Hole interactions and the spin-spin correlation function in a CuO2 plane with completely localized oxygen holes.

Abstract

A model of an oxygen-hole-doped ${\mathrm{CuO}}_{2}$ plane is studied in the low-dopant-concentration limit where the approximation is made that the O holes are completely localized. An effective Hamiltonian is found that represents a nonbipartite lattice where only antiferromagnetic exchange interactions are present, thus leading to geometrical frustration (triangular coordination) that reduces the magnetic order. The diminution of antiferromagnetic order has been studied quantum mechanically for small clusters, and classically for much larger lattices, at zero temperature. The dramatic reduction of 〈\ensuremath{\Omega}^ $^{2}\mathrm{〉}$ with doping found in experiments is mimicked in this model, although the finite-size effects in such cluster studies are demonstrated to be very large. Ground states including quantum fluctuations substantially reduce the order parameter in comparison to systems of classical Heisenberg spins. The obtained ground-state configurations of the clusters composed of classical Heisenberg spins are not planar. When the behavior of the spin-spin correlation function with increasing dopant levels is studied, it is found that for all cluster sizes and dopant concentrations (x\ensuremath{\le}0.05) the correlation function is extremely well fitted by ${\mathit{C}}_{\mathit{r}}$\ensuremath{\sim}${\mathit{r}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$exp(-r/\ensuremath{\xi}).The implied correlation length \ensuremath{\xi} at such x agrees well with the measured correlation length of short-ranged antiferromagnetic order present in the spin-glass phase. As the O holes are added to the plane we also obtain results characterizing the interaction of such holes. Quantum mechanically we find that pairs of O holes can interact to form local spin-triplet ground states when the dipole moments characteristic of the individual holes are aligned in real space. When these interactions are studied for classical spins, it is found that these holes can add constructively to augment the dipolar distortion that would have been generated by two individual dipoles; also, they may interfere destructively so as to eliminate the dipolar field, and subsequently produce a shorter-ranged distortion with quadrapolar symmetry. All of the interacting hole behavior is consistent with the semiclassical theory of Shraiman and Siggia [Phys. Rev. Lett. 61, 467 (1988)] when the dipolar distortions are attributed not to mobile vacancies, but rather to quenched O holes.