Origin of the orbital polarization of ${\mathrm{Co}}^{2+}$ in ${\mathrm{La}}_2{\mathrm{CoTiO}}_6$ and (${\mathrm{LaCoO}}_3{)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$: A $\mathrm{DFT}+U$ and DFMT study

The unequal electronic occupation of localized orbitals (orbital polarization), and associated lowering of symmetry and degeneracy, play an important role in the properties of transition metal oxides. Here, we examine systematically the underlying origin of orbital polarization, taking as exemplar the $3d$ manifold of ${\mathrm{Co}}^{2+}$ in a variety of spin, orbital, and structural phases in the double perovskite ${\mathrm{La}}_2{\mathrm{CoTiO}}_6$ and the (001) superlattice ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ systems. Superlattices are of specific interest due to the large experimentally observed orbital polarization of their Co cations. Based on first principles calculations, we find that robust and observable orbital polarization requires symmetry reduction through the lattice structure; the role of local electronic interactions is to greatly enhance the orbital polarization.

Figure 1

Schematics energy levels of ${\mathrm{Co}}^{2+}$ ions in ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ superlattices with majority spin up for (a) low-spin state with $e_g$ orbital polarization and (b) high-spin state with $t_{2g}$ orbital polarization. This schematic picture holds for both $P_4/mmm$ and $P{2}_1/n$ phases. The yellow shaded states are those determining the orbital polarization in each case.

Figure 2

Schematics of the atomic-scale structures of structures studied in this work. Each structure is labeled by its space group and octahedral rotation pattern. (a) Ideal ${\mathrm{La}}_2{\mathrm{CoTiO}}_6$ double perovskite. (b), (c), and (d) show one repeat of $\text{LCO}+\text{LTO}$ (001) superlattices where the (001) direction is vertical. La atoms are not shown for clarity.

Figure 3

(a)–(d) show the $\text{DFT}+U$ orbital polarization $P\left(e_g\right)$ of the LS state structures with different space groups versus $U\left(\mathrm{Co}\right)$ and as a function of $U\left(\mathrm{Ti}\right)$. Empty and filled points indicate metallic and insulating phases, respectively. Insets show schematic side views of the octahedral tilts and distortions of the ${\mathrm{CoO}}_6$ and ${\mathrm{TiO}}_6$ oxygen octahedra. (e) and (f) present Co-O bond lengths along the $c$ axis of the $P_4/mmm$ and $P{2}_1/n$ phases, respectively. Dashed lines represent in-plane Co-O bond lengths which depend weakly on $U$.

Figure 4

$\text{DFT}+\text{DMFT}$ spectral functions of Co $d$ Wannier orbitals for the low-spin state: (a) ${\mathrm{La}}_2{\mathrm{CoTiO}}_6$ ($Fm\overline{3}m$ structure) and (b) $\text{LTO}+\text{LCO}$ superlattice ($P_4/mmm, a=b=c$ structure). The calculations use $U=3$ eV, $J=0.5$ eV, and a temperature of 300 K.

Figure 5

(a) Co $3d$ and (b) Ti $3d$ projected density of states for low-spin (LS) Co in the ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ superlattice with $U_{\mathrm{Ti}}=0$ and $U_{\mathrm{Co}}=0$. (c) Co $3d$ and (d) Ti $3d$ projected density of states with $U_{\mathrm{Ti}}=5$ eV and $U_{\mathrm{Co}}=0$. The in-plane lattice parameters $a$ and $b$ are fixed to 3.891 Å, and the atomic structure has the $P_4/mmm$ ($a=b\ne c$) space group.

Figure 6

(a)–(c) show the $\text{DFT}+U$ orbital polarization $P\left(e_g\right)$ of the LS state structures with different space groups versus $U\left(\mathrm{Co}\right)$ and as a function of $U\left(\mathrm{Ti}\right)$ when the magnetic order between Co is antiferromagnetic (AFM). Empty and filled points indicate metallic and insulating phases, respectively. Insets show schematic side views of the octahedral tilts and distortions of the ${\mathrm{CoO}}_6$ and ${\mathrm{TiO}}_6$ oxygen octahedra.

Figure 7

(a),(c) Orbital polarization from $\text{DFT}+U$ of the $\text{LCO}+\text{LTO}$ superlattice ($P_4/mmm, a^0{a}^0{a}^0$, and $a=b\ne c$) with different in-plane lattice parameters. Empty and filled points indicate metallic and insulating phases, respectively. (b),(d) Out-of-plane Co-O bond lengths with different in-plane lattice parameters. Dashed lines represent the in-plane Co-O bond lengths, which are weak functions of $U$.

Figure 8

(a)–(c) $t_{2g}$ HS orbital polarization from $\text{DFT}+U$ of the $\text{LCO}+\text{LTO}$ heterostructures with different space groups; as shown by the insets, (b) has $a=b=c$ and (c) has $a=b\ne c$. Filled and empty points indicate metallic and insulating phases, respectively. (d) Out-of-plane Co-O bond lengths of the $P_4/mmm$ phase with $a=b\ne c$. The dashed line represents the in-plane Co-O bond lengths, which are robust versus $U$.

Figure 9

$\text{DFT}+\text{DMFT}$ spectral functions of Co $d$ Wannier orbitals for (a) HS ${\mathrm{La}}_2{\mathrm{CoTiO}}_6$ ($Fm\overline{3}m$) and (b) the HS $\text{LTO}+\text{LCO}$ superlattice ($P_4/mmm, a=b=c$). The calculations use $U=3$ eV, $J=0.9$ eV, and a temperature of 300 K.

Figure 10

(a)–(f) Co $3d$ projected density of states for low-spin (LS) Co in the ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ superlattice as a function of $U_{Co}$ in eV. The value of $U_{Ti}$ is 5 eV, the in-plane lattice parameters $a$ and $b$ are fixed to 3.811 Å, and the atomic structure has the $P{2}_1/n$ space group.

Figure 11

(a)–(f) Co $3d$ projected density of states of high-spin (HS) Co in the ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ superlattice as a function of $U_{Co}$ in eV. The value of $U_{Ti}$ is 5 eV, the in-plane lattice parameters $a$ and $b$ are fixed to 3.811 Å, and the atomic structure has the $P{2}_1/n$ space group. Note that for $U_{Co}=0$ and 0.5, the HS state is not even metastable, so the Co has the LS state. The in-plane lattice parameters $a$ and $b$ are fixed to 3.811 Å.

Figure 12

Comparison of the Co $3d$ projected density of states of the two different self-consistent solutions for low-spin (LS) Co in the double perovskite ${\mathrm{La}}_2{\mathrm{TiCoO}}_6$ (cubic, $Fm\overline{3}m$ space group) as a function of $U_{Co}$ and $U_{Ti}$ in eV. (c) and (e) are for the $d_{z^2}$-occupied LS states, and (d) and (f) are for the $d_{x^2-y^2}$-occupied LS states. Lattice parameters are fixed to $a=b=c=3.891$ Å, obtained by minimizing the stress of ${\mathrm{La}}_2{\mathrm{TiCoO}}_6$ with $U_{Co}=U_{Ti}=3$.

Figure 13

Comparison of the Co $3d$ projected density of states of the two different self-consistent solutions for low-spin (LS) Co in the ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ superlattice ($P_4/mmm$ space group, $a=b=c$) as a function of $U_{Co}$ and $U_{Ti}$ in eV. (c) and (e) are for the $d_{z^2}$-occupied LS states, and (d) and (f) are for the $d_{x^2-y^2}$-occupied LS states. Lattice parameters are fixed to $a=b=c=3.891$ Å.

Figure 14

Comparison of the Co $3d$ projected density of states of the two different self-consistent solutions for low-spin (LS) Co in ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ ($P_4/mmm$ space group, $a=b\ne c$) as a function of $U_{Co}$ and $U_{Ti}$ in eV. (c) and (e) are for the $d_{z^2}$-occupied LS states, and (d) and (f) are for the $d_{x^2-y^2}$-occupied LS states. In-plane lattice parameters are fixed to $a=b=3.891$ Å, while $c$ is different due to the relaxation.

Figure 15

Ti $3d$ projected density of states for the low-spin (LS) Co in the ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ superlattice ($P_4/mmm$ space group, $a=b\ne c$) as a function of $U_{Co}$ and $U_{Ti}$ in eV. In-plane lattice parameters are fixed to $a=b=3.891$ Å, while $c$ is different due to the relaxation.

Figure 16

Co $3d$ projected density of states for the low-spin (LS) Co in the ${\left({\mathrm{LaCoO}}_3\right)}_1+{\left({\mathrm{LaTiO}}_3\right)}_1$ superlattice ($P{2}_1/n$ space group, $a=b\ne c$) as a function of $U_{Co}$ and $U_{Ti}$ in eV. In-plane lattice parameters are fixed to $a=b=3.891$ Å, while $c$ is different due to the relaxation.

Figure 17

Co $3d$ projected density of states (PDOS) for high-spin (HS) Co in the double perovskite ${\mathrm{La}}_2{\mathrm{TiCoO}}_6$ (cubic, $Fm\overline{3}m$ space group) as a function of $U_{Co}$ and $U_{Ti}$ in eV. (a) For $U_{Ti}=U_{Co}=0$, the minority (down) $t_{2g}$ states are equally occupied showing no orbital polarization. For $U_{Ti}=U_{Co}=5$ eV, three physically equivalent different minority $t_{2g}$ configurations can be stabilized: (b) $d_{xy\downarrow }^1{d}_{xz\downarrow }^0{d}_{yz\downarrow }^1$, (c) $d_{xy\downarrow }^1{d}_{xz\downarrow }^1{d}_{yz\downarrow }^0$, and (d) $d_{xy\downarrow }^0{d}_{xz\downarrow }^1{d}_{yz\downarrow }^1$. Lattice parameters are fixed to $a=b=c=3.891$ Å.