Exploring battery cathode materials in the Li-Ni-O phase diagrams using structure prediction

The Li-Ni-O phase diagram contains several electrochemically active ternary phases. Many compositions and structures in this phase space can easily be altered by (electro-)chemical processes, yielding many more (meta-)stable structures with interesting properties. In this study, we use ab initio random structure searching (AIRSS) to accelerate materials discovery of the Li-Ni-O phase space. We demonstrate that AIRSS can efficiently explore structures (e.g. LiNiO2) displaying dynamic Jahn-Teller effects. A thermodynamically stable Li2Ni2O3 phase which reduces the thermodynamic stability window of LiNiO2 was discovered. AIRSS also encountered many dynamically stable structures close to the convex hull. Therefore, we confirm the presence of metastable Li-Ni-O phases by revealing their structures and properties. This work will allow Li-Ni-O phases to be more easily identified in future experiments and help to combat the challenges in synthesizing Li-Ni-O phases.


Introduction
Lithium-ion batteries (LIBs) are popular energy storage technologies to power portable electronics. While there exist several working principles for LIBs, the principle based on the intercalation chemistry is better understood and many intercalation LIBs have been commercialized [1,2]. The cathode components in those intercalation LIBs greatly dictate the energy densities and possess the largest occupancy of the raw materials cost (∼22%) [3,4]. To promote a wide-scale adoption of LIBs in other sectors (e.g. transportation) requires developing cathode materials with lower cost and higher energy densities.
The earliest studied oxide cathode is the layered LiCoO 2 [3]. With the rising prices of cobalt and the geopolitical issues with its supply, nickel became a popular ingredient for cathodes since it is cheaper and more abundant [3,5]. This led to the discoveries of ternary Li-Ni-O compounds such as LiNiO 2 and Li 2 NiO 3 which eliminate the use of Co (see figure 1) [5][6][7][8]. The compositions and structures of several Li-Ni-O phases have very specific growth conditions. A subtle change in the Li/Ni ratio and oxygen partial pressure can lead to different structures [9]. This is because: (1) many of those structures are related as they are based on the same oxygen cubic close-packed (ccp) sublattice [9]; (2) Ni 3+ and Ni 4+ are difficult to stabilize and can be easily converted to Ni 2+ (e.g. via disproportionations) [9][10][11][12][13]; (3) cation mixing becomes more favorable due to the similar ionic radii of Ni 2+ (0.69 Å) and Li + (0.76 Å) and the magnetic frustration/interactions [5,14,15]. The superexchange interaction between neighboring Ni cations that are bridged by O anions is stronger in a (linear) interplane fashion between the antisite Ni 2+ (upon Ni 2+ /Li + exchange) and Ni cations in the transition metal (TM) layer, compared to the 90 • intraplane superexchange between Ni in the TM layer. The interplane superexchange also relieves the magnetic frustration that arises from the intraplane interaction within the TM layer by providing different exchange paths [16]. Many more (meta)-stable structures could have been generated during (electro)-chemical processes. Some of them may exist in small quantities, which becomes difficult to isolate. However, there have been successful discoveries of new phases in this way. For example, the metastable P3m1 Li 2 NiO 2 (1T-Li 2 NiO 2 ) was discovered by electrochemical Li insertion from LiNiO 2 , charge compensated by Ni 2+ [11,17].
Another way to discover new structures is to use computational methods, where hundreds of structures can be evaluated in a shorter time and more cheaply compared to experiments [18]. AIRSS has become an increasingly popular method to access new structures [19][20][21][22][23][24]. It works by sampling the potential energy surface (PES) of various structural configurations in a random fashion. This method is highly parallel and there is a small number of easy-to-understand parameters to control the search efficiency. Chemical knowledge can be used to help the generation of sensible random structures, followed by geometry optimization using DFT calculations. This process is repeated until low energies structures are encountered multiple times. To reduce the computational cost of DFT calculations and structure search time, one can employ crystal symmetry to bias search towards lower local energy minima and/or selectively perform geometry optimizations with various convergence criteria [19].
In this work, we use AIRSS to explore structures in the Li-Ni-O phase diagram as we believe our chosen search space is enriched with many yet undiscovered (meta)-stable structures. To achieve this, we first re-explored the experimentally known Li-Ni-O systems to illustrate that AIRSS can effectively sample the PES and locate known structures. We then searched structures in 11 different hypothetical Li x Ni y O z stoichiometries and investigated the lowest energy structures by evaluating their thermodynamic and dynamical stabilities (see table 1). Among many (meta-)stable phases we identified, the layered Li 2 Ni 2 O 3 and Li 4 NiO 3 phases have comparable theoretical energy densities to LiNiO 2 , and they share some structural features with 1T-Li 2 NiO 2 . The newly discovered thermodynamically stable Li 2 Ni 2 O 3 phase is a limiting phase which reduces the thermodynamic stability window of LiNiO 2 .

Methods
Charge neutral Li x Ni y O z compositions were enumerated by the SMACT code [25], assuming there are no mix-valent Ni ions. Compositions were restricted to have a total number of atoms in the empirical (reduced) formula ⩽12 and the total number of atoms in the full formula ⩽28. This gives a total of 14 compositions to be searched, of which three compositions (LiNiO 2 , Li 2 NiO 3 and Li 2 NiO 2 ) have experimentally known structures. For the three experimentally known compositions, at least 2000 structures with number of formula units in the unit cell (controlled by the NFORM keyword) between 1 and 4 were generated and geometry optimized. For each of the 11 hypothetical compositions, around 500 structures (with NFORM ⩽2) were generated and optimized. Search with NFORM = 4 was also performed on Li 3 NiO 3 in search of  [19,20]. Sensible random structures were first generated with pre-assumed ferromagnetic (FM) spin states and then geometry optimized by the plane-wave DFT code CASTEP [26] (version 19.11). Spin-polarized calculations were performed using the PBEsol exchange-correlation XC functional [27] with Hubbard U correction (U eff = 6 eV) for Ni d electrons (i.e. PBEsol + U) [28,29]. The U value was taken from a previous study where it was fit to empirical oxidation energies [29,30]. Results are expected to be insensitive to the exact values of U used, thus the same qualitatively trend should be observed unless a widely different U is used. Geometry optimizations were initially performed with a loose convergence setting (i.e. crude search). Monkhorst-Pack [31] meshes of k-points with a spacing of 0.07 2π Å −1 was used with plane-wave cutoff energy of 340 eV. The on-the-fly generated (OTFG) ultrasoft pseudopotentials QC5 were used [32,33]. A subset of structures was then reoptimized using harder OTFG C19 pseudopotentials and with tighter convergence settings (700 eV energy cutoff and Monkhorst-Pack k-point spacing of 0.05 2π Å −1 ) to refine the structures and their energetics (i.e. refined search).
Further structural optimization and property calculations were performed with the VASP code [34-36] on a selection of CASTEP optimized refined structures using the PBEsol + U (U Ni = 6 eV) with a cutoff energy of 550 eV and a k-point spacing of 0.05 2π Å −1 on gamma-centered Monkhorst-Pack grids. A maximum force tolerance of 0.01 eV Å −1 was applied during geometry optimization. The PAW pseudopotentials [37,38] (version PBE 5.4) of Li_sv, Ni_pv and O were used unless otherwise stated. Lattice dynamics calculations were performed on the known phases in the Li-Ni-O phase diagrams and the lowest energy structure in each hypothetical composition with the ground magnetic ordering using the Phonopy package [39] with the finite-displacement method at an increased cutoff energy of 800 eV and a tighter force tolerance of 0.001 eV Å −1 during geometry optimization (table S1). Non-analytical term corrections [40] were applied on all oxides to treat the long range electrostatic interactions. The thermodynamic stability of a selection of structures were evaluated with other functionals (PBE + U [41], LDA + U [42], SCAN [43] and HSE06 [44,45]) using the VASP code and Hubbard U for Ni was kept at 6 eV. All structures were optimized in the same functional as that for the final static (property) calculations. The screening parameter ω = 0.207 Å −1 was consistently used for all HSE06 calculations. The energy cutoff for different functionals were specified in table S2. The PAW pseudopotentials (version LDA 5.4) of Li_sv, Ni_pv and O were used for the LDA + U calculations. For thermodynamically stable phases, the atomic chemical potentials ranges that a target phase is stable were determined using CPLAP [46,47]. The average voltages and voltage profiles were calculated using the FM ordering spin arrangements using the PBEsol + U functional, since the magnetic ordering generally gives small differences to the calculated voltages. More details on the voltage calculations can be found on the "Voltage calculations" section in the supplementary information document. The band gaps were obtained from the HSE06 calculations. The climbing image NEB method [48,49] was used to study Li-ion diffusion in Li 2 Ni 2 O 3 . A 4 × 4 × 2 supercell (224 atoms) was used to ensure that the periodic image of the migrating Li is at least 12 Å apart. Additional electrons have been added to supercells to avoid the complexity associated with charge localization upon incorporating the Li vacancies.
The ase [50], pymatgen [51] packages were used for manipulating structures and general analysis. Sumo [52,53] was used for plotting electronic and phonon band structures. The AiiDA framework [54,55] was used to manage calculation provenance. Crystal structure diagrams were produced using VESTA [56].

Re-exploring known systems
We initially performed searches on compositions with experimental structures. This ensures that sensible parameters were used to operate AIRSS effectively. We used the species pair-wise minimum separations, averaged volume per atom from known structures and symmetry to bias the search towards 'chemically sensible regions' enriched with low energy structures [22]. To validate the searches, we compare our searched structures with experimental structures that were optimized under identical settings. We also considered theoretical structures from materials project (MP) database [57,58] in the analysis, although it is unclear how these structures were generated [23].

LiNiO 2
The layered LiNiO 2 structure is based on the ccp layers of oxygen anion lattice, with Li and Ni occupying in the octahedral sites, and was initially identified as an isostructural compound to the R3m LiCoO 2 cathode [6]. The (low spin) t 2g 6 e g 1 configuration of Ni 3+ leads to pronounced Jahn-Teller (JT) distortions, resulting in larger volume changes during cycling compared to the layered LiCoO 2 [59]. Since it is demanding to capture the dynamic JT distortion experimentally, the layered LiNiO 2 structure has commonly been described as an average high symmetry R3m structure, with six identical Ni-O bond lengths [60]. First-principles studies identified a few distorted LiNiO 2 structures and several studies found the P2 1 /c structure with JT distorted octahedra arranged in a zigzag fashion as the ground state [6,[60][61][62]. The bond lengths of the distorted structure agree with the extended x-ray absorption fine structure data, which showed two sets of distinct bond lengths of 1.91 Å and 2.09 Å [63]. The R3m structure, on the other hand, is not dynamically stable and gives an erroneous electronic structure (i.e. showing a half-metal rather than a small band gap semiconductor) [62,[64][65][66]. Therefore, it is important to use the correct ground state structure in computational studies.
We performed searches using the minimum separations from the experimental average high symmetry R3m structure and the ground state P2 1 /c structure separately (see table S3) and notably both cases found the P2 1 /c as the ground state structure. The distribution of structures optimized from refined search is shown in figure 2. The energy ordering of the frequently mentioned LiNiO 2 structures relative to the ground state P2 1 /c structure follows: C2/m (+2.54 meV atom −1 ) < R3m (+28.17 meV atom −1 ), consistent with previous studies [60][61][62]64]. A low energy distorted spinel-like structure (+0.31 meV atom −1 ) was encountered by AIRSS (figure S1). This structure was also identified by a previous computation study and the parent undistorted spinel-like structure was seen in low-temperature synthesis of LiCoO 2 [62,67]. Many other higher energy structures were also encountered by AIRSS, and some have lower predicted energies than the theoretical structures from the MP database. Most of the AIRSS structures have Ni in the distorted octahedral coordination and Li in octahedral or prismatic coordination. Overall, this search shows that AIRSS is an effective tool for exploring compounds that exhibit dynamic JT distortion.

Li 2 NiO 3
Li 2 NiO 3 is a Li-excess system, where Ni is (theoretically) in the +4 oxidation state, with electrochemistry driven by anionic redox. It is isostructural to Li 2 MnO 3 , with a ccp oxygen sublattice and alternating layers of pure Li and [Ni 2/3 Li 1/3 ] [8,9,11,68]. The Li 2 NiO 3 structure has been previously assigned to space groups C2/m with 2 f.u. and C2/c with 4 f.u., respectively [9]. We relaxed both the C2/m and C2/c structures and they give identical energies, confirming that they are identical ground state structures (figure 3). The ground state structures can also be accessed by AIRSS. An unreported low energy dynamically stable polymorph (+1.10 meV atom −1 ) with a rock salt framework was also encountered (figure S2). AIRSS also found many other higher energy polymorphs and most of them are rock salts.

Li 2 NiO 2
Assuming 2e − (Ni 2+ /Ni 4+ ) redox, the Li 2 NiO 2 composition gives a high theoretical capacity (table 2). There are two Li 2 NiO 2 experimental structures (figure 4): (1) Immm Li 2 NiO 2 (I-Li 2 NiO 2 ), (2) P3m1 Li 2 NiO 2 (1T-Li 2 NiO 2 ). The I-Li 2 NiO 2 is isostructural to the orthorhombic Li 2 CuO 2 , and can be described as a structure with chains of edge-shared square planar [NiO 4 ] units that is not based on close-packed oxygen layers [69,70]. The 1T-Li 2 NiO 2 structure has oxygen anions arranged in hexagonal close-packed (hcp)  layers, with layers of Ni in octahedral sites and layers of Li in tetrahedral sites distributing between the oxygen layers alternately [71]. The ground state Li 2 NiO 2 structure is the I-Li 2 NiO 2 as heating the 1T-Li 2 NiO 2 gives I-Li 2 NiO 2 . [70] While I-Li 2 NiO 2 can be accessed by solid-state synthesis method, the metastable 1T-Li 2 NiO 2 can hardly be synthesized via solid-state reaction directly [69,72]. The unfavorable electrostatic repulsion between the tetrahedrally coordinated Li was considered as the primary factor in affecting their stability ordering [69]. The minimum separations between Li are 2.48 Å and 2.28 Å for the optimized structures of I-Li 2 NiO 2 and 1T-Li 2 NiO 2 , respectively (see figure S3). The latter structure has Li more closely packed, giving greater repulsion.
The energy ordering of the I-Li 2 NiO 2 and 1T-Li 2 NiO 2 phases predicted by DFT at 0 K varies with choices of functionals. The PBEsol + U and HSE06 results suggest that the 1T-Li 2 NiO 2 phase is the ground state, whereas results from the SCAN functional indicate that the I-Li 2 NiO 2 is the ground state (table 3). The energy difference between the FM ordering and antiferromagnetic ordering (AFM) of 1T-Li 2 NiO 2 is   negligible. However, the AFM ordering is always lower in energy than that of the FM ordering, which is consistent with its magnetic behavior at low temperatures [71]. Despite that the energy ordering at 0 K does not necessarily equate to the energy ordering observed in experiments at some finite temperatures, the discrepancies between different functionals partly originate from the inherent limitations of DFT in approximating electron interactions and the Hubbard U plays a role in biasing the results [28,44,45,73,74].
Using the PBEsol + U functional, the predicted ground state structure is I-Li 2 NiO 2 when U = 0-4 eV, however, the 1T-Li 2 NiO 2 becomes lower in energy when U > 4 eV ( figure S4). We also considered the lattice dynamics in both phases and evaluated the vibrational contribution to the constant volume Helmholtz free energy (A) using the PBEsol + U functional to predict the thermodynamic stability following [75]: U latt is the athermal lattice energy. U vib and S vib denote the vibrational internal energy and vibrational entropy from phonon contribution under the harmonic model, respectively. Figure S5 shows the Helmholtz free energy difference ∆A relative to the lower energy phase as a function of temperature. Between 0 and 1000 K, the 1T-Li 2 NiO 2 is always lower in energy and higher temperature gives greater stabilization over the I-Li 2 NiO 2 phase. Therefore, this method failed to describe the phase stability of Li 2 NiO 2 or to predict any temperature-induced phase transitions with the chosen functional. Some forms of anharmonicity (e.g. associated with the square planar [NiO 4 ] geometry), may have been neglected in the calculations.
Although there exist two experimental Li 2 NiO 2 phases with distinctive structural framework, we simply use the minimum separations from Li 2 NiO 3 to perform the search (table S3). This is an example of an explorative search, where the structural information from a relevant composition is used to generate sensible random structures. Regardless of the oxidation states on Ni, the minimum separations from different (Li x Ni y O z ) compositions set constraints that avoid cation-cation and anion-anion being too close, hence the separations do not need to be exact values to achieve efficient searching. Figure 5 shows the structures optimized using the PBEsol + U and SCAN functionals. Both the I-Li 2 NiO 2 and 1T-Li 2 NiO 2 phases were encountered by AIRSS. We also identified two higher energy polymorphs and phonon calculations showed that they are located at local energy minima (figures S6 and S7). Table 1 shows the 11 new systems performed with explorative searches according to the pair-wise minimum separations from table S3. Further property calculations (e.g. band gaps, average voltages) were performed on dynamically stable structures (see figure S12) with distances to the convex hull of the Li-Ni-O phase diagram within 100 meV atom −1 (i.e. E above_hull < 100 meV atom −1 ) from PBEsol + U calculations, as the chosen energy range generally covers that for experimental synthesizable structures [76].

Exploring new systems
From table 1, Li 2 Ni 2 O 3 is a thermodynamically stable phase on the Li-Ni-O convex hull. The next low energy phase is the metastable Li 4 NiO 3 , with E above_hull < 10 meV atom −1 . Interestingly, the structures of Li 2 Ni 2 O 3 and Li 4 NiO 3 (see figure 6) resemble 1T-Li 2 NiO 2 ( figure 4(b)). In Li 2 Ni 2 O 3 , for every layer of (tetrahedral) Li and there are two (octahedral) Ni layers arranged repeatedly. The Li 4 NiO 3 structure, however, has two Li layers for every Ni layer. The additional layering of Li in Li 4 NiO 3 gives more unfavorable electrostatic repulsion, making the structure less stable. Overall, there is a consistent energy (E above_hull ) ordering of phases predicted using the PBEsol + U and HSE06 functionals. The HSE06 functional tends to better describe the electronic structures for a wider range of materials in different chemical environments and hence their results are generally more reliable [44,45].
Assuming only cationic redox during delithiation, the average voltages, theoretical capacities and theoretical energy densities for dynamically stable phases are benchmarked against experimentally known phases (table 2). The two Li 2 NiO 2 phases have the highest theoretical energy densities, suggesting that they are potentially good cathode candidates. Unfortunately, the practical performance of I-Li 2 NiO 2 is poor, as it was found to experience an irreversible structural change and a significant capacity loss during the first charge-discharge cycle [69]. The 1T-Li 2 NiO 2 , on the other hand, was reported to be stable over 100 cycles  [70]. Below half lithiation, the 1T-Li 2 NiO 2 becomes the layered LiNiO 2 , and its practical performance is limited due to the existing issues with cycling LiNiO 2 (e.g. the anisotropic lattice change and the various phase transformations during lithium deintercalation) [6,8,77,78].
For layered intercalation cathode materials, volume reductions (>5%) are commonly observed when >75% of Li have been extracted from the structures [79]. The cell volume change at different Li contents has contributions from: (1) steric factors (e.g. the number of Li that are required to sterically sustain the interlayer spacing); (2) electrostatic factors (e.g. strength of Van der Waals interactions vs other coulombic interactions) [80]. The experimental percentage volume reduction for the layered LiNiO 2 to the H3 phase (approaching a dilute limit of Li) is around 7%, which is close to the calculated result [6]. Structures with layers of tetrahedral Li generally have a larger volume reduction than those with octahedral Li (e.g. the layered LiNiO 2 vs 1T-Li 2 NiO 2 ), as the Li slab thickness (see table S4) is generally greater for structures with layers of tetrahedral Li to accommodate additional Li.
The I-Li 2 NiO 2 experiences a greater volume change compared the 1T-Li 2 NiO 2 , and their Ni coordination can explain this. Kang et al [69] indicated that the square planar [NiO 4 ] coordination in I-Li 2 NiO 2 is extremely unfavorable in the delithiated state, which led to structural collapse upon cycling. Significant volume changes (>17%) are also observed on the Li 3 NiO 3 and Li 5 NiO 4 phases, where Ni coordination is exclusively based on square planar/bipyramid. According to the crystal field theory, electron configurations of d 7 low spin (Ni 3+ ) and d 8 (Ni 2+ ) would encourage square planar Ni. However, there are only few experimental structures with Ni in square planar coordination. Many ternary Li-Ni-O compounds are in either the layered or rock salt framework [6,9]. The preference for Ni to adopt in the square planar coordination is therefore not greatly driven by the crystal field stabilization energy but other (long range) electrostatic interactions between the ions [69]. At a high Li content, square planar Ni provides greater stabilization by increasing the separation between the Li. When some Li are removed from the structure, the driving force for Ni to adopt in the square planar coordination to stabilize the structure is reduced and octahedral NiO 6 becomes more favorable. Consequently, structures with square planar Ni can transform more easily, leading to irreversible structural change and large volume change.
The electronic conductivities generally decrease with increasing band gaps. The band gaps of 1T-Li 2 NiO 2 , Li 2 Ni 2 O 3 and Li 4 NiO 3 are ⩾5 eV (see table 2), suggesting that the structural feature of layers of tetrahedral Li + and octahedral Ni 2+ is associated with poorer electronic conductivity. Previous studies indicated that the charge transfer mechanism in layered LiNiO 2 are dominated by small polaron hopping mechanisms [13,81]. The electron transport in similar materials can be further calculated using the small polaron migration model [82].
From the projected density of states of the ternary Li-Ni-O phases calculated with HSE06 functional (figures S13-S15), the valence band maxima (VBM) of all phases are composed of Ni 3d states and O 2p states, and the conduction band minima are dominated by Ni 3d states. Since late 3d TM tend to have increased bonding with oxygen (i.e. more hybridized), the contribution of O states at VBM are generally higher compared to other compounds (e.g. LiCoO 2 ) [81]. There is a general trend that O states contribution increases with the formal oxidation states of Ni. The proportion of O states at the VBM for a given Li-Ni-O phase could indicate its oxygen thermal stability and the potential for anionic redox. Materials with electrochemistry driven by anionic redox (e.g. Li 2 NiO 3 ) generally have poorer oxygen thermal stability [8,9]. The larger O states contribution of 1T-Li 2 NiO 2 at the VBM compared to that of I-Li 2 NiO 2 could be related to the poorer oxygen thermal stability of 1T-Li 2 NiO 2 . Overall, the layered Li 2 Ni 2 O 3 and Li 4 NiO 3 have comparable and/or better theoretical capacity and energy density to LiNiO 2 . The volume shrinkage behavior could be improved by elemental substitution [6,79]. Depending on the site preference of the extrinsic species on the host, foreign species with a large ionic radius (e.g. Mg 2+ , W 6+ ) could act as a pillaring ion which either sit on the Li/Ni layers to reduce the shrinking of the Li/Ni layers [83,84]. The structural instability induced by the JT effect due to the formation of Ni 3+ during delithiation can also be alleviated by introducing impurities [79].  figure S16(b)) has a layer of octahedral Li, analogous to that in LiNiO 2 . Therefore, the tetrahedral Li arrangement (in Li 2 Ni 2 O 3 ) is for accommodating additional Li in the structure. A precursor to prepare the Li 2 Ni 2 O 3 could be the layered LiNi 2 O 3 phase with octahedral Li.
To verify the existence of the layered LiNi 2 O 3 structure, we performed AIRSS search on this mix-valent composition. This structure, which exists in the P3m1 space group, was encountered by AIRSS. Figure S17 shows that it is at a local energy minimum. Interestingly, the lowest energy structure with the LiNi 2 O 3 composition is not the P3m1 LiNi 2 O 3 , and the energy difference between the P3m1 LiNi 2 O 3 and the lowest energy Immm LiNi 2 O 3 structure is 27.63 meV atom −1 . Within this energy difference window, there also exist several polymorphs. Some of the polymorphs are layered, with pure NiO 6 octahedra arranged in a single layer of rather than two consecutive layers as in Li 2 Ni 2 O 3 . This suggests that the structural feature of NiO 6 units arranged in two consecutive layers is stabilized by excess Li. Figure S18 shows the volume change for the Li 2−x Ni 2 O 3 lattice on the convex hull. The volume decreased rapidly between x = 0 → 1. This has contributions from the decreasing thickness of the Li layers and the shortening of Ni-O bond due to Ni oxidation. There exists a small window (1.0 < x < 1.5) with small volume variation, but further Li extraction collapses the Li layer. The final delithiated Ni 2 O 3 is not thermodynamically stable structure on the Li-Ni-O convex hull, with E above_hull = 264.88 meV atom −1 according to PBEsol + U calculations. The Cmcm Ni 2 O 3 phase from the MP database is lower in energy by 9.15 meV atom −1 . Figure 7(b) shows that the average potential window is between 2.6 and 4.8 V, but the low voltage part dominates, which is not a desirable feature for a cathode. We note that the calculated voltages may contain systematic errors, since the voltages from DFT calculations is sensitive to the choice of functional and that not all the structural transformations can be captured or correctly predicted [22].

Li diffusion
Ionic conductivities in intercalation cathodes (e.g. LiNiO 2 ) are mostly governed by the mobility of Li + [13]. The Li slab thickness, the electrostatic interactions between diffusing ions and the surrounding cations are considered to greatly influence ion mobility [59,85]. Despite the fact that Li slab thickness is increased in Li-excess systems (e.g. 1T-Li 2 NiO 2 ) to hold more Li (see table S4), it was speculated that Li diffusion is not promising as the overall available space for diffusion is reduced [72]. The electrostatic repulsion between more closely arranged Li is a contributing factor [8,86].
To estimate the activation energies for Li migration in Li 2 Ni 2 O 3 , we identified two plausible diffusion paths with a supercell containing a single Li + vacancy: (1) a direct hop from one tetrahedral site to the nearest neighboring tetrahedral site, with a hopping distance of 2.26 Å; (2) an indirect hop from one tetrahedral site to the next nearest tetrahedral site (∼3.03 Å away) via an interstitial site that is ∼1.68 Å away from the initial/end positions of the diffusing Li. Figure 8 shows that a direct hop between the nearest Li tetrahedral sites is more favorable, with a barrier height of 0.34 eV. The direct hop maximizes the distance between the diffusing Li with the surrounding Li, therefore minimizes the unfavorable electrostatic/steric contributions from closely arranged Li.
We also considered divacancy-like migration paths by introducing two vacancies in the supercell [87]. The energy of the configuration with two clustered Li vacancies (∼2.26 Å apart) is (∼0.40 eV) higher than that with two vacancies being ∼3.03 Å apart (figure S19). Nevertheless, the energy barrier for a Li + to hop directly to a vacant site is lower than that of an indirect hop. The barrier height for the divacancy indirect hop is 0.17 eV lower than that of the monovacancy case and we attribute this reduced barrier to the lowering of electrostatic repulsion between fewer adjacent Li and the diffusing Li + . The barrier for Li + diffusion would theoretically decrease further when the system approaches a more Li dilute limit. However, this assumes that the structures remain stable in their highly delithiated states.

The stability field of LiNiO 2
We have discussed the properties of predicted structures for battery cathode applications. A new stable structure, however, may have implications beyond its properties. Here, we use an example to demonstrate how previously undiscovered stable phases can be used explain the chemistries of other related phases.
The synthesis-defect chemistry-property relationships of several complex oxide battery cathode materials have been investigated by theoretical studies [81,88]. One key term for calculating the defect formation energies is the atomic chemical potential, which can be considered as the available atomic species responsible for generating the defects [88]. The ranges of atomic chemical potentials where the target compound is stable reflect the ranges of possible growth environments for the material to be synthesized experimentally. Under the assumption that a ternary compound Li x Ni y O z forms in an equilibrium reaction with constituent elements' standard phases requires to be satisfied, where ∆H f is the formation enthalpy. To avoid decomposition to the elemental phases, µ Li ⩽ 0, µ Ni ⩽ 0 and µ o ⩽ 0 are required. The competing phases impose additional thermodynamic constraints. For example, to avoid the formation of NiO: µ Ni + µ O ⩽ ∆H f (NiO). For a system containing n number of species with m competing phases, combinations of n − 1 linear equations are solved at one time from a total of 2n + m − 1 equations automatically using CPLAP [47]. Only solutions that simultaneously satisfy the conditions given by all constraints are kept. The final solutions yield a region of stability in the plane defined by the independent variables µ Li and µ Ni . A large atomic chemical potential stability field corresponds to a wider range of growth environments for the host. The defect formation energies, therefore, can vary over a larger range. Since defect concentrations scale exponentially with the defect formation energies, the predicted concentrations can be orders of magnitude different. To correctly calculate the atomic chemical potentials term, it is essential to consider all the stable phases in the corresponding phase diagrams [47]. Hoang and Johannes presented a detailed DFT study on the intrinsic point defects for the layered LiNiO 2 [13]. Their atomic chemical potential diagram shows that the thermodynamic stability window of LiNiO 2 is constrained by four other limiting phases, which gives five unique sets of atomic chemical potentials. We re-evaluated the stability field of LiNiO 2 by incorporating the Li 2 Ni 2 O 3 phase in the calculations. Figure 9 shows that Li 2 Ni 2 O 3 is an additional limiting phase which affects the thermodynamic stability of LiNiO 2 . We repeated the analysis using several other functionals (LDA + U, PBE + U, PBEsol + U, SCAN) and all calculations (see figure S20) suggest the Li 2 Ni 2 O 3 is a limiting phase for LiNiO 2 , despite that it is yet a hypothetical structure. Therefore, the defect chemistry of LiNiO 2 could potentially be different.

The likelihood of synthesizing AIRSS predicted phases
Our calculations have revealed the structures of many (meta-)stable Li-Ni-O compounds. However, the experimental synthesis of those compounds may be challenging (e.g. require out-of-equilibrium conditions) or impossible. It is likely that the predicted phases exist in the disordered forms due to the cation mixing behaviors of Li/Ni. The predicted phase stability at 0 K does not account for temperature effects which also encourage disordering. The challenges of getting a precise control of the known compositions/structures in the Li-Ni-O phase space have also been discussed in previous works [6,9]. We tried to incorporate finite temperature effects in our PBEsol + U convex hull analysis. This includes computing the constant volume Helmholtz free energies from phonon calculations (see figure S21), where the entropy term of O 2 gas was taken from the thermochemical tables [89]. Additionally we estimated the Gibbs free energies using the machine-learned descriptor (see figure S22) [90]. Unfortunately, neither approaches correctly capture the stability of known phases. The DFT + U method, with U calibrated to materials properties (e.g. band gaps, oxidation enthalpies) are limited known experimental phases such as NiO and NiO 2 . Therefore, more advanced theoretical methods (e.g. linear response calculations of DFT + U(+V) with onsite U (and intersite V) term determined self-consistently) may be needed to improve the phase stability prediction [30,[91][92][93][94].
Finally, we evaluated the thermodynamic stability of Li 2 Ni 2 O 3 using HSE06 (see figure 10), LDA + U, PBE + U, PBEsol + U and SCAN functionals (see figure S23). The narrow stability field suggests the difficulty in synthesizing this compound. It is predicted to form under an extremely O-poor and Li-rich environment. The limiting phases are LiNiO 2 , 1T-Li 2 NiO 2 /I-Li 2 NiO 2 and NiO (figure 10).

Conclusions
We have explored Li x Ni y O z in the Li-Ni-O phase diagram and predicted many (meta-)stable structures using AIRSS. The LiNiO 2 search shows the capability of AIRSS to explore structures displaying dynamic JT distortions. The predicted energy ordering of known Li 2 NiO 2 phases is sensitive to the choice of density functionals and Hubbard U. Among the low energy Li x Ni y O z structures, Ni can adopt octahedral and square planar/bipyramid coordination, with octahedral coordination being the most common. Previously unknown thermodynamically stable layered Li 2 Ni 2 O 3 and metastable layered Li 4 NiO 3 (<10 meV atom −1 above the hull), are predicted to have comparable theoretical energy densities to the layered LiNiO 2 and both have layers of octahedral Ni and edge-sharing tetrahedral Li to accommodate excess Li, like that in 1T-Li 2 NiO 2 . The structural feature is associated with a large band gap (∼5 eV) and large volume change upon delithiation. Li diffusion in Li 2 Ni 2 O 3 would take place along paths that are farthest to the surrounding Li + to minimize electrostatic repulsion. Several density functionals consistently predicted that the Li 2 Ni 2 O 3 phase further reduces the thermodynamic stability ranges of LiNiO 2 , which is known to be difficult to be synthesized stoichiometrically. By revealing the structures and properties of our predicted phases, these phases may become more identifiable in future experiments. Finally, we show that structure prediction is a valuable approach to explore the chemical space with complex chemistry.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: https://doi. org/10.5281/zenodo.8022111 [95].