Statistics of the Charge Distribution for a Localized Flaw in a Semiconductor

A localized flaw such as a vacancy-interstitial pair, gold atom, or small void is considered. It is found that the charge situation can be described by a set of energy levels which are independent of the Fermi level but which are temperature-dependent. If the flaw is electrically neutral for a certain position of the Fermi level, then as the Fermi level is raised from the neutral level successively above each higher-lying level of the set, the flaw acquires an additional electron unit of negative charge. These levels are called the first-acceptor level, second-acceptor level, etc. The energy levels are essentially equal to $\mathrm{kT}\ln \left(\frac{Z_1}{Z_2}\right)$, where $Z_1$ and $Z_2$ are the contributions of the flaw to the state sum for the system for the two conditions of charge. Similarly, as the Fermi level is lowered below the neutral point, it passes the first-donor level, second-donor level, etc., and the flaw acquires charges of plus one, plus two, etc. The statistics are derived for the distribution for the various conditions of charge, referred to as first-donor condition, neutral condition, first-acceptor condition, etc.