Theoretical and Experimental Study of Raman Scattering from Coupled LO-Phonon-Plasmon Modes in Silicon Carbide

Abstract

A general semiclassical derivation is given of the efficiency of the scattering of light from a longitudinal-optical (LO) phonon coupled to a single-component plasma in a semiconductor. The efficiency is the sum of two terms. One is due to the modulation of polarizability by atomic displacements and by the macroscopic longitudinal field; it obeys ordinary Raman polarization selection rules. The other term is due to free-electronic-charge-density fluctuations, is proportional to the square of the wave vector transferred, and obeys different selection rules. When the exciting light is near resonance with the band gap, the second term may have two interfering parts, one due to the free-electron-charge-density fluctuations and the other due to photon-induced, virtual, bound-electronic-charge-density fluctuations coupled to the macroscopic field (Fröhlich mechanism). New data are presented on coupled plasmon-phonon scattering in heavily nitrogen-doped ($n$-type) $6H$ silicon carbide. The plasmon is overdamped (${\omega }_p\tau \approx 0.2-0.3$). The dominant light-scattering mechanism is shown to be the polarizability-modulation mechanism. The density mechanism is predicted to be 3500 times weaker. The ratio of polarizability derivatives in dimensionless form is ${\gamma }_{\mathrm{xxz}}=+2.60\mathrm{}$. From this and from published data on the nonlinear optical constants of $6H$ SiC we obtain the following value for the derivative of the polarizability with respect to relative atomic displacement: $\frac{\partial {\alpha }_{\mathrm{xx}}}{\partial u_z}=(\mp 10.4\pm 1.2)\times {10}^{-16\mathrm{}}$ ${\mathrm{cm}}^2$. Absolute efficiencies are also calculated for the coupled mode spectrum in doped materials and for the $A_1\left(\mathrm{LO}\right)$ phonon in lightly doped crystals. Sum rules are derived for the differential efficiencies integrated with respect to the frequency squared.