Stability analysis of special-point ordering in the basal plane inYBa2Cu3O7−δ

It is shown that the Cu-O basal plane of $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ can undergo three types of ordering wave instabilities at the $k$ points $〈0, 0〉$, $〈\frac{1}{2}, 0〉$, or $〈\frac{1}{2}, \frac{1}{2}〉$ depending on the values of the effective pair interactions between filled and empty oxygen sites. The $〈0, 0〉$ instability gives rise to the observed chains of oxygen atoms along either the $a$ or the $b$ axis of the crystals. The $〈\frac{1}{2}, 0〉$ and $〈\frac{1}{2}, \frac{1}{2}〉$ instabilities produce doubling and quadrupling of the unit cell, respectively. Comparison with experiments suggests the necessity for interactions beyond nearest neighbors.