# Spectrometer-Free Graphene Plasmonics Based Refractive Index Sensor

^{*}

## Abstract

**:**

^{−1}figure of merit in the optical mode of operation and a 713.2 meV/RIU sensitivity, a 246.8 RIU

^{−1}figure of merit in the electrical mode of operation. This performance outlines the optimized operation of this spectrometer-free sensor that simplifies its design and can bring terahertz sensing one step closer to its practical realization, with promising applications in biosensing and/or gas sensing.

## 1. Introduction

## 2. Results

#### 2.1. Sensing Mechanism and Sensor Design

^{2}/(V·s), which is a reasonable value for graphene with good quality [30] (see Section 3.3.) The metallic reflector is chosen to be made from gold with permittivity calculated using the Drude’s model [31].

#### 2.2. Performance Analysis and Simulation Results

^{®}) with designed $\beta $ and $p$. The sensitivities from both equations and simulations are obtained when the relative permittivity of the analyte medium is varied around the central value 3.5. Practically, in Equations (3) and (5) both $g$ and $s$ cannot be solutions expressed in a direct analytical formula. Therefore, during the solving process, the values of both sides of Equation (A1) are calculated with different sets of variables. By finding the minimum error between both sides, the relation between variables can be acquired numerically.

^{−1}, respectively. For the electrical mode of operation, the sensitivity and FoM are 713.21 meV/RIU and 246.8 RIU

^{−1}, respectively. Figure 6c plots the electric field intensity of the device at the resonance peak, shown for one period of the grating (there are four wave nodes corresponding to two wavelengths of the GSPR mode). When the device is operating at the resonance peak, the fields are, as expected, highly confined within a narrow area around the graphene layer due to the property of GSPR.

^{−1}, and 713.4 meV/RIU, 250.5 RIU

^{−1}, respectively, which are very close to those calculated from Figure 6a,b. Additionally, from Figure 7c, it can be clearly seen that the finite structure device has the same optical mode as the infinite periodic one. Generally, it can be concluded that the device with finite periodic structure operates in the same way as the infinite periodic structure and possesses the same performance (in terms of sensitivity and FoM.)

## 3. Discussion

#### 3.1. Geometrical Optimization

#### 3.2. Performance Comparison

^{−1}. In [36], the performance of the proposed sensor is not outstanding, however, it is insensitive to the incident wave polarization. In [37,38,39], the sensors come with good sensitivity but relatively low FoM.

#### 3.3. Mobility, Gating, and Advantages of the Sensor

^{2}/(V·s) was assumed in this work [27,40,41,42], however, if graphene features a relatively lower mobility (lower quality), the SPR will be degraded and may exhibit a broader reflection dip, which leads to a lower FoM. Also, the size of the gratings has an effect on the device performance. Since the SPR mode is highly confined around the graphene layer, the height of the grating will not have an obvious effect on the device operation. However, the width of the grating will influence the SPR mode. Therefore, the grating width used in the simulations is well traded-off between device performance and nanofabrication technology limits.

- (1)
- Spectrometer-free: The design does not require the use of a spectrometer, which relaxes the design and reduces its complexity, via the electrical mode of operation.
- (2)
- Higher performance: The GSPR sensor possesses simultaneously a high FoM and sensitivity. This equips the proposed design with a higher precision in detection, in the terahertz range.
- (3)
- Geometry robustness: The operation is insensitive to the height of the grating, which gives more flexibility in the design and robustness regarding the imperfections in nanofabrication.
- (4)
- Dynamic tunability: The spectral location of the EM absorption peak can be tuned by changing ${\mu}_{c}$. This shows that the same device can be optimized for different frequency ranges (with reconfigurability, by use of electrostatic biasing).

## 4. Conclusions

^{−1}, respectively were demonstrated. On the other hand, for the electrical mode of operation, these were 720.75 meV/RIU and 287.2 RIU

^{−1}. In addition, a finite-size device (consisting of only 40 unit-cells) is also characterized and it is subsequently shown that it has almost the same performance as the infinitely periodic structure. The robustness of the sensor with respect to geometric variations was further investigated, and the results show that this device can be further optimized with a smaller width but at the expense of the complexity of the fabrication process. Last but not least, a comparison of the proposed sensor’s capabilities with similar works is drawn, and it is demonstrated numerically that the combination of the four advantages (mentioned in the discussion section) significantly increases the sensitivity and FoM of the proposed device in comparison with the listed designs.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Dispersion Relation and Graphene Conductivity

^{−7}+$6.06\times $10

^{−4}i S/m under our working frequency of 46.05 THz with ${\mu}_{c}=1500\mathrm{meV}$) represent intra-band conductivity, inter-band conductivity, and total conductivity of graphene, respectively. ${e}_{0}$ is the elementary electronic charge, ${\mu}_{c}$ is the chemical potential of graphene, $\mu $ is the electron mobility of graphene, ${\u03f5}_{0}$ and ${\u03f5}_{g}$ (−2.$36\times $10

^{2}− 1.1$0\times $10

^{−1}i under our working frequency of 46.05 THz with ${\mu}_{c}=1500\mathrm{meV}$) are the vacuum permittivity and graphene relative permittivity, respectively. With both Equations (A1) and (A2), the GSPR properties can be analyzed. From a qualitative point of view, some simple conclusions can be made about how the conductivity and effective permittivity of graphene sheet may affect the sensor performance. If we keep the same resonance mode (i.e., fixed $\beta $ defined by the gratings) and consider that $\beta $ is much larger than ${k}_{0}$, we can conclude that ${k}_{1}$, ${k}_{g}$, ${k}_{2}$ remain almost unchanged, then the effective permittivity of the graphene sheet ${\u03f5}_{g}$ has to remain unchanged. Therefore, a higher conductivity will lead to a higher resonant frequency. Also, a higher conductivity will lead to a resonance with less damping. (i.e., smaller FWHM).

## Appendix B. Device Performance Under Finite Band Excitation

Mode of Operation | Sigma (THz) | FoM (RIU^{−1}) | Sensitivity |
---|---|---|---|

Optical | 0.02 | 149.7 | 1557.1 nm/RIU |

0.01 | 204.8 | 1556.6 nm/RIU | |

Electrical | 0.02 | 148.9 | 714.5 meV/RIU |

0.01 | 204.1 | 714.4 meV/RIU |

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**Figure 1.**Cross-view of the proposed sensor structure, with its different geometrical parameters, and the normally incident plane wave excitation (polarization and wavevector.).

**Figure 3.**Simulated (triangles and stars) and analytically calculated (solid lines) sensitivity versus graphene chemical potential for (

**a**) the optical mode of operation and (

**b**) the electrical mode of operation.

**Figure 4.**Simulated results versus graphene chemical potential for (

**a**) FWHM in the optical mode of operation, (

**b**) FWHM in the electrical mode of operation, and (

**c**) FoM in both optical and electrical mode of operations.

**Figure 5.**(

**a**) Absorption spectrum for different scenarios. (

**b**) Reflection coefficient (${S}_{11}$) for the device with reflector working in the optical sensing mode with ${\mu}_{c}=700\mathrm{meV}$ for different permittivities. (

**c**) Reflection coefficient for the device with reflector working in the electrical sensing mode with the same ${\mu}_{c}=700\mathrm{meV}$.

**Figure 6.**(

**a**) Reflection coefficient for the sensor (with reflector) working in the optical mode of operation with ${\mu}_{c}=1500\mathrm{meV}$. (

**b**) Reflection coefficient for the same device working in the electrical mode with the same chemical potential. (

**c**) Simulated electric field intensity at GSPR peak for ${\u03f5}_{1}=3.5$ (${n}_{1}=1.87$).

**Figure 7.**(

**a**) SCS for a device with 40 unit-cells and reflector working in the optical mode of operation with ${\mu}_{c}=1500\mathrm{meV}$. (

**b**) Scattered power density for the same device working in the electrical mode of operation with the same chemical potential. (

**c**) Simulated electric field intensity at the GSPR peak when ${\u03f5}_{1}=3.5$ (${n}_{1}=1.87$).

**Figure 8.**(

**a**) Reflection coefficient for the device with infinite periodic structure and reflector working in the optical mode of operation with ${\mu}_{c}=1500\mathrm{meV}$ under large medium refractive index change. (

**b**) Peak wavelength versus corresponding medium refractive index and its linear fitting.

**Figure 9.**Performance comparison of FoM and sensitivity of the sensor proposed in this study to similar other sensors. The star denotes the current study.

**Table 1.**Designed sensor geometry (See Figure 1).

Geometry Dimensions | Value (nm) |
---|---|

p | 384.8 |

w | 40 |

h | 50 |

$\delta $ | 50 |

d | 1244.4 |

Grating Width (nm) | FWHM (nm) | Sensitivity (nm/RIU) |
---|---|---|

40 | 6.3 | 1566.03 |

26 | 4.5 | 1603.8 |

18 | 3 | 1641.5 |

Grating Height (nm) | FWHM (nm) | Sensitivity (nm/RIU) |
---|---|---|

50 | 6.3 | 1566.03 |

25 | 6.25 | 1566.04 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, L.; Farhat, M.; Salama, K.N.
Spectrometer-Free Graphene Plasmonics Based Refractive Index Sensor. *Sensors* **2020**, *20*, 2347.
https://doi.org/10.3390/s20082347

**AMA Style**

Zhang L, Farhat M, Salama KN.
Spectrometer-Free Graphene Plasmonics Based Refractive Index Sensor. *Sensors*. 2020; 20(8):2347.
https://doi.org/10.3390/s20082347

**Chicago/Turabian Style**

Zhang, Li, Mohamed Farhat, and Khaled Nabil Salama.
2020. "Spectrometer-Free Graphene Plasmonics Based Refractive Index Sensor" *Sensors* 20, no. 8: 2347.
https://doi.org/10.3390/s20082347