Vibrational modes of nanolines

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.