# Oxygen Vacancies in Zirconia and Their Migration: The Role of Hubbard-U Parameters in Density Functional Theory

## Abstract

**:**

## 1. Introduction

## 2. Computational Methods

`Zr.pbe-spn-rrkjus_psl.1.0.0.UPF`and

`O.pbe-n-rrkjus_psl.1.0.0.UPF`for Zr and O, respectively. The Zr pseudopotential treats explicitly 12 electrons as valence electrons, generated in the configuration $4{s}^{2}4{p}^{6}5{s}^{2}4{d}^{2}$, while the oxygen pseudopotential has a valence of 6 electrons, generated in the configuration $2{s}^{2}2{p}^{4}$. Both pseudopotentials have originally been developed for the

`pslibrary`version 1.0.0 [20]. Regarding the plane–wave basis set, we used a kinetic energy cutoff of 40Ry for the Kohn–Sham orbitals and of 400 Ry for the charge density. The convergence of the energies with respect to these parameters has been carefully verified and is shown in the Supplemental Information.

## 3. Results

#### 3.1. Bulk c-ZrO_{2}

#### 3.2. Oxygen Vacancies in c-ZrO_{2}

## 4. Conclusions

## Supplementary Materials

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) The density of states (DOS) of c-ZrO${}_{2}$ calculated at the PBE level of theory; (

**b**) DOS of the same system, calculated with U(Zr-4d) = 2.85 eV and U(O-2p) = 9.16 eV. See text for a discussion of these values which have been calculated by linear response. The Fermi energy is set to zero in both panels.

**Figure 2.**Calculated band gap of c-ZrO${}_{2}$ as a function of the Hubbard-U parameters. The black line indicates the experimental band gap of 5.7 eV.

**Figure 3.**Localization of the excess electrons upon formation of an oxygen vacancy. Zr atoms are shown in grey, oxygen atoms in red. The isosurface of the excess electron density is shown in light blue and is localized mainly around the oxygen vacancy in the middle of the graph.

**Figure 4.**DOS for c-ZrO${}_{2}$ with an oxygen vacancy: (

**a**) for the case of V${}_{\mathrm{O}}$; (

**b**) for the case of V${}_{\mathrm{O}}$${}^{2+}$. The Fermi level has been set to 0 eV.

**Figure 5.**Nudged Elastic Band (NEB) calculation of the transition path for a vacancy migration: (

**a**) for the case of V${}_{\mathrm{O}}$; (

**b**) for the case of V${}_{\mathrm{O}}$${}^{2+}$.

**Figure 6.**Calculated V${}_{\mathrm{O}}$ diffusion barrier as a function of the Hubbard-U parameters. The black line indicates the barrier height from a hybrid DFT calculation (using PBE0).

ecutwfc | ecutrho | n${}_{\mathit{k}}$ | Lattice Parameter |
---|---|---|---|

40 Ry | 400 Ry | 4 | 5.09 Å |

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**MDPI and ACS Style**

Gebauer, R.
Oxygen Vacancies in Zirconia and Their Migration: The Role of Hubbard-U Parameters in Density Functional Theory. *Crystals* **2023**, *13*, 574.
https://doi.org/10.3390/cryst13040574

**AMA Style**

Gebauer R.
Oxygen Vacancies in Zirconia and Their Migration: The Role of Hubbard-U Parameters in Density Functional Theory. *Crystals*. 2023; 13(4):574.
https://doi.org/10.3390/cryst13040574

**Chicago/Turabian Style**

Gebauer, Ralph.
2023. "Oxygen Vacancies in Zirconia and Their Migration: The Role of Hubbard-U Parameters in Density Functional Theory" *Crystals* 13, no. 4: 574.
https://doi.org/10.3390/cryst13040574