# Design of Inverted Nano-Cone Arrayed SERS Substrate for Rapid Detection of Pathogens
## 3 Results
### Nanocone Array Geometry Optimization for Tight Target Cell-Hot Spot Contact  
By optimizing the nanocone coating material selection via electric field intensity simulation, the nanorods with Si coated cavity wall can have an enhanced Raman sensitivity by a factor of two over that of the nanorods on flat PDMS substrate. Now it is the key to bring the target cell in a very close proximity to the extremely small hot spots near nanorods. This can be achieved by pulling the magnetic beads toward the cone array in between the cones by applying magnetic field from the bottom. As a result, a cell with 5-6 magnetic beads bonded will be pulled down to the cone array. To have a tight contact between the cone arrays and bacterial cell wall, the cone's height needs to be taller than the magnetic bead diameter which is usually in the size \(\sim\)500 nm [20], but not too tall which may exert to much force to break the bead-cell connection while being pulled down. We don't expect the PDMS cone array will not collapse by this pulling force. The cell wall fluctuates (with possible wrinkles) in the order of \(\sim\)50 nm, where the dynamic biochemical reaction drives the cell wall to move up and down [26]. The height is chosen as 700 nm to leave 200 nm indentation of cone into the cell wall. From the literature [31] one can see that bacterial cell walls can typically tolerate 200 nm indentation without cell wall ruptures. Figure 3: 758 nm electric field in cross sections of (**a**) Si coated (**b**) Au coated inverse cavity without nanorod (**c**) Electric filed enhancement spectrum comparison for 3 structures (Si-coated nanocone with one top nanorod, Au-coated nanocone, Nanorod on flat PDMS) (**d**) Illustration of the nanocone cross section with one nanorod. This 200 nm indentation is bigger than the wall fluctuation as well as to make a stable contact, while avoiding the breakage of the cell wall. In order to increase the number of cones in touch with a single cell, the cone array packing density needs to be maximized, thereby the cone array is arranged into a dense hexagonal pattern in our design as shown in Figure 4a. In order to accept magnetic beads between cones when pulled down, the center of every honeycomb cell is left open as shown in Figure 4b. The second step is to find the optimal slope and bottom diameter of cones. The relative intensity \(I_{relative}\) has the relationship [11]:  
\[I_{relative}\propto\sum EF_{top}\times N_{molecule} \tag{5}\]  
\(N_{molecule}\) is the number of molecules at the top of the cone and is proportional to the top surface area \(S_{top}\). The enhancement at the top of cone \(EF_{top}\) is positively related to the slope of the cone while \(S_{top}\) is negatively related. To find the trade-off between \(EF_{top}\) and \(S_{top}\), a function is defined as:  
\[f_{cone}=\frac{EF_{top}\times S_{top}}{\sum S_{i}} \tag{6}\]  
where \(\sum S_{i}\) is the total area, a single cone takes up, on average. This function is actually the ratio between the signal intensity of the cone array to that of the flat surface at the same nanorod deposition density. For a flat surface, \(f_{cone}\) is 1, and there is no funneling effect. The effort is to maximize \(f_{cone}\). The \(EF_{top}\) is calculated through:  
\[\sqrt{EF_{top}}=\frac{S_{top}+\left(\pi D^{2}-S_{top}\right)\times\underline{ R}(0.5\pi-\theta)}{S_{top}} \tag{7}\]  
\(\underline{R}\) is the polarization averaged reflection of the film, obtained from [PERSON]'s equations. In the calculation, given that the film thickness is 15 nm, much smaller than the wavelength in the film, a thin film approximation is introduced where wavevectors in PDMS and water are close to free space and the wavevector in film is nearly independent of radius _a_[32]. In the calculation, only the first order of reflection at the surface of the film is taken into account. In Figure 5, knowing that the top diameter cannot be smaller than 100 nm, we can find an optimal geometry for the cone as bottom diameter 550 nm, top diameter 110 nm and height 700 nm.