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Cite This: Chem. Mater. 2018, 30, 607−618

Simulating Charge, Spin, and Orbital Ordering: Application to Jahn− Teller Distortions in Layered Transition-Metal Oxides Maxwell D. Radin and Anton Van der Ven* Materials Department, University of California, Santa Barbara, Santa Barbara, California 93106, United States S Supporting Information *

ABSTRACT: The degrees of freedom associated with orbital, spin, and charge ordering can strongly affect the properties of many crystalline solids, including battery materials, high-temperature superconductors, and naturally occurring minerals. This work reports on the development of a computational framework to systematically explore the ordering of electronic degrees of freedom and presents results on orbital ordering associated with Jahn−Teller distortions in four layered oxides relevant for Li- and Na-ion batteries: LiNiO2, NaNiO2, LiMnO2, and NaMnO2. Our calculations reveal a criterion for the stability of orbital orderings in these layered materials: each oxygen atom must participate in two short and one long transition-metal/oxygen bond. The only orderings that satisfy this stability criterion are row orderings, such as the “zigzag” ordering. The near degeneracy of such row-orderings in LiNiO2 suggests that boundaries between domains with distinct but symmetrically equivalent Jahn−Teller distortions will be relatively low in energy. Based on this result, we speculate that a microstructure consisting of nanoscale Jahn−Teller domains could be responsible for the apparent absence of a collective distortion in experiments on LiNiO2.

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INTRODUCTION

Unanswered questions about charge, spin, and orbital ordering motivate the use of first-principles modeling to explore electronic degrees of freedom. While the ordering of chemical species is in some sense straightforward to study computationally, the ordering of electronic degrees of freedom is less amenable to simulation. This is primarily because imposing a particular charge, spin, or orbital ordering within first-principles calculations is nontrivial: one cannot coerce electrons into a particular state. Additionally, orbital ordering is more complex than chemical ordering because different possible electronic states of an atom may be related by symmetry. (In contrast, two distinct chemical species are not related by symmetry.) A number of methods have been developed to navigate the energy landscape associated with electronic degrees of freedom in first-principles simulations. Techniques for biasing density functional theory calculations toward particular magnetic states include the initialization of atomic magnetic moments, constraints on the total spin in the simulation cell, and energy penalties for atomic moments.28 A common strategy for imposing charge/orbital ordering is to perturb the initial atomic positions26,29,30 based on an educated guess. Charge ordering can also be induced by the application of different U parameters to different sites.31−33 Gradually increasing the U parameter (U ramping) has also been suggested as a route toward finding the

The properties of many crystalline materials are strongly affected by electronic degrees of freedom associated with each site, resulting in charge ordering, spin ordering, and orbital ordering. (As discussed below, orbital ordering is often intimately related to the Jahn−Teller effect.) Magnetite (Fe3O4) has played an important role historically: it was the first material found to be magnetic1 and, some two millennia later, the first material found to be charge ordered.2−4 The ground-state charge and orbital ordering, however, has only recently been determined.5 The “stripe” charge-ordering in perovskite manganates6,7 is another well-known example of simultaneous charge, spin, and orbital ordering; similar stripe ordering phenomena are thought to play an important role in cuprate high-temperature superconductors.8 Layered alkali transition-metal oxides, widely studied because of their ubiquitous application as battery electrode materials (e.g., LiCoO2),9,10 also exhibit complex charge, spin, and orbital ordering phenomena. In particular, the layered Ni11−14 and Mn15−18 oxides exhibit strong Jahn−Teller activity associated with the eg electrons of Ni3+ and Mn3+ which results in orbital ordering.19 However, a number of questions remain unanswered. For example, the nature of the ground state of LiNiO2 remains controversial, despite a large number of experimental12,20−24 and theoretical studies.19,25−27 Furthermore, the energy spectrum of excited states in the layered oxides (which determines the finite-temperature behavior) is not well understood. © 2017 American Chemical Society

Received: July 21, 2017 Revised: December 20, 2017 Published: December 24, 2017 607

DOI: 10.1021/acs.chemmater.7b03080 Chem. Mater. 2018, 30, 607−618

Article

Chemistry of Materials global minimum of the energy landscape associated with electronic degrees of freedom.34 Arguably the most general strategy, however, is the imposition of occupation matrices (occupation matrix control),35,36 as in principle it can be used to impose spin, orbital, and charge ordering. However, a systematic methodology for exploring electronic ordering is absent. This report details a new framework that combines occupation matrix control (OMC) with classical lattice alloy theory to enable the systematic study of orderings of electronic degrees of freedom with first-principles calculations. As a case study, we apply this occupation-matrix lattice-alloy framework to orbital, spin, and charge ordering in the four layered transition-metal oxides: LiNiO2, LiMnO2, NaNiO2, and NaMnO2. In these systems, spin ordering can arise from the unpaired electrons on Ni and Mn; charge ordering can result from the disproportion of Ni3+ into Ni2+ and Ni4+; and orbital ordering occurs as a consequence of the Jahn−Teller activity of Ni3+ and Mn3+ in an octahedral environment. (The partial deintercalation of these compounds can also result in charge ordering;37,38 however, the calculations presented here are restricted to the fully intercalated phases.) We have chosen to focus on LiNiO2, LiMnO2, NaNiO2, and NaMnO2 because of the extensive interest in them from both the solid-state physics community17,18,23,24,39 and the battery community.9,10,40−42 Much of the interest from the physics community stems from the potential for these materials to realize exotic ground states associated with magnetic frustration and orbital degeneracy. For the battery community, layered LiNiO 2 and LiMnO 2 represent end members of the LiNixMnyCo1−x−yO2 (NMC) and LiNixCoyAl1−x−y O2 (NCA) families, which are extensively used for intercalation electrodes in commercial Li-ion batteries.9,10 The first main result is a criterion for the stability of orbital orderings associated with Jahn−Teller distortions in the Ni and Mn layered oxides: each oxygen must participate in two short M−O bonds and one long M−O bond. Configurations which did not satisfy this criterion were found to be locally unstable: they would spontaneously relax to different orderings in the absence of any constraints. The second main result is the large number of nearly degenerate row orderings of orbital occupancy in LiNiO2. This suggests that the domain boundary energy is quite low and that the apparent absence of a collective Jahn−Teller distortion in LiNiO2 might be due to a highly domained nanostructure. The high stability of such boundaries appears to be related to interactions between the Li+ ions and occupied eg* orbitals.

Figure 1. Crystal structure of a layered oxide AMO2 with the O3 (αNaFeO2) structure.

t2g states into states with a1g and eg symmetry. (The two eg states of the Oh point group, on the other hand, remain degenerate when the octahedra are compressed.) Because the orbital degeneracy of Ni3+ and Mn3+ does not directly involve t2g states, their splitting does not appear to play a major role in the compounds under consideration presently. (However, the splitting of t2g states does play an important role in layered Ti43,44 and V45 oxides, where the t2g manifold is only partly occupied.) The presence of a single electron in the doubly degenerate eg level leads to a local Jahn−Teller distortion in the Ni and Mn layered oxides.19 In essence, the distortion causes one of the eg levels to go down in energy and the other to go up; this leads to an overall lowering of the energy because only one of the eg states is occupied. The distortion observed in layered Ni and Mn oxides is a “positive” Jahn−Teller distortion: two metal− oxygen bonds lengthen while the other four contract. There are three symmetrically equivalent positive distortions, as shown in Figure 2. Figure 2 also shows the shapes of the eigenstates in the distorted octahedron that are derived from the eg orbitals.

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CRYSTALLOGRAPHY The structures of LiNiO2, LiMnO2, NaNiO2, and NaMnO2 are all derived from the α-NaFeO2 structure (shown in Figure 1), also referred to as O3. This structure is a cation-ordered rocksalt, with the alkali and transition metals residing in alternating (111) planes. All of the cations are coordinated octahedrally by oxygen. In a perfectly octahedral environment, the five transition-metal d levels are split into two degenerate eg states whose lobes point toward the anions and three degenerate t2g states whose lobes point between anions. A further subtlety is that the ordering of cations breaks the cubic symmetry of the rocksalt host, and as a result the octahedra are slightly compressed or stretched along the (111) rocksalt axis. This reduces the point group symmetry of the cation sites from Oh to D3d, resulting in a further splitting of the

Figure 2. Three symmetrically equivalent positive Jahn−Teller distortions of an octahedral complex.

This picture is complicated by the hybridization of transitionmetal eg orbitals with oxygen 2p orbitals, which results in the formation of bonding and antibonding states with eg symmetry. In the Ni and Mn oxides, the bonding states are entirely filled and the Jahn−Teller effect arises from the antibonding states (often referred to as eg*).46 Antibonding states with lobes pointed along lengthened bonds will be lower in energy than those with lobes pointed along contracted bonds, as indicated in Figure 2. A contraction of two bonds and lengthening of four (not shown) is referred to as a negative distortion. Such negative distortions have been reported, for example, in the case of 608


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Chemistry of Materials degeneracies in the t2g states in layered NaVO2.45 However, since degeneracies in the eg states generally lead to a positive distortion in octahedral environments,19,47−49in this study we do not consider negative distortions. The six orbitals in Figure 2 are not linearly independent: they span a vector space of dimension two. To illustrate this graphically, Figure 3 shows how any of these six orbitals ψn(r)

Figure 2 and Figure 3 illustrate the relationship between the Jahn−Teller distortions and orbital ordering. The term distortion refers to a displacement of atoms, whereas orbital ordering refers to the occupation of different eg orbitals at each site. One describes the state of the ions while the other describes the state of the electrons. When the electrons reside in their ground state, there is a one-to-one mapping between distortions and orbital occupancy, as illustrated in Figure 2. In this work, we assume that electrons reside in the ground state with regards to orbital occupancy; this is justified when the energy splitting induced by the distortions is significantly larger than the thermal energy. Figure 4 shows several important orbital orderings (or equivalently, orderings of Jahn−Teller distortions) in the layered crystal structure. The collinear ordering, shown for a single MO2 layer in Figure 4a, has the same orbital occupancy at each transition-metal site; this corresponds to the local environment of each site being distorted in the same manner. The simplest noncollinear ordering within a layer is the zigzag (i.e., herringbone) ordering, which consists of alternating rows of two different orientations as shown in Figure 4b. Note that there are actually two zigzag orderings which exhibit the inplane ordering of Figure 4b; these two differ in the relative alignment of Jahn−Teller orderings from one layer to the next. Another important ordering is the trimer ordering, Figure 4c.

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Figure 3. Illustration of how any eg orbital can be constructed as a linear combination of two reference eg orbitals. The circle represents orbitals that have a norm of unity.

COMPUTATIONAL METHODS

The orbital, charge, and spin orderings are considered here in the context of lattice alloy theory.50−53 Within this model, each possible orbital, charge, or spin state is considered as a discrete species. For example, six species that represent distorted high-spin Mn3+ in an octahedral environment arise from the three orientations for a positive Jahn−Teller distortion and the two possible spin states, s = 2 and s = −2. (These spin states arise from the four unpaired electrons each with spin 1/2.) The use of such a methodology requires a mapping between first-principles simulations and the lattice model: we must be able to impose (or at least bias) first-principles calculations toward an

can be expressed as ψn(r) = aψ1(r) + bψ2(r), where ψ1(r) and ψ2(r) are the two orbitals associated with distortion 1. The arrows indicate the basis formed by ψ1(r) and ψ2(r). The points in Figure 3 that correspond to the six ψn(r) of Figure 2 are marked with black dots; the six points corresponding to −ψn(r) are also marked.

Figure 4. Examples of Jahn−Teller orderings within the layers of a layered oxide. Only the lengthened M−O bonds are shown. The dashed black lines indicate the unit cell of each ordering, while the short solid black lines indicate the orientation of the Jahn−Teller distortion at each transitionmetal site. 609


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Chemistry of Materials −1

arbitrary ordering of electronic states and also must be able to determine if, after relaxation, the configuration has retained this state or relaxed into a different one. In the present work, we employ occupation matrices as the foundation for such a mapping. Occupation matrices characterize the electronic state of each site in terms of the single-particle orbitals obtained from electronic structure calculations. For a given orbital angular momentum channel S , the spin-up and spin-down occupation matrices are each described by a (2S + 1) × (2S + 1) matrix defined by54

nijσ =

∑ f bσk ⟨ψbσk|ϕi⟩⟨ϕj|ψbσk⟩ b,k

(Sn̂ )σ = D(S ̂ )nt(S)̂ σ D(S)̂

where D(Ŝ) is the (2S + 1) × (2S + 1) representation that describes how the reference atomic orbitals are transformed by symmetry operations, and t(Ŝ) = ±1 indicates whether or not a symmetry operation includes time reversal (which has the effect of swapping spin-up and spin-down states). Algorithms developed by Thomas for the determination of the permutation representation from the action of symmetry on n and use of the permutation representation in enumerating unique orderings will be described in a forthcoming publication.55 In order to check whether electronic degrees of freedom changed during relaxation, we define a distance metric between occupation matrices:

(1)

where |ϕi⟩ and |ϕj⟩ are reference atomic orbitals, σ = ±1 is the spin channel, ψσbk is the Kohn−Sham wave function of band b with spin σ at a particular k point, f σbk is the occupation (between zero and one) of that state. The value of nijσ in essence measures to what extent any electrons with spin σ possess both |ϕi⟩ and |ϕj⟩ character. We adopt the following notation: while n is a (2S + 1) × (2S + 1) × 2 array (i.e., a pair of matrices), nσ will refer to the (2S + 1) × (2S + 1) matrix corresponding to spin channel σ. Within the lattice alloy picture, each possible “species” (i.e., electronic state of an atom) is defined by a pair of occupation matrices describing the spin-up and spin-down electron configuration. We will denote the pair of occupation matrices for a species p as np. The application of a symmetry operation Ŝ that is respected by the host structure can transform one species p into a different species p′: Ŝnp =

d(n , n′) =

1 0 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0⎞ ⎟ 0⎟ 0⎟ ⎟ 1⎟ 0⎟ ⎟ 0⎠

σ

(4)

Although other distance metrics can be defined, the metric employed here is physically motivated because it is invariant with respect to rotations and reflections (see Supporting Information). We assign the occupation of each site after relaxation based on which reference occupation matrix is closest to the relaxed occupation matrix. In the calculations presented in this study, we considered three types of lattice alloy models. The first case represents different orbital orderings of M3+ ions with ferromagnetic spin ordering; this is similar to a ternary alloy with the three species representing the three symmetrically equivalent orientations of the positive Jahn−Teller distortion. The second case allows for different magnetic orderings and has six species arising from the three orbital orientations and two spin states. (We presently assume the spin to be collinear.) The third case considered was a disproportionated model, corresponding to a binary alloy between M2+ and M4+. The symmetrically inequivalent orderings were enumerated with the Clusters Approach to Statistical Mechanics (CASM) code.56−59 This enumeration accounted for the permutation of species upon application of symmetry operations to eliminate symmetrically equivalent orderings. Time-reversal symmetry was not explicitly included in this process, but enumeration was restricted to orderings where the number of spin-down sites was no greater than the number of spin-up sites. Once enumerated, the electronic states were imposed in DFT calculations using occupation-matrix control (OMC), as implemented by Allen and Watson.35,36 OMC adds a nonlocal bias potential (of the same form as the U on-site Coulomb correction) that invites the electrons to relax into a target electronic configuration, specified in terms of an occupation matrix. The target occupation matrices used for the different Jahn−Teller orientations in each compound were taken to be the occupation matrices obtained by fully relaxing the three variants of the collinear Jahn−Teller ordering of that compound. Because energies obtained with the OMC bias potential on are not physically meaningful, a second DFT calculation was performed after relaxing with the bias potential. We tried both performing full relaxations starting from the geometries and wave functions obtained with OMC and performing static calculations keeping the OMC geometry fixed. For structures that retained the same electronic ordering upon full relaxation, we found that the single-point and fully relaxed energies were nearly identical, as shown in Figures S1 and S2 of the Supporting Information. DFT calculations were performed with the Vienna Ab initio Simulation Package (VASP)60−63 with projector augmented-wave (PAW) pseudopotentials.64 All calculations were spin polarized, employed plane-wave basis sets with 600 eV energy cutoffs, and sampled the Brillouin zone with a k-point mesh density of 26 Å or more. Relaxations used Gaussian smearing of width 0.1 eV and were converged to within a force convergence criterion of 0.02 eV/Å; these were followed by single-point calculations using the tetrahedron method with Blöchl corrections65 in order to obtain accurate energies. Except where otherwise stated, calculations employed the PBE GGA exchange-correlation functional66 with the rotationally invariant

and Ppp′(Ŝ) = 0 if Ŝnp ≠ np′. P(S)̂ is an M × M matrix, where M is the number of species. For example, a 3-fold rotation acting on the six states of Mn3+ would be represented by

0 0 1 0 0 0

∑ (nijσ − nij′σ )2 = ∑ tr[(nσ − nσ′ )† (nσ − nσ′ )] i,j,σ

np′. In other words, unlike a conventional lattice alloy model, the species considered here can be anisotropic. For example, consider the effect of a 3-fold rotation operation on the six different Jahn−Teller orientations and spin states of Mn3+ described above: this rotation will cyclically permute the three spin-up states and also cyclically permute the three spin-down states (cf. Figure 2). The anisotropy of the species fundamentally changes the enumeration of inequivalent orderings: configurations that are equivalent in an alloy of isotropic components may be inequivalent in an alloy of anisotropic components, and orderings that are equivalent in the alloy of anisotropic components may be inequivalent in an alloy of isotropic components. For example, consider the volume 1 configurations obtained by decorating the α-NaFeO2 structures with the six states of Mn3+. There are six such configurations, but all are symmetrically equivalent; therefore, only one needs to be calculated. On the other hand, if six unique isotropic species (e.g., different chemical elements) are permitted to occupy the transition-metal site in the α-NaFeO2 structure, then there are six unique configurations with the periodicity of the volume 1 primitive cell. The enumeration of symmetrically unique orderings can be accomplished once it is determined how the symmetry operations permute the species. The action of symmetry operations on the species is described by the permutation representation Ppp′(Ŝ) = 1 if Ŝnp = np′

⎛0 ⎜ ⎜1 ⎜0 P(Ĉ3) = ⎜ ⎜0 ⎜0 ⎜ ⎝0

(3)

(2)

where the first three species are the spin-up states and the second three the spin-down states. The permutation representation can be determined from the action of the symmetry operations on the elements of the occupation matrices. As discussed in the Supporting Information, the occupation matrices transform according to 610


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Chemistry of Materials on-site Coulomb corrections of Dudarev et al.67 A U − J value of 5 eV was used, representing typical values applied to Mn and Ni in oxides.67−69 For comparison, the energies of Jahn−Teller orderings in LiNiO2 with a smaller U − J of 2 eV are provided in Figure S3 of the Supporting Information. The smaller value of U − J has little effect on the relative energies of different orderings, although it does make the distorted structures somewhat less stable compared to the undistorted geometry; this result is in agreement with prior calculations examining the effect of U on Jahn−Teller distortions in LiNiO2.25 A few calculations were performed with the HSE hybrid functional70,71 as spot checks, as discussed below. These employed a slightly coarser kspace grid density of 18 Å and lower plane-wave cutoff of 530 eV.

this is 12 meV/LiNiO2 lower in energy than the collinear ordering, consistent with prior DFT+U studies.25,26 Other zigzag-type orderings involve repeat units larger than two. For example, the second lowest energy structure found for LiNiO2 repeats every three rows. The high stability of the zigzag orderings relative to the collinear ordering appears to be connected to interactions between eg* states and lithium ions. This Ni−O−Li interaction has previously been observed to play an important role in the relative energies of lithium/vacancy orderings in LixNiO2,37 and the site-projected density of states (not shown) confirms that the eg* states hybridize to some degree with Li s and p states. The first point to consider is how Li 2s and 2p orbitals can couple occupied eg* states in different layers. In the collinear structure, the two eg* states that bond with a Li site are 180° apart (cis configuration); as shown in Figure 6, this can lead to

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RESULTS AND DISCUSSION We now apply this framework to study spin and Jahn−Teller orderings in two nickel oxides (LiNiO2 and NaNiO2) and two manganese oxides (LiMnO2 and NaMnO2). For the nickel oxides, we consider different orderings of Jahn−Teller orientations but restrict the magnetic ordering to ferromagnetic because the intraplane interactions are ferromagnetic in LiNiO 2 72,73 and NaNiO 2 . 13,74−76 On the other hand, LiMnO219,77−79 and NaMnO217,18,39,80,81 possess antiferromagnetic in-plane interactions, and so it is necessary to include spin ordering in order to capture the ground state. LiNiO2. Figure 5 shows the single-point energies for structures that were relaxed with OMC. The type of symbol

Figure 6. Illustration of interaction between Li and eg* orbitals in the collinear Jahn−Teller ordering. The dotted lines indicate the Li−O bonds which desire to contract because of interactions between the Li site and the electron in the eg* state.

a favorable bonding interaction between the oxygen and the lithium orbitals. A similar 180° interaction is present in the ground state zigzag structure. There is a higher energy structure which has the same in-plane zigzag ordering but differs in the stacking of the layers. Because this structure has a 90° Ni−O− Li−O−Ni interaction (trans configuration), we expect bonding between layers to be less favorable. (We shall refer to these two structures as cis- and trans-zigzag orderings.) The importance of Ni−O−Li bonding is supported by the fact that the transzigzag structure is higher in energy than the collinear and ciszigzag orderings, as shown in Table 1.

Figure 5. Energies of Jahn−Teller orderings in LiNiO2 relative to the undistorted state. The energies represent the single-point energy at the geometry obtained with occupation-matrix control (OMC). Blue circles represent structures that retained the ordering when further relaxed without OMC, whereas orange crosses represent structures that spontaneously relaxed into different orderings.

indicates whether a structure is locally stable when the OMC bias potential is removed: blue circles indicate that the structure retained its Jahn−Teller ordering, while orange crosses indicate that it relaxed to a different ordering. (For the orderings that are stable, the single-point and relaxed energies are essentially identical; see Figure S1 in the Supporting Information.) Figure 5 shows that there are many competitive low energy states that involve zigzag-type orderings, with rows of Ni being distorted in the same direction. Many of the enumerated orderings have the same in-plane ordering but differ in how the in-plane ordering is stacked along the c-axis of the layered crystal. DFT+U predicts the lowest energy ordering to be a zigzag ordering that repeats every two rows; after full relaxation,

Table 1. Fully Relaxed Energies Bond Lengths in Selected LiNiO2 Jahn−Teller Orderings collinear energy (meV/LiNiO2) Ni−O distance (Å) Li−O distance, (Å)

611

along occupied eg * along unoccupied eg *

zigzag (cis)

zigzag (trans)

−108 2.137, 1.892 2.139

−120 2.116, 1.900 2.035

−103 2.105, 1.907 2.033

2.102

2.096, 2.238

2.101, 2.243


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Chemistry of Materials

structures with the HSE functional (Table 2).70,71 Like PBE+U, HSE predicts the collinear and zigzag Jahn−Teller orderings to

Further geometric considerations suggest that the cis-zigzag ordering may have more favorable Ni−O−Li interactions than the collinear ordering. Table 1 shows the lengths of Li−O bonds in the collinear, cis-zigzag, and trans-zigzag structures. In the collinear case, the Li−O bonds aligned with the occupied eg* orbitals are slightly longer than those aligned with the unoccupied eg* orbitals (2.139 Å vs 2.102 Å). That the bonds aligned with the occupied orbitals are longer can be attributed to the strain in the plane of the layers arising from the collective Jahn−Teller distortion. In contrast, in the zigzag orderings the Li−O bonds aligned with the occupied eg* states are significantly shorter than those aligned with unoccupied eg* states. We speculate that the shorter Li−O distances along occupied eg* orbitals may contribute to the stability of the ciszigzag structure, as these shorter bonds will enhance Ni−O−Li bonding. For the collinear and cis-zigzag orbital orderings, we additionally considered several antiferromagnetic spin orderings (seven in the collinear structure and nine in the cis-zigzag structure). For a given orbital ordering, the antiferromagnetic spin orderings were within 3 meV/LiNiO2 of the ferromagnetic spin ordering−an energy difference that is comparable to the numerical errors in the calculation. None of the antiferromagnetic configurations was found to be lower in energy than the ferromagnetic cis-zigzag configuration. We also explored the previously proposed hypothesis that LiNiO2 could disproportionate, resulting in a mixture of Ni2+ and Ni4+.25 Figure 7 shows the energies calculated for all Ni2+/

Table 2. Fully Relaxed Energies of Selected LiNiO2 Structures relative energy (meV/LiNiO2) collinear Jahn−Teller zigzag Jahn−Teller disproportionated

PBE+U, U = 5 eV

HSE06

0 −11.9 −5.2

0 −0.2 68.9

be nearly degenerate. However, while PBE+U predicts the disproportionated cell to be nearly degenerate with these orderings, HSE predicts the disproportionated structure to be nearly 70 meV higher in energy than the Jahn−Teller structures. That the disproportionated structure is less stable than the Jahn−Teller distorted structures is supported by neutron and EXAFS experiments, which show that there are more short Ni−O bonds than long ones.11,12 (A positive Jahn− Teller distortion results in two long bonds and four short, while disproportion results in an equal number of short and long.) However, thermal excitations may nevertheless induce the formation of a substantial concentration of Ni2+ and Ni4+. Furthermore, crystallographic defects such as Li vacancies or Ni in the Li layer may induce the formation of Ni2+ and Ni4+.82,83 NaNiO2. DFT calculations indicate that the ordering of Jahn−Teller distortions in NaNiO2 is rather different from that in LiNiO2. As shown in Figure 8, the collinear distortion is

Figure 7. Energies of disproportionated Ni2+/Ni4+ orderings in LiNiO2 relative to the undistorted state having a uniform distribution of Ni3+. The energies represent the single-point energy at the geometry obtained with occupation-matrix control (OMC). Blue circles represent structures that retained the ordering when further relaxed without OMC, whereas orange represents structures that spontaneously relaxed into different orderings.

Figure 8. Energies of Jahn−Teller orderings in NaNiO2 relative to the undistorted state. The energies represent the single-point energy at the geometry obtained with occupation-matrix control (OMC). Blue circles represent structures that retained the ordering when further relaxed without OMC, whereas orange crosses represent structures that spontaneously relaxed into different orderings.

Ni4+ orderings with unit cells containing up to four formula units. Our findings confirm the results of Chen et al.:25 a zigzag arrangement of Ni2+ and Ni4+ is quite close in energy to the lowest energy Jahn−Teller distorted phase. After full relaxation, the lowest energy disproportionated structure is 7 meV/LiNiO2 higher in energy than the lowest energy Jahn−Teller distorted structure. To further resolve the relative energies of competing groundstates in LiNiO2, we calculated the energies of selected

predicted to be the ground state of NaNiO2, in agreement with experimental observation.13 The zigzag orderings are substantially higher in energy than the collinear ordering, and unlike LiNiO2, the trans-zigzag is preferred over the cis-zigzag by 11 meV/NaNiO2. This suggests that the bonding between alkali ions and occupied eg* states is weaker in NaNiO2 than LiNiO2, although other factors such as intercalant size could play a role as well.84 612


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Chemistry of Materials LiMnO2. Figure 9 shows the relative energies of orbital and magnetic orderings of LiMnO2 versus the average magnetic

Figure 10. Energies of Jahn−Teller orderings in NaMnO2 relative to the ferromagnetic undistorted state. The energies represent the singlepoint energy at the geometry obtained with occupation-matrix control (OMC). Blue circles represent structures that retained the ordering when further relaxed without OMC, whereas orange crosses represent structures that spontaneously relaxed into different orderings.

Figure 9. Energies of Jahn−Teller orderings in LiMnO2 relative to the ferromagnetic undistorted state. The energies represent the singlepoint energy at the geometry obtained with occupation-matrix control (OMC). Blue circles represent structures that retained the ordering when further relaxed without OMC, whereas orange crosses represent structures that spontaneously relaxed into different orderings.

structure has a ferromagnetic spin ordering, and Mn−O−Na bonds in the trans configuration.

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GENERAL DISCUSSION We have introduced an approach that enables a systematic enumeration of charge, spine, and orbital orderings in crystalline solids. This has enabled us to perform an in-depth study of the interactions between different localized electronic degrees of freedom in layered transition metal oxides. Applied to LiNiO2, NaNiO2, LiMnO2, and NaMnO2, we have been able to identify several general trends. First, the strength of the Jahn−Teller distortion associated with orbital ordering varies substantially with chemistry. As shown in Table 3, the strength

moment. (An average magnetic moment of zero corresponds to an antiferromagnetic spin ordering, while a magnetization of 4 μB/LiMnO2 corresponds to a ferromagnetic ordering.) The calculations predict the collinear Jahn−Teller distortion to be the most stable orbital ordering for any given net magnetization, in agreement with the experimentally observed crystal structure.15,16 The lowest energy ordering is predicted to be antiferromagnetic, consistent with the experimental observation of antiferromagnetism in LiMnO2.79 In the ground state, the two short Mn−Mn bonds are antiferromagnetic while half of the four long Mn−Mn bonds are antiferromagnetic and half are ferromagnetic; this is in agreement with previous first-principles calculations.19,77,78 Compared to the LiNiO2 and NaNiO2, the strength of the Jahn−Teller distortion is much larger in LiMnO2 in the sense that the difference in energy between the ground state and the undistorted state is much larger. The lowest energy zigzag configuration in LiMnO2 (excluding zigzig-zag orderings) is 59 meV/LiMnO2 higher in energy than the ground state. It has an antiferromagnetic spin ordering, and Mn−O−Li bonds in the cis configuration. NaMnO2. Figure 10 shows the energies of spin and orbital orderings in NaMnO2. Like LiMnO2, NaMnO2 favors a collinear orbital ordering with antiferromagnetic in-plane magnetic ordering, in agreement with prior experiments17,18,39,85,86 and calculations.80,81 This magneto-elastic coupling breaks the equivalence between two nearest-neighbor Mn−Mn distances. In the calculated ground state structure, these two distances differ by 0.17%, a value similar to that obtained by low-temperature neutron diffraction (0.23%).17 The higher energy configurations of NaMnO2, however, differ from those of LiMnO2. The lowest energy zigzag configuration in NaMnO2 (excluding zig-zig-zag orderings) is 103 meV/NaMnO2 higher in energy than the ground state. Unlike the lowest energy zigzag structure in LiMnO2, this

Table 3. Energies of Jahn−Teller and Magnetic Orderings, Relative to the Ferromagnetic Undistorted (rhombohedral) State

material

ground state Jahn−Teller ordering

ground state in-plane magnetic ordering

distortion energy (meV/AMO2)

E[zigzag] − E[collinear] (meV/AMO2)

LiNiO2 NaNiO2 LiMnO2 NaMnO2

zigzag collinear collinear collinear

FM FM AFM AFM

−120 −131 −354 −350

−12 22 59 103

of a Jahn−Teller distortion is much stronger in terms of energy in Mn oxides vs Ni oxides. Second, orderings where an oxygen site does not participate in exactly two short M−O bonds and one long M−O bond are mechanically unstable. This trend is consistent with a bond valence sum picture: too many long bonds makes an oxygen anion undercoordinated; too many short bonds makes it overcoordinated. As shown in the Supporting Information, the only orderings satisfying this oxygen bond criterion are row orderings consisting of any combination of “zigs” and “zags”, as illustrated in Figure 11. 613


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found to be unstable when the OMC bias potential was removed. This result indicates that LiNiO2 will not form a trimer-domain structure. Based on the calculated energies for Jahn−Teller orderings, we hypothesize another possible explanation for the apparent absence of a collective Jahn−Teller distortion in LiNiO2: a nanostructure with Jahn−Teller domains too small to be readily observed by diffraction. Figure 12a illustrates a hypothetical Figure 11. Examples of structures that satisfy the two-short/one-long criterion for M−O bonds. The lines show the Jahn−Teller axis at each transition-metal site within a layer.

Although DFT predicts collective Jahn−Teller distortions to be thermodynamically favorable in LiNiO2, no collective distortion (zigzag or collinear) has been experimentally observed in LiNiO2, even at temperatures as low as 1.4 K.87,88 While thermal fluctuations resulting in a dynamic Jahn− Teller distortion can explain the absence of a collective distortion at room temperature, the apparent absence of a collective distortion at low temperatures is a long-standing mystery. (In contrast, NaNiO2, LiMnO2, and NaMnO2 are experimentally observed to exhibit collective Jahn−Teller distortions, even at room temperature.13,15−17,85) Even though no collective distortion occurs in LiNiO2, the observation of two distinct Ni−O bond lengths with EXAFS20 and neutron diffraction12 experiments indicates that local Jahn−Teller distortions occur. Several possible explanations for the absence of a collective distortion at low temperatures have been hypothesized: 1. The disruption of long-range orbital ordering by point defects, especially Ni atoms substituting on the Li layer. This defect chemistry results in a composition of Li1−zNi1+zO2, with z typically being on the order of several percent.82,83 2. The disproportionation of Ni3+ into Ni2+ and Ni4+, rather than Jahn−Teller activity.25 3. Trimer Jahn−Teller orderings that, at least locally, preserve 3-fold symmetry.12,22 The first possibility (point defects) lies outside the scope of this work and is not considered further. The second possibility (disproportionation) seems unlikely to explain the absence of a collective distortion: as discussed above, DFT calculations suggest that Jahn−Teller distortions are more favorable than disproportionation. Furthermore, experimental data shows that there are more short Ni−O bonds than long ones;12,20 this is not consistent with full disproportionation, which would result in equal numbers of short and long bonds. The third possibility, trimer orderings, can be ruled out based on the results presented here. Figure 5 shows that the trimer model is locally unstable and furthermore has the highest single-point energy of all orderings considered. The instability of the trimer ordering is consistent with prior DFT calculations.25,26 The trimer orderings represent the most egregious violation of the criterion for M−O bonds: instead of having one long and two short bonds, each oxygen in the trimer ordering has either three long bonds or three short bonds. It was previously suggested such trimers may be more stable when arranged into antiphase domains.12 To test this trimerdomain model, we constructed a large supercell model containing 12 formula units as shown in Figure S6 of the Supporting Information. While relaxation with OMC did result in the formation of antiphase trimer domains, this ordering was

Figure 12. Hypothetical boundaries between Jahn−Teller distorted domains. (a) An antiphase boundary between zigzag domains. (b) A twin boundary between collinear domains. (c) An antiphase boundary between antiferromagnetic domains that have the same collinear Jahn−Teller distortion. (d) A twin boundary between antiferromagnetic domains that have the same collinear Jahn−Teller distortion but different variants of antiferromagnetic order. In (c) and (d), the dark and light lines represent spin-up and spin-down transition-metal sites.

boundary between two zigzag domains. The near degeneracy of different zigzag-type orderings suggests that the energy to create such boundaries in LiNiO2 is quite low. Similar domain formation could also occur if a collinear distortion, rather than zigzag ordering, were the ground state; Figure 12b illustrates such a boundary between two collinear domains. (Note that the interface in Figure 12a is an antiphase boundary because the two domains are related by a translation, while the interface in Figure 12b is a twin boundary because the two domains are related only through rotation/reflection.) In addition to explaining the absence of an observable collective distortion, the formation of such Jahn−Teller domains could explain the anomalous behavior observed in neutron PDF wherein longrange ordering becomes weaker as the temperature is lowered.12 To further explore the structure of such domain boundaries, explicit models between zigzag domains in LiNiO2 and collinear domains in NaNiO2 were calculated, as discussed in the 614


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M−O bond, but the transition-metal ion at the center of the junction (represented by a black dot) possesses no long M−O bonds. We speculate that topological defects such as those shown in Figure 13 would tend to form around chemical point defects. For example, the junction shown in the right panel of Figure 13 might be centered on a transition-metal ion that is not Jahn− Teller active. In the case of LiNiO2, this could be a Ni2+ which serves to charge compensate Ni defects in the Li layer82,83 or a Ni4+ which serves to charge compensate Li vacancies. In the case of an alloy, other chemical species may reside at the center of topological defects. For example, previous DFT calculations on Li(NixCoyAlz)O2 found the ordering of Co and Al to be coupled to the orientation of Jahn−Teller distortions on nearby Ni sites.89

Supporting Information. In these models, each domain consisted of four rows. We considered several different boundary orientations and found the lowest energy domain superstructure in LiNiO2 to be 10 meV/supercell higher in energy than the LiNiO2 ground state (a single cis-zigzag domain). In contrast, the lowest energy domain superstructure in NaNiO2 was 65 meV/supercell higher in energy than the NaNiO2 ground state (a single collinear domain). That the domain boundary energy is significantly lower in LiNiO2 than NaNiO2 supports the hypothesis that the absence of a collective Jahn−Teller distortion in LiNiO2 is due to a Jahn−Teller domain nanostructure Recently, an analogous phenomenon has been suggested to occur in NaMnO2: the formation of antiferromagnetic domains within collinearly distorted crystallites.86 Once the direction of the collinear Jahn−Teller distortion is fixed, there are four symmetrically equivalent ground state magnetic orderings; these arise from the two equivalent directions in which the ferromagnetic chains can run (resulting in different magnetoelastic distortions) and then two translational variants for each of these (i.e., variants where the spins are reversed). The microstructure of NaMnO2 was previously suggested to consist of domains of translational variants,86 whose interfaces represent antiphase boundaries as shown in Figure 12c. We speculate that domains with differing magnetoelastic distortions may also coexist. The interfaces between such domains represent twin boundaries, as shown in Figure 12d. To test the feasibility of antiphase- and twin-boundary formation in NaMnO2, we calculated the energy of a zigzag spin ordering superimposed on a collinear Jahn−Teller distortion, as shown in Figure S7 of the Supporting Information. This structure is within 1 meV/NaMnO2 of the ground state structure (the row ordering of spins), indicating that the energy to create the boundaries shown in Figure 12c,d is very small. The near degeneracy of the zigzag and row orderings of spins can be rationalized by noting that, in both structures, each Mn exhibits two antiferromagnetic short Mn−Mn bonds, two antiferromagnetic long Mn−Mn bonds, and two ferromagnetic long Mn−Mn bonds. A final point of consideration is the nature of topological defects in the ordering of Jahn−Teller distortions, which can manifest as junctions between three or more domains. Figure 13 illustrates two hypothetical examples of topological defects. The left panel shows a junction between three collinear domains. In this structure, every oxygen atom participates in two long bonds and one short bond, except for the oxygen site at the center of the junction; this site has three short bonds. The right panel shows a junction between six collinear domains. In this structure, every oxygen retains two short and one long

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CONCLUSION A systematic investigation of charge, spin, and orbital orderings in the layered oxides reveals two criteria for the stability of Jahn−Teller orderings. First, each transition metal must exhibit two long bonds and four short, due to the well-known preference for positive Jahn−Teller distortions in octahedral environments.19,47−49 The second criterion is that each oxygen will exhibit one long M−O bond and two short. These two constraints do not uniquely determine the orbital ordering: there is an infinite number of zigzag-like row orderings that satisfy both criteria. Whether the ground state will be a zigzag ordering or the collinear ordering will depend on the details of the system. Importantly, in LiNiO2 the zigzag-type orderings are very close in energy to the collinear ordering, possibly due to strong interactions between eg* orbitals and Li+ ions. The near degeneracy of LiNiO2 orbital orderings suggests that LiNiO2 crystallites may exhibit a complex microstructure with nanoscale domains. Such a nanostructure could explain why a collective Jahn−Teller distortion is not observed by diffraction in LiNiO2. Similar phenomena have been widely studied in the context of martensitic transformations in alloys,90 where twinned microstructures emerge to accommodate the strain accompanying the transformation from a high symmetry phase to the low temperature, low symmetry phase. We conclude by discussing the applicability of the occupation-matrix lattice-alloy framework presented here to other problems. The results above show that the energy landscape of the layered oxides has many local minima that were not previously identified (e.g., the trans-zigzag ordering); that the simulations were able to find these minima suggests that the occupation-matrix lattice-alloy framework would be effective in exploring electronic degrees of freedom in other materials. Another natural extension of this work would be to construct a lattice-alloy cluster expansion to predict the energies of charge/spin/orbital orderings.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.7b03080. Derivation of symmetry properties of occupation matrices, comparison of single-point and relaxed energies, results obtained with different values of U, and topological analysis of Jahn−Teller orderings (PDF)

Figure 13. Hypothetical triple and sextuple junctions between collinear Jahn−Teller domains in a layered oxide. The black dot in the right panel represents a transition-metal site with no Jahn−Teller distortion. 615


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AUTHOR INFORMATION

Corresponding Author

*(A. Van der Ven) E-mail: [email protected]. ORCID

Maxwell D. Radin: 0000-0002-8989-8114 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.D.R. would like to thank John C. Thomas for valuable discussion and assistance in implementing the enumeration of anisotropic occupants. M.D.R. would also like to thank Ram Seshadri for valuable discussion. This work was supported as part of the NorthEast Center for Chemical Energy Storage (NECCES), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0012583. Computational resource support was provided by the Center for Scientific Computing at the CNSI and MRL: an NSF MRSEC (DMR1121053) and NSF CNS-0960316. Computational resources provided by the National Energy Research Scientific Computing Center (NERSC), supported by the Office of Science and U.S. Department of Energy, under Contract DE-AC0205CH11231, are gratefully acknowledged.

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