First principles calculations on order and disorder in La2Ce2O7 and Nd2Ce2O7

Liv-Elisif Kalland a and Chris E. Mohn b
aDepartment of Chemistry, Centre for Materials Science and Nanotechnology, University of Oslo, FERMiO, Gaustadalléen 21, Norway. E-mail:
bCentre for Earth Evolution and Dynamics, Department of Geosciences, University of Oslo, Norway

Received 18th February 2020 , Accepted 7th May 2020

First published on 11th May 2020

In this paper, we highlight the connection between the local structure and collective dynamics of the defective fluorites La2Ce2O7 and Nd2Ce2O7. The local and average structure is explored by investigating a large number of different structural models and snapshots from Born–Oppenheimer Molecular dynamics calculations. Both compounds show a strong preference for local oxygen vacancy order similar to that found in the C-type structure. This suggests that previous studies, where Nd2Ce2O7 and La2Ce2O7 are viewed as disordered defective fluorites, or as a pyrochlore for the latter, did not capture the nature of local order in the disordered phase. We observe more collective chains of migrating oxygen in Nd2Ce2O7 – a manifestation of a stronger preference for a dynamic local oxygen vacancy order – than in La2Ce2O7. The stronger preference for 〈210〉 vacancy–vacancy alignments can explain why long range ordering is identified by distinct C-type like superlattice peaks in neutron diffraction patterns for Nd2Ce2O7 whereas they appear to be almost invisible in La2Ce2O7.

1. Introduction

The local structure of a disordered oxide is of key importance in order to understand many of its chemical and physical properties, such as ionic conductivity and hydration. Popular structural models, however, often assume that the structure of a crystalline disordered material can be represented by an “average” model where the disordered ions are distributed randomly over a sublattice. Although such models are useful as a starting point for many properties, they do not capture changes in the environment a diffusing ion will experience as it jumps from one position to another one. Such changes in the local structure are therefore essential to understanding diffusion processes in disordered oxides.1,2 Many A2B2O7 compounds, where A is a trivalent lanthanide and B is a tetravalent cation, are conveniently classified as fully ordered perfect pyrochlore structures or as oxygen deficient disordered flourites (see Fig. 1(c) and (a), respectively). Minervini et al.3 suggested that a tolerance factor, R = rA/rB (rA is the ionic radius for an 8-fold coordinated trivalent A cation and rB is the ionic radius for a 6-fold coordinated tetravalent B cation), can be used to predict whether an A2B2O7 compound should be classified as a perfect pyrochlore structure (R > 1.4) or if it will be disordered (and hence be classified as a disordered fluorite). La2Ce2O7, for example, has attracted considerable interest since it displays both high oxygen ion and proton conductivity,4,5 but its crystal structure is poorly understood. Since the tolerance factor for La2Ce2O7 is 1.33, which is slightly less than 1.4, one would expect that it displays a disordered (oxygen deficient) fluorite structure. Although Vanpoucke et al.6 suggested that La2Ce2O7 exhibits an ordered pyrochlore structure,6 most works support this prediction4,7–9 and the relatively high oxygen conductivity of “undoped” La2Ce2O7 also suggests that it has a highly disordered oxygen structure.
image file: d0cp00921k-f1.tif
Fig. 1 Polyhedral representations of: (a) the fluorite structure for CeO2, (Z = 2, space group Fm[3 with combining macron]m) where Ce4+ is 8 fold coordinated, (b) the primitive oxygen cube where the oxygen sits at the 8c site in the fluorite structure, (c) the pyrochlore structure (A2B2O7/Ln2Ce2O7, Z = 8, space group Fd[3 with combining macron]m) where Ce4+ is 6-fold and Ln3+ is 8-fold coordinated and (d) the C-type fluorite (Ln2O3, Z = 16, space group Ia[3 with combining macron]) where Ln is 6-fold coordinated. The cubic C-type structure and the pyrochlore structure are both oxygen deficient ordered superstructures of the perfect cubic fluorite and structurally very similar. The “cube” in (b) is used to define the alignments of vacancy–vacancy pairs as 〈100〉, 〈110〉 and 〈111〉. Note that since the “oxygen cubes” alternate between having a cation in the centre and not, the 〈111〉 vac–vac pairs may be aligned with a cation in the centre or not.

Oxygen disordered oxides often display high oxygen ion conductivity, and it is therefore interesting to see how their structure changes when incorporating a smaller lanthanide cation and how in turn these structural changes affect ionic conductivity. The tolerance factor decreases to about 1.22 when La3+ is replaced with Nd3+ and this suggests that Nd2Ce2O7 is a disordered fluorite as well (all Ln2Ce2O7 compounds are actually predicted to be disordered fluorites). However, this prediction contradicts several X-ray studies where the presence of weak superlattice peaks indicates structural deviation from the fluorite structure.7 Neutron diffraction studies confirmed this and observed distinct Bragg peaks for Nd2Ce2O7, which indicates long range crystalline order.10,11 The peak positions are consistent with the C-type structure (see Fig. 1d) which could possibly be explained by the presence of strong but partial oxygen vacancy interactions as suggested for heavily yttria doped ceria.12,13 Long range oxygen order is thus observed once La3+ is substituted by Nd3+ in La2−xNdxCe2O7, but interestingly, notable modulations of the background scattering between the fluorite peaks of La2Ce2O7 are found in the same region as the superlattice Bragg peaks were found for Nd2Ce2O7.10 This suggests that La2Ce2O7 may also have some local order that resembles the order found in the C-type structures.

In this work, we shall investigate the local structure and ionic conductivity of La2Ce2O7 and Nd2Ce2O7 using density functional theory (DFT). We will attempt to provide a local structural view on the nature of vacancy ordering in La2Ce2O7 and Nd2Ce2O7 and briefly discuss how these local motifs affect the conductivity of La2Ce2O7 and Nd2Ce2O7.

To investigate the structural properties of these two compounds, a large number of different configurations of cations and oxygen are analysed representing possible ordered phases or structural “snapshots” of the disordered phase. We investigate both the “static” structure found by searching for the lowest energy minimum of a given configuration of cations and oxygen ions, and the dynamic structure that are “snapshots” taken directly from the molecular dynamic (MD) trajectories. The static structure we investigate includes well known structural models (see the next section for a description) as well as quenched configurations from high temperature MD runs to search for new (low energy) structures.

2. Computational methods and details

The DFT calculations in this work were performed using the generalized gradient approximation (GGA) represented by the Perdew–Burke–Ernzerhof (PBE) functional14 together with a projector augmented wave (PAW)15 as implemented in the VASP code.16 In all calculations, we have used a 3 × 3 × 3 Monkhorst–Pack for the integration in the Brillouin zone and an energy cut-off of 500 eV for the structural optimisations. In the MD simulations, we used the gamma point only, an energy cut-off of 400 eV and a step length of 2 fs.

We present results obtained from structural optimisations performed using a 88 atoms supercell (i.e. a 2 × 2 × 2 cubic supercell) constructed from fluorite unit cells. This supercell has the same size as a single unit cell of the pyrochlore and the C-type structure. The MD runs are carried out in a 3 × 3 × 3 (297 atoms) supercell in the NVT ensemble. To obtain sufficient statistics to calculate the diffusion coefficient from the mean square displacement (MSD), the MD simulations ran for 45.8 ps for the La2Ce2O7 system and for 76.2 ps for Nd2Ce2O7.

The nature of oxygen and vacancy configurations are explored by identifying vacant tetrahedral cavities centred at the 8c site of the cubic fluorite structure and by calculating the distance and direction between pairs of vacancies. The notation 〈100〉, 〈110〉, 〈111〉, 〈200〉, 〈210〉, 〈211〉 and 〈220〉 is used to label the crystallographic different vac–vac alignments, as illustrated in Fig. 1(b).

The configurations (before optimisation) are grouped together by structural similarities in the oxygen and cation sublattices. A “random” oxygen sublattice is labelled “OrandX”, where X = 1–4 represents different configurations. Similarly, “CtypeX” denotes different ordered oxygen sublattices that are crystallographically related to the C-type structure, as shown in Fig. 2 and 3. The “Vac111” configuration is also related to the C-type structure, but the plane contains only one vacancy per ab plane and thus has only 〈111〉 pairs (see Fig. 4(c)). The oxygen configuration, termed “Withers”, is an ordered oxygen configuration suggested by Withers et al. for YxCe1−xO2−δ (x = 0.5)13 (see Fig. 4(e)). In addition, the configuration denoted as “Vac200” is constructed by repeating the oxygen configuration of a single fluorite unit cell with one vacant 8c site in all crystallographic directions. This configuration will consequently contain only 〈200〉 alignments of vacancies. The cation sublattice has been labelled using a similar notation, where “RandX” represents different random cation configurations with X = 1–4. “Fluorite” is an ordered configuration with equal cations in the crystallographic (100)-planes and the “Pyro” configuration has the cations ordered as in the pyrochlore structure.

image file: d0cp00921k-f2.tif
Fig. 2 Different crystallographic ab planes (or (001) planes) in the C-type structure (space group Ia[3 with combining macron]). The different vac–vac pair alignments found in the C-type, i.e. the 〈110〉, 〈111〉 and 〈210〉 motifs, are shown at the bottom left.

image file: d0cp00921k-f3.tif
Fig. 3 Different stacking sequences (along the c-axis) of the C-type related oxygen configurations “CtypeX” constructed by combining the ab/(100)-planes shown in Fig. 2.

image file: d0cp00921k-f4.tif
Fig. 4 Oxygen sublattice (shown after relaxation in our DFT calculations) of (a) the C-type structure where all 16c sites are vacant, (b) the C-type related structure termed “Ctype1”, where half of the 16c sites are filled, (c) the most stable (lowest energy) configuration for La2Ce2O7, which has a pyrochlore cation structure, (d) the “Vac111” with vac–vac pairs with a 〈111〉 alignment where half of the 16c sites are filled, and (e) the “Withers” – model.

We constrained the simulation cell to remain cubic during the geometry optimisation, which also makes structural analysis17 and comparison between the different configurations, easier. Calculations show that the relative order of the different optimised configurations does not change when we compare with the results obtained by full structural optimisation allowing both the cell volume and the cell shape to relax.

The Hubbard type +U correction is frequently used when investigating compounds containing 4f electrons since GGA and LDA may fail to describe the correlated nature of the f-electrons. DFT+U is essential if the goal is to study the electronic conductivity and charge transfer processes. However, in this work, our objective is to study the structural properties rather than the electronic properties, and it can be computationally challenging to use DFT+U if “+U” has to be (re)optimised for different compositions and configurations. Test calculations carried out by VanPoucke et al.18 using a +U correction term for cerium, did not change the relative stability of the investigated configurations for La2Ce2O7. We have, nevertheless, performed DFT+U calculations on a few configurations for Nd2Ce2O7 and La2Ce2O7, to ensure that the relative energies between the different configurations calculated using GGA are qualitatively in agreement with those from GGA+U calculations. In these tests, we used U = 5 eV for Ce and U = 6.5 eV for Nd, which are the same values as in several previous studies (for Ce18–20 and for Nd21–23). The energy difference between GGA and GGA+U is similar for all configurations. Some low energy configurations of La2Ce2O7 became even lower in energy and closer to the lowest energy configuration, but the relative order between the energies of the configurations was in general not changed (see Tables 1 and 2 in the additional information). We therefore do not use a +U correction term to DFT in our structural investigation for La2Ce2O7 and Nd2Ce2O7.

3. Results and discussion

Nature of vacancy order in La2Ce2O7 and Nd2Ce2O7

Comparison of the energy-minima of the different groups of oxygen configurations in Fig. 5 shows that the oxide ions prefer to order in both La2Ce2O7 and Nd2Ce2O7. Although we do not show all optimised configurations in Fig. 5, the oxygen-order is similar for both compounds arranged with increasing energy: E(Ctype1) < E(Vac111) < E(Withers) < E(Ctype2) < E(Ctype3, Ctype4) ≈ E(OrandX).
image file: d0cp00921k-f5.tif
Fig. 5 Total energies relative to the lowest energy configuration, for different arrangements of oxygens and cations for (a) Nd2Ce2O7 and (b) La2Ce2O7 calculated using a supercell of 88 ions. The structure configurations are grouped based on the initial oxygen configuration (before structural optimisation). The 6 first groups (“Ctype1”, “Vac111”, “Withers”, and “Ctype2-4”) are crystallographically ordered, and the next 4 groups (“Orand1-4”) represent “disordered” oxygen sublattices with P1 symmetry in the oxygen sublattice. The last group, “Vac 200”, is initially ordered but relaxes to a disordered oxygen configuration in combination with almost all cation sublattices and is therefore grouped together with the disordered configurations. Configurations where the initial oxygen configuration relaxes to a new configuration during the structural optimisation are shown as open symbols whereas those that retain their ordered pattern during the structural optimisations are shown as filled symbols. The lowest energy configuration for each compound is highlighted by a bright green colour. The numbers of vacancy pairs in the 88 ion simulation cell are listed.

All random oxygen configurations, such as “Orand1”, have a high energy of >kBT (even at T = 2000 K) and several of them relax to a different oxygen-configuration. The ordering in the oxygen sublattice is more pronounced than the ordering in the cation sublattice (as seen in Fig. 5), which is not surprising since exchanging a vacancy with an oxygen distorts the local structure to a greater extent compared to exchanging two (aliovalent) cations.

Many of the “unstable” high energy configurations shown in Fig. 5 have either cations with coordination numbers below 6 or contain several nearest neighbour 〈100〉 vacancy pairs. These are particularly unfavourable in agreement with what we reported earlier.10 Configurations with a high fraction of 〈210〉 alignments, on the other hand, are found to be very favourable as there are many 〈210〉 vacancy pairs in all low energy configurations. More difficult to predict is the energy of configurations with many 〈200〉 or 〈220〉 vacancy pairs, because even though they appear to be “strained” and therefore often tend to relax to 〈210〉 pairs, we find that the ordered oxygen configuration called 〈Withers〉, which contains both <200> and <220> motifs, is surprisingly low in energy! However, the 〈Withers〉 configuration also contains many 〈210〉 vacancy pairs, which lowers its total energy, and so in general the 〈200〉 and 〈220〉 vacancy pairs appear to be energetically unfavourable.

The two oxygen configurations with lowest energy (when omitting, for now, the lowest energy configuration of La2Ce2O7), “Ctype1” and “Vac111”, are similar in the sense that they both can be described as ordered oxygen excess C-type structures. We can visualize the crystallographic connection between these two oxygen structures and the C-type structure by filling up 8 of the 16 vacant oxygen positions in the C-type structure and aligning the 8 remaining vacant 16c site in an ordered manner (as shown in Fig. 4(b) and (d)). We can recognize the patterns of vacancies in the “Ctype1” and “Vac111” through a comparison with the C-type structure. “Ctype1”, for example, contains ab-planes with vacancies aligned as they are in the C-type structure (see Fig. 4(b) and Fig. 2 for comparison). Both “Ctype1” and “Vac111” contain a large number of 〈210〉 vacancy pairs, but “Ctype1” has more 〈210〉 vac–vac pairs than “Vac111” and has the lowest energy of the two. The next two oxygen configurations in Fig. 5, “Withers” and “Ctype2”, contain both 〈200〉 and 〈220〉 vacancy motifs, which might be one of the main reasons why their energies are slightly higher than the “Ctype1” and “Vac111” configurations. The four most favourable ordered configurations presented (“Ctype1, “Vac111”, “Withers” and “Ctype2”) have the vacancies more homogenously distributed compared to “Ctype3” and “Ctype4”. The latter two have all the vacancies clustered together to one side of the simulation box (see Fig. 2 and 3). This again results in fewer favourable 〈210〉 vac–vac alignments but in more 〈110〉 and 〈111〉 pairs giving a low coordination number of some cations that is not favourable (i.e. more than 8 〈110〉 or 〈111〉 pairs in our 88 ion super cells).

It may seem surprising that the “Vac111” configuration is so low in energy since the pyrochlore structure, which contains only 〈111〉 motifs, is a highly unfavourable configuration, as we will discuss more in detail later. It is thus important to note that the “Vac111” configuration presented in the graphs only has 〈111〉 vacancy pairs aligned in “oxygen cubes” without a cation in the cube centre, which is opposite to the pyrochlore structure shown in Fig. 1(c). If the oxygen sublattice of the “Vac111” configuration is shifted, and the 〈111〉 pairs are aligned with cations in the cube centre positions (i.e. if we transform the entire anion lattice by 1/4 × 〈100〉), the energy of the configuration formed by such a translation is increased substantially by about 1 to 1.5 eV per supercell of La2Ce2O7 and by about 1.5 to 2 eV for Nd2Ce2O7. This energy increase is largely independent of the type of cation in the cube centre, but reflects that the electrostatic interaction between the cation and anion sublattices, in some cases, may strongly influence the total energy. When the 〈111〉 pairs are aligned through an oxygen cube with a cation in the cube centre, the remaining oxygens in the cube will relax forming an octahedron around the cation. However, this oxygen configuration will still be very strained since the octahedron is deformed and stretched to fit into an otherwise cubic oxygen sublattice. The energy of these configurations is therefore much higher compared to the “Vac111” configurations with 〈111〉 pairs that are aligned without cations in the cube centre positions.

To sum up, from a comparison of low and high energy groups of configurations, we identified a number of constraints on local oxygen/vacancy order for both La2Ce2O7 and Nd2Ce2O7: (1) a high fraction of 〈210〉 vacancy pairs is beneficial and is best achieved when vacancies are ordered in C-type related “long range” patterns, (2) 〈100〉 pairs should be avoided, and (3) 〈111〉 vac–vac alignments are only favourable when aligned in an oxygen cube without a cation in the cube centre. (4) The vacancies should also be evenly “spread out” in a way that is consistent with cation coordination numbers between 6 and 8. (5) C-type related ordering of vacancies is found to be energetically favorable independent of the cation arrangement.

Cation ordering

We find that there is one (ordered) cation arrangement that is clearly favoured over others for both La2Ce2O7 and Nd2Ce2O7. This may seem surprising since most structural analyses do not capture any cation ordering. For Nd2Ce2O7, the lowest energy cation configuration is an “ordered fluorite” (shown as diamonds in Fig. 5), where the cations are evenly distributed in such a way that they all have the same local oxygen environment regardless of cation type. The lowest energy configurations for La2Ce2O7 have the pyrochlore cation sublattice (squares in Fig. 5) (see Fig. 1(c) for a description of the pyrochlore structure). The reason why the cation sublattice with the lowest energy for La2Ce2O7 and Nd2Ce2O7 is different, is due to the difference in size mismatch between Nd/Ce and La/Ce: La2Ce2O7 does not order in an ordered “fluorite” cation configuration because the large size mismatch between the cations would create a substantial strain along the {100}-planes in the direction where identical cations are aligned. The pyrochlore cation configuration is a better “packing alternative” for La2Ce2O7 as well as for any A2B2O7 compound with an even larger tolerance factor R. However, the tolerance factor R for La2Ce2O7 is smaller than 1.4 and, as predicted by Minervini et al.,3 we confirm that the pyrochlore structure is unfavourable since the oxygen sublattice is not pyrochlore ordered.

By comparing all configurations, we found an average coordination number close to 7 for all cations in the group of low energy configurations (see Fig. A in additional information where we plot the minimized energies versus coordination number of the cations). This supports that the ordering schemes found in the cation and oxygen sublattices are not directly linked through specific preferences in coordination numbers of the cations. In contrast, Liu et al.24 found that differences in cation oxygen bond lengths indicate a lower coordination number for Gd than for Ce in Gd2Ce2O7 (comparable to the Nd2Ce2O7 compound) and for Ce than for La in La2Ce2O7. From Fig. A in the additional information, there is a possible indication that Nd has a slightly lower average coordination number than Ce4+, as can be seen by comparing a group of low energy configurations. In this group, Nd has an average coordination number close to 6.9. However, in the configuration with lowest energy, both cations still have an average coordination number of 7. Thus, the possible preference of Nd having a lower coordination number than Ce is not strong and is not linked to a specific ordering between cations and anions. Also, for La2Ce2O7, several of the low energy configurations where cations are pyrochlore ordered including the lowest energy configuration have an average coordination number of 6.5 for Ce and 7.5 La. This indicates that Ce better accommodates a lower coordination number than La in La2Ce2O7 when the cations are pyrochlore ordered. We will now turn to discuss the nature of the oxygen sublattice when the cations are ordered in the pyrochlore manner.

Why Ln2Ce2O7 does not exhibit the pyrochlore structure

A number of previous structural analyses of La2Ce2O7 have started with the perfect pyrochlore structure and explored various anti-Frenkel defects.6,25,26 The most favourable Frenkel defect is created by moving a vacancy from the 8a to the 48f site:
image file: d0cp00921k-t1.tif(1)
We therefore map, in Fig. 6, the energy for the different oxygen configurations with a fixed cation pyrochlore sublattice, as a function of the number of vacant crystallographic 8a positions of the pyrochlore (see the description of the structure in Fig. 1(c)). From Fig. 6, we immediately see that there are several anion configurations that are more stable than the pyrochlore structure (marked as a black star) for La2Ce2O7, as well as for Nd2Ce2O7. It is evident that the anti-Frenkel defect formation described in eqn (1) is exothermal (marked as a black arrow in Fig. 6). However, the lowest energy configuration for each compund is not found by creating one or two such defects, but requires the creation of several defects. This underlines that the structure of La2Ce2O7 should not be viewed as a Frenkel defective pyrochlore because these defects are too extended to provide a meaningful description of its crystal structure. In fact, the perfect pyrochlore structure is more than 2 eV higher (per 88 ion supercell) in energy than the configuration with the lowest energy shown in the figure and is therefore not a representative structural model for La2Ce2O7 at any temperature. The pyrochlore structure is even more energetically unfavourable for Nd2Ce2O7 since its energy is more than 5.5 eV higher (per supercell) than the configurations with the lowest energy.

image file: d0cp00921k-f6.tif
Fig. 6 Energies per 88 ion super cell as a function of the number of vacant 8a positions for (a) La2Ce2O7 and (b) Nd2Ce2O7. The cation sublattice is fixed to that of a pyrochlore. The filled black star represents the energy of the perfect pyrochlore structure where the oxygen sublattice also has a pyrochlore structure. Starting with the pyrochlore structure (space group Fd[3 with combining macron]m, see Fig. 1(c)) with 8 vacant 8a positions, we can either move a vacancy to a 48f site (indicated by a black arrow for a single anti-Frenkel defect) or to a 8b site (indicated by a red arrow for a single anti Frenkel defect). The black filled squares represent configurations where one or several vacancies are moved from the 8a site to a 48f site only, whereas the open square represents configurations where at least one of the vacancies is moved to an 8b site. The red squares represent cations with less than 6 oxygens in the 1st coordination shell, and the open star represents the inverse pyrochlore structure where all 8b positions are vacant.

The lowest energy configuration for La2Ce2O7 (marked as a filled green square in both Fig. 5(b) and 6(b)) has an oxygen sublattice that strongly resembles that found in the C-type structure. The decrease in energy when moving a vacancy from an 8a site to an adjacent 48f site in the pyrochlore structure is easily understood since 〈111〉 vacancy pairs are effectively replaced by more energetically favourable 〈210〉 motifs (as illustrated in Fig. 7). We can thus link the favourable formation of several anti-Frenkel defects to the C-type related ordering of vacancies. The lowest energy configuration for La2Ce2O7 is obtained when moving half of the vacancies from the 8a position: 8 〈110〉, 8 〈111〉 and 24 〈210〉 vacancy pairs (in our 88 ion supercell) compared to 32 〈111〉 and 24 〈220〉 vacancy pairs (and 0 〈210〉 motifs) in the pyrochlore structure. The change in energy is even larger for Nd2Ce2O7 when we move half of the vacancies from the 8a site, and here we find that the well-ordered “Ctype1” configuration has the lowest energy with 8 〈110〉 vacancy pairs, no 〈111〉 motifs and the highest possible number of 〈210〉 alignments (which is 32).

image file: d0cp00921k-f7.tif
Fig. 7 Schematic illustration for the formation of an oxygen Frenkel defect where a vacancy is moved from an 8a site to a neighbouring 48f site. In this example, the number of 〈111〉 vac–vac alignments is strongly decreased and replaced by an increasing number of 〈110〉 and 〈211〉 vacancy pairs. The number of 〈220〉 pairs is decreased and the number of 〈210〉 alignments is increased. Alternatively, if the vacancy would have been moved to the 8b position (instead of a 48f site), a number of 〈111〉 vacancy pairs aligned in cubes containing Ce4+ in the centre position would have been replaced with 〈111〉 vacancy pairs located in empty cubes, and some 〈220〉 pairs would be substituted by unfavourable 〈200〉 vacancy pairs.

The oxygen sublattice in the lowest energy configuration of La2Ce2O7 is structurally more similar to the “Ctype1” configuration than it is to the oxygen structure in pyrochlore. Any similarity to the pyrochlore structure in the oxygen sublattice is dictated by the cation being ordered in the pyrochlore manner. In fact, the energy is ∼2.5 eV higher per 88 atom supercell if this particular oxygen configuration is combined with a random cation sublattice instead of a pyrochlore cation sublattice. The coupling between the two sublattices in the lowest energy configuration of La2Ce2O7 is also seen by the lower average coordination number for Ce4+ than for La3+ as previously discussed, which is not found for any other cation configuration. The (larger) size mismatch between La3+ and Ce4+ seems to favour the pyrochlore packing of the cations, and may provide a possible explanation for why the oxygen arrangement in the lowest energy configuration of La2Ce2O7 is different from the “Ctype1” configuration. It is probably seen by the surprising stability of 〈111〉 vacancy pairs when aligned with a Ce4+ ion in the cube centre. The resulting octahedra around Ce4+ are thus accommodated more easily in the oxygen sublattice when the cations are ordered in the pyrochlore manner.

The size mismatch between La3+ and Ce4+ in La2Ce2O7 is, however, obviously not large enough to favour the perfect pyrochlore structure and a C-type ordering of the vacancies emerges as a consequence of the strongly favourable 〈210〉 motifs. This explains why a pyrochlore structure is not a representative model for La2Ce2O7 (or Nd2Ce2O7) and we also stress that a perfect pyrochlore is not a suitable starting point for defining anion defects because the oxygen sublattice has an entirely different structure! For La2Ce2O7, it seems that the lowest energy oxygen configuration entails a good compromise between the C-type related ordering of oxygens/vacancies and a pyrochlore cation ordering. However, we believe that the ordering of cations is difficult to capture experimentally due to kinetics, as we will discuss below.

Kinetic trapping limits cation ordering

When one synthesizes these compounds, the anion lattice will be able to relax and equilibrate fairly quickly upon cooling or quenching with relaxation times on the same order of magnitude as the residence time (∼0.01 to 0.1 ns, see more in the next section when we calculate residence times of the oxygen ions from the MD trajectory). The diffusion of the cations is expected to be much slower.27 The energetic gain of relaxing to the cation configuration with the lowest energy, is smaller compared to the enthalpic gain of forming a C-type vacancy order, especially for Nd2Ce2O7. Also, since the C-type related ordering of vacancies appears to be low in energy for most cation sublattices, the presence of oxygen order does not provide a driving force for relaxing the cation sublattice to the most favourable cation configurations. A combination of the small enthalpy gain of locating the lowest energy cation configuration and slow cation diffusion, suggests that there is an extremely low probability of relaxing to a single ordered cation configuration at the timescales of the experiment. Such kinetic trapping has been discussed for various complex oxides with the fluorite-, pyrochlore- and perovskite structure.28–30 Although we cannot rule out that the cations may order locally or even be quite long range ordered as expected from the ND (or XRD) diffraction patterns,31 the findings in our previous experimental report on La2Ce2O7 and Nd2Ce2O7, do strongly suggest that the cations are disordered. We found no evidence for different coordination numbers for the two cations nor did we find any indications of order in the cation sublattice.10 Also, test calculations carried out on both ordered and disordered cation sublattices using MD (within a modest sized 88 ion supercell) indicate that cation order does not strongly affect the anion mobility. Therefore, we decided to further investigate the nature of the oxygen structure and diffusivity in randomly chosen cation sublattices representing a plausible “frozen in cation disorder” scenario.

Nature of diffusion in La2Ce2O7 and Nd2Ce2O7

In Fig. 8, we plot the MSDs from MD runs at 1500 K for 3 × 3 × 3 super cells (297 atoms) of La2Ce2O7 and Nd2Ce2O7 where anions diffuse within a randomly chosen cation sublattice. The figure clearly shows that oxygen diffusion is faster in La2Ce2O7 than in Nd2Ce2O7.
image file: d0cp00921k-f8.tif
Fig. 8 Calculated tracer image file: d0cp00921k-t4.tif where r is the position of atom i and N is the number of atoms i, and collective image file: d0cp00921k-t5.tif, where image file: d0cp00921k-t6.tif, in 3 × 3 × 3 super cells of La2Ce2O7 and Nd2Ce2O7 at 1500 K in a random configurations of cations.

From the MSDs we can calculate the diffusion coefficient for single ion diffusion and collective diffusion, Dtracer and Dcollective:

image file: d0cp00921k-t2.tif
where t is the time and MSD is defined in the caption of Fig. 8. Dtracer for La2Ce2O7 and Nd2Ce2O7 is found to be 2.7 × 10−10 m2 s−1 and 1.4 × 10−10 m2 s−1 at 1500 K. A higher value for La2Ce2O7 is in agreement with conductivity measurements showing a higher mobility of oxygen ions in La2Ce2O7 than in Nd2Ce2O7.7 The diffusion coefficient can often be correlated with the degree of ion disorder (configurational entropy). However, when comparing two structurally similar compounds, such as La2Ce2O7 and Nd2Ce2O7, the diffusion coefficient cannot be used directly to measure the extent of local order since the lattice parameter is markedly larger in La2Ce2O7 than in Nd2Ce2O7. A larger lattice parameter leads to a longer hopping distance for an oxygen/vacancy and thus results in a higher value of diffusivity in La2Ce2O7 than in Nd2Ce2O7. The hopping frequency, Γ, on the other hand, found from a simple hopping model image file: d0cp00921k-t3.tif, where n is the number of jumps and a is the lattice parameter, is a more meaningful parameter to measure the extent of local order in the two compounds. Γ is found to be 2.1 × 1010 s−1 for La2Ce2O7 and 1.1 × 1010 s−1 for Nd2Ce2O7 showing that oxygen jumps much more often in La2Ce2O7 suggesting that La2Ce2O7 has a higher mobility and is more disordered than Nd2Ce2O7.

Also, the Haven ratio, H = Dtracer/Dcollective, is correlated with local order (non-ideality) as it measures the extent of collective diffusion of oxygen ions. H quantifies the ratio between isolated single particle jumps and the centre of mass diffusion, which also include collective groups of migrating oxygens. To calculate the Haven ratio, we need Dcollective, which is found from the MSDcollective in Fig. 8 to be 4.5 × 10−10 m2 s−1 and 3.2 × 10−10 m2 s−1 for La2Ce2O7 and Nd2Ce2O7. The resulting H is about 0.60 and 0.44 for La2Ce2O7 and Nd2Ce2O7, respectively, and suggests that the oxygens move rather collectively in both compounds, but more so in Nd2Ce2O7. This is entirely consistent with more anion order in Nd2Ce2O7 than in La2Ce2O7.

Dynamic disorder

To analyse the nature of dynamic disorder in La2Ce2O7 and Nd2Ce2O7, we plot in Fig. 9 oxygen vacancy pairs sampled from the MD runs at 1500 K within a randomly chosen sublattice of cations. The dynamically disordered structure has very similar distributions of vac–vac motifs compared to the “static” structures shown in Fig. 5. That is, the ones that are low in energy in the static picture are more frequently found in the MD runs and those that are high in energy are more rarely seen in the MD runs. Both compounds have a high number of 〈210〉 motifs compared to that expected from a random distribution of vacancies in our simulation box, which is consistent with results from the static optimisations that show that low energy configurations always contain many 〈210〉 vac–vac pairs linked to C-type related ordering. The number of 〈210〉 vacancy alignments is higher, and there are fewer 〈220〉 alignments in Nd2Ce2O7 compared to La2Ce2O7, showing a weaker tendency in the latter. Few 〈100〉 and 〈200〉 vacancy pairs are sampled for both compositions, which is also relatable to the C-type structure, however there are more of them in La2Ce2O7 than in Nd2Ce2O7, and they are stable for a longer time in Nd2Ce2O7. This is consistent with La2Ce2O7 being more oxygen disordered and possessing higher oxygen ion mobility as we discussed earlier.
image file: d0cp00921k-f9.tif
Fig. 9 Sampled vac–vac alignments in a 3 × 3 × 3 supercell of (a) La2Ce2O7 and (b) Nd2Ce2O7 fluorites during the first 32 ps of the MD simulations at 1500 K with a randomly chosen initial configuration of the oxygens and cations. The horizontal dashed lines with matching color codes represent the number expected from a random distribution of oxygen.

Since an oxygen jumps in the crystallographic 〈100〉 direction in the fluorite structure, there will be many more possibilities for single particle jumps in La2Ce2O7 than in Nd2Ce2O7 since we see many more unfavourable 〈100〉 configurations in La2Ce2O7. There is also a higher average number of 〈110〉 and 〈111〉 vacancy pairs in La2Ce2O7. Finally, all other vacancy-pairs than 〈100〉 and 〈200〉 seem to be longer lived in Nd2Ce2O7, as seen by the more and longer “plateaus” in the plot, which is in accordance with more collective diffusion and more long range order.

Linking the extent of local order and kinetic trapping

The nature of dynamical disorder is closely connected to the extent of local (static) order and non-ideality, which can be measured from the number of thermally populated low-energy configurations. Although La2Ce2O7 appears to be more disordered (less non-ideal) than Nd2Ce2O7 as discussed in the previous chapter from the MD runs, this is not evident when comparing directly the energy spectra of La2Ce2O7 and Nd2Ce2O7 since the spectra are quite similar. The main difference between the two energy spectra, as shown in Fig. 5 and 6, is that the energy gap between the lowest and next lowest energy configuration is much larger in La2Ce2O7, which could indicate that Nd2Ce2O7 would be more disordered than La2Ce2O7. However, Fig. 5 and 6 do not show the multiplicity of each configuration and since the lowest energy configuration for La2Ce2O7 has a lower symmetry than the lowest energy configuration for Nd2Ce2O7, it will have several symmetrically equivalent oxygen configurations and La2Ce2O7 will have more thermally accessible configurations than Nd2Ce2O7. This implies in turn that La2Ce2O7 has higher configurational entropy than Nd2Ce2O and is thus more disordered. In addition to this, GGA+U calculations reported in Table 2 in the additional information show that the configurations with the next lowest energies for La2Ce2O7 configurations lie much closer in energy to the lowest energy one than what was found using GGA (without +U). These configurations will be more thermally accessible for La2Ce2O7, which provides additional support for the ND results, showing that La2Ce2O7 is the most disordered of the two.10

However, the argument above assumes that the cations are able to fully relax to reach the equilibrium cation configuration. If we assume that the cations are entirely disordered due to kinetic trapping, we should remove all ordered cation configurations including the lowest energy configuration for La2Ce2O7, which is the green square in Fig. 5(b). Then, the two compounds should actually have very similar diffraction spectra, but this contrasts the experimental observations mentioned above.10 In this case, static disorder therefore does not explain why La2Ce2O7 is more disordered than Nd2Ce2O7 and the higher degree of disorder in La2Ce2O7 observed experimentally can only be explained by dynamic oxygen disorder. In Nd2Ce2O7, we observe more configurations with a large number of 〈210〉 vacancy pairs during the MD runs, and these motifs are natural “building blocks” to form partial long range order connectivity patterns consistent with the C-type structure. In La2Ce2O7, we suggest that C-type related oxygen ordering is more short ranged in nature, with 〈110〉 and (empty) 〈111〉 vacancy pairs occurring more often during the MD runs. This could explain why diffraction peaks characteristic for C-type order are seen in Nd2Ce2O7 whereas such order is only visible as modulations of the (diffuse) background scattering for La2Ce2O7.10

4. Concluding remarks

Here, we explored the local structure of the fluorite structured La2Ce2O7 and Nd2Ce2O7 through a comparison of a large number of cation configurations in the static limit and from Born–Oppenheimer Molecular dynamics calculations using DFT. We found that anion ordering is more pronounced than cation ordering. Both compounds have a strong preference towards a C-type related order of oxygen vacancies, and this order is largely independent of the ordering or disordering of the cations. The C-type like order is identified by a high fraction of 〈210〉 vacancy pairs, and 〈100〉 pairs are unfavorable as opposed to the 〈110〉 and 〈111〉 vacancy pairs. However, 〈111〉 vac–vac configurations are only favorable when aligned in an oxygen cube without a cation in the cube centre. The vacancies should also be distributed in a way that is consistent with cation coordination numbers between 6 and 8.

Lattice static calculations show that there is an energetic advantage of particular ordering in the cation sublattice which is explained by maximising the close packing of the cations. Whereas the lowest energy configuration of Nd2Ce2O7 has the ordered configuration named “fluorite”, the larger size difference between La3+ and Ce4+ in La2Ce2O7 is better suited to the pyrochlore cation structure (see Computational methods and details for description). However, we argue that the cations might be “frozen in” and hence disordered under experimental conditions. We also stress that although the cation sublattice in the lowest energy configuration of La2Ce2O7 possesses the pyrochlore structure, the perfect pyrochlore is not the most stable configuration for La2Ce2O7 (nor for Nd2Ce2O7).

Both long range and short range vacancy interactions will influence the properties such as conductivity, and the nature of oxygen diffusion is here studied within a random cation sublattice based on our assumption of a “frozen in” disordered cation sublattice. La2Ce2O7 is here found to have higher oxygen diffusion than Nd2Ce2O7. Collective chains are more dominant in Nd2Ce2O7 than in La2Ce2O7 with Haven ratios – which measure the single particle to collective diffusion – of about 0.44 and 0.60, respectively. A lower Haven ratio is consistent with stronger order in the former.

Our present results show that previous computational models where La2Ce2O7 has been viewed as a pyrochlore with one or two Frenkel defects, are not representative structural models of this compound. On the other hand, when modelling La2Ce2O7 or Nd2Ce2O7 as a disordered fluorite with a random distribution of vacancies, one ignores the fact that these compounds have a preference towards (local or long range) C-type related order of the oxygen sublattice. C-type oxygen order is found to be more dominant in Nd2Ce2O7 than in La2Ce2O7, and the observed higher amount of 〈210〉 vacancy pairs in the former suggests that the stacking of “Ctype1” (and other low energy) configurations forms more long range order in Nd2Ce2O7 than in La2Ce2O7.

Conflicts of interest

There are no conflicts of interest to declare.


The authors gratefully knowledge the Norwegian Metacentre for Computational Science (Notur) for providing computational resources under the project number nn4604k and nn2916k. This work was partly supported by the Research Council of Norway through its Centres of Excellence funding scheme project 223272.


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Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp00921k
See Figure A in additional information for energy versus coordination number of the cations.

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