Vibrational modes of nanolines

Abstract

Brillouin-light-scattering spectra previously have been shown to provide information on acoustic modes of polymeric lines fabricated by nanoimprint lithography. Finite-element methods for modeling such modes are presented here. These methods provide a theoretical framework for determining elastic constants and dimensions of nanolines from measured spectra in the low gigahertz range. To make the calculations feasible for future incorporation in inversion algorithms, two approximations of the boundary conditions are employed in the calculations: the rigidity of the nanoline/substrate interface and sinusoidal variation of displacements along the nanoline length. The accuracy of these approximations is evaluated as a function of wavenumber and frequency. The great advantage of finite-element methods over other methods previously employed for nanolines is the ability to model any cross-sectional geometry. Dispersion curves and displacement patterns are calculated for modes of polymethyl methacrylate nanolines with cross-sectional dimensions of 65 nm × 140 nm and rectangular or semicircular tops. The vibrational displacements and dispersion curves are qualitatively similar for the two geometries and include a series of flexural, Rayleigh-like, and Sezawa-like modes.

Propagation of Elastic Waves in a One-Dimensional High Aspect Ratio Nanoridge Phononic Crystal

“…From a general point of view, one can notice that the highlighted black modes coincide with the Lamb modes of an infinite plate with respectively xy and z polarizations, as mentioned by Heyliger et al [25]. This conclusion is supported by the calculated dispersion curves of an infinite epoxy plate (red dotted lines) with a thickness of w = 250 nm where we can notice a good similarity between the modes belonging to the nanoridge structure with the Lamb modes of the plate of epoxy.…”

Elastic constants and dimensions of imprinted polymeric nanolines determined from Brillouin light scattering

“…We employed the finite-element methods of Heyliger et al [11] to model the displacement patterns and dispersion curves of the vibrational modes of the nanolines. In this approach, the displacements are approximated as varying sinusoidally in phase along the length, and the nanoline/substrate interface is approximated as rigid.…”

“…Heyliger et al [11] presented a detailed description of the FE method for modeling vibrational modes of nanolines, performed calculations for rectangular and rounded-top cross sections, determined the form of additional vibrational modes, and evaluated the accuracy and range of validity of the approximations that were employed. A higher-order flexural mode, not detected in BLS measurements, was found to have a dispersion curve close to that of the Rayleigh-like mode.…”

“…Inversion of BLS data from an array of imprinted PMMA nanolines is performed here using the FE approach of Heyliger et al [11], which enables an arbitrary cross-sectional geometry to be incorporated in the model. Information on the dimensions of the nanolines is simultaneously determined under the simplifying assumption of a cross-sectional shape with only two adjustable parameters.…”

The effect of nickel nanostructure on surface waves propagation in silicon support

“…Recently of great interest are the elastic properties of a silicon support loaded with nanostructures of different types [1][2][3]. 2D periodic nanostructure made of materials of different elastic properties have been an attractive subject of research over the past decade.…”