Hybrid functionals applied to rare-earth oxides: The example of ceria

We report periodic density functional theory (DFT) calculations for ${\mathrm{CeO}}_2$ and ${\mathrm{Ce}}_2{\mathrm{O}}_3$ using the Perdew-Burke-Ernzerhof (PBE0) and Heyd-Scuseria-Ernzerhof (HSE) hybrid functionals that include nonlocal Fock exchange. We study structural, electronic, and magnetic ground state properties. Hybrid functionals correctly predict ${\mathrm{Ce}}_2{\mathrm{O}}_3$ to be an insulator as opposed to the ferromagnetic metal predicted by the local spin density (LDA) and generalized gradient (GGA) approximations. The equilibrium volumes of both structures are in very good agreement with experiments, improving upon the description of the LDA and GGA. The calculated ${\mathrm{CeO}}_2$ (O $2p$–Ce $5d$) and ${\mathrm{Ce}}_2{\mathrm{O}}_3$ $\left(\mathrm{Ce}4f\text{−}5d4f\right)$ band gaps are larger by up to 45% (PBE0) and 15% (HSE) than found in experiments. Furthermore, we calculate atomization energies, heats of formation, and the reduction energy of $2{\mathrm{CeO}}_2\to {\mathrm{Ce}}_2{\mathrm{O}}_3+(1∕2){\mathrm{O}}_2$. The latter is underestimated by $\sim 0.4–0.9\mathrm{eV}$ with respect to available experimental data at room temperature. We compare our results with the more traditional DFT+$U$ (LDA$+U$ and PBE$+U$) approach and discuss the role played by the Hubbard $U$ parameter.

Figure 1

(Color online) Fluorite-type structure (face-centered cubic, $Fm\overline{3}m$) of ${\mathrm{CeO}}_2$ (left) and the sesquioxide $A$-type structure (hexagonal, $P\overline{3}m1$) of ${\mathrm{Ce}}_2{\mathrm{O}}_3$ (right), respectively. Large blue and small red balls indicate Ce and O atoms, respectively.

Figure 2

(Color online) Equilibrium lattice constants of ${\mathrm{CeO}}_2$ and ${\mathrm{Ce}}_2{\mathrm{O}}_3$ as a function of $U_{\mathrm{eff}}$. The lattice constants are given in percentages with respect to the experimental values ($a_0=5.41\mathrm{\AA }$ for ${\mathrm{CeO}}_2$; $a_0=3.89\mathrm{\AA }$, $c_0=6.06\mathrm{\AA }$ for ${\mathrm{Ce}}_2{\mathrm{O}}_3$). The stars indicated the results obtained with the calculated $U_{\mathrm{eff}}$ using the linear-response approach of Ref. 43.

Figure 3

(Color online) Total and local density of states (TDOS and LDOS) calculated with different XC energy functionals. Zero in the energy $x$ axis indicates the top of the occupied valence band. All DOSs are calculated at the optimized geometries. For ${\mathrm{Ce}}_2{\mathrm{O}}_3$, only the spin-up component of the AF spin solution is shown.

Figure 4

(Color online) ${\mathrm{CeO}}_2$ and ${\mathrm{Ce}}_2{\mathrm{O}}_3$ band gap $E_g$ and position of the $\mathrm{Ce}4f$ states (see text) $E_{g\text{−}f}$ at the $\Gamma$ point, calculated as a function of $U_{\mathrm{eff}}$. Results are obtained for the theoretical equilibrium volumes. The stars indicate the results obtained with the calculated $U_{\mathrm{eff}}$.

Figure 5

(Color online) Reaction energy of reduction $2{\mathrm{CeO}}_2\to {\mathrm{Ce}}_2{\mathrm{O}}_3+(1∕2){\mathrm{O}}_2$ as a function of $U_{\mathrm{eff}}$. Energies are determined at the optimized geometries. The stars indicated the results obtained with the calculated $U_{\mathrm{eff}}$. The horizontal dashed line corresponds to the most recent experimental result in Ref. 69.