An effective single-band model for the cuprates is derived by a cell-perturbation method from a five-band model which includes ${\mathit{d}}_{3{\mathit{z}}^2\mathrm{-}{\mathit{r}}^2}$ orbitals on copper and ${\mathit{p}}_{\mathit{z}}$ orbitals on apical oxygen. In addition to the usual Zhang-Rice singlets of ${\mathit{A}}_1$ symmetry, there are two-hole cell states of ${\mathit{B}}_1$ symmetry, which can become low in energy and depend sensitively on the apical oxygen ions. Provided that hybridization with the apical oxygen orbital is sufficiently weak to permit reduction to a t-${\mathit{t}}^{\prime }$-J model, the main effect of the ${\mathit{B}}_1$-symmetry states is to renormalize the effective next-nearest-neighbor hopping (${\mathit{t}}^{\prime }$) of doped holes. This effect can be quite large and may even change the sign of ${\mathit{t}}^{\prime }$. The variation of ${\mathit{t}}^{\prime }$ between various compounds due to differences in crystal structure is shown to correlate with ${\mathit{T}}_{\mathit{c}}^{\max }$, the critical temperature at optimum doping, suggesting that ${\mathit{t}}^{\prime }$ may be a crucial parameter for the low-energy physics, which moreover differentiates between the various cuprates. The effective single-band model is shown to break down when the apex level approaches the in-plane oxygen level, and to describe that situation, which cannot be ruled out completely for the cuprates with present experimental evidence, we propose a specific minimal effective (two-band) model. © 1996 The American Physical Society.

• Received 21 November 1995

DOI:https://doi.org/10.1103/PhysRevB.53.8774