Spin-Phonon Coupling in the Single-Layer Extended t-J Model

We consider the implications of spin-phonon coupling within the slave-boson, mean-field treatment of the extended t-J model of a high-temperature superconductor. In bilayer cuprates such as YBaCuO, where the $CuO_2$ plane is buckled, this interaction is linear in $O$ displacement along the c-axis, and the coupling constant is found to be large. The formation of a spin singlet causes additional contributions to the phonon self-energy, and we calculate from these the superconductive phonon anomalies. The magnitude and sign of the frequency shift and linewidth broadening for various mode symmetries correspond well with Raman and infra-red light scattering experiments, and with neutron scattering studies. In the t-J model, spin singlet formation and superconductivity do not coincide in the low-doping regime, giving rise to spin-gap features and a variety of temperature scales in the spin response observed by NMR and neutron investigations. Phonon anomalies in underdoped YBaCuO compounds indeed show evidence of spin-gap phenomena with the same characteristic temperature, suggesting that the theory may offer the possibility of a unified understanding of the anomalies in magnetic and lattice properties. While the origin of the superconducting interaction is electronic, this spin-phonon coupling affords the possibility of a small isotope effect, and our estimate is in good agreement with recent site-selective $O$-substitution experiments.