Analysis of morphologically stable horizontal ribbon crystal growth

Abstract

The HRG process (also known as the low-angle silicon sheet [LASS] process) has been analysed from the standpoint of a quasi steady-state heat flow analysis, subject to the assumption of a parabolic ribbon cross-section. It is demonstrated that under reasonable operating conditions the convective heat transport term in the moving ribbon can be neglected. A solution to LaPlace’s equation in parabolic coordinates is found which satisfies the solidification boundary conditions corresponding to the so-called heat clamp case, wherein the free surface of the ribbon is subject to a fixed temperature below the fusion point. Operating states were found corresponding to a Péclet number, which is defined as the pull speed, V, multiplied by the ribbon tip radius, R, divided by the thermal diffusivity of the solid a, i.e. Pé ≡ VR/2α. The analysis shows that Pé is proportional to the temperature difference between the solid-liquid interface and the heat clamp, and only slowly varying with position along the ribbon. The Péclet number is singular near the ribbon tip, but this behavior merely reflects the artificial nature of the boundary conditions at the ribbon tip. The growth conditions described imply inherent morphological stability (except for the singular behavior near the tip) which indicates that smooth, nondendritic ribbon should in principle be a possible solidification product of the HRG process.