Crystal interface, growth

The approach used in these studies follows idezus from bifurcation theory. We consider the structure of solution families with a single evolving parameter with all others held fixed. The lateral size of the element of the melt/crystal interface appears 2LS one of these parameters and, in this context, the evolution of interfacial patterns are addressed for specific sizes of this element. Our approach is to examine families of cell shapes with increasing growth rate with respect to the form of the cells and to nonlinear interactions between adjacent shape families which may affect pattern formation. 

Molecular recognition of crystal interfaces makes possible the control of crystal growth processes in that suitably designed auxiliary molecules aci as promoters or inhibitors of crystal nucleation inducing, for instance, the resolution of enantiomers or Ihe crystallization of desired polymorphs and crystal habits. 

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). 

Convection in Melt Growth. Convection in the melt is pervasive in all terrestrial melt growth systems. Sources for flows include buoyancy-driven convection caused by the solute and temperature dependence of the density surface tension gradients along melt-fluid menisci forced convection introduced by the motion of solid surfaces, such as crucible and crystal rotation in the CZ and FZ systems and the motion of the melt induced by the solidification of material. These flows are important causes of the convection of heat and species and can have a dominant influence on the temperature field in the system and on solute incorporation into the crystal. Moreover, flow transitions from steady laminar, to time-periodic, chaotic, and turbulent motions cause temporal nonuniformities at the growth interface. These fluctuations in temperature and concentration can cause the melt-crystal interface to melt and resolidify and can lead to solute striations (25) and to the formation of microdefects, which will be described later. 

A major complication in the analysis of convection and segregation in melt crystal growth is the need for simultaneous calculation of the melt-crystal interface shape with the temperature, velocity, and pressure fields. For low growth rates, for which the assumption of local thermal equilibrium is valid, the shape of the solidification interface dDbI is given by the shape of the liquidus curve Tm(c) for the binary phase diagram ... 

Although the balance equations are linear, in the absence of bulk convection, the unknown shape of the melt-crystal interface and the dependence of the melting temperature on the energy and curvature of the surface make the model for microscopic interface shape rich in nonlinear structure. For a particular value of the spatial wavelength, a family of cellular interfaces evolves from the critical growth rate VC(X) when the velocity is increased. 

Results from a quasi steady-state model (QSSM) valid for long crystals and a constant melt level (if some form of automatic replenishment of melt to the crucible exists) verified the correlation (equation 39) for the dependence of the radius on the growth rate (144) and predicted changes in the radius, the shape of the melt-crystal interface (which is a measure of radial temperature gradients in the crystal), and the axial temperature field with important control parameters like the heater temperature and the level of melt in the crucible. Processing strategies for holding the radius and solid-... 

Here, Cp is the concentration of the dissolved solute in the bulk of the liquid, Cp is the concentration of the solute at the liquid-crystal interface, and Cp is the solubility. Note that the nucleation rate (Jn) and the linear growth rate (G) have been transformed into molar units by using appropriate multiplying factors. It should be emphasized that, while these equations capture the phenomena under consideration, to be correct, they should be expressed in terms of activities in stead of concentrations. 

Figure 3. Streamlines (on right) and isotherms (on left) for growth of Si in a prototype Czochralski system. The volume of the melt, at the bottom in each drawing, changes among the calculations, affecting the qualitative form of the convection cell and the shape of the crystal interface. From Theory of Transport Processes in Single Crystal Growth from the Melt, by R. A. Brown, AIChE Journal, Vol. 34, No. 6, pp. 881 -911, 1988 [29]. Reproduced by permission of the American Institute of Chemical Engineers copyright 1988 AIChE.

Fig. 5. Schematic illustrating the growth of the growth of the crystal interface in a system with the present of (A) disrupter and (B) blocker (Ref. 24).

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