Shape and stability of Menisci in czochralski growth and comparison with analytical approximations

Nonlinear model-based control of the Czochralski process I: Motivation, modeling and feedback controller design

“…Under steady state conditions the shape of the meniscus is described by the Euler-Laplace equation (cf. [66,67]). There is no analytic solution of this expression, so one needs to solve the Euler-Laplace equation numerically in order to obtain the meniscus height for a specific growth configuration defined by the crystal radius r i and the slope angle a i .…”

“…In this case wetting of the crucible can be neglected. The approximation describes the height of the meniscus at the interface with sufficient accuracy compared to the height calculated from numerically solving the Euler-Laplace equation, as has been investigated thoroughly in [66,9]. An advantage of approximation (6) is the fact that it can be explicitly solved for a i .…”

On the pressure difference ranges which assure a specified gap size for semiconductor crystals grown in terrestrial dewetted Bridgman

“…The earlier investigations aimed at solving practical problems such as sessile drops, rod-in-free-surface meniscus, menisci between parallel and nonparallel solid surfaces [16][17][18][19][20][21]. For shaped crystal growth, the stability of the capillary surfaces was investigated theoretically and numerically by Mika, Uelhoff and Tatarchenko for Czochralski growth [22,23], by Coriell and Cordes for floating zone [24], by Balint et al for tubes and ribbons grown by edge-defined film-fed technique [25,26]. In these investigations, the menisci are found considering variational problem of the total free energy minimum of a liquid column, and their static stability with respect to small perturbations, is studied via the conjugate point criterion of the calculus of variations.…”

“…satisfying the boundary conditions Zðr a Þ ¼ 0; Z 0 ðr a Þ ¼ 1; has no conjugate points [21][22][23][24][25][26][27], i.e., the solution ZðrÞ of the Jacobi equation is no null for any r belongs to ðr c ; r a Þ.…”

Crystal Shape Stability in Meniscus-Controlled Growth Processes

“…[25]) in Czochralski crystal growth systems which are useful for the present analysis. Figure 5 shows the variations of 0 C{J lot and 0 C{J 10 s with diameter t in the growth of silicon crystals; these curves were calculated from the numerical data of Mika and U elhoff [22] on the meniscus shape in Czochralski growth (CfJo = 10 0 is assumed for these curves in order to match the data in the reference). Since 0 C{J /0 t > 0 for all values of t, it can be conc1uded that the crystal shape in Czochralski growth would be unstable for the case where the thermal effects are excluded from the analysis (i. e., eq.…”