s+d pairing in orthorhombic phase of copper-oxides

2 Pages·0.07 MB·English

by  N. M. Plakida

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]. Quite a large ”s”-
c
the gap in the form (2) with ∆ (k ,k )=∆η(k)
d x y wavecomponentinthegap(2)isinaccordwitha
and ∆ (k ,k )=∆γ(k). By taking into account
s x y nonzero tunnelling along the c-axis in YBCO [8].
theconstraintofnodoubleoccupancy,Eq.(1),we
estimate the weight ǫ of the s-component. The REFERENCES
criticaltemperatureT (δ) (inunits oft)is shown
c
on Fig.1 in the tetragonal, α = 0.0, (bold line) 1. C. C. Tsuei, et al., Science, 271 (1996) 329;
and orthorhombic, α = 0.1, (dashed line) phases J. R. Kirtley, et al. Science 285 (1999) 1373.
′
forJ =0.4t, ξ =2, t =0.0. SuppressionofT in 2. N.M.Plakidaetal.,PhysicaC160(1989)80.
c
the orthorhombic phase is due to a deformation 3. M. Sigrist et al., Z. Phys. 68 (1987) 9.
of the FS resulting in a less favourable electron 4. N.M. Plakida and V.S. Oudovenko, Phys.
pairing by the AFM spin fluctuations. Increas- Rev. B 59 (1999) 11949.
ing of AFM interaction due to larger J or/and ξ 5. C.O’Donovan and J.P. Carbotte, Phys. Rev.
stronglyenhances T thoughdoes notchangethe B 52 (1995) 4568.
c
shape of the curve. Its maximum at δ ≃ 0.35 is 6. T.Goko,F.NakamuraandT.Fujita,J.Phys.
due to an interplay between the shape of the FS Soc. Jpn. 68 (1999) 3074.
(defined by the quasiparticle spectrum E ) and 7. U.Welpetal.,Phys.RevLet.69(1992)2130.
q
the coherent spectral weight Z2 in Eq.(3). Par- 8. A.G. Kouznetsov, A.G. Sun, B. Chen, et al.,
′
ticularly, for t/t = −0.1 (+0.1) we observed a Phys. Rev. Let. 79 (1997) 3050.