Raman Spectra of Nonpolar Crystalline Nanoparticles: Elasticity Theory-like Approach for Optical Phonons

A simple way to investigate theoretically the Raman spectra (RS) of nonpolar nanoparticles is proposed. For this aim, we substitute the original lattice optical phonon eigenproblem by the continuous Klein-Fock-Gordon-like equation with Dirichlet boundary conditions. This approach provides the basis for the continuous description of optical phonons in the same manner how the elasticity theory describes the longwavelength acoustic phonons. Together with continuous reformulation of the bond polarization model, it allows one to calculate the RS of nanoparticles without referring to their atomistic structure. It ensures a powerful tool for interpreting the experimental data, studying the effects of particle shape and their size distribution. We successfully fit recent experimental data on very small diamond and silicon particles, for which the commonly used phonon confinement model fails. The predictions of our theory are compared with recent results obtained with the dynamical matrix method-bond polarization model approach, and an excellent agreement between them is found. The advantages of the present theory are its simplicity and the rapidity of calculations. We analyze how the RS are affected by the nanoparticle faceting and propose a simple power law for Raman peak position dependence on the facet number. The method of powder RS calculations is formulated, and the limitations on the accuracy of our analysis are discussed.