Cooper-pair and Bose-Einstein condensations in two dimensions: A critical analysis based on the Nozières and Schmitt-Rink formalism

The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of ``di-electronic'' molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozi\`eres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential {μ} so calculated is classified into two classes as decreasing temperatures; i.e., class (a) where {μ} approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of ``di-electronic'' molecules, and class (b) where {μ} diverges positively along the line of BCS-type mean-field pair condensation. This feature is rather universal irrespective of strength V of the attractive interaction of the s-wave type. While the former class (a) has been found by Schmitt-Rink, Varma, and Ruckenstein, existence of the latter class (b) is recognized here. In the case where V is fixed, class (a) is realized for electron number density N smaller than Ncr, which is an increasing function of V, and class (b) is realized for N larger than Ncr. If N{}Ncr in particular, there exists a regime, where the Fermi-liquid-like description is valid, between the BCS-type mean-field transition temperature and the Fermi temperature. In the situation where V is changed with N being fixed, low-temperature states for the strong-coupling case belong to class (a) while those for the weak-coupling case belong to class (b). Therefore, with decreasing V, the chemical potential {μ}(T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V=Vcr(N) corresponding to the transition from class (a) to (b). However, this is in contradiction to a physical picture that the chemical potential should smoothly cross over between the above two limits unless the liquid-gas transition occurs. This shows in turn a necessity of improving the Nozi\`eres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.