Theory of superconductivity. II. Excited Cooper pairs. Why does sodium remain normal down to 0 K?

Abstract

Based on a generalized BCS Hamiltonian in which the interaction strengths (V11,V22,V12) among and between “electron” (1) and “hole” (2) Cooper pairs are differentiated, the thermodynamic properties of a type-I superconductor below the critical temperatureTc are investigated. An expression for the ground-state energy,W-W0, relative to the unperturbed Bloch system is obtained:W-W0=−1/4[N1(0)Δ 1 2 +N2(0)Δ 2 2 ], whereNj(0) represent the electron and hole densities of states at the Fermi energy eF, and Δj are the solutions of the simultaneous equations,\(\Delta _j = \tfrac{1}{2}V_{j1} N_1 (0)\Delta _1 {\text{ sinh}}^{ - {\text{1}}} (\hbar \omega _D /\Delta _1 ) + \tfrac{1}{2}V_{j2} N_2 (0)\Delta 2_2 {\text{ sinh}}^{ - {\text{1}}} (\hbar \omega _D /\Delta _2 )\) with ωD denoting the Debye frequency. The usual BCS formulas are obtained in the limits: (all)Vjl=V0,N1(0) =N2(0). Any excitations generated through the BCS interaction Hamiltonian containingVjl must involve Cooper pairs of antiparallel spins and nearly opposite momenta. The nonzero momentum orexcited Cooper pairs belowTc are shown to have an excitation energy band minimum lower than the quasi-electrons, which were regarded as the elementary excitations in the original BCS theory. The energy gapeg(T) defined relative to excited and zero-momentum Cooper pairs (whenVjl>0) decreases fromeg(0) to 0 as the temperatureT is raised from 0 toTc. If “electrons” only are available as in a monovalent metal like sodium (V12=0), the energy constant Δ1 is finite but the energy gap vanishes identically for allT. In agreement with the BCS theory, the present theory predicts that a pure nonmagnetic metal in any dimensions should have a Cooper-pair ground state whose energy is lower than that of the Bloch ground state. Additionally it predicts that a monovalent metal should remain normal down to 0K, and that there should be no strictly one-dimensional superconductor.