초록
When concentration of vacancies in a CZ silicon crystal is defined by molar fraction XB, the degree of super-saturation $\sigma$ is given by [XB-XBS]/XBS=XB/XBS-1=ln (XB/XBS) because XB/XBS is nearly equal to unity. Here, XBS is the saturated concentration of vacancies in a silicon crystal and XB is a little larger than XBS. According to Bragg-Williams approximation, the chemical potential of the vacancies in the crystal is given by ${\mu}$B=${\mu}$0+RT ln XB+RT ln ${\gamma}$, where R is the gas constant, T is temperature, ${\mu}$0 is an ideal chemical potential of the vacancies and ${\gamma}$ is an adjustable parameter similar to the activity of solute in a solution. Thus, $\sigma$(T) is equal to (${\mu}$B-${\mu}$BS)/RT. Driving force of nucleation of the vacancy agglomeration will be proportional to the chemical potential difference (${\mu}$B-${\mu}$BS) or $\sigma$(T), while growth of the vacancy agglomeration is proportional to diffusion of the vacancies and grad ${\mu}$B.