Abstract
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 Ti:sapphire laser. It is believed that FOM is proportional to the ratio of the concentration of isolated Ti$^{3+}$ ions to the concentration of Ti$^{3+}$-Ti$^{4+}$ pairs. We find that generally this ratio is in inverse proportion to the concentration of Ti$^{4+}$ isolated substitutional defects with the coefficient of proportionality that depends on the temperature at which the thermodynamically equilibrium concentration of defects is reached. We argue that in certain cases the inverse proportion between concentrations of Ti$^{3+}$-Ti$^{4+}$ and Ti$^{4+}$ may be violated. 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 Ti$^{4+}$ 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.
Highlights
INTRODUCTIONDoping of insulating crystals with active ions is a widely used method of obtaining functional materials (active laser medium, luminescent materials, scintillators, and many others) with required properties
Doping of insulating crystals with active ions is a widely used method of obtaining functional materials with required properties
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
Summary
Doping of insulating crystals with active ions is a widely used method of obtaining functional materials (active laser medium, luminescent materials, scintillators, and many others) with required properties. We derive the general relation between equilibrium concentrations of isolated and complex defects and show that this relation takes place if the total number of dopant atoms is fixed. Substituting Eq (10) into Eq (5) and redefining the Lagrange multiplier λq = λq − μe one can exclude the electron chemical potential from the problem This means that equilibrium concentrations of defects can be expressed through the quantities independently of μe. If the total number of Ti atoms is fixed, the chemical potential of Ti can be excluded as well In the latter case equilibrium concentrations of defects do not depend on μTi. Applying Eq (9) to Ti:sapphire we find that the ratio of the concentration of isolated Ti3+ ions to the concentration of Ti3+-Ti4+ pairs is in inverse proportion to the concentration of isolated Ti4+ ions: c3 e−.
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