It
is also common to introduce dopant atoms into the melt so that a particular
resistivity wafer can be made. To do this, one can simply weigh the melt,
determine the number of impurity atoms that would be needed, and add that
weight of impurity. The process is complicated, however, by the fact that
impurities tend to segregate at solid/liquid interface.
Every
impurity has a solid solubility in silicon and different equilibrium solubility
in the melt. An equilibrium segregation coefficient k0 may be
defined as
ko=CS/Cl
Where
Cs and Cl are the equilibrium concentrations of the
impurity in the solid and liquid near the interface, respectively. [2]
The
distribution of an impurity in the grown crystal can be described mathematically
by the normal freezing relation.
CS = k0C0(1-X)k0-1
Where
X is the fraction of the melt solidified, C0 is the initial melt
concentration, Cs is the solid concentration, and k0 is
the segregation coefficient.