Mohamed El Barghouti

Abdellatif Akjouj

Abdellah Mir

Effect of MoS2 layer on the LSPR in periodic nanostructures

Introduction

Plasmonic response of the metal nanoparticles is highly dependent on their size [START_REF] Larsson | Sensing characteristics of NIR localized surface plasmon resonances in gold nanorings for application as ultrasensitive biosensors[END_REF][START_REF] Scaffardi | Size dependence of refractive index of gold nanoparticles[END_REF][START_REF] Hanarp | Optical properties of short range ordered arrays of nanometer gold disks prepared by colloidal lithography[END_REF], their form [START_REF] Chen | Shape-and size-dependent refractive index sensitivity of gold nanoparticles[END_REF][START_REF] Bukasov | Highly tunable infrared extinction properties of gold nanocrescents[END_REF][START_REF] Aizpurua | Optical properties of gold nanorings[END_REF] and the material constituting the chemical 2D composition [START_REF] Langhammer | Localized surface plasmon resonances in aluminum nanodisks[END_REF][START_REF] Ye | Observation of plasmonic dipolar anti-bonding mode in silver nanoring structures[END_REF][START_REF] Zoric | Gold, platinum, and aluminum nanodisks plasmons: material independence, subradiance, and damping mechanisms[END_REF]. The understanding of their properties is even more complex when they are deposited on a substrate [START_REF] Larsson | Sensing characteristics of NIR localized surface plasmon resonances in gold nanorings for application as ultrasensitive biosensors[END_REF][START_REF] Pinchuk | Substrate effect on the optical response of silver nanoparticles[END_REF][START_REF] Valamanesh | Substrate effect on the plasmon resonance of supported flat silver nanoparticles[END_REF] or when they interact with each other [START_REF] Pinchuk | Nanoparticle optical properties: far-and near-field electrodynamic coupling in a chain of silver spherical nanoparticles[END_REF][START_REF] Funston | Plasmon coupling of gold nanorods at short distances and in different geometries[END_REF][START_REF] Rechberger | Optical properties of two interacting gold nanoparticles[END_REF]. It appears from the theoretical and the experimental studies already completed that there are many parameters governing their behavior [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostructures: experimental and theoretical studies[END_REF]. As part of our study, we will investigate the influence of the MoS 2 layer on the LSPR. The complex refractive index of the MoS 2 monolayer obtained from the experimental measurement data by Castellanos-Gomez et al. [START_REF] Zhang | Measuring the refractive index of highly crystalline monolayer MoS 2 with high confidence[END_REF] is n MoS2 = 5.9 + 0.8i at 632.8 nm and the thickness of the MoS 2 layer is d MoS2 = N × 0.65 nm [START_REF] Zenga | Graphene-MoS 2 hybrid nanostructures enhanced surface plasmon resonance biosensors[END_REF][START_REF] Mishra | MgF 2 prism/rhodium/graphene: efficient refractive index sensing structure in optical domain[END_REF], where N is the number of coatings of MoS 2 [START_REF] Sharma | Fuchs Sondheimer-Drude Lorentz model and Drude model in the study of SPR based optical sensors: a theoretical study[END_REF][START_REF] Mishra | Graphene and beyond graphene MoS 2 : a new window in surface-plasmon-resonance-based fiber optic sensing[END_REF][START_REF] Wu | Highly sensitive graphene biosensors based on surface plasmon resonance[END_REF][START_REF] Lertvachirapaiboon | Gold nanoparticles synthesis used for sensor applications[END_REF], deposited on the metal nanoparticles of gold. Molybdenum disulfide (MoS 2 ) holds great promise for optical applications due to the variations of optical transition depending on the atomic thickness of the film [START_REF] Sharma | Fuchs Sondheimer-Drude Lorentz model and Drude model in the study of SPR based optical sensors: a theoretical study[END_REF]. A gradual red shift of the absorption bands the background absorption. Hence, Increasing MoS 2 film thickness is observed to allow tuning the absorption properties of such films. The optical processes in a low dimensional materials T 1 such as MoS 2 can furthermore be changed by the presence of resonance cavities and plasmonic nanostructures [START_REF] Zhang | Measuring the refractive index of highly crystalline monolayer MoS 2 with high confidence[END_REF]. In biosensing applications, the gold is not well adapted because of its poor absorption of biomolecules, causing limitations on the sensitivity of the biosensor [START_REF] Mayer | Localized surface plasmon resonance sensors[END_REF]. As a solution to this limitations, the gold layer is substituted for the gold nanoparticles (AuNPs). Gold nanoparticles have a unique property called localized surface plasmon (LSPR). The AuNPs-based LSPR is widely used for the detection [START_REF] Szunerits | Sensing using localised surface plasmon resonance sensors[END_REF].I n addition, the dielectric environment of the Localized surface plasmons resonance (LSPR) has been extensively studied in recent years [START_REF] Touahir | Plasmonic properties of silver nanostructures coated with an amorphous silicon-carbon alloy and their applications for sensitive sensing of DNA hybridization[END_REF][START_REF] Malinsky | Chain length dependence and sensing capabilities of the localized surface plasmon resonance of silver nanoparticles chemically modified with alkanethiol self-assembled monolayers[END_REF] 2D networks nanoparticles of gold (AuNPs). In this work, we theoretically study the impact of the MoS 2 substrate thickness on the plasmonic signal of gold nanoparticle (AuNPs) arrays (Fig. 1B). Throughout this paper, the geometrical parameters of the AuNPs is fixed with l is the particles length, a is the lattice parameter (along x-axis), and h is the particles height. It is well established that AuNPs strongly absorb within narrow frequency bands in the visible range as their localized surface plasmon resonances get excited. Besides, the effect of the dielectric environment surrounding the AuNPs has been widely studied [START_REF] Haes | Nanoscale optical biosensor: short range distance dependence of the localized surface plasmon resonance of noble metal nanoparticles[END_REF][START_REF] Xu | Modeling the optical response of nanoparticle-based surface plasmon resonance sensors[END_REF]. A part of this work consists of investigating numerically the influence of MoS 2 coatings (0-7.15 nm) on the localized surface plasmon resonance signal of AuNPs arrays (Fig. 1A). In the first part, the idea is to highlight the role of a few layers of MoS 2 as one of the finest collection thickness of layers deposited on MoS 2 nanostructures. The considered ranging is from 0 nm to 3.90 nm (strong red shift of the resonance). The layers of the dielectric have been deposited on a substrate of SiO x (n 1 = 1.45) over which two types of nanostructures have been investigated. In the second part, numerical simulations of structures SiO x /AuNPs/MoS 2 /water and SiO x /MoS 2 /AuNPs/water with a normal bias following the z-axis, show significant sensitivity ∼300 (nm RIU -1 ) Fig. 1(A) and ∼171 (nm RIU -1 ) Fig. 1(B) for the eight layers MoS 2 5.20 nm [START_REF] Haes | Nanoscale localized surface plasmon resonance biosensors[END_REF][START_REF] Murray | Overlayers on silver nanotriangles: field confinement and spectral position of localized surface plasmon resonances[END_REF][START_REF] Rindzevicius | Long-range refractive index sensing using plasmonic nanostructures[END_REF][START_REF] Gupta | Raman scattering from high-frequency phonons in supported n-graphene layer films[END_REF][START_REF] Lei | Revealing plasmonic gap modes in particle-on-film systems using dark-field spectroscopy[END_REF][START_REF] Noual | Nanoscale plasmon waveguide including cavity resonator[END_REF]. In addition, the plasmonic response of metal nanoparticles with various geometrical parameters gold nanoparticles l = 125 nm, a = 300 nm, and h = 15 nm. These results are consistent with other reports in the literature showing long row detecting refractive index on plasmonic nanostructures [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostructures: experimental and theoretical studies[END_REF][START_REF] Mishra | Graphene and beyond graphene MoS 2 : a new window in surface-plasmon-resonance-based fiber optic sensing[END_REF][START_REF] Lertvachirapaiboon | Gold nanoparticles synthesis used for sensor applications[END_REF]. The distributions of the electric field structures excited to resonance at normal incidence and represented by the maps of the 2D and 3D electrical fields which indicate that this mode is a localized surface plasmon dipole whose hot spots (areas of high intensities field) are pushed to the top and bottom corners of AuNPs for the structure SiO x /AuNPs/MoS 2 /water, and top corners of AuNPs to layer MoS 2 under the nanoparticles to the structure SiO x / MoS 2 /AuNPs/water to the lengths wave resonances (λ max ).

One observes a significant enhancement of the sensitivity of localized surface plasmon resonances of coated AuNPs with layers of MoS 2 to the surrounding environment as compared to bare nanoparticles. Hence, the design of Au nanoparticles/MoS 2 based-plasmonic bio-sensors with more sensitive could be reached. Indeed, we show that AuNPs coating with few layers of MoS 2 leads to approximately a 26% higher sensitivity. The optimal thickness of MoS 2 layers is numerically investigated to achieve a high LSPR sensitive system.

Theoretical methods

The optical properties of gold nanoparticles are solved numerically, in the frequency domain, using the scattered field formulation. Field analysis was performed using a commercially available Finite Elements Method (FEM) method package (COMSOL Multiphysics 4.4) [START_REF] Knight | Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core-shell nanoparticles beyond the quasistatic limit[END_REF][START_REF] Jin | The Finite Element Method in Electromagnetics[END_REF]. The simulation method has been well documented [START_REF] Zhan | Plasmonic Fano resonances in nanohole quadrumers for ultra-sensitive refractive index sensing[END_REF][START_REF] Rakic | Analysis of optical channel cross talk for free-space optical interconnects in the presence of higher-order transverse modes[END_REF][START_REF] Powell | Analysis of optical-and inelastic-electron-scattering data. II. Application to Al[END_REF]. A layer of gold nanoparticles of diameter (l), height (h) and interparticle distance (a), is coated with a MoS 2 thin-layer and immersed in a homogeneous matrix. A transparent glass of SiO x (refractive index n 1 = 1.45) is used as a substrate. The frequency-dependent complex permittivity of metal (gold) is described by the Lorentz-Drude model (Eq. ( 1)) [START_REF] Castellanos-Gomez | Optical identification of atomically thin dichalcogenide crystals[END_REF][START_REF] Rindzevicius | Long-range refractive index sensing using plasmonic nanostructures[END_REF].

∑ =+ -+ ∞ = εω ε fω ωωj ω () Γ r m M m p m m , 0 2 2 2
(1)

where ε r,∞ is the relative permittivity at infinite frequency, ω p the plasma frequency, and ω m , f m and Γ m the resonance frequency, strength and damping frequency, respectively, of mth oscillators. The Lorentz-Drude model uses M damped harmonic oscillators to describe the small resonances observed in the metals frequency response. The values of the constants in Eq. ( 1) are taken from reference [START_REF] Castellanos-Gomez | Optical identification of atomically thin dichalcogenide crystals[END_REF], with the value of the dielectric constant of infinite frequencies being ε ∞ = 1. The values of the other parameters are listed in Table 1: 3. Results and discussion

Localized surface plasmon for periodic nanostructures

The evolution of the localized surface plasmon modes in 2D periodic nanostructures on two different surface architectures are investigated (Fig. 1). The morphology of the periodic gold nanoparticle such as particle diameter (l), height (h), and period (a) is kept the same. The difference between the two structures is the position of the dielectric MoS 2 layer, being either around the Au NPs (Fig. 1A) or below the Au NPs (Fig. 1B). A transparent glass substrate SiO x with n 1 = 1.45 is used in both cases and the assembly immersed in a non-absorbing medium such as water (n 2 = 1.333).

For the refractive index of MoS 2 , we have used a constant value n MoS2 = 5.9 + i 0.8 (at 632.8 nm) [16,17], the thickness of the MoS 2 layer being estimated by d MoS2 = N × 0.65 nm, where N is the number of MoS 2 layers, and we have not used the function of the wavelength n MoS2 = εω () [42], because in our nanostructures the absorption spectra either varies in the thickness of the MoS 2 layer or in the detection refractive index medium. This simulation demonstrates a single peak that varies and the two skins remain fixed. So, in our work we focus on the plasmon resonance of localized surface which changes according to the parameters of nanostructures. In other different computational checks we have noticed that we are using a fixed value of n MoS2 .

The LSPR spectrum of a glass/AuNPs interface where the AuNPs parameters were set to a length of 25 nm, a height of 20 nm and a period of 70 nm, shows a plasmon resonance at 525 nm Fig. 2(A). As expected, an increase in the thickness of MoS 2 shifts the peak wavelength maximum λ max to the red, from λ max = 538 nm for d MoS2 = 0.65 nm (monolayer) to λ max = 611 nm for d MoS2 = 3.90 nm (six layers). In addition, the presence of MoS 2 results in a considerable increase in the overall absorption intensity in the near-infrared Fig. 2(B). It also seems that some additional badly defined bands are occurring in the presence of MoS 2 thicknesses above 1.3 nm.

In Fig. 2B, we have plotted the evolution of the plasmon resonance wavelength in relation to the thickness of the dielectric layer MoS 2 calculated for interfaces SiO 2 /AuNPs/MoS 2 /water. Was found that an increase in the thickness of MoS 2 (d MoS2 ) shifts the peak wavelength (λ max ) to red almost linearly from λ max = 538.21 nm for d MoS2 = 0.65 nm up to λ max = 611.25 nm for d MoS2 = 3.90 nm, due to the interaction of AuNPs/MoS 2 with a normal incident light pulse.

In the case of MoS 2 present on the glass interface rather than the Au NPs (Fig. 1(B), the plasmonic behavior is rather different. With increasing MoS 2 thickness, next to a band at 525 nm, a second strongly absorbing plasmon band is observed and shifts to higher wavelength with increase of the thickness of MoS 2 Fig. 2(C).

For the structure SiO x /MoS 2 /AuNPs Fig. 2(C), we observe that we add the MoS 2 layer between the substrate and the AuNPs, the refractive index of medium under the nanoparticles increase SiO x (n 1 = 1.45) < n 2 to SiO x /MoS 2 (n MoS2 = 5.9 + 0.8i)>n 2 leads to an increase in the resonant wavelength rapidly, an enlargement of the plasmon band, as well as the appearance of a multimodal response [START_REF] Saison-Francioso | Plasmonic nanoparticles array for high-sensitivity sensing: a theoretical investigation[END_REF]. The evolution of the amplitude of the oscillations thus seems to result from the competition between several phenomena. Two of them help to reduce the amplitude of the oscillations when n 1 /n MoS2 becomes larger than the detection medium refractive index. This is the decrease in the contrast of the standing wave system inside the cavity on the AuNPs, and the appearance of a second peak plasmonic response (multimodal). The last phenomenon is the increase of the spectral width of the main plasmonic mode with the increase in the thickness of the MoS 2 layer, which tends to increase the amplitude of surface plasmon oscillations.

The position of the resonant wavelength plasmon λ max shifts almost linearly with the increase of MoS 2 thickness to λ max = 865 nm for (d MoS2 ) = 3.90 nm Fig. 2(D). In contrast to glass/AuNPs/MoS 2 , the position of to λ max is unchanged for glass/MoS 2 /AuNPs interface when changing the refractive index of the immersion medium (from n 2 =1ton 2 = 1.45) Fig. 3. In the case of glass/AuNPs/ MoS 2 , a shift is observed for all thicknesses of MoS 2 .

The influence of refractive index at the interface of the structure SiO x /AuNPs/MoS 2 was investigated by recording the wavelength shift when it is changed. In different proportions giving refractive indices (n 2 = 1.333 for water, 1 for air and 1.45 SiO x ), the response of the absorption spectrum of LSPR is generally described in Fig. 3(A). It shows that the position of λ max moves the higher wavelength 530 nm (air) 542 nm (SiO x ) to a layer of MoS 2 (0.65 nm) while increasing the refractive index of the detection medium [START_REF] Mishra | Infrared SPR sensitivity enhancement using ITO/TiO 2 /silicon overlays[END_REF][START_REF] Mishra | An SPR-based sensor with an extremely large dynamic range of refractive index measurements in the visible region[END_REF]. For the SiO x /MoS 2 /AuNPs structure, conservation λ max has been observed for the change in refractive index (n 2 = 1, 1.333 and 1.45) Fig. 3(B), followed by a very large red shift of the length wave between 0.65 nm ≤ d MoS2 ≤ 3.90 nm. In the purpose of comparing the absorption spectrum of results calculated with the model of L-D by the FEM method (for 0.65 nm ≤ d MoS2 ≤ 3.90 nm thickness of MoS 2 layer) are structures that were studied in Fig. 2. Which shows the evolution of the normalized absorption resonance wavelength of the two structures according thick MoS 2 , after the interaction of the studs with a light pulse incident [400-1000 nm]. The parameter values selected structures are: l = 25 nm, a = 70 nm and h = 20 nm. Giving the structure SiO x /MoS 2 /AuNPs/water a very significant growth of the resonance wavelength of 333.7 nm plasmon which passes from 531.35 nm to 865.05 nm (Fig. 2B). Relative that obtained in the structure SiO x /AuNPs/MoS 2 /water we see dramatic growth of the 3(A), λ max presents a spectral shift to red when the nanoparticles are started to be covered for deferments n 2 media.

It was investigated by recording that the wavelength shifts when it is changed, small proportions giving 518.13 nm for air, 531.35 nm for water and 539.96 nm for SiO x at d MoS2 = 0 nm, with the sensitivity of S = 39.68 nm/RIU, and 776.40 nm for air, 780.49 nm for water and 781.25 nm for SiO x at d MoS2 = 2.60 nm with the sensitivity S = 12.28 nm/RIU. It is found that the sensi- tivity, like the resonance wavelength, shows a weak oscillation when n 2 increases. The studied system thus has a non-zero sensitivity, but this is small and always less than the sensitivity of the nanoparticles covered by MoS 2 , for the n 2 values to be considered. Indeed, for n 2 = 1.000, 1.333 and 1.45, Almost a small variation of the resonance wavelength is observed. Moreover, for the thickness of MoS 2 increases, the resonance wavelength is shifted towards the red as Fig. 3B. These behavioral differences are simply due to the way the refractive index (n 2 ) by the nanoparticles varies and geometrical parameters Au nanoparticles. Fig. 4, it is deduced that whatever the value of n 2 , and for large geometrical parameters of the structure l = 125 nm a = 300 nm and h = 15 nm, the sensitivity and the resonance wavelength of the structures studied are presented extremely for the same thicknesses of d MoS2 . Note also that the sensitivity and the resonance wavelength oscillate as a function of d MoS2 . However, it is now known that the value of the refractive index of the detection medium (n 2 ) and the geometric characteristics of the nanoparticles have an impact on the displacement of LSPR and the sensitivity. Moreover, it can be deduced that the sensitivity is oscillated for two reasons. On the one hand, the fact of depositing the layer of MoS 2 under the nanoparticles is detrimental to the sensitivity, and causes a decrease in the latter for the thicknesses of d MoS2 (0.65-7.15 nm). On the other hand, the refractive index of the dielectric covering the nanoparticles (n 2 ) is lower than the refractive index of the medium which deposits the AuNPs (n 1 = 1.45/n MoS2 = 5.9 + 0.8 i), and the geometric parameters of system, that's what we can check in Fig. 4 and Table 2.

This variation in the effect of geometric parameters of gold nanoparticles leads to the increase of detection optic properties. Increasing the geometric parameters of the AuNPs first generates an increase in the wavelength of plasmon resonance. The increase in λ max is consistent with the evolution of the electric field contact within the gold nanoparticles. These results are consistent with other reports in the literature showing long row detecting refractive index on the plasmonic nanostructures [START_REF] Galopin | Sensitivity of plasmonic nanostructures coated with thin oxide films for refractive index sensing: experimental and theoretical investigations[END_REF] have studied these effects, which show a linear dependence as a function of the refractive index of the surrounding medium.

The refractive index sensitivity, defined as the ratio of the variation of the position of the plasmon band during the variation of the refractive index (Eq. ( 2)) and determinable from the slop of a quasi-linear plot of λ max vs. n 2 [START_REF] Haes | A nanoscale optical biosensor: sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles[END_REF].

= - S λ n ∆ ∆ (nm RIU ) max 2 1 (2)
Of both interfaces us shown in Table 2. Two circles used to determine the sensitivity for the same interface indexes in the n 2 = 1.45 material and air n 2 =1 (Δn 2 = 0.45).

Fig. 4, shows the change of λ max with the refractive index of different dielectrics used to put some dielectric layers (MoS 2 ) and gold nanostructures.

Several numerical simulations were done to find out how changing the sensitivity depend on the thickness of MoS 2 filed on nanoparticles (AuNPs/MoS 2 ) or between the substrate and the nanoparticles (MoS 2 /AuNPs).

These results are in agreement with other reports, in the field of study, showing long rank refractive index of detection of plasmonic nanostructures [START_REF] Galopin | Sensitivity of plasmonic nanostructures coated with thin oxide films for refractive index sensing: experimental and theoretical investigations[END_REF] Table 1, exposes this evolution of sensitivity according to the layer thickness of MoS 2 .

We have initially determined the sensitivity of the system described in Fig. 1, without the thickness of MoS 2 (d MoS2 = 0 nm) above or below the gold nanoparticles. The wavelength of resonance of such a system is λ max = 810.81 nm for n 2 = 1.000 and λ max = 917.03 nm for n 2 = 1.45, we find S = 236.04 nm/RIU.

On the other hand, for (d MoS2 ), either when the nanoparticles are covered by different MoS 2 slips or when the nanoparticles are covered with 8 layers of MoS 2 (d MoS2 = 5.20 nm), the resonance wavelength is λ max = 937.5 nm for n 2 = 1.000 and λ max = 1071.43 nm for n 2 = 1.45, which corresponds to a sensitivity of S = 297.62 nm/RIU.

For a thickness of MoS 2 (from 0.65 to 7.15 nm) we vary in sensitivity by changing the thickness of MoS 2 .W efind that the sensitivity is proportional to the thickness of MoS 2 . First of all, there has been an increase in sensitivity (glass/AuNPs/MoS 2 ) for thicknesses of MoS 2 between 1.95 and 5.20 nm. Indeed, S = 206.95 nm/RIU for 2 layers (1.30 nm) and S = 297.62 nm/RIU to 5.20 nm, after we noticed a decrease in sensitivity from thickness 9 layers (5.85 nm).

Indeed, the sensitivity that corresponds to the structure (Fig. 1B) is decreasing when the thickness of the layer of MoS 2 diminished, S = 202.52 nm/RIU for d MoS2 = 0.65 nm at S = 144.43 nm/RIU for d MoS2 = 7.15 nm [START_REF] Mishra | Graphene and beyond graphene MoS 2 : a new window in surface-plasmon-resonance-based fiber optic sensing[END_REF][START_REF] Lertvachirapaiboon | Gold nanoparticles synthesis used for sensor applications[END_REF][START_REF] Mishra | Gas sensing in Kretschmann configuration utilizing bi-metallic layer of Rhodium-Silver in visible region[END_REF][START_REF] Mishra | Doped single-wall carbon nanotubes in propagating surface plasmon resonance based fiber optic refractive index sensing[END_REF]. The enhancement of sensitivity for the AuNPs/MoS 2 based LSPR biosensor is calculated to have more than 26% of when compared the sensors LSPR conventional without the layer of MoS 2 , with the same parameters of AuNPs. These characteristics should make these biosensors a preferred choice for bio-detection applications, compared to other contemporary biosensors.

Influence of dielectric MoS 2 on the distribution of electric field

The sensitivity of the interface can be understood from the electrical field maps in Fig. 5 shows the distribution of the electric field to the resonance near nanoparticles. It can be seen by comparing Fig. 5(A) that when the nanoparticles (Au NPs) are covered with MoS 2 a substantial part of the field is found inside of the latter. In the case of MoS 2 coated glass Fig. 5(B) that when the nanoparticles covered with a non-absorbing dielectric (water) the field spreads out more and more deeply into the substrate as that its refractive Maps of calculated electric field Fig. 5(A) to the resonance wavelength plasmon λ max = 585.48 nm for d MoS2 = 2.60 nm show a strong improvement of the field located on the four corners of the nanoparticles within the confined field. The corners of AuNPs with strong localization at the upper corners of AuNPs compared to the structure of Fig. 5(B) calculated at the resonant wavelength plasmon λ max = 865.05 nm d MoS2 = 3.90 nm a strong localization is observed at the upper corners and at the lower surface of the AuNPs in contact with the layer of MoS 2 [START_REF] Cao | Enhancement of broad-band light absorption in monolayer MoS 2 using Ag grating hybrid with distributed Bragg reflector[END_REF][START_REF] Viste | Enhancement and quenching regimes in metal-semiconductor hybrid optical nanosources[END_REF][START_REF] Hutter | Interaction of metallic nanoparticles with dielectric substrates: effect of optical constants[END_REF]. It is thus seen in Fig. 5(B), that in the MoS 2 nanoparticles, the present two hotspots electric field at the upper corners of AuNPs and the surface in contact with the substrate. In addition, Fig. 5(A) shows that the nanoparticles coated with 2.60 nm of MoS 2 have hot spots in each of their corners, and the two hot spots in contact with the substrate have spread spatially. Furthermore, the addition of MoS 2 led to an increase of the amplitude of the electric field in both sides of AuNPs in contact with this medium recouvrement.

Fig. 6 shows the change in the resonance wavelength depending on the thickness of d MoS2 MoS 2 layers deposited below the gold nanoparticles to study the structure SiO x /MoS 2 /AuNPs/water, The effect of the allowance index of dielectric substrate. Fig. 6(A) shows the change in resonance wavelength as a function of the thickness d MoS2 , studied for the structure SiO x /MoS 2 / AuNPs/water. It is found that the λ max of shift for the same detection medium is different d MoS2 value from 0 to 50 nm. Fig. 6(A) shows the influence of the thickness of the dielectric layer on the plasmonic response of the system studied SiO x /MoS 2 /AuNPs/water. Indeed, the resonant wavelength is not shifted towards the red but remains constant. It is recalled that the refractive index of the substrate is n 1 = 1.45 (SiO x ), and which covers the nanoparticles with a dielectric having a refractive index n 2 = 1 (water). For the variation of the thickness of This is what highlights Fig. 6. We can see the evolution of the position of resonance, when the dielectric substrate of refractive index changes to the value of d MoS2 = 1.95 nm distinct.

As before, we mapped the absorption spectra of the studied MoS 2 /AuNPs system for d MoS2 = 1.95 nm (3 layers of MoS 2 ), in order to analyze the influence of n 1 on the plasmonic response of the nanoparticles (see Fig. 6(C)). It appears, similar to what was found for Fig. 2(C), that the growth of the amplitude of absorption spectra when n 1 increases, coincides with the appearance of new modes of plasmonic resonance in addition to the main mode [START_REF] Wu | Highly sensitive graphene biosensors based on surface plasmon resonance[END_REF][START_REF] Saison-Francioso | Plasmonic nanoparticles array for high-sensitivity sensing: a theoretical investigation[END_REF]. This is what highlights Fig. 6. We can see the evolution of the position of resonance, when the dielectric substrate of refractive index changes to the value of d MoS2 = 1.95 nm distinct. In Fig. 6(C), one can verify that λ max is red-shifted when the substrate refractive index increases, it therefore becomes independent from you the value of the refractive index of the substrate. Moreover, Fig. 6(D) depicts the change in the localised surface plasmonic resonance (λ max ) with the substrate refractive index, there is a very significant linear increase λ max = 704.23 nm to λ max = 862 nm, with an amplitude of 0.36-0.55 calculated for the substrate refractive index increases from n 1 =1ton 1 = 3. This observation is important to search for a plasmonic sensor to optimal properties.

Conclusion

We have theoretically studied the optical properties of the influence of the dielectric film with MoS 2 nanoparticles of gold structure on plasmonic properties and sensitivity. The Lorentz-Drude model was used to calculate the optical signal of localized surface plasmon resonance (LSPR) gold nanostructures SiO x /AuNPs/MoS 2 /water and SiO x /MoS 2 /AuNPs/water; this study provides a better understanding resonance coupled to systems based on gold nanoparticles and MoS 2 layers. Due to this research put into the development of the surface plasmon functionalization of hybrid interfaces applications. Developing several sensing components, the offset to the wavelength of red, localization of the electric field and sensitivity of the interface hybrid of the LSPR depends on thickness of the MoS 2 layer. A general conclusion can be drawn that the interest in these multilayer interfaces is the possibility of sensitive long detection range with important implications for biological detection studies.

Fig. 1 .

 1 Fig. 1. Schematic representation of the plasmonic structures studied in this work. (A) Periodic gold nanoparticle (Au NPs) array on SiO x (n 1 = 1.45) and coated with a layer of MoS 2 (SiO x /AuNPs/MoS 2 ); (B) SiO x (n 1 = 1.45) coated with a thin film of MoS 2 onto which a periodic gold nanoparticle (Au NPs) was deposited (SiO x /MoS 2 /Au NPs). The height (h = 20 nm), width (l = 25)) and lattice parameter (a = 70 nm) defined as the closest distance between two adjacent AuNPs is the same in both cases. The input source is placed in the substrate (SiO x ) and the detector in the dielectric, being water (n 2 = 1.333).

Fig. 2 .

 2 Fig. 2. (A) Variation in of absorption spectrum of the glass/Au NPs/MoS 2 interface (Fig. 1A) as a function of the thickness of MoS 2 (d MoS2 =0-3.9 nm) when immersed in water. (B) Evolution of the position of the LSPR band as a function of the thickness of the MoS 2 coating; (C) Variation in of absorption spectrum of a glass/MoS 2 /Au NPs interface (Fig. 1B) as a function of the thickness of MoS 2 (d MoS2 =0-3.9 nm) when immersed in water; (D) Evolution of the position of the LSPR band as a function of the thickness of the MoS 2 underlying coating.

Fig. 3 .

 3 Fig. 3. Evolution of λ max with the thickness of the overcoat MoS 2 , comparison upon change of the refractive index of the surrounding medium: (A) for structure "Fig. 1(A)", air (blue square), water (red circle) and SiO x (green triangle). (B) for structure "Fig. 1(B)", air (red square), water (olive circle) and SiO x (yellow star). The nanoparticles are characterized by l = 25 nm, h = 20 nm and a = 70 nm. The refractive index of the substrate on which they are deposited is n 1 = 1.45. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .

 4 Fig. 4. Evolution of λ max with the thickness of the overcoat MoS 2 comparison of different refractive index of the medium surrounding structure (Fig. 1(A)), (A) air n 2 = 1.000 and (B) SiO x n 2 = 1.45. Comparison with the structure (Fig. 1(B)), (C) air n 2 = 1.000), and (D) and SiO x n 2 = 1.45. The nanoparticles are characterized by l = 125 nm, h = 15 nm and a = 300 nm. The refractive index of the substrate on which they are deposited is n 1 = 1.45.

  MoS 2 from 0 to 19.25 nm resonance wavelength plasmon red-shifted from 531.35 nm to 1168.83 nm. Then from d MoS2 = 29.25 nm, there is almost no variation observed in the resonant wavelength LSPR. Fig. 6(B) shows that the resonant wavelength plasmon increases when the height decreases, λ max = 531.3 nm for h = 20 nm and λ max = 571.4 nm for h =10nmtod MoS2 = 0 nm, and λ max = 865 nm for h = 20 nm and λ max = 926 nm for h = 10 nm to six layers of MoS 2 d MoS2 = 3.90 nm. This red shift of LSPR when the height (h) decreases when d MoS2 remains constant increases, is varied from 40 to 60 nm.

Fig. 5 .

 5 Fig. 5. Electric field maps on (A) glass/AuNPs/MoS 2 for d MoS2 = 2.60 nm under monochromatic incident radiation at 585 nm; (B) glass/MoS 2 / AuNPs for d MoS2 = 3.90 nm under monochromatic incident radiation at 865 nm.

Fig. 6 .

 6 Fig. 6. (A) Evolution of LSPR peak with change in thickness of the layer of MoS 2 (d MoS2 ) of the structure (Fig. 1B) SiO x /MoS 2 /AuNPs/water, for a = 70 nm, l = 25 nm and h = 20 nm. (B) Evolution of λ max with the thickness of the overlay MoS 2 (SiO x /MoS 2 /AuNPs/water) to compare two values of the height (h) of gold nanoparticles (h = 10 nm red square) and (h = 20 nm green circle), the other parameters are a = 70 nm, l = 25 nm. (C) Absorption spectrum corresponding to the structure substrate/MoS 2 /AuNPs/water for defeated refractive index of the substrate. (D) Change in the resonant wavelength in plasmon function of refractive index of the dielectric substrate. The geometric parameters of the gold nanoparticles are: l = 25 nm, h = 20 nm, a = 70 nm and d MoS2 = 1.95 nm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1

 1 Gold (Au) Lorentz-Drude model parameters.

	Term	f m (rad/s)	ω p (rad/s)	ω m (rad/s)	Γ m (rad/s)
	m = 0	0.760	13.72 × 10 15	0.00	0.08052 × 10 15
	m = 1	0.024	13.72 × 10 15	0.6305 × 10 15	0.3661 × 10 15
	m = 2	0.010	13.72 × 10 15	1.261 × 10 15	0.5241 × 10 15
	m = 3	0.071	13.72 × 10 15	4.511 × 10 15	1.322 × 10 15
	m = 4	0.601	13.72 × 10 15	6.538 × 10 15	3.789 × 10 15
	m = 5	4.384	13.72 × 10 15	20.24 × 10 15	3.364 × 10 15

Table 2

 2 Comparison of the evolution of sensitivity acquired theoretically for glass/AuNPs/MoS 2 and glass/MoS 2 /AuNPs. The nanoparticles are characterized by l = 125 nm, h = 15 nm and a = 300 nm. The refractive index of the substrate on which they are deposited is n 1 = 1.45.

		MoS 2	Thickness (nm)	Sensitivity (nm/RIU)
	Glass/AuNPs/MoS 2	0	-	236.04
		1	0.65	210.46
		2	1.30	206.95
		3	1.95	207.64
		4	2.60	226.90
		5	3.25	252.38
		6	3.90	264.23
		7	4.55	285.53
		8	5.20	297.62
		9	5.85	288.28
		10	6.50	281.55
		11	7.15	264.82
	Glass/MoS 2 /AuNPs	0	-	236.04
		1	0.65	202.52
		2	1.30	189.76
		3	1.95	187.79
		4	2.60	175.97
		5	3.25	171.19
		6	3.90	174.51
		7	4.55	177.94
		8	5.20	170.94
		9	5.85	161.91
		10	6.50	152.59
		11	7.15	144.43


Mohamed El Barghouti
email: mohamed.elbarghouti@gmail.com

Abdellatif Akjouj

Abdellah Mir

Effect of graphene layer on the localized surface plasmon resonance (LSPR) and the sensitivity in periodic nanostructure

Introduction

As part of our study, we analyze the evolution of the localized surface plasmon resonance (LSPR) that has been extensively studied over the last year [START_REF] Mayer | Localized surface plasmon resonance sensors[END_REF][START_REF] Szunerits | Sensing using localised surface plasmon resonance sensors[END_REF] in a periodic nanostructure 2D metal nanoparticle. We study in this work the influence of a few thin layers of graphene on the LSPR behavior of gold nanoparticles (AuNPs).

The idea is to highlight the role of a few layers of graphene as one of the finest collections. The optical properties are significantly dependent on the chemistry surface [START_REF] Szunerits | Short-and long-range sensing on gold nanostructures, deposited on glass, coated with silicon oxide films of different thicknesses[END_REF][START_REF] Doron-Mor | Ultrathin gold island films on silanized glass. Morphology and optical properties[END_REF] on which they are deposited, and on the refractive index of the dielectric environment surrounding these nanostructures [START_REF] Malinsky | Chain length dependence and sensing capabilities of the localized surface plasmon resonance of silver nanoparticles chemically modified with alkanethiol self-assembled monolayers[END_REF][START_REF] Zhao | Interaction of plasmon and molecular resonances for rhodamine 6g adsorbed on silver nanoparticles[END_REF]. Namely that apart from low graphene layer thicknesses (strong red shift of the resonance), the plasmon oscillation amplitude is increasing almost linearly with up to 5 nm, compared to the case where the gold nanoparticles are directly related to the detection of dielectric, the surface plasmon resonance of AuNPs deposited on a glass (SiO x ) substrate (n 1 = 1.45) and covered with layers of a dielectric material [START_REF] Szunerits | Short-and long-range sensing on gold nanostructures, deposited on glass, coated with silicon oxide films of different thicknesses[END_REF][START_REF] Haes | Nanoscale optical biosensor: short range distance dependence of the localized surface plasmon resonance of noble metal nanoparticles[END_REF][START_REF] Xu | Modeling the optical response of nanoparticle-based surface plasmon resonance sensors[END_REF][START_REF] Willets | Nanoscale Localized Surface Plasmon Resonance Biosensors[END_REF][START_REF] Murray | Overlayers on silver nanotriangles: field confinement and spectral position of localized surface plasmon resonances[END_REF][START_REF] Rindzevicius | Long-range refractive index sensing using plasmonic nanostructures[END_REF][START_REF] Gupta | Raman scattering from highfrequency phonons in supported n-graphene layer films[END_REF].

Numerical simulations of the structure (SiO x /AuNPs/Graphene/ SiO x ) show a high sensitivity of 59.81 (nm RIU -1 ) for the monolayer graphene 0.34 nm [START_REF] Haes | Nanoscale optical biosensor: short range distance dependence of the localized surface plasmon resonance of noble metal nanoparticles[END_REF][START_REF] Gupta | Raman scattering from highfrequency phonons in supported n-graphene layer films[END_REF], and response of plasmonic metal nanoparticles of various geometrical parameters gold nanoparticles, these findings are consistent with other reports in the literature [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostrucures: experimental and theoretical studies[END_REF] demonstrating long row detection of refractive index on plasmonic nanostructures.

Gold nanoparticles [START_REF] Wu | Highly sensitive graphene biosensors based on surface plasmon resonance[END_REF] are of great interest for the development of chemical and biological nanosensors and their use in the detection area based on the localized surface plasmon resonance (LSPR) [START_REF] Mayer | Localized surface plasmon resonance sensors[END_REF][START_REF] Szunerits | Sensing using localised surface plasmon resonance sensors[END_REF].

Numerical simulations of our structure indicate that the resonance wavelength of the electric field mode is the surface plasmon, the hot spots of which (areas of high field intensities) are localized to the lower corners of the AuNPs, deferens by compared to the structure SiO x / AuNPs/SiO x , published in literature [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostrucures: experimental and theoretical studies[END_REF][START_REF] Maurer | Enhancing LSPR sensitivity of au gratings through graphene coupling to au film[END_REF] and that on SiO 2 substrates [START_REF] Viste | Enhancement and quenching regimes in metal-semiconductor hybrid optical nanosources[END_REF][START_REF] Hutter | Interaction of metallic nanoparticles with dielectric substrates: effect of optical constants[END_REF].

Theoretical methods

Theoretical model

The optical properties of gold nanoparticles are solved numerically, in the frequency domain, using the scattered field formulation. Field analysis was performed using a commercially available finite-elementmethod. The simulation method has been well documented in [START_REF] Knight | Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core-shell nanoparticles beyond the quasistatic limit[END_REF][START_REF] Lei | Revealing plasmonic gap modes in particle-on-film systems using dark-field spectroscopy[END_REF][START_REF] Zhan | Plasmonic fano resonances in nanohole quadrumers for ultra-sensitive refractive index sensing[END_REF].

A layer of gold nanoparticles of diameter (l), height (h) and interparticle distance (a), are coated with a graphene thin-layer-thickness; and immersed in a homogeneous matrix, a transparent glass substrate T of SiO x (refractive index n 1 = 1.45). The frequency-dependent complex permittivity of metal (gold) is described by the Lorentz-Drude model [START_REF] Tsai | Analysis of optical channel cross talk for free-space optical interconnects in the presence of higher-order transverse modes[END_REF][START_REF] Powell | Analysis of optical-and inelastic-electron-scattering data. II. Application to Al[END_REF].

∑ =+ -+ ∞ = εω ε fω ωωj ω Γ () r m M m p m m , 0 2 2 2
(1

)
where ε r,∞ are the relative permittivity at infinite frequency, ω p the plasma frequency, and ω m , f m and Γ m are the resonance frequency, strength and damping frequency, respectively, of mth oscillator. The Lorentz-Drude model uses M damped harmonic oscillators to describe the small resonances observed in the metal's frequency response. The values of the constants in Eq. ( 1) are taken from reference [START_REF] Tsai | Analysis of optical channel cross talk for free-space optical interconnects in the presence of higher-order transverse modes[END_REF]. The value of the dielectric constant of infinite frequencies in reference [START_REF] Tsai | Analysis of optical channel cross talk for free-space optical interconnects in the presence of higher-order transverse modes[END_REF] is ε ∞ = 1. The values of the other L-D parameters are given in Table 1.

Refractive index

Covering the nanoparticles with a no-absorbent and no-dispersive is not the only option to provide them with chemical protection. Thus, a material as graphene, in addition to protecting the nanoparticles, provides advantages for their sensitivity. It has indeed been shown to intercalate graphene sheets between the metal film and the medium detection delocalized surface plasmon resonance sensors generates an increased sensitivity [START_REF] Verma | Sensitivity enhancement of a surface plasmon resonance based biomolecules sensor using graphene and silicon layers[END_REF][START_REF] Wijaya | Graphene-based high-performance surface plasmon resonance biosensors[END_REF]. This increase is not only due to the optical properties of graphene, but also to its excellent adsorption properties of biomolecules [START_REF] Song | Graphene on au (111): a highly conductive material with excellent adsorption properties for high-resolution bio/ nanodetection and identification[END_REF].

The layer of our model is monolayer graphene (d = 0.34 nm) and its complex refractive index (n g ) in the visible range is an absorbent and dispersing material, the refractive index of which is given by the following formula (2) [START_REF] Bruna | Optical constants of graphene layers in the visible range[END_REF]:

=+ n C λ 3.0 3 g 1 (2)
where the constant C 1 = 5.446 μm -1 [START_REF] Nair | Fine structure constant defines visual transparency of graphene[END_REF], and is λ the wavelength of the incident light in μm.

Results and discussion

The effect of graphene layers deposited on the gold nanoparticles

Fig. 1 shows schematically the studied system. The network of nanoparticles is characterized by l = 25 nm, h = 20 nm and a = 70 nm. It is excited by a plane wave generated in the substrate refractive index n 1 = 1.45 (SiO x ), propagating along the Oy-axis and the electric field is polarized along the Ox-axis. Calculating is constructed as to simulate a network infinitely periodic nanoparticles and gold, constituting the nanoparticles is described by the Lorentz-Drude model [START_REF] Tsai | Analysis of optical channel cross talk for free-space optical interconnects in the presence of higher-order transverse modes[END_REF][START_REF] Powell | Analysis of optical-and inelastic-electron-scattering data. II. Application to Al[END_REF].

These values of nanoparticles parameters chosen following the investigation described above, on the basis of their experience [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostrucures: experimental and theoretical studies[END_REF], show a sectional view of the overall shape of the particles. We can note the rather random shape of the particles, which made it necessary to determine statistics relating to the dimensions of the nanoparticles in the plane (inter-particle diameter and distance), l and p = al (respectively). The results of such statistics, respectively for l and p, are obtained thanks to SEM [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostrucures: experimental and theoretical studies[END_REF]. We then see that the most probable values are 25 ± 8 nm for the diameter l, and 16 ± 8 nm for the inter-particle distance p. For the third dimension which is the height h of the values obtained is 13.6 ± 3 nm [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostrucures: experimental and theoretical studies[END_REF]. To do this, the thickness of the layer of the dielectric (graphene) deposited on the particles at d = 5 nm was fixed on the one hand; and on the other hand the length l (Fig. 9), the grating period of the particles a, and then the height of the pads h have been varied. These variations were made with the experimental constraints, a one-dimensional filling ratio for the theoretical structure l/ a = 0.37; and the choice of the values of l,a( a = l + p) and h in the same orders of quantities provided by the experiment [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostrucures: experimental and theoretical studies[END_REF][START_REF] Maurer | Enhancing LSPR sensitivity of au gratings through graphene coupling to au film[END_REF]. The effect of the nanoparticles length in the plane of the substrate supporting them l, on the nanostructure plasmonic response was looked at keeping the other geometrical parameters constant, that is to say a = 70 nm, h = 20 nm and d = 5 nm. In Fig. 9, there is a fairly large displacement of the wavelength corresponding to the plasmonic response of the nanoparticles to the long wavelengths (low frequencies). This is done by increasing progressively from 15 nm to 55 nm. There is also a higher absorption of the pads with the increase of l. This can be related, from one side, to more confined energy with the increase of the particle-dielectric interfaces at resonance, from the other side, the increase of the nanoparticles. Also, we have seen previously that in high inter-AuNPs coupling regime [START_REF] Maurer | Enhancing LSPR sensitivity of au gratings through graphene coupling to au film[END_REF][START_REF] Haes | A nanoscale optical biosensor: sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles[END_REF][START_REF] Jain | Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine[END_REF]. There is also a sharp decrease in the width of the resonances when the nanoparticles are distant from each other. The widening of the resonances results from the fact that the studied system becomes more and more close to a system composed of a metallic film when the width of the nanoparticles is increased while keeping the period of the constant network. Metallic gold nanoparticles characterized by the geometrical parameters, (l) to the diameter (h) to the height, and (a) the grating period of the nanoparticles, the nanoparticles are coated with a layer of graphene to thin thickness (d) in the context of this study, the refractive index graphene n g , and the whole is immersed in a homogeneous matrix of SiO x (refractive index n 1 = 1.45).

Comparing the results of the absorption spectrum calculated with the L-D model (d = 5 nm thick graphene layer) with the absorption spectrum of graphene results without (d = 0) that you had already published [START_REF] Galopin | Short-and long-range sensing using plasmonic nanostrucures: experimental and theoretical studies[END_REF] to the same structure.

We recall that in the frequency range of this study, we get the more given graphene layer that gives without graphene.

Fig. 2 shows the theoretical calculation results of the normalized absorption of the metal pins (structure SiO x /AuNPs/Graphene/SiO x )as Fig. 3. Maps of the electric field for monochromatic incident radiation at the wavelength of maximum absorption as calculated by numerical analysis of metal studs, on (a) SiO x /AuNPs/SiO x at λ max = 574.71 nm, (b) SiO x /AuNPs/Graphene/SiO x at λ max = 657.90 nm, (c) SiO x /AuNPs/Air at λ max = 537.63 nm, and (d) SiO x /AuNPs/ Graphene/Air at λ max = 636.94 nm, where l = 25 nm, a = 80 nm and h = 20 nm, and that for the thickness value graphene d = 5 nm. a function of frequency, and the result of the interaction of the studs with a light pulse incident [400-1000 nm] the values of the structural parameters are chosen: l = 30 nm, a = 80 nm and h = 15 nm.

In order to compare the results calculated with the layer of graphene and graphene-free, we can see a pretty good increase between the theoretical spectra with and without graphene layer, there has been a dramatic decrease in the resonance wavelength of the plasmon studs passing from 636.94 nm observed on SiO x /AuNPs/Graphene (red line) to 537.63 nm observed on SiO x /AuNPs (green line) (with an energy decay absorbed 0.62 to 0.37), for a thickness of the air detection medium. In the case where the pads (metal particles) are covered with 5 nm of graphene (pink line), as in the case of pads uncovered graphene (blue lines) with thickness SiO x detection medium. There has been a dramatic decrease in the resonance wavelength of the plasmon studs passing from 657.90 nm on SiO x /AuNPs/Graphene to 574.71 nm on SiO x /AuNPs (with a decrease of energy absorb from 0.64 to 0.57).

Furthermore, beyond the comparison results, with and without a layer of graphene, it is noted that in both there is cases a strong absorption of the studs to the 574.71 nm wavelength (without the graphene layer) and 657.90 nm when those are covered (d = 5 nm) to (h = 15 nm, a = 80 nm, and l = 30 nm). These wavelengths correspond to the excitation of the plasmon-polariton mode particle surface, confined at the interface between the particles and the surrounding dielectric.

Besides, in Fig. 2 are the absorption spectra of the system studied for different cases without graphene layer and with the graphene layer (5 nm). It is observed that the addition of graphene generates a loss of symmetry of the resonance bands due to their widening for the large wavelengths (λ > λ max ). This enlargement arises from the fact that the absorption of graphene increases with the wavelength ( =+ n C λ 3.0 3

g 1 , C 1 = 5.446 μm -1 ) [26]
. This is important for finding plasmon sensors with optimum properties. Indeed, sensitivity performance is not only correlated with a strong spectral shift in resonance due to a refractive index variation. They are also dependent on the spectral width of the resonance bands, since it is easier to spot the movement of the latter when they are fine.

The excited electrical field distributions on resonance, at a normal incidence structures, are represented by cards of electric field in Fig. 3(a-d), which indicate that this mode is a dipolar localized surface plasmon, whose hot spots (areas of high field strengths) are pushed to the top and lower corners of the AuNPs of the structure lacking the Graphene layer, and to lower corners of higher AuNPs graphene layer of the structure with graphene (Fig. 3a). The confined fields within the corners of the structure AuNPs of SiO x /AuNPs/SiO x with a strong localization on the lower surface of the AuNPs, compared to the structure AuNPs/SiO x /Air in (Fig. 3c) in which a strong location is observed on the upper and lower corners of the AuNPs [START_REF] Lee | Gold and silver nanoparticles in sensing and imaging: sensitivity of plasmon response to size, shape, and metal composition[END_REF][START_REF] Zeng | Graphene-gold metasurface architectures for ultrasensitive plasmonic biosensing[END_REF].

For the structure SiO x /AuNPs/Graphene/SiO x in Fig. 3b the fields are, basically, limited to the space between the graphene layer and the dielectric of detection which are lower than the AuNPs. It can be seen by comparing Fig. 3(a andb) that when the nanoparticles covered by 5 nm of graphene have hot spots at each of their corners and that the two hot spots in contact with the substrate have spread spatially. Moreover, the addition of graphene has resulted in an increase in the amplitude of the electric field in the interparticle space followed by a decrease in peak value intensity. This observed decrease in field strength is due to the coupling between the plasmon mode and the Fabry-Perot cavity modes connected to the presence of the graphene and SiO x layer presented above the gold nanoparticles [START_REF] Haes | A nanoscale optical biosensor: sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles[END_REF][START_REF] Zeng | Graphene-gold metasurface architectures for ultrasensitive plasmonic biosensing[END_REF][START_REF] Leem | Mechanically self-assembled, three-dimensional graphene-gold hybrid nanostructures for advanced nanoplasmonic sensors[END_REF][START_REF] Sherrott | Experimental demonstration of > 230 phase modulation in gate-tunable graphene-gold reconfigurable mid-infrared metasurfaces[END_REF].

In the structure (Fig. 3d) SiO x /AuNPs/Graphene/Air fields are supposed to have a change in the lower refractive index. On the one hand, the fields are partially confined to the dielectric detection and inside the AuNPs. On the other hand, this structure shows a lack of field distribution at the place where the top corners of the graphene bound in the middle of detection, and therefore, at the interface layer with air. The lowest optical index of air (n 2 = 1) compared to higher graphene (n g ) explains why this mode shows a lower location of the electric field in comparison with similar manufactured networks in mid-detection SiO x (n 1 = 1.45) [START_REF] Sherrott | Experimental demonstration of > 230 phase modulation in gate-tunable graphene-gold reconfigurable mid-infrared metasurfaces[END_REF][START_REF] Rechberger | Optical properties of two interacting gold nanoparticles[END_REF][START_REF] Goffard | Plasmonic engineering of spontaneous emission from silicon nanocrystals[END_REF][START_REF] Rahman | Design and numerical analysis of highly sensitive Au-MoS2-graphene based hybrid surface plasmon resonance biosensor[END_REF].

In this study, the effect of the graphene layer, registered on gold nanoparticles was studied by gradually increasing the thickness of the graphene layers of d = 0.34 nm at 5 nm.

Fig. 4a shows the range of absorption of the plots in function. The gold nanoparticles of the values of d = 0.34, 0.68, 1, 1.34, 2, 3, 4, and 5 nm, keeping all other parameters constant (a = 70 nm, h =20nm and l = 25 nm). There is an absorption 0.38 (d = 0.34 nm) to 0.56 (d = 5 nm).

To better understand this oscillation observed in Fig. 4a, the wavelength plotted plasmon response as a function of thickness (d)o f graphene layer, which has been calculated theoretically with the Drude-Lorentz model [START_REF] Zhan | Plasmonic fano resonances in nanohole quadrumers for ultra-sensitive refractive index sensing[END_REF] (Fig. 4b), Shows a significant linear increase in the resonant wavelength plasmon LSPR with the thickness of layer of Indeed, the wavelength remains constant at 537.63 nm for the starting value of the resonance wavelength on the curve representing LSPR does not depend on the graphene layer (d = 0 nm), and for d = 0.34 nm (single layer) and d = 0.68 nm (bi-layer). The increase of (0.68 nm ⩽ d ⩽ 5 nm) is solely due to the red shift of LSPR when keeping the other parameters constant nanoparticles.

The effect of the refractive index of the interface SiO x /AuNPs/ Graphene was studied by recording the wavelength shift when it is changed, in different proportions giving the refractive indices (n 2 = 1.00 square symbols for air, n 2 = 1.333 round symbols for water, and SiO x n 2 = 1.45 triangular symbols), the response of LSPR spectrum is usually described by Fig. 5 shows that the position of λ max moves to higher wavelengths with an increase in the refractive index. The variation of the position of λ max , shows a linear dependence in accordance with the refractive index of the surrounding medium. Sensitivity, defined as the ratio of the variation of the Plasmon's position band in the variation of the refractive index [START_REF] Haes | A nanoscale optical biosensor: sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles[END_REF][START_REF] Kaya | Sensitivity comparison of localized plasmon resonance structures and prism coupler[END_REF].

= - S λ n ∆ ∆ (nm RIU ) max 2 1 (3)
(Nanometer change unit by refractive index) is determined from the slope of Fig. 5 and increases if the thickness of the graphene layer interface decreases plasmonic (Table 2).

Fixing the length of particles l = 25 nm, the grating period a = 70 nm and height h = 20 nm particles and then varying the refractive index n 2 of the dielectric layers deposited on the thinnest layer of the graphene. Fig. 5 shows the change in resonance wavelength as a function of the thickness of graphene for different detection media n 2 = 1.00 (air), n 2 = 1.333 (water), and n 2 = 1.45 (SiO x ). There is the offset of the resonance wavelength LSPR (λ max ) for a same variation of n 2 is different according to the value of d. For d = 0.34 nm is noted that the resonance wavelength plasmon particles as a function of refractive index is 526 nm (air), 539 nm (water) and 546 nm (SiO x ). It is observed in Table 2, the sensitivity changes when going from d = 0 nm to d = 5 nm. Adding that in the case of depositing thinner thickness of graphene, there is a strong shift towards the red resonance, plasmon oscillation's amplitude increases almost linearly with d. Furthermore, it was explained that the sensitivity of nano-sensors based on this phenomenon is related to the displacement of the resonance wavelength of the plasmon response of the particles with d, that is to say in the regions of d where the variation of the oscillation amplitude is large, the sensitivity is optimal. This indicates a greater effectiveness of bio-detection with these structures.

The objective of this part is to study quantitatively using the FEM method, the influence of refractive indices n 1 and n 2 on the oscillation of the sensitivity of nanostructure to the plasmon resonance when d (graphene) varies. To do this, we start from the same system as before. The curve representing the evolution of the sensitivity (Eq. ( 3)) According to the thickness (d) can be characterized by the two nanostructures defined in Fig. 6:

In a second phase, several numerical simulations were carried out to determine how the sensitivity evolves according to the thickness of graphene deposited on the nanoparticles. The results obtained appear in Fig. 6. First, an increase in sensitivity is observed for graphene thicknesses between 0 and 2 nm for both nanostructures.

In fact, (Fig. 6a) S = 32.92 nm/RIU for d = 0 nm and S = 41.6 nm/ RIU for d = 2 nm, corresponding to an increase of about 26.37%. Fig. 6b, S = 199.76 nm/RIU for d = 0 nm and S = 376.85 nm/RIU for d = 2 nm, corresponding to an increase of about 88.65%. Then the sensitivity decreases from d = 2 nm, to reach 15.1 nm/RIU (Fig. 6a) and 287.96 nm/RIU (Fig. 6b) When d = 4 nm.

Fig. 6c represents the absorption spectra of the system studied for different thicknesses of graphene. It is observed that the addition of graphene generates a loss of symmetry of the resonance bands due to their widening for large wavelengths (λ > λ max ). This enlargement arises from the fact that the absorption of the graphene increases with the wavelength. This is important for finding a plasmon sensor with optimum properties. Indeed, sensitivity performance is not only correlated with a strong spectral shift in resonance due to a refractive index variation. They are also dependent on the geometric parameters of the gold nanoparticles (h, l and a) and the refractive index (n 1 and n 2 ).

Study of plasmonic response of metal particles as a function of geometric parameters of the gold ribbons, a, l and h

The behavior of the optical response of the particles, in particular, the plasmon resonance according to the parameters l, a, and h. The procedure was as follows: we first fixed the particle length l =25nm, and the grating period a = 70 nm, then varied the height h of the particles between 15 and 27 nm. This is a spectacular non-linear decay of the dabs plasmon resonance wavelength that goes from 876.71 nm to 535.56 nm (for a displacement of 341.15 nm), where h varies only from 5 nm to 40 nm.

This corresponds with the frequency at which the incident wave is strongly couples the free electrons of the particles, at the interface between the surface of the nanoparticles covering dielectric (graphene). That is to say, gold particle surface plasmon-polariton mode. The wavelength corresponding to this resonance is greatly shifted to the low lengths (toward the blue) wavelength with increasing h.

Second, plasmon mode behavior was studied according to the particle network period, and by setting all other parameters (l =25nm, h = 20 nm and d = 5 nm). Fig. 8a shows the study of the plasmonic response of the pads according to a. In this figure, the absorption spectrum is calculated a = 50, 55, 60, 70 and 80 nm.

In Fig. 8b, is carried changing the resonant wavelength of the plasmon pads, depending a. The lattice parameter was varied between 35 nm and 120 nm. There has been a dramatic non-linear decrease in the resonant wavelength plasmon studs passing from 988.41 nm to 575.28 nm (for a displacement of 413.13 nm), where a is only 35 nm to 120 nm.

For a < 50 nm, the resonance wavelength increases greatly due to the coupling of the nanoparticles by their evanescent fields (increased confinement of the electromagnetic field in the space inter-ribbons). On the other hand, for a > 110 nm resonance wavelength hardly varies. Ribbons are far enough apart so that their evanescent fields will almost cover more and tends to the plasmonic response of a single tape.

Third, plasmon mode behavior was studied according to the length of the tape, keeping constant the other geometric parameters, that is to This is a significant displacement of the wavelength of absorption corresponding to reply Plasmon of the studs to the long wavelength (low frequencies) from 545.84 to 977 nm.

In Fig. 9b, it is reported the development of the frequency of the absorption zeros, observed in Fig. 9a (plasmon response), according to the l. in a wider range of possible values between l = 15 nm and l = 55 nm. This figure shows more precise information on the behavior of the plasmon l. It can be seen that between l = 15 nm and l = 45 nm, that is to say between a situation where the inter-distance blocks is 55 nm and 25 nm respectively; the resonance wavelength increases almost linearly with l. Starting, of l = 45 nm, there is an abrupt nonlinear growth very important to the resonance wavelength plasmonic, up from 760 nm to 977 nm (near infrared), an increase of 10 nm only. These two points may be explained in part by the fact that between l = 15 nm and l = 45 nm. When the nanoparticles are weakly coupled, it results in a linear variation with LSPR. While that from l = 45 nm, when the nanoparticles are strongly coupled, and therefore, a non-linear variation of LSPR depending on l.

Conclusion

We have theoretically demonstrated the structural effect of gold nanoparticles covered with thin layers of graphene [0.34 nm, 5 nm] on the optical properties. The Lorentz-Drude model was used to calculate the optical signal plasmon resonances localized surface (LSPR), gold nanostructures (AuNPs) on the glass and covered with a thin dielectric layer of graphene and SiO x , indicates that this study provides a better understanding of systems based on resonant metal nanoparticles coupled with graphene layers. To validate the model, we compared the spectrum obtained through numerical simulations of LSPR to those previously released without a layer of graphene. From a practical standpoint, it opens the door to nanotechnology in a controlled and predictable manner. The spectral properties of systems based on metalgraphene nanoparticles to enhance their particular applicability for sensing applications. This study paves the way to highly sensitive sensors and improved biosensors.

Fig. 1 .

 1 Fig. 1. Typical diagram of the modeled structure, namely gold ribbons apart and characterized by a length l and height h. the tapes are covered with a thin layer of graphene.

Fig. 2 .

 2 Fig. 2. Computed absorption spectra of AuNPs, in the case where l = 25 nm, a = 80 nm, h = 20 nm and the thickness of the graphene layer (d = 5 nm), for two detection Milieus, SiO x (pink) and air (red). Or without a layer of graphene (d = 0 nm) for SiO x (blue) and air (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Fig. 4 .

 4 Fig. 4. (a) Change of absorption spectra versus the wavelength of SiO x /AuNPs/ Graphene/SiO x . The absorption is calculated for AuNPs of different thickness of a graphene d. The period of AuNPs is fixed at a = 70 nm, the height of AuNPs is fixed at h = 20 nm and the length of AuNPs at l = 25 nm. (b) Evolution of LSPR peak with change in graphene film thickness d. a = 70 nm, l = 25 nm and h = 20 nm.

Fig. 5 .

 5 Fig. 5. Shifting the LSPR peak wavelength depending on the refractive index of the surrounding medium, air (blue square), water (green circle) and SiO x (red triangle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Fig. 6 .

 6 Fig.6. Evolution of the sensitivity and absorption spectrum of the system studied according to the thickness of graphene covering the nanoparticles of gold. (a) Nanowires are characterized by l = 25 nm, h = 20 nm and a = 70 nm, the refractive index of the substrate on which they are deposited is n 1 = 1.45. The sensitivity was approximated by n 2 is the refractive index of the detection medium. The two media used to determine the sensitivity were indices n 2 = 1.000 and 1.45. (b and c) The nanowires are characterized by l = 100 nm, h = 15 nm and a = 125 nm, the refractive index of the substrate on which they are deposited is n 1 = 2.00 (ITO), the two media used to determine the sensitivity have n 2 = 1.000 and 1.333.

Fig. 7 .

 7 Fig. 7. (a) the absorption spectrum corresponding to the structure SiO x /AuNPs/Graphene/SiO x , for values of h = 15, 18, 20, 25 and 27 nm (height of the studs), other parameters are kept constant l =25nm a = 70 nm and d = 5 nm (graphene thickness deposited on the pads). (b) Change in the resonant wavelength corresponding to the absorption observed in (a), according to the height h of the metal pads on a wider range of values.

Fig. 8 .

 8 Fig. 8. (a) The absorption spectrum corresponding to the structure SiO x /AuNPs/Graphene/SiO x , for the values of a = 50, 55, 60, 70 and 80 nm (grating period), other parameters are kept constant l = 25 nm, h = 20 nm and d = 5 nm (graphene thickness deposited on the pads). (b) Change in the resonant wavelength corresponding to the absorption observed in (a), depending on the period of the array of metal contacts a, over a wider range of values.

  Fig. 7a gives the absorption spectrum calculated for several values of h = 15, 18, 20, 25 and 27 nm, a thickness of 5 nm deposited graphene. Fig. 7b shows the change in the resonance wavelength of the plasmon of the particles in terms of h, in a wider range of possible values between 5 and 40 nm.

Fig. 9 .

 9 Fig. 9. (a) The absorption spectrum corresponding to the structure SiO x /AuNPs/Graphene/SiO x , for values of l = 25, 30, 35, 40 and 45 nm (length studs), other parameters are kept constant have a = 70 nm, h = 20 nm and d = 5 nm (graphene thickness deposited on the pads). (b) Change in the resonance wavelength for the absorption observed in (a), depending on the length of the metal pads l on a wider range of values.

Table 1

 1 Gold (Au) Lorentz-Drude model parameters.

	Term	f m [rad/s]	ω p [rad/s]	ω m [rad/s]	Γ m [rad/s]
	m = 0	0.760	13.72 × 10 15	0.00	0.08052 × 10 15
	m = 1	0.024	13.72 × 10 15	0.6305 × 10 15	0.3661 × 10 15
	m = 2	0.010	13.72 × 10 15	1.261 × 10 15	0.5241 × 10 15
	m = 3	0.071	13.72 × 10 15	4.538 × 10 15	1.322 × 10 15
	m = 4	0.601	13.72 × 10 15	6.538 × 10 15	3.789 × 10 15
	m = 5	4.384	13.72 × 10 15	20.24 × 10 15	3.364 × 10 15

Table 2

 2 Simulated wavelength and sensitivity to gold nanoparticles for different graphene layer thicknesses of structure SiO x /AuNPs/Graphene/SiO x , other parameters are kept constant l = 25 nm, h = 20 nm and a = 70 nm.

	Graphene thickness (d) (nm)	λ max (nm)	Normalised absorption	⎜ ⎛ ⎝ Sn m R I U = λ ∆ max / n ∆ 2	⎟ ⎞ ⎠
	0	537.63	0.38	32.93
	0.34	537.63	0.38	59.81
	0.68	537.63	0.38	26.47
	1	540.54	0.39	39.92
	1.34	543.48	0.40	8.92
	2	549.45	0.41	41.60
	3	558.66	0.43	36.31
	4	568.18	0.44	15.10
	5	578.03	0.45	23.76