Metal-insulator transition in a two-band model for the perovskite nickelates

Motivated by recent Fermi-surface and transport measurements on LaNiO${}_3$, we study the Mott metal-insulator transitions of perovskite nickelates, with the chemical formula $R$NiO${}_3$, where $R$ is a rare-earth ion. We introduce and study a minimal two-band model, which takes into account only the e${}_g$ bands. In the weak to intermediate correlation limit, a Hartree-Fock analysis predicts charge and spin order consistent with experiments on $R=\text{Pr}$, Nd, driven by Fermi surface nesting. It also produces an interesting semimetallic electronic state in the model when an ideal cubic structure is assumed. We also study the model in the strong-interaction limit and find that the charge and magnetic order observed in experiment exist only in the presence of very large Hund’s coupling, suggesting that additional physics is required to explain the properties of the more insulating nickelates, $R=\text{Eu}$, Lu, Y. Next, we extend our analysis to slabs of finite thickness. In ultrathin slabs, quantum confinement effects substantially change the nesting properties and the magnetic ordering of the bulk, driving the material to exhibit highly anisotropic transport properties. However, pure confinement alone does not significantly enhance insulating behavior. Based on these results, we discuss the importance of various physical effects and propose some experiments.

Figure 1

Fermi surfaces for the tight-binding model. In (a) and (b), we show the conduction and valence band Fermi surfaces, respectively, for $t^{\prime }/t=0$. For larger $t^{\prime }/t$, the conduction band Fermi surfaces become large and holelike, as shown in (c) and (d) for $t^{\prime }/t=0.15$. The approximate nesting in the latter case is indicated schematically in (d).

Figure 2

Zero-frequency spin susceptibility for the tight-binding Hamiltonian for $t^{\prime }/t=0.05$, 0.1, and 0.15, as a function of momentum $\mathbf{k}$ in the cubic Brillouin zone. Note that for the best nested situation, $t^{\prime }/t=0.15$, the susceptibility is sharply peaked close to the wavevector $2\pi (\frac{1}{4},\frac{1}{4},\frac{1}{4})$.

Figure 3

Spin configurations depending on the phase of $\psi$, $\theta$, along $\widehat{x}$ axis. (a) Shows “site-centered” spin ordering for $\theta =0$, (b) for intermediate $\theta =\pi /8$, and (c) is “bond-centered” ordering for $\theta =\pi /4$.

Figure 4

MIT bulk phase diagram for the model with cubic symmetry as a function of $U/t$ and $J_H/t$, with $t^{\prime }/t=0.15$. Here, $U$ is the on-site Coulomb interaction, $J_H$ is the Hund’s coupling, and $t$ is the nearest-neighbor hopping magnitude. The main phases that appear are paramagnetic metallic state (M), metallic SDW (wavy region close to M), insulating site-centered SDW (S-SDW), and semimetallic bond-centered SDW (B-SDW). In between the S-SDW and B-SDW state, one observes an off-center SDW phase with $0<\theta <\pi /4$ (shaded region). The black colored shapes show the points for which the optical conductivity is plotted in Sec. 6.

Figure 5

Dispersion [panels (a), (b), and (c)] and density of states [DOS, panels (d), (e), and (f)] for the e${}_g$ tight-binding model on honeycomb lattice. In (a) and (d), $t=1$ and $t^{\prime }=0$, we observe Dirac points with clear linear dispersion and corresponding linear DOS. The Dirac cone is stable to small $t^{\prime }=0.15$ as shown in (b) and (e). In (c) and (f), an orbital splitting induced by the orthorhombic distortion of the lattice is included, with $t^{\prime }=0.15$ and $\mathbf{D}=1.5/\sqrt{3}(1,1,1)$. An induced gap is clearly seen.

Figure 6

Plot of the single-particle gap $\Delta$ vs the orbital field $\left|D\right|$, for the honeycomb model with nearest-neighbor $t=1$ and $t^{\prime }=0.15$. Here, we have arbitrarily taken the orbital field of the form $\mathbf{D}=(D,D,D)/\sqrt{3}$.

Figure 7

Phase diagram of the classical ground state as a function of two dimensionless parameters $\alpha \equiv U/J_H$ and $\beta \equiv t^{\prime }{J}_H/t^2$.

Figure 8

Panel (a) shows the zero-temperature phase diagram for a single layer $L=1$ with the nesting vector ${\mathbf{Q}}_{\text{sdw}}^{2\mathrm{d}}=2\pi (\frac{1}{4},\frac{1}{4})$. As in the bulk case, the paramagnetic metallic phase (M) is stable for weak interactions. The wavy region indicates a metallic SDW state, and the shaded region indicates an insulating off-center SDW. Panel (b) shows the phase diagram for three layers, $L=3$, with the nesting vector ${\mathbf{Q}}_{\text{sdw}}^{L=3}=2\pi (\frac{1}{4},0,0)$. This nesting vector leads to a metallic B-SDW phase, which persists even for large $U/t$. The dark gray region between the S-SDW and B-SDW phases is a metallic off-center SDW.

Figure 9

The zero-frequency spin susceptibility, ${\chi }_0\left(\mathbf{k}\right)$, for finite thickness slabs of $L$ layers, for the free-electron tight-binding Hamiltonian with $t^{\prime }/t=0.15$. Plot (a) shows the purely two-dimensional single-layer case. Here, ${\chi }_0\left(\mathbf{k}\right)$ is sharply peaked at $\mathbf{k}={\mathbf{Q}}_{\text{sdw}}^{2d}=\pi /2\left(11\right)$. Plot (b) shows several cases with varying thickness with $2\le L\le 30$, compared with the bulk case $L=\infty$. One sees that large $L=30$ (filled circles) agrees well with the bulk susceptibility (black solid line). For smaller $L$, we see that the nesting properties change considerably. This is especially pronounced for $L=3$ (filled squares), for which ${\chi }_0\left(\mathbf{k}\right)$ is sharply peaked at $\mathbf{k}=\mathbf{Q}\approx \pi /2\left(100\right)$ and for $L=4$ (open squares), for which it is peaked at $\mathbf{k}\approx 0$.

Figure 10

The Fermi surfaces for different subbands in the $k_x$-$k_y$ plane with discretized $k_z=\pi l_z/(L+1)$ for the cases $L=3,4$, and 5. The dotted (solid) lines correspond to conduction (valence) subbands. In the case $L=4$, we see two valence subband Fermi surfaces at $k_z=0.2\pi$ ($l_z=1$), which is responsible for an enhancement of the DOS at Fermi energy.

Figure 11

Bulk conductivity anisotropy in the B-SDW state as a function of the amplitude $\left|\psi \right|$ of the SDW order parameter.

Figure 12

Electrical conductivity for $L=3$ and bulk with fixed $\tilde{t}=1,{\tilde{t}}^{\prime }=0.15,\theta =\pi /4$, and $\Phi =0$. The isotropic conductivity in the bulk case (dotted line), ${\sigma }^{\text{Bulk}}$, decreases to zero as a gap in the DOS develops with increasing SDW order. For the three-layer case, $L=3$, the conductivity shows a large anisotropy in the $xy$ plane once the SDW develops.

Figure 13

The real part of the optical conductivity, $\sigma \left(\omega \right)$, for each phase (black star) in the bulk phase diagram (see Fig. 4). In (a), the paramagnetic metallic phase shows a large Drude peak and a small hump (see inset plot). The hump is related to a region of large DOS for interband transitions (see Ref.11). In (b), the metallic SDW phase has a reduced but nonzero Drude peak and a small second peak due to the SDW gap. Plot (c) shows the case of the semimetallic B-SDW phase, for which a linear increase of $\sigma \left(\omega \right)$ for small frequency $\omega$ is found, related to the linear dispersion near the Fermi level. It also shows strong anisotropy between the conductivity ${\sigma }_{{\left[111\right]}_{\parallel }}$ (along the [111] direction) (solid line) and ${\sigma }_{{\left[111\right]}_{\perp }}$ (perpendicular to [111]) (dashed line). In plot (d), a large gap is visible in the S-SDW phase.