Modeling the effects of the vertical temperature gradient in the furnace in an edge-defined film-fed growth technique

In this paper, the mathematical model for the growth of cylindrical bars described in Refs. [1,2] is considered. Using MathCAD 11 Enterprise Edition and mathematical tools, the asymptotically stable steady-states (r*, h*) of the nonlinear system of differential equations which governs the evolution of the bar radius r = r(t) and the meniscus height h = h(t), for different values of the pulling rate v , the melt temperature T0 at the meniscus basis and the vertical temperature gradient k in the furnace, respectively, are found. given k, the range of the stable growth regions in the ( v , T0) plane (i.e. those couples ( v , T0) for which (r*, h*) has physical sense) are determined. The effects of the changes of the vertical temperature gradient k are investigated and it is shown that if v and T0 are constant, and k increases, then the bar radius r increases and the meniscus height h decreases. cal results are given for the silicon bar grown in an edge-defined film-fed growth (E.F.G.) system with a die radius r0e = 20 (cm × 10−2).