# Hot Electron Plasmon-Resonant Grating Structures for Enhanced Photochemistry: A Theoretical Study

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## Abstract

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## 1. Introduction

_{2}films by plasmonic near-field coupling from Au nanoparticles [6,7]. Hot electrons generated by the surface plasmon decay from Au nanoparticles were used to drive H

_{2}and D

_{2}dissociation at room temperature [8,9]. Theoretical studies have revealed that hot carriers are generated via direct transitions above the interband transition threshold in commonly used plasmonic materials gold, silver, copper, and aluminum [10,11]. The lifetime of these carriers was measured as a few picoseconds by ultra-fast, pump-probe spectroscopy measurements [12].

_{2}O

_{3}/graphene) heterostructures [15]. Corrugated Au gratings have also been used to inject photoexcited hot electrons into an aqueous solution and to drive water splitting reactions [7,15]. Photocurrent enhancement has been demonstrated with hot electrons generated in corrugated Ag grating structures [13]. These measurements use a fixed wavelength and scan the incident angle, which enables tuning through the plasmon resonance. In these grating structures, the plasmon-resonant excitation can be distinguished from bulk interband transition simply by rotating the polarization of the incident light.

## 2. Methods

^{2}, referred to as the steepness factor.

_{ref}). Absorbed power (P

_{abs}) was calculated as P

_{abs}= 1 − P

_{ref}. The electric field intensity was monitored with a 2D field monitor. The broadband absorption spectra were calculated for different incident angles for each metal, as shown in Figure 2 (top row).

## 3. Theory

_{r}is the resonance wavelength; and V

_{eff}is the effective mode volume. A

_{c}is the effective aperture of the microstructure, which is determined by the radiation pattern of the whole grating array [20,21]. The total radiation pattern of the large area grating structure can be regarded as a plane wave, and therefore A

_{c}is equal to incident planewave cross section. Q

_{rad}and Q

_{abs}are the radiation and absorption quality factors of the structure, respectively, and are related to the resonant wavelength and loss rates in the structure as follows [20]:

_{eff}is not expected to depend strongly on the geometry when the dimensions are small compared to wavelength [22]. The field enhancement increases with ${\Gamma}_{\mathrm{rad}}$, reaches maximum when ${\Gamma}_{\mathrm{rad}}={\Gamma}_{\mathrm{abs}}$, and then decreases with further increase in ${\Gamma}_{\mathrm{rad}}$. This condition (${\Gamma}_{\mathrm{rad}}={\Gamma}_{\mathrm{abs}}$) is called the critical coupling. The critical coupling could be understood as analogous to the impedance matching principle. When an electromagnetic wave couples with the resonant structure, the maximum energy is stored in the resonator when the loss rates are matched [19]. The maximum field enhancement at critical coupling varies inversely proportional to the absorption losses (${\Gamma}_{\mathrm{abs}}$). Therefore, the maximum field enhancement is obtained when ${\Gamma}_{\mathrm{rad}}={\Gamma}_{\mathrm{abs}}$ and ${\Gamma}_{\mathrm{tot}}={\Gamma}_{\mathrm{rad}}+{\Gamma}_{\mathrm{abs}}=2{\Gamma}_{\mathrm{abs}}$ is as close to minimum as possible.

## 4. Corrugated Grating Structure

^{0}to 10

^{0}. The absorption spectra obtained from the FDTD simulations for each metallic grating were then fitted to the analytical equation (6) using non-linear curve fitting with the NLopt library [25] in MATLAB to deduce the fitting coefficients (${\Gamma}_{\mathrm{rad}},{\Gamma}_{\mathrm{abs}},{\mathsf{\omega}}_{0},{\mathrm{r}}_{\mathrm{p}},\mathsf{\varphi}$). Figure 3 shows the simulated absorption spectra (markers) and fitted absorption spectra using the analytical equation (solid lines). The field profile corresponding to the resonant angles at wavelengths 633 nm and 785 nm is also plotted. For fitting the spectra with multiple resonant modes, the linear superposition of Lorentzian-like coupled surface plasmon polariton modes are considered [26]. It can be seen from Figure 3 that the analytical equation agrees well with the simulated spectra.

## 5. Tuning the Corrugation for Each Material

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Cross-sectional SEM image of the fabricated corrugated grating. (

**b**) The refractive index profile of the structure modelled by Equation (1).

**Figure 2.**Simulated absorption of gratings plotted as a function of incident angle and wavelength (

**top**) and resonant wavelength for different incident angles for each mode using Equation (1) (

**bottom**).

**Figure 3.**(

**top row**) The absorption spectra numerically obtained from finite difference time domain (FDTD) simulations (markers) and fitted spectra from Equation (6) (solid lines) (

**top row**). Corresponding resonant field enhancements at 633 nm (

**middle row**) and 785 nm (

**bottom row**).

**Figure 4.**Loss rates of resonant modes at 633 nm and 785 nm for the initial grating geometry of different material systems.

**Figure 5.**Field enhancement at 633 nm and 785 for different metal gratings under consideration as a function of A and $\mathsf{\sigma}$. The white dot represents the initial grating structure and yellow star represents the optimized geometry.

**Figure 6.**(

**top row**) The absorption spectra numerically obtained from FDTD simulations (dots) and fitted spectra from Equation (3) (solid lines) for the optimum structure of each metal. Corresponding resonant field enhancements at 633 nm (

**middle row**) and 785 nm (

**bottom row**).

**Figure 7.**Loss rates of resonant modes at 633 nm and 785 nm for the optimized geometry of different material systems.

**Table 1.**Values of corrugation amplitude ($\mathrm{A}$) and steepness factor ($\mathsf{\sigma}$) that yield maximum field enhancement at 633 nm and 785 nm.

Original Gratings | Optimized Geometry | |||||||
---|---|---|---|---|---|---|---|---|

A (nm) | σ (nm) | ${\mathbf{I}}_{\mathbf{avg}}633\mathbf{nm}$ | ${\mathbf{I}}_{\mathbf{avg}}785\mathbf{nm}$ | A (nm) | σ (nm) | ${\mathbf{I}}_{\mathbf{avg}}633\mathbf{nm}$ | ${\mathbf{I}}_{\mathbf{avg}}785\mathbf{nm}$ | |

Ag | 58.9 | 125 | 7.05 | 6.04 | 41.4 | 235 | 13.00 | 23.00 |

Au | 58.9 | 125 | 4.37 | 4.85 | 46.2 | 198 | 5.90 | 17.60 |

Al | 58.9 | 125 | 5.14 | 5.82 | 59.5 | 198 | 7.06 | 7.44 |

Cu | 58.9 | 125 | 3.65 | 4.77 | 50.5 | 215 | 4.47 | 12.20 |

Pt | 58.9 | 125 | 1.50 | 1.97 | 95.7 | 178 | 1.73 | 2.48 |

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**MDPI and ACS Style**

Aravind, I.; Wang, Y.; Cai, Z.; Shen, L.; Zhao, B.; Yang, S.; Wang, Y.; Dawlaty, J.M.; Gibson, G.N.; Guignon, E.;
et al. Hot Electron Plasmon-Resonant Grating Structures for Enhanced Photochemistry: A Theoretical Study. *Crystals* **2021**, *11*, 118.
https://doi.org/10.3390/cryst11020118

**AMA Style**

Aravind I, Wang Y, Cai Z, Shen L, Zhao B, Yang S, Wang Y, Dawlaty JM, Gibson GN, Guignon E,
et al. Hot Electron Plasmon-Resonant Grating Structures for Enhanced Photochemistry: A Theoretical Study. *Crystals*. 2021; 11(2):118.
https://doi.org/10.3390/cryst11020118

**Chicago/Turabian Style**

Aravind, Indu, Yu Wang, Zhi Cai, Lang Shen, Bofan Zhao, Sisi Yang, Yi Wang, Jahan M. Dawlaty, George N. Gibson, Ernest Guignon,
and et al. 2021. "Hot Electron Plasmon-Resonant Grating Structures for Enhanced Photochemistry: A Theoretical Study" *Crystals* 11, no. 2: 118.
https://doi.org/10.3390/cryst11020118