First-principles study of Ti-doped sapphire. I. Formation and optical transition properties of titanium pairs

Titanium sapphire is one of the most important laser crystals suitable for widely tunable and ultrashort pulsed lasers with high gain and high power outputs, but its performance is limited by the residual infrared absorption at the operation wavelength region of the laser. Although studies have been made over decades in improving the laser performance and the solutions to eliminate this residual absorption, there still remain some controversies for the binding tendency and charge-transfer transition energy of Ti related pairs, which is supposed to be the culprit of this residual absorption. In this paper, we clarify that previously predicted strong binding tendencies in ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{3+}$ and ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{4+}$ pairs should be artificial and are blamed on the intrinsic delocalization error in general approximate density functionals. We show that such errors can be eliminated by Hubbard $U$ corrected generalized-gradient-approximation method with $4\le U\le 5\mathrm{eV}$ or hybrid density functionals such like HSE06 and PBE0, which approximately satisfy the generalized Koopmans' condition. Our calculations reveal that the equilibrium geometry and electronic structure of ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{4+}$ pairs can be ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{4+}$-type or ${\mathrm{Ti}}^{3.5+}\text{−}{\mathrm{Ti}}^{3.5+}$-type configurations with very small energy differences ($\lesssim 0.1\mathrm{eV}$), and both of them have residual infrared absorption, which are predicted to be about three orders of magnitude stronger in oscillator strength per defect than the pump absorption of ${\mathrm{Ti}}^{3+}$ dopants. Regarding ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{3+}$ pairs, it is shown that they do not contribute to the residual infrared absorption in the wavelength range of laser operation, but their charge-transfer transitions can explain the residual UV band at 270 nm and $E$ band at 400 nm in the absorption spectra of Ti:${\mathrm{Al}}_2{\mathrm{O}}_3$ crystals. Furthermore, the charge-transfer transition energies and Stokes shift of ${\mathrm{Ti}}^{4+}$ and ionization energies for ${\mathrm{Ti}}^{3+}$ are also well interpreted by our calculations. The calculation method developed together with the predicted optical properties forms the basis for exploration on eliminating the residual infrared absorption in titanium sapphire.

Figure 1

Three octahedral units in $\alpha \text{−}{\mathrm{Al}}_2{\mathrm{O}}_3$ are drawn to show the local environment of Ti ions. The white balls are Al atoms and red balls are oxygen atoms. Two cyan octahedral units are connected by the face-sharing mode where these two units share three oxygen atoms, and the cyan and pink octahedrons are connected in edge-sharing mode by sharing two oxygen atoms.

Figure 2

The deviations of total energy function $E_{\mathrm{tot}}\left({\mathrm{Ti}}^{3+q}\right)$ between exact functional and $\mathrm{GGA}+U$ results, where $U=0,3,4,5$ (in units of eV). Several HDFT methods in $\mathrm{HSE}(\alpha ,\mu$) functional are also used as references, where $\alpha$ is the HF mixing parameter and $\mu$ is the screen parameter. $N-q$ is the total number of electrons.

Figure 3

The optimized nearest configurations of ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{4+}$ pair using the GGA (a), $\mathrm{GGA}+U$ (b), and HSE(0.40, 0.2) (c) methods. Their corresponding density distribution of $3d$-electron wave functions are plotted in (d)–(f). The average Ti1-O and Ti2-O length in (a) are both 1.987 $\AA$, while in (b) are 2.034 and 1.986 $\AA$, and in (c) are 2.023 and 1.964 $\AA$, respectively.

Figure 4

Schematic configuration coordinate diagram for ${\mathrm{Ti}}^{4+}$ charge transfer and ${\mathrm{Ti}}^{3+}$ photoionization (PI) processes. Here $E_{\mathrm{g}}^{\mathrm{opt}}$ and $E_{\mathrm{g}}$ denote, correspondingly, the optical and fundamental band gaps of ${\mathrm{Al}}_2{\mathrm{O}}_3$.

Figure 5

The $3d^1$ levels obtained with Molcas on ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{4+}$ structure obtained via the $\mathrm{GGA}+U$ method and ${\mathrm{Ti}}^{3.5+}\text{−}{\mathrm{Ti}}^{3.5+}$ structure optimized via the HSE06 method, both nearest and next configurations are considered. ${\mathrm{Ti}}_1$–$3d$ and ${\mathrm{Ti}}_2$–$3d$ components dominate all the states shown, and the color of a state is an indication of proportion of the orbital ${\mathrm{Ti}}_1$–$3d$ normalized to total $3d$ components.

Figure 6

The energy levels obtained with Molcas for different electronic configurations in isolated and the nearest pair of ${\mathrm{Ti}}^{3+}$ equilibrium configurations, which are denoted as ${\left({\mathrm{Ti}}^{3+}\right)}_{\mathrm{c}}$ and ${\left({\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{3+}\right)}_{\mathrm{c}}$, respectively. Energy levels denoted as $d^1$ in the first column belong to ${\mathrm{Ti}}^{3+}$ ions, and the second and third columns are energy levels of ${\mathrm{Ti}}^{2+}$, which is named as $d^2$. The fourth and fifth columns are energy levels of the ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{3+}$ pair where two $3d$ electrons are localized on two Ti atoms separately and we denote this situation as $d^1\otimes d^1$, while levels in the last two columns are also energy levels of the ${\mathrm{Ti}}^{3+}\text{−}{\mathrm{Ti}}^{3+}$ pair, but two $3d$ electrons are now localized on one Ti ion where actually the electronic structure is ${\mathrm{Ti}}^{2+}\text{−}{\mathrm{Ti}}^{4+}$ and we denote it as $d^2\otimes d^0$. The lowest vertical $d^1\otimes d^1$ to $d^2\otimes d^0$ spin-allowed transition (CT absorption) is derived as $E_{\mathrm{CT}}^{S=1}=2.80\mathrm{eV}$ and $E_{\mathrm{CT}}^{S=0}=4.26\mathrm{eV}$.