Charge Disproportionation without Charge Transfer in the Rare-Earth-Element Nickelates as a Possible Mechanism for the Metal-Insulator Transition

We study a model for the metal-insulator ($M\text{−}I$) transition in the rare-earth-element nickelates $R{\mathrm{NiO}}_3$, based upon a negative charge transfer energy and coupling to a rocksaltlike lattice distortion of the ${\mathrm{NiO}}_6$ octahedra. Using exact diagonalization and the Hartree-Fock approximation we demonstrate that electrons couple strongly to these distortions. For small distortions the system is metallic, with a ground state of predominantly $d^{8}L$ character, where $\underset{\_}{L}$ denotes a ligand hole. For sufficiently large distortions ($\delta d_{\mathrm{Ni}\text{−}\mathrm{O}}\sim 0.05–0.10\AA$), however, a gap opens at the Fermi energy as the system enters a periodically distorted state alternating along the three crystallographic axes, with $(d^8{\underset{\_}{L}}^2{)}_{S=0}(d^8{)}_{S=1}$ character, where $S$ is the total spin. Thus the $M\text{−}I$ transition may be viewed as being driven by an internal volume “collapse” where the ${\mathrm{NiO}}_6$ octahedra with two ligand holes shrink around their central Ni, while the remaining octahedra expand accordingly, resulting in the ($1/2$, $1/2$, $1/2$) superstructure observed in x-ray diffraction in the insulating phase. This insulating state is an example of charge ordering achieved without any actual movement of the charge.

Figure 1

(a) ${\mathrm{Ni}}_2{\mathrm{O}}_{10}$ quasi-1D cluster with periodic boundary conditions along the $y$ axis, studied with ED. Both Ni $3d$ $e_g$ orbitals are included at each Ni site, but only one orbital per Ni is depicted here to establish our phase convention. (b) ${\mathrm{Ni}}_2{\mathrm{O}}_6$ unit cell used for our 3D calculations using HFA. The arrows in both panels indicate the displacement pattern for the rocksaltlike oxygen distortion.

Figure 2

A summary of ED results for the ${\mathrm{Ni}}_2{\mathrm{O}}_{10}$ cluster. (a) Electronic contribution to the ground state energy in the ($3\uparrow 3\downarrow$) (blue circle) and ($4\uparrow 2\downarrow$) (red square) sectors as a function of the static O displacement. (b),(c) Average hole occupancy at Ni and O sites, respectively. Sites on the left (right) side of the ${\mathrm{Ni}}_2{\mathrm{O}}_{10}$ cluster correspond to the compressed (expanded) ${\mathrm{NiO}}_6$ octahedron for $\delta d>0$ (see Fig. 1), and are represented by the solid (open) symbols.

Figure 3

(a) A comparison of the HFA [red (light)] and ED [black (dark)] ground state energies as a function of the oxygen displacement for the ${\mathrm{Ni}}_2{\mathrm{O}}_6$ cluster, containing only $\mathbf{k}=0$. (b)–(e) HFA results for the 3D lattice with ${80}^3/2$ $\mathbf{k}$ points in the Brillouin zone: (b) Ground state electronic energy, (c) Average hole occupation of the oxygen (blue) sites and the compressed (black) and expanded (red) Ni sites. Note that the lattice distortion retains the local cubic symmetry about each Ni site so no orbital polarization of the Ni $e_g$ orbitals occurs. (d) Magnetization $m_z=(n_{\uparrow }-n_{\downarrow })$ of the Ni orbitals. (e) single particle density of states for several oxygen displacements. The spectrum has been lifetime broadened with a Lorentzian (half width at half maximum $=25\mathrm{meV}$). The inset shows the gap as a function of displacement.