COLLECTIVE MODES IN THE ELECTRONIC POLARIZATION OF DOUBLE-LAYER SYSTEMS IN THE SUPERCONDUCTING STATE

Standard weak-coupling methods are used to study collective modes in t he superconducting state of a double-layer system with intralayer and interlayer interaction, as well as a Josephson-type coupling and singl e-particle hopping between the layers by calculating the electronic po larization function perpendicular to the layers. Analytical results ar e derived for the mode frequencies corresponding to fluctuations of th e relative phase and amplitude of the layer order parameters in the ca se of interlayer pairing and finite hopping t. An effect is found for finite k-dependent hopping: then the amplitude and phase fluctuations are coupled. Therefore two collective modes may appear in the dynamica l c-axis conductivity below the threshold energy for breaking Cooper p airs. With help of numerical calculations we investigate the temperatu re dependence of the collective modes and show how a plasmon correspon ding to charge fluctuations between the layers evolves in the normal s tate.