# The UV Plasmonic Behavior of Rhodium Tetrahedrons--A Numerical Analysis
## 3 Results and Discussion
### Rh Tetrahedra: Comparison with Other Fabricated Rh Geometries  
Apart from Rh tetrahedra, other Rh NPs geometries have been reported in the literature with demonstrated applicability in photocatalysis and surface-enhanced spectroscopies. These geometries correspond to Rh tripod stars [6; 8] and nanocubes [7; 9; 10]. Here, we will compare the near-field enhancement and spectral position of the LSPR peaks in perfect tetrahedra (\(L=20\) nm), and tripod stars and cubes of equivalent volume. The tripod stars have length \(l=12\) nm, width \(w=6\) nm and height \(h=6\) nm [8]. The cube has an edge length of \(L=10\) nm. Additionally, we consider in the comparison a concave tetrahedron with size \(L=20\) nm, and a concavity parameterized by \(t=5\) nm and \(d=4\) nm (smaller volume). For the comparison, all the geometries have been illuminated under normal incidence with a plane-wave polarized along the \(\mathbf{x}\)-axis. Finally, it is worth mentioning that the TD is the polyhedron with highest surface-to-volume ratio (0.72 nm\({}^{-1}\) for the TD \(vs\) 0.60 nm\({}^{-1}\) for the cube). The tripod star has a surface-to-volume ratio of 0.76 nm\({}^{-1}\). Geometries with high values of this magnitude are desired in photocatalysis. It is worth mentioning that this magnitude is increased when a concavity is added to the particle ( 1.27 nm\({}^{-1}\) for the concave TD \(vs\) the 0.72 nm\({}^{-1}\) of the perfect TD).  
Figure 10 plots (a) \(C_{abs}\) and (b) \(\langle|\mathbf{E}|^{2}\rangle\) for the aforementioned geometries: the cube, tripod star, tetrahedron and concave tetrahedron. The near-field distribution on their surfaces when illuminated at resonance is shown in Figure 10c.  
Figure 10a shows big differences in the LSPR peaks of the different analyzed geometries. Both tripod star and the concave tetrahedron present well-defined peaks at approximately the same photon energy (\(\approx 3.75\) eV) with a difference in amplitude caused by the reduced volume of the concave TD. The cube also presents a well-defined peak with an amplitude slightly higher than that of the concave TD, although in this case the peak is considerably broader and at higher photon energy (\(\approx 6.5\) eV). The perfect TD shows a wide plateau covering the 4.5-6.5 eV spectral range and an amplitude similar to that of the concave TD. Regarding the values of \(\langle|\mathbf{E}|^{2}\rangle\), all the analyzed geometries show very similar maximum values, ranging between 7.5-8.5. Again, both the tripod star and the concave TD show well-defined peaks at lower energies (\(\approx 3.7\) eV) than those of the TD and cube. The broad peaks of the TD and the cube are formed by two overlapping components--for the  
Figure 10: (a) Absorption cross-section (\(C_{abs}\)) and (b) local field enhancement (\(|\mathbf{E}|^{2}\)) of a cube (\(L=10\) nm), tripod star (\(l=12\) nm, \(w=6\) nm and \(h=6\) nm), tetrahedron (\(L=20\) nm) and concave tetrahedron (\(L=20\) nm, \(t=5\) nm and \(d=4\) nm) illuminated under normal incidence with a \(\mathbf{x}\)-polarized plane wave. (c) Near field distribution on the surface of the particles when illuminated at resonance.  
TD at 4.4 and 6.2 eV, and for the cube at 5.0 and 6.2 eV. Whereas for the TD the dominant component is the one at low photon energy (4.4 eV), for the cube, the dominant one is that at higher photon energies (6.2 eV). Therefore, the TD and the cube produce near-field enhancements comparable to those of the tripod star and concave TD but in a broader spectral range, allowing a less precise spectral tuning of the illuminating beam in order to reach a resonant behavior.