Superexchange in the cuprates

It is shown that the usual three-center cation-anion-cation model or fourth-order pertrubation expression (in the Cu-O hopping parameter ${\mathit{t}}_{\mathit{p}\mathit{d}}$) does not give a useful (even qualitative) estimate for the superexchange J, even though, by coincidence, it predicts a magnitude that is reasonable. There are two additional contributions, both due to the oxygen-oxygen hopping ${\mathit{t}}_{\mathit{p}\mathit{p}}$, which are estimated to be responsible for about 2/3 of the total exchange interaction. The first causes a strong enhancement of the usual fourth-order J and is conventional in the sense that it goes like 1/U for a large on-site Coulomb interaction U on copper and oxygen, and is always antiferromagnetic (AFM). Its importance is due to a large prefactor in the perturbation series in ${\mathit{t}}_{\mathit{p}\mathit{p}}$/Δ, where Δ is the charge-transfer energy. The second AFM term is of a topological nature in the sense that its sign is determined by the signs of the different hopping parameters and the arrangement of the Cu and O sites in the ${\mathrm{CuO}}_2$ planes. This term does not involve doubly occupied sites, and is a consequence of the extra degrees of freedom formed by the oxygen 2p orbitals. Starting from the three-band model, these two effects are examined by means of fifth-order perturbation theory and perturbation theory that involves oxygen bands explicitly. The results are compared with numerical estimates of J from finite-size clusters.