content_image/383803.jpg content_image/383806.jpg content_image/383807.jpg content_image/383810.jpg content_image/383802.jpg
model interaction [27] leads to an approximate linear energy-vs-CMM relation 1 $_{2}$v$_{F}$ ℏ K for excited CPs propa- gating in the Fermi sea in 3D. This kind of dispersion re- lation is not unique to the Cooper model interaction. For example, an attractive interfermion delta potential [28] in 2D (imagined regularized [29] to support a single bound state of binding energy B$_{2}$ ⩾ 0) leads [30] (for a review see Ref.[24]) to the general CP dispersion relation E$_{K}$ = E$_{0}$ + c$_{1}$ ℏ K + [ 1 − (2 − 16 /π $^{2}$) E$_{F}$ /B$_{2}$ ] ℏ $^{2}$K$^{2}$/ 4 m $^{∗}$+ O ( K $^{3}$), where c$_{1}$ = 2 v$_{F}$ /π precisely as with the Cooper model in- teraction [31]. Hence, the leading-order linearity is not induced by the particular interfermion interaction bind- ing the CPs but is a consequence of the Fermi sea with v$_{F}$ ̸ = 0 and in which a CP by definition propagates. Only in the vacuum limit v$_{F}$ → 0 ⇒ E$_{F}$ ≡ 1 $_{2}$m $^{∗}$v 2 F → 0 does that general dispersion relation reduce by inspection to the expected quadratic form E$_{K}$ = E$_{0}$ + ℏ $^{2}$K$^{2}$/ 4 m ∗ for a composite object of mass 2 m $^{∗}$. For either inter- electron interaction model, the linear term is a conse- quence of the presence of the Fermi sea. The forma- tion of a BEC of CPs in 2D does not violate Hohen- berg’s theorem [32] as this holds only for quadratically- dispersive particles. The predicted 2D BEC temperature is T$_{c}$ ∝ ( n 2 $^{D}$) 1 / 2 ∝ (Θ$_{D}$ T$_{F}$ ) 1 / 2 where n 2 D is the CP number per unit area. This leads to values of T$_{c}$ that are too high compared with empirical values. However, these schemes provide a correct description of other rele- vant physical properties of HTSCs such as a short coher- ence length, a type II magnetic behavior, and the tem- perature dependence of the electronic heat capacity [14]. They also lead to excellent fits of the condensate frac- tion curves for quasi-2D cuprates just below T$_{c}$ [33], as well as for 3D and even quasi-1D SCs. To go beyond the simple s -wave interaction, an l -wave formulation of BCS theory was discussed by Schrieffer [26] himself and studied in considerable detail by Anderson and Morel [34] in the weak-coupling limit. This has been success- fully employed [6, 35] to describe thermodynamic and transport properties of high- T$_{c}$ cuprates. The d -wave ex- tension in strong-coupling Eliashberg theory is reported in Refs.[36]. 

Here we develop a general l -wave BCS-type theory which is then applied in a quasi-2D BEC picture with either l = 0 or l = 2 pairing symmetry. In § II the l -wave BCS theory within the framework of the present model is discussed. In § III we study a quasi-2D BEC of linearly- dispersive, massless-like CPs and we evaluate the number density. In § IV the areal density n 2 D of charge carriers is estimated by calculating the magnetic penetration depth arising from the CPs. In § V an analytical expression for the critical BEC temperature is derived, which is then applied in § VI for various superconducting materials in- cluding YBCO under different doping levels. Discussion and conclusions are given in § VII. 

## II. BCS THEORY WITH l -WAVE PAIRS 

Some aspects of the l -wave BCS theory [6, 26, 34] rel- evant to our HTSC model follow. Consider a system of electron- (or hole-) pairs formed via a two-fermion isotropic potential V near the Fermi surface and with kinetic energies ϵ$_{k}$ ≡ ℏ $^{2}$( k 2 − k 2 $_{F}$) / 2 m ∗ (with ℏ k$_{F}$ the Fermi momentum) taken relative to the Fermi energy. The Pauli principle prevents background fermions in elec- tron states just below (above) the Fermi surface from participating in the interaction. In the absence of ex- ternal forces each pair propagates freely within a layer of finite width δ along the z direction and infinite ex- tent on the x − y plane so that its total momentum ℏ K = ( ℏ K$_{‖}$ , ℏ K$_{z}$ ) is a constant of motion. By neglecting spin-dependent interactions the total spin S is conserved too and for a spin singlet S = 0 configuration the Pauli principle requires that the orbital wavefunction be of the form Ψ( r$_{1}$ , r$_{2}$ ) = exp( i K$_{‖}$ · R$_{‖}$ ) cos( K$_{z}$z )Φ( r ), where the relative coordinate r = r$_{1}$ − r$_{2}$ , R$_{‖}$ is the horizontal projection of the CM coordinate R = ( r$_{1}$ + r$_{2}$ ) / 2, and K$_{z}$ = nπ/δ (with n integer). The z -dependence of the wavefunction ensures that the vertical flux of the elec- tron (hole) pair across the layer boundary is null. Since the relative-coordinate problem is isotropic then Φ( r ) is an eigenfunction of angular momentum with quantum numbers l = 0 , 1 , 2 , · · · . The total spin S = 0 singlet eigenstates of the system satisfy the Schr¨ odinger equa- tion 

where H$_{0}$ is the free Hamiltonian V the interaction po- tential and E the energy eigenvalue. For a given CMM wavevector K , we may expand the wave function as 

<img src='content_image/383803.jpg'>

In momentum space (1) thus becomes 

<img src='content_image/383806.jpg'>

with V$_{kk}$ ′ = < k , − k | V | k $^{′}$, − k > . Since the interaction potential V depends only on r it admits the expansion 

<img src='content_image/383807.jpg'>

For small coupling amplitudes V$_{l}$ ( | k | , | k $^{′}$| ) the contribu- tions of different l spherical harmonics Y m l (Ω$_{k}$) in (4) can with good accuracy be considered relatively independent [34]. In that case (3) yields an analytical solution by as- suming that V$_{l}$ is separable, i.e., V$_{l}$ ( | k | , | k $^{′}$| ) = V ( l ) 0 f l $_{k}$f l ∗ k ′ so that (3) becomes 

<img src='content_image/383810.jpg'>

<img src='content_image/383802.jpg'>

