Abstract
Atomisticlevel understanding of ion migration mechanisms holds the key to design highperformance solidstate ion conductors for a breadth of electrochemical devices. Firstprinciples simulations play an important role in this quest. Yet, these methods are generally computationallyintensive, with limited access to complex, lowsymmetry structures, such as interfaces. Here we show how topological analysis of the procrystal electron density can be applied to efficiently mitigate this issue. We discuss how this methodology goes beyond current state of the art capabilities and demonstrate this with two examples. In the first, we examine Liion transport across grain boundaries in Li_{3}ClO electrolyte. Then, we compute diffusion coefficients as a function of charge carrier concentration in spinel LiTiS_{2} electrode material. These two case studies do not exhaust the opportunities and might constitute motivations for still more complex applied materials.
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Introduction
Solidstate ionic conductors find application in many materials science domains. For example, they are deemed to play a key role in the swiftly evolving energy storage and conversion technologies^{1}, as well as the development of functional materials for sensing^{2,3}. Understanding and controlling ionic transport in these systems is essential for their successful application in highperformance electrochemical devices. This typically involves identifying ion occupancies, ion migration paths, charge carrier types, point defects and iondoping sites at both bulk and interfacial structures such as grain boundaries and heterophases^{4,5}.
Quantum mechanics is a wellestablished theoretical approach for accessing these aspects at the atomistic level. In particular, density functional theory (DFT) offers a good tradeoff in terms of accuracy and efficiency, with a successful track record^{6,7}. Yet, gathering indepth information about ion migration processes and their underlying mechanisms in complex systems requires exhaustive explorations of tortuous potential energy landscapes or the evaluation of atomic dynamics, which are computationally expensive tasks. This limits the use of DFTbased calculations in highthroughput screening campaigns to discover ionic conductors or the assessment of the role of charge carrier concentration and interfacial structures in performance. And less computationally intensive alternatives to DFTbased computations^{8,9,10} often come at the expense of structural detail, quantitative predictability, and transferability.
Workflows for the computation of kinetic properties (i.e., ion diffusivity) require not only finding the lowest energy geometries of initial and final states, but also considering all plausible migration paths and computing their transition states. And, from a general perspective, such workflows should consider multistep diffusion mechanisms and the role of possible metastable sites. Current DFTbased methods are well suited to perform such tasks in relatively simple bulk materials and provide accurate predictions^{7,11}. The two most successful approaches in this context are ab initio molecular dynamics (AIMD) simulations^{12,13,14,15} and nudged elastic band (NEB) methods^{16,17,18}. Despite important advances to enhance these approaches by, for example, exploiting crystal symmetry^{19,20} or using machine learning techniques^{21,22,23}, the main bottleneck associated to their high computational time remains. And this limitation is largely amplified when dealing with complex, lowsymmetry systems (such as interfaces) and the systematic exploration of large compositional and configurational spaces, all those key aspects for truly applied materials development.
Here, we implement a topological analysis of procrystal electron density (TAPED) to compute the kinetic properties of ion conductors effectively and efficiently. Compared with conventional AIMD and NEB techniques, this approach requires much less computational time, while allowing for the consideration of interface models and large supercells containing thousands of symmetrically nonequivalent ion migration pathways. Furthermore, the implemented algorithm is fully automated and requires minimal user involvement. We show that the approach can be calibrated to yield quantitative agreement with DFTbased NEB calculations by modeling two representative case systems (one solid electrolyte and one electrode material) for which high quality theoretical and experimental data is available. Accurate estimates of the ionic kinetic properties can be achieved in both cases at a relatively low computational cost.
Results and discussions
In previous works we have reported the general concept of the TAPED approach^{24,25,26}. Here, we provide a detailed description of the underlying algorithms and the overall methodology. And most importantly, we expand the approach for the study of ion transport phenomena in complex systems such as polycrystalline or disordered ion conductors. Specifically, we illustrate the capabilities and reach of this approach by examining two case studies. First, we focus on the grain boundary Liion transport phenomena considering a polycrystalline Li_{3}ClO solid electrolyte. And second, we build on a kinetic Monte Carlo (KMC) model to compute diffusion coefficients as a function of charge carrier concentration in spinel LiTiS_{2}.
TAPED vs. DFT
We validate this approach by thoroughly comparing ion trajectories and diffusion barriers obtained using TAPED with the DFTbased climbing image nudged elastic band (DFTCINEB) method^{27}. In total, we consider five Liion migration pathways in the antiperovskite bulk Li_{3}OCl structure (Fig. 1a–e) and three Liion hops in the spinel LiTiS_{2}, with one, two, and three Li vacancies along the pathway (Fig. 1f–h). We find that the minimal electron density paths (MEDPs) predicted by the TAPED approach are in very good agreement with the corresponding minimum energy paths (MEPs) for Liion migration yielded by the DFTCINEB method. Specifically, the root mean square deviation (RMSD) between the MEDP and MEP for Li_{3}OCl and LiTiS_{2} are 0.17 Å and 0.05 Å, respectively, with the greatest RMSD of 0.23 Å observed for the 3.9 Å Li hop in Li_{3}OCl (Fig. 1c). Interestingly, when comparing the MEDP and MEP barriers (i.e., Δρ = ρ_{max}ρ_{min} and ΔE = E_{max}E_{min}, respectively, where ρ is the procrystal electron density at some point of crystal space and E is the potential energy of the system) we find a linear correlation (Fig. 2a, d). Furthermore, the electron density profile reproduces accurately the shape of the energy profile (Fig. 2b, c, e, f). There are some divergences in the high energy barriers for Li_{3}OCl, but the overall correlation is high, with coefficients of determination (R^{2}) of 0.953 and 0.996 for Li_{3}OCl and LiTiS_{2}, respectively.
To understand the origin of the observed MEDP deviations from the corresponding MEP trajectories and profiles, we need to consider that the TAPED approach neglects geometrical changes in the local atomic environment of the diffusing Liion. The approximation leads to an intrinsic error in the predicted geometry and barriers of the migration pathways. And this error is expected to become larger as the trajectory of the diffusing Liion is closer to nearby atoms of the structure, especially when there are anisotropic geometrical changes in the local atomic environment along the pathway. To illustrate this, consider a diffusing ion trajectory that goes through a narrow window that is smaller than the size of the ion (Suplementary Fig. 1a, b). To squeeze through the window the ion should push the atoms out of their equilibrium positions. In the case of isotropic deformation, the bottleneck positions of the equilibrium and the deformed window coincide, while for anisotropic deformation, the positions differ. The same happens with the whole Liion migration trajectory. For example, the DFTCINEB calculation of the 3.9 Å Liion diffusion (Fig. 1c) with frozen atomic coordinates yields a trajectory equivalent to the MEDP (Supplementary Fig. 1d). In contrast, gradient paths calculated for the transition state geometry coincide with the DFTCINEB trajectory (Supplementary Fig. 1c). Thus, for the ‘rigid’ structures, with relatively high binding energy like oxides, we could expect a better correlation of Δp with ΔE than for the ‘soft’ structures such as sulfides. Yet, if the contributions of structural relaxation during ion migration in ΔE for different pathways are close, the correlation should be high even for a structure with soft phonon modes. But this contribution is often different for distinct structures and, therefore, the Δp vs. ΔE linear equation can be expected to show poor transferability.
Note that this limitation is shared by all methods that neglect geometrical relaxation in the modeling of the diffusion process, like bond valence^{10,28,29}, pinball^{8}, pathfinder^{30}, chargebased analysis^{31,32}, or Voronoibased methods^{33,34}. However, even with this limitation, the results yielded by the TAPED approach are in good quantitative agreement with those obtained using the DFTCINEB method. Furthermore, the TAPEDcomputed ion migration pathways, barriers, and shape of the energy profiles can readily be used to improve the effectiveness of DFTCINEB simulations by guiding the setting of the initial position of each image along the diffusion paths; this is especially useful when tackling complex multistep diffusion processes.
Progress beyond the stateoftheart
One of the main disadvantages of NEBbased methods is a critical dependency on the initial guess of the diffusion pathway. Generally, the pathway geometry is approximated by linear interpolation between equilibrium initial and final atomic positions. However, for some pathways (e.g., Fig. 1b, e) such crude interpolation can cause atom overlapping that leads to slow convergence or simulations to fail. To overcome this issue various algorithms exist, such as the pathfinder^{30} which preoptimizes the initial trajectory using an electrostatic force field. Yet, the problem of searching for the global MEP remains. In other words, a NEB optimization yields a local MEP, which can differ from the global MEP. This occurs when the initial pathway and global MEP are separated hills on the potential energy surfaces (PES) as illustrated in Supplementary Fig. 2a. For example, when the NEB optimization, starts from the linear interpolation of one of the Kion hops in K_{2}SbPO_{6}, the energy barrier is overestimated by about 7.5 eV due to this issue^{24}. Thus, finding the global MEP requires scanning the entire PES, which is unfeasible at the DFT level.
One alternative strategy is to use simple interatomic force fields to approximate the complete PES. In this way, the MEPs can be found applying topological analysis of the field critical points (minima and adjacent saddle points) and the gradient lines connecting them. Followup DFTCINEB calculations can then be used to refine such approximated MEPs, achieve better accuracies and maximize computational resources. For example, Adams et al. have implemented the bond valence pathway analyzer (BVPA) program for automatically finding ion migration paths based on scanning bond valence energy landscapes (BVELs)^{10}. In this method, the critical points of a BVEL are calculated using a gridbased algorithm, and the path geometries determined by means of a modified Dewar, Healy, and Stewart algorithm^{35}. To reduce the number of potential paths under consideration, they introduced geometrical restrictions, a cutoff distance of 3.5 Å between the minima and the saddle point, and an angle range of 90–270 degrees for the two minima and adjacent saddle points. These criteria are justified by the absence in available literature of reported ion jumps with longer distances between the site and the saddle point; and the assumption that straightlike hops are the most relevant, even if such pathways imply slightly higher energy barriers than other competing paths. However, the lack of reports is not solid proof that long ion hops can be systematically ruled out. And ignoring possible lowenergy transition states contradicts the transition state theory^{36}, according to which the hop rate depends only on the activation energy, temperature, and an effective frequency associated with vibration of the migrating ion in the direction of the transition state (Eq. 3 in the Methods section). Furthermore, the BVPA approach neglects the interaction between mobile ions; this can cause significant underestimation of migration energy barriers in nondilute systems. The TAPED approach, which explicitly includes these interactions, shows indeed that there is almost twice difference in the Liion migration energy barriers in LiTiS_{2} with one and three vacancies along the pathway (Fig. 2f). This limitation applies to similar implementations of BV methods^{37,38}. Replacing BVEL by electron density and avoiding geometrical constraints enable eliminating these disadvantages. Shen and coworkers proposed an alternative method based on topological charge density analysis that considers the electrostatic repulsion of mobile ions^{31}. The method provides information about possible sites of mobile ions, but it does not allow to approximate pathways geometry and shape of the energy profile without involving expensive NEB calculations. Our TAPED approach eliminates the bottlenecks of the methods mentioned above and provides a straightforward way for finding the global MEP.
Another potential issue with the NEB method is the proper identification of transition states within multistep MEPs, because accurate representations of energy profiles require many images. However, increasing the number of images drastically increases the computation time of NEB calculations. In this regard, the CINEB approach helps mitigate this issue and guarantees to find the transition state and corresponding activation energy using a minimal number of images. The CINEB algorithm forces the intermediate image that has maximum energy to climb the hill of PES along the MEP until it reaches the top. Specifically, the algorithm maximizes the energy of the climbing image towards the elastic band and minimizes it perpendicular to the band^{27}. Nevertheless, in the case of an energy profile with more than one maximum (i.e., at least one intermediate local minimum between the initial and final states), there is a probability that the method will converge to a lowenergy transition state (Supplementary Fig. 2b). The best practical solution is to split the multistep diffusion path into a series of individual steps, ensuring that each CINEB calculation only involves one transition state. In principle, the CINEB optimization of such elementary pathways only needs one intermediate image and, therefore, the spring forces are not applied, which enhances convergence. Yet, the prerequisite to split the diffusion pathways into suitable singletransitionstate steps implies having a good approximation for the energy profile which is generally unknown. In this regard, the TAPED approach can provide all required information to split the pathways (i.e., the positions of the migrating ion at local minima and transition states), which can be obtained from the MEDP and the corresponding electron density profiles.
Overall, the TAPED approach efficiently solves the critical issues of NEBbased calculations, namely the choice of starting diffusion geometry and image sampling. Since the approach is much less computationallyintensive than DFTCINEB, it can be used as an individual tool for screening for potential ion conductors. Unlike the methods based on visual analysis of maps and isosurfaces of electron density^{32,39,40} or BVEL^{28}, the TAPED approach is fully automated. The only required input information is the crystallographic data and the type of migrating ion. We should point out that the approach is only rigorously applicable to isolated ion hops; accounting for concerted migrations of multiple ions^{41}, which might be relevant to explain superionic conductivity, is beyond the reach of the current implementation. The fact that the migration energy barriers and the maxima of electron density for the MEDPs are linearly correlated (Fig. 2a, d) enables the implementation of quantitative simulations of complex systems, as demonstrated in the following.
Grain boundary Liion transport in polycrystalline Li_{3}ClO
We have considered four lowenergy periodic grain boundary (GB) models of antiperovskite Li_{3}OCl generated using the aimsgb online service^{42} and geometrically optimized using DFT. The Liion migration pathways were then calculated between all neighboring Li sites, with a total of 1536, 1956, 1408, and 2988 computed pathways for Σ3(111), Σ3(112), Σ5(210), and Σ5(310) GB models, respectively. The diffusion barriers through the GBs were found by analysis of the Liion migration maps as described in the Methods section.
Figure 3 shows the Liion migration map of the Σ3(111) GB model, where we see highenergy migration barriers (0.63 eV) perpendicular to the interface (red pathways) and lowenergy barriers within the grains. Notice how the lowestenergy barrier pathways (0.28 eV) form 1periodic maps (blue); and these maps are connected to the 2periodic map parallel to the GB by 0.29 eV pathways (violet). Since the grains have finite size this 2periodic map corresponds to a thin high conductive layer close to the surface of the grains. However, polycrystalline Liion diffusion is limited by the highenergy barrier (0.63 eV) through GBs. Supplementary Figs. 3–5 show the Liion migration maps of the Σ3(112), Σ5(210), and Σ5(310) GB models. In general, we find that the Liion migration energy barriers through and along the GBs are consistently higher than that of the bulk structure (Table 1). This is in good agreement with previous MD calculations of ref. ^{43} and available experimental data^{44,45}. The limiting energy barriers for Σ3(111), 0.63 eV, and Σ3(112), 0.57 eV, are very close to the experimental value of 0.59 eV for polycrystalline Li_{3}OCl. Note that the concentration of Σ5 GBs and, therefore, their contribution to the total Liion diffusion characteristics should be much less than Σ3 GBs because Σ5 GBs are less energetically favorable than Σ3 GBs^{43}. Thus, in the following we focus the discussion on the results for the Σ3(111) and Σ3(112) models.
Further analysis of the Σ3(112) GB model leads to a more fundamental understanding of the migration mechanism. Its migration map reveals pathways with relatively lowenergy barriers within the interface region (Fig. 4a, Supplementary Fig. 3): 0.23 eV (blue) and 0.25 eV (purple). Therefore, one would expect an increase in ionic conductivity at the GB with respect to the bulk. However, in practice these pathways cannot be realized due to a special structural feature of the Σ3(112) GB. Notice the electron density profile of the pathway that crosses the interface (Fig. 4a) has a concave shape (Fig. 4b). This is a consequence of the repulsive forces between neighboring Li ions located in the same cage of the framework. Thus, the formation of a vacancy on one side of the pathway forces a Liion on the opposite side to fell into the potential energy well, which is in the center of the cage (site II in Fig. 4b, c). The geometrical DFT optimization of the Σ3(112) GB model with the vacant Li site leads to the displacement of the opposite Li to the interface (Fig. 4c). Since this position is energetically more favorable than the initial site, the effective energy barrier for a diffusing Liion to cross the GB is 0.79 eV, as depicted in Fig. 4b. Among the alternative diffusion routes through the GB (Supplementary Fig. 3), we find the lowest barrier is 0.57 eV. Thus, as in the case of Σ3(111) GBs, the Σ3(112) GBs also exhibits higher ionic resistance than the bulk.
It is important to highlight that the identified redistribution of Li ions and vacancies at the Σ3(112) GBs should also be observed in MD simulations. However, the detection of this effect might be cumbersome because it requires a detailed analysis of MD trajectories. For example, the MD investigation of Li_{3}OCl GBs by ref. ^{43} mainly focused on Liion diffusion properties but omitted a complementary analysis capable of revealing such subtle dynamics. In contrast to MD, the TAPED approach provides comprehensive information on all possible Liion diffusion trajectories inside and between the grains, as well as values of migration barriers for each pathway. This approach can systematically examine the effect of the structural features of GB on ionic conductivity, which could help gain a better understanding of ion transport phenomena in polycrystalline ionic conductors. Moreover, from an efficiency viewpoint, it is important to point out that we explicitly accessed here the Liion migration energy barriers of over a thousand of nonequivalent trajectories for a range of GB models−a problem that would be intractable using a plain DFTCINEB approach. And for this, we only needed to compute five DFTCINEB pathways in a 2 × 2 × 2 supercell of the bulk Li_{3}OCl to obtain the linear correlation shown in Fig. 2a for converting density barriers into energy barriers. The good agreement between our results and alternative MD simulations and agreement with experiment (Table 1) indicates that the approach can be quantitative and scalable.
Liion diffusion coefficients in LiTiS_{2}
Now we consider the application of the TAPED approach for computing Liion diffusion coefficients in LiTiS_{2} using KMC simulations. For this task, we first need to explore the complete configurational space of the Li_{x}TiS_{2} system as a function of Li content (0 ≤ x ≤ 1), which was performed by discretizing the Li content using Δx steps of 0.0625. This corresponds to a total of 531 Li_{x}TiS_{2} structures within the unit cell with symmetrically different Livacancy arrangements, which were relaxed using DFT. Then, to build the KMC model, we have applied the TAPED approach to compute migration trajectories and migration energy barriers for 12504 Liion hops between the closest pairs of occupied and vacant Li sites within the unit cell. The diffusion coefficients were averaged over 100 independent KMC trajectories, including 10^{7} steps.
The distribution of Liion migration energy barriers shows a diverse dependency with the Li content (Fig. 5a). This is due to a strong sensitivity of the migration barriers to the local atomic environment. The distribution has three distinct energy ranges corresponding to 1, 2 and 3vacancy Liion migration mechanisms^{46}. Regardless of the Li content, the highest and lowest barrier pathways correspond to the 1 and 3vacancies ones, respectively. Obviously, the number of possible low barrier pathways increases as the Li content decreases. Consequently, diluted systems should yield the highest ionic conductivities. This is confirmed by the calculated diffusion coefficients (Fig. 5b). Starting from ~10^{−12.4} cm^{2} s^{−1} at a Li content of 6.25%, the trace and jumpdiffusion coefficients reached their maxima (~10^{−11.9} cm^{2} s^{−1} and ~10^{−11.4} cm^{2} s^{−1}, respectively) at a Li content of 25%. Within this Li range (less than 25%), there are significant changes in the unit cell volume (Fig. 5c). The lowering of the diffusion coefficient as the Li content approaches to zero is related to the contraction of the unit cell. A further increase in the Li content from 25% to 93.75% leads to a decrease in the diffusion coefficient to ~10^{−18.4} cm^{2} s^{−1} and ~10^{−17.3} cm^{2} s^{−1} for trace and jump diffusion, respectively. The swift decline of the diffusion coefficients at 50% and 80% Li are associated with the significant reduction of available 3 and 2vacancies Li hops, respectively (Fig. 5a).
These results are consistent with the previous clusterexpansionbased KMC simulation of the spinel Li_{x}TiS_{2} by Bhattacharya and Van der Ven^{46}. However, we find a systematic difference of ~0.2–0.3 eV between the energy barriers calculated here (Fig. 2f) and in the previous study which we attribute to the underestimation of the energy barriers by the standard NEB method used in the previous work. This also leads to a deviation in the values and shape of the curves of the diffusion coefficients as a function of the Li content (Fig. 5b, d). Specifically, the diffusion coefficients calculated by Bhattacharya and Van der Ven are several orders of magnitude higher than ours because of the mismatch in the energy barriers. And their dependencies with the Li content are smoother than ours due to the smaller energy gaps between 1, 2, and 3vacancies pathways. Nevertheless, the underlying trend of the dependency and following conclusions agree qualitatively. Thus, the TAPED approach is an efficient method for exploring Liion migration energy barriers in the configuration space of disordered structures, requiring a significantly lower number of NEBbased computationallyintensive calculations than the clusterexpansionbased method. Furthermore, we put forward that combining TAPED with KMC can allow the computation of Liion diffusion coefficients in complex systems. This opens new perspectives in assessing the impact varying the charge carrier concentration has not only in pristine bulk materials, but also in GBs, doped materials, and other simulations of practical interest.
Methods
Calculation of diffusion trajectories and migration barriers
The TAPED method approximates the PES by a field to obtain an analytical or numerical function of the potential energy from coordinates of the migrating ion. Based on this representation of the PES, it is possible to identify all the MEPs between the initial and final positions of the migrating ion and find the global MEP. In particular, the equilibrium and metastable sites of the ion are associated with the minima of the field, so the MEPs correspond to gradient paths connecting these minima. These gradient paths starting from minima end in saddle points representing the transition state between two adjacent minima. The value of the field in the saddle point characterizes the migration barrier. The MEP can include one or more transition states. Given an equilibrium structure, the ratio between the number (N) of the saddle points, minima, and gradient paths within the MEP is N: N + 1: 2 N, with the only exception of posttransition state bifurcations^{47}. Finding the global MEP requires the iteration over all MEPs to identify the one with the lowest values of the barrier. Due to the periodicity of the crystal structure, the number of MEPs is infinite, but the number of transition states is finite. Thus, this task can be solved by a simple graphbased algorithm, as illustrated in Fig. 6. Initially, the topology of the field is represented as a weighted graph in which vertices, edges, and weights correspond to the critical points, gradient paths, and field value at the saddle points. Next, the ends of the MEPs are associated with the closest minima, saddle points, or points of the gradient paths. Accounting for the last two involves considering nonequilibrium structures, without available geometry optimization. An example is II´ pathways in Fig. 4. In the case of equilibrium structures, the initial and final position of the migrating ion matches with the minima of the PES. Further steps of the algorithm search for the route with the lowest value of the barrier (Fig. 6c), which is solved using a Dijkstralike algorithm^{48,49}. Starting from one of the two given vertices, the algorithm iterates over all possible routes, discarding variants with a higher weight than the most optimal route found in the previous steps. As a result, we obtain the ion migration trajectory, the field profile of the pathway, as well as the positions of local minima and transition states (Fig. 6d). The detailed information about the implementation of the approach can be found in the IonExplorer2 code (https://github.com/mcscice/IonExplorer2). The calculation of the gradient paths and the critical points was carried out using the Critic2 program^{50}. Visualization of ion diffusion trajectories and crystal structures was performed by means of Vesta^{51}.
To approximate the PES, we use procrystal electron density^{39}, which yields good agreement with experimental^{52} and ab initio derived electron density distributions^{53}. The process of ion migration is considered as the movement of a single negative point charge in the constant electron field of the structure (Fig. 6b). The approximation assumes that the main forces that affect the ion trajectory are the electrostatic repulsion of electron shells of the ion and atoms of the structure. In this case, the equilibrium and metastable ion positions are defined as minima of electron density: the center of intrastructural cages, namely cage critical points (CCP). The positions of the migrating ion at transition states are in the maxima of electron density along the gradient path: the center of the atomic window, namely ring critical points (RCP). Notice that the algorithm to calculate the procrystal electron density distribution only requires information about the atomic types and coordinates.
One of the input data required for calculating the lowestbarrier ion migration pathway is the initial and final positions of the ion. In ion conductors, mobile ions hop between nearest vacant atomic positions. In other words, the possible final positions of any diffusion trajectory should correspond to the neighbor sites of the ions. A common practice to choose the final positions of the pathways is by simple visual analysis of the structure or subjective distance cutoff criteria. However, such approach cannot be automated for calculation of a large data set and complex geometries. Alternatively, potential ion hops can be found using information about neighbor sites derived from the Voronoi partition of the ionic sublattice^{33}. For a set of M geometrical objects in Euclidean space, the Voronoi partition divides the space into M regions (domains). Each domain surrounds one of the objects so that the distance from any point of the domain to the enclosed object is no greater than the distance to any other object of the set^{54}. Thus, the domain allocates a region of space that belongs to the object. Depending on the geometry and arrangement of the objects, Voronoi domains take various shapes. In the case of ion sublattice approximated by a set of points or spheres with an equal radius, the atomic domains are convex polyhedra (Supplementary Fig. 6a). Atoms whose domains share common faces, edges, or vertices are neighbors (Supplementary Fig. 6b). Nearest (direct) neighbors are adjacent by faces, while more distant (indirect neighbors) by edges or vertices of the polyhedra. In most cases, ion hops between direct neighbor positions are the most energetically favorable ones, but there are exceptions. For the highthroughput screening of ion conductors, considering only hops between direct neighbors can significantly decrease the computational cost. However, the detailed analysis of the diffusion requires the calculation of all possible trajectories. The whole set of neighbor pairs forms a graph that reflects the map of potential ion hops (Supplementary Fig. 6c). Due to the periodicity of the crystal structure, this graph is always 3periodic. Depending on the distribution of ions, graph vertices can have a quite high degree. For example, each atom of the Li sublattice in the Li_{3}OCl structure has 8 direct and 26 indirect neighbors (Supplementary Fig. 6b). This means that in the unit cell with 3 Li atoms there are about 100 possible hops for neighbor sites. However, among all of them, there are only five symmetrically nonequivalent ones. Thus, taking symmetry into account can drastically decrease the numbers of calculations of the lowestbarrier ion migration trajectories.
In summary, the implemented searching scheme of initial and final position migration pathways include three main steps:

1.
Calculation of Voronoi partition for the sublattice of ions involved in diffusion process.

2.
Search for the pairs of ions whose Voronoi polyhedra are adjacent by faces, edges, or vertices and build the graph of neighbors.

3.
Identification of symmetrically different edges of the graph.
In the case of studying ion conductors at various levels of ion content (e.g., Li_{x}TiS_{2} system), the graph of possible hops is calculated for a structure with fully occupied positions. The graph for a partially occupied structure is then generated by removing all edges connecting two occupied or two vacant positions.
Analysis of ion migration map
The presence of low barrier migration pathways is not a sufficient criterion for high ionic conductivity. Since ion transport is limited by the highest barrier along the diffusion path, low barrier pathways should form a periodic migration map allowing ions to diffuse through the structure. The preference for certain diffusion crystallographic directions and their associated barriers can be found based on analysis of the ion migration map. For this purpose, we iteratively removed pathways in decreasing order of the barriers until the map loses periodicity. At each iteration, nets of pathways and their periodicity are determined, and all nonperiodic parts of the migration map are removed. The maximal barrier at the step before the map loses the periodicity in a certain direction corresponds to the diffusion barrier in this direction. For example, in the case of LiTiS_{2} (Fig. 7a, b), the migration map of Li hops between neighboring sites includes three symmetrical nonequivalent pathways. The density barriers calculated by the TAPED approach equal to 18.7 a.u., 7.7 a.u., and 3.0 a.u. Removing first two pathways does not change the periodicity of the map (Fig. 7a); thus, the lowest barrier Li migration map of the LiTiS_{2} structure is 3periodic and formed by one type of the pathway. And the diffusion barrier in all directions is equal to 3.0 a.u. In the case of structures with lower symmetry the algorithm can pass through 2 and 1periodic maps. Moreover, there is a possibility of splitting the migration map into several independent subnets of pathways, which may have equal or lesser periodicity relative to the parent net. The migration maps at the step before changing the periodicity or splitting the net of pathway are milestones, which provide comprehensive information about diffusion barriers in all crystallographic directions. Furthermore, this iterative procedure shows the evolution of the topology of the ion migration map depending on the amount of energy transferred to the system to overcome the diffusion barriers, for example, by heating or applying an electric potential.
The periodicity of the net is determined by the number of independent translation vectors. These vectors can be found using the breadthfirst search algorithm. For this purpose, we represent the net using a periodic graph^{55,56}. In this notation, each vertex (Pi) of the graph is associated with its translation image by the (r,s,t) symbol, which indicates the ra + sb + tc translation from Pi(0,0,0), where a, b, and c are basis translation vectors (Fig. 7c). Starting from some arbitrary vertex, the algorithm traverses graph vertices by the neighbors. Before moving on the next vertex, the algorithm explores all vertices at the current depth level. Once a vertex Pi is traversed the algorithm does not explore the translational equivalent images Pi(r,s,t), but keeps the information about the vectors between these vertices. The procedure stops after traversing all translationally nonequivalent vertices.
Diffusion coefficient and KMC simulations
The KMC methods are based on stochastic simulations of dynamic processes within a predefined set of states of the system and transition rates between them^{57}. Detailed information of the methodology for calculating ion diffusion coefficients in undiluted systems can be found in refs. ^{46,58} by Van der Ven and coworkers. Specifically, we use here the Bortz–Kalos–Lebowitz algorithm^{59,60} to estimate ionic diffusion coefficients and ion hop frequencies, where the selection of a transition to a new state and time increment is computed using the following two formulas:
where Γ_{tot} is the sum of individual mth hop frequencies, Γ_{m}, and p is a random number from 0 to 1; and
where ∆t and ξ are the time increment and a random number from 0 to 1, respectively.
The hop frequencies are calculated as:
where ΔE is the energy barrier for the hop of the ion and ν* is the vibrational preexponential factor. Here we assumed ν* = 10^{13 }Hz as a commonly accepted approximate value.
We compute ΔE using the TAPED approach. To convert TAPED derived density barriers into actual energy barriers, we use a linear equation of Δρ vs. ΔE (Fig. 2d) obtained through a combination of DFT calculation and the CINEB method, considering a few different diffusion pathways. For the calculation of ion selfdiffusion from KMC trajectories, we use the following two formulas for jump and tracediffusion coefficients:
and
where t stands for time, d is the dimension of the ion migration map, N is the number of ions involved in the diffusion process, and r_{i} is the vector that connects the position of the ith ion at initial (t_{0}) and final (t) times.
DFT calculations
The DFT calculations were performed using the Vienna ab initio simulation package (VASP) 5.4 code^{61,62,63}. We use the Perdew−Burke−Ernzerhof (PBE) exchange−correlation functional^{64} to calculate total energies and electron densities, together with a planewave kinetic energy cutoff of 400 eV and a Γpoint kpoint sampling. The Li (1 s, 2 s, 2p), S (3 s, 3p), Cl (3 s, 3p), O (2 s, 2p), Ti (3p, 4 s, 3d) electrons were treated explicitly. The remaining electrons were replaced by PBEbased projectoraugmented wave potentials^{65,66}. Structural optimization of ground states was performed by allowing the lattice parameters, cell shape, and atomic positions to relax until the maximum force on each atom was less than 0.05 eV Å^{−1}. The CINEB calculations were carried out employing the transition state tools (TST) for VASP (https://theory.cm.utexas.edu/vtsttools/references.html).
Data availability
All the data supporting the findings of this work are available from the authors upon reasonable request. The DFTrelaxed structures of the Li_{3}OCl (upload Id: YgSjkRuvTduzU1HSpX4Ymg) and LiTiS_{2} (upload Id: 9exhy71TSwKctviNKmYAYA) systems can be found in the NOMAD public repository https://nomadlab.eu/.
Code availability
The program implementation of algorithms reported in this work is available at https://github.com/mcscice/IonExplorer2.
References
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Acknowledgements
This work was supported by the European Commission through the H2020 program under Grant agreement number 875028 (SUBLIME Project). The authors gratefully acknowledge the technical support provided by the Barcelona Supercomputing Center and the computer resources from Calendula (RESQS202110003 and RESQS202120004) and SGI/IZOSGIker UPV/EHU.
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A.G.: Conceptualization, methodology, investigation, writing – original draft, writing – review and editing. J.C.: Conceptualization, funding acquisition, supervision, writing – review and editing.
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Golov, A., Carrasco, J. Enhancing firstprinciples simulations of complex solidstate ion conductors using topological analysis of procrystal electron density. npj Comput Mater 8, 187 (2022). https://doi.org/10.1038/s41524022008776
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DOI: https://doi.org/10.1038/s41524022008776