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 ${\mathrm{Ti}}^{3+}\text{\ensuremath{-}}{\mathrm{Ti}}^{3+}$ and ${\mathrm{Ti}}^{3+}\text{\ensuremath{-}}{\mathrm{Ti}}^{4+}$ 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 $4\ensuremath{\le}U\ensuremath{\le}5\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ 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 ${\mathrm{Ti}}^{3+}\text{\ensuremath{-}}{\mathrm{Ti}}^{4+}$ pairs can be ${\mathrm{Ti}}^{3+}\text{\ensuremath{-}}{\mathrm{Ti}}^{4+}$-type or ${\mathrm{Ti}}^{3.5+}\text{\ensuremath{-}}{\mathrm{Ti}}^{3.5+}$-type configurations with very small energy differences ($\ensuremath{\lesssim}0.1\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$), 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 ${\mathrm{Ti}}^{3+}\text{\ensuremath{-}}{\mathrm{Ti}}^{3+}$ 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.