1 s+d pairing in orthorhombic phase of copper-oxides N.M. Plakida and V.S. Oudovenko Joint Institute for Nuclear Research, 141980 Dubna, Russia 0 0 Amicroscopical theoryofelectronicspectrumandsuperconductivityisformulatedwithinthetwo-dimensional 0 anisotropict−J modelwitht 6=t andJ 6=J . Renormalization ofelectronicspectrumandsuperconductivity x y x y 2 mediatedbyspin-fluctuationsareinvestigatedwithintheEliashbergequationintheweakcouplingapproximation. n Thegap functionhasd+ssymmetrywiththeextendeds-wavecomponentbeingproportionaltotheasymmetry a ty−tx. Some experimentalconsequences of theobtained results are discussed. J 7 2 Recently the d-wave symmetry of supercon- theory developed by us in [4], both in tetrago- 1 ducting pairing in cuprates was unambiguously nal and orthorhombic phases. The orthorhom- v 1 confirmedbyobservationofhalf-integermagnetic bic (D2h) distortion is taken into account by in- 0 fluxquantum[1]. Inatetragonalphaseofcopper troducingtheasymmetrichoppingparameterst ij 4 oxides the s-wave component must be strongly and the exchange interaction J for the nearest ij 1 suppressed due to on-site Coulomb correlations. neighbors (n.n.) in the form: t = t(1 ±α), 0 x/y For the 2D t-J model it follows from the con- J =J(1±β)wheretheasymmetryparameters 0 x/y 0 straint of no double occupancy on a single site aresupposedtobesmallquantities: α∼β ∼0.1. ′ / given by the identity [2]: For the next n.n., t =t. t ij a The Dyson equations for the matrix Green 1 m haˆi,σaˆi,−σi= Xhaˆk,σaˆ−k,−σi=0. (1) function (GF) in the Nambu notation was ob- N - kx,ky tained by the equation of motion method for the d n for the projected electron operators aˆi,σ = Hubbard operators as described in [4]. For esti- o ai,σ(1−ni,−σ). The anomalous correlation func- mation of the role of orthorhombic deformation c tionhaˆk,σaˆ−k,−σiisproportionaltothegapfunc- we consider here only the weak-coupling approx- v: tion, ∆(k ,k ), multiplied by a positive func- imation. However, to take into account strong x y Xi tionsymmetricinrespecttothe D4h pointgroup electroniccorrelationsinthet-J modelduetore- which defines the symmetry of the Fermi sur- stricted hopping in the singly occupied subband ar face (FS). Therefore to satisfy the condition (1) wewritethesingle-electronspectraldensityinthe the gap function should have a lower symme- form: A(k,ω)≃Zkδ(ω+µ−Ek)+Ainc(ω). The try, e.g. B1g, ”d-wave” symmetry (see, e.g. [3]): quasiparticle weight Zk and the incoherent part ∆d(kx,ky)=−∆d(ky,kx). Ainc(ω)arecou+pl∞edbythe sumruleforthespec- In the orthorhombic phase the FS has a lower tral density: R−∞ dωA(k,ω) = 1− n/2. To fix symmetry,e.g.,D2h,andthecondition(1)canbe the value of Zk we assume that the FS for quasi- fulfilled for a gap function of the same symmetry particles with spectrum Ek obeys the Luttinger (withintheE1g irreduciblerepresentation)which theorem: theaveragenumberofelectronsisequal can be written in a general form (”d+s”): to the number of states in k-space below the chemicalpotential µ: n=(1/N) {exp[E − ∆(k ,k )=∆ (k ,k )+ǫ∆ (k ,k ). (2) Pk,σ k x y d x y s x y µ)/T]+1}−1.¿Fromtheseconditionswehavees- where ∆s(kx,ky) = ∆s(ky,kx) is ”the extended timations: Z ≃(1−n)/(1−n/2) and Ainc(ω)≃ s-wave” component. (n/2)2/(1 − n/2)(W − Γ) where we have sug- In the present paper we calculate supercon- gested that the coherent band lies in the range ′ ducting T for the 2D t-t-J model within the c 2 −Γ≤ω ≤Γwhiletheincoherentbandliesbelow strongenhancement(suppression)ofT (δ)dueto c the coherent band in the range −W ≤ ω ≤ −Γ. change of the FS. In the orthorhombic phase the Bytakingintoaccounttherenormalizationofthe s-wave component with ǫ ≤ −1 (depending on coherent part of the spectral weight by Z we the doping) appears in Eq.(2) shifting 4 nodes of k write the equation for the gap in the weak cou- the gap at the FS from the diagonals k = ±k x y pling approximation in the form: in a tetragonal phase as in Ref. [5]. ∆k = 1 XK(q,k−q)Zq2 ∆q tanh Ωq. (3) 0.012 N 2Ω 2T q q α=0.0 α=0.1 where Ω = [(E − µ)2 + ∆2]1/2 and E ≃ q q q q 0.008 ′ ′ −t [γ(q) + αη(q)] − t γ (k) with renormal- eff eff ized due to strong correlations hopping param- Tc eters and γ(q) = (1/2)(cosq +cosq ), η(q) = x y 0.004 ′ (1/2)(cosq − cosq ),γ (q) = cosq cosq [4]. x y x y K(q,k −q) = {2g(q,k − q)−λ(q,k − q)} with the vertex g(q,k − q) = t(q) − J(k − q)/2 and λ(q,k−q)=g2(q,k−q)χ(k−q). The first term 0.0000.0 0.2 0.4 0.6 0.8 in the vertex, t(q), is due to the kinematical in- δ teraction caused by constraints and the second Figure 1. T (δ) in the tetragonal (bold line) and c one, J(q), is the exchange coupling. They have orthorhombic (dashed line) phases (t′ =0.0). different q-dependence and are effective at dif- To conclude, in the present paper we estimate ferent doping. The spin-fluctuation coupling in the role of orthorhombicdeformationin cuprates λ(q,k −q) is defined by the static spin suscep- ′ within the t-t-J model by solving the equation tibility χ(k − q) for which we used the model (3) for a gap of general symmetry (2) with con- χ(q)=χ0/[1+ξ2(1+γ(q))] where the antiferro- straint of no double occupancy, Eq.(1). The ob- magnetic (AFM) correlation length ξ is a fitting tained dependence T (δ) in Fig.1 can explain a c parameter while χ0 is normalized by the condi- suppressionofT in the orthorhombicphase that tion: 1/N hS S i=(3/4)n. c Pi i i observedinLSCO[6]andanisotropicpressurede- WeperformednumericalsolutionofEq.(3)for pendence of T in YBCO [7]. 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