Abstract
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 $\mathrm{Cu}3d$ and $\mathrm{O}2p$ orbitals and between $\mathrm{O}2p$ 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 $J.$ The same mechanism is applied to hopping matrix elements in the ladder cuprates. The elements between $\mathrm{O}2p$ orbitals are found to be responsible for the anisotropic $J$'s along leg and rung of the ladder; i.e., ${J}_{leg}g{J}_{\mathrm{rung}}.$ We find a unique dependence of the electronic structure of cuprates on the dimensionality.
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