# The UV Plasmonic Behavior of Rhodium Tetrahedrons--A Numerical Analysis
## 1 Introduction  
The UV has been elusive for plasmonics due to the difficulty of finding metals with strong and ambient-stable plasmonic response in this spectral range. Common plasmonic metals (i.e., Ag or Au) cannot be used for this purpose because of the presence of interband transitions in the UV that inhibit the excitation of localized surface plasmon resonances (LSPRs) in this spectral region. [PERSON] et al. [1] initiated a quest for metals able to support LSPRs in the UV. From this study, magnesium (Mg), aluminum (Al), gallium (Ga) and rhodium (Rh) were presented as the most promising candidates. However, an issue that should be carefully addressed when dealing with the applicability of these materials in real applications is their stability when exposed to ambient conditions. For example, Mg, rapidly oxidizes when exposed to air, consuming the metal in the nanoparticles (NPs) while quenching completely the LSPRs [1; 2]. Al and Ga form dense oxide layers that dramatically slows down subsequent oxidation reactions after only a few monolayers preventing further oxygen diffusion. In the case of Al, the thickness of the Al\({}_{2}\)O\({}_{3}\) layer is of the order of several nanometers [1; 3]. Ga, a liquid metal just above room temperature, forms a self terminating Ga\({}_{2}\)O\({}_{3}\approx 1\) nm thick that maintains the shape of the nanoparticles and protects the inner metal from environmental degradation [4]. The formation of an oxide shell on the surface of the NPs made of these metals affects both the spectral position and near-field enhancement of the LSPRs, hampering a reproducible plasmonic response [5].  
Rhodium was another of the recently proposed promising metals for UV plasmonics by [PERSON] et al. [1]. A lot of attention is being paid to this metal in UV plasmonics for several reasons:(i) the lack of a native oxide coating [6], (ii) the possibility of enhancing rhodium's already favorable catalytic activity by UV illumination near its plasmonic resonance [7] and (iii) the variety of shapes in which Rh NPs can be synthesized through chemical methods with nanometer size control. Already, size-controlled Rh tripod stars and nanocubes have been grown with their corresponding systematic numerical study on their plasmonic response, and it has been demonstrated their activity in UV surface-enhanced Raman spectroscopy (SERS), surface-enhanced fluorescence (SEF) and photoinduced degradation of p-aminothiophenol [6; 8; 9; 10]. Another Rh NPs shape that has been synthesized is the tetrahedron (TD). Interestingly, this polyhedron is the one with highest surface-to-volume ratio, a desired characteristic in photocatalytic systems. Not only perfectly shaped Rh TDs have been synthesized [11], but also concave ones with defect-rich surfaces [12; 13]. These defects have been shown to enhance the plasmon assisted photocatalytic energy conversion [13]. Other active catalytic sites are corners and edges, where adsorbed chemical intermediates may have higher coordination with surface atoms there [14].  
Here, we report a detailed numerical analysis of the UV plasmonic behavior and charge distribution of Rh TDs, a geometry easily fabricated with precise size control even with concave surfaces [11; 12; 13]. It is shown that the highest hot-spots are located at their edges and corners, and that their intensity decreases when they are progressively rounded. This deformation of the perfect shape commonly appears when TDs, and other NPs, are repeatedly used in photocatalytic cycles [10]. Due to the possibility of synthesizing concave TDs, we will also show how the concavity of TDs can be optimized by varying its depth and size to achieve, simultaneously, a maximum surface charge density on it and on its external corners and edges. In this way, we provide the path for designing of concave TDs with optimum electromagnetic behavior for photocatalysis applications.