Defect chemistry of Ti and Fe impurities and aggregates in Al 2 O 3

We report a theoretical evaluation of the properties of iron and titanium impurities in sapphire (corundum structured a -Al 2 O 3 ). Calculations using analytical force ﬁ elds have been performed on the defect structure with the metals present in isolated, co-doped and tri-cluster con ﬁ gurations. Crystal ﬁ eld parameters have been calculated with good agreement to available experimental data. When titanium and iron are present in neighbouring face and edge-sharing orientations, the overlap of the d-orbitals facilitates an intervalence charge transfer (Fe III /Ti III / Fe II /Ti IV ) with an associated optical excitation energy of 1.85 eV and 1.76 eV in the respective con ﬁ gurations. Electronic structure calculations based on density functional theory con ﬁ rm that Fe III /Ti III is the ground-state con ﬁ guration for the nearest-neighbour pairs, in contrast to the often considered Fe II /Ti IV pair. Homonuclear intervalence charge transfer energies between both Fe III /Fe II and Ti IV /Ti III species have also been calculated, with the energy lying in the infra-red region. Investigation of multiple tri-clusters of iron and titanium identi ﬁ ed one stable con ﬁ guration, Ti III – (Ti IV /Fe II ), with the energy of electron transfer remaining unchanged.


Introduction
2][3] The presence of metal impurities in a-Al 2 O 3 , predominantly iron and titanium, dramatically alters its optical properties.5][6][7][8][9] These studies, however, have not connected the multiple optical processes that occur within and between these cation impurities in a complete analysis.
Al 2 O 3 formally consists of Al 3+ and O 2À ions and adopts a trigonal lattice symmetry in its ground-state corundum structure. 10The Al centres are surrounded by six oxide ligands in a distorted octahedral geometry lling 2/3 of all available octahedral holes.These octahedral units are linked in both facesharing and edge-sharing orientations as parallel and perpendicular to the c-axis, respectively. 11The relative orientation of the metal centres causes a pseudo Peierls distortion, resulting in neighbouring metal centres that are rotated at an angle of 64.3 away from each other.Elongation in pairs of the surrounding oxide ligands (Fig. 1) results in a pentagonal bipyramidal geometry belonging to the space group R 3c. 12 The tetrahedrally coordinated oxide ligands form a close-packed sub-lattice.The material is largely ionic in nature with a wide band gap of 9.25 eV.
To induce the colours characteristic of minerals formed from Al 2 O 3 (e.g.blue sapphire and red ruby), the presence of impurities is required.As a consequence to doping, electronic energy levels are inserted into the band gap of the material.Lower energy transitions within the visible and/or thermal range can occur, giving rise to colour and conducting behaviour in the insulating and transparent host material. 13,14e have focused on two of the most common and researched impurities that increase the absorption of visible light.These impurities can be found in naturally occurring minerals or can be incorporated synthetically during synthesis or post-annealing.][20] The recognition that Ti III , a d 1 cation, gives a tunable laser was reported in 1982. 21It is the intra-valence d-d transitions occurring within the metal and the vibrational interactions with the sapphire lattice, which initiate the electronic transitions as a consequence of photon absorption.The interactions between the oxide host and titanium impurity are therefore crucial with regard to determining the optical properties of the doped material. 1,18In addition to efficient emission of near IR (infrared) light, the ceramic material with a signicant thermal conductivity also has a high damage threshold. 18,22nce present in the a-Al 2 O 3 structure, Ti will substitute Al and remain on the same lattice site (eqn (1)).In addition to the isovalent Ti III impurity, titanium has also been reported as Ti IV .This tetravalent cation will cause a larger lattice distortion and therefore is found at lower concentrations in natural sapphire.Compensation mechanisms, such as those shown in eqn ( 2) and (3), are required for the aliovalent substitution in order to maintain overall charge neutrality.The binding energies between the isolated point defects and more complex charge compensating clusters have previously been calculated. 5,23he standard Kröger-Vink notation has been used to describe the charge compensation mechanisms, following the standard notation A Y B , where A represents the dopant, B the host lattice site and Y the charge relative to the host lattice site; Â indicates no net charge.B can also be i, representing an interstitial defect and A can also be V, representing a vacancy defect. 24Eqn (1)-(3) refer to reactions involving Ti III , Ti IV and Ti IV respectively, with the atomic chemical potentials determined by the respective binary oxides.
Iron-doped sapphire has not been as intensely studied as the Ti III -Al 2 O 3 system, but is still known to produce interesting optical properties.The increase in conductivity when doped with iron has applications including thin-lm stress sensors and planar optical waveguides. 25Sapphire doped only with iron has been shown to result in yellow crystals, demonstrating visible light absorption. 26With homonuclear intervalence charge transfer and interactions with other dopants in the structure, iron-doped sapphire has the potential to display more complex optical properties.Again, two oxidation states can be incorporated, Fe II (from FeO) and Fe III (from Fe 2 O 3 ): Iron is present as Fe II (eqn ( 5) and ( 6)) at a much lower concentration than Fe III (eqn (4)); however, the concentration of divalent iron can be increased by the application of heat. 27ehmann and Harder (1970) gave an extensive review of the effect of the concentration of Fe III and Fe II on the colour of corundum crystals, also conrming in natural samples the isovalent Fe III to be the dominant oxidation state. 28The transfer of an electron between the two valence states of iron can also be initiated by light (eqn (7)).The energy required for the described homonuclear charge transfer between neighbouring Fe III and Fe II will be calculated and compared to the experimentally determined values ranging from 1.40-1.55eV (800-886 nm). 26,29 The substitution of both iron and titanium in Al 2 O 3 results in the vibrant blue colour of sapphire crystals.The energy and intensity of light absorbed by the material has been shown to be dramatically affected by the concentration of either cation. 30,31ith multiple charge transfer and intra-valence excitations occurring between and within these cations, spectroscopic analysis of natural sapphire samples is oen disputed and inconclusive.
An optical charge transfer process between iron and titanium is understood to be the origin of colour in blue sapphire.The overlap of metal d-orbitals of the neighbouring metals facilitates the transfer of an electron between species. 32The transfer is initiated by visible light absorption, resulting in the alternate valence states of the metals.
4][35][36] As a consequence of heating, the blue colour of a sapphire crystal intensies, making this a standard industrial process to increase their market value. 27n this paper, following our initial communication, 37 we report the calculated optical transition energies of Fe and Ti in the Al 2 O 3 lattice and offer suggestions as to the assigning of the absorption spectra of natural sapphires.Our predictions are compared to available experimental evidence.The investigation includes isolated titanium and iron substitutions, the co-doped systems with both metals present in differing valence states, and metal tri-clusters that may form in the lattice.

Methodology
We employ Mott-Littleton defect calculations, based on analytical pairwise potentials, to probe the potential energy landscape associated with inter-atomic charge transfer and intra-atomic crystal eld excitations.Further electronic structure calculations are performed using a range of density functionals in order to conrm the ground-state magnetic structures.

Interatomic potentials
Calculations within the Born model of ionic solids were performed using the GULP (General Utility Lattice Programme) code. 38The interatomic interactions are represented in the form of the pairwise Buckingham potential, inclusive of a long-range Coulombic term (eqn (9)) evaluated via an Ewald summation and a short-range potential active within 15 Å. [39][40][41] The potential parameters (A, r and C) have been taken from Catlow and Lewis to model the pure and doped systems (Table 1). 40,41The shell model of Dick and Overhauser has been employed to model the electronic polarisation of the system, 42 with the harmonic spring constant (k) tted to reproduce the experimental highfrequency (3 N ) and static (3 0 ) dielectric tensors.It should be noted that the relatively large error in the calculated elastic constants is common in pair potential methods and does not inuence the properties of interest (Table 2).

Embedded crystal defect calculations
The defect formation energies were calculated within the Mott-Littleton approach. 44The simulated lattice surrounding the central defect is divided into multiple regions, the details of which have been reviewed extensively elsewhere. 5Sizes were set at 10 Å (950 ions) and 25 Å (14 278 ions) for regions 1 and 2, respectively.The interatomic potentials used to model the extrinsic transition metals are listed in Table 3.Two local minimisation approaches were required for the calculation of the energies associated with intervalence charge transfer.The rst, full optimisation, results in the lowest energy local conguration of the ions and shells in region 1, calculated via a Newton-Raphson procedure.The second minimisation technique is 'shell only'.During this optimisation, the cores of the ions are xed in location and only the shells are allowed to relax, representing the polarisation of the electrons due to the charge transfer process.As a consequence, a vertical optical transition as dened by the Franck-Condon principle can be modelled, in addition to the adiabatic (thermal) excitation energy.The pairs of ions involved in the intervalence charge transfer (Fe II /Ti IV and Fe III /Ti III ), which are charge compensating when substituted into corundum, were modelled separately.
A congurational coordinate diagram can be constructed using the results of the two energy minimisation techniques, as illustrated in Fig. 2. The diagram is a representation of the relative potential energy of different ionic congurations as a function of the defect geometry; the structure of the system is Table 1 Interatomic potentials used to model corundum as parameterised by Lewis and Catlow 40,41 Parameter a -Al 2 O 3  described by a collective conguration coordinate.Each parabola represents the pair of cations involved in the intervalence charge transfer.The defect energies are labelled 1-4, which have been calculated using the techniques described above.The local minima (1 and 4) are calculated by full geometry optimisation of each charge state.The adiabatic energy E thermal is the difference between these minima.The alternate valence pair is then substituted into the previously optimised positions to model vertical optical excitations (2 and 3).The calculation of the absolute charge transfer energy between iron and titanium requires an additional term to align their ionisation potentials (E diff ), as determined from the lattice energies (DE LE ) of the respective binary oxides: An alignment term is not required for homonuclear intervalence charge transfer as the ionisation potentials and electron affinity of the two valence states cancel out.

Ligand eld splitting
The AOLM (angular overlap model) developed and implemented by Woodley was employed to obtain the crystal eld parameters of the transition metal cations. 46The inclusion of angular functions into the purely radial Buckingham and Coulomb potentials allows non-spherical cations to be modelled and additional electronic properties to be calculated.8][49] The augmented short-range potentials have been applied to 3d impurities to calculate the energies of the dorbitals in the distorted environments of the sapphire lattice.
The distortion of the cation environments in sapphire from perfect octahedra, as shown in Fig. 1, reects their anti-prismatic geometry.In this geometry the octahedral units rotate from the c-axis.The pairwise elongation of the oxide ligands results in a pinching and consequent destabilisation of the d xy orbitals.In this conguration the d z 2 and d x 2 Ày 2 orbitals are degenerate (Fig. 3).
The crystal eld parameters were tted through the calculations of hexa-aqua complexes in order to obtain the absolute energy splittings for each cation.The interatomic potentials (Table 1 and 3) were used to model these complexes, with the splittings compared to available reference values. 50Spin-orbit coupling and Jahn-Teller distortions are not considered in this study; however, their effects should be minor.In particular, Jahn-Teller effects will not change the vertical excitation energies.

Electronic structure calculations
2][53][54] Calculations were initially performed by treating the exchange and correlation effects within Density Functional Theory (DFT) [55][56][57] using the PBEsol generalised gradient functional and a 2 Â 2 Â 2 k-point grid for the supercell.A quasi-Newton relaxation was performed for local structure optimisation, which was converged the forces to 0.005 eV ÅÀ1 or lower.The wave functions were constructed via a plane wave basis set, with a cut off of 500 eV.The projector augmented wave method was employed to represent the interactions between the core and valence electrons of the ions. 53t should be noted that even within an excited-state approach such as time-dependent DFT, the description of charge-transfer  excitations remains a signicant challenge.Therefore, the focus of the electronic structure studies here is to probe the groundstate electron and spin distributions.9][60] Images of chemical structures and electron densities were made using the VESTA soware. 61e employ the standard supercell approach to investigate Ti and Fe incorporation into Al 2 O 3 .The conversion from the hexagonal unit cell to an orthorhombic supercell expansion reduces the interactions between defect in neighbouring supercells. 62The resulting cuboid was of dimensions 8.27 Â 9.55 Â 13.01 Å and contained 120 atoms.This was formed by via an anisotropic expansion of the form: 63 0 @ 2 0 0 1 2 0 0 0 1 For these calculations we have maintained the equilibrium lattice parameters of Al 2 O 3 , relaxing only the internal coordinates during the defect calculations, which is appropriate for perturbations in the dilute limit.As shown in Table 4 our calculated crystal eld parameter of 2.47 eV is in good agreement. 65The three underlying optical processes are labelled in Fig. 3.

Results and discussion
With a 13% increase in ionic radius, Ti III incorporation causes a distortion to the local structure.Bond lengths for the face and edge-sharing congurations are 2.81 Å and 2.90 Å, respectively.We have calculated the solutional substitutional enthalpy of each metal dopant in the sapphire lattice, representing the incorporated isolated defect at innite dilution (see eqn (1)-( 6)).This enthalpy was calculated by the subtraction of the total energy of the defective system (with the dopant present) from that of the pure system (with no defects present), which was balanced thermodynamically using the lattice energies of the binary oxides of the 3d metal dopants (eqn (1)).
In addition to a substitution energy of 3.41 eV, we have found that isovalent titanium disrupts the local environment around the lattice site.However, the cations remain in a trigonal antiprismatic geometry (Fig. 1).Typical intensities of d-d transitions range between 1-10 3 M À1 cm À1 (Table 5), but due to the distortion of the 3d cations the intensities of these transitions should be greater.The solutional substitution energy for the optically inactive Ti IV is calculated to be 2.78 eV and 3.05 eV from eqn (2) and (3), respectively.

Fe-doped sapphire
Fe is found in sapphires in both its divalent and trivalent oxidation states. 36,67Sapphires doped only with Fe, predominately in the trivalent state, are yellow in colour.It is still under debate if this is due to intravalence d-d transitions within a single impurity, or to an intervalence charge transfer between the divalent and trivalent valences of iron. 26,31,68This colour arises from visible light absorption in the energy region of between 2.88 eV and 3.26 eV.
The energy calculated for the intra-valence transition in trivalent Fe is 2.94 eV (Table 4), which is within the measured energy range. 26The results suggest, in support of Ferguson and Fielding, that visible light absorption in Fe-doped sapphire is due to intra-valence d-d transitions in the dopant cation even though all electronic transitions in these dopants are formally spin forbidden.Typical intensities for such transitions are given in Table 5.As shown they range between 10 À5 -10 0 M À1 cm À1 .The electron transfer energy between divalent and trivalent Fe has also been calculated to further support this conclusion (see Section 3.3).
The calculated enthalpy of solution for Fe III (eqn (4)) is 0.40 eV, while for Fe II the two compensation mechanisms (eqn ( 5) and ( 6)) were calculated to be 1.12 eV and 1.53 eV, respectively.The larger ionic radii of Fe II and Fe III in comparison to Al III is 7% and 1%, respectively, which contributes to the smaller solution energy of the latter.The isovalent Fe III substitution also avoids the formation of charge compensating point defects.When considering the difference in enthalpy of solution for isolvalent Ti and Fe, the increased energy calculated for Ti is purely due to the steric affect of the size of either cation.As Ti is larger than Fe the distortion of the lattice with its substitution is greater, which has an associated energetic penalty.3.3 Co-doping sapphire 3.3.1 Fe/Ti: ionic potentials.Experimental evidence for the presence of bi-particles (neighbouring substitutional impurities) in sapphire was rst reported by Eigenmann (1972) and has since been of great interest as to the effect on pleochroism and the colour of materials. 30,69When neighbouring in the corundum structure an overlap of the metal d-orbitals, i.e. d z 2 and d x 2 Ày 2, facilitates the efficient transfer of an electron between the cations.We have shown that the Fe III and Ti III charge states are thermodynamically favored for the isolated ions; we have also calculated that this is true for co-doping.
When neighbouring, Fe III and Ti III are at a distance ranging from 2.88 to 2.99 Å.An electron can be transferred between the species, initiated by the absorption of light, resulting in the Fe II /Ti IV congurations being formed (eqn ( 8)).This intervalence charge transfer is not limited by symmetry or spin selection rules, resulting in optically allowed transitions.
The energy absorbed during the intervalence optical transfer has been speculated to be responsible for the intense blue colour of sapphire, with this initial suggestion dating back to 1967. 31Alternative mechanisms are still debated, e.g. that Ti is a charge compensating cation and Fe is the colour centre. 28owever, it is more popularly considered that the Fe II /Ti IV conguration represents the ground state and Fe III /Ti III the excited state. 26,29,69,70We conrm that intervalence charge transfer is responsible for colour, but this instead occurs for the Fe III /Ti III pairs. 37For this to be a viable mechanism, an energy between 1.99 and 2.14 eV must be absorbed.
We have calculated the energy required for the transfer of an electron between Fe III and Ti III (the principal absorption) in both edge and face-sharing congurations.There is a third possible environment where the cations could be substituted parallel to the c-axis, resulting in a separation of 3.37 Å.The transfer of an electron would be less probable, due to smaller dorbital overlap and substitution at this site has therefore not been investigated.As shown in Fig. 3, the absorption occurs with the transfer of an electron from the more stable Fe III /Ti III to Fe II /Ti IV .The Fe II /Ti IV conguration represents a metastable excited state; thermal relaxation (1.31 eV) to the Fe III /Ti III conguration is favoured; however, we cannot comment on the timescale for this process. 37e have also investigated the energy associated with defect clustering.For the Fe II /Ti IV conguration there is a binding energy of 0.64 eV owing to the strong Coulomb attraction of oppositely charge defects, which may stabilise neighbouring impurities during synthesis or annealing: Absorption spectroscopy shows a strong, broad band with maximum between 2.14 and 2.17 eV. 29,70We can assign this peak as the intervalence charge transfer for the co-doped systems between Ti III and Fe III (eqn ( 8)).The measured signal is broadened by the changes in environment that are characteristic of an intervalence charge transfer.
3.3.2Fe/Ti: electronic structure.Due to the importance of the "blue" transition and the assignment of the ground state Ti/Fe conguration, further calculations were carried out based on spin DFT to ensure that the correct ground-state had been identied.
2][53][54] The dopants were then incorporated in adjacent sites for the substitution of Al in both orientations, as had previously been modelled with the ionic potentials.The relative energies were assessed based on the self-consistent total energies.
Different magnetic and charge-state congurations can be stabilised accordingly to the initialisation of the calculation in terms of total and local magnetic moments.Spin density proles (r [ À r Y ) were plotted to conrm the location and topology of the magnetisation.These results were compared to the congurational coordinate diagram (Fig. 2).The d occupations associated with each of the metals in their formal oxidation states are d 0 (Ti IV ), d 1 (Ti III ), d 5 (Fe III ) and d 6 (Fe II ).For Ti, the local spin moment is well dened; however, Fe may access both low (LS) and high (HS) spin congurations.][73] As shown by the spin-density proles, independent of the magnetic initialisation, when the metals were in high spin, the Fe II /Ti III conguration was always observed (Table 7).This phenomenon conrms the instability of the Fe II /Ti IV conguration and supports the congurational coordinate diagram.This spontaneous electron transfer demonstrates that the Fe II /Ti IV conguration is less stable, in agreement with the pair potential results.Calculations based upon hybrid and meta-DFT, as well as HF, conrm the absence of electron transfer (Table 7).
The energies of all congurations relative to the calculated ground-state high spin Fe III /Ti III are presented in Table 7.In low spin, when the Fe II /Ti IV pair is stabilised, the resultant conguration is 1.18 eV higher than the ground-state.The energy difference between the two optimised congurations (1.18 eV) also closely correlates to the adiabatic electron transfer reported in the previous section (1.31 eV).We note that in order to fully probe the optical excitations from electronic structure theory, an explicit time dependent approach should be applied.

3.3.3
Fe/Fe and Ti/Ti aggregates.The energies associated with homonuclear Fe and Ti impurity pairs have been calculated using the ionic potentials, DFT and HF methods.The ionic potentials were employed to obtain the intervalence charge transfer energies, while the electronic structure approaches were used to probe the local spin congurations.
Both Fe III /Fe II and Ti III /Ti IV pairs were considered.The calculated charge transfer energies are given in Table 6.The energy of homonuclear intervalence charge transfer has been calculated to occur in the thermal range.These have no direct contribution to the colour of the crystal, but will inuence the thermoelastic properties.
Table 7 Self-consistent spin density profiles from electronic structure calculations for the Fe/Ti co-doped sapphire.Neighbouring Ti (left) and Fe (right).Spin (S) refers to the total spin moment that was fixed for the supercell, LS indicates a low spin configuration for Fe, while E refers to the total energy of the supercell relative to the most stable configuration.Positive spin density (yellow) and negative spin density (blue) are shown for an isolated fragment of the periodic supercell.The same spin topology and energetic ordering was found for meta-GGA and DFT + U functionals (not shown) Fe II -Fe III 0.00 Ti III -Ti IV À0.02 The mirror symmetry of the transitions (E opt and E opt 2 ) are typical of a homonuclear system, with each conguration representing the same valence states of cations but occupying the opposite sites in the lattice.From these calculated results of 1.2-1.4eV, we can further conclude the origin of the colour of yellow sapphires to be due to the intra-valence transitions in the Fe III cations (Table 4) and not due to homonuclear intervalence charge transfer (Table 6).
Binding energies of 0.00 eV and À0.02 eV were calculated for Fe II /Fe III and Ti III /Ti IV pairs with bond lengths of 2.92 Å and 3.05 Å, respectively.Results suggest that there is no strong driving force for impurity aggregation and hence a low probability of homonuclear charge transfer.
We have also performed electronic structure analysis of the homonuclear systems for neighbouring aliovalent pairs.Spin density proles have been plotted to illustrate the magnetic structure and are presented in Table 9.The limitations of the underlying DFT and HF techniques should be noted, which are based on a single determinant when calculating the selfconsistent eld, which means that the electronic minimisation procedure is unable to distinguish between symmetric homonuclear cations.As a consequence, the models may be biased towards electron delocalisation between the two metal centres.
For the homonuclear Fe system, the spin density is similarand almost sphericalon each ion.Results are shown for the most stable high-spin conguration, which suggests a preference for two Fe III (d 5 ) ions with the occupation of the minority spin t 2g orbital split between them.An equivalent electron count is found around each metal.The same results were observed at the GGA, hybrid DFT and HF levels of theory; the latter two results are shown in Table 9. Results including explicit breaking of the homonuclear symmetry are also presented.A model isoelectronic heteronuclear system was constructed: Co III /Fe III , i.e. d 6 /d 5 .As shown, the topology of the Co/Fe spin density is similar, but analysis of electronic charge and magnetisation for each ions conrms the occupation of the minority spin t 2g state on Co.Therefore, neighbouring d 6 /d 5 centres are possible.
Analysis of the spin density for the homonuclear Ti system displays a similar delocalisation when modelled at all levels of theory (Table 9).Instead of a d 1 /d 0 conguration, a single d band on each Ti centre is partially occupied by one electron.Within HF theory, which fully describes the electron-exchange term, the density is also depicted as delocalised (positive spin density on both Ti cations), suggesting this effect is not due to the electron self-interaction, which was further conrmed using the electron localisation function. 74,75A model calculation was again conducted to remove the mirror symmetry, via the substitution of one Ti for Y: Ti III /Y III , i.e. d 1 /d 0 .Here, the single d electron is forced to localise on Ti, resulting in the asymmetric spin density shown in Table 9.These results conrm that neighbouring d 1 /d 0 centres are also possible, but appear not to be accessible for the homonuclear system, at least not within a single-determinant framework.

Tri-clusters in sapphire
In addition to isolated defects and bi-particles, larger defect clusters are also possible, especially at high impurity concentrations.The attraction and consequent binding of charge compensating defects, with the substitution of Al for an aliovalent cation has previously been reported. 23,76e have modelled possible metal tri-clusters to assess their optical properties and to see if they can further stabilise Fe II /Ti IV pairs (Table 10).All tri-clusters have been modelled as L-shaped, with each cluster containing edge and face-sharing metals (Fig. 5).The co-doped edge-sharing Fe II /Ti IV congurations were unstable, this is also true when incorporated into a tri-cluster conguration.We therefore modelled the tri-clusters involving Fe II /Ti IV as face-sharing, with the third metal in an edge-sharing conguration.
The calculations demonstrate that the intervalence charge transfer energy between trivalent Fe and Ti remains within the visible range when present in a tri-cluster.Absorption energies remain almost unchanged compared to the isolated pairs.There is only one cluster, [Ti III -(Ti IV -Fe II )], which gains stability by forming this aggregate (Table 8).It is unlikely that the remaining clusters would form to any signicant concentration.
Table 9 Self-consistent spin density profiles (yellow isosurface) from calculations for (Fe/Fe) À and (Ti/Ti) + doped sapphire.The systems (Co/Fe) and (Y/Ti) were also investigated as isoelectronic analogues.The spin (S) refers to the total spin moment fixed for the supercell The [Ti III -(Ti IV -Fe II )] cluster would offer a greater stability to the Fe II /Ti IV pairs and may extend their lifetime in the crystal.However, even if formed, the optical process would remain largely unchanged by the third neighbouring cation.We propose that experimental detection of these aliovalent cations can be due either to the isolated Ti IV -Fe II pairs, or their presence within the tri-cluster as described.Homonuclear intervalence charge transfer was also modelled in these tri-clusters and the energies were found to remain unaffected (Table 10).
Preliminary attempts to model more complex clusters consisting of these metal dopants with charge compensating intrinsic point defects such as vacancies and interstitials have been made.However, these larger clusters have large positive binding energies and therefore will not form under conditions of thermodynamic equilibrium in sapphire.

Conclusions
We have investigated the structural, energetic and optical properties of Ti and Fe in the corundum Al 2 O 3 lattice using two materials modelling techniques.Our insights can be summarised as follows: 1. Crystal eld parameters have been calculated for the trivalent cations showing their intra-valence optical transitions to occur within the visible light region.
2. The dominant intervalence charge transfer resulting in the absorption of visible light occurs between Fe III /Ti III to produce the metastable Fe II /Ti IV conguration.
3. Electronic structure calculations have conrmed the instability of the Fe II /Ti IV pairs in sapphire, with total energies more than 1 eV higher than the Fe III /Ti III conguration.
4. Homonuclear iron and titanium charge transfer has been shown to occur within the IR range and will have no contribution to visible light absorption of the material.
5. Electronic structure calculations, based on DFT and HF, have been unable to stabilise nearest neighbour Fe II /Fe III and Ti III /Ti IV pairs.Delocalisation of the excess electron occurs between the two metal centres, which would prohibit nearestneighbour intra-valence charge transfer.
6. Possible Ti and Fe tri-clusters in differing nearest neighbour congurations were modelled.The calculated binding energies between these species identied one conguration to be thermodynamically accessible.[Ti III -(Ti IV -Fe II )] is the most stable combination, which may be formed in doped sapphire samples.
Moving beyond low concentrations of defects, ternary and quaternary compounds formed of these elements should display a similar behaviour.Ilmenite (FeTiO 3 ) also adopts the corundum structure, consisting of edge and face-sharing iron and titanium cations throughout the crystal structure. 77,78A description of charge transfer processes in these systems, within a rst-principles framework, would represent a major development to their materials chemistry.10.

Fig. 1
Fig. 1 Face-sharing and edge-sharing AlO 6 octahedral units in the corundum structure of Al 2 O 3 (hexagonal crystal setting).The two sets of oxide ligands in the trigonal antiprismatic geometry (D 3d symmetry) are shown.

Fig. 2
Fig. 2 Configuration coordinate diagram used to illustrate the charge transfer process between two neighbouring cations in the Al 2 O 3 structure.It contains both an adiabatic charge transfer energy (E thermal ), which corresponds to the barrier to thermal activation, and two vertical processes (E opt and E opt 2).The shape of the curves is schematic.

Fig. 3
Fig. 3 Configuration coordinate diagram for the intervalence charge transfer between face-sharing iron and titanium bi-particles in the corundum structure (left).The curvature is schematic.An intravalence crystal field diagram of transition metal impurities in sapphire (D 3d ) symmetry is also drawn (right).

Fig. 4
Fig. 4 Schematic configuration coordinate diagrams for the homonuclear charge transfers between face-sharing iron (left) and titanium (right) bi-particles in the corundum structure.Note that the thermal ionisation energy is zero and E opt ¼ E opt 2 by symmetry.

3. 1
Ti-doped sapphire Sapphires doped only with Ti in the trivalent state are pale pink in colour representing the absorption of energy between 2.21 and 2.53 eV, as conrmed by optical spectroscopy.More specically, absorption maxima are given as 2.25 eV and 2.56 eV by Johnson et al.

Fig. 5
Fig. 5 Illustration of a Ti III -(Ti IV -Fe II ) tri-cluster.The L-shaped geometry of metal tri-clusters is shown, where X-Y are edge-sharing and Y-Z are face-sharing.The notation X-(Y-Z) is used in Table10.

Table 2
43lculated properties (structural, dielectric and elastic) for the rhombohedral primitive unit cell of corundum (experimental values taken from the CRC Handbook)43

Table 5
66pical absorption intensities for optical processes as governed by the given selection rules66

Table 4
Calculated crystal field transition energies (in eV) for isolated Ti III (d 1 ) and Fe III (d 5 ) impurities in the corundum structure.1, 2 and 3 refer to the first three optical transitions for D 3d symmetry (see Fig.3), while (SF) denotes a spin forbidden transition

Table 6
Intervalence charge transfer energies (vertical optical transitions) between substitutional bi-particles in the corundum structure.UNS refers to a configuration unstable with respect to optimisation

Table 10
Binding energy and charge transfer energy (in eV) for the most stable metal tri-clusters calculated using interatomic potentials.Metals in parentheses are parallel to the central metal in the cluster Cluster Binding E opt FeÀTi Fe III -(Ti III -Fe III ) À0.13 1.84 Fe III -(Fe III -Ti III ) À0.15 1.91 Fe III -(Ti IV -Fe II ) À0.33 À0.05 Fe III -(Fe II -Ti IV ) À0.44 0.17 Ti III -(Ti IV -Fe II )