Impurity modes due to a single and a pair of defects in germanium

A detailed study of impurity modes due to a single and a pair of point defects in germanium crystal has been made using the Green's-function technique. For a pair of substitutional impurities, the two triply degenerate infrared- and Raman-active $F_2$-symmetry modes split into four symmetry modes $A_{1g}$, $A_{1u}$, $E_g$, and $E_u$ of point group $D_{3d}$. The $A_{1u}$-symmetry motion involves only the masses of the defects whereas the remaining symmetry motions $E_u$, $A_{1g}$, and $E_g$ involve both the masses and the force constants between the impurity atoms. The results are quite different from those of the earlier workers. Numerical computations have been made by utilizing the results of a second-neighbor-force model for the host lattice. Calculations have also been performed for resonance modes arising due to heavy and light impurity atoms. The results obtained are discussed and compared with the recent infrared and Raman scattering experiments in Ge: Si alloys. A correct assignment of the observed bands has been made with a new set of couplings for Si-Si atoms. An increase of 16% and a decrease of 8% in the central and angular force constants, respectively, over those in a Si crystal have been observed. The calculated three lines at 318, 440, and 445 ${\mathrm{cm}}^{-1\mathrm{}}$ have been detected in the experiments. The fourth line appearing at 520 ${\mathrm{cm}}^{-1\mathrm{}}$ may be observable in Raman measurements.