Binding enthalpies and entropies of neutral Zn,In And Sn In Ge

A statistical thermodynamical model is developed for a doubly ionizable acceptor with excited states in an elemental semiconductor. The special cases of a singly ionizable acceptor, no excited acceptor states and a neutral solute are readily extracted. The equations obtained describe not only the distribution of an acceptor among its various electronic states but also the chemical potentials of the acceptor atom and of the solvent atom. The equations are applied, with and without the excited states included, to obtain about equally good fits to the low-temperature, low carrier concentration Hall data for Zn in Ge. For the low impurity concentrations involved, the excited states are present and so should be included in the analysis. The ability to obtain an equally good fit with the excited states omitted is most likely due to the fact that the concentrations of the major Zn impurity and the Sb counter dopant are not fixed independently of the Hall measurements. The additional flexibility obtained by treating these concentrations as adjustable parameters is sufficient to compensate for the error made in neglecting the excited states. In contrast, the solubilities of the acceptors Zn and In are large enough that the excited states should be pushed out of the band gap. Consequently, the equations are applied without the excited states to the experimental chemical potential-solubility data for Zn, In and Sn in Ge to obtain the Gibbs energy, enthalpy and entropy of binding for the neutral (un-ionized) solute at 950 K. With a solute reference phase consisting of an ideal monatomic gas at 0 K, the entropy of binding is close to that for Ge itself, $\text{14.6 cal}\text{K g}$ atom. This implies that, when they are in the neutral state, these substitutional solute atoms behave very much like the Ge atom itself in contributing to the vibrational spectrum. However, the enthalpy of binding varies significantly and is ?3.8, ?39 and ?65 $\text{kcal}\text{g}$ atom for neutral Zn, neutral In and Sn, respectively, compared to $\text{?84 kcal}\text{g}$ atom for Ge.