small drop at the Fermi level which is generic for strongly correlated systems (the calculations are done at finite temperature and the character of the drop cannot be disclosed). Figure 7. Left panel: occupation numbers N(k) = (1/Q)〈Xσ0 k X0σ k 〉. Right panel: Tc(δ) for the AFM correlation length ξ = 1 (full line) and ξ = 3 (dashed line) [5]. The superconducting Tc was calculated from a linearized gap equation which was solved by direct diagonalization in (k, ω)-space: Φσ(k, iωn) = T N ∑ q ∑ m {J(k − q) + λ12(q,k − q \| iωn − iωm)} × G11 σ (q, iωm)G11 σ̄ (q,−iωm)Φσ(q, iωm), (29) where the interaction function λ12(q,k − q \| iων) = g2(q,k − q)D−(k − q, iων) and the Matsubara frequencies iωn = iπT (2n + 1) were introduced. The doping dependence of superconducting Tc(δ) is shown in the right panel of figure 7 for the AFM correlation length ξ = 1 (full line) and ξ = 3 (dashed line). Also the eigenfunctions Φσ(k, iωn) of the equation (29) were determined which unambiguously demonstrated the d-wave character of superconducting pairing (for details see [5]). By comparing the Tc(δ) dependence for the Hubbard model: figure 5 (left panel) 855 N.M.Plakida with Tmax c ∼ 280 K, and for the t-J model: figure 7 (right panel) with Tmax c ∼ 180 K, we observe a strong reduction of Tmax c in the latter model due to taking into consideration a large contribution from the ImΣ11 σ (k, ω) (see figure 6, right panel). At the same time, the large value of the δopt ' 0.33 in the t-J model in comparison with the experimentally observed δopt ' 0.16 and δopt ' 0.12 in the Hubbard model shows that the t-J model is incapable of properly reproducing the doping dependence of Tc since in the model the weight transfer between the Hubbard subbands under doping is neglected. To conclude, the present investigations provide a microscopic theory for the superconducting pairing mediated by the AFM exchange interaction and spin-fluc- tuations induced by the kinematic interaction, characteristic of the Hubbard model. The singlet dx2−y2-wave superconducting pairing was proved both for the original two-band p − d Hubbard model and for the reduced effective one-band t-J model. These mechanisms of superconducting pairing are not observed in the fermionic models (for a discussion, see Anderson [19]) and turn out to be generic for cuprates. We believe that the proposed superconducting pairing is the relevant mechanism of high-temperature superconductivity in copper-oxide materials. 856 A theory of superconductivity in cuprates References 1. Plakida N.M. High-Temperature Superconductivity. Springer-Verlag, Berlin, Heidel- berg, 1995. 2. Anderson P.W., Science, 1987, 235, 1196; Anderson P.W. The Theory of Supercon- ductivity in the High-Tc Cuprates. Princeton University Press, Princeton, 1997. 3. Plakida N.M., Yushankhai V.Yu., Stasyuk I.V., Physica C, 1989, 160, 80; Yushankhai V.Yu., Plakida N.M., Kalinay P., Physica C, 1991, 174, 401. 4. Izyumov Yu.A., Letfulov B.M., Intern. J. Modern Phys. B, 1992, 6, 321. 5. Plakida N.M., Oudovenko V.S., Phys. Rev. B, 1999, 59, 11949. 6. Plakida N.M., JETP Letters, 2001, 74, 36. 7. Emery V.J., Phys. Rev. Lett., 1987, 58, 2794; Varma C.M., Schmitt-Rink S., Abra- hams E., Solid State Commun., 1987, 62, 681. 8. Plakida N.M., Hayn R., Richard J.-L., Phys. Rev. B, 1995, 51, 16599. 9. Feiner L.F., Jefferson J.H., Raimondi R., Phys. Rev. B, 1996, 53, 8751. 10. Yushankhai V.Yu., Oudovenko V.S., Hayn R., Phys. Rev. B, 1997, 55, 15562. 11. Plakida N.M., Anton L., Adam S., Adam Gh., ZhETF, 2003, 124, 367 (in Russian); [JETP, 97, 331]. 12. Zubarev D.N., Usp. Fiz. Nauk, 1960, 71, 71 (in Russian); [Sov. Phys. Usp., 3, 320]. 13. Beenen, J., Edwards D.M., Phys. Rev. B, 1995, 52, 13636. 14. Avella A., Mancini F., Villani D., Matsumoto H., Physica C, 1997, 282–287, 1757; Di Matteo T., Mancini F., Matsumoto H., Oudovenko V.S., Physica B, 1997, 230– 232, 915. 15. Stanescu, T.D., Martin, I., Phillips Ph., Phys. Rev. B, 2000, 62, 4300. 16. Lokshin, K.A., Pavlov D.A., Putilin, S.N. et al., Phys. Rev. B, 2001, 63, 064511. 17. Harrison W.A. Electronic structure and the properties of solds. W.H.Freeman and Company, San Francisco, 1980. 18. Zhao G.-M., Singh K.K., Morris D.E., Phys. Rev. B, 1994, 50, 4112. 19. Anderson P.W., Adv. in Physics, 1997, 46, 3. 857 N.M.Plakida Теорія надпровідності в купратах Н.M.Плакіда Об’єднаний інститут ядерних досліджень, 141980 Дубна, Росія Отримано 11 липня, 2005 Розроблено мікроскопічну теорію надпровідного спарювання через антиферомагнітний (АФМ) обмін та спін-флуктуації в рамках ефек- тивної p − d моделі Хаббарда для площини CuO2. Доведено, що запізнюючі ефекти для АФМ обмінної взаємодії є неважливими і приводять до спарювання всіх електронів у зоні провідності та високої Tc, пропорційної до енергії Фермі. Спін-флуктуації, спричинені кінематичною взаємодією, дають додатковий внесок у спарювання d-типу. Досліджується залежність Tc від концентра- ції дірок та сталої гратки (чи тиску). Пояснено малий зсув Tc від ізотопів кисню. Отримані дані порівнюються з результатами для t − J моделі. Ключові слова: високотемпературна надпровідність, сильні електронні кореляції, модель Хаббарда, антиферомагнітна обмінна взаємодія, спін-флуктуації PACS: 74.20.-z, 74.20.Mn, 74.72.-h 858

A theory of superconductivity in cuprates

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