We have simulated the phase diagram and structural properties of the oxygen ordering in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ testing simple extensions of the asymmetric next-nearest-neighbor Ising (ASYNNNI) model. In a preliminary paper [Phys. Rev. B 60, 110 (1999)] we demonstrated that the inclusion of a single further neighbor interaction gave results that could account for several hitherto unexplained structural observations. In this paper we make an extensive study showing that these results are robust against further extensions. Additional inclusion of three-dimensional or infinite range interactions does not obviate the finding of the following results: the existence of ortho-III, ortho-V, and ortho-VIII phase correlations; suppression of the ortho-I–ortho-II transition temperature relative to that of the tetragonal–ortho-I transition; no ortho-II Bragg peaks, but a crossover from Lorentzian toward Lorentzian squared line shapes; and a finite average chain length even at low temperatures. Simulations with the extended (ASYNNNI) model yield a realistic picture of the oxygen order of importance for the understanding of the hole doping. The experimental facts are consistent with a low temperature structure broken up on a nano scale into box-like domains and anti-domains of typical average dimension $\mathrm{}(10a,30b,2c).$ Theory and model simulations demonstrate that the distribution of such domains causes deviations from Lorentzian line shapes, and not the Porod effect. Analytic theory is used to estimate the effect of a range of values of the interaction parameters used, as well as the effect of an extension to include infinite ranged interactions. In the experiments a large gap is found between the onset temperatures of the ortho-I and ortho-II orders at $x=\mathrm{0.5.}\mathrm{}$ This cannot be fully reproduced in the simulations. The simulations yield a quite symmetric phase diagram around $x=0.5\mathrm{}$ with respect to the low temperature phases, whereas experimentally it has not been possible to detect such phases for small-$x$ values.

• Received 27 February 2001

DOI:https://doi.org/10.1103/PhysRevB.64.224520