First-principles study for low-spin ${\text{LaCoO}}_3$ with a structurally consistent Hubbard $U$

In this paper, we use the local density $\text{approximation}+\text{Hubbard}$ $U$ method to calculate the structural and electronic properties of low-spin ${\text{LaCoO}}_3$. The Hubbard $U$ is obtained by first principles and consistent with each fully optimized atomic structure at different pressures. With structurally consistent $U$, the fully optimized atomic structure agrees with experimental data better than the calculations with fixed or vanishing $U$. A discussion on how the Hubbard $U$ affects the electronic and atomic structures of ${\text{LaCoO}}_3$ is also given.

Figure 1

Hubbard $U$ as a function of LS ${\text{LaCoO}}_3$ rhombohedral unit-cell volume $V$. The Hubbard $U$ of the LDA-relaxed structure is referred to as $U_o$ (dashed line), and the structurally consistent $U$ (see text) is referred to as $U_{\text{sc}}$ (solid line). While $U_o$ decreases with $V$, $U_{\text{sc}}$ has a minimum of 8.20 eV at $V=814\text{a}\text{.u}{\text{.}}^3$. The different volume dependence affects the estimate of pressure, as later shown in Fig. 3.

Figure 2

(Color online) Total energies $E$ as functions of LS ${\text{LaCoO}}_3$ unit-cell volume $V$ obtained from different choices of Hubbard $U$: volume-dependent $U_o$, $U_{\text{sc}}$, and fixed $U=8.46\text{eV}$ (average of $U_o$) and $U=8.33\text{eV}$ (average of $U_{\text{sc}}$). The $E\left(V\right)$ curves obtained from the two fixed $U$’s have very similar shape, and the $E\left(V\right)$ curves obtained from the volume-dependent $U_o$ and $U_{\text{sc}}$ have different shapes.

Figure 3

(Color online) Calculated zero-temperature volume at different pressures. At $P=0\text{GPa}$, the volume given by $\text{LDA}+U$ with $U_{\text{sc}}$ agrees with the experimental data taken at $T=5\text{K}$, $P={10}^{-4}\text{GPa}$ (orange square) (Ref. 16) better than other calculated results. Experimental data taken at room temperature (green triangles) (Ref. 24) are shown as a reference.

Figure 4

(Color online) Lattice constant $a$ (top) and rhombohedral angle $\alpha$ (bottom) of LS ${\text{LaCoO}}_3$ at different pressures. At $P=0\text{GPa}$, the results of $\text{LDA}+U$ with $U_{\text{sc}}$ agree best with the experimental data taken at $T=5\text{K}$ (Ref. 16). The discrepancy with respect to the $T=298\text{K}$ (Ref. 24) data is a signature of spin excitations.

Figure 5

(Color online) Structural parameters of ${\text{LaCoO}}_3$, Co-O distance (top), Co-O-Co bond angle (middle), and O-Co-O bond angle (bottom), at different pressures. At $P=0\text{GPa}$, the results of $\text{LDA}+U$ with $U_{\text{sc}}$ agree best with the experimental data taken at $T=5\text{K}$ (Ref. 16). The discrepancy with respect to the $T=298\text{K}$ data (Ref. 24) is a signature of spin excitations.

Figure 6

(Color online) Total and partial densities of states of LS ${\text{LaCoO}}_3$ at $V=744.65\text{a}\text{.u}{\text{.}}^3$ obtained using LDA (experimental equilibrium $V=745.16\text{a}\text{.u}{\text{.}}^3$). (a) Total density of states; (b) partial density of states of different atoms; and (c) partial density of states of $e_g$ and $t_{2g}$ states. LDA gives metallic ${\text{LaCoO}}_3$ at low temperature. The occupation of $t_{2g}$ state is 5.66/Co.

Figure 7

(Color online) Total and partial densities of states of LS ${\text{LaCoO}}_3$ at $V=744.13\text{a}\text{.u}{\text{.}}^3$ obtained using $\text{LDA}+U$ (experimental equilibrium $V=745.16\text{a}\text{.u}{\text{.}}^3$). (a) Total density of states; (b) partial density of states of different atoms; and (c) partial density of states of $e_g$ and $t_{2g}$ states. Applying $U$ opens an indirect gap of 1.43 eV. The occupation of $t_{2g}$ state is 5.91/Co.

Figure 8

(Color online) Charge density of LS ${\text{LaCoO}}_3$ around Co given by (a) LDA at $V=744.65\text{a}\text{.u}{\text{.}}^3$ and (b) $\text{LDA}+U$ at $V=744.13\text{a}\text{.u}{\text{.}}^3$ Co is placed at the center and surrounded by six O sites. The charge density on the Co site given by $\text{LDA}+U$ has deeper dimples and a more cubiclike shape. This shows a more confined and $t_{2g}$-like charge density. The isosurface value is 0.09.

Figure 9

(Color online) MLWFs of LS ${\text{LaCoO}}_3$ at $V=744.65\text{a}\text{.u}{\text{.}}^3$ given by LDA. Green, light blue, and red spheres represent La, Co, and O sites, respectively. The isosurface values are $\pm 0.60$. [(a)–(e)] MLWFs associated with the $\text{Co}3d$ band; [(f)–(h)] MLWFs associated with the $\text{O}2p$ band. In order to see how the $\text{O}2p$ electrons bond with the atoms in the adjacent unit cell, each $\text{O}2p$ MLWF in panels (f)–(h) is duplicated and plotted at the equivalent atomic site in the adjacent cell.

Figure 10

(Color online) MLWFs of LS ${\text{LaCoO}}_3$ at $V=744.13\text{a}\text{.u}{\text{.}}^3$ given by $\text{LDA}+U_{\text{sc}}$. Green, light blue, and red spheres represent La, Co, and O sites, respectively. The isosurface values are $\pm 0.60$. [(a)–(e)] MLWFs associated with the $\text{Co}3d$ band; [(f)–(h)] MLWFs associated with the $\text{O}2p$ band. In order to see how the $\text{O}2p$ electrons bond with the atoms in the adjacent unit cell, each $\text{O}2p$ MLWF in panels (f)–(h) is duplicated and plotted at the equivalent atomic site in the adjacent cell.

Figure 11

(Color online) Lattice constant $a$ (top) and rhombohedral angle $\alpha$ (bottom) of LS ${\text{LaCoO}}_3$ unit cell as functions of unit-cell volume $V$. At $V=745\text{a}\text{.u}{\text{.}}^3$, the results of $\text{LDA}+U$ agree with the experimental data taken at $T=5\text{K}$, $P={10}^{-4}\text{GPa}$ (Ref. 16) better than LDA. The discrepancy with respect to experimental data (Refs. 16, 24) is a signature of spin excitations.

Figure 12

(Color online) Structural parameters of ${\text{LaCoO}}_3$, Co-O distance (top), Co-O-Co bond angle (middle), and O-Co-O bond angle (bottom), as a function of unit-cell volume $V$. At $V=745\text{a}\text{.u}{\text{.}}^3$, the results of $\text{LDA}+U$ agree with the experimental data taken at $T=5\text{K}$, $P={10}^{-4}\text{GPa}$ (Ref. 16) better than LDA. The discrepancy with respect to the experimental data (Refs. 16, 24) is a signature of spin excitations.