Electronic and magnetic properties of the Ti{}_{5}O{}_{9} Magnéli phase

Electronic and magnetic properties of the TiO Magnéli phase

Abstract

Structural, electronic and magnetic properties of TiO have been studied by ab initio methods in low-, intermediate- and high-temperature phases. We have found the charge and orbital order in all three phases to be non-stable, and the formation of Ti-Ti bipolaronic states less likely as compared to TiO. Several quasidegenerate magnetic configurations were calculated to have different width of the band gap, suggesting that the reordering of the unpaired spins at Ti ions might at least partially be responsible for the changes in conductivity of this material.

pacs:
31.15.A-, 71.20.-b, 75.25.Dk, 71.30.+h, 75.47.Lx

I Introduction

For several decades binary transition-metal oxides and related compounds have attracted increasing attention as resistive-switching materials. These are materials which can be switched between a high- and a low-resistance states and are very promising candidates for the next generation non-volatile resistive random access memory (RRAM) . The resistive switching in many of these technologically relevant oxides is claimed to be based on the formation and disruption of highly-conductive filaments, through which the current flow is realized . However, still very little is known about the composition, structure and dimensions of these filaments.

Recent low-temperature conductivity and in situ current-voltage measurements confirmed that the conducting filaments in Pt/TiO/Pt Kwon_2010 (), as well as Fe-doped SrTiO Waser_2010 () are composed of TiO and TiO Magnéli phases and their mixtures. These compounds belong to the homologous series TiO () of Ti oxides with TiO and TiO as the end members. Structurally they are related with rutile: there are adjacent rutile-like chains of edge-sharing TiO octahedra, running parallel to the pseudorutile axis (see Fig.1); the octahedra at the end of each of the chains share their faces with the next pseudorutile chain. With the exception of TiO, these compounds posses a triclinic structure and they are mixed-valence compounds with two Ti (3 electronic configuration) and () Ti () ions per formula unit (in an ionic picture). Several members of this homologous series, as well as the isostructural vanadium Magnéli phases VO, are known to exhibit phase transitions with temperature, at which their conductivity changes drastically (it is the largest for and 5 and reduces for the members with higher ). These transitions were interpreted in terms of the localization of the electrons in the cation sublattice. Indeed, the presence of both Ti and Ti ions provides several possibilities of charge distribution at cation sites, resulting in various charge-ordered states.

Take TiO for example, which undergoes two consecutive phase transitions with temperature: a semiconductor-semiconductor transition at 130 K and a semiconductor-metal transition at 150 K Marezio_1972 (). X-Ray diffraction studies Marezio_1973 () on TiO single-crystals revealed that at low temperatures Ti ions are covalently bonded to form the so-called bipolarons – singlet electron pairs (Ti-Ti pairs). In the low-temperature (LT) semiconducting phase these pairs order so that they occupy alternate pseudorutile chains. In the high-temperaure (HT) phase the pairing is believed to be absent and all the Ti-cations are in the same oxidation state Ti, resulting in further increase of the conductivity and metallic behavior. The nature of the intermediate temperature (IT) phase remains rather unclear, however it is commonly believed that the (semiconductor-semiconductor) transition from LT to IT phase is accompanied with the reordering of the Ti-Ti pairs Marezio_1973 (). It was assumed that in the IT phase bipolaros distribute in Ti-chains with a long-range order that requires a fivefold superstructure (bipolaron liquid state Lakkis_1976 ()). Later, the structure of the IT phase was revisited and suggested to be more complicated with the presence of bipolarons and a long-range order of Ti valences Le_Page_1984 (). Recent photoemission experiments Taguchi_2010 () interpreted the IT phase in TiO as a highly anomalous state sandwitched between the mixed-valent Fermi liquid (HT phase) and charge ordered Mott-insulating phase (LT phase).

Two distinct phase transitions were also observed in TiO, at 128 K and 139 K, accompanied by the conductivity decrease by a factor of 3 (Ref. Marezio_1977, ) (or 50 according to Bartolomew and Frankl Bartolomew_1969 ()). While LT and IT phases are found to be semiconducting, the conductivity in the HT phase increases with increasing temperature, which means that this phase is not a true metal and its nature is not clear. Compared to TiO, for which a stable charge-ordered state was shown to exist at low temperature, the charge order in TiO is rather uncertain, since the number of Ti ions in the pseudorutile chains is odd, preventing the formation of the Ti long-range ordered pairs . X-ray studies performed by Marezio et al Marezio_1977 (), did not reveal any localization of charges and/or formation of covalent bonds between the Ti cations in TiO. However, in order to explain the semiconducting properties of the LT and IT phases, which in TiO are related with the formation of Ti-Ti pairs, they suggested a microdomain structure of the material with Ti mostly present at domain walls, making the detection of the charge localization and Ti-Ti bonds formation by x-ray techniques rather impossible. The observed EPR signal the authors associated with the presence of unpaired Ti. Watanabe Watanabe_2009 () has recently studied the structural changes across the phase transitions in Magnéli phases by means of Raman spectroscopy. He observed two semicondutor-semiconductor transitions, at 137 K and 130 K, and revealed a fine structure of the LT phase, somewhat resembling the IT phase of TiO. He suggested a considerable disorder for the LT phase of TiO and a rather itinerant nature of the 3 electrons in the IT and HT phases as in the HT phase of TiO. The author did not observe any clear tendency as to the nature of the charge ordering in the considered Magnéli phases and concluded that in the compounds with odd like TiO the charge order is unstable.

Figure 1: (color online) The crystal structure of the TiO: spatial view (a), schematic view of the chain structure (b) and two adjacent Ti chains (c). The Ti atoms in the adjacent chains, which run parallel to the pseudorutile c axis, are shown in grey and blue, while oxygen atoms are shown in red. The black dashed line indicates the shear plane.

The effect of pressure on the phase transitions in TiO (=4, 5, 6) Magnéli phases was studied in Ref. Ueda_2002, . In contrast to a commonly accepted bipolaronic picture of the metal-insulator transition in these compounds, which is attributed to the strong electron-lattice coupling, the authors proposed their own explanations of the physics behind the phase transitions. The remarkably enhanced pressure effect on the phase transition temperatures as goes from 6 to 4 the authors explained by a delicate interplay between the electron-lattice coupling and electron correlations. In this respect charge ordering in TiO is explained to be due to a dominant electron correlations with a subsequent bipolarons formation by an electron-lattice coupling. TiO, on the contrary, is believed to be a small polaron insulator due to a prevailing electron-lattice coupling. TiO is suggested to be in between these two extreme cases. The formation of small polarons, localized on one atomic site and randomly distributed over the lattice, explains rather well the unclear charge distribution in higher Magnéli phases, observed in earlier experimental works Marezio_1977 (); Le_Page_1983 (). The substantial Curie contribution, observed in TiO at low temperatures Mulay_1970 (), which can be associated with the presence of upaired localized moments, is also in line with the proposed small polaron picture. A distinct pressure dependence of the resistivity in the HT phase of TiO allowed the authors to assume that the system is just at the crossover region from large to small polarons. Such crossover in the systems with intermediate electron-coupling is often associated with a metal-insulator transition.

As for the magnetic properties, TiO, as well as TiO and TiO phases show antiferromagnetic (AFM) behavior with Néel temperature () of about 130 K Keys_66 (). According to magnetic susceptibility measurements in TiO, there are two transitions which occur at 128 and 138 K, in contrast to TiO with only one transition in magnetic susceptibility . The is very close to the phase transition temperature, which allows one to relate the magnetic ordering with the phase transitions observed in this compound. At low temperatures ( K) the measurements show a Curie behavior, while for the 40 K128 K the temperature-independent susceptibility might be due to Van Vleck paramagnetism. Susceptibility in the HT phase is clearly temperature independent. Based on the measured molar susceptibility in Ti oxides, the effective magnetic moment for TiO was calculated to be of 0.245  and it was concluded that a relatively small percentage of the unpaired electrons (7.4%) contributes to the Curie-Weiss behavior, while a large percentage (92.6%) contributes to its metallic (free electron gas) behavior; the small Weiss constant indicates that these spins do not interact strongly Danley_1972 ().

Although rather well studied experimentally, there is very little known about the mechanism behind the transitions in Magnéli phases theoretically. Up to today, there are only a few DFT calculations on TiO Eyert_2004 (); Leonov_2006 (); Liborio_2009 (); Weissmann_2011 (); Padilha_2014 (), and to our knowledge, they are completely lacking for higher- phases. Therefore, in this work we aimed to develop a fundamental understanding of the mechanisms that underlie the phase transitions in Magnéli phases like TiO, which, due to its high electrical conductivity and chemical stability, has a promising application not only in electrochemical engineering, but even in medicine as a neural stimulation electrode Canillas_2013 (). With this aim we performed DFT calculations in order to illuminate the changes in its electronic structure on a microscopic level and establish relations between the structural, electronic and magnetic properties and their implications for conductive phenomena. We believe that the obtained results will be helpfull in interpreting the physics of resistively switching Ti oxides.

Ii Computational details

The electronic and magnetic properties of the different phases of TiO were studied within the DFT, using the Vienna ab initio Simulation Package (VASP) Kresse1_1996 (); Kresse2_1996 () with projector augmented potentials (PAW) Blochl_1994 (); Kresse_1999 (). Kinetic energy cut-off of 450 eV and k-points mesh was used for the unit cell of 28 atoms. The exchange correlation functional was treated within the generalized gradient approximation (GGA) Perdew_1996 (); Perdew_1997 (). GGA+ approximation in Dudarev’s approach Dudarev_1998 () with applied to states of Ti was used to take into account the electronic correlations.

The structural data for all three phases have been taken from the Ref. Marezio_1977, . The structure optimization was performed only for the LT phase with the aim to find the ground state of the system under study. For the sake of simplicity, we do not consider the spin-orbit coupling in our calculations, and we restrict ourselves to the collinear spins only.

Iii Results and discussion

iii.1 Correlation effects

It is known that oxygen deficient rutile TiO can be transformed into a Magnéli phase, when oxygen nonstoichiometry is sufficiently high ( Bursil_1969 ()). At such concentrations, oxygen vacancies arrange into extended defects, resulting in the shear plane structure (see Fig.1(c)). As a result, the defect states appear in the band gap close to the conduction band minimum, which are occupied by the electrons of Ti ions. Numerous studies on TiO, including recent hybrid functional calculations Janotti_2010 (), have reported on the effect of the electronic correlations on the position of these defect states and the distribution of Ti ions. For TiO or TiO with similar peculiarities of the electronic structure, approach or hybrid functionals were used to correctly account for the correlation effects and were able to reproduce the ground state properties of the phases Eyert_2004 (); Leonov_2006 (); Liborio_2009 (); Weissmann_2011 (); Padilha_2014 (). However, the correlations might be less effective in the phases with larger (i.e., with smaller density of 3 electrons), as it was assumed in Ref. Watanabe_2009, based on the observed lack of the systematic trend in charge ordering in different Magnéli phases.

In order to ilucidate the importance of electron correlations for the charge localization in TiO, we have peformed calculations within GGA+ approach, assuming the unit-cell model for the order/disorder of the Ti and Ti ions. To cover the limits of weak and strong electron interactions, the on-site Coulomb energy parameter was varied within the range  eV. The evolution of the density of electronic states (DOS) with in the LT phase with the experimental structure is presented in Fig.2; these calculations were performed assuming an AFM spin coupling with a zero total magnetic moment per cell.

Figure 2: (color online) The GGA+ spin-polarized DOS, calculated for the LT phase of TiO at different for the AFM spin allignment. The Fermi level is at zero energy (shown by red dashed line). Blue and red shading denotes the majority and minority spins, correspondingly.
Figure 3: (color online) valence charge distribution in the LT phase of TiO: (a) general view with Ti-Ti bonds; (b) zoomed view with Ti-O bonds. The calculated charge density isosurfaces correspond to the Ti gap states. The presented results were obtained at  eV for the AFM spin configuration with . Blue and grey colors distinguish between the Ti atoms in the adjacent pseudorutile chains. Note different orientation of the -like orbitals.

As follows from Fig.2, for  eV the DOS remains metallic with Ti states merged into the conduction band. A small band gap of about 0.1 eV between the occupied and unoccupied states of Ti opens in both spin channels at  eV, and it increases up to 0.7 eV at  eV (see Tab.1). It should be noted that an energy gap of 0.7 eV was also obtained for TiO at  Ry (5.4 eV) in Ref.Weissmann_2011, (the choice of this value of was justified by comparing the total energy difference between the three phases at different with the transition temperatures). The topology of the insulating DOS very much resembles that of the TiO Leonov_2006 (); Liborio_2009 (); Weissmann_2011 (). The well-localized states just below the Fermi level are formed by the orbitals of Ti possessing non-zero magnetic moment. Figure 3 shows the spatial distribution of these localized gap states, calculated at  eV. It is clear that the system is not only charge ordered, but also orbitally-ordered: occupied orbitals at different Ti sites are of and (or ) symmetry, if we define them in the local frame of the octahedra. While the conduction band is mainly contributed by Ti states, the valence band originates from the oxygen states (with a small admixture of Ti states due to Ti-O bonds formation) and it is separated from the gap states by an energy interval, which is much larger than the fundamental gap. This gap decreases with the increasing from around 2.8 eV at to 1.9 eV at  eV. The width of the split-off -band is decreasing as the electron localization increases with the increased . Following the evolution of the DOS as a function of parameter, it is difficult to make certain conclusions on the appropriate value of in the case of TiO, since the existing experimental data about the electronic band structure of Ti Magnéli phases is quite diverse. According to Mulay et al Mulay_1970 (), who studied the cooperative magnetic transitions in titanium oxides, the band gap of TiO in the LT phase is found to be of 0.035 eV. The authors also report on the gap of 0.041 eV for TiO, while optical measurements in Ref. Kaplan_1977, gave a band gap of 0.25 eV; later photoemission spectroscopy experiments revealed a gap of only 0.1 eVTaguchi_2010 (). One is clear: even if the electron-electron interactions are less pronounced in TiO than in the other Magnéli phases with higher concentration of electrons, they play an important role in the stabilization of its insulating ground state.

=2 eV =3 eV =4 eV =5 eV =6 eV
Ti(1) 0/0.21 0/0.23 0/0.15 0/0.14 0/0
Ti(2) 0/0.47 0/0.52 0/0.50 0/0.55 0/0.44
Ti(3) 0.29/0.30 0.37/0.43 0.38/0.45 0.28/0.44 0/0.56
Ti(4) 0.19/0.30 0.43/0.36 0.58/0.43 0.71/0.37 0.81/0.43
Ti(5) 0.27/0.19 0.32/0.47 0.37/0.49 0.53/0.61 0.77/0.59
Ti(6) 0/0.09 0/0.26 0/0.22 0/0.17 0/0.12
0/0 0/0 0.1/0 0.4/0 0.7/0
55 47 23
Table 1: The GGA+ absolute values of magnetic moments (in ) at Ti sites in the LT phase of TiO, calculated for different (at eV) for the AFM/FM spin allignment. The energy gaps (in eV) and the total energy differences (in meV/f.u.) are aslo listed.

iii.2 Structural effects

115 K 295 K 115 K 295 K
Ti(1)-O(2) 1.949 1.955 Ti(4)-O(5) 1.978 1.987
Ti(1)-O(3) 1.963 1.970 Ti(4)-O(6) 2.085 2.095
Ti(1)-O(4) 2.009 2.015 Ti(4)-O(9) 2.086 2.092
Ti(2)-O(1) 2.013 2.008 Ti(5)-O(5) 1.849 1.842
Ti(2)-O(2) 1.973 1.974 Ti(5)-O(6) 2.055 2.053
Ti(2)-O(5) 2.004 2.007 Ti(5)-O(7) 2.145 2.155
Ti(3)-O(1) 1.903 1.906 Ti(5)-O(7) 2.002 1.992
Ti(3)-O(2) 1.939 1.929 Ti(5)-O(8) 1.956 1.943
Ti(3)-O(3) 1.991 1.976 Ti(5)-O(9) 2.045 2.040
Ti(3)-O(6) 2.029 2.035 Ti(6)-O(4) 1.879 1.883
Ti(3)-O(7) 2.075 2.071 Ti(6)-O(6) 2.143 2.133
Ti(3)-O(8) 2.081 2.086 Ti(6)-O(7) 1.991 2.003
Ti(4)-O(1) 1.942 1.943 Ti(6)-O(8) 1.855 1.856
Ti(4)-O(3) 1.902 1.907 Ti(6)-O(9) 2.196 2.186
Ti(4)-O(4) 2.007 1.995 Ti(6)-O(9) 1.976 1.981
Table 2: Ti-O interatomic distances (in Å) in the LT (115 K) and HT( 295 K) phases of TiO according to structural parameters from Ref.Marezio_1977,
115 K 135 K 295 K
Ti(1)-Ti(3) 2.930 2.932 2.928
Ti(2)-Ti(4) 2.906 2.908 2.924
Ti(3)-Ti(5) 2.979 2.989 3.004
Ti(4)-Ti(6) 3.037 3.043 3.041
Ti(5)-Ti(6) 2.827 2.816 2.829
Table 3: Ti-Ti interatomic distances (in Å) in the LT (115 K), IT (135 K) and HT( 295 K) phases of TiO according to structural parameters from Ref. Marezio_1977,

As mentioned earlier, TiO crystallizes in the triclinic structure with two formula units per unit cell. There are six crystallographically independent Ti atoms in its unit cell, surrounded by distorted oxygen octahedra (see Fig.1 and Tab.2). Among them Ti(1) and Ti(2) are at the center of symmetry, while the other eight atoms are related by inversion symmetry, namely Ti(3)Ti(7), Ti(4)Ti(8), Ti(5)Ti(9) and Ti(6)Ti(10). These atoms form two crystallographycally independent pseudorutile chains: Ti(5)Ti(3)Ti(1)Ti(3)Ti(5) and Ti(6)Ti(4)Ti(2)Ti(4)Ti(6) Marezio_1974 (). The Ti(5) and Ti(6) atoms are the nearest to the shear planes and their environment is not rutile-, but sesquioxide-like, similarly to the TiO. The Ti(5)-Ti(6) distance is 2.83 Å and is the shortest in the lattice (see Table 3). The Oxygen octahedra around these atoms are severely distorted (the difference in Ti-O distances is 0.3 Å ) (see Table 2). According to Ref. Marezio_1977, , the Ti(5)Ti(3)Ti(1)Ti(3)Ti(5) chains are more distorted than the Ti(6)Ti(4)Ti(2)Ti(4)Ti(6) one, and thus have a slightly larger separation of charge. However, as can be seen from Tab.3, these distortions are very small and structurally the low-temperature phases are almost identical to the HT phase. This is not the case for TiO, where the formation of bipolarons in the LT phase and their rearrangement into polarons and bipolarons in the IT phase with the consequent collapse in the HT phase are clearly related with the electron-lattice coupling . The transition from HT to LT phase in TiO is accompanied by an essential decrease in the Ti-Ti distances (0.2 Å) with a consequent charge localization at Ti ions, involved in longer Ti-O and shorter Ti-Ti bonds. Interestingly, in the isostructural VO no changes in cation-cation distances larger than 0.06 Å are observed at the phase transition, despite the clear separation of charge into and in the alternate chains. Similar behavior applies to another member of the vanadium series, VO, which undergoes a metal-insulator transition at 135 K Marezio_1974 (). Although incomplete, V-V dimerization is clearly observed in VO Botana_2011 (), while it is negligible in VO Obara_2010 (). It was speculated that there are many different bonding patterns in the latter two compounds, and that an average of these patterns is observed by the classical x-ray diffraction methods Marezio_1974 ().

Figure 4: (color online) The GGA+ spin-polarized density of electronic states (DOS), calculated for the LT, IT and HT phases of TiO at for the AFM spin allignment (the lowest-energy configuration on Fig.5). Positive (negative) values indicate spin-up (spin-down) DOS.

Obviously, if the insulating behavior was induced by the lattice distortions only, the crystalline symmetry below the phase transition would be lower than in the metallic region. However, this is not the case in the TiO crystal, although its LT phase is slightly more distorted than the HT one. To find out how these small lattice distortions at the phase transitions affect the electronic properties of TiO, we performed total-energy calculations for the experimental structures Marezio_1977 () of all three phases. Figure 4 shows the density of states (DOS) of TiO, calculated at =5 eV for the LT, IT and HT structures (as reported by Marezio et al Marezio_1977 ()), assuming an AFM spin ordering with zero total moment per cell. One can clearly see that the DOS for all the three phases is very similar to each other. They all are insulating with the band gap in the LT phase of about 0.43 eV and only slightly smaller gap of 0.38 eV in the IT and HT phases. The corresponding charge distribution is similar from phase to phase with only a slight difference in the Ti magnetic moments (see Tables 4 and 5 for ). Therefore, although the peculiarities of the crystalline structure of TiO are certainly reflected in the rich variety of the possible charge order paterns, the lattice distortions at the phase transitions are rather insignificant to have a crucial effect on the conducting properties of the material.

iii.3 Magnetic and charge order

To find the ground state magnetic ordering and which Ti atoms participate in it, we have considered an antiferromagnetic and a ferromagnetic (FM) spin configuration along with a non-magnetic one (which turned out to be much higher in energy for all three phases). First, we have considered the effect of electron correlations on the magnetic behavior of the crystal. For the experimental structures we have found that at  eV the LT phase is FM and metallic, while for larger it tends to be an AFM insulator (see Table 1). Depending on the chosen and the resulting spin configuration, the calculated magnetic moments on each of the inequivalent Ti atoms vary within the range 0.1-0.8 . This is in contrast to TiO, for which the ionic magnetic moments in the LT phase were calculated to be robust with respect to their relative orientation and thus were linked to the local structure around Ti Weissmann_2011 (). As follows from Table 1, even in the FM state the magnetic moment is distributed among all the Ti atoms somewhat inhomogeneously. In the AFM state the charge disproportination is more pronounced within the whole range of values and non-zero moment is observed only at Ti(3), Ti(4) and Ti(5) sites. While Ti(4) and Ti(5) magnetic moments increase with increasing , the moment at Ti(3) site starts to decrease for and is completely suppressed at  eV (nominal Ti and Ti ions become clearly distinguishable). When Ti(3) moment becomes zero, the charge order pattern is symmetric with respect to the inversion. At the FM spin allignment the behavior is somewhat different: the charge order pattern is less symmetric, but more homogeneous, and it is less sensitive to the changes of value. As a result, the energy gap is zero for the whole range of values.

We have performed fixed spin moment calculations for all three phases by calculating the total energy at the constrained magnetic moment of the unit cell. The results, obtained for  eV, are presented in Tab.4 and Tab.5. It is clear from the listed data that there are several stable magnetic configurations with a very small relative energy difference, namely the configurations with total moment of 0 (AFM), 2 and 4 /cell. Moreover, for each of the magnetic configurations we have obtained a whole set of different possible charge oder patterns (see Fig.5); therefore, we present here only those which in our calculations have the lowest energy.

Low-temperature phase

For the experimental structure of the LT phase the configuration with the total magnetic moment /cell is practically degenerate with the AFM configuration with (the energy difference is only 6 meV/f.u.) (see Table 4). The FM configuration with /cell is 20 meV/f.u. higher in energy than the AFM one. We observe stronger localization of the charge, and hence, larger band gap, for the configurations where more Ti moments are alligned antiparallel to each other: the band gap is the largest for the AFM state and it decreases to zero for the FM one. Therefore, opening of the gap in TiO when going from high to low temperatures might be closely related with the AFM ordering of the unpaired spins. This is in line with the fact that experimentally the Néel temperature and the semiconductor-metal transition temperatures in this material nearly coincide.

Figure 5: (color online) A schematic diagram showing the relative stability of several lowest-energy charge and magnetically ordered states, obtained for the LT phase taking into account structural relaxations. Shaded area within a red-dashed oval depicts Ti-Ti singlet pairs (bipolarons). Dark blue and grey colors are used to distinguish between Ti in adjacent pseudorutile chains. Big (small) balls denote Ti (Ti) ions.
M /cell /cell /cell
exp. relaxed exp. relaxed exp. relaxed
Ti(1) 0 0 0 0 0.14 0
Ti(2) 0 0 0.31 0 0.54 0
Ti(3) 0 0 0.44 0
Ti(4) 0.71 0.83 0.68 0.84 0.37 0.85
Ti(5) 0.61 0.84
Ti(6) 0 0 0 0 0.17 0
Ti(7) 0 0.32 0 0.44 0
Ti(8) 0.67 0.84 0.37 0.85
Ti(9) 0.55 0.83 0.61 0.84
Ti(10) 0 0 0 0 0.17 0
6 0 0 2 26 7
0.46 0.9 0.19 0.82 0 0.67
Table 4: Magnetic moments on Ti atoms (in ), total energies with respect to the lowest-energy magnetic configuration (in meV/f.u.) and the band gap (in eV), calculated for the experimental and relaxed (theoretical) structures of LT phase.

Interestingly, as follows from Table 4, when structural relaxations are taken into account, the lowest energy charge distribution is the same for all the three magnetic configurations. It is such that Ti ions (polarons) are at the position of Ti(4) and Ti(5) cations. Within a pseudorutile chain they are well separated by either one or three Ti ions (Fig.3), retaining the inversion symmetry. While Ti in the Ti(5) position has a common face with the Ti at Ti(6) site across the shear plane, the Ti in the Ti(4) position is close to the middle of the pseudorutile chain. It is clear from Fig. 3(b) that there is no direct overlap between the orbitals at Ti(4) and Ti(5) sites and the interaction between the corresponding electrons is realized through the hopping via bridging oxygens.

Several decades ago, James and Catlow James_1977 () theoretically predicted that the electrons created by the reduction of rutile or higher Magnéli phases should be trapped by the shear plane. In TiO this would result in charge-order state with Ti ions occupying Ti(5) and Ti(6) sites. However, in our calculations such state is 148 meV/f.u. less stable than the lowest-energy one with Ti at Ti(5) and Ti(4) sites. This means that not all electrons occupy the sites adjacent to the shear planes, but prefer the localization at the nearest neighbor sites.

In contrast to the experimental structures, the localization of the charges in the relaxed structures is much stronger and it results in the insulating behavior even for the FM spin allignment. The total energy difference between the states with , 2 and /cell is less than 10 meV/f.u. The fact that it is so small indicates that the charge and spin order is unstable. A large separation between the Ti ions results in rather weak magnetic interactions, which is reflected in the small energy difference between the states with the same charge distribution, but different number of parallel/antiparallel spins. This is in line with the conclusions made by Danley and Muley Danley_1972 (), who observed small Weiss constant and hence weak interaction between the unpaired spins in TiO. This is different from TiO, for which the AFM interaction between the spins within the Ti-Ti pairs was found to be rather strong Leonov_2006 (), although the inter-pair coupling was calculated to be much weaker. We have also considered the states, where four unpaired spins at Ti sites are distributed in the way to form bipolarons within the same pseudorutile chain, as well as the state with coexisting bipolarons and polarons. These states were obtained to be 11 meV/f.u. and 34 meV/f.u., correspondingly, higher in energy than the ground state and are both insulating. Therefore, the ground state of the LT phase of TiO is semiconducting, however its charge order pattern is not unique.

Intermediate- and high-temperature phases

Since our ab initio calculations are performed at  K temperature, for the IT and HT phases we have considered only the experimental structures as reported by Marezio et al Marezio_1977 (). In analogy to the LT phase, we observe several stable magnetic configurations for the IT and HT phases, which are almost degenerate within our calculation error: these are the configurations with the total magnetic moment of 0, 2 and 4 /cell. For each of these magnetic configurations our fixed-spin moment calculations revealed several possible charge order patterns, but in Table 5 we present only the lowest-energy ones. For the IT phase the calculated band gap is the largest for the configuration with /cell (about 0.4 eV) and it is almost the same as the gap calculated for the experimental structure of the LT phase. The gap is somewhat smaller for the configuration with /cell and completely vanishes when the spins are alligned ferromagnetically due to more homogeneous distribution of the valence charge. Similar tendency is also observed for the HT phase with the only difference being that the gap (although very small) is non-zero for the FM configuration. It should be noted that for the HT phase there is another FM state, which is metallic and has a more homogeneous charge distribution, but this state is about 30 meV/f.u. higher in energy.

The magnetic moments do not vary much from phase to phase, which once again indicates a secondary role of the crystal structure in the electronic properties of the TiO. The charge distribution for the considered configurations resembles the one observed in the LT phase, however, it is slightly more localized for the FM configuration in the HT phase than in the LT or IT phases. However, the magnetic moment distribution remains inhomgeneous even for the FM configuration in the HT phase and does not correspond to the suggested Ti picture. Therefore, the local environment and distortions play a visible role in the magnetic moment distribution in this material.

M /cell /cell /cell
IT HT IT HT IT HT
Ti(1) 0 0 0 0 0.20 0
Ti(2) 0 0 0.35 0.27 0.68 0.29
Ti(3) 0.43 0.41
Ti(4) 0.71 0.73 0.67 0.70 0.19 0.70
Ti(5) 0.69 0.47
Ti(6) 0 0 0 0 0.19 0.14
Ti(7) 0.33 0.29 0.46 0.42 0.40
Ti(8) 0.64 0.69 0.19 0.70
Ti(9) 0.57 0.39 0.69 0.47
Ti(10) 0 0 0 0 0.19 0.14
9 8 0 2 2 0
0.43 0.38 0.15 0.25 0 0.04
Table 5: Magnetic moments on Ti atoms (in ), total energies with respect to the lowest-energy magnetic configuration (in meV/f.u.) and the band gap (in eV), calculated for the experimental structures of the IT and HT phases.

Iv conclusions

In summary, we have performed first-principles calculations to study the electronic and magnetic propeties of the TiO crystal, which belongs to the homologous series of TiO oxides, known as Magnéli phases. From our results it is difficult to conclude on which charge distribution is realized in this material at different temperatures, since for all three phases we find several quasidegenerate magnetic solutions. We therefore assume that the charge order within the cation sublattice in TiO is not unique and thus unstable, which makes its experimental detection at finite temperatures difficult. We suggest that in contrast to TiO, the formation of Ti-Ti pairs in this compound is less likely, although not completely impossible, since a bipolaronic state is only slightly higher in energy than the ground state. Our calculations perfomed for all three phases showed that crystal structure changes at the transitions make a minor influence on the electronic structure of the compounds. Although the electronic correlations are important for the stabilization of the insulating ground state, the phase transitions in this compound are related to a greater extent to a reordering of the unpaired spins. This conclusion agrees well with the antiferromagnetism model presented by Adler in Ref.Adler_1968, , where the author considered different scenarios for the metal-nonmetal transitions in transition metal oxides and sulfides. Thus, the mechanism behind the phase transitions in TiO must be a complex interrelation between the electronic correlations, electron-lattice, as well spin-lattice coupling.

We appreciate the support of the Jülich Supercomputing Centre (Project JIFF38) and gratefully acknowledge the support of the SFB917-Nanoswitches and Young Investigators Group Program of the Helmholtz Association, Contract VH-NG-409.

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