Small polarons and magnetic antiphase boundaries in Ca2-xNaxCuO2Cl2 (x=0.06, 0.12):: Origin of striped phases in cuprates

We report hybrid density functional theory calculations on hole doped Ca2-xNaxCuO2Cl2 performed in 4 x 4, 4 root 2 x 4 root 2, and 8 x 2 supercells with hole concentrations x=0.0625 and x=0.125. Holes at the lower concentration form small polarons, in which the hole is mainly localized on four oxygen ions surrounding one copper ion. The polaron is a spin one-half ferromagnetic polaron (Cu5O4), in which the moment on the central copper ion is parallel to those on the four neighboring copper ions and the moment on the oxygen ions is opposed to that on the copper ions. This is therefore an Emery-Reiter spin polaron rather than a Zhang-Rice singlet. At the higher hole concentration (x=0.125), many cuprates form stripes. Hybrid density functional theory calculations on linear chains of spin polarons separated by 4a(0) show a group of bands localized mainly on the stripe. Spins on neighboring copper ions in the stripe are parallel and so the stripe forms a magnetic antiphase boundary between antiferromagnetically ordered blocks of copper spins. Stripes of this kind, which run in one direction only, may explain recent scanning tunneling microscopy data from Ca2-xNaxCuO2Cl2 by Kohsaka et al. [Science 315, 1380 (2007)]. We also consider an ordered spin polaron phase where magnetic antiphase boundaries intersect at right angles. In this case, sets of four copper ions in squares at stripe intersections have parallel spins. This phase may be the 4 x 4 checkerboard pattern reported by Hanaguri et al. [Nature (London) 430, 1001 (2004)].