(1) Defect Formation Energies
The stability of the different group IV-doped fluorite CeO2 systems is investigated through the comparison of the defect formation energies Ef, which are calculated as follows:
with Ndf the number of substituent atoms in the system, the total energy of the relaxed-doped system, the total energy of a CeO2 fluorite supercell of equal size, ECe and EZ the energy per atom of bulk α-Ce and the bulk phase of the substituent element Z. Larger positive values of Ef indicate that more energy is required for doping. Pure Ce metal has several different phases depending on the external pressure and temperature. In our case, we chose to use the α-phase, as the calculations are performed at zero temperature and pressure. The used crystal structure of the dopant element bulk phase is presented in Appendix A, Table V. Defect formation energies can also be calculated with regard to dopant-oxides, and are presented in Appendix B. The obtained results remain qualitatively the same as those presented below.
Table 1 shows that Ef varies only slightly with dopant concentration. For cells with a dopant concentration of 3.125% Ef is also calculated within the DFT + U framework, using the local density approximation (LDA)/Perdew, Burke, and Ernzerhof (PBE) optimized geometry and a Hubbard U = 5 eV for the Ce f electrons. As can be seen in Table 1, the formation energies are slightly larger in the DFT + U framework, although they do not change the relative stability of the systems compared with the pure LDA/PBE results.
Table 1. Defect Formation Energy for Group IV Dopants at Different Concentrations
| ||Ef (eV)|
| ||LDA||LDA + Ua|
| ||PBE||PBE + Ua|
For different substituent elements, the formation energies show a spread over quite a large range, with roughly 20 eV/atom required to imbed a C atom, down to merely 0.1 eV/atom required to imbed a Hf atom. This shows that none of the group IV elements increase the stability of CeO2. A direct comparison with experiment is not straightforward, as ab initio calculations do not account for effects of pressure and temperature. In addition, during the experimental preparation of a compound there are often several steps (thermal and/or mechanical treatment, etc.,) involved which provide additional energy to the system that enable the formation of metastable-doped systems. For example, in experiments one observes that oxygen vacancies form spontaneously in CeO2 cubic fluorite, although ab initio simulations show them always to require energy. In an attempt to compare with experiment, we use our calculated oxygen vacancy formation energy as a reference. Substitutions that require the same ab initio computed energy as this reference, or less, are considered more likely under experimental conditions, while those requiring more, less likely. For the LDA and PBE functionals, we calculated the O vacancy formation energy to be 4.035 and 3.097 eV per vacancy, respectively, at a vacancy concentration of 1.5625%, in excellent agreement with values found in literature.[42, 43] Comparing the 4.035 eV found for pure LDA to the 3.61 eV for LDA + U presented by Andersson et al., shows that the introduction of a Hubbard U term does not qualitatively alter the stability difference observed for the group IVa and IVb dopants. Note, however, that the pure LDA value is closer to the experimental value of 4.65–5.00 eV for bulk reduction. For higher O vacancy concentrations higher formation energies are found, up to 5.006 and 4.145 eV for LDA and PBE, respectively, at 12.5%, which is within the range of experimentally derived values.
Based on these reference energies, the dopants in Table 1 nicely split in the “more likely bulk dopants” (group IVb) and the “less likely bulk dopants” (group IVa). In this context, group IVb dopants are likely to remain (homogeneously) dispersed in the bulk of the CeO2 grains and crystals, whereas the group IVa dopants would likely segregate to the surface regions of the CeO2 grains, or the interface regions with other materials, or cluster to form small domains of the dopant element inside CeO2 crystals and grains. This makes the latter of interest for applications where tuning of surface effects is important. The former, however, are expected to remain distributed in the CeO2 bulk, making them well suited for applications where CeO2 bulk parameters need to be modified, or in oxide mixing experiments. These results also agree with the observation that Zr-doped CeO2 is widely used and easily produced in experiments, whereas it is much harder to form Si- and Sn-doped CeO2 in a controlled way.[32, 34-36]
For the group IVb elements, the formation energy decreases with increasing atomic size, showing a better size-match with Ce results in reduced system strain. The formation energies for the group IVa-doped systems show a similar globally decreasing trend. The additional oscillation is interesting, and is retained even with the introduction of the Hubbard U correction. The slight increase in the formation energy for Ge and Pb coincides with the introduction of a “new” filled shell, the d shell in case of Ge, and the f shell in case of Pb. This shows that not only the outer valence electrons play a role in the stability of the system but also the presence of filled shells near the Fermi level.
(2) Density of States
We investigated the influence, at low concentration (x = 3.125%), of group IV dopants on the density of states (DOS) of Ce1−xZxO2, within the DFT + U framework. Due to the low concentration, the general shape of the DOS is comparable to the DOS of pure CeO2; there is a conduction band of unoccupied Ce 4f states in the band gap between the O 2p valence band and Ce 5d conduction band (e.g., Figs. 1 and 2). Table 2 shows that the band gap between the unoccupied Ce 4f states and the O 2p valence band is always smaller than for pure CeO2. Excluding C, we find for both group IVa and IVb dopants an increase in the band gap size with the atomic number (within each group). The same behavior is observed for the O 2p–Ce 5d band gap. Comparison of the LDA + U and PBE + U results in Table 2, shows that the trends are independent of the functional used. In case of the group IVa elements, there is an atomic band roughly 2 eV below the O 2p band (e.g., Fig. 1). From the local DOS (LDOS), it is clear that this localized state is a combination of the dopant s state and the O 2p state of the O ions surrounding the dopant. This is shown for Si doping in Fig. 1, and agrees well with the work of Andersson et al. where a symmetry broken configuration was studied. In contrast to that work, we also observe such a localized combined state in the 2p–5d band gap (cf., Fig. 1). Closer investigation reveals that both bands each integrate to two electrons, indicating that this is a pair of bonding and antibonding states. To investigate this discrepancy the LDOS for a symmetry broken configuration (Ce0.96875Si0.03125O2) was calculated. With the exception of the Ce 4f gap state, due to the presence of an oxygen vacancy in the structure of Andersson et al., we obtained qualitatively the same LDOS as was reported by these authors. This indicates that our observed s–p gap state originates from the different chemical environment for the Si dopant. However, it remains unclear to us if this state simply vanished upon symmetry breaking, or merely merged with the Ce 5d band. The work of Tang et al. on Sn-doped CeO2 also shows the appearance of such atomic states below the O 2p band and in the band gap. Furthermore, Tang et al. show that the gap state lies under the Fermi level when an oxygen vacancy next to the dopant is created, showing the excess electrons to transfer to the group IVa dopant. Table 2 shows the position of these gap states with regard to the valence band edge. This shows that only for C doping this state is occupied and is located at the edge of the valence band (hence the negative value). For the other IVa elements, the band is either located above or below the unoccupied Ce 4f band. The relation between these gap states and the atomic state below the O 2p valence band becomes even clearer due to the strong correlation of their position, placing the atomic states for the Ge- and Pb-doped systems roughly 0.7 eV below those of the Si- and Sn-doped systems.
Table 2. Band Gaps in Group IV-doped CeO2a
| ||LDA + U||PBE + U|
|2p 4f||2p 5d|| gs ||2p 4f||2p 5d|| gs |
Figure 1. The total DFT + U DOS for Ce0.96875Si0.03125O2, indicating important features (top). The 3s and 3p Si LDOS, showing the origin of the gap state and atomic state below the O 2p band (middle). The 2p O LDOS for the eight nearest-neighbor (NN) O atoms surrounding the Si dopant, and an O atom far away from the dopant (bottom). The upper and lower panels present the spin up and spin down channels, respectively.
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Figure 2. The total DFT + U DOS for Ce0.96875Zr0.03125O2, indicating important features (top). The 5s and 4d Zr LDOS, showing the origin of the gap state and atomic state below the O 2p band (middle). The 2p O LDOS for the eight nearest-neighbor (NN) O atoms surrounding the Zr dopant, and an O atom far away from the dopant (bottom). The upper and lower panels present the spin up and spin down channels, respectively.
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In the LDOS of the heavier group IVa, elements also filled d (Ge, Sn) and f (Pb) states near the Fermi level are observed, in line with the behavior expected from the calculated formation energies (cf., Table 1), gap state positions (cf., Table 2), and mechanical properties (cf., Table 3).
Table 3. Calculated Bulk Moduli and Thermal Expansion Coefficients for Group IV-Doped CeO2a
| ||B0 (Mbar)||α (10−6 K−1)|
For the group IVb elements no atomic state below the O 2p valence band is present, only a gap state above the unoccupied Ce 4f band is observed (e.g., Fig. 2). This state moves toward the Ce 5d band, with increasing ionic size of the dopant (cf., Table 2). For the Ti and Zr dopants, the band integrates to six and four electrons, respectively. In case of the Hf dopant this band shows a small overlap with the Ce 5d band, and the number of electrons is estimated to be in the range of that found in Ti- and Zr-doped ceria. Just as for the IVa elements, this localized state is a combination with O 2p states of the O ions surrounding the dopant, as is shown for Zr in Fig. 2.
In addition, the DOS and LDOS of systems with a dopant concentration of 25% were investigated within the pure DFT framework. We found qualitatively similar results, showing the presence of d and f states just below the Fermi-level for group IVa dopants with filled d and f shells.
For the group IVa dopants (except C), the presence of s-type conduction states results in a serious reduction in the 2p–4f band gap, compared with the low concentration systems. Furthermore, the position of the maximum of these s bands appears correlated with the defect formation energies.
In case of the group IVb dopants, on the other hand, the 2p–4f band gap does not change with the dopant concentration. Similar as for the low concentration systems, the unoccupied Ce 4f band contains a significant contribution of the group IVb dopant d-state (cf. Fig. 2).
As a result, the clearly better stability of the group IVb-doped systems, may be attributed to the better resemblance of the dopant LDOS to the LDOS of Ce. In addition, the reduction in the 2p–4f band gap width in the group IVa-doped systems, and its correlation with the defect formation energy suggests that either the 2p–4f band gap is important for the stability of the system, or that the presence of s states near the Fermi-level is detrimental for its stability.
(3) Thermal Expansion Coefficients and Bulk Moduli
We investigated the influence of group IV dopants on the mechanical properties of CeO2, more specifically, the TEC and the BM. To reduce the computational cost, cells with a dopant concentration of 25% are used.
The linear (α) and volumetric (β) TECs of the doped systems can clearly be distinguished, as is seen in Fig. 3. Although group IVa dopants result in a significant increase in the TEC, group IVb dopants result in a status quo or a very small decrease in the CeO2 TEC. In Table 3, the calculated BM and linear TEC at 500 K of doped CeO2 are compared, as both the TEC and the BM give a measure for how easily a material deforms under external conditions. Doping of CeO2 only has a small influence on the system's BM, except for C. As experimentally most often lower concentrations are used, even a smaller influence is expected.
Figure 3. Volumetric (top) and linear (bottom) TECs for LDA calculations. The left panels zoom in on the 0–300 K interval, the right panels show the same data over a 0–1500 K interval. Dopant concentrations of 25% are used.
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Similar as for the defect formation energies, an oscillation is seen for the BM and TEC values of group IVa-doped ceria, showing the importance of filled d or f shells near the Fermi-level. For the group IVa elements, either the introduction of a filled d or f shell near the Fermi-level or the gap states in the O 2p–Ce 4f band gap reduces the resistance to compression (i.e., lower BM).
Comparison of the BMs and TECs in Table 3 shows an inverse relation between the two: an increase in the BM corresponds to a decrease in the TEC, and vice versa. However, the relative positions of Si and Sn appear reversed, as do those of Ge and Pb. Examining the TEC curves presented in Fig. 3 shows that around 250 K the order of the Si and Sn curves switches (the same for Ge and Pb), restoring the inverse behavior of the TEC and BM for the group IVa elements at lower temperatures. Because the BM is calculated at 0 K the trend of the inverse behavior is found to be a general one. These findings agree with our expectations: First, as the BM is a measure of the resistance of a material against uniform compression/expansion, it stands to reason that a large BM will lead to a small TEC. Second, it is well-known that both the BM and cohesive energy relate to the melting temperature, implying that the BM and cohesive energy correlate as well. On the other hand, the linear TEC α and the cohesive energy show an inverse correlation. As a result, an inverse correlation between the BM and linear TEC α is expected.
For practical applications, e.g., to reduce interfacial strain in a layered system through BM matching of the different layers, a linear interpolation of the presented BM could be used to obtain a first-order approximation of the optimal dopant concentration. For the elements of group IVb, Table 3 shows a BM that is only slightly larger than that of CeO2 and that increases with the dopant atomic number. The group IVa elements show a more complex behavior (cf., above), and with the exception of Si, all lower the BM of CeO2. As a result, group IV elements are not suitable as dopants if an increase in the CeO2 BM is required. However, group IVa elements could be beneficial to obtain a lowered BM. In addition, the nature of the group IVa elements [cf., Section II(1)] may limit the effects to the grain boundaries or interfaces, allowing one to create a BM gradient using group IVa-doped CeO2.
(4) Atomic Charges and Charge Transfer
The introduction of dopants in CeO2 not only influences the electronic structure at the level of bands and the DOS but also at the level of the local charge density distribution. To investigate this effect we calculate the atomic charges in the doped systems using the Hirshfeld-I method.[47-49] Table 4 shows the calculated charges on the dopant elements. The charge of the O and Ce atoms in pure CeO2 is shown as reference.
Table 4. Hirshfeld-I Charges in Ce1−xZxO2, with Z a Group IV Elementa
| || M ||O||Ce|
The atomic charges for all dopants show an increase of no more than 0.05e when increasing the dopant concentration from 3.125% to 25%. This is a first indication that the influence of the dopants on the electron density distribution is quite localized. Mainly the atomic charge of the nearest-neighbor O atoms changes. The atomic charges of the next nearest-neighboring Ce atoms change very little due to doping (cf. Fig. 4 and Table 4). The O atoms farther from the dopant in the c222 cell show atomic charges that differ only slightly from those in pure CeO2.
Figure 4. Ball-and-stick model of a c222 CeO2 supercell doped with 3.125% of a group IV element. The red/yellow/green balls give the positions of the O/Ce/dopant(Z) atoms. The nearest-neighbor O atom in one octant is labeled NN, whereas the next nearest neighboring Ce atoms in a single octant are labeled NNN. Atoms “far” away from the dopant are indicated by the blue disk.
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Although all dopants are tetravalent the atomic charges vary significantly. The difference in atomic charge of the dopants and Ce is almost entirely compensated by the change in atomic charge on the nearest-neighbor O atoms, showing that even the nearby O–Ce bonds are barely affected by these dopants. For all group IV elements, the amount of noncompensated charge is less than 0.05e per O atom in the nearest-neighbor shell.
All dopants behave similarly, with carbon being the sole exception. This may be an indication of the much weaker bonds of the C atom with the surrounding O atoms than the other dopants, which is in agreement with the fact that the small C atom prefers short bonds. The small size of C (Shannon crystal radius[50, 51] of 0.29 (0.3) Å for four (six) coordinated C) might make it very suited as an interstitial dopant or as substitutional dopant on an oxygen site, which are interesting scenarios for a study focussing on C doping of CeO2, but are beyond the scope of this work.
Note, however, that even in the C case the effects remain localized to the C dopant and the surrounding O atoms in the nearest-neighbor shell. The fact that the C atom loses roughly two electrons less (i.e., the two electrons remain localized on the C atom) than the other dopants and Ce to the surrounding O atoms is consistent with the recent findings of Hellman et al. They observed that for reduced C-doped ceria, the two electrons, which provided the formal charge of -II on the removed O atom, localize in C p orbitals.
From the results in Table 4, it appears that the group IV dopants cause no significant changes in the atomic charges beyond the nearest-neighbor shell.