content_image/403981.jpg content_image/403982.jpg content_image/403991.jpg content_image/404000.jpg content_image/403983.jpg content_image/403985.jpg content_image/403994.jpg content_image/403996.jpg
gular momentum dependence. This is given by [6] 

<img src='content_image/403981.jpg'>

In the low-temperature limit (17) yields for l = 0 an exponential T -dependence 

<img src='content_image/403982.jpg'>

while for l = 2 it gives the linear T -dependence 

Experiments [8, 9] on the temperature variation of the magnetic penetration depth λ$_{ab}$ ( T ) are consistent with the quasi-linear behavior (19) which is a signature of d - wave symmetry. However, its asymptotic value λ$_{ab}$ ( T → 0) is independent of l , a result that we apply below. 

## III. BOSE-EINSTEIN CONDENSATION 

We assume that charge carriers are an ideal binary mixture of non-interacting unpaired fermions plus break- able bosonic linearly-dispersive CPs [14, 22, 23, 26]. Let the fermion number per unit area be n$_{f}$ = n$_{f}$$_{1}$ + n$_{f}$$_{2}$ where n$_{f}$$_{1}$ and n$_{f}$$_{2}$ are the number densities of unpairable and pairable fermions, respectively. Unpairable fermions lie outside the interaction region of (8) unlike the pairable fermions whose T -dependent density n$_{f}$$_{2}$ ( T ) is 

<img src='content_image/403991.jpg'>

Here CMM wavenumber K = 0, n 2 D 0 <K<K$_{0}$ that with 0 < K < K$_{0}$ , and n u f 2 the number density of pairable but unpaired fermions. By asserting that in thermal equilibrium these kinds of fermions arise precisely from broken CPs [22] we identify n u f $_{2}$( T ) = 2 n 2 D K$_{0}$<K<K $_{max}$( T ). On the other hand, at T = 0 all pairable fermions should belong to the con- densate (Ref.[38], p. 122) so that n$_{f}$$_{2}$ (0) = 2 n 2 D 0 (0) ≡ 2 n 2 D where n 2 D is the total boson number per unit area. The number equation for pairable fermions may thus be reexpressed in terms of boson quantities alone, namely n 2 D = n 2 D 0 ( T )+ n 2 D 0 <K ≤ $_{K$_{0}$}$( T )+ n 2 D K$_{0}$<K ≤ K $_{max}$( T ) ≡ n 2 D 0 ( T ) + n 2 D 0 <K ≤ K $_{max}$( T ). Thus 

<img src='content_image/404000.jpg'>

where β ≡ 1 /k$_{B}$T , µ the boson chemical potential and z ≡ exp βµ is the fugacity (0 ≤ z ≤ 1). On in- troducing (21) the energy-shifted boson dispersion re- lation E ( l ) K = ℏ c$_{1}$K for K > 0 the integral can eval- uated by changing to the variable x ≡ β ℏ c$_{1}$K . Since 

<img src='content_image/403983.jpg'>

<img src='content_image/403985.jpg'>

c$_{1}$ = 2 v$_{F}$ /π and K$_{max}$ = k$_{F}$ (1 + k 2 $_{D}$/k 2 $_{F}$), the upper inte- gration limit x$_{max}$ in (21) is then be very large, namely x$_{max}$ = β ℏ v$_{F}$ k$_{F}$ = 2 E$_{F}$ /k$_{B}$T ≫ 1. The last inequality is consistent with the maximum empirical value for the ratio k$_{B}$T$_{c}$/E$_{F}$ ≤ 0 . 05 reported [20] for all SCs includ- ing cuprate SCs. Given the rapid convergence of Bose integrals the upper integration limit x$_{max}$ may safely be taken as infinite in (21) so that the integrals can then be evaluated exactly by expanding the integrand in powers of z exp( − x ) and integrating term by term. The number density (21) becomes 

The critical BEC temperature T$_{c}$ is now determined by solving (22) for n 2 D 0 ( T$_{c}$ ) = 0 and z ( T$_{c}$ ) = 1. One obtains 

where ζ (2) = π $^{2}$/ 6. 

## IV. CHARGE CARRIER DENSITY 

The areal density of charge carriers was formerly es- timated from measurements of the London penetration 

<img src='content_image/403994.jpg'>

<img src='content_image/403996.jpg'>

