Dependence of the Yield Percentage and Orientation Distribution on the Growth Condition in Growing Single Crystals from the Melt

Abstract

Firstly, the thermal factor or the temperature distribution, in particular, the temperature gradients in the solid and in the liquid at the solid-liquid interface, G_so and G_lo, the crystallographic factor or the direction of preferential crystal growth, and the impurity factor or the constitutional supercooling, and their dependence on the crystallization velocity, R_i, and the furnace temperature gradient, G_f, are discussed for the case of solidification. It is shown that the conditions necessary for growing successfully single crystals from the melt are the high purity of the material and the high value of the ratio G_f⁄R_i as already known empirically. Secondly, the yield percentage, S, of growing single crystals from the melt as dependent on R_i and G_f is considered. It is found that single crystals grow practically only when G_lo is positive and that S is higher as G_lo is larger. If, as the first approximation, S is taken to be proportional to G_lo, the relation between S and G_lo is given by S = S_o [1 - D {1 - (S/A)^1/2(σ_l/k_l)^1/2}^-1(R_i/G_f)], where D = (A/S)^1/2 ρ_sL/{(k_l σ_l))^1/2 + (k_s σ_s)^1/2}. Here S_o is the value of S for R_i→0, A and S the cross-sectional area and circumference of the specimen (their variation accompanying solidification are neglected here for simplicity), respectively, k_l and k_s the thermal conductivities of the liquid and of the solid, respectively, σ_l and σ_s the thermal emissivities of the liquid and of the solid, respectively, ρ_s the density of the solid, and L the latent heat of solidification. This expression indicates that S is higher as G_f⁄R_i is larger, as is known empirically, and it agrees, in the relation between S and R_i, with the experimental results obtained preirously by the author and Watanabé in growing single crystals of zinc, bismuth, and white tin by the Bridgman method. Finally, it is shown that, if the orientation distribution of single crystals grown from the melt is determined mainly by the counterbalance of the crystallographic and thermal factors of the solidification process, the probability of single crystal growth is the same for all orientations for small R_i⁄G_f and, as R_i⁄G_f becomes larger, it becomes larger for crystals with orientations along the preferred directions of growth, which accords with the observed facts.