Unconventional Magnetism and Band Gap Formation in LiFePO4: Consequence of Polyanion Induced Non-planarity

Oxygen plays a critical role in strongly correlated transition metal oxides as crystal field effect is one of the key factors that determine the degree of localization of the valence d/f states. Based on the localization, a set of conventional mechanisms such as Mott-Hubbard, Charge-transfer and Slater were formulated to explain the antiferromagnetic and insulating (AFI) phenomena in many of these correlated systems. From the case study on LiFePO4, through density-functional calculations, we demonstrate that none of these mechanisms are strictly applicable to explain the AFI behavior when the transition metal oxides have polyanions such as (PO4)3−. The symmetry-lowering of the metal-oxygen complex, to stabilize the polyanion, creates an asymmetric crystal field for d/f states. In LiFePO4 this field creates completely non-degenerate Fe-d states which, with negligible p-d and d-d covalent interactions, become atomically localized to ensure a gap at the Fermi level. Due to large exchange splitting, high spin state is favored and an antiferromagnetic configuration is stabilized. For the prototype LiFePO4, independent electron approximation is good enough to obtain the AFI ground state. Inclusion of additional correlation measures like Hubbard U simply amplifies the gap and therefore LiFePO4 can be preferably called as weakly coupled Mott insulator.

There are some other TMOs, mostly involving weakly localized d electrons (e.g. Sr 2 IrO 4 ) 23 , which undergo metal-insulator transition at the Neel temperature (T N ). The antiferromagnetic ordering stabilizes as electrons with opposite spins move in different potentials 20 below T N . As a consequence each Brillouin zone is reduced by half and each energy level splits into two with a gap in the middle as shown in Fig. 1c.
In the case of LFPO, we show that the planar and octahedral symmetries among Fe and O ions are lost as phosphorus strongly attracts the oxygens to form stable (PO 4 ) 3− tetrahedral polyanions in the system. This lowering in symmetry introduces a completely anisotropic and inhomogeneous crystal field to create multiple non-degenerate d-states which are devoid of any covalent interaction with the O-p orbitals. These atomic like d-states ensure a gap at the Fermi level to drive the insulating behavior in LFPO as we see in zero dimensional systems such as clusters and nano dots 24 . Also these states undergo large spin-splitting below T N to create multiple spin half states which mediate Heisenberg type antiferromagnetic interaction in the system. The correlation correction, made through Hubbard U, simply increases the magnitude of the gap as demonstrated in Fig. 1d.

Structural and Computational Details
Olivine Crystal Structure. Crystal structure of LiFePO 4 is orthorhombic and it belongs to the Olivine family of compounds 1,25 with space group Pnma (No. 62). The two deterministic features of the crystal structure of LFPO are: (i) the presence of perfect PO 4 tetrahedras (Fig. 2a) and (ii) the presence of FeO 6 complexes (Fig. 2b), where the octahderal symmetry is lost due to unequal bond lengths and bond angles as shown in Fig. 2c. The crystal axes differ from the axes of the FeO 6 complexes and neighboring FeO 6 complexes are tilted with each other.
Computational Details. Density functional calculations are performed using the Vanderbilt ultra-soft pseudo-potentials 26 and plane wave basis sets as implemented in Quantum Espresso (QE) 27 . Exchange-correlation potential is approximated through PBE-GGA functional 28 . Some of the calculations are performed using LDA for comparison and analysis purpose. Strong correlation is a natural phenomena in transition metal oxides. To account for this, parameterized Hubbard U is included in our ab initio calculations. The kinetic energy cutoff to fix the number of plane waves is taken as 30 Ry. However, the kinetic energy cutoff for charge density is taken as 250 Ry. A 6 × 10 × 12 k-mesh of the BZ, yielding 456 irreducible k-points, for the regular unit cell is found to be sufficient to calculate the total energy with reasonable accuracy within pseudopotential approximation. shows that the DFT optimized structure is quite close to the experimental one. Hence it is expected that the DFT calculations will reproduce the low temperature experimental properties of LFPO.

Results
Non-Planarity and Structural Stability. Crystals with planar geometry are highly symmetric and therefore many compounds, particularly the TMOs prefer to stabilize in a planar (layered) crystal structure. Some of the well known planar TMOs are monoxides like MnO 29 , NiO 30 , perovskites (AMO 3 ) 31,32 , cuprates 33 and Ruddlesden-Popper series: A n+1 M n O 3n+1 34,35 . In these TMOs, the transition metal and oxygen ions lie (almost) on a plane. As a consequence, the O ligands of the M-O complex create a symmetric crystal field to split the fivefold degenerate atomic M-d states into multiple degenerate states. For example if there is an octahedral symmetry, as in the case of perovskites, the d-states split into triply degenerate t 2g and doubly degenerate e g states 21,36 . These degenerate states, depending on their occupancies and strong correlation due to localization, exhibit many exotic phenomena. These include half-metallicity 37 , itinerant magnetisms 38 , colossal magneto resistance (CMR) 39 , Mott-Hubbard and Charge-transfer insulators 10,19 and high-T C superconductivity 40 .
Unlike the TMOs discussed above, LFPO is highly non-planar, yet stable and shows AFI behavior. Therefore, it is paramount to study the link between non-planarity and structural stability which, in return, will give useful insight to the electronic and magnetic structure of this compound. In this context, we have carried out a virtual structural deformation experiment as demonstrated in Fig. 3. We start with a perfectly planar hypothetical structure and gradually deform it to the experimental non-planar structure and in each step we performed the ab initio calculations. The hypothetical planar structure may be realized experimentally through atomic layer deposition (ALD) and molecular beam epitaxy (MBE) approaches.
Though for the virtual experiment several intermediate structures between the planar and the experimental structure were studied, only three of them, whose Wyckoff positions are listed in Table 2, are discussed here to avoid the redundancy. The stability of these structures are measured through the total energy calculations using the ground state antiferromagnetic ordering of the experimental structure.
A comparison of the structures and the corresponding total energies (Fig. 3) suggests that the lowering in symmetry through planar deformation brings stability to the structure. In fact the planar structure is unstable by ~20 eV, with respect to the experimental structure, which is too high for a crystal. During the process of deformation, FeO 8 complexes give rise to FeO 6 complexes and concurrently perfect tetrahedral (PO 4 ) 3− polyanions are formed. The octahedral symmetries, C 3 , C 2 , C 4 and C ' The relation between structural stability and formation of PO 4 tetrahedra is quite evident from the energy comparison of structure III and IV. As we move from III to IV, the PO 4 complex takes the shape of a tetrahedra by attracting two O ions from the FeO 8 complex. In this process the system gains stability by ~11 eV. Further perfection of the tetrahedral shape (structure V) makes the system most stable. The consequence is very significant in realizing LFPO as a cathode material. As Li ion has minimal role for the structural stability, it can be easily diffused and absorbed to facilitate charging and discharging process respectively. The non-planarity brings a big change in the electronic and magnetic properties of LFPO as well. For the planar structure (I), i.e. in the absence of PO 4 tetrahedra, the band structure is metallic with widely dispersed bands crossing the Fermi level (E F ). By lowering the symmetry the bands at E F become less and less dispersed. For the experimental structure they are almost flat to induce insulating behavior in this compound. Detailed discussion on the mechanisms that lead to the AFI behavior in LFPO is made in the following two sections. We note that weak electron conductivity is one of the major disadvantages of LFPO as cathode material 7,16 . Bottom two structures in Fig. 3 suggest that a small distortion in tetrahedra enhances the band dispersion substantially. The distortion may be achieved by external pressures or doping.
Electronic and Magnetic Ground State. The objective of this section is to see how octahedral asymmetry due to structural non-planarity affects the electronic and magnetic properties of LFPO. It is well known that a small distortion of the MnO 6 octahedra breaks the e g degeneracy in LaMnO 3 and as a consequence A-type AFI  Table 2. The Wyckoff positions for the structures shown in Fig. 3. The numbers written in bold indicate the changes made with respect to the previous structure.
ground state emerges in this compound 36 . However, in most of the theoretical studies 11,13-15 on LFPO the primary structural assumption is that the FeO 6 complex is not distorted to the extent that t 2g and e g symmetries of the Fe-d states are broken. While such an assumption does not affect the energetics and hence stability of the system, it lacks in explaining the electronic and magnetic behavior. Also, as LFPO has 3d electrons, it is necessary to investigate the correlation effect on the electronic properties of this system. To our knowledge there are few literature which have partly discussed the correlation effect using DFT + U calculations 13,15-17 and using DFT + dynamical mean-field theory (DMFT) calculations 41,42 . While the DMFT studies are restricted to the paramagnetic phase, DFT + U calculations were carried out to match the theoretical band gap with the experimental band gap 15,16 . Therefore, a definitive mechanism to explain the experimental AFI behavior has not evolved so far.
Electronic structure of LiFePO 4 within GGA. Electronic structure of LFPO can be well understood from  temperature 7,8 . Therefore, it contradicts the Slater mechanism 20 . As the band gap exists even without additional correlation (Hubbard U), it is not appropriate to define LFPO as a conventional Mott insulator 18 . The charge -transfer mechanism 19 is ruled out as we have d-d gap instead of a p-d gap at E F . The band structures in Fig. 4 further reveal that irrespective of the magnetic order, non-dispersive bands at E F are basically non-degenerate Fe-d states. The distinction of the non-degenerate states is very prominent in the AFM structures. The three dimensional electron density of these non-dispersive bands for the AFM(GS) structure, plotted in the bottom panel of Fig. 4, clearly shows that these bands are basically the individual atomic d orbtials. Therefore, the p-d covalent interaction is either absent or negligible which can be further confirmed from the bandwidth of these states.
A careful observation of Fig. 4, tells us that the band width of the non-degenerate states lies in the range of 0.24 to 0.4 eV for the unstable NM, 0.22 to 0.84 eV for the FM configurations and 0.08 to 0.52 eV for the stable AFM configuration. If we map these bandwidths to that of a oversimplified nearest-neighbor tight-binding band dispersion (2t cos ka) for a one-dimensional lattice of length 'a' , then the hopping parameter will lie in the range 0.06 to 0.1eV for NM, 0.055 to 0. 21 eV for FM and 0.016 to 0.13 eV for AFM configuration. Such a hopping strength is very negligible to assume any covalent p-d or d-d interactions in the system.
The non-interacting d-orbitals in the presence of a weak crystal field are filled following the Hund's rule and hence Fe 2+ favors high spin (HS) state. This agrees well with the DFT prediction as the AFM (GS) band structure shows that except four spin-minority d states, the rest are lying below E F . In fact the NM configuration represents the low spin (LS) state for Fe 2+ and is highly unstable, approximately by 0.5 eV, than that of the magnetic configurations. At higher temperature it will give rise to paramagnetic insulating phase. We note that for odd number of d electrons (Fe 3+ ), as in the case of the de-lithiated compound FePO 4 , the hypothetical NM phase may have states pinned on the Fermi level, but the paramagnetic phase is always insulating. This is discussed in detail in Fig. 1

Effects of Exchange Correlation and Onsite Correlation on the Electronic Structure.
Appropriate exchange correlation functionals are vital for accurate DFT prediction of the electronic structure of solids, particularly for the correlated TMOs due to presence of localized d states. A simple case study is the family of monoxides. While LDA predicts accurately the ground state magnetic ordering for NiO and MnO, it gives a smaller band gap than the experimental values 10,43 . For FeO and CoO, LDA suggests a metallic solution while in reality these are insulators with wide band gaps 43 . The errors are attributed to the inability of LDA to account for the correlation. One way to correct the error is through a Hubbard term H 1 where i represents the site index and ν = (m, σ ) collectively represents the orbital (m) and spin index (σ ) of the state. U serves the purpose of onsite repulsion between the states ν and ν′ . Within mean field solution, the pairs ν and ν′ split to produce a gap ~ U between them. While one of the states gets occupied, other remains above E F. Based on the strong correlation effect, two of the mechanisms (Mott-Hubbard and Charge-transfer) that have evolved to explain the AFI behavior in TMOs are schematically shown in Fig.1.
Earlier DFT calculations [11][12][13] using LDA predict ferromagnetic and metallic ground state for LFPO. Inclusion of U makes it AFI and the gap resembles to that of a Mott insulator 17 . However, the electronic structure with GGA, presented in the last section, predicts the AFI behavior even without U. To see whether the LDA really provides a metallic solution, we performed the calculations using a relatively highly dense k-mesh (~3350 irreducible k-points) and large number of plane waves (~22000) and we found a zero band gap as shown in Fig. 5a. A very small value of U opens up the gap. Therefore the metallic solution is to some extent a computational inaccuracy rather than a correlation problem.
To further investigate the effect of correlation, we have plotted U dependent DOS in Fig. 5b. As expected, with increasing U, the occupied d states are pushed below and unoccupied d states are pushed above to open up a larger band gap. Also with increasing U, the d states penetrate the O-p spectrum and induce a reverse hybridization with the latter. The band gap vs U plot of Fig. 5c shows that for small values of U, E g ~ U for LDA and E g ~ U + E' g for GGA. Here E' g is the gap without U. Such a situation arises for extreme localization of the states. For large value of U (> 5 eV) E g saturates as O-p states occupy the valence band maximum. We note that experimental band gap is (~3.8 eV) 16 which is obtained theoretically when U is close to 4.5 eV.
It is important to compare the electronic structure of FeO and LFPO as in both the cases Fe is in 2 + charge states as well as in high spin state. The difference is in the structure. In FeO, the octahedral symmetry is present while it is completely broken in LFPO as discussed in the early part of this paper. In this context we have schematically summarized the electronic and magnetic structure of both the compounds in Fig. 6. The electronic structure of FeO is understood from the DFT results reported by Anisimov et al. 10 and Terakura et al. 43 Within LDA, the degenerate t 2g and e g states of the FeO 6 octahedra have larger band width in FeO since the covalent p-d and d-d interactions are maximized in this case. Fe 2+ being in high spin state, the spin-down t 2g band remains partially occupied. Therefore, without U, FeO will always have a metallic solution. On the contrary in LFPO, the five non-degenerate d states with negligible p-d and d-d covalent interactions resemble to the electronic states of clusters and are localized enough to produce a narrow gap at E F . The Hubbard term in the Hamiltonian (Eq. 1) amplifies this gap. If the Hamiltonian involves both intra-orbital and inter-orbital onsite Coulomb repulsions, ordering of the d orbitals with respect to the Fermi level might also change along with the gap amplification 41 .
The mechanism that leads to the formation of AFI ground state in LiFePO 4 can be extended to other members of the LiMPO 4 (M = Cr, Mn, Co and Ni) (see Fig. 4    study the stability of each magnetic ordering by calculating the spin exchange interactions (J). Experimentally, magnetocrystaline anisotropies are observed in the family of LiMPO 4 due to spin-orbit coupling [44][45][46] . For LiFePO 4 it is found to be modereate for (Δ g/g ~ 0.1, g is the Lande g factor) 44 . This may be significant for the magnetoelectric effect 47 but the Dzyaloshinskii-Moriya (DM) exchange interaction strength is found to be weak and its contribution to T N is very negligible for LFPO 44,48 . Therefore, we have neglected the DM term in this work. To estimate J in LFPO we have applied the extension of the Noodleman's broken-symmetry method 49 given by Dai and Whangbo 50 . According to this method, the energy difference between the high spin state and low spin state for a spin dimer can be approximated as: where J is related to the spin-dimer Hamiltonian, = ⋅ˆĤ J S S E hs and E ls are energies of the high and low spin states respectively which can be estimated from the DFT calculations as discussed below. In Eq. 3, N (= 4) represents the number of unpaired spins in each monomer. The net spin arrangement can be considered as a sum of individual spin-dimers of the lattice. The spin-dimers considered in this paper are shown in Fig. 7.
To evaluate the exchange interaction parameters from DFT results, we have considered seven magnetic configurations out of which one of them is ferromagnetic (FM). In other configurations (A1 -A6) at least one of the spin dimers is antiferromagnetic. The configurations for A1 (AFM(GS)) and A2 (AFM(ES)) are shown in Fig. 4 and are already discussed in the context of electronic structure. The rest are not shown in this paper, but can be mapped from the expression of their total spin-exchange energy which, using Eq. 3, can be written as: The energies, E, of Eq. 4 are now equated to the DFT calculated total energy of the respective magnetic configuration to estimate the J i values 51 and the results are listed in Table 3. Small values of J i (0-1 meV) suggest that LFPO is a magnetically weak system. Our estimated exchange interaction parameters are comparable with the previously reported values 44,50 . We attribute the weak magnetic interaction of the spin dimers to the extremely non-linear super exchange paths shown in Fig. 7. A pair of localized spins (dimer) always prefers to be antiferromagnetic which cannot be full filled when the spin-dimers are not isolated. For example an antiferromagnetic J 1 would prefer ferromagnetic J 4 and J 6 as can be observed from Fig. 7. Therefore, the system may exhibit incommensurate where Z i represents the number of equivalent magnetic neighbors corresponding to J i (see Table 3). K B is the Maxwell-Boltzmann constant and μ is a mean field constant. From the experimental studies 55 θ WC and T N are found to be − 115 K and 51 K respectively and hence μ ~ 2.25. Now taking S as 2, we have estimated θ WC and T N for the optimized and experimental structures and the results are listed in Table 3.
Our DFT calculations predict the Neel temperature in the range 54.04 to 64.36 K which agree well with the experimental value. However, one has to be careful while predicting the exchange interactions from DFT. The mean-field based calculations, such as DFT, overestimate the exchange interaction strengths. Also we would like to note that in the present case, a change in the total energy by 1 meV can change the T N by 5 K on either side. For better estimation of exchange interactions and T N , more comprehensive methods at the atomic scale such as DFT + DMFT 56 and atomistic spin dynamics 57 may be used.

Summary
In summary, we have studied the electronic and magnetic structure of the cathode material LiFePO 4 with the objective to study the role of a polyanion in transition metal oxides. We find that the stable (PO 4 ) 3− polyanion makes the compound non-planar and the degree of non-planarity determine the electronic and magnetic stability of the system. Here, the FeO 6 complex is formed to stabilize the (PO 4 ) 3− polyanion. The former is largely distorted where the octahedral symmetry is completely absent. Therefore, the crystal field induced on the Fe-d orbitals is strongly asymmetric and splits the degenerate d states into multiple non-degenerate states. These states are localized, in the absence of strong d-d and p-d and p-p covalent interactions, and resemble to that of zero-dimensional systems such as clusters and nano-dots. The crystal field splitting is sufficient to introduce a narrow band gap which is unlikely for the conventional Mott insulators. The increase in Hubbard U simply enhances the magnitude of the gap. The spin-dimer analysis presented in this paper suggests a weak antiferromagnetic ordering and the estimated Neel temperature is found to be lying in the range [54-64 K] which agrees very well with the experimental value of 51 K. The mechanism presented in this article to explain the antiferromagnetic and insulating ground state of LFPO is extendable to other members of transition metal olivine phosphates and is expected to be a prototype for many transition metal oxides having polyanions.  Table 3. Exchange interaction parameters (J i ), in meV, are obtained by equating the DFT calculated energies to that of Eq. 4. The corresponding exchange paths are shown in Fig. 7. Z i represents the number of equivalent magnetic neighbors for J i exchange path. Positive and negative values of J i indicate antiferro and ferro ordering respectively. The Curie-Weiss temperature Θ CW and the Neel temperature T N are evaluated using Eq. 5.