Combined Theoretical Approach for Identifying Battery Materials: Al3+

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A combined theoretical approach for identifying battery materials: Al mobility in oxides 3+

Tina Nestler, Falk Meutzner, Artem A. Kabanov, Matthias Zschornak, Tilmann Leisegang, and Dirk Carl Meyer Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.8b03631 • Publication Date (Web): 20 Jan 2019 Downloaded from http://pubs.acs.org on January 20, 2019

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Chemistry of Materials

A combined theoretical approach for identifying battery materials: Al3+ mobility in oxides Tina Nestler,∗,† Falk Meutzner,†,‡ Artem A. Kabanov,‡,¶ Matthias Zschornak,†,§ Tilmann Leisegang,†,‡,¶ and Dirk Carl Meyer† †TU Bergakademie Freiberg, Institute of Experimental Physics, Leipziger Str. 23, 09596 Freiberg, Germany ‡Samara National Research University, Moskovskoye Shosse 34, Samara 443086, Russia ¶Samara State Technical University, Molodogvardeyskaya Ulitsa, 224, 443001 Samara, Russia §Helmholtz-Zentrum Dresden-Rossendorf e.V. (HZDR), Institute of Ion Beam Physics & Materials Research, Bautzner Landstraße 400, 01328 Dresden, Germany E-mail: [email protected]

Abstract In this work, we take a significant step forward in the Al-ion battery material search by screening already existing aluminum compounds, not being considered in this regard before. A novel combination of different established theoretical methods to filter structural databases is applied here. The presented high-throughput analysis minimizes the computational time, while still providing reliable results. Starting with VoronoiDirichlet partitioning of 4,346 aluminum oxides listed in the Inorganic Crystal Structure Database, bond valence and density functional theory calculations are subsequently

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performed. AlVO3 is the most promising candidate for cathode materials found. Limitations of the filter are discussed, with emphasis being placed on the comparison of the data derived from the different methods. The broad coincidence of the found migration networks and trend in migration barriers validates the screening algorithm. In further studies the filter can be applied to rapidly find crystalline electrolytes and electrodes for other mobile species as well.

Introduction Due to the high abundance and high theoretical capacity of aluminum, its application in electrochemical energy storage is receiving increasing attention. Currently, for the attempts to build rechargeable Al-ion batteries (AIBs) mostly expensive ionic liquids are used that are, however, highly sensitive to moisture and may react with other cell components. 1 Furthermore, it is still under debate, if there are rechargeable AIBs at all: 2–4 those cells that actually do show reversible intercalation are most of the time definitely known to insert AlCl4– instead of Al3+, and thus can actually not be named AIB. This in turn lowers the energy density. Hence, finding solid electrolytes and Al3+ intercalation cathodes will push the development of competitive AIBs. Up to now, the hope to be able to transfer compounds or at least structural motifs from the lithium-ion-battery to high-valent battery materials such as magnesium 5 and aluminum 1 often failed, demonstrating the need to start from scratch. For singly and doubly charged ions like Li+, Na+, K+, 6 Ag+, Cu+, Mg2+, 7,8 and O2–, different solid electrolytes are available. In contrast, for ions with a charge higher than two, here so-called high-valent ions, their existence is highly controversial. 4 For instance, X 2(BO4)3-type compounds (X = Sc, Al, In, Lu, Yb, Tm, Er; B = Mo, W) have been proven by both theoretical 9–14 and experimental work 15 to show anion (dominantly O2–) instead of X 3+ conduction. On the other hand, there are strong indicaions that β”-alumina is indeed capable to conduct specific trivalent ions such as Gd3+, as was claimed by the work group of Farrington in the 80s of the last century. 4,16 Additionally, there are strong indications 2


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that Al3+ ions can intercalate in the chevrel phase (Mo6S8 17,18 ), suggesting that finding solid electrolytes with considerable Al mobility is plausible, despite the expected strong Coulomb interactions with the host lattice and low polarizability of Al3+. Given the lack of systematic investigations in this field, it seems promising to perform high-throughput analysis of already existing aluminum compounds. This appears reasonable, taking into account that widely-spread lithium cathode materials have been known for decades, before they were considered for the use in batteries. 19 This has already motivated several studies to screen structural databases for new battery materials and already lead to promising discoveries (e.g. Refs. [ 20–26 ]). In this work, for the first time a combined approach of different established theoretical methods is employed for this purpose: Voronoi-Dirichlet partitioning (VDP), bond-valence site energy (BVSE), and density functional theory (DFT) calculations. The most promising compounds are discussed in detail, while the emphasis is placed on the comparison of the results of the used methods and the elaboration of their predictive power. The presented results aim to encourage more theoretical and experimental work on aluminum battery materials to enable reliable batteries with outstanding volumetric energy density in the future.

Methodology The sequential use of VDP, BVSE, and DFT represents a filter (Figure 1, the mentioned exclusion criteria are explained in the following sections), quickly picking out promising candidates first, which are subsequently evaluated with increasingly accurate, but more timeconsuming approaches. This procedure allows for a fast and reliable screening of thousands of compounds to find intercalation materials and solid electrolytes for Al3+. Of course, this algorithm can be applied for other active species as well. For this study, only aluminum-containing oxides are considered for two reasons: On the one hand, the applied methods have been proven to work well for oxides and data mining of

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Figure 1: Scheme of the suggested screening approach for crystalline materials with fast ionic transport and the data handling and processing for Al-ion battery materials. Simulation methods with different accuracy levels and thus computational effort, as indicated by the clock symbol, are performed in succession.

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necessary input values for both VDP and BVSE is statistically sound due to the huge variety of oxygen-containing compounds. On the other hand, oxide compounds are assumed to be more stable and easier to handle in air in comparison to the other chalcogenides or halides and are less prone to form insulating interfaces with cathode materials (e.g. [ 27 ]). VDP constructs smallest possible polyhedra by creating planes perpendicular to interatomic line segments. Their features can be geometrically interpreted as describing crystalchemical parameters. Due to specific division coefficients, this works only if there is one anion of a kind in the structure, as different anions may change the interaction width between anions and cations. 28 The VDP approach has been already shown to reliably predict intercalation or electrolyte materials for Li+ 20 and Na+, 26,29 while analyzing thousands of structures within hours. Essentially, VDP identifies voids in the crystal structure that can host the respective ion. If these voids are connected, the compound is likely to permit ionic diffusion. Findings encompass both types of battery materials: intercalation electrodes and solid electrolytes. For a more detailed theoretical background, please see the Supporting Information. In order to rank the identified candidates, the BVSE approach, developed by Adams and Rao, is subsequently applied. 30,31 An extensive collection of BV based methods can be found in Ref. [ 32 ], while a short overview is provided in Refs. 4,19 Similar to the VDP methodology, bond-valence methods are based on the analysis of thousands of structures and can indicate sites accessible to mobile ions. Additionally, the BVSE method can give a rough estimation of the migration barrier and has thus been used to analyze solid electrolytes and intercalation electrodes for mono- and divalent ions so far. 19,22,24,32–38 However, this advantage over VDP comes with the price of a higher time-consumption. One compound can be computed in the range of minutes to several hours, depending on the analyzed structure and used program. Since BVSE cannot consider structural adjustments during ion migration, DFT calculations are performed for those compounds with reasonably small BVSE-derived migration energies. Electronic conductivities are considered as well, which allows to distinguish be-

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tween potential intercalation hosts and solid electrolytes. The following paragraphs describe the application of the different methods in detail.

Voronoi-Dirichlet Partitioning As the first step to find prospective structures with considerable Al3+ mobility, all compounds from the Inorganic Crystal Structure Database (ICSD; 2014/1 39 ) containing at least both Al and O (#10077) were analyzed by the VDP method (Figure S3). Both the data mining and the actual VDP approach were carried out using the crystal-chemical program suite ToposPro 40 since it allows applying the methodology to whole databases. According to the results of data mining, Al shows a four- or sixfold coordination in most cases (Table S1). The averaged second moments of inertia, G3 , exhibit values in very good agreement with the theoretical values for a tetrahedron (0.104 theoretical vs. 0.103 data-mined) in the case of coordination number four as well as a cube/octahedron in the case of coordination number six (0.083 theoretical VS. 0.084 data-mined). 41 This suggests mostly undistorted oxygen environments for Al. This is emphasized by the fact that all datamined values for the most important parameters used in the VDP approach show rather low standard deviations (Table S1). For the actual VDP approach, from the database of 4,346 compounds all those entries are omitted containing group 1, 2, 11 and 12 elements. Due to their lower oxidation states compared to aluminum it is suggested that their diffusion would be more probable. The reduced database of 724 entries has all VDP polyhedra—excluding all Al sites—constructed. Furthermore, those structures which show disorder on Al-sites with another element are removed as well. If there were any non-Al3+-ions in the migration network, they may hinder Al3+ ions from migrating through those channels. 42 If Al sites are disordered with vacancies, however, it would be advantageous. According to the VDP approach, voids have to show a minimum size to be occupiable by cations. The best-conducting compounds should have voids with very similar sizes. The 6


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same holds true for the channel sizes between voids. Small deviations of these values suggest a flat energetic profile in the conduction path. The more structure-immanent Al-ions coincide with VDP voids the more the Al content will be used (this will be called “coincidence” in the following). Some structures show diffusion paths without any coincidence (as indicated in Figure 1 as Al×Al 6⊂ migration path). Although this is in principle possible, it should be questioned if migration is likely, since additional charge would need to be introduced to an already electroneutral structure. In this case, a DFT study should be carried out to clarify this possibility. In order to optimize time, those compounds already stable and showing promising features “as is” are chosen and structures exhibiting no coincidence are neglected for this study. As an electrode material, enough multivalent cations should be available for reduction when the triply charged Al enters to assure neutrality. 4 Generally speaking, ions of high oxidation numbers are advantageous for stabilization of the structure. The stronger oxygen is bound to the matrix of the compound, the easier aluminum can move through the structure and the lower should be the migration barriers for Al. This is why binary materials are less probable to exhibit high mobility since, if all Al ions moved, all O ions would have to stay fixed.

Bond-valence site energy calculations The program 3DBVSMAPPER 33 written for materials studio 43 is used for the calculation of BVSEs. Corresponding to the resolution set to 0.2 ˚ A, BVSE values are calculated for every created voxel in the unit cell. For the calculation, all inherent Al-ions are removed first and on each voxel an imaginary Al-ion is placed to generate its BVSE. Regions with low BVSE in the cell are likely to host Al-ions. Thus, mapping these regions can reveal potential ion pathways. The already implemented BVSE-parameters are used, i.e. predominantly the ones originally published by Adams and Rao. 31 Calculations with higher resolution only showed centesimal variations, showing a sufficient convergence, which is in accordance with literature. 22 Applied covalent radii are specified in the program as given by private communication 7



with Stefan Adams. For compounds with equivalent atomic sites occupied by different ion species, whether it is different elements or only different valences, the BVSE calculations are done by weighting the contribution of the different ion species by their occupancy. In general, a flat energetic landscape for the migrating ion is the condition for ionic conduction. Small deviations between saddle points and hopping sites are thus the key factor for such a material. Hence, the conduction is not dependent on the absolute values of energy in the path, but the relative ones. Therefore, the changes in coordination within the conduction path should be minimal, either passing many unfavorable or many low-energy sites. 7,44–46 In order to exclude oxygen conduction from the most promising compounds, all candidates with Al migration energies lower than 3 eV are tested for oxygen mobility with BVSE as well. This step is crucial, since oxygen has a lower absolute valence than Al, which makes it in general more probable for rapid migration. For the same reason other ions with a valence of |2| were investigated, while compounds with monovalent ions have been already sorted out in the first filter step.

DFT calculations DFT calculations together with the nudged elastic band (NEB) method 47 were used to estimate ionic migration barriers in the selected structures. The Vienna ab initio simulation package (VASP) 48 with Perdew-Burke-Ernzerhof generalized gradient approximation 49 (PBE-GGA) and the projector-augmented wave (PAW) method 50 were adopted for all firstprinciples calculations. Vacancy formation energies EV have been calculated according to Refs. [ 51,52 ]. For computational details see the Supporting Information. To minimize periodic interactions between neighboring cells the following supercells were used for calculations: 2×2×1 for YAlO3, 1×2×1 for AlFe2O4 and 1×1×3 for AlVO3. The supercells have been chosen to accommodate the partial Al stoichiometries on the one hand and to keep the computational demand reasonable on the other hand. To account for the 8


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high degree of disorder in the case of AlVO3 in the Al-V sublattice, exemplarily 100 different configurations have been created (Figure S1) and the structure with the lowest total energy was chosen for subsequent analysis, following the methology used in Refs. [ 53,54 ]. To identify these promising, non-equivalent migration paths, VDP analysis was employed. 20,40 Sequentially, the NEB method was used for the calculation of the migration barriers for the selected channels. Besides the topology of the pathway, this analysis provides information about the perturbation of the lattice during ion hopping. Within the selected AlVO3 supercell, only the paths which form an infinite diffusion channel throughout the structure were selected for NEB analysis to limit the number of investigated migration channels (Figure 3). Taking into account the disordered nature of the compound and the great number of possible diffusion pathways (ca. 100 Al- and more than 500 O-diffusion paths), here we can only exemplarily show the theoretical possibility of preferred Al-diffusion. A full consideration of all diffusion pathways requires considerable computational resources and will be published later. However, first results show that there are many low-energy Al-pathways in this compound (Figure S2).

Results and discussion Material screening The biggest group of compounds that potentially show considerable Al mobility according to VDP analysis comprises 17 structures that belong to the garnet family. Apart from their composition they mostly vary slightly in their lattice parameters. As they are all cubic, they exclusively show a three-dimensional migration network that consists of two main sites. In general, many 3D compounds are identified, some 2D, the least are 1D (Table S2). The fact that the few different coordinations that Al shows are all very close to ideal coordination shapes could be the reason why predominantly cubic compounds have been identified. In the cubic crystal class such “perfect” four- and sixfold coordinations can be found: according to 9



Neumann’s principle, migration channels must exhibit at least a sub-symmetry of the point group of their host structure. Al The majority of the garnet structures show very high estimated migration barriers EBV

of around 3.5 eV, while the most promising compound is a spinel: AlVO3 with the space Al group Fd 3m exhibits an estimated energy barrier of EBV = 0.52 eV (Table 1). It is also the O Al only compound with a higher barrier for oxygen EBV than for EBV and thus potentially

showing a high Al3+ transfer number. Except for YAlO3, which is the compound with the Al within those listed, the calculated migration paths are three-dimensional in highest EBV

accordance with the VDP results, again, hailing from their cubic crystal class. Thus, it appears that low migration energies in already Al-containing compounds are found mainly in highly symmetrical structures. Table 1: Compounds with Al3+ migration barriers lower than 2 eV obtained by BVSE calculations a .

a

b

Al EBV

O Dim. EBV

Space Crystal Group System 55 AlVO3 0.52 3 1.19 49645 Fd 3m cubic b,c 56 cubic AlFe2O4 0.73 3 0.38 769772 Fd 3m cubic Gd2.91Sc1.8Al3.15O11.8 0.84 3 0.37 78052 57 Ia3d 58 Y3Sc2Al3O12 1.53 3 0.91 67055 Ia3d cubic 59 YAlO3 1.82 2 0.99 27100 P 63 /mmc hexagonal Al = 0.55 eV as The screening also ranked the compound Ni1.4Al4.67O8.7 (#60314) with EBV interesting, but it is not considered here due to inconsistent structure data. For the calculation, the structure earlier assigned in the ICSD was used (see text). The current ICSD entry has been corrected. c The barrier for Fe2+ migration is estimated to be 7.49 eV. Compound

ICSD #

The migration pathway in AlVO3 can be described as a straight connection of the aluminum 8a sites, coordinated by four oxygen atoms (Figure 2). During hopping between these tetrahedral sites, they pass an octahedral site that represents a local energy minimum. Therefore, the ions have to overcome two energy barriers of the same absolute height upon hopping between two sites due to symmetric equivalency, which correspond to crossing a three-coordinated site, as has been discussed for the migration of high-valent ions in spinels

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before. 44 Additionally, this compound seems favorable as the migration barrier equals the activation energy for migration: Since only 2/3 of the 8a sites are randomly occupied by Al, 55 for moving ions there is a high probability to find a vacant adjacent site to jump into. Thus, in an ideal case where there is no vacancy ordering there is no need for aluminum vacancy formation. Within the migration channel considered with DFT (Table 2, Figure 3), the pathway with the most beneficial local configuration shows a migration barrier of 1.0 eV (green), which represents the lowest energy for which we can expect to see considerable Al3+ mobility. While pathway 1, 3, and 4 clearly pass an intermediate tetrahedral site, pathway 2 appears distorted and bypasses it. In this last case, the tetrahedral sites in the vicinity are fully occupied with Al, while for the other pathways only one is occupied and can interact with the mobile species (Figure 3b vs. c). Thus, it appears that the interaction with other mobile cations cannot be neglected in this case and the morphology and the energy barrier is significantly influenced. Parasitic O2– transport can be excluded to play a role, due to a higher migration barrier (Table 2) and the need for vacancy formation. The vacancy formation energies EV(O) for ten randomly selected oxygen positions ranges from 4.2 eV up to 5.7 eV, which is even higher than for the Al atoms that are involved in the calculated infinite diffusion pathway (EV(Al) varies from 3.5 eV up to 4.3 eV). Table 2: Migration barriers Emig for non-equivalent paths for O2– and Al3+ calculated by DFT. All paths for DFT-NEB analysis were selected according to VDP and BVSE results. AlFe2O4 Emig Al path 1 Al path 2 O path 1 O path 2

(eV) 1.00 5.55 5.94 1.51

AlVO3 Emig Al path 1 Al path 2 Al path 3 Al path 4 O path 1 O path 2 O path 3 O path 4

(eV) 1.02 2.99 2.67 1.77 2.11 0.37 0.49 1.59

YAlO3 Al path O path

Emig (eV) 3.17 2.57

Considering the electronic conductivity of AlVO3, Reid and Sabine, who were the first to synthesize AlVO3 in 1970, reported that the “electronic spectrum did not indicate the 11



Figure 2: Bond valence energy landscape (orange) for Al (light blue) at 0.52 eV in AlVO3 (ICSD #49645). (a) The dark blue polyhedra denote the 16d site, which is occupied by V (2/3) and Al (1/3) and coordinated by eight oxygens. Half of all Al in the structure occupies the 8a site and is suggested to be part of a 3D pathway. This network coincides with the hopping positions calculated by VDP (grey). (b) For aluminum migration an octahedral site (grey) has to be passed during migration. presence of any considerable number of conduction electrons”. 55 However, we do not know more from literature, since this is the only publication known to us about this material. On the other hand, DFT calculations on the density of states already exist, pointing out that AlVO3 should be metallic (aflowlib.org, 60 AUID# ef6eeb5a366979e1). Hence, AlVO3 is thus thought to be a potential cathode material for AIBs with a theoretical capacity of 639 mAh/g, which is comparatively high for an intercalation electrode. In order to have more results to compare to and thus to be able to verify the functionality of the filter algorithm, migration path energy calculations are also carried out for AlFe2O4 and YAlO3 (Table 2). The latter was chosen as VDP and BVSE suggest a 2D pathway in contrast to the other 3D systems. The migration barrier for YAlO3 was confirmed to be too high to allow for ionic transport. AlFe2O4 was picked, even though the ICSD structure (#76977) was not correct: it differed from the original publication 56 in the way that the 8a site is assumed to be fully occupied by Al, while in experiment it is only 0.1 Al3+ and

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Figure 3: Different Al3+ pathways investigated by NEB analysis in the 1×1×3 AlVO3 supercell. All pathways lead via a nearest-neighbor, unoccupied Al site. Blue octahedra represent VO6, while grey octahedra are AlO6. Dark blue tetrahedra denote occupied Al sites, while blue balls represent the start and end positions in the investigated paths.

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0.9 Fe2+ 1 . It could be useful as a model system, since it is the structure with the lowest Al (Table 3). Literature does not tell about antisite disorder of Al and Fe in this material, EBV

however, by the comparison with FeAl2O4, which possesses the same space group and base but different atomic occupancies, it can be estimated to normally occur in a one-percentage range. 61 The falsely assumed structure with most Al being on the tetrahedral site might be stabilized experimentally first by doping. Secondly, these 8a sites need to show a considerable amount of vacancies, which is improbable for the pristine compound given the high V(Al) . Thus, aliovalent doping might approach both prerequisites. Table 3: Electronic band gaps Eg from the AFLOW database a , 60 lowest migration barriers for Al3+ Emig(Al) and O2– Emig(O) transport and vacancy formation energies EV for the considered ions in the pathways from ab initio calculations b . Compound Eg (eV) Emig(Al) (eV) Emig(O) (eV) EV(Al) (eV) EV(O) (eV) AlVO3 0.00 1.02 2.11 3.5 – 4.3 4.2 – 5.7 AlFe2O4 0.00 1.00 1.51 3.0 3.9 YAlO3 3.50 3.17 2.57 a AUID# (top down): ef6eeb5a366979e1; 3c83439071d39f2b; 901d512ca4ebdee5. b The vacancy formation energies for YAlO3 are not given here due to its already high Emig(Al) .

Filter evaluation Comparison VDP-BVSE The dimensionality of the conduction channels identified by VDP is the same as for those determined with BVSE. For most compounds and especially for those showing highestsymmetry space groups with 3D conduction channel networks the results of VDP and BVSE are in very good agreement, not least due to the symmetry. Deviations are observed for structures that show large intrastructural voids or pores. These are not identified unambiguously. There are basically two main problems for VDP: Lacking coincidence between Furthermore, the 16d site was assumed to be occupied by 0.5 Fe2+ and 0.5 Fe3+, while the correct version states 0.55 Fe3+ and 0.45 Al3+. This error has been recently corrected. 1

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voids identified by VDP and structure-immanent Al-ions and pores too large for hosting ions. The first phenomenon can be observed for example in YAlO3. VDP does not identify the large void of the structure in the position 2/3, 1/3, 1/4. Instead, two “satellite” voids (grey, Figure 4) just next to the higher symmetric site are determined at 2/3, 1/3, 0.3365 (generating two voids by symmetry). The actual hopping site is more probable to lie in-between the identified voids. Similarly, the structure-immanent Al-ion is shown not to directly coincide with the VDP vertices.

Figure 4: Voronoi-Dirichlet partitioning of the O1 site of YAlO3 (ICSD #27100) in the edges of the unit cell: Y is green, O red and Al blue and the identified voids are the vertices of the red Voronoi-Dirichlet polyhedra. Due to the geometry of the structure, the Al site is not recognized directly but “split” up into three different vertices (small blue circles). The hopping site is also split into two void sites (grey). The second dissonance can be illustrated at the example of Al(PO4) (ICSD #97551), which shows large pores. VDP identifies and usually interprets those as beneficial. Nevertheless, from an energetic point of view, large voids and pores are not favorable. The hopping site between the structure-immanent Al-positions is identified by VDP as in the middle of a large pore. In contrast, BVSE shows the migration to occur on the inner surface of the pore (see Figure 5). As BVSE is determined by a Morse-type potential, energy does not increase linearly with the distance of two atoms. Therefore, the optimized conduction path is not found in the middle of the pore (all distances are very high) but close to the 15



pore-constituting atoms.

Figure 5: Void network (grey) as identified by VDP and the BVSE (orange) of Al(PO4) (ICSD #97551): PO4 is green, Al blue. Both conduction networks are 3D but do not completely coincide. VDP recognizes the large intrastructural void but BVSE suggests a movement along the inner surface of the pore. In order to compare VDP and BVSE methodologies based on the numeric values generated, the void sizes identified through VDP are compared with their respective bond valence site energies. For this purpose, the voids of all most-promising compounds are calculated with ToposPro. All voids with cationic coordination partners of solid angles larger than 5 % are deleted, since the resulting interaction between such close structure-immanent cations and Al3+ would give a high energy barrier. For all the remaining voids, their respective BVSEs are determined from the calculations. The radius of the spherical domain rSD is a simplification of the three-dimensional volume of a VDP. It translates a 3D into a 1D value that can be interpreted as a soft sphere mean distance of the central atom to its coordinated atoms. On the other hand, the bond valence energy on a given site is the summation of the different contributions of the coordination shell atoms. In order to compare these two values, the site energy is regarded similar to the q volume of the VDP and a scalar value called Θ = 3 3E4πsite is defined as an analogue to rSD . 16


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These two values describe a generalized distance and energy between the central atom and its “merged” neighboring atoms. The plot of these values looks similar to the plot of the interatomic distance and BVSEs and thus similar to a Morse potential (Figure 8a). For the fitting, a Morse-like function is used from Ref. [ 31 ], exchanging the variables taken within: 0

E = D0

h

i 2 eα(rSD,min −rSD ) − 1 − 1 .

(1)

Here, the hard-sphere distance R is exchanged for the soft-sphere rSD . For the rSD,min of minimum energy, the data-mined value is taken, while D0 is determined as the energy value at around rSD,min . A value between 2 and 2.25 for α describes the data very well. In comparison to the data accumulated by S. Adams, the values for R, D0 and α are smaller. This can be attributed to the difference of hard and soft spheres, the difficult assignment 0

of coordination numbers and a smaller dataset. The shift in dissociation energy D0 can be attributed to the lacking possibility to calculate contributions of asymmetric coordinations and lacking Coulomb interactions in VDP. For this analysis, the data is split depending on the oxygen coordination numbers of the voids, where again four-, five-, and six-fold coordination is mainly observed. There are some compounds with even higher values. They do not allow for a statistically reliable analysis, though. The data can be interpreted with a baseline: as already pointed out, asymmetric coordination is not explained well through rSD values alone. The more asymmetric and the higher the influence of neighboring cations, the higher should be the energy on a position with the same rSD . Anyway, a minimum can be seen in the data that corresponds very well with the data-mined rSD values from the ICSD (Figure 8a). Furthermore, neglecting the coordination numbers, the data can still be reasonably described using the fit for coordination number four only. The data is majorly influenced by one compound (Al5Ge1O9.5 with 55 data points). Anyway, a similar trend between its rSD and BVSE-values is observable compared to the

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data as a whole. This means that the rSD value could be used as a (soft) measure for the energy of an ion put into a void inside of a compound. As opposed to how the method was used up to now, an approach similar to BV sum mismatch could be applied and all voids larger/smaller than the data-mined value of rSD by a certain factor would be a much better approximation to evaluate potential ion mobility. Nevertheless, the compounds should be checked afterward either with other geometric parameters (e.g. G3 and ratio of circumscribed and inscribed sphere of the VDP) or (semi-) energetic methods like BVSE and DFT.

Comparison BVSE-DFT Slight deviations of DFT and BVSE pathways are expected since BVSE neither takes relaxational effects nor repulsion between the mobile species and the mobile ions and immobile ions (if it is the same species and not part of the pathway) into account. In general, BVSE migration barriers are thought to be higher than the experimental value since such atomic rearrangements would release local stress, which lowers the total energy. 32 For monovalent ions there is enough data to compare and it appears there is a good correlation between experimental and BVSE activation energies. An extensive study, which gives an exact formula for the correlation, is waiting to be published soon. 62 Nevertheless, for Li+, 24 Na+, 34,37 and recently Mg2+ 38 the strong correlation of BVSE and DFT barriers and pathways has already been demonstrated, which is a more suitable comparison, as experimental data might also contain contributions from e.g. grain boundaries and other real structure imperfections of the crystal (defects, texture etc.). While for Li there is a substantial overestimation of the migration barrier height relative to DFT 24 (more than twice), in the case of Mg the correlation is close to 1:1. 38 Thus, it appears that the correlation is dependent on the investigated ion, but may also be influenced by the computational details (BVSE correlation factor, DFT functional and basis). For instance, the program utilized here uses fixed screening factors, while in 38 they are adapted for each structure. Furthermore, it should be kept in mind that the BVSE method’s inability to include specific three- and many-particle interactions as well as

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the anisotropy of electronic charge distributions and charge transfer during migration could cause additional errors in the barrier estimation. A direct comparison between the pathways obtained by BVSE and DFT is in general hampered due to the at least slightly different unit cell and atomic positions during DFT relaxation. For the sake of spatial and quantitative comparability, the fractional atomic coordinates of the DFT intermediate steps were treated as if the underlying supercell would be simply a multiple of the original experimentally determined cell used for BVSE calculations. In accordance with the cited preceding investigations, in this work there is a close match of the pathways derived from both approaches in terms of the route, interstitial sites and bottle neck position for O2– and Al3+ for AlFe2O4 (Figure 6) and YAlO3 (Figure 7). Only for the Al pathway in YAlO3 a rather strong spatial discrepancy of the obtained bottle neck position can be found. This is explained by the significant shift of the framework atom positions upon aluminum migration (Figure 7), which is not surprising for a 2D material. In the case of AlVO3, the spatial comparison is not feasible, since the relaxed structure differs significantly from the published one used for BVSE calculations. This is due to the fact that the measured positions are only an average of the actual highly disordered local structure with partial occupancies. This is why BVSE calculations have been performed for the DFT-relaxed cell as well. As opposed to most results in literature, in this work, all DFT derived barriers are higher than the ones from BVSE, which is consistent with our previous work on AIB cathode materials. 4 It is stated in the literature that, normally, ab initio models would lead to a slight underestimation of the energy barrier due to an overestimation of unit cell volumes, 63 while BVSE analysis provides higher barriers as discussed before. However, the cell volumes of the relaxed structures employed for DFT analysis of the pathways are smaller than the experimentally measured ones (Table S3). Since the latter have been used for BVSE, this fact appears to be a potential reason for the inversed correlation to DFT values. Indeed, BVSE Al Al calculations for the relaxed DFT-super-cell of AlVO3 satisfy EBV > EDFT (2.8 eV vs. 1.0 eV),

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Figure 6: (a) Spatial comparison of the BVSE (orange) and DFT path (light blue) in the AlFe2O4 structure (former ICSD #76977): Dark blue denotes the Al position, green is Fe and red O. (b) Migration energy barrier at different positions in the path derived from DFT and the respective bond valence energies calculated for the respective site. however, this is not the case for YAlO3. Eventually, only the comparison with experimental values can show if there is indeed an overestimation by DFT or an underestimation by BVSE. As this inverse behavior is also true for the oxygen migration calculations, the opposed behavior is not plausible to emerge solely from the three valences of Al and/or specific BVSE parameters connected with it. Rather, energy contributions from many-body effects as well as neglected Al–Al repulsion, which are considered in DFT calculations, may cause this deviation. Due to the high charge of Al3+, the influence of Coulomb interactions between mobile ions (as well as mobile and structure immanent Al) are expected to be higher as in the so far predominantly studied monovalent ion conductors. Indeed, also in the study of Mg oxides 38 higher BV energies for interstitial sites and even migration barriers appeared for some compounds, which has been ascribed to the disregarded Mg–Mg repulsion. In line with the previous studies on mono- and divalent battery materials, there is a strong correlation of the migration barriers derived from BVSE and DFT for Al3+ (Figure 8b). To add more data points, the BVSE results from our previous work 4 and DFT calculations cited therein are included in the graph. A roughly linear relationship appears plausible, presumably with different slopes for O2– and Al3+. Solely the pairs of values for

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Figure 7: Spatial comparison of the BVSE (orange) and DFT (light blue) migration path in YAlO3 (ICSD #27100): Dark blue denotes the Al position, green is Y and red O. The lighter-colored atoms in (b) denote the atom positions of the relaxed DFT structure during migration when Al is in the bottle neck position. compounds with presumably one-dimensional pathways deviate significantly. Again, the migration barrier estimation for low-dimensional pathways is thought to be much less accurate, due to expectedly more pronounced relaxation in comparison to 3D frameworks.

Limitations of the filter Despite the high number of screened compounds, we only found one promising candidate for a cathode material among Al-containing oxides. The question is whether the filter is too rough, or, if there are indeed only few suitable compounds within this material group, as has been suggested by the ab initio study by Rong et al. 44 Thus, in the following section the limits of the applied methods are discussed. VDP works most reliably on compounds with only one anion present, thus excluding multiple-anion-compounds like oxyfluorides. The geometrical routines perform best on compounds of low complexity, showing high symmetry and no large intrastructural voids. As already mentioned, the biggest problem of the methodology is vortex clustering. Due to how the methodology works, these clusters are interpreted to be complex and many-step diffu-

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Figure 8: (a) Plot of rSD (from VDP) vs. Θ (from BVSE, energy value analogue to rSD ) of void sites from the resulting compounds of the screening. The trend between single coordination numbers can be described according to a Morse-type potential (lines). (b) The migration barriers for Al3+ and O2– calculated by BVSE vs. DFT in this work as well as taken from the literature (Ref. [ 4 ] and Refs. cited therein [ 44,64,65 ]). The solid line represents a linear fit to the data, which serves as guide to the eyes to compare to the dotted perfect 1:1 correlation (corresponding to EBV =EDFT ). sion paths. They should be described rather as either single step paths (merging vertices to their centers of gravity) or bend paths, though. This might be seen as a disadvantage of the methodology, restricting its applicability. Anyway, we expect those compounds of high symmetry and unambiguous “simple” 3D diffusion paths to show the most promising ionic conductors since they should feature flat energy profiles for the conduction ion through the structure. 4,29,45 In contrast to the findings upon the analysis of claimed AIB cathode materials, 4 which do not contain Al in the first place, suitable structures identified here do not exhibit slightly distorted but highly symmetric octahedral or tetrahedral sites. On the other hand, the additionally found supporting structural motifs (3D connected pathways surrounded by anions building face-sharing commonly octa- or tetrahedral environments) do match with our screening results. Furthermore, the screening might have contributed to an overrepresentation of the highly symmetric cubic structures for another reason: both VDP and BVSE do not include local distortion of the lattice during migration. As mentioned before, this is thought to lead to the calculation of more unfavorable specific values and higher activation energies, respectively. 22


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However, this should influence the calculated migration barrier for layered materials more, since 2D conductivity commonly involves a widening of the conduction layer, while structures showing 3D ionic transport are generally more rigid. In consequence, the overestimation of energy barriers for 2D conductivity should be more pronounced compared to 3D. Adams and Rao estimated a scaling factor of ca. 0.4 for layered structures and 0.8 for dense frameworks, however, without providing supporting data. 32 This is why our approach might inherently sort out some two-dimensional ion conductors or cathode materials that could have been of interest. Moreover, it has been shown e.g. for an excellent Li+ conductor that not only distortion but also rotational disorder in the immobile sublattice can promote fast ion transport. 66 Such effects cannot be taken into account with our screening methods, yet. Likewise, for DFT calculations of the migration barrier a simple vacancy/interstitial mechanism is considered. More complex ion movements, involving simultaneous hops or even different ionic species as seen for Sc2 (WO4 )3 , 14 cannot be predicted. Another concern, especially for spinel materials, are cation antisite defects, which is beyond the scope of this work, but will be the subject of future research. It is known for layered structures that cation disorder can lower the mobility by decreasing the interlayer distance. 67 However, 3D diffusion networks are expected to be less impacted by cation disorder than layered structures, since it represents a rather rigid framework. Moreover, due to the high vacancy density in the AlVO3 pathway, antisite disorder (inversion) should not be able to fully block Al-ion transport. Nevertheless, inversion can significantly shape the energetic landscape of the migration pathway: As has been already reported for Li- 68 and Mg 69 -spinels, the migration barriers can be lowered and increased with respect to the specific local configuration induced by inversion. The degree of inversion will determine the percolation of ion pathways through the crystal structure. 69 Furthermore, as AlVO3 is a potential cathode material, further studies will need to shed light on the influence of Al extraction during cycling on the percolation behavior and stability, since this can significantly influence the amount of Al that can be extracted and thus the reachable capacity.

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Conclusion and Outlook To find new electrode and electrolyte materials from scratch, we created a three-step computational filter, combining simulation methods with different accuracy: VDP, BVSE and DFT. As a first step we applied it to all Al-containing oxidic compounds in the ICSD. VDP identified mainly cubic compounds showing 3D ionic pathways, with close to ideal four- and six-fold coordination of Al. The ionic migration networks found with VDP and BVSE coincide well in most cases. However, BVSE may identify more complex pathways and can provide a rough Emig estimation. Among the candidates identified by VDP, BVSE sorted out the garnet type structures, while some spinels appeared promising. DFT calculations consistently yielded a higher Emig for Al3+, which is opposed to the reported behavior for monovalent ions. This is assumed to be due to the high impact of strong repulsive Al–Al Coulomb interactions that come with the high valence, which cannot be considered with VDP and BVSE. Nevertheless, the comparison of bond valence and ab initio results showed that BVSE yields a similar trend in Emig and can thus be used as rather quick screening filter. The quality of the semi-quantitative prediction, however, strongly depends on the dimensionality of the ionic pathway of the investigated structure. The filter is thought to may oversee some 2D compounds, mainly due to neglected structural rearrangements. On the other hand, the promising structural motifs from our previous work 4 have been confirmed and AlVO3 has been identified as promising cathode material. The confirmation of the reliability of the filter is an important step to find a significant amount of potential candidates in further studies: we currently consider anions that are more easily polarized (S, Se), as this may result in a weaker bonding of Al3+ to the skeleton framework and, thus, potentially higher mobility. Furthermore, the search radius has to be expanded by also taking into account crystals that do not contain Al in the first place. Once VDP and BVSE have found sites and pathways for aluminum, DFT calculations will examine whether the promising Al-inserted compounds would exist and how Al insertion influences the crystal structure and thus pathways. Furthermore, as only crystalline structures have 24


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been examined in this manuscript, having a look at glasses seems to be promising, too. However, the lack of an extensive database on amorphous compounds hinders a wide search. Having an idea about reasonable building blocks and prelaminary Molecular Dynamic or Monte Carlo calculations seems to be the only way to find possible amorphous candidates, which would cause long calculation times. The effort seems worth it, still, since amorphous compounds can be produced without high-temperature steps and are thus more attractive to industry, as e.g. LIPON. Supporting Information. Theoretical background for VDP and DFT; computational details for all employed methods; graphical representation for VDP processing; cell parameters for relaxed cells used for the NEB-calculations; and additional low-energy Al-pathways for AlVO3

Acknowledgement The authors thank Stefan Adams for a discussion on BVSE results, Andrey Golov (SCTMS) for helpful advises concerning the identification of Al paths, and Mateo de Vivanco for preparing the Sankey diagrams. AK, FM, and TL thank Vladislav A. Blatov for providing infrastructure for the calculations and for discussions. TN, FM, MZ and TL are grateful for financial support of the Federal Ministry of Education and Research (CryPhysConcept (03EK3029A) and R2RBattery (03SF0542A)). AK thanks the Russian Ministry of Science and Education for support with grant No. 3.7626.2017/9.10.

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Combined Theoretical Approach for Identifying Battery Materials: Al3+

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