Pair susceptibility and mode propagation in superconductors: A microscopic approach

We derive the full microscopic set of equations governing small oscillations: (1) in the magnitude of the superconducting order parameter (the Schmid mode), (2) the phase of the order parameter in a neutral superfluid (the Anderson-Bogoliubov mode), and (3) the coupled oscillations in the phase of the order parameter and in the electric field (the transverse, or Carlson-Goldman mode). The derivation is not limited by the restrictions of previous papers. No limitations are required for the magnitude of the frequency, the concentration of impurities, or the magnitude of the temperature. Special attention is given to the Carlson-Goldman (CG) mode, whose dispersion law frequency (Ω) vs. wave vector (k) and damping is calculated. The velocity of the CG mode in the propagation region {Mathematical expression} is found to equal {Mathematical expression}, where D is the diffusion constant and χ is the function appearing in the theory of superconducting alloys. In the dirty (l « ξ₀) and clean (l ≫ ξ₀) limits, this expression reduces to those previously derived by Schmid and Schön, and by Artemenko and Volkov, respectively. At large values of k, the frequency of the CG mode approaches a limiting value of 2δ. The damping is small in this limit and tends to zero as |k| increases. p ]Our results are obtained by calculating the linear response of a superconductor to a perturbation in the magnitude and phase of the order parameter, and the electromagnetic potentials. The response of the superconductor to these perturbations is calculated by properly continuing the thermodynamic perturbation function of linear response from imaginary frequencies to the real ones, then inserting into the self-consistency BCS equation and Poisson's equation. The derivation is based on the self-consistent BCS scheme. No kinetic equations are introduced at any stage of the calculation.