Low-energy limit of the three-band model for electrons in a CuO$_{2}$ plane

TL;DR

This paper derives the low-energy effective models from the three-band model of electrons in CuO2 planes, including the Heisenberg and t-J models, and calculates relevant parameters and corrections for different doping levels.

Contribution

It provides a systematic reduction procedure from the three-band model to low-energy models and computes key parameters, including superexchange and hopping terms, with doping-dependent differences.

Findings

The three-band model reduces to the Heisenberg model at unit filling.

Electron doping leads to a t-J model with specific hopping parameters.

Hopping parameters differ by approximately 40% between electron and hole doping.

Abstract

The three-band model with the O-O direct hopping near to unit filling is considered. We present the general procedure of reduction of this model to the low-energy limit. At unit filling the three-band model in the charge-transfer limit is reduced to the Heisenberg model and we calculate the superexchange constant. For the case of the small electron doping the three-band model is reduced to the $t-J$ model and we calculate electron hopping parameters at the nearest and next neighbors. We derive the structure of corrections to the $t-J$ model and calculate their magnitude. The values of the hopping parameters for electron- and hole-doping differ approximately at 40 %.