The role of exchange–correlation functional on the description of multiferroic properties using density functional theory: the ATiO₃ (A = Mn, Fe, Ni) case study

Abstract

In this study, ab initio density functional theory calculations were performed on ATiO₃ (A = Mn, Fe, Ni) materials for multiferroic applications. Structural, magnetic, electronic and topological analysis were investigated as regard the A-site cation effect. Concerning the prediction of new multiferroic candidates in silico, the role of different exchange-correlation functionals was reviewed. From such properties was verified that PBE0 functional performs the best for ATiO₃ compounds overcoming the deficiency of standard LDA and PBESol functionals and get the better of B3LYP, PBE0+D and B3LYP+D. Regarding the A-site cation effect, calculated structural parameters follow the ionic radii trend moving from Mn to Ni, resulting in more ordered AO₆ clusters; while, the TiO₆ distortion increases. Using magneto-structural and magneto-electronic parameters, the antiferromagnetic exchange coupling constant was rationalized from GKA rules. The increase of the antiferromagnetic order along the transition metal series was explained from the 3d orbital occupancy and A–O bond interactions. DOS and band structure profiles clarify the semiconductor behavior as regard the 3d crystal field splitting for both A-site and Ti cations. Further, electronic results suggest the existence of intermetallic connection (A–O–Ti–O–A); whereas, topological analysis in the framework of quantum theory of atom in molecules was carried out to evaluate the A–O and Ti–O bond interaction through the analyses of ∇²ρ sign at bond critical point and of the charge-transfer parameter obtained from net charges.

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1 Introduction

Recently, the multiferroics materials have assumed a remarkable importance as research topic on the design of new technological devices. The most complete definition for this class involves two or more forms of ferroic orders (i.e. ferroelectricity, ferroelasticity, ferromagnetism, ferrotoroidicity) combined into a single crystalline structure. However, the main studies in this field are strictly related to candidates that combine a magnetic order (ferromagnetism, antiferromagnetism) with ferroelectricity, being denominated magnetoelectric multiferroics.^1–3 The renewed interest on multiferroic materials was motivated from theoretical explanations about the physical reasons behind the scarcity of such compounds, as well to clarify the unusual phenomena and successfully predict new candidates.^4–8

The theoretical understanding of the multiferroic materials depends of the precise description of structural, electronic and magnetic properties. In the last decades, the first-principles calculations, mainly Density Functional Theory (DFT), have played a fundamental role on solid-state investigations which notice the reliable description for different kind of properties.^5,9–14 However, the multiferroic materials class corresponds to a challenging topic on first-principles calculations due to the strong correlation, which results from the unpaired electrons. In such materials, the partial filling of d or f orbitals induces a high localization summed to the strong interaction causing failures on the electronic structure representation, mainly for standard LDA or GGA exchange–correlation functionals.¹⁵ Such incoherency is associated with the Self-Interaction Error (SIE) that provides a spurious interaction of an electron with itself resulting in underestimated band gaps, erroneous localization for d–f orbitals, and the tendency toward magnetism.^16–19 Countless approaches to correct the SIE limitation were proposed, being the GW approximation, self-interaction correction (SIC) and the DFT plus Hubbard potential (U) the most widely reported. In GW and SIC methods, the SIE is corrected through a formal procedure to calculate the self-energy, which is included in the splitting of occupied and unoccupied bands.^20,21 In the other hand, DFT+U introduce a local correction to DFT based on a mean-field (HF) treatment of the Hubbard model characterized by two parameters, the on-site coloumb potential (U) and the exchange interaction (J) providing a correct interpretation for Jahn–Teller distortions and exchange interactions.^15,22–25 Despite the notorious picture for the electronic structure of some transition metal oxides,^15,25 the performance of such method is related to the system-dependent U and J parameters making the choice often arbitrary; besides, the wrong solutions for under-pressure systems or phase-transitions investigations.²⁶

Hybrid functionals are an alternative approach to significantly improve the accuracy behind local (LDA) and semi-local (GGA) methods, corresponding to a mixing between the exact nonlocal Fock exchange and LDA/GGA exchange–correlation.²⁷ Regarding the theoretical investigation of multiferroic materials, the hybrid Hamiltonian overcome the LDA and GGA approaches through a significant improvement of the band-gap description for semiconductors, as well through the correct localization of unpaired electrons in open-shell systems.^15,28–32 Recently, these functionals have been widely applied to different multiferroic materials. Goffinet and co-workers report a systematic comparison between different exchange–correlation formalism to predict the structural, magnetic, electronic and dynamical properties of BiFeO₃, showing the powerful prediction of hybrid functionals for multiferroic candidates.³³ In addition, Stroppa and Picozzi confirmed this argument through a hybrid DFT/HF investigation of proper (BiFeO₃) and improper (HoMnO₃) multiferroic materials.³⁴ Further, McDonnell and co-authors showed the agreement between hybrid DFT and experimental findings for optical properties.³⁵ Subsequent studies in this field reveal that hybrid functionals calculations, mainly based on HSE formalism, are useful to predict new multiferroic candidates such as PbNiO₃,³⁶ YMnO₃,³⁷ BaMnO₃,³⁸ PbVO₃,³⁹ Bi₃FeCrO₃,⁴⁰ among others.^41,42 On the other hand, Heifets and collaborators investigated the thermodynamic stability of BiFeO₃ by means of B3PW hybrid functional and noted a underestimated formation energy at 0 K suggesting that the further investigation about the performance of available hybrid functionals is needed.⁴³

From this point of view, the present work proposes a systematic theoretical investigation comparing local (LDA), semilocal (GGA-PBESol), pure and energy dispersion augmented hybrid functionals (PBE0, PBE0+D, B3LYP and B3LYP+D) to evaluate the structural, magnetic and electronic properties of ATiO₃ (A = Mn, Fe, Ni) multiferroic materials with LiNbO₃-type structure. In the past few years, the scientific interest on such materials has increased due to the possibility of controlling the magnetism from antisymmetric Dzyaloshinskii–Moriya interaction. This interest was motivated from a theoretical structural–chemical investigation proposed by Fennie, where symmetry arguments and DFT+U calculations were combined to discuss the strong polarization-magnetism coupling in ATiO₃ (A = Mn, Fe, Ni) ilmenite based materials when stabilized in R3c phase.⁴⁴ Such predictions were validated by experimental reports about the synthesis of the high-pressure form of FeTiO₃,⁴⁵ MnTiO₃,^46,47 and NiTiO₃ candidates.⁴⁸ Despite the current theoretical studies developed for these materials,^49–51 both exchange–correlation and A-site cation modification effect on structural, magnetic and electronic properties remains unclear.

2 Computational methodology

ATiO₃ (A = Mn, Fe, Ni) materials crystallize in the acentric LiNbO₃-type structure (R3c) under high pressure and temperature conditions through a cation reordering phase transition from ilmenite polymorph.^48,52,53 The rhombohedral primitive unit cell for this polymorph contains two ATiO₃ unit formulas (10-atoms) with A (A = Mn, Fe, Ni), Ti and O atoms located at (0; 0; u), (0; 0; 0) and (x, y, z), respectively, as showed in Fig. 1. Theoretical models proposed in this study are based on the experimental structural parameters obtained by X-ray measurements.^48,53,54 The A, Ti, and O atoms were described by atom centered all-electron gaussian basis set such as 86-411d41G, 86-411(d31)G and 8-411G, respectively.^55–57

Fig. 1 Primitive (a) and conventional (b) crystallographic unit cell (R3c) for ATiO₃ (A = Mn, Fe, Ni) materials. The black, blue, and red balls represent A, Ti, and O atoms, respectively. The blue and black polyhedral represent the [TiO₆] and [AO₆] clusters, respectively.

The ATiO₃ (A = Mn, Fe, Ni) structures were fully-relaxed (lattice parameters and atomic positions) in the framework of Density Functional Theory (DFT) employing local (LDA),⁵⁸ semilocal (PBESol),⁵⁹ and hybrid (B3LYP,^60,61 PBE0 (ref. 62)) exchange–correlation functionals. A general drawback of all local, semilocal or hybrid exchange–correlation functionals is the inability to describe long-range electron correlations (dispersive interactions) as van der Waals forces. These interactions correspond to a balance between electrostatic and exchange-repulsion forces, playing a fundamental role on the theoretical description of ferroelectric materials, such as ATiO₃ (A = Mn, Fe, Ni) in LiNbO₃-type structure. In this study, we include the dispersive effects on hybrid B3LYP and PBE0 exchange-correlation functionals through a London-type pairwise empirical correction proposed by Grimme.⁶³ In this approach, the total energy (eqn (1)) and the empirical correction dispersion (eqn (2)) were described by the following equations:

E_XC+D = E_XC + E_disp (1)

Here, the summation involves all atom pairs and lattice vectors (g) excluding the i = j contribution for g = 0; s₆ is a scaling factor that depends only of exchange–correlation functional used (1.05 for B3LYP, 0.6 for PBE0), c₆^ij is the dispersion coefficient for the atom pair ij, R_ij,g is the interatomic distance between atoms i in the reference cell and j in the neighboring cells at distance |g| and f_dmp is the damping function used to avoid near-singularities for small interatomic distances consisting of summed atomic van der Waals radii (i.e. R_vdw = R_vdwⁱ + R_vdw^j) and the steepness set to 20.^63,64

All calculations were carried out using the periodic ab initio code CRYSTAL09.⁶⁵ The convergence criteria for mono and bielectronic integrals were both set to 10⁻⁸ Hartree, while the RMS gradient, RMS displacement, maximum gradient and maximum displacement were set to 3 × 10⁻⁵, 1.2 × 10⁻⁴, 4.5 × 10⁻⁵ and 1.8 × 10⁻⁴ a.u., respectively. Regarding the density matrix diagonalization, the reciprocal space net was described by a shrinking factor set to 8, corresponding to 65 k points in accordance with the Monkhorst–Pack method.⁶⁶ The accuracy in evaluating the Coulomb and exchange series was controlled by five thresholds, for which adopted values are 10⁻⁸, 10⁻⁸, 10⁻⁸, 10⁻⁸ and 10⁻¹⁶.

Experimental reports suggest a G-type antiferromagnetic ground-state for all ATiO₃ (A = Mn, Fe, Ni) materials consisting of spins ferromagnetically ordered within (111) planes, while the adjacent planes are antiferromagnetically coupled.^45,46,48 To determine the performance of the exchange–correlation functional in the magnetic ground-state description for strongly correlated materials, we consider two collinear magnetic configurations using the primitive (10-atoms) unit cell: (i) ferromagnetic (FEM), where the spins for all neighbors are parallel ordered; (ii) antiferromagnetic (AFM) for which the spins on the nearest neighbors are antiparallel ordered to each other. The exchange coupling constant was computed from the energy difference between the magnetic configurations using the Ising model:

Ĥ_ising = ∑J_ijŜ_izŜ_jz (3)

Here J_ij is the magnetic exchange coupling constant between A²⁺ neighbors, while Ŝ_z correspond to the operator for the z-component of spin momentum.

In order to understand the electronic structure of ATiO₃ (A = Mn, Fe, Ni) multiferroic materials, theoretical results for density of states (DOS), band structure profiles and topological analysis were investigated from the optimized wavefunction. Topological analysis of the electron density ρ(r) and atom basin properties were obtained with TOPOND98 package⁶⁷ within the framework of the quantum theory of atom in molecules (QTAIM) developed by Bader.⁶⁸ This code provides a comprehensive evaluation of electron density (ρ) and of its Laplacian, as well as atomic basin properties and 2D–3D plots of periodic systems. In this study, we focus on the determination of so-called bond critical points (BCP), which are powerful tools to classify the chemical bonds. The critical points (CP) in the electron density and its Laplacian are the points where of the ∇ρ(r) vanishes. The CP can be classified in terms of their type (r, s), where r correspond to the number of non-zero eigenvalues (λ_1–3) of the Hessian matrix, while s describe the difference between positive and negative eigenvalues. The BCP correspond to (3, −1) in terms of the (r, s) notation, indicating two negative curvatures associated with the interatomic surface orthogonal to the BCP, and one positive curvature related to the bond path at the BCP. Further, another section of TOPOND deals with atomic basin boundaries enabling the evaluation of several atomic properties such as Bader's net charge, atomic volume and others. A more detailed description of TOPOND98, QTAIM and topological analysis can be found in the ref. 69–73 and the references there in.

3 Results and discussions

3.1 Structural properties

In order to assess the validity of different exchange–correlation functionals to compute structural properties, the lattice parameters and unit cell volume obtained from DFT benchmark for ATiO₃ (A = Mn, Fe, Ni) materials were investigated, as summarized in Table S1.† Despite the fact that non-local PBESol functional slightly overcomes local (LDA) formalism for all ATiO₃ material, it was observed that standard exchange–correlation functional underestimate the experimental results, while pure and dispersion augmented hybrid formalism are the closest. Surprisingly, B3LYP exhibits the worst performance among the hybrid functionals, resulting in overestimated structural parameters for all ATiO₃ materials. Similar results were reported for other ferroelectric titanates, where Bilc and co-authors argue that the origin of B3LYP failures, mainly for lattice parameters and unit cell volume, is the Becke's GGA exchange part.³¹ On the other hand, PBE0 performs the best among the investigated exchange–correlation functionals. Further, such results are closer to the experimental measurements even when compared to the literature DFT+U results for ATiO₃ (A = Mn, Fe, Ni) materials.^44,49,50

From the results reported in Table 1, it can be seen that the inclusion of dispersive effects on hybrid B3LYP and PBE0 functionals induces a large unit cell contraction mainly for the c lattice parameter, resulting in a worse performance than non-augmented hybrid formalism. This error can be attributed to the nature of empirical R_vdw and c₆^ij parameters that does not take in account the chemical environment resulting from the crystalline ordering in solid state materials, where the coordination number controls the van der Waals radii. One way to overcome this problem consists of a empirical search of system-dependent R_vdw and c₆^ij parameters as function of desired properties, as performed by Albuquerque for TiO₂ anatase.⁷⁴

Table 1 Theoretical results for local magnetic moment (S – μ_B), superexchange coupling constant (J_A–A – in K) and A–O–A bond angle (Θ) of the ATiO₃ (A = Mn, Fe, Ni) materials as function of exchange–correlation functionals

	MnTiO3			FeTiO3			NiTiO3
	S	J	ΘMn–O–Mn	S	J	ΘFe–O–Fe	S	J	ΘNi–O–Ni
B3LYP	4.755	−8.74	118.44	3.762	−13.87	122.35	1.719	−9.30	126.56
B3LYP+D	4.748	−6.62	115.80	3.754	−10.45	120.90	1.716	−5.13	126.13
PBE0	4.788	−6.89	117.19	3.791	−11.34	121.82	1.751	−10.06	126.49
PBE0+D	4.785	−5.90	115.72	3.787	−10.08	121.05	1.749	−6.99	126.26
PBESol	4.597	−28.62	119.27	3.672	−7.83	125.82	1.528	21.30	128.21
LDA	4.524	−43.53	120.09	3.607	−12.49	126.43	1.480	6.16	128.85

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Let us now briefly discuss the performance of different exchange–correlation functionals on the main bond distances in R3c multiferroic titanates. The LiNbO₃-type polymorph of ATiO₃ (A = Mn, Fe, Ni) can be described by two sub-units centered on A- and Ti-site cations, as presented in Fig. 1. Unlike the common ABO₃ materials, both sub-units in R3c structure are centered on six-fold coordinated metals (AO₆ and TiO₆) linked up to oxygen atoms by short (3) and long (3) paths, enabling an A–O–Ti–O–A intermetallic connection via face and edge-sharing. Further, both AO₆ and TiO₆ clusters share the corners with other octahedrals through a A–O–A or Ti–O–Ti path. To balance the electrostatic repulsion between the metals, both A- and Ti cations are displaced from the octahedral central position, resulting in structural distortion that controls the ferroelectricity. The Table S2† summarizes the M–O (M = Mn, Fe, Ni, Ti) bond distances and octahedral distortion (Δ = 1/6Σ[(d − d_av)/d_av]²) obtained from different exchange–correlation functionals.

Bond lengths calculated for both AO₆ and TiO₆ clusters show the general behavior observed for the lattice parameters and unit cell volume: (i) local (LDA) and semilocal (PBESol) functionals underestimates the experimental values; (ii) whereas the pure and dispersion augmented hybrid functionals exhibit a reasonable agreement. However, both B3LYP+D and PBE0+D functionals perform slightly worse than non-augmented hybrid functionals clarifying the dependence of R_vdw and c₆^ij parameters as function of chemical environment. Despite the same coordination number for A and Ti cations, the experimental results shown that both clusters exhibit different distortions, which are originated from distinct mechanisms. Inasmuch as the values proposed by Grimme⁶³ for the 3d transition metals are equal in DFT-D2 method, the calculated structural properties show an incorrect picture as regard the crystalline arrangement. For instance, the octahedral distortion calculated with B3LYP+D and PBE0+D functionals suggests a reduced role of TiO₆ clusters in the structural deformation, while overestimates the A-site contribution.

The latter investigation in this section involves the A-site cation effect on the structural properties of ATiO₃ (A = Mn, Fe, Ni) materials in LiNbO₃-type structure. For this purpose, the lattice parameters, unit cell volume and bond lengths obtained with hybrid PBE0 exchange–correlation functional were selected. Theoretical results presented in Table S1† reveal that the unit cell volume of ATiO₃ materials decrease in order MnTiO₃ > FeTiO₃ > NiTiO₃ following the A²⁺ ionic radii series. Such contraction is accompanied by a shortening of the A–O bond distances resulting in more ordered AO₆ clusters. This tendency can be explained by the 3d orbital occupancy moving from the Mn to Ni, where the progressive filling of t_2g levels induces an orbital contraction summed to the large crystal field splitting for high-spin configurations in order Mn²⁺ (t³_2g − e²_g) < Fe²⁺ (t⁴_2g − e²_g) < Ni²⁺ (t⁶_2g − e²_g). Such results are similar to the general trend observed for Radtke and co-authors as regard the ATiO₃ (A = Mn, Fe, Ni) materials in the ilmenite polymorph.⁷⁵ On the other hand, the distortion on TiO₆ clusters increases moving from Mn to Ni suggesting the fundamental role of second-order Jahn–Teller distortion (SOJT) on the stabilization of R3c polar structure for ATiO₃ (A = Mn, Fe, Ni) materials in agreement with the polarization results proposed by Fennie.⁴⁴

3.2 Magnetic properties

As previously discussed, the magnetic ordering of ATiO₃ (A = Mn, Fe, Ni) materials in LiNbO₃-type structure can be described by a G-type arrangement consisting of alternated (111) planes antiferromagneticly ordered. In such geometry each magnetic A²⁺ cation is surrounded by six neighbors through a A–O–A superexchange coupling constant, which is controlled by the 3d orbital occupancy and A–O–A bond angle following the Goodenough–Kanamori–Anderson (GKA) rules, as presented in Fig. 2. In order to investigate the performance of different exchange–correlation functionals on the description of magnetic ground-state for ATiO₃ (A = Mn, Fe, Ni) materials, the exchange coupling constant along the A–O–A path (J_A–A) was computed from the energy differences between FEM and AFM configurations without spin–orbit contribution, as summarized in Table 1.

Fig. 2 Hexagonal unit cell (R3c) for ATiO₃ (A = Mn, Fe, Ni) materials and G-type antiferromagnetic spin ordering representation (orange vectors). The black polyhedra represent seven (AO₆) clusters denominated as (AO₆)₇ magnetic complex with A–O–A superexchange coupling constant J_A–A.

Regarding the magnetic ground state all exchange–correlation functionals successfully predict the antiferromagnetic exchange coupling constant for ATiO₃ (A = Mn, Fe, Ni) materials, except for NiTiO₃ where LDA and PBESol predict a ferromagnetic ordering in disagreement with the experimental predictions. Further, it was observed that LDA and PBESol provide smallest values for the local magnetic moment (S) comparing with hybrid formalism. If we apply a fully ionic picture, the A-site metal is a divalent cation resulting in Mn²⁺ (3d⁵), Fe²⁺ (3d⁶) and Ni²⁺ (3d⁸) valence orbital configuration, which would result in a magnetic moment of 5, 4 and 2 μ_B, respectively. Table 1 shows that pure and dispersion augmented hybrid functionals results exhibit a better agreement with expected values than local and semilocal formalism, which can be explained by the spurious self-interaction error presented by local and semilocal formalism, resulting in inaccurate pictures toward magnetism of strongly correlated materials.^16–19 Furthermore, the inclusion of dispersion effect on hybrid functionals induces smallest values for J_A–A suggesting a magneto-structural relation. Thus, DFT+D formalism overestimates the distortion degree for AO₆ clusters resulting in longer A–O–A paths (Table S2†) with high Θ_A–O–A (Table 1) bond angle. Such distortion reflects on different covalent bond character for the different hybrid formalism, as presented in Table S3.† It was observed that the inclusion of dispersive effects on hybrid functionals induces a higher ionic character for A–O interactions than pure formalism, in excellent agreement with A–O–A bond angle (Table 1).

The superexchange J_A–A coupling constant follows the GKA rules through a virtual electron transfer between magnetic A²⁺ (A = Mn, Fe, Ni) cation using the molecular orbital originated from the mixing between transition metal 3d and oxygen 2p orbitals. Therefore, the covalent/ionic character of A–O chemical bonds play a fundamental role on the magnetic ground-state stabilization, since the GKA theory predict that the increase of covalent character for A–O bonds extends out the 3d metal wavefunction over the oxygen atoms facilitating the electron transfer responsible for the antiferromagnetic ordering.⁷⁶ In this way, the high AO₆ octahedral distortion obtained with dispersive augmented hybrid functionals induces an increased ionic character for A–O bonds (Table S3†) resulting in more positive values for J_A–A.

Looking at the chemical dependence of J_A–A along the transition metal series (A = Mn, Fe, Ni) it was observed that the antiferromagnetic ordering strength increases with 3d orbital occupancy in agreement with theoretical results obtained for AO oxides, as well reported by Fennie using DFT+U.^44,77 This tendency can be explained by the A–O covalent character that increases moving from Mn to Ni (Table S3†), meaning an extended 3d–2p overlap stabilizing the antiferromagnetic ordering.

3.3 Electronic properties

In order to investigate the electronic structure of ATiO₃ (A = Mn, Fe, Ni) materials, theoretical results for band structure profiles and density of states (DOS) were analyzed. Firstly, it was analyzed the effect of different exchange–correlation functionals on the band-gap description of strongly correlated ATiO₃ materials, as presented in Fig. S1.† It was noted that conventional exchange–correlation functionals (LDA and PBESol – Fig. S1m–r†) underestimate the semiconductor behavior of ATiO₃ materials, suggesting band-gap values around 0–1.5 eV. This failure can be clearly noted for the band structure of FeTiO₃ obtained with LDA and PBESol functionals (Fig. S1n and q†), where a metallic behavior was found. These results are originated from the self-interaction error presented in such functionals, as well observed for other strongly correlated oxides.^19,30,49,78 On the other hand, pure and dispersive augmented hybrid functionals successfully predict the semiconductor behavior for all ATiO₃ (A = Mn, Fe, Ni) materials without meaningful differences between B3LYP+D(PBE0+D) and B3LYP(PBE0) results, as presented in Table 2. Further, comparing our results with reported DFT+U one for MnTiO₃ and NiTiO₃, we noted that Hubbard corrected DFT results are very similar to the standard exchange–correlation functionals and smaller than hybrid formalism. It is important to clarify that experimental measurements of electronic band-gap for ATiO₃ (A = Mn, Fe, Ni) materials have not been reported to date; however, the existence of ferroelectric properties in such materials suggest a semiconductor–insulator behavior following the electronic results reported for the ilmenite polymorph, where all ATiO₃ candidates are semiconductors with band-gap around 2.0–3.5 eV.⁷⁹

Table 2 Theoretical results for electronic band-gap (in eV) of ATiO₃ (A = Mn, Fe, Ni) materials at different exchange–correlation functionals

	Mn	Fe	Ni
a Ref. 50.b Ref. 51.c Ref. 49.
B3LYP	3.89 (L–Γ)	2.79 (T–Γ)	4.42 (L–Γ)
B3LYP+D	3.95 (Γ–Γ)	2.77 (T–Γ)	4.43 (L–Γ)
PBE0	4.48 (Γ–Γ)	3.30 (T–Γ)	5.12 (L–Γ)
PBE0+D	4.51 (Γ–Γ)	3.30 (T–Γ)	5.13 (L–Γ)
PBESol	1.37 (L–Γ)	Metallic	1.78 (FB–Γ)
LDA	1.12 (L–Γ)	Metallic	1.57 (FB–Γ)
DFT+U	1.79^a	—	2.35^c
	0.85^b

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Concerning the A-site cation effect on the electronic band-gap of the titanate materials, it was observed that the smallest excitation energy (Table 2) follows the order FeTiO₃ < MnTiO₃ < NiTiO₃. Such effect can be explained from the 3d site orbital occupation and the crystal field splitting for t_2g and e_g levels. Moving from Mn to Ni, the electrons are included in t_2g (t³_2g, t⁴_2g and t⁶_2g), while e_g remains constant (e²_g) increasing the crystal field splitting. In this way, the valence band region (VBR) is expected to be sensibly different along the transition metal series, in agreement with theoretical results presented in Fig. S1.† These results show that the VBR for NiTiO₃ has the higher degeneration character, while MnTiO₃ and FeTiO₃ show an internal gap around 1.0 and 1.5 eV, respectively. This increased internal band-gap for FeTiO₃ is originated from the energy difference between double-occupied t²_2g and degenerated single-occupied t_2g orbitals. Further, the t²_2g orbitals moves back in energy, the single-occupied t_2g states are displaced to higher energy resulting in a reduced band-gap comparing to nickel and manganese titanates. Regarding the electronic excitation nature, theoretical results obtained with hybrid formalism show a direct band-gap for MnTiO₃ between Γ–Γ points, while FeTiO₃ and NiTiO₃ have indirect band-gaps at T–Γ(L–Γ) points.

In order to understand the electronic structure of the ATiO₃ candidates, total and atom-resolved density of states were investigated, as presented in Fig. 3. From now, all results are based on PBE0 formalism, which best performs the structural, magnetic and band-gap results for strongly correlated materials proposed in this work. Toward the electronic state contribution to both valence (VB) and conduction band (CB) regions, it was noted that transition metal titanates posses the same pattern of distribution. The general order can be described by a VB predominantly composed by oxygen states highly hybridized with A-site cation valence orbitals (A = Mn, Fe, Ni), while the CB region was mostly based of empty valence orbitals from titanium atoms. However, it was observed on the A-site an increasing cation contribution to the formation of CB moving from Mn to Ni due to sequential filling of 3d orbitals, suggesting a possible electron transfer between 3d orbitals. Further, these results clearly indicate that 3d orbitals for both A-site cation and titanium atoms are divided in low-(t_2g) and high-lying (e_g) energy levels under the trigonal crystal field. However, the splitting energy for Ti(3d) orbitals is higher that of the A(3d) orbitals for MnTiO₃ and NiTiO₃ cases (Fig. 3a and b), which can be explained by the more severe distortional degree observed for TiO₆ cluster comparing to (Mn/Ni)O₆ clusters (see Table S2†), in excellent agreement with theoretical and experimental results.^79,80 Unlike the (Mn/Ni)TiO₃ materials, FeTiO₃ (Fig. 3c) exhibits a larger crystal field splitting for Fe(3d) orbitals than Ti(3d), mainly due to the 3d orbital occupancy. In trigonal systems, like R3c crystalline structure, the distortion on MO₆ octahedral induces the splitting of t_2g levels into non-degenerated a_1g and double-degenerated e^π_g states, while the e_g becomes e^σ_g. Thus, the 3d⁶ orbital occupancy of Fe²⁺ cation induces the electron pairing in non-degenerated a_1g orbital, while e^π_g states remains partially occupied, which allows an internal splitting gap between such states summed to the e^π_g and e^σ_g contribution, resulting in higher crystal field splitting energy. Furthermore, such electronic configuration changes the upper VB region from highly hybridized 3d_xy/yz–2p_y/z states in (Mn/Ni)TiO₃ (Fig. 3a and b) materials to non-bonding 3d_z² states for FeTiO₃ (Fig. 3c), in agreement with the reduced band-gap (Table 2) and stronger antiferromagnetic coupling (Table 1).

Fig. 3 Total and atom-resolved density of states (DOS) for MnTiO₃ (a), FeTiO₃ (b) and NiTiO₃ (c) materials at PBE0. In all cases the Fermi level was set to zero.

Regarding the bonding overlap, Fig. 3 allows to point out that A–O bond interaction is originated from anion-metal overlap between 2–0 eV, 3–0 eV, 2–0 eV energy range of upper VB in MnTiO₃, FeTiO₃ and NiTiO₃, respectively. On the other hand, Ti–O bonding interactions are concentrated along the CB region at 4–6 eV, 3–5 eV, 5–7 eV energy range moving from Mn to Ni. From the combination between bonding overlap it is possible to suggest the existence of A–O–Ti–O–A intermetallic connection, as reported for the ilmenite structure.⁸¹ However, in this case the structural-induced charge ordering occurs in both ab plane and c-axis direction.

3.4 Topological analysis

The topological analysis of the electron density (ρ) is a powerful tool to investigate the electronic properties associated with the M–O bond paths in crystalline materials. In this study, it was focused on the properties of ρ(r) at A–O (A = Mn, Fe, Ni) and Ti–O BCPs, as presented in Table 3. A simple discussion for bond interactions within the framework of the QTAIM may be customarily performed in terms of the dichotomous classification based on the sign of ∇²ρ, which is equal to 4 × (2G(r) + V(r)), where G is the positive definite electron kinetic energy density and V is the electron potential energy density (which is always a negative quantity). Thus, the sign of ∇²ρ reflects the dominant energy contribution on the local expression of the virial theorem. The Laplacian of ρ is given by the sum of the three principal curvatures of the electron density ρ, where λ_1,2 are the negative curvatures and define, through their associated eigenvectors, a plane perpendicular to the bond path at BCP, while λ₃ is positive and its associated eigenvector is tangent to the bond path. Resuming, ∇²ρ expresses the competition between charge accumulation toward the bond path (λ_1,2) for shared-shell (covalent and polar) bonds, and the contraction of electronic charge within the atomic basins (λ₃) for closed-shell (ionic, H-bonds and vdW) interactions.⁷³ However, from Sc to Ge the outermost N-shell of charge depletion and concentration is missing in the atomic Laplacian description and the ∇²ρ is always positive for 4th-row atoms (including Ti, Mn, Fe, Ni), making the dichotomous classification inadequate.⁸²

Table 3 Electron charge density (ρ × 100), Laplacian (∇²ρ × 100), Hessian eigenvalues (λ_n × 100), ellipticity, and Bader's charge obtained by topological analysis in the framework of quantum theory of atoms in molecules for ATiO₃ (A = Mn, Fe, Ni) multiferroic materials. All results are in atomic units. R_x correspond to the atomic d_BCP for M–O (x–y) interactions

		Topological properties									Bader's Charge
		dM–O	Rx	Rx/dM–O	ρ	∇^2ρ	λ1	λ2	λ3	ε	A	Ti	O
MnTiO3	Mn–O	2.116	1.047	0.495	6.2	30.0	−7.7	−7.6	45.3	0.009	1.559	2.402	−1.322
	Ti–O	1.873	0.942	0.503	7.2	31.2	−10.3	−10.2	51.7	0.010
FeTiO3	Fe–O	2.087	1.031	0.495	6.5	32.3	−8.5	−8.2	49.1	0.031	1.493	2.400	−1.297
	Ti–O	1.867	0.939	0.503	6.9	29.6	−9.7	−9.7	48.9	0.007
NiTiO3	Ni–O	2.042	0.988	0.517	6.6	38.4	−7.3	−6.8	52.5	0.078	1.376	2.411	−1.260
	Ti–O	1.856	0.934	0.503	6.8	28.7	−9.6	−9.5	47.8	0.010

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Concerning the BCP distance, obtained R_x/d_M–O relations for Mn–O and Fe–O bond interactions indicate that the critical point is almost equidistant to the involved atoms, whereas for Ni–O is dislocated toward oxygen, suggesting a negative charge population for nickel atoms according to the increased bonded radius. On the other hand, reported R_x/d_M–O values for Ti–O are invariant along the transition metal series, clarifying that the A-site cation modification induces a local charge disorder in the ATiO₃ materials. Furthermore, it was observed that the BCP electron density (ρ) increases at A–O bonds moving from Mn to Ni, while reduces for Ti–O, indicating that oxygen atoms prefer to localize their valence electrons in the AO₆ cluster at the expense of TiO₆, according with the increase of the A-site cation electronegativity.

Then, the integral atomic basin properties in the framework of QTAIM were calculated, as presented in Table 3. Theoretical results show that the A-site cation net charge decreases moving from Mn to Ni, in accordance with the increased BCP electron density (ρ – Table 3), suggesting a higher covalent character for A–O bond interactions. Meanwhile, the Ti charge remains almost constant along the series, enabling to point out that the A-site cation replacement has a primary effect on oxygen atoms that withdraw an almost constant number of electrons from Ti. Such statement allows to interpret the effect of A-site cation on the charge balance through a local disorder that modifies, mainly, the A–O bond character, which is responsible for several properties of the ATiO₃ candidates, e.g. the antiferromagnetic order enhancement along the transition metal series from the extended 3d–2p overlap.

Further, we classify the A–O and Ti–O bond interaction using the integral atomic basin properties (charge) obtained from Bader's theory. For this purpose, we use an independent coordinate through a charge-transfer scale proposed by Mori-Sanchez and co-workers, as follows:⁸³

N represents the number of involved atoms, q_b and q_o correspond to the net charge from Bader's analysis and nominal oxidation state in fully ionic picture (A²⁺, Ti⁴⁺ and O²⁻), respectively. Mori-Sanchez and co-workers developed and applied such tool to classify a large number of crystalline materials as ionic, covalent or metallic.⁸³ In this study, we use the charge-transfer scale to classify both A–O and Ti–O bond interactions, comparing the general trends with theoretical analysis of ρ (Table 3). The obtained c parameter for A–O bond interaction are 0.68 (Mn), 0.66 (Fe) and 0.64 (Ni), suggesting an increased covalent character along the series in agreement with the reported values of the electron density (Table 3). For Ti–O bonds the c parameters were calculated as 0.65 (Mn), 0.64 (Fe) and 0.63 (Ni), indicating an enhancement of the covalent character along the series, also for A–O bonds. Such effect can be explained by the charge balance spread along the A–O–Ti–O–A intermetallic connection, once the Ti charge is stable along the series and the A-site cation replacement reflects in less negative oxygen anions. The calculated charge-transfer constant (c) for the entire crystal are in 0.64–0.67 range, proving the existence of polar component along the M–O interactions, as expected for ferroelectric materials.

4 Conclusions

Density functional theory calculations employing local (LDA), semilocal (PBESol), pure (B3LYP and PBE0) and dispersive augmented (B3LYP+D and PBE0+D) hybrid functional were carried out to investigate the structural, magnetic and electronic properties of ATiO₃ (A = Mn, Fe, Ni) multiferroic materials at LiNbO₃-type structure.

Regarding the role of exchange–correlation functionals on the description of multiferroic properties, local (LDA) and semilocal (PBESol) formalism fails to describe the structure, as well the antiferromagnetic ordering and semiconductor behavior of ATiO₃ materials. Indeed, hybrid DFT/HF functionals are better for all investigated properties of the strongly correlated ATiO₃ materials. However, the inclusion of dispersive effects at hybrid Hamiltonians induces higher deviations when compared to the pure one. Thus, PBE0 best performs the investigated properties along the employed exchange–correlated functionals. Further, comparing such results with reported DFT+U is possible to argue that hybrid functionals overcome the Hubbard corrected DFT formalism, widely used for multiferroic investigations.

Calculated lattice parameters shown that unit cell volume for ATiO₃ (A = Mn, Fe, Ni) materials is controlled by the A-site cation ionic radius. Following such behavior, the AO₆ cluster becomes more ordered along the transition metal series in agreement with the 3d orbital contraction resulting from the increased orbital occupancy. However, the distortion degree of TiO₆ increases, suggesting the enhancement of SOJT effect – a fundamental stabilization criterion of polar structures as R3c polymorphs.

The magnetic properties were evaluated by the A²⁺ next neighbors exchange coupling constant. All investigated transition metal titanates are antiferromagnetic following the GKA rules through a virtual electrons transfer between 3d orbitals of the A²⁺ cation and oxygen 2p states. Further, we verify the dependence of J_AA as regard the covalent character of A–O–A bond interactions, which increases moving from Mn to Ni resulting in more antiferromagnetic materials.

Electronic properties section clarifies the semiconductor behavior of ATiO₃ (A = Mn, Fe, Ni) multiferroic materials with band-gap around 3.3–5.2 eV. Furthermore, the nature of band-gap excitation was rationalized as direct for MnTiO₃, while (Fe,Ni)TiO₃ materials posses indirect excitation nature. Regarding the A-site cation effect on band-gap, the calculated values increase along the transition metal series in accordance with the 3d orbital occupancy and crystal field splitting (Δ_o). Such effects were deeply investigated from DOS analyses showing that both A-site and Ti cations 3d orbitals are divided in a_1g, e^π_g and e^σ_g levels, being the Δ_Ti-3d higher than Δ_A-3d due to the octahedral distortion degree. Bonding overlap was also observed in DOS results, suggesting the existence of A–O–Ti–O–A intermetallic connection, as widely reported for the ilmenite polymorph. Therefore, we can argue that hybrid HF/DFT formalism are the most suitable choice to investigate the strongly correlated multiferroic candidate, especially to design new candidates in silico.

Topological analysis of ρ and Bader atomic charges reveal the main effect behind the A-site cation replacement along the ATiO₃ candidates, which corresponds to a charge concentration in the AO₆ clusters at the expense of TiO₆, following the electronegativity trend along the transition metal series. Such effect was discussed as a local charge disorder, which has a primary effect on A–O bond interactions, being responsible to control several properties of LiNbO₃-type materials.

In this way, all computed parameters are useful to understand the nature of intriguing properties of multiferroic materials class. Moreover, the systematic exchange–correlation functional investigation validates the mightiness of DFT plus hybrid functionals to successfully predict new candidates in this field.

Acknowledgements

The authors acknowledge UEPG, Capes and Araucaria Foundation for financial support.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra21465g

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