## 3 Results
### FEM Modelling  
Finite element method modelling in _COMSOL_ was performed on representative models of substrates B and C, built from fragments of their SEM images (Figure 6). In order to assess the influence of irregularities in the nanostructure on the enhancement factor, a perfectly uniform configuration of nanospheres was also modelled. Here, the nanospheres had diameters and interparticular distances equal to the average particle size and interparticular distance of the modelled fragment of substrate C. The resulting electric field calculations were used to determine the ratio between the electric field around an unstructured substrate and a structured one (\(r_{electric}=\frac{E_{pub}}{E_{B}}\)). This ratio for both models is mapped in Figure 7 and it clearly shows the formation of hotspots in locations where the nanostructures are in close proximity to one another.  
Figure 4: The 1358 cm\({}^{-1}\) Raman peak of Rhodamine B intensity dependence on concentration (substrates A, B, and C). The incident laser wavelength is 532 nm @ 2.55 mW, 1 s exposure.  
Alongside electric field enhancement maps, the average and maximum electromagnetic enhancement of the Raman signal was calculated, using the well-established fourth power approximation with [4]:  
\[EF_{EM}=\left|\frac{E_{enh}}{E_{0}}\right|4 \tag{3}\]  
The maximum EF was identified in the whole domain of calculation.  
The maximum electromagnetic enhancement from the _COMSOL_ modelling results was predicted to be around \(10^{6}\) for substrate B and \(10^{3}\) for substrate C, with the uniform configuration reaching only around \(10^{2}\) (Figure 8). The enhancement factor in the more chaotic and fractal-like substrate B was predicted to be higher, which seems to be supporting our hypothesis about small-scale detail improving the SERS performance of a substrate. Looking at the field enhancement maps, it is quite clearly visible that there are more hotspots in the geometry based on substrate B. That is despite the fact that substrate  
Figure 5: Spectral intensity ratio maps of the 1649 cm\({}^{-1}\) Raman peak of Rhodamine B at a concentration of \(10^{-5}\) M on substrates (**a**) A; (**b**) B; and (**c**) C. The incident laser wavelength is 532 nm @ 0.452 mW, 1 s exposure.  
Figure 6: The chosen input images for contour drawing in _COMSOL_: (**a**) substrate B; (**b**) substrate C.  
C features on average smaller interstructural distances--the random protrusions and imperfections in the nanostructure of substrate B allow for locations where the interstructural distances are tiny, leading to a large EF in them. This is especially clearly underlined by comparison to the uniform configuration where the large amount of small gaps between the nanospheres cannot outperform the enhancement obtained from the more chaotic real-like substrates where smaller gaps, though not as abundant, are enough to yield a higher Raman spectral enhancement.  
Figure 8: Maximum EM enhancement \(EF_{EM}=\left|\frac{E_{m}}{\epsilon_{0}}\right|^{4}\) dependency on incident wavelength for three different FEM models based on the real substrates (substrate B and substrate C), as well as a uniform configuration of spheres.  
Figure 7: Electric field enhancement maps at 530 nm excitation for three different FEM models representing the real substrates, as well as a uniform configuration of spheres: (**a**) substrate B; (**b**) substrate C; (**c**) uniform configuration.