Anharmonic Contributions to the Heat Capacities of Silicon and Germanium

The heat capacities of silicon and germanium have been analyzed, in terms of their equivalent Debye temperatures at fixed volume, to obtain the coefficients of the leading anharmonic contribution to the free energy. The analysis is based on a temperature-dependent frequency distribution in which the anharmonicity is treated in terms of frequency shifts in the appropriate quasiharmonic expression. At the highest temperatures (300°K) the anharmonic heat capacities become proportional to the absolute temperature. Thus $\Delta C^{\mathrm{anh}}=3\mathrm{}\mathfrak{N}kA\left(V_0\right)T$; where $3\mathfrak{N}$ is the number of modes: for Si we find $A\left(V_0\right)=6.6\mathrm{}\pm 2.0\times {10}^{-5\mathrm{}}$ ${\mathrm{deg}}^{-1\mathrm{}}$ and for Ge $A\left(V_0\right)=5.2\mathrm{}\pm 1.0\times {10}^{-5\mathrm{}}$ ${\mathrm{deg}}^{-1\mathrm{}}$.