We study the half-breathing phonon in the three-band model of a high temperature superconductor, allowing for vibrations of atoms and resulting changes of hopping parameters. Two different approaches are compared. From the three-band model a $t\text{‐}J$ model with phonons can be derived, and phonon properties can be calculated. To make contact to density functional calculations, we also study the three-band model in the Hartree-Fock (HF) approximation. The paramagnetic HF solution, appropriate for the doped cuprates, has similarities to the local-density approximation (LDA). However, in contrast to the LDA, the existence of an antiferromagnetic insulating solution for the undoped system makes it possible to study the softening of the half-breathing phonon under doping. We find that although the HF approximation and the $t\text{‐}J$ model give similar softenings, these softenings happen in quite different ways. We also find that the HF approximation gives an incorrect doping and $\mathbf{q}$ dependence for the softening and too small a width for the (half-)breathing phonon.

• Received 21 June 2004

DOI:https://doi.org/10.1103/PhysRevB.70.224518