#### 2.1. Sensing Mechanism and Sensor Design

The sensing mechanism of the proposed sensor in this work is based on graphene surface plasmon resonance (GSPR). Under the SPR, a strong electromagnetic field, highly confined around graphene leads to a high absorption peak. Thanks to the GSPR mode’s sensitivity to small changes of the surrounding optical properties, the proposed graphene sensor may sense the refractive index of the neighboring medium.

Consider a typical GSPR based structure, e.g., a thin layer of graphene located in the middle and covered by two dielectric materials from the top and the bottom (these materials can be different.) By assuming a simple plane wave solution form and applying boundary conditions, we can obtain the dispersion relation of this sandwiched structure [

28] (See

Appendix A.) Also, graphene may be treated as a current sheet with corresponding boundary conditions, when the dispersion relation is derived [

29].

Assuming that the medium to be characterized is on top of the graphene layer, Equation (A1) (See

Appendix A) can be generalized into the dispersion relation with the permittivity (or equivalently the refractive index) of the analyte medium

${\u03f5}_{1}$ (

${n}_{1}$) as a parameter:

In Equation (1), $f$ is a function of $\beta $ (the wavenumber along the propagation direction) with fixed ${\u03f5}_{1}$. Equation (1) summarizes the basic sensing mechanism of a simple GSPR sandwich structure, whereas, a change of the medium’s refractive index leads to different dispersion relations.

Generally, coupling techniques are required to overcome the wavenumber mismatch between incident electromagnetic (EM) wave and GSPR mode. In our study, we make use of gratings in order to realize the required coupling [

28]. Owing to phase-matching, the grating adds an extra term to the propagation constant, i.e.,:

In Equation (2),

${k}_{\mathrm{inc}}$ is the incident wavenumber,

$\theta $ is the angle of incidence,

$n$ is an integer number, and

$p$ is the period of the grating. In our study, a normally incident plane wave is used (i.e.,

$\theta =0$). Once the period and integer number

n are chosen, the propagation constant is fixed, which leads to a specific GSPR mode. With a fixed

$\beta $, we can rewrite Equation (1) as:

Equation (3) shows the direct relationship between the incident wave frequency $\omega $ and the sensed medium permittivity ${\u03f5}_{1}$, which is the basic sensing mechanism for this sensor to work in the optical mode of operation, that requires a spectrometer.

Thanks to the tunability of graphene conductivity, one has extra free variables and parameters in the dispersion relation of the GSPR structure. In [

13], the conductivity and permittivity of a graphene sheet are expressed, leading to the relation between quantities e.g., mobility, chemical potential, and permittivity (see

Appendix A). Then, the relationship between graphene chemical potential (

${\mu}_{c}$) and its permittivity can be written as:

Here

$h$ is a function of both

$\omega $ and

${\mu}_{c}$. If we have a monochromatic incident plane wave, we can rewrite Equation (3) by using Equation (4) as:

In Equation (5), $s$ is a function of ${\u03f5}_{1}$ with fixed $\beta $ and $\omega $. Equation (5) shows the possibility of spectrometer-free sensing, using the tunability of graphene through its chemical potential. In fact, ${\mu}_{c}$ can be tuned by different techniques, e.g., electrostatic biasing and chemical doping. If the former biasing is used (as done in this work) to tune the chemical potential, this sensor operates in the electrical mode of operation and leads to spectrometer-free sensing.

The graphene sensor structure is shown in

Figure 1. This sensor consists of a grating coupled to a GSPR structure with a reflector, at the bottom. The basic structures of this sensor (responsible for the sensing mechanism) are the SPR sandwiched structures (the analyte medium on the top, a graphene layer, and a grating in the middle with a dielectric substrate, and the reflector on the bottom). The reflector located under the GSPR structure will reflect the transmitted energy back so that a higher absorption peak will appear which is easier to be detected by photodetectors. The geometrical parameters of this sensor are defined as, grating period

$p$, grating height

$h$, grating width

$w$, and distance between the reflector and the graphene layer

$d$. The thickness of the reflector will not affect the device performance significantly, as long as it is thick enough to reflect the transmitted energy back (the thickness

$\delta $ in this study is chosen to be 50 nm), therefore, it can be designed flexibly according to the fabrication process capabilities. This device is designed to work for a normally-incident, transverse-magnetic (TM) polarized electromagnetic wave, as schematized in

Figure 1.

In this work, we first fix the grating period

$p\approx 2{\lambda}_{\mathrm{GSPR}}=384.8\mathrm{nm}$ and GSPR mode propagation constant

$\beta \approx 70{k}_{0}$ (for

${\mu}_{c}=700\mathrm{meV}$), which corresponds to a feasible surface plasmonic resonance mode with regards to the physical dimensions of the structures to be used. Furthermore, the grating width

$w$ and height

$h$ are set to 40 nm and 50 nm, respectively. A summary of the designed dimensions is shown in

Table 1. The material properties (permittivity, mobility, etc.) of the designed devices are set with reasonable values from the common nanofabrication process. For the dielectric substrate, the relative permittivity is set to 3.5. As for graphene, mobility is set to 50,000 cm

^{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

To quantitatively evaluate the performance of the sensor, sensitivity, figure of merit, limit of detection and specificity are commonly used. Since our sensor is a general RI sensor that does not involve molecular detection, specificity is not applicable in this study.

In the optical mode of operation, we use the common definition of sensitivity, figure of merit (

FoM) and detection limit (

DL) [

32] for optical RI sensor, i.e.,:

where the sensitivity

$S$ is defined as the ratio between the peak shift in wavelength and its corresponding refractive index change and the

$FoM$ is defined as the sensitivity over the peak full width at half maximum (

$FWHM$). From the very definition of

FoM, one can see that it is easier to distinguish between two resonance peaks under a certain RI change provided that the sensor has a higher

FoM. Therefore,

FoM is a critical characteristic related to the resolution of the sensor and a high

FoM is desirable for enhanced sensitivity. Except from

FoM, the resolution of the spectrometer and the noise parameters also determine the resolution of the sensor [

32], which describe the ability to accurately determine the spectral shift. With the resolution of the sensor, we can define the detection limit of the sensor which equals the resolution divided by the sensitivity.

As this graphene sensor has an additional spectrometer-free working mode (i.e., the electrical mode of operation), we use a similar approach to define the counterpart sensitivity,

FoM and

DL in that scenario, i.e.,:

In this study, only calculations and simulations are done to examine the sensor performance. Hence, the noise level (including noise from the excitation laser, photodetector and auxiliary electric circuit) and spectral resolution (including the resolution of the tunability of excitation laser and the spectrometer) may not be acquired without an experimental implementation. Moreover, without loss of generality, only sensitivity and figure of merit will be calculated to evaluate the performance of the sensor.

After setting up the basic definitions, we can start by analyzing the sensitivity under different graphene chemical potentials by analytical means [solving Equations (3) and (5)] and finite-elements based simulations (using COMSOL Multiphysics^{®}) 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.

For the simulations, we treat the whole sensor structure as a repeated sequence of unit-cells (shown in

Figure 2). To characterize the whole structure, we make use of periodic boundary condition (PBC) at both sides of the unit-cell. In addition, the top and bottom boundaries are set as port 1 and port 2, respectively and the plane wave excitation is enforced at port 1. The scattering parameters (

S-parameters) are calculated to evaluate the relationship between incident and reflected wave intensity. The simulations are performed both with reflector and without reflector, respectively. For the simulations with a reflector, a preliminary simulation is done in order to optimize the distance between graphene and the reflector so that we obtain the highest absorption peak.

As can be seen from

Figure 3a, when

${\mu}_{c}$ is varied from 700 meV to 1600 meV, the calculated optical mode of operation sensitivity for the ideal GSPR structure decreases. Meanwhile, the simulated optical mode of operation sensitivity for the device structure (GSPR and grating) shows the same trend as the calculated one. However, the existence of gratings slightly changes the GSPR mode, which leads to the small discrepancy between simulations and analytical results by solving Equation (A1). Besides, the results with and without the metallic reflector are identically the same, which shows that the inclusion of the reflector will not evidently influence the sensitivity of the device. The sensitivity results for the electrical mode of operation are shown in

Figure 3b. On the contrary, the sensitivity for the electrical mode of operation increases with increasing chemical potential. The simulation results also have the same trend as the calculated results with deviation due to the existence of gratings. In order to have perfect agreement between the simulation results and analytical results, one needs to add the effect of the grating, but this will make the problem insolvable analytically. However, the results of

Figure 3 illustrate the sensing mechanism of the proposed device and give the trend of the sensitivity versus

${\mu}_{c}$.

Afterwards, all the simulated sensitivities and

FWHMs,

FoMs for both optical mode of operation and electrical mode of operation under different graphene chemical potentials are calculated and shown in

Figure 4. For both modes, the

FoMs increase linearly with the graphene’s chemical potential. Different from the sensitivities for both modes, the

FWHMs and

FoMs become slightly deteriorated due to the existence of the reflector.

The

FWHM is also simulated versus different graphene chemical potentials.

Figure 4a shows that the

FWHM decrease as

${\mu}_{c}$ increases for both cases, with and without reflector. Different from the optical mode of operation, as can be seen from

Figure 4b, the

FWHM of the electrical mode of operation is almost independent of the change of

${\mu}_{c}$.

Figure 5a shows the absorption spectrum for different scenarios: for a structure with graphene’s chemical potential

${\mu}_{c}=700\mathrm{meV}$ and no reflector, the absorption peak corresponding to the design

$\beta $ is denoted by ①. For the case of

${\mu}_{c}=1500\mathrm{meV}$ and no reflector, the corresponding peak is ②. After including the reflector, the peak ② increases dramatically and transforms into peak ③, which almost reaches perfect-absorption [

33]. Generally, by adding the gold reflector and simultaneously by increasing

${\mu}_{c}$, we can realize stronger absorption. The simulation results after adding the reflector to the device in the case of

$700\mathrm{meV}$ chemical potential are shown in

Figure 5b,c, for the optical and electrical mode of operations respectively. Since we add the reflector (transmission through port 2 is 0), the reflection equals the total incident power subtracted by the absorption, therefore, the reflector results in stronger absorption and converts the absorption peak to a reflection dip, which is easier to measure practically.

Summarizing all the observations above, a higher graphene’s chemical potential leads to better FoMs for both optical mode of operation and electrical mode of operation and better sensitivity for the electrical mode of operation, while decreasing the sensitivity for the optical mode of operation. (i.e., for the optical mode of operation, the choice of chemical potential will lead to a trade-off between sensitivity and FoM.) The inclusion of the reflector will not affect the sensitivity and FoM alike, but it helps to realize stronger absorption peak (or weak reflection dip), which may reduce the constraint on sensitivity of the detecting devices such as photodiodes. Therefore, for the final design, graphene chemical potential is chosen to be 1500 meV with a gold reflector on the bottom structure.

The simulation results of the chosen design, based on the abovementioned assumptions, are shown in

Figure 6a,b where the reflection coefficient of the optical mode of operation and the electrical mode of operation are given, respectively. It is clear that the minimum reflection coefficients for both cases are nearly zero (perfect-absorption), which are even better than the results of

Figure 5b,c. The perfect-absorption is realized by optimizing the distance between the graphene and reflector (here

$d=1244$ nm). Next, we can calculate the sensitivity and

FoM from

Figure 6a,b. The calculated sensitivity and

FoM for the optical mode of operation are 1566.03 nm/RIU and 250.6 RIU

^{−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.

To further investigate the feasibility of this device for a realistic configuration, an additional simulation is performed with a finite device structure (finite number of unit-cells of the grating). In this scenario, the device consists of only 40 units. A background normally-incident plane wave impinges on the device and the total scattered power density is calculated (by integrating the Poynting vector in the far-field over a closed surface enclosing the structure). The simulation results for the scattering cross-section (SCS) are shown in

Figure 7.

From

Figure 7a, it can be observed that with the finite structure, the peaks in the optical mode operation are slightly redshifted in comparison to those of

Figure 6a (i.e., for the infinite periodic structure), while the peaks in the electrical mode of operation shown in

Figure 7b are almost the same as those of

Figure 6b. The sensitivities and

FoMs calculated from

Figure 7a,b are 1561.5 nm/RIU, 269.2 RIU

^{−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.)

In order to investigate the sensor’s performance under significant RI changes, an additional simulation is performed with large analyte medium refractive index change.

Figure 8a depicts the peak-shift for different medium RIs. As can be seen, when the RI increases, the resonance peak redshifts (i.e., towards lower frequencies). Further, when the RI has a value that is very far from the original design value (

n = 1.87), the absorption strength drops, because the reflector position is fixed and optimized to the original design. In other words, we can optimize the reflector distance according to different sensing cases.

Figure 8b shows the relation between the peak wavelength and the corresponding medium refractive index. This sensor can retain a relative linear performance spanning different RIs. The sensitivity under large RI change is 1286.68 nm/RIU according to the linear fitting, shown in

Figure 8b.