Superexchange interaction in cuprates

Recent experiments have shown that the superexchange interaction $J$ in one-dimensional (1D) cuprates is larger than that in two-dimensional (2D) ones. We investigate a microscopic origin of the difference of $J$ in the 1D and 2D cuprates. The hopping matrix elements between $\mathrm{Cu}3\mathrm{}d$ and $\mathrm{O}2\mathrm{}p$ orbitals and between $\mathrm{O}2\mathrm{}p$ orbitals are considerably influenced by the Madelung potential, which is a function of crystal structure, and these values in the 1D cuprates are enhanced as compared with those for the 2D ones, resulting in larger values of $J.\mathrm{}$ The same mechanism is applied to hopping matrix elements in the ladder cuprates. The elements between $\mathrm{O}2\mathrm{}p$ orbitals are found to be responsible for the anisotropic $J$’s along leg and rung of the ladder; i.e., $J_{leg}>J_{\mathrm{rung}}.$ We find a unique dependence of the electronic structure of cuprates on the dimensionality.