Theoretical Modeling of Czochralski Crystal Growth
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MRS BULLETIN/OCTOBER1988
tion regarding melt crystal growth modeling. The following sections analyze the physical transport mechanisms of CZ growth, present sample results for semiconductor and oxide growth simulations, assess the state-of-the-art of modeling, and project future developments. Analysis Offen the successful practice of crystal growth relies on experiential intuition; however, the underlying science of crystal growth is based on the strong foundation of first principles. Our goal is to obtain a fundamental understanding of the Controlling features of CZ growth by assessing the underlying physical phenomena. These features can vary widely for different materials
and processes, leading to the dilemma in analyzing crystal growth Systems that few theoretical generalizations are both practically useful and universally valid. In spite of this difficulty, two typical Systems are appraised which are representative of the major materials produced by the Czochralski method, a semiconductor, Silicon, and an oxide, gadolinium gallium garnet (Gd3Ga5012, hereafter referred to as GGG). The Czochralski method is illustrated schematically in Figure la for a typical oxide growth process. A single-crystal seed is dipped into a crucible filled with molten material, allowed to equilibrate, and slowly withdrawn upward. On the successful initiation of growth from the seed, suitable manipulations of the process parameters, such as pull rate and heater power, prompt the crystal to grow out and maintain a constant diameter as the melt is depleted from the crucible. At the end of the growth run, the crystal is slowly cooled to ambient temperature and then removed for device processing. A typical semiconductor CZ growth System is very similar to that depicted in Figure la, but the crucible is offen heated by a surrounding resistance h e a t e r . The l i q u i d - e n c a p s u l a t e d Czochralski (LEC) method, in which the crystal is grown through a floating layer of inert material designed to contain volatile species in the melt, is an important means of producing Compound s e m i c o n d u c t o r s such as gallium arsenide (GaAs).
Ambient Crystal shape
Tri-junction
Meniscus
(a) Entire furnace
(b) Crystal and melt
(c) Tri-junction
Figure 1. Schematk diagram of the Czochralski growth of an oxide crystal showing relevant length scales.
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Theoretical Modeling of Czochralski Crystal Growth
Understanding Czochralski crystal growth involves analysis on several disparate length and time scales. Characteristic scales are listed in Table I, and three length scales used to classify the CZ process are shown schematically in Figure 1. The macro-scale comprises the entire growth Station, the intermediate scale includes the crystal and melt, and the micro-scale embodies phenomena occurring at phase boundaries. Ultimately, the quality of the crystal is expressed in micro-scale quantities such as concentration distribution, stress distribution, and dislocations. However, attempts to influence the process necessarily occur at
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