Analysis of finite length dynamics in Czochralski crystal growth

In the analysis of energy transport within crystals grown by the Czochralski process, time dependence appears in both the boundary conditions, through the changing crystal shape, and in the energy equation, by inclusion of the unsteady term. Commonly, these phenomena are neglected by utilization of the quasi-steady state approximation. The present study examines the validity of the approximation with respect to changing crystal length. A one dimensional model, neglecting phenomena occuring at the crystal-melt interface, is developed so that the influence of time dependent changes in crystal length on the dynamics of this process can be evaluated. Finite difference solutions of this relatively simple model are found and used to predict crystal temperature distribution and pull rate. It is shown that use of the quasi-steady state approximation can lead to errors in calculating both the crystal pull rate and the "infinite", or critical, crystal length. Also, it was found that the choice of initial condition can have a significant effect on early-time crystal growth behavior.