**1. Introduction**

Surface plasmon resonance (SPR) biosensors are based on the principle of surface plasmon generation, which is sensitive to the changes in the refractive index of the sensor surface structure along with the sensing medium. Surface plasmon (SP) is the collective oscillation of free electrons at the surface of a metal conduction band, which is excited by the electromagnetic field at the metal-dielectric interface. The SP exponentially decays into the surrounding media, and the decay depth is usually in the range of hundred nanometers, which is the key reason why the sensor surface of SPR-based sensors is in the nanometer range. Because the SP penetration depth is in the nanometer range makes them sensitive enough to detect changes in the refractive

index in the milli to micro-range, making them the suitable and preferable sensors for detection of biomolecules in the micro-to-pico concentration levels. A highly sensitive biosensor is defined as one that can provide a measurable output signal for even the smallest changes in the refractive index of the sample under testing, and is often desirable.

#### **1.1 Basic SPR sensor system**

A basic (conventional) SPR sensor surface consists of a thin film, which interacts with the biomolecule material under testing also referred to as sensing medium. A change in the biomolecule concentration causes a local change in the refractive index near the metal surface of the SPR sensor, which results in a change in the propagation constant of the excitation optical wave. This change in the propagation constant can be optically measured [1]. Many different metals such as gold (Au), silver (Au), copper (Cu), aluminum (Al), etc. have been used to form the sensor surface and support the excitation of the surface plasmon usually dictates that the thickness of these metal films is in the 45–55 nm range. **Figure 1a** shows a typical SPR setup in a Kretschmann configuration. A beam of *p*-polarized laser light is incident on the metal-dielectric interface and the intensity of the reflected light is detected as a function of the angle of incidence. At a particular angle of incidence, called the surface plasmon resonance angle, *θspr* a resonance condition is satisfied and the surface plasmon is excited. The relation that governs the excitation of the surface plasmon is given as [2]:

$$\theta\_{spr} = \sin^{-1}\left(\frac{\mathbf{1}}{n\_p}\sqrt{\frac{n\_m^2 n\_d^2}{n\_m^2 + n\_d^2}}\right),\tag{1}$$

where *np*, *nm*, and *nd* are the refractive indices of the prism, metal, and dielectric (sensing) medium, respectively. The excited surface plasmon is observed as a sharp drop in reflectance as shown in **Figure 1b**. As the refractive index of the sensing medium changes the resonance angle changes. SPR-based sensors essentially detect the shifts in the resonance angle for different materials under test. The "angular sensitivity" of an SPR setup is commonly defined as the ratio of the shift in the SPR angle, Δ*θspr* to a given change in the refractive index, Δ*n* of the sensing medium and is given as [3]:

$$\mathcal{S}\_n = \frac{\Delta \theta\_{spr}}{\Delta n}. \tag{2}$$

Along with the parameter, *Sn*, sensitivity, two other parameters figure of merit (FOM), and quality factor (QF) are used to evaluate the performance of an SPR system and are defined as [2]:

$$FOM = \frac{\Delta\theta\_{gr}}{FWHM} \tag{3}$$

$$QF = \frac{S\_n}{FWHM} \tag{4}$$

*Application of Electric Bias to Enhance the Sensitivity of Graphene-Based Surface Plasmon… DOI: http://dx.doi.org/10.5772/intechopen.106556*

**Figure 1.**

*(a) A SPR system layout in Kretschmann configuration showing Au-Gr layers as the sensor surface. (b) A typical normalized reflectance curve showing the surface plasmon excitation as a drop in reflected intensity at 37.59 deg incident angle.*

## **1.2 SPR angle dependence on the refractive indices** *np***,** *nm***, and** *nd*

$$\theta\_{spr} = \sin^{-1}\left(\frac{\mathbf{1}}{n\_p}\sqrt{\frac{n\_d^2 n\_m^2}{n\_d^2 + n\_m^2}}\right) \tag{5}$$

where, *np*, *nd*, and *nm* are the refractive indices of the prism, dielectric (analyte), and metal. Given this relation, we can look at the variation of the resonance angle, with respect to the changes in the refractive index of the metal layer.

$$\frac{d\theta\_{spr}}{dn\_m} = \frac{n\_d^3}{n\_d^2 + n\_m^2} \times \frac{1}{\sqrt{n\_p^2 \left(n\_d^2 + n\_m^2\right) - n\_d^2 n\_m^2}}\tag{6}$$

Given the above relation, we can see that the change in the resonance angle is a function of the refractive indices of the dielectric (sample under test), prism material, and also the metal. Now, we can also find the values of *np*, *nd*, and *nm* for which the maximum resonance angle occurs when *dθSPR=dnm* ¼ 0. This gives us two possible scenarios when either of the two terms in the equation can be zero.

#### *1.2.1 Case #1*

Consider the first term of Eq. (2),

$$\frac{n\_d^3}{n\_d^2 + n\_m^2} = 0,\tag{7}$$

this can happen when

$$n\_d = 0; n\_d^2 + n\_m^2 \to \infty,\tag{8}$$

the condition *nd* ¼ 0 dictates that the dielectric material (sample under testing) needs to have its refractive index to be zero, which is not a practical case. The condition, *n*<sup>2</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> *<sup>m</sup>* ! ∞, combined with the earlier condition *nd* ¼ 0, dictates that:

$$n\_d \to 0; n\_m > n\_d; n\_m \to \infty. \tag{9}$$

*1.2.2 Case #2*

Now, consider the second term of Eq. (2),

$$\frac{1}{\sqrt{n\_p^2 \left(n\_d^2 + n\_m^2\right) - n\_d^2 n\_m^2}} = \mathbf{0},\tag{10}$$

this term shows that the maximum resonance angle occurs when, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *n*2 *<sup>p</sup> n*<sup>2</sup> *<sup>d</sup>* þ *n*<sup>2</sup> *m* � � � *<sup>n</sup>*<sup>2</sup> *dn*2 *m* q ! ∞. This stipulates that to achieve maximum resonance angle, the product *n*<sup>2</sup> *<sup>p</sup> n*<sup>2</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> *m* � � ! <sup>∞</sup> while *<sup>n</sup>*<sup>2</sup> *dn*2 *<sup>m</sup>* ! 0. These two conditions combined with the conditions in Eq. (5) indicate that the most favorable conditions to achieve the maximum plasmon resonance angle for better sensitivity are:

$$n\_d \to 0; n\_m \to \infty; n\_p \to \infty. \tag{11}$$

Up till now, we have used *nm* to represent the refractive index of the metal layer used to excite the surface plasmon, however, there can be many possible combinations of multi-layers of different materials, and the *nm* would be the effective refractive index of all these combined multi-layers used to excite the surface plasmon. As has been the case in the past decades where many researchers have proposed varied types of sensor surfaces with combinations of Au, Ag, Al, Si, SiO2, MoS2, Black Phosphorus (BP), etc., with varying thicknesses optimized for high sensitivity.

#### **1.3 Ultra-sensitive SPR sensor design**

Many researchers have proposed and shown that other metals, or combinations thereof, can be used to improve the sensitivity of the SPR system. Many methods have been demonstrated to improve the SPR biosensor sensitivity, such as, utilizing metal nanoparticles and nanoholes [4], metallic nanoslits [5], and colloidal gold nanoparticles [6] in buffered solutions. In the conventional SPR system, gold is commonly preferred due to its resistance to oxidation and corrosion, however,

#### *Application of Electric Bias to Enhance the Sensitivity of Graphene-Based Surface Plasmon… DOI: http://dx.doi.org/10.5772/intechopen.106556*

biomolecules adsorb poorly on gold, causing less interaction with the metallic surface and this limits the sensitivity of the SPR biosensor. One of the attractive alternatives to improve sensitivity by increasing the adsorption of biomolecules on the gold surface is to use graphene (Gr). Among these materials, graphene (Gr) due to its carbon composition has proven to be a more preferable material due to its affinity and absorptive nature to hydrocarbons. This not only helped in the binding process of biomolecules under testing but also helped improve the sensitivity compared to the conventional system [7, 8]. Gold-graphene SPR biosensors have been shown to stably adsorb biomolecules with carbon-based ring structures, causing an increase in sensitivity to changes in refractive index of the biomolecules, leading to increased detection and identification [9, 10]. However, the precise control over the geometry and the optical properties of nanostructures is challenging. Many methods were demonstrated to improve the sensitivity, by either utilizing metal nanoparticles and nanoholes [4], metallic nanoslits [5], and colloidal gold nanoparticles [6] in buffered solutions. However, still improvements can be made using other techniques or methods. Recently, we proposed an idea to increase an SPR biosensor sensitivity by changing the refractive index of the graphene layer, which is a part of the sensor surface by the application of electrical bias voltage across metal-graphene system [11]. Our numerical investigation has shown that apart from providing an increase in sensitivity compared to other techniques and sensor surface structures, we also realized that this technique can also address another issue: control over the optical properties of the sensor surface [12].

#### **1.4 Novelty and necessity of the present idea**

In a typical SPR sensor setup, for a particular type of sensing medium, a particular sensor surface configuration is used. This means that for a different type of sensing medium and for an enhanced sensitivity, a different sensor surface may be required. This is combined with the fact that once a sensor surface configuration for use in the sensor is selected, it would not be possible to make changes to it, and the only option would be to switch sensor surface with another one. For example, if a gold-graphene sensor surface with certain thickness combination is chosen as sensor surface, but a particular sensing medium requires a different thickness combination that could result in better sensor sensitivity output, then the only option is to switch out the sensor surface. The reason for such a need to switch the sensor surface film is based on the fact that the effective refractive index of the sensor surface is constant for a particular wavelength of the plasmon excitation light source. We believe that if the sensor surface properties could be tuned dynamically, we could eliminate the need for a different sensor surface combination as would be required for different sensing media. As mentioned earlier, we have already presented our numerical results on the application of electrical bias across gold-graphene sensor surface structure in Ref [11], and gold-SiO2-graphene in Ref [12], and here we extend our numerical investigation to the Au-MoS2-Gr sensor surface structure.

## **2. Electrical bias across the metal-Gr system**

Graphene is a thin two-dimensional layer of graphite, an allotrope of carbon, where the carbon atoms are arranged in a honeycomb lattice structure. The valence and conduction bands in graphene touch at six points known as the Dirac points,

where the chemical potential of graphene is located for undoped samples. The position of the chemical potential can easily be shifted above or below the Dirac point, thus allowing the carrier concentration in the material to be tuned by applying an electrical voltage [13]. In this Au-Gr system, the carrier concentration in graphene, *ng* due to an applied voltage, *Vg* can be calculated using the relation [14]:

$$n\_{\rm g} = \frac{V\_{\rm g} \varepsilon\_0 \varepsilon\_r}{q d\_{sub}}\tag{12}$$

where *<sup>ε</sup>*<sup>0</sup> <sup>¼</sup> <sup>8</sup>*:*<sup>85</sup> � <sup>10</sup>�12*F=<sup>m</sup>* is the permittivity of vacuum, *<sup>ε</sup><sup>r</sup>* is the relative permittivity of the substrate (50 nm thick gold film), *q* is the electron charge, and *dsub* is the substrate thickness. Based on the carrier concentration of the system, the chemical potential *μ<sup>c</sup>* can be calculated using [15]:

$$
\mu\_c = \hbar v\_f \sqrt{\pi n\_\text{g}} \tag{13}
$$

where ℏ is the reduced Planck's constant and *vF*≈10<sup>6</sup> cm/s is the Fermi velocity. Graphene optical conductivity, *σ* is related to the intra-band electron-photon scattering, *σintra* and the inter-band electron transition conductivity, *σinter* as function of radiation frequency, *ω* and given as:

$$
\sigma(a) = \sigma\_{\text{intra}}(a) + \sigma\_{\text{inter}}(a) \tag{14}
$$

which can be calculated using the Kubo formula [16]:

$$\sigma\_{\rm intra}(\boldsymbol{\alpha}) = i \frac{q^2}{\pi \hbar (\boldsymbol{\alpha} + i\boldsymbol{\tau}^{-1})} \left[ \mu\_c + 2K\_B T \times \ln \left\{ e^{\left(\frac{-\mu\_c}{K\_B T}\right)} + 1 \right\} \right] \tag{15}$$

$$\sigma\_{inter}(\boldsymbol{\alpha}) = i \frac{q^2}{4\pi\hbar} \ln\left[\frac{2|\mu\_c| - \hbar(\boldsymbol{\alpha} + i\boldsymbol{\tau}^{-1})}{2|\mu\_c| + \hbar(\boldsymbol{\alpha} + i\boldsymbol{\tau}^{-1})}\right] \tag{16}$$

where *KB* is the Boltzmann's constant, *<sup>T</sup>* is the temperature, *<sup>τ</sup>* <sup>¼</sup> *<sup>μ</sup>cmu=qv*<sup>2</sup> *<sup>F</sup>* is the momentum relaxation time, and *mu* <sup>¼</sup> <sup>10</sup><sup>4</sup>*cm*<sup>2</sup>*=<sup>V</sup>* � *<sup>s</sup>* is the impurity-limited direct current mobility.

The complex conductivity of graphene may be expressed as:

$$
\sigma(\alpha) = \sigma\_R(\alpha) + i\sigma\_I(\alpha) \tag{17}
$$

where *σR*ð Þ *ω* and *σI*ð Þ *ω* are the real and imaginary parts of the graphene conductivity and can be derived from Eqs. (5)–(7) and given as:

$$\sigma\_R(\boldsymbol{\alpha}) = \frac{\boldsymbol{\pi}^{-1}\boldsymbol{q}^2}{(\boldsymbol{\alpha}^2 + \boldsymbol{\tau}^{-2})\pi\hbar^2} \times \left[\mu\_c + 2k\_B T \times \ln\left\{e^{\left(\frac{-\kappa}{k\_B T}\right)} + 1\right\}\right],\tag{18}$$

$$\sigma\_{l}(\boldsymbol{\omega}) = \frac{\alpha q^{2}}{(\boldsymbol{\alpha}^{2} + \boldsymbol{\tau}^{-2})\pi\hbar^{2}} \times \left[\mu\_{c} + 2k\_{B}T \times \ln\left\{e^{\left(\frac{\boldsymbol{\omega}\_{c}}{k\_{B}T}\right)} + 1\right\}\right] + \frac{q^{2}}{4\pi\hbar}\ln\left[\frac{2|\mu\_{c}| - \hbar(\boldsymbol{\alpha} + i\boldsymbol{\tau}^{-1})}{2|\mu\_{c}| + \hbar(\boldsymbol{\alpha} + i\boldsymbol{\tau}^{-1})}\right].\tag{19}$$

*Application of Electric Bias to Enhance the Sensitivity of Graphene-Based Surface Plasmon… DOI: http://dx.doi.org/10.5772/intechopen.106556*

The thickness of a single graphene layer is 0.34 nm, and for a given number of graphene layers with thickness, *dGr* the real *nGr*,*<sup>R</sup>* and imaginary *nGr*,*<sup>I</sup>* parts of the graphene refractive index can be expressed as:

$$m\_{Gr,R} = \sqrt{\frac{\sqrt{\left(\sigma\_I - a\varkappa\_0 d\_{Gr}\right)^2 + \sigma\_R^2} - \left(\sigma\_I - a\varkappa\_0 d\_{Gr}\right)}{2a\varkappa\_0 d\_{Gr}}}\tag{20}$$

$$n\_{Gr,l} = \sqrt{\frac{\sqrt{\left(\sigma\_I - a\epsilon\epsilon\_0 d\_{Gr}\right)^2 + \sigma\_R^2} + \left(\sigma\_I - a\epsilon\epsilon\_0 d\_{Gr}\right)}{2a\epsilon\epsilon\_0 d\_{Gr}}}\tag{21}$$

From Eqs. (3)–(13), it is evident that the refractive index of the metal-graphene sensor surface system can be varied by the application of an electrical bias voltage. Using the N-layer model [1], the reflectance of the incident light in the Kretschmann configuration of SPR setup was calculated for this Au-Gr sensor surface system. **Figure 2** shows a top-view kind of perspective to how the SPR angle shifts at different levels of chemical potential. Using chemical potential as the basis to evaluate the shift in the SPR angle for a change in the sample refractive index, we defined a new measure of sensitivity *Sμ<sup>c</sup>* , which is the ratio of the change in the SPR angle due to a change in the chemical potential applied across the metal-graphene system [11],

$$\mathcal{S}\_{\mu\_{\epsilon}} = \frac{\Delta \theta\_{spr,\mu\_{\epsilon}}}{\Delta \mu\_{\epsilon}} \tag{22}$$

#### **Figure 2.**

*A top view kind of perspective showing how the SPR curves are changing wrt the chemical potential. With prism material as SF10 (np = 1.723), the sensor surface configuration is: Au (30 nm thick), MoS2 (3 layers), Gr (10 layers) with μ<sup>c</sup> at 0.0 eV, and the incident radiation wavelength λ = 632.8 nm.*

along with the figure of merit wrt chemical potential as:

$$FOM\_{\mu\_c} = \frac{\Delta\theta\_{spr,\mu\_c}}{FWMHM} \tag{23}$$

#### **2.1 Modeling of the multi-material sensor surface structure**

In this section let us look at how we can model the presence of more than one material such that it can be biased using an electrical voltage. To accomplish this, we can still consider the Au film as the base on which the graphene layers are deposited, as discussed in the Section 2, and the other material layers may be considered to be positioned either between the Au and Gr layers or exterior to the Au and Gr layers, and the following set of equations may be used for any novel sensor structures. **Figure 3** shows such a layout where MoS2 layer(s) are sandwiched between the Au and Gr layers. We will use this configuration to investigate the effect of electrical bias on the sensitivity of an SPR system.

MoS2 is a semiconductor material with an ultrathin direct bandgap and belongs to the transition metal dichalcogenide group, with characteristics similar to graphene. A strong coupling can be induced at the metal/MoS2 interface because of the effective charge transfer and large field enhancement resulting in improved SPR sensitivity [8, 17]. The nominal thickness of a single layer of MoS2 is 0.65 nm.

Consider the relation for capacitance of a capacitor:

$$\mathbf{C} = \frac{\varepsilon\_0 \varepsilon\_r A}{d} = \frac{Q}{V},\tag{24}$$

#### **Figure 3.**

*Diagram showing the three layers of the sensor surface and biased using ohmic contacts to a voltage source. Shown here are Au-MoS2-Gr layers forming, which can be modeled as capacitors in series. Note that the thicknesses of the layers shown are not to scale.*

*Application of Electric Bias to Enhance the Sensitivity of Graphene-Based Surface Plasmon… DOI: http://dx.doi.org/10.5772/intechopen.106556*

#### **Figure 4.**

*Diagram showing the three layers of the sensor surface and biased using ohmic contacts to a voltage source. Note that the thicknesses of the layers shown are not to scale.*

where *C*, *A*, and *d* are the capacitance, surface area, and the separation distance between the plates of a capacitor, respectively. Modeling the Au-MoS2-Gr layers as two capacitors in series (**Figure 4**, where the equivalent capacitance of such system is:

$$\frac{1}{C\_{eq}} = \frac{1}{C\_1} + \frac{1}{C\_2} \tag{25}$$

We can find the carrier concentration, *ng* in Gr on top of this Au-MoS2 layers, due to an applied voltage, *Vg* given by:

$$m\_{\rm g} = \frac{V\_{\rm g}}{q} \frac{\varepsilon\_0}{\left(\frac{d\_{\rm Mo\rm S2}}{\varepsilon\_{\rm fr\rm Sa2}} + \frac{d\_{\rm Gr}}{\varepsilon\_{\rm Gr}}\right)}\tag{26}$$

where *<sup>ε</sup>*<sup>0</sup> <sup>¼</sup> <sup>8</sup>*:*<sup>85</sup> � <sup>10</sup>�<sup>12</sup> F/m is the permittivity of vacuum, *<sup>q</sup>* is the electron charge, *dMoS*2, *dGr*, *εrMoS*<sup>2</sup> , and *εrGr* are the thickness and relative permittivity of MoS2 and Gr layers respectively.

Eq. (26) combined with Eqs. (13)–(21) can be used to calculate the refractive index of Gr as the electric bias *Vg* is varied from 0 to 10 eV. The SPR resonant angle and reflection values can be found using relations given in Ref [1].

## **3. Results and discussion**

**Figure 5** shows the SPR curves as the chemical potential is varied from 0 to 8 eV, for the case of sensor configuration with prism material, MgF2 (*np* ¼ 1*:*377), followed by Au layer (30 nm thickness, *nAu* ¼ 0*:*1726 þ 3*:*4218 i), MoS2 (1 layer, *nMoS*<sup>2</sup> ¼ 4*:*8046 þ 0*:*84395 i, [18]), Gr (10 layers), and the sample considered is urea (10 nm thickness, *nUrea* ¼ 1*:*335), the wavelength used for these calculations is *λ* ¼ 632*:*8 nm. To better understand the impact of the applied electrical bias across the Au-MoS2-Gr system and the shift in the SPR angle with respect to change in the refractive index of the urea sample, the SPR curves are referenced to the SPR curve when the urea sample has refractive index value n = 1.332. We will base our discussion, using *nurea* ¼ 1*:*332 as our reference, while all other values of the sensor surface material are the same. In the

#### **Figure 5.**

*SPR Curves for the sensor configuration: MgF2 (np* ¼ 1*:*377*), followed by Au layer (30 nm thickness, nAu* ¼ 0*:*1726 þ 3*:*4218 *i), MoS2 (1 layer, nMoS*<sup>2</sup> ¼ 4*:*8046 þ 0*:*84395 *i, [18]), Gr (10 layers), and the sample considered is urea (10 nm thickness, nurea* ¼ 1*:*335*).*

**Figure 5**, we can see that the value of the SPR dip, Rmin = 0.15 with SPR angle = 49.1 deg. for n = 1.332 and for n = 1.335, the Rmin = 0.13 with SPR angle = 48.9 deg., both at 1 meV applied chemical potential, which again will be our reference applied chemical potential value for the rest of our discussion of the results throughout this text. As seen in **Figure 2**, we also see in **Figure 5**, that the SPR curves

#### *Application of Electric Bias to Enhance the Sensitivity of Graphene-Based Surface Plasmon… DOI: http://dx.doi.org/10.5772/intechopen.106556*

move toward lower SPR angle from 49.1 deg. to 47.06 deg. at 2.5 eV; and then show up at the extreme right (90 deg) mark and move to the lower angles before settling down between 51.49 deg. at 4.5 eV to 50.1 deg. at 8 eV and it stays almost the same even as the chemical potential is increased further till 10 eV. We also see that the Rmin for all these *μ<sup>c</sup>* values from 0.001 eV to 8 eV is below 0.05 (a.u.), which shows that there is a strong excitation of surface plasmon facilitated by the application of electrical bias. A shift of more than 1 deg. in the SPR angle is a very strong indication of the high sensitivity this technique provides as the sample changes its refractive index value from n = 1.332 to 1.335, which gives a sensitivity value of 333.33°/*RIU*. It should be noted that optimized graphene-based sensor surface structure for a particular biomolecule can be further optimized for higher sensitivity by application electrical bias and without any changes to the sensor structure.

**Figure 6** shows the variation of the SPR angle as the chemical potential varied. It has already been shown that the number of graphene layers usually results in enhancement of the electrical, optical, chemical, and mechanical properties of device [19], but here we specifically want to highlight the impact of graphene layers on the SPR angle shift, which is what is done in this figure. This figure shows that when Gr layers = 12, at *μ<sup>c</sup>* ¼ 3 eV, there is a very large shift of 28.91 deg. (from 49.46 at 1 meV to 20.55 deg. at 3 eV) whereas the SPR angle shift is 9.22 deg. and 7.48 deg. for Gr = 8 & 10 layer structure (refer to **Table 1**), which in their own right they are much larger than other optimized sensor structures reported for urea detection.

**Figure 7** shows the variation of the Rmin and FWHM of the SPR curves with respect to the variation of chemical potential across the sensor surface. We see that the Rmin = 0.9 and FWHM = 16.05 deg. for Gr = 12 layers at *μ<sup>c</sup>* ¼ 3 eV, which eliminates this as a not least favorable case for SPR measurement, and so we will have to choose the values obtainable at *μ<sup>c</sup>* ¼ 2*:*5 eV, which provides Rmin = 0.04 and FWHM = 0.46 deg. However, if compared to the case of Gr = 8 and Gr = 10 layers we see that these two cases are far more favorable for SPR measurement because of their

#### **Figure 6.**

*SPR Curves for the configuration: MgF2 (np* ¼ 1*:*377*), followed by Au layer (30 nm thickness, nAu* ¼ 0*:*1726 þ 3*:*4218 *i), MoS2 (1 layer, nMoS*<sup>2</sup> ¼ 4*:*8046 þ 0*:*84395 *i, [18]), Gr (10 layers), and the sample considered is urea (10 nm thickness, nurea* ¼ 1*:*335*).*


*The maximum value of the SPR angle achievable, along with the corresponding Rmin and FWHM, are highlighted in red color. These maximum SPR angle values occur at 2.5, 3.0 and 3.5 eV for the case of 8, 10 and 12 Gr layers in the sensor system, which also indicates that maximum SPR angle occurs at higher applied electrical potential values as the Gr layers increase. The SPR angle, Rmin and FWHM values are highlighted in blue color as comparison with the values for the Au (50 nm) - Gr sensor system as described in [11].*

#### **Table 1.**

*Chemical potential (eV) versus SPR angle (deg), Rmin (a.u.), and the FWHM (deg) for different graphene layers for the sensor structure Au (30 nm) - MoS2 (3 layers) - Gr, with the urea as sample (thickness 10 nm, n = 1.335).*

low Rmin and FWHM when the maximum SPR angle shift occurs. So, one cannot only use the SPR angle shift as the sole parameter to understand the sensitivity of the SPR sensors, and so it is common to use FOM and QF to better evaluate the sensitivity of the SPR sensors.

**Figure 8** shows the variation of Sn and FOM for three different cases of Gr = 8, 10 &12 layers while Au-MoS2 thicknesses are 30 nm and 3 layers, respectively. The curves shown, here can be seen in **Table 2**. **Figure 9** shows the variation of *Sn*,*μ<sup>c</sup>* and *FOM<sup>μ</sup><sup>c</sup>* for Gr = 8, 10 &12 layers for the same Au-MoS2 thicknesses of 30 nm and 3 layers, respectively.

**Table 2** provides the values of Sn and FOM for varying chemical potential values using urea sample with n = 1.332 as reference and comparing the values to urea sample with n = 1.335. Also, note that the negative values of the Sn and FOM indicate the fact *Application of Electric Bias to Enhance the Sensitivity of Graphene-Based Surface Plasmon… DOI: http://dx.doi.org/10.5772/intechopen.106556*

#### **Figure 7.**

*R min and FWHM curves for the sensor-sample: MgF2 (np* ¼ 1*:*377*), followed by Au layer (30 nm thickness, nAu = 0.1726 + 3.4218 i), MoS2 (1 layer, nMoS*<sup>2</sup> ¼ 4*:*8046 þ 0*:*84395 *i, [18]), Gr (10 layers), and the sample considered is urea (10 nm thickness, nurea* ¼ 1*:*335*).*

#### **Figure 8.**

*Sn and FOMn curves for the sensor-sample: MgF2 (np* ¼ 1*:*377*), followed by Au layer (30 nm thickness, nAu = 0.1726 + 3.4218 i), MoS2 (1 layer, nMoS*<sup>2</sup> ¼ 4*:*8046 þ 0*:*84395 *i, [18]), Gr (10 layers), and the sample considered is urea (10 nm thickness, nurea* ¼ 1*:*335*).*


*Application of Electric Bias to Enhance the Sensitivity of Graphene-Based Surface Plasmon… DOI: http://dx.doi.org/10.5772/intechopen.106556*

*Note that θspr values for n = 1.335 are provided in Table 1 for Gr: 8, 10, and 12 layers and not repeated here. The best achievable values for SPR angle, Rmin and FWHM are highlighted in red color. The SPR angle, Rmin and FWHM values are highlighted in blue color as comparison with the values for the Au(50 nm) - Gr sensor system as described in [11].*

#### **Table 2.**

*Chemical potential (eV) versus sensitivity and FOM for the sensor structure Au (30 nm) - MoS2 (3 layers) - Gr, with the urea as sample thickness 10 nm, n = 1.335 compared with same sample urea at 10 nm thickness and n = 1.332.*

that the SPR angle shift happens toward the lower angles at lower *μ<sup>c</sup>* values, and for comparison at different *μ<sup>c</sup>* only the magnitude values should be used. Referring to FOM value of 1564 at 2.5 eV for Gr = 12 layers is a better indicator of the sensor's capability to detect the urea sample than only using the Δ*θSPR*, which clearly has a larger SPR angle shift value at 3 eV. This clearly indicates that a smaller FWHM is needed for a better measurable and distinguishable SPR curve signal. We see this same trend for Gr = 8 layers with FOM = 1267 and Gr = 10 layers with FOM = 845, where the FOM is a better indicator of the SPR signal detection. We see that at 8 eV, the FOM

#### **Figure 9.**

*Suc and FOMuc curves for the sensor-sample: MgF2 (np* ¼ 1*:*377*), followed by Au layer (30 nm thickness, nAu = 0.1726 + 3.4218 i), MoS2 (1 layer, nMoS*<sup>2</sup> ¼ 4*:*8046 þ 0*:*84395 *i, [18]), Gr (10 layers), and the sample considered is urea (10 nm thickness, nurea* ¼ 1*:*335*).*

values are rather in an upward trend from 115, 197 to 303 for Gr = 8, 10 &12 layers, which also indicates that the more number of graphene layers and the better sensitivity of the SPR system. **Table 3** shows the values of *S<sup>μ</sup><sup>c</sup>* and *FOM<sup>μ</sup><sup>c</sup>* for Gr = 8, 10 &12


*Application of Electric Bias to Enhance the Sensitivity of Graphene-Based Surface Plasmon… DOI: http://dx.doi.org/10.5772/intechopen.106556*

*The best achievable values for SPR angle, Rmin and FWHM are highlighted in red color. The SPR angle, Rmin and FWHM values are highlighted in blue color as comparison with the values for the Au(50 nm) - Gr sensor system as described in [11].*

#### **Table 3.**

*Sμ<sup>c</sup> (sensitivity wrt μc), and FOMμ<sup>c</sup> (figure of merit wrt μc) values with reference to chemical potential for the sensor structure: Au (30 nm) - MoS2 (3 layers) - Gr, with the urea as sample thickness 10 nm, n = 1.335.*

layers and with reference to *μ<sup>c</sup>* ¼ 1 meV. Highlighted in red color are the maximum values obtained. Here again, we see that *FOM<sup>μ</sup><sup>c</sup>* is a better indicator of the measurability of the SPR effect in a graphene-based SPR sensor, but with an added advantage of not having to change the sensor structure to achieve ultra-sensitivity to measure changes in the sample concentration upto 3/1000.
