Control of charge state of dopants in insulating crystals: Case study of Ti-doped sapphire

We study mechanisms of control of charge state and concentration of different point defects in doped insulating crystals. The approach is based on the density functional theory calculations. We apply it to the problem of obtaining of Ti-doped sapphire crystals with high figure-of-merit (FOM). The FOM of a given sample is defined as the ratio of the coefficient of absorption at the pump frequency to the coefficient of absorption at the working frequency of Ti:sapphire laser. It is believed that FOM is proportional to the ratio of the concentration of isolated Ti$^{3+}$ ions to the concentration of Ti$^{3+}$-Ti$^{4+}$ pairs. We find that generally this ratio is in inverse proportion to the concentration of Ti$^{4+}$ isolated substitutional defects with the coefficient of proportionality that depends on the temperature at which the thermodynamically equilibrium concentration of defects is reached. We argue that in certain cases the inverse proportion between concentrations of Ti$^{3+}$-Ti$^{4+}$ and Ti$^{4+}$ may be violated. We show that codopants that form positively (negatively) charged defects may decrease (increase) the concentration of positively charged defects formed by the main dopants. To evaluate the effect of codoping it is important to take into account not only isolated defects but defect complexes formed by codopants, as well. In particular, we show that codoping of Ti:sapphire with nitrogen results in an essential increase of the concentration of Ti$^{4+}$ and in a decrease of the FOM, and, consequently, growth or annealing in the presence of nitrogen or its compounds is unfavorable for producing Ti:sapphire laser crystals. The approach developed can be used for determining appropriate growth and annealing conditions for obtaining doped crystals with the required characteristics.


I. INTRODUCTION
Doping of insulating crystals with active ions is a widely used method of obtaining functional materials (active laser medium, luminescent materials, scintillators and many others) with required properties. Usually, in such materials dopant ions should be a certain charge state, occupy certain crystallography positions, and not form (or, instead, form) complexes with other dopant ions or intrinsic defects. The problem of control of the state of the dopant in a host matrix remains very important.
Density functional theory (DFT) is a powerful tool for evaluation of efficiency of different methods of such control. Using the results of DFT calculations and thermochemical data one can analyze finite-temperature properties. In particular, one can calculate equilibrium concentrations of different defects at a given temperature. Due to the condition of overall charge neutrality of defects the concentration of a given charged defect species cannot be calculated independently: in the general case this concentration depends on formation energies of all charged defects. One can imply that equilibrium (or almost equilibrium) concentrations of defects are reached under annealing. Therefore the temperature that enters into equations for equilibrium concentrations of defects can be associated with the temperature of annealing. For as-grown samples it can be replaced with the melting temperature T m . The situation becomes more complicated in a situation where concentration of dopants that enter into the crystal under its growth is smaller than their equilibrium concentration at T = T m . In this case the concentration of a given specie depends on the formation energies of all charged and uncharged defects.
In this paper we consider the problem of control of the charge state of dopants with reference to Ti-doped sapphire crystals. Some elements of our approach were already presented in our previous paper [1], where the results of DFT study of defect complexes of Ti-doped sapphire were reported.
We start with a short introduction where we describe the problem as it is formulated in the material science community.
Ti-sapphire is a widely used active laser medium. Operation of a Ti:Al 2 O 3 tunable laser was first reported by Moulton [2] in 1982. An active ion in Ti:sapphire is Ti 3+ substituted for the octahedrally coordinated Al 3+ . This ion has a single 3d electron above a closed shell. Five d-electron levels are split by the crystal field into an e g doublet and t 2g triplet. Transitions between the t 2g and e g levels are responsible for the absorption of visible light and nearinfrared fluorescence [3,4]. Titanium substituted for the Al ion can also be in the Ti 4+ state. A charge-transfer transition between O 2− and Ti 4+ causes ultraviolet (UV) absorption in Ti:sapphire [5]. Measurement of UV spectral characteristics gives the information on the concentration of Ti 4+ ions in Ti-sapphire samples [5,6].
Ti:sapphire exhibits weak near-infrared (NIR) absorption [3,4,[7][8][9][10] that results in losses at the wavelength of the laser emission. To qualify the performance of Ti:sapphire as a laser crystal the ratio of the absorption α m at the pump wavelength (λ m ≈ 500 nm) to the absorption α r at the laser emission wavelength (λ r ≈ 800 nm) is used. This ratio is known as a figure-of-merit (FOM) characterization of commercial materials. To calculate the FOM one should specify exact values of λ m and λ r . One of the accepted choices is λ m = 514 nm and λ r = 820 nm [11]. Slightly different λ m and λ r are also accepted [9].
NIR absorption is associated with Ti 3+ -Ti 4+ pairs. A correlation between NIR absorption and the concentration of Ti 3+ -Ti 4+ pairs was demonstrated in [9] where the dependence of α r on α m was measured. A partially oxidized sample studied in [9] was clear near the surface and pink inside that visualized a variation of the concentration of Ti 3+ across the sample. The total concentration of Ti c Ti was a constant. The obtained dependence of α r on α m is of a bell-like shape. This dependence is described by the formula α r ∝ α m (α 0 − α m ), where α 0 is some constant. It is believed that the concentration of Ti 3+ -Ti 4+ pairs (c Ti 3+ −Ti 4+ ) is proportional to the product of concentrations of isolated Ti 3+ and Ti 4+ ions. The first one is proportional to α m , and the second, to c Ti − c Ti 3+ ∝ α 0 − α m . Therefore the observation [9] correlates with the expectation that the adsorption coefficient α r depends linearly on c Ti 3+ −Ti 4+ . The pair mechanism of NIR absorption is supported by calculations of energy levels of the Ti 3+ Ti 4+ O 2− 9 cluster [12]. The energy difference [12] between the ground state and the first excited level of such a cluster corresponds to the wavelength λ = 813 nm that is in excellent agreement with the experimental value of λ at which the maximum of NIR absorption is observed.
The problem of NIR absorption was revisited recently in [13] where the absorption data in the range of wavelengths from 190 nm to 2000 nm were analyzed with reference to many Ti:sapphire samples of different origin. A sample with a large fraction of Ti 4+ (that was confirmed by a very weak absorption by this sample at λ = 490 nm) was used as a reference. The Ti 4+ scaling factor was defined as the absorption at λ = 225 nm by a given sample related to the absorption by the reference sample. It was implied that this factor is proportional to the concentration of Ti 4+ . If NIR is caused by Ti 3+ -Ti 4+ pairs, the FOM will be in inverse proportion to the Ti 4+ scaling factor. Nevertheless, some samples demonstrated strong deviation from this law. Based on this observation the authors of Ref. [13] arrived at the conclusion that the Ti 3+ -Ti 4+ pair model of NIR absorption needs a revision.
Defect energetics in Ti:sapphire were investigated within the DFT approach in [14,15]. It was shown that in the oxidized conditions Ti 4+ ions substituted for Al 3+ ions together with charge compensating vacancies of Al (V 3− Al ) are the most stable defects. In the reduced conditions the formation energy of substitutional Ti 3+ ions is of the smallest value. In the intermediate range of the oxygen potential the substitutional Ti 3+ and Ti 4+ defects exhibit similar formation energies, indicating that they can coexist. It was established that Ti 3+ ions demonstrate a tendency to form pairs and larger clusters. The binding energy of the Ti 3+ -Ti 3+ pair strongly depends on the distance between Ti 3+ ions. For the first nearest neighbors this energy is about 1.2 eV, but for the third nearest neighbors it is less than 0.2 eV. The binding energy of Ti 3+ triples and quadruples is about 2 eV and 3 eV, correspondingly. In addition, positively charged Ti 4+ ions bind in pairs with negatively charged Al vacancies V 3− Al with almost the same binding energy as one of Ti 3+ -Ti 3+ pairs. Energetics of defect complexes in Ti-doped sapphire was studied in detail in Ref. [1]. The formation energies and the binding energy of pairs, triples and quadruples formed by Ti 3+ , Ti 4+ and V 3− Al were calculated. It was shown that equilibrium concentrations of complex defects can be on the same order of or even larger than the concentration of isolated Ti 3+ or Ti 4+ defects. It was found that complex defects in Ti:sapphire influence significantly the balance between charged defects.
In this paper we present a thorough investigation of the problem of control of the charge state of titanium in Ti:sapphire. Our investigation compliments the study [14,15] and our recent study [1].
The method for the calculation of defect formation energies and of dopant concentration presented in Secs. III-V is the central result of this paper. It can be applied to dopant and defect complexes in different technologically relevant materials. In Secs. V and VI we answer a number of questions that arose in [13] and clarify the role of several factors that may increase the FOM such as an appropriate choice of the temperature of annealing, adding of certain compounds into the atmosphere or into the melt under the growth or annealing, co-doping with nonisovalent atoms. This analysis is another principal result of the paper. The approach developed is quite general and can be used for determining appropriate growth and annealing conditions to control the charge state of dopants in various insulating crystals not restricted to Ti:sapphire.

II. EQUILIBRIUM CONCENTRATION OF DEFECTS AND UNIVERSAL RELATION BETWEEN CONCENTRATIONS OF SIMPLE AND COMPLEX DEFECTS
In this section we derive the general relation between equilibrium concentrations of isolated and complex defects and show that this relation also takes place if the total number of dopant atoms is fixed.
Equilibrium concentrations of defects can be obtained from the condition of the minimum of the free energy. The free energy depends on the defect formation energies E i and the configurational entropy ln W , where W is the number of ways to place defects in the crystal: In Eq. (1) F 0 is the free energy of the perfect crystal, the sum is taken over all possible defect species, n i is the number of defects of the i-th specie, T is the temperature, and k B is the Boltzmann constant. We imply that the total number of defects n tot = n i1 + n i2 + . . . is much smaller than the number of lattice sites and approximate W by the equation where N i is the number of different positions and orientations for the i-th defect specie. We take into account that overall electric charge of all charged defects is zero. This results in the constraint where q i is the electrical charge of the i-th defect (below we use elementary charge units for q i ). If the total number of dopant in the crystal is fixed we have the additional constraint Here we specify the case of Ti dopants. In Eq. (4) k Ti,i is the number of Ti atoms per an i-th defect and n Ti is the total number of Ti ions in the crystal. The constraint (4) should be taken into account if an actual concentration of dopant differs from its equilibrium concentration at T = T m (for as-grown samples) or at the temperature of annealing. The free energy minimum with the additional constraints can be found by the method of Lagrange multipliers. The constraints (3) and (4) can be taken into account by two additional terms with two Lagrange multipliers λ q and λ Ti . The extremum condition yieldsñ Substituting Eq. (5) into Eqs. (3) and (4) we obtain two equations for λ q and λ Ti . One can find the general relation between the concentrations of isolated and complex defects using Eq. (5). Let us consider a complex defect i c composed of r 1 + r 2 + . . . r s simple defects: For such a defect Eq. (5) can be rewritten as where is the binding energy of the complex defect i c . Charge and particle number conservation requires q ic = s j=1 r j q ij and k Ti,ic = s j=1 r j k Ti,ij . Thus Eq. (7) reduces tõ Considering the problem with only one constraint (3) we put λ Ti = 0 from the beginning and again arrive at the relation (9). In the DFT approach the defect formation energy is given by the equation [16,17] where E def,i is the energy of the supercell with a given defect, E perf is the energy of the perfect supercell, p X,i is the number of atoms of type X (host or impurity atoms) that have been added to (p X,i > 0) or removed from (p X,i < 0) the supercell to form the defect of the i-th species, µ X is the chemical potential of the atom of the type X, µ e is the electron chemical potential, and E (c) i is the correction that excludes electrostatic interaction caused by periodic coping of charged defects in the supercell calculations.
Substituting Eq. (10) into Eq. (5) and redefining the Lagrange multiplierλ q = λ q − µ e one can exclude the electron chemical potential from the problem. This means that equilibrium concentrations of defects can be expressed through the quantities independently of µ e . If the total number of Ti atoms is fixed, the chemical potential of Ti can be excluded as well. In the latter case equilibrium concentrations of defects do not depend on µ Ti .
Applying Eq. (9) to Ti:sapphire we find that the ratio of the concentration of isolated Ti 3+ ions to the concentration of Ti 3+ − Ti 4+ pairs is in inverse proportion to the concentration of isolated Ti 4+ ions: Here and below we use the notations 3 ≡ Ti 3+ ≡ Ti 0 Al , 4 ≡ Ti 4+ ≡ Ti + Al , and 3 − 4 ≡ Ti 3+ -Ti 4+ . The factor of 2 in the denominator of Eq.

III. CALCULATION OF DEFECT FORMATION ENERGIES
To calculate defect formation energies we use the Kohn-Sham density functional method in the generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization for the exchange-correlation functional and doublezeta basis with polarization orbitals as implemented in the open source SIESTA code [18]. The pseudopotentials were generated with the improved Troullier-Martins scheme. The Al-3s 2 3p 1 , O-2s 2 2p 4 and Ti-4s 1 3d 3 electronic states are considered as valence ones. We do not include 3d semicore states into the valence ones due to the following reason. The heats of formation of crystal phases used as references for building the Al-Ti-O phase diagram (TiO 2 , Ti 3 O 5 , Ti 2 O 3 ) calculated with the use of the 4-electron state of Ti are in better coincidence with experimental quantities than the ones of the 10-electron state of Ti. In addition, calculations with 10 valence electrons per atom are slower than with 4. Actually there is no unambiguous answer as to where semicore states should be included [19]. To study co-doping with carbon, nitrogen and fluorine we set C-2s 2 2p 2 , N-2s 2 2p 3 , and F-2s 2 2p 5 valence electronic states.
Lattice parameters and atomic positions are optimized until the residual stress components converge to less than 0.1GPa and the residual forces are less than 0.01 eV/Å. The plane-wave cutoff energy of 250 Ry is used to calculate the total energy of the system. A 120-atom supercell of α-Al 2 O 3 is built of 4 optimized unit cells. One simple or complex defect is placed in the supercell. The optimization of atomic positions in the supercell with a defect is fulfilled again. Numerical integrations over the supercell Brillouin zone are performed at the Γ point. To check convergency of the Γ point integration we calculate several formation energies integrating over 3 × 3 × 3 and 5 × 5 × 5 grids of k-points. Integration over 3 × 3 × 3 grid yields Ti 3+ , Ti 4+ and V 3− Al formation energies higher than ones for the Γ point integration by 0.02 eV, 0.027 eV and 0.018 eV, correspondingly. The difference of energies given by integration over 3 × 3 × 3 and 5 × 5 × 5 grids of k-points is less than 10 −3 eV. Based on this results we conclude that restriction with integration over the Γ point (which considerably speed up the calculations) mainly results in unessential overestimate (about 10 %) of the total equilibrium concentration of Ti. The error in relative concentrations of different defects is less than 3 percents.
Chemical potentials of atoms that enter into Eq. (10) are calculated from chemical potentials of materials related to the Al − Ti − O system. The chemical potential of a given material is the sum of the zero-temperature DFT energy and a temperature correction part: where µ 0 (X) is the DFT energy of a crystal (per formula unit), or of an isolated molecule if the material is a gas at the standard conditions. The temperature correction is determined by the equation [20,21] ∆µ where H X (T, p) and S X (T, p) are the enthalpy and entropy at the temperature T and pressure p. We put p = 0.1 MPa (standard conditions) and take the values of H X (T, p) and S X (T, p) from the thermochemical tables [22]. The heat of formation H f for a compound with the general formula X u Y v , where X, Y = Al, Ti, C is calculated as Equations (14) and (15) give heats of formation per one atom. Chemical potentials in Eqs. (14) and (15) take into account the temperature correction Eq. (13). Using calculated heats of formation (14) and (15) Table I. Chemical potentials of atoms are calculated for all reference points using the equations that relate these potentials with the potentials Eq. (12). For instance, for the point A of the ternary system we have I. Phases in equilibrium with Al2O3 and the oxygen chemical potential in reference points of the ternary and fourcomponent phase diagrams at T = Tm. The potential µO is counted from the energy of an isolated oxygen atom.
TiO2, Ti3O5 TiO2, Ti3O5, CO2, TiO2, Ti3O5, N2 TiO2, Ti3O5, AlF3 -8.37 The expression for the defect formation energies (10) contains the difference of the energies of a defect and of the perfect supercell E d−p,i = E def,i − E perf . Within the DFT method we calculate this difference at T = 0. At finite T one should take into account a temperature correction ∆E d−p,i (T ) to this difference. The origin of ∆E d−p,i (T ) is differences of finite temperature enthalpy and entropy of a defect and a perfect supercell. We evaluate ∆E d−p,i (T ) from the temperature corrections Eq. (13) for relevant materials in the crystal state.
The difference ∆E d−p,Ti Al (T ) is evaluated from ∆µ T for pure Al 2 O 3 and Ti 2 O 3 : To evaluate the temperature correction for the supercell with a vacancy we imply that Equation (20) can be justified as follows. If one removes an integer number of formula units from the supercell and rearranges atoms one can obtain another perfect supercell. The formation energy of such a "defect" is equal to zero and the temperature correction to E i is equal to zero as well. The temperature correction to E i contains the corrections to µ X and to E def,i − E perf and they should cancel each other. Equation (20) and Eq. (20). Temperature corrections to the energies of substitutional N, C, and F defects are evaluated from ∆µ T (AlN), ∆µ T (Al 4 C 3 ) and ∆µ T (AlF 3 ), correspondingly. For instance, to evaluate ∆E d−p,NO (T ) we use the relation Equation (23)  in the expression for the defect formation energy (10) is evaluated by a method similar to the one proposed in [23,24]. DFT supercell calculations give the formation energies E (0) i with spurious electrostatic interaction between periodically arranged charged defects. Calculations of the charge distribution show that the defect charge is strongly localized. Therefore, the main contribution to the electrostatic energy comes from thr monopolemonopole interaction, while the contribution of the monopole-dipole and dipole-dipole interaction [25,26] is much smaller. This allows us to equate E (c) i with the Madelung energy taken with a negative sign. We imply that the Madelung energy for a supercell composed of n × m × k unit cells can be presented in the form where q ef f is the effective charge of the defect, ε ef f is the effective dielectric constant, andẼ M (n, m, k) is the electrostatic energy of a periodic structure of q = 1 point charges in the medium with ε = 1 and with a charge compensating background. For a cubic latticeẼ M (n, n, n) = −2.837/na, where a is the lattice parameter. In the general caseẼ M (n, m, k) depends on the size and the form of the supercell. For elongated supercells it can be positive.
For each defect specie we calculate the supercell defect formation energy E (0) i for five different supercells: (2, 2, 1), (2, 3, 1), (3, 3, 1), (2, 2, 2) and (2,2,3). Each supercell is built from hexagonal unit cells with lattice parameters a = b = 4.86Å and c = 13.19Å obtained for pure Al 2 O 3 by optimization of lattice vectors. Then, we calculate the Madelung energyẼ M by the Ewalds method for the same supercells, define the dimensionless quantity v M (n, m, k) = aẼ M (n, m, k), and fit the energy E As two examples we present the result of fitting for Ti 4+ and V 3− Al in Fig 2. Taking into account that for infinitely large supercell v M (n, m, k) = 0, we evaluate the corrected formation energy as E i = c 0,i and the correction E i (2, 2, 1) for the considered charged defects are given in Table II. Note that if we set ε ef f = 10 we obtain the effective charges |q Ti 4+ ,ef f /e| = 0.76 and |q V 3− Al ,ef f /e| = 2.52 which looks reasonable.
for several supercells versus vM and linear fit (25). For concreteness, we set µe equals to the valence band maximum.

IV. EQUILIBRIUM CONCENTRATIONS OF DEFECTS IN Ti:SAPPHIRE
As was already mentioned in the Introduction the equilibrium concentration of a given charged defect depends on the formation energies of all charged defects. If one considers the additional constraint (4) one should know the formation energies of all uncharged defects as well. In practice defect species with large formation energies can be ignored since their contribution in Eqs. (3) and (4) is negligible. to the defect formation energy for the 2 × 2 × 1 supercell We have calculated formation energies of different defects including substitutional and interstitial ions of Ti, C, N, and F in different charge states, native defects (Al and O vacancies and interstitials), and complexes of such defects. We have found that most of defects have rather large formation energies. We restrict our analysis with several defects species with the smallest formation energies. In the case of Ti:sapphire without co-dopants we consider three species of isolated defects ( Ti 3+ , Ti 4+ and V 3− Al ), three pairs (Ti 3+ -Ti 3+ , Ti 3+ -Ti 4+ and Ti , and a quadruple complex Ti 4+ -Ti 4+ -Ti 4+ -V 3− Al . For Ti:sapphire with co-dopants we add to this list two or three defect species (see below) formed by co-dopants.
In this section we concentrate on Ti:sapphire without co-dopants. If the total number of Ti atoms is not fixed (the constraint (4) is not applied) the concentrations of charged defects are found from the following system of equations: where we use the notation V ≡ V 3+ Al . In Eq. (27) we take into account the relation (9). The factors C ic are the coefficients in the relation (9) averaged over different orientations and configurations: where E ic,f is the binding energy of the complex i c and K ic,f is the number of configurations and orientations with the same energy (different f label configurations and orientations with distinct energies). Calculated binding energies and the numbers K ic,f are given in Table III. Binding energies are independent of the chemical potentials of atoms. Binding energies given in Table III include the electrostatic correction E (c) i . They differ from ones calculated in Refs. [1,15], where this correction was not taken into account. The concentration of electrically neutral defects depends only on its own formation energy:ñ The formation energies of Ti 3+ and of the electrically neutral combination of three Ti 4+ and one V 3− Al at T = T m are given in Table IV for all reference points of the Al − O − Ti phase diagram. Using Eqs. (26), (27) and (29) and the data from Tables III and IV we         that of Ti 3+ -Ti 3+ pairs. The concentrations of the Ti 3+ -Ti 3+ -V 3− Al , and Ti 3+ -Ti 4+ -V 3− Al complexes are not displayed either. They are generally much lower than the concentration of Ti 4+ -Ti 4+ -V 3− Al complexes (they are comparable only in the reduced conditions).
One can see from Fig. 3 that the overall concentration of Ti varies in the range from 6 · 10 19 cm −3 to 3.5 · 10 20 cm −3 that corresponds to the range from 0.12 wt% to 0.7 wt% of Ti. This concentration is rather high. For instance, in samples investigated in [13] the concentration of Ti was in the range from 0.006 to 0.2 wt%. Samples used in Refs. [3-5, 8, 11] had the concentration of Ti less than 0.1 wt%. The concentration of Ti in samples grown in [6] was in the range from 0.1 to 0.25 wt%.
If the total concentration of Ti differs from the equilibrium one we should take into account the constraint (4). Then we arrive at a system of three equations:ñ . The quantity 3E  To obtain crystals with a low concentration of Ti 4+ and high FOM it is desirable to grow them in the reduced conditions. To do that one can add certain substances or compounds to the melt or to the atmosphere or anneal samples in their presence. A side effect of such a treatment is codoping of Ti:sapphire. Charged defects formed by dopants may change a balance between different Ti defects which results in a change of the ratio of concentration of isolated Ti 3+ ions to the concentration of Ti 3+ −Ti 4+ pairs. The mechanism is quite simple. If codoping results in an appearance of additional positively charged defects, some negatively charged vacancies V 3− Al compensate codopants and the concentration of Ti 4+ decreases. Consequently, according to Eq. (11) (valid for codoped samples, as well) the ratio c Ti 3+ /c Ti 3+ −Ti 4+ increases. If codoping results in the appearance of additional negatively charged defects, the concentration of isolated Ti 4+ ions increases to provide charge compensation. Consequently, the ratio c Ti 3+ /c Ti 3+ −Ti 4+ decreases. To determine the role of a given co-dopant one should take into account not only simple charged defects formed by co-dopant atoms but complex defects as well. Among such complex defects it is important to consider pairs and triples formed by co-dopants with vacancies V 3− Al and with Ti 4+ . In addition to a shift of the charge balance, co-doping may create additional impurity levels in the band gap that will influence optical properties of crystals.
In this section we consider co-doping of Ti:sapphire with carbon, nitrogen and fluorine. Our results correlate with ones obtained for C-doped α-Al 2 O 3 in [27][28][29], where the formation energies of substitutional and interstitial carbon defects were calculated. It was shown in [27][28][29] that in the reduced conditions the substitutional C O defects have the smallest formation energy, while in the oxidized conditions the substitutional C Al defects are energetically preferable. The formation energies of interstitial carbon defects are large both in the reduced and in the oxidized conditions. Here we do not consider C Al but find that in the oxidized conditions the concentrations of C O are extremely low ones (see Fig. 7 below). We cannot compare directly the results of [27][28][29]   with our results since in [27][28][29] the chemical potential of C was set to be the same as in diamond [27] or in graphite [28,29] irrespective of the value of the oxygen chemical potential. In Fig. 6 we display equilibrium concentrations of Ti defects and Al vacancies at the reference points of the Al−Ti−O−C phase diagram at T = T m . The concentrations are obtained from Eqs. (26) and (27), where we neglect the contribution of carbon defects.
The concentrations of carbon defects are found from the relations  Table IV, and the binding energy of the C 2− O − Ti 4+ pair, in Table III. Calculated concentrations of carbon defects are shown in Fig. 7.
In Fig. 8 Table IV. In Fig. 9 we present calculated concentrations of Ti defects and Al vacancies at T = T m at the reference points of the Al − Ti − O − N phase diagram. One can see that the presence of nitrogen defects results in a considerable increase of the concentration of Ti 4+ and in an increase of the overall concentration of Ti in the reduced and intermediate condition.
Negatively charged nitrogen defects can bind in pairs with Ti 4+ . The binding energy of such a pair is given in Table  III. The calculated concentration of isolated N − O defects and N − O − Ti 4+ pairs, and the total concentration of nitrogen are shown in Fig. 10. In the oxidized conditions the concentration of nitrogen defects decreases and they do not influence the concentration of Ti 4+ . The concentration of N − O − Ti 4+ pairs is slightly higher than the concentration of isolated N − O defects, but since these pairs are electrically neutral, they do not influence the concentration of charged defects.
The increase in the concentration of Ti 3+ − Ti 4+ pairs results in a decrease of the c Ti 3+ /c Ti 3+ −Ti 4+ ratio and a decrease of the FOM.
which follows from Eq. (5). The difference E F + O − E 4 is given in Table IV. The binding energies of the complexes Table III  formation of F + O − F + O − V 3− Al triples. We note that such a reduction is not a general rule. It would not happen if the binding energy of complexes was smaller.

D. Influence of co-doping on FOM
To determine the impact of carbon, nitrogen and fluorine compounds on the FOM of Ti:sapphire we calculate the c Ti 3+ /c Ti 3+ −Ti 4+ ratio at all reference points of Table I using the data presented in Figs. 6, 9 and 12. The result is shown in Fig. 14. Since the concentration of carbon defects is low, at the points A, B, D, I ,J the result is the same for Al−Ti−O−C and Al−Ti−O systems. At the same time under conditions that corresponds to the points G and L one can to reach the much larger ratio of c Ti 3+ /c Ti 3+ −Ti 4+ . The latter is connected with the low total equilibrium concentration of Ti at these points. This conclusion is in agreement with experimental study [30] where it was shown that thermal carbon treatment of raw materials (Al 2 O 3 and TiO 2 ) makes it possible to decrease the concentration of Ti 4+ in Ti:Al 2 O 3 crystals. With this, Fig. 14 illustrates that nitrogenization provokes considerable reduction of the c Ti 3+ /c Ti 3+ −Ti 4+ ratio in the reduced and intermediate conditions, and that fluoridization leaves this ratio almost unchanged.
It is instructive to evaluate the relation between the FOM and c Ti 3+ /c Ti 3+ −Ti 4+ . To do that we use the data of [13]. One of the samples investigated in [13] (labeled as SY1b) had FOM = 12. Concentrations of Ti 3+ and Ti 4+ in this sample were estimated as 1.3 · 10 18 cm −3 and 2.3 · 10 18 cm −3 , correspondingly. Using Eq. (11) and taking T = 2000 K we obtain c Ti 3+ −Ti 4+ ≈ 10 16 cm −3 and c Ti 3+ /c Ti 3+ −Ti 4+ ≈ 100. Thus one can estimate that  Table I Co-doping may result in the appearance of additional defect levels in the band gap. To consider this effect we calculate the band structure of the system with one given defect per supercell. Strictly speaking, from such calculations one obtains impurity bands of a crystal with periodically arranged defects. But since such bands are very narrow they can be associated with impurity levels connected with a given defect. The position of obtained impurity levels in the band gap and separation between them weakly depend on the supercell size. This can be seen from comparison of the results of Refs. [31,32] and our calculations [1].
In Figs. 15, 16 and 17 we present the band structure of a sapphire crystal with carbon, nitrogen and fluorine defects. For comparison, in Fig. 15d and Fig. 17d we reproduce the band structures of crystals with Ti 4+ substitutional defects and with V 3− Al vacancies [1]. One can see that fluorine defects do not cause additional impurity levels in the band gap. Formation of complexes of fluorine defects with Al vacancy results in a minor modification of Al vacancy levels. In contrast, negatively charged carbon and nitrogen defects (C 2− O , C − O and N − O ) reveal themselves in an appearance of additional impurity levels. Binding of such defects with Ti 4+ causes splitting of Ti impurity levels.
Due to small equilibrium concentration of carbon defects one can expect that carbon impurities will not influence significantly optical properties of Ti:sapphire. Equilibrium concentration of nitrogen defects is much higher. Therefore growth or annealing in the presence of nitrogen compounds may result in an essential modification of optical properties of Ti:sapphire not only due to a change of the balance between Ti 3+ and Ti 4+ ions, but also due to the appearance of impurity levels caused by nitrogen defects. In this section we discuss whether the observed relations between the concentrations of isolated Ti 3+ and Ti 4+ defects, and Ti 3+ − Ti 4+ pairs confirm or put in question the pair model of NIR absorption.
According to Eq. (11), for samples where the equilibrium concentration of defects was reached at the same temperature, e.g. the samples are annealed at the same T , the factor exp(−E (b) 3−4 /k B T ) will be the same and the ratio of the concentration of Ti 3+ to the concentration of Ti 3+ − Ti 4+ will be in inverse proportion to the concentration of Ti 4+ . Assuming that the c Ti 3+ /c Ti 3+ −Ti 4+ ratio determines the FOM one can expect that the FOM would be in inverse proportion to c Ti 4+ as well. At the same time samples annealed at different temperatures will have different coefficients of proportionality between these quantities and their FOM may not demonstrate such a proportionality. Let us consider the following example. We imagine that someone obtained five samples labeled as c, d, f, i and j. These samples were grown in the conditions that correspond to the reference points C, D, F, I and J of the Al−Ti−O−C phase diagram, respectively. According to our calculations (see Fig. 6) these samples should have approximately the same total concentration of Ti, but different concentrations of Ti 4+ . Then we imagine that each sample was divided into seven parts and 30 samples were annealed in the conditions that correspond to the reference points C, G and L of  the Al − Ti − C phase diagram at two different temperatures, T = 2100 K and T = 2000 K. The reference points for the Al − Ti − C diagram and the value of µ O at three different temperatures are given in Table V (under lowering in temperature two reference points, C and G, disappear and are replaced with the point C 1 ). We imply that the total concentration of Ti is not changed under annealing. As a result there were obtained 35 samples: 5 parent samples c, d, f, i and j; 15 samples annealed at T = 2100 K in the conditions that correspond to the points C, G and L of the Al−O−C phase diagram and labeled as xCa, xGa, and xLa (x=c,d,f,i,j); and 15 samples annealed at T = 2000 K and labeled as xCb, xGb, and xLb, correspondingly). We calculate the c Ti 3+ /c Ti 3+ −Ti 4+ ratio for these samples and plot it against 1/c Ti 4+ (Fig. 18). One can see that the points in Fig. 18 belong to three zero origin straight lines with different slopes. If one takes randomly several samples from this 35-sample set and measures the FOM one may find that the FOM is not proportional to 1/c Ti 4+ . But this does not mean that the pair model of NIR absorption is incorrect. This example demonstrates that it is important to investigate samples annealed at the same temperature to verify the pair model of NIR absorption. The dependencies presented in Fig. 18 also demonstrate that annealing in the reduced conditions (point L) should increase the FOM and the effect is larger at lower temperature of annealing. At the same time annealing in the intermediate conditions (points G and C) may decrease the FOM. Let us also discuss the possible scenario of violation of proportionality between the FOM and 1/c Ti 4+ even for samples annealed at the same temperature. We imply that at rather low temperature the diffusion coefficient for certain defects is so small that such defects can be considered as frozen ones. Let Ti ions be such defects while Al vacancies remain mobile ones. To calculate concentrations of Ti defects we take into account two additional constraints These constraints forbid formation of new Ti clusters and dissociation of existing Ti clusters. The concentrations of defects with one Ti (c 1 ) and with two Ti ions (c 2 ) calculated for the same imaginary samples c, d, f, i and j are shown in Fig. 19. For the system with the constraints (36) and (37) the relation (9) is not satisfied. Therefore the relation (11) is violated which can be demonstrated by direct calculations.
We calculate equilibrium concentration of Ti 3+ , Ti 4+ and Ti 3+ − Ti 4+ under the constraints (36) and (37) at T = 1600 K (we imply that the frozen regime is reached at this temperature). We fix µ O that corresponds to the point C 1 (Table V). The quantities c 1 and c 2 are fixed to ones as in the samples c, d, f, i and j. The results of the calculations are shown in Fig. 18 by triangles labeled as c1, d1, f1, i1 and j1. One can see that indeed c Ti 3+ /c Ti 3+ −Ti 4+ is not proportional to 1/c Ti 4+ . Comparing Fig. 18 and Fig. 19 we find that c Ti 3+ /c Ti 3+ −Ti 4+ correlates with the c 1 /c 2 ratio. Note that the violation of the relation (11) and more general relation (9) is caused solely by the additional constraints (36) and (37), and therefore this effect most probably holds true for a wide range of "freezing" temperatures.
To conclude this analysis we discuss one more feature observed in [13]. It was found in [13] that for some sets of samples the absorption coefficient α 820 is proportional to the square of the absorption coefficient α 490 . If NIR absorption is caused by Ti 3+ − Ti 4+ pairs such a behavior means that the concentration of Ti 3+ − Ti 4+ pairs is proportional to the square of the concentration of Ti 3+ isolated ions. It cannot be the general property. Nevertheless our calculations show that for the samples annealed at the same µ O this property is satisfied at least approximately. We specify µ O = −10.0 eV and calculate equilibrium concentrations of defects at T = T m , T = 2100 K and T = 2000 K assuming that the total concentration of Ti c Ti is fixed. We consider 20 different c Ti in the range from 5 · 10 18 cm −3 to 10 20 cm −3 . Obtained concentrations of Ti 3+ − Ti 4+ are plotted in Fig. 20 against the concentration of Ti 3+ . One can see that for given T the law c Ti 3+ −Ti 4+ = κc 2 Ti 3+ is satisfied. The coefficient κ weakly depends on temperature. Thus the observed in [13] correspondence between α 820 and α 490 does not contradict the pair model of NIR absorption.

VII. CONCLUSION
In conclusion, with reference to Ti:doped sapphire we analyzed factors that determine the charge state of dopants in insulating crystals. Basing on DFT calculation combined with thermochemical data we found that Ti atoms enter For laser application it is important to reduce the concentration of Ti 3+ − Ti 4+ pairs keeping concentration of unpaired Ti 3+ ions at high level. To increase c Ti 3+ /c Ti 3+ −Ti 4+ ratio to the level of 10 3 or higher the crystals should be grown or annealed in the reduced conditions. Alternatively one can fix the total concentration of Ti at a relatively low level ( < ∼ 10 18 cm −3 ). Annealing in the conditions that correspond to intermediate values of the oxygen chemical potential may result in a decrease of the c Ti 3+ /c Ti 3+ −Ti 4+ ratio.
We have shown that codopants that form charged defects may change relative concentrations of the main dopants in different charge states. In particular, we have found that growth or annealing of Ti:sapphire in the presence of nitrogen compounds results in the appearance of a large number of negatively charged nitrogen defects that cause a decrease of the c Ti 3+ /c Ti 3+ −Ti 4+ ratio.
We have demonstrated that growth or annealing of doped crystals in the presence of additional compounds may influence the charge state of dopants indirectly. Such a situation is realized in Ti:sapphire grown or annealing in the presence of carbon or its compounds. Carbon defects have large formation energies and their equilibrium concentrations are quite low. Nevertheless the presence of carbon and its compounds may increase the c Ti 3+ /c Ti 3+ −Ti 4+ ratio due to lowering of the oxygen chemical potential and increasing of the titanium chemical potential.
We analyze the general relation between the concentrations of isolated and complex defects. We find that normally the concentration of complex defects is proportional to the product of concentrations of isolated defects which form the complex defect. The coefficient of proportionality depends on the binding energy and on the temperature of annealing or the melting temperature (for as-grown samples). This relation is violated if some defects are frozen. Such a situation is expected at rather low temperature of annealing. In application to Ti:sapphire it means that the c Ti 3+ /c Ti 3+ −Ti 4+ ratio and FOM are proportional for the inverse concentration of Ti 4+ only for samples annealed at the same temperature and even in that case such a proportionality can be violated if the temperature of annealing is quite low.