# The Design and Optimization of Plasmonic Crystals for Surface Enhanced Raman Spectroscopy Using the Finite Difference Time Domain Method

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Comparison of FDTD Simulated SERS Responses for Nanowell and Nanopost Plasmonic Crystals

#### 2.2. Optimization of FDTD Simulated SERS Responses for Nanowell and Nanopost Plasmonic Crystals

#### 2.3. Optimization of FDTD Simulated SERS Responses for Novel Plasmonic Crystals

## 3. Discussion

## 4. Materials and Methods

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Unit cell geometries for (

**a**) nanowells, (

**b**) cylindrical nanoposts, (

**c**) square nanoposts, and (

**d**) particle imbedded nanowells. Plasmonic crystals result from the use of periodic boundary conditions.

**Figure 2.**Comparison of experimental SERS response (exp) and calculated SERS responses (${G}_{\mathrm{SERS}}^{4}$ and ${G}_{\mathrm{mid}}^{4}$) for the (

**a**) NW and (

**b**) NP geometries. Experimental SERS response for the NW and NP geometries are taken from Refs. [13,14], respectively. Experimental values are plotted with filled circles. SERS responses using Equation (1) are plotted with open circles. SERS responses calculated using Equation (2) are plotted with open squares. Solid and dashed/dotted lines correspond to spline interpolation between data points and are meant for ease of visualization only. Calculated SERS responses are scaled such that maximum values equal the experimental maximum.

**Figure 4.**Plot of optimization factors ($O.F.={G}_{\mathrm{SERS}}^{4}/{G}_{\mathrm{control}}^{4}$) as nanowell (

**a**) relief depth, (

**b**) diameter, (

**c**) periodicity, and (

**d**) metal thickness were sequentially varied. The final optimization factor is 6.0. Dashed lines correspond to spline interpolation between data points and are meant for ease of visualization only.

**Figure 5.**Plot of dielectric constant and time-averaged electric field intensity enhancements (${g}^{2}={\left|E\right|}^{2}/{\left|{E}_{0}\right|}^{2}$) at $\lambda =785$ nm and $\lambda =857$ nm for (

**a**) unit cell of control NW geometry ($D=456$ nm, $P=730$ nm, $RD=360$ nm, and $MT=40$ nm) and (

**b**) unit cell of optimized NW geometry ($D=500$ nm, $P=730$ nm, $RD=160$ nm, and $MT=70$ nm). Only enhancements in the 50–500 range are plotted.

**Figure 6.**Plot of optimization factor, O.F., as a function of periodicity for cylindrical NP geometry with $RD=210$ nm, $D=200$ nm, and $MT=24$ nm in the range 680–750 nm. Dashed lines correspond to spline interpolation between data points and are meant for ease of visualization only.

**Figure 7.**Plot of electric field intensity enhancement (${g}^{2}={\left|E\right|}^{2}/{\left|{E}_{0}\right|}^{2}$) at $\lambda =785$ nm for optimized square NP. The optimal parameters are $D=150$ nm, $RD=190$ nm, $P=730$ nm, and $MT=24$ nm with an $O.F.=6.3$. Only enhancements in the 50–500 range are plotted.

**Figure 8.**Plot of optimization factors as a function of particle diameter for spherical gold nanoparticles imbedded in an NW plasmonic crystal. Dashed lines correspond to spline interpolation between data points and are meant for ease of visualization only.

**Figure 9.**Extinction efficiency, ${Q}_{\mathrm{ext}}$, for a 300 nm diameter spherical gold nanoparticle.

**Figure 10.**Plot of electric field intensity at (

**a**) $\lambda =785$ nm and (

**b**) $\lambda =857$ nm for optimized NW with imbedded 300 nm diameter nanoparticles. The optimal parameters are $D=650$ nm, $RD=100$ nm, $P=700$ nm, and $MT=40$ nm with an $O.F.=2400$.

Diameter (nm) | Periodicity (nm) |
---|---|

174 | 490 |

224 | 584 |

256 | 658 |

456 | 730 |

500 | 800 |

514 | 760 |

616 | 1000 |

685 | 1100 |

**Table 2.**Optimization factors for nanowell geometries imbedded with 300 nm spherical gold nanoparticles.

P (nm) | D (nm) | RD (nm) | O.F. | O.F. |
---|---|---|---|---|

($\mathit{MT}=40$ nm) | ($\mathit{MT}=70$ nm) | |||

700 | 550 | 100 | 1163 | 917 |

700 | 550 | 140 | 401 | 366 |

700 | 550 | 180 | 320 | 213 |

700 | 600 | 100 | 1815 | 1171 |

700 | 600 | 140 | 619 | 751 |

700 | 600 | 180 | 452 | 460 |

700 | 650 | 100 | 2358 | 1405 |

700 | 650 | 140 | 887 | 966 |

700 | 650 | 180 | 627 | 616 |

730 | 550 | 100 | 870 | 600 |

730 | 550 | 140 | 223 | 307 |

730 | 550 | 180 | 147 | 181 |

730 | 600 | 100 | 1538 | 852 |

730 | 600 | 140 | 381 | 411 |

730 | 600 | 180 | 184 | 211 |

730 | 650 | 100 | 1920 | 1156 |

730 | 650 | 140 | 889 | 897 |

730 | 650 | 180 | 647 | 656 |

760 | 550 | 100 | 463 | 561 |

760 | 550 | 140 | 52 | 143 |

760 | 550 | 180 | 16 | 47 |

760 | 600 | 100 | 837 | 753 |

760 | 600 | 140 | 201 | 248 |

760 | 600 | 180 | 164 | 195 |

760 | 650 | 100 | 1050 | 880 |

760 | 650 | 140 | 347 | 437 |

760 | 650 | 180 | 292 | 351 |

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**MDPI and ACS Style**

Bigness, A.; Montgomery, J.
The Design and Optimization of Plasmonic Crystals for Surface Enhanced Raman Spectroscopy Using the Finite Difference Time Domain Method. *Materials* **2018**, *11*, 672.
https://doi.org/10.3390/ma11050672

**AMA Style**

Bigness A, Montgomery J.
The Design and Optimization of Plasmonic Crystals for Surface Enhanced Raman Spectroscopy Using the Finite Difference Time Domain Method. *Materials*. 2018; 11(5):672.
https://doi.org/10.3390/ma11050672

**Chicago/Turabian Style**

Bigness, Alec, and Jason Montgomery.
2018. "The Design and Optimization of Plasmonic Crystals for Surface Enhanced Raman Spectroscopy Using the Finite Difference Time Domain Method" *Materials* 11, no. 5: 672.
https://doi.org/10.3390/ma11050672