Compositional phase stability of strongly correlated electron materials within DFT+U

Compositional phase stability of strongly correlated electron materials within DFT+

Eric B. Isaacs eric.isaacs@columbia.edu    Chris A. Marianetti chris.marianetti@columbia.edu Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027
July 26, 2019
Abstract

Predicting the compositional phase stability of strongly correlated electron materials is an outstanding challenge in condensed matter physics. In this work, we employ the density functional theory plus (DFT+) formalism to address the effects of local correlations due to transition metal electrons on compositional phase stability in the prototype phase stable and separating materials LiCoO and olivine LiFePO, respectively. We exploit a new spectral decomposition of the DFT+ total energy, revealing the distinct roles of the filling and ordering of the orbital correlated subspace. The on-site interaction drives both of these very different materials systems towards phase separation, stemming from enhanced ordering of the orbital occupancies in the and species, whereas changes in the overall filling of the shell contribute negligibly. We show that DFT+ formation energies are qualitatively consistent with experiments for phase stable LiCoO, phase separating LiFePO, and phase stable LiCoPO. However, we find that charge ordering plays a critical role in the energetics at intermediate , strongly dampening the tendency for the Hubbard to drive phase separation. Most relevantly, the phase stability of LiCoO within DFT+ is qualitatively incorrect without allowing charge ordering, which is problematic given that neither charge ordering nor the band gap that it induces are observed in experiment. We demonstrate that charge ordering arises from the correlated subspace interaction energy as opposed to the double counting. Additionally, we predict the Li order-disorder transition temperature for LiCoO, demonstrating that the unphysical charge ordering within DFT+ renders the method problematic, often producing unrealistically large results. Our findings motivate the need for other advanced techniques, such as DFT plus dynamical mean-field theory, for total energies in strongly correlated materials.

pacs:
71.15.Mb, 71.27.+a, 81.30.Bx, 82.47.Aa

I Introduction

Strongly correlated materials (SCMs), in which density functional theory (DFT)Hohenberg and Kohn (1964); Kohn and Sham (1965) calculations break down for selected observables due to strong electron-electron interactions, are at the forefront of condensed matter physics.Kotliar and Vollhardt (2004); Morosan et al. (2012) Phenomenologically, SCMs exist in a ground state which is in the vicinity of a Mott transition Mott (1968); Imada et al. (1998) whereby electronic hopping may be overwhelmed by local interactions, resulting in an insulating state. Realizations of SCMs often contain atoms with open-shell or electrons, such as the high-temperature superconducting cupratesBednorz and Müller (1986), colossal magnetoresistance manganitesRamirez (1997), and heavy fermion actinide based materials.Coleman (2007)

Predicting the properties of strongly correlated materials is an outstanding problem in solid state physics. The standard approach of DFT, which is the most generic theory of electronic structure for materials physics, is in principle an exact theory for the ground state energy and electron density of any many-electron system. However, the exact exchange-correlation functional is unknown and must be approximated; and in practice DFT is often unable to capture critical aspects of the physics of strongly correlated materials.Jones and Gunnarsson (1989); Jones (2015); Kotliar and Vollhardt (2004) The difficulty of constructing increasingly intelligent exchange-correlation density functionals led to pursuit of additional variables that could be more easily approximated; one such variable is the local Green function for a set of correlated orbitals in the material (e.g orbitals)Kotliar et al. (2006). A corresponding approximation for this density plus local Green functional theory leads to DFT plus dynamical mean-field theory (DFT+DMFT). A further Hartree approximation to the DMFT impurity problem leads to the more ubiquitous DFT+ approach, which can be counterintuitive given that the latter preceded the former historicallyAnisimov et al. (1997); Kotliar et al. (2006). The long-term goal of these techniques is their application in an unbiased manner to any material containing localized orbitals, nominally or otherwise. Compositional phase stability is a particular stringent test since it relies on total energies from multiple compositions, making it more difficult to cancel errors by varying the few still maturing aspects of the formalism.

One of many important contexts highlighting the need for accurate total energy methods for strongly correlated materials is that of doped transition metal oxides, which include doped Mott insulators. From a technological perspective, Li-based transition metal oxides form the basis of rechargeable battery cathodes. Such cathodes typically employ transition metals and oxygen or oxygen-containing anion groups and, since they have open electron shells, are susceptible to strong correlation physics. One fundamental characteristic of a cathode material is whether there is a stable solid solution for intermediate Li concentration (), as dictated by the formation energy of the compound. The formation energy has a strong impact on the charge/discharge mechanism and also has implications for the voltage and capacity of the battery.

Figure 1: Crystal structures of (a) LiCoO and (b) LiFePO. The large green, medium blue, medium gold, small purple, and small red spheres represent ionic positions of Li, Co, Fe, P, and O, respectively. The black lines indicate for (a) the unit cell of the lowest-energy LiCoO configuration and for (b) the primitive unit cell. Images of crystal structures are generated using vesta.Momma and Izumi (2011)

Many of the current cathode materials utilized today are based on the material LiCoO, shown in Fig. 1(a) for , which can store Li ions in between its CoO layers.Mizushima et al. (1980) LiCoO has several stable intermediate phases.Reimers and Dahn (1992); Van der Ven et al. (1998); Wolverton and Zunger (1998) In contrast, olivine LiFePO Padhi et al. (1997a, b) [see Fig. 1(b)] does not have any stable compounds for intermediate . It instead exhibits phase separation into its fully lithiated () and fully delithiated () endmembers, which may contribute to slow charge/discharge kinetics. Padhi et al. (1997a); Delacourt et al. (2005); Dodd et al. (2006)

Previous work demonstrates that DFT predicts very stable compounds of intermediate in LiFePO, in stark disagreement with experiment.Zhou et al. (2004a) This is a flagrant failure of the DFT method. In contrast, DFT does properly capture the phase stability of LiCoO.Van der Ven et al. (1998); Wolverton and Zunger (1998) A beyond-DFT approach combining DFT and Hartree-Fock exchange, called a hybrid functional, also fails to capture the total phase separation in LiFePO.Ong et al. (2011) DFT+ calculations, which include an explicit on-site Coulomb interaction for the transition metal electrons, were found to predict phase separation in LiFePO for sufficiently high values of .Zhou et al. (2004a) In later work, a cluster expansion based on such DFT+ energetics rationalized this phase separation in terms of strong Li–electron interactions.Zhou et al. (2006) A lingering question, which we will address in this work, is whether DFT+ can be applied to LiCoO Andriyevsky et al. (2014); Aykol and Wolverton (2014); Aykol et al. (2015); Seo et al. (2015) without degrading the already reasonable DFT results, such that DFT+ can give a robust description for both of these prototype phase stable and phase separating systems.

The DFT+ study for LiFePO claims that the instability of compounds of intermediate relates to energy penalties stemming from charge ordering (CO), a symmetry breaking in which the number of electrons on different transition metal sites differs.Zhou et al. (2004a) However, it is unclear if the CO is physical since static ordering is the only way in which a static mean-field theory like DFT+ can mimic strong electronic correlations. Furthermore, the phase separation in LiFePO is fundamentally puzzling since with reasonable parameters the canonical model Hamiltonians describing strong correlations such as the Hubbard and models do not exhibit phase separation for extended regions of the phase diagram.Visscher (1974); Marder et al. (1990)

To explore such issues, in this work we employ extensive DFT+ calculations to understand in detail the impact of electronic correlations on phase stability in correlated intercalation materials. We focus on phase stable LiCoO and phase separating LiFePO and explicitly investigate the role of CO and structural relaxations. A new energy decomposition is elucidated to quantitatively analyze the impact of on the overall filling and on the ordering of orbitals within the shell. Comparison is made to the LiCoPO system, which is isostructural to LiFePO but does have a stable compound of intermediate . We also investigate another physical observable, the Li order-disorder transition temperature, for LiCoO to provide another clear benchmark of DFT+ theory. The tendency for CO in DFT+, how it impacts these thermodynamic properties, and whether it is physical or not is investigated and discussed. A thorough understanding at the DFT+ level of theory is critical as we advance to more sophisticated methods for studying phase stability in strongly correlated materials.

For all the materials we have investigated, we find that destabilizes compositions at intermediate . This results because most strongly affects the endmembers by enhancing the deviations in the orbital occupancies from their average value; compounds of intermediate experience the same effect but are unable to orbitally polarize to the same extent, which directly drives them towards phase separation as increases. CO and structural relaxations serve to dampen, but not eliminate, this fundamental effect. However, CO leads to the formation of a band gap for LiCoO that is not observed experimentally, suggesting it is an unphysical artifact of DFT+. Furthermore, CO leads to highly erratic estimations of the Li order-disorder transition temperature for LiCoO, demonstrating a serious shortcoming of the DFT+ method. Our calculations demonstrate that the interaction term rather than the double counting term in DFT+ is responsible for the CO, which suggests that more accurate approaches such as DFT plus dynamical mean-field theory (DFT+DMFT) may be necessary to accurately describe this class of materials.

The rest of this paper is organized as follows. Section II discusses the DFT+ approach to materials with strong electronic correlations and elucidates our new energy decomposition in this framework. Section III describes the computational details for the simulations performed in this work. The electronic structure of the endmembers of LiCoO and LiFePO within DFT are described in Sec. IV.1. The impact of on the electronic structure of LiCoO is discussed in Sec. IV.2. The origin of the tendency for CO in DFT+, taking LiCoO as an example, is the focus of Sec. IV.3. The impact of on the phase stability of LiCoO is discussed in Sec. IV.4. The electronic structure and phase stability of LiFePO within DFT+ are described in Sec. IV.5 and Sec. IV.6, respectively. Section IV.7 discusses the formation energy trends for LiCoPO. Section IV.8 elucidates the role of the double counting on the formation energy trends. The average intercalation voltages for LiCoO and LiFePO are presented in Sec. IV.9. Section IV.10 focuses on the Li order-disorder transition temperature of LiCoO and its dependence on CO. Finally, conclusions are presented in Sec. V.

Ii Methodology

ii.1 Dft+ approach for correlated materials

The idea of DFT+Himmetoglu et al. (2014) is to provide an improved treatment of electronic correlations by using not only the density as a primary variable of the energy functional, but also the single-particle density matrix of a relevant set of local orbitals associated with strong correlations. This set of orbitals, which form the correlated subspace, are typically localized electronic states having or character and in practice are defined using Wannier functions or atomic orbitals labeled by ionic site and angular momentum projection .

Having defined the correlated subspace, one needs to construct an approximation for the energy as a functional of the density and the single-particle density matrix of the correlated subspace. This is typically approximated using two separate additive functionals:

(1)

where is the local single-particle density matrix for spin projection and is the usual Kohn-Sham DFT energy functional using one of the many possible approximations to the exchange-correlation energy such as the local density approximation (LDA)Ceperley and Alder (1980); Vosko et al. (1980); Perdew and Zunger (1981); Perdew and Wang (1992) or the generalized gradient approximation (GGA).Perdew et al. (1992, 1996) The functional of the density matrix , to be defined mathematically below, is given by the Hartree-Fock interaction energy based upon a set of screened interactions within the correlated subspace. There is a clear double counting (dc) problem with such a formulation, as the LDA or GGA already accounts for some interactions of the density arising from the correlated subspace, and therefore a double-counting energy must be defined and subtracted.

It is common to employ the spin-dependent formulation of DFT (SDFT) rather than pure DFT, in which case the total energy expression becomes

(2)

where is the density for spin projection . In the simplified rotationally-invariant formalism of Dudarev et al.Dudarev et al. (1998) the interaction term, which we always write in the diagonal eigenbasis of the density matrix, is

(3)

where is the th eigenvalue of and represents the screened on-site Coulomb energy between different spin-orbitals within the correlated subspace. This approach is equivalent to the full rotationally-invariant formalism by Liechtenstein et al.Liechtenstein et al. (1995) if the on-site exchange parameter is set to 0, which can be justified by previous work indicating that SDFT already contains an intrinsic .Chen et al. (2015); Park et al. (2015) The most commonly used dc is the fully-localized-limit (FLL) type

(4)

in which is the total correlated occupancy on a site. In this work we focus on materials in which states form the correlated subspace, so at times we refer to this quantity as (or if there is only a single site). This double counting energy is equal to in the limit in which all take on values of 0 or 1.

Using this dc form, the total energy expression can be rewritten as

(5)

This form illustrates that DFT+ penalizes fractional occupancy of the correlated orbitals and serves to restore the derivative discontinuity of the total energy with respect to total particle number that is missing in approximate DFT.Perdew et al. (1982); Anisimov et al. (1993); Himmetoglu et al. (2014); Cococcioni and de Gironcoli (2005)

It is important to emphasize the connection between DFT+ and DFT+DMFT to in order to appreciate the limitations of the DFT+ approach.Kotliar et al. (2006) DFT+DMFT corresponds to a functional of the density and the local Green function of the correlated orbitals, which contains the single-particle density matrix in addition to information about single-particle excitations. This functional is approximated using, in part, DMFT, which requires the solution of the DMFT impurity problem. Since the DMFT impurity problem itself is a difficult (though tractable) many-body problem, one approach is to make further approximations and solve the DMFT impurity problem within the Hartree-Fock approximation, which results in the DFT+ approach. Alternatively, it may be possible to exactly solve the DMFT impurity problem using quantum Monte Carlo (QMC) techniquesGull et al. (2011); Haule (2007), which can have a profound influence on the energetics as compared to DFT+Park et al. (2014a, b); Chen et al. (2015). In particular, the error associated with DFT+ was shown to strongly depend on the spin state of Fe in a spin crossover moleculeChen et al. (2015), which may suggest that DFT+ errors will strongly depend on Li concentration in cathode materials and therefore affect phase stability and battery voltages. The quality of DFT+DMFT (solved using QMC) is superior to DFT+ in every sense, but the former is far more computationally expensive and complex. This current study will give us a detailed understanding of how DFT+ performs in the context of phase stability, sometimes exposing obvious deficiencies but also demonstrating enhancements over DFT, and provide a baseline on which DFT+DMFT may improve. A forthcoming publication will then extend the current study using DFT+DMFT.

ii.2 Filling and ordering energy decomposition in DFT+

Two important aspects of DFT+ calculations in correlated materials are (1) the total number of correlated ( or ) electrons per site and (2) the ordering of electrons within the correlated subspace based on spin (magnetism), angular momentum (orbital ordering), or site (CO). These two effects can be labeled as the “filling” and “ordering” of the correlated orbitals, respectively. Here we elucidate a decomposition of the interaction and dc energies that enables a useful decoupling of these two effects for analysis of DFT+ results. This spectral decomposition was also very briefly introduced in our previous work.Isaacs and Marianetti (2016)

To represent the average filling on a site we define the orbital occupancy mean

(6)

where is the number of correlated spin-orbitals per site (e.g. 10 for the shell). We then rewrite the interaction and dc energies to contain only terms containing the mean or the deviation from the mean .

(7)

where

(8)

is the standard deviation of the occupancies for site . We call the filling factor and the ordering factor, and at times in our discussion of results we drop the subscript on and for convenience. is thus rewritten as with:

(9)

and

(10)

These terms encapsulate the two ways in which a system can avoid paying the Coulomb energetic penalty . From the filling term, the system can minimize by moving towards completely empty () or completely filled () correlated shells on average. From the ordering term, the system can maximize by enhancing the spread in via some type of ordering. Note that values for nominally-unoccupied orbitals still contribute to , so this quantity is distinct from other measures of orbital ordering that are determined from low-energy orbitals (e.g. Wannier orbitals defined from a small energy window). One can rewrite and in terms of and as

(11)

and

(12)

A function of only , takes the same form as except the prefactor on the quadratic term is instead of 1; depends on both and .

In this work we are primarily interested in the formation energy, which is defined as

(13)

where is the cohesive energy of a system with intercalant concentration . The formation energy indicates whether a species of intermediate has a higher or lower cohesive energy than the corresponding linear combination of those of the endmembers. Therefore, only energy terms beyond linear order in contribute. The formation energy dictates the stability of such a species in the limit of low temperature (); it is stable if negative and unstable if positive. One can separately construct the formation energy contributions stemming from , , , using expressions analogous expressions to Eq. 13, allowing one to understand the contribution of each term to the formation energy. Note that will be negative if is lower than the endmember linear combination, while will be negative if is higher than the endmember linear combination due to the negative sign in the definition of in Eq. 10. All formation energies reported in this work are normalized per formula unit (f.u.). We also compute the average intercalation voltage via

(14)

where is the elementary charge, is the cohesive energy of body-centered-cubic Li, and the energies are normalized to the number of f.u.Aydinol et al. (1997)

Iii Computational Details

DFT+ calculations based on the spin-dependent generalized gradient approximation,Perdew et al. (1996) and the rotationally-invariant Hubbard interactionLiechtenstein et al. (1995) with set to 0, and FLL dc are performed using the Vienna ab initio simulation package (vasp).Kresse and Hafner (1994, 1993); Kresse and Furthmüller (1996a, b) The projector augmented wave methodBlöchl (1994); Kresse and Joubert (1999) is employed and the single-particle equations are solved with a plane-wave basis set with a kinetic energy cutoff of 500 eV. We use a () -point grid for the rhombohedral (orthorhombic) primitive unit cell of LiCoO (LiFePO and LiCoPO) and -point grids with approximately the same -point density for supercell calculations. The bulk lithium calculation is performed using a -point grid. For structural relaxations in metals we employ the first-order Methfessel-Paxton methodMethfessel and Paxton (1989) with a 50 meV smearing and for all other calculations the tetrahedron method with Blöchl correctionsBlöchl et al. (1994) is used. The ionic forces, stress tensor components, and total energy are converged to 0.01 eV/Å, 10 GPa, and 10 eV, respectively.

The disordered (solid solution) phase of LiCoO is modeled via the special quasirandom structure (SQS) approachZunger et al. (1990) using the Alloy Theoretic Automated Toolkit (atat)van de Walle et al. (2002); van de Walle (2009) using point, pair, triplet, and quadruplet clusters. Candidate structures are generated based on the correlation functions of clusters with a maximum inter-site distance up to the in-plane 2nd nearest neighbor and out-of-plane 1st nearest neighbor distance. To evaluate the structures, we consider the minimal values of the following figure of merit: the root-sum-square of the differences between the cluster correlation functions and those of the random structure. In this evaluation, the correlation functions of clusters containing a maximum inter-site distance up to the in-plane 5th nearest neighbor and inter-plane 6th nearest neighbor are taken into account.

Iv Results and Discussion

iv.1 Endmember electronic structure within DFT

We begin by reviewing the electronic structure of the endmembers of LiCoO and LiFePO within DFT, comparing with the latest experimental understanding of these materials. LiCoO crystallizes in the layered structure illustrated in Fig. 1(a) consisting of layers of edge-sharing CoO octahedra and layers of Li. It has a rhombohedral primitive unit cell with the space group and A-B-C (O3) oxygen stacking.Johnston et al. (1958); Orman and Wiseman (1984) CoO has a very similar structure with a hexagonal unit cell and A-B (O1) oxygen stacking in the space group.Amatucci et al. (1996) Here for convenience we consider CoO in the O3 structure, which has a very similar electronic structure to that of the O1 structure within DFT (see the Supplementary Material).

In LiCoO the octahedral coordination of Co is slightly distorted due to the ability of the oxygens to relax in the out-of-plane direction, resulting in a symmetry lowering with . Nominally LiCoO is in the configuration with filled and levels, while CoO has one fewer electron ().

Figure 2: Projected and density of states for (a) CoO, (b) LiCoO, (c), FePO, and (d) LiFePO within DFT. For antiferromagnetic FePO and LiFePO only a single spin channel is shown. The dashed gray lines indicate the valence band maximum for insulators and the Fermi energy for metals. Insets are the nominal transition metal 3 orbital fillings from crystal field theory.

The and projection of the electronic density of states within DFT for CoO and LiCoO are shown in Fig. 2(a) and 2(b), respectively. LiCoO is found to be a band insulator in agreement with experiments.van Elp et al. (1991); Ménétrier et al. (1999, 2008) The computed band gap of 1.1 eV underestimates the experimental value of 2.7 eV as is typical for DFT.van Elp et al. (1991) The occupancies of , and are 0.94, 0.95, and 0.41 for LiCoO, demonstrating that hybridization between Co and O states leads to appreciable occupation of the nominally-unoccupied states. CoO has a ferromagnetic low-spin metallic ground state with a Co magnetic moment of 0.8 . In experiments on CoO, there is evidence for Fermi liquid behavior and Pauli paramagnetism without any long-range magnetic ordering.de Vaulx et al. (2007); Motohashi et al. (2007); Kawasaki et al. (2009) For CoO the occupation of , and are 0.96 (0.67), 0.96 (0.76), and 0.54 (0.52) for spin up (down) electrons, showing the significant degree of covalency in this system.

LiFePO and FePO both take on the olivine structure, which has an orthorhombic primitive unit cell containing 4 formula units in the space group.Santoro and Newnham (1967); Rousse et al. (2003) The structure consists of corner-sharing FeO octahedra layers connected via PO tetrahedra as shown in Fig. 1(b). For LiFePO there are one-dimensional chains of Li ions.

LiFePO has significantly distorted FeO octahedra, though for convenience we still crudely discuss the orbitals as and . Experimentally LiFePO and FePO are high-spin antiferromagnetic (AFM) insulators with Néel temperatures of 52 and 125 K, respectively.Santoro and Newnham (1967); Rousse et al. (2003) While FePO () nominally has all the orbitals on a given site singly occupied with aligned spins (i.e., ), for LiFePO () there is one additional minority-spin electron in the manifold.Tang and Holzwarth (2003); Shi et al. (2005) Fe linked via corner-sharing octahedra in the same layer have anti-aligned magnetic moments, while those laterally adjacent in different layers linked via PO have aligned magnetic moments.Santoro and Newnham (1967); Rousse et al. (2003)

The projected density of states for the olivine endmembers are shown in Fig. 2(c) and 2(d). Due to the antiferromagnetism, both spin channels are identical so only one is shown. FePO can be viewed as a charge transfer type insulator since the gap is in nature, whereas in LiFePO states form both the valence and conduction bands and the electronic bandwidths near the Fermi energy are extremely narrow (as little as 0.1 eV). Although LiFePO has an even number of electrons, the local Coulomb interaction can play a strong role in developing or enhancing the insulating behavior. Within DFT, the band gaps of FePO and LiFePO are 1.0 eV and 0.4 eV, respectively. These values are brought much closer to agreement with the experimental band gaps of 1.9 eVZaghib et al. (2007) and 3.8 eVZhou et al. (2004b); Zaghib et al. (2007) using the DFT+ approach.Zhou et al. (2004b); Seo et al. (2015)

iv.2 Impact of on electronic structure of LiCoO

Figure 3: Electronic band gap as a function of for (a) CoO, (b) LiCoO, and (c) LiCoO with relaxations (filled red squares) and with frozen structures (large open black circles). The corresponding plots of Co magnetic moment are shown in panels (d)–(f). For panels (b) and (e), the additional small filled blue circles correspond to calculations with frozen structures and CO suppressed.

The on-site interaction has been computed as 4.9 and 5.4 eV for LiCoO and CoO, respectively.Zhou et al. (2004c) The -dependence of the band gap and Co magnetic moment for LiCoO are shown in Fig. 3. The band insulator LiCoO has no magnetic moment and its band gap increases roughly linearly with from 1.1 eV at to 2.9 eV for eV. Structural relaxations with have little impact on the electronic structure. Note that we do not explore eV since in this regime we find DFT+ predicts a high-spin state for LiCoOAndriyevsky et al. (2014) in contradiction with experimental observationMénétrier et al. (1999, 2008).

CoO is semimetallic at lower values of with the and states both partially occupied. Beyond eV (or eV when including structural relaxations), an orbital ordering occurs in which completely fills and becomes a nominally half-filled state. This opens up a band gap of 0.5–1.2 eV and increases the Co magnetic moment to 1.1–1.4 , as compared to 0.7–0.8 for lower .

The lowest-energy structure of LiCoO has an in-plane ordering of Li and vacancies corresponding to the unit cell shown in Fig. 1(a) with the Li1 ion removed.Reimers and Dahn (1992); Van der Ven et al. (1998); Wolverton and Zunger (1998) Experimental studies suggest LiCoO is a paramagnetic metal with small Co magnetic moments of around 0.25–0.35 .Motohashi et al. (2009); Miyoshi et al. (2010); Ou-Yang et al. (2012) Within our calculations LiCoO is a ferromagnetic metal for eV with equal Co magnetic moments of 0.4 . For larger values a new ground state with CO emerges in which the first site takes on a CoO-like configuration with a moment of 1.1–1.4 and the second takes on a LiCoO-like configuration with no moment (the actual charge difference between sites is small; see Fig. 4(a)). CO opens an electronic band gap that increases with of 0.6–1.2 eV (0.5–0.9 eV without structural relaxations). Ignoring structural relaxations, the metallic state without CO is metastable and the Co magnetic moments are 0.4–0.5 and remain roughly constant for all values of considered.

iv.3 Tendency for charge ordering in DFT+

To understand the tendency for CO in DFT+, which is not observed in experiment, we investigated whether its origin is the interaction () term or the dc term. Given that the dc energy has a term which is a negative quadratic of , it has a disposition towards charge ordering. Alternatively, the interaction term may indirectly drive CO as a means to reduce the interaction penalty, arising at the expense of . To clearly identify the origin of CO, we implemented a modified DFT+ approach in which both the single-particle potential and total energy contributions stemming from the interaction term or the dc term are averaged over correlated sites. We call this site-averaged interaction and site-averaged dc, respectively.

Figure 4: (a) Co N, (b) Co magnetic moments, and (c) total energy for LiCoO with the frozen 0 eV structure for standard DFT+ (blue lines) as well as with the interaction (filled red circles) or double counting (open black squares) terms averaged over correlated sites. Total energies in panel (c) are with respect to those of the standard DFT+.

Figure 4 illustrates the results of this computational experiment for ordered LiCoO using the frozen structure. With site-averaged dc (black squares) we still find a CO transition, for eV. The magnitude of the CO in terms of is slightly reduced, but the deviations in Co magnetic moment are the same or even more substantial than the standard DFT+ results (blue lines). The total energy shown in panel (c) for this case is only slightly (on the order of 10 meV/f.u.) higher than in the case of standard DFT+, which indicates that the dc energetics are not very much lowered via CO. With site-averaged interaction (red circles), however, no CO can be obtained and the total energy is massively penalized as increases (by hundreds of meV/f.u.). Therefore, we conclude that it is the interaction term and not the dc term that is responsible for the CO in DFT+. This suggests that more accurate solutions to the interaction problem such as DMFT may be critical to resolving the issues associated with spurious CO.

iv.4 Impact of on phase stability of LiCoO

Figure 5: (a) Total LiCoO formation energy (open black circles) and its DFT (filled red squares) and –dc (filled blue circles) components as a function of (b) LiCoO –dc (filled blue circles) formation energy component and its orbital filling (open red circles) and orbital ordering (open black squares) components as a function of (c) Number of electrons per Co as a function of for CoO (filled black circles), LiCoO (open red squares), and LiCoO (open blue circles). All data correspond to the case of frozen structures and CO suppressed in LiCoO.

We first consider the formation energy of LiCoO in the frozen structure (i.e., no structural relaxations when imposing ) and without allowing CO, in order to purely see the effects of in the absence of CO and lattice distortions. For the experimental Li ordering in LiCoO, the two Co atoms in the unit cell are equivalent by point symmetry. Therefore CO is a spontaneously broken symmetry, which enables one to precisely investigate various observables with and without CO. We will return to the effects of both CO and structural relaxations after thoroughly explaining the role of in their absence.

As shown in Fig. 5(a), LiCoO is phase stable with a total formation energy of meV for . The formation energy increases monotonically with and for eV it becomes positive, corresponding to a prediction of phase separation. This indicates that the trend of a destabilization of compounds of intermediate found previously in LiFePOZhou et al. (2004a) also occurs for LiCoO. Furthermore, it demonstrates that such a trend is found even in the absence of CO.

To illustrate the origin of this behavior, we also examine the DFT and –dc components of the total formation energy in Fig. 5(a). For eV the –dc component is negligible ( eV) and the slightly less negative value of formation energy ( meV) stems from the DFT component. Compared to the values, for eV the DFT energy increases by 17 meV for LiCoO compared to only 7 meV for CoO and 14 meV for LiCoO. For larger the –dc component is strongly positive and increases roughly linearly with at a rate of around 130 meV per eV, leading to a more rapid increase in total formation energy. The DFT component has the opposite trend of becoming more negative with , largely since the DFT energy of CoO is strongly penalized by the orbital ordering, but the overall effect on the formation energy is smaller with changes of around 15–30 meV per eV. Therefore, it is the –dc component that is responsible for the destabilization of LiCoO.

The number of electrons per Co site () is plotted for , , and as a function of in Fig. 5(c). We note that the difference in between CoO and LiCoO is only around 0.1 even though Li is nominally donating a full electron, which is due to the rehybridization effect in LiCoO.Wolverton and Zunger (1998); Marianetti et al. (2004) While the behavior of is roughly constant at 7.3 for LiCoO and smoothly increasing for LiCoO between 7.25 and 7.28, there is discontinuous behavior for CoO in which drops from 7.22 to 7.16 from to eV corresponding to the orbital ordering. This change in electronic structure in the endmember is responsible for the change in behavior in the –dc formation energy contribution.

Since the dc term in Eq. 4 is a negative quadratic function, one might expect that the dc is responsible for the trend towards phase separation. This simple line of reasoning immediately becomes more complicated given that is a nonlinear function of as demonstrated in Fig. 5(c), and a careful analysis in Sec. IV.8 shows that it is not the dc that drives phase separation. Alternatively, we proceed to understand the contributions of both the and terms simultaneously in a different framework using the energy decomposition described in Sec. II.2. In Fig. 5(b) we break down the –dc formation energy contribution into the filling and ordering contributions. Remarkably, the magnitude of the filling contribution contributes negligibly, only being at most tens of meV in magnitude, whereas essentially all of the –dc formation energy comes from the ordering term. Therefore, it is the ordering rather than the filling of the correlated orbitals that drives phase separation.

Figure 6: Filling factor as a function of for different in LiCoO for (a) frozen structures and CO suppressed, (b) frozen structures and CO allowed, and (c) relaxed structures. Panels (d)–(f) show the corresponding plots for the ordering factor . The two lines per in some plots correspond to the two distinct Co sites in LiCoO. The line thickness increases for increasing values of .

The individual filling factor and ordering factor as a function of Li concentration for different are shown for this case in Fig. 6(a) and 6(d), respectively. We note that based on nominal electron counting will be 0.25 for Co and 0.24 for Co. Consistent with this expectation, we observe that the filling factor is highest for and lowest for . The actual values for LiCoO are lower in magnitude (around 0.2) due to the substantial covalent nature of the bonding, in particular the hybridization of O states with Co states. Although this filling factor magnitude is around half an order of magnitude higher than that of the ordering factor, the relative deviations of the value compared to the average of the and values are tiny (around electrons). This is why filling term leads to a negligible contribution to the formation energy.

The values of increase with for all in agreement with the expectation that the and dc parts of the total energy functional will penalize fractional orbital occupancy (i.e., ). For eV is nearly linear in , thus leading to no appreciable formation energy contribution. However, once CoO undergoes the orbital ordering, its value of significantly increases from 0.03 to 0.08. After this phase transition, for LiCoO is substantially lower than the average of those of CoO and LiCoO. For example, for eV is 0.277 for CoO and 0.282 for LiCoO but only 0.239 for LiCoO. This lower-than-average is translated to a positive formation energy contribution via the negative sign in prefactor of Eq. 10.

Figure 7: (a) Total LiCoO formation energy (open black circles) and its DFT (filled red squares) and –dc (filled blue circles) components as a function of (b) LiCoO –dc (filled blue circles) formation energy component and its orbital filling (open red circles) and orbital ordering (open black squares) components as a function of (c) Number of electrons per Co as a function of for CoO (filled black circles), LiCoO (open red squares), and LiCoO (open blue circles). All data correspond to the case of frozen structures and CO allowed and the two lines in panel (c) for LiCoO correspond to the two distinct Co sites.

The fundamental behavior we find is that drives phase separation via enhanced ordering of the correlated orbitals for the endmembers relative to the species with intermediate Li concentration. Given that we restricted the possibility of CO, LiCoO has no low energy means to orbitally order and keep pace with CoO as is increased. We will articulate this important point in a bit more detail. For eV is much smaller for than ; CoO has a smaller range of due to its semimetallic behavior and enhanced hybridization with O states. Once it has orbitally ordered at eV and the hole in the manifold is localized in the minority-spin state, however, CoO has 5 very occupied orbitals () and 5 much less occupied orbitals ( of around 0.15 for minority-spin and higher values of 0.49–0.59 for due to hybridization). LiCoO is a band insulator, so there is no abrupt change in as a function of . In terms of , LiCoO behaves very similarly to CoO in the regime of in which CoO is orbitally ordered. In LiCoO there are 6 very occupied orbitals ( for and states) and 4 much less occupied orbitals ( for states). This gives a large spread () in orbital occupancies for LiCoO in addition to CoO. In contrast, for LiCoO we have a metallic state with a nominally half-filled minority-spin level () and thus smaller . This lower-than-average for LiCoO is what results in a positive contribution to the formation energy from the –dc energetics.

This same type of behavior is preserved even when we now allow CO in LiCoO, though CO will allow LiCoO to order and completely avoid phase separation for all (we still restrict the possibility of structural relaxations until later in this analysis). Fig. 7 shows the results with the frozen structures but now allowing for CO in LiCoO. Here again the formation energy increases with . For small values of the increase is small and stems from the DFT contribution. After CoO orbitally orders and opens a band gap, the –dc energetics are a phase separating contribution to the total formation energy. As before, essentially all of the –dc contribution comes from the ordering, not the filling, of the orbitals. When LiCoO charge orders for greater than 2 eV, the –dc phase separating contribution is significantly dampened but there is always still a positive phase separating contribution (53–84 meV). CO also leads to an increase in the DFT formation energy contribution from eV at eV to eV at eV.

As illustrated in Fig. 6(b) and 6(e), the variations in filling factor are again negligible so the only appreciable component to the –dc energetics stems from the changes in ordering factor . The CO of LiCoO significantly increases the average , but it still always lags behind the average of those of CoO and LiCoO. For example, for eV is 87 meV for and 90 meV for the average for the and values. In this case we still end up with a total formation energy that steadily increases with , though now CO dampens the process substantially such that for eV the value remains negative consistent with experiment.

Figure 8: (a) Total LiCoO formation energy (open black circles) and its DFT (filled red squares) and –dc (filled blue circles) components as a function of (b) LiCoO –dc (filled blue circles) formation energy component and its orbital filling (open red circles) and orbital ordering (open black squares) components as a function of (c) Number of electrons per Co as a function of for CoO (filled black circles), LiCoO (open red squares), and LiCoO (open blue circles). All data correspond to the case of fully relaxed structures and the two lines in panel (c) for LiCoO correspond to the two distinct Co sites.

When we include CO in LiCoO and full structural relaxations, we find the same fundamental effect as in the previous case only allowing CO: structural relaxations only provide a further dampening. As shown in Fig. 8, the total formation energy increases with and the positive contribution stems from the ordering component of the –dc energetics. Here the CO and CoO orbital ordering both occur for eV. The increase in the DFT formation energy contribution upon CO is dampened due to relaxations. The total formation energy remains negative at meV at eV. The plots of and are shown in Fig. 6(c) and 6(f), respectively. Structural relaxations serve to slightly enhance for CoO and decrease the difference in for the distinct Co sites in LiCoO.

These results have significant implications on the accuracy and robustness of the DFT+ description of strongly correlated materials. Without CO, which is not found in experiments on this class of materials, DFT+ incorrectly predicts that LiCoO phase separates once becomes appreciable. This is true even if we allow structural relaxations while suppressing CO, which is possible since CO is a spontaneously broken symmetry in LiCoO, and in this case the total formation energy values are -40 meV, +67 meV, and +182 meV for of 3, 4, and 5 eV, respectively. That DFT+ requires artificial CO to correctly capture the phase stable nature of LiCoO is a significant weakness of this approach.

iv.5 Impact of on electronic structure of LiFePO

Figure 9: Electronic band gap as a function of for (a) FePO, (b) LiFePO, (c) LiFePO, (d) LiFePO, and (e) LiFePO with relaxations (filled red squares) and with frozen linearly interpolated experimental structures (open black circles). The corresponding plots of total magnetization and Fe magnetic moment magnitudes are shown in panels (f)–(j) and (k)–(o), respectively. For panels (k)–(o), the multiple symbols per correspond to the 4 distinct Fe sites.

The interaction has been computed as 3.7 and 4.9 eV for LiFePO and FePO, respectively.Zhou et al. (2004c) The variation of the band gap, total magnetization, and Fe magnetic moment as a function of for LiFePO is illustrated in Fig. 9. The endmembers are both AFM so there is zero net magnetization. The Fe magnetic moment, which increases approximately linearly with , is 4.0–4.4 for FePO and 3.6–3.8 for LiFePO for 0 eV 6 eV. The endmembers are both insulating. For this range the band gaps increases from 1.0 to 2.7 eV for FePO and 0.4 to 4.5 eV, a much larger range, for LiFePO. Very similar results for the endmembers are found when one freezes the ions to the experimental structures.

For intermediate we consider the lowest-energy configurations that fit within the primitive unit cell shown in Fig. 1(b). The structures correspond to removing Li2, Li3, and Li4 for , Li3 and Li4 for , and Li4 for .Zhou et al. (2004a) Calculations on the two other possible primitive cell LiFePO structures are always found to be higher in energy, so we do not discuss them here. In addition to the case in which structures are fully relaxed, we also perform calculations on structures constructed via linear interpolation of the experimental and structures. This is an ideal manner to isolate the effect of lattice relaxations given that appreciable lattice distortions arise in the intermediate compounds even for .

For the intermediate species of LiFePO are AFM metals without any CO. The Fe magnetic moments are 3.9, 3.8, and 3.7 for , , and , respectively. The magnetic moment magnitude gradually increases with , and above critical values a CO transition occurs leading to distinct Fe-like () and Fe-like () sites within the primitive unit cell. This symmetry breaking leads to distinct local magnetic moments and the opening of an electronic band gap. Unlike LiCoO for which differences among correlated sites are 0.15-0.18 electrons in the CO state, LiFePO has CO states with substantially larger differences around 0.4 electrons due to the highly localized nature of this system.

For LiFePO using the frozen structure (linear interpolation of endmember experimental structures), the CO transition at eV yields a ferrimagnetic state with total magnetization 2 ; there are 2 Fe magnetic moments of 3.7 (Fe-like) and 2 Fe magnetic moments of (Fe-like). It opens a band gap of 0.6 eV for eV that is further increased with to a value of 1.1 eV for eV. Including structural relaxations has little effect on the electronic structure in the regime of before the onset of CO. However, relaxations assist the CO transition and result in a lower critical value of eV above which the CO state remains AFM with Fe magnetic moments of 3.7 and 4.2 . In addition, relaxation serves to enhance the band gap of the CO state to values of 0.4–2.2 eV.

LiFePO and LiFePO show similar behavior in which at critical values of (lower when structural relaxation is included) a CO transition opens an electronic band gap. For LiFePO with (without) relaxations a band gap opens at eV (5.3 eV) when CO yields 1 Fe-like site and 3 Fe-like sites. For LiFePO with (without) relaxations a band gap opens at eV (5.3 eV) when CO yields 3 Fe-like sites and 1 Fe-like site. For and the magnitude of the band gap is around 0.1–0.6 eV for the frozen structures and a larger values of 0.5–1.8 eV including relaxations. For these , the CO transition always leads to a ferrimagnetic state with magnetization of 1 . The Fe magnetic moments of the distinct sites are around 3.8 and 4.4 and slowly increase with as in the case of . Differences in the magnetic moments between the frozen and relaxed structures in the CO regime are small. We note that for these , unlike in , we find partial CO for intermediate values of in which the Fe magnetic moments begin to take on slightly different values without the presence of a band gap. For example, including relaxations for eV LiFePO is metallic with Fe magnetic moments of 3.9, , , and 4.0 . This indicates DFT+ is driving LiFePO towards CO even for small , which is to be expected given the very narrow bandwidths.

Unlike for LiCoO (see Fig. 3), for LiFePO the CO is stabilized by structural relaxations as evidenced by the lower critical values for CO. The stabilization of CO by structural relaxations suggests stronger coupling between the electronic and lattice degrees of freedom in LiFePO as compared to LiCoO. This is consistent with the evidence for polarons in LiFePO.Ellis et al. (2006); Zaghib et al. (2007)

iv.6 Impact of on phase stability of LiFePO

Figure 10: Total LiFePO formation energy (open black circles) and its DFT (filled red squares) and –dc (filled blue circles) components as a function of for (a) , (b), , and (c) . (d)–(f) show the corresponding plots of LiFePO –dc (filled blue circles) formation energy component and its orbital filling (open red circles) and orbital ordering (open black squares) components as a function of . (g)–(i) show the corresponding plots of number of electrons per Fe as a function of for FePO (black line), LiFePO (open red squares), and LiFePO (blue line). All data correspond to the case of frozen linearly interpolated experimental structures.

The formation energy behavior for LiFePO as a function of without allowing for the effect of structural relaxations is summarized in Fig. 10. The behavior shares many similarities and a few differences to that of LiCoO, which we will describe. As illustrated in panels (a)–(c), for all intermediate the total formation energy increases with as in the case of LiCoO and changes sign from negative to positive for sufficiently high . For example, for the formation energy is eV at and increases to +0.21 eV for eV. For and , there are only slightly negative values of around eV for , which increase to as much as +0.26–0.27 eV as increases. We note that the formation energies are higher for this frozen structure case since the intermediate species will exhibit more significant structural relaxations than the endmembers.

For most of the range, that below the critical for CO, the DFT component of the formation energy is approximately constant and does not strongly influence phase stability. For example, for this component varies by only 25 meV for 0 eV 4.7 eV. In contrast, the –dc component of the formation energy undergoes significant changes as it increases roughly linearly with ; the change is 0.21 eV over the same range for . This –dc contribution is positive and therefore, as in the case of LiCoO, is what drives the total formation energy towards phase separation in LiFePO in this regime before CO. One caveat to this characterization is that the DFT component varies more considerably (by 90 meV) for LiFePO in this regime. In this case it is positive for 2 eV 3 eV and thus can be described as partially responsible for the positive total formation energy.

Upon CO there is a drastic change in the formation energy components, as in the case of LiCoO. The –dc component drops steeply to much smaller values of around 0.06–0.08 eV. For , for example, the value is 0.33 eV for eV and only 0.07 eV for eV due to the CO transition; the drop in the –dc contribution is substantial though smaller in magnitude for and . Here as in LiCoO we also find an increase in the DFT formation energy contribution due to CO. For example, for LiFePO the DFT formation energy contribution jumps from meV to +122 meV across the CO phase boundary. Unlike in LiCoO [see Fig. 7(a)], for LiFePO the increase in the DFT component is enough to make it positive. It is even larger in magnitude than the –dc component. For and the DFT component is already positive but it becomes substantially more positive after CO. Despite how sharp the changes in formation energy components are, the total formation energy changes less abruptly from CO. For example, for the total formation energy changes only by 3 meV between eV and eV. After the CO transition, the formation energy and its components are relatively flat versus . For , for example, the changes are on the order of only 15–25 meV.

The breakdown of –dc formation energy into filling and ordering contributions is shown in Fig. 10(d)–(f). As in LiCoO, it can be seen that the average filling contributes negligibly to the phase separation; the maximum contribution is at most 25 meV and typically much smaller. Therefore, again the impact of and the dc is essentially entirely contained within the ordering contribution. The ordering contribution tracks the behavior of the total –dc term and it can be as much as +0.3 eV.

The plots of on each Fe site versus in Fig. 10(g)–(i) illustrate that there is also a rehybridization mechanism in LiFePO similar but smaller than that in LiCoO with differences in around 20–30% of a full electron between and . Like the Fe magnetic moment data in Fig. 9, these values illustrate the partial and full CO that occurs as increases. We note that without structural relaxations the spread in values after CO for the intermediate can be larger than the difference between of and . We find the behavior of with is complex and system dependent: in this case for intermediate is generally lower than the endmember linear interpolation, whereas in LiCoO the opposite is true.

Figure 11: Filling factor as a function of averaged over Fe sites for different in LiFePO for (a) linearly interpolated experimental structures and (b) relaxed structures. Panels (c) and (d) show the corresponding plots for the ordering factor . The line thickness increases for increasing values of .

To investigate the origin of the positive –dc phase-separating contribution to the total formation energy for LiFePO, in Fig. 11 we plot the individual [panel (a)] and [panel (c)] values as a function of for different . For clarity, we take the average over the 4 Fe sites in the primitive unit cell. As in the case of LiCoO, the magnitude of the filling factor is high (around 0.24) but the changes with respect to the average of the endmember values are very small (on the order of ). This is responsible for the negligible contribution of the filling component to the total formation energy. As for LiCoO, the filling factor is highest for and . Compared to those of LiCoO, the values of LiFePO are much nearer to the range expected from nominal electron counting for Fe and Fe (0.24–0.25) since there is less hybridization between and states. This is also the reason for the enhanced magnitude of compared to that of LiCoO.

Compared to , has a smaller but still significant magnitude (around 0.18) for LiFePO. The values monotonically increase with (by around 0.003–0.006 per eV) for all , which as in the case of LiCoO is expected since the and dc terms serve to enhance orbital polarization in the correlated subspace. Unlike in , there are substantial deviations of the values for intermediate compared to the endmember average. For example, for at eV =0.172 as compared to 0.187 for the endmember average. The ordering factor is consistently lower than the endmember linear interpolation, thus leading to the positive formation energy contribution.

One can observe a moderate increase in values for intermediate upon CO. For example, for jumps from 0.177 to 0.194 upon CO at eV. However, as in the case of LiCoO the values still are always lower than the linear interpolation of and values even after CO. For example, the endmember average is 0.196 for eV so it is still slightly larger than the value for LiFePO. Therefore, we find that CO alleviates but does not eliminate the tendency for phase separation with derived from the ordering of correlated orbitals.

Figure 12: Total LiFePO formation energy (open black circles) and its DFT (filled red squares) and –dc (filled blue circles) components as a function of for (a) , (b), , and (c) . (d)–(f) show the corresponding plots of LiFePO –dc (filled blue circles) formation energy component and its orbital filling (open red circles) and orbital ordering (open black squares) components as a function of . (g)–(i) show the corresponding plots of number of electrons per Fe as a function of for FePO (black line), LiFePO (open red squares), and LiFePO (blue line). All data correspond to the case of relaxed structures.

Figure 12(a)–(c) shows the formation energy behavior of LiFePO including full structural relaxations. In this case, since there are more significant total energy lowerings from relaxing the intermediate structures compared to those from relaxing the endmembers, the total formation energy values are significantly lower. For example, for LiFePO the maximum formation energy is 52 meV as opposed to 262 meV in the case of linearly interpolated endmember experimental structures. Otherwise, the behavior is generally similar to the case without relaxations. For all , the total formation energy increases monotonically with and for sufficiently high switches from negative to positive. We note that the formation energy LiFePO is only slightly positive (+6.7 meV) for the largest we considered. This is in quantitative disagreement with the original DFT+ workZhou et al. (2004a) for reasons which are not clear, but is consistent with a more recent report.Ong et al. (2011)

Here again the DFT formation energy contribution is approximately constant (deviations of at most 25 meV and typically less) before the CO transition with a slight tendency to increase with . In this case is no longer an exception to the general trend. In contrast, the –dc contribution is positive and significantly increases (roughly linearly) with . The largest value it takes on is 151 meV for at eV. Therefore, again it is the positive –dc contribution that destabilizes the compounds of intermediate in this regime.

After CO, the –dc formation energy contribution is significantly dampened but remains positive. For example, for it decreases from 106 to 18 meV at the CO phase boundary. The effect is similarly substantial for all . At the same time, the DFT formation energy component abruptly increases due to CO and becomes positive ( and ) or much less negative (). The magnitude of this increase is more substantial than that of LiCoO, which suggests that in LiFePO CO constitutes a more significant rearrangement of charge density. The net effect is that upon CO the increase in total formation energy with is slowed. For the case, for example, the total formation energy increases only by 12 meV from to eV as opposed to the lower- regime in which the same change in yields increases of around 46 meV. In the CO state both the DFT and –dc components of the formation energy are nearly constant with respect to . As such, the total formation energies saturate to around +50, +6, and +50 meV for , , and in the regime of large .

As illustrated in Fig. 12(d)–(f), in this case again the positive –dc formation energy contribution stems entirely from the ordering contribution (from the spread in orbital occupancies); the filling component (from the average orbital occupancy) is negligible with values less than 10 meV. CO lowers the filling contribution even further to no more than 1 meV. In contrast, the ordering contribution is dampened but still positive and appreciable; it tracks the behavior of the total –dc formation energy contribution.

The evolution of the values into two discrete groups (Fe- and Fe-like) due to CO is shown in Fig. 12(g)–(i). A similar magnitude of - rehybridization is observed based on the overall range of . As in the case without structural relaxations, here for and though not there is a regime of intermediate in which there is partial CO in metallic states. After the CO transition the two groups of values closely match those of and , which suggests the CO is very complete and the local environments of Fe for intermediate mimic those of the endmembers.

The filling and ordering factors for relaxed LiFePO are shown in Fig. 11(b) and 11(d), respectively. We find similar results when including structural relaxations. The deviations in with respect to the linear interpolation of endmember values are negligible. gradually increases with and the values for intermediate always lag behind the endmember linear interpolation; the effect is substantially dampened but not entirely eliminated when CO increases for intermediate . We note that with structural relaxations after CO the versus curves are nearly linear, which illustrates that relaxations provide a further dampening of the general tendency towards phase separation with stemming from the ordering term.

iv.7 Phase stability of LiCoPO

To further validate our general understanding of the impact of on phase stability, we investigate LiCoPO. This material is isostructural to the olivine LiFePO structure shown in Fig. 1 with (nominally) an additional electron on the transition metal site. LiCoPO is of interest since it has been shown to have a very high voltage (4.8 V) as a cathode material.Amine et al. (2000) It is intriguing physically since unlike LiFePO it does have a stable intermediate compound, for .Bramnik et al. (2007); Ehrenberg et al. (2009); Strobridge et al. (2014)

We consider the lowest-energy configuration of LiCoPO deduced by Strobridge et al.Strobridge et al. (2014) and study the formation energy as a function of with the frozen structures. Experimentally the LiCoPO system is AFM,Santoro et al. (1966); Ehrenberg et al. (2009) so our calculations consider the endmembers in the AFM state. We base our calculations on the magnetic configuration of LiCoPO, which is identical to that of LiFePO.Santoro et al. (1966); Santoro and Newnham (1967) For LiCoPO we find the AFM state to be unstable and devolves into a ferrimagnetic state. In this state one of the Co sites has no magnetic moment unlike the values of the other 11 leading to a total magnetization of .

We find total formation energies of , , and eV for values of 0, 2, and 5.48 eV, respectively. This corresponds to the same trend of formation energy increasing with . We again find the positive formation energy contribution is the –dc component, and here also the filling contribution is very small (magnitude of at most 20 meV) so the origin is increased ordering of the endmember states relative to those of the intermediate species.

For the above analysis we have restricted our attention to the AFM or AFM-like states as they are the experimental magnetic structure for LiCoPO. However, we note that the magnetic ground state of LiCoPO predicted by DFT+ changes as a function of , which is likely the origin of a previous study finding the formation energy becomes more negative as a function of .Strobridge et al. (2014) For LiCoPO we find a ferromagnetic (FM) ground state within DFT with the AFM state 37 meV higher in energy, whereas for CoPO we find a non-spin-polarized (NSP) ground state with the FM and AFM states 57–61 meV higher in energy. At the ground state of LiCoPO becomes antiferromagnetic, and by eV the ground state for both endmembers is AFM consistent with experiments. For we find the FM state is unstable within DFT and the NSP state is 0.54 eV higher in energy than the ferrimagnetic state. At eV the FM state becomes metastable only 7 meV above the ferrimagnetic state, and for eV the FM state becomes the ground state with the ferrimagnetic state 68 meV high in energy. In any case, DFT+ does predict the phase to be stable, qualitatively consistent with experiment.

iv.8 Impact of double counting on phase separation trend

Our preceding analysis shows that it is valuable to recast the –dc energy as the sum of ordering and filling terms, as the filling term was shown to have negligible impact on formation energies. However, there is still utility in analyzing what can be learned by separately inspecting and , given that the term specifically is handled much more precisely in the context of DFT+DMFT.

The simplest possible explanation for the positive –dc formation energy contribution comes from the form of the dc term Eq. 4. We expect the Li in LiCoO and other intercalation materials will be ionized and donate some amount of charge (depending on the degree of hybridization with O states) to Co. Therefore, we might make the assumption that is linear in . Since energy contributions linear in do not contribute to the formation energy by the definition of Eq. 13, this means the term linear in in the dc term cannot contribute to the formation energy. The other part of the dc, however, yields a term proportional to (a negative quadratic in ) in the total energy. With the assumption of a linear relationship between and , this term is a negative quadratic in that necessarily provides a positive formation energy contribution.

This simple kind of argument fails to fully describe the observed behavior for the following reasons, which we illustrate using the simplest example of LiCoO without CO considering the frozen structures. The first two reasons come from the fact that the variation of with is in fact substantially nonlinear, as can be easily seen for the eV data in Fig. 5(c). In this regime, for is much closer to the value than the value. For example, for eV the values are 7.19, 7.28, and 7.30 for , , and , respectively. First of all, this means that the contribution of the dc energy that is proportional to will contribute to the formation energy. For example, for this case it takes on the value of +89 meV for eV.

Secondly, the quadratic part of the dc energy no longer necessarily gives a positive formation energy contribution for that is nonlinear in . One can parametrize the deviation from linearity using the additive form , where is the value for Li concentration , is the linear value of , and is the deviation from linearity. Here we assume a single site for clarity. It then can be shown that the formation energy contribution stemming from the quadratic part of the dc energy is

(15)

Note that the term proportional to