Solubility of Flaws in Heavily-Doped Semiconductors

The solubility of a charged impurity in a semiconductor depends upon the Fermi level. This dependence may be understood in terms of a conceptual model in which an impurity is allowed to diffuse in a specimen containing a $p-n$ junction, so that the Fermi level varies in respect to the band edges. If the impurity can exist in many states of charge (i.e., is a "flaw"), then the concentration of flaws with charge $r$ times the electronic charge varies as the $r\mathrm{th}$ power of the hole density. Summing the concentrations for the different states of charge gives the solubility and its dependence upon hole concentration, and, hence, Fermi level.