Comparative studies on the room-temperature ferrielectric and ferrimagnetic Ni3TeO6-type A2FeMoO6 compounds (A = Sc, Lu)

First-principles calculations have been carried out to study the structural, electric, and magnetic properties of Ni3TeO6-type A2FeMoO6 compounds (A = Sc, Lu). Their electric and magnetic properties behave like room-temperature ferrielectric and ferrimagnetic insulators where polarization comes from the un-cancelled antiparallel dipoles of (A(1), Fe3+) and (A(2), Mo3+) ion groups, and magnetization from un-cancelled antiparallel moments of Fe3+ and Mo3+ ions. The net polarization increases with A’s ionic radius and is 7.1 and 8.7 μCcm−2 for Sc2FeMoO6 and Lu2FeMoO6, respectively. The net magnetic moment is 2 μB per formula unit. The magnetic transition temperature is estimated well above room-temperature due to the strong antiferromagnetic superexchange coupling among Fe3+ and Mo3+ spins. The estimated paraelectric to ferrielectric transition temperature is also well above room-temperature. Moreover, strong magnetoelectric coupling is also anticipated because the magnetic ions are involved both in polarization and magnetization. The fully relaxed Ni3TeO6-type A2FeMoO6 structures are free from soft-phonon modes and correspond to stable structures. As a result, Ni3TeO6-type A2FeMoO6 compounds are possible candidates for room-temperature multiferroics with large magnetization and polarization.

Single phase polar materials with ferromagnetic (ferrimagnetic) properties have drawn much attention 1-3 recently due to their applications in developing spintronic devices for nonvolatile memories and in achieving electric-field control of magnetization in realistic information storage [4][5][6][7] . Therefore, searching for multiferroic materials becomes an important research direction in material physics. Up to now, various mechanisms have been proposed to explain the electric polarization in magnetic compounds. Among others, the off-center displacement of lone-pairs 6s electrons 8,9 , the chiral spin-density-wave driven polarization [10][11][12] , the charge ordering [13][14][15] , and the strain-induced polarization are mechanisms being discussed most [16][17][18][19] . Although great progresses have been made in developing single phase multiferroic materials, many important issues remain unsolved. For example, compounds with both large magnetization and polarization are still rare; the ferro(ferri)magnetic transition temperatures are usually below room temperature and restricted their applications; even if the requirements of large magnetization and polarization are fulfilled, enhancing magnetoelectric coupling is still a big challenge.
In order to meet these crucial requirements, searching for the multiferroics which have magnetic ions contributing simultaneously to electric polarization can be a good choice. Thus in this report, we have analyzed the structural, electric, and magnetic properties of two corundum-derived oxides A 2 FeMoO 6 (A = Sc, Lu). All of them are found to be multiferroic materials and have the same polar structure as Ni 3 TeO 6 3 , ZnTiO 3 20 , and FeTiO 3 21 . The general crystal structure displayed in Fig. 1(a) is described by a chemical formula ′ A B O B 2 6 ( ≠ ′ B B for Ni 3 TeO 6 type; = ′ B B for LiNbO 3 or FeTiO 3 type). The structural advantage is its ability to incorporate different magnetic transition-metal ions on all cation sites for realizing magnetoelectric coupling. The common feature of the structures is the small A-site cation in six oxygen coordination, such as Sc 3+ (0.69 Å) and Lu 3+  , r B is an averaged radius of B-and B′ -site ions) 22 , and O ions, respectively. As found in most systems, for < t 1 R , the cubic perovskite changes its symmetry by the BO 6 octahedral rotation or tilting or Jahn-Teller distortion. Good examples are the Ni 3 TeO 6 (Ni 2 NiTeO 6 ) compound with nonhysteretic colossal magnetoelectricity 3 , ScFeO 3 (Fe takes both the B-and B′ -sites) 23,24 and Mn 2 FeMO 6 (M = Nb, Ta, Mo, and W) 25-27 compounds with polar structure and antiferromagnetic or ferrimagnetic structure. Due to the strong antiferromagnetic superexchange coupling between nearest neighbors Fe 3+ ( ) d 5 in ScFeO 3 , the Néel temperature is well above room-temperature (545 K). To improve the net magnetization of ScFeO 3 , constructing ferrimagnetic structure by replacing one of the B-site Fe 3+ by a d n (n < 5) ion is a possible way, which was done in the synthesized Bi 2 FeCrO 6 (Cr 3+ : d 3 compound 28,29 ).

Results
To accomplish this goal, we have carried out comprehensive first-principles study on Ni 3 TeO 6 -type A 2 FeMoO 6 compounds (A = Sc, Lu) where one of the B-site Fe 3+ is replaced by Mo 3+ ( ) d 3 . The structural, electric, and magnetic properties of Ni 3 TeO 6 -type A 2 FeMoO 6 have been systematically analyzed as a function of A-site cation radius. We found that the ferrimagnetic state is indeed the ground-state with net magnetic moment of 2 μ B /f.u. forming between the antiparallel Fe 3+ (5 μ B ) and Mo 3+ ( 3 μ B ) ions. The polarization increases with A's ionic radius and is 7.1 and 8.7 μCcm −2 for Sc 2 FeMoO 6 and Lu 2 FeMoO 6 , respectively. Moreover, strong magnetoelectric coupling is achieved since the electric polarization comes partly from the same magnetic ions. The robust antiferromagnetic coupling is sustained, ensuring a Néel temperature well above room temperature. The structural analyses suggest that Ni 3 TeO 6 -type A 2 FeMoO 6 compounds are free from soft-phonon modes and correspond to stable structures.

Discussion
Let us start with the Ni 3 TeO 6 -type structures of A 2 FeMoO 6 compounds as shown in Fig. 1(a). The structures are obtained after the full relaxation of lattice parameters and atomic positions with effective on-site Coulomb repulsion U eff = 4.0, 1.0, and 5.0 eV for Fe-3d, Mo-4d and Lu-5f electrons, respectively. The structures can be constructed in two steps: (1) (A(1)O 6 , FeO 6 ) ( Fig. 1(b)) and (A(2)O 6 , MoO 6 ) ( Fig. 1(c)) octahedral pairs form face-sharing structures along c-axis; (2) the two face-sharing structures then form zigzag chains by edge-sharing A(1)O 6 /FeO 6 and A(2)O 6 /MoO 6 octahedral pairs in the ab-plane. Due to the strong electrostatic repulsion among the neighboring cations in the centers of the face-shared octahedral pairs, large antiferro-polar displacements take place along c-axis for (A(1), Fe), and (A(2), Mo) ion pairs (see Fig. 1(b,c)). Thus, antiparallel electric moments are formed for each face-sharing A(1)O 6 /FeO 6 and A(2)O 6 /MoO 6 octahedral pairs, and ferrielectric polarization is generated along c-axis. The fully optimized structural parameters and atomic positions of ferrimagnetic Ni 3 TeO 6 -type A 2 FeMoO 6 are listed in Table 1 together with those of antiferromagnetic ScFeO 3 as a reference. In Table 1, the lattice parameters, atomic positions, and bond angles are highly accurate. A relative error less than 1% is achieved between our calculated data and the available experimental data 23,24 . The spontaneous polarization was computed using the Berry phase method 30 . The total polarization of ScFeO 3 is 2.0 and 1.6 μCcm −2 for theoretically optimized and the experimentally measured structures, respectively. These results are close to the value 1.4 μCcm −2 observed experimentally 23 . The computed polarization is 7.1 and 8.7 μCcm −2 for ferrimagnetic Sc 2 FeMoO 6 and Lu 2 FeMoO 6 , respectively. The polarization increases with A's ionic radius. Larger radius, probably, strengthens the repulsive force between neighboring ions in the centers of face-sharing A(1)O 6 /FeO 6 and A(2)O 6 /MoO 6 octahedral pairs. Our study shows that the ferrimagnetic structures not only greatly improved the magnetization property, but also significantly enhanced the polarization of A 2 FeMoO 6 regarding the reference compound ScFeO 3 . The incompatibility between ferroelectricity and ferromagnetism gets nicely reconciled in the ferrielectric and ferrimagnetic A 2 FeMoO 6 31 . In addition, strong magnetoelectric coupling between the polarization and magnetization is also intrinsically embedded in the structures.
Having investigated the structural and electric properties of Ni 3 TeO 6 -type A 2 FeMoO 6 compounds, we are now in position to discuss their electronic and magnetic properties. For Ni 3 TeO 6 -type A 2 FeMoO 6 , the orbital configurations of ( ) + d Fe 3 5 and ( ) + d Mo 3 3 are similar to those of La 2 FeCrO 6 according to previous study 32 . The schematic diagram for the relevant atomic energy levels is illustrated in Fig. 2. The spin-up and spin-down d-orbitals are separated by spin exchange energy Δ S , d(e g ) and d(t 2g ) orbitals are separated by a crystal-field-splitting energy 10Dq. The nature of the superexchange coupling between ( ) + d Fe 3 5 and ( ) + d Mo 3 3 ions is quite complicated because of the orbital degeneracy and two possible hybridization schemes. pdσ hopping favors ferromagnetic superexchange coupling while pdπ hopping favors antiferromagnetic superexchange coupling. The subtle competition between the two determines the magnetic ordering of ground state. Our first-principles calculations show that the ferrimagnetically ordered state is consistently lower in energy than that of the ferromagnetically ordered state in Ni 3 TeO 6 -type A 2 FeMoO 6 . Thus the polar state with ferrimagnetic ordering can be the favored ground state.
To have an overall picture of the electronic and magnetic properties of A 2 FeMoO 6 , the spin-resolved partial densities of states (DOS) are plotted in Fig. 3 for both ferromagnetically and ferrimagnetically ordered structures. To distinguish between the two types of transition-metal ions associated with A(1)O 6 /FeO 6 and A(2)O 6 /MoO 6 octahedral pairs, the DOSs of Fe and Mo are represented by solid (black) and dashed (red) lines, respectively. As shown in Fig. 3, the positions of extended e g and localized t g 2 orbitals are in accord with the atomic level scheme in      Fig. 1(b,c). The electronic structural patterns can be understood from the level scheme of Fig. 2 together with hybridization processes. In particular, the valence and conduction bands near the Fermi energy is mainly resulted from the t g 2 orbitals of Fe and Mo. For the ferromagnetically ordered state, the up-spin ( ) t d g 2 orbitals form the Fe and Mo dominated valence bands while the down-spin ( ) t d g 2 orbitals form the Fe and Mo dominated conduction bands. The hybridization with oxygen orbitals pushes the Mo dominated ( ) t d g 2 valence band edge upwards and pulls the Fe dominated ( ) t d g 2 conduction band edge downwards. This makes the ferromagnetic band-gap extremely small. For the ferrimagnetically ordered state, the band structure in the vicinity of the Fermi energy is mainly determined by down-spin ( ) t d g 2 orbitals of Fe and Mo across the Fermi energy. The difference in energy level essentially determines the band-gap between Fe dominated conduction band and Mo dominated valence band. This also explains why the overall features of DOSs for A 2 FeMoO 6 (A = Sc, Lu) look rather similar. In addition, above discussion suggests that ferrimagnetically ordered state mainly involves hybridizing down-spin t g 2 orbitals of Fe and Mo across the Fermi energy. The resulting band splitting, thus, can significantly lower the binding energy. This is also the basic mechanism dictating the ferrimagnetically ordered ground state. The similar scenario also takes place in the double perovskite La 2 FeCrO 6 as proved by the GGA electronic structure calculation 32 .
It is known that the choice of the Coulomb interaction U eff has a notable impact on the electronic structure, and thus affects the relative stability of different magnetically ordered states. To investigate such effect, we have also performed GGA + U simulations for other = , , . The choice of parameter values are based on the fact that the Coulomb interaction is typically weaker for spatially more extended 4d electrons than for more localized 3d electrons. The computed energy difference Ferri between the ferromagnetically and ferrimagnetically ordered states are shown in Fig. 4

as functions of U eff
Fe and U eff Mo . It has been found that Δ E is a monotonic deceasing function with increasing U eff Fe or U eff Mo , which varies from 0.65 to 0.3 eV, but the ferrimagnetically ordered state is consistently lower than that of the ferromagnetically ordered state. The monotonic decreasing behavior of energy difference originates from the superexchange interaction, ∝ /U 1 eff , for ferrimagnetic state since the ferromagnetic state is less sensitive to U eff . The energy difference decreases slightly as A's ionic radius increases, because large A's ionic radius reduces the effective hopping integral between Fe and Mo ions and so is that of the antiferromagnetic superexchange coupling. However, large A's ionic radius expands the oxygen octahedra and favors the polar distortion. To estimate the magnetic transition temperature for A 2 FeMoO 6 and ScFeO 3 , we adopt the single parameter Heisenberg spin by assuming the same exchange parameter for all the nearest-neighbor couplings. Using S = 5/2 for Fe 3+ and S = 3/2 for Mo 3+ , one can determine the exchange coupling J by matching the energy differences obtained from the Heisenberg model and first-principles calculations. Then magnetic transition temperature T C is related to the energy difference Δ E by = ∆ / T E k C B which best reproduced the experimentally observed lattice parameters, T C of ScFeO 3 is 661 K. The mean-field estimated T C is higher than the measured value 545K 26   for Lu 2 FeMoO 6 . This is consistent with Lu's results on AlScFeMoO 6 (space group: R3) 33 . Therefore, we have shown that the A 2 FeMoO 6 not only have large magnetization and polarization, but also possess room-temperature magnetic transition temperature T C . These encouraging properties make A 2 FeMoO 6 a promising candidate for future multistate memory applications.
It remains to be verified that the structure of ferrielectric and ferrimagnetic Ni 3 TeO 6 -type A 2 FeMoO 6 (A = Sc, Lu) insulators are robust structures and can be prepared by the usual laboratory method. Therefore, the phonon dispersion spectra are calculated using the frozen-phonon method. The calculated phonon dispersions are plotted in Fig. 5 for both the reference compound ScFeO 3 and Ni 3 TeO 6 -type A 2 FeMoO 6 (A = Sc, Lu). The overall dispersion curves are quite similar for the three compounds except that the phonon frequency scales with the inverse square root of transition metal ion mass. It is clear that the soft-phonon modes are absent in the entire Brillouin Zone, which indicates that the Ni 3 TeO 6 -type A 2 FeMoO 6 structure does correspond to stable structures.
To further check the stability of Ni 3 TeO 6 -type A 2 FeMoO 6 (A = Sc, Lu) (R3 structure) against other common structures, we have also considered R3, P21/c, and C2 structures. After the full structural relaxation with respect to the atomic positions and lattice constants, the initial trial C2 structure may converge either to C2/m, C2, C2/c, or Imma structure depending on the material composition. The calculated energies of different structures are summarized in Table 2. Only those of ferrimagnetic (antiferromagnetic) states are shown because they always have lower energy than those of ferromagnetic state. One finds that Ni 3 TeO 6 -type A 2 FeMoO 6 (R3 structure) consistently has lower energy than other structures. However, for large ionic radius of Y atom, the stable structure of Y 2 FeMoO 6 takes P21/c space group rather than the R3 space group. This suggests that Ni 3 TeO 6 -type A 2 FeMoO 6 is stable with respect to P21/c structure only for small ionic radius of A atoms (see Supplementary Information  Table S4). The paraelectric to ferrielectric transition temperature can also be estimated from the energy difference between the structurally connected polar (R3) and nonpolar ( ) R3 structures. As shown in Table 2, the energy difference is 1.378 eV/2f.u. for ScFeO 3 , 0.408 and 0.542 eV/f.u. for Sc 2 FeMoO 6 and Lu 2 FeMoO 6 , respectively. Scaling the energy with that of ScFeO 3 and considering its polar structure being stable above 1400 K 1,2 yield a paraelectric-ferrielectric transition temperature    In the view that ScFeO 3 , Mn 2 FeMO 6 (M = Nb, Ta, Mo, and W), and Ni 3 TeO 6 , all with smaller A-site ions, can be synthesized under the high temperature and high pressure environment 34 , we expect that the Ni 3 TeO 6 -type A 2 FeMoO 6 can also be synthesized under similar conditions. If so, one expects that other room-temperature ferrielectric and ferrimagnetic insulators may also be realized in the corundum-derived transition metal oxides. Through incorporating different magnetic transition metal ions on the cation sites, one can easily tune the superexchange interaction and polar distortion, so that the polarization, magnetization, magnetoelectric coupling as well as critical temperature can be optimized for potential applications.
In summary, comprehensive first-principles calculations have been carried out for the structural, electronic, and magnetic properties of Ni 3 TeO 6 -type A 2 FeMoO 6 (A = Sc, Lu). All of them show the ferrielectric and ferrimagnetic insulator properties with large magnetization (2μ B /f.u.) and polarization (> 7 μCcm −2 ). The strong antiferromagnetic superexchange interaction between Fe and Mo yields a mean-field critical temperature above room-temperature. Strong intrinsic magnetoelectric coupling is also ensured because the magnetic ions are involved in both the magnetic moment formation and polarization. The Ni 3 TeO 6 -type Sc 2 FeMoO 6 and Lu 2 FeMoO 6 are also proved to be stable structures because they have lower energies than other possible structures. Thus, one expects that these materials and other related ones can be synthesized in experiments.

Methods
The study has been carried out using the generalized gradient approximation + U (GGA + U) method 35 with Perdew-Becke-Erzenhof exchange-correlation functional 36 as implemented in the Vienna ab Initio simulation package (VASP) 37,38 . To account for the population imbalance on localized transition metal d-and rare earth f-orbitals, the effective on-site Coulomb interactions U eff = 4.0, 1.0, and 5.0 eV are adopted for Fe-3d, Mo-4d and Lu-5f electrons, respectively 39 . The projector augmented wave (PAW) potentials 40 explicitly include three valence electrons for Sc (3d 1 4s 2 ), 11 for Y (4s 2 4p 6 4d 1 5s 2 ), and 25 for Lu (5s 2 5p 6 4f 14 5d 1 6s 2 ), 14 for Fe (3p 6 3d 6 4s 2 ), 12 for Mo (4p 6 4d 5 5s 1 ), and six for O (2s 2 2p 4 ) atoms. The same result is also obtained for the PAW potential excluding f electrons for Lu. The wave function is expanded in a plane wave basis with an energy cutoff of 600 eV. The crystal unit cell includes two formula units for ScFeO 3 , and one formula unit for Sc 2 FeMoO 6 and Lu 2 FeMoO 6 . A 7 × 7 × 7 Γ -centered k-points sampling is used for reciprocal space integrations. Each self-consistent electronic calculation is converged to 10 −6 eV and the tolerance force is set to 0.005 eV/Å for ionic relaxation. The convergence checks with respect to the k-points sampling have been made for the total energy, densities of states as well as the phonon dispersion curves (see Supplementary Information Figures S1-S3).
To calculate the electric polarization of Ni 3 TeO 6 -type A 2 FeMoO 6 (A = Sc, Lu) with space group R3, we choose the structure with space group R3 as a reference state 41 . The R3 structure displayed in Figure S1 has space inversion symmetry. It is a non-polar insulator and has zero electric polarization (see Supplementary Information). Since the electric polarization is along 3-fold rotational axis, a 30-atom hexagonal unit cell is chosen, so that the in-plane polarization is zero. In calculating the electric polarization, a 7 × 7 × 4 Γ -centered k-points sampling is used for the self-consistent loop and 14 k-points sampling is adopted for parallel direction integration in Berry phase method. As shown in Figure S5, 14 k-points sampling is almost convergent for electric polarization calculation.
To calculate the phonon dispersion of Ni 3 TeO 6 -type structure of A 2 FeMoO 6 (A = Sc, Lu) and ScFeO 3 , the structures are firstly atomically relaxed with a higher accuracy using the 8× 8× 8 Γ -centered k-points sampling and the tolerance force of 0.0001 eV/Å. The phonon dispersion is then calculated using the Phonopy code 42 with a 2 × 2 × 2 supercell composed of ten-atom rhombohedral unit cell. The force constants are calculated by VASP using a 4 × 4 × 4 Γ -centered k-points sampling for the supercell.  . The energy is given in unit of eV with R3 phase taken as the reference structure.