Microscopic origin of diagonal stripe phases in doped nickelates

We investigate the electron density distribution and the stability of stripe phases in a realistic two-band model with hopping elements between $e_g$ orbitals at Ni sites on the square lattice, and compare these results with those obtained for the doubly degenerate Hubbard model with two equivalent orbitals and diagonal hopping. For both models we determine the stability regions of filled and half-filled stripe phases for increasing hole doping $x=2-n$ in the range of $x<0.4$, using the Hartree-Fock approximation for large clusters. In the parameter range relevant to the nickelates, we obtain the most stable diagonal stripe structures with filling of nearly one hole per atom, as observed experimentally. In contrast, for the doubly degenerate Hubbard model the most stable stripes are somewhat reminiscent of the cuprates, with half-filled orbitals at the domain wall sites. This difference elucidates the crucial role of the off-diagonal $e_g$ hopping terms for the stripe formation in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Ni}{\mathrm{O}}_4$. The influence of the crystal field is discussed as well.

Figure 1

(Color online) Fermi surfaces of the higher (top) and lower (bottom) bands, as obtained for different values of the chemical potential $\mu$ in the two-band tight-binding model for degenerate $e_g$ orbitals $(E_z=0)$. Note a different scale for both Fermi surfaces.

Figure 2

(Color online) Static spin-spin correlation functions ${\chi }_S\left(\mathbf{Q}\right)$ for $e_g$ electrons for increasing doping, as obtained for: (a) $E_z=0$ and (b) $E_z=t$. Different lines correspond to $\mathbf{Q}=(0,0)$ (solid line), $(0,\pi )$ (dotted line), and $(\pi ,\pi )$ (dashed line). van Hove singularity for $\mathbf{Q}=(0,0)$ is obtained at doping higher than that realized experimentally in nickelate LSNO oxides: (a) $x\simeq 1.49$ and (b) $x\simeq 1.68$.

Figure 3

(Color online) Free-energy gains $\delta F$ (19) of the VBC (top) and DBC (bottom) stripe phases with respect to the AF phase (19) as functions of doping $x$, obtained at temperature $\beta t=100$ for the $e_g$ model. Cluster sizes, distances between DWs $d$, and parameters are the same as in Table .

Figure 4

(Color online) Free-energy gains $\delta F$ (19) as in Fig. 3, but with finite crystal-field splitting $E_z=t$. Cluster sizes, distances between DWs $d$, and parameters are the same as in Table .

Figure 5

(Color online) Free-energy gains $\delta F$ (19) of the VBC (top) and DBC (bottom) stripe phases with unit cells of length $d$, as a function of doping $x$, obtained for the DDH model. Parameters: $U=8t$, $J_H=1.5t$, $E_z=0$, and $\beta t=100$.

Figure 6

(Color online) Doping dependence of stripe phases for the VBC (left) and DBC (right) ground states: (a) and (b) magnetic incommensurability $ϵ$; (c) and (d) stripe filling $\nu$; (e) and (f) chemical potential $\mu$ (in units of $t$), as deduced from the data of Figs. 3, 4, 5. Filled (open) symbols for $e_g$ (DDH) model; circles and squares for $E_z=0$ and $t$. Solid lines in (a) and (b) show the experimental behavior of $ϵ$ in LSNO for $x⩽1∕8$, as given in Ref. 4.

Figure 7

(Color online) Charge and magnetization distribution in the filled VBC (left) and DBC (right) stripe phase found in either the $e_g$ model (filled circles), or the DDH model (open circles) on a $64\times 64$ cluster at doping $x=1∕8$: local hole densities $n_h\left(l_x\right)$ (top row); modulated magnetization densities $m_{\pi }\left(l_x\right)$ (second row); local hole $o_h\left(l_x\right)$ (third row); and local modulated magnetic $p_{\pi }\left(l_x\right)$ orbital polarization (bottom row). For more clarity, the data points for the DDH model are shifted by one lattice constant from the origin of the coordinate system. Parameters: $U=8t$, $J_H=1.5t$, $E_z=0$, and $\beta t=100$.

Figure 8

(Color online) Charge and magnetization distributions in the half-filled HVBC (left) and HDBC (right) stripe phases found in either the $e_g$ (filled circles) model, or in the DDH model (open circles) on a $64\times 64$ cluster at doping $x=1∕8$. The meaning of the different panels, shift of the DDH data, and parameters as in Fig. 7.

Figure 9

(Color online) Double occupancy distribution in the filled VBC (left) and DBC (right) stripe phase shown in Fig. 7: intraorbital double occupancy $D\left(l_x\right)$ (top row), and two local interorbital $D_{xz}^{\sigma \overline{\sigma }}\left(l_x\right)$ (middle) and $D_{xz}^{\sigma \sigma }\left(l_x\right)$ (bottom) double occupancies. Filled (open) circles denote the results found in the $e_g$ (DDH) model, respectively. Parameters and filled (empty) circles as in Fig. 7.

Figure 10

(Color online) Double occupancy distribution in the half-filled stripe phases shown in Fig. 8: HVBC (left) and HDBC (right). Different double occupancies as in Fig. 9. Parameters and filled (empty) circles as in Fig. 8.

Figure 11

Densities of states for the DBC stripe phase in the $e_g$ model at $x=1∕8$ doping: (a) partial density of states $N_x\left(\omega \right)$ for $\mid x⟩$ orbital, (b) partial density of states $N_z\left(\omega \right)$ for $\mid z⟩$ orbital, and (c) total density of states $N\left(\omega \right)$. (d) shows for comparison the total density of states $N\left(\omega \right)$ as found in the DBC stripe phase within the DDH model. Parameters as in Fig. 7.

Figure 12

Densities of states for the HDBC stripe phase in the $e_g$ model at $x=1∕8$ doping. (a)–(c) as in Fig. 11. (d) shows for comparison the total density of states $N\left(\omega \right)$ as found in the HDBC stripe phase within the DDH model. Parameters as in Fig. 8.