First- and second-order resonant Raman scattering from ${\mathrm{Mg}}_2$Si,${\mathrm{Mg}}_2$Ge, and ${\mathrm{Mg}}_2$Sn were measured with laser frequencies covering the $E_0$ and $E_1$ gaps for ${\mathrm{Mg}}_2$Si, the $E_1$ gap for ${\mathrm{Mg}}_2$Ge, and the $E_1$ and $X$ gaps for ${\mathrm{Mg}}_2$Sn. Each component of the Raman tensor is separated using polarized light. The resonance of the Raman allowed $F_{2g}$ phonon for these three materials near the $E_1$ and $E_1+{\Delta }_1$ gaps is well described by three band terms in the dielectric theory of the Raman tensor. The resonance of the $F_{2g}$ phonon near the $E_0$ gap of ${\mathrm{Mg}}_2$Si was observed as a shoulder of the strongly resonating $E_1$ gap. The observed resonance intensity is about one-half the value expected from ${\left|\frac{d\epsilon }{d\omega }\right|}^2$ when this function is fitted to the maximum of the $E_1$ resonance. A pseudopotential calculation is presented, showing that the corresponding deformation potential (7.5 eV) is small. The resonance in the $X$-gap region for ${\mathrm{Mg}}_2$Sn is found to be analogous to that of the $E_1$ gap. The $F_{2g}$ allowed phonon resonates broadly and weakly; however, Fröhlich-interaction-induced $F_{1u}\left(\mathrm{L}\mathrm{O}\right)$ and its overtone resonate sharply and show structure due to the spin-orbit splitting of the $X_{5^{\prime }}$ valence band. The spin-orbit splittings observed in the resonances at $E_1$ gaps are 0.14 eV for ${\mathrm{Mg}}_2$Ge and 0.27 eV for ${\mathrm{Mg}}_2$Sn; at the $X$ gap the splitting for ${\mathrm{Mg}}_2$Sn is 0.14 eV. The deformation potentials involved in the several resonances observed are discussed and an attempt to determine their relative values is made. The forbidden $F_{1u}\left(\mathrm{L}\mathrm{O}\right)$ and its overtone resonate very sharply at the $E_1$, $E_1+{\Delta }_1$, and $X$ gaps: These resonances are well described by the second derivative of the dielectric constants ${\left|\frac{d^2\epsilon }{d{\omega }^2}\right|}^2$ and are strongly polarized in parallel-parallel scattering configurations. Sharp structure due to $F_{2g}+F_{1u}\left(\mathrm{LO}\right)$ combination of phonons is also observed; it resonates in a way analogous to the forbidden $F_{1u}\left(\mathrm{LO}\right)$ phonon. These characteristic features suggest that the normally forbidden $F_{1u}\left(\mathrm{LO}\right)$ and its overtone for the materials investigated are induced by the intraband Fröhlich interaction. The $F_{2g}+F_{1u}\left(\mathrm{LO}\right)$ combination is due to combined Fröhlich-deformation-potential interaction.

• Received 1 April 1976

DOI:https://doi.org/10.1103/PhysRevB.14.3520