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Orbitally relieved magnetic frustration in NaVO2

The magnetic properties of NaVO2 are investigated using full-potential linearized augmented plane wave method. We perform calculations for three structures. For the rhombohedral structure at 100 K, the t2g orbitals of V ions are split into upper a1g and lower e'g orbitals by a trigonal distortion of compression. For the monoclinic structure at 91.5 K, the system behaves like a frustrated spin lattice with spatially anisotropic exchange interactions. For another monoclinic structure at 20 K, the magnetic frustration is relieved by a lattice distortion which is driven by a certain orbital ordering, and the long-range magnetic ordering is thus formed. Moreover, the small magnetic moment originates from the compensation oforbital moment for the spin moment.


I. INTRODUCTION
The compounds with a common chemical formula ATO 2 (A=Na or Li, T=3d transition metals) = -U ef f (n σ -1 2 I) [22], wheren σ is the orbital occupation matrix of spin σ. This type of doublecounting correction (DCC) has been called the fully localized limit [23,24]. For NaVO 2 , we used U ef f =3.6 eV which has been used in its sister compound LiVO 2 [5]. Note also that the conclusion made in this paper is not affected for U ef f =2-6 eV [25]. To explore the origin of small magnetic moment observed in the LT phase, we performed LSDA+SOC+U calculations, where the SOC is included by the second-variational method with scalar relativistic wave functions [18]. The easy magnetization direction was set along (110) direction (short V-V bonds in the LT phase) observed in the experiment [14].
In order to investigate different magnetic patterns, 2×2×2 supercell was used in our calculations. We took into account two AFM structures in V-V plane as described in Fig. 2

A. HT phase
As the paramagnetic behavior of NaVO 2 has been determined from the magnetic susceptibility measurements in the HT phase [14,15], we just focus on the electronic structure instead of its magnetic properties.
The band structures obtained from LSDA and LSDA+U calculations are shown in Fig. 3.
Within LSDA (Fig. 3(a)), the bands near the Fermi level (E F ) are mainly derived from V 3d states. Since straight V-O-V paths are not present in layered NaVO 2 and instead only nearly 90 • V-O-V bonds exit, the V 3d states are quite narrow. In the approximately octahedral crystal field, the 3d orbitals are split into upper e g and lower t 2g states. As shown in Fig. 3(a), e g derived bands range from 1.5 to 2.5 eV and t 2g derived bands lie between -1.5 and 0.5 eV. The splitting between t 2g and e g bands is about 1 eV. Under the trigonal crystal field, the t 2g orbitals are further split into one a 1g and two degenerate e' g orbitals. However, the splitting is much less than the band widths so that the t 2g orbitals still cross the E F , which denotes a metallic state within LSDA. That is to say, LSDA calculations can not reproduce the insulating nature of NaVO 2 from experiment [14].
The LSDA+U scheme [20] is thus used to count the strong correlation of V 3d electrons. As shown in Fig. 3(b), the empty a 1g band is pushed upwards by about 1 eV, and a gap is opened near the E F . The system is hence an insulator due to electron correlation and NaVO 2 is indeed a good candidate for Mott-Hubbard insulator.
According to the pure crystal field theory, the a 1g orbital is of lower energy than the e' g orbitals under the trigonal distortion of compression, which is opposite to our LSDA+U results. So it is necessary to discuss the controversy on relative order of a 1g -e' g in such trigonal distortions.
In Ref. [26], Landron and Lepetit pointed out that this relative order is strongly influenced by the e g -e' g hybridization. The e g and e' g orbitals belong to the same irreducible representation (E g ) and can thus mix despite the large t 2g -e g energy difference. Such a mix may be small but it modulates large energetic factors: the on-site Coulomb repulsions. When the e g -e' g hybridization is taken into account, the energy difference ∆E between the a 1g and e' g orbitals depends on two competitive parts: ∆E = ∆E 1 + ∆E 2 = ε(a 1g ) -ε(e' g ). ∆E 1 includes the kinetic energy, the electron-charge interaction, and the interaction with the core electrons. ∆E 2 denotes the repulsion and exchange terms within the 3d shells. Additionally, ∆E 1 and ∆E 2 both depend on the amplitude of the trigonal distortion and are of opposite effect with each other. Under a trigonal distortion of compression, if we only consider the crystal field effect (∆E 1 ), the a 1g orbital is of lower energy than the e' g orbitals (∆E < 0). Whereas if we take ∆E 2 into account, the relative order between the a 1g and e' g orbital is reversed (∆E > 0), comparing with the crystal field prediction.
Therefore, LSDA+U calculations predict that the a 1g orbital is of higher energy than the e' g orbitals in NaVO 2 . In fact, such a controversy has been presented in another compressed triangular system Na x CoO 2 [27,28,29]. From the crystal field theory, some authors [27] proposed that the energy of a 1g orbital is lower than the e' g orbitals. However, the LDA+U method [28,29] yielded an a 1g orbital of higher energy than the e' g orbitals. Later, the experimental results [30] showed that the Fermi surface of the CoO 2 layers issues from the a 1g orbital, not at all from the e' g orbitals, supporting the LDA+U results.

B. IT phase
The triangular lattice of NaVO 2 exhibits magnetic frustration and spatially anisotropic exchange interactions in the IT phase. As shown in Table II,  In order to describe the magnetic frustration and spatially anisotropic exchange interaction more clearly, we estimate the exchange interactions along one of the triangle directions (J 1 ) and the other two (J 2 ) ( Fig. 2(a)(i)). Since all the configurations exhibit insulator characteristics, the spin size of V is stable, and the difference of total energy between C-AFI and G-AFI (C-AFII and G-AFII) configurations (See Table II) is so small that the system exhibits a 2D characteristic, a nearest neighbor Heisenberg-like Hamiltonian may be a good primary approximation for the in-layer magnetic energy. The corresponding 2D spin Hamiltommian can be written as where (i,j) denotes a nearest-neighbor pair (short V-V bond) and (k,l) denotes a next-nearestneighbor pair (long V-V bond). By mapping the obtained total energies for each magnetic state to the Heisenberg model, the exchange interactions J 1 and J 2 were calculated within this approximation: With S=1, we get J 2 =2.1 meV and J 1 =6.1 meV for NaVO 2 in the IT phase, which reflects strong spatial anisotropy. The AFM chains are established along the (110) direction (J 1 ) and the interchain coupling (J 2 ) is frustrated. Moreover, the value of J 2 /J 1 =0.3 is so small that this magnetic structure can be described as so-called weakly coupled zigzag (S=1) chains model [31].
The integer spins (S=1) are able to weaken the frustration effects in the frustrated systems, as in kagomé lattice [32]. Such a lattice has four nearest neighbors with the adjacent triangles on the lattice sharing only one lattice point. Interestingly, the triangular lattice can be composed of four kagomé lattices [27]. Thus, there are some analogous properties in these two frustrated systems.
Therefore, it is reasonable to suppose that the triangular lattice also has the rule that the halfodd-integer spins are more highly frustrated than integer ones. For example, NaTiO 2 (S = 1 2 )[5] and LiCrO 2 (S = 3 2 )[6] with half-odd-integer spins are always frustrated even at low T, while the magnetic frustration of NaMnO 2 (S=2) is clearly lifted by a structural distortion [2]. Besides NaMnO 2 , NaVO 2 is another typical triangular lattice with integer spins (S=1). Therefore, we can presume that the magnetic frustration in NaVO 2 can be lifted in some way.

C. LT phase
From the discussion above, we expect that NaVO 2 with S=1 will show a long-range magnetic ordering or a finite ground-state magnetization at low T. As shown in Table II, G-AFII is only 0.3 meV lower in total energy than C-AFII within LSDA, and both G-AFII and C-AFII have the same lowest total energy from LSDA+U results for the LT phase. This reflects the obvious 2D characteristic of NaVO 2 : the interlayer interaction is much weaker than the intralayer one. As stacking antiferromagnetically between layers is observed in the experiment [14], G-AFII state should be more favorable at low T. Such a magnetic state denotes the long range 3D magnetic ordering with AFM coupled FM chains in V cation layers and interlayer AFM coupling. Obviously, the magnetic frustration is lifted in G-AFII state by the first-order transition at 91.5 K: the lattice distorts to another monoclinic (C2/m) with four long and two short V-V bonds reversed, comparing with IT phase.
The lattice distortion, which relieves the frustration, is driven by the formation of OO in the LT phase. Fig. 4 shows the orbital characteristic of V 3d in G-AFII state. Since the orbital occupancies of the two inequivalent V (V1 and V2 in Fig. 2(a) This OO relieves the magnetic frustration and stabilizes the long-range magnetic ordering state.
In view of the weak superexchange interaction resulted from the nearly 90 • angle of V-O-V [33] as well as the weak magnetic interaction between adjacent VO 2 planes interleaved by a layer of Na ions, the V-V direct exchange interaction in-plane should be dominant in such a particular crystal structure. Particularly, we only consider the σ overlap in V-V direct exchange, which is much stronger than the π overlap. It means that each orbital in a V ion only hybridizes with the same orbitals in the two nearest-neighboring V ions. That is to say, d yz orbital hybridizes with two neighboring d yz orbitals in the (010)  In the LT phase, another important aspect is that the SOC turns out to be crucial for the small magnetic moment of 0.98 µ B per V 3+ observed experimentally [14]. As shown in Table II, the magnetic moments from LSDA+U calculations are much larger than the ones observed in the experiment. Further to investigate the magnetic moments changing with the particular choice of U ef f , we calculate the moments for U ef f =2-6 eV and find that the moments are not sensitive to U ef f : as shown in Table III, the magnetic moments stay constant within 0.2 µ B as long as the system is an insulator. Since an easy magnetization direction (110) is observed in the exprement [14], the SOC may play an important role in determining the total magnetic moment. Thus, the SOC is included to reinvestigate the magnetic moment.
Then, we perform LSDA+SOC+U calculations for the favorable magnetic configuration (G-AFII), and obtain a local moment of 0.89 µ B per V 3+ with 1.65 µ B spin and -0.77 µ B orbital contributions. This value is half the expected moment (2 µ B ), but very close to the observed one (0.98 µ B ). The DOS projected on (2,m) space shown in Fig. 5 reveals the origin of orbital moment. Note that the z axis is set to the direction of easy magnetization along (110) now. Since d 1 and d −1 have nearly the same occupancies, the orbital moment only comes from the contribution of different occupancies between d 2 and d −2 . By further analysis, the d 2 occupancy is less than one half of the d −2 one, which should give an orbital moment between 1 µ B and 2 µ B . Nevertheless, there is no surprising that the calculated orbital moment is 0.77 µ B here, because some hybridization effects are neglected in above analysis, e.g., the covalence effects with O 2p. Thus, the inclusion of SOC leads to a surprising but experimentally sound results.

IV. CONCLUSIONS
In Summary, we have investigated the electronic structure and magnetic properties of NaVO 2 by first-principles calculations. The t 2g orbitals are split into upper a 1g and lower e' g states by a trigonal distortion of compression in the HT phase, which is similar to the splitting in Na x CoO 2 [28]. In the IT phase, the crystal symmetry is lowered to C2/m, under which the system behaves like a frustrated spin lattice with spatially anisotropic exchange interactions. Finally, a long-range ordered AFM ground state is formed when the magnetic frustration is relieved by another lattice distortion resulted from a certain ordering of occupied orbitals at low T. The small magnetic moment observed originates from the compensation of orbital moment for the spin moment. It is obvious that so many physical phenomena in the triangular lattice are reflected in NaVO 2 , suggesting that NaVO 2 is a very good model material for studying 2D triangular lattice systems.  x,y,z(Na) 3a(0, 0, 0) 2a(0, 0, 0) 2a(0, 0, 0) x,y,z(V) 3b(0, 0, 0.5) 2d(0, 0.5, 0.5) 2d(0, 0.5, 0.5) x,y,z(O) 6c(0, 0, 0.2339(0)) 4i(0.2368 (7), 0, 0.6989 (5)) 4i(0.2296 (5), 0, 0.7005(4))