Structural, Electronic and Magnetic Properties and Hyperfine

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Structural, Electronic and Magnetic Properties and Hyperfine Interactions at the Fe Sites of the Spinel TiFeO. Ab Initio, XANES and Mössbauer Study 2

4

Azucena M. Mudarra Navarro, Arles V. Gil Rebaza, Karen Lizeth Lizeth Salcedo Rodríguez, Jhon J. Melo Quintero, Claudia E. Rodríguez Torres, Mariana Weissmann, and Leonardo Antonio Errico J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b06550 • Publication Date (Web): 13 Aug 2019 Downloaded from pubs.acs.org on August 13, 2019

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The Journal of Physical Chemistry

Structural, Electronic And Magnetic Properties And Hyperfine Interactions At The Fe Sites Of The Spinel TiFe2O4 . Ab Initio, XANES And Mössbauer Study A. M. Mudarra Navarro1, *, A. V. Gil Rebaza1,2, K. L Salcedo Rodríguez1, J. J. Melo Quintero1, C. E. Rodríguez Torres1, M. Weissmann3 and L. A. Errico1,4. 1IFLP

y Departamento Física, Facultad de Ciencias Exactas, Universidad

Nacional de La Plata-CCT La Plata CONICET, C.C. 67, CP 1900, La Plata, Argentina. 2

Grupo de Estudio de Materiales y Dispositivos Electrónicos (GEMyDE), Dpto. de Electrotecnia, Fac. de Ingeniería, UNLP. 3Departamento

de Física, Comisión Nacional de Energía Atómica, Av. del

Libertador 8250, 1429 Buenos Aires, Argentina 4

Universidad Nacional del Noroeste de la Provincia de Buenos Aires –

UNNOBA, Monteagudo 2772, Pergamino, CP 2700, Buenos Aires, Argentina. * Corresponding author: Dra. A. M. Mudarra Navarro ([email protected]) Abstract We present here an experimental and theoretical study of the Ti-ferrite (TiFe2O4, ülvospinel). The theoretical study was performed in the framework of Density Functional Theory using the Full-Potential Linearized Augmented Plane Waves (FPLAPW) method and employing different approximations for the exchange and correlation potential. In order to discuss the magnetic ordering and the electronic structure of the system we considered different distributions of the Fe/Ti atoms in the two cationic sites of the structure and, for each distribution, different spin arrangements (ferromagnetic, ferrimagnetic and antiferromagnetic cases). We found that the equilibrium structure corresponds to an inverted spinel structure with an antiferromagnetic spin configuration in which the magnetic moments of the Fe ions in both A and B sublattices are ferromagnetically ordered, while the magnetizations of these two sublattices are antiparallel with respect to each other. . Our calculations predict that TiFe2O4 is a wide-band gap semiconductor (band-gap in the order of 2.3 eV) and successfully describe the hyperfine properties (isomer shift, magnetic

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hyperfine field and quadrupole splitting) at the Fe sites that are seen by Mössbauer Spectroscopy (MS) experiments at 4.2 K reported in the literature and MS performed at 300 K in the present study. We also measured and simulated the X-ray Absorption Near Edge Spectroscopy (XANES) spectra of TiFe2O4 at both Ti and Fe K-edges. Our calculations correctly reproduce the XANES spectra and enable us to separate the contribution of each site to the experimental spectra. All these studies enable us to obtain a complete structural, electronic, magnetic and hyperfine characterization of TiFe2O4. 1. Introduction The most important rock-forming magnetic minerals are oxides of Fe and Ti. Particularly, the principal magnetic minerals that crystallize from basaltic magmas forming the Earth's crust are titanomagnetites, solid solutions of magnetite (Fe3O4) and ülvospinel (TiFe2O4) in different proportions that crystallize in an spinel-like structure. Titanomagnetites are also present as a rock in Moon's and Mars' crust, being then of petrological and geophysical interest in rock-magnetism and geo-magnetism1-3. For this reason, and for the technological importance of magnetic spinel compounds4-11, Titano-magnetites have been the subject of a significant number of studies and experiments and their crystalline structure, crystal chemistry, mechanic properties, magnetic behaviour and relationships between atomic structure and physical properties have been investigated by different experimental techniques such as x-ray diffraction, neutron diffraction, magnetic measurement, and Mössbauer experiments12-19. While one of the end member of the series, magnetite, was widely studied and their structural, electronic and magnetic properties and the cation distribution in the structural sites are clearly understood, the magnetic structure of the other member, ülvospinel remains ambiguous. Experimental results show that TiFe2O4 (Ti-ferrite) crystallizes in the inverse spinel structure8. This structure is characterized by two cationic sites, the A-sites (tetrahedral oxygen coordination) and the B sites (octahedral oxygen coordination). Fe2+ ions occupy the A-sites and the B ones are occupied by Fe2+ and Ti4+ in equal proportion8. The room temperature structure is cubic (Fd-3m). At 163 K (Tct) a cubic  tetragonal phase transition takes place1. Below Tct the structure (I41/amd) is slightly elongated along the c-axis1,

8.

Neutron diffraction studies showed that TiFe2O4 becomes weakly


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ferromagnetic below 140 K and antiferromagnetic below 80 K with an easy c-axis of magnetization8, 17. The magnetic structure is of the Néel type with the sublattice magnetic moments of 4.22 µB on both octahedral and tetrahedral sites at 4.2 K. Readman20 reported the hysteresis loops at low temperature (100-80 and 4 K) and observed that this material did not show a saturation magnetization (Ms), even at high applied field (24 kOe). In addition to neutron diffraction and magnetization studies, different works reported the hyperfine properties at the Fe sites of TiFe2O4. In most cases these studies were not conclusive due to the complex structure of the spectra21-23. One of the factors that hinder the analysis is the formation of additional spurious phases, such as TiFe2O4-Fe3O4 (titanomagnetites),

FeTiO3-Fe2O3

(titanomaghemites)

and

Fe2TiO5-FeTi2O5

(ferropseudobrokite). The formation and amount of these phases depends on the growth conditions of the samples1. Maybe the most relevant studies are those reported by Vanleerberghe and Vandenrberghe23 and Nakamura and Fuwa24. Vanleerberghe and Vanderberghe performed a

57Fe-Mössbauer

spectroscopy (MS) study of the paramagnetic

phase of TiFe2O4 applying an external magnetic field in order to separate the contributions to the spectra of Fe at sites A and B of the structure. Nakamura and Fuwa performed MS studies in the temperature range 16-500 K. The authors found multiple interactions that were attributed (based in a simple point-charge model) to Fe at sites A and B24. From the theoretical side, very few studies of TiFe2O4 have been published. We can cite the work of Jun Lui et al.25, who reported a Density Functional Theory-based ab initio study of the electronic, structural and magnetic properties of TiFe2O4 that shows that this oxide presents a half-metallic behaviour25. But this result raises many doubts because the authors assumed that TiFe2O4 adopts the normal spinel structure (in this normal structure the Ti atoms are located at tetrahedral A-sites, whereas the Fe atoms occupy solely the octahedral B-sites). Additionally, the authors employed a 14-atoms unit cell (only one antiferromagnetic configuration can be studied) and the Perdew et al. parameterization of the General Gradient Approximation26 (GGA) for the computation of the exchange and correlation (XC) potential. It is well-known that the exchange and correlation effects included in the GGA formalism are insufficient to describe 3-d transition oxides (as an example, the GGA approximation fails in the prediction of the correct nature of the



ZnFe2O4 ferrite27). In the case of hyperfine parameters at the Fe sites of TiFe2O4, as far as we known no realistic calculations were published. In this work we will present a combined Densitity-Functional Theory (DFT28, 29) based ab initio and experimental study of the structural, electronic, and magnetic properties of TiFe2O4. The calculations were performed using the Full-Potential Linearized Augmented Plane Waves method (FPLAPW30-32). In order to obtain a good description of the electronic structure of the system, different approximations for the exchange and correlation potential were employed: GGA26, GGA+U33, the Tram-Blaha modified BeckeJohnson exchange potential (TB-mBJ34) and the state-of-the-art hybrid exchangecorrelation functional proposed by Heyd, Scuzeria and Erzenhorf (HSE0635). We considered in our study the normal and inverted structures. In this second case we study different distributions of Ti and Fe in the B sites of the structure. For each distribution of Ti and Fe in the B sites we considered the ferromagnetic and different antiferromagnetic spin configurations and we computed the total energy of each configuration in order to find the equilibrium structural and magnetic configuration of TiFe2O4. The band structures and hyperfine parameters (isomer shift, quadrupole splitting, and hyperfine magnetic field) at the Fe sites were calculated. Also, we simulated, for the lower energy structure, the Ti- and Fe-K edges X-ray Absorption Near-Edge Spectroscopy (XANES) spectra. All these results are compared with new Mossbauer Spectroscopy (MS) results performed by us at 300 K and results at low temperatures (below 16 K) reported in the literature and XANES experiments. These comparisons give confidence to our predicted structural and magnetic equilibrium structure and enable us to separate the contributions of each site to the hyperfine and XANES spectra. The paper is organized as follows: in section 2, we will briefly present the method of calculation, in section 3 the sample preparation and the details of the MS and XANES experiments performed are presented. In section 4 the results obtained for the structural, magnetic and hyperfine properties of TiFe2O4 are presented, discussed and contrasted to the experimental data. Finally, in section 5 the main conclusions of the work are presented.


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2. Method of calculation 2.1. The system under study The titanium ferrite (TiFe2O4) crystallizes in the spinel structure, space group Fd3m (Oh7). The structure is characterized by a face-centered cubic lattice with a closepacked arrangement of oxygen ions, with Ti and Fe ions at two different crystallographic sites, the A-sites (tetrahedral oxygen coordination, Wyckoff position 8a (1/8, 1/8, 1/8)) and the B sites (octahedral oxygen coordination, 16d positions, (1/2, 1/2, 1/2))4. In the case of TiFe2O4 the Fe2+ ions occupy the A-sites and the B ones are occupied by Fe2+ and Ti4+ in equal proportion8, i.e., an inverted spinel structure that can be described by the formula (Fe)[FeTi]O4, where parenthesis and square bracket refers to sites A and B, respectively. The structure is characterized by a lattice parameter a=8.52973 at 300 K (8.520520 at 173 K1). The oxygen atoms are located at 32e (u, u, u) positions of the F.C.C. structure. For TiFe2O4, u=0.25617 at room temperature and u=0.25589 at 173 K1. Below Tct=163 K the stable structure is tetragonal (space group I41/amd). In this low temperature phase, the structure is slightly elongated along the c-axis (a=8.494645, c= 8.524734), see Ref. [1]. The u parameter at 4.2°K is also 0.2561 (see Ref. 8). The magnetic structure of TiFe2O4 (atomic coordinates with corresponding magnetic moment projections) was not reported in the literature. One of the goals of our work is to find the symmetry relations between the Fe atoms with their corresponding magnetic moments within the cell. 2.2. Theoretical methodology. Density Functional Theory (DFT28,

29)

based ab initio calculations have been

performed in order to study the structural, magnetic and hyperfine properties of TiFe2O4. The Kohn-Sham self-consistent equations were solved employing the full-potential linearized augmented plane waves (FPLAPW) method30-32 in the scalar relativistic formalism implemented in the Wien2k code36. The exchange and correlation potential was treated using the Wu-Cohen parameterization37 of the Generalized Gradient Approximation (GGA). It is well known that the exchange and correlation effects included in GGA are insufficient to describe 3-d transition oxides, GGA plus the Hubbard U term (GGA+U) in the self-interaction correction (SIC) scheme was employed. In the DFT+U methodology, a



value for the on-site Coulomb energy U must be selected prior to the calculations. In this study we took U=5.0 eV for the 3d-Fe orbitals. This value was selected after the study of a set of Fe oxides as a function of U27. Other possibility to determine the better value for U for a given system is the method based on the linear-response approach developed by I. Timrov, N. Marzari and M. Cococcioni38 and M. Cococcioni and S. De Gironcoli39 to calculate the interaction parameters entering the GGA+U functional in an internally consistent way. Using this approach, we obtained U=5.0 eV for the 3d-Ti and 3d-Fe orbitals. In order to better improve the description of the electronic structure of TiFe2O4, we have also performed calculations considering the state-of-the art hybrid exchangecorrelation functional proposed by Heyd, Scuzeria and Erzenhorf (HSE0635), and the metaGGA exchange potential proposed by Becke-Jhonson and modified by Tran-Blaha (TBmBJ34). In the FPLAPW method, the unit cell is divided into non-overlapping spheres with radius Ri and the interstitial region. The atomic spheres´ radii used for Ti and Fe were 1.06 Å and for the oxygen atoms the radii were 0.8 Å. The parameter RKmax controls the size of the basis set and was set to 7. Here R is the smallest muffin-tin radius and Kmax the largest wave number of the basis set. We introduced local orbitals to better describe O-2s, Fe-3p and Ti-3p orbitals36. Refinements of the structures were obtained by calculating the forces acting on the atoms and moving them accordingly in the way described in Ref. 40. The procedure was repeated until the forces on the ions were below a tolerance value of 0.1 eV/Å. This energy-minimization procedure was performed for all the Ti and Fe distributions and for each spin arrangement considered in the present study. Integration in reciprocal space was performed using the tetrahedron method, taking a dense k-mesh points (50 k-points) in the first Brillouin zone. From a careful study we can ensure that for the parameters previously reported, our calculations are extremely well converged (relative numerical errors smaller than 2%). It is well established that TiFe2O4 is an antiferromagnetic inverted spinel with an ordering temperature TN in the order of 140 K1. However, the magnetic structure was not described. In order to determine the magnetic configuration of the antiferromagnetic structure of TiFe2O4 we have considered different antiferromagnetic spin arrangements in the TiFe2O4 unit cell (the ferromagnetic and some ferrimagnetic configurations were also


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studied). For the description of the structure of crystals with magnetic ordering, in which the positions and also the orientation of the magnetic moments of each atom must be specified, symmetry is of prime importance. For this reason, to take into account different spin configurations and also different distributions of Ti and Fe atom in the B-sites of the structure the Fd-3m symmetry was broken and a cell with 56 atoms was constructed from the primitive cell. Theoretical XANES spectra at the Fe and Ti K-edge were obtained as the product of the projected partial density of state (PDOS) for unoccupied 3d Fe/Ti orbitals in the final state and the radial part of the transition matrix elements. For the calculations two cases were computed: the ground state and the transition state that is characterized by the excitation of a 1s core electron of Fe/Ti to an unoccupied 3d Fe/Ti band (core-hole correction41). In the present calculations the core-hole correction were taken into account by removing one electron from the 1s-shell of Fe/Ti and adding one electron as a uniform charge background. In the present calculations, the minimum distance among adjacent atoms with the core-hole correction (one in the unit cell) is in the order of 8.5Å. Although a small interaction cannot be excluded, such separation is typical of FPLAPW calculations including the core-hole correction42. Precise and accurate information related to the local structure at cationic sites of the structure can be obtained from the experimental determination of the hyperfine interactions at the site of a probe-nucleus. These interactions can be measured by using nuclear techniques such as Mössbauer Spectroscopy (MS43,

44).

This nuclear technique is

particularly suited for the study of ferrites because one of the Fe-isotopes (57Fe) is the most commonly used probe-nucleus in MS experiments. MS enables the simultaneous measurement of charge symmetry as well as magnetic related properties. As a result, a fingerprint of the electronic and magnetic configuration near and at the probe nucleus can be obtained. All the information contained in the experimental MS results can be extracted comparing it with ab initio predictions for the hyperfine parameters at each site of the structure under study. In this work we will compare our calculations with three experimentally determined hyperfine parameters: the isomer shift (IS), which provides information on the local chemical bond of the Fe atoms, the quadrupole splitting (QS), which is directly related to the local symmetry around the probe nucleus and the hyperfine



magnetic field (BHF), that is originated in the magnetic configuration and the spin polarization near and at the

57Fe

probe-nucleus43,

44.

A detailed description of the

calculation of the IS, QS, and BHF in the DFT framework can be found in Ref. 27.

2.3. Experimental methodology Powder samples of TiFe2O4 were synthetized by conventional solid state reaction from FeO, TiO2 and 0.072% of metallic Fe powders, searching for a stoichiometry Fe/Ti=2. The powders were pressed into a pellet and calcinated at 1000 ºC during 10 hours under argon atmosphere. X-Ray Diffraction (XRD) measurements (not shown here) were recorded using a θ-2θ scan with Cu-Kα radiation (λ=0.15406 nm). XRD confirms that the samples correspond to the crystalline spinel structure of TiFe2O4. X-ray absorption spectroscopy (XAS) measurements at K-Fe and Ti edges were performed at the XAFS2 beam-line of the Brazilian Synchrotron Light Laboratory-LNLS (Campinas, Brazil) in transmission mode at room temperature. This structural analysis was complemented with a 57Fe-Mössbauer

Spectroscopy (MS) study using the same samples in order to obtain (by

measuring the hyperfine interactions at the Fe sites) information of the electronic structure in the close vicinity of Fe probes. MS measurement was carried out in transmission geometry at room temperature with a 57Co source in a Rh matrix.

4. Results 4.1 Structural and magnetic configuration In order to study the structural and magnetic ground state of TiFe2O4 we considered the normal and inverted structures. In the second case a problem appears: the distribution of the 8 Fe and 8 Ti atoms in the 16 sites B of the 56 un-equivalent atoms cell. To deal with this problem, we performed calculations for different distributions of the Ti and Fe atoms in these sites. For each cation distribution we considered different spin alignment of the Fe atoms in the A and B sites. Only parallel and antiparallel alignments were considered. We found that the inverted and antiferromagnetic structures have the lower energies, in perfect agreement with the expected inverted and antiferromagnetic nature of TiFe2O4.


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The lowest energy case corresponds to an inverted structure (see Figure 1) with the Fe atoms allocated at the positions indicated in Table 1. For this structural and magnetic configuration our GGA+U (U=5 eV) calculations predict a=8.556 Å and u=0.2504. These theoretical results are nearly independent of the cation distribution and the antiferromagnetic spin arrangement considered and are in very good agreement with the experimental ones1,

2, 8.

The equilibrium lattice parameter is also independent of the

inversion degree (for the normal structure, the predicted a=8.554 Å). Concerning the magnetic moments at the Fe sites (μ(Fe)) a value of +3.60 μB for Fe at the sites B and -3.60 μB for Fe atoms at the sites A were predicted (see Table 1). No spin polarization was found at the Ti sites, while the magnetic moment at the oxygen atoms is smaller than ±0.05 μB. So, for the lower energy structure of TiFe2O4 the magnetic moments of Fe at the A sites are aligned antiparallel to those of Fe at B sites. Our predictions for the magnetic moments at the Fe sites are in good agreement with the experimental results obtained in neutron diffraction and MS experiments12-20, 22 performed at 4.2 K (|μ(Fe)|=4.04.2 μB. Additionally, magnetization measurement at 4.2K and nearly 0

K20, 45-47

report

values of a net magnetic moment between 0.0 and 0.5 μB /u.f., in agreement with the antiferromagnetic alignment predicted by our FPLAPW calculations. Finally, TB-mBJ and HSE06 calculations predict magnetic moments at the Fe sites that are very similar to those obtained in the GGA+U calculations: ±3.7 μB (TB-mBJ) and ±3.5 μB (HSE06). At this point we can make a first comparison with our experimental results. The determination of the degree of cationic inversion could be not simple in the case of commonly used techniques such as Rietveld refinement of X-ray diffraction data. On the other hand, XANES experiments provide information about the electronic state and the chemical bond and reflect the coordination of a specific atom of the system under study. So, this X-ray absorption technique could be a very valuable tool for the determination of the inversion degree. In Figure 2.a. we present our experimental XANES spectrum obtained at the Ti Kedge. In order to determine the Ti localization we compare this experimental result with the ab initio spectra predicted for a Ti atom located at the A-site or at the B-site of the spinel structure of TiFe2O4. For the simulation of the Ti allocated at the A-site we considered the lowest energy antiferromagnetic structure obtained for the normal structure of TiFe2O4. The



XANES spectra for Ti at the B-site were obtained considering the lowest energy inverted structure reported in Table 1. As can be seen the simulated XANES spectra for Ti atoms located at the B sites is in excellent agreement with the experimental result, confirming that the Ti atoms are located at the B-sites i.e, a structure with a high degree of inversion for TiFe2O4 is revealed from the XANES experiments. The pre-peak in the experimental spectra close to 4940 eV could be related with a small fraction of Ti atoms with tetrahedral coordination. In Figure 2.b. we present the experimental XANES spectrum of Ti ferrite at the Fe K-edge. As can be seen, the spectrum is like the corresponding one to magnetite, where Fe cations are in A and B sites. To analyze this spectrum we performed simulations of the XANES spectra considering that Fe is located at the A-sites and at the B-sites. For these calculations, we considered the lower energy structure of TiFe2O4 predicted by FPLAPW (Table 1). As can be seen the experimental spectrum can be successfully reproduced by a linear combination of the simulated spectra of Fe at the A-sites and the B-sites in equal proportions. This result shows that Fe atoms are located at both tetragonal and octahedral sites of the spinel structure of TiFe2O4 in nearly equal proportions and then confirms that our sample presents a high degree of cationic inversion. 4.2. Electronic structure. The total density of states (DOS) of normal and inverted TiFe2O4 (lower energy structural and magnetic configuration) obtained in the GGA calculations are shown in Figure 3. As can be seen, a band gap in the order of 0.15 eV is predicted for the normal structure. On the other hand, a half-metallic behavior is predicted for the inverted structure, being the band gap for the negative spin channel in the order of 0.05 eV. These results are nearly independent of neither the spin configuration nor the distribution of the Fe and Ti atoms in the B sub-lattice of the structure (in the case of the inverted structure). This result is in clear contradiction with the experimental ones that showed that TiFe2O4 is an insulator with a band gap in the order of 2.1 eV48, 49. To obtain a correct description of the electronic structure of TiFe2O4 we applied approximations beyond GGA. Initially we performed calculations using the GGA+U approximation with U = 5.0 eV. By applying this formalism a significant opening of the


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band gap was found. In these calculations a band gap in the order of 1.7/1.5 eV was obtained for the positive/negative spin channel (see Figure 4.a.) for the inverted structure in good agreement with the experimental results. In the case of the normal structure, all the general features of the DOS are very similar to those presented for the inverted case. Only an increment of the band gap is predicted for the normal structure (2.1 eV). In conclusion, GGA+U predicts that TiFe2O4 is a wide band gap semiconductor and not a half-metallic oxide as it was previously claimed in the literature25. In Figure 4.b to 4.e. we also present the partial DOS (pDOS) of each of the constituent atoms of the system under study. As can be seen, the valence band of TiFe2O4 (located in the energy range -7.0 eV-0 eV) is dominated by the O-2p and Fe-3d states. Above the Fermi Level, the conduction band has predominantly Fe-3d and Ti-3d character with small hybridization with O-2p states. These hybridization contributions evidence the covalent nature of TiFe2O4. Even though the GGA+U formalism is simple and computationally cheaper it is not a fully ab initio method because an external and arbitrary parameter must be selected a priori, the on-site Coulomb energy U. In order to avoid the use of arbitrary external factors we performed additional calculations using two different approximations for the exchange and correlation functional that do not depend on an arbitrary parameter to explore the band gap correction: the TB-mBJ functional34 and the state-of-the-art hybrid exchange and correlation functional proposed by Heyd, Scuseria and Ernzerhof (HSE06) with 25% of mixing of the exact Hartree-Fock exchange35. The TB-mBJ and the HSE06 formalism for the exchange and correlation potential lead to band gaps of 2.3 eV (see Figure 5). This result confirms the insulating nature of TiFe2O4 obtained using a U value of 5.0 eV and is in agreement with the experimental works that reported band gaps in the order of 2.1 eV. Again, a slightly larger band gap is obtained (for both calculations) in the case of the normal structure of TiFe2O4. In conclusion, from our GGA+U, TB-mBJ and HSE6 calculations, the predicted lowest energy inverted and antiferromagnetic configuration of TiFe2O4 has an insulator nature with a band gap in the order of 2.3 eV. This result is independent of the spin configuration and the distribution of Fe and Ti atoms in the B sites of the structure. Also,



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the insulating character of TiFe2O4 is independent of the inversion degree (the band gap value slightly depends on the inversion degree). 4.3. Hyperfine characterization and Mössbauer measurements One of the best suited tools for the investigation of structural, electronic, and magnetic properties at the atomic scale are the hyperfine techniques43,

44.

Among the

hyperfine techniques, MS spectroscopy is particularly adequate to study Fe-compounds because one of the Fe isotopes (57Fe) can be used as a probe. MS enables the determination of both charge symmetry related properties and magnetic properties, making it possible to obtain a complete picture of the electronic and magnetic configuration near and at the probe nucleus43, 44. All the information that the experimental MS results contain can be obtained by confronting it with an accurate calculation of these electronic and magnetic properties27, 50.

As we said before, different MS experiments were reported2,

22, 23, 48, 51-55,

but in

most cases the experimental results were not conclusive. So we will center the discussion in the MS experimental results obtained by Nakamura and Fuwa24 and our results obtained at 300 K. Nakamura and Fuwa performed MS experiments in the temperature range 16-500 K. In the whole temperature range, the spectra were fitted with four hyperfine interactions. The hyperfine parameters that characterized these interactions are shown in Table 2. In this table, the reported QS values were obtained by extrapolation of the experimental results from 500 – 163 K to 0 K (below 163 K the system transforms from the cubic spinel structure to a tetragonal I41/amd phase24). The magnetic interactions are observed at temperatures below 130 K, where the system presents a magnetic ordering. For the calculation of the BHF we considered two structures: first, we obtained the BHF for Fe atoms at sites A and B considering the cubic spinel structure (Fd-3m). After that, we performed calculations in the tetragonal I41/amd structure, the stable one below 163 K. We found that the BHF at sites A and B of the spinel structure differ in less than 1 T from those corresponding to the sites of the tetragonal structure. So, the effect of the structural phase transition is not significant for the study of the BHF in the present case. The experimental values of BHF are presented in Table 2 and correspond to the extrapolation to 0 K of the experimental results reported in reference24.


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Two of the fitted interactions reported in Ref. 24 are characterized by IS=1.2 mm/s, QS 0f 1.9-2.3 mm/s and BHF of about 30-40 T. Based in a simple point charge model the authors assigned these two interactions to Fe atoms located at the A sites of the structure (Fesite A ). The other two interactions, characterized by IS=1.2 mm/s, QS in the order of -3.0 mm/s and BHF of 20 T were assigned to Fe at sites B (Fesite B). We will discuss this assignment of the hyperfine interactions in the framework of our ab initio calculations. The existence of a QS for Fe at the A-sites also deserves a comment. In the normal structure the symmetry of the A-site is cubic (-43m) and therefore the QS must be 0.0 mm/s. The non-zero QS at Fesite A is explained by the fact that the cationic inversion produces an asymmetric distribution of Fe and Ti around Fe in A sites. This breaks down the cubic symmetry of the A-site, inducing a charge asymmetry and, in consequence, a QS at this site. In Figure 6.a we present our experimental Mössbauer spectrum obtained at 300 K. In this experiment we employed the samples that were characterized by XANES. This spectrum was fitted with four doublets originating from quadrupole interactions with hyperfine parameters (see fit 1 in Table 3) similar to those reported by Nakamura and Fuwa24 (the sign of the QS was not determined in our experiment). According to the site assignment proposed by Nakamura et al. two of the interactions correspond to Fe atoms at sites A and the others to Fe at sites B. Now we can compare the experimental results with our ab initio predictions obtained for the different structural and magnetic scenarios. Initially, we performed calculations for the normal structure of TiFe2O4 (all the Fe atoms are located at the octahedral B-sites). This situation does not correspond to the real situation, as we discussed in previous sections but is of interest in order to evaluate the changes in the hyperfine parameters at the B-sites originated by the cationic inversion. In this configuration our GGA+U calculations predict for Fesite

B

IS=1,08mm/s, QS=-2.5 mm/s and BHF=43.6T,

values that are nearly independent of the spin configuration. This interaction presents hyperfine parameters (HF) that are very similar to those of the interaction associated by Nakamura and Fuwa24 to Fe atoms located at the A-sites. This result is a first indication that the assignment of the experimentally observed interactions done in Ref. 24 could be wrong. Now we consider the inverted structures. In Table 2 we present our predictions for the lower energy structure (Fe distribution and magnetic arrangement presented in Table 1).



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These results are nearly independent of the model employed for the XC potential and the spin arrangement. We also note that the interaction at Fesite B in the normal and inverted structures are very similar, showing that the hyperfine interaction at Fesite

B

is “short-

sighted” and is not affected by the change of a Fe atom by a Ti one in its second coordination shell (distance Fesite

B

to its first Fe/Ti neighbors in the inverse structure is

3.00 Å). As it was expected, for the inverted structure the break-down of the cubic symmetry of the A-site originated by the cationic inversion induces a QS at this site. An important result is that the IS values for Fe atoms in the A and B sites are very similar. This means that the nature of the Fe-O bonding does not change although these atoms have different O atom coordination (tetrahedral and octahedral). The obtained values of the IS and the QS are characteristic of Fe2+ ions43. We now compare our results with the MS experimental ones at 0 K. As can be seen in Table 2 there is an excellent agreement between theory and MS experimental results. But contrary to the analysis of Nakamura and Fuwa24, the hyperfine interaction that presents negative QS and a large BHF corresponds to Fe probes located at the A-sites. The second hyperfine interaction (positive QS, BHF larger than those of the other hyperfine interaction) corresponds to Fe at the B-sites. The failure of the previous assignment is associated to the use of a simple model for the structure of TiFe2O4 (the point charge model and antishielding factors) for the calculation of the QS. The electronic structure of this oxide is too complex to be described by this simple model and in consequence the predictions obtained are erroneous. Mossbauer spectra measured below TN reported in literature21-23 present extremely broad line widths due to the distribution in the hyperfine magnetic field. The possible origin of such hyperfine field distribution is the disorder in the Ti4+ and Fe3+ distribution in the B sites. In order to explore the influence of disorder in the HF parameters we study different cation distributions (among B sites). The resulting calculated HF parameters of Fe in both sites corresponding to two disordered distribution together with the ordered one are shown in Figure 7. As can be seen, cation disorder does not influence hyperfine parameters of Fe atoms at the B sites, but strongly affects the corresponding to Fe at sites A. In the case of sites B we determined a dispersion of the hyperfine parameters of less than 0.1 mm/s, 0.8 mm/s and 4 T for IS, QS and BHF, respectively. For Fe atoms located at the sites A


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components with smaller IS (around 0.7 mm/s) and QS (between 0.5 and 1 mm/s) could appear as a consequence of cation disorder in the sites B. Based in these results we performed a new fitting of the room temperature spectrum adding an interaction with smaller IS and QS (see Figure 6.b). We observed that the fit noticeably improves and the hyperfine parameters of the added component (see A3 site in Table 3) are in good agreement with the expected ones for the site A for a random distribution of Fe and Ti in sites B. We conclude that our ab initio calculations considering the lowest energy of structure of TiFe2O4 correctly reproduces the experimental MS results. Considering the sensitivity of the hyperfine interactions to fine details of the electronic and magnetic structure in the sub-nanoscopic vicinity of the probe atoms, the excellent agreement theoryexperiment gives support to our model for the structural and magnetic configuration of TiFe2O4. Finally, the experimental results at 300 K can only be “crudely” compared with the ab initio predictions because all possible thermal effects are not included in the calculations. Moreover, at 300 K the Ti-ferrite behaves as a paramagnet and it is not possible to perform an ab initio simulation of a paramagnetic system, which is indeed a weakness of the approach. But taking this comment into account our calculations correctly reproduce the Mössbauer results at 300 K.

5. Conclusions. In this work we have studied both theoretically and experimentally the structural, electronic, magnetic and hyperfine properties of the TiFe2O4 ferrite. In order to obtain an accurate description of the electronic structure and the band gaps of TiFe2O4 system DFT based FPLAW calculations were performed employing different exchange and correlation potentials: GGA, GGA+U and the state-of-the-art Heyd-Scuceria-Ernzerhof (HSE06) and Tran-Blaha modified Becke-Johnson potentials (TB-mBJ). We showed here that GGA+U (provided that an adequate U value is used), HSE06 and TB-mBJ predict very similar results for the structural, electronic, electronic, magnetic, and hyperfine properties of



TiFe2O4, showing that the simple and computationally inexpensive GGA+U scheme is accurate for the study of ferrites. To obtain the structural and magnetic configuration of the system under study the calculations were performed considering the normal and the inverted structure of TiFe2O4. In the second case, we considered different distributions of the Fe and Ti atoms in the Bsites of the spinel structure. For each configuration, different spin arrangements were considered. From the whole set of calculations, we determine the lowest energy structural and magnetic configuration, that resulted to be inverted and antiferromagnetic. For these structures, the magnetic moments of the Fe atoms in both A and B sub-lattices are ferromagnetically ordered, while the magnetizations of these two sub-lattices are antiparallel with respect to each other. XANES experiments reported here confirmed this theoretical result. In the equilibrium structure and similarly to the case of magnetite the magnetic moments of Fe atoms at the A sites are aligned antiparallel to those of Fe atoms at the B sites. For this configuration a lattice parameter of 8.556 Å and magnetic moments at the Fe sites of 3.60 μB were predicted by our GGA+U calculations, in very good agreement with the reported values. Concerning the electronic structure of TiFe2O4 we show that the GGA calculations fail in the prediction of the band gap of TiFe2O4 (a metallic behavior is predicted by GGA). Models beyond GGA (GGA+U TB-mBJ, HSE06 hybrid functional) are essential for a good description of the electronic structure of TiFe2O4. TB-mBJ and HSE06 calculations predict that TiFe2O4 is an insulator with a band gap of 2.3 eV, in excellent agreement with the experiments (2.1 eV). GGA+U (using a U value of 5 eV) predicts a band gap in the order of 1.7 eV. These results refute the half-metallic nature of TiFe2O4 reported in the literature. The hyperfine interactions reported by MS experiments at room temperature and at low temperature are also successfully reproduced by our GGA+U, TB-mBJ and HSE06 calculations. From the comparison with the experiments we can associate the observed hyperfine interactions to Fe located at the two cationic sites, confirming that Fe populates both cationic sites of the spinel structure of TiFe2O4. Moreover, our results clearly show that a previous assignment of the hyperfine interactions (based in a simple point-charge model and antishielding factors) is incorrect. Since the hyperfine interactions are extremely


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sensitive to fine details of the electronic structure in the close vicinity of the probe site (Fe in the present case), the excellent agreement theory-MS experiment gives confidence for our model for the structural and magnetic configuration of TiFe2O4.

Acknowledgments This research was partially supported by CONICET (Grant No. PIP0747, PIP0720), UNLP (Grant No. 11/X678, 11/X680, 11/X708, 11/X788, 11/X792), ANPCyT (Grant No. PICT PICT 2012-1724, 2013-2616, 2016-4083) and UNNOBA (Grant No. SIB0176/2017), and “Proyecto Acelerado de Cálculo 2017”, Red Nacional de Computación de Alto Desempeño (SNCAD-MINCyT) - HPC Cluster, Rosario. Argentina. References 1

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observed by maximum entropy method at high pressure and low temperature. Phys. Rev. B 2009, 80, 134120. 2

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Evolution in the Fe3O4-Fe2TiO4 Pseudobinary System and its Implications for Remanent Magnetization in Martian Minerals. IEEE T. Magn. 2011, 47, 4124-4127. 3

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Chemistry: Plenum: New York, 1989. 7

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Fe2TiO4, J. Phys. Soc. Jpn. 1971, 31, 452-460. 9

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epitaxial ZnFe2O4 films: An experimental and first-principles study. J. Applied Phys. 2014, 115, 213908. 11

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