Vibration of nanostructures: from atomic to continuum scales

Abstract

One of the most important and challenging characteristics of nanostructures is the size-dependence that the properties of these materials and components exhibit from what is known as the quantum size effect. The most suitable models to study the behavior of particles at this very small scale are the computationally expensive methods of quantum mechanics and molecular dynamics. At the other end of the scale spectrum, continuum mechanics methods are computationally much faster, but they do not explicitly consider atomic and electronic interactions that may affect the properties of these novel structures which, below a certain scale, cannot be considered as continuum solids. In this study, vibrational frequency spectra of sphere-, cube-, tetrahedron-, and pyramid-shape nanoparticles made of silicon, germanium, and carbon were computed using three different models. The three methods span the range of length scales starting from molecular mechanics at the atomic level, passing through a modified molecular dynamics approach, and continuing on to the continuum level in which the nanoparticles are considered as solid elastic structures. The dependence of the natural frequencies of particles on the size at the nanoscale was confirmed, the magnitude of the frequencies and the number and location of degenerate frequencies change with particle size. Normalized frequencies increase non-linearly as the number of atoms forming the particle is increased, while they increase linearly as a function of the particle size. The latter indicates that the scale invariance of the normalized frequencies in continuum elasticity does not hold for the range of particles sizes considered in this study. First and second order polynomials were proposed to describe the variation of the lowest normalized frequency as a function of the size, and of the number atoms forming the particles, respectively. The converged frequency values using the second order polynomials only differ about -2% for silicon particles compared to continuum mechanics results for cubic material symmetry, while the differences are about +20% and -15% for germanium and carbon nanostructures. Therefore, the applicability of the continuum mechanics assumption at this scale depends not only on the size, but also on the material of which the nanostructures are composed.