Structures and electronic states of nickel-rich oxides for lithium ion batteries

Abstract

A new superstructure of layered pristine LiNiO₂ (LNO) was obtained by first replicating 16 times the optimized structure of the 12-atom primitive cell of the R3̄m space group and then relaxing both cell parameters and internal positions. The structural optimization is accompanied by a very significant lowering of the internal energy. The R3̄m space group is the space group observed experimentally by XRD. In contrast to the structure commonly proposed in the literature, this new crystal structure, which still belongs to the R3̄m space group, shows size and charge disproportionation of the NiO₆ octahedra. The charge disproportionation results in a change in the Ni oxidation state from Ni³⁺ calculated in the 12-atom primitive unit cell, to Ni⁴⁺ and almost Ni²⁺. This is the first time that such charge disproportionation has been calculated for the R3̄m crystal structure of LNO. The Ni–O bond length distribution of the new structure agrees well with experimental values. Our results show that the choice of the simulation unit cell is important for determining the energetics of this class of oxide material, proposed for cathodes in lithium ion batteries (LIBs). We used this new structure as a template for the study of the structural and electronic changes induced by delithiation and by Mn for Ni cation substitution, originating the solid solutions LiNi_yMn_(1−y)O₂ (LNMO). Our results, surprisingly, agree well with existing experiments and explain observed trends better than previous studies.

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1. Introduction

The transition to a green economy and environmental protection needs the development of new Li-ion batteries with higher energy density, faster recharge, and greater safety. Layered oxides have been the materials of choice for cathodes in Li ion batteries. The cathodes of commercial Li ion batteries are made of LiCoO₂ (LCO) and contain cobalt, a rare and toxic element. The theoretical capacity of LCO is 270 mA h g⁻¹,¹ which reduces to an actual reversible charge capacity of about 145 mA h g^−1 2 because of thermal instability issues. As a consequence, LCO needs to be substituted for large power applications. In the last few years, a renewed interest in the isostructural LiNiO₂ (LNO) compound, based on the alternative nickel transition metal, has emerged.³ This layered material is attractive since it has a lower cost for the same theoretical energy density as LCO. However, despite much research effort being devoted to this material, it has not yet been commercialized because of its many drawbacks.

First, despite much care during synthesis, the material turns out to be always nonstoichiometric with Ni ions occupying Li sites, so the compound is actually Li_1−xNi_1+xO₂ with x ranging typically between 2% and 10%.⁴ The off-stoichiometry is thought to be due to the similarity of the Ni²⁺ and Li⁺ ionic radii (r(Ni²⁺) = 0.69 Å, r(Li⁺) = 0.76 Å). LNO stabilizes in the rhombohedral R3̄m Bravais lattice⁵ despite a number of first-principles calculations having found monoclinic crystal structures to be the ground states for the layered crystal structures.^6,7

Other instabilities arise during the battery charge–discharge cycles, in the form of phase transitions, in which Li–Ni cation mixing leads to the formation of a spinel (LiNi₂O₄) phase and/or rock-salt-type phases.^8,9 Other forms of instability include surface reactivity affecting the electrode–electrolyte interface with the formation of NiO.^10,11 Another critical factor governing the degradation of LNO is the multistep phase transformation process occurring on Li insertion/extraction. Four different phases have been described in the literature, namely the hexagonal H1, the monoclinic M, the hexagonal H2, and the hexagonal H3 phases,^3,4,12–14 where the H2–H3 transformation at a high state of charge (4.15–4.25 V) has the strongest negative effect on the intrinsic stability of LNO, as it is accompanied by a sudden collapse of the structure along the crystallographic c-axis.¹²

Despite the large number of studies, the structure of LNO at full Li content is still a subject of considerable debate. The structure determined experimentally is rhombohedral, of R3̄m symmetry, with O3 stacking, where the oxygen atoms occupy the 6c sites, and lithium and nickel occupy the octahedral 3a and 3b sites.¹⁵ In this structure, the Ni cations are supposed to have Ni³⁺ oxidation states in a low-spin configuration, with a single electron in the e_g orbital. This electronic configuration is subject to Jahn–Teller (JT) distortion of the NiO₆ octahedra, which has been measured through EXAFS and neutron diffraction.^16,17 The measurements support the JT distortion: four bond lengths are grouped as long bonds (2.04 Å and 2.06 Å with an average length of 2.05 Å) and short bonds (1.90 Å and 1.96 Å with an average length of 1.93 Å) suggestive of the 2 : 1 ratio of short-to-long bonds expected for JT distortion. However, the measurements did not indicate the existence of long range order. Indeed, a collective distortion has not been observed for LNO, as is the case for LiNaO₂, so it has often been assumed that in LNO the JT orbitals are randomly oriented. Experiments later excluded a random arrangement, pointing, instead, to some sort of local ordering of a trimer kind with the three 3d_z²−r^2/3 orbitals pointing towards the shared oxygen site.^16,18 First-principles calculations have confirmed that JT-distorted configurations of NiO₆ octahedra in LNO are more stable than the R3̄m undistorted structure. Chen et al. have compared the energies of differently oriented JT-distorted phases, finding that the structure of the P2₁/c space group with JT-distorted octahedra in a zig-zag arrangement is more stable than the C2/m structure with a collinear arrangement.⁷ A similar result was obtained by Das et al.⁶ and Zhang et al.¹⁹ However, the zig-zag arrangement of the octahedra has been excluded by experiments, which always finds the rhombohedral R3̄m symmetry for the complete lithiated phase. Nowadays, a non-cooperative and dynamic JT effect is generally accepted for LNO,^3,20 although disproportionation based on Ni–O bond lengths has also been argued.²¹ From the measured Ni–O bond lengths and by comparing them with other nickelates, competition between charge ordering and orbital ordering for the ground state is expected for LNO.

In this paper, we intend first to revisit the problem of the LNO structure using highly accurate first-principles calculations. All the simulations of recent years studying doping and defects in LNO invariably assume pristine LNO model structures containing only Ni³⁺ cations. In this paper, we show that the structural optimization of a large supercell of the R3̄m space group for layered LNO leads to an ordered crystal structure again belonging to the R3̄m space group, but presenting structural and charge disproportionation. Structural optimization leads to a large energy gain, much larger than the energy gain obtained for the C2/m monoclinic space group in a similarly large supercell. The monoclinic crystal structure has still a lower energy than the hexagonal one, but the difference between their energies becomes much smaller. This result is obtained for pristine LNO without Ni in the Li layers. The new ordered crystalline structure presents properties in nice agreement with the experimental data and, when delithiated, can explain the driving force behind the observed H1 to M phase transition.

As shown here, the issue of the choice of the unit cells for calculating the energy of different crystal structures of the LNO layered system is a delicate one since their formation energies and their relative values are usually used to calculate the phase diagrams and voltage-capacity curves through the cluster-expansion (CE) method.^6,22 In this method, the energies of a large number of Li_xNiO₂ ordered crystal structures in small unit cells are calculated and fitted to extract the cluster (pairs, triplets, and quadruplets formed by lattice points) contributions to the energy. In a given ordered crystal structure, each cluster is associated with a “correlation function” calculated by associating each lattice point with a “spin” value dependent on the actual ion occupying that lattice point. This paper shows that, at least in the case of LiNiO₂, the choice of the basic cells needed to calculate the input total energies for CE may have a great influence, not only on the determination of the ground structures and phase diagrams, but also on the structural, electronic, and magnetic properties associated with the input structures.

Since LNO suffers from the above-mentioned structural instabilities and maintaining structural stability during cycling is critically important to achieve a long-term cycling performance, the new crystalline structure is used to study the properties of Mn-modified LNO. Indeed, the instability of LNO has originated research efforts aimed at stabilizing the material through the addition of other metal cations. Other layered oxides with general LMO formula have been synthesized in which M stands for other transition metals inserted in partial substitution of Ni to reduce the quantity of cobalt.^23,24 Among these promising layered oxide compounds is nickel-rich NMC, where N stands for nickel, M for manganese, and C for cobalt. To eliminate cobalt, layered Li_xNi_(1−y)Mn_yO₂ (LNMO) solutions have been synthesized and tested. Only few papers have focused on the LNMO material system. Layered manganese-substituted LiNi_(1−x)Mn_xO₂ solid solutions can only be formed when x is less than 0.5 and they have the same α-NaFeO₂ layered structure as R3̄m LNO in a nickel-rich regime (x ≤ 0.5). For x ≥ 0.5, a phase transition occurs to a cubic spinel symmetry. In most LNMO samples, some Li is found in the transition metal layer, Li_TM, hence the solution is indicated as Li[Ni_xLi_(1−2x)/3Mn_(2−x)/3]O₂ (0 ≤ x ≤ 0.5). Manganese incorporated in pristine LNO, Li(Ni_(1−x)Mn_x)O₂, has been found in the tetravalent Mn⁴⁺ oxidation state,^25,26 so that more Ni²⁺ cations are supposed to be generated to conserve the charge, in the fully lithiated solid solutions Li(Ni²⁺_xNi³⁺_(1−2x)Mn⁴⁺_x)O₂.²⁷ Obviously, this consideration stems from the assumption that in lithiated LNO all the Ni cations are Ni³⁺, an assumption which is challenged in the present paper.

The presence of Mn was also found to induce an increase in Li/Ni cation mixing. The lithium ions in the transition metal layer, Li_TM, are electrochemically active, resulting in the creation of tetrahedral sites for lithium occupancy when, during cathode charging, Li_TM are removed.²⁸ First-principles calculations have shown that the Li/Ni exchange promoted by Mn tends to increase the activation energy for Li ion diffusion.²⁹ Other authors³⁰ have pointed out that the formation energy of an extra Ni defect in the Li layer in LiNi_0.5Mn_0.5O₂ increases in the presence of a high-valence Mn⁴⁺ dopant. This fact, together with the observation that the presence of Mn⁴⁺ reduces Ni³⁺ to Ni²⁺, thus reducing the lattice distortion due to the Ni³⁺ JT effect, is thought to be at the origin of the observed stabilization of the crystal structure.

In this paper, we model pristine layered LNMO (e.g. without Li_TM or other kinds of defects) using large optimized supercells of pristine LNO of R3̄m space group. The aim is to study the evolution of the structural and electronic properties of the solid solutions versus Mn content and to investigate the effects of Mn introduction, in substitution of Ni, on the structural and electronic properties of the delithiated structures. Our study confirms that Mn ions are predominantly in the Mn⁴⁺ state and thus, do not participate in the redox reactions occurring during delithiation. They have, however, a role in determining the oxidation states of the Ni ions, and thus, of the Ni²⁺/Ni³⁺ and Ni³⁺/Ni⁴⁺ redox couples which determine the battery charge and discharge potentials, and the intercalation voltage curves.

2. Computational method

Total energy calculations were performed within the spin-polarized density functional theory (DFT) approach using the plane-wave pseudopotential method as implemented in the Quantum Espresso Package.³¹ The exchange and correlation density functional was expressed through the GGA approximation of Perdew–Becke–Ernzerhof (PBE).³² We used norm-conserving (NC) pseudopotentials and all plane waves with energy less than 80 Ry were included in the basis set. The high cutoff of 80 Ry ensured good convergence of the total energies. The Hellman–Feynman forces were converged to 0.005 eV Å⁻¹. The rotationally invariant scheme proposed by Dudarev et al.³³ was used to include an energy term based on the Hubbard U model, facilitating electron localization in the transition metal d orbitals. We chose U = 3 eV since using this value, the calculations predicted structural, magnetic and electronic properties of LiNiO₂ and LiMnO₂ in the rhombohedral (R3̄m), monoclinic (C2/m), spinel (Fd3̄m), and orthorhombic (Pmmn) crystal structures in reasonable agreement with the available experiments. We also checked other values of U and found that the physical interpretation of the results was not affected substantially. As shown in Fig. 1, the only relevant effect of the value of U on the total density of states (TDoS) of rhombohedral layered LiNiO₂ in its primitive unit cell, is to push the energy of the empty states upward and, in this specific case, to widen the gap between the spin-down bands. However, in this paper, we are interested in ground-state properties, while experimental energy gaps are better predicted by excited-state theories. Evidence that the results of the calculations do not depend strongly on the U value is also given in the literature.³⁴

Fig. 1 Effect of U on the TDoS of LNO.

The calculation of the mixing energy against phase separation in a completely lithiated and completely delithiated phase was used to compare the stability of structures on the same lattice with different numbers of Li vacancies. The formula used to calculate it is:

E_mixing = E(Li_xNiO₂) − xE(LiNiO₂) − (1 − x)E(NiO₂) (1)

where, E(LiNiO₂) is the total energy of the fully lithiated structure, E(Li_xNiO₂) is the total energy of the structure with lithium concentration x; and E(NiO₂) is the total energy of the fully delithiated structure. We consider only the internal energy since, as usually assumed in literature, the enthalpic and entropic contributions can be neglected when considering the systems at low temperature.

Analogously, the stability of Li_xNi_1−yMn_yO₂ is assessed by defining the mixing energy as:

E_mixing = E(Li_xNi_1−yMn_yO₂) − x(1 − y)E(LiNiO₂) − xyE(LiMnO₂) − (1 − x)(1 − y)E(NiO₂) − (1 − x)(y)E(MnO₂) (2)

Here, E(Li_xNi_1−yMn_yO₂) is the internal energy of the Mn/Ni-substituted structure with a concentration x of lithium, [(1 − x) Li vacancies] in the Li layers, and a concentration y of Mn [(1 − y) of Ni] in the transition metal layers. E(LiNiO₂), E(LiMnO₂), E(NiO₂) and E(MnO₂) are the energies of the fully lithiated LNO and LMO and the fully delithiated NO₂ and MO₂, respectively. All the structures are considered in the same crystal structure. This choice for the stability assessment of Li_xNi_(1−y)Mn_yO₂ was previously used³⁵ to investigate how the stability of different arrangements of the two kinds of transition metal ion depends on their mutual interaction, rather than to determine the absolute stability of the compounds from constituent elements.

Ferromagnetic alignment of Ni spins was adopted throughout all the calculations since it was shown that the spin-interaction energies are smaller than those between different structural phases and a ferromagnetic ordering corresponds to the ground state of LNO and Li-rich and Mn-poor LMNO.^21,36

3. Fully lithiated LiNiO₂

3a. Structural properties

LiNiO₂ can exist as a layered oxide with an arrangement of oxygen atoms forming a cubic close packed structure. In the crystalline rhombohedral R3̄m space group, the transition metal and lithium ions occupy octahedral sites. In the O3 stacking, as shown in Fig. 2, the primitive 12-atom cell (Fig. 2(a)) contains three formula units, shown in the figure as A, B, and C, consisting of parallel planes of lithium, oxygen, nickel, and oxygen atoms. In each plane, the atomic positions are shifted along the a- and b-axes occupying different sites. The stacking sequence along the c-axis of the planes is abcabca…, where the a, b, c, a planes constitute the formula A; b, c, a, b the formula B; and c, a, b, c the formula C. A, B, C in Fig. 2 refer also to the Li planes positions.

Fig. 2 (a)–(c) Ball and stick representation of LNO in the R3̄m space group with 12, 48 and 192 atoms unit cells, respectively. Li atoms are represented by green balls, oxygen balls by red, and nickel atoms by grey.

The Li ions diffuse, intercalate and de-intercalate between the planes of NiO₆ octahedra. Typically, the rhombohedral space group is represented in the corresponding hexagonal (H) cell with lattice parameters a, b, and c (orthogonal to a and b).

We performed calculations using simulation cells obtained by increasing the a and b in-plane lattice parameters of the hexagonal cell twice and four times, containing 48 and 192 atoms, respectively. The unit cells are shown in Fig. 2. The structural relaxation of the larger cells was performed starting from the cell parameters and atom positions of the optimized primitive 12-atom cell.

In the 12-atom (1 × 1 × 1) and 48-atom (2 × 2 × 1) cells, the NiO₆ octahedra do not change appreciably, but in the 192-atom unit cell (4 × 4 × 1) the structural relaxation leads to size disproportionation of the octahedra with expanded and compressed octahedra having different Ni–O bond lengths. The octahedra show also a small JT distortion. Furthermore, the cell lattice parameters change with the cell dimensions, as shown in Table 1. The expanded and compressed octahedra are related to disproportionation of the Ni ion oxidation states, as shown in Table 2, where the bond lengths of the different kinds of octahedra in the 192-atom supercell are also reported.

Table 1 Optimized values of structural parameters in fully lithiated LiNiO₂

No. of atom cell		Lattice parameters (Å)			Other theories	Expt.
		a	b	c
R3̄m	12	2.89	2.89	14.25	(2.84, 14.29),^37 (2.89,14.20),^37 (2.90, 14.18),^38 (2.88,14.27)^39	(2.88, 14.18),^15 (2.87,14.21),^14 (2.88, 14.19),^40 (2.90, 14.2)^41
	48	2.86	2.86	14.05
	192	2.87	2.87	14.01
C2/m	8	4.96	2.96	5.74	(5.16, 2.79, 5.13)^7
	192	9.83	11.71	5.74

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Table 2 Calculated Ni–oxygen bond lengths in the different atom cells

System	No. of atoms in cell	Metal–oxygen bond	Bond length (Å)
a Here, average is the arithmetic average and Δ is the difference between highest and lowest values.
R3̄m	12	Ni^3+–O
	192	Ni^2+–O
		Ni^4+–O
C2/m	8	Ni^3+–O

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Thus, the use of the larger 192-atom unit cell has shown that the undistorted NiO₆ octahedra calculated by optimizing the atomic coordinates in the smaller 12- and 48-atom unit cells are actually unstable towards disproportionation. The calculation for the 192-atom unit cell starts by replicating 16 times the optimized 12-atom unit cell, so the initial octahedra are exactly the same as those obtained by optimizing the 12-atom unit cell. The relaxation occurs spontaneously and is accompanied by a relevant diminishing of the total energy. Thus, the structures calculated using the 12- and 48-atom unit cells, the ones always considered so far in the literature for the R3̄m space group of LiNiO₂, are shown to be unstable when represented using a larger unit cell.

A phase consisting of a mixture of expanded, compressed, and JT-distorted octahedra corresponding to Ni²⁺, Ni⁴⁺ and Ni³⁺, respectively, was assumed from experimental neutron-pair distributions, which could agree with most of the existing experimental data. The authors claimed that such a disordered metastable “charge-glass” phase could be stabilized by entropic effects.²¹

Here, we have found a structure consisting of expanded and compressed octahedra with Ni charge disproportionation by optimizing the R3̄m O3 structure in the 4 × 4 × 1 supercell, while the monoclinic structures, of which the C2/m is one representative, have JT-distorted octahedra with four short and two long Ni–O bonds and Ni³⁺ cations at their center (see Table 2). Indeed, there are two modes of JT distortion occurring in layered LNO, Q2 and Q3, discussed in detail in the literature.⁴² In the C2/m structure, only the Q3 mode occurs, leading to collinear JT distortion, where all the long bonds are similarly oriented. We define a JT distortion parameter Δ, which is the average difference between the longest Ni–O and shortest Ni–O ones. The values obtained for the hexagonal and monoclinic structures are given in Table 2.

This result is the first to report charge disproportionation and relative NiO₆ octahedra size disproportionation in the fully lithiated O3 layered hexagonal crystal structure.

Our results are in qualitative agreement with the neutron diffraction data¹⁶ where the peak broadening was found to be more consistent with a continuous distribution of Ni–O bond lengths, due to nonuniform distortion than with the two different fixed Ni–O bond lengths expected in the case of a monoclinic lattice. Our calculated average long and short Ni–O bond lengths are 2.03 and 1.90 Å which nicely compare with the experimental values of 2.05 and 1.93 Ź⁶ and with the X-ray absorption data finding Ni–O bonds of 2.08 and 1.91 Å, respectively.⁴³

3b. Electronic and magnetic properties

The electronic structure is discussed by inspecting the Total Density of Satets (TDoS) and Partial Density of States (PDoS) of the considered structures. In Fig. 3(a), we can see that, using the 12-atom cell for the R3̄m crystal structure, the spin-up TDoS has a half-occupied band at the Fermi level, while the spin-down TDoS has an electronic gap. In this case, the Ni atoms are at the center of similarly perfect NiO₆ octahedra (see Table 2). Since the crystal field splits the five d states into a three-fold degenerate t_2g state and a two-fold degenerate e_g state, the Ni states should be all in a Ni³⁺ oxidation state which adopts the low-spin configuration t⁶_2g(|↑↓|↑↓|↑↓|) e¹_g(|↑| |), where the t_2g states are occupied leaving one electron in the doubly degenerate e_g state at higher energy. In the crystal, the Ni d states are hybridized with the oxygen p states, thus the half-occupied feature at the Fermi level corresponds to the antibonding band. The unpaired e_g Ni electrons occupy half of the spin-up band, while the spin-down band remains empty. This behavior is reflected in the A peak, which corresponds to the spin-up orbitals and is half occupied, as shown in Fig. 3(b). The B peak corresponds to the spin-down states.

Fig. 3 (a) Total density of states, and (b) atom-centered partial density of states in 12-atom fully lithiated LiNiO₂.

The Ni³⁺ cations have an estimated magnetic moment of 1 μ_B (based on the calculated difference of a Löwdin partition of the up and down spin charge on atoms) and the preferred phase is ferromagnetic. The estimated local magnetic moments on the transition metal cations are reported in Table S1 in the ESI,† under the heading 12 atom.

These results compare well with the calculations reported in the literature, which invariably use the 12-atom cell to describe the fully lithiated R3̄m LNO.^37,39,44

The Ni³⁺ cation with only one electron in the orbitals is JT unstable. However, the distortions cannot take place in the small 12- and 48-atom cells, thus the octahedra have equal Ni–O bond lengths (see Table 2).

As seen above, a structural distortion takes place instead in the 192-atom supercell, and, as a consequence, the A peak feature at the Fermi level in the TDoS is significantly changed, as shown in Fig. 4(a). The spin-up A band splits into a mostly occupied A₁ peak and an empty A₂ peak, hinting at the formation of a band gap, and similarly the spin-down B peak splits into B₁ and B₂. The states just below the Fermi energy show predominant oxygen and, to a lesser extent, nickel character, as shown in Fig. S1 (ESI†). These hybridized (O p-Ni d) energy states are of particular interest, as these are the states from which electrons are removed during the delithiation process.

Fig. 4 TDoS in (a) R3̄m and (b) C2/m; nickel-atom-centered partial density of states in R3̄m (c) and (d) and in C2/m (e) and (f) in fully lithiated LiNiO₂.

Analyzing the partial density of states, we find that there are two kinds of Ni ions, as shown in Fig. 4(c) and (d). The spin-up state of the first kind is almost fully occupied, while that of the second kind is mainly unoccupied. This situation is reflected in their different magnetic moments, as reported in Table S1 (ESI†). From the values reported in the table under the heading “192 Atom cell” and x = 1, we can see that the magnetic moments acquire different values: magnetic moments close to 0 indicate Ni⁴⁺ cations, while those around 1.4 are characteristic of Ni²⁺ (or close to Ni²⁺) cations.

Thus, Ni ions have two oxidation states: Ni⁴⁺ with the spin-up level completely empty and magnetic moment zero and Ni ions close to the Ni²⁺ oxidation state with an almost full spin-up band and an estimated magnetic moment of about 1.4 μ_B, much larger than the 1 μ_B associated with Ni³⁺. We name these second Ni ions simply as Ni²⁺. The features of the PDoS can be assigned neatly to these two different Ni cations: the A₁ peak is the occupied part of the 3d-e_g orbital of Ni²⁺ (Fig. 4(c)), as well as the empty peak B₂, while the empty peaks A₂ and B₁ belong to Ni⁴⁺ (see Fig. 4(b)). The existence of Ni⁴⁺ ions in LNO has been calculated previously only when some Li vacancies are present⁴⁵ and never for the fully lithiated ideal structure. This was due to the use of the smaller unit cells to describe LNO in the layered R3̄m space group. Such charge disproportionation into Ni²⁺ and Ni⁴⁺ was calculated before only for the monoclinic P2/c structure³⁴ and in monoclinic C2/m LNO only after one additional Ni ion was inserted into the Li plane.³⁰

The distorted NiO₆ octahedra of the 192-atom supercell lead to the breaking of the e_g and t_2g orbital degeneracies. The Ni⁴⁺ cations (Fig. 4(c)), where the spin-up and spin-down bands are empty, are at the center of the compressed octahedra with short Ni–O bond lengths (see Table 2). In this case, the (Ni d–O p) hybridization increases and the antibonding bands are shifted to higher energies, remaining empty. The Ni²⁺ cations, instead, are at the center of the expanded octahedra having longer Ni–O bond lengths (Fig. 4(d)). The hybridization in this case decreases and the spin-up bands shift down in energy, becoming mostly occupied. The presence of Ni²⁺ cations in an ideally layered LNO would explain why it is so difficult to obtain samples without Ni in the Li layer, given the very low energy barrier for Ni²⁺ to replace Li⁺.

The assignment of the oxidation states of the Ni ions is also validated by calculating the Bader charges of the Ni ions in the 12-atom cell, where Ni are all equivalent and surely Ni³⁺, obtaining a charge of 16.74 e, and of the Ni ions in the 192-unit cell where we have found two values: a larger charge of 16.80 e and a lower charge of 16.64 e, for the ions we attributed to Ni²⁺ (or almost Ni²⁺) and Ni⁴⁺.

The TDoS of the C2/m structure shown in Fig. 4(b), depicts band gaps in both spin channels, as found by other authors.⁷ The PDoS of the two Ni atoms in the primitive cell, shown in Fig. 4(e, f), have the same 0.25 eV band gap which is due to the splitting of the bands for the JT effect. There is no charge disproportionation in this case, so the Ni oxidation states are Ni³⁺. For both Ni³⁺ ions, only the spin-up electron goes in the lowest energy JT-split band below the Fermi energy, while the other spin-up band and both the spin-down bands, closer in energy, are empty.

To provide a clear understanding of the difference in the electronic configurations associated with the spin-up bands in the hexagonal 192-atom supercell and in C2/m, starting from the undistorted octahedra case of the R3̄m structure in the primitive 12-atom cell, we show in Fig. 5 a scheme of the different splittings and occupations of the spin-up e_g states on the Ni atoms. The e_g spin-down orbitals are higher in energy and unoccupied.

Fig. 5 Schematic diagram of Ni d e_g spin-up states in the hexagonal 192-atom structure and C2/m structure.

The arrangements of the Ni⁴⁺ and Ni²⁺ ions form in-plane ordered patterns with 9 Ni²⁺ and 7 Ni⁴⁺, as shown in Fig. 6. If the Ni ions are all in the Ni³⁺ oxidation state, then there will be 16 electrons in the bands per plane, one per Ni atom. If Ni⁴⁺ ions have no electrons in , then the 9 Ni²⁺ need to have a total of 1.78 electrons in . Assuming that no charge is transferred to other bands, their average oxidation state is actually +2.22. The Ni²⁺ ions form two kinds of hexagons: one regular and the other irregular with 9 ions on the perimeter. The hexagons surround three Ni⁴⁺ ions. The symmetry of the structure belongs to the hexagonal space group, with 12 symmetry operations, like the structure described with the 12-atom unit cell, but the structural, electronic and magnetic properties are quite different.

Fig. 6 Pattern formed by the Ni⁴⁺ (grey) and Ni²⁺ (blue) cations in the transition metal layers of the 192-atom supercell.

Our calculations have shown the strong dependence of the physical properties of the hexagonal fully lithiated LNO on the choice of unit cell. Since the octahedra have common edges, they cannot distort independently; thus, it is quite possible that the use of even larger unit cells could allow more freedom for octahedral distortion, which could lower the energy even further and lead, possibly, to the opening of a gap between the spin-up states as in the case of the C2/m monoclinic point group, in accordance with the experimental observation of a 0.5 eV gap for the almost stoichiometric LNO in the hexagonal phase.⁴⁶ However, gaps exhibit excited-state properties and cannot always be correctly predicted by DFT methods. The total energy per atom of LNO in a hexagonal 192-atom cell and C2/m in its primitive 8-atom unit cell are almost the same, within the accuracy error of the calculations.

4. Partially lithiated LNO

4a. Li vacancy mixing energies and structural properties

The structural models of the delithiated Li_xNiO₂ structures are obtained starting from the relaxed hexagonal supercell with 192 atoms used as a template. We know from the literature that during the charge and discharge cycles the delithiated systems undergo various phase transitions (H1, M, H2, and H3, where M stays for Monoclinic and H for Hexagonal) at certain values of Li concentration x, but maintain the layered structure.^3,4,12–14

To obtain structural models for delithiated Li_xNiO₂, we considered the three layers A, B, and C shown in Fig. 2 and removed the Li atoms from a different number of layers.

Li cations were removed from only one of the three Li planes characterizing the O3 stacking, say the A layer, and named these delithiated configurations as single-layer structures.

For these single-layer structures, eight different Li vacancy configurations were calculated realizing four concentrations: x = 0.75, 0.625, 0.5, and 0.25. These configurations, named 1c and 1b (x = 0.75), 2a and 2b (x = 0.625), 3a and 3b (x = 0.5), and 4a and 4b (x = 0.25) are shown in Fig. 7(a).

Fig. 7 Delithiated structures in the 192-atom cell: (a) single-layer structures: 1c and 1b (8.33% Li vacancies in total), 2b and 2a (12.5%), 3a and 3b (16.67%), 4a and 4b (25%); (b) multilayer structures: 5a and 5d (25% Li vacancy), 6c and 6b (33% Li vacancy), 7c and 7a (50% Li vacancy) and 8c and 8a (75% Li vacancy).

Other delithiated configurations were obtained by removing Li from more than one layer and are named multilayer structures. In the case of the multilayer structures, there are numerous ways in which the Li ions can be removed from each layer. Here, we have considered only some of the possible combinations, since our aim is to obtain a general overview of the trends in the properties of the delithiated structures as a function of Li content x. Some of the structures are shown in Fig. 7(b) while a complete list of the multilayer structures is reported in Table S2 (ESI†), where the positions of the removed Li cations in each layer are given.

For the multilayer structures, we generated a Li vacancy configuration in each Li layer (named a sequence), then we stacked the sequences in different ways on the A, B, and C planes, sometime just repeating the same sequence in all planes, at other times using different sequences in the three planes. We generated 22 different multilayer configurations covering Li concentrations x = 0.916, 0.833, 0.75, 0.667, 0.58, 0.5, 0.41, 0.33, 0.25, 0.167, and 0.083. For each Li content, we compared the energies of two configurations. The two configurations were chosen to have the Li vacancies close to one another or far apart. Li vacancy configurations have been used by other authors^5,47,48 using smaller monoclinic unit cells, in which only two oxidized Ni states (Ni³⁺ and Ni⁴⁺) were found as a consequence of delithiation.

The mixing energies of these partially lithiated Li_xNiO₂ were calculated using eqn (1), and only those whose energy was found to be the lowest are plotted in Fig. 8 (henceforth, we will refer to these structures as “most stable”, but since we have not considered all the possible vacancy configurations over the 48 Li sites in the 192-atom supercell, the structures are just the most stable among those calculated). Nevertheless, even with this limitation, interesting trends can be found. For the same Li content, the multilayer structures generally have lower energy than the single-layer ones, showing that delithiation occurs preferably by removing Li simultaneously from all the layers.

Fig. 8 Mixing energies of Li_xNiO₂.

At high values of x, the more stable configurations are characterized by an ordering of the Li vacancies that tends to maximize their mutual distance. This result suggests the existence of a repulsive interaction between the Li-vacancies, which decays with distance. At lower x, when the vacancies need to be closer, the most stable configurations present rows of vacant and occupied Li sites. These chain-like arrangements of Li atoms and Li vacancies seem to characterize favorable Li vacancy arrangements. This could be due to the strong interaction between Li cations of different planes along the direction of the Li–O–Ni–O–Li chains⁵ that in the presence of Li vacancies will induce JT distortion of the NiO₆ octahedra. Indeed, the presence of Li ions and vacancies in ordered interlayer chains favors the collinear formation of JT distortion with smaller lattice strain since channels are formed, as shown in Fig. 7.

The optimized lattice parameters of the delithiated structures are reported in Table S3 (ESI†).

In Fig. 8 we show, together with the mixing energies of the lowest energy delithiated configurations, the energies of the fully lithiated hexagonal and C2/m LNO structures, both calculated using the primitive unit cells (12 atoms for R3̄m and 8 atoms for C2/m) and 192-atom supercells. The energies are all relative to the energy of the hexagonal 192-atom supercell. We can see that the energy of the hexagonal structure decreases with atomic position optimization, passing from the primitive cell to the supercell, much more than the energy of the monoclinic structure, thus showing a more efficient structural relaxation. The relaxed structures in the 192-atom supercell are much closer in energy: 8.75 meV per atom.

In Fig. 9, we report the variation of the lattice parameters a = b and c versus Li concentration x. The lattice parameters of the lowest energy structures are reported for each x. The decreasing trend in a is due to the progressive increase in the number of Ni⁴⁺ ions, which have a smaller ionic radius (0.46 Å) than Ni²⁺ (0.69 Å) and Ni³⁺ (0.56 Å) and are associated with the compressed octahedra. This trend is consistent with the experimental results.^16,43 Sharp contraction in the c vertical lattice parameter within the range of 1 < x < 0.80 is due to the emergence of JT-active Ni³⁺ cations out of the Ni²⁺ cations. The number of Ni³⁺ cations and the corresponding JT distortion parameter Δ increase, as shown in Table 3, and they are largest in the Li concentration region 0.50 < x < 0.75 where the layered structure undergoes the observed structural phase transition from the hexagonal H1 to the monoclinic M phase and remains in the monoclinic structure (M) (see Fig. 9). The other dip in the c-axis for x < 0.4 is due to the dominance of Ni⁴⁺ ions and low Li content, hence the octahedra become compressed, JT distortion reduces, and interaction of the Ni ions with the O ions increases. The Li interlayer distance reduces as less Li is present and the oxygen atoms do not repel any more as the oxygen electronic −2e charge is now more localized inside the octahedra. The reduction of the JT distortion leads to a new phase transition back to a hexagonal phase. At high delithiation levels, the H3 phase was found to be unstable, probably due to oxygen loss with electrochemically inactive NiO formation. This trend in the c-axis with Li content, similar to that calculated for Li_xCoO₂, was obtained by other authors only after using the SCAN and SCAN + D3 functionals in place of the PBE+U functional used in this work. In that work, PBE+U produced only an ever-decreasing c lattice parameter.⁴⁹ In our case, the expected trend has been obtained using PBE+U with the larger supercell.

Fig. 9 Variation of the crystal lattice parameters, a and c, as a function of Li concentration x. The vertical dashed lines correspond to the Li content at which phase coexistence and phase transition between the hexagonal H and monoclinic M phases have been evidenced by XRD during the charging of the LNO cathode.¹⁴

Table 3 Number of different Ni cations and Δ parameters in Li_xNiO₂

Li concentration	No. of Ni cations			Jahn–Teller distortion parameter Δ in LixNiO2
x	Ni^2+	Ni^3+	Ni^4+	Ni^2+	Ni^3+	Ni^4+
1	27	0	21	0.06	—	0.04
0.75 (chain)	12	12	24	0.11	0.20	0.03
0.75 (square)	12	12	24	0.03	0.21	0.04
0.5	4	18	26	0.12	0.18	0.03
0.25	2	9	37	0.11	0.16	0.02
0	0	0	48	0	0	0

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4b. Electronic and magnetic properties

Table 3 reports the number of Ni²⁺, Ni³⁺, and Ni⁴⁺ ions present in the delithiated Li_xNiO₂ structures for x = 0, 0.25, 0.5, 0.75, and 1.

The numbers of different Ni cations were derived by inspection of the PDoS of the single Ni ions (see below) and from the calculated atomic magnetic moments reported in Table S1 (ESI†) under the headings x = 0.25, x = 0.5, and x = 0.75, respectively. From Table S1 (ESI†), we can see that the magnetic moments close to 0 again indicate Ni⁴⁺ cations, while those around 0.8 are characteristic of the emerging Ni³⁺ cations. Finally, the magnetic moments around 1.4 again indicate Ni²⁺ (or close to Ni²⁺) cations.

We can see from the table that the number of Ni²⁺ ions decreases with x, while the number of Ni⁴⁺ increases. The number of Ni³⁺ ions increases until x = 0.5, then decreases. The active electrons are those in the orbitals and we can see that the numbers of these electrons are approximately 48 (54), 36, 24 (26), 12 (13), and 0, (considering the nominal values of 2 and 1 for Ni²⁺ and Ni³⁺, respectively), where in the parentheses are given the actual numbers compared with the nominal ones. These numbers indicate that roughly all the electrons lost by the cell in delithiation are lost by the Ni cations.

The second part of Table 3 reports the Δ parameters relative to the octahedra of the different Ni ions. These parameters are calculated as the difference between the length of the longest and shortest Ni–O bonds in the NiO₆ octahedra, as explained in the text and at the bottom of Table 2, and they are an indication of the Jahn–Teller distortion of the octahedron. We can see that the JT distortion is largest for the Ni³⁺ ions.

The different Ni oxidation states in the delithiated Li_xNiO₂ structures contribute to the electronic TDoS shown in Fig. 10, for x = 0, 0.25, 0.50, 0.75, and 1. The peaks and features within the energy range from −1 eV to 3 eV correspond, as for the fully lithiated case, to orbitals. To compare the evolution and shifts of the features, we have chosen for all the partially delithiated structures similar configurations for the arrangement of Li vacancies, specifically a chain-like pattern, with one chain for x = 0.75, two chains for x = 0.50 (see, for example, Fig. 7 case 3a), and three chains for x = 0.25.

Fig. 10 Total density of states (TDoS) of Li_xNiO₂.

In the case of x = 0.75, the chain arrangement of Li vacancies is not the most stable. Thus, we compare in Fig. 11 the TDoS of the chain arrangement (Fig. 11(a)) with the TDoS of the more stable square arrangement (Fig. 11(b)). The two TDoS are qualitatively similar, since the spin-up and spin-down states fall within the same ranges of energy. The main difference stems from the different heights of the peaks. Peaks are more pronounced for the chain arrangement. Thus, in the case of the chain arrangement the energies of the Ni–O orbitals fall preferably in narrower intervals, while for the square arrangement their energies are more smoothly distributed over their energy range. Both delithiated configurations have the same number of Ni²⁺, Ni³⁺, and Ni⁴⁺ cations (see Table 3). However, the chain arrangement has three kinds of oxygen ions: 24 O′′ ions, first neighbors to two Li vacancies, 24 O′ ions, first neighbors to one Li vacancy and 48 O ions bonded to three Li ions. These different kinds of oxygen ions are also arranged in similar chains. The less charged O′′ tend to bond strongly to Ni ions that become Ni⁴⁺. This structure has more groups of similar Ni–O bonds than the “square” pattern that has 24 O ions with no first-neighbor vacancies and 72 O′ ions with only one first-neighbor Li vacancy. The O ions are also arranged in a square pattern. In this case, the different kinds and lengths of Ni–O bonds are more uniformly distributed over the structure. This situation is shown in Fig. 12, where in the case of the chain arrangement the groups of chain-aligned O′′–Ni⁴⁺ bonds are clearly visible.

Fig. 11 Comparison of TDoS of two delithiated patterns at x = 0.75. (a) Chain delithiation model; (b) square delithiation model.

Fig. 12 Ball and stick representation of the two delithiated structures at x = 0.75 showing the change of oxidation states of Ni in relation to the Li vacancies: (a) chain model, (b) square model. Li is represented by green balls, while Ni is represented by three different colors depending on the oxidation state. Black represents Ni⁴⁺, blue Ni²⁺ and magenta Ni³⁺.

The main features in the TDoS of Fig. 10 are named A₁, , A₂, B₁, , , and B₂. The A peaks are constituted of spin-up orbitals and the B peaks of spin-down orbitals. A₁ and are occupied and A₂ is empty. We can see that the occupied part of progressively reduces with an increase in the number of Li vacancies due to the reduction of available electrons, and the A₁ peak height reduces until it disappears at x = 0.25.

To further investigate the relation between the features of the TDoS of Li_xNiO₂ and the oxidation states of the Ni cations, it is instructive to look at the Ni PDoS.

We can clearly distinguish the contributions of the differently oxidized Ni cations by looking at the Ni PDoS shown in Fig. 13. In the fully lithiated LNO, only Ni²⁺ and Ni⁴⁺ are present. The peaks A₁ and, to a much lesser extent, in the occupied part and B₂ in the unoccupied part are due to Ni²⁺ (see Fig. 4(d)), while Ni⁴⁺ (Fig. 4(c)) is characterized by only two wide empty features A₂ (spin up) and B₁ (spin down) in roughly the same energy range (0.3–1.2) eV. As Li vacancies are created, the number of Ni²⁺ cations decreases and that of Ni³⁺ increases. We can see from Fig. 13(b) that the Ni³⁺ cations contribute to the spin-up occupied peak near the Fermi level and to the wide empty spin-up feature A₂ at about 0.5 eV. Also the empty spin-down split peaks and in the energy range from 1 eV to 2 eV are attributed to Ni³⁺. These features can be compared with the Ni PDoS of the C2/m structure (Fig. 4(e) and (f)), where only Ni³⁺ cations are present. The and peaks widen and slightly shift with delithiation, but are recognizable up to x = 0.25, after which they start to disappear towards the full delithiation limit where only fully oxidized Ni⁴⁺ cations are present. With progressive delithiation we see that the occupied spin-up features change: the A₁ peak (Ni²⁺) decreases, while (Ni³⁺) increases. At Li concentration lower than x = 0.50, the peak decreases and disappears. The PDoS of Ni⁴⁺ (Fig. 12(c)) shows wide empty spin-up and spin-down features extending from 0 to 1.5 eV, similar to the TDoS of NiO₂ (Fig. 10 at x = 0). The peaks and features of Ni PDoS change with increasing delithiation; in particular, the Ni²⁺ and Ni³⁺ PDoS at x = 0.25 undergo considerable modification. One main reason is the shrinking of the cell volume (see Fig. 9), leading to shortening of the Ni–O bonds and related increase in Ni–O interaction. For the antibonding occupied orbitals this means a shift towards higher energies. Furthermore, the presence of residual JT-distorted orbitals in the shrunken volume leads to enlarged splitting of the bands. Indeed, the presence of Li vacancies introduces further differences in the strengths of the Ni–O bonds, which also depend on the different charges on the oxygen ions, with the less charged oxygens being those bonded to fewer Li ions.

Fig. 13 (a)–(c) Projected DOS (PDOS) of Ni-3d with different magnetic moments in Li_xNiO₂.

5. Mn for Ni substitution in Li_xNiO₂

5a. Substitutional Mn mixing energies and structural properties

We substituted 6, 12, and 18 Ni ions over the 48 transition metal sites in the template cell with Mn, realizing Mn concentrations y = 0.125, 0.25, and 0.374 of Li_xNi_(1−y)Mn_yO₂, all within the concentration range for which LNMO compounds have been synthesized in the α-NaFeO₂ layered structure.^50–52

The different Mn in-plane arrangements are reported in Fig. 14. Mn ions are arranged in the same way in all the transition metal layers of O3 stacking.

Fig. 14 Configurations of Mn ions in the transition metal planes. The purple dots show the positions of the Mn ions.

A, B, and C correspond to 12.5% concentration of Mn atoms, A₁, B₁, and C₁ to 25%, and A₂, B₂, and C₂ to 37.5%. The lowest energy configurations for each Mn concentration in the fully lithiated case were found to be A, B₁ and B₂, and we used them to optimize the 192-atom supercell for different concentrations of Li vacancies to obtain the optimized lattice parameters.

For comparison with the results of the previous section, we considered the same delithiated patterns. In Fig. 15, we report the same trend of the calculated mixing energies as for eqn (2). It is clear that Mn substitution is particularly favorable at low concentrations of Li vacancies, while below x = 0.25 (75% Li vacancies) the presence of Mn becomes less advantageous and for x < 0.2 the compounds with highest Mn concentration are unstable against phase separation. These results are qualitatively in agreement with the trend of previous calculations which considered only the x = 1 and x = 0 cases.⁵³ This result was interpreted in terms of an attractive interaction between the Ni and Mn ions at full lithiation becoming a repulsive one at full delithiation. This result also seems to be contrary to experimental findings where the presence of Mn seems to have a beneficial effect only at high cathode charge voltages. The discrepancy is apparent since in the literature the stabilization effect of Mn is attributed to the pillar effect, that is, to the fact that Mn at low concentrations would induce more Ni²⁺ defects in the Li layer than in pristine LNO and would favor Li⁺/Ni²⁺ mixing. It was shown⁵⁴ that the number of these Ni²⁺ defects in the Li layer remains unchanged during charge and discharge cycles, and their presence would avoid the collapse of the Li intercalation layers at high delithiation levels. Our calculations do not consider the presence of such defects, thus, the trend shown in Fig. 15 refers to the case of ideal cathodes without Ni/Li defects.

Fig. 15 Mixing energies of L_xNMO for different values of Li concentrations x.

The changes in the lattice parameters of the supercell as a function of Li and Mn concentrations x and y, respectively are shown in Fig. 16. The a and c parameters follow the same trends with x as L_xNO. Both a and c become longer with increasing Mn concentration. This agrees with the experimental results for LiNi_0.8−xCo_0.1Mn_0.1+xO₂, with 0.0 ≤ x ≤ 0.08, where the lattice parameters, especially the c lattice parameter, increase with increasing Mn content. The increase was said to be due to the substitution of Ni³⁺ cations by Mn⁴⁺ ions, which have a higher ionic radius, and the consequent reduction in the corresponding number of Ni³⁺ ions into larger Ni²⁺ (0.69 Å) to maintain charge neutrality.^55,56

Fig. 16 Crystal lattice parameters versus Li concentration x of Li_xNi_(1−y)Mn_yO₂ for the three Mn concentrations: y 12.5%, 25% and 37.5%, respectively.

This interpretation again stems from the assumption that in pristine LNO all Ni is in the Ni³⁺ oxidation state. Ni²⁺ or almost Ni²⁺ are already present in pristine LNO in our case; thus, the larger parameters are attributed to, as we will see below, to the partial substitution of Ni⁴⁺ by the larger Mn⁴⁺ ions and the increase in the total number of electrochemically active (Ni²⁺ + Ni³⁺) cations that are associated with larger octahedra (see Table 4). Apart from the very first value (x = 1.0), the a parameter of LNO (y = 0) follows this trend with the shortest a values, while the behavior of c does not. Indeed, only for y = 0.375 is the c lattice parameter of Li_xNi_(1−y)Mn_yO₂ longer than that in the system without Mn (y = 0). This non-linear trend in the c parameter with y is most likely due to the fact that the A, B₁, and B₂ Mn arrangements are inserted in lattices with different delithiation patterns for each x. As we will see later, the delithiation pattern and Mn distribution pattern strongly affect each other in determining the electronic configuration of the system. In this paper, we have not searched for the lowest energy structures where the total energy is minimized for both x and y simultaneously, thus, we do not expect a very precise trend in the calculated parameters.

Table 4 Number of different Ni ions and Δ parameters in Li_xNi_1−yMn_yO₂

LixNi1−yMnyO2	No. of oxidized Ni at Mn substitution				Jahn–Teller distortion parameter Δ in LixNi1−yMnyO2
y	x	Ni^2+	Ni^3+	Ni^4+	Ni^2+	Ni^3+	Ni^4+	Mn^4+
0	1	27	0	21	0.065	—	0.04	—
0.125	1	18	12	12	0.06	0.21	0.04	0.06
	0.75	10	16 (14 up, 2 down)	16	0.1	0.21	0.03	0.05
	0.5	2	20	20	0.03	0.19	0.06	0.06
	0.25	0	12	30	—	0.19	0.05	0.05
0.25	1	12	24	0	0.02	0.22	—	0.01
	0.75 (chain)	2	32 (20 up, 12 down)	2	0.07	0.25	0.06	0.05
	0.75 (square)	12	12	12	0.04	0.27	0.06	0.05
	0.5	2	20 (2 up, 18 down)	14	0.1	0.23	0.06	0.05
	0.25	0	12 (1 up, 11 down)	24	—	0.17	0.03	0.05
0.375	1	18	12	0	0.05	0.21	—	0.05
	0.75	9	18	3	0.07	0.21	—	0.08
	0.5	4	15	11	0.1	0.25	0.03	0.09
	0.25	2	8	20	0.06	0.18	0.06	0.09

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5b. Electronic and magnetic properties

First, looking at the estimated magnetic moments of the transition metals (Table S4, ESI†), we have verified that, as is generally assumed in the literature, the Mn ions are in the Mn⁴⁺ inactive oxidation state, in agreement with other calculations⁵³ and most experiments.⁵⁶

Table S4 (ESI†) reports the calculated magnetic moments of Ni ions for three configurations of LNMO corresponding to Mn content y = 0.125, 0.25, and 0.375 (headings Li_xNi_0.875Mn_0.125O₂, Li_xNi_0.75Mn_0.25O₂, LixNi_0.625Mn_0.375O₂). For each Mn concentration, different delithiation levels were considered: x = 1 (no Li vacancies), x = 0.75 (25% Li vacancies), x = 0.5 (50% Li vacancies), and x = 0.25 (75% Li vacancies). Mn⁴⁺ has magnetic moments around 2.6 μ_B, while for the Ni ions we again find the usual values.

Table 4 reports the evolution of the number of differently oxidized transition metals as a function of the Li and Mn concentrations x and y, summarizing the data in Table S4 (ESI†). The table also reports the corresponding Δ parameters. Again, the NiO₆ octahedra of Ni³⁺ show the largest value of Δ.

From Table 4, we can obtain the following information:

(i) The number of active electrons N in the orbitals is determined by the Li concentration. Indeed, if we assign two electrons to Ni²⁺ and one to Ni³⁺, we can see that for x = 1 we have about N = 48 electrons in , for x = 0.75 N = 36, for x = 0.50 N = 24, and for x = 0.25 N = 12, for all y. This result means that the missing electrons due to delithiation are again taken mainly from the Ni cations.

(ii) The number of Ni⁴⁺ cations diminishes as the number of Mn⁴⁺ cations increases. Their sum remains approximately the same and increases with an increase in delithiation.

(iii) Comparing similarly delithiated structures, at x = 0.75 and 1, with and without Mn/Ni substitution, we can see an increase in electrochemically active Ni ions, particularly in Ni³⁺.

(iv) Unlike the results of Table 3, for y ≠ 0, the particular delithiation arrangement matters in the redistribution of electrons and the numbers of Ni²⁺, Ni³⁺, and Ni⁴⁺ cations. This is shown by the two entries corresponding to x = 0.75 and y = 0.25, where only the in-plane arrangement of Li vacancies is different, for the chain and the square patterns examined above. This means that the arrangement of Mn ions and the arrangement of Li vacancies interact. The square Li vacancy arrangement together with the square arrangement of Mn ions produces the most stable of the structures studied for the same x and y concentrations.

(v) The relationship between Mn⁴⁺ ions and Li vacancies can also be seen in a few antiferromagnetic couplings arising in the case of chain delithiation and low Mn concentration. In the case of chain delithiation and low Mn concentration, structural optimization produces a number of singly occupied spin-down orbitals on Ni ions which were previously Ni⁴⁺ ions in the absence of Mn. The antiferromagnetic coupling between the Mn and Ni ions in this case is mediated by the less charged O′′ oxygen atoms, first neighbors to two Li vacancies, present only in the chain delithiation model. The Ni³⁺ cations with occupied spin-down orbitals are, in absence of Mn ions, Ni⁴⁺ ions arranged in the chains bonded to four O′′ ions (two below and two above) shown in Fig. 12(a). In this case, electron transfer occurs from and to the orbitals through the pσ orbitals of O′′.^36,57 However, for Li concentrations as high as x = 0.75, the chain delithiation model is less stable than the square model, where this superexchange effect does not take place. On the other hand, at high levels of delithiation, the O′′ oxygen ions could play an important role in mediating the interaction between Ni and Mn, which is supposed to become repulsive in this limit.

To analyze the global effect on the electronic structure of Mn doping, we calculated the TDoS and local magnetic moments. In Fig. 16, we report the modification of the TDoS with the degree of delithiation. For the occupied orbitals near the Fermi level, we again see a progressive diminishing of the A peaks. Between the B₁ and B₂ empty peaks, a number of new structures emerge, some due to the emergence of Ni³⁺ ions as seen above and others due to the Mn⁴⁺ ions. It is shown in Fig. 17 that the Mn orbitals (hybridized Mn d–O p) are empty and fall above 1 eV, with a spin-up feature between 1 eV and 2 eV (in correspondence with the spin-down peaks due to N³⁺) and two negative features between 2 eV and 3.5 eV (the first one in correspondence with the spin-down structures of N²⁺). These features become larger and merge with an increase in Mn concentration. There are no occupied Mn orbitals, in agreement with the assignment of the Mn⁴⁺ oxidation state to the Mn ions. Around the Fermi level, the active electronic orbitals are still provided by the hybridized Ni d–O p orbitals, where the Ni ions are Ni²⁺ and Ni³⁺. Inspection of the Ni ion PDoS around the Fermi level shows similar features to those already reported in Fig. 13. These results confirm that Mn in Li_xNi_(1−y)Mn_yO₂ is electrochemically inactive and its only role is to stabilize the structure. Indeed, it was found that the disordered Ni/Li tends to decrease during cycling and not to increase, which is one reason for the greater stability provided by Mn ions where the fixed number of Ni²⁺ in the Li layer sustains the structure from collapse at high charge levels.^54,58,59

Fig. 17 (a) and (b) Total density of states (TDoS) of Mn-substituted LNO as a function of Li concentration x for y = 0.125 and 0.375. The TDoS of LiMnO₂ refers to ferromagnetic ordering for comparative reasons. Antiferromagnetic ordering between Mn ions is indeed more stable.

It is generally thought that Mn⁴⁺ are responsible for the generation in LNO of Ni²⁺ cations since, for charge neutrality, in layered LiNiO₂ and LiMnO₂ the transition metal cations have to be in the Ni³⁺ and Mn³⁺ states, respectively. Thus, the presence of Mn⁴⁺ ions should be balanced with the simultaneous reduction of Ni³⁺ to Ni²⁺ to compensate for the change of Mn³⁺ to Mn⁴⁺. This observation has aroused questions about how to explain the fact that the discharge potentials measured in layered LiNiO₂ and LiNi₀₅Mn_0.5O₂ in the hexagonal phase are practically identical^60,61 given that different redox reactions are expected in LiNiO₂, where only Ni³⁺/Ni⁴⁺ is supposed to be active, and in LiNi_0.5Mn_0.5O₂, where both Ni²⁺/Ni³⁺ and Ni³⁺/Ni⁴⁺ should occur.⁵³ The implication of only Ni³⁺ ions being present in LNO is due to the fact that the calculations were performed using small supercells for rhombohedral and monoclinic symmetries and these calculations invariably produce only Ni³⁺ ions, as we have seen above. Moreover, the presence of a few percent of Ni²⁺ defects in the Li layer of LNO cannot explain the discrepancy, but the charge disproportionation we have found using the supercell can, since the numbers of Ni²⁺ are similar in LNO and in LiNi₀₅Mn_0.5O₂ and the charging will create similar numbers of Ni³⁺ and Ni⁴⁺ ions in both systems.

6. Summary and conclusions

In summary, this paper deals with the layered oxide LNO and LMNO materials proposed for Ni-rich cathodes. We first replicated 16 times the primitive cell of the R3̄m space group, the space group of LNO observed experimentally by XRD, and relaxed both cell parameters and internal positions. The supercell relaxation leads to a large decrease in the total energy accompanied by charge and size disproportionation of the NiO₆ octahedra. The charge disproportionation has a large effect on the Ni oxidation states, since there are only Ni²⁺ and Ni⁴⁺ ions, whereas in the smaller primitive unit cells, considered in previous computational studies, all Ni ions are Ni³⁺. The Ni²⁺ and Ni⁴⁺ ion arrangement forms an ordered superstructure in the ab layers and the superstructure still belongs to the observed hexagonal symmetry. The Ni–O bond lengths form a distribution, as found experimentally, around two average bond lengths whose values are also in agreement with experiment and contrary to the behavior expected for a monoclinic crystal structure. The gain in energy obtained in the atomic relaxation of the supercell compared to that of the primitive cell is much larger than that obtained with an analogous supercell of monoclinic symmetry, although the monoclinic structure still remains more stable. We conclude that the choice of unit cell is important for this class of material and an even larger supercell, or perhaps the inclusion of Ni defects in the Li layer (always present in real samples) may favor the observed hexagonal phase over the monoclinic one at full lithiation. We have found that optimizing larger unit cells leads to the emergence of new structural and electronic behaviors, which remain hidden if using primitive or smaller cells.

Further, we used the calculated superstructure as a template for a study of delithiation in the hexagonal LNO structure. We calculated the energies of a number of structures for different concentrations of Li and studied the evolution of the structural parameters and the electronic structure with the number of Li vacancies. The changes in the a and c lattice parameters are in agreement with previous calculations, and with experiment, at least for the Li concentration for which LNO is in the hexagonal structure. We have found that delithiation leads to changes in the number of Ni ions; in particular, Ni²⁺ become Ni³⁺ and Ni⁴⁺, the dominant Ni ion at high battery charging levels. Ni³⁺ and its related JT octahedral distortions become the dominant ions at Li concentrations of LNO in the monoclinic phase.

Finally, we analyzed the changes in the properties of the delithiated structures when 12.5%, 25% and 37.5% of Ni ions are substituted with Mn ions. We found that the way Mn affects the lattice parameters and electronic structure depends on the arrangement of the Mn ions in the transition metal layers and Li vacancies in the Li layers. However, in all cases, the Mn ions are always in the Mn⁴⁺ oxidation state and do not participate actively in the redox reactions determining the battery charge and discharge potentials, since the important electronic states around the Fermi level involved in the reactions belong only to the Ni²⁺ and Ni³⁺ ions. The presence of a large number of Ni²⁺ ions in pristine LNO can explain the similarity between the LNO and LiNi₀₅Mn_0.5O₂ discharge potentials found experimentally, since both Ni²⁺/Ni³⁺ and Ni³⁺/Ni⁴⁺ redox pairs can be present in the initial stages of charging in both systems.

Author's contributions

Saleem Yousuf performed calculations. Contribution in writing the paper. Made figures and tables. Interpretation of simulation results. Md Maruf Mridha performed calculations. Rita Magri Conceptualization. Validation of calculations. Interpretation of simulation results. Writing of paper. Supervision. Acquisition of funding.

Data availability

The raw data required to reproduce these findings are available upon request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This research was developed under the framework of the BAT4EVER project that has received funding from the European Union's Horizon 2020 research and innovation program under Grant Agreement No. 957225. The authors would also like to acknowledge the CINECA HPC facility for the approved ISCRA B (IscrB_CATCH22 and IscrB_LION-CAT). R. M. acknowledges the PNRR MUR project ECS_00000033_ECOSISTER. R. M. acknowledges the PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR) – MISSIONE 4 COMPONENTE 2, “Dalla ricerca all’impresa” INVESTIMENTO 1.4, Potenziamento strutture di ricerca e creazione di “campioni nazionali di R&S” su alcune Key Enabling Technologies, finanziato dall’Unione europea – NextGenerationEU – Progetto identificato con codice CN00000023. Titolo “Sustainable Mobility Center (Centro Nazionale per la Mobilità Sostenibile – CNMS)” – Spoke 12 – Avviso MUR 3138/2021 modificato con DD 3175/2021. This manuscript reflects only the authors’ views and opinions, neither the European Union nor the European Commission can be considered responsible for them.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3ma00906h

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