Finite-Difference Time-Domain (FDTD) Simulations

tions package (versions 8.6 and 8.7). As in previous work [108, 107], simulations were performed by placing rounded cones on a polystyrene core with a diameter of about 95 nm. The core is slightly expanded in order to fill in small gaps in the interface between gold and polystyrene. These gaps did not affect the results of the simula- tion. Spiky nanoshells were simulated in dielectrics modeling air (n=1) and water (n=1.333).

The geometry of the spiky shell was estimated by examining TEM and SEM images of the particles. We note that using this approach we are not able to extract a true cross-sectionional image of the particle, but rather we can extract the perimeter of the projection of the particles. There are potential errors associated with estimating the parameters this way. For example, the lowest valley of a particle may be obscured by a spike closer to the viewer than the valley, and large spikes will obscure small ones, making an estimate of the distribution of spike heights very difficult. As a result, an exact duplication isn’t possible.

We can, however, produce useful bounds on the typical spike size. Our analysis method provides that the average radius of the particles is less than 105.6 nm. With a core diameter of about 95 nm, the shell thickness should then be on average less than 58.1 nm. Likewise, the average largest spike on the particles examined is estimated to be 82 nm, and among the sampled particles the smallest observed largest spike on a single particle is 68 nm. This does not mean that a typical spike is 68 nm, but rather that a typical spike should be less than 68 nm. The lowest valley typically has an upper bound of 39 nm. Experimentally we estimate that there is about 2.99*106

It is clear from this that there are large outliers in the shape, and that it will be very difficult to produce a completely accurate shape. Since not all spiky nanoparticles contained large outlier spikes, and the results of the Raman scattering were very consistent, we believe that the largest outlier spikes are not dominant in the behavior of the particles. We therefore kept our variation below this level in order to make the modeling simpler. With that in mind, we sought to produce a model which qualitatively captured the behavior of the system, with a reasonable volume of gold. Sharp features are also most likely underestimated as they are numerically difficult to simulate and the spikes were rounded in order to remove artificially sharp points. Within these constraints, our models were selected to reproduce the far field scattering and extinction behavior observed on average from the particles in both air and water. The ordered structure was constructed by hand in order to have a desirable level of symmetry for ease of computational analysis. The ordered particle was modeled as 66 spikes with a 58 nm height, a 5 nm tip radius and a 60 nm cone angle. This shape occupies 2.04*106 nm3 gold, making it a bit smaller than a typical experimental shell, but still in reasonable agreement. The difference in gold volume is partially due to the decision not to model the large outlier spikes. Given the high level of reproducibility in experimental results, the models should still capture all of the behavior of the spiky nanoshells.

To produce the disordered structure we used a force algorithm [15], where the cone locations were evenly distributed on the surface of the core by a r−2 repulsion force. This repulsion force results in asymmetric but evenly placed seed positions once the seeds reach equilibrium. Additionally, variations in the cone parameters were introduced to the disordered model. The disordered particle was modeled as 60 spikes with randomly chosen spike heights between 50-65 nm, tip radii between 2-6

nm and cone angles of 30-75 degrees. All random distributions were uniform on the range described. While this model more closely represents the experimental variations in the nanoshell shape and size, we expect our predictions of the field enhancement to be lower than the experimental field enhancement due to lack of sharp corners, edges, and defects that are present in the real nanoshells. This shape occupies 2.14*106 _nm3

gold, making it 27% smaller than a typical experimental shell, but still in reasonable agreement given the choice to not include large outlier spikes in the model. Using the same amount of gold, the smooth shell was modeled as gold on polystyrene with an inner diameter of 98 nm and a resulting outer diameter of 172 nm respectively. The slightly different core diameter is due to the gold cones in the ordered and disordered model occupying the outer few nanometers of the core, thereby replacing it with gold. This was done to prevent any pockets of air from potentially influencing the simulation.

In order to validate the models, calculated backscattering spectra in air (back- ground n=1) (Figure 5.5b) was compared to the experimental dark field spectrum and the calculated extinction cross section with n=1.333 was compared to the ex- perimentally measured extinction in aqueous conditions (Figure 5.6). Both of these are seen to be a reasonable match. Additionally, studies found that the addition of a dielectric substrate near the particle had little effect on the scattering, and as a result the glass slide used during experimental measurements was not necessary to model. From those studies, the substrate would have at most resulted in a very small red shift. We conclude from their matching the average spectrum in Figure 5.6 and Figure 5.5b, and from the spread of the distribution of the LSPR in Figure 5.4, that these models represent typical particles in the experimental spectrum. In the course of these studies, several randomized versions of the disordered particle were produced

as discussed below, and although the exact spectrum differed the resulting conclu- sions were shared: the yz and zz quadrupoles result in spectrally broad and strong QERS, with similar ratios of Raman signal at 633 nm and 785 nm excitation.

The additional families of disordered structures were produced using previously reported parameters [107] for spikes as the mid points for randomization. Each data set was produced by generating new spike positions and re-randomizing the spike parameters ten times. Breadth was set to be approximately consistent with the breadth chosen for the original disordered model. For the short models, 40 spikes with lengths between 38 and 53 nm, cone angles between 45 and 85 degrees and tips between 2 and 6 nm were used. The medium models was as described above: 60 spikes with randomly chosen spike heights between 50-65 nm, tip radii between 2-6 nm and cone angles of 30-75 degrees. The long models were chosen to have 80 spikes with length 58 to 73 nm, cone angle 15 to 45 degrees, and tips between 2 and 6 nm in radius.

The electric field was monitored during the simulations and used to measure the near field parameters including both electric field enhancement and polarization cur- rent. To estimate enhancement the index of refraction at each point in space was sampled and an isosurface of refractive index produced surrounding the gold. The distance from that surface to every field sample was then computed, and anything outside of the surface and less than 2.1 nm from that surface was included as a sample point. The sample points have then formed a thin shell of 2.1 nm surrounding the nanoshell. Sample points were then multiplied in order to produce the appropriate en- hancement (taking into account both enhancement frequency and Stokes frequency). Field maps as shown in Figure 5.12 were produced by projecting the thin shell used for integration onto the surface of the structure. In particular, each position on the

surface of the nanoshell, as determined by a Matlab isosurface, was assigned a field value by interpolation from the near field shell used earlier using a nearest neighbor method. For Figure 5.5b, and Figure 5.6, backscattering cross section refers specifi- cally to the component of the electric field projected alongk,i.e., directly backwards. In contrast, the extinction, absorption, and regular scattering measurements include the field in all directions.

Mesh settings in Lumerical play an important role in the validity of these sim- ulations. During simulation, mesh refinement was not used near the metal surface. Index assignment on the mesh introduces sharp artifacts along the edge of the metal- lic structure, but these most likely are still an underestimate of the true disorder of the structure and should not affect the validity of the simulation. The mesh spacing in the region around the particle (the override region in FDTD Solutions) was set to 2 nm, the shortest distance possible with our current computational constraints. This limits the near field measurement to a single layer of pixels surrounding the object. This will cause an underestimation in the field as the field drops strongly away from the metal surface, while in Raman experiments the analyte molecule is expected to be closer to the surface than this distance.