# Propagation of Elastic Waves in a One-Dimensional High Aspect Ratio Nanoparticle Phononic Crystal
## 3 Results and Discussion  
### Propagation Parallel to the 1D Phononic Nanoridge  
Figure 2 represents the dispersion curves for the propagation direction parallel to the nanoridges (along \(x\)). Since the structure does not present any periodicity in this direction, we chose an arbitrary \(q\) limit of 0.025 nm\({}^{-1}\) in accordance with typical BLS experiments [24]. The dispersion curves of the nanoridge structure are calculated with the following parameters: \(a\) = 630 nm, \(h\) = 2.7 \(\upmu\)m, and \(w\) = 250 nm. The calculated dispersion curves show a huge number of branches, mainly due to the high aspect ratio of the nanoridges (\(AR\) = 10.2). Figure 1: (**a**) Three-dimensional (3D) schematic representation of the one-dimensional (1D) periodic nanoridges phononic crystal (**b**) Elementary unit cell used for the finite infinite method (FEM) calculation where the lattice constant is a, the height of the nanoridge is \(h\), and the width is \(w\). Figure 2: Dispersion curves calculation for the direction parallel to the nanoridges, limited to the 300 first eigenmodes for the set of geometrical parameters: \(a\) = 630 nm, \(h\) = 2.7 \(\upmu\)m and \(w\) = 250 nm. The red (green) dashed and solid lines correspond respectively to the longitudinal and transverse velocities of the glass (epoxy). We report in color on the same diagram the bulk phase velocities (straight lines) of the involved materials. The red (green) dashed and solid lines correspond respectively to the longitudinal and transverse velocities of the glass (epoxy). In the following, our interest will be focused on the modes below the transverse sound lines of the glass substrate (red solid line), which are expected to be localized at the surface of the substrate and/or localized in the nanoridge phononic structure. To go further and learn more about the localization and polarization of the surface acoustic modes, we proceed to a selection of modes using a band-sorting method, which enables us to keep only the modes that are highly confined inside the nanoridges. In addition, we divide them into two sets according to the following relations:  
\[P_{xy}=\frac{\int\int\int_{Nanwalls}\left(\left|u_{x}\right|^{2}+\left|u_{y} \right|^{2}\right)dv}{\int\int\int_{unit\ cell}\left(\left|u_{x}\right|^{2}+ \left|u_{y}\right|^{2}+\left|u_{z}\right|^{2}\right)dv} \tag{1}\]  
\[P_{z}=\frac{\int\int\int_{Nanwalls}\left(\left|u_{z}\right|^{2}\right)dv}{\int \int\int_{unit\ cell}\left(\left|u_{x}\right|^{2}+\left|u_{y}\right|^{2}+ \left|u_{z}\right|^{2}\right)dv} \tag{2}\]  
The expression \(P_{xy}\) (resp. \(P_{z}\)) gives the contribution of the modes with the components \(u_{x}\) and \(u_{y}\) (resp. \(u_{z}\)), which are localized only inside the nanoridge. In particular, modes with high \(u_{x}\) and \(u_{y}\) (resp. \(u_{z}\)) components display a \(P_{xy}\) (resp. \(P_{z}\)) close to 1. The black curves in Figure 3a,d give respectively the dispersion branches obtained from the dispersion curves of Figure 2 that satisfy the conditions \(P_{xy}\geq 80\%\) and \(P_{z}\geq 80\%\). 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 [PERSON] 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. The reason for this similarity is due to the high aspect ratio of the nanoridge, which brings the vibration modes of the nanoridge in close connection to those of a single plate.