# Propagation of Elastic Waves in a One-Dimensional High Aspect Ratio Nanoparticle Phononic Crystal
## 3 Results and Discussion
### Effect of the Geometrical Parameters  
The previous calculations have been realized in the particular case of a high aspect ratio of the nanoridges. In what follows, we investigate the effect of the geometrical parameters of the nanoridges for the directions that are parallel and perpendicular to the nanoridges. Figure 6 shows the dispersion curves for the parallel propagation direction to the nanoridges for three values of height: \(h=68\) nm (i.e., \(AR=0.3\)), \(h=135\) nm (i.e., \(AR=0.5\)), and \(h=270\) nm (i.e., \(AR=1.087\)). The band diagrams are represented with a band-sorting calculation, which highlights the modes inside the nanoridges as a function of their 'in-plane' \(P_{xy}\) polarization with a color bar, following Equation (1). Two features can be noticed from the analysis of the band diagrams. First, one can see that, as predicted, the number of branches increases with the height \(h\) of the nanoridges, leading to the  
Figure 5: (a) Dispersion curves for the propagation perpendicular to the nanoridges for the set of geometrical parameters: \(a=630\) nm, \(h=2.7\) μm, and \(w=250\) nm; (b) Eigenfrequencies calculation of a single nanoridge with a rigid boundary condition at the bottom; (c) Representation of the displacement fields at the frequencies 1, 2, and 3 of the ridges deposited on a substrate, and for the frequencies 4, 5, and 6 of one ridge with a rigid bottom surface. Figure 6: Dispersion curves for the direction parallel to the nanoridges with a fixed width of \(w=250\) nm and for different heights (a) \(h=68\) nm, (b) \(h=135\) nm, and (c) \(h=270\) nm. (d) Calculation of the first 10 branches at \(q=0.0167\) nm\({}^{-1}\) versus the height of the nanoridges. The color bar selects the modes of \(P_{xy}\) polarization inside the nanoridges. limiting case that is presented in Figure 2 (\(AR\) = 10.2). For the low aspect ratios, the behavior of the branches cannot be compared to A\({}_{0}\) and S\({}_{0}\)-like modes of a single plate, as shown previously for high AR. Second, one can see that the polarizations of the branches generally change when the height increases. For example, when \(h\) = 68 nm, branch 2 is red, indicating that the mode is polarized in the (\(x\), \(y\)) plane; meanwhile, when the height increases, the second branch becomes green for \(h\) = 135 nm and blue for \(h\) = 270 nm, meaning that the polarization becomes along \(z\) (\(P_{z}\) close to 1). Figure 6d reports in detail the evolution of the frequency of the first 10 modes and their polarization as a function of the height \(h\) for a fixed wave vector \(q\) = 0.0167 nm\({}^{-1}\). Focusing specifically on branch 2, one can notice a progressive conversion of the polarization from \(xy\) to \(z\) (red to blue color) as far as the height of the nanoridge increases. At the same time, the polarization of branch 1 remains unchanged. Such a physical behavior can have a considerable effect on the BLS experimental measurements spectra where the polarization plays a central role [27]. One can note that branch 1 corresponds to the A\({}_{0}\)-like mode, and branch 2 corresponds to one of the edge modes. Figure 7 sketches the evolution of the dispersion curves as a function of the geometrical parameters of the nanoridge for the direction perpendicular to the nanoridge. When the height of the nanoridge increases from 68 nm to 540 nm, with the the thickness \(w\) = 250 nm being constant, one can see that the number of flat branches increases and shifts to low frequencies. As mentioned in the previous section, the flat branches correspond to eigenmodes of the clamped nanoridge. In the band structures of Figure 7, one can see the occurrence of small band gaps where the surface acoustic waves cannot propagate at the surface of the substrate (green area). Similarly, we proceed to the calculation of the dispersion curves as a function of the width \(w\) from 150 nm to 300 nm, for a constant value of the height \(h\) = 540 nm (Figure 7b).