Thermodynamic Calculations of Ti Ion Concentrations and Segregation Coefficients during Ti:Sapphire Crystal Growth

Abstract

The theoretical phase diagrams of the binary Al 2 O 3 −Ti 2 O 3 and Al 2 O 3 −TiO 2 , as systems calculated using FactSage software and its thermodynamic databases, are reoptimized. This has been performed on the basis of a literature survey on experimental precipitation limits of Ti 2 O 3 and TiO 2 in the Al 2 O 3 solid solution. The appropriate oxygen partial pressure (p O 2 ) range for controlling the valence state of Ti ions during Ti:sapphire crystal growth is obtained with respect to the calculated Ellingham type predominance diagram. Factors affecting the p O 2 such as furnace gas, crucible materials, and graphite heating elements are analyzed. The influence of p O 2 on Ti 3+ and Ti 4+ concentrations in the solid and liquid phases is derived from thermodynamic equilibrium calculations. The equilibrium segregation coefficients of Ti 3+ and Ti 4+ in sapphire are calculated to be 0.046 and 0.011, respectively. Taking into account the data extrapolated from experiments, a value of 0.06 ± 0.02 is proposed for the equilibrium segregation coefficient of total Ti under low p O 2 .

On the way to modelling dopant valence in ionic crystal growth: the Ti:Al2O3 case

“…where k Ti 3+ and k Ti 4+ are the segregation coefficients of Ti 3+ and Ti 4+ , which values have been computed to be 0.046 and 0.011 through thermodynamic calculations. 12 The subscripts "S" and "L" represent species concentrations in the solid and liquid phases. Considering the Arrhenius expressions of aluminum vacancy diffusion coefficient (16), reaction rate constant (17) and boundary condition ( 23) are function of temperature, the temperature field in the crystal needs to be taken into account.…”

“…8) The reaction occurring on crystal surface is very fast 13 so that the p O 2 released by surface reaction is in equilibrium with the surrounding atmosphere; 9) Enthalpies of point defect reactions are negligible; 10) The oxygen activity in the liquid is controlled by the Mo crucible. 12 During Ti:sapphire growth process, p O 2 is so low that local reactions ( 2) and (3) occur towards the left. The governing equations during Ti:sapphire growth process are the same as eqn ( 3)-( 5) while the reaction rates for Ti 3+ , Ti 4+ and V Al ‴ are: 9…”

“…With the mass action law and charge neutrality condition , the equilibrium constant of reaction (19) can be written as:During Czochralski growth of Ti:sapphire, p O 2 is very low, around 10 −20 atm, due to the use of Mo crucible and graphite casing in the furnace, then is very low and negligible compared to the total Ti concentration [Ti] in the crystal. 12 So, eqn (20) can be simplified as:…”

“… [Ti × Al ] S = [Ti 3+ ] S = k Ti 3+ [Ti 3+ ] L , [Ti˙ Al ] S = [Ti 4+ ] S = k Ti 4+ [Ti 4+ ] L , where k Ti 3+ and k Ti 4+ are the segregation coefficients of Ti 3+ and Ti 4+ , which values have been computed to be 0.046 and 0.011 through thermodynamic calculations. 12 The subscripts “S” and “L” represent species concentrations in the solid and liquid phases. Taking into account the potential reaction between Mo crucible and the melt (Al 2 O 3 + TiO 2 ) at 2054 °C, 8 the initial concentration ratio in the liquid phase is [Ti 4+ ] L /[Ti 3+ ] L = 4.04.…”

“…According to the Brouwer diagram, 11 and holes are dominant for oxygen partial pressure ( p O 2 ) larger than 10 −20 atm, which is consistent with the p O 2 obtained from thermodynamic calculations of Ti:sapphire growth process. 12 At lower p O 2 , oxygen vacancies and electrons should be considered. 5 Based on previously proposed physico-chemical model, 9 the set of reactions (1)–(3) are considered in the present contribution in order to make explanations simpler.…”