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We study mechanisms of control of charge state and concentration of differentpoint defects in doped insulating crystals. The approach is based on thedensity functional theory calculations. We apply it to the problem of obtainingof Ti-doped sapphire crystals with high figure-of-merit (FOM). The FOM of agiven sample is defined as the ratio of the coefficient of absorption at thepump frequency to the coefficient of absorption at the working frequency ofTi:sapphire laser. It is believed that FOM is proportional to the ratio of theconcentration of isolated Ti$^{3+}$ ions to the concentration ofTi$^{3+}$-Ti$^{4+}$ pairs. We find that generally this ratio is in inverseproportion to the concentration of Ti$^{4+}$ isolated substitutional defectswith the coefficient of proportionality that depends on the temperature atwhich the thermodynamically equilibrium concentration of defects is reached. Weargue that in certain cases the inverse proportion between concentrations ofTi$^{3+}$-Ti$^{4+}$ and Ti$^{4+}$ may be violated. We show that codopants thatform positively (negatively) charged defects may decrease (increase) theconcentration of positively charged defects formed by the main dopants. Toevaluate the effect of codoping it is important to take into account not onlyisolated defects but defect complexes formed by codopants, as well. Inparticular, we show that codoping of Ti:sapphire with nitrogen results in anessential increase of the concentration of Ti$^{4+}$ and in a decrease of theFOM, and, consequently, growth or annealing in the presence of nitrogen or itscompounds is unfavorable for producing Ti:sapphire laser crystals. The approachdeveloped can be used for determining appropriate growth and annealingconditions for obtaining doped crystals with the required characteristics.

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