Renormalized mean-field theory of neutron scattering in cuprate superconductors

The magnetic excitation spectrum of the $t-t^{\prime }-J$ model is studied in mean-field theory and compared to inelastic neutron-scattering (INS) experiments on ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+y}$ (YBCO) and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (BSCCO) superconductors. Within the slave-particle formulation the dynamical spin response is calculated from a renormalized Fermi liquid with an effective interaction $\sim J$ in the magnetic particle-hole channel. We obtain the so-called “41-meV resonance” at wave vector $(\pi ,\pi )$ as a collective spin-1 excitation in the d-wave superconducting state. It appears sharp (undamped), if the underlying Fermi surface is holelike with a sufficient next-nearest-neighbor hopping $t^{\prime }<0.\mathrm{}$ The double-layer structure of YBCO or BSCCO is not important for the resonance to form. The resonance energy ${\omega }_{\mathrm{res}}$ and spectral weight at optimal doping come out comparable to experiment. The observed qualitative behavior of ${\omega }_{\mathrm{res}}$ with hole filling is reproduced in the underdoped as well as overdoped regime. A second, much broader peak becomes visible in the magnetic excitation spectrum if the 2D wave vector is integrated over. It is caused by excitations across the maximum gap, and in contrast to the resonance its energy is almost independent of doping. At energies above or below ${\omega }_{\mathrm{res}}$ the commensurate resonance splits into incommensurate peaks, located off $(\pi ,\pi ).\mathrm{}$ Below ${\omega }_{\mathrm{res}}$ the intensity pattern is of “parallel” type and the dispersion relation of incommensurate peaks has a negative curvature. This is in accordance with recent INS experiments on YBCO.