Crystal growth convection-controlled

The solubility concentration is c which is smaller than the bulk concentration c. The total concentration drop Ac = c - c is split up into two contributions. The first part (c - Cj) within the concentration boundary layer is the driving force for diSusion and convection whereas the second part (cj - c ) in the very thin layer where the integration step takes place is effective for this step. The index I means Interface. In the case of growth completely controlled by diffusion and convection, (cj - c ) (c - Cj) or (Cj - c )/(c - Cj) l is valid. Contrary to this with the ratio (c - Cj)/ cj - c ) l crystal growth is controlled by the integration step. The molar flux density h directed toward the crystal srrrface is... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

Figure 1-11 Concentration profile for (a) crystal growth controlled by interface reaction (the concentration profile is flat and does not change with time), (b) diffusive crystal growth with t2 = 4fi and = 4t2 (the profile is an error function and propagates according to (c) convective crystal growth (the profile is an exponential function and does not change with time), and (d) crystal growth controlled by both interface reaction and diffusion (both the interface concentration and the length of the profile vary).

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

One-Dimensional, Diffusion-Controlled Crystal Growth. Neglecting bulk convection leads to an idealized picture of diffusion-controlled solute transport of a dilute binary alloy with the solute composition c0 far from an almost flat melt-crystal interface located at z = 0 (I, 21). 

First, the role of system design on the details of convection and solute segregation in industrial-scale crystal growth systems has not been adequately studied. This deficiency is mostly because numerical simulations of the three-dimensional, weakly turbulent convection present in these systems are at the very limit of what is computationally feasible today. New developments in computational power may lift this limitation. Also, the extensive use of applied magnetic fields to control the intensity of the convection actually makes the calculations much more feasible. 

Here, we show only a bare outline of the individual components in the overall system. This SYSTEM is capable of operation in inert atmosphere or vacuum. A melt/seed contact monitor is provided as well as a CCTV camera for observing and controlling the crystal diameter as it grows. Note that both the crucible and crystal rotation can be controlled. In order to control the heat-convection patterns which normally appear in the melt, an external cryomagnet is supplied. Its magnetic field controls heat losses, plus it maintains a better control of the crystal growth. A slave micro-processor controls both crystal diameter and meniscus-contact of the growing crystal. 

Hydrodynamics and computer simulation of crystal growth In the crystal pulling growth technique, two types of convection have to be taken into consideration, namely natural and forced convections (Dupret and Van Den Bogaert 1994). The behavior of the meniscus is also important for the control of the diameter and... 

In Bridgman-type crystal-growth configurations one possibility to control convection and therefore the shape of the solid/liquid interface, as well as the dopant distribution, is the so-called accelerated cmcible rotation technique (ACRT). This technique was developed by Scheel [56] and has been applied especially to the Bridgman growth of CdHgTe and CdZnTe crystals, e.g. [57, 58). 

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