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

We study mechanisms of control of charge state and concentration of different point defects in doped insulating crystals. The approach is based on the density functional theory calculations. We apply it to the problem of obtaining of Ti-doped sapphire crystals with high figure of merit (FOM). The FOM of a given sample is defined as the ratio of the coefficient of absorption at the pump frequency to the coefficient of absorption at the working frequency of a Ti:sapphire laser. It is one of standard specifications of commercial Ti: sapphire laser crystals. It is believed that the 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 thermodynamic 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.