The effect of octahedral distortions on the electronic properties and magnetic interactions in O3 NaTMO2 compounds (TM = Ti–Ni & Zr–Pd)

The interplay between the coordination environment and magnetic properties in O3 layered sodium transition metal oxides (NaTMO2) is a fascinating and complex problem. Through detailed and comprehensive density functional investigations on O3 NaTMO2 compounds, we demonstrate that the TM ions in O3 NaMnO2, NaFeO2 and NaCoO2 adopt a high spin state. Structurally, NaMnO2 and NaPdO2 undergo Jahn–Teller distortions while NaNbO2 undergoes puckering distortion. Furthermore, in addition to Jahn–Teller distortion, NaPdO2 exhibits charge disproportionation as it contains Pd2+, Pd3+ and Pd4+ species. These distortions stabilize the inter-plane ferromagnetism. Additionally, the inter-plane ferromagnetic coupling is stabilized by kinetic p–d exchange mechanism in undistorted NaCoO2, NaNiO2 and NaTcO2. The intra-plane coupling in this family of compounds on the other hand was found to be generally weak. Only NaMnO2, NaNiO2 and NaTcO2 are predicted to show bulk ferromagnetism with estimated Curie temperatures below ∼50 K.


Introduction
Layered hexagonal compounds with formula NaTMO 2 in which TM is a transition metal oen exhibit interesting magnetic, 1-3 thermoelectric, 4 and electrochemical properties. 5 One obstacle in studying these compounds is the rich variety of their polymorphs each with distinct symmetry and local coordination for the transition metal ions which complicates nding general property trends for this family of materials. One such area of research that lacks a concise overview is the magnetic properties of the NaTMO 2 compounds. For instance, reports of conicting experimental observations of the magnetic properties for the same compound is not unheard of. 6,7 Such contradictions oentimes stem from coarse structural characterization, restricted by instruments' range and resolution, which falls short in capturing the delicate structural details that dictate the magnetic ground state in these compounds. 8,9 This study therefore presents a detailed and focused density functional insight into one important family of such layered materials, namely O3 NaTMO 2 compounds in which TM is a fourth or h row transition metal element.
We start our investigations with compounds of R 3m symmetry which is a common symmetry group among layered compounds. 10 As shown in Fig. 1(a) and (b), the hexagonal representation of this structure consists of three alternating TMO 2 and Na layers. The notion "O3" indicates that oxygen ions are stacked in ABCABC order and Na ions occupy the octahedral site with respect to the surrounding O ions. The primitive cell of the R 3m O3 structure, presented in Fig. 1(c), has rhombohedral symmetry and the TM ion is located in the center of the primitive cell coordinated by six oxygen ions. The O-TM-O angles, marked h in Fig. 1(c), depend on the lattice parameters of the rhombohedral primitive cell (a and a) and the fractional coordinates of oxygen. If this angle is not exactly 90 , then it follows that the O-TM-O angles alternate between values smaller and larger than 90 resulting in a rhombohedral distortion to the TMO 6 octahedra. These angles are marked h and q in Fig. 1(d). This distortion decreases the octahedral symmetry and splits the energy levels of the t 2g orbitals of the TM ions into a single a 1g orbital and doubly degenerate e 0 g orbitals. The sequence of stabilization of the a 1g and e 0 g orbitals is not however trivial. 11 In addition to the rhombohedral distortion which is inherent to the R 3m symmetry, NaTMO 2 compounds may also experience additional distortions that further reduce the overall symmetry and inuence the electronic structure. We will thoroughly examine all such distortions and determine how they inuence the electronic and magnetic properties of O3 NaTMO 2 compounds.

Computational and system settings
Spin polarized density functional theory (DFT) calculations were carried out within augmented plane-wave potential 12 formalism as implemented in VASP code. 13,14 Brillouin zone was sampled using a mesh generated by Monkhorst-Pack scheme with spacing of $0.02Å À1 among k points while the energy cutoff was set to 550 eV. The threshold for energy convergence was set to 10 À7 eV per atom. Orbital population and bonding characteristics were examined using LOBSTER code. 15 The exchange-correlation functional was approximated by the Perdew-Burke-Ernzerhof method. 16,17 To improve the electronic description of the compounds in term of localizing the d shell electrons of the transition metal elements, an orbital dependent Hubbard term 18 was applied to the d orbitals. The value of U eff (U À J) was set to 5 eV for 3d TM elements and 2 eV for 4d TM elements. Weaker localization effects of the 4d electrons justies the smaller U eff value for the 4d elements. Among the different elements of the 3d and the 4d rows, a small variation in U eff is naturally expected. However, the choice of constant U eff for each row allows a more straightforward comparison. 19 This procedure is further justied by the fact that the localization effects in NaVO 2 are not affected by slight variation of U eff . 11 Furthermore, as shown in Table 1, the applied U eff values reproduce the lattice constants reported in earlier experiments within $1% deviation indicating the adequacy of the chosen values.
O3 NaTMO 2 structure with R 3m symmetry in hexagonal representation, as shown in Fig. 1(a) and (b), was initially used for all compounds. To nd the nal geometries of NaTMO 2 compounds, the lattice parameters and all internal coordinates of the primitive cell were allowed to relax to forces smaller than 0.001 eVÅ À1 . Furthermore, the geometry optimization was repeated with 2a Â 2a Â 1c, 3a Â 3a Â 1c and 4a Â 4a Â 1c supercells with symmetry restrictions turned off, to detect any possible distortion that may lower the total energy by breaking the symmetry.
The magnetic phase stability was examined by comparing the total energies of the ferromagnetic system (E t FM ) with those of competing antiferromagnetic phases. The energy of the ferromagnetic state was calculated by aligning the spin of the all TM ions in the hexagonal cell parallel. The total energy of the Ctype antiferromagnetic state (E t CAFM ) was calculated by aligning the spin of adjacent TM ions within the basal planes of a 2a Â 1a Â 1c supercell antiparallel. DE CAFM is dened as the difference between total energies E t CAFM and E t FM the per TM ion: Here, n is the total number of the TM ions in the ferromagnetic supercell which is 3 for systems without distortions but larger for distorted systems. The energy of the A-type antiferromagnetic states (E t AAFM ) calculated by aligning the spin of TM ions in a 1a Â 1a Â 2c supercell antiparallel in alternating manner. DE AAFM is dened as the difference between E t AAFM and E t FM the per TM ion: Positive DE CAFM values indicate the preference of TM ion to align ferromagnetically within a TMO 2 plane (inter-plane) while positive E t AAFM indicate the preference of ferromagnetic coupling across TMO 2 planes (intra-plane).

TM spin state
Based on the calculated spin populations presented in Table 2, the early 3d TM ions in NaTiO 2 , NaVO 2 and NaCrO 2 generally conform to the octahedral crystal eld splitting t 2g e g . However, as we will see later, there are ner splittings among t 2g states caused by symmetry considerations. Later TM ions in NaMnO 2 , NaFeO 2 and NaCoO 2 compounds stabilize in high spin conguration. We found that the total energy of the NaMnO 2 compound rose by 1.306 eV/f.u. (f.u. is formula unit) when the Mn ion was set to low spin conguration (t 2g 4 e 0 g ). Similarly, the total energy of low spin NaFeO 2 (t 2g 5 e 0 g ) rose by 0.889 eV/f.u. and the total energy of low spin NaCoO 2 (t 2g 6 e 0 g ) rose by 0.157 eV/f.u. with respect to their corresponding high spin congurations. Ni ions in NaNiO 2 , nonetheless, are stabilized in low spin conguration as setting Ni to high spin conguration (t 2g 5 e g 2 ) raised the total energy by 0.777 eV/f.u. Our calculations for Ni is agreement with the experimental observation of low spin NaNiO 2 . 24 Unlike their 3d counterparts, early 4d TM ions in NaZrO 2 and NaNbO 2 , deviate from t 2g e g splitting as Zr bears no magnetic moment and Nb adopts two distinct magnetic moments both signicantly smaller than the anticipated t 2g 2 e 0 g . NaZrO 2 in which Zr set to t 2g 1 e 0 g was 0.317 eV/f.u. higher in energy than non-magnetic NaZrO 2 while NaNbO 2 with Nb xed to t 2g 2e 0 g conguration was 0.649 eV/f.u. higher than the presented ground state. Moreover, contrarily to the 3d TM ions, the later 4d TM ions in NaTMO 2 stabilized in low spin conguration. The total energy of NaTcO 2 rose by 1.086 eV/f.u. when Tc was set to high spin conguration (t 2g 2 e g 2 ). Similarly, the high spin NaRuO 2 (t 2g 3 e g 2 ) and NaRhO 2 (t 2g 4 e g 2 ) were higher in energy than their low spin counterpart by 1.937 eV/f.u. and 4.642 eV/ f.u. respectively. Fig. 2 shows the total and partial density of states (DOS) of 3d TM containing NaTMO 2 compounds. In the NaTiO 2 , rhombohedral distortion in the TiO 6 octahedra splits the t 2g orbitals of the spin-up channel into lower single fold a 1g orbital which is occupied by Ti 3+ 's lone 3d electron and higher empty e 0 g orbitals. Furthermore, a 1g orbital is detached from the lower O 2p states and creates a pseudo-gap within the valence band. Consequently, the complete separation of Ti 3d states from O 2p states implies that Ti-O bond is highly ionic. In NaVO 2 , the rhombohedral splitting is still dominant. However, contrary to the Ti case, the occupied e 0 g orbitals have lower energy than the empty a 1g orbital. Moreover, since the gap between e   the merging of the e 0 g and a 1g states in the spin-up channel, the band structure resembles conventional octahedral splitting where the spin-up t 2g states in the valence band are all occupied while the empty e g states constitute the bottom of the conduction band. The DOS of NaMnO 2 corresponds to the elongated Jahn-Teller distortion. The lower region of the valence band ($À7 eV < E < $À4 eV) is occupied by d xz and d yz states while the middle part ($À4 eV < E < $À1.2) is occupied by d xy states. The top of valence band is nevertheless occupied by d z 2 states. As inferred from the DOS, the proximity of d xy and d z 2 favors the high spin conguration for the Mn ions.

Electronic structures
The DOS of the half-lled Fe 3d shell (t 3 2g e 2 g ) in NaFeO 2 exhibits a different arrangement when compared to earlier compounds. Here, due to strong electron-electron repulsion between the half-lled Fe 3d 5 states and O 2p states, all of the occupied Fe 3d states are shied downwards below O 2p states.
The proximity of the t 3 2g and e 2 g states in the spin-up channel to one another favors the high spin conguration for Fe ions as the spin-down t 2g states are $11 eV higher in energy than spin-up e g states. In NaCoO 2 , the t 2g states of the spin-up channel, although mainly concentrate at the bottom of the valence band, still stretch over the entire valence band width and strongly hybridize with O 2p states. Furthermore, the tale of the spin-up t 2g states crosses the Fermi level creating half metallic conduction. Similarly, in NaNiO 2 , the spin-up t 2g states stretch over the valence band and cross the Fermi level while the spindown t 2g states and d z 2 states remain conned within the middle of the valence band without crossing the Fermi level. Fig. 3 show the total and partial DOS in 4d TM containing compounds. NaZrO 2 exhibits strong metallic character as its Fermi level intercepts the Zr 4d states in the conduction band. Metallicity in NaZrO 2 is facilitated by a metallic Zr-Zr bond which is caused by extraordinarily large Zr 3+ ionic radius of Fig. 2 The density of states (DOS) of NaTMO 2 compounds in which TM is forth row transition metal element. The black, red and purple lines correspond to total, O 2p and Na states respectively. The blue and green shaded areas correspond to TM 3d elements with net spin-up and spin-down electronic population respectively. Compounds with only blue shaded areas are either non-magnetic or those with interplane ferromagnetic coupling. Fig. 3 The density of states (DOS) of NaTMO 2 compounds in which TM is fifth row transition metal element. The black, red and purple lines correspond to total, O 2p and Na states respectively. The blue and green shaded areas correspond to TM 4d elements with net spin-up and spin-down electronic population respectively. Compounds with both blue and green shaded areas are G-type antiferromagnetic. $0.89Å (ref. 25) and the Zr-Zr distance of 3.21Å which is comparable to that in Zr metal. The metallic character of NaZrO 2 explains the lack of magnetic moment as there is no signicant hybridization between Zr with O. NaNbO 2 also exhibits metallic conduction as the Fermi level crosses through the 4d states in the conduction band. However, as we will discuss later, due to puckering distortion, there are two distinct Nb species in this compound each with different levels of metallicity. The band structure of the NaMoO 2 shows a conventional octahedral distortion where the half-led t 2g states constitute the top of the valence band while the empty e g states are separated by $1 eV above the Fermi level. In NaTcO 2 , NaRuO 2 and NaRhO 2 compounds the spin-down channel of the t 2g states is progressively lled as expected for the TM ions in low spin conguration. As will be discussed in detail in the next section, Pd in NaPdO 2 undergoes charge disproportionation and form Pd 2+ , Pd 3+ , Pd 4+ species. The t 2g states of all Pd species occupy the lower part of the conduction band while the lled e g occupy the top of the valence band.

Octahedral distortions
The geometry optimization conducted with larger supercells revealed that NaMnO 2 , NaNbO 2 and NaPdO 2 compounds, each to a different extent, exhibits additional distortions in their TMO 6 octahedra. In NaMnO 2 , as indicated by purple arrows in Fig. 4, two out of six Mn-O bonds in all MnO 6 octahedra are elongated causing a deviation from perfect R 3m symmetry. This distortion is similar to the Jahn-Teller distortion depicted in Fig. 1(e). The long Mn-O bond is 2.26Å while the short Mn-O bonds is 2.01Å implying a 12.4% elongation. We found that perfectly R 3m symmetric NaMnO 2 primitive cell with no elongation had a total energy 0.763 eV/f.u. higher than the distorted compound indicating that this distortion leads to signicant stabilization. NaNbO 2 showed puckering distortion [ Fig. 1(f)] in half of its NbO 6 octahedra. As marked by blue arrows in Fig. 5(a), NbO 6 octahedra on every second row in [100] direction are alternately puckered to the le and the right along [010] direction while the octahedra on the adjacent row only had rhombohedral distortion. In the puckered NbO 6 octahedra, the short Nb-O bond was 2.18Å while the long Nb-O bond was 2.23Å indicating a 2.2% puckering distortion in bond lengths. The bond length in unpuckered NbO 6 octahedra had a median value of 2.22Å. The puckering altered the electronic structure of the NaNbO 2 compound as ions in the puckered and unpuckered octahedra had distinct spin populations of 1.051e and 0.350e respectively. According to Fig. 5(b), Nb ions in the puckered octahedra has a signicantly larger spin-up population (marked with red arrow) and smaller spin-down population (marked with blue arrow) compared to the Nb ions in unpuckered octahedra. To examine the stability induced by this distortion, we once set all Nb ions to low magnetization equal to that in the unpuckered octahedra and once again to high magnetization equal to that in the puckered octahedra and recalculated the total energy. The earlier setting raised the total energy of NaNbO 2 compound by 0.230 eV/f.u. while the latter setting raised the total energy by 0.649 eV/f.u. demonstrating the stabilizing effect of puckering distortion. Given that Nb's total electronic population does not signicantly depend on the puckering of NbO 6 octahedra, we infer that this relatively minor distortion does not cause charge disproportionation but rather only alters the magnetization of Nb ions.
The distortion in PdO 6 octahedra in NbPdO 2 were accompanied with charge disproportionation among Pd ions. As  shown in Fig. 6(a) octahedra, on the other hand, showed elongated Jahn-Teller distortion with long bonds of 2.32Å and short bonds of 2.09Å accounting for an elongation of 11.0%. The stability of this distortion was veried by the fact that the NaPdO 2 in which all Pd ions were xed to +3 oxidation state had a total energy 0.381 eV/f.u. higher than the presented ground state. Furthermore, this distortion pattern and the accompanying charge disproportionation prevailed in larger 4a Â 4a Â 1c supercell and persisted under different U eff values. Fig. 6(b) shows how Pd e g states are occupied as charge disproportionation occurs. For Pd 2+ , e g 's spin-up states are all occupied while the empty spin-down states are entirely located $1 eV above the Fermi level. For Pd 3+ , the spin-up peak decreases (marked with a red arrow) while an empty spin-up e g peak appears above the Fermi level. Finally, for Pd 4+ , all e g states are now located above the Fermi level.

Magnetic coupling
The magnetic ground state of a compound can be predicted by comparing the total energies corresponding to different spin alignments among the TM ions that is DE CAFM and DE AAFM . 26 If different spin alignments result in the same energy, the compound is paramagnetic. 27 Furthermore, DE CAFM and DE AAFM are functions of the magnetic exchange integrals (J) which determine the Curie and Néel temperatures (T C and T N ) in compounds with long range magnetic ordering. 28 According to the mean eld approximation these critical temperatures depends linearly on the J. 29 For instance, room temperature ferromagnetism requires positive DE CAFM and DE AAFM values of $12 meV. 30 As presented in Table 2, NaMnO 2 , NaCoO 2 , NaNiO 2 , NaNbO 2 NaTcO 2 and NaPdO 2 have positive DE CAFM values indicating inter-plane ferromagnetism which is dened as the ferromagnetic coupling among TM ions within the basal TMO 2 planes. This ferromagnetic coupling can be attributed to one of two distinct mechanisms: the kinetic p-d exchange interaction and the superexchange interaction. The density of states in Fig. 2(f)-(g) and Fig. 3(d) reveals a special p-d hybridization in NaCoO 2 , NaNiO 2 and NaTcO 2 compounds. Because of this hybridization, the spin majority p states are shied to higher energies, while the spin minority p states are shied to lower energies. This hybridization scheme therefore creates spin polarized p states which mediate the ferromagnetic coupling. 29 In NaMnO 2 , NaNbO 2 and NaPdO 2 , on the other hand, positive DE CAFM values are caused by ferromagnetic superexchange. The prerequisite for ferromagnetic superexchange is a $90 TM-O-TM angle which stabilizes the ferromagnetic coupling through p TM-O bonds in TM-O-TM trimer. 31 The octahedral distortions in these compounds orient the TM-O-TM angles in these compounds to $90 . Under perfect R 3m symmetry, as shown in Fig. 1(c), the TM-O-TM angle is determined by O's fractional coordinates and alternates between the supplementary angles h and q [dened in Fig. 1(c) and (d)] preventing the stabilization of the ferromagnetic phase. 9,11,32,33 If a distortion, however, breaks the symmetry and brings the TM-O-TM angle closer to 90 , ferromagnetic superexchange can prevail. One should note that, as indicated in Table 1, the octahedral distortions in these compounds basically bring the O-TM-O angle closer to 90 . However, since these compounds do not have any octahedral tilting, the TM-O-TM angle, at least in certain planes, also approaches 90 due to the similar distortion in neighbouring TMO 6 octahedra. Those TM-O-TM angles assisting the ferromagnetic superexchange are marked a in Fig. 4-6. a is 91.67 in NaMnO 2 , 89.86 in NaNbO 2 and 88.48 in NaPdO 2 . Contrary to our results, earlier DFT calculation of the NaMnO 2 compound using a small supercell restricted to C2/m symmetry, predicted weak frustrated antiferromagnetic ground state. 34 This discrepancy shows the importance of taking into account the octahedral distortions that stabilizes the ferromagnetic ground state. Inferred from DE CAFM values, the kinetic p-d exchange interaction seems to be generally stronger than the ferromagnetic superexchange interaction.
The magnetic coupling across the TMO 2 planes or intraplane coupling, in principle, can be mediated a by second nearest neighbour superexchange interaction through TM-O-Na-O-TM chain via by O's p orbitals and Na sp 2 hybrid orbitals. 35 Because of anisotropy in the R 3m crystal structure which prevents the hybridization of TM d states with the p states of adjacent TMO 2 layers, p-d kinetic exchange is not expected to result in signicant intra-plane coupling. Among all compounds only NaMnO 2 , NaNiO 2 and NaTcO 2 had small positive DE AAFM values indicating weak ferromagnetic intraplane coupling resulting in T C values lower than $50 K. This prediction, particularly for the NaNiO 2 compound, is agreement with the earlier observation that measured a T C of $20 K. 7 In the case of NaNbO 2 , NaPdO 2 and NaCoO 2 compounds, the negative DE AAFM values along with positive DE CAFM values predict A-type antiferromagnetic ground state. For the rest of compounds for which both DE AAFM and DE CAFM are negative, G-type antiferromagnetic ground state prevails. Such antiferromagnetism has been observed in NaCrO 2 with T N ¼ 40-50 K, 36,37 NaVO 2 (ref. 11 and 38) and NaTiO 2 (ref. 39).
Last, note that relativistic spin-orbit interaction can play a signicant role in determining the structural and magnetic properties of isolated TM octahedral complexes. [40][41][42][43] However, in the context of bulk NaTMO 2 compounds that have been studied here, the role of spin-orbit interaction on the calculated DE CAFM and DE AAFM values is negligibly small. Spin-orbit interaction constant is proportional to the mass of the interacting ions and can be signicant in 5d TM oxides such as iridates. 44 However, experimental studies have shown that the magnitude of the spin-orbit interaction in 3d and 4d TM oxides such as cobaltates 45 and rhoates 46 is generally small. To verify this notion, we recalculated the DE CAFM and DE AAFM for NaPdO 2 with the inclusion of the spin-orbit calculation and obtained DE CAFM ¼ 11.492 meV and DE AAFM ¼ À2.841 meV. These values differ only by $0.1 meV from the values of Table 2 which have been obtained without including spin-orbit interaction. The role of spin-orbit interaction is expected to be even smaller for the rest of the compounds, especially for 3d NaTMO 2 , as their molecular mass is considerably smaller than that of NaPdO 2 .

Conclusions
We demonstrated that the rhombohedral distortion inherent to the 3 Rm symmetry stabilizes G-type antiferromagnetism in NaTiO 2 , NaVO 2 , NaCrO 2 , NaFeO 2 , NaMoO 2 and NaRuO 2 compounds. Inter-plane ferromagnetism however can be stabilized if the 3 Rm symmetry breaks due to additional octahedral distortions which is the case for NaMnO 2 , NaNbO 2 and NbPdO 2 . Here, because of favorable orbital orientation, the superexchange interaction stabilizes ferromagnetism instead of antiferromagnetism among the TM ions of the same TMO 2 plane. Additionally, strong p-d hybridization, as in NaCoO 2 , NaNiO 2 and NaTcO 2 can also result in inter-plane ferromagnetism. In this case, due to its strength, the kinetic p-d exchange mechanism overcomes the underlying inter-plane antiferromagnetism. The intra-plane ferromagnetic coupling is mediated by a weak second neighbor coupling which prevails only in NaMnO 2 , NaTcO 2 and NaNiO 2 giving rise to bulk ferromagnetism with low T C .
The weak intra-plane coupling appears to be a general feature of O3 compounds. This is in contrast to the P2 structures in which the magnitudes of inter-plane and intra-plane coupling are of the same order. 35 This is probably because this interaction strongly depends on Na's coordination environment. This line of enquiry warrants further research.

Conflicts of interest
The authors declare no competing nancial interest.