The electrochemical thermodynamic and kinetic characteristics of rechargeable batteries are critically influenced by the ordering of mobile ions in electrodes or solid electrolytes. However, because of the experimental difficulty of capturing the lighter migration ion coupled with the theoretical limitation of searching for ordered phases in a constrained cell, predicting stable ordered phases involving cell transformations or at extremely dilute concentrations remains challenging. Here, a group-subgroup transformation method based on lattice transformation and Wyckoff-position splitting is employed to predict the ordered ground states. We reproduce the previously reported Li_{0.}_{75}CoO_{2}, Li_{0.}_{8333}CoO_{2}, and Li_{0.8571}CoO_{2} phases and report a new Li_{0.875}CoO_{2} ground state. Taking the advantage of Wyckoff-position splitting in reducing the number of configurations, we identify the stablest Li_{0.0625}C_{6} dilute phase in Li-ion intercalated graphite. We also resolve the Li/La/vacancy ordering in Li_{3x}La_{2/3−x}TiO_{3} (0 <

corrected publication 2022

The ever-growing demands for electrical energy storage have led to the higher performance requirements for rechargeable batteries^{1–5}. Tremendous efforts have been devoted to the study of (i) high voltage or capacity cathodes (e.g., Li[Ni_{1−y−z}Mn_{y}Co_{z}]O_{2}, NMC compounds)^{6,7}, (ii) solid electrolytes (e.g., garnet, perovskite, and NASICON family)^{8–15}, and (iii) post-lithium battery chemistries (e.g., Na, K, and Mg batteries)^{16–22}. A key commonality of the above electrolytes and electrodes is that their properties (e.g., ionic conductivity for electrolytes, phase stability, and voltage for electrodes) are closely linked to the concentrations of mobile ions and the corresponding ordered ground states during either the preparation or ion-intercalation process. Unfortunately, it is difficult to determine the ordered ground states in these systems because of the low sensitivity of current spectroscopic techniques to the light elements (e.g., H and Li)^{13}. For example, even neutron scattering cannot detect the precise occupation of Li^{+} in Li-containing compounds, only giving a “disordered” distribution of these ions in an averaged manner. Thus, it is difficult to directly obtain the precise arrangements of mobile ions at the atomic scale. For example, the exact ordered ground states of the lithium graphite intercalation compounds (LGICs), a commercially successful graphite anode, at their dilute limit (viz., Li_{x}C_{6} with 0 < _{3x}La_{2/3−x}TiO_{3} (LLTO, 0 < ^{23,24}. Therefore, it is critical to resolving the ordered ground states of the electrodes or solid electrolytes during the operation with the help of foresighted calculations.

In this context, the theoretical lattice-gas model (LGM)^{25,26} is widely used to determine the Li^{+} occupation ordering in guest-host intercalation electrochemical systems, where the host is an ordered-sites containing network and each site can be occupied by a guest or vacancy. Various configurations associated with rechargeable battery chemistries can be obtained by enumerating possible mobile ion/vacancy arrangements on the given supercell, which is produced by replication of the unit cell with a specified integer number^{27–29}. To overcome the enormous configurational space challenge or avoid the large-scale first-principles calculations, several so-called “parameter-constructing” methods, including sampling-based Metropolis Monte Carlo and fitting effective-cluster-interaction-based cluster expansion, etc., have been developed^{30–32}. The most representative method is the cluster expansion (CE). It sets up a mathematical framework where the energy is expanded as a series of cluster basis functions that can be multiplied by effective cluster interactions (ECIs). However, prediction errors will inevitably occur when a large lattice mismatch among configurations and small training sets of configurations are encountered^{33}.

The problem of the ordered arrangements of alkali-ion/vacancy in alkali-ion batteries mentioned above can be treated as a group−subgroup transformation (viz., reducing symmetry from the parent structure) since all possible arrangements can be classified into one of the subgroups. Different from supercells which are determined ad hoc, structures obtained by group−subgroup transformation have their lattices which are determined by the transformation matrix. Thus, all possible supercells can be included. Moreover, unlike randomizing or enumerating the arrangements of alkali-ion/vacancy, group−subgroup transformation assigns alkali-ion/vacancy into distinct subsets of Wyckoff positions in each subgroup to avoid enumeration^{34–36}. Therefore, ordered phases within diverse supercells and dilute concentrations can be formulated rigorously.

Previously, this method has been employed to find the relationship among the high symmetric original structures in existing ordered phases^{34–36}. In this context, we describe a prediction framework based on such a group−subgroup transformation for generating possible ordered phases of electrolyte/electrodes with variable concentrations of mobile ions during the typical preparation/ion-intercalation process. High-precision first-principles formation energies are further employed to determine the ordered ground states. Potential variation, e.g., stacking of the host lattice, is also included for comparison. By searching through a comprehensive range of supercells, we identify the commonly accepted ordered ground states of Li_{x}CoO_{2} and propose Li_{0.875}CoO_{2} as a new ordered ground state at _{0.0625}C_{6} phase can be identified. By extending our group−subgroup transformation method to uncover the joint ordering of Li/vacancies with immobile La in the solid electrolyte Li_{3x}La_{2/3−x}TiO_{3}, we reveal that the Li-ion diffusion anisotropy is caused by the blocking effect of La ions.

Because the ordered ground states in Li_{x}CoO_{2} determine the 0 K equilibrium voltage, understanding the ordering of Li/vacancy in Li-de-intercalated phases is important for tailoring this material to the specific electrochemical application. In the exploration of Li_{x}CoO_{2} ordered ground states during the charging/discharging process, the first and most important step is to determine the possible configurations. To cover as many configurations as possible, the most common method is to search within several ad hoc supercells. However, the results may conflict with each other when different supercells are used. For example, Van der Ven et al. concluded that Li_{0.8333}CoO_{2} is one of the ordered ground states predicted by the cluster-expansion method^{37}, whereas Wolverton et al. suggested that Li_{0.8571}CoO_{2} is the ordered ground state^{38}. This discrepancy might stem from the use of different sets of supercells. Herein, using the group–subgroup transformation method, the ordering of Li/vacancy in Li_{x}CoO_{2} is studied by searching for configurations with different Li concentrations via enumerating different sizes of supercells and obtaining ordered phases from transformed lattices, thus avoiding possibly missing stable configurations. In principle, the group–subgroup transformation method should enable us to obtain diverse supercells based on the symmetry reduction and lattice transformation for Li_{x}CoO_{2}, where different Li concentrations are examined.

It is worth noting that the rearrangement of the LiCoO_{2} host structure occurs by altering the stacking sequences of oxygen during charge and discharge. The only difference between the original and rearranged structures is how the O–Co–O slabs of LiCoO_{2} relate to each other across the Li planes. Because ordered phases are determined within a specific host structure, we must consider these different host structures as an additional variable in addition to the Li concentrations. Experimentally, three hosts have been confirmed. The first is O3^{39}, which has ABC oxygen stacking while the second is O1 and has an ABAB oxygen stacking^{40}. The stability of the O1 host is demonstrated to be restricted to zero Li concentration. The third is referred to as ^{41,42}, which features the characteristics of both O3 and O1. Li is assumed to prefer the octahedral sites of O3 to those of O1. Thus, the maximum Li concentration that can be obtained in this host with _{x}CoO_{2}. Supplementary Table ^{35}. Finally, formation energies of 377 nonidentical configurations are used to determine the ordered ground states.

As illustrated in Fig. _{2} host is more stable with lower formation energy (60 meV per f.u.) than that of the O3 host at zero Li concentration. This result is in good agreement with the previously reported values of 40^{37} and 50 meV^{38}. This is why in electrochemical experiments, it is difficult to obtain single O3−CoO_{2} with 0 Li in the structure. Several ground states at various Li concentrations are identified, as indicated by the convex hull formed by seven stable phases at

_{f}) of Li_{x}CoO_{2}. Ordered ground states at

To ensure the validity of the group–subgroup transformation method, we present the ordering pattern of Li/vacancies in Li_{0.5}CoO_{2}, which has in-plane 2 × 1 ordering with the Li-ions arranged in rows and separated by rows of vacancies (Supplementary Table ^{43} and was later confirmed by Shao−Horn based on electron diffraction experiments^{35}. Van der Ven et al.^{37} also obtained this arrangement using the cluster-expansion method and confirmed this ordering. In our work, this arrangement is obtained by subgroup _{1}^{35,37,38}. Numerous arrangements within different supercells are compared, and they all indicate that

It is worth noting that at ^{37,38}. This inconsistency results from the differences in the sizes of ad hoc supercells. Here, ordered phases in different supercells with sizes up to eight times of the primitive cell are conducted for the systematic investigation. The ordered phases at _{0.8333}CoO_{2} and Li_{0.875}CoO_{2}. The energy of Li_{0.8571}CoO_{2} slightly leaves the tie line between

Figure _{2} to the Li_{0.875}CoO_{2} within a _{2} with a _{2} and Li_{0.875}CoO_{2} indicates that the phase transition occurs after the extraction of Li at 3

_{2} original structure.

We also investigate the ordered ground state at ^{34}. By considering different supercells, both orderings are obtained by the group–subgroup transformation. Our calculation shows that the 2 × 2 ordering phase is less stable with an energy of 20 meV per f.u. higher than that of the most stable one. Thus, the 2 × 4 ordering is the ground state.

Figure _{2}, we obtained ^{37,41}.

_{2} (_{6} polyhedron.

The intercalation process of lithium in the layered materials often results in the formation of “stages”, such as LGIC. These stages describe the 2D stacking sequence of the lithium layers between the graphene layers, e.g., stage _{4}, KC_{8}, etc. using XRD and electrochemical techniques^{44,45}. However, detailed structural information on stages with different Li concentrations from both experimental and theoretical studies is still scarce, leading to a non-unified description of such stages, especially for extremely dilute Li concentrations^{46–48}. For example, the precise arrangement of atoms in each stage has rarely been reported either computationally or experimentally.

To evaluate the thermodynamic stability of the LGIC with various concentrations and stages, sequences with AA and AB stacking have been considered because of the relative gliding during charge and discharge processes^{49}. For the former, graphene layers have a stacking order of –A–A–A–A–, while another stacking pattern is formed by shifting to a zigzag shape (–A–B–A–B–) (Supplementary Table _{x}C_{6} (0 <

Firstly, we use group–subgroup transformation to search for possible configurations in the Li_{x}C_{6} (0 < _{6} with a stage of I. The composition-induced stages occur at well-defined

_{f}) of Li_{x}C_{6} as a function of Li composition. Ordered ground states determined by the convex hull at

Of these ordered ground states, the experimentally and computationally reported Li_{0.5}C_{6} compound is well known for its stage II ordering^{50}. Using the group–subgroup transformation method, we can compare the arrangements in several different supercells nearing the convex hull, such as ^{47,51}.

In the dilute concentrations, because research on LGIC leads to a large number of configurations, brute force computation is not realistic. In previous research, the most stable dilute phase with AB stacking was observed at Li_{0.0833}C_{6}, which has a stage of IV within a ^{49}. This phase is verified by our group–subgroup transformation when the 1^{4} configurations within the dilute region (Supplementary Fig. _{0.0833}C_{6}, we identify that a more dilute phase occurs at Li_{0.0625}C_{6} with a stage of IV.

_{6}_{0.0833}C_{6} and Li_{0.1667}C_{6} are obtained by _{0.0625}C_{6} is obtained by 2

The ordered ground state of Li_{0.0625}C_{6} has a subgroup of _{0.0625}C_{6} and Li_{0.0833}C_{6}, we calculate the charge−density differences using the 2 × 1 × 1 supercell (Supplementary Fig. _{0.0625}C_{6}, which suggests that the Li-atom interaction disappears in Li_{0.0625}C_{6}. These findings confirm that the Li_{0.0625}C_{6} with extremely low Li concentration is the final extremely dilute concentration phase during the charging/discharging process.

We also note that there is non-consensus on the ground state at a Li concentration of ^{52,53}. Using group–subgroup transformation, possible arrangements under different stages for Li_{0.3333}C_{6} are determined within different supercells as shown in Fig. _{6}, respectively, indicating that the stage II phase of the Li_{0.3333}C_{6} compound is more stable. This computational observation is consistent with the experimental results of Yazami et al.^{54}. When comparing with the stage II structure of Li_{0.5}C_{6}, the Li layers of stage II are not fully occupied. Similar conflicts also appear at Li_{0.1667}C_{6}. This phase has the same subgroup as Li_{0.0833}C_{6}, with another 1^{49}, stage IV has lower formation energy of 20 meV per C_{6}., which indicates that a higher stage is preferred in dilute Li concentrations.

Subgroup structures and Li/vacancy arrangements of stage II, stage III, and stage II−IV at Li_{0.3333}C_{6}.

Solid electrolytes have been widely studied because they are applicable in energy-dense solid-state batteries and other electrochemical devices. The elementary process of ionic transport is known to be strongly affected by the distribution of local ordering, i.e., the Li/vacancy ordering for Li-stuffed garnets^{13} and the more complicated coupled ordering of Li/vacancies with immobile La in Li_{3x}La_{2/3−x}TiO_{3}. In the latter case, the diffusion pathways are strongly affected by the La ordering^{55–57}. In previous work, to elucidate the effect of La on Li diffusion, the initial Li/La/vacancy ordering has been specified ad hoc^{55,58}. To address the La ordering rigorously, an effective guiding tool is needed. Inspired by the effectiveness in predicting ordered phases within transformed lattices, the group–subgroup method is naturally employed to determine the blocking effect of La on the Li diffusion pathway. Here, the starting structures are generated by simultaneously assigning Li/La/vacancy to perovskite A-cages so that configurations with different dimensionalities of the Li^{+} diffusion pathway are included (see an example in Supplementary Fig.

It is worth mentioning that the La order is accompanied by a change of symmetry in previous studies. For example, Li-poor compositions exhibit orthorhombic symmetry with high La-site occupancy in the La-rich layer, whereas the Li-rich composition has tetragonal symmetry, and the occupancy of La becomes homogeneous^{59–61}. However, recent studies have shown that the symmetry with Li-rich composition is also orthorhombic with the ^{58,62,63}. In order to understand the relationship between the La ordering and symmetry, configurations at 3

As illustrated in Fig. ^{63,64}. Based on these ordered ground states (Supplementary Table ^{62}.

Once the ordered ground states are determined, an analysis of the blocking effect of La on the migration pathway is performed. To further reveal the Li diffusion behavior in the Li_{3x}La_{2/3−x}TiO_{3} systems, we calculate the energy barriers using the bond valence site energy (BVSE, see in ^{65,66}. At low Li concentrations of 3^{67,68}. This is caused by La ordering along the ^{69}. As discussed above, lithium-ion diffusion for all the Li_{3x}La_{2/3−x}TiO_{3} systems is strongly anisotropic owing to the blocking effect of La. To eliminate the effect of diffusion anisotropy, it is useful to introduce another fast ion conductor (such as lithium silicate) into the grain boundary^{69}.

BVSE energy barriers (eV) of Li_{3x}La_{2/3−x}TiO_{3} along with different directions.

0.125 | 0.234 | 0.830 | 9.268 |

0.2 | 0.205 | 0.840 | 8.965 |

0.25 | 9.102 | 0.840 | 0.215 |

0.3125 | 9.277 | 0.293 | 0.205 |

0.35 | 9.0527 | 0.293 | 0.293 |

3

In this work, we present that ordered ground states formed either during the preparation or the ion-intercalation process in several rechargeable battery materials, especially with transformed lattices or dilute alkali-ion concentrations, could be predicted using group–subgroup transformation. In LiCoO_{2}, we solve the ordering problem for different sizes and shapes of supercells, including confirming the ordered ground states at _{x}CoO_{2} at _{0.875}CoO_{2} ground state is identified at _{6}, utilizing the Wyckoff splitting rule, we identify the new stable Li_{0.0625}C_{6} with a stage IV structure as the most dilute phase, which has not been previously demonstrated in experimental and computational studies. Besides, this method also reveals the blocking effect of La on the diffusion anisotropy of Li. This method has also been successfully applied in the prediction of ordered phases in K_{x}Mn_{7/9}Ti_{2/9}O_{2}^{70}. Moreover, partial replacement, such as substituting Co^{3+} with Ni^{2+} and Mn^{4+} in Li(Ni_{x}Mn_{y}Co_{1−x−y})O_{2} layered oxides (coined NMC) would be an interesting topic for further investigation. It is worth mentioning that in the cases where framework ions can leave their original site and migrate to the site of mobile ions, such as the transition of LiCoO_{2} into cubic Li_{0.5}CoO_{2}, the phase transition is more complicated and thus not discussed here. In addition, constraints on crystal systems and

In addition to atomic ordering, this method is also valuable for other ordering problems such as systems with magnetism, which is closely related to the ordering of charges. It is also potentially applicable to materials such as light-emitting Cs_{2}In^{I}In^{III}Cl_{6} with the charge ordering of In^{I}/In^{III} and pristine Cr_{2}Ge_{2}Te_{6} (two-dimensional van der Waals material) with ferromagnetic ordering^{71,72}. Identifying the charge ordering, ferroelectricity, or magnetic ordering and their evolutions would be comparatively difficult but desired. Thus, in addition to rechargeable batteries, we believe the group–subgroup transformation can also be applied to other areas such as ferroelectrics and other charge-ordering-related materials.

Group–subgroup transformation starts from a space group of a highly symmetrical parent structure (_{1} > _{2} > ... >

_{x}C_{6}).

For each step, the transformation matrix that determines new lattices of the subgroup is compiled in the International Table of Crystallography, and can also be accessed by the Bilbao Crystallographic Server^{73,74}. Finally, ^{75}.

In addition, the transformation of the lattice may occur because symmetry operations in the subgroup inevitably change. In each of the transformation matrix–column pairs (^{74}. The column

In the parent structure with high symmetry, atoms are symmetrically equivalent if they share the same Wyckoff position under the manipulation of symmetry. However, in the subgroup, because of the reduction of symmetry, these atoms may become non-equivalent. This allows the high-symmetry Wyckoff position of alkali-ion which corresponds to the high-symmetry group of the prototype to split into different sets of positions in the subgroup (Fig. ^{74}.

After configurations are obtained by assigning alkali-ion/vacancy to the independent sets of positions, the StructureMatcher utility in Pymatgen is employed to exclude identical arrangements^{76}. It compares two structures by reducing them to primitive cells and evaluates whether the maximum root means square displacement is less than a predefined tolerance cutoff. This method can effectively distinguish the nonidentical structures and has proven useful in many previous works^{77,78}. First-principles calculations are then performed to obtain the formation energies and determine the ordered ground states (for calculation details, see ^{29,79}. Taking the Li_{x}CoO_{2} compound as an example, the formation energy of a given Li/vacancy configuration with content _{x}CoO_{2} is defined as_{x}CoO_{2}) is the total energy of the configuration per Li_{x}CoO_{2} formula unit, and _{2}) and _{2}) are the energies of LiCoO_{2} and CoO_{2} in the O3 host, respectively. The formation energy reflects the relative stability of that structure concerning phase separation into a fraction _{2} and a fraction (1 – _{2}.

Procedures of group–subgroup transformation and bond valence site energy calculations are implemented in the high-throughput computational platform for battery materials^{80}.

This work is supported by the National Natural Science Foundation of China (Nos. 11874254, 51622207) and the National Key Research and Development Program of China (No. 2017YFB0701600). All the computations are performed on the high-performance computing platform provided by the High-Performance Computing Center of Shanghai University.

Y.R. prepares the manuscript and analyzes the data. D.W., Z.Z., B.P., and Z.L. help to perform the analysis with constructive discussions. B.L., Y.R., and W. S perform the first-principles calculations and check the data. P.M. and B.H. develop the BVSE tool. Y.L., Z.L., X.L., and B.H.L. help to revise the manuscript. S.S. is the leader in this manuscript and contributes to the conception of the study. All authors participate in discussing the results and comments on the manuscript.

The authors declare that the main data (3779 and 5811 ordered phases, obtained by group−subgroup transformation, during ion-intercalation/extraction processes of Li_{x}CoO_{2} (0 < _{x}C_{6} (0 <

All source codes of the method that are implemented in Python are uploaded to the repository:

The authors declare no competing interests.

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Supplementary Information

The online version contains supplementary material available at