Orbital Order and Superconductivity in Bilayer Nickelate Compounds

Abstract

We propose a theory for bilayer nickelate materials, where a large tetragonal field - intrinsic or induced by epitaxial strain - lifts the orbital degeneracy and localizes the $3z^2-r^2$ orbital states. These states host local spins $S=1/2$ bound into singlets by strong interlayer coupling, and their dynamics is described by weakly dispersive singlet-triplet excitations ("triplons"). The charge carriers occupy the wide bands of $x^2-y^2$ symmetry, and their Cooper pairing is mediated by the high-energy triplon excitations. As the $x^2-y^2$ band filling increases, i.e., moving further away from the Ni$^{3+}$ valence state, the indirect Ruderman-Kittel-Kasuya-Yosida interactions between local spins induce spin-density-wave order via triplon condensation. Implications of the model for compressively strained La$_3$Ni$_2$O$_7$ films and electron doped oxychloride Sr$_3$Ni$_2$O$_5$Cl$_2$ are discussed.

Figures (4)

• Figure 1: Schematic of the model for a bilayer nickelate, where large $e_g$ orbital splitting $\Delta$ (see inset) stabilizes the $3z^2\!-\!r^2$ states hosting local spins $\boldsymbol{S}_1$ and $\boldsymbol{S}_2$. The strong interlayer exchange coupling $J_c$ (red wavy line) binds $\boldsymbol{S}_1$ and $\boldsymbol{S}_2$ into a singlet pair, forming a nonmagnetic background for a motion of the conduction electrons in the $c_1$ and $c_2$ bands of $x^2\!-\!y^2$ symmetry. The itinerant spins $\boldsymbol{S}^c_1$ and $\boldsymbol{S}^c_2$ couple to local spins by ferromagnetic interaction $-J_{\boldsymbol{q}}$ (blue wavy lines).
• Figure 2: (a) Momentum dependence of the singlet-triplet excitation energy $\omega_{\boldsymbol{q}}$, calculated with $t=0.6\:\mathrm{eV}$, $U=4.5\:\mathrm{eV}$, and $J_\mathrm{H}=0.6\:\mathrm{eV}$ along $\Gamma(0,0)$---$\mathrm{X}(\pi,0)$---$\mathrm{M}(\pi,\pi)$ path. (b) Schematic of the bonding $\alpha$ and antibonding $\beta$ band energies $\varepsilon_{\boldsymbol{k}}$ split by interlayer hopping $t_{\perp}$TBnote. The chemical potential $\mu$ is determined by the $x^2\!-\!y^2$ band filling $\delta$. (c) Fermionic self-energy; the wavy line is the triplon propagator $\Phi$ describing singlet-triplet fluctuations, and the blue dots represent the spin exchange coupling $J_{\boldsymbol{k},\boldsymbol{k}'}$ between the local and itinerant electrons. (d) Pairing interaction mediated by the singlet-triplet fluctuations.
• Figure 3: (a) Triplon propagator $\Phi$ including the coupling $J_{\boldsymbol{k},\boldsymbol{k}'}$ (blue dots) between the local and itinerant spins. The self-energy depends on the $x^2\!-\!y^2$ band spin susceptibility $\widetilde{\chi}_{\boldsymbol{q},\omega}$, enhanced by Hubbard correlations (shaded vertex) between conduction electrons. (b) The triplon self-energy at $\omega=0$ for different dopings $\delta$ away from the Ni$^{3+}$ valence state. At $\delta_\mathrm{crit}\simeq 0.4$, the self-energy reaches the dashed curve $\Lambda_{\boldsymbol{q}}$ at some $\boldsymbol{q}$ between $\Gamma$ and $\mathrm{M}$ points, and triplons condense into SDW order. Parameters used: $J_\mathrm{H}=0.6\:\mathrm{eV}$, $\kappa=0.5$, and $t=0.6\:\mathrm{eV}$. (c) Phase behavior of the $3z^2\!-\!r^2$ orbital-ordered bilayer nickelate as a function of Ni valence. Close to the Ni$^{3+}$ end, the $3z^2\!-\!r^2$ local spins are bound into the interlayer singlet pairs, while the $x^2\!-\!y^2$ band electrons form a Fermi-liquid; singlet-triplet excitations (triplons) mediate their Cooper pairing. At large doping, $\delta\!>\!\delta_\mathrm{crit}$, the intralayer RKKY interactions between local spins, mediated by the band electrons, induce magnetic order via triplon condensation.
• Figure S1: Band dispersions and Fermi surfaces for the tight-binding model \ref{['eq:TB']}: (a) Basic nearest-neighbor model with $t_1=0.45\:\mathrm{eV}$ and $t_\perp=0.1\:\mathrm{eV}$. (b) Model including further-neighbor hoppings to reproduce the observed Fermi surfaces: $t_1=0.45$, $t_2=-0.12$, $t_\perp=0.14$, $t_{2\perp}=-0.07$, and $t_{3\perp}=0.035$ (in units of eV). The band filling corresponds to $0.5$ electron per site.