Theory of transport processes in single crystal growth from the melt

Abstract

The quality of large semiconductor crystals grown from the melt for USE! in electronic and optoelectronic devices is strongly influenced by the intricate coupling of heat and mass transfer and melt flow in growth systems. This paper reviews the present state of understanding of these processes starting from the simplest descriptions of solidification processes to detailed numerical calculations needed for quantitative modeling of processing with solidification. Descriptions of models for the vertical Bridgman-Stockbarger and Czochralski crystal growth techniques are included as examples of the level of understanding of industrially important methods.

Convection effects on radial segregation and crystal–melt interface in vertical Bridgman growth

“…The precise multiplicative decomposition of the fliot number (q, into q, and ( is unimportant since in the final results, only the product fql appears. The same is true for fq 2 , f q 1 , and fq2,.…”

“…The concentration of solute (one of the two components of the binary mixture) at the top of the cylinder , = h is c = c 2 . The top of the cylinder is maintained at temperature P = T. 2 , while the bottom of the cylinder is at temperature T = T 1 , which is significantly smaller than the melting temperature To of a planar interface. The density of the melt is assumed to be A = Oo at temperature to and concentration c 2 .…”

“…No azimuthal angular variable is necessary as the flow variables are assumed to be axi-symmetric. The mathematical equations for the steady solution in the melt are (1) v .Vc = DV 2 c, (2) 67.V' = -Vp + {aT + fi(c-c 2 ))g2 + vV 2 …”

“…Since this paper concerns situations which in the absence of horizontal heat transfer is a thermally stable configuration, natural convection is not relevant in this context unlike that of an otherwise thermally or solutally unstable arrangement where possibility exists of a "resonant" response leading to vigorous convection. There are many papers dealing with the Bridgman problem itself (see Brown, 1988 for references). Nonetheless, the first set of fully consistent calculations that account for convection in the fluid, and its coupling with thermo-solutal field, heat diffusion in the solid and a non-planar interface appear to be due to Chang & Brown 16 and Adornato & Brown"'.…”

“…We ignore such terms in P1. Similarly, for 5 < 1 :_ 8, the elements M,,ke-Wk(I'I) scale as e'(-z (2) for j = 3,4,5 and is transcendentally small in RT. This leaves us with the contribution from terms multiplying…”