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meV [38], and δ = 2 . 15 ˚ A [39, 40] to get the relation 

Figure 1 is adapted from Ref.[37] and compares theoret- ical predictions (29) with experimental data, as well as with data pertaining to higher doping regimes. We see that (29) gives an excellent fit to the experimental data. The same functional dependence has been observed in single YBCO crystals near the optimally-doped regime [41]. More recently, Broun et al. [9] found that their samples of high-purity single-crystal YBCO followed the rule T$_{c}$ ∝ λ − 1 ab ∝ n 1 / 2 s ∝ ( p − p$_{c}$ ) 1 / 2 where the doping p is the number of holes per copper atom in the CuO$_{2}$ planes and p$_{c}$ the minimal doping for superconductivity onset. The measured value of the penetration length in YBCO crystals is an order of magnitude bigger than in thin films [9, 41], so that the specific values of T$_{c}$ s de- rived from (28) are not in such good agreement as in the YBCO films. However, one should expect variations of parameters such as the energy gap associated to crys- tals and film systems. It has been pointed out [37] that YBCO films seem to behave more like other cuprates such as BiSrCaCuO or LaSrCuO than do YBaCuO crys- tals . Furthermore, a different approach [42] based on measurements of the lower critical magnetic field H$_{c}$$_{1}$ ( T ) for highly underdoped YBCO indicates that experimen- tal data may be consistently described only by assuming T$_{c}$ ∝ n 0 . 61 s , in close agreement with studies mentioned above. 

Theoretical values of T$_{c}$ for superconducting cuprates with different compositions have been also calculated using (28). Here we report on these seven layered-cuprate superconducting com- pounds: (La$_{.}$$_{925}$Sr$_{.}$$_{075}$)$_{2}$CuO$_{4}$; YBa$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{60}$; YBa$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{95}$; Tl$_{2}$Ba$_{2}$Ca$_{2}$Cu$_{2}$O$_{8}$; Tl$_{2}$Ba$_{2}$Ca$_{2}$Cu$_{3}$O$_{10}$; Bi$_{2}$Sr$_{2}$Ca$_{2}$Cu$_{3}$O$_{10}$; and Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$. Characteristic parameters for these materials were taken from tables compiled in Ref.[38] (see also [43–45]). Concerning the layer width δ no direct experimental data are available. We have employed results derived from energy band-structure calculations for cuprates. Contour plots [39, 40] of the charge distribution for La$_{2}$CuO$_{4}$, YBa$_{2}$Cu$_{3}$O$_{7}$, and BiCa$_{2}$SrCu$_{2}$O$_{8}$ suggest that charge carriers in each of these systems are concentrated within slabs of average width δ ≃ 2 . 61 ˚ A , 2 . 15 ˚ A , and 2 . 28 ˚ A , respectively, about their CuO$_{2}$ planes. As c$_{int}$ denotes the average separation between adjacent CuO$_{2}$ planes, it follows from crystallographic data [38] that the yttrium and bismuth compounds give δ ≃ 0 . 64 c$_{int}$ and 0 . 68 c$_{int}$ , respectively. Taking into account that BiSr$_{2}$Ca$_{n}$Cu$_{n}$$_{+1}$O$_{6+}$$_{n}$ compounds possess the same layering scheme as their TlBa$_{2}$Ca$_{n}$Cu$_{n}$$_{+1}$O$_{6+}$$_{n}$ counter- parts [38], we assumed that the condition δ ≃ 0 . 68 c$_{int}$ 

<img src='content_image/404329.jpg'>

holds also for the thallium compounds. The former esti- mations are congruent with Uemura’s surmise [20] that SC charge carriers in layered cuprates are concentrated within slabs of width δ = c$_{int}$ . 

Table I shows results obtained using the foregoing as- sumptions, together with the physical parameters in- volved in the calculation. In most cases we find rather satisfactory agreement between predicted and measured values of T$_{c}$ . We also find very good agreement between theoretical and experimental gap-to- T$_{c}$ ratios 2Δ$_{0}$ /k$_{B}$T$_{c}$ . Average theoretical and experimental such ratios presented in Table I are (2Δ$_{0}$ /k$_{B}$T$_{c}$ ) th ≃ 4 . 45 and (2Δ$_{0}$ /k$_{B}$T$_{c}$ ) exp ≃ 4 . 59, respectively. Both are consis- tent with the ratio 2Δ (2) 0 /k$_{B}$T$_{c}$ ≃ 4 . 28 predicted by the l = 2 BCS theory in (16). We have not attempted es- timate uncertainties of our theoretical results since the accumulated data of the physical parameters involved in the calculation, particularly Δ$_{0}$ and λ$_{ab}$ , show a wide scatter. 

## VI. DISCUSSION AND CONCLUSIONS 

We have shown that layered-cuprate HTSC can be de- scribed by means of an l -wave BCS theory for a quasi-2D BEC of Cooper pairs. The theory involves a linear , as opposed to quadratic, dispersion relation in their total or CM momenta. The theory yields a simple formula for the critical transition temperature T$_{c}$ with a functional relation T$_{c}$ ∝ 1 /λ$_{ab}$ ∝ n 1 / 2 s which applies to a variety of cuprate SCs over a wide range of dopings. Although this behavior apparently disagrees with the phenomenological Uemura relation T$_{c}$ ∝ 1 /λ 2 ab [20], different experimen- tal studies [9, 37, 41] show consistency with the inverse linear dependence of T$_{c}$ . Additional consistency is also seen with the reported dependence T$_{c}$ ∝ n 0 . 61 s arising from measurements of the lower critical magnetic field [42]. When averaged over a cylindrical Fermi surface, the physical quantities involved in the theory show small dependence on the angular momentum state l . However, the gap-to- T$_{c}$ ratio 2Δ$_{0}$ /k$_{B}$T$_{c}$ is closer to that predicted by the extended BCS theory for l = 2 than for l = 0. It is shown elsewhere [46] that all relevant 2D expressions derived here arise in the limit k$_{B}$T δ/ ℏ c$_{1}$ → 0 of a more general 3D BCS-BEC theory for layered materials. ∗ Permanent address. 

Acknowledgments We thank M. Fortes, S. Fu- jita, L.A. P´erez, and M.A. Sol´ıs for fruitful discus- sions. MdeLl thanks UNAM-DGAPA-PAPIIT (Mexico) IN106908 as well as CONACyT (Mexico) for partial sup- port. He thanks D.M. Eagles and R.A. Klemm for e- correspondence and is grateful to W.C. Stwalley for dis- cussions and the University of Connecticut for its hospi- tality while on sabbatical leave. 

