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The BCS-BE crossover phase diagram at T = 0 K for a d-wave superconductor: The importance of the Debye frequency and the tight binding band structure

Abstract:

We consider the phase diagram of the BCS (Bardeen-Cooper-Schrieffer)-BE (Bose-Einstein) crossover in the ground state (r = 0 K) of a d <inf>x2</inf>-<inf>y2</inf>-wave superconductor, with a nearest neighbour tight binding structure, when we take into account the Debye (phononic) frequency around the chemical potential, μ. This approach is a continuation of the work of den Hertog (1999 Phys. Rev. B 60 559) and that of Scares et al (2002 Phys. Rev. B 65 174506). The latter authors considered the influence of the second-nearest neighbours, but neither set of authors took into account the effect of the Debye frequency, ω<inf>D</inf>, or the influence of the next nearest neighbour matrix hopping element. We have found the following results: (1) there is not a metallic phase - that is, δ/4t → 0 when V/4t → 0, Vω<inf>D</inf>/4t, ∀αε (-1/2,+1/2), and ∀ n, where n is the carrier density per site, V is the attractive interaction, t is the nearest neighbour hopping integral, and α′ is the next nearest neighbour hopping ratio; (2) the BCS-BE crossover line is strongly affected by the presence of ω<inf>D</inf>/4t and that of α′ - actually, the values of V/4t needed to achieve the Bose-Einstein regime become extremely large for small values of ω<inf>D</inf>/4t; and (3) both δ/4t and μ/4t strongly depend on the values of ω<inf>D</inf>/4t and α′. The results (1) are in agreement with the ones found by Perali et al (2003 Phys. Rev. B 68 066501 (Preprint condmat/0211132)) and Rodríguez-Nunez et al (2003 Phys. Rev. B 68 066502), and in disagreement with those of den Hertog and Soares et al.