Mathematical modeling for assorted problems in crystal growth

Abstract

Crystal growth is a field that is ripe with opportunities for mathematical modeling to elucidate interesting phenomena. Important process parameters such as solute concentration, interface shape and location, and temperature field are uniquely difficult to observe \textit{in-situ} for many high temperature melt crystal growth systems. Additionally, the slow process of growing large, industrially-relevant single crystals can be prohibitive in time, material cost, and labor for tedious repeated experimental studies that are likely to be destructive. Modeling provides an efficient way for researchers to quickly gain an understanding of the physics underlying a crystal growth system. In this thesis, we examine three different cases where mathematical modeling can be utilized to interrogate crystal growth systems. First, we investigate the transport of oxygen in Czochralkski-grown silicon by posing a simple lumped-parameter model. The lumped-parameter model tracks transport of oxygen into and out of the melt without specifying its spatial distribution, relying only on estimated fluxes from various surfaces. The lumped-parameter model offers a near-instantaneous way to obtain a coarse estimate of oxygen given process parameters such as crystal/crucible rotation scheme, melt height, and melt overheating. Second, we examine a past experiment involving Europium-doped BaBrCl monitored \textit{in-situ} via Energy-Resolved Neutron Imaging. Europium acts as a strong neutron attentuator, allowing visualization of its migration in both the solid and melt phases. A prior experiment was conducted to perform \textit{in-situ} imaging of a melt crystal growth system, and we realized this presented an opportunity to use modeling to extract additional data from this past experiment. A 1D model of europium migration in both phases was formulated and solve via Finite Fourier Transforms and Finite Difference Method. The Finite Difference Method, being more flexible, allowed us to deduce the apparent solid and liquid diffusion coefficients of Eu as well as its segregation coefficient. This coupling of \textit{in-situ} imaging and modeling presents an exciting new way to measure physical properties and extract additional value from past experiments. Last, we analyze the curious phenomenon of Temperature Gradient Zone Melting (TGZM), whereby a solute-rich liquid particle migrates through a solid crystal under a thermal gradient. While this phenomenon has been studied in the past, prior models failed to give practical predictions in the time-evolution behavior of such migrating particles. We pose analytical and numerical models of 1-dimensional TGZM, which agree well with each other. The numerical model, solved via Finite Element Method, shows reasonable agreement with experimental data on Te-rich second-phase particles migrating in CdTe. It additionally shows excellent agreement with another physical system, NaCl brine particles in water ice, providing a far more accurate description of the particle's migration than previous theoretical models. Considerations are made for extending the model to higher dimensions in order to understand changes in particle morphology during migration. Different types of modeling using various analytical and numerical techniques are employed for each of these case studies. These three example cases show different scenarios in which mathematical modeling can be utilized to help researchers gain insight in crystal growth systems.