Incorporation of Impurity in Crystals

There are several impurities in mc-Si, including carbon, oxygen, nitrogen, and iron. Carbon is one of the major impurities in silicon feedstock. When the carbon content exceeds its solubility limit in silicon, it will precipitate to form SiC particles in a directional solidification process. It has been experimentally 

The author focused on modeling the primary phase of SiC particles, which are precipitated in the molten silicon and then incorporated into the solidified ingot [19]. The carbon precipitation in solid is neglected due to the small diffusion constant in a solid. 

The units for carbon concentration and temperature in (4.2) are 1017 atoms cm-3 and I, respectively. With the molten silicon being solidified and the solidification interface moving upward in the crucible, during the solidification process, the carbon concentration in the melt increases due to the small segregation coefficient of carbon in silicon. If the carbon concentration exceeds the local solubility limit, excessive carbon precipitates and the following chemical reaction occurs  

the substitutional carbon is reduced and the same amount of SiC particles is generated in the melt. The formation rate of SiC particles and the destruction rate of substitutional carbon are equal and proportional to the super-saturation degree of substitutional carbon and the speed of the chemical reaction (4.3)  

With these assumptions, the governing equations for the concentrations of substitutional carbon and SiC particles in the melt can be written as follows  

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