Determination of the angle ${\mathbf \alpha }_o(x_1,h_1;p)$ appearing in the ribbon growth, when the profile curve is concave

Abstract

In this paper a procedure for the determination of the angle $\alpha _o(x_1,h_1;p)$, between the horizontal axis 0X and the tangent line to the free surface of a meniscus (having concave profile curve), at the three-phase point, of coordinates $(x_1,h_1)$, is presented. This angle appears in the system of differential equations which describes the evolution of the half thickness \textit{$x_1$} of the ribbon, grown from the melt by edge-defined film-fed growth (EFG) method. The angle can fluctuate during the growth, due to the fluctuations of the ribbon half-thickness \textit{$x_1$}, the fluctuations of the crystallization front height \textit{$h_1$} and the pressure fluctuations \textit{p} (of the gas flow introduced in the furnace for release the heat). The knowledge of the function $\alpha_o $ is necessary for the simulation of the half thickness evolution of the ribbon to be grown, with a given pulling rate \textit{v} in specified thermal and pressure conditions. This kind of results can be useful in experiment and manufacturing technology design. Numerical illustration is given for an InSb ribbon.