Volume-dependent electron localization in ceria

We have performed a numerical study of the process of electron localization in reduced ceria. Our results show that different localized charge distributions can be attained in a bulk system by varying the lattice parameter. We demonstrate that the effect of electron localization is mainly determined by lattice relaxation and an accurate account for the effects of electronic correlation is necessary to achieve localized charge distribution.

Figure 1

(a) A cubic supercell (small balls: O; large balls: Ce) used to model reduced ceria with $(2\times 2\times 2)$ oxygen vacancy periodicity (periodical images are also shown). For a given oxygen vacancy (yellow ball), nearest-neighbor Ce ions (dark gray) and next-nearest-neighbor Ce ions (gray) are labeled. (b) ${\text{Ce}}^{3+}$ ions (differentiated by size and grayscale level) which belong to different groups of ${\text{Ce}}_1^{3+}{\text{Ce}}_{m^{\prime }}^{3+}$ configurations. (c) ${\text{Ce}}^{3+}$ ions (differentiated by size and grayscale level) which belong to different groups of ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{m^{\prime }}^{3+}$ configurations.

Figure 2

(a) Energy vs volume for all distinct reduced ceria structures relative to the energy of the ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{2^{\prime }}^{3+}$ configuration. (b) Dependence of enthalpy on pressure for ${\text{Ce}}_1^{3+}{\text{Ce}}_2^{3+},{\text{Ce}}_1^{3+}{\text{Ce}}_{{10}^{\prime }}^{3+}$ structures with respect to the enthalpy of the ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{2^{\prime }}^{3+}$ configuration.

Figure 3

(a) Dependence of energy on volume for all distinct reduced ceria structures relative to the energy of the ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{2^{\prime }}^{3+}$ configuration for the $\text{LDA}+U$ type of calculations. (b) Enthalpy vs pressure for ${\text{Ce}}_1^{3+}{\text{Ce}}_2^{3+},{\text{Ce}}_1^{3+}{\text{Ce}}_{{10}^{\prime }}^{3+}$ structures with respect to the enthalpy of the ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{2^{\prime }}^{3+}$ configuration.

Figure 4

(a) Dependence of energy vs volume for all distinct reduced ceria structures relative to the energy of the ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{2^{\prime }}^{3+}$ configuration for the CSM calculations. (b) Dependence of enthalpy vs pressure for ${\text{Ce}}_1^{3+}{\text{Ce}}_2^{3+},{\text{Ce}}_1^{3+}{\text{Ce}}_{{10}^{\prime }}^{3+}$ structures with respect to the enthalpy of the ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{2^{\prime }}^{3+}$ configuration.

Figure 5

Contributions of the ionic energy term (blue), $\Delta E_{\mathrm{ion}}$, and of the electronic subsystem (red), $\Delta E_{\mathrm{el}}$, to the total energy difference (black), $\Delta E$, between two reduced ceria configurations, $\Delta E=E_{{\mathrm{Ce}}_{1^{\prime }}^{3+}{\mathrm{Ce}}_{2^{\prime }}^{3+}}-E_{{\mathrm{Ce}}_1^{3+}{\mathrm{Ce}}_2^{3+}}$, for different types of calculations: (a) $\text{PBE}+U$ $\left(\blacksquare \right)$ and $\text{LDA}+U$ $\left(▴\right)$, (b) GGA CSM $(•)$.

Figure 6

Displacements of various ions within the simulation cell with respect to the position of the O vacancy for relaxed ${\text{Ce}}_1^{3+}{\text{Ce}}_2^{3+},{\text{Ce}}_1^{3+}{\text{Ce}}_{{10}^{\prime }}^{3+}$, and ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{2^{\prime }}^{3+}$ configurations at (a) $a_{\mathrm{latt}}=5.35\text{Å}$ and (b) $a_{\mathrm{latt}}=5.50\text{Å}$. Displacements corresponding to ${\text{Ce}}^{+3},{\text{Ce}}^{+4}$, and ${\text{O}}^{-2}$ ions are shown by squares, triangles, and circles, respectively (some data overlap). For $\delta R<0(\delta R>0)$, ions move towards (away from) the vacancy site.

Figure 7

(a) Ce-O distances within the first coordination sphere of the O vacancy (creamy sphere) for different ${\text{Ce}}^{3+}$–${\text{Ce}}^{3+}$ configurations relaxed at $a_{\mathrm{latt}}=5.35\text{Å}$. (b) Ce-O distances within the first coordination of the O vacancy for the initial structure (same for all ${\text{Ce}}^{3+}$–${\text{Ce}}^{3+}$ configurations, except for the positions of ${\text{Ce}}^{3+}$ ions). Ce-O bonds are colored according to their length: red-shifted (blue-shifted) if O ions move towards (away from) Ce ions. (c) Ce-O distances within first coordination of the O vacancy for different ${\text{Ce}}^{3+}$–${\text{Ce}}^{3+}$ configurations relaxed at $a_{\mathrm{latt}}=5.50\text{Å}$.

Figure 8

(a) Volume dependence of the energy of relaxation for three reduced ceria structures: different symbols and colors correspond to different ${\text{Ce}}^{3+}$–${\text{Ce}}^{3+}$ configurations. (b) Contributions to the relaxation energy from different groups of ions $(▴$: Ce and O in the first coordination sphere of the O vacancy; $▾$: Ce in the first coordination sphere and O within the first two coordination shells of the vacancy; $•$: Ce and O within the first two coordination shells of the O vacancy; $\blacksquare$: Ce in the first two coordination shells and O within the first four coordination shells of the defect) in the vicinity of the O vacancy for different reduced ceria structures (red: ${\text{Ce}}_1^{3+}{\text{Ce}}_2^{3+}$; green: ${\text{Ce}}_1^{3+}{\text{Ce}}_{{10}^{\prime }}^{3+}$; blue: ${\text{Ce}}_{1^{\prime }}^{3+}{\text{Ce}}_{2^{\prime }}^{3+})$.