Chiral plaquette polaron theory of cuprate superconductivity

Abstract

Ab initio density functional calculations on explicitly doped ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ 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 $\mathrm{O}\phantom{\rule{0.2em}{0ex}}{p}_{z}$, planar $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}{d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$, and planar $\mathrm{O}\phantom{\rule{0.2em}{0ex}}{p}_{\ensuremath{\sigma}}$. This is in contrast to the assumption of hole doping into planar $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and $\mathrm{O}\phantom{\rule{0.2em}{0ex}}{p}_{\ensuremath{\sigma}}$ orbitals as in the $t\text{\ensuremath{-}}J$ 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 $x\ensuremath{\approx}0.05$ for ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$, a $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and planar $\mathrm{O}\phantom{\rule{0.2em}{0ex}}{p}_{\ensuremath{\sigma}}$ band is formed. The computed percolation doping of $x\ensuremath{\approx}0.05$ equals the observed transition to the ``metallic'' and superconducting phase for ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$. 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, ${J}_{dd}\ensuremath{\sim}0.1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, 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 ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$. The integrated imaginary susceptibility, observed by neutron spin scattering, satisfies $\ensuremath{\omega}∕T$ 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 $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}{d}^{9}$ spins is computed for classical spins on large two-dimensional lattices and is found to be incommensurate with a separation distance from $(\ensuremath{\pi}∕a,\ensuremath{\pi}∕a)$ given by $\ensuremath{\delta}Q\ensuremath{\approx}(2\ensuremath{\pi}∕a)x$, where $x$ is the doping. When the perturbed ${x}^{2}\ensuremath{-}{y}^{2}$ band energy in mean field is included, incommensurability along the Cu-O bond direction is favored. A resistivity $\ensuremath{\sim}{T}^{\ensuremath{\mu}+1}$ arises when the polaron energy separation density is of the form $\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{\mu}}$ due to Coulomb scattering of the ${x}^{2}\ensuremath{-}{y}^{2}$ band with polarons. A uniform density leads to linear resistivity. The coupling of the ${x}^{2}\ensuremath{-}{y}^{2}$ band to the undoped $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}{d}^{9}$ 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.