Abstract
Links between dynamical Frenkel defects and collective diffusion of fluorides in β-PbF2 are explored using Born–Oppenheimer molecular dynamics. The calculated self-diffusion coefficient and ionic conductivity are 3.2 × 10−5 cm2 s−1 and 2.4 Ω−1 cm−1 at 1000 K in excellent agreement with pulsed field gradient and conductivity measurements. The calculated ratio of the tracer-diffusion coefficient and the conductivity-diffusion coefficient (the Haven ratio) is slightly less than unity (about 0.85), which in previous work has been interpreted as providing evidence against collective ‘multi-ion’ diffusion. By contrast, our molecular dynamics simulations show that fluoride diffusion is highly collective. Analysis of different mechanisms shows a preference for direct collinear ‘kick-out’ chains where a fluoride enters an occupied tetrahedral hole/cavity and pushes the resident fluoride out of its cavity. Jumps into an occupied cavity leave behind a vacancy, thereby forming dynamic Frenkel defects which trigger a chain of migrating fluorides assisted by local relaxations of the lead ions to accommodate these chains. The calculated lifetime of the Frenkel defects and the collective chains is approximately 1 ps in good agreement with that found from neutron diffraction.
This article is part of the Theo Murphy meeting issue ‘Understanding fast-ion conduction in solid electrolytes’.
1. Introduction
Fast-ion conductivity in the solid state was first observed by Faraday himself in the early eighteenth century when the resistivity of a solid piece of lead fluoride, PbF2, suddenly dropped when heated to high temperature [1,2]. This opened up the field of solid-state ionics and the β phase of PbF2 is now widely considered a parent of a broad class of ‘type II’ superionics that on heating undergo a rapid and continuous increase in the ionic conductivity [1,2]. These compounds include many fluorite structured halides (XF2 with X = Ca, Sr, Ba) as well as functional oxides such as CeO2, UO2, ThO2 and ZrO2 [2]. In contrast to ‘type I’ superionics (e.g. δ-Bi2O3), where the transition to the superionic phase is first order due to disordering of the inherent vacancies, β-PbF2 contains no such vacancies, and the superionic transition is associated with transient and extended Frenkel defects [2–4]. The ionic conductivity in β-PbF2 increases continuously by eight orders of magnitude when heated from room temperature to the superionic transition temperature, K [5], levelling off to ∼3 cm−1 just above [5,6], where it remains nearly constant until the melting point at 1158 K. The β-phase of PbF2 thus possesses one of the highest known values of ionic conductivity at intermediate temperature, more than three orders of magnitude higher than that of stabilized zirconia frequently used as the electrolyte in solid oxide fuel cells (SOFC). The exceptionally high conductivity makes β-PbF2 particularly attractive as a reference for designing the next generation of fuel cells and gas separation membranes to lower the current high operating temperature of these devices. There is, however, at present very little atomistic understanding of the nature of the Frenkel defects in β-PbF2 and how they trigger diffusion processes. Such knowledge should give crucial input into the design and development of new fast ionic conductors in future devices and links between structure and conductivity therefore warrant further attention. Here, we use ab initio Born–Oppenheimer molecular dynamics simulations (MD) to investigate the nature of ionic transport within β-PbF2 which allows us to learn more about collective (multi-ion) dynamics. In particular, we aim to probe the links between the dynamical Frenkel defects, the nature of fluoride migration and the role played by the immobile lattice in promoting fast ion behaviour.
The crystal structure of β-PbF2 is shown in figure 1. It has a cubic fluorite structure (space group Fmm) in which the lead ions form a face centred ‘immobile’ cubic sublattice. The fluorides are located in the tetrahedral holes/cavities, but crystallographic interpretations from different neutron diffraction (ND) experiments report large variations in the average fluoride positions within these cavities (e.g. [2]) reflecting that the extensive thermal ‘activity’ of the mobile sublattice is not so easy to capture using simple average crystallographic models. From these ND experiments, different models of the dynamic and highly transient Frenkel defects have been suggested. The best fit to ND data was found using a defect model with as many as nine vacant cavities with an equal amount of fluorides significantly displaced from their regular tetrahedral site [3]. These extended Frenkel clusters are highly transient with lifetime s which is of the magnitude of a jump-time of the fluorides [5]. This, in turn, indicates that formation of Frenkel defects is intimately linked to, or are manifestations of, collective migrating fluorides.
Figure 1. Crystal structure of β-PbF2 (). The green and purple spheres are the fluorides and the lead ions located at the tetrahedral 8c and the 4a sites, respectively, whereas the small black spheres are the (empty) octahedral 4b site. A tetrahedral hole (which we also refer to as a tetrahedral ‘cavity’ in this work) is the centre (8c site) of the tetrahedron formed by the four nearest neighbour lead ions at the 4a site. (Online version in colour.)
Atomistic insight into the nature of diffusion can be obtained from experiment by the comparison of the single particle self-diffusion or tracer-diffusion coefficient (i.e. or ) and the diffusion coefficient calculated (indirectly) from conductivity experiment, , which carries information about the net displacement of all mobile species.1 The self-diffusion coefficient can be determined from different techniques such as nuclear magnetic resonance (NMR) which requires some a priori knowledge of the mechanism of diffusion and also more directly from pulsed field gradient (PFG) experiments. The ratio of tracer-diffusion coefficient to the conductivity-diffusion coefficient is the Haven ratio, . Using from Ref. [7] and from conductivity measurement [6], a value of between 0.7 and 0.9 (at around 1000 K) has been taken as evidence for little collective diffusion or for the presence of a simple non-collinear interstitialcy mechanism [7] shown in figure 2a. Here an interstitial fluoride, F1, at the octahedral 4b site of the fluorite structure kicks one of its nearest neighbours, F2, at the regular tetrahedral 8c site, into a new octahedral site and replaces the atom that has been kicked out. Such a mechanism is consistent with H being slightly less than 1, but appears to be inconsistent with results from ND which show that the fluorides do not occupy the interstitial 4b site [3]. Other possible mechanisms, not considered previously, are ‘kick-out’ cooperative events in which a fluoride, F1, at the regular tetrahedral 8c site of the cubic fluorite structure enters a neighbouring occupied tetrahedral cavity and pushes the fluoride in that cavity, F2, into a new cavity. This new cavity, C, is normally occupied but can also be ‘instantaneously’ empty if a fluoride has just left it. If C is empty, as shown in figure 2c, the collective chains of fluorides is terminated by the jump into cavity C. By contrast, if C is occupied, as shown in figure 2f, longer ‘caterpillar’ chains may form. Short collinear strings such as that visualized in figure 2c have been shown to play an important role in understanding fast-ion conduction in e.g. δ-Bi2O3 [8] and could also contribute to collective diffusion in β-PbF2. Indeed with longer multi-ion chains a combination of many different mechanisms can contribute to the conductivity in superionic PbF2. In addition to the ‘caterpillar’ mechanism mentioned above (figure 2f), ‘rings’ (figure 2g), for example, can also add to the conductivity. Interestingly, an eight-ion ring similar to that shown in figure 2g has been suggested to contribute strongly to diffusion in β-PbF2 [9], but such closed ‘loops’ cannot alone explain its high value of conductivity since the mechanism involves no net charge transport! In principle, combinations of different mechanisms can be consistent with the value of the Haven ratio which is just an average value of different possible collective paths. As long as the net mean square displacement of the cooperative diffusing chain of fluorides is longer than that of uncorrelated diffusion, the Haven ratio is less than unity.
Figure 2. Different possible mechanisms of collective diffusion in β-PbF2 suggested in the literature and investigated in this work. (a) Is an example of a non-collinear interstitialcy mechanism suggested in [7]. Here a fluoride, F1, is assumed to sit at the interstitial octahedral site and then it migrates to the neighbouring tetrahedral site and kicks F2 into a new octahedral site. (b) Shows a pair of fluorides exchanging positions. We also investigate different correlated events shown in (c–g). In (c,e,f) a fluoride, F1, jumps to another occupied cavity and kicks/pushes the resident fluoride, F2, out of its cavity. These ‘kick-out’ or ‘caterpillar’ mechanisms can be collinear, as shown in (c), or non-collinear as visualized in (e). (d) Shows a fluoride, F1, that jumps to an instantaneously empty cavity B, which triggers a jump of a neighbouring atom F2 from C to D. The mechanisms in (f) and (g) are examples of longer collinear ‘caterpillar-like’ chains and ‘ring-like’ pattern representing extreme examples of mechanisms with very low and very high Haven ratios, respectively. (Online version in colour.)
2. Theory
To investigate the collective dynamics of β-PbF2 from the atomic Born–Oppenheimer MD trajectories, we first calculate the ionic conductivity by integrating the charge-current correlation function
Following Castiglione & Madden [10], we also calculate the conductivity form the mean-square displacement of the centre of mass
To calculate we also need which can be obtained from equation (2.3) by ignoring correlations in the atomic positions of the fluorides. We can then write down the well-known Nernst–Einstein equation
(a) Single particle and collective dynamics from a jump diffusion model
We can also calculate and hence from a jump diffusion model as follows. Since vibrations and jump frequencies occur over different time scales, we can decompose the ionic trajectory into vibrations and jump events. We divide the simulation cell into space-filling primitive cubes [11–14] with the cube-centre located at the tetrahedral 8c site of the fluorite structure, and the octahedral 4b site and the lead atoms at alternate cube-corner positions. This division allows us to calculate the jump frequency by counting the number of times an ion leaves a primitive cube, and hence a tetrahedral cavity, and enter a new one. It should be noted that the fluorides vibrate with large amplitudes that sometimes extend into a neighbouring fluoride tetrahedral cavity. To avoid counting such vibrations as two consecutive jumps (jump relaxations) we require that the fluoride must stay within the new cavity for at least before we label the event as a jump. The value of should be at least the period of a low frequency vibrational mode, but not much larger either because then two consecutive rapid jumps could be identified as a single long jump. We find a reasonable choice of ps which we confirm below by the comparison of the diffusion coefficient calculated from a jump diffusion model (using this value of ) with that calculated from the MSD.
We also record correlations between two successive jumps of a single fluoride and distinguish between different jump direction. Back-jumps (jump-relaxations) is defined as movements in which an atom jumps back to where it came from, sideway jumps involve two jumps in different crystallographic directions (i.e. a jump in the direction is followed by a jump in either the or the directions), whereas forward jumps are two consecutive jumps in the same crystallographic direction, i.e. a jump in the direction is followed by a jump in the direction. We then calculate the self-diffusion coefficient using
We also investigate collective diffusion from a hopping model where correlation between diffusing neighbouring pairs are identified. We label two nearest neighbouring cavities A and B and assume that these cavities are occupied by fluorides F1 and F2. We identify events where F1 jumps to B and where F2 jumps to a third cavity C. If the cavities A, B and C involved in the diffusion of the two fluorides are all aligned in the same direction as shown in figure 2c, the mechanism is collinear. If the three cavities are not aligned along the same crystallographic direction (as shown in figure 2e), the mechanism is non-collinear whereas if A is identical to C, the fluorides exchange positions which gives no net charge transport (figure 2b). Longer chains of collective diffusing fluorides can form a range of different patterns, such as the caterpillar mechanism (figure 2f), arcs and also loops (figure 2g).
(b) Single particle and collective dynamics from dynamic heterogeneities
Analysing the nature of dynamical heterogeneity of the highly mobile species is another possible approach to understand the nature of particle dynamics and, in contrast to the hopping model, there is no reference to the underlying crystallographic lattice. Instead, a ‘rearrangement indicator’ [15,16] of a fluoride , is used to measure the variance in the fluoride positions within a time window as
3. Computational details
Ab initio Born–Oppenheimer MD simulations were carried out using density functional theory (DFT) in the ensemble with a cubic simulation box containing 768 atoms with Å)3 and K. The time step was 2.5 fs and a Nosé-Hoover thermostat was used. During a 120 ps simulation, we counted more than 10 000 fluoride jumps, which ensured sufficient statistics for a qualitative comparison with the ionic (d.c.) conductivity obtained experimentally. Such a long trajectory is required to converge the centre of mass diffusion to sufficient precision since only one point (i.e. the centre of mass of the ions) is collected at each time step. The d.c conductivity was thus calculated from the collective MSD after the ballistic regime from ps to ps. At each step along the trajectory, the total energy was minimized using the Perdew–Burke–Ernzerhof functional [17], an energy cut-off of 300 eV and sampling the Brillouin zone at the gamma point only, as implemented in VASP [17–19].
4. Results and discussion
After the ballistic regime, we calculate the single particle and collective diffusion coefficients from the slopes of the MSD (shown in figure 3). The calculated cm2 s−1 at 1000 K is in good agreement with results from a PFG experiment ( cm2 s−1) [7]. From the collective MSD, we find that is cm2 s−1 which is also in good agreement with the experimental ionic (d.c.) conductivity using equation (2.2). Our calculated conductivity from equation (2.3), cm−1, is in good agreement with that measured experimentally ( cm−1 at around 950 K) [5] and with that from an MD study at 1000 K using pair potentials [20].
Figure 3. The collective (thick green line) and single-particle Nernst–Einstein (thin red line) mean-square displacements of the fluorides at 1000 K from an ab initio Born–Oppenheimer MD simulation. (Online version in colour.)
The Haven ratio is when equation (2.3) is used to calculate the collective diffusion coefficient and when equation (2.1) is used to calculate . The good agreement between these two methods indicates that our MD run is sufficiently long to ensure convergence of the integral in equation (2.1). Our Haven ratio is markedly higher than the Haven ratio estimated from MD calculations using pair potentials [10], but in very good agreement with that reported by Carr et al. [6], in which the self-diffusion coefficient from the PFG experiment [7] and extracted from conductivity measurement gave a Haven ratio between 0.7 and 0.9 at around 1000 K [6]. Bearing in mind the modest precision in the PFG data and also the difficulty of measuring the conductivity above 950 K due to sublimation losses of the sample [6], the overall agreement is satisfactory.
Our calculated Haven ratio is lower than 1 but larger than those of other (type 1) superionic conductors with a high concentration of inherent vacancies such as δ-Bi2O3 ( [8]), indicating that possibly collinear events are less important in type II superionic conductors, such as β-PbF2, than in type I. To provide some mechanistic insight into the diffusion of the fluorides, we now carry out an atomistic investigation of the mechanism of diffusion in β-PbF2.
(a) Single particle dynamics
Analysis of the ab initio MD trajectories at 1000 K using a jump diffusion model shows that about 80% of the fluorides migrate to the nearest neighbouring cavity aligned in the direction, about 17% migrate to the next nearest neighbours aligned in the direction, whereas only about 4% jump in the direction. This is remarkably consistent with MD results for CaF2 where 80, 16 and 3% of the jumps were in the and directions, respectively [21]. The fluorides vibrate mostly within the tetrahedral cavities centred at the 8c site, as seen in figure 4, where we plot the ionic density and two snapshots of the MD trajectory. The average residence time, , is about 5.1 ps and the jump time, , between the tetrahedral cavities are typically between 0.5 and 1 ps. The fluorides follow irregular and highly curved pathways as highlighted in figure 4 and the anions sometime reside, for a very short time, near an octahedral site or near the edge between two nearest neighbour tetrahedral cavities. The octahedral site, however, is not a resident site for the fluorides but a site avoided. If we attempt to quench dumped MD configurations along the trajectory by a steepest descent (structural optimization), all fluorides in the simulation box end up at the regular tetrahedral site and none at the octahedral site. This confirms that the octahedral site is not a resident site for the fluorides. The lead ions also relax appreciable distances ∼0.5 Å to accommodate the jumping fluoride. This highlights the important role played by the ‘soft’ cation sublattice in promoting fast-ion conductivity.
Figure 4. The top picture shows ionic density contours of the fluorides from an MD simulation at 1000 K. The bottom (left) picture shows trajectories of the fluorides (green lines) in a 5 ps time window and the bottom (right) picture shows two typical examples of two jump events. The jump of F1 follows a highly curved trajectory in the crystallographic direction whereas F2 jumps several times including a jump in the direction. (Online version in colour.)
We find evidence for correlations between successive jumps only when these are between nearest-neighbour tetrahedral cavities, i.e. there is a slightly higher probability of back jumps (prob. ≈0.2) compared to that of two uncorrelated consecutive jumps (prob. ≈0.167). Inserting and the calculated jump frequencies into equation (2.5), we find that cm2 s−1 (with ps) which is in very good agreement with calculated above from the single particle MSDs.
A preference for backwards jumps is consistent with previous experimental and computational results for other fast-ion conductors [22–24]. However, jump-relaxations seem to be less pronounced in compounds without inherent vacancies than in ‘type I’ fast-ion conductors with a high concentration of vacancies such as δ-Bi2O3 [23], α-CuI [13], Ba2In2O2.50 [25], La2Ce2O7 [26] etc.
(b) Collective diffusion from a jump diffusion model and dynamical heterogeneities
An interstitialcy mechanism (figure 2a) suggested in [7] is consistent with a Haven ratio slightly less than unity. However, since the interstitial 4b octahedral site is not a resident site for the fluoride, as is evident from the ionic density plot and the trajectories shown in figure 4, this mechanism cannot be used to explain the nature of collective dynamics within β-PbF2. Instead, our MD trajectories show that the fluorides jump rapidly and directly between tetrahedral cavities. We start to analyse the nature of collective migrating fluorides from a hopping model by identifying all events in which an fluoride, F1 in cavity A, jumps to cavity B (which is occupied by F2) and kicks/pushes F2 out of its resident cavity. The fate of F2 can be captured by classifying these ‘kick-out’ events as collinear (figure 2c), non-collinear (figure 2e) or exchange (figure 2b). In this analysis, we only count pair events where F1 and F2 resides in the same cavity, B, for at least 0.1 ps.
Compared to a random diffusion process, we find a marked preference for collinear ‘kick-out’ pairs (figure 2c) over non-collinear (figure 2e) and very few exchanges (figure 2b). Indeed about 40% of the ‘kick-out’ events are collinear (compared to 1/6 for uncorrelated diffusion), slightly less than 60% are non-collinear ones (compared to 2/3 for uncorrelated diffusion) and less than 2% involve exchange of positions (compared to 1/6 for uncorrelated diffusion). This suggests that, together, a preference for collinear ‘kick-out’ events and very few exchanges explains, at least in part, why the Haven ratio is less than unity. This analysis does not include events where F1 and F2 do not share a common cavity such as that shown in figure 2e, i.e where a jump to cavity B can possibly ‘trigger’ a neighbouring fluoride, F1 in C to jump to D. Analysis of the trajectories shows that these events are much less correlated than the ‘kick-out’ pair mechanism. The main contribution to collective dynamics is therefore probably collinear ‘kick-out’ chains (figure 2) but diffusion processes in PbF2 clearly involve a range of different mechanisms. Many of these are hard to capture in an jump diffusion analysis since we cannot always distinguish jump relaxations and vibrations with large amplitudes.
Difficulties in analysing the nature of collective chains with more than two atoms from a hopping model due to large amplitudes of vibrations and local distortions of the sub-lattices has motivated us to investigate the collective diffusion of fluorides from a cluster size analysis using equation (2.6). The cluster sizes are plotted in figure 5 with different cut-off values, , chosen to capture different crystallographic alignments of fluorides involved in collective transport. If we count clusters along the MD trajectory using a cut-off value of , we will only include events in which two nearest neighbouring fluorides are jumping at about the same time, i.e. the time difference between the two jumps is shorter than . Thus this includes ‘kick out’ events such as those shown in figure 2c,e, whereas events where the two fluorides are aligned further apart, i.e. as next-nearest neighbours before they jump, as shown in figure 2d, are not counted. If is set to 1.5ann in the cluster analysis, we also include many of the events displayed in figure 2d and if we increase further to , collective chains in which two fluorides are third-nearest neighbours are also included in the analysis. With an even higher value of we include so many uncorrelated events that these start to outnumber the correlated events included in the analysis. Thus the red and grey lines in figure 5 are therefore very similar when . This provides an upper constraint for the correlation length of the collective chains of migrating fluorides. When we find, compared to uncorrelated events, about 30% fewer isolated jumps (cluster size = 1) and, twice the number of three-atom chains. By contrast, clusters with less than four or five atoms involving nearest neighbours () are rare, which rules out the long caterpillar-like mechanisms shown in figure 2c. When we find that the average cluster size is 1.78 (compared to 1.38 for a distribution of uncorrelated jumps) and when the calculated average cluster size is about 2.45 (compared to 2.03 for a distribution of uncorrelated events). In figure 6, representative snapshots of the geometries of the collective chains of migrating fluorides are shown. These illustrate that a range of different mechanisms are at play.
Figure 5. The red full lines are the fraction of different cluster sizes containing fluorides that move distances of a typical jump i.e. with , where is calculated from equation (2.6). The grey dashed lines are the corresponding values found if the rearranging fluorides do not correlate. (Online version in colour.)
Figure 6. Different snapshots of fluoride positions during an MD run at 1000 K with 768 atoms in a cubic simulation box. The green spheres are fluorides with a rearrangement indicator, , representing jumping fluorides (the arrow marks the jump-direction), whereas the grey are those with (i.e. the non-migrating fluorides). (Online version in colour.)
(c) Linking the nature of Frenkel defects and collective dynamics
Strings of collective migrating fluorides are intimately connected to the formation of short-lived Frenkel defects seen in neutron diffraction [5]. A jump into an occupied cavity such as that in figure 2c leaves behind a vacant cavity and the migrating fluoride, F1, shares for a short time a cavity with F2, thereby creating a Frenkel defect. These Frenkel defects are highly dynamic. If we attempt to capture these in a local minimum configuration on the potential energy surface by quenching the MD configurations by a steepest descent energy minimization, the fluorides always end up in their regular tetrahedral sites. Formation of dynamic Frenkel defects distorts both the mobile and immobile lattices, suggesting that these defects are highly extended in agreement with defect models suggested from neutron diffraction [3]. The average lifetime of a vacant cavity is about 0.5 ps, but about 5% of these are vacant for more than 1 ps. The lifetime of a doubly occupied cavity (where two fluorides are resident within the same tetrahedral cavity) is less than 0.4 ps. If we assume that the average lifetime of the vacancy is about the same as the lifetime of the collective fluoride chains, we roughly estimate the lifetime of the Frenkel defects to be around 1–2 ps. This is consistent with the lifetime of the highly dynamic Frenkel defects seen in neutron diffraction ≈1 ps [5]. These Frenkel defects are manifestations of collective diffusion in β-PbF2.
5. Conclusion
Links between dynamical Frenkel defects and collective diffusion of fluorides in β-PbF2 are investigated from a hopping model and from the nature of dynamical heterogeneities using ab initio molecular dynamics simulations. The ionic density plots and the MD trajectories show that the octahedral holes/cavities of the fluorite structure are empty and that the fluorides stay within the tetrahedral cavities for 5 ps, on average, before they jump rapidly within 1 ps to an occupied tetrahedral cavity. This can initiate a chain of migrating fluorides and a range of different collective mechanisms contributes to the conductivity. Analysis of correlations between pairs of simultaneously jumping fluorides shows that a collinear ‘kick-out’ mechanism (figure 2c) occurs twice as frequently as a random collinear ‘kick-out’ event and that direct exchange (figure 2b) is rare. Other pair mechanisms investigated include that in figure 2d where a jump into an occupied cavity can also trigger a nearest neighbour fluoride to jump. Such events, however, are much less correlated than the ‘kick-out’ events which also rules out correlations between two even more distant jumps that take place at about the same time. A combination of collinear kick-out chains and few exchanges can thus explain why the Haven ratio is slightly lower than 1 [6]. Large local distortions of the ‘immobile’ cation sub-lattice and rapid fluoride back-jumps make it difficult to capture all collective fluoride chains from a jump diffusion model, in particular if these chains contain several fluorides.
This has motivated us to identify the nature of collective chains from the dynamical heterogeneity: we first calculate the ‘rearrangement indicator’ of all fluorides within a short time interval using equation (2.6). We then identify the fluorides with a large rearrangement indicator typical of a jump and calculate the distances between them. All rearranging fluorides within a given cut-off distance that jump at about the same time (<1 ps) are members of the same cluster. We find that these clusters are more compact than those found by distributing the rearranging fluorides at random over all fluoride positions, providing additional evidence for collective dynamics within β-PbF2. When compared to a mean of a random distribution, we find fewer isolated rearranging atoms, twice as many three-atom clusters and equally many clusters with four or more atoms.
The nature of collective diffusion in β-PbF2 is strikingly similar to that seen in many other fast-ion conductors [27–36]). Oxygen diffusion in δ-Bi2O3, fast Li+ diffusion in Li3PS4 and silver ion diffusion in AgI, for example, all involve ‘multi-ion’ chains with . Collective diffusion in the fast-ion regime is advantageous since the nature of energetically preferred local motifs and their connectivity patterns are kept intact as the chain of ions migrates, minimizing bond breaking. The ‘immobile’ sublattice also plays important roles as the ions in this sublattice relax appreciable distances away from their average lattice positions generating sufficient ‘local space’ to accommodate the diffusing multi-ion chains. Thus, experimental and atomistic studies are about to unveil the ‘universal’ nature of diffusion in the superionic regime, an important next step towards designing new devices for future energy technology.
Data accessibility
The simulation output file (which also contains input parameters) can be downloaded from https://www.doi.org/10.17635/lancaster/researchdata/.
Authors' contributions
C.M. performed calculations, analysis and wrote the paper. M.K. did analysis and contributed to discussion. N.L.A. and W.K. contributed to analysis, discussions and writing of the paper. All authors read and approved the manuscript.
Competing interests
We declare we have no competing interests.
Funding
The Centre for Earth Evolution and Dynamics is funded by CoE-grant no. 223272 from the Research Council of Norway. The computing time was provided by the Norwegian Metacenter for Computational Science (NOTUR) through a grant of computing time (nn2916k and nn9329k).
Acknowledgements
W.K. is a senior member of the Institut Universitaire de France.