# A note on nonlinear diffusive equations on Poincaré half plane

Roberto Garra, Francesco Maltese

###### Abstract

In this paper we show some explicit results regarding non-linear diffusive
equations on Poincaré half plane. We obtain exact solutions by using the
generalized separation of variables and we also show the meaning of these
results in the context of the general theory of the invariant subspace method.

Keywords: Nonlinear diffusion equation, Hyperbolic geometry, Exact solutions.

##  1 Introduction

The analysis of exact solutions for nonlinear diffusive equations plays a
central role for the applications, for example in the physics of porous media
[8]. There are many different methods to find particular exact interesting
solutions for nonlinear PDEs, for example by using the invariant subspace
method [1] or the generalized separation of variables [5, 6]. In the recent
paper [2], the authors study the non-linear time-fractional equation

$\frac{\partial^{\nu}}{\partial
t^{\nu}}u(\eta,t)=\Delta_{H}u^{n}-u=\frac{1}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u^{n}}{\partial\eta}-u,\quad n>1,$ (1.1)

involving the time-fractional derivative $\partial^{\nu}/\partial t^{\nu}$ in
the sense of Caputo. Motivated by this study, here we consider two general
class of nonlinear equations in the hyperbolic plane, showing that it is
possible to construct interesting exact solutions for these equations by using
simple methods.
We first consider a more general formulation of the equation (1.1), that is

$\widehat{O}_{t}u(\eta,t)=\Delta_{H}u^{n}-u=\frac{1}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u^{n}}{\partial\eta}-u,\quad n>1,$ (1.2)

where $\widehat{O}_{t}$ is a linear differential operator acting on the
variable $t$. We show that this equation admits elementary exact solutions
that can be constructed by separation of variables. These results can be
proved rigorously by using the invariant subspace method for example. We
consider some particular interesting cases in the wide family of nonlinear
equations (1.2), when the linear operato $\widehat{O}_{t}$ is a first order
time-derivative or a Laguerre derivative or a fractional derivative. In the
second part of the paper we consider the following family of nonlinear
equations in the hyperbolic plane

$\widehat{O}_{t}u(\eta,t)=u\Delta_{H}u=\frac{u}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u}{\partial\eta}.$ (1.3)

This case is particularly interesting, since it admits useful non trivial
exact solutions.
Essentially there are few investigations about nonlinear fractional equations
on the hyperbolic plane. Our main aim is to provide some ideas for the
construction of exact solutions for some family of interesting nonlinear
equations. By using similar methods, many other problems can be solved.

##  2 Construction of exact solutions for nonlinear diffusive equations on
Poincaré half plane

###  2.1 A first interesting case

There are many families of nonlinear equations that can be solved by means of
the invariant subspace method (see [1] for the details). As far as we know,
there are few studies about the application of this method to solve nonlinear
equations in the hyperbolic space. We show the potential utility of this
method, starting from a generalization of the time-fractional nonlinear
diffusive equation on Poincairé half plane studied in [2], that is

$\widehat{O}_{t}u(\eta,t)=\Delta_{H}u^{n}-u=\frac{1}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u^{n}}{\partial\eta}-u,\quad n>1.$ (2.4)

We have the following result.

######  Theorem 1.

The equation (2.4) admits a solution of the form

$u(\eta,t)=f(t)\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}},$ (2.5)

where $f(t)$ is a solution of the equation

$\widehat{O}_{t}f(t)=-f(t)$

and $c_{1}$ e $c_{2}$ are real constants such that
$c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}>0$.

###### Proof.

We observe that

$\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial}{\partial\eta}\bigg{[}f(t)\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}}\bigg{]}^{n}=\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial}{\partial\eta}\bigg{[}f^{n}(t)(c_{1}\ln(\tanh\frac{\eta}{2})+c_{2})\bigg{]}$

$=f^{n}(t)\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial}{\partial\eta}\bigg{[}c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}\bigg{]}=0$

Thus in this case $u(\eta,t)$ belongs to kernel of $\Delta_{H}$. From this
observation, we can insert the function $u(\eta,t)$ in the equation (2.4) and
we have

$\widehat{O}_{t}\bigg{(}f(t)\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}}\bigg{)}=\frac{1}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial}{\partial\eta}\bigg{[}f(t)\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}}\bigg{]}^{n}-f(t)\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}}$

$\left(\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}}\right)\cdot\widehat{O}_{t}f(t)=-f(t)\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}}$

$\widehat{O}_{t}f(t)=-f(t)$

By hypothesis on $f(t)$ we obtain the claimed result. ∎

By means of Theorem 2.1 we can provide an exact solution for (2.4) in
different interesting cases. For example if $\widehat{O}_{t}$ coincides with a
Caputo fractional derivative of order $\beta\in(0,1)$ , i.e. in this case we
have that

$\widehat{O}_{t}u(\eta,t)=\frac{\partial^{\beta}u}{\partial
t^{\beta}}=\frac{1}{\Gamma(1-\beta)}\int_{0}^{t}(t-\tau)^{-\beta}\frac{\partial
u}{\partial\tau}d\tau,$ (2.6)

we have that the function

$u(\eta,t)=E_{\beta}(-t^{\beta})\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}},$
(2.7)

is a solution for the non-linear time-fractional equation

$\frac{\partial^{\beta}}{\partial
t^{\beta}}u(\eta,t)=\frac{1}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u^{n}}{\partial\eta}-u,\quad n>1.$ (2.8)

In (2.7) we denoted by $E_{\beta}(-t^{\beta})$ the one-parameter Mittag-
Leffler function, i.e. the eigenfunction of the Caputo fractional derivative
(see e.g. [3]).
For $\beta=1$ we recover the solution

$u(\eta,t)=e^{-t}\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}},$ (2.9)

for the classical equation

$\frac{\partial}{\partial
t}u(\eta,t)=\frac{1}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u^{n}}{\partial\eta}-u,\quad n>1.$ (2.10)

Another interesting case is when $\widehat{O}_{t}=\frac{\partial}{\partial
t}t\frac{\partial}{\partial t}$ that is the so-called Laguerre derivative (see
for example [7]). In this case we have that the function

$u(\eta,t)=C_{0}(t)\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}},$ (2.11)

is a solution for the equation

$\frac{\partial}{\partial t}t\frac{\partial}{\partial
t}u(\eta,t)=\frac{1}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u^{n}}{\partial\eta}-u,\quad n>1.$ (2.12)

In the Equation (2.11) the function $C_{0}(t)$ is a Bessel-type function, i.e.

$C_{0}(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{k!^{2}}$ (2.13)

By using this approach we can construct exact solutions for many different
nonlinear PDEs involving the hyperbolic Laplacian. For example, an interesting
outcome inspired by [4] is given by the following result

######  Proposition 1.

The nonlinear equation

$\frac{\partial u}{\partial
t}-\frac{i\omega}{\alpha}u=\frac{1}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u^{n}}{\partial\eta},$ (2.14)

admits the completely periodic separating variable solution

$u(x,t)=\exp\bigg{(}\frac{i\omega}{\alpha}t\bigg{)}\sqrt[n]{c_{1}\ln(\tanh\frac{\eta}{2})+c_{2}}.$
(2.15)

The proof can be simply obtained by direct substitution.

###  2.2 The second case

######  Theorem 2.

The nonlinear diffusion equation

$\widehat{O}_{t}u(\eta,t)=\frac{u}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u}{\partial\eta}$ (2.16)

admits an explicit solution of the form

$u(\eta,t)=f_{1}(t)\ln(\sinh\eta)+f_{2}(t)\ln(\tanh\frac{\eta}{2})+f_{3}(t),$
(2.17)

where $f_{1}(t)$ is a solution of the nonlinear equation

$\widehat{O}_{t}f_{1}(t)=f_{1}^{2}(t),$ (2.18)

while $f_{2}$ and $f_{3}$ satisfy the equations

$\widehat{O}_{t}f_{2}=f_{1}f_{2},\quad\widehat{O}_{t}f_{3}=f_{1}f_{3}.$ (2.19)

###### Proof.

We observe that

$\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial}{\partial\eta}\bigg{[}f_{2}(t)\ln(\tanh\frac{\eta}{2})+f_{3}(t)\bigg{]}=0,$
(2.20)

and

$\frac{f_{1}(t)\ln(\sinh\eta)}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
f_{1}\ln(\sinh\eta)}{\partial\eta}=f_{1}^{2}\ln(\sinh\eta).$ (2.21)

Therefore, if we search a solution of the form

$u(\eta,t)=f_{1}(t)\ln(\sinh\eta)+f_{2}(t)\ln(\tanh\frac{\eta}{2})+f_{3}(t)$
(2.22)

we have, by substitution

$\ln(\sinh\eta)\widehat{O}_{t}f_{1}(t)+\ln(\tanh\frac{\eta}{2})\widehat{O}_{t}f_{2}(t)+\widehat{O}_{t}f_{3}(t)=f_{1}^{2}\ln(\sinh\eta)+f_{1}f_{2}\ln(\tanh\frac{\eta}{2})+f_{1}f_{3}.$
(2.23)

Therefore the function (2.22) is a solution if

$\widehat{O}_{t}f_{1}(t)=f_{1}^{2},\quad\widehat{O}_{t}f_{2}=f_{1}f_{2},\quad\widehat{O}_{t}f_{3}=f_{1}f_{3}.$

as stated. ∎

######  Remark.

Following the theory of the invariant subspace method, we have used the fact
that the equation (2.16) admits the invariant subspace
$W^{3}:<1,\ln(\sinh\eta),\ln(\tanh\frac{\eta}{2})>$.

We now consider some interesting particular cases that are related to
classical mathematical models. First of all we consider the classical
nonlinear diffusive case

$\frac{\partial u}{\partial
t}=\frac{u}{\sinh\eta}\frac{\partial}{\partial\eta}\sinh\eta\frac{\partial
u}{\partial\eta}.$ (2.24)

In view of the previous Theorem, we can construt an exact solution by solving
the following simple ODEs

$\frac{d}{dt}f_{1}(t)=f_{1}^{2},\quad\frac{d}{dt}f_{2}=f_{1}f_{2},\quad\frac{d}{dt}f_{3}=f_{1}f_{3}.$

Therefore we have that a solution of the equation (2.24) is given by

$u(\eta,t)=\frac{\ln(\sinh\eta)}{t_{0}-t}+c_{1}\frac{\ln(\tanh\frac{\eta}{2})}{t_{0}-t}+\frac{c_{2}}{t_{0}-t}.$
(2.25)

## References

  * [1] Galaktionov, V. A., & Svirshchevskii, S. R. (2006). _Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics_. CRC Press.
  * [2] Garra, R., Maltese, F., Orsingher, E., A note on generalized fractional diffusion equations on Poincaré half plane, _Fractional Differential Calculus_ , 11(1), 111-120, (2021)
  * [3] Gorenflo, R., Kilbas, A. A., Mainardi, F., Rogosin, S.V., _Mittag-Leffler functions, related topics and applications_ , 2d Edition, Berlin: Springer (2020)
  * [4] Mariani, M., Calogero, F., Isochronous PDEs, _Physics of Atomic Nuclei_ , 68.5 (2005): 899-908.
  * [5] Polyanin, A. D., Zaitsev, V. F. (2004). _Handbook of nonlinear partial differential equations_. CRC press.
  * [6] Polyanin, A. D., Zhurov, A. I. (2021). _Separation of variables and exact solutions to nonlinear PDEs_. CRC Press.
  * [7] Ricci, P. E. (2020). Laguerre-type exponentials, Laguerre derivatives and applications. A survey. _Mathematics_ , 8(11), 2054.
  * [8] Vázquez, J. L. (2015). Fundamental solution and long time behavior of the porous medium equation in hyperbolic space. _Journal de Mathématiques Pures et Appliquées_ , 104(3), 454-484.


# Control of charge state of dopants in insulating crystals: case study of Ti-
doped sapphire.

L. Yu. Kravchenko1, D. V. Fil1,2<EMAIL_ADDRESS>1Institute for Single
Crystals, National Academy of Sciences of Ukraine, 60 Nauky Avenue, Kharkiv
61072, Ukraine
2V.N. Karazin Kharkiv National University, 4 Svobody Square, Kharkiv 61022,
Ukraine

###### Abstract

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 one of standard specifications of
commercial Ti:sapphire laser crystals. It is believed that FOM is proportional
to the ratio of the concentration of isolated Ti3+ ions to the concentration
of Ti3+-Ti4+ pairs. We find that generally this ratio is in inverse proportion
to the concentration of Ti4+ 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
Ti3+-Ti4+ and Ti4+ may be violated. The role of codoping in the control of the
charge state of dopants is analyzed. 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 Ti4+ 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:Al2O3
tunable laser was first reported by Moulton [2] in 1982. An active ion in
Ti:sapphire is Ti3+ substituted for the octahedrally coordinated Al3+. 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 near-infrared fluorescence [3, 4]. Titanium
substituted for the Al ion can also be in the Ti4+ state. A charge-transfer
transition between O2- and Ti4+ causes ultraviolet (UV) absorption in
Ti:sapphire [5]. Measurement of UV spectral characteristics gives the
information on the concentration of Ti4+ 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
$\alpha_{m}$ at the pump wavelength ($\lambda_{m}\approx 500$ nm) to the
absorption $\alpha_{r}$ at the laser emission wavelength ($\lambda_{r}\approx
800$ 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 $\lambda_{m}$ and $\lambda_{r}$. One of the accepted
choices is $\lambda_{m}=514$ nm and $\lambda_{r}=820$ nm [11]. Slightly
different $\lambda_{m}$ and $\lambda_{r}$ are also accepted [9].

NIR absorption is associated with Ti3+ -Ti4+ pairs. A correlation between NIR
absorption and the concentration of Ti3+-Ti4+ pairs was demonstrated in [9]
where the dependence of $\alpha_{r}$ on $\alpha_{m}$ 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 Ti3+ across the sample. The
total concentration of Ti $c_{\mathrm{Ti}}$ was a constant. The obtained
dependence of $\alpha_{r}$ on $\alpha_{m}$ is of a bell-like shape. This
dependence is described by the formula
$\alpha_{r}\propto\alpha_{m}(\alpha_{0}-\alpha_{m})$, where $\alpha_{0}$ is
some constant. It is believed that the concentration of
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$ pairs
($c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$) is proportional to the product of
concentrations of isolated $\mathrm{Ti}^{3+}$ and $\mathrm{Ti}^{4+}$ ions. The
first one is proportional to $\alpha_{m}$, and the second, to
$c_{\mathrm{Ti}}-c_{\mathrm{Ti}^{3+}}\propto\alpha_{0}-\alpha_{m}$. Therefore
the observation [9] correlates with the expectation that the adsorption
coefficient $\alpha_{r}$ depends linearly on
$c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$. The pair mechanism of NIR absorption
is supported by calculations of energy levels of the
$\mathrm{Ti}^{3+}\mathrm{Ti}^{4+}\mathrm{O}_{9}^{2-}$ cluster [12]. The energy
difference [12] between the ground state and the first excited level of such a
cluster corresponds to the wavelength $\lambda=813$ nm that is in excellent
agreement with the experimental value of $\lambda$ 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 Ti4+ (that was confirmed by a very weak
absorption by this sample at $\lambda=490$ nm) was used as a reference. The
Ti4+ scaling factor was defined as the absorption at $\lambda=225$ 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 Ti4+. If NIR is
caused by $\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$ pairs, the FOM will be in
inverse proportion to the Ti4+ 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
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$ 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 Ti4+ ions substituted
for Al3+ ions together with charge compensating vacancies of Al
($V_{\mathrm{Al}}^{3-}$) are the most stable defects. In the reduced
conditions the formation energy of substitutional Ti3+ ions is of the smallest
value. In the intermediate range of the oxygen potential the substitutional
Ti3+ and Ti4+ defects exhibit similar formation energies, indicating that they
can coexist. It was established that Ti3+ ions demonstrate a tendency to form
pairs and larger clusters. The binding energy of the
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$ pair strongly depends on the distance
between Ti3+ 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 Ti3+ triples and quadruples is about 2 eV and 3 eV, correspondingly.
In addition, positively charged Ti4+ ions bind in pairs with negatively
charged Al vacancies $V_{\mathrm{Al}}^{3-}$ with almost the same binding
energy as one of $\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$ 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 $\mathrm{Ti}^{3+}$, $\mathrm{Ti}^{4+}$ and $V_{\mathrm{Al}}^{3-}$
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 $\mathrm{Ti}^{3+}$ or $\mathrm{Ti}^{4+}$ 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:

$F=F_{0}+\sum_{i}E_{i}n_{i}-k_{B}T\ln W.$ (1)

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 $n_{tot}=n_{i_{1}}+n_{i_{2}}+\ldots$
is much smaller than the number of lattice sites and approximate $W$ by the
equation

$W\approx\prod_{i}\frac{N_{i}!}{(N_{i}-n_{i})!n_{i}!},$ (2)

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

$\sum_{i}q_{i}n_{i}=0,$ (3)

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

$\sum_{i}k_{\mathrm{Ti},i}n_{i}=n_{\mathrm{Ti}}.$ (4)

Here we specify the case of Ti dopants. In Eq. (4) $k_{\mathrm{Ti},i}$ is the
number of Ti atoms per an $i$-th defect and $n_{\mathrm{Ti}}$ 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 $\lambda_{q}$
and $\lambda_{\mathrm{Ti}}$. The extremum condition yields

$\tilde{n}_{i}=\frac{n_{i}}{N_{i}}=\exp\left(-\frac{E_{i}-\lambda_{q}q_{i}-\lambda_{\mathrm{Ti}}k_{\mathrm{Ti},i}}{k_{B}T}\right).$
(5)

Substituting Eq. (5) into Eqs. (3) and (4) we obtain two equations for
$\lambda_{q}$ and $\lambda_{\mathrm{Ti}}$.

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 $r_{1}+r_{2}+\ldots r_{s}$ simple defects:

$i_{c}=\underbrace{i_{1}\ldots
i_{1}}_{r_{1}\,\mathrm{times}}\underbrace{i_{2}\ldots
i_{2}}_{r_{2}\,\mathrm{times}}\ldots\underbrace{i_{s}\ldots
i_{s}}_{r_{s}\,\mathrm{times}}.$ (6)

For such a defect Eq. (5) can be rewritten as

$\displaystyle\tilde{n}_{i_{c}}=\exp\left(\frac{E_{i_{c}}^{(b)}}{k_{B}T}\right)\prod_{j=1}^{s}\left[\exp\left(-\frac{{E}_{i_{j}}-\lambda_{q}q_{i_{j}}-\lambda_{\mathrm{Ti}}k_{\mathrm{Ti},i_{j}}}{k_{B}T}\right)\right]^{r_{j}}$
(7)
$\displaystyle\times\exp\frac{\lambda_{q}\left(q_{i_{c}}-\sum_{j=1}^{s}r_{j}q_{i_{j}}\right)}{k_{B}T}\exp\frac{\lambda_{\mathrm{Ti}}\left(k_{\mathrm{Ti},i_{c}}-\sum_{j=1}^{s}r_{j}k_{\mathrm{Ti},i_{j}}\right)}{k_{B}T},$
(8)

where

$E^{(b)}_{i_{c}}=\sum_{j=1}^{s}r_{j}E_{i_{j}}-E_{i_{c}}$ (9)

is the binding energy of the complex defect $i_{c}$. Charge and particle
number conservation requires $q_{i_{c}}=\sum_{j=1}^{s}r_{j}q_{i_{j}}$ and
$k_{\mathrm{Ti},i_{c}}=\sum_{j=1}^{s}r_{j}k_{\mathrm{Ti},i_{j}}$. Thus Eq. (7)
reduces to

$\tilde{n}_{i_{c}}=\exp\left(\frac{E_{i_{c}}^{(b)}}{k_{B}T}\right)\left(\tilde{n}_{i_{1}}\right)^{r_{1}}\left(\tilde{n}_{i_{2}}\right)^{r_{2}}\ldots\left(\tilde{n}_{i_{s}}\right)^{r_{s}}.$
(10)

Considering the problem with only one constraint (3) we put
$\lambda_{\mathrm{Ti}}=0$ from the beginning and again arrive at the relation
(10).

In the DFT approach the defect formation energy is given by the equation [16,
17]

$E_{i}=E_{\mathrm{def},i}-E_{\mathrm{perf}}-\sum_{X}\mu_{X}p_{X,i}+\mu_{e}q_{i}+E^{(c)}_{i},$
(11)

where $E_{\mathrm{def},i}$ is the energy of the supercell with a given defect,
$E_{\mathrm{perf}}$ 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, $\mu_{X}$ is the chemical potential of the atom of the
type X, $\mu_{e}$ is the electron chemical potential, and $E_{i}^{(c)}$ is the
correction that excludes electrostatic interaction caused by periodic coping
of charged defects in the supercell calculations.

Substituting Eq. (11) into Eq. (5) and redefining the Lagrange multiplier
$\tilde{\lambda}_{q}=\lambda_{q}-\mu_{e}$ 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 $\mu_{e}$. 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 $\mu_{\mathrm{Ti}}$.

Applying Eq. (10) to Ti:sapphire we find that the ratio of the concentration
of isolated $\mathrm{Ti}^{3+}$ ions to the concentration of
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ pairs is in inverse proportion to the
concentration of isolated $\mathrm{Ti}^{4+}$ ions:

$\frac{c_{3}}{c_{3-4}}=\frac{e^{-\frac{E^{(b)}_{3-4}}{k_{B}T}}}{2\tilde{n}_{4}}.$
(12)

Here and below we use the notations
$3\equiv\mathrm{Ti}^{3+}\equiv\mathrm{Ti}_{\mathrm{Al}}^{0}$,
$4\equiv\mathrm{Ti}^{4+}\equiv\mathrm{Ti}_{\mathrm{Al}}^{+}$, and
$3-4\equiv\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$. The factor of 2 in the
denominator of Eq. (12) is the number of different positions and orientations
of $\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$ pairs per one Al site. The
coefficient of proportionality depends on the temperature at which the
equilibrium concentration of defects is reached. For defects with positive
binding energy (DFT calculations show that $E^{(b)}_{3-4}>0$) the coefficient
of proportionality decreases under lowering in this temperature. The relation
(12) is fulfilled for any total concentration of Ti (equilibrium at given
temperature or fixed to a certain level). At the same time the relation (12)
may be violates under additional constraints on the concentrations of defects
(see, Sec. VI).

##  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 double-
zeta basis with polarization orbitals as implemented in the open source SIESTA
code [18]. The pseudopotentials were generated with the improved Troullier-
Martins scheme. The $\mathrm{Al}$-$3s^{2}3p^{1}$, $\mathrm{O}$-$2s^{2}2p^{4}$
and $\mathrm{Ti}$-$4s^{1}3d^{3}$ 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 (TiO2, Ti3O5, Ti2O3) 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
$\mathrm{C}$-$2s^{2}2p^{2}$, $\mathrm{N}$-$2s^{2}2p^{3}$, and
$\mathrm{F}$-$2s^{2}2p^{5}$ 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/$\AA$. The plane-wave cutoff energy of 250 Ry is used to
calculate the total energy of the system. A 120-atom supercell of
$\alpha$-Al2O3 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 $\Gamma$ point. To check
convergency of the $\Gamma$ point integration we calculate several formation
energies integrating over $3\times 3\times 3$ and $5\times 5\times 5$ grids of
$k$-points. Integration over $3\times 3\times 3$ grid yields Ti3+, Ti4+ and
$V_{\mathrm{Al}}^{3-}$ formation energies higher than ones for the $\Gamma$
point integration by 0.02 eV, 0.027 eV and 0.018 eV, correspondingly. The
difference of energies given by integration over $3\times 3\times 3$ and
$5\times 5\times 5$ grids of $k$-points is less than $10^{-3}$ eV. Based on
this results we conclude that restriction with integration over the $\Gamma$
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. (11) are calculated from
chemical potentials of materials related to the
$\mathrm{Al}-\mathrm{Ti}-\mathrm{O}$ system. The chemical potential of a given
material is the sum of the zero-temperature DFT energy and a temperature
correction part:

$\mu(X)=\mu_{0}(X)+\Delta\mu_{T}(X),$ (13)

where $\mu_{0}(X)$ 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]

$\Delta\mu_{T}(X)=H_{X}(T,p)-H_{X}(0,p)-TS_{X}(T,p),$ (14)

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 ${X},{Y}=\mathrm{Al},\mathrm{Ti},\mathrm{C}$ is
calculated as

$H_{f}({X}_{u}{Y}_{v})=\frac{\mu({X}_{u}{Y}_{v})-u\mu({X})-v\mu({Y})}{u+v}$
(15)

or, if ${Y}=\mathrm{O},\mathrm{N},\mathrm{F}$, as

$H_{f}({X}_{u}{Y}_{v})=\frac{\mu({X}_{u}{Y}_{v})-u\mu({X})-\frac{v}{2}\mu({Y}_{2})}{u+v}.$
(16)

Equations (15) and (16) give heats of formation per one atom. Chemical
potentials in Eqs. (15) and (16) take into account the temperature correction
Eq. (14). Using calculated heats of formation (15) and (16) we build the
$\mathrm{Al}-\mathrm{O}-\mathrm{Ti}$ phase diagram and define points in which
$\alpha-\mathrm{Al}_{2}\mathrm{O}_{3}$ is in equilibrium with two other
phases. The obtained phase diagram at $T=T_{m}=2327$K (the melting temperature
of $\alpha$-Al2O3) is shown in Fig. 1. The same procedure is done for four-
component systems $\mathrm{Al}-\mathrm{O}-\mathrm{Ti}-X$, where
$X=\mathrm{C},\mathrm{N},\mathrm{F}$. The reference points in the phase
diagrams, where two (three) phases are in equilibrium with
$\mathrm{Al}_{2}\mathrm{O}_{3}$ are listed in Table 1.

Figure 1: Phase diagram of the Al$-$Ti$-$O system at $T=T_{m}$. The points A, B, D, I and J denoted by arrows correspond to the vertices of the three-phase region around Al2O3. Table 1: Phases in equilibrium with $\mathrm{Al}_{2}\mathrm{O}_{3}$ and the oxygen chemical potential in reference points of the ternary and four-component phase diagrams at $T=T_{m}$. The potential $\mu_{\mathrm{O}}$ is counted from the energy of an isolated oxygen atom. Point | $\mathrm{Al}-\mathrm{O}-\mathrm{Ti}$ | $\mathrm{Al}-\mathrm{O}-\mathrm{Ti}-\mathrm{C}$ | $\mathrm{Al}-\mathrm{O}-\mathrm{Ti}-\mathrm{N}$ | $\mathrm{Al}-\mathrm{O}-\mathrm{Ti}-\mathrm{F}$ | $\mu_{\mathrm{O}},eV$
---|---|---|---|---|---
A | O2, TiO2 | O2, TiO2, CO2 | O2, TiO2, N2 | O2, TiO2, AlF3 | -7.29
B | TiO2, Ti3O5 | TiO2, Ti3O5, CO2, | TiO2, Ti3O5, N2 | TiO2, Ti3O5, AlF3 | -8.37
C | $-$ | Ti3O5, CO2, CO | $-$ | $-$ | -9.24
D | Ti3O5, Ti2O3 | Ti3O5, Ti2O3, CO, | Ti3O5, Ti2O3, N2 | Ti3O5,Ti2O3, AlF3 | -9.75
E | $-$ | $-$ | Ti2O3, N2, TiN | $-$ | -9.88
F | $-$ | Ti2O3, CO, TiC | $-$ | $-$ | -10.06
G | $-$ | CO, TiC, C | $-$ | $-$ | -10.24
H | $-$ | $-$ | N2, TiN, AlN | $-$ | -10.30
I | Ti2O3, TiO | Ti2O3, TiO, TiC | Ti2O3, TiO, TiN | Ti2O3, TiO, AlF3 | -10.36
J | TiO, Al | TiO, Al, TiC | TiO, Al, TiN | TiO, Al, AlF3 | -10.45
K | $-$ | $-$ | Al, TiN, AlN | $-$ | -10.45
L | $-$ | Al, TiC, C | $-$ | $-$ | -10.45

Chemical potentials of atoms are calculated for all reference points using the
equations that relate these potentials with the potentials Eq. (13). For
instance, for the point A of the ternary system we have

$\displaystyle 2\mu_{\mathrm{Al}}+3\mu_{\mathrm{O}}$ $\displaystyle=$
$\displaystyle\mu(\mathrm{Al}_{2}\mathrm{O}_{3}),$ (17)
$\displaystyle\mu_{\mathrm{Ti}}+2\mu_{\mathrm{O}}$ $\displaystyle=$
$\displaystyle\mu(\mathrm{Ti}\mathrm{O}_{2}),$ (18) $\displaystyle
2\mu_{\mathrm{O}}$ $\displaystyle=$ $\displaystyle\mu(\mathrm{O}_{2}).$ (19)

The expression for the defect formation energies (11) contains the difference
of the energies of a defect and of the perfect supercell
$E_{\mathrm{d-p},i}=E_{\mathrm{def},i}-E_{perf}$. Within the DFT method we
calculate this difference at $T=0$. At finite $T$ one should take into account
a temperature correction $\Delta E_{\mathrm{d-p},i}(T)$ to this difference.
The origin of $\Delta E_{\mathrm{d-p},i}(T)$ is differences of finite
temperature enthalpy and entropy of a defect and a perfect supercell. We
evaluate $\Delta E_{\mathrm{d-p},i}(T)$ from the temperature corrections Eq.
(14) for relevant materials in the crystal state.

The difference $\Delta E_{\mathrm{d-p},\mathrm{Ti}_{\mathrm{Al}}}(T)$ is
evaluated from $\Delta\mu_{T}$ for pure $\mathrm{Al}_{2}\mathrm{O}_{3}$ and
$\mathrm{Ti}_{2}\mathrm{O}_{3}$:

$\Delta
E_{\mathrm{d-p},\mathrm{Ti}_{\mathrm{Al}}}(T)=\frac{1}{2}\left[\Delta\mu_{T}(\mathrm{Ti}_{2}\mathrm{O}_{3})-\Delta\mu_{T}({\mathrm{Al}_{2}\mathrm{O}_{3}})\right].$
(20)

To evaluate the temperature correction for the supercell with a vacancy we
imply that

$3\Delta E_{\mathrm{d-p},V_{\mathrm{O}}}(T)+2\Delta
E_{\mathrm{d-p},V_{\mathrm{Al}}}(T)=-\Delta\mu_{T}(\mathrm{Al}_{2}\mathrm{O}_{3}).$
(21)

Equation (21) 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 $\mu_{X}$ and to
$E_{\mathrm{def},i}-E_{perf}$ and they should cancel each other. Equation (21)
provides the fulfillment of this condition. Assuming that $\Delta
E_{\mathrm{d-p},V_{X}}(T)$ (with $X=\mathrm{Al},\mathrm{Ti},\mathrm{O}$), are
the same for $\mathrm{Al}_{2}\mathrm{O}_{3}$, $\mathrm{Ti}_{2}\mathrm{O}_{3}$
and $\mathrm{Ti}\mathrm{O}_{2}$, we calculate $\Delta
E_{\mathrm{d-p},V_{\mathrm{Al}}}(T)$ and $\Delta
E_{\mathrm{d-p},V_{\mathrm{O}}}(T)$ using equations

$\displaystyle 3\Delta E_{\mathrm{d-p},V_{\mathrm{O}}}(T)+2\Delta
E_{\mathrm{d-p},V_{\mathrm{Ti}}}(T)=-\Delta\mu_{T}(\mathrm{Ti}_{2}\mathrm{O}_{3}),$
(22) $\displaystyle 2\Delta E_{\mathrm{d-p},V_{\mathrm{O}}}(T)+\Delta
E_{\mathrm{d-p},V_{\mathrm{Ti}}}(T)=-\Delta\mu_{T}(\mathrm{Ti}\mathrm{O}_{2}),$
(23)

and Eq. (21).

Temperature corrections to the energies of substitutional N, C, and F defects
are evaluated from $\Delta\mu_{T}(\mathrm{AlN})$,
$\Delta\mu_{T}(\mathrm{Al}_{4}\mathrm{C}_{3})$ and
$\Delta\mu_{T}(\mathrm{Al}\mathrm{F}_{3})$, correspondingly. For instance, to
evaluate $\Delta E_{\mathrm{d-p},\mathrm{N}_{\mathrm{O}}}(T)$ we use the
relation

$\Delta
E_{\mathrm{d-p},\mathrm{N}_{\mathrm{O}}}(T)=\Delta\mu_{T}(\mathrm{Al}\mathrm{N})+\Delta
E_{\mathrm{d-p},V_{\mathrm{Al}}}(T)+\Delta
E_{\mathrm{d-p},V_{\mathrm{O}}}(T).$ (24)

Equation (24) is based on the fact that the $\mathrm{N}_{\mathrm{O}}$ defect
can be created by removing of one Al and one O atom from the
$\mathrm{Al}_{2}\mathrm{O}_{3}$ crystal and adding one formula unit of
$\mathrm{Al}\mathrm{N}$. Similar arguments were used to evaluate $\Delta
E_{\mathrm{d-p},\mathrm{C}_{\mathrm{O}}}(T)$, $\Delta
E_{\mathrm{d-p},\mathrm{C}_{\mathrm{Al}}}(T)$, and
$E_{\mathrm{d-p},\mathrm{F}_{\mathrm{O}}}(T)$. The temperature correction
$\Delta E_{\mathrm{d-p,i_{c}}}(T)$ for a complex defect is taken as the sum of
$\Delta E_{\mathrm{d-p},i}(T)$ for simple defects which form this complex
defect.

The correction $E^{(c)}_{i}$ in the expression for the defect formation energy
(11) is evaluated by a method similar to the one proposed in [23, 24]. DFT
supercell calculations give the formation energies $E_{i}^{(0)}$ 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 monopole-monopole interaction, while the contribution of
the monopole-dipole and dipole-dipole interaction [25, 26] is much smaller.
This allows us to equate $E_{i}^{(c)}$ with the Madelung energy taken with a
negative sign. We imply that the Madelung energy for a supercell composed of
$n\times m\times k$ unit cells can be presented in the form

$E_{M}(n,m,k)=\frac{q_{i,eff}^{2}}{2\epsilon_{eff}}\tilde{E}_{M}(n,m,k),$ (25)

where $q_{eff}$ is the effective charge of the defect, $\varepsilon_{eff}$ is
the effective dielectric constant, and $\tilde{E}_{M}(n,m,k)$ is the
electrostatic energy of a periodic structure of $q=1$ point charges in the
medium with $\varepsilon=1$ and with a charge compensating background. For a
cubic lattice $\tilde{E}_{M}(n,n,n)=-2.837/na$, where $a$ is the lattice
parameter. In the general case $\tilde{E}_{M}(n,m,k)$ 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
$E_{i}^{(0)}$ 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 $a=b=4.86$ $\AA$ and $c=13.19$ $\AA$ obtained for pure
$\mathrm{Al}_{2}\mathrm{O}_{3}$ by optimization of lattice vectors. Then, we
calculate the Madelung energy $\tilde{E}_{M}$ by the Ewalds method for the
same supercells, define the dimensionless quantity
$v_{M}(n,m,k)=a\tilde{E}_{M}(n,m,k)$, and fit the energy $E_{i}^{(0)}$ by a
linear function:

$E_{i}^{(0)}(v_{M})=E_{i}+\frac{q_{i,eff}^{2}}{2\epsilon_{eff}a}v_{M}=c_{0,i}+c_{1,i}v_{M}.$
(26)

As two examples we present the result of fitting for $\mathrm{Ti}^{4+}$ and
$V_{\mathrm{Al}}^{3-}$ 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 $E_{i}^{(c)}$, as
$E_{i}^{(c)}(n,m,k)=c_{0,i}-E_{i}^{(0)}(n,m,k)$. Calculated electrostatic
corrections $E_{i}^{(c)}(2,2,1)$ for the considered charged defects are given
in Table 2. Note that if we set $\varepsilon_{eff}=10$ we obtain the effective
charges $|q_{\mathrm{Ti}^{4+},eff}/e|=0.76$ and
$|q_{V_{\mathrm{Al}}^{3-},eff}/e|=2.52$ which looks reasonable.

Figure 2: Calculated defect formation energies $E^{(0)}_{i}$ (in eV) for $\mathrm{Ti}^{4+}$ (a) and $V_{\mathrm{Al}}^{3-}$ (b) for several supercells versus $v_{M}$ and linear fit (26). For concreteness, we set $\mu_{e}$ equals to the valence band maximum. Table 2: Electrostatic correction $E_{i}^{(c)}$ to the defect formation energy for the $2\times 2\times 1$ supercell Defect | q |  $E_{i}^{(c)}$, eV
---|---|---
$\mathrm{Ti}^{4+}$ | $+1$ | 0.11
$\mathrm{V}_{\mathrm{Al}}^{3-}$ | $-3$ | 1.21
$\mathrm{Ti}^{4+}$-$\mathrm{V}_{\mathrm{Al}}^{3-}$ | $-2$ | 0.42
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$ | $+1$ | 0.05
$\mathrm{Ti}^{4+}$-$\mathrm{Ti}^{4+}$-$\mathrm{V}_{\mathrm{Al}}^{3-}$ | $-1$ | 0.21
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$-$\mathrm{V}_{\mathrm{Al}}^{3-}$ | $-2$ | 0.68
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$-$\mathrm{V}_{\mathrm{Al}}^{3-}$ | $-3$ | 1.48
$\mathrm{F}_{\mathrm{O}}^{+}$ | +1 | 0.14
$\mathrm{F}_{\mathrm{O}}^{+}$-$\mathrm{V}_{\mathrm{Al}}^{3-}$ | $-2$ | 0.54
$\mathrm{F}_{\mathrm{O}}^{+}$-$\mathrm{F}_{\mathrm{O}}^{+}$-$\mathrm{V}_{\mathrm{Al}}^{3-}$ | $-1$ | 0.16
$\mathrm{N}_{\mathrm{O}}^{-}$ | $-1$ | 0.17
$\mathrm{C}_{\mathrm{O}}^{2-}$ | $-2$ | 0.60
$\mathrm{C}_{\mathrm{O}}^{-}$ | $-1$ | 0.15
$\mathrm{C}_{\mathrm{O}}^{2-}$-$\mathrm{Ti}^{4+}$ | $-1$ | 0.16

##  IV Equilibrium concentrations of defects in $\mathbf{Ti}$: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.

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 ( $\mathrm{Ti}^{3+}$,
$\mathrm{Ti}^{4+}$ and $V_{\mathrm{Al}}^{3-}$), three pairs
($\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$, $\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$
and $\mathrm{Ti}^{4+}$-$V^{3-}_{\mathrm{Al}}$), four triples
($\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$,
$\mathrm{Ti}^{4+}$-$\mathrm{Ti}^{4+}$-$V^{3-}_{\mathrm{Al}}$,
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$-$V^{3-}_{\mathrm{Al}}$ and
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$-$V^{3-}_{\mathrm{Al}}$), and a quadruple
complex
$\mathrm{Ti}^{4+}$-$\mathrm{Ti}^{4+}$-$\mathrm{Ti}^{4+}$-$V^{3-}_{\mathrm{Al}}$.
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:

$\displaystyle\tilde{n}_{4}^{3}\tilde{n}_{V}=e^{-\frac{3E_{4}+E_{V}}{k_{b}T}},$
(27)
$\displaystyle\tilde{n}_{4}+2C_{34}\tilde{n}_{3}\tilde{n}_{4}-3\tilde{n}_{V}-8C_{4V}\tilde{n}_{4}\tilde{n}_{V}-18C_{44V}\tilde{n}_{4}^{2}\tilde{n}_{V}-36C_{34V}\tilde{n}_{3}\tilde{n}_{4}\tilde{n}_{V}-54C_{33V}\tilde{n}_{3}^{2}\tilde{n}_{V}=0,$
(28)

where we use the notation $V\equiv V^{3+}_{\mathrm{Al}}$. In Eq. (28) we take
into account the relation (10). The factors $C_{i_{c}}$ are the coefficients
in the relation (10) averaged over different orientations and configurations:

$C_{i_{c}}=\frac{\sum_{f}K_{i_{c},f}e^{\frac{E_{i_{c},f}^{(b)}}{k_{B}T}}}{\sum_{f}K_{i_{c},f}},$
(29)

where $E_{i_{c},f}^{(b)}$ is the binding energy of the complex $i_{c}$ and
$K_{i_{c},f}$ 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 $K_{i_{c},f}$ are given
in Table 3. Binding energies are independent of the chemical potentials of
atoms. Binding energies given in Table 3 include the electrostatic correction
$E_{i}^{(c)}$. They differ from ones calculated in Refs. [15, 1], where this
correction was not taken into account. The concentration of electrically
neutral defects depends only on its own formation energy:

$\tilde{n}_{i}=e^{-\frac{E_{i}}{k_{B}T}}.$ (30)

Table 3: Binding energies $E_{i_{c},f}^{(b)}$ and the numbers of different orientations and configurations $K_{i_{c},f}$ with the same energy. Complexes of Al vacancy surrounded by two Ti and Ti surrounded by Ti and Al vacancy are notated as $\mathrm{Ti}-V-\mathrm{Ti}$ and $\mathrm{Ti}-\mathrm{Ti}-V$, correspondingly. For complexes $\mathrm{Ti}-\mathrm{Ti}-\mathrm{Ti}-V$ we consider only configurations with the vacancy in the center. Complex defect $i_{c}$ | Notation | $K_{i_{c},f}$ |  $E_{i_{c},f}^{(b)}$, eV
---|---|---|---
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$ | 33 | 3 | 1.36
1 | 1.22
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$ | 333 | 6 | 2.03
$\mathrm{Ti}^{4+}$-$\mathrm{Ti}^{4+}$-$\mathrm{Ti}^{4+}$-$V_{\mathrm{Al}}^{3-}$ | 444V | 4 | 4.47
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$ | 34 | 3 | 0.76
1 | 0.68
$\mathrm{Ti}^{4+}$-$V_{\mathrm{Al}}^{3-}$ | 4V | 3 | 2.05
1 | 1.85
$\mathrm{Ti}^{4+}$-$V_{\mathrm{Al}}^{3-}$-$\mathrm{Ti}^{4+}$ | 44V | 3 | 3.36
3 | 2.91
$\mathrm{Ti}^{4+}$-$\mathrm{Ti}^{4+}$-$V_{\mathrm{Al}}^{3-}$ | 6 | 2.51
6 | 2.40
$\mathrm{Ti}^{3+}$-$V_{\mathrm{Al}}^{3-}$-$\mathrm{Ti}^{4+}$ | 34V | 3 | 2.69
3 | 2.51
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$-$V_{\mathrm{Al}}^{3-}$ | 6 | 2.55
6 | 2.49
$\mathrm{Ti}^{3+}$-$V_{\mathrm{Al}}^{3-}$-$\mathrm{Ti}^{3+}$ | 33V | 6 | 0.92
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$-$V_{\mathrm{Al}}^{3-}$ | 12 | 1.88
$\mathrm{N}_{\mathrm{O}}^{-}$-$\mathrm{Ti}^{4+}$ | - | 3 | 1.35
3 | 1.10
$\mathrm{C}_{\mathrm{O}}^{2-}$-$\mathrm{Ti}^{4+}$ | - | 6 | 2.95
$\mathrm{F}_{\mathrm{O}}^{+}$-$V_{\mathrm{Al}}^{3-}$ | FV | 3 | 1.61
3 | 2.03
$\mathrm{F}_{\mathrm{O}}^{+}$-$V_{\mathrm{Al}}^{3-}$-$\mathrm{F}_{\mathrm{O}}^{+}$ | FVF | 3 | 3.44
3 | 2.87
3 | 3.79
6 | 3.30

The formation energies of $\mathrm{Ti}^{3+}$ and of the electrically neutral
combination of three $\mathrm{Ti}^{4+}$ and one $V_{\mathrm{Al}}^{3-}$ at
$T=T_{m}$ are given in Table 4 for all reference points of the
$\mathrm{Al}-\mathrm{O}-\mathrm{Ti}$ phase diagram. Using Eqs. (27), (28) and
(30) and the data from Tables 3 and 4 we calculate $\tilde{n}_{3}$,
$\tilde{n}_{4}$ and $\tilde{n}_{V}$. Concentrations of complex defects are
found from Eq. (10) in which the factor $\exp(E^{(b)}_{i_{c}}/k_{B}T)$ is
replaced with the coefficient $C_{i_{c}}$ Eq. (29). The concentration of an
$i$-th defect specie is calculated as
$c_{i}=K_{i}\tilde{n}_{i}c_{\mathrm{Al}}$, where $c_{\mathrm{Al}}=4.7\cdot
10^{22}$ cm-3 is the concentration of Al in the
$\mathrm{Al}_{2}\mathrm{O}_{3}$ crystal, and $K_{i}=\sum_{f}K_{i,f}$ is the
total number of configurations and orientations of the $i$-th defect specie
($K_{i}=1$ for $\mathrm{Ti}^{3+}$, $\mathrm{Ti}^{4+}$ and
$V_{\mathrm{Al}}^{3-}$).

Table 4: Defect formation energies (in eV) at reference points of $\mathrm{Al}-\mathrm{O}-\mathrm{Ti}$ and $\mathrm{Al}-\mathrm{O}-\mathrm{Ti}-X$ ($X=\mathrm{C},\mathrm{N},\mathrm{F}$) phase diagram at $T=T_{m}$. The energies of electrically neutral combinations are given per one defect. Point | $E_{3}$ | $(3E_{4}+E_{V})/4$ | $(E_{4}+E_{\mathrm{N}^{-}_{\mathrm{O}}})/2$ | $(2E_{4}+E_{\mathrm{C}_{\mathrm{O}}^{2-}})/3$ | $(E_{4}+E_{\mathrm{C}_{\mathrm{O}}^{-}})/2$ | $E_{\mathrm{F}_{\mathrm{O}}^{+}}-E_{4}$
---|---|---|---|---|---|---
A | 2.25 | 1.57 | 3.21 | 5.64 | 7.24 | 0.29
B | 1.71 | 1.57 | 2.40 | 4.19 | 5.33 | 0.29
C | 1.57 | 1.79 | - | 3.23 | 3.97 | -
D | 1.48 | 1.92 | 1.59 | 2.83 | 3.41 |  $-$0.17
E | 1.48 | 1.96 | 1.53 | - |  | -
F | 1.48 | 2.03 | - | 2.62 | 3.09 | -
G | 1.92 | 2.43 | - | 2.80 | 3.14 | -
H | 2.11 | 2.59 | 1.64 | - |  | -
I | 1.48 | 2.14 | 1.65 | 2.67 | 3.17 |  $-$0.47
J | 1.52 | 2.21 | 1.68 | 2.70 | 3.19 |  $-$0.56
K | 2.11 | 2.65 | 1.68 | - |  | -
L | 2.23 | 2.74 | - | 2.94 | 3.19 | -

The obtained equilibrium concentrations of defects and the overall
concentration of Ti are shown in Fig. 3. To make this figure more readable we
do not display the concentration of
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$ triples which is
lower than that of $\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$ pairs. The
concentrations of the
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{3+}$-$V^{3-}_{\mathrm{Al}}$, and
$\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$-$V^{3-}_{\mathrm{Al}}$ complexes are not
displayed either. They are generally much lower than the concentration of
$\mathrm{Ti}^{4+}$-$\mathrm{Ti}^{4+}$-$V^{3-}_{\mathrm{Al}}$ complexes (they
are comparable only in the reduced conditions).

Figure 3: Equilibrium concentrations of defects in Ti:sapphire at $T=T_{m}$ in
the conditions that correspond to reference points of
$\mathrm{Al}-\mathrm{Ti}-\mathrm{O}$ phase diagram Fig. 1 (see also Table 1).
Lines are guides to the eye. Oxygen chemical potential is given in the inset.

One can see from Fig. 3 that the overall concentration of Ti varies in the
range from $6\cdot 10^{19}$ cm-3 to $3.5\cdot 10^{20}$ 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, 4, 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:

$\displaystyle\tilde{n}_{4}^{3}\tilde{n}_{V}=\tilde{n}_{3}^{3}e^{\frac{3E_{3}-3E_{4}-E_{V}}{k_{b}T}},$
(31)
$\displaystyle\tilde{n}_{4}+2C_{34}\tilde{n}_{3}\tilde{n}_{4}-3\tilde{n}_{V}-8C_{4V}\tilde{n}_{4}\tilde{n}_{V}-18C_{44V}\tilde{n}_{4}^{2}\tilde{n}_{V}-36C_{34V}\tilde{n}_{3}\tilde{n}_{4}\tilde{n}_{V}-54C_{33V}\tilde{n}_{3}^{2}\tilde{n}_{V}=0,$
(32)
$\displaystyle\tilde{n}_{3}+\tilde{n}_{4}+4C_{34}\tilde{n}_{3}\tilde{n}_{4}+4C_{33}\tilde{n}_{3}^{2}+18C_{333}\tilde{n}_{3}^{3}+4C_{4V}\tilde{n}_{4}\tilde{n}_{V}+36C_{44V}\tilde{n}_{4}^{2}\tilde{n}_{V}$
(33)
$\displaystyle+36C_{34V}\tilde{n}_{3}\tilde{n}_{4}\tilde{n}_{V}+36C_{33V}\tilde{n}_{3}^{2}\tilde{n}_{V}+12C_{444V}\tilde{n}_{4}^{3}\tilde{n}_{V}=\frac{n_{\mathrm{Ti}}}{n_{\mathrm{Al}}}.$
(34)

.

The quantity $3E_{3}-3E_{4}-E_{V}$ in Eq. (31) and the coefficients
$C_{i_{c}}$ in Eqs. (32), and (33) are independent of $\mu_{\mathrm{Ti}}$.
Consequently, the solution of the system (31), (32), and (33) is also
independent of $\mu_{\mathrm{Ti}}$ and concentrations can be presented as
continuous functions of $\mu_{\mathrm{O}}$. The oxygen chemical potential
varies in the range $[\mu_{\mathrm{r}},\mu_{\mathrm{o}}]$, where
$\mu_{\mathrm{r}}=[\mu(\mathrm{Al}_{2}\mathrm{O}_{3})-2\mu(\mathrm{Al})]/3$
($\mu_{\mathrm{o}}=\mu(\mathrm{O}_{2})/2$) is the value of the oxygen chemical
potential in the reduced (oxidized) limit. The boundaries $\mu_{\mathrm{r}}$
and $\mu_{\mathrm{o}}$ depend on temperature.

Calculated concentrations of simple and paired defects at fixed overall
concentration of Ti ($c_{\mathrm{Ti}}$) are presented in Fig. 4. We consider
three different $c_{\mathrm{Ti}}$ (1020 cm-3, $10^{19}$ cm-3 and $10^{18}$
cm-3) and two temperatures: $T=T_{m}=2327$ K and $T=T_{a}=2000$ K. We consider
that $T=T_{m}$ concentrations correspond to as-grown samples and
$T=T_{a}<T_{m}$ ones, to samples annealed at $T=T_{a}$. One can see from Fig.
5 that annealing may result in an increase or a decrease of the concentration
of a given defect species depending on the oxygen chemical potential and the
total concentration of Ti. As was expected, a relative fraction of isolated
defects increases under lowering in the total concentration of Ti.

Figure 4: Equilibrium concentrations of isolated and paired defects at
$T=2327$ K (solid curves) and $T=2000$ K (dashed curves) at fixed overall
concentration of Ti $c_{\mathrm{Ti}}=10^{20}$ cm-3 (a,b),
$c_{\mathrm{Ti}}=10^{19}$ cm-3 (c,d) and $c_{\mathrm{Ti}}=10^{18}$ cm-3 (e,f).

##  V Concentration of defects in Ti:sapphire grown in the presence of carbon,
nitrogen, fluorine and their compounds

To obtain crystals with a low concentration of $\mathrm{Ti}^{4+}$ 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 Ti3+ ions to the concentration of Ti${}^{3+}-$Ti4+ pairs. The
mechanism is quite simple. If codoping results in an appearance of additional
positively charged defects, some negatively charged vacancies
$V_{\mathrm{Al}}^{3-}$ compensate codopants and the concentration of
$\mathrm{Ti}^{4+}$ decreases. Consequently, according to Eq. (12) (valid for
codoped samples, as well) the ratio
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ increases. If
codoping results in the appearance of additional negatively charged defects,
the concentration of isolated $\mathrm{Ti}^{4+}$ ions increases to provide
charge compensation. Consequently, the ratio
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ 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 $V_{\mathrm{Al}}^{3-}$ and with
$\mathrm{Ti}^{4+}$.

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.

###  V.1 $\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{C}$ system

Using calculated heats of formation of relevant materials we build the four-
component $\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{C}$ phase diagram (Fig.
5). The calculations were done for graphite phase of carbon.

Figure 5: The phase diagram of the Al$-$Ti$-$O$-$C system at $T=T_{m}$ and
reference points.

There are nine points at the diagram Fig. 5 (labeled as A, B, C, D, F, G, I,
J, L), where $\mathrm{Al}_{2}\mathrm{O}_{3}$ is in equilibrium with three
other phases.

We have calculated the formation energies of substitutional
$\mathrm{C}_{\mathrm{O}}$ and $\mathrm{C}_{\mathrm{Al}}$ defects and carbon
interstitial in different charge states. Most of these energies are quite
large. In the intermediate and reduced conditions two substitutional defects,
$\mathrm{C}_{\mathrm{O}}^{2-}$ and $\mathrm{C}_{\mathrm{O}}^{-}$, dominate.
Negatively charged carbon defects may bind in pairs and triples with
$\mathrm{Ti}^{4+}$ and $\mathrm{Ti}^{3+}$. Binding energies of such complexes
are large and complex defects prevail over isolated ones. Among complex
defects the concentration of $\mathrm{C}_{\mathrm{O}}^{2-}$-$\mathrm{Ti}^{4+}$
pairs is the highest one. It can reach the level of $10^{16}$ cm-3. But since
the concentration of isolated $\mathrm{Ti}^{4+}$ is at least two orders
higher, carbon defects practically do not influence the concentration of
$\mathrm{Ti}^{4+}$ and $\mathrm{Ti}^{3+}$-$\mathrm{Ti}^{4+}$.

Our results correlate with ones obtained for C-doped $\alpha$-Al2O3 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 $\mathrm{C}_{\mathrm{O}}$ defects have
the smallest formation energy, while in the oxidized conditions the
substitutional $\mathrm{C}_{\mathrm{Al}}$ 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
$\mathrm{C}_{\mathrm{Al}}$ but find that in the oxidized conditions the
concentrations of $\mathrm{C}_{\mathrm{O}}$ 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. (27) and (28), where we neglect the
contribution of carbon defects.

The concentrations of carbon defects are found from the relations

$\tilde{n}_{\mathrm{C}_{\mathrm{O}}^{-}}=\frac{e^{-\frac{E_{\mathrm{C}_{\mathrm{O}}^{-}}+E_{4}}{k_{B}T}}}{\tilde{n}_{4}},\quad\tilde{n}_{\mathrm{C}_{\mathrm{O}}^{2-}}=\frac{e^{-\frac{E_{\mathrm{C}_{\mathrm{O}}^{2-}}+2E_{4}}{k_{B}T}}}{(\tilde{n}_{4})^{2}}.$
(35)

Equations (35) are obtained from Eq. (5) (with $\lambda_{\mathrm{Ti}}=0$). The
concentration of pairs $\mathrm{C}_{\mathrm{O}}^{2-}-\mathrm{Ti}^{4+}$ is
found from Eq. (10). The energies $(E_{\mathrm{C}_{\mathrm{O}}^{-}}+E_{4})/2$
and $(E_{\mathrm{C}_{\mathrm{O}}^{-}}+2E_{4})/3$ are given in Table 4, and the
binding energy of the $\mathrm{C}_{\mathrm{O}}^{2-}-\mathrm{Ti}^{4+}$ pair, in
Table 3. Calculated concentrations of carbon defects are shown in Fig. 7.

Figure 6: Equilibrium concentrations of defects in Ti:sapphire at $T=T_{m}$ in
the conditions that correspond to reference points of
$\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{C}$ phase diagram Fig. 5 Figure 7:
Equilibrium concentrations of carbon defects at $T=T_{m}$. Lines are guides to
the eye.

###  V.2 $\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{N}$ system

In Fig. 8 we present the $\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{N}$ phase
diagram at $T=T_{m}$ that was built using calculated heats of formation of
related materials. This diagram contains eight reference points (A, B, D, E,
H, I, J, K).

Figure 8: The phase diagram of the Al$-$Ti$-$O$-$N system at $T=T_{m}$ and
reference points.

The formation energy of $\mathrm{N}_{\mathrm{O}}^{-}$ defects is small and
their concentration is high. It correlates with the results of [27] where the
formation energies of nitrogen defects in Al2O3 were calculated. It was shown
[27] that even in the oxidized conditions the substitutional
$\mathrm{N}_{\mathrm{O}}^{-}$ defect has the smallest formation energy. The
$\mathrm{N}_{\mathrm{O}}^{-}$ defects shift the balance between other charged
defects in Ti:sapphire. To evaluate this shift we add the term
$-(3/2)\tilde{n}_{\mathrm{N}}$ to the left-hand side of Eq. (28), where
$\tilde{n}_{\mathrm{N}}$, is the number of $\mathrm{N}_{\mathrm{O}}^{-}$
defects normalized to the total number of oxygen atoms. The quantity
$\tilde{n}_{\mathrm{N}}$ is expressed through $\tilde{n}_{4}$ using Eq. (5)
with $\lambda_{\mathrm{Ti}}=0$:

$\tilde{n}_{\mathrm{N}}=\frac{e^{-\frac{E_{\mathrm{N}_{\mathrm{O}}^{-}}+E_{4}}{k_{B}T}}}{\tilde{n}_{4}}.$
(36)

The energy $(E_{\mathrm{N}_{\mathrm{O}}^{-}}+E_{4})/2$ is given in Table 4. In
Fig. 9 we present calculated concentrations of Ti defects and Al vacancies at
$T=T_{m}$ at the reference points of the
$\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{N}$ phase diagram. One can see
that the presence of nitrogen defects results in a considerable increase of
the concentration of $\mathrm{Ti}^{4+}$ 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 $\mathrm{Ti}^{4+}$.
The binding energy of such a pair is given in Table 3. The calculated
concentration of isolated $\mathrm{N}_{\mathrm{O}}^{-}$ defects and
$\mathrm{N}_{\mathrm{O}}^{-}-\mathrm{Ti}^{4+}$ 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 $\mathrm{Ti}^{4+}$. The concentration of
$\mathrm{N}_{\mathrm{O}}^{-}-\mathrm{Ti}^{4+}$ pairs is slightly higher than
the concentration of isolated $\mathrm{N}_{\mathrm{O}}^{-}$ defects, but since
these pairs are electrically neutral, they do not influence the concentration
of charged defects.

The increase in the concentration of $\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ pairs
results in a decrease of the
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ ratio and a
decrease of the FOM.

Figure 9:  Equilibrium concentrations of defects in Ti:sapphire at $T=T_{m}$
in the conditions that correspond to reference points of
$\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{N}$ phase diagram Fig. 8. Figure
10:  Equilibrium concentrations of nitrogen defects at $T=T_{m}$.

###  V.3 $\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{F}$ system

Keeping in mind that fluorine ions enter into Ti:sapphire mostly in the form
of substitutional $\mathrm{F}_{\mathrm{O}}^{+}$ defects one would expect that
co-doping with fluorine decreases the concentration of $\mathrm{Ti}^{4+}$. But
the situation appears to be more complicated due to formation of negatively
charged complexes of $\mathrm{F}_{\mathrm{O}}^{+}$ and $V_{\mathrm{Al}}^{3-}$.

The calculated four-component $\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{F}$
phase diagram at $T=T_{m}$ is shown in Fig. 11. This diagram contains five
reference points (A, B, D, I, J). In all these points Al2O3 is in equilibrium
with AlF3. Two other compounds are the same as for the reference points of the
$\mathrm{Al}-\mathrm{Ti}-\mathrm{O}$ diagram (Fig. 1).

Figure 11: The phase diagram of the Al$-$Ti$-$O$-$F system at $T=T_{m}$ and
reference points.

Calculations show that together with positively charged
$\mathrm{F}_{\mathrm{O}}^{+}$ defects two types of negatively charged
complexes formed by fluorine have small formation energies. They are the pairs
$\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$ and the triples
$\mathrm{F}_{\mathrm{O}}^{+}-\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$.
To calculate equilibrium concentrations of defects we take into account three
fluorine defect species and add to the left-hand-part of Eq. (28) the terms
$({3}/{2})\tilde{n}_{F}-12C_{\mathrm{F}V}\tilde{n}_{\mathrm{F}}\tilde{n}_{V}-15C_{\mathrm{F}V\mathrm{F}}\tilde{n}_{\mathrm{F}}^{2}\tilde{n}_{V}$,
where $\tilde{n}_{\mathrm{F}}$ is the number of $\mathrm{F}_{\mathrm{O}}^{+}$
defects normalized to the total number of oxygen atoms. The quantities
$\tilde{n}_{\mathrm{F}}$ and $\tilde{n}_{4}$ satisfy the equation

$\tilde{n}_{\mathrm{F}}=\tilde{n}_{4}e^{-\frac{E_{\mathrm{F}_{\mathrm{O}}^{+}}-E_{4}}{k_{B}T}},$
(37)

which follows from Eq. (5). The difference
$E_{\mathrm{F}_{\mathrm{O}}^{+}}-E_{4}$ is given in Table 4. The binding
energies of the complexes $\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$
and
$\mathrm{F}_{\mathrm{O}}^{+}-\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$
are presented in Table 3.

The calculated equilibrium concentrations of defects are shown in Figs. 12 and
13. One can see from Fig. 13 that the concentrations of positively charged
$\mathrm{F}_{\mathrm{O}}^{+}$ and negatively charged
$\mathrm{F}_{\mathrm{O}}^{+}-\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$
are close to each other. Negatively charged defects almost compensate
positively charged ones and the concentrations of $\mathrm{Ti}^{4+}$ and
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ are changed unessentially. Thus, our
expectation on a positive role of fluorine is not confirmed. It appears that
triples
$\mathrm{F}_{\mathrm{O}}^{+}-\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$
have rather small formation energy and the positive effect caused by isolated
$\mathrm{F}_{\mathrm{O}}^{+}$ ions reduces due to formation of
$\mathrm{F}_{\mathrm{O}}^{+}-\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$
triples. We note that such a reduction is not a general rule. It would not
happen if the binding energy of complexes was smaller.

Figure 12:  Equilibrium concentrations of defects in Ti:sapphire at $T=T_{m}$
in the conditions that correspond to reference points of
$\mathrm{Al}-\mathrm{Ti}-\mathrm{O}-\mathrm{F}$ phase diagram Fig. 11. Figure
13:  Equilibrium concentrations of fluorine defects at $T=T_{m}$.

###  V.4 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
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ ratio at all
reference points of Table 1 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
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$. 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 (Al2O3 and TiO2) makes it
possible to decrease the concentration of Ti4+ in Ti:Al2O3 crystals. With
this, Fig. 14 illustrates that nitrogenization provokes considerable reduction
of the $c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ 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
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$. To do that we
use the data of [13]. One of the samples investigated in [13] (labeled as
SY1b) had $\mathrm{FOM}=12$. Concentrations of $\mathrm{Ti}^{3+}$ and
$\mathrm{Ti}^{4+}$ in this sample were estimated as $1.3\cdot 10^{18}$ cm-3
and $2.3\cdot 10^{18}$ cm-3, correspondingly. Using Eq. (12) and taking
$T=2000$ K we obtain $c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}\approx 10^{16}$
cm-3 and $c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}\approx
100$. Thus one can estimate that $\mathrm{FOM}\sim 0.1\cdot
c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$. Note that,
normally, commercial samples have FOM $=100\div 300$ and FOM of the best
samples is up to 1000.

Figure 14: The ratio of concentrations of Ti3+ to the concentration of
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ pairs at reference points of
Al$-$Ti$-$O$-X$ phase diagrams (see Table 1) at $T=T_{m}$. Filled circles
correspond to $X=\mathrm{C}$, open circles, to $X=\mathrm{N}$ and open
squares, to $X=\mathrm{F}$. Lines are guides to the eye.

###  V.5 Band structure of Ti:sapphire co-doped with carbon, nitrogen and
fluorine

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 $\mathrm{Ti}^{4+}$
substitutional defects and with $V_{\mathrm{Al}}^{3-}$ 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 ($\mathrm{C}_{\mathrm{O}}^{2-}$,
$\mathrm{C}_{\mathrm{O}}^{-}$ and $\mathrm{N}_{\mathrm{O}}^{-})$ reveal
themselves in an appearance of additional impurity levels. Binding of such
defects with $\mathrm{Ti}^{4+}$ 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 $\mathrm{Ti}^{3+}$ and $\mathrm{Ti}^{4+}$
ions, but also due to the appearance of impurity levels caused by nitrogen
defects.

Figure 15: Band structure of $\mathrm{Al}_{2}\mathrm{O}_{3}$ with
$\mathrm{C}_{\mathrm{O}}^{2-}$ (a), $\mathrm{C}_{\mathrm{O}}^{-}$ (b),
$\mathrm{C}_{\mathrm{O}}^{2-}-\mathrm{Ti}^{4+}$ (c) and $\mathrm{Ti}^{4+}$ (d)
defects. The valence band maximum is set at 0 eV, and the arrow indicates the
position of the highest occupied level.

Figure 16: The same as in Fig. 15 for $\mathrm{N}_{\mathrm{O}}^{-}$ (a) and
$\mathrm{N}_{\mathrm{O}}^{-}-\mathrm{Ti}^{4+}$ (b)

Figure 17: The same as in Fig. 15 for a $\mathrm{F}_{\mathrm{O}}^{+}$ (a),
$\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$ (b),
$\mathrm{F}_{\mathrm{O}}^{+}-\mathrm{F}_{\mathrm{O}}^{+}-V_{\mathrm{Al}}^{3-}$
(c) and $V_{\mathrm{Al}}^{3-}$ (d)

##  VI Universal relations between the concentrations of $\mathrm{Ti}^{3+}$,
$\mathrm{Ti}^{4+}$ and $\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$

In this section we discuss whether the observed relations between the
concentrations of isolated $\mathrm{Ti}^{3+}$ and $\mathrm{Ti}^{4+}$ defects,
and $\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ pairs confirm or put in question the
pair model of NIR absorption.

According to Eq. (12), 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 $\mathrm{Ti}^{3+}$ to the concentration of
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ will be in inverse proportion to the
concentration of $\mathrm{Ti}^{4+}$. Assuming that the
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ ratio determines
the FOM one can expect that the FOM would be in inverse proportion to
$c_{\mathrm{Ti}^{4+}}$ 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 $\mathrm{Ti}^{4+}$. 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
$\mathrm{Al}-\mathrm{Ti}-\mathrm{C}$ phase diagram at two different
temperatures, $T=2100$ K and $T=2000$ K. The reference points for the
$\mathrm{Al}-\mathrm{Ti}-\mathrm{C}$ diagram and the value of
$\mu_{\mathrm{O}}$ at three different temperatures are given in Table 5 (under
lowering in temperature two reference points, C and G, disappear and are
replaced with the point C1). 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 $c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$
ratio for these samples and plot it against $1/c_{\mathrm{Ti}^{4+}}$ (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 $1/c_{\mathrm{Ti}^{4+}}$. 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.

Table 5: Oxygen chemical potential at reference points of $\mathrm{Al}-\mathrm{O}-\mathrm{C}$ phase diagram at three different temperatures. $T$,K | Reference points, phases in equilibrium, $\mu_{\mathrm{O}}$ in eV
---|---
| A
---
$\mathrm{Al}_{2}\mathrm{O}_{3}$, $\mathrm{O}_{2}$, $\mathrm{CO}_{2}$
| C
---
$\mathrm{Al}_{2}\mathrm{O}_{3}$, $\mathrm{CO}_{2}$, CO
| C1
---
$\mathrm{Al}_{2}\mathrm{O}_{3}$, $\mathrm{CO}_{2}$, C
| G
---
$\mathrm{Al}_{2}\mathrm{O}_{3}$, CO, C
| L
---
$\mathrm{Al}_{2}\mathrm{O}_{3}$, C, Al
2100 | -6.97 | -9.11 | - | -9.73 | -10.38
2000 | -6.83 | -9.06 | - | -9.50 | -10.35
1600 | -6.28 | - | -8.73 | - | -10.25

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 $1/c_{\mathrm{Ti}^{4+}}$ 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

$\displaystyle
c_{\mathrm{Ti}^{3+}}+c_{\mathrm{Ti}^{4+}}+c_{\mathrm{Ti}^{4+}-V_{\mathrm{Al}}^{3-}}$
$\displaystyle=$ $\displaystyle c_{1}=const,$ (38) $\displaystyle
c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{3+}}+c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}+c_{\mathrm{Ti}^{4+}-\mathrm{Ti}^{4+}-V_{\mathrm{Al}}^{3-}}+c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}-V_{\mathrm{Al}}^{3-}}+c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{3+}-V_{\mathrm{Al}}^{3-}}$
$\displaystyle=$ $\displaystyle c_{2}=const.$ (39)

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 (38) and
(39) the relation (10) is not satisfied. Therefore the relation (12) is
violated which can be demonstrated by direct calculations.

We calculate equilibrium concentration of Ti3+, Ti4+ and
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ under the constraints (38) and (39) at
$T=1600$ K (we imply that the frozen regime is reached at this temperature).
We fix $\mu_{\mathrm{O}}$ that corresponds to the point C1 (Table 5). 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
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ is not
proportional to $1/c_{\mathrm{Ti}^{4+}}$. Comparing Fig. 18 and Fig. 19 we
find that $c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$
correlates with the $c_{1}/c_{2}$ ratio. Note that the violation of the
relation (12) and more general relation (10) is caused solely by the
additional constraints (38) and (39), and therefore this effect most probably
holds true for a wide range of ”freezing” temperatures.

Figure 18: The $c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$
ratio versus the inverse concentration of $\mathrm{Ti}^{4+}$. Open circles
show equilibrium concentrations at $T=T_{m}$, filled circles, at $T=2100$K and
$T=2000$K, labeled as xYa and xYb, respectively
($\mathrm{x}=\mathrm{c,d,f,i,j}$ refers to the parent sample, and
$\mathrm{Y}=\mathrm{C,G,L}$, to the reference point in the Al$-$O$-$C phase
diagram). Triangles correspond to equilibrium concentration at $T=1600$ K
reached under condition that diffusion of Ti is suppressed. Dashed lines show
linear fit. A rectangle in (a) indicates the area shown in (b) in another
scale. Figure 19: Sum of equilibrium concentrations of isolated Ti and pairs
Ti${}^{4+}-V_{\mathrm{Al}}$ ($c_{1}$) and sum of concentrations of Ti$-$Ti
pairs and Ti$-$Ti$-V_{\mathrm{Al}}$ triples ($c_{2}$) at $T=T_{m}$ in the the
conditions that correspond to the point C, D, F, I, J of the Al$-$Ti$-$O$-$C
phase diagram (Fig. 5).

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
$\alpha_{820}$ is proportional to the square of the absorption coefficient
$\alpha_{490}$. If NIR absorption is caused by
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ pairs such a behavior means that the
concentration of $\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ pairs is proportional to
the square of the concentration of $\mathrm{Ti}^{3+}$ isolated ions. It cannot
be the general property. Nevertheless our calculations show that for the
samples annealed at the same $\mu_{\mathrm{O}}$ this property is satisfied at
least approximately. We specify $\mu_{\mathrm{O}}=-10.0$ 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 $c_{\mathrm{Ti}}$ is fixed. We
consider 20 different $c_{\mathrm{Ti}}$ in the range from $5\cdot 10^{18}$
cm-3 to $10^{20}$ cm-3. Obtained concentrations of
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ are plotted in Fig. 20 against the
concentration of $\mathrm{Ti}^{3+}$. One can see that for given $T$ the law
$c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}=\kappa c^{2}_{\mathrm{Ti}^{3+}}$ is
satisfied. The coefficient $\kappa$ weakly depends on temperature. Thus the
observed in [13] correspondence between $\alpha_{820}$ and $\alpha_{490}$ does
not contradict the pair model of NIR absorption.

Figure 20: Dependence of the concentration of
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ on the concentration of Ti3+ calculated
for the case where equilibrium concentrations are reached at fixed total Ti
concentration at $\mu_{\mathrm{O}}$=-10.0 eV and $T=T_{m}$ (diamonds),
$T=2100$ K (squares) and $T=2000$ K (circles). Lines show the quadratic fit.
Total concentration of Ti is varied from $5\cdot 10^{18}$ cm-3 to $10^{20}$
cm-3

##  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
into the Al2O3 crystal predominantly in the form of substitutional
$\mathrm{Ti}^{3+}$ and $\mathrm{Ti}^{4+}$ ions. An essential fraction of such
defects bind in pairs, triples etc., or form complexes with Al vacancies.

For laser application it is important to reduce the concentration of
$\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}$ pairs keeping concentration of unpaired
Ti3+ ions at high level. To increase
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ 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 ($\lesssim 10^{18}$ cm-3). Annealing in the conditions
that correspond to intermediate values of the oxygen chemical potential may
result in a decrease of the
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ 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
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ 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
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ 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
$c_{\mathrm{Ti}^{3+}}/c_{\mathrm{Ti}^{3+}-\mathrm{Ti}^{4+}}$ ratio and FOM are
proportional for the inverse concentration of $\mathrm{Ti}^{4+}$ 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.

## Acknowledgment

This work was performed using computational facilities of the Joint
computational cluster of the State Scientific Institution ”Institute for
Single Crystals” and the Institute for Scintillation Materials of the National
Academy of Sciences of Ukraine incorporated into the Ukrainian National Grid.

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