Physica C 282-287 (1997) 1645-1646
Local Dynamical d-wave pairing in Layered Quantum Paraelectric Superconductors Subodh R. Shenoy a , V. Subrahmanyam b, and A. R. Bishop c International Centre for Theoretical Physics, P.O.Box 586, Trieste 34100, Italy b Department of Physics, Indian Institute of Technology, Kanpur-208016, India ¢ Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory Los Alarnos, NM87545, U.S.A. Elastic c-axis coupling between a (GuO) buckled a-b plane of buckling angle s << 1, and a ferroelectric (oxygen) layer, produces novel w~:(q') modes, from s-induced mixing of 'a-b' and 'c' direction vibrations. The Cooper interaction coupling, ,-, (w~. -w2._)-I ~,, s -1 >> 1, is strongly resonance-enhanced and anisotropic (,-, d-wave), yielding dynamical local pairing.
1. I N T R O D U C T I O N High temperature superconductors (HTS), with a low carrier concentration x << 1, and peaks/plateaus in T¢(z), are structurally similar to perovskite ferroelectrics [1,2] with ubiquitous CuO layer bucklings ; and exhibit pyro/piezoelectricity . On the other hand, x << 1 doped paraelectrics like SrTi03 show  a peaked To(z), <0.1 K. 'Doublet' modes in SrTiOa, that almost cross at small wavevectors, have been reported . It is natural to explore if HTS pairing can arise from possibly novel modes related to HTS structures , and their potentially ferroelectric  character. 2. B U C K L I N G MODES
displacements. For s = 0, there is an acoustic mode w 2 -- 2 g a s i n 2 ( q a / 2 ) / ( M c u + M o ) , (q in the a-b plane, of lattice constant a); and a flat mode, w ~ = K J ( M c u + M o ) (vibration along the c-axis). Here Mvt,, M o are the masses of the copper (H layer), and oxygen (H or B layer); and Ks, Kc are the Cu-O spring constants in the H layer and between H and B layers, respectively. With nonzero s, these (otherwise orthogonal) modes are mixed, and for q in the a-b plane,
2w~ = A + B + x / ( A - B) 2 + 4s2AB~te/M
We consider the modes of a CuO buckled a-b plane (H), coupled elastically along the c-axis to a Bilz model  layer (B) of harmonically coupled double-well ferroelectric sites (e.g., apical oxygen, in Y oxides). These double wells are smeared by c-axis zero-point motion to broad single wells of curvature D, with a self-consistent harmonic solution D > 0 i.e. a quantum paraelectric state. The (sine of the ) buckling angle is nonzero  s --~ 0.07, <<1; s acts as a 'gear-wheel', interconverting a-b plane and c-axis 0921-4534/97/$17.00 © Elsevier Science B.V. All rights reserved. Pll S0921-4534(97)00906-4
where/~e = (1 + M o / M c u ) 2, and M -- M e . + M o . For q -+ 0,w 2_ .., A .., Kaq 2, while w~_ /~Kc, with the flat mode softened by a weak anchor point,/} -- D / ( D + K¢) < < 1. For q ... ~r/a, the w+ and w_ limits are interchanged, with a sharp N s anticrossing region in between, around q -- qB "~ ( B K c / K a ) - l / 2 / a , t h a t turns out to be ,-, (10Angstroms) -1, a s / } ,~ 0.05 , from solution of a selfconsistent harmonic equation. The modes involve fluctuating B-layer dipoles, that can interact electrically with a-b plane charged carriers NX.
S.R. Shenoy et al./Physica C 282-287 (1997) 1645-1646
3. d - W A V E P A I R I N G A N D Tc The dynamical effective pairing is attractive for ~2 < ~ ( q ) and for ~ ( q = 0) < ~2 < w~(q). It has an anticrossing factor ~ (w~_w t ) - l , ~ ( s ~ ) - i >> 1 at q ~ qB, showing strong resonance enhancement, and anisotropy in pairing, with qU = (~: - ~1)2, even without anisotropy of the fermi surface (therefore taken to be circular, to illustrate this). Frequency averaging yields a static Cooper pairing potential of square wells up to some Debye cutoff. This pairing potential can be expanded in angular momentum (1) components, and the transition temperature calculated. For buckling angle s --~ 0, Tc is nonanalytic, ~ exp(--1/ls log sl).The figure shows Tc vs z for d-wave pairing, with s-wave pairing suppressed by a large repulsive Hubbard U . Since the dominant interaction is from nonzero w and q , the pairs are neither static bipolarons, nor extended BCS pairs, but dynamical local pairs, mediated by novel modes, in a quantum paraelectric model of HTS.
Figure 1. Transition temperature (scaled in the cutoff) vs carrier doping fraction, for d-wave pairing, and for two sets of parameters, showing a peak/plateau, as in La/Y oxides. R E F E R E N C E S
1. J. D. Jorgenson, D.G. Hinks, B.A. Hunter, R.L. Hitterman, A.W. Mitchell,P.G. Ra~aelli,
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B. Dabrowski, J.L. Wagner, H. Takahashi,and E.C.Larson, in 'Lattice Effects in High Temperature Superconductors', Eds.. Y. BarYam, T. Egami, J. Mustre de Leon and A. R. Bishop, World Scientific, Singapore, 1992. H. Bilz, G. Benedek, and A. Bussman-Holder, Phys. Rev. B35, (1984) 4840. D. Mihailovic and A. Seeger, Solid State Comm., 75 (1990) 319. M. L. Cohen in 'Superconductivity' (ed) P~. D. Parks, Marcel-Dekker, N.Y., 1964.) E. Courtens, B. Hehlen, G. Coddens, and B. Hennion, Physica B, 219+220 (1996) 577. S. R. Shenoy, V. Subrahmanyam and A. R. Bishop, (unpublished). P.W. Anderson and P. Morel, Phys. Rev. 123 (1961), 1911.