Microscopic theory of longitudinal sound velocity in CDW and SDW ordered cuprate systems

Research highlights: → Reported the study of the interplay of the CDW and SDW interactions in the high-Tc cuprates. → The longitudinal velocity of sound is studied in the under-doped region. → The velocity of sound exhibits suppression in both the CDW and SDW phases. → Strong electron-phonon interaction is observed in normal phases. - Abstract: We address here the self-consistent calculation of the spin density wave and the charge density wave gap parameters for high-Tc cuprates on the basis of the Hubbard model. In order to describe the experimental observations for the velocity of sound, we consider the phonon coupling to the conduction band in the harmonic approximation and then the expression for the temperature dependent velocity of sound is calculated from the real part of the phonon Green's function. The effects of the electron-phonon coupling, the frequency of the sound wave, the hole doping concentration, the CDW coupling and the SDW coupling parameters on the sound velocity are investigated in the pure CDW phase as well as in the co-existence phase of the CDW and SDW states. The results are discussed to explain the experimental observations.