5 TABLE I: Physical parameters of cuprate superconductors and predicted values for T c , and the ratio 2Δ$_{0}$ /k B T c according to (28). Debye temperature is Θ D ≡ ℏ ω D /k B . Parameters taken from from Ref.[38] (see also references cited therein): a) table 4.1, b) table 6.1, c) table A.1, d) table A.2, e) table A.1, f) table 6.1, g) estimated from band-structure calculations [39, 40], h) estimated as δ = 0 . 64 c int , and i) estimated as δ = 0 . 68 c int , where c int is the CuO$_{2}$ interlayer separation for a given cuprate. For YBCO Θ D , Δ$_{0}$, and δ are assumed the same for different dopings. λ − 1 ab (0) = λ − 1 a (0) + λ − 1 b (0) is the geometric mean of this parameter measured along crystallographic in-plane di- rections a and b . As shown in § II, this parameter is in- dependent of the explicit value of the angular momentum l . By substituting the dispersion relation (11) to elimi- nate K$_{0}$ from λ$_{0}$ and imposing the relation n 2 D = δn 3 D the 2D charge carrier density becomes depth λ$_{L}$ which is the distance over which an external magnetic field decays within the superconductor. For superelectrons with a 3D density n$_{s}$ , charge e , and effec- tive mass m $^{∗}$, one has the well-known relation 1 /λ 2 L = 4 πe $^{2}$n$_{s}$ /m $^{∗}$c $^{2}$. By introducing [20] the average interlayer spacing c$_{int}$ between CuO$_{2}$ planes in HTSCs it follows that n 2 D ≃ c$_{int}$n$_{s}$ . Penetration-depth data spanning a wide range of critical temperatures are consistent with the phenomenological Uemura relation T$_{c}$ ∝ 1 /λ 2 L ∝ n 2 $^{D}$/m ∗ [20]. Within the present model we evaluate the magnetic penetration depth due to linearly-dispersive CPs with charge 2 e , and constrained to move within a thin layer of width δ with a uniform CM speed c$_{1}$ . Thus, we first con- sider the expression for the 3D supercurrent of excited CPs [14] J$_{s}$ = n 3 $^{D}$(2 e ) c$_{1}$ ˆ K with ˆ K ≡ K /K . We now introduce the contour integral of the CP wavefunction phase within a homogeneous medium and in the pres- ence of an external magnetic field B = ∇ × A . The following integral along any closed path vanishes n 2 D = e 2 c 2 δ |E ( l ) 0 | 16 πc 2 1 1 λ 2 ab . (26) This latter expression can be reformulated by considering the relation (15) and the weak-coupling limit of (12). It follows that |E ( l ) 0 | = (Δ ( l ) 0 ) $^{2}$/ 2 ℏ ω$_{D}$ so that n 2 D = e 2 32 πc 2 $_{1}$c 2 δ Δ 2 0 ℏ ω$_{D}$ 1 λ 2 ab (27) where the approximate relation Δ ( l ) 0 / Γ ( l ) ≃ Δ$_{0}$ as justi- fied in § II was used. V. CRITICAL TEMPERATURE The final explicit expression for the critical BEC tem- perature T$_{c}$ is now obtained by substituting (27) in (23). This leaves T$_{c}$ = ℏ c 2 πk$_{B}$e ( 3 δ 2 ℏ ω$_{D}$ ) 1 / 2 Δ$_{0}$ λ$_{ab}$ (28) ∮ ( ℏ K + 2 e c A ) · d r = 0 (24) where c is the speed of light in vacuum. By expressing K in terms of J$_{s}$ and using Stoke’s theorem to evalu- ate (24) we get a modified version of London’s equation J$_{s}$ = − Λ$_{p}$ A where Λ$_{p}$ ≡ 4 e $^{2}$c$_{1}$ n 3 $^{D}$/ ℏ cK . Taking now the curl of this modified London equation and introduc- ing Ampere’s law ∇ × B = (4 π/c ) J$_{s}$ , it follows that the magnetic induction B satisfies the Helmholtz equation ∇ $^{2}$B = λ − $^{2}$B , where 1 λ 2 ≡ (2 e ) 2 c 2 4 πc$_{1}$n 3 D ℏ K . (25) which is independent of the CP speed c$_{1}$ . We observe that for fixed values of ω$_{D}$ , Δ$_{0}$, and δ , the critical tem- perature increases linearly with λ − 1 ab . This dependence has been observed by Zuev et al. [37] in experiments in underdoped YBCO films with T$_{c}$ s ranging from 6 to 50 K . They conclude that, within some noise, their data fall on the same curve ρ$_{s}$ ∝ λ − 2 ab ∝ T 2 . 3 ± 0 . 4 c , irrespective of an- nealing procedure, oxygen content, etc. Thus, by assum- ing that except for λ$_{ab}$ the other YBCO parameters are approximately independent of the doping level, we intro- duced in (28) the values: Θ$_{D}$ = 410 K [38], Δ$_{0}$ = 14 . 5 Note that London’s result is recovered for quasi-particles with density n 3 D → n$_{s}$/ 2, momentum ℏ K → 2 m $^{∗}$c$_{1}$ , and charge 2 e → e . This expression for λ varies be- tween its minimum value λ = 0 when K = 0 (perfect diamagnetism), and its maximum, say λ$_{0}$ , for K = K$_{0}$ (CP breakup). It seems natural to identify λ$_{0}$ with the experimentally observed value of the in-plane penetra- tion depth at T = 0, namely λ$_{0}$ = λ$_{ab}$ ( T = 0). Here, superconductor Θ D (K) a Δ$_{0}$ (meV) b λ$_{ab}$ (nm) c δ ( ˚ A ) d T exp c (K) e T th c (K) (2Δ$_{0}$ /k B T c ) exp ( f ) (2Δ$_{0}$ /k B T c ) th (La . $_{925}$Sr . $_{075}$)$_{2}$ CuO$_{4}$ 360 6.5 250 4.43 g 36 36.4 4.3 4.14 YBa$_{2}$Cu$_{3}$O$_{6}$ . 60 410 15.0 240 2.15 h 59 56.0 5.90 6.09 YBa$_{2}$Cu$_{3}$O$_{6}$ . 95 410 15.0 145 2.15 g 93.2 92.6 4.0 3.68 Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$ 250 16.0 250 2.24 g 80 72.2 4.64 4.85 Bi$_{2}$Sr$_{2}$Ca$_{2}$Cu$_{3}$O$_{10}$ 260 26.5 252 2.24 i 108 109.2 5.7 4.99 Tl$_{2}$Ba$_{2}$Ca$_{2}$Cu$_{2}$O$_{8}$ 260 22.0 221 2.14 i 110 104.1 4.5 4.47 Tl$_{2}$Ba$_{2}$Ca$_{2}$Cu$_{3}$O$_{10}$ 280 14.0 200 4.30 i 125 105.5 3.1 2.96