content_image/389515.jpg content_image/389510.jpg content_image/389512.jpg content_image/389503.jpg content_image/389505.jpg content_image/389509.jpg
<img src='content_image/389515.jpg'>

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 

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 

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, 

<img src='content_image/389510.jpg'>

<img src='content_image/389512.jpg'>

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 

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 

where the approximate relation Δ ( l ) 0 / Γ ( l ) ≃ Δ$_{0}$ as justi- fied in § II was used. 

The final explicit expression for the critical BEC tem- perature T$_{c}$ is now obtained by substituting (27) in (23). This leaves 

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 

## V. CRITICAL TEMPERATURE 

<img src='content_image/389503.jpg'>

<img src='content_image/389505.jpg'>

<img src='content_image/389509.jpg'>

