# Magnetic Frustration Model and Superconductivity on Doped Lamellar CuO_{2} Systems

## Extended Abstract

In most of the doped lamellar CuO_{2} based high temperature superconductors, the behavior varies strongly as function of the dopant concentration, which is directly related to the concentration of the electronic holes. In the present paper we emphasize the interplay among the various phases which appear in the temperature - concentration phase diagram.

Without doping, these systems are insulating antiferromagnets. The antiferromagnetic exchange interactions between the localized spin-1/2 copper ions are strong in the CuO_{2} planes, and weak among the planes. The quantum spin fluctuations can be renormalized, and the behavior in each plane is excellently described by an effective classical two dimensional Heisenberg antiferromagnetic model.^{1} Recently,^{2} we have shown that the weak coupling between the planes is very well described by a mean field theory. In La_{2}CuO_{4}, the orthorhombic rotation of the CuO_{6} octahedra generates an additional antisymmetric Dzyaloshinskii-Moriya interaction between the Cu spins in the planes. This results in a bilinear coupling between the staggered magnetization perpendicular to the plane, yielding a canting of the magnetic moments out of the planes. At zero magnetic field, the ferromagnetic moments of neighboring planes order antiparallel to each other. However, external magnetic fields cause a variety of spin flip transitions._{2,3} The bilinear coupling between the staggered and the uniform magnetizations implies an indirect coupling between the external uniform field and the staggered magnetization, explaining the unusual sharp peak in the uniform susceptibility at the Neél point.^{2} Our mean field theory for weakly coupled planar Heisenberg models fits the susceptibility data, with practically no adjsutable parameters.^{2} Similar theories should work for all other properties of these weakly coupled planar systems, e.g. in the superconducting phase.

Doping introduces quenched randomness, as well as deviations from stoichiometry, usually adding electronic holes, which reside mainly on the oxygen ions in the planes. At low concentration, these holes are localized, exhibiting variable range hopping conductivity.^{4} The holes always remain localized in the planes, with a small perpendicular localization length. Within the plane, the localization length at low doping is of order 10Å (about 2.5 lattice constants). In La_{2}CuO_{4}, the hole conductivity is very sensitive to the external magnetic field, changing by about a factor 2 at the spin flip transition.^{2,3} This shows a strong coupling between the hole states and the magnetic ordering.

In order to discuss the coupling of the hole to the underlying magnetic ordering of the copper spins, we considered^{5} a single hole, localized at an oxygen site. There is a strong superexchange coupling of the spin of this hole to those of the two neighboring copper ions, resulting in an effective ferromagnetic exchange interaction between these copper spins. The exchange interaction between the hole spin and the copper spins will be ferromagnetic (antiferromagnetic) if the hole sits mainly on a p_{π} (p_{σ}) oxygen state, with the wave function mainly perpendicular (parallel) to the bond. The resulting three-spin state will have total spin 3/2 (or 1/2). It is not yet completely clear which state wins. However, both situations create ferromagnetic Cu-Cu interactions, and frustrate the antiferromagnetic copper state. We believe that this frustration is responsible to the fast decrease of the Ne61 temperature with doping. We also predicted^{5} that at higher doping, the antiferromagnetic phase should be replaced by a spin glass phase, as indeed confirmed by many experiments. Alternative pictures, which couple the hole into a singlet state with one copper ion, do not contain the frustration necessary for the spin glass.

The frustration model implies that the hole spin prefers to be perpendicular to the copper spins, and may help in understanding the sensitivity of the hole mobility to the spin ordering. In addition, the disturbance to the copper antiferromagnetic ordering is smaller if two holes sit on neighboring or next nearest neighboring (nnn) oxygens. Since the former seems to be excluded by Coulomb repulsion, we considered the attractive pairing potential between two holes on nnn oxygens.^{6} This potential decays with increasing doping, as the antiferromagnetic correlation length decreases. Assuming that the holes move freely in the p-band, we used a BCS theory with the cutoff set at the Fermi energy (proportional to the number of mobile holes).^{6} Adjusting only the temperature scale, we were able to reproduce the concentration dependence of the superconducting transition temperature of La_{2-x}Sr_{x}CuO_{4}, including the peak at around x ≃ 0.15 and the decay for larger x.^{7} More recently,^{8} we also considered the tight binding band structure for the motion of the Cu-O-Cu. “polaron”, and the results are qualitatively similar.

### Keywords

Thio La2CuO4### References

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