The ratio of composition in the solid and the melt is a prominent quantity during crystallization of single crystals. This ratio which is known as the effective distribution coefficient is defined by k_eff=C_S/C_L. In fact, achieving a precise value of distribution coefficient could help to grow a crystal with high accurate dopant concentration and high optical quality.   

In many crystal growth processes by Czochralski method, the distribution of solute can be approximately given by the Scheil equation,C_s=k_eff(1-f_s)^keff-1,[1]. So, the effective distribution coefficient (k_eff) can be derived as the form of k_eff=W(z)/Ln(1-f_s) which z equals to (Cs/C₀)(1-f_s)Ln(1-f_s). Where, the Lambert W function [2], represented by W(z), is defined as the inverse of the function f(z) = ze^z, satisfying W(z)e^W(z) = z for any complex number z. 

In this study, four Nd:YAG crystals were grown by Czochralski method using induction heating equipment and auto diameter control software (ADC). The crystals were grown along the <111> crystallographic direction utilizing pure YAG seed crystal. They were pulled with a rate of 0.4 to 0.6 mm/h. The crystal rotation rate was varied from 16 to 20 rpm in order to improve structural and stoichiometric uniformity in the melt during all crystal growth processes. The initial concentrations of neodymium (C₀) into YAG melt were 2.73 at% and 3.33 at%. Each crystal was cut into 1.0 mm thick slices along growth direction of the boul. All samples have been polished on top and bottom planes to achieve good transparency. In the end, Neodymium concentration of samples was measured by Spectrometer instrument. 

Because of interface flip, a sudden and uncontrollable interface inversion occurs during the growth of crystal with constant crystal rotation rate and increasing crystal radius (or with constant crystal diameter and decreasing melt level), Nd:YAG crystal is grown almost under convex interface. Utilizing the growth parameters (weight signal, level of melt and crystal diameter) recorded by ADC software is a new idea to calculate interface deflection and solid fraction (f_s). This technique eliminates the error created by Archimedes force during the measurement of crystal weight and causes more accurate k_eff using the Scheil equation. In conclusion, our results indicate that the average of effective segregation coefficients of neodymium in YAG crystals, respectively, are 0.28 and 0.24 for C₀=2.72 at% Nd⁺³ and C₀=3.33 at% Nd⁺³.

References

[1] E. Scheil, Z. fur Metall., 34 (1942) 70

[2] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, Advances in Computational Mathematics 5 (1996) 329

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