Spin-phonon coupling, q-dependence of spin excitations and high-T$_C$ superconductivity from band models

An understanding of spin excitations in cuprates is essential since the mechanism of high-T$_C$ superconductivity might be linked to spin fluctuations. Band calculations for long "1-dimensional" unit cells of La$_2$CuO$_4$ show that the coupling between antiferromagnetic spin waves and phonons is larger for distortions of oxygens than for Cu or La. When this result is applied to a 2-dimensional, free-electron like band, it leads to a q-dependent spin excitation spectrum in good agreement with recent experiments. It is argued that important parameters for spin-phonon coupling, which comes out from the comparison between experiment and theory, are relevant for the mechanism of superconductivity, and are large enough to explain a high T$_C$.