Orbital Ordering, New Phases, and Stripe Formation in Doped Layered Nickelates

Ground-state properties of layered nickelates are investigated based on the orbital-degenerate Hubbard model coupled with lattice distortions, by using numerical techniques. The Néel state composed of spin $S=1$ ions is confirmed in the undoped limit $x=0$. At $x=1/2$, novel antiferromagnetic states, called CE- and E-type phases, are found by increasing the Hund’s coupling. $(3x^2-r^2/3y^2-r^2)$-type orbital ordering is predicted to occur in a checkerboard-type charge-ordered state. At $x=1/3$, both Coulombic and phononic interactions are found to be important, since the former stabilizes the spin stripe, while the latter leads to the striped charge order.

Figure 1

(a) Spin correlation $S(\mathit{q}\mathbf{)}$ vs $\Delta$ for $x=0$. (b) Two kinds of local $e_g$-electron arrangements for $x=0$. (c) AFM spin pattern theoretically determined for $\Delta \lesssim 3$.

Figure 2

(a) Ground-state phase diagram at $x=1/2$. (b) $S(\mathit{q}\mathbf{)}$ for the CE- and E-type phases, at the couplings indicated. (c) Spin and charge patterns for the CE- and E-type phases. These are schematic views, since local charge densities in practice are not exactly 1 and 2. (d) Numerically obtained cooperative distortion pattern for an 8-site lattice at $x=1/2$. Solid and open circles indicate Ni and O ions, respectively. Open symbols indicate $e_g$ orbitals in the optimized state. (e) Total ground-state energy vs $d$ for $x=1/2$. (f) Orbital densities $⟨{\tau }_{z\mathbf{i}}⟩$ and $⟨{\tau }_{x\mathbf{i}}⟩$ for sites 1–4. See (d) for the site labels. Optimized orbitals at $d=0.3$ for sites 1 and 3 are also shown.

Figure 3

(a) Zigzag 6-site cluster covering the 2D lattice. Solid circles denote Ni ions, and dashed lines indicate hole positions. (b) Phase diagram at $x=1/3$. Each phase is characterized by the momentum that shows a peak in $S(q)$. (c) $S(q)$ and (d) $C(q)$ vs $J$ for $U^{\prime }=6$ and $\Delta =0.5$. (e) Cooperative distortion pattern for the zigzag 6-site cluster at $x=1/3$. (f) Total ground-state energy and (g) $C(q)$ vs $d$ for $x=1/3$.