Unraveling the Negative Role of Oxygen-Vacancy Cluster in Ionic

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Article

Unravelling the Negative Role of Oxygen Vacancy Cluster on Ionic Conductivity in CeO: Hybrid Functional Study 2

Xiaoping Han, Noureddine Amrane, Zongsheng Zhang, and Maamar Benkraouda J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b11300 • Publication Date (Web): 27 Feb 2018 Downloaded from http://pubs.acs.org on March 6, 2018

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The Journal of Physical Chemistry

Unravelling the Negative Role of Oxygen Vacancy Cluster on Ionic Conductivity in CeO2: Hybrid Functional Study

Xiaoping Han1, Noureddine Amrane1, Zongsheng Zhang2,†, and Maamar Benkraouda1,*

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Department of Physics, United Arab Emirates University, Al-Ain, P.O.Box 15551, U.A.E.

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College of International Education, North University of China, Taiyuan 030051, China

* E-mail: [email protected] †

E-mail: [email protected]

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ABSTRACT:

Oxygen vacancy formation and migration play a critical role in the high performances of CeO2 as a highly promising material for solving the environmental and energy issues. However, most of associated works were directed towards the single or isolated oxygen vacancies. In this contribution, the formation and migration of oxygen vacancy cluster in CeO2 have been presented in detail using the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional method. The results demonstrate that oxygen vacancies exhibit a strong tendency to cluster in the direction of CeO2. The detailed analyses of formation energy reveal the favorability of forming such a vacancy cluster under Opoor conditions. By means of the climbing-image nudged elastic band (cNEB) method and molecular dynamics (MD) simulations, the vacancy cluster is found to have a high kinetic stability and low mobility, thus becomes a challenge for achieving high ionic conductivity in CeO2. Attempts have been made to unravel the negative effect of vacancy clustering on ionic conductivity at an atomistic level, and possible means for avoiding or eliminating the vacancy clustering are proposed. The current work has implications for tailoring or optimizing the ionic conductivity in CeO2-based materials or devices for environmental friendly applications.

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I. INTRODUCTION CeO2 and related compounds have attracted much interest, due to their extensive applications to dealing with the global environmental and energy issues.1-4 For example, CeO2 has been used as anode and electrolytic materials within the solid oxide fuel cell (SOFC), which is expected to become high-efficiency electrical power generators that enable clean energy production and support sustainable development.5-7 In the purification of automobile exhausts, CeO2 often acts as three-way catalysts (TWC), oxidizing CO and hydrocarbons to CO2 and reducing NOx to N2.5-10 Also, materials based on CeO2 are efficient to catalytically bind active species like ionic Au in the water-gas-shift reaction,5,7,11-17 thus promoting activity of the catalysts. These outstanding performances are largely due to the good oxygen storage capacity (OSC) of CeO2, i.e., the ability to store and release oxygen.2,13,18,19 This originates from the unique redox properties of CeO2 that, with a fast Ce4+/Ce3+ balance, CeO2 can store O2 under O-rich conditions and provide oxygen when O2 partial pressure decreases, which is directly related to the oxygen vacancy formation and migration properties. Therefore, the favorability of oxygen vacancy formation and diffusion has become critical for enhancing the OSC and ionic conductivity and hence the efficiency of CeO2based materials or devices in environmental friendly applications. It is well known that pure CeO2 does not possess a good ionic conductivity because of very low concentration of oxygen vacancy. To achieve a good ionic conductivity, a general approach is to introduce more vacancies into CeO2 through doping. Previous experimental and theoretical

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studies showed that doping CeO2 with transition metals like Mn,20-25 Zr26-28 and rare earth metals (like Y, Pr, Gd, Sm and Yb)20,21,29-35 effectively promotes the oxygen vacancy formation and migration, thus enhancing the OSC and ionic conductivity and making CeO2 attractive for a wide variety of applications. However, these investigations were mostly related to the single or isolated vacancies. Actually, in some literatures36-38 heavy doping with rare earth elements Sm, Y, Yb, Gd and Dy in CeO2 was reported to cause the oxygen vacancies to cluster in the 36 or 37,38 directions, which was found to lower the ionic conductivity instead. This indicates the clustered vacancies influence the ionic conductivity in a different way from the single or isolated vacancies. Apart from doped CeO2, there have been some explorations performed on the clustered oxygen vacancies in undoped and surface structures of CeO2. Hull et al39 used Monte Carlo modelling to show that the oxygen vacancies preferentially align as pairs in the direction of CeO2 as the degree of nonstoichiometry increases. A recently published paper40 also reported that the oxygen vacancies tend to cluster in the direction, and the positive effect of such clustering on the photovoltaic, photocatalytic and ferromagnetic properties was proposed. Besides the direction, the was suggested by Murgida et al41 to be a possible direction for the oxygen vacancies to cluster, while another investigation exhibited that oxygen vacancies prefer to align along than directions.42 Also, a substantial number of associated examinations have been done on CeO2(111) surface. The vacancy clustering has also been proposed in reduced CeO2(111) surfaces in theoretical explorations.43-45 Esch et al used high-resolution scanning

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tunneling microscopy (STM) to investigate the oxygen vacancies on the CeO2(111) surface, and found that two or more vacancies tend to form linear clusters.46 The similar observations on CeO2(111) surfaces were also obtained using atom-resolved noncontact atomic force microscopy (AFM)47 and the dynamic force microscopy (DFM).48 However, in either undoped or surface structures of CeO2, there is a notable lack of systematic studies on the formation and migration properties of clustered vacancies. Even though in CeO2 doped with rare earth elements36-38 the existence of clustered vacancies was found to induce the ionic conductivity to lower, little was done to probe into the fundamental mechanism underlying the lowered ionic conductivity. Such studies are necessary since the oxygen vacancy distributions and transport properties directly control the functionality of CeO2-based materials and devices in many applications for solving environmental and energy issues. Understanding the atomistic mechanism of oxygen vacancy clustering and how the distributions affect the properties is not only of fundamental interest, but also highly desirable for tailoring or optimizing the functionality of CeO2 for a wide variety of applications. In the present work, we use hybrid functional method to make detailed theoretical explorations of the formation and migration of clustered oxygen vacancies in bulk CeO2, considering the more fundamental significance of undoped bulk CeO2 compared to the doped and surface structures. To authors’ best knowledge, this is the first attempt to theoretically elaborate the formation and migration of the clustered vacancies in CeO2. Interestingly, systematic thermodynamic and kinetic investigations display the vacancy clustering in direction, and the favorability of creating

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this kind of vacancy cluster under the reducing condition as well. The presence of such oxygen vacancy clustering is proven to block the migration and transportation of oxygen vacancies, yielding a big challenge to achieve high ionic conductivity. The present work aims to provide a blueprint for enhancing the functionality of CeO2-based materials or devices through unravelling the mechanism of vacancy cluster to negatively influence the ionic conduction.

II. METHOD All theoretical explorations of virgin and oxygen-deficient CeO2 are performed using the Vienna Ab initio Simulation Package (VASP)49,50 in terms of the projector augmented wave (PAW) method.51,52 We employ the Heyd, Scuseria and Enrzerhof (HSE) hybrid method53 to consider the nonlocal effect in the exchange-correlation (XC) functionals. With the HSE, the exchange potential is separated into a long-range and a short-range part, and a part of Hartree-Fock (HF) exchange is mixed with the Perdew-Burke-Ernzerhof (PBE) functional54 only in the short-range part and the long-range part of the exchange potential is described by the PBE alone. The (4f, 5s, 5p, 5d, 6s) electrons of Ce atom and (2s, 2p) electrons of O atom are treated as valence states with a planewave energy cutoff of 400 eV. The Brillouin zone is sampled using a 8×8×8 Monkhorst-Pack grid for the primitive cell of CeO2, and a 2×2×2 grid for the 2×2×2 supercell of CeO2 unit cell (which is used to construct the oxygen-deficient structures). A series of tests have been conducted to ensure convergence with respect to the number of k-points and energy cutoff. All structures are fully optimized first at the PBE level and further at the HSE level until the total force on each ion is less 6


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than 0.02 eV/Å. In choosing the computational framework, we note that the standard density functional theory (DFT) using the local density approximation (LDA) or generalized gradient approximation (GGA) for XC functional cannot give a satisfactory description of CeO2, due to failing to capture the localization of the Ce 4f electrons. This deficiency can be partly solved by adding the Hubbard U potential into the Ce 4f orbitals in LDA (LDA+U) or GGA (GGA+U) to account for localized Ce 4f electrons. However, there is little guidance for the choice of the Hubbard U parameter. In many literatures the modest U terms (4-6 eV) have been used in CeO2 compounds.6,9,30,41,43,44,55-64 In practice, the U plays the role of an adjustable parameter, i.e., the appropriate U is usually got through using a series of calculations to reproduce or fit a certain experimental values (although some studies pointed out that the Hubbard U parameter can be determined non-empirically65,66). Unfortunately, as shown below, either LDA+U or GGA+U is very hard to use a single U value to simultaneously reproduce more than one crucial parameters (like lattice constant, band gap and formation energy), exhibiting the notable defectiveness of DFT+U in describing systems like CeO2. A more sophisticated treatment is the GW method,67,68 which introduces a one-particle Green’s function G and a dynamically screened Coulomb interaction W into the many-body effects, remarkably improving the description of excited state properties. But it is too cost-expensive to treat the f-electron systems. Comparatively, the HSE method is an alternative moderate-cost method. As shown below, the HSE offers a reliable theoretical description for CeO2.

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III. RESULTS AND DISCUSSIONS Starting from the primitive cell of CeO2, we use the different XC functionals, including the LDA+U, GGA(PBE)+U and HSE functionals, to consider the non-local effect. In the LDA+U and GGA+U formalisms, the Coulomb repulsion U and exchange interaction J are treated by a single effective parameter Ueff = U - J.69 Here, a wide range of Ueff from 0 (LDA or GGA) to 10 eV are employed to make correction to the 4f electrons in Ce atoms. For the HSE, the proportion of 25% HF exchange is applied in the short-range mixing. Their calculated results for the lattice constant, band gaps (as shown in the following figures of band structure and densities of state, there exist two typical gaps in the band structure of pure CeO2: O2p-Ce4f and O2p-Ce5d) and formation energy of CeO2 are shown in Figure 1. In the LDA+U and GGA+U cases, all calculated parameters linearly increase with Ueff, being consistent with the linear response of electronic and structural properties of correlated systems in the DFT functional with the correction of Hubbard U.65 However, it is clear from Fig. 1 that, even though a certain Ueff is used to successfully reproduce one of these parameters, the same Ueff cannot yield others close to their experimental values. In contrast, the HSE-calculated values for lattice constant, band gaps and formation energy agree excellently with their corresponding experimental data, showing the effectiveness of the HSE in describing CeO2. Furthermore, we use the HSE to calculate the band structure and partial density of states (PDOS) for the primitive cell of CeO2, as shown in Figure 2. One can see that both O2p-Ce4f and O2pCe5d band gaps are indirect, which is consistent with two recent theoretical explorations.73,74 The

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calculated valence band (VB) is mainly attributed to O 2p states and its width is about 4.1 eV, being in excellent agreement with the experimental value of 4.0 eV.71,75 The VB is separated by a gap of 3.07 eV from the narrow unoccupied band predominantly composed of Ce 4f states and by a gap of 5.96 eV from the dispersive band with the most contributions from Ce 5d states. These are consistent with the experimental71 and theoretical76 results. Having confirmed that the utilization of HSE functional enables us to pursue the convincing investigations on CeO2, let us examine 2×2×2 supercells with oxygen vacancy pair clusters over a short-range scale. Here the vacancy-vacancy separation is restricted to be no more than the lattice constant of a unit cell of CeO2 (with the further increased vacancy-vacancy separation, the system is predicted to become more and more unstable, and the vacancies tend to be isolated not clustered). Figure 3 displays five configurations of distributing vacancy pairs in oxygen sub-lattice. We fully optimize all of them and find that the cluster is the most energetically stable, indicating that this vacancy cluster is favorably formed compared to other configurations. This result is confirmed by two neutron diffraction studies on undoped CeO2-x, where the vacancy pairs prefer to align in the direction.39,77 Furthermore, we calculate the vacancy-vacancy interaction in the above five configurations using the equation in the literature78 [ 𝐸𝐸int = 𝐸𝐸(CeO2 : 2𝑉𝑉𝑂𝑂 ) + 𝐸𝐸(CeO2 ) − 2𝐸𝐸(CeO2 : 1𝑉𝑉𝑂𝑂 ), where 𝐸𝐸(CeO2 : 1𝑉𝑉𝑂𝑂 ) and 𝐸𝐸(CeO2 : 2𝑉𝑉𝑂𝑂 ) mean the total energies of the structures with

one and two oxygen vacancies, and 𝐸𝐸(CeO2 ) represents that of perfect structure]. Results shows that the vacancies are attractive to each other while there are repulsions in other clustered

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vacancies. Evidently, such vacancy clustering is driven by the repulsion, which makes vacancies minimize their interaction by ordering along the direction. Energetically, it is necessary to assess the likelihood of forming the vacancy cluster considering the energetics of vacancy formation is of great interest. The formation energy of oxygen vacancies in CeO2 can be expressed as79 𝑞𝑞

𝛥𝛥𝛥𝛥𝑓𝑓 (nV𝑂𝑂 ) = 𝐸𝐸(CeO2 : nV𝑂𝑂 ) − 𝐸𝐸(CeO2 ) + 𝑛𝑛𝑛𝑛𝑂𝑂 + 𝑞𝑞(𝐸𝐸𝐹𝐹 + 𝐸𝐸VBM + 𝛥𝛥𝛥𝛥VBM ),

(1)

where 𝐸𝐸(CeO2 : nV𝑂𝑂 ) and 𝐸𝐸(CeO2 ) are the total energies of 2×2×2 supercells with and without nVO in the charge state q. EF is the Fermi level with respect to the VBM (EVBM), and ΔEVBM aligns the VBM in the supercells with and without VO. The chemical potential 𝜇𝜇𝑂𝑂 = 𝜇𝜇𝑂𝑂elem + 𝛥𝛥𝛥𝛥𝑂𝑂 , where 𝜇𝜇𝑂𝑂elem refers to the atomic total energy of O and the extraneous chemical potential 𝛥𝛥𝛥𝛥𝑂𝑂 is subject to the surrounding environment, i.e., O-poor and O-rich conditions. Here 𝛥𝛥𝛥𝛥𝑂𝑂 is limited by the constraints: 𝛥𝛥𝛥𝛥𝑂𝑂 ≤ 0 and 𝛥𝛥𝛥𝛥Ce + 2𝛥𝛥𝛥𝛥𝑂𝑂 = 𝐸𝐸𝑓𝑓 (CeO2 ), where 𝐸𝐸𝑓𝑓 (CeO2 ) refers to the formation energy of CeO2 (its

HSE-calculated Ef(CeO2) is -11.32 eV per molecule unit, agreeing excellently with the experimental value of -11.29 eV72). Under the O-rich condition, 𝛥𝛥𝛥𝛥𝑂𝑂 is 0 eV, and 𝛥𝛥𝛥𝛥Ce = 𝐸𝐸𝑓𝑓 (CeO2 ). Under the O-poor condition (where CeO2 is reduced into Ce2O3), 𝛥𝛥𝛥𝛥𝑂𝑂 and 𝛥𝛥𝛥𝛥Ce are limited by the constraints: 𝛥𝛥𝛥𝛥Ce + 2𝛥𝛥𝛥𝛥𝑂𝑂 = 𝐸𝐸𝑓𝑓 (CeO2 ) and 2𝛥𝛥𝛥𝛥Ce + 3𝛥𝛥𝛥𝛥𝑂𝑂 = 𝐸𝐸𝑓𝑓 (Ce2 𝑂𝑂3 ). Here Ef(Ce2O3) is the formation energy of Ce2O3, which is calculated to be -18.54 eV per molecule unit, very close to the experimental value of -18.63 eV.72 Figure 4a illustrates 𝛥𝛥𝛥𝛥𝑂𝑂 and 𝛥𝛥𝛥𝛥𝐶𝐶𝐶𝐶 between the O-rich limit (𝛥𝛥𝛥𝛥𝑂𝑂 = 0 eV and 𝛥𝛥𝛥𝛥𝐶𝐶𝐶𝐶 = −11.32 eV) and the O-poor limit (𝛥𝛥𝛥𝛥𝑂𝑂 = −4.10eV and 𝛥𝛥𝛥𝛥𝐶𝐶𝐶𝐶 = −3.12 eV), 10


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i.e., the thermodynamically allowed range of CeO2. Based on the above formula and data, we calculate the formation energies of a single vacancy and the vacancy cluster in different charge states, and the transition levels EF(q/q’) between charge states q and q’ [which can be derived from the equation 𝛥𝛥𝛥𝛥𝑓𝑓 (𝑉𝑉𝑂𝑂𝑞𝑞 ) = 𝛥𝛥𝛥𝛥𝑓𝑓 (𝑉𝑉𝑂𝑂𝑞𝑞′ )] as well

.

Figure 4b shows the calculated formation energies of VO in different charge states (0, +1 and 2+). Clearly, the 1+ state is unstable for all values of EF: when EF is less than 1.75 eV, the 2+ state is stable; beyond this value, the neutral state is stable. Namely, its transition level EF(2+/0) is at 1.75 eV, or 1.32 eV below the conduction band minimum (CBM, ECBM). The formation energies of the vacancy cluster in 0, +1, +2, +3 and +4 states are shown in Figure 4c. Here, the 1+, 2+ and 3+ states of the vacancy cluster are unstable, and its transition level EF(4+/0) is at 1.81 eV. In order to simplify the presentation, only the stable states for a single vacancy (1VO) and a vacancy cluster (2VO) under the O-poor and O-rich conditions are illustrated in Figure 4d. Apparently, the O-poor condition encourages the formation of oxygen vacancies while the O-rich chemical potential is energetically inferior due to the very high formation energy. Under the Opoor condition, the highest formation energy of a single vacancy is just a small positive value, suggesting that it would be readily surmountable. More interestingly, the formation energies of the vacancy cluster under the same condition are negative, demonstrating that it is highly favourable to form a vacancy cluster. These results reveal that, if a single vacancy forms first, it can easily grow into vacancy cluster even without further requiring the external force.

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After understanding the favourability of forming the oxygen vacancy cluster, let us turn to its migration. First we calculate the migration barrier of the oxygen vacancy hopping from one lattice position to adjacent sites using the climbing-image nudged elastic band (cNEB) method.80,81 Figure 5 shows the four migration paths of a single vacancy in the , and directions (it is generally accepted that oxygen vacancy in CeO2 can migrate along these three directions), and the corresponding migration barriers as well. Obviously, the migration barriers along the and directions are much larger than that along the direction, meaning that a negligible contribution to the ionic conductivity is expected for the and directions, i.e., the migration of a single vacancy in CeO2 is strongly anisotropic. This result gets strong support from previous literatures. For example, the ionic conduction in CeO2-x was reported to be a consequence of anion motion via a vacancy diffusion mechanism between nearest neighbour sites in the directions.34,82 In a recently published paper,83 Nilsson et al used the molecular dynamics method to show that the migration of oxygen vacancy in Gd-doped CeO2 occurs in the simple oxygen sublattice in the direction. The same migration direction was also found in Y-doped CeO2 using neutron powder diffraction combined with the maximum entropy method.84 Here the cNEBcalculated migration barrier of 0.42 eV (in Path 1) is remarkably closer to the experimental value of 0.40 eV85 than that of 0.36 eV40 obtained using the nudged elastic band (NEB) method,86 indicating that the cNEB method allows for the more accurate finding of saddle points than the NEB method.

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Furthermore, we use the cNEB method to examine the migration of the vacancy cluster along the , and directions. Evidently, with respect to a single vacancy, the vacancy cluster has a greatly increased number of possible migration paths, which makes it much more complicated to investigate the migration of the vacancy cluster. In general, it is energetically much cheaper for a vacancy pair to migrate by successive jumps of individual vacancies than by the simultaneous diffusions of both vacancies, so we focus on the successive monovacancy diffusions in inspecting the migration of the vacancy cluster. For the migration along the directions, four typical migration paths (as illustrated in Figure 6) are examined, and their corresponding migration barriers are shown in Figure 7. Apparently, Path A is the most favourable among four paths investigated (it is worth stressing that there may be other diffusion paths which has the lower migration barriers than Path A. But this will not influence the subsequent conclusion that the vacancy cluster prefers to migrate along the direction. As will be detailed below, the much higher migration barriers in and directions compared to the existent values in direction reveal that it is very hard for the vacancy cluster to migrate along the and directions). In the same way, we investigate the migration of the vacancy cluster along the and directions, and their minimum-energy diffusion paths and corresponding migration barriers are shown in Figure 8, where the migration barriers in the and directions are found to be at least two times as high as that in direction (Path A in Figure 7). As a result, the vacancy cluster is predicted to predominantly migrate along the direction.

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However, the significantly enhanced migration barriers of the vacancy cluster (see Figures 5 and 7) indicate that the vacancy cluster has a lower mobility and higher kinetic stability with respect to a single vacancy, negatively influencing the ionic conduction in CeO2. To further gain more insight into the microscopic nature of oxygen vacancy motion and better characterize the negative effect of vacancy cluster on ionic conductivity at an atomistic level, we run ab initio molecular dynamics (AIMD) simulations on bigger 3×3×3 CeO2 supercells (containing 108 Ce cations and 216 O anions) with a single vacancy and a vacancy cluster. Considering the increasingly expensive computation for AIMD simulations, the PBE (not HSE) is chosen for the XC functional. In all simulations, we employ a NVT ensemble with a constant volume and temperature (1000 K), integrating the equations of motion at 1 fs time intervals and controlling temperature via the Nosè-Hoover thermostat. At each time-step, the total energy is evaluated to an accuracy of 10-4 eV/atom with a plane-wave energy cutoff of 400 eV, while sampling the Γ-point in the Brillouin zone. For both systems with a single vacancy and a vacancy cluster, a 2-ps simulation is first performed for equilibration, followed by the 4-ps simulations for statistical analysis. We have observed the frequent diffusions of the single vacancy. Most of these diffusions manifest themselves in the jumps of an oxygen anion into the vacancy site along the direction, which is consistent with the above calculated results of migration barriers. In contrast, the clustered vacancies are observed to mainly vibrate around their equilibrium positions, suggesting that vacancies are trapped and hard to diffuse. Occasionally, one vacancy of

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the cluster is observed to jump, but often followed by a successive jump back to its original position, and seldom followed by the jump of the other. Understandably, it is the correlation between clustered vacancies that limits the mobility of vacancies, hindering ion conduction. Another unneglectable factor influencing the ion conduction in the case of vacancy cluster is the existence of more Ce3+ ions. It is generally accepted that creating an O vacancy by the removal of an O atom from CeO2 causes two excess electrons to transfer to two Ce4+ ions near the vacancy, reducing them into Ce3+ ions. Though the distribution of these two Ce3+ ions is still under intense debate (they were reported to prefer to be nearest neighbors to the vacancy by some studies6,9,30,40,5557,87-89

while others suggested the next nearest neighbors are energetically more favorable41,58,60,90),

the bigger-size Ce3+ ions (compared to Ce4+ ones) definitely influence the vacancy migration through appearing near the migration path. In order to elaborate this point, we have investigated the migration of a single vacancy with different distributions of oxygen vacancy and Ce3+ ions. As shown in Figure 9A, two adjacent tetrahedra composed of Ce4+ ions are used to define the path of oxygen vacancy migration along the direction, with an oxygen vacancy being in one of two tetrahedra and an oxygen ion in the other. Here we use an approach by Zacherle et al91 to control the positions of Ce3+ ions, where an electron polaron is prepared to make the additional electron well localized on the chosen Ce ion. The configurations of oxygen vacancy relative to Ce3+ ions are listed in Figure 9(B-H), displaying 7 possible models with 1 and 2 Ce3+ ions near the migration path. Their migration barriers have been calculated and shown in Fig. 10 (for the sake of

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comparison, the barrier for Model A is also listed together with those of Models B and C in Figure 10a). Apparently, the vacancy migration barrier has a strong dependence on the amount of Ce3+ ion around the vacancy: the migration barrier remarkably increases with the amount of Ce3+ ion (from 0 to 1 and 2), making the hopping probabilities of oxygen vacancy greatly reduced. In the case of a single vacancy, the amount of Ce3+ is very limited, which makes Ce3+ ions impossible of appearing near all lowest-energy migration paths. Comparatively, the clustered vacancies induce the increased occurrence of Ce3+ even Ce3+ pairs near the migration paths, significantly enhancing the difficulty of vacancy cluster in diffusing. Indeed, it should be fairly effective to hinder the migration of vacancy cluster as long as a single Ce3+ ion or Ce3+ pairs just appear near one vacancy of the cluster. The negative effect that the vacancy clustering has on ionic conductivity accounts well for the presence of a maximum in the ionic conductivity of CeO2-δ in theoretical92 and experimental85 literatures (where the ionic conductivity increases initially with the non-stoichiometry δ, and reaches a maximum at a certain δ then falls): At low concentrations, oxygen vacancies are isolated or randomly distributed and the ionic conductivity increases with δ; when δ is increased up to a critical value, the oxygen vacancies start to cluster, thus inducing the ionic conductivity to decrease. The similar phenomenon was reported in the systems of Y-doped CeO2 (Ce1-xYxO2-x/2, x is the dopant concentration),36 which is also closely related to the oxygen vacancy clustering in CeO2 doped with high concentrations of Y, although other effects may also contribute to this, such as the

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interaction between the dopant Y and oxygen vacancy. In another studies37,38 using electron energy loss spectroscopy and select area electron diffraction (SAED) on CeO2 doped with rare-earth metals Y, Sm, Gd, Dy and Yb, an evident decrease in ionic conductivity was observed as the concentration of each dopant increased from 15 to 25 at%. Clearly, the common phenomenon for different dopants, i.e., the decrease in ionic conductivity, is regardless of dopants. What really takes effect is the higher degree of the clustering of the dense oxygen vacancies originating from the higher concentration of dopants (although the different dopants have different effect on the clustering degree of vacancies). Therefore, the consideration for improving ionic conductivity of CeO2-based materials should be taken at the range of lower vacancy concentrations. Overall, the vacancy cluster has become the challenge for achieving a high ionic conductivity in CeO2, leading to the degradation in the related performances. Once the vacancy cluster forms, higher additional energies are required for overcoming its negative influence on diffusion. It is therefore for one to try to avoid or eliminate the clustering of oxygen vacancies. First, a strongly reducing condition like high-temperature atmosphere ought not to be used when preparing or processing CeO2. A short-range ordering of the anion vacancies has been reported in theoretical39 and experimental93 investigations on CeO2 under the condition of the high temperature (1273K). Another experimental and molecular-dynamic studies of Y-doped CeO2 also showed that at 1073K the vacancies would rather cluster along the direction than distribute randomly while the vacancy clustering becomes less pronounced at 873K.36 On the other hand, strict control should be

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made on the concentration of dopants which are used to promote the ion conduction. Although many transition metals or rare-earth metals can be used to dope CeO2 for enhancing ionic conductivity, it is only effective when the vacancies are isolated or randomly distributed at low dopant concentrations. Higher dopant concentrations induce the dense vacancies to cluster,36-38 which in turn hinders the ion conduction. In addition, some external processing treatments have been found to notably influence oxygen vacancy clustering in real applications. For instance, photoexcitation can effectively induce the dissociation of clustered vacancies at the LaAlO3/SrTiO3 interface94 while the tensile strain imposed on CaMnO3 is beneficial for oxygen vacancy clustering.95 Undoubtedly, these are useful for weakening or adjusting the clustering of oxygen vacancies. Actually, apart from CeO2, the strong ordering of oxygen vacancies along the direction has been found in other fluorite oxides. Two experimental literatures reported that the vacancy pairs cluster in the direction of PrO296 and TbO2.97 The same clustering of anion vacancies in ZrO2 compounds was also confirmed using the neutron diffraction, Monte Carlo and impedance spectroscopy measurements.98-101 Taking these into account, it appears that the presence of the vacancy pair is a common feature of oxygen-deficient fluorite oxides. Accordingly, the outcome of this work is expected to be generalized to fluorite oxides.

IV. CONCLUSIONS

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In summary, we have investigated the formation and migration of oxygen vacancy cluster in bulk CeO2 using the HSE hybrid functional method, and found that oxygen vacancies energetically cluster in the direction. The detailed analyses of formation energy show that it is rather easy to form such vacancy cluster under the O-poor condition. The explorations using the cNEB and MD simulation demonstrate that the vacancy cluster is kinetically stable and hard to diffuse, indicative of a low ionic conductivity. Efforts have been done to fundamentally unravel the negative effect of vacancy clustering on the ionic conduction in CeO2. Appropriate means should be taken to eliminate or weaken oxygen vacancy clustering, so that the functionality of CeO2-based materials for solving the environmental and energy issues can be maximized. The outcome of this work is expected to be generalized to other fluorite oxides.

ACKNOWLEDGEMENT This work was supported by United Arab Emirates University through the University Program for Advanced Research (No. 31S109-UPAR and 31R109-Research Center-ECEER-9-2016). Part of computing time was provided by North University of China.

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Figure 1. Effect of Ueff on the lattice constant a (a), O2p-Ce4f band gap (b), O2p-Ce5d band gap (c) and formation energy (d) of CeO2. The Hubbard correction is applied to the 4f electrons. The dot lines represent the corresponding experimental values (the experimental lattice constant of 5.41 Å is from Ref. 70, the experimental O2p-Ce4f band gap of 3.0 eV and O2p-Ce5d band gap of 6.0 eV are from Ref. 71, and the experimental formation energy of -11.29 eV is from Ref. 72). For the sake of comparison, the corresponding HSE values are marked using red squares.

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Figure 2. Calculated band structure (a) and partial density of states (b) for the primitive cell of CeO2 using the HSE formalism. The dot-dashed lines at energy zero represent the Fermi level.

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Figure 3. Schematic configurations of short-range vacancy pairs in CeO2, where the vacancyvacancy separation is set to be no more than the lattice constant of a unit cell of CeO2. Each dashed cube corresponds to a unit cell of CeO2 and each solid cube represents an O sublattice. Here only the oxygen vacancies are marked with red cubes. The , and structures represent the vacancy pairs within the unit cell of CeO2 along , and directions, while the e and e ones represent the vacancy pairs along the and directions extending out of the unit cell, respectively. For the e, the vacancy-vacancy separation equal to the unit-cell lattice constant. The e structure, in spite of having the same vacancy-vacancy separation to the above structure, includes a face-centered Ce atom halfway between.

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Figure 4. (a) The Ce chemical potential 𝛥𝛥𝛥𝛥Ce as a function of the O chemical potential 𝛥𝛥𝛥𝛥𝑂𝑂 , as

derived from the stability conditions of different oxide stoichiometric CeO2 and Ce2O3. Two vertical dot lines indicate the O-rich and O-poor limit for the growth of CeO2, respectively. (b) The calculated formation energies as a function of the Fermi level for a single vacancy in CeO2 under O-poor conditions. (c) The calculated formation energies as a function of the Fermi level for a vacancy cluster in CeO2 under O-poor conditions. (d) The formation energies for a single vacancy (1VO) and a vacancy cluster (2VO) in CeO2 under the O-poor (in red) and O-rich (in black) conditions.

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Figure 5. Migration barriers of a single oxygen vacancy. On the left, four paths for the migration of a single oxygen vacancy in the directions of , and are illustrated. Here only the O sublattice is shown. The O vacancy and O atoms are represented by a red cube and a yellow sphere, respectively. Note Paths 3 and 4 show two different diffusion paths in the direction, distinguished by whether or not the path contains a Ce atom (represented by a blue sphere).

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Figure 6. Schematic for four typical paths (Paths A-D) for the migration of the vacancy cluster along the direction, including their corresponding initial, intermediate and final configurations (Configurations 1-4). Here only the O sublattices along the migration direction are shown. The O vacancy, O atom and Ce atom are represented by a red cube, yellow and blue spheres, respectively.

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Figure 7. The migration barriers of the vacancy cluster along the paths described in Fig. 6.

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Figure 8. The migration barriers of the vacancy cluster along (a) and (b) directions, with the corresponding minimum-energy paths listed along the horizontal axis. The O vacancy, O atom and Ce atom are represented by a red cube, yellow and blue spheres, respectively.

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Figure 9. Possible migration models of a single vacancy in the cases of 0 (A), 1 (B,C) and 2 (D-H) Ce3+ ions. The O vacancy is marked with a red cube, and O2-, Ce4+ and Ce3+ ions are represented by red, yellow and purple spheres, respectively.

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Figure 10. (a-b) The migration barriers of a single vacancy in the models shown in Figure 9.

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TOC Graphic

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Unraveling the Negative Role of Oxygen-Vacancy Cluster in Ionic

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