Controlling spoof plasmons in a metal grating using graphene surface plasmons

Controlling spoof plasmons in a metal grating using graphene surface plasmons Eduardo J. C. Dias∗and N. M. R. Peres∗ Department of Physics and Center of Physics, and QuantaLab, University of Minho, 4710–057, Braga, Portugal E-mail: edua[email protected]; p[email protected] Abstract Spoof plasmons mimic noble metal plasmons. The equivalent of the plasma frequency is an energy scale imposed by the geometry of the metal grating upon which they propagate. In this paper we show that the dispersion of spoof plasmons in the THz can be controlled placing a doped graphene sheet on the proximity of a metallic grating, adding more versatility to this type of system. We develop a semi-analytical model, based on a perfect-metal diffraction grating. This model allows to reproduce well FDTD calculations for the same problem but with much less computer time. We discuss the optical properties of the system covering a spectral range spanning the interval from the THz to the mid-IR. It is shown that the system can be used as both a perfect absorber and a sensing device. For illustrating the latter property we have chosen different alcohols as analytes. The frequency at which perfect absorption appears can be controlled by the geometric parameters of the grating and by the value of the Fermi energy in graphene. The theoretical results predicted throughout this work can be verified experimentally in the future. 1 arXiv:1708.01794v1 [cond-mat.mes-hall] 5 Aug 2017 Keywords Spoof plasmons, Graphene plasmons, Perfect absorber, Sensing 1 Introduction It is a well known result that a perfect flat conducting surface does not support surface plasmon polaritons (SPPs).1A real metal/dielectric interface, on the other hand, does support SPPs, but their energy is of the order of the plasma energy of the metal, which, for most good plasmonic metals (i.e. with relatively low losses, like gold or silver) is in the ultraviolet spectral range,2which is an energy scale too high for many sensing applications. An usual method to excite low-energy plasmons in a perfect metal is to introduce periodic structures (namely grooves or holes) in its surface. Under these conditions, the corrugated system supports surface modes which, strictly speaking, are not SPPs (because those are excited in flat interfaces), but effectively mimic their properties, and for that reason they are usually referred to as spoof plasmons (or spoof surface plasmons, SSPs). These plasmons were firstly studied and named by Pendry et al. in 2004, who showed that these solutions were equivalent to the ones retrieved from a flat metal/dielectric interface with the permittivity of the metal being modelled by an effective permittivity dependent on the geometry of the grooves.3Using that method, the authors were able to calculate an approximate dispersion relation for these plasmons. Later, this method was generalized to show that spoof plasmons are supported by real periodically-grooved metals,4and different authors have also suggested that spoof plasmons can be excited in small structures with only a few unit cells.5 The advantage of this type of plasmons is that their energy depends highly on the geometry of the grooves, and particularly on their depth. In fact, using the effective permittivity method, one finds that the effective plasma energy of the corrugated perfect metal —which 2 allows the estimation of the spoof plasmons energy— is given by6 ωeff p=πc 2h√ε,(1) where his the depth of the grooves and εthe permittivity of the medium inside the grooves. This result (which will be rederived in this work, using a different method) suggests that, in general, the scale of the grooves’ dimensions has a substantial impact on the frequency of the spoof plasmons. For this reason, these plasmons have been regarded recently as a promising alternative to traditional SPPs for applications that include waveguides,7–10 couplers/decouplers,11,12 leaky wave antennas13–15 and sensing devices,16–18 specially in the THz spectral range (which is very useful, since many fundamental excitations —like phonons in a lattice or molecule vibrations in a gas— have frequencies which lie in the THz and mid-IR spectral regions19). Although the optimization of the geometry of the grooves allows an effective tuning of the frequency of the spoof plasmons, it has the disadvantage of being immutable for some certain system —and many times, building many different systems with slightly different geometric parameters may not be the most practical solution. In traditional plasmonics, the usual solution for this limitation is the introduction of graphene, because its Fermi energy (that strongly controls its plasmonic response20) can be easily varied either through chemical doping or the application of a gate potential.21 Other advantages of graphene include its high carrier mobility (what translates in small losses, compared to noble metals22) and a strong light confinement in its surface.23 Following these features, we propose in this work the usage of a doped graphene sheet to effectively tune the plasmonic properties of the spoof plasmons, by controlling graphene’s Fermi energy and the distance to the grooved metallic surface. A first approach to this problem has been carried out by Ding et al.,24 for a 2D metal grating, using the effective medium approach to describe the periodic region of the structure; we, on the other hand, 3 consider a 1D metal grating, and have employed a mode-matching method considering the metal to be perfect, what will prove to be a good approximation. Moreover, we have accounted for the non-local effects in the graphene conductivity, using Mermin’s formula (see the Supporting Information for further details). The addition of graphene is shown to be very efficient when the energy of the uncoated diffraction grating SSPs is close to the energy of the graphene plasmons, especially in the THz spectral range. Afterwards, we will propose how this tunnability can be used for waveguiding and sensing applications. 2 Spoof plasmons in a grooved metallic surface coated with graphene Let us consider a semi-infinite grooved perfect-conducting surface coated by a graphene sheet parallel to its surface, like the one depicted in Figure 1. a w d h s I II III ε1 ε2 ε3 z x y Figure 1: Front-view schematic representation of the system presently under study. The grey area depicts a substract where the the perfect metal thin film, in gold color, is deposited. The light-blue areas depict the dielectric regions and the dark-blue thin area is the graphene sheet. All the different geometrical parameters and dielectric functions in either region I–III are marked in the figure. The system is assumed to be infinite in the z-direction and periodic in the x-direction. 4 We divide the dielectric area in three different regions (I, II and III) which can, in general, have different permittivities (ε1,ε2and ε3, respectively). The grooves in the metal are assumed to be rectangular, with width a, depth h, and period d; the metal itself is assumed to be perfect. The graphene sheet is lying at a distance sfrom the top of the grooves, and is described by a non-local conductivity σ(which will be discussed later) dependant on its Fermi energy EFand relaxation energy Γ. 2.1 Dispersion Relation The procedure we adopted to characterize the spoof plasmons consists on the modal decomposition of their fields in a Fourier series. This method has been used in the literature before,1,25,26 but never in the presence of graphene. The starting point is the solution of the wave equation retrieved from Maxwell’s equations in either region I, II or III. Considering explicitly p-polarized modes with a frequency ωand a harmonic time-variation e−iωt, the magnetic field in region λmust have the form Bλ(r, t) = Bλ(x, y)e−iωtˆz (omitting the z-dependence) with Bl(x, y) = ∞ X n=−∞ Bneiβnxeiκ(1) ny,(2) Bll(x, y) = ∞ X n=−∞ eiβnxhC+ neiκ(2) ny+C− ne−iκ(2) nyi,(3) Blll(x, y) = ∞ X n=0 Ancoshnπ ax−a 2icosκ(3) n(y+h),(4) and the respective electric fields are given by Eλ(r, t) = ic2 ωελ∇×Bλ(r, t).(5) Note that equations (2)–(4) were written explicitly to ensure that (i) the fields in region I do not diverge as y→ ∞; and (ii) the tangential component of the electric field in region III always vanishes in the surface of the metal. This is a consequence of the fact that 5 a perfect metal does not admit non-null electric fields on its inside, and the tangential electric field is continuous across any interface. The coefficients An,Bnand C± nare, for the moment, unknown, while the electromagnetic wave equation imposes that κ(1) n=κ(βn, ε1), κ(2) n=κ(βn, ε2) and κ(3) n=κ(nπ/a, ε3), with κ(q, ε)≡pεω2/c2−q2. Furthermore, since the grooved system is periodic, the corresponding fields must obey the Bloch’s Theorem, B(r+dˆx) = eiqdB(r), which means that βn=q+ 2nπ/d (qbeing the momentum of the plasmons in the x-direction). Another consequence of Bloch’s Theorem is that we need only to determine the fields in an unit cell |x|< d/2 of the system, and the fields elsewhere are then totally determined. In order to find the coefficients An,Bnand C± n, we need to evaluate the boundary conditions at the interfaces I/II and II/III. The detailed derivation from this point onwards is presented in the Supporting Information (SI), where we show that the dispersion relation of the spoof plasmons in this system is given by the matrix equation det(M−I) = 0, where I is the unit matrix with elements [I]`m =δ`m and Mis a matrix whose elements [M]`m =M`m are given by M`m = iε2 ε3 a d2 1 + δ`0∞ X n=−∞χ+ n+χ− n χ+ n−χ− nκ(3) m κ(2) n sinκ(3) mh cosκ(3) `hS∗ n`Snm,(6) with χ± n=1 2"1 + σ(βn, ω)κ(1) n ωε0ε1±ε2κ(1) n ε1κ(2) n#eiκ(1) nse∓iκ(2) ns(7) and S`n =1 aRa/2 −a/2dxe−iβ`xcosnπ ax−a 2in an integral with an analytical solution. In the previous expression, σ(q, ω) is the non-local conductivity of the graphene sheet (consult the SI for further details); note therefore that all the influence of the graphene in the dispersion is contained in the factors (χ+ n+χ− n)/(χ+ n−χ− n) in each term of the sum, which are equal to 1 in its absence. At this point, it is interesting to observe that a simple approximation we could have done 6 to simplify our calculations was to consider that, in region III (the grooves), only the lowest mode n= 0 was non-zero (that is, An=A0δn0). In doing so, the corresponding equation for the dispersion relation of the spoof plasmons becomes 1 = iε2 ε3 a d √ε3ω ctan√ε3ω ch∞ X n=−∞χ+ n+χ− n χ+ n−χ− n|Sn0|2 κ(2) n ,(8) where we have used the explicit expression for κ(3) 0. The first thing to note in the above expression is that, in the absence of graphene and dispersive dielectric media, it can only have solutions when the momentum of the leading mode in region II, κ(2) 0, is imaginary, because otherwise the RHS would be imaginary while the LHS is real. This means that the plasmonic solutions are only allowed in the region q > √ε2ω/c, what is not surprising, and is only a consequence of the bound nature of surface modes. On the other hand, the above equation can only have a solution when the tangent function present on its RHS is positive, what means that these solutions can only appear below a certain maximum frequency ωmax given by ωmax =πc 2h√ε3 .(9) Comparing this frequency with the effective plasma frequency given by equation (1), we conclude that these are exactly the same, meaning that we recover the solution from the effective permittivity model with a completely different model, when we make the same approximation. Note that this expression has some limitations, namely the fact that it is only valid for frequency-independent permittivities, but nonetheless it is very useful to make a rough estimation of the order of magnitude of the plasmons frequency, although it tends to overestimate it; following the analogy to the perfect metal/dielectric interface, a better definition of a reference value for the spoof plasmons’ fundamental mode frequency is ωref =ωmax/√1 + ε1. Apart from allowing the determination of ωmax, equation (8) has the additional advantage of being much easier to solve than the exact equation det(M−I) = 0, which gets increasingly 7 demanding with the number of modes we introduce in the fields description in region III. Although this approximation looks somewhat naive, we will show later on that it produces really accurate results. 2.2 Optical Properties If we add an impinging field in our description of the fields in region I [eq. (2)], the formalism described in the previous section allows for the calculation of the optical properties of the system (namely its reflectance). From a practical point of view, this result is particularly useful, because the reflectance of the system is easily measurable and allows the indirect measurement of other quantities like the absorbance spectrum and the identification of plasmonic resonances (note that there is no transmittance in this system, since a perfect metal is a perfect reflector). Assuming that the impinging electromagnetic wave has a frequency ωand makes some angle θwith the y-axis, the field in region I must be rewritten as Bl(x, y) = B0 ∞ X n=−∞ eiβnxhe−ikyyδn0+rneiκ(1) nyi,(10) where B0is the intensity of the impinging magnetic field, and we have redefined the coefficients Bn≡B0rn, so that the rncoefficients have the meaning of reflectance amplitudes. In the previous expression, kyis the momentum of the impinging wave in the y-direction, given by ky=kcos(θ), k=√ε1ω/c. On the other hand, Bloch’s Theorem now imposes that βn=kx+ 2nπ/d,kx=ksin(θ). From this point onwards, the procedure is completely analogous to the previous one. Following the already described steps (with the differences noted in the SI), we arrive at another matrix equation, this time with the form (M−I)·A=F, where the matrices Mand Iare the same as before, Ais a column with elements [A]`=A` and Fis a column with elements [F]`=φ`, defined in equation (24). Unlike the previous case —where we found the solution (M−I)·A= 0, with no source 8 term—, the source introduced by the impinging wave allows the immediate determination of the coefficients rn, upon the resolution of the previous equation [and using equation (21)]. From these coefficients, the reflectance of the system is simply defined as R(ω) = X n∈PM Re(κ(1) n ε1)Re(ε1 κ(1) 0)|rn|2,(11) where Re{x}stands for the real part of x, and both summations are performed strictly over the propagating modes (PM). For the energy scale of interest in this problem (up to a few THz), typically only the fundamental n= 0 mode is propagating, and hence the reflectance takes the simpler form R(ω) = |r0|2. The absorbance, on the other hand, corresponds to the fraction of energy which is not reflected, being thus given by A= 1 −R. Besides enabling the calculation of the reflectance and absorbance spectra, the determination of the rncoefficients has the additional advantage of allowing the representation of the loss function20 of this problem, defined as L(ω, q)≡ −PnIm{rn}(Im{x}stands for the imaginary part of x), which allows for the indirect determination of the dispersion relation of the spoof plasmons —even without the approximation considered in the previous section. Occasionally, we will be using a slightly different definition for the loss function, ˜ L= sgn(L) log(1 + |L|), aimed at highlighting the dimmer dispersion curves in the presence of brighter ones. In the previous definition, sgn(x) stands for the sign of x, and |x|stands for its absolute value. 3 Results and Discussion For the results that will be presented henceforth, graphene’s conductivity has been calculated through Mermin’s non-local formula,20,27 synthetically presented in the SI. Moreover, although the results presented in the previous section were derived for isotropic media only, we will now occasionally consider hBN, which is anisotropic; the generalization of the previous results to this case is discussed in the SI as well. Finally, hBN’s dielectric function 9 (a) (b) (c) (d) Figure 5: Top: Loss function spectrum of two different systems whose spoof plasmons lie on the (a) THz and (b) mid-IR spectral range, in the presence of doped graphene. Overlaid to the loss function are the dispersion of the spoof plasmons in the absence of graphene (‘NG SSP’, dashed yellow) and the dispersion of the graphene surface plasmons (‘GSP’, dashed red) in a air/graphene/air configuration. In (b) is also plotted the dispersion of the acoustic plasmons in a flat metal/air/graphene/air configuration (‘MDGD’, dashed black). The white dot-dashed curve is the dispersion of the light in the air. Bottom: electric field intensity in the vicinity of a groove for (c) the THz and (d) the mid-IR corresponding systems, with and without graphene. Both distributions were calculated for q= 0.9π/d (for the corresponding din each system), what translates into the frequencies (c) 1.94 THz and 2.86 THz and (d) 46.3 THz and 47.3 THz, respectively without and with graphene. All remaining parameters are the same as disclosed in (a) and (b). The color-scale is the same in both panels of each figure (c) or (d). 16 surfaces have been studied recently as an alternative to traditional metallic surfaces for optoelectronic applications, due to its much higher ability to resist oxidation and corrosion38,39 (what is a recurring problem in plasmonics), while keeping (or even improving) the surface’s characteristics. 3.3 Application to filtering and sensing Coming back to the THz spectral range, spoof plasmons have a particularly interesting effect on the reflectance spectrum of the grating, which is visible in Figure 6. In this figure, the reflectance spectra has been plotted for the case of a flat metal surface and a for two different geometries of a grooved metal. Comparing the three plots, one sees that the introduction of the grooves changes dramatically the reflectance spectrum of the system, which goes from an almost perfect reflector to exhibiting well-defined resonances corresponding to the excitation of spoof plasmons. These results were verified for several different materials in the spacer. Unsurprisingly, the position of these resonances is strongly controlled by the geometric dimensions of the system, since they are intrinsically connected to the dispersion relation of the spoof plasmons. This becomes clear in Figure 6(d), where is represented the loss function of the air/Al2O3/air system. Comparing the position of the resonances in either case to the respective dispersion curves, one sees that, qualitatively, the resonance occurs where the dispersion intersects the Brillouin zone boundary. In the plots of the previous figure, a neutral graphene sheet has been placed in the system in order to add some damping which brightens the dispersion curves in Figure 6(d); however, its conductivity is very low and therefore the behaviour of the system does not differ very much from the no-graphene case. It has the additional advantage of allowing the excitation of the spoof plasmons in the case where neither dielectric region is dispersive —without the graphene, the reflectance spectrum of the air/air/air configuration would be identically equal to 1 even for the grooved system. However, a much more useful behaviour arises when the graphene is doped, as presented 17 0246810 0.0 0.2 0.4 0.6 0.8 1.0 (a) 0246810 0.0 0.2 0.4 0.6 0.8 1.0 (b) 0246810 0.0 0.2 0.4 0.6 0.8 1.0 (c) (d) Figure 6: Reflectance spectrum of (a) a flat and (b),(c) a grooved metallic surface, with air above the graphene and inside the grooves, and several different materials in the spacer. Systems in plots (b) and (c) have different groove depths, what deeply influences the reflectance spectrum. In figure (d) is represented the loss function of the systems in (b) and (c) with Al2O3. 18 in Figure 7(a). Since the graphene doping strongly controls the dispersion curves of the system, it indirectly also controls the position of the resonances, which can be adjusted at will by varying the graphene’s Fermi energy. Furthermore, the results show that the graphene sheet strongly increases of absorbance of the system even for low doping. This translates in very strong resonances, whose width is controlled by the graphene damping Γ, which can have their minimum as low as R= 0. This situation of full-absorbance can be very useful for the development of actively-tunnable filters in the THz spectral range. Figure 7(b) shows that this tuning may be superior than 1 THz, and for dopings above 0.3 eV the energy at the resonance is fully absorbed. 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 (a) 2.2 2.4 2.6 2.8 3.0 3.2 3.4 0.0 0.1 0.2 0.3 0.4 0.5 0. 0.2 0.4 0.6 0.8 (b) Figure 7: (a) Reflectance Spectrum of a graphene-coated metallic grating for several values of the Fermi energy of the graphene sheet. ‘NG’ stands for ‘no graphene’, whereas ‘CNP’ stands for ‘charge neutral point’, equivalent to EF= 0.0 eV. (b) Position of the resonance (top) and minimum reflectance (bottom) of the reflectance spectrum, for several different spacer materials. On an alternative perspective, the difference between the spectra for different spacers in Figure 6 shows that the reflectance spectrum of the metal grating is highly sensitive to changes in the dielectric function of its composing materials, what suggests its utility for sensing applications. This approach has already been explored in the literature, with different authors proposing its usage to discern media with different refractive indexes18,40 placed inside the grooves of a corrugated surface. In this work, however, we propose a different 19 configuration in which the material to be sensed is placed above the graphene sheet in a layer with some thickness b. This approach is preferable when the aim is to sense very thin layers of fluids, in the order of 10 µm. This thin layers are smaller than the wavelength of the THz radiation, so no Fabry-Perot oscillations occur, what poses a more challenging problem. In Figure 8(a) is plotted the reflectance spectrum of an air/fluid/hBN/air configuration, where the fluid being sensed is either an alcohol (ethanol, methanol, 1,2-propanol), water, or none (air). 0. 0.5 1. 1.5 2. 2.5 3. 0.0 0.2 0.4 0.6 0.8 1.0 (a) 2.0 2.5 0.0 0.1 0.2 0.3 0.4 0.5 0. 0.1 (b) Figure 8: (a) Reflectance Spectrum of a graphene-coated metallic grating for several different fluids placed above the (neutral) graphene. (b) Position of the resonance (top) and minimum reflectance (bottom) of the reflectance spectrum, for the different alcohols being sensed. This resonances visible in that spectrum, with neutral graphene, already allow to discern whether the material being sensed is water, an alcohol, or none, but the resonance position of the different alcohols is very similar, what may cause difficulties to discern them from each other. A possible solution to overcome this difficulty is to dope the graphene sheet and trace the resonance position and minimum reflectance with the graphene’s Fermi energy, what is done in Figure 8(b). On the one hand, the graphene doping increases the separation between the resonance position of the different alcohols up to 30%; however, the greatest effect occurs in the minimum reflectance, since the total-absorption point for each alcohol arises for a different value of graphene doping, what provides an effective method to discern 20 them. These results show how adding graphene to the system can also be used for sensing purposes. 4 Conclusions In this work, we were able to show that the widely-studied tunnability of graphene can be successfully applied to a metallic grating in order to control its dispersion relation, especially in the THz spectral range. We also showed that this feature can have several applications, ranging from optoelectronic waveguides to filters and to THz sensing, with the additional bonus of providing graphene-protected metallic surfaces. The model we employed proved to be accurate when benchmarked against FDTD calculations, but has some limitations that should be properly emphasized. The most important one is the fact that the metal is considered ideal, which means that its validity is restricted to situations where the metal’s skin depth is negligible, such as the case of the THz and mid-IR radiation in most good plasmonic metals. For the same reason, although this model accounts for non-local effects in the graphene, it cannot account for non-local effects in the metal surface, that should be important when the graphene sheet is very close (a few nanometers) from the metal surface; under those conditions, further corrections need to be employed to ensure experimentally accurate results.41,42 However, in our case, the distance between graphene and the metal grating is large enough for non-local effects to negligible compared to that length scale. Acknowledgments E J Dias and N M R Peres acknowledge support from the European Commission through the project ”Graphene-Driven Revolutions in ICT and Beyond” (Ref. No. 696656) and the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Financing UID/FIS/04650/2013. 21 Supporting Information Available The following files are available free of charge. A Calculation details of the dispersion relation of the spoof plasmons The coefficients An,Bnand C± nare determined using the boundary conditions across interfaces I/II and II/III. Starting by the former, these are the continuity of the tangential electric field, Ex l(x, s) = Ex ll (x, s), and the discontinuity of the magnetic field proportional to the density of surface currents in the graphene sheet, K, given by the expression Bl(x, s)−Bll(x, s) = µ0K·ˆx. According to Ohm’s Law, the density of surface currents is proportional to the electric field in that surface, K(r, q, ω) = σ(q, ω)Ek(r, q, ω), where σ(q, ω) is the surface’s conductivity. This function is assumed to be uniform in every point of the sheet, but we allow it to be dependent on the momentum of the EM field, in order to account for non-local effects. Note that this discussion is valid for any 2D material in the interface between regions I and II (not only graphene, but also an electron gas or a doped 2D transition metal dichalcogenide, for example) as long as the adequate conductivity function for that material is considered. Since the electric fields in the neighbouring regions of this interface are composed by the superposition of several modes with different momenta, each mode that composes the total electric field is effectively influenced by a different conductivity, and the overall current density can be written as Kx(x) = ic2 ωε1∞ X n=−∞ iκ(1) nBneiβnxσ(βn, ω)eiκ(1) ns.(12) Using the above expression and the mentioned boundary conditions, one finds that the C± n 22 and the Bncoefficients are related by the expression C± n=χ± nBn, with χ± n=1 2"1 + σ(βn, ω)κ(1) n ωε0ε1±ε2κ(1) n ε1κ(2) n#eiκ(1) nse∓iκ(2) ns.(13) On the other hand, at the interface II/III, the tangential electric field is continuous, Ex ll (x, 0) = Ex lll(x, 0), as well as the magnetic field, Bll(x, 0) = Blll(x, 0), but only in the region |x|< a/2; in the complementary region a/2<|x|< d/2, the tangential electric field must vanish, Ex ll (x, 0) = 0. Therefore, when multiplying the first condition by e−iβ`xand integrating it in |x|< a/2, we can take advantage of the fact that the integrand must vanish at a/2<|x|< d/2 to conclude that Za/2 −a/2 dxe−iβ`xEx lll(x, 0) = Za/2 −a/2 dxe−iβ`xEx ll (x, 0) = Zd/2 −d/2 dxe−iβ`xEx ll (x, 0).(14) Further noting that the last term in the previous equation yields the integral Rd/2 −d/2dxe−iβ`xeiβnx= dδ`n, this boundary condition can be written as B`= iε2 ε3 a d1 χ+ `−χ− `∞ X n=0 κ(3) n κ(2) ` sinκ(3) nhS`nAn,(15) where the S`n is defined as the integral (with an analytical solution) S`n ≡1 aZa/2 −a/2 dxe−iβ`xcoshnπ ax−a 2i.(16) The second boundary condition may, in turn, be multiplied by cos`π ax−a 2and integrated it in |x|< a/2, yielding the equation A`=2 1 + δ`0∞ X n=−∞ S∗ n` cosκ(3) `hχ+ n+χ− nBn.(17) where we have used the integral Rd/2 −d/2dxcos`π ax−a 2cosnπ ax−a 2=a 2(1 + δ`0)δ`n. 23 Equations (15) and (17) relate reciprocally the coefficients Anand Bn; merging the two equations, we arrive at A`= ∞ X m=0   iε2 ε3 a d2 1 + δ`0∞ X n=−∞χ+ n+χ− n χ+ n−χ− nκ(3) m κ(2) n sinκ(3) mh cosκ(3) `hSnmS∗ n`   Am.(18) Defining the expression in brackets in the previous expression as M`m, we may write it as P∞ m=0(M`m −δ`m)Am= 0, what may be rewritten in the matrix form       M00 −1M01 ··· M10 M11 −1··· . . .. . ....      ·      A0 A1 . . .       =      0 0 . . .       .(19) It is now obvious that the previous equation can only admit solutions if the determinant of the square matrix in the LHS (designated hereby as M−I, where Iis the unit matrix) vanishes. That equation, det(M−I) = 0, is therefore the equation that sets the dispersion relation of the spoof plasmons allowed in this system. B Calculation details of the reflectance amplitudes The calculation procedure for this case is completely analogous to the previous one, using now the field in region I given by equation (10). This slightly changes the solutions of the boundary conditions at the interface I/II, which now yield the equations C± `=χ± `r`+δ`0Λ±, with the χ± `being the same as before, and Λ±≡1 21−σ(βn, ω)ky ωε0ε1∓ε2ky ε1κ(2) ne−ikyse∓iκ(2) ns.(20) Because of these new terms, the relations between the coefficients Amand rmare slightly 24 changed to the equations r`= iε2 ε3 a d1 χ+ `−χ− `∞ X n=0 κ(3) n κ(2) ` sinκ(3) nhS`nAn−δ`0Λ+−Λ− χ+ 0−χ− 0,(21) A`=2 1 + δ`0∞ X n=−∞ S∗ n` cos(κ(3) `h)χ+ n+χ− nrn+2 1 + δ`0S∗ 0`(Λ++ Λ−) cos(κ(3) `h).(22) Merging once again the previous two equations, we arrive at an expression with the form P∞ m=0(M`m −δ`m)Am=φ`, or, in the matrix form,       M00 −1M01 ··· M10 M11 −1··· . . .. . ....      ·      A0 A1 . . .       =      φ0 φ1 . . .       ,(23) with φ`≡ −2 1 + δ`0S∗ 0` cos(κ(3) `h)Λ+1−χ+ 0+χ− 0 χ+ 0−χ− 0+ Λ−1 + χ+ 0+χ− 0 χ+ 0−χ− 0.(24) This is a readily solvable equation which allows the direct calculation of the Ancoefficients, and the calculation of the rncoefficients using equation (21). C Graphene’s Conductivity For the conductivity of the graphene sheet, we have used Mermin’s formula, which includes non-local effects. Let x≡q/kFand y≡~ω/EFbe dimensionless variables constructed from qand ω, respectively. EFrefers to the graphene’s Fermi energy, kF=EF/(~vF) is the Fermi momentum (vF≈c/300 is the Fermi speed) and Γ is the material’s relaxation energy. The formula we have used was retrieved from Gon¸calves and Peres,20 σ(q, ω) = 4iσ0 ~ω q2χτq kF ,~ω EF,(25) 25 (36) Otto, A. Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection. Zeitschrift f¨ur Physik 1968,216, 398–410. (37) Zhan, T. R.; Zhao, F. Y.; Hu, X. H.; Liu, X. H.; Zi, J. Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies. Physical Review B 2012,86, 165416. (38) Kravets, V. G. et al. Graphene-protected copper and silver plasmonics. Scientific Reports 2014,4, 5517. (39) Ansell, D.; Radko, I. P.; Han, Z.; Rodriguez, F. J.; Bozhevolnyi, S. I.; Grigorenko, A. N. Hybrid graphene plasmonic waveguide modulators. Nature Communications 2015,6, 8846. (40) Ng, B.; Wu, J.; Hanham, S. M.; Fern´andez-Dom´ınguez, A. I.; Klein, N.; Liew, Y. F.; Breese, M. B.; Hong, M.; Maier, S. A. Spoof plasmon surfaces: a novel platform for THz sensing. Advanced Optical Materials 2013,1, 543–548. (41) Luo, Y.; Fernandez-Dominguez, A.; Wiener, A.; Maier, S. A.; Pendry, J. B. Surface plasmons and nonlocality: a simple model. Physical Review Letters 2013,111, 093901. (42) Luo, Y.; Zhao, R.; Pendry, J. B. van der Waals interactions at the nanoscale: The effects of nonlocality. Proceedings of the National Academy of Sciences 2014,111, 18422–18427. 32