Before discussing the XANES spectra for the doped and non-stoichiometric ZrO2, its is important to describe the relaxed structures of S0, S1, and S2 containing Ni4+ ions (Figure 2a,b), Ni3+ ions (Figure 2c,d), and Ni2+ ions (Figure 2e,f), respectively, since these are the structures used for the XANES calculations. We have found that in the relaxed structure of S0, ferromagnetic spin alignments are more stable compared to the antiferromagnetic configuration, and the local environment around Ni dopants assumes to a tetrahedral symmetry following the relaxation (Figure 2a,b). In the case of S1, the ferromagnetic configuration is still the most stable while the site symmetries of Ni1 and Ni2 show a coexistence of distorted tetrahedral and trigonal bipyramidal geometries, respectively (Figure 2c,d). However, in the case of S2, the antiferromagnetic spin configuration is the most stable in which the nickel atoms Ni1 and Ni2 coexist in distorted octahedral and trigonal bipyramidal geometries, respectively (Figure 2e,f).

Figure 2: Relaxed nickel geometries and crystal-field splitting diagrams for different Ni-doped zirconia containing varying oxygen vacancy concentrations. (a, b) Nickel dopant atoms in tetrahedral geometry in the structure S0 with Ni1 and Ni2 being spin-up polarized. It should be noted that S0 has no oxygen vacancies. (c, d) Nickel dopant atoms in trigonal bipyramidal and tetrahedral geometries in the structure S1 with Ni1 and Ni2 being spin-up polarized. Structure S1 has one oxygen vacancy. (e, f) Nickel dopant atoms in octahedral and trigonal bipyramidal geometries in the structure S2 with Ni1 being spin-up and Ni2 being spin-down polarized. Structure S2 has two oxygen vacancies. In the figures, blue and red spheres represent Ni and O atoms, respectively. Black filled arrows represent occupied electron states while the hollow arrows indicate empty electron states.

We present XANES spectra for defect-free zirconia, that is, without Ni dopant (x = 0 atom %), and the doped structures S0, S1, and S2 in Figure 3a. In the case of defect-free zirconia, the XANES spectrum is needed only for one oxygen atom site since the lattice sites of all the oxygen atoms are equivalent. The resulting spectrum shows no peak in the pre-edge region where the Fermi level is considered as zero energy in the plot (Figure 3a). This observation is in agreement with our previous observations in the case of iron-doped zirconia [27]. As noted earlier, when the two nickel atoms are introduced (structure S0), the ferromagnetic structure is the more stable configuration, and the O K-edge spectrum, calculated as the average of contributions from all oxygen atoms in the supercell (i.e., spectra from the 64 O atoms), shows a pre-edge peak in the energy range between 0 and 1.45 eV (Figure 3a). This pre-edge peak is ascribed to dipole transitions from 1s to 2p states that are hybridized with unoccupied nickel 3d states as we demonstrated previously in the case of iron-doped zirconia [27]. For comparison with experimental data, we have also plotted the experimental O K-edge spectrum of iron-doped zirconia (ZrO2:Fe) at x = 6 atom % Fe dopant concentration from [27] (Figure 3a). This previous experimental spectrum is very similar to the current theoretical ones since the Fe has oxidation states similar to Ni, although Fe has two electrons less in the d shell. It has to be noticed that, in the previous study, we have considered iron atoms in the oxidation state +3.

Figure 3: (a) O K-edge spectra of pure zirconia (blue curve) and of nickel-doped zirconia (ZrO2:Ni) at x = 6.25 atom % Ni concentration and varying number of vacancies. The notations S0, S1, and S2 correspond to the doped structures containing zero, one, and two oxygen vacancies, respectively. Note the presence of the pre-edge peak, which decreases when the number of oxygen vacancies increases. Also, the black, red, and green curves are the XANES spectra obtained from the averages of 64, 63, and 62 individual O K-edge spectra in the zirconia supercell, while the spectrum E (black dashed line) corresponds to the experimental O K-edge spectrum in iron-doped zirconia (ZrO2:Fe) at x = 6 atom % Fe concentration from [27]. (b) The mean contributions of the first oxygen shells around the nickel atoms Ni1 (solid lines) and Ni2 (dashed lines) to the O K-edge spectra of the structures S0, S1, and S2.

Assuming the nickel atoms to be in the oxidation state +4 since they substitute Zr4+ cations and considering the nickel atom Ni1, we observe that its contribution to the pre-edge peak is mainly due to its 3dxy, 3dzx, and 3dzy spin-down orbitals, as well as its 3dzy spin-up orbitals, as shown by the PDOS in Figure 4a. Ni1 with Ni4+ oxidation state and 3d6 orbital configuration has a tetrahedral coordination, which suggests that it is in low spin-polarization with an electron configuration according to ligand field theory [46-48] (cf. Figure 2). This is because the two orbitals e and t2 participate together to the transitions occurring in the pre-edge region (Figure 4a). The atom Ni2 has similar site symmetry and electronic configurations with low spin-polarization, as well as an oxidation state similar to that of the Ni1 atom. Also, its orbital contribution to the pre-edge peak is similar to that of Ni1.

Figure 4: Projected densities of states (PDOS) of nickel atoms Ni1 and Ni2. The solid line corresponds to spin-up polarization, the dashed line corresponds to spin-down. (a, b) Nickel atoms Ni1 (spin-up) and Ni2 (spin-up) in tetrahedral geometries, belonging to the doped structure S0. (c, d) Nickel atoms Ni1 (spin-up) and Ni2 (spin-up) in trigonal bipyramidal and tetrahedral geometries, respectively, belonging to the doped structure S1. (e, f) Nickel atoms Ni1 (spin-up) and Ni2 (spin-down) in octahedral and trigonal bipyramidal geometries, respectively, belonging to the doped structure S2.

For the structure S1, the O K-edge spectrum (calculated from the average of 63 O K-edge spectra) shows a pre-edge peak in the same energy range as in the case of S0, but with lower intensity. Here, nickel atoms coexist in two different geometry sites, which are a distorted trigonal bipyramid for Ni1 and a distorted tetrahedron for Ni2. The PDOS of Ni1 (spin-up) shows that the contribution to the O K pre-edge peak is mainly due to , 3dxy and spin-down orbitals (Figure 4c). This orbital arrangement suggests that the nickel atom has an oxidation state of +3 and is in high spin-polarization with the valence electron configuration according to ligand field theory (cf. Figure 2). In the case of nickel atom Ni2, the contribution to the O K pre-edge is mainly due to 3dxy, 3dzx and 3dzy spin-down orbitals (cf. Figure 4d). In this case, the distorted tetrahedron symmetry can not explain the nickel oxidation state +4 as initially assumed. However, the electron configuration of the nickel atom is the oxidation state +3 in high spin-polarization with the valence electron configuration according to the ligand field theory (cf. Figure 2). This is because both e and t2 orbitals participate together in the transitions occurring in the pre-edge region.

When the second oxygen vacancy is introduced, that is, structure S2, the antiferromagnetic configuration becomes the more stable structure, and the geometry sites of the nickel atoms change to a distorted octahedron for Ni1 and a trigonal bipyramid for Ni2. The O K-edge spectrum (resulting from the average of 62 O K-edge spectra) shows a weaker pre-edge peak (in intensity) than in the case of S0 and S1. The nickel Ni1 contribution to the pre-edge peak is mainly due to and orbitals of spin-down electrons as shown in Figure 4e. Again, such an orbital configuration can not explain the oxidation state +4 for a nickel atom in octahedral coordination according to ligand field theory. However, this situation may correspond to a nickel having oxidation state +2 with the valence electron configuration (cf. Figure 2). In the case of nickel Ni2 with spin-down electrons, the contribution to the pre-edge peak is mainly due to and spin-up orbitals (Figure 4f), which also cannot explain the oxidation state +4 for a nickel atom in trigonal bipyramidal coordination. However, this configuration matches with a nickel atom having the oxidation state +2 in high spin-polarization, with the valence electron configuration according to ligand field theory. Furthermore, the peaks in the PDOS plots (immediately above the Fermi level) that correspond to the pre-edge peaks in the O K-edge XANES spectra show sensitivity to increasing oxygen vacancy concentration. These PDOS peaks show change in intensity and splitting in response to the distortions from ideal geometric arrangements of oxygen shells surrounding the nickel atoms Ni1 and Ni2. This leads to a partial lifting of the symmetry of the ideal structures, resulting in overlap or hybridization of the d states (Figure 4).

The above analysis demonstrates that oxygen vacancies affect not only the coordination environment around nickel atoms, but also impact strongly its oxidation state. To have a deeper understanding of various contributions to the pre-edge peak with varying oxygen vacancies concentration, we focus on the O atoms neighboring nickel atoms in the three structures S0, S1, and S2. For these structures, we plotted the O K-edge spectrum representing the mean contributions of oxygen first nearest-neighbor shell around each of the nickel atoms Ni1 and Ni2, since the pre-edge peak originates mainly from the oxygen atoms belonging to that shell, as we demonstrated previously for Fe-doped zirconia [27]. The resulting spectra are presented in Figure 3b, where the solid and dashed lines correspond to oxygen atoms belonging the first shells around nickel atoms Ni1 and Ni2, respectively. In the case of S0 (black lines in Figure 3b), the contributions to the pre-edge peak correspond to two intense peaks (solid and dashed lines correspond to Ni1 and Ni2, respectively), which are mainly located at 0.82 eV above the Fermi energy. These two pre-edge peaks are almost equivalent since in S0, nickel atoms Ni1 and Ni2 adopt tetrahedral geometries (point group Td), with almost identical Ni–O bond lengths and small standard deviations as shown in Table 1. The latter indicates negligible distortions of the bond lengths.

Table 1: Ni–O bond lengths (di, i = 1, 2, 3, 4, 5, 6) for nickel atoms Ni1 and Ni2 in the structures S0, S1, and S2 containing zero, one, and two oxygen vacancies, respectively. Ni1 and Ni2 are in tetrahedral (T), trigonal bipyramidal (TB), and octahedral (O) geometries. σd is the standard deviation of Ni–O bond lengths for Ni1 and Ni2 within different geometries. The standard deviation increases with increasing number of oxygen vacancies, indicating how distorted the geometry becomes.

structures S0 S1 S2 oxygen vacancy 0 1 2 nickel atom Ni1 Ni2 Ni1 Ni2 Ni1 Ni2 geometry T T TB T O TB point group Td Td D3h Td Oh D3h d1 (Å) 1.873 1.875 2.013 1.903 2.448 1.980 d2 (Å) 1.874 1.872 1.971 1.933 1.997 2.100 d3 (Å) 1.873 1.872 2.181 1.891 2.192 1.982 d4 (Å) 1.873 1.872 1.933 1.890 2.011 2.140 d5 (Å) – – 1.981 – 1.942 2.144 d6 (Å) – – – – 2.593 – ⟨d⟩ (Å) 1.873 1.873 2.016 1.904 2.197 2.069 σd 0.00043 0.0013 0.08646 0.01737 0.24477 0.07364

In the case of structure S1 (red lines in Figure 3b), the contributions to the pre-edge peak are two peaks of which the first one, corresponding to Ni2, is mainly located at 0.82 eV above the Fermi energy (dashed red line), that is, at the same position than the pre-edge peaks in S0. This is expected since in S1, nickel atom Ni2 also adopts the tetrahedral geometry (point group Td). The second peak is mainly located at 1.65 eV above the Fermi energy (solid red line) and corresponds to nickel Ni1 having a trigonal bipyramidal geometry (point group D3h). However, these two peaks are less intense compared to their equivalents in S0. The corresponding Ni–O bond lengths have stronger variations as shown by their standard deviations presented in Table 1.

For the S2 structure (green lines in Figure 3b), the contributions to the pre-edge peak correspond to two peaks, of which the first one is mainly located at 0.41 eV above the Fermi energy (solid green line). The corresponding nickel atom Ni1 has an octahedral symmetry (point group Oh). The second peak is mainly located at about 1.64 eV above the Fermi energy (dashed green line) and corresponds to nickel atom Ni2 in trigonal bipyramidal symmetry (point group D3h). This second peak is almost at the same position as that of Ni1 in the structure S1 since they have similar site symmetry. Here, nickel sites are strongly distorted since the Ni–O bond lengths for the two nickel atoms Ni1 and Ni2 present much higher variations from their equivalents in the structures S1 and S0, as shown by the corresponding standard deviations (Table 1).

The aforementioned analyses show that the combination of dopant symmetry sites and structural disorder due to the presence of oxygen vacancies in S1 and S2 may be responsible for the decrease in the pre-edge peaks with increasing concentration of oxygen vacancies.