Charge ordering in ${\mathrm{Ni}}^{1+}/{\mathrm{Ni}}^{2+}$ nickelates: ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ and ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$

Ab initio calculations allow us to establish a close connection between the Ruddlesden-Popper layered nickelates and cuprates not only in terms of filling of $d$ levels (close to $d^9$) but also because they show ${\mathrm{Ni}}^{1+}(S=1/2)/{\mathrm{Ni}}^{2+}(S=0)$ stripe ordering. The insulating charge-ordered ground state is obtained from a combination of structural distortions and magnetic order. The ${\mathrm{Ni}}^{2+}$ ions are in a low-spin configuration $\left(S=0\right)$ yielding an antiferromagnetic arrangement of ${\mathrm{Ni}}^{1+} S=1/2$ ions like the long-sought spin-1/2 antiferromagnetic insulator analog of the cuprate parent materials. The analogy extends further with the main contribution to the bands near the Fermi energy coming from hybridized Ni $d_{x^2-y^2}$ and O $p$ states.

Figure 1

Level scheme showing the possible spin states in the trilayer Ni compound La438 (three Ni atoms per formula unit (f.u.) are represented). On the left side, the purely ionic picture of the low-spin charge-ordered state, ${\mathrm{Ni}}^{2+}$ (low spin $S=0$) and $2 {\mathrm{Ni}}^{1+}\left(S=1/2\right)$, gives rise to an insulating state with a gap between the $d_{x^2-y^2}$ subbands as in the cuprates. In the center, the high-spin state gives rise to an insulating molecular state with a gap between $d_{z^2}$ subbands. On the right, the metallic low-spin state is shown.

Figure 2

Left panel: proposed charge/spin ordering pattern inside the ${\mathrm{NiO}}_2$ planes in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$, with ${\mathrm{Ni}}^{1+}$ in blue and ${\mathrm{Ni}}^{2+}$ in gray. Arrows correspond to up/down spins for the ${\mathrm{Ni}}^{1+}$ ions. Right panel: structure after relaxation showing the buckling in the outer layers and the Ni-O bond lengths.

Figure 3

Band structure of La438 obtained within GGA. Left panel: unrelaxed structure. Right panel: relaxed structure. The gap opening is caused by structural distortions and is formed solely within the $d_{x^2-y^2}$ manifold of states. Note that both panels are for the charge-ordered supercell.

Figure 4

Calculated orbital resolved $e_g$ density of states for Ni atoms in the trilayer for La438 obtained within GGA. Top curves, spin-up; bottom curves, spin-down. Left panel: Ni atoms in the outer layers (Ni1, Ni2, and Ni3, as shown in the central panel). Right panel: Ni atoms in the inner layer (Ni4, Ni5, and Ni6, as shown in the central panel). The central panel shows a 3D plot of the spin density in the striped ground state, with an isosurface at 0.1 e/Å${}^3$ obtained using the XCrysDen program [21]. Ni1 and Ni4 are the nonmagnetic ${\mathrm{Ni}}^{2+}$ ions. Different colors (shades of gray) represent the spin-up (spin-down) density.

Figure 5

Left panel: proposed charge and spin ordering pattern in the ${\mathrm{NiO}}_2$ planes for La326, with ${\mathrm{Ni}}^{1+}$ in blue, ${\mathrm{Ni}}^{2+}$ in gray. Arrows correspond to up/down spins for the ${\mathrm{Ni}}^{1+}$ ions. Right panel: Structure after relaxation showing the Ni-O bond lengths in plane and the buckling of the ${\mathrm{NiO}}_2$ layers.

Figure 6

Calculated orbital resolved $e_g$ density of states for La326 obtained within LDA+$U$ ($U=4.75$ eV). Top panel: ${\mathrm{Ni}}^{2+}$ atoms. Bottom panel: ${\mathrm{Ni}}^{1+}$ atoms. Top and bottom curves in each panel are majority and minority spin, respectively. ${\mathrm{Ni}}^{2+}$ is nonmagnetic. The ${\mathrm{Ni}}^{1+} d_{x^2-y^2}$ states are separated by the Mott-Hubbard gap $U$.