Numerical simulations of Czochralski silicon growth

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Abstract
The present study involves the development and application of a numerical technique to the most common method of growing silicon crystals—the Czochralski process. The equations solved for the melt phase are the conservation equations for the mass, momentum, and energy. For the crystal phase, the energy balance equation is solved. The governing equations are formulated using primitive variables and are not thereby restricted to two‐dimensional symmetry. The crystal and melt phases are coupled through thermal conditions applied along the melt–crystal interface. The shape of the melt–crystal interface and meniscus are determined through solutions to differential equations that govern the kinematic conditions at the specific interfaces, and are not assumed a priori. The numerical procedures are obtained by a well‐documented procedure known as the consistently split linearized block implicit (LBI) scheme. Calculations, including one three‐dimensional case, are performed for various growth conditions suitable for silicon crystal growth. The present study has successfully demonstrated the numerical capabilities for Czochralski crystal growth simulations.

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