Ti, V, Cr Fe ) Al

L T Le

J Goniakowski

C Noguera

Properties of mixed transition metal oxides: MM'O 3 in corundum-type structures (M, M' =

The growing interest of modern technologies for oxide materials is in part due to the possibility of their doping and/or mixing, which often leads to artificial compounds with tunable properties, such as, e.g., the gap width or the band edge positions. However, engineering of mixed oxide materials requires a better fundamental understanding of their structural, electronic, and ordering properties, well beyond low doping levels, and with a reference to their parent materials. In the present work, a series of mixed MM'O3 oxides (M, M' = Al, Ti, V, Cr, Fe) in three corundum-type structures: ilmenite, LiNbO3, and C is studied by means of first principles DFT+U simulations. We find that, regardless the precise atomic structure, the local structural and electronic characteristics of most of the compounds are very close to those of their parent corundum M2O3 oxides. The two noticeable exceptions are TiVO3 and TiFeO3, for which the structural, electronic, and magnetic characteristics are consistent with a mixing-induced change of the cation oxidation states. We show that this actual oxydo-reduction process can be rationalized by analyzing the relative band edge positions of the parent oxides. The formation energies of these mixed oxides correlate well with the experimental evidence and, within a mean field approximation, allow predicting the thermodynamics of solid solutions M1-xM'xO3 at finite temperature.

I. INTRODUCTION

Oxides are the most abundant compounds in our natural environment and, since long, have been used in catalysis, electronics, optoelectronics, or as thermal or electrical insulating barriers. Nowadays, modern technologies attempt to fine tune their properties by means of nano-structuration or doping, producing "artificial" oxide compounds of required characteristics, which often have no natural counterparts. For example, doping ZnO by magnetic atoms allowed obtaining magnetic semiconductors 1,2 . Mixing two oxides has often been considered as a way to engineer the band gap, as exemplified by the Zn 1-x Mg x O alloy, which is a transparent material active in the ultraviolet region of the optical spectrum with applications in optoelectronics 3 , or by Ti 1-x Sn x O 2 alloy 4 . Similarly, doping perovskites (e.g., BaTiO 3 ) with iron enables engineering the band edge positions, which makes these materials useful as anodes in photo-assisted water-electrolysis 5 .

The interest in mixed oxide compounds has called for a better understanding of their structural, electronic, and ordering properties, not only at low doping but also in a wider composition range. In the past, several studies of phase diagrams of mixed rocksalt 6,7 or corundum 8 oxides have been performed, based on pair-potential approaches. First principles simulations have been mainly devoted to studies on the dilute limit, where extremely few -often a single -doping cation or anion are introduced in the repetitive unit cell [9][10][11][12] . The focus in that case is principally on the local lattice distortions around the impurity, the position of the defect electronic states in the band gap, and the resulting magnetic structure.

The goal of the present study is a first principles analysis of the strong doping limit, which has hardly been tackled in the past 13 , especially in the case of multi-valent cations, for which changes of the oxidation state with respect to the pure parent compounds may occur. To this goal we have considered a series of corundum MM'O 3 transition metal mixed oxides (M, M' = Ti, V, Cr, and Fe), which enables establishing links between the structural, electronic, and energetic characteristics of the mixed compounds and the corresponding parent oxides. The series contains ilmenite TiFeO 3 , commonly found in metamorphic and igneous rocks 14 , but also TiVO 3 , which does not naturally exist but has been recently synthesized as a supported thin film 15 . As to broaden the perspective over a case of a larger lattice mismatch, larger bond ionicity, and sp cation character, we also included the TiAlO 3 compound, obtained by mixing corundum Al 2 O 3 and Ti 2 O 3 .

The paper is organized in the following way. After a description of the computational details (Section II), in Section III, we first summarize the reference data obtained for the pure parent M 2 O 3 oxides and then present our findings on the structural and electronic properties of MM'O 3 in three different bulk atomic structures. A discussion follows (Section IV) highlighting the physical origin of the behavior displayed by these mixed oxides, before a conclusion.

II. COMPUTATIONAL DETAILS

DFT calculations were performed with the Vienna Abinitio Simulation Package (VASP) 16,17 using the Projector Augmented Wave (PAW) method 18,19 to represent the electron-core interaction and a 400 eV energy cutoff in the development of Kohn-Sham orbitals on a planewave basis set. Transition metal (TM) 3p states were systematically considered as semi-core states. Dispersioncorrected (optB88-vdW) [20][21][22] exchange-correlation func- tional was employed, within the DFT+U approach proposed by Dudarev 23,24 . We have used U values close to those reported in the literature: U = 1 eV for Ti 2 O 3 25 , U = 1.7 eV for V 2 O 3 26,27 , and U = 3 eV for Cr 2 O 3 and Fe 2 O 3 28 . All calculations were spin-polarized and the relative stability of simple non-magnetic (NM), ferro-magnetic (FM), and alternative anti-ferromagnetic (AF) solutions was systematically tested. Ionic charges were estimated with the partition scheme proposed by Bader 29,30 and magnetic moments were obtained by integration of the spin density within the Bader's volumes. Atomic configurations were plotted with VESTA 31 .

Compounds of the MM'O 3 stoichiometry may adopt the ilmenite, lithium niobate (LiNbO 3 ) or perovskite structures, depending upon thermodynamic conditions. For example, ilmenite is the most stable TiFeO 3 polymorph at ambient pressure and temperature. It transforms into the lithium niobate phase at elevated temperature and into the perovskite phase at high pressures 32 . Targeting principally the ambient pressure conditions, we have focused on mixed MM'O 3 oxides in corundum-type structures and have considered three different cation orderings, Fig. 1. In the ilmenite structure M and M' cations are ordered into pure M or M' layers alternating along the hexagonal c axis, while in the lithium niobate structure they form mixed {M,M'} layers. Moreover, in this latter case, the short cation-cation bond along the c axis may involve either dissimilar (genuine LiNbO 3 structure) or similar (structure labeled C in the following) cations. The C structure, characterized by two inequivalent anions, has already been discussed in the case of MnTiO 3 compound 33 , but, to our knowledge, has not yet been observed experimentally.

An hexagonal corundum unit cell containing six formula units was used in the simulations of all parent and mixed oxides and the sampling of its Brillouin zone was performed with a Γ-centered (8 × 8 × 3) Monkhorst-Pack mesh 34 .

The formation energy of the mixed compounds with respect to their parent oxides is calculated as:

E form = E MM O3 - E M2O3 + E M 2 O3 2 (1)
where E M2O3 , E M 2 O3 and E MM O3 are the total energies of one formula unit of bulk M 2 O 3 , M' 2 O 3 , and MM'O 3 , respectively.

III. RESULTS

Before presenting the computational results on the mixed MM'O 3 compounds, we first briefly summarize the reference data on the corresponding pure parent M 2 O 3 oxides (Section III A). For the sake of clarity, we then provide a comprehensive description of the mixed compounds in their most stable ilmenite structure (Section III B), before highlighting the principal similarities and differences between the three alternative structures (Section III C).

A. Pure M2O3 oxides

The structural (lattice parameters, inter-atomic distances) and electronic properties (charges, band gaps, magnetic moments and structures) of the five corundum parent oxides M 2 O 3 (M = Al, Ti, V, Cr, Fe) are summarized in Tab. 1 and their ion-projected densities of states (LDOS) are given in Fig. 2. TABLE 1. Calculated properties of corundum M2O3 oxides (M = Al, Ti, V, Cr, Fe): lattice parameters a and c ( Å), cation-oxygen bondlengths dM-O ( Å), Bader charges (e) on cations QM and anions QO, cation magnetic moment µM (µB), gap G (eV), and ground state magnetic ordering MS (when two labels are given, the first one refers to in-plane coupling, and the second to inter-plane coupling along the short cation-cation bond In all cases, the overall agreement between the calculated and experimental characteristics is fairly satisfac- tory. In particular, the calculated lattice parameters of all oxides differ by less than 2% from the experimental ones 35,36 . Al 2 O 3 is correctly found as a non-magnetic charge-transfer insulator, with the top of the valence band (VBM) essentially made of oxygen states, and the bottom of the conduction band (CBm) made of aluminum states, but its gap suffers from the usual DFT underestimation. Cr 2 O 3 and Fe 2 O 3 are correctly found as anti-ferromagnetic semiconductors, with a mixed chargetransfer-Mott-Hubbard character 37 . The states at the top of their VB have an hybridized oxygen-cation character (stronger in Fe 2 O 3 ), while the bottoms of their CB are mostly cationic. Thanks to the DFT+U correction, their gaps are only little underestimated. The cation magnetic moments reveal a high spin state (three and five d electrons, respectively), corresponding to the formal oxidation state +3. Their spin orderings are in agreement with experiment, i.e. G-type AFM for Cr Ti 2 O 3 is a spin-paired metal at high temperature and displays a transition (MI) towards a Mott-Hubbard semiconducting state around 470 K, without modification of its rhombohedral structure and without apparition of long range magnetic order 37,38 . In its low temperature phase, the opening of a small gap (0.1 eV) is assigned to the formation of Ti-Ti pairs along the short bond parallel to the c axis, which shifts the bonding a g state below the conduction band. Our calculated non-magnetic metallic structure of Ti 2 O 3 is fully consistent with this picture. There exists a filled Ti a g bonding state just below the Fermi level, consistent with the Ti +3 oxidation state. However, since the calculated Ti-Ti distance (2.63 Å) is slightly larger than the experimental one at low temperature (2.58 Å36 ), the a g bonding-antibonding gap is slightly too small in our simulation.

V 2 O 3 has a paramagnetic metallic phase at high temperature and displays a MI transition of the Peierls-Mott type at 150 K towards a monoclinic anti-ferromagnetic Mott-Hubbard insulating state with an optical gap of 0.5-0.6 eV 37 . Polarization-dependent x-ray absorption measurements 39 indicate that the spin is S=1 in both phases. Consistently with experiments, and despite the imposed corundum structure, we find a G-type antiferromagnetic semi-conducting ground state. The two V filled states below the Fermi level are consistent with the +3 oxidation state of vanadium.

In summary, despite the approximate character of the DFT+U approach, the present results account correctly for both the structural and electronic characteristics of the M 2 O 3 oxides under consideration. Importantly, while the band gaps suffer from some underestimations, the calculations reproduce satisfactorily the character of the electronic structure in the Fermi level vicinity as well as the experimental magnetic orderings.

B. Mixed MM'O3 compounds: Ilmenite structure

Table 2 summarizes the structural, electronic, magnetic, and energetic properties of the series of mixed transition metal oxides in the ilmenite structure. Their projected densities of states are displayed in Figs. 3 and4.

In the following we will systematically refer to M and M' as the first and the second cation in the MM'O 3 formula, respectively.

Based on these results, the mixed compounds can be split into two groups, depending on how much their struc- tural and electronic characteristics differ from those of their parent materials. While in most cases the mixing induces relatively small modifications, TiVO 3 and TiFeO 3 stand out by a substantial change of their local properties. We will see in the following that their structural and electronic characteristics concomitantly provide a clear evidence of a change of cation oxidation states.

From the structural point of view, mixed oxides experience small elastic distortions with respect to their parents. The largest relative difference of lattice parameters a and c is equal to 7% between Al 2 O 3 and Ti 2 O 3 , but does not exceed 2% between the transition metal oxides. We note that, while the inter-atomic distances 2. Properties of mixed MM'O3 compounds in the ilmenite structure (M, M' = Al, Ti, V, Cr, Fe): lattice parameters a and c ( Å), Bader charges (e) on cations QM , Q M and anions QO, gap G (eV), magnetic moments µM , µ M (µB), magnetic ordering MS (the three shortest distance magnetic couplings are given: in-plane, inter-plane along the short cation-cation bond, and inter-plane between second neighbors), and formation energy E form (eV/formula unit) with respect to pure corundum parents.

Ilmenite

Al 3+ Ti 4+ V 2+ Ti 3+ Cr 3+ Ti 4+ Fe 2+ V 3+ Cr 3+ V 3+ Fe 3+ Cr 3+ Fe 3+
From the electronic point of view, Bader oxygen charges Q O (MM'O 3 ) are systematically very close to the average of their parent oxides

(Q O (M 2 O 3 ) + Q O (M' 2 O 3 ))/2.
Since in the ilmenite structure each oxygen is bound to three M and three M' cations, this may suggest that there is a negligible change of iono-covalent character of the cation-oxygen bonds upon formation of the mixed oxide. However, while consistently with this picture, the Bader charges of cations Q M , Q M change little with respect to their parents in most of the mixed compounds, this is not the case for TiVO 3 or TiFeO 3 in which a noticeable increase of the Ti charge and a decrease of the Fe and V ones take place. Although the Bader charges cannot straightforwardly give information on the formal charges, the large δQ M values corroborate the change of the oxidation states in these two oxides. As a consequence, the relationship Also the values of the cation magnetic moments are consistent with the distinct character of these two materials. Indeed, in VCrO 3 , VFeO 3 , and CrFeO 3 , µ M and µ M differ little from those in the parent oxides. In TiCrO 3 and TiAlO 3 , the Ti magnetic moment of the order of 1 µ B is consistent with the presence of a single electron (Ti +3 state) and the absence of the Ti-Ti paring which exists in Ti 2 O 3 . Tab. 2 indicates the nature of the in-plane and inter-plane magnetic couplings. The in-plane Ti-Ti, Cr-Cr (except in VCrO 3 ) and Fe-Fe magnetic coupling is ferromagnetic and the V-V one is anti-ferromagnetic, similarly to those in the parent oxides for V, Fe, and Cr in the case of VCrO 3 . Inter-plane V-Fe, Cr-Fe and V-Cr are anti-ferromagnetic while the Ti-Cr one is ferromagnetic, similar to the V-Cr along the short bond parallel to the c axis. At variance, in TiVO 3 and TiFeO 3 , the Ti magnetic moments are close to zero and the magnetic moments of V and Fe are significantly changed compared to those in V 2 O 3 and Fe 2 O 3 , congruent with an electron transfer from Ti towards V and Fe in these two compounds. The vanadium magnetic moment has increased by nearly one Bohr magneton, as expected from the addition of one electron in this strong Mott-Hubbard oxide. The change in iron magnetic moment is smaller but its similarity with the value obtained in rocksalt FeO within the same simulation set-up points towards an actual Fe 2+ state (see also Tab. 5 in the appendix). Due to the odd number of Fe layers in our simulation cell, we were not able to reproduce the full TiFeO 3 magnetic structure with alternating positive and negative magnetic moments in the Fe layers, as experimentally determined in Reference 41. However, the ferromagnetic in-plane order is well reproduced.

Q O (MM'O 3 )∼ (Q O (M 2 O 3 )+Q O (M' 2 O 3 ))/2
The scenario deduced from the changes of atomic structure, ionic charges, and cation magnetic moments is further validated by the LDOS characteristics, Figs. 3 and4. Indeed, in VCrO 3 and VFeO 3 the cation projected DOS are close to those of their parent materials and in CrFeO 3 it is nearly a rigid superposition of those of Cr 2 O 3 and Fe 2 O 3 . In TiAlO 3 , a well-defined isolated Ti majority spin peak exists below Fermi level filled by an unpaired electron (Ti +3 state). An analogous Ti peak can also be identified just below E F in TiCrO 3 , where however it is strongly hybridized with Cr states. At variance, the LDOS of TiVO 3 and TiFeO 3 are substantially modified with respect to their parent oxides. The most important change concerns the Ti-projected DOS, in which the peak located just below E F in Ti 2 O 3 is shifted above the Fermi level and depopulated, thus indicating a change from 3+ to 4+ oxidation state. Si-multaneously, the weight of V and Fe states just below E F is enhanced, consistent with a change from 3+ to 2+ oxidation state. We note that the present assignment of Ti 4+ and Fe 2+ oxidation states in TiFeO 3 is in agreement with a previous simulation of ilmenite 42 .

As far as formation energies are concerned, Tab. 2, three qualitatively different behaviors can be defined: E form is large and positive in TiAlO 3 , large and negative in TiVO 3 and TiFeO 3 , and small (positive or negative) in the remaining mixed oxides. The large and positive E form of TiAlO 3 is likely driven by the large lattice misfit and the large difference of bond iono-covalency between Al 2 O 3 and Ti 2 O 3 . At variance, the two mixed transition metal oxides for which the change of the cation oxidation state takes place are characterized by strongly negative mixing energies.

Before concluding this description, let us stress two more points. First, TiVO 3 and TiFeO 3 display large and negative formation energies not only with respect to their M 2 O 3 parent oxides, but also with respect to the TiO 2 and VO or FeO oxides (-0.49 and -0.20 eV per formula unit, respectively, TiO 2 , VO and FeO being simulated in their rutile and rocksalt structures). Second, ilmenite TiFeO 3 indeed exists as a naturally occurring mineral, at the origin of the generic name 'ilmenite' given to ternary oxides which crystallize in its structure. To our knowledge, the naturally existing mixed Ti-V oxide is Berdesinskiite TiV 2 O 5 (Ti and V in +4 and +3 oxidation states, respectively) 35 rather than TiVO 3 , but a corundum-structured thin film of TiVO 3 has recently been synthesized on c-cut sapphire 15 , for which a strong contribution of Ti 4+ was evidenced by X-ray photo-electron spectroscopy. Finally, vanadomagnetite V-Fe oxide compounds exist, but only at small vanadium concentrations. The most stable ordered ternary mineral containing these two cations is Coulsonite FeV 2 O 4 in which Fe and V are in +2 and +3 oxidation states, respectively 35 .

In summary, despite the arbitrary character of the projection scheme employed in the evaluation of atomic charges, magnetic moments, and LDOS, our computational results clearly reveal a different behavior of TiVO 3 and TiFeO 3 as compared to the other mixed compounds upon consideration. Indeed, while in most cases the changes induced by mixing are small or even negligible, the local structural, electronic, and magnetic characteristics of these two compounds differ substantially from those of their parent oxides. The mixing-induced modifications concomitantly point towards an increase of the Ti oxidation state (3+→ 4+) and a simultaneous decrease of those of V and Fe (3+→ 2+). This sound modification of the electronic structure correlates with a pronounced tendency for mixing (large negative E form ), and is consistent with the experimental evidence.

C. Mixed MM'O3 compounds: LiNbO3 and C structures

The structural, electronic, and magnetic characteristics of the mixed oxides in the two alternative LiNbO 3 and C-structures are systematically very close to those reported for the ilmenite structure. The full results are thus moved to the Appendix, Tabs. 6 and 7, while, in the following, we restrict the presentation to the most pronounced differences and focus more particularly on the formation energetics.

Indeed, compared to ilmenite, from a structural point of view, in most compounds, the lattice parameters a and c of the three structures differ by less than 0.02 Å and 0.1 Å, which corresponds to relative differences of less than 1% and 2%, respectively, and all cation-oxygen distances d M -O and d M -O differ by less than 0.05 Å only. Similarly, the electronic characteristics are very similar -the anion and cation charges and the cation magnetic moments are nearly identical (to within 0.02 e and less than 0.1 µ B ), and the LDOS in the vicinity of the Fermi level are practically the same. Consequently, the mixing-induced changes of the cation oxidation states reported for TiVO 3 and TiFeO 3 in the ilmenite structure also take place for these two compounds in the LiNbO 3 and C structures.

The unique noticeable dissimilarity between the three structures concerns TiCrO 3 , for which δc = 0.25 Å (relative difference of 2%), δQ Cr = 0.08 e and δµ Cr = 0.6 µ B . The characteristics of the mixed compound in the LiNbO 3 and C structures are visibly closer to those of the parent Cr 2 O 3 and Ti 2 O 3 , while the ilmenite structure is characterized by a more pronounced modification. By inspecting the projected densities of states of TiCrO 3 in the three structures, Fig. 5, this difference can be assigned to a more pronounced hybridization between Cr and Ti in the peak just below Fermi level.

As far as the relative stability of the three structures is concerned, Tab. 3 shows that the specificities of formation energies reported for the ilmenite structure are preserved: E f orm are large and positive for TiAlO 3 , large and negative for TiVO 3 and TiFeO 3 and smaller (either positive or negative) for all other mixed oxides. Ilmenite is the most stable polymorph, characterized by the most negative or the smallest positive formation energies. CrFeO 3 constitutes the unique exception, with the LiNbO 3 structure being the most stable. We note however that energy differences between the two most stable structures may be as small as 0.02 eV per formula units (TiVO 3 , VCrO 3 , CrFeO 3 ), likely close to the precision of the present estimation. Similarly, the LiNbO 3 structure is more stable than the C structure in all mixed transition metal oxides. The reverse is true in TiAlO 3 only, where the formation of short Ti-Ti bonds along the c-axis enables a small energy gain. Despite small differences between the various polymorphs, the formation energies of the mixed oxides follow a general trend. In the case of TiVO 3 and TiFeO 3 , in which a redox reaction takes place, the large negative mixing energy is mainly driven by the gain of band energy, associated with electron transfers from Ti states (higher on the energy scale) towards V and Fe states (lower on the energy scale) and the associated gain of electrostatic energy. In other mixed transition metal oxides, in the absence of such a strong stabilizing effect, the formation energies are much smaller (in absolute values) and mostly positive as a result of the necessary adjust-ments of the atomic structure with respect to the local environments in the parent oxides, but also due to the incompatibilities between the parent magnetic structures.

In TiAlO 3 , a system in which no redox reaction takes place, the large difference in cation ionic radii between Ti 3+ and Al 3+ induces strong structural distortions upon mixing, leading to a large positive mixing energy. Formation energies of the three structures enable an estimation of effective short-range in-plane and inter-plane cation-cation interactions. The shortest in-plane cationcation distance d 1 is of the order of 3 Å (three neighbors), and is much smaller than the second-neighbor one (≈ 5 Å). The shortest inter-plane distance d 2 is of the order of 2.8-2.9 Å (a single neighbor). It is followed by secondneighbor inter-plane distances d 3 in the range 3.4-3.7 Å (nine neighbors). By introducing effective mixing parameters

W i = 2V M M i -V M M i -V M M
i associated to interaction energies V i between cation pairs at distances d i , it is possible to write the formation energies of the three structures as:

E f orm ilm = W 2 + 9W 3 E f orm LN O = 3W 1 + W 2 + 3W 3 E f orm C = 3W 1 + 6W 3 (2)
The values of W i deduced from the DFT formation energies of the mixed compounds are given in Tab. 3. Positive values of W correspond to preferential formation of pure M-M and M'-M' bonds and thus to a tendency for phase separation, while negative W favor mixed M-M' bonds and thus a tendency for mixing.

Let us first note that the three parameters W i display a similar behavior along the considered series. They increase progressively from more negative (TiFeO 3 ), through close to zero (VCrO 3 ), up to large positive (TiAl 3 ) values. This behavior parallels that of E f orm and shows that the latter is due to a concomitant effect of the first, second, and third neighbor cation-cation interactions, rather than a competition between them. Indeed, if W 2 tends to be in most cases somewhat more negative than W 1 and W 3 , its overall effect remains small because it concerns a single cation-cation pair.

Conversely, parameters W 1 and W 3 play the major role in the overall energetics due to the large number of cation-cation neighbors they are associated with. Indeed, both parameters are negative for TiFeO Finally, if the ilmenite structure is favored in all cases but CrFeO 3 , this preference is relatively small and cannot be easily assigned to a cation-cation interaction between a precise pair of neighbors, but results rather from a complex interplay between in-plane W 1 and out-of-plane W 3 interactions. In this context, one could note that the W i decrease relatively slowly as a function of cation-cation distance d i . This is particularly clear when comparing W 1 (d 1 ≈ 3 Å) and W 3 (d 3 ≈ 3.4 -3.7 Å) and may suggest that taking into account effective interactions between more distant cation-cation pairs could yield a finer description of the mixing energetics.

In summary, the structural and electronic characteristics found for the mixed oxides in the ilmenite structure are preserved in the alternative LiNbO 3 and C ones. Due to the large number of corresponding cation-cation pairs, the oxide mixing characteristics are driven by concomitant effect of in-plane W 1 and inter-plane W 3 effective cation-cation interactions. If ilmenite structure is favored in most cases, this preference cannot be assigned to the interaction between a specific pair of neighbors. This is partially due to a relatively slow decrease of cationcation interaction strength as a function of distance. Finally, the decomposition into pair interactions reveals a different nature of interactions in TiAlO 3 , in which all cation-cation pairs contribute to the particularly strong preference for phase separation.

IV. DISCUSSION

In this last section, we discuss the microscopic mechanisms behind the electronic structure characteristics of mixed compounds and analyze how they can be predicted from those of the parent materials. Furthermore, relying on the mixing energies E f orm and parameters W i , we sketch the expected behavior of mixed oxides at finite temperatures and for compositions different from 1/2.

Change of oxidation states

The analysis of the electronic structure (Section III) has highlighted the existence of strong cation-to-cation electron transfers in some mixed oxides (TiVO 3 and TiFeO 3 ) and its absence in the other compounds. The fact that this result does not depend on the precise ordered structure suggests that it is due to intrinsic properties of the parent oxides and that it can be rationalized by general arguments similar to those used, for example, at semiconductor interfaces. Electron transfers at semiconductor interfaces have been rationalized in the past by aligning the electronic structures of the two parent materials (their bulks or their surfaces) with respect to a common reference energy which, depending on the authors, was the vacuum level in the "electron affinity rule" model 43 , the mean electrostatic potential 44,45 , the valence band positions [START_REF] Harrison | Electronic Structure and the Properties of Solids[END_REF] , the core level positions 47 , the point of zero charge [48][49][50] , or the oxygen 2p states at perovskite-perovskite interfaces 51 .

As in this last case, in the mixed corundum MM'O 3 compounds the oxygen sub-lattice is shared.

We have thus performed an alignment of the oxygen 1s states (estimated by the electrostatic potential at the ionic cores) of the two parent M 2 O 3 and M' 2 O 3 oxides and have quantified their band-offsets with two parameters ∆ While for the former, ∆ 1 > 0 is consistent with the calculated absence of charge transfer in VFeO 3 , the negative ∆ 1 < 0 for the latter is not consistent with the full calculation results on TiCrO 3 . However, we stress that the band alignment of the parent materials only determines the initial state for an electron exchange and neglects all the subsequent exchange-induced processes. Such initial state approximation may be questionable whenever the ∆ i ∼ 0, as illustrated by the present case of Ti 2 O 3 -Cr 2 O 3 .

1 =E CBm (M' 2 O 3 ) -E V BM (M 2 O 3 ) and ∆ 2 =E CBm (M 2 O 3 ) -E V BM (M' 2 O 3 ,
As to further validate our results in VFeO 3 and TiCrO 3 , we have checked to what extent they are sensitive to the values of U F e and U Cr used in the DFT+U approach. Fig. 7 shows that, despite substantial variations of the Fe 2 O 3 and Cr 2 O 3 gap widths, there is no change of sign of ∆ 1 when U varies in a wide range 1 eV < U <6 eV. Interestingly, the formation energies of these two oxides also keep a constant sign.

In order to give a rationale for the values and trends of ∆ 1 and ∆ 2 , let us note that they involve only the difference of oxygen binding energies (BE) ∆ BE = E V BM (M 2 O 3 ) -E V BM (M' 2 O 3 ) and the gap widths G(M 2 O 3 ) and G(M' 2 O 3 ) of the two parent materials, Fig. 6: density of states at the gap edges. This is not the case in the oxides upon consideration. The states at the gap edges are narrow d states, highly localized on cations and associated to a high density of states. A band overlap thus easily induces correlated shifts of whole d states across the Fermi level and their associated population and depopulation by one electron. In that case, an actual oxydo-reduction process takes place and it is legitimate to talk of a change of the cation oxidation states.

∆ 1 = -∆ BE + G(M 2 O 3 ) ∆ 2 = +∆ BE + G(M 2 O 3 ) (3
Finally, let us note that ∆ 1 =E CBm (M' 2 O 3 ) -E V BM (M 2 O 3 ) can also provide an estimation of the band gap in the mixed compounds in which no change of oxidation state takes place. Indeed, we find that ∆ 1 for TiCrO 3 , VFeO 3 , VCrO VBM and an M' character of their CBm. In the case of TiAlO 3 , for which both the VBM and CMm display an Ti character, the zero gap is reminiscent of that in Ti 2 O 3 .

A. Thermodynamics of mixing

The formation energies that we have calculated characterize the stability of mixed M 2-2x M' 2x O 3 oxides at T = 0 K and for equal M and M' concentrations (x = 1/2). However a qualitative discussion of their thermodynamics at finite temperatures and in a wider range of composition is possible.

In the case of Ti 2-2x V 2x O 3 and Ti 2-2x Fe 2x O 3 , it has been found at x = 1/2 that all cations experience a change of oxidation state. At compositions x = 1/2, such a global change will not be possible and Ti 4+ and Ti 3+ cations, as well as M' 2+ and M' 3+ cations (M' = V or Fe) will coexist. Taking into account the large negative ilmenite TiVO 3 and TiFeO 3 formation energies, this will very likely lead to a phase separation between Ti 4+ V 2+ O 3 and Ti 3+ V 3+ O 3 and between Ti 4+ Fe 2+ O 3 and Ti 3+ Fe 3+ O 3 .

Ordered MM'O 3 mixed oxides with positive formation energies (VCrO 3 , TiCrO 3 , CrFeO 3 , and TiAlO 3 ) are not stable at zero temperature, but they may be stabilized at finite temperatures as disordered (substitutional) solid solutions thanks to disorder effects. Retaining only the entropy contribution of configurational disorder in the Gibbs free energy of formation, the latter reads:

G f orm = E f orm -4k B T ln 2 (4)
for a x = 1/2 solid solution (k B is the Boltzmann constant). G f orm becomes negative when T > T max = E f orm /(4k B ln 2). Based on the calculated values of E f orm , T max is equal to 3, 200, 250, 1300 K, for VCrO 3 , TiCrO 3 , CrFeO 3 , and TiAlO 3 , respectively. Solid solutions of these mixed oxides at x = 1/2 can thus be stable above T max . Moreover, for the same family of mixed oxides within a mean field approach, one can use the zero of the Gibbs free energy :

G f orm = 4x(1-x)E f orm -4k B T (x ln x+(1-x) ln(1-x))
(5) to estimate above which temperature T a solid solution of composition x can be stable. Figure 8 displays the resulting phase diagram, with a miscibility gap below the T (x) curve, which is symmetric with respect to x = 1/2 and which disappears at x = 1/2 for T > T max . This approximate treatment suggests that V 2-2x Cr 2x O 3 , Ti 2-2x Cr 2x O 3 and Cr 2-2x Fe 2x O 3 solid solutions should form in the whole composition range already at ambient temperatures. We note that the present estimation for Cr 2-2x Fe 2x O 3 is in a good agreement with the results of an existing computational study on the mixing thermodynamics in the corundum Fe 2 O 3 -Cr 2 O 3 system 8 .

V. CONCLUSION

Within the DFT+U approximation, we have studied a series of mixed MM'O 3 compounds (M, M' = Al, Ti, V, Cr, Fe) in three corundum-type structures: ilmenite, LiNbO 3 , and C. We find that, regardless the precise atomic structure, the local structural and electronic characteristics of most of the compounds are very close to those of their parent corundum M 2 O 3 oxides. The two noticeable exceptions are TiVO 3 and TiFeO 3 , for which the structural, electronic, and magnetic characteristics are consistent with a mixing-induced change of the cation oxidation states. We show that this actual oxydo-reduction process can be rationalized by the relative positions of VBM and CBm of the parent oxides, with respect to a common reference. The mixed oxide formation energies that we find are consistent with experimental evidence and, within a mean field approximation, allow to estimate the thermodynamics of mixing of M 2-2x M' 2x O 3 solid solutions at finite temperature. Aside its direct interest for either the considered bulk oxides, or their thin supported films, the present study provides also a reference understanding for a future analysis of low dimensional and/or nano-scale mixed oxide objects. Indeed, the control of the band off-sets of the parent materials by reducing the dimensionality and/or the size of the systems may enable to design and fabricate mixed oxides of required composition and properties.

VI. APPENDIX

FIG. 1 .

 1 FIG. 1. Bulk unit cells of mixed MM'O3 oxides in the ilmenite (left), LiNbO3 (middle) and C (right) structures. Cations and anions are represented as large blue (light and dark) and small red balls, respectively.

FIG. 2 .

 2 FIG. 2. Projected densities of states in corundum M2O3 oxides (M = Al, Ti, V, Cr, Fe) on anions (blue lines) and cations (red lines in non-magnetic Al2O3 and Ti2O3, red and green lines in anti-ferromagnetic V2O3, Cr2O3, and Fe2O3). Black dashed lines indicate the position of the Fermi level.

  2 O 3 [(AF,AF) in Tab. 1 ] and C-type AFM for Fe 2 O 3 [(FM,AF)].

FIG. 3 .

 3 FIG. 3. MM'O3 (TiAlO3, TiVO3, TiCrO3 and TiFeO3) density of states, projected on cations (red and green lines for M and M', respectively) and oxygen atoms (blue lines) in the ilmenite bulk structure. Black dashed lines indicate the position of the Fermi level.

FIG. 4 .

 4 FIG. 4. Same as Fig. 3 for VCrO3, VFeO3 and CrFeO3.

  d M -O and d M -O remain close to their parent values in most MM'O 3 mixed compounds, d T i-O exhibit a noticeable decrease and d M -O (M'=V, Fe) a noticeable increase in TiVO 3 and TiFeO 3 . According to the behavior of cation ionic radii: 40 r M 4+ < r M 3+ < r M 2+ , such structural modification upon mixing is consistent with an increase of the Ti oxidation state and its decrease for V and Fe. It is further confirmed by a comparison with inter-atomic distances in TiO 2 , VO and FeO (Tab. 5 in the appendix).

TABLE

  in TiVO 3 and TiFeO 3 does not indicate a negligible change of iono-covalent character of the cation-oxygen bonds but rather results from a compensation between those of Ti-O and M'-O bonds.

FIG. 5 .

 5 FIG. 5. Projected densities of states of TiCrO3 [Ti (red), Cr (green), and O (blue and cyan)] in the ilmenite (left), LiNbO3 (middle), and C (right) structures.

3 and TiVO 3 (

 3 large negative E f orm ), and positive for VCrO 3 , TiCrO 3 and CrFeO 3 (positive E f orm ), with values which increase progressively along the series. TiAlO 3 behavior is somewhat different. It is the only case in which a large and positive W 2 is met. Since W 3 is nearly vanishing, those are the values of W 1 and W 2 which explain the large positive E f orm value.

Fig. 6

 6 and Tab. 4. Within this definition, negative values of ∆ 1 and ∆ 2 indicate an overlap between VB and CB of the parent materials, consistent with an electron transfer M→M' and M'→M, respectively.

FIG. 6 .

 6 FIG. 6. Relative positions of the valence band maximum (VBM) and conduction band minimum (CBm) of two parent oxides after alignment of their oxygen 1s levels. Parameters ∆1, ∆2, and ∆BE (see text) are indicated.

)

  FIG. 7. Calculated band gaps of the parent materials G (eV) (top), ∆1 (eV) (middle), and formation energies E f orm (eV) (bottom) for Ti2O3-Cr2O3 (full red) and V2O3-Fe2O3 (dashed blue) systems as a function of UCr,F e (eV).

  FIG. 8. Estimated miscibility gaps for the M2-2xM'2xO3 solid solutions of mixed oxides with no tendency for mixing (E f orm > 0): VCrO3 (blue), TiCrO3 (green), CrFeO3 (brown), and TiAlO3 (red).

TABLE 3 .

 3 Formation energies E f orm (eV per formula unit) of mixed MM'O3 compounds in the ilmenite, LiNbO3, and C structures together with the deduced values of effective cation-cation mixing parameters W1, W2 and W3 (see text) (eV).

		TiFeO3 TiVO3 VFeO3 VCrO3 TiCrO3 CrFeO3 TiAlO3
	Ilmenite -0.79	-0.35 -0.10	0.00	0.04	0.08	0.29
	LiNbO3 -0.73	-0.33 -0.01	0.02	0.16	0.06	0.60
	C	-0.56	-0.28	0.01	0.04	0.19	0.15	0.40
	W1	-0.

TABLE 4 .

 4 Values of the parameters ∆1, ∆2, and ∆BE (eV) deduced from the band structures of the parent oxides M2O3 and M'2O3, after the alignment of their oxygen 1s levels, Fig.

	6.
	M2O3-M'2O3 ∆1 (eV) ∆2 (eV) ∆BE (eV)
	Ti2O3-Fe2O3	-1.1	2.8	2.8
	Ti2O3-V2O3	-1.1	1.5	1.5
	Ti2O3-Cr2O3	-0.2	2.9	2.9
	V2O3-Fe2O3	0.4	1.7	1.4
	V2O3-Cr2O3	1.3	1.8	1.5
	Cr2O3-Fe2O3	1.6	2.7	-0.1
	Ti2O3-Al2O3	2.4	4.1	4.1

TABLE 5 .

 5 Comparison of the bulk properties of transition metal oxides MO, M2O3 and MO2 in the rocksalt, corundum and rutile structures, obtained with the simulation set-up described in Section II: cation magnetic moment (µB) (top) and average M-O bondlength < dM-O > ( Å) (bottom) .

	µ	MO M2O3 MO2
	Ti	0	0	0
	V	2.51 1.75 1.04
	Cr	3.71 2.85 2.31
	Fe	3.60 4.05 3.62
	< dM-O > MO M2O3 MO2
	Ti	2.16 2.06 1.98
	V	2.17 2.04 1.93
	Cr	2.22 2.02 1.91
	Fe	2.19 2.01 1.98

TABLE 6 .

 6 Same as Table 2 for the LiNbO3 structure. The three shortest range magnetic interactions are given in the following order: in-plane MM', inter-plane MM', inter-plane MM/M'M'. .11 1.94;2.12 2.00;2.10 1.89;2.14 1.99;2.08 1.97;2.06 2.00;2.03 d M -O 1.88;2.01 2.06;2.10 2.02;2.05 2.08;2.17 1.99;2.06 1.97;2.20 1.96;2.15 Al 3+ Ti 4+ V 2+ Ti 3+ Cr 3+ Ti 4+ Fe 2+ V 3+ Cr 3+ V 3+ Fe 3+ Cr 3+ Fe 3+

	LiNbO3 TiAlO3 TiVO3	TiCrO3	TiFeO3	VCrO3	VFeO3	CrFeO3
	a	5.05	5.09	5.10	5.17	5.04	5.06	5.07
	c	13.32	14.12	13.84	13.80	13.98	13.97	13.71
	dM-O 2.04;2QM 1.95	2.09	2.00	2.18	1.86	1.92	1.78
	Q M	2.49	1.64	1.69	1.43	1.74	1.68	1.74
	QO	-1.48	-1.24	-1.23	-1.20	-1.20	-1.20	-1.17
	µM	0.8	0.0	0.7	0.0	1.7	1.6	2.8
	µ M	0	2.2	2.7	3.7	2.9	4.1	4.1
	MS	(-,-,AF) (-,-,AF) (AF,AF,F) (-,-,AF) (AF,F,AF) (AF,AF,F)
	E form	0.60	-0.33	0.16	-0.73	0.02	-0.01	0.06
	Config Ti 3+

TABLE 7 .

 7 Same as Table 6 for the C structure. .10 1.98;2.06 2.02;2.09 1.95;2.05 1.99;2.08 1.96;2.06 1.98;2.06 d M O 1.89;1.98 2.04;2.13 2.00;2.05 2.05;2.28 2.00;2.06 1.99;2.15 1.97;2.11 Al 3+ Ti 4+ V 2+ Ti 3+ Cr 3+ Ti 4+ Fe 2+ V 3+ Cr 3+ V 3+ Fe 3+ Cr 3+ Fe 3+

	C-structure TiAlO3 TiVO3	TiCrO3	TiFeO3	VCrO3	VFeO3	CrFeO3
	a	5.05	5.07	5.09	5.20	5.05	5.05	5.07
	c	13.19	14.20	13.92	13.81	13.96	14.01	13.73
	dM-O 2.04;2QM 1.90	2.09	1.97	2.19	1.86	1.90	1.77
	Q M	2.48	1.65	1.73	1.42	1.75	1.67	1.76
	QO	-1.46	-1.25	-1.23	-1.20	-1.20	-1.19	-1.18
	µM	0.0	0.1	0.7	0.0	1.8	1.6	2.8
	µ M	0.	2.2	2.7	3.7	2.9	4.0	4.1
	MS	NM	(-,-,AF) (AF,AF,F) (-,-,F) (AF,F,AF) (AF,AF,F) (F,AF,AF)
	E form	0.40	-0.28	0.19	-0.56	0.04	0.01	0.15
	Config	Ti 3+