First-principles electronic-structure calculations of impurities in semiconductors are extended to finite temperatures. We first calculate the vibrational free energy in cubic C, Si, and Ge and hexagonal GaN and show that reliable phonon densities of state, specific heats, and other thermodynamic quantities can be obtained in the same (64-atom) supercells commonly used to study impurities. Then, we use examples in Si to quantify various free energy contributions. Only the vibrational free energy plays a role in the temperature dependence of the relative energy of the two $\mathrm{C}{\mathrm{H}}_2^{*}$ complexes, and the configurational entropy dominates when calculating the binding energy of copper pairs, while the rotational free energy is critical to interstitial ${\mathrm{H}}_2$, HD, and ${\mathrm{D}}_2$. The contribution to the free energy of electrons (holes) originating from occupied (empty) localized impurity levels in the gap is estimated and found to be very small.

• Received 10 March 2004

DOI:https://doi.org/10.1103/PhysRevB.70.125209