# 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

## References

- Khattab, E.S.R.; Abd El Rehim, S.S.; Hassan, W.M.I.; El-Shazly, T.S. Band Structure Engineering and Optical Properties of Pristine and Doped Monoclinic Zirconia (m-ZrO
_{2}): Density Functional Theory Theoretical Prospective. ACS Omega**2021**, 6, 30061–30068. [Google Scholar] [CrossRef] [PubMed] - Lindgren, M.; Panas, I. Oxygen Vacancy Formation, Mobility, and Hydrogen Pick-up during Oxidation of Zirconium by Water. Oxid. Met.
**2017**, 87, 355–365. [Google Scholar] [CrossRef][Green Version] - Zhu, H.; Li, J.; Chen, K.; Yi, X.; Cheng, S.; Gan, F. Nature of charge transport and p-electron ferromagnetism in nitrogen-doped ZrO
_{2}: An ab initio perspective. Sci. Rep.**2015**, 5, 8586. [Google Scholar] [CrossRef] [PubMed][Green Version] - Raza, M.; Cornil, D.; Cornil, J.; Lucas, S.; Snyders, R.; Konstantinidis, S. Oxygen vacancy stabilized zirconia (OVSZ); a joint experimental and theoretical study. Scr. Mater.
**2016**, 124, 26–29. [Google Scholar] [CrossRef] - Fabris, S. A stabilization mechanism of zirconia based on oxygen vacancies only. Acta Mater.
**2002**, 50, 5171–5178. [Google Scholar] [CrossRef][Green Version] - Oka, M.; Kamisaka, H.; Fukumura, T.; Hasegawa, T. Density functional theory-based ab initio molecular dynamics simulation of ionic conduction in N-/F-doped ZrO
_{2}under epitaxial strain. Comput. Mater. Sci.**2018**, 154, 91–96. [Google Scholar] [CrossRef] - Lee, Y.L.; Duan, Y.; Morgan, D.; Sorescu, D.; Abernathy, H.; Hackett, G. Density functional theory modeling of cation diffusion in bulk tetragonal zirconia. Ceram. Trans.
**2019**, 266, 95–110. [Google Scholar] [CrossRef] - Schultze, T.K.; Arnold, J.P.; Grieshammer, S. Ab Initio Investigation of Migration Mechanisms in La Apatites. ACS Appl. Energy Mater.
**2019**, 2, 4708–4717. [Google Scholar] [CrossRef] - Koga, H.; Hayashi, A.; Ato, Y.; Tada, K.; Hosokawa, S.; Tanaka, T.; Okumura, M. Facile NO-CO elimination over zirconia-coated Cu (110) surfaces: Further evidence from DFT + U calculations. Appl. Surf. Sci.
**2020**, 508, 145252. [Google Scholar] [CrossRef] - Mueller, M.P.; Pingen, K.; Hardtdegen, A.; Aussen, S.; Kindsmueller, A.; Hoffmann-Eifert, S.; De Souza, R.A. Cation diffusion in polycrystalline thin films of monoclinic HfO
_{2}deposited by atomic layer deposition. APL Mater.**2020**, 8, 081104. [Google Scholar] [CrossRef] - Lee, Y.L.; Duan, Y.; Sorescu, D.C.; Morgan, D.; Abernathy, H.; Kalapos, T.; Hackett, G. Density functional theory modeling of cation diffusion in tetragonal bulk ZrO
_{2}: Effects of humidity and hydrogen defect complexes on cation transport. Phys. Rev. Res.**2021**, 3, 013121. [Google Scholar] [CrossRef] - Kumar, N.; Seriani, N.; Gebauer, R. DFT insights into electrocatalytic CO
_{2}reduction to methanol on α-Fe_{2}O_{3}(0001) surfaces. Phys. Chem. Chem. Phys.**2020**, 22, 10819–10827. [Google Scholar] [CrossRef] [PubMed] - Ulman, K.; Poli, E.; Seriani, N.; Piccinin, S.; Gebauer, R. Understanding the electrochemical double layer at the hematite/water interface: A first principles molecular dynamics study. J. Chem. Phys.
**2019**, 150, 041707. [Google Scholar] [CrossRef] [PubMed] - Ahamed, I.; Ulman, K.; Seriani, N.; Gebauer, R.; Kashyap, A. Magnetoelectric e-Fe
_{2}O_{3}: DFT study of a potential candidate for electrode material in photoelectrochemical cells. J. Chem. Phys.**2018**, 148, 214707. [Google Scholar] [CrossRef] [PubMed] - Nguyen, M.T.; Gebauer, R. Graphene Supported on Hematite Surfaces: A Density Functional Study. J. Phys. Chem. C
**2014**, 118, 8455–8461. [Google Scholar] [CrossRef] - Walker, B.G.; Hendy, S.C.; Gebauer, R.; Tilley, R.D. Application of Lanczos-based time-dependent density-functional theory approach to semiconductor nanoparticle quantum dots. Eur. Phys. J. B
**2008**, 66, 7–15. [Google Scholar] [CrossRef] - Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G.L.; Cococcioni, M.; Dabo, I.; et al. QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter
**2009**, 21, 395502. [Google Scholar] [CrossRef] - Giannozzi, P.; Andreussi, O.; Brumme, T.; Bunau, O.; Buongiorno Nardelli, M.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Cococcioni, M.; et al. Advanced capabilities for materials modelling with Quantum ESPRESSO. J. Phys. Condens. Matter
**2017**, 29, 465901. [Google Scholar] [CrossRef][Green Version] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef][Green Version] - Dal Corso, A. Pseudopotentials periodic table: From H to Pu. Comput. Mater. Sci.
**2014**, 95, 337–350. [Google Scholar] [CrossRef] - Perdew, J.P.; Ernzerhof, M.; Burke, K. Rationale for mixing exact exchange with density functional approximations. J. Chem. Phys.
**1996**, 105, 9982–9985. [Google Scholar] [CrossRef] - Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys.
**1999**, 110, 6158–6170. [Google Scholar] [CrossRef] - Mulliken, R.S. Electronic Population Analysis on LCAO–MO Molecular Wave Functions. I. J. Chem. Phys.
**1955**, 23, 1833–1840. [Google Scholar] [CrossRef][Green Version] - Bader, R.F.W. Atoms in Molecules—A Quantum Theory; Oxford University Press: Oxford, UK, 1990. [Google Scholar]
- Sit, P.H.L.; Car, R.; Cohen, M.H.; Selloni, A. Simple, unambiguous theoretical approach to oxidation state determination via first-principles calculations. Inorg. Chem.
**2011**, 50, 10259–10267. [Google Scholar] [CrossRef] - Krukau, A.V.; Vydrov, O.A.; Izmaylov, A.F.; Scuseria, G.E. Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys.
**2006**, 125, 224106. [Google Scholar] [CrossRef] [PubMed] - Yang, Y.L.; Fan, X.L.; Liu, C.; Ran, R.X. First principles study of structural and electronic properties of cubic phase of ZrO
_{2}and HfO_{2}. Phys. B Condens. Matter**2014**, 434, 7–13. [Google Scholar] [CrossRef] - Cococcioni, M.; de Gironcoli, S. Linear response approach to the calculation of the effective interaction parameters in the LDA+U method. Phys. Rev. B
**2005**, 71, 035105. [Google Scholar] [CrossRef][Green Version] - Timrov, I.; Marzari, N.; Cococcioni, M. Hubbard parameters from density-functional perturbation theory. Phys. Rev. B
**2018**, 98, 085127. [Google Scholar] [CrossRef][Green Version] - Li, J.; Meng, S.; Niu, J.; Lu, H. Electronic structures and optical properties of monoclinic ZrO
_{2}studied by first-principles local density approximation + U approach. J. Adv. Ceram.**2017**, 6, 43–49. [Google Scholar] [CrossRef][Green Version] - Henkelman, G.; Uberuaga, B.P.; Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys.
**2000**, 113, 9901–9904. [Google Scholar] [CrossRef][Green Version]

**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