Reentrant softening in copper-oxide superconductors

On the basis of the reentrant-softening model, sound-velocity results in La/sub 1.85/Sr/sub 0.15/CuO/sub 4/ and YBa/sub 2/Cu/sub 3/O/sub 7-x/ are shown to be neither dissimilar nor inconsistent with thermodynamic requirements. In both materials, as temperature decreases, the lattice softens just above T/sub c/. Below T/sub c/, this softening is offset by increased stiffness associated with the developing superconducting phase. This model agrees with results from other physical-property measurements, and it predicts that elastic stiffness is higher in the normal state than in the superconducting state. Recently reported sound-velocity measurements in the perovskite-like superconductors La/sub 1.85/Sr/sub 0.15/CuO/sub 4/ and YBa/sub 2/Cu/sub 3/O/sub 7-x/ appear to be anomalous as well as qualitatively different. The La/sub 1.85/Sr/sub 0.15/CuO/sub 4/ results indicate elastic softening at temperatures near T/sub c/, whereas the YBa/sub 2/Cu/sub 3/O/sub 7-x/ results give no such indication. Below T/sub c/, the increases in the bulk modulus and the elastic stiffness with decreasing temperature in these materials have been reported to be anomalous and inconsistent with thermodynamic requirements. The La/sub 1.85/Sr/sub 0.15/CuO/sub 4/ and YBa/sub 2/Cu/sub 3/O/sub 7-x/ results raise two issues. First, since the normal-superconducting transition is second order, general macroscopic thermodynamics requires the elastic bulk modulus to soften for temperatures at and below T/sub c/. Second, stiffening with no associated softening is difficult to reconcile with Mossbauer-spectroscopy measurements, which indicate lattice softening below T/sub c/ in EuBa/sub 2/(Cu/sub 1-y/Sn/sub y/)/sub 3/O/sub 7-x/. Also, it is difficult to reconcile with reasonable values of model parameters such as the Debye temperature, θ/sub D/, which affects the electron-phonon coupling constant