# The UV Plasmonic Behavior of Rhodium Tetrahedrons--A Numerical Analysis
## 2 Methods  
The electromagnetic interaction between the Rh TDs and light has been modeled using finite element simulations implemented using COMSOL Multiphysics 5.2 (COMSOL Inc., Burlington, MA, USA). Its electromagnetic waves module allows solving of the differential form of [PERSON]'s equations together with initial and boundary conditions. The equations are solved using the finite element method (FEM) with numerically stable edge element discretization in combination with state-of-the-art algorithms for preconditioning and solving the resulting sparse systems of equations. The spectral dependence of three parameters has been studied--the absorption cross-section (\(C_{abs}\)), the local field enhancement of the electric field (\(|\mathbf{E}|^{2}\)) and the surface charge density (\(\sigma\)), all of which depend on the photon energy of the illuminating electromagnetic field. For particles much smaller than the wavelength, \(C_{abs}\) can be associated with the far-field extinction efficiency. \(|\mathbf{E}|^{2}\) represents the local field enhancement. The areas with the highest enhanced local field intensity (i.e., hot-spots) are the regions where the hot-electron generation process takes place. The evaluation of \(|\mathbf{E}|^{2}\) allows us to identify the regions over the NP surface where hot electrons are being generated, this information being useful for photocatalysis research. A deeper analysis of this aspect is made through the evaluation of \(\sigma\): the regions with the highest charge concentrations are the most reactive having the highest probability of charge transfer. We have performed finite element simulations on Rh TDs and have compared them with identical ones made of Pt and Pd. These nanostructures were illuminated with a monochromatic linearly polarized plane wave within the interval of interest (i.e., 2-6 eV). The illumination geometry is depicted in Figure 1a: the TDs are illuminated at normal incidence (wave vector \(\mathbf{k}\) along the \(z\)-axis) with polarization \(\mathbf{E_{x}}\) parallel to one of the edges. Two different TD deformations have been studied stating from the perfect TD: (i) the rounded and (ii) the concave TDs. Each of them has their own parameters in order to characterize it. Figure 1b schematizes each of the aforementioned deformations and their characterization parameters. Starting by the perfect TD, it is defined by the side length \(L\). The rounded TD is defined by the side length \(L\) of the perfect TD in which it is inscribed and the radius of curvature of the corners \(R_{c}\). Finally, the concave TD is defined by the side length \(L\), the depth of the concavity \(d\) and the distance between the corner of the TD and the deformation \(t\). In this work, we have used the frequency dependent complex dielectric functions (\(\epsilon(\omega)=\epsilon_{1}(\omega)+ie_{2}(\omega)\)) of Rh, Pt and Pd taken from References [15; 16; 17]. The complex dielectric functions of these materials are plotted in Figure 2. The Frohlich frequency (\(\epsilon_{1}=-2\)) for these metals are 8.0 eV for Rh, 4.5 eV for Pt and 5.7 eV for Pd. At those energies, the figure of merit \(Q_{LSP}\)[18], typically used to evaluate the efficiency to generate LSPRs, and defined as \(Q_{LSP}=-\epsilon_{1}/\epsilon_{2}\), takes the following values: 2.25 for Rh, 0.35 for Pt and 0.66 for Pd. Therefore, Rh is able to support higher quality LSPRs reaching higher photon energies. Figure 1: (**a**) Scheme of simulations geometry: the tetrahedrons (TDs) are illuminated with monochromatic plane waves with orthogonal polarization \(\mathbf{E_{x}}\) and \(\mathbf{E_{y}}\) under normal incidence \(\mathbf{k}\) (along the \(z\)-direction). (**b**) Scheme of the different deformations introduced to the perfect TD. With red arrows are indicated the parameters that characterize each deformation.