Simple model for the vibrations of embedded elastically cubic nanocrystals

Abstract

The purpose of this work is to calculate the vibrational modes of an elastically anisotropic sphere embedded in an isotropic matrix. This has important application to understanding the spectra of low-frequency Raman scattering from nanoparticles embedded in a glass matrix. First some low frequency vibrational modes of a free cubically elastic sphere are found to be nearly independent of one combination of elastic constants. This is then exploited to obtain an isotropic approximation for these modes which enables to take into account the surrounding isotropic matrix. This method is then used to quantatively explain recent spectra of gold and copper nanocrystals in glasses.

Vibrational density of states of silicon nanoparticles

“…[3][4][5] In addition to this, Raman scattering experiments have been used in numerous studies of the vibrational properties of silicon nanostructures. [6][7][8][9][10][11][12][13][14][15][16] In many of these studies it is found that the first-order peak due to the transverse optical (TO) phonon at the point is broadened and its maximum shifted to lower energies. The phononconfinement model 6,17 attributes this behavior to the fact that in nanostructures more vibrational modes can become Raman active than in bulk crystals since the translational symmetry is broken and consequently no k = 0 selection rule applies.…”

“…Heino has calculated phonon dispersion relations for silicon thin films using moleculardynamics simulations, 18 whereas Saviot et al have studied low-frequency vibrational modes in silicon nanoparticles using an elastic continuum model. 13 Hu et al and Valentin et al employed empirical atomistic force models to calculate the VDOS of silicon nanoparticles by diagonalization of the dynamical matrix. 19,20 The latter two studies did not include relaxation effects from the particle surfaces.…”

Raman Scattering from Acoustic Phonons Confined in Spherical Nanoparticles

“…Matrix can disable movement on the surface of nanoparticle. If there is no displacement at the particle surface, it is the case of so called rigid boundary conditions [6]. If properties of a matrix make it possible for * e-mail: rkostic@phy.bg.ac.yu a nanoparticle to vibrate without restrictions we can assume that there is no force acting on the particle surface, i.e.…”

“…transverse v T and longitudinal v L sound velocities. As we attend to describe a confined acoustic phonon in a small crystal we must consider the boundary conditions at the surface of crystal, in our case sphere, and combine them with equation of motion in spherical coordinates [4][5][6][7][8].…”

Low-Frequency Raman Scattering from ZnO(Fe) Nanoparticles

“…If we assume that nanoparticles are small spheres, equation of motion must be solved in spherical coordinate. It is useful to introduce dimensionless variables η = ωR/v T = ωd/2v T and [7][8][9][10][11].…”