Origin of strong correlations and superconductivity in ${\text{Na}}_{\mathbf{x}}{\text{CoO}}_2$

We propose a minimal model resolving a puzzle of enigmatic correlations observed in sodium-rich ${\text{Na}}_x{\text{CoO}}_2$ where one expects a simple, free motion of the dilute $S=1/2$ holes doped into a band insulator ${\text{NaCoO}}_2$. The model also predicts singlet superconductivity at experimentally observed compositions. The model is based on a key property of cobalt oxides—the spin-state quasidegeneracy of ${\text{CoO}}_6$ octahedral complex—leading to an unusual physics of, e.g., ${\text{LaCoO}}_3$. We show that correlated hopping between $t_{2g}$ and $e_g$ states leads to the spin-polaron physics at $x\sim 1$, and to an extended $s$-wave pairing at larger doping when coherent fermionic bands are formed.

Figure 1

(a) Electron hopping from $t_{2g}$ to $e_g$ orbital via oxygen atoms in the case of $90°$ bonds creating $S=1$ $t_{2g}^5{e}_g$ configuration of ${\text{Co}}^{3+}$. The $t_{2g}$ orbital laying in the plane (here the $xy$ orbital) couples to the out-of-plane $e_g$ $\left(3z^2-r^2\right)$ orbital. (b) The coupling to the planar orbital is zero because of the destructive interference of two channels. (c) Bond directions and corresponding angles in the hexagonal lattice of Co ions and $\tilde{t}$-active orbitals on these bonds.

Figure 2

(a) Spectral functions of a ${\text{Co}}^{4+}$ hole doped in ${\text{NaCoO}}_2$ along $M\text{−}\Gamma \text{−}K$ path in the Brillouin zone [inset (ii)]. Bare and renormalized dispersions, measured both from the chemical potential for better comparison, are shown. (The polaron binding-energy shift is $E_b\simeq 2.6t$). (i) The diagram describing the dressing of a hole by $S=1$ $\mathcal{T}$ excitations. (b) Density of states compared to that of the bare band. Virtual processes associated with the polaronic spectral features are sketched: Every use of $\tilde{t}$ hopping channel generates $S=1$ states of ${\text{Co}}^{3+}$ behind the hole.

Figure 3

(a) The physical picture behind Eq. (3). $t_{2g}$ electron of a ${\text{Co}}_i^{3+}$ ion moves to $e_g$ level of the NN ${\text{Co}}_j^{4+}$ ion (process 1) and then to the $t_{2g}$ level of the next ${\text{Co}}_k^{4+}$ neighbor (process 2). This is depicted in (b) as a motion of the hole pair. (c) Singlet-pair motion via an intermediate state composed of $S=1$ ${\text{Co}}_j^{3+}$ ion ($\mathcal{T}$ exciton) and triplet state of the two holes on $⟨ik⟩$ bond. The relative amplitude $A$ resulting from spin algebra is indicated. (d) The triplet-pair hopping amplitude $A$ is 3 times smaller because of the destructive interference of contributions involving singlet and triplet $⟨ik⟩$ bond.

Figure 4

(a) Diagrammatic representation of RPA equations for the spin susceptibilities involving the interaction equation (5) between ${\mathit{s}}_{\mathit{q}}$ (local) and ${\tilde{\mathit{s}}}_{\mathit{q}}$ (nonlocal) spin densities that reside on sites and bonds, respectively. Bare and RPA enhanced susceptibilities are represented by empty and shaded bubbles, respectively. (b) Map of bare ${\chi }_s^{″}/\omega$ for $n_d=0.5$ (the average Co-valence 3.5) at $T=0.025t$ and $\omega =0.005t$. (c) Corresponding RPA-enhanced susceptibility calculated at ${\tilde{t}}^2/E_T=3t$. The interaction enhances the susceptibility at the $2k_F$ ring which (at given density $n_d=0.5$) nearly matches the Brillouin zone boundary.

Figure 5

(a) $T_c$ in the extended $s$-wave and $p$-wave channels. The complete profile of the dominant, $s$-wave $T_c$ curve is shown in the left inset together with ${\gamma }_{\mathit{k}}^2$ (in arbitrary units) on the Fermi surface. The dashed $\left(\beta V_C=1.5\right)$ and dotted $\left(\beta V_C=3\right)$ $T_c$ curves are calculated including NN Coulomb repulsion which reduces the pairing interaction $V$ at large $n_d$. Shaded regions indicate the observed competing orderings (including the spin-charge order at $n_d=0.5$). (b) Probability ratio $p\left(n_d\right)$ (see text for definition) renormalizing the pairing interaction at different values of NN Coulomb repulsion relative to the effective temperature $1/\beta \propto$ bandwidth. The feature at $n_d=1/3$ for large $V_C$ manifests a honeycomb-lattice formation where each ${\text{Co}}^{3+}$ $(○)$ has the maximum possible number of neighboring ${\text{Co}}^{4+}{\text{-Co}}^{4+}$ pairs $\left(●–●\right)$. Above $n_d=2/3$, ${\text{Co}}^{4+}$ holes can avoid each other completely if $V_C$ is sufficiently large.

Figure 6

(a) Band structure for different values of $t^{\prime }/t$ $\left(n_d=0.6\right)$ and the corresponding effect on the $s$-wave transition temperature. As the band dispersion near $\Gamma$ point comes closer to the Fermi level, it can exploit the larger form factor and $T_c$ increases as shown in the inset. The increase is also partially induced by the decreased Fermi velocity along $\Gamma \text{−}M$ direction. (b) Form factor of the extended $s$-wave pairing interaction. The inset shows the contours for ${\gamma }_{\mathit{k}}^2=1,2,3,4,5$.