Effects of reduced dimensionality, crystal field, electron-lattice coupling, and strain on the ground state of a rare-earth nickelate monolayer

Motivated by the potential for cupratelike superconductivity in monolayer rare-earth nickelate superlattices, we study the effects of crystal field splitting, lattice distortions, and strain on the charge, magnetic, and orbital order in undoped two-dimensional (2D) nickelate monolayers $R\mathrm{Ni}{\mathrm{O}}_3$. We use a two-band Hubbard model to describe the low-energy electron states, with correlations controlled by an effective Hubbard $U$ and Hund's $J$. The electrons are coupled to the octahedral breathing-mode lattice distortions. Treating the lattice semiclassically, we apply the Hartree-Fock approximation to obtain the phase diagram for the ground state as a function of the various parameters. We find that the 2D confinement leads to strong preference for the planar $d_{x^2-y^2}$ orbital even in the absence of a crystal-field splitting. The $d_{x^2-y^2}$ polarization is enhanced by adding a crystal field splitting, whereas coupling to breathing-mode lattice distortions weakens it. However, the former effect is stronger, leading to $d_{x^2-y^2}$ orbital and antiferromagnetic order at reasonable values of $U,J$ and thus to the possibility to realize cupratelike superconductivity in this 2D material upon doping. We also find that the application of tensile strain enhances the cupratelike phase and phases with orbital polarization in general, by reducing the $t_2/t_1$ ratio of next-nearest to nearest neighbor hopping. On the contrary, systems with compressive stress have an increased hopping ratio and consequently show a preference for ferromagnetic (FM) phases, including, unexpectedly, the out-of-plane $d_{3z^2-r^2}$ FM phase.

Figure 1

$T=0$ phase diagram of the 2D layer when ${\epsilon }_b={\mathrm{\Delta }}_{\text{CF}}=0$. $U$ and $J$ are scaled by $W$, the bandwidth of the noninteracting system. Solid (red) contours show the CD $\delta$. Dashed (purple) contours show the orbital polarization $O_{\text{FO}}$. Resolution is $30\times 30$. The four black symbols indicate the cases analyzed in more detail in Figs. 2, 3, 4, 5.

Figure 2

(a) Total density of states (DOS) per Ni site (in units of $1/t_1^{\left(0\right)}$), plotted against energy $E$ (in units of $t_1^{\left(0\right)}$), and (b) partial DOS per Ni site, projected onto the two orbitals, for the paramagnetic metallic phase $00,\overline{z}\overline{z}$ at $U/W=0.2,J/W=0.1$ (marked with a solid black circle in Fig. 1). The vertical red line marks the Fermi energy. In panel (b), the histograms' colors are labeled in the legend. Where they overlap, their transparency allows both histograms to be seen.

Figure 3

For the cupratelike, insulating AFM phase labeled $\uparrow \downarrow ,\overline{z}\overline{z}$ in Fig. 1, we show (a) the total DOS per Ni site, (b) the partial DOS projected onto the two orbitals, and (c), (d) the partial DOS projected by spin component, for the two inequivalent Ni sites. Both orbital contributions are included in the spin-resolved PDOS histograms, but one can infer the orbital character of the various bands from the results in panel (b). The vertical red line marks the Fermi energy. Histogram colors are labeled in the respective legends. There is no charge order in this phase, and the AFM order is clear from panels (c) and (d) as each site preferentially has either $\uparrow$ or $\downarrow$ occupied states below the Fermi energy. The parameters are $U/W=0.8,J/W=0.05$ (black star in Fig. 1).

Figure 4

For the half-metal phase $\uparrow \uparrow ,\overline{z}\overline{z}$: (a) total DOS per Ni site, (b) partial DOS with orbital projections, and (c) partial DOS with spin projections. Parameters are $U/W=0.8,J/W=0.1$ (black square in Fig. 1). The orbital polarization already seen in the metallic phase in Fig. 2 is joined by FM polarization, as seen in panel (c).

Figure 5

(a) Total DOS per Ni site (averaged over two inequivalent sites) in the charge-disproportionated insulator $\uparrow \uparrow ,$ CD. Parameters are $U/W=0.5,J/W=0.2$ (black diamond in Fig. 1). (b) Site-averaged orbital DOS and (c), (d) spin-projected DOS for two inequivalent sites. Note that, due to charge disproportionation, at site 1 the portion of the DOS below the Fermi level is larger, as electron density is shifted from site 2 to 1 for $\delta >0$.

Figure 6

Combined effect on the $T=0$ 2D phase diagram of the crystal field splitting ${\mathrm{\Delta }}_{\text{CF}}/W$ [increasing vertically from 0 (bottom row) to 0.05 (middle row) and 0.1 (top row)] and of the electron-lattice coupling ${\epsilon }_b/W$ [increasing horizontally from 0 (left column) to 0.05 (middle column) to 0.1 (right column)]. Solid contours and corresponding red numbers indicate the magnitude of charge disproportionation; dashed contours with purple numbers indicate the magnitude of orbital polarization. The cupratelike phase with nearly complete orbital order is favored by increasing ${\mathrm{\Delta }}_{\text{CF}}$ and suppressed by increasing ${\epsilon }_b$.

Figure 7

Change (in $\%$) of the area occupied by the cupratelike phase in the $U-J$ diagram for the specified $({\mathrm{\Delta }}_{\text{CF}},{\epsilon }_b)$ values vs that of Fig. 1 (i.e., for ${\mathrm{\Delta }}_{\text{CF}}={\epsilon }_b=0$). See text for more details. The red contour marks an unchanged area and the black diagonal shows the line of equal strengths ${\mathrm{\Delta }}_{\text{CF}}={\epsilon }_b$.

Figure 8

Combined effect on the $T=0$ 2D phase diagram of the crystal field splitting ${\mathrm{\Delta }}_{\text{CF}}$ and electron-lattice coupling ${\epsilon }_b$ when a very large tensile strain $\epsilon =+1$ is applied. Note that here the noninteracting bandwidth $W_c$ is roughly $e$ times smaller than in Fig. 6, according to Eq. (5). Contours are as in Fig. 6.

Figure 9

Combined effect on the $T=0$ 2D phase diagram of the crystal field splitting ${\mathrm{\Delta }}_{\text{CF}}$ and electron-lattice coupling ${\epsilon }_b$ when a very large compressive strain $\epsilon =-1$ is applied. Note that here the noninteracting bandwidth $W_c$ is roughly $e$ times larger than in Fig. 6, according to Eq. (5). Contours are as in Fig. 6.

Figure 10

Euclidean distance between mean-field solutions (solid black line) and difference in ground state energies per site (dashed blue line) shows convergence as a function of $N$, the number of momentum points per dimension. The hopping parameters are $t_1^{\left(0\right)}=1,t_2^{\left(0\right)}=0.15,t_3^{\left(0\right)}=0.05$, and the remaining model parameters were chosen to be $U=J={\mathrm{\Delta }}_{\mathrm{CF}}={\epsilon }_b=0$.

Figure 11

Comparing the mean-field parameters during the iterative process obtained from (a) using a fixed mixing rate ${\alpha }_{\mathrm{mix}}=0.5$ vs (b) using the scheduled mixing rate. The iteration dependence of the scheduled mixing rate used in panel (b) is shown in panel (c). As can be seen from panel (a), using a fixed mixing rate results in the iteration “cycling” between the same few points, until the iteration sequence is aborted upon reaching the maximum iteration limit. By comparison, panel (b) shows that the scheduled mixing rate at the same value of parameters leads to rapid convergence in merely 15 iterations. Model parameters are the same as in Fig. 10, except $J=1.2$. The number of momentum points per dimension $N$ is set to 100.

Figure 12

Ground state energy of the noninteracting model, with the same model parameters as in Fig. 10. The number of momentum points per dimension $N$ is set to 120.