Ab initio density functional calculations on explicitly doped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ find that doping creates localized holes in out-of-plane orbitals. A model for cuprate superconductivity is developed based on the assumption that doping leads to the formation of holes on a four-site Cu plaquette composed of the out-of-plane $A_1$ orbitals apical $\mathrm{O}{p}_z$, planar $\mathrm{Cu}{d}_{3z^2-r^2}$, and planar $\mathrm{O}{p}_{\sigma }$. This is in contrast to the assumption of hole doping into planar $\mathrm{Cu}{d}_{x^2-y^2}$ and $\mathrm{O}{p}_{\sigma }$ orbitals as in the $t\text{−}J$ model. Allowing these holes to interact with the $d^9$ spin background leads to chiral polarons with either a clockwise or anticlockwise charge current. When the polaron plaquettes percolate through the crystal at $x\approx 0.05$ for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$, a $\mathrm{Cu}{d}_{x^2-y^2}$ and planar $\mathrm{O}{p}_{\sigma }$ band is formed. The computed percolation doping of $x\approx 0.05$ equals the observed transition to the “metallic” and superconducting phase for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$. Spin exchange Coulomb repulsion with chiral polarons leads to $d$-wave superconducting pairing. The equivalent of the Debye energy in phonon superconductivity is the maximum energy separation between a chiral polaron and its time-reversed partner. This energy separation is on the order of the antiferromagnetic spin coupling energy, $J_{dd}\sim 0.1\mathrm{eV}$, suggesting a higher critical temperature. An additive skew-scattering contribution to the Hall effect is induced by chiral polarons and leads to a temperature dependent Hall effect that fits the measured values for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$. The integrated imaginary susceptibility, observed by neutron spin scattering, satisfies $\omega ∕T$ scaling due to chirality and spin-flip scattering of polarons along with a uniform distribution of polaron energy splittings. The derived functional form is compatible with experiments. The static spin structure factor for chiral spin coupling of the polarons to the undoped antiferromagnetic $\mathrm{Cu}{d}^9$ spins is computed for classical spins on large two-dimensional lattices and is found to be incommensurate with a separation distance from $(\pi ∕a,\pi ∕a)$ given by $\delta Q\approx (2\pi ∕a)x$, where $x$ is the doping. When the perturbed $x^2-y^2$ band energy in mean field is included, incommensurability along the Cu-O bond direction is favored. A resistivity $\sim T^{\mu +1}$ arises when the polaron energy separation density is of the form $\sim {\Delta }^{\mu }$ due to Coulomb scattering of the $x^2-y^2$ band with polarons. A uniform density leads to linear resistivity. The coupling of the $x^2-y^2$ band to the undoped $\mathrm{Cu}{d}^9$ spins leads to the angle-resolved photoemission pseudogap and its qualitative doping and temperature dependence. The chiral plaquette polaron leads to an explanation of the evolution of the bilayer splitting in Bi-2212.

• Received 2 June 2006

DOI:https://doi.org/10.1103/PhysRevB.76.014514