Models melt crystal growth

Figure 3. Three spatial scales for modeling melt crystal growth, as exemplified by the vertical Bridgman system.

The governing equations and boundary conditions for modeling melt crystal growth are described for the CZ growth geometry shown in Figure 6. The equations of motion, continuity, and transport of heat and of a dilute solute are as follows ... 

Figure 2. Three spatial scales for modeling melt crystal growth, as exemplified by the vertical Bridgman process. From Theory of Transport Processes in Single Crystal Growth from the Melt, by R. A. Brown, AJChE Journal, Vol. 34, No. 6, pp. 881-911, 1988, [29]. Reproduced by permission of the American Institute of Chemical Engineers copyright 1988 AIChE.

G. Muller, D.T.). Hurle, and H. Wenzl, 1989, (eds.) Proceedings of the first NATO workshop on computer modeling in crystal growth from the melt, Parma, Italy,... 

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. 

Keywords Chain folding Computer modeling Crystal growth Crystal-melt interfaces Molecular dynamics Polymer crystallization... 

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. 

Kobayashi (143) presented the first computer simulations that considered the determination of the crystal radius as part of the analysis but avoided the capillary problem by considering a flat melt-ambient surface, which is consistent with <)>o = 99°. Calculations were performed for a fixed crystal radius, and then the growth rate was adjusted to balance the heat flux into the crystal. Crowley (148) was the first to present numerical calculations of a conduction-dominated heat-transfer model for the simultaneous determination of the temperature fields in crystal and melt and of the shapes of the melt-crystal and melt-ambient surfaces for an idealized system with a melt pool so large that no interactions with the crucible are considered. She used a time-dependent formulation of the thermal-capillary model and computed the shape of an evolving crystal from a short initial configuration. 

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-... 

Figure 19. Sample isotherms and interface shapes computed for the QSSM for CZ crystal growth by Atherton et al. (153). The model includes detailed radiation between the surfaces of the melt, crystal, and crucible. Isotherms are spaced at 10 K increments in the melt and 30 K increments in the other...

Integration of a time-dependent thermal-capillary model for CZ growth (150, 152) also has illuminated the idea of dynamic stability. Derby and Brown (150) first constructed a time-dependent TCM that included the transients associated with conduction in each phase, the evolution of the crystal shape in time, and the decrease in the melt level caused by the conservation of volume. However, the model idealized radiation to be to a uniform ambient. The technique for implicit numerical integration of the transient model was built around the finite-element-Newton method used for the QSSM. Linear and nonlinear stability calculations for the solutions of the QSSM (if the batchwise transient is neglected) showed that the CZ method is dynamically stable small perturbations in the system at fixed operating parameters decayed with time, and changes in the parameters caused the process to evolve to the expected new solutions of the QSSM. The stability of the CZ process has been verified experimentally, at least... 

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

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