Acoustic vibrations of anisotropic nanoparticles

Abstract

Acoustic vibrations of nanoparticles made of materials with anisotropic elasticity and nanoparticles with non-spherical shapes are theoretically investigated using a homogeneous continuum model. Cubic, hexagonal and tetragonal symmetries of the elasticity are discussed, as are spheroidal, cuboctahedral and truncated cuboctahedral shapes. Tools are described to classify the different vibrations and for example help identify the modes having a significant low-frequency Raman scattering cross-section. Continuous evolutions of the modes starting from those of an isotropic sphere coupled with the determination of the irreducible representation of the branches permit some qualitative statements to be made about the nature of various modes. For spherical nanoparticles, a more accurate picture is obtained through projections onto the vibrations of an isotropic sphere.Comment: 11 pages, 9 tables, 6 figure

Acoustic vibrations of metal nano-objects: Time-domain investigations

“…In the case of the lowest frequency mode of a homogeneous sphere formed by a cubic crystal material, embedded in a homogeneous isotropic matrix, taking into account crystallinity leads to partial lifting of the degeneracy of the fundamental quadrupolar (n = 0, l = 2) mode. It splits into two modes of lower and higher energy as compared to the polycrystalline case, which are 2 and 3 times degenerated, respectively [65,79,115,116], in agreement with Raman experiments on single-crystal gold nanoparticles [65,115]. In contrast, the nondegenerated radial modes of a sphere are weakly altered, their frequencies computed using either the Visscher's method [38,116] or with FEM simulations [79] being almost identical for monocrystalline and polycrystalline spheres.…”

“…It splits into two modes of lower and higher energy as compared to the polycrystalline case, which are 2 and 3 times degenerated, respectively [65,79,115,116], in agreement with Raman experiments on single-crystal gold nanoparticles [65,115]. In contrast, the nondegenerated radial modes of a sphere are weakly altered, their frequencies computed using either the Visscher's method [38,116] or with FEM simulations [79] being almost identical for monocrystalline and polycrystalline spheres. Similar results are obtained using different approaches for obtaining the averaged elastic constants of a polycrystal from those of a single crystal [75][76][77][78][79], all leading to a less than 1% deviation on the computed frequency for gold and silver nanospheres.…”

Simple model for the vibrations of embedded elastically cubic nanocrystals

“…12 For spheres whose diameter is small compared to the wavelength of light, the Ramanactive vibrations are S 0 and S 2 (for every m and n). 15 In the present work, RUS calculations have been used to model the vibrations of elastically anisotropic spheres as in a previous work 12 by expanding the displacements onto x i y j z k functions with i + j + k ≤ 20.…”

“…In the following, we focus on calculations using the pseudo-modes obtained with the complex frequency model (CFM). 12 Other models such as the core-shell model (CSM) 12 would be more suitable for the calculation of Raman spectra but won't be considered here as we focus only on the positions of the Raman bands. Details about the validity of the CFM approach using the isotropic approximations are presented in Appendix A.…”

“…When the elastic constants are not isotropic, a numerical approach such as the one known as Resonant UltraSound (RUS) 9 can find mode frequencies and displacement fields of free nanoparticles. [10][11][12] What has been missing up until now is a description of the vibrations of an elastically anisotropic sphere embedded in an isotropic matrix. The case of anisotropic elasticity has been discussed for the problem of the scattering of acoustic waves 13,14 but none of these approaches provides the eigendisplacements required for modeling the coupling of the vibrations with electrons which is at work in all the optical techniques used to detect such vibrations.…”