Surface plasmons in suspended graphene : launching with in-plane gold nanoantenna and propagation properties

Graphene physics and plasmonics are two fields which, once combined, promise a variety of exciting applications. One of those applications is the integration of active nanooptoelectronic devices in electronic systems, using the fact that plasmons in graphene are tunable, highly confined and weakly damped. A crucial challenge remains before achieving these active devices: finding a platform enabling a high propagation of Graphene Plasmons Polaritons (GPPs). Suspended graphene presenting ultrahigh electron mobility has given rise to increasing interest. We numerically studied the plasmonic properties of suspended graphene. We propose a hybrid configuration and a set of conditions to launch graphene plasmons via an in-plane gold nanoantenna, for micrometric propagation of surface plasmons in suspended graphene. 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Introduction
Electronic devices, used in telecommunications and information processing, exhibit inherent limitations due to the materials electronic losses and noises which have consequences on their conductivity.To remove this barrier and keep on increasing electronic devices performances, the technological trend is to find another information carrier that is able to replace or complete electrons, among which photons appear to be the ideal candidate.Until now, however, despite recent promising demonstrations of strong near-field confinement in hybrid Silicon on Insulator structures, e.g [1], photonic devices have been diffraction-limited by the light wavelength which leads to some difficulties when components are scaled down to the nanoscale.On the other hand, surface plasmons photonic has turned out to constitute a promising bridge between nanophotonics and nano-electronics resulting in many application in energy transfer [2], in molecular chemistry [3], in biology [4] etc.
Recently, graphene [5] has been proposed as a good candidate to bridge the gap between electrons and photons for a new generation of optoelectronic devices [6].While many promising properties like its ultimate thickness, transparency, ultra-high electronic mobility or mechanical strength have concentrated the efforts of numerous groups, a few pioneering groups have recently started considering and studying graphene for its optoelectronic [7] and plasmonic properties [8], opening new routes to ultimate optoelectronic nanodevices based on electron-photon interactions.
From all the pioneering theoretical [9][10][11][12] and experimental [13,14] studies, it now appears feasible to use light as an information carrier by manipulating it through Graphene Plasmons Polaritons (GPPs).The promises of graphene for plasmonics lie in the high confinement and weak damping [15] of sustained plasmons as well as in the possibility to electrostatically tune them simply by applying a voltage.The main asset would be to enable miniaturization of photonic component without facing usual thermal limitations thanks to low graphene electronic losses.The applications could therefore be numerous.
However, two challenges remain.In most of the cases cited above, the studies assumed having an excellent control over the amount of charge carriers located in different part of the layer, by applying an electric field.Unfortunately, this remains a big challenge since the electric field needed locally is extremely strong.Therefore, many research groups are working on finding an experimental system that could lead to on-demand variations of graphene optical and electronic properties at the micrometric scale.This would typically requires sufficient gate tuning using a high dielectric strength substrate [36].
A second challenge is to find a platform that also enables GPPs to propagate over long distances.The usual substrate used to study graphene (SiO 2 ) induces perturbations and corrugations of the graphitic lattice [37], trapping charges [38], and inducing external and/or internal scattering [39][40][41][42].In the framework of waveguides, it is known that the traditional Insulator/Metal/Insulator (IMI) heterostructure is able to support long distance plasmon propagation [43] notably due to the structure symmetry.There exist several ways to fabricate such systems in a laboratory.Recent studies on encapsulated graphene in h-BN, a substrate known to be suitable for graphene [44,45], showed promising results [14,46,47].
An easy and effective way to obtain such systems could be to suspend graphene [40,48] as this would reduce the losses of GPPs [49].This approach is expected to lead to an improvement of the electronic properties, increasing the mobility [50,51], although an intrinsic limitation related to flexural phonons may occur [52,53].Nevertheless, mobility as high as 38 m 2 s −1 V −1 has been achieved in suspended graphene for low Fermi energies [54].Enhancing mobility in graphene has been giving rise to a lot of interests and works.Inducing strain in graphene [55], suspending graphene in liquid with high static dielectric constant [56], introducing polymer nano dots spacers [57] or functionalizing graphene oxide [58] are so many routes that could lead to a significant improvement of the mobility.
So far, the observation of plasmonics behaviors in suspended graphene has been shown through Raman measurements [59], or with Fourier-Transform-InfraRed spectroscopy on silicon pillars and on aluminium oxide [60,61].Quasi free standing graphene, achieved by placing a nanometric spacer between the graphene and the substrate, has also been studied through scattering-type scanning near-field optical microscopy (s-SNOM), and an improvement of the GPPs propagation length has been demonstrated [57].
In this paper, we present a full numerical study allowing us to discuss the possibility of launching, via in-plane gold nanoantennae, GPPs on suspended graphene.We discuss the impact of the antenna geometry with and without graphene and propose a realistic experimental configuration that could permit GPPs to propagate over microns.The challenge is to provide photonic functions to suspended graphene whose electronic and mechanical properties have already been widely investigated.
We first expound the different ways to compute graphene in Finite Difference Time Domain (FDTD) method, as well as the strong influence of graphene doping and mobility to make possible propagation over micrometric distances with supended graphene.Then, we investigate and optimize the launcher of plasmons onto the GPPs propagation.Finally, we propose an experimental configuration which should in the near future confirm the possibility of making GPPs propagate over long distances, and thus making possible the design of optoelectronic devices based on suspended graphene.

GPPs and the three main parameters ruling the plasmons propagation
Graphene is a 2D material, and as such it is not trivial to simulate using FDTD.Its optical properties are usually presented in terms of optical conductivity, which can be expressed using the Kubo formula [10].Two main methods exist to implement it in FDTD.
The first one attempts to use full Yee cells to describe it, but difficulties arise due to their inherent 3D nature.This approach requires converting the 2D conductivity into a volume anisotropic permittivity, assuming that graphene has a finite thickness.A study [62] has been done in order to know if this procedure is safe or if it will profoundly change the optical properties of the graphene sheet.We first tried following this path, and numerical results confirmed the accuracy of this approach.However, this method requires a local spatial discretization finer than the nanometer, while our simulations involve wavelengths and structures bigger than the micron, leading to huge computational domains and non-uniform meshing.Moreover, the stability criterion of the FDTD then requires an extremely short time discretization, and a huge number of iterations is thus needed to complete a single simulation.Therefore, this method is highly CPU-intensive and numerical aberration may occur.
A second approach aims at describing graphene as a charged surface boundary condition applied only to a fraction of a Yee cell [12,63].This condition is of the general form: Where n is the vector normal to the graphene sheet, H 1 and H 2 the magnetic fields on each side of the sheet, σ the conductivity and E t the tangent electric field.Through this, there is no need to discretize space (and time) to extreme sizes, and this was found to lead to an increased stability of our simulations.This is the method that will be used in this paper.The conductivity, σ(ω, Γ, T, µ c ), is defined as the sum of two terms, namely the intraband term σ intra (ω, Γ, T, µ c ) and the interband term σ inter (ω, Γ, T, µ c ), which can be expressed as [12] : With f d the Fermi-Dirac distribution, E the energy, µ c the chemical potential, e the elementary charge, Γ the scattering rate and ω the excitation pulsation.The scattering rate is a phenomenological physical quantity related to the relaxation time τ of electron in graphene by 2Γ = τ −1 with the latter approximated as follows [64]: With v f the Fermi velocity and µ the carrier mobility.The interband term is determined numerically whereas the intraband term of the conductivity is solved analytically as: This 2D graphene model is convenient because it allows us to simply change the model parameters (eg.Fermi energy, scattering rate or relaxation time, number of graphene layers, temperature) in order to fit the characteristics of samples.Graphene plasmons (GPs), which are similar to Surface Plasmons in ordinary metals, are associated to a electromagnetic surface wave that can be excited by matching the two main parameters of the mode: the energy and the wave-vector.The modes are usually described thanks to a dispersion equation.In the case of a free current lying in between two dielectrics media, the Maxwell's equations, restricted to the TM modes, lead to [65]: Where 1,2 the dielectric constants of media sandwiching the graphene, 0 the vacuum permittivity, k p the GP wavevector, and k 0 = 2π/λ 0 the wavevector in vacuum where λ 0 is the wavelength in vacuum.Considering the non-retarded regime, which is suitable for 2D Electrons Gas in general, and particularly for graphene, we can get the dispersion relation of graphene plasmons in the case of TM modes: For a sake of clarity, it is possible to insert into Equ.(7) a simplified Drude-like model of the conductivity.Although this approach is restricted to the infrared spectrum range, it gives a more intuitive link between the different parameters that play a role in the propagation of plasmons in graphene.An important remark is that the ensuing expression (Equ.8) is valid only for µ c >> k B T, which is the case for these simulations, and we can approximate µ c ≈ E F , where E F is the Fermi Energy.We get the following dispersion relation: Where α ≈ e 2 /4π 0 c ≈ 1/137 is the fine structure constant.From this perspective, it is relevant to have a look at the key parameters which are the wavelength of the GPPs and the propagation length.They are given by λ p = 2π/Re(k p ) and L p = 1/(2Im(k p )) respectively.Considering suspended graphene, i.e. a layer of graphene sandwiched in air, we get: From this set of equations, we can notice that the propagation length depends on three main parameters: the Fermi level E F and the electron scattering relaxation time τ that is directly proportional to the mobility of the charge carrier µ.It is interesting to notice that the wavelength of the plasmons in suspended graphene is directly proportional to the Fermi energy in the infrared region.S-SNOM measurements, which have been a relevant tool for accessing the wavelength of the plasmons in graphene, can actually be used to probe locally the Fermi energy at high doping in the infrared region.

Use of plasmonic antenna for GPPs excitation
Originally, GPPs excitations were achieved by the use of a laser illuminated metal tip in the scanning near-field optical microscopy configuration [16,17].Although the resulting scattering angular spectrum was broad, only a small part of it was exploited for successfully launching the GPs.
Another way of launching plasmons at metal/dielectric interfaces relies on the use of prisms or gratings [66] to efficiently convert the incident wave-vector into the in plane graphene plasmon one.In our simulations, we use Au nanorods as nanoantennae to study the GPPs launched perpendicularly to the main axis of the nanorod in order to anticipate the possibility to maximize the power of the GPPs launched using a grating.It is indeed possible to place side by side many nanoantennae together, and obtain an enhancement effect thanks to the grating configuration.As a first step, in this paper, we will only rely on the angular spectrum generated by a single rod.Compared to an out-of-plane tip, the in-plane resonant launchers allow one to envisage future integrated chips.Additionally, the resonance enables the enhancement of the angular spectrum components.
As a preliminary study, the local near field enhancement of single rods, with and without graphene, is investigated for a wavelength of 11 microns.It has been shown that the plasmons damping from SiO 2 substrate is the lowest [16] around this wavelength.The configuration with graphene is depicted in Fig. 1.A gold nanorod having a rectangular cross section lays on a Si0 2 /Si substrate.The nanorod is surrounded by both suspended and supported graphene for comparison.The nanorod was illuminated with a pulse that propagates perpendicular to the sample plane, along z.The incident light was linearly polarized with the electric field parallel to the rod long axis (x axis).This pulse has a wide-band spectrum going from 8 µm to 12 µm, with a central wavelength around 10 µm.The optical properties of the gold nanoantenna are characterized numerically in order to fit the energy and momentum conditions reminded above.The field analysis was performed using the Finite Difference Time Domain method (Lumerical FDTD).While the optical properties of graphene are described by the above equations, the optical properties used for gold, Si and SiO 2 are obained from Palik handbook.
Following [67,68], the length L and the width W of the rod are first set to 2.9µm and 0.6µm respectively.The nanoantenna section is therefore rectangular, and its thickness is set to 100 nm.Figures 2 show electric field intensity around the rod under illumination, without any graphene (Fig. 2(a)), and with graphene (Fig. 2(b)).The near-field distributions displayed on these figures unambiguously show the dominant mode excited at this wavelength is dipolar in nature.On Figure 2(b), two calculated top-view half-maps of the electric field intensity at the graphene layer plane are shown.For y > 0, graphene is suspended, meaning that the graphene is surrounded by air.For y < 0, graphene is sandwiched between air and the SiO 2 substrate.The rod is localized at y=0 nm, acting as a resonant dipole along L. The graphene is defined via its 2D conductivity by a Fermi level of 0.2eV, a temperature of 300K and a mobility 1 m 2 s −1 V −1 .The parameters set for the graphene will be discussed in the next section.
In figure 2(b) the spatial decay of the intensity of propagating plasmons in graphene can be discussed.Whereas suspended graphene leads to clear propagation of GPPs (Fig. 2(b), top), it is not possible to distinguish this propagation in the case of graphene supported by SiO 2 substrate (Fig. 2(b), bottom).In the latter case, we clearly see a dipole-like resonance of the antenna at 11 µm, comparable to Figure 2(a), that does not launch any efficient GPP (in terms of propagation length).
It is interesting to notice that the intensity tends to vanish along a vertical line (x=0) where destructive interference occurs between the phase shifted graphene plasmons that have been launched by the two out-of-phase antenna extremities.Such effects could open new routes to GPPs designing and engineering.Over a second phase, the question of the dimension of the nanostructure have been addressed.So far, the geometrical parameters were chosen by considering recent studies [67,68], where the main parameter used for optimizing the coupling is the electric field average.This factor reflects the significant near-field enhancement, consequence of the Local Surface Plasmons Resonance (LSPR) of the antenna, that is most likely to launch strong plasmons in graphene.As shown before, the current geometry could be used for our purpose, but [67] only optimized the GPPs coupling efficiency through variations of the antenna length.The rod width is another parameter that could be explored to increase the coupling efficiency even further.As such, we followed the same path as [67], but performed a parametric study of the near-field average |E|/|E 0 | for different lengths and widths of rods lying on a Si0 2 /Si substrate without graphene.The results are shown on Fig. 3.The near-field average |E|/|E 0 | was taken at a 10 nm distance from the rod surface.The big axis L was variating from 1.6 to 3.2µm, whereas the width W was variating from 0.05 to 1.15 µm.A maximum of the electric near-field average is found for a micro antenna length of 2.5µm and a width of 50nm.Taking advantage of this results, simulations with suspended graphene are done to compare the new geometrical parameters coupling efficiency to the literature, and to study GPPs modes in details.
Figure 4 shows the spatial Fourier Transform of the near-field for the case of two different antenna geometry, with suspended graphene.Figure 4(a) shows the case of geometry dimension taken from [67,68], that is the Fourier transform of the near field plotted in Fig. 2(b) top-half, for positive (k x ,k y ).It turns out that the nanorod launches two main surface plasmons: one in the x direction (peak 2), parallel to the antenna axis, and one in the y direction (peak 1), perpendicular to the nanorod axis.They are illustrated by the two maxima along the quarter-circle of radius k 2 x + k 2 y /k 0 = |k p |/k 0 = λ 0 /λ p = 43.These results were easily predictable from Fig. 2(b), where the two main GPPs propagations can also be observed, parallel and perpendicular to the rod long axis.
Figure 4(b) is using the geometry found in our parametric study (Fig. 3).It can be observed a strong enhancement of the peak 1, corresponding to GPPs launched in the y direction, whereas the peak 2 disappears (GPPs launched in the x direction).It is concluded that the new dimensions of the rod, following the investigation of the electric near field average (Fig. 3), lead to a stronger coupling to the GPPs launched in the y direction, resulting in an increased effective propagation length in this direction.In the future, the coupling efficiency of graphene plasmon propagating along the y direction could be also strongly enhanced by using a set of parallel nanoantennae to form a resonant grating.
However, the coupling may still not be optimal.After all, the optimization was performed without graphene, by using the average field exaltation as a criterion.Yet as shown here, the true value of the coupling efficiency can only be found by actually computing the whole system, and investigating the Fourier transform.As such, we decided to refine our optimisation once again.For this purpose, spatial Fourier Transforms of the electric fields maps at 11 microns were performed for different lengths and widths of nanorod (the height remained fixed at 100 nm).The results of this final parametric study are shown in Fig. 5. Figure 5(a) displays the intensity of the Fourier transform peak (Fig. 4(a), peak 2) of the graphene plasmons launched towards the x direction and Fig. 5(b) is the one for the y direction (figure 4(a) and 4(b), peak 1).We conclude that the best shape for the gold nanoantenna is to be as thin as possible (needle-like, see white crosses in Fig. 5), 50 nm here, with a length around 2.75 µm, in order to efficiently launch a plasmon perpendicular to rod long axis.While the average near field intensity enhancement of the antenna may not be the highest for this shape (see Fig. 3), it gives a stronger coupling with the plasmons in y direction.For comparison, the black crosses in Fig. 5 indicate the geometry used in the papers [67,68], Fig. 4(a), while the grey crosses correspond to Fig. 4(b) and are the result of our first parametric study (Fig. 3).As far as plasmon propagation is concerned, the initial geometry (black cross) constitutes an intermediate case where both x and y directions are possible, and the field energy is thus distributed into both.4(a) as an example ), in the x direction and in the y direction respectively, as a function of the length and the width of the nanorod.The black cross indicates the dimension of nanoantenna used in papers [67,68], picked up from Fig. 4(a).Grey cross corresponds to the antenna dimensions resulting from the parametric study Fig. 3. White cross is showing optimum dimensions for launching plasmon in y direction.
In the case of suspended graphene, the theory, from Equ. ( 8), shows a large wavevector mismatch between the incident light (characterized by k 0 ) and the Graphene Plasmons (characterized by k p ) in this range of frequency, illustrating the need for an antenna.At 11 µm, we can calculate the localization parameter, or effective index, λ 0 / λ p , which varies as a function of the Fermi level of the graphene layer.For E F = 0.2eV and a mobility of 1 m 2 s −1 V −1 , we find real (n e f f ) = 42.3 from the dispersion relation (Equ.( 7)), whereas the FDTD simulation gives an average value of 43 (±1.1) from the Fourier transform maps.Moreover, the propagation length L p is about 558nm theoretically (from expression ( 7)) whereas we find 532nm in our simulation.It is interesting to notice that we have here already a longer propagation length as compared to SiO 2 supported graphene.Using data from [69], the longest propagation length would be less than 300nm for SiO 2 -supported graphene.
Figure 6 shows the optimized system for GPPs launched along the y axis, with the antenna geometry defined in Fig. 5 (white crosses).Compared to figure 2(b), figure 6 shows clearly that a rod with a length of 2.75µm and a width of 50nm can launch stronger GPPs that propagates on suspended graphene over a distance nearly 3µm away from the rod, normalized on the incident excitation.Again, supported graphene (y<0) does not enable significant GPP propagation.
As we have shown in this study, the geometry used in [67,68] does allow for a non-negligible Fig. 6.Map of the electric near-field intensity, for an excitation of 11µm, with a limit up on the color scale in order to see the plasmon propagating through graphene.The nanoantenna is on the center of the map.The graphene is set with a fermi energy about 0.2eV and a mobility of 1 m 2 s −1 V −1 .The dimensions of the rod are optimized for strong plasmons in y-direction : L=2750nm and W=50nm.
GPP coupling efficiency.Yet, that geometry had a fixed rod width, which could be optimized.Ultimately, our work showed that an accurate optimization of the coupling efficiency requires simulating the full system, and investigating the energy the plasmon actually carries through Fourier Transforms of the near field.Quantitatively speaking, this method is superior to investigating the average field only, as the average near field contains many k components that cannot be coupled to a GPP.However, the average field criterion needs not to be dismissed entirely.As we showed here, it still leads to a good first order approximation of the optimal coupling efficiency.Unlike our criterion that requires numerical methods to compute the full system, the average field criterion can be used with analytical or pseudo-analytical models, or numerical methods or softwares that are unable to simulate the full system, graphene being fairly hard to simulate.
A third parameter that has not been explored is the nanoantenna thickness, but this parameter also affects the distance between the graphene sheet and the substrate, and can then not only affect the antenna behavior, but also the properties of graphene itself.

Designed system based on GPPs
In equation ( 9), we can notice the importance of the Fermi energy for achieving a long propagation.However, it has been shown that suspended graphene is free of intrinsic doping [70].Suspended graphene is thus characterized by a limited value for E F while high mobility constitutes a clear asset that has been discussed in many published papers [54,71,72].The design of a nano-opto-electronic system based on suspended graphene should take into account this issue.Therefore, it is important to engineer a system offering the possibility to inject charges through electrostatic gating.On SOI substrates with 300nm of SiO 2 layer, it has been shown the possibility to apply a difference of potential Vg higher than 100 V [13].Modeling a capacitor between graphene and a Si-doped substrate, with two dielectric layers in between (SiO 2 300 nm, air 100 nm, see inset of Fig. 7), we can estimate a Fermi energy through: Where t air and t SiO 2 are the thicknesses of the air gap and the SiO 2 layer respectively and C g is the total capacitance.We find E F ≈ 0.2eV, which justifies the value used in the previous section.
In addition, the gate potential that is applied between the graphene and the substrate will bend the graphene, resulting in a deflection height h 0 that is a function of C g , and therefore of the dielectric used.It is also a function of the potential applied V g , and of the length of suspension L. At high V g , we can obtain the deflection as [6] : With E the Young's modulus in graphene which is about 1TPa, L the length of the trench, t the thickness of graphene, which is taken as 3.410 −10 m, and P the electrostatic pressure.Let us consider the case where the graphene is suspended straight over a 100 nm deep gap, (therefore with a maximum deflection h 0 of 100nm), and a gate voltage V g of 100V.This enables us to design a trench of L=1.5µm, using Equ.(14).Taking into account the penetration depth of the plasmons in air which is 1/Re(k p ) ≈ 41nm, we have a maximum deflection depth h 0 = 60 nm, and a maximum suspension length around L = 1µm so that the plasmon experiences damping from the subtrate that is as weak as possible.
As a result, we propose the design (see Fig. 7) of a realistic sample taking into account the different experimental parameters discussed above.On a SiO 2 (300nm)/Si substrate are lying gold structures, which will also play the role of suspension support for graphene, including a nanorod (antenna) for launching GPPs.The thickness of the gold support and the gold antenna is set at 100nm.The gold nanoantenna is embedded in a gold nano-slots system perpendicular to the rod long axis.Gold slots, of 300nm of width, are designed to both suspend graphene and guide GPPs perpendicular to the rod long axis, opening the route to future nanophotonic circuitry based on suspended graphene.
Four slots are designed: two per extremities, in positive and negative y-directions.The extreme left and right parts of the gold constituting the slots are designed to be used as electrodes, inbetween which voltage can be applied to generate current in gold.High temperature induced in graphene by high current (joule effect) have shown an improvement of the mobility in suspended graphene, by evaporating the thin layer of pollutants stuck on the carbon sheet after graphene transfer.In fact, electric current is one of the most efficient way to heat the graphene up and clean it, in order to get the best mobility in suspended graphene [54].Finally, a difference of potential V g is applied between the Si doped substrate and the graphene, in order to raise the Fermi energy.Fig. 8. Map of the electric field intensity, for an excitation of 11µm, with a limit up on the color scale in order to see the plasmon propagating through graphene.The nanoantenna is on the center of the map (y=0).We have set the graphene with a Fermi energy of 0.2eV and a mobility of 1 m 2 s −1 V −1 .Suspended graphene should be free of charge impurities, strain, corrugation and remote interfacial phonons induced by the substrate.The mobility, which is the physical value taking into account those effects, would increase up to one order of magnitude [51] in suspended graphene, as compared to the one on Si/SiO 2 substrate.However, studies predicted that the graphene mobility at T>10K could not exceed 4 m 2 s −1 V −1 [52,53] due to flexural phonon.Recent reports measured a mobility of 1.5 m 2 s −1 V −1 [71], up to 38 m 2 s −1 V −1 [54] in CVD suspended graphene at low Fermi level.Therefore, 1 m 2 s −1 V −1 for a Fermi energy of 0.2eV in suspended graphene is reliable.
Similarly to Fig. 5, we simulated this system, and obtained the map of the electric field intensity shown in Fig. 8.In the middle of the map is laying a gold nanorod of 2.75µm of length, 80 nm of width, and 100nm high which is launching plasmons perpendicular to the long axis ( x axis ) of the antenna, i.e. along the y axis.As we have seen in Fig. 5, the width of the rods should be as small as possible.However, the height of the nanorods is set at 100nm in order to safely suspend graphene.For mechanical stability reasons when the rod is fabricated from e-beam lithography, it is better to consider a width comparable to the height of the structure.Therefore, we choose a width of about 80nm.The length of the rod corresponds to a resonance around 11 microns, where graphene plasmons exhibit an optimal propagation.While figure 8 reveals the micronic propagation of the total photonic intensity, Fig. 9 provides more informations.Two modes are appearing in the channel.
Figure 9(a) shows a TE-like mode where we have a strong intensity of the x component of the electric field.This mode is partly supported by the edges of the gold slots and is likely to result from coupling between GP and sharp metal edges.Properly speaking, it does not correspond to any pure GP modes.
Figure 9(b) shows a TM-like mode, which is the GP mode, with a strong intensity of the y component of the electric field.That is the surface plasmon mode of interest.It is worth noticing that the associated wavefronts, shown in the bottom of Fig. 9(b), are clearly defined by the phase giving a GPPs wavelength λ p ≈ 265nm, that is to say an effective index n e f f ≈ 41.5.Again, the plasmons propagation length can be calculated using the exponential decay of the electric field along the slab.In case of Fig. 9(b) we find L p higher than 525nm.The reduction of the propagation length is due to the confinement of the GPs path by the golden slots.Reducing the width of the slot decrease the propagation length.
As pointed out above, higher propagation length in supported CVD graphene, fabricated on industrial scale, could be achieved with higher Fermi energy.In the case of chemically doped graphene, taking data from [69], we find a propagation length of 1 micron in suspended graphene, as compared to the 300nm with graphene on SiO 2 , using the same data.
Moreover, it worth noticing that the propagation length is a conservative value.For example, the propagation length for the system described above is about 540nm.However, as we can observe Fig. 9(b), the intensity of the electric field decreases to the intensity of the incident light at a distance of 2 microns, that is to say about 4 times the propagation length.This phenomenon results from the field enhancement at the launching resonant nanoantenna.
Finally, it is important to notice that the simulations here do not take into account the interaction between graphene and metal.It is known that metal can transfer charges at the vicinity of graphene, which could lead locally to highly doped graphene, as high as 0.4eV [73].As we have seen in Equ.(9), the propagation length of graphene plasmons is directly proportional to the Fermi energy.Therefore, it could lead to a strong increase of the GPPs propagation.

Conclusion and perspective
The initial aim of this article is to lay the foundations of plasmon propagation for the development of future graphene-based optoelectronic devices.Ideally, graphene plasmons in the infrared suffer low metallic losses as compared to noble metals.This is due to its excellent electronic properties.However, substrate implies impurities and defects that affect the transport properties of graphene.Also, the use of a substrate induces a strong damping due to a high optical energy dissipation.In this context, suspended graphene appears to be a promising route for the design of future opto-electronic devices.
Different configurations were studied.They are based on the use of a gold resonant antenna that launches graphene plasmons.Using a realistic value for the Fermi energy, we have shown that the plasmon propagation length in suspended graphene is significantly l arger t han i t is in supported graphene.As a result, a realistic design was proposed.It is based on suspended graphene.Obtained results illustrate the concept of plasmon propagation over large distances and the opening of technological perspectives in optoelectronics.In particular, the designed device is based on gold slots whose role is threefold: i) they are used as as electrodes to clean suspended graphene sheet by Joule effect, ii) they mechanically support the suspended graphene, iii) they present an in-plane structuration to spatially guide and squeeze the propagating plasmon wave.
In the near future, using heterodyne scattering type IR-SNOM [74], we expect to observe plasmons launching and to measure the propagation length.We also expect to tune the GPPs through gate doping in order to create photonic switches and logic gates.

Fig. 1 .
Fig.1.Illustration of a plasmonic antenna lying on a SiO 2 /Si substrate + graphene.A trench is etched at the vicinity of the golden rod through the SiO 2 /Si substrate.A monolayer of graphene is deposited on the top of the trench, and a gold nanoantenna is deposited on the edge of the trench.A part of the graphene is therefore suspended.

Fig. 2 .
Fig. 2. Comparison of the response of a gold nanoantenna, lying on Si/SiO 2 substrate, without (a) and with (b) graphene, through electric field intensity maps.(b) The graphene is suspended on the half-top, and supported on SiO 2 substrate for the half-bottom.The color scale is clamped in order to see the plasmon propagating through graphene.The 2.9 µm long, 600nm width, gold nanoantenna is in the center of the maps, at y=0, from x=-1.45 to x=1.45 µm.The wavelength excitation is 11µm.

Fig. 3 .
Fig. 3. Map of the electric near-field average, |E|/|E 0 |, calculated at the vicinity of rods of length L and width W, without graphene, for an excitation at λ 0 = 11µm.

Fig. 4 .
Fig. 4. Maps representing |f f t(E x )| 2 + | f f t(E y )| 2 + | f f t(E z )| 2 ,the spatial Fourier Transform of the near field in the reciprocal space, with positive (k x ,k y ), for suspended graphene, in the case of two different rods.(a) The rod length and width are set as L=2.9µm and W=600nm, according to[67,68].(b) A rod of length L=2.5µm and width W=50nm is chosen from the parametric study Fig.3.The graphene is set with a Fermi energy about 0.2 eV and a mobility of 1 m 2 s −1 V −1 .

Fig. 5 .
Fig. 5. Maximum intensity of (a) the peak 2 and (b) the peak 1, the plasmons wavenumber k p (see Fig.4(a) as an example ), in the x direction and in the y direction respectively, as a function of the length and the width of the nanorod.The black cross indicates the dimension of nanoantenna used in papers[67,68], picked up from Fig.4(a).Grey cross corresponds to the antenna dimensions resulting from the parametric study Fig.3.White cross is showing optimum dimensions for launching plasmon in y direction.

Fig. 7 .
Fig.7.Illustration of the system studied.A plasmonic antenna is lying on a SiO 2 /Si substrate.Gold nano-slots gather and propagate the plasmons on suspended graphene.The slab has also a function of electrode for doping suspended graphene.Gold, Si and Si0 2 are represented in yellow, gray and blue respectively.Inset : a schematic side view along the x axis of the graphene, golden slot and substrate.The deflection height h 0 is also represented.

Fig. 9 .
Fig. 9. Maps of the intensity (on the top) and the phase (on the bottom) of the (a) x component of the electric field, and the (b) y component of the electric field of the system described in Fig. 7