Activity Coefficients of Electrons and Holes at High Concentrations

The use of the mass-action law in connection with solid-state equilibria involving conduction electrons or holes assumes that the chemical potential of the species is related to its concentration by μ=μ0(T)+kT lnn. This approximation can fail at concentrations which are still very small by ordinary chemical standards. The formalism may be preserved in principle by introducing a suitable activity coefficient γ. It is shown theoretically, however, that γ increases almost exponentially with increasing n when n/λ≳0.1, where λ is the partition function in the quasi-free-particle approximation. As a consequence, the proper exponent of n in mass-action equations becomes much larger than expected from stoichiometry considerations alone. The use of such equations becomes difficult, therefore, when n/λ≳0.5, and entirely unjustified when n/λ≳10. The determination of λ, which lies between 1016 and 1021 cm—3 for most semiconductors is discussed from an experimental viewpoint.