Effect of dZ2‐Cu and p‐O Orbital Mixing in the High‐Tc Superconducting Materials. A Three‐Dimensional Anisotropic Interaction

A three-dimensional (3D) anisotropic interaction is obtained for the oxide high-Tc superconductors when we take into account the mixing of dZ2-Cu with the p-O orbitals. Two hopping matrix elements, tl, on the plane, and t2, along the z-axis, are used to study anisotropy in the original Hubbard Hamiltonian. The derived interaction contains a local Hubbard term, U, and two non-local superexchange contributions, J1, and J2. Both J1 and J2 strongly depend on the anisotropy ratio (α ≡ t2/tl). However, for α = 0, J2, 0, and J1, the superexchange on the plane, coincides with results previously published in 2D. Furthermore, when α = 1, J1 ⟺ J2. The latter result is also known in the literature. The effective Hamiltonian presented here is more physical for studying high-Tc superconducting materials, since the third dimension is playing an increasing role as it has been pointed out by several authors. It is also found that the hole occupation on the p,-orbital (〈npz〉), for α ≪ 1, is related to the hole occupation on the px- and py-orbitals, by means of the relationship 〈npz〉 ≈ α2 〈npx〉, which is in qualitative agreement with experimental and theoretical findings that 〈npz〉 is very small. The calculated Tc versus n shows agreement with recent experiments in La2−xSrxCuO4.