Electron-phonon interaction in the three-band model

TL;DR

This paper investigates the electron-phonon interaction in a high-temperature superconductor model, comparing approaches like the three-band, t-J, and Hartree-Fock methods to understand phonon softening and limitations.

Contribution

It introduces a comparative analysis of phonon behavior using three different theoretical frameworks within the three-band model.

Findings

HF approximation shows incorrect doping dependence

t-J and HF models predict similar phonon softening but via different mechanisms

HF approximation underestimates phonon width and q dependence

Abstract

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-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-J model give similar softenings, these softenings happen in quite different ways. We also find that the HF approximation gives an incorrect doping and q dependence for the softening and too small a width for the (half-)breathing phonon.