First-principles study of Ti-doped sapphire. II. Formation and reduction of complex defects

Eliminating the effect of residual infrared absorption in titanium sapphire remains a crucial task to fulfill, despite that this kind of laser crystal has been developed for decades. The ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{4+}$ (${\mathrm{Ti}}_{\mathrm{Al}}^0\text{−}{\mathrm{Ti}}_{\mathrm{Al}}^{1+}$) pair model is the most widely accepted explanation to this residual absorption, but theoretical analyses based on first-principles calculation have not yet depicted a clear picture for the variation of the ${\mathrm{Ti}}_{\mathrm{Al}}^0\text{−}{\mathrm{Ti}}_{\mathrm{Al}}^{1+}$ pair and a variety of other potentially important defects in titanium sapphire when fabricating conditions change. Here, we extend the work in [W. Jing, M. Liu, J. Wen, L. Ning, M. Yin, and C.-K. Duan, Phys. Rev. B 104, 165103 (2021)] about binding tendencies of ${\mathrm{Ti}}_{\mathrm{Al}}\text{−}{\mathrm{Ti}}_{\mathrm{Al}}$ pairs and optical transition properties to providing a comprehensive understanding of the formation and processing condition dependence of various potentially significant defects and complexes besides the (${\mathrm{Ti}}_{\mathrm{Al}}\text{−}{\mathrm{Ti}}_{\mathrm{Al}}{)}^{0,1+}$ pairs. Apart from the complexes composed of $V_{\mathrm{Al}}^{3-}$ and ${\mathrm{Ti}}_{\mathrm{Al}}^{0,1+}$ defects, two new significant negatively charged defects, ${\left(V_{\mathrm{Al}}\text{−}{\mathrm{Al}}_i\text{−}{V}_{\mathrm{Al}}\right)}^{3-}$ and ${\left(V_{\mathrm{Al}}\text{−}{\mathrm{Ti}}_i\text{−}{V}_{\mathrm{Al}}\right)}^{2-}$ are revealed. These two complexes are correspondingly an interstitial ${\mathrm{Al}}^{3+}$ and ${\mathrm{Ti}}^{4+}$ surrounded by two neighboring aluminium vacancies $V_{\mathrm{Al}}^{3-}$. We show that $V_{\mathrm{Al}}^{3-}$ plays the key role to stabilize ${\mathrm{Ti}}_{\mathrm{Al}}^0\text{−}{\mathrm{Ti}}_{\mathrm{Al}}^{1+}$ and form various stable complexes with ${\mathrm{Ti}}_{\mathrm{Al}}^{1+}$ and ${\mathrm{Ti}}_{\mathrm{Al}}^0\text{−}{\mathrm{Ti}}_{\mathrm{Al}}^{1+}$ in weak and moderate reductive atmospheres. Thus, besides annealing at strong reductive atmosphere at elevated temperatures, Al ion injection or annealing in Al vapor is a potential method to eliminate the harmful residual infrared absorption, which is pinpointed at the reduction of the concentration of $V_{\mathrm{Al}}^{3-}$ and its variant ${\left(V_{\mathrm{Al}}\text{−}{\mathrm{Al}}_i\text{−}{V}_{\mathrm{Al}}\right)}^{3-}$. While in extremely strong reductive atmosphere, isolated Ti substitution dominates over those complexes containing $V_{\mathrm{Al}}^{3-}$, but there still remains a tiny amount of ${\mathrm{Ti}}_{\mathrm{Al}}^0\text{−}{\mathrm{Ti}}_{\mathrm{Al}}^{1+}$ and ${\mathrm{Ti}}_{\mathrm{Al}}^{1+}$, which are charge compensators to balance the charge of trace amount ${\mathrm{Ti}}_{\mathrm{Al}}^{1-}$ further reduced from ${\mathrm{Ti}}_{\mathrm{Al}}^0$. And this effect can be further eliminated by lower temperature reductive-atmosphere annealing. In addition, we obtain a simple numerical expression to predict the achievable figure of merit when the concentration of ${\mathrm{Ti}}_{\mathrm{Al}}^{1+}$ is given. Formation energies for simple defects and binding energies for complexes obtained in this work may serve as the basis for simulations and design various quenching and annealing processes to further reduce harmful defect species in titanium sapphire.

Figure 1

Top left: the modified primitive cell of ${\mathrm{Al}}_2{\mathrm{O}}_3$ containing a ${\mathrm{Ti}}_i$ (or ${\mathrm{Al}}_i$) in the center. Bottom right: the basic unit to visualize multiple complex defects. The primitive cell without ${\mathrm{Ti}}_i$ (or ${\mathrm{Al}}_i$) contains four Al atoms in blue color along the $C_3$ axis and six oxygen atoms in red forming a distorted octahedron. The basic unit includes the center Al site (blue and denoted as $c$ site) and four surrounding neighbors of Al site including one nearest neighbor (green and denoted as $u$ site) and three next neighbors (purple and denoted as $o$ site). The arrow from the center Al to its nearest neighbor Al is the $C_3$ axis. It is noted that there is only one inequivalent single Al site and $c, u$ and $o$ are just used to distinguish inequivalent positions in a complex.

Figure 2

The relative concentration at equilibrium of the dominant (a) and other less dominant but important (b) defects and complexes in Ti:${\mathrm{Al}}_2{\mathrm{O}}_3$ crystal. The total Ti concentration is fixed at 0.50 at.% ($2.34\times {10}^{20}{\mathrm{cm}}^{-3}$ or 0.47 wt.%) and is used to normalize the concentrations of others. The annealing temperature is set at 1873 K. It is noted that Eq. (11) suggests a rescaling of the pressure by a factor $2.2\times {10}^{-4}$. For compact and accurate display, the equilibrium concentrations of similar complexes are merged by ignoring the difference between $u$ and $o$ site in configuration, i.e., by treating $u$ and $o$ sites as the same. In addition, relationships between formation energies of these defects and complexes and the Fermi level in a given oxygen partial pressure can be referred in Sec. S4 [41].

Figure 3

The defects' thermodynamic transition levels and the Fermi levels at different oxygen partial pressures of $p_0$ (O rich), ${10}^{-6}{p}_0$ (O moderate), and ${10}^{-12}{p}_0$ (O poor). The two integers $(q,M)$ are the charge $q$, and the multiplicity of (near) degenerate ground states, $M$. We note that all charge transition levels in the band gap are calculated with GGA$+U$ method, the HDFT band gap is obtained via PBE0-type hybrid density functional with HF mixing parameter $\alpha =0.29$, and the VBM of HDFT is 1.954 eV lower than GGA one according to the alignment of average electron potential in the supercell. Such alignment leads to the estimation of the threshold of ${\mathrm{Ti}}^{4+}$ charge transfer transition $E_{\mathrm{CT}}\left(0\right)=4.48\mathrm{eV}$ and ${\mathrm{Ti}}^{3+}$ photoionization $E_{\mathrm{PI}}\left(0\right)=4.68\mathrm{eV}$, which are correspondingly consistent to the slightly different values of 4.27 and 4.89 eV obtained by our previous PBE0 calculation [24].

Figure 4

Dependence of the chemical potential $\mathrm{\Delta }{\mu }_{\mathrm{Ti}}$ relative to Ti metal and the Fermi level $\mathrm{\Delta }{\varepsilon }_F$ relative to VBM of HDFT on the relative oxygen chemical potential $\mathrm{\Delta }{\mu }_{\mathrm{O}}$ and oxygen partial pressure in Fig. 2. The Fermi level relative to VBM of GGA compatible with Table 3b is $\mathrm{\Delta }{\varepsilon }_{\mathrm{F}}^{\prime }=\mathrm{\Delta }{\varepsilon }_{\mathrm{F}}-1.954\mathrm{eV}$.

Figure 5

The $c_{{\mathrm{Ti}}_{\mathrm{Al}}^0}/c_{{\mathrm{Ti}}_{\mathrm{Al}}^0-{\mathrm{Ti}}_{\mathrm{Al}}^{1+}}\sim \mathrm{FOM}·R$ versus the oxygen partial pressure when (a) the total doped Ti concentration is fixed at 0.5 at.% and (b) the annealing temperature is at 1873 K. $R\sim 1/3$ is the radio of effective absorption bandwidth between ${\mathrm{Ti}}_{\mathrm{Al}}^0$ and ${\mathrm{Ti}}_{\mathrm{Al}}^0\text{−}{\mathrm{Ti}}_{\mathrm{Al}}^{1+}$ and $f_r$ has been taken as ${10}^{-3}$. It is noted that the unit of $\mathrm{\Delta }{\mu }_{\mathrm{O}}$ in (a) is $kT$, which equals to 0.1442, 0.1614, 0.1786, and 0.1959 eV for annealing temperature $T=1673$, 1873, 2073, and 2273 K, respectively. Similar to Fig. 2, Eq.(11) suggests a rescaling of the pressure by the factors $0.80\times {10}^{-4}$ ($1673\mathrm{K}$), $2.2\times {10}^{-4}$ ($1873\mathrm{K}$), $4.9\times {10}^{-4}$ ($2073\mathrm{K}$), and $9.7\times {10}^{-4}$ ($2273\mathrm{K}$).