# BCS-BEC model of high-T c superconductivity in layered cuprates with unconventional pairing 

C. Villarreal 1 and M. de Llano 2 

$^{1}$Instituto de F´ısica, Universidad Nacional Aut´ onoma de M´ exico Ciudad Universitaria, 04510 M´ exico, DF, MEXICO 

$^{2}$Physics Department, University of Connecticut, Storrs, CT 06269 USA and $^{∗}$Instituto de Investigaciones en Materiales, Universidad Nacional Aut´ onoma de M´ exico Apdo.Postal 70-360, 04510 M´ exico, DF, MEXICO 

(Dated: August 3, 2017) 

High- T c superconductivity in layered cuprates is described in a BCS-BEC formalism with linearly- dispersive s- and d-wave Cooper pairs moving in quasi-2D finite-width layers about the CuO$_{2}$ planes. This yields a closed formula for T c determined by the layer width, the Debye frequency, the pairing energy, and the in-plane penetration depth. The new formula reasonably reproduces empirical values of superconducting T c s for seven different compounds among the LSCO, Y BCO, BSCCO and TBCCO layered cuprates. 

It seems to be well-established that central to high- T$_{c}$ superconductivity (HTSC) is the layered two- dimensional (2D) structure of copper oxides and that su- perconducting pairing occurs mainly on the CuO$_{2}$ planes. However, the precise nature of the pairing is still the sub- ject of intense research. Recent experiments based on angle-resolved photon emission spectroscopy (ARPES) of underdoped cuprates suggest that bound fermion Cooper pairs (CPs) form already at and below temperatures higher than the critical transition temperature T$_{c}$ [1– 4]. Furthermore, ARPES studies of the electron spectral function of optimally doped Bi2212 samples show that the magnitude of the isotope effect correlates with the superconducting gap [5], thus suggesting a role of lat- tice phonons in the superconducting pairing. ARPES data also suggest that the energy gap Δ (a measure of the energy needed to break a CP) displays an unconven- tional d$_{x}$ $^{2}$− y 2 orbital pairing symmetry, with a functional dependence Δ = Δ$_{0}$ cos 2 θ where θ = tan − $^{1}$K$_{y}$ /K$_{x}$ is the angle between the total or center-of-mass momen- tum (CMM) ℏ K = ( ℏ K$_{x}$, ℏ K$_{y}$ ) of paired electrons in the CuO$_{2}$ plane and the a - (or x -) axis while Δ$_{0}$ is the value of the superconducting gap at the antinode ( θ = 0 , π/ 2) [6]. This behavior is also apparent in studies based on electronic Raman scattering [7] and in determinations of the in-plane magnetic penetration depth λ$_{ab}$ [8, 9]. 

Although the majority viewpoint in the high- T$_{c}$ com- munity seems to argue for such non-s-wave pairing sym- metry there are compelling dissenting views, particularly work within the past few years, by M¨ uller [10], Harshman [11], Klemm [12], and many others. In particular, M¨ uller concludes that “...recent experiments probing the surface and bulk of cuprate superconductors [show that their] character is d on the surface and substantially s in the bulk.” This conclusion has been bolstered by muon-spin relaxation ( µ SR) experiments with YBCO reported and interpreted in Ref.[11]. Several authors [13, 14] have pro- posed that the doping process could modify the electron- 

PACS numbers: 05.30.Fk, 74.20.-z, 74.72.-h 

## I. INTRODUCTION 

phonon interaction and the Fermi surface with a con- comitant shift from d - to s -type coupling as doping in- creases. The strongest evidence for an s-wave order pa- rameter in a cuprate is reviewed in Ref.[12] where several c -axis twist experiments on BSCCO along with earlier c -axis tunneling between BSCCO/Pb junctions are sur- veyed. Ref.[15] summarizes many of the problems with the so-called “phase-sensitive” tests [16] in YBCO. Addi- tionally, predictions made in Ref.[17] that a vortex in a d - wave superconductor would exhibit a measurable density of states in a four-fold pattern emanating from the core have not been observed [18] in either YBCO or BSCCO. On the contrary, the vortex cores appear consistent [18] with isotropic s -waves. 

Replacing the characteristic phonon-exchange Debye temperature Θ$_{D}$ ≡ ℏ ω$_{D}$/k$_{B}$ of around 400 K (with k$_{B}$ the Boltzmann constant and ω$_{D}$ the Debye frequency) by a characteristic magnon -exchange temperature of around 1000 K can lead to a simple model interaction such as the BCS one but associated with spin-fluctuation-mediated pairing [19]. 

The so-called “Uemura plot” [20] of data from µ SR, neutron and Raman scattering, and ARPES measure- ments exhibit T$_{c}$ vs Fermi temperatures T$_{F}$ ≡ E$_{F}$/k$_{B}$ where E$_{F}$ the Fermi energy and k$_{B}$ the Boltzmann con- stant. Empirical T$_{c}$ s of many cuprates straddle a straight line parallel to the Uemura-plot line associated with the simple BEC formula T$_{B}$ ≃ 3 . 31 ℏ $^{2}$n 2 / 3 B /m$_{B}$k$_{B}$ ≃ 0 . 218 T$_{F}$ corresponding to an ideal gas of bosons of mass m$_{B}$ = 2 m ∗ and number density n$_{B}$ = n$_{s}$/ 2 where m ∗ is the effective mass and n$_{s}$ the number density of individual charge carriers. The parallel line is shifted down from T$_{B}$ by a factor 4-5. This has been judged [21] as a “funda- mental importance of the BEC concept in cuprates.” 

Previous theoretical papers on the possible origin of HTSC [14, 22–24] proposed that the phenomenon might be rooted in a 2D Bose-Einstein condensate (BEC) of CPs pre-existing above T$_{c}$ and coupled through a BCS- like phonon mechanism [25], originally taken as s -wave. As apparently first reported by Schrieffer [26], the Cooper 

