Bose–Einstein Condensation of Nonideal Cooper Pairs in the Hartree–Fock–Popov Theory

Abstract

The Hartree–Fock–Popov theory of interacting Bose particles is generalized to the Cooper-pair system with a screened Coulomb repulsive interaction in high-temperature superconductors. At zero temperature, it is found that the condensate density \(n_c(0)\) of Cooper pairs is of the order \(n_c(0)\simeq 10^{18}\) cm\(^{-3}\), consistently with the fact that a small fraction of the total p holes participate in pairing. We find that the phonon velocity c(0) at zero temperature is of the order \(c(0)\simeq 10\) km s\(^{-1}\). The computation shows that the transition temperature \(T_c\) is a dome-shaped function of the p hole concentration \(\delta \), which is consistent with experiments. At finite temperature, we find that the condensate fraction \(n_c(T)/n\) decreases continuously from \(n_c(0)/n\) to zero as the temperature increases from zero to the transition temperature \(T_c\). For higher temperatures, we find that the repulsive interaction between Cooper pairs drives more Cooper pairs into the condensate. The computation reveals that the phonon velocity c(T) decreases continuously from c(0) to zero as the temperature increases from zero to the transition temperature \(T_c\). The Cooper-pair system undergoes a first-order phase transition from the normal state to the BEC state.