True equilibrium in a system means that the system is isothermal and that the impurity distributions in both the melt and crystal are uniform, without concentration gradients in either phase. In practice, the impurity distribution in the solid phase is not homogeneous, depending on crystal-growth conditions, since it takes a definite time for impurities to distribute themselves uniformly. Therefore, a true equilibrium state is practically impossible to achieve. Impurity segregation in silicon is conventionally described in terms of the ratio of impurity concentration in the solid and in the molten silicon, or by its equilibrium segregation coefficient, keq, defined as,
k — _S.
where Cs is the impurity concentration in the crystal silicon at the crystal/melt interface, and CL the impurity concentration in the molten silicon. At a thermal equilibrium state, the impurity concentrations in the solid and molten silicon can be obtained by the binary phase diagrams of impurity-silicon systems.
According to the phase diagram, the equilibrium segregation of impurities must be dependent on their solubility in silicon, and therefore, at least qualitatively, by their atomic sizes. Figure 3.3 shows the correlation of the equilibrium impurity segregation coefficients in silicon as a function of their atom sizes.