Probing nonlocal effects in metals with graphene plasmons

Probing Nonlocal Effects in Metals with Graphene Plasmons Eduardo J. C. Dias,1, ∗David Alcaraz Iranzo,2P. A. D. Gon¸calves,3, 4, 5 Yaser Hajati,6Yuliy V. Bludov,1 Antti-Pekka Jauho,7, 5 N. Asger Mortensen,3, 8, 5 Frank H. L. Koppens,2, 9 and N. M. R. Peres1, † 1Department of Physics and Center of Physics, and QuantaLab, University of Minho, PT–4710–057, Braga, Portugal 2ICFO - The Institute of Photonic Sciences, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 3Center for Nano Optics, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark 4Department for Photonics Engineering, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 5Center for Nanostructured Graphene, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 6Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran 7Department of Microand Nanotechnology, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 8Danish Institute for Advanced Study, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark 9ICREA - Instituci´o Catalana de Recera i Estudis Avan¸cats, Barcelona, Spain (Dated: April 20, 2018) In this paper we analyze the effects of nonlocality on the optical properties of a system consisting of a thin metallic film separated from a graphene sheet by a hexagonal boron nitride (hBN) layer. We show that nonlocal effects in the metal have a strong impact on the spectrum of the surface plasmon-polaritons on graphene. If the graphene sheet is shaped into a grating, we show that the extinction curves can be used to shed light on the importance of nonlocal effects in metals. Therefore, graphene surface plasmons emerge as a tool for probing nonlocal effects in metallic nanostructures, including thin metallic films. As a byproduct of our study, we show that nonlocal effects lead to smaller losses for the graphene plasmons than what is predicted by a local calculation. We show that these effects can be very well mimicked using a local theory with an effective spacer thickness larger than its actual value. I. INTRODUCTION Nanoplasmonics is a field of optics dealing with the interaction of electromagnetic radiation with metallic nanostructures and nanoparticles1–4. Over the last decade, the characteristic size of plasmonic structures has been steadily approaching the few-nanometer scale5, with concomitantly ultra-confined plasmonic modes6–9. The most common description of the electrodynamics governing plasmonic systems typically ignores that the response of a metallic nanostructure is controlled by a nonlocal dielectric tensor10,11. Indeed, the general linear-response expression for the electric displacement vector reads12 D(r, ω) = ε0Zdr0¯ ε(r,r0, ω)E(r0, ω),(1) where D(r, ω) is the electric displacement vector, E(r, ω) is the electric field, and ¯ ε(r,r0, ω) is the nonlocal dielectric tensor. For a translational invariant system, we have ¯ ε(r,r0, ω) = ¯ ε(r−r0, ω). Equation (1) embodies the statement that the electric displacement field at point rdepends on the electric field at all points r0. In general, and in particular for systems with translational invariance, it is convenient to transform Eq. (1) to momentum space, obtaining D(k, ω) = ε0¯ ε(k, ω)E(k, ω), where translational invariance has been assumed and k denotes the wave vector. The local response approximation (LRA) is equivalent to neglecting the wave vector dependence of the dielectric tensor by taking the long-wavelength limit (k→0). However, when the system’s characteristic length scales approach the nanometer range, the wave vector dependence of the dielectric tensor has profound consequences on the spectrum of plasmonic resonances. It follows from Maxwell’s equations that the wave equation in momentum space reads3 −k×[k×E(k, ω)] = −k[k·E(k, ω)] + k2E(k, ω) =ω2 c2¯ ε(k, ω)E(k, ω).(2) Equation (2) has two types of solutions. The divergencefree solution (k·E= 0) corresponds to the usual wave equation k2E=¯ ε(ω/c)2E; in this context, this solution is usually dubbed as the transverse mode. However, within the nonlocal framework, there is an additional curl-free solution (k×E=0) which, for a given frequency ω, is obtained when the condition ε(k, ω) = 0 is satisfied. This solution is traditionally referred to as the longitudinal mode, although one should bear in mind that this mode is also perpendicular to the direction of propagation of the field. The longitudinal solution is generally overlooked within the LRA, which may reveal itself as an inaccurate approximation when either the size of the metallic nanostructures or the separation between two metallic surfaces fall below a couple of tens of nanometers.7,13,14. Here, we consider a geometry similar to the latter, namely a configuration in which a graphene sheet is placed parallel to and extremely close to a metal surface—see Fig. 1. The spectrum of the aforementioned structures can be computed either analytically or numerically— depending of the geometry—and such calculations show that the resonances associated with the excitation of surarXiv:1804.06478v2 [cond-mat.mes-hall] 19 Apr 2018 2 graphene air hBN metal air ε1 ε2 ε3 ε4 I II III IV s h zx y FIG. 1. Schematic representation of one type of layered heterostructure investigated in this work. We assume the system to be invariant in the x– and z–directions. face plasmon-polaritons (either localized or propagating) exhibit strong deviations from the ones obtained within the LRA15,16. Additionally, the local-response calculation predicts field-enhancements which are larger than in the nonlocal case17–19. This is particularity true when we have two nanoparticles in close proximity2,17,20–22. Experimentally, one can investigate nonlocal effects using diverse techniques, such as electron energy-loss spectroscopy8,20 and far-field spectroscopy23. Therefore, a proper theoretical description of nonlocal plasmonics requires the calculation of a suitable nonlocal dielectric function (or, equivalently, the nonlocal conductivity). A possible approach is using ab-initio methods24,25. However, typical (albeit small) plasmonic structures involve a large number of atoms which render those methods difficult to use in a routinely fashion, since they quickly become time-consuming and computationally demanding. Hence, it is natural to seek for an alternative approach to the calculation of the nonlocal dielectric function. As early as in the 1970s, it was observed that the plasmonic properties of thin metallic films did not follow the prediction of the local-response theory26,27. The need to reconcile theory and experiment required a description of an electron gas subjected to external fields introduced by Bloch28,29 and Jensen30. This model of an electronic fluid came to be known as the hydrodynamic model13,27,31–33. This model combines Maxwell’s equations with Newton’s second law of motion for a charged particle supplemented with a term taking into account the statistical pressure in the electron gas due to charge inhomogeneity13,26,27. The revival of plasmonics unburied the hydrodynamic model and applied it to the optical response of metallic nanoparticles with great success7,22,32. As already mentioned, both the position and width of the plasmon resonance, and the field enhancement measured experimentally do not agree with the approach using the LRA, whereas the hydrodynamic model is able to explain the experimental results7. For this reason, the hydrodynamic model has now become a popular approach for nonlocal optics11,13,34. More recently, the birth of 2D crystals35 introduced another possibility for studying nonlocal effects in the optical response of materials. In particular, the emergence of graphene plasmonics3constitutes a new playground for studying these effects down to the one atom thick limit9,36. The combination of graphene with twodimensional (2D) insulators—such as hexagonal Boron Nitride (hBN)—has allowed an unprecedented control of the distance that a 2D material can be placed in the vicinity of metallic films or nanostructures. In fact, the placement of a graphene sheet at a distance of a few nanometers away from a metal surface has recently been experimentally demonstrated9, in a similar setting as depicted in Fig. 1. This geometry provides a suitable way to obtain deep subwavelength confinements down to the atomic limit. This is possible due to the interplay of graphene plasmons and the screening exerted by the nearby metal film. This approach has revealed that the nonlocal properties of graphene’s conductivity have to be included in a proper account of the experimental data9,37. In graphene, it is more useful to describe the nonlocal response via the material’s nonlocal dynamical conductivity, which within linear-response theory relates the surface current, Ks(r, ω), and the in-plane electric field, Ek(r, ω), via3 Ks(r, ω) = Zdr0¯ σ(r−r0, ω)Ek(r0, ω).(3) Notice that this statement is merely a reformulation of Eq. (1). Here, ¯ σ(r, ω) refers to the nonlocal (surface) conductivity tensor of graphene. This quantity can be probed experimentally using graphene plasmons when the material is in the proximity of a metallic surface, and varying the graphene-metal distance, thereby retrieving the Fourier transform of ¯ σ(r, ω), ¯ σ(q, ω)9. In the absence of strain, graphene is isotropic and thus ¯ σ(q, ω) = ¯ σ(q, ω), where qlabels the in-plane wave vector. If the graphene sheet is transformed into a grating (or an external grating is deposited on the graphene sheet), the properties of the plasmons depend sensitively on the lenght of the imposed period3,38,39. Therefore, adjusting the grating period can be used to tune the graphene plasmons’ frequency to the desired spectral range. Either graphene itself38 or a grating may be patterned39. When graphene is placed in the vicinity of a metallic surface, e.g., as illustrated in Fig. 1, nonlocal effects arising from the metal’s response may influence indirectly the behavior of the fields in the region between graphene and the metal, namely through a larger penetration of the field inside the metal due to a less effective screening induced by nonlocality. The quantification of the importance of such nonlocal effects constitutes the goal of this work. Here, we investigate the influence of the nonlocal response of a metal on the optical response of systems based on graphene lying in close proximity to a metallic film or surface. In what follows, we consider both the case in which the metal is constituted of Gold and of Titanium. We shall focus on the latter, as this metal exhibits a very strong nonlocal behavior (compared, for example, with Gold). The nonlocal effects arising from 3 the metal’s response are evaluated both on the plasmonic (near-field physics) and on the optical properties (far-field spectroscopy) of these structures. Throughout the manuscript, graphene’s conductivity is taken as being nonlocal3. The aim of this work is therefore to use graphene plasmonics to probe nonlocal effects within the metal, in opposition to the study of nonlocality in graphene performed in a recent publication9. Specifically, we study the impact of nonlocal effects due to the metal as a function of the graphene-metal separation, and discuss its implications on the field distribution and plasmonic losses. Finally, we consider a prospective system in which the graphene sheet is replaced by a graphene diffraction grating made of a periodic array of graphene ribbons. In this case, we compute the system’s response in the far-field and determine the influence of the metal’s nonlocality on the measured spectra. We take realistic parameters for the dielectric functions of the materials that constitute each system, which allows us to compare our results directly to the experiments. II. MATHEMATICAL DETAILS Throughout this work, we consider nonmagnetic (µ= 1) media. The system under study consists in a multilayered heterostructure, with either dielectric or metallic layers stacked along the y–direction, as portrayed in Fig. 1. The structure is further assumed to be infinite in the xz–plane. We look for TM-modes with a harmonic time dependence in the form of e−iωt, and thus the corresponding magnetic field may be written as H(r, t) = Hz(x, y)e−iωtˆz.(4) The form of Hz(x, y) follows from Maxwell’s equations, and, in general, is different of the regions defined in Fig. 1. As we consider nonlocal effects only in graphene and in the metal, it is useful to distinguish the description of the fields in the dielectric(s) and metal regions. A. Dielectric Regions In the dielectric media (source-free)the magnetic field obeys Helmholtz’s equation, ∇2H+εk2 0H= 0, where k0=ω/c. Assuming a magnetic field in the form of Eq. (4), the wave equation admits the following (transverse) solution: Hz T(x, y) = C+eikTy+C−e−ikTyeiqx,(5) where the in-plane wave vector, q(assumed to be along the x–direction without loss of generality), and the perpendicular wave vector, kT, are related by kT=qεk2 0−q2.(6) The ‘T’ subscript was introduced to make explicit the transverse nature of these solutions. For uniaxial media (characterized by different permittivities in the xz– plane, εx, and in the y–direction, εy), such as hBN, q and kTare connect through an alternative condition, i.e., kT=pεxω2/c2−q2εx/εy. Maxwell’s equation ET= (−iωε0ε)−1∇×HTenables us to write the corresponding electric field components as Ex T(x, y) = −kT ωε0εxC+eikTy−C−e−ikTyeiqx,(7) Ey T(x, y) = q ωε0εyC+eikTy+C−e−ikTyeiqx.(8) Naturally, if the dielectric is isotropic, then ε≡εx=εy. B. Metal Region Within the metal, we assume a Drude dielectric function εmof the form εm(ω) = ε∞−ω2 p ω2+ iγω ,(9) where ωpand γare the plasma and damping frequencies, respectively, and ε∞a background permittivity to account for interband polarization effects. We take nonlocal effects in the metal within the framework of the hydrodynamic model—see Refs. 31,33 for details. When taking nonlocality into account, Amp`ere’s law becomes31,33 ∇×H=−iωε0εm[E−ξ∇(∇ · E)] ,(10) with the parameter ξdefined as ξ=βqω2 p/ε∞−ω2−iγω, (11) where βis a nonlocal parameter proportional to the Fermi velocity vFof electrons in the metal. In this work, we will take the most usual definition33 β=p3/5vF. Naturally, the local limit is recovered upon setting β= 0. As discussed above, Maxwell’s equations can be shown to admit two kinds of solutions31,33: divergence-free and curl-free fields. The former are the usual transverse waves that exist in the local regime.Within the LRA, the electromagnetic fields in the metal are then described similarly to the fields in the dielectric case, upon replacing ε→εm(ω)—see Eqs. (5)–(8). On the other hand, curl-free waves do not have an associated magnetic field, as imposed by Faraday’s law. However, unlike the local case, the electric field has a non-trivial solution given by the vanishing of the term in square parenthesis figuring in Eq. (10), equivalent to the wave equation ∇2E−(1/ξ)E= 0. This equation describes longitudinal solutions of the form EL= (Ex Lˆx +Ey Lˆy)e−iωt, with Ex L(x, y) = D+eikLy+D−e−ikLyeiqx,(12) 4 Ey L(x, y) = kL qD+eikLy−D−e−ikLyeiqx,(13) with (longitudinal) wave vector kL=p−ξ−2−q2=β−1qω2+ iγω −ω2 p/ε∞−q2. (14) Furthermore, Ex Land Ey Lare related by the curl-free condition, ∂Ey L/∂x =∂Ex L/∂y = iqEx L. The general solution for the fields inside the metal is therefore Hz=Hz Tand Ex/y =Ex/y T+Ex/y L. C. Boundary Conditions The coefficients which characterize the fields in each layer are determined by the boundary conditions (BCs). For bound modes (like surface plasmons) we have q > √εω/c. This renders kimaginary, and the signal must be chosen judiciously to ensure that the fields satisfy Sommerfeld’s radiation condition. For an interface between two dielectrics, the usual BCs apply, that is, the continuity of the tangential component of the electric field (Ex) and the (dis)continuity of the magnetic field (Hz) in the (presence) absence of a surface current density at the interface. The presence of a finite surface current density is needed in order to describe a graphene sheet (or any other 2D material) placed at an interface between the two media , and enters through Ohm’s law Ks(q, ω) = σ(q, ω)Ex(q, ω)ˆx; see also Eq. (3).Note that here σ(q, ω) entails both frequency and momentum dependencies in order to account for nonlocal effects in graphene. We employ graphene’s nonlocal conductivity using Mermin’s particle-conserving prescription, which is detailed in Appendix C. Finally, for a dielectric/metal interface (without graphene40), although the BCs described above remain valid, these need to be augmented by an additional BC (due to the existence of a longitudinal mode within the metal). Typically, this additional BC dictates that the normal component of the polarization vector vanishes at metal’s surface, given by31 P= (i/ω)∇×H−ε0ε∞E.(15) Note that this polarization field refers to the one associated with the free electrons in the surface of the metal, responsible for the transport of electric currents. Therefore, from a physical perspective, this condition merely imposes that cannot exist currents flowing from the metal to the neighboring dielectric regions (e.g., the interface is a hard wall). In possession of this additional BC the amplitudes of the fields may now be determined; see Appendices A and B for a more detailed account. III. NONLOCAL EFFECTS IN THE PLASMONIC PROPERTIES We consider a system composed by a thin metallic layer (with thickness h) with a graphene sheet lying at a distance sfrom its surface. In the spacer region, i.e., between graphene and the metal, we assume to have slab of hBN; however, our formalism is general and thus allows for the consideration of different dielectric media. Below the metal and above the graphene, we assume to have air. For the sake of clarity, the system has been divided into four regions, I–IV, as depicted in Fig. 1. The calculation of the nonlocal plasmonic properties of the system follows the guidelines discussed in the previous section, and can be consulted with a greater degree of detail in the Appendix A. In order to assess the influence of the metal’s nonlocal effects in the plasmonic properties of the system, we compare our nonlocal results with the corresponding predictions of the local-response theory. A. Nonlocal Effects in the Plasmon Dispersion For the study of the metal’s nonlocal effects in the dispersion relation of graphene plasmons, we considered here two different metals—Gold (Au) and Titanium (Ti)—described by the parameters presented in Table I. The dielectric function of hBN, on the other hand, is adopted from Refs. 36 (out-of-plane direction) and 41 (in-plane direction). TABLE I. Drude model parameters used for Titaniuma(Ti) and Gold (Au). The superscripts indicate the corresponding references. Ti Au ωp[eV] 2.8042 8.8443 γm[meV] 82.042 103.043 ε∞2.244 9.8445 vF/c 0.0059746 0.0046447 aNote that the authors of Ref. 42 described Titanium using both Drude and Lorentz terms, but, for the purpose of this work, we only considered the Drude contribution, what nonetheless provides a very good approximation. The influence of the nonlocal effects arising from the metal’s response is investigated by computing the dispersion relation, ω(q), of graphene plasmons and comparing the ensuing spectra obtained assuming a localand a nonlocal-response. The outcome is presented in Fig. 2 (white curves) for the allowed plasmonic modes in the structure pictured in Fig. 1. Since there is dissipation in the system (both in the metal, hBN and graphene), either qor ωneeds to be regarded as a complex quantity in order to fulfill the boundary conditions. In what follows, we have chosen qto be a real number and hence ωis complex. For the time being, we focus on the real 5 part of the frequency, Re{ω}, and denoting it by ωfor simplicity. The corresponding imaginary part, Im{ω}, associated with the plasmonic losses, will be discussed at a later stage. FIG. 2. Dispersion relation (DR) of SPPs in an air/metal/hBN/graphene/air configuration, both for a local (dashed white) and nonlocal (solid white) metal, in the case of Gold (top) and Titanium (bottom). The black dot-dashed line corresponds to the light dispersion in vacuum, whereas the green dot-dashed line corresponds to the graphene’s electron-hole continuum boundary, ω=vFq, with vF≈c/300. The color-plot shows the respective loss function, obtained nonlocally (see Appendix B). The remaining parameters are: s= 1 nm, h= 10 nm, EF= 0.5 eV, and Γ = 16 meV. It is apparent from Fig. 2 that the dispersion curves are not continuous, but rather present two asymptotes around 750 and 1350 cm−1; this feature is shared in both local and nonlocal frameworks. These frequencies correspond to the beginning of the hBN’s Restrahlen bands, where this material is hyperbolic3, and are associated with the excitation of surface phonon-polaritons (which in this case hybridize with graphene plasmons)3. Moreover, Fig. 2 clearly demonstrates that nonlocal effects (associated with the metal) impacts the plasmon dispersion very differently depending on the metal: for Au, the nonlocal and local curves lay very close, whereas in the case of Ti, there is a significant blueshift of the dispersion of graphene plasmons due to the nonlocal effects of the Ti (we stress that graphene is treated as being nonlocal in both cases). Quantitatively, the respective blueshifts are around 2% and 20%. We have determined that this difference in the magnitude of the nonlocal effects originates from the cumulative influence of two main factors: the Ti’s larger nonlocal parameter with respect to Au’s (βTi/βAu ≃1.29), and Ti’s smaller plasma and relaxation frequencies (specially the former, ωp,Ti/ωp,Au ≃0.32). Both quantities—which are intrinsic to each metal—are crucial for the enhancement of the nonlocal effects in Ti. For this reason, we shall focus on Titanium henceforth in order to illustrate a case in which nonlocal effects are pivotal. It should be emphasized that, apart from the specific characteristics of the metal, the dielectric environment in its vicinity also plays a significant role on the impact of nonlocal effects. In the particular case considered here, the proximity between the metal and graphene (that is, the thickness sof the spacer) is determinant, as Fig. 3(a) plainly shows. The figure shows the difference between the local and nonlocal calculations increases as the thickness of the spacer region is decreased. In particular, while for s= 10 nm the difference is negligible, nonlocal effects become clearly perceptible for s= 5 nm, and for s= 1 nm nonlocal effects become of paramount importance for an accurate description of the system’s plasmonic response. This behavior is due to a greater spatial confinement of the fields inside narrower spacers, which is enhanced owing to the screening of graphene plasmons by the metal, and in turn gives rise to an acoustic-like graphene plasmons with very high momenta3,9 and thus more susceptible to nonlocality. Figure 3(a) also shows that an increment of the spacer thickness induces a blueshift in the dispersion relation of the plasmonic modes.For that reason, a na¨ıve approach to account for nonlocal effects while carrying out a local calculation is to consider an effective spacer thickness, seff , larger than the actual value, s, in a similar fashion to what has been proposed in earlier works45,48,49. Although this method can indeed be regarded as somewhat na¨ıve version of quantum-corrected boundary conditions50,51, it can mimic the proper nonlocal calculation as illustrated in Fig. 3(b). In particular, the nonlocal dispersion is can be reproduced using a local formalism with an effective parameter α=seff /s between 1.75 and 1.80. However, it should be stressed that the 6 0 50 100 150 200 250 1400 1600 1800 2000 2200 2400 (a) 0 50 100 150 200 250 0 500 1000 1500 2000 2500 (b) FIG. 3. (a) Uppermost branch of the dispersion relation (ω > 1350 cm−1) of the plasmonic modes in an air/Ti/hBN/graphene/air configuration, for different values of the hBN’s slab thickness, s, calculated for a local (dashed) and nonlocal (solid) metal. (b) Dispersion relation in the same configuration, calculated for local and nonlocal metals for different values of the effective parameter α=seff /s, with s= 1 nm. The remaining parameters, for both panels, are: h= 10 nm, EF= 0.5 eV and Γ = 16 meV. value of the effective parameter highly depends both on the considered materials (particularly the metal’s properties) and on the configuration itself. For this reason, this procedure is generally hard to implement because the particular value of α=seff /s that reproduces the nonlocal effects is difficult to predict theoretically, which renders this approach unsuitable. Instead, it can be merely regarded as a fitting parameter when describing an experiment through local calculations. Before concluding the present section, we investigate how nonlocal effects vary with the thickness of the metal film. To that end, in Fig. 4(a) we have plotted the plasmon dispersion relation for an air/Ti/hBN/graphene/air structure using three different values for the Titanium thickness: 1, 10 and 100 nm. The plasmonic spectrum was obtained both nonlocally (left panel) and locally (right panel). In both cases, the graphene is treated as a nonlocal medium. The figure demonstrates that the 10 and 100 nm cases produce the same dispersion, whereas the 1 nm curve lies toward smaller frequencies (for the same q). This results suggest that, above a certain threshold thickness, the plasmonic properties of a system with a finite-thickness metal are equivalent to those of a configuration with a semi-infinite metal. This behavior becomes particularly evident upon inspection of Fig. 4(b), where the plasmon wavevector, for a given frequency, is shown as a function of the metal’s thickness. Clearly, for h&3 nm, the metallic film is well approximated by a semi-infinite metal (this threshold of ∼3 nm is consistent with the penetration depth of the fields in the nonlocal regime, as will be discussed in Section III B). 1. Comparison to the semi-infinite metal case We have seen that for metal thicknesses h&3 nm, the thin-film can be well approximated by a semiinfinite metal. Such a scenario is in fact the most relevant under realistic experimental conditions, which typically employ metals (acting as a gate) with thicknesses in excess of 10 nm9,52. This convenient because it not only simplifies the analysis, but it also allows us to write a closed-form expression for the plasmon dispersion in the heterostructure—now a dielectric/graphene/dielectric/metal configuration—, reading εx 4 κ(4) T +εx 3 κ(3) T +iσ ωε0! 1 + εmκ(3) T εx 3κm T +δnl!= = εx 4 κ(4) T−εx 3 κ(3) T +iσ ωε0! −1 + εmκ(3) T εx 3κm T−δnl!e−2κ(3) Ts, (16) where κ(ν) T=pq2εx ν/εy ν−εx νk2 0for ν={3,4},κm T= pq2−εmk2 0, and δnl is a nonlocal correction term given by33 δnl =q2 κm Lκm T εm−ε∞ ε∞ ,(17) with κm L=qq2−ω2+ iγω −ω2 p/ε∞/β2. Indeed, our calculations demonstrate that Eq. (16) is able to reproduce extremely well the plasmon dispersion presented, for instance, in Figs. 2 and 3. Furthermore, it is instructive to note that by taking the εm→ ∞ limit in Eq. (16), 7 0.00 0.05 0.10 0.15 0.20 0 500 1000 1500 2000 0.05 0.10 0.15 0.20 (a) 1 10 100 1000 0.15 0.20 0.25 0.30 (b) FIG. 4. Effect of the variation of the dispersion relation of the plasmonic modes in a air/Ti/hBN/graphene/air structure with the metal thickness, h. (a) Nonlocal (left, solid) and local (right, dashed) dispersions for three distinct values of the metal thickness (the orange curve is behind the green one). (b) Plasmon wave vector, for a fixed frequency ω= 2000 cm−1(dot-dashed on the top panel) as a function of the metal thickness. In both panels, the remaining parameters are: s= 1 nm, EF= 0.5 eV and Γ = 16 meV. which corresponds to the case where the metal becomes a perfect conductor, one obtains (neglecting nonlocal effects) εx 4 κ(4) T coth hκ(3) Tsi+εx 3 κ(3) T +iσ ωε0 = 0 .(18) Equation (18) coincides with the dispersion relation of the acoustic plasmon branch in double-layer graphene3 in a symmetric dielectric environment, where the individual graphene layers are separated by a distance 2s. This result reflects the scenario in which the screening exerted by the perfect conductor mirrors exactly the charges induced in the graphene sheet. Lastly, notice that for large separations, i.e., s→ ∞, the plasmon dispersion (16) reduces to that of an isolated graphene sheet between two dielectric media εx 4and εx 3. B. Nonlocal Effects in the Field Distributions Having discussed the influence of nonlocal effects in the plasmon dispersion, we now study their impact in the spatial distribution of the fields associated with the plasmonic modes. The fields amplitudes follow from the BCs, and we determine the y–dependence of the fields at a given (q, ω)–point which satisfies the plasmon dispersion shown in the previous sectio. In this spirit, Fig. 5 depicts the variation of the absolute value of the magnetic and electric field (in logarithmic units), along the heterostructure, for a (real part of the) frequency ω= 2000 cm−1. It is apparent from Fig. 5(a) that magnetic field distribution remains essentially unchanged under nonlocal corrections. This is a natural consequence of the hydrodynamic model, in which nonlocality only enters in the longitudinal components. Since the magnetic field is purely transverse, nonlocal effects do not influence it, and the only differences between the local and nonlocal cases arise from small differences between the coefficients (and the value of qfor the same frequency) that describe the magnetic field on each case. In contrast to the magnetic field, the inclusion of nonlocal effects in the metal’s response renders significant changes in the spatial distribution of the electric field, as can be seen from Fig. 5(b)–(c). Naturally, the differences between the local and nonlocal calculations arise mostly in fields within the metal region, as could be anticipated, due to the fact that we have allowed the existence of a longitudinal mode inside the metal, in the nonlocal case. The figure demonstrates that the nonlocality introduced by this additional longitudinal wave has profound implications in the eletric field’s spatial distribution. Specifically, notice that the electric field inside the metal in the local case is significantly smaller than the field in the spacer region (as expected for a good metal). However, when nonlocal effects are taken into account, the electric field is increased (when compared with the LRA) by several orders of magnitude in the vicinity of the metal’s surface. This feature is consequence of the smearing of the electron density introduced by nonlocality, which translates into a larger penetration of the fields inside the metal. This penetration depth is practically negligible when taking the local approximation, but becomes important when taking nonlocality into account, as Fig. 5 shows. In the latter, the penetration length reaches a couple nanometers, and can become comparable to the metal’s thickness for ultra-thin films. Another feature visible in Fig. 5(b)–(c) is the pres- 8 -15 -10 -5 0 5 10 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 (a) -15 -10 -5 0 5 10 -4 -3 -2 -1 0 (b) FIG. 5. Spacial distribution of (a) the magnetic field and (b) the electric field components, for a air (blue)/Ti (gray)/hBN (green)/graphene (white)/air (blue) structure, calculated using local and nonlocal metal models, for plasmonic modes with the same real part of the frequency ω= 2000 cm−1. H0and E0are respectively the nonlocal magnetic and total electric fields calculated at x= 0. The parameters used are: h= 10 nm, s= 1 nm, EF= 0.5 eV and Γ = 16 meV. ence, in the nonlocal case, of a sharp dip in the electric field magnitude around y=−2.5 nm, increasing again towards the metal/air interface. This is due to a node of its y–component, which is caused by a destructive interference between the transversal and longitudinal modes inside the metal. Note that both the transversal and longitudinal modes have xand y–components, with the total field being given by Ex/y =Ex/y T+Ex/y L; the node occurs when Ey T+Ey L= 0. For that reason, across the node there is a change of the sign of the y-component of the electric field, dividing the regions where the longitudinal mode amplitude is higher (to the right of the node) or lower (to the left) than the transversal mode amplitude. C. Nonlocal Effects on the Plasmonic Losses We conclude the study of the system portrayed in Fig. 1 with an analysis on the effect of the nonlocality in the losses associated with the plasmonic modes. As mentioned above, in the presence of losses and for a realvalued q, the ensuing condition for the plasmon dispersion requires a complex-valued ω. So far we have limited our discussing to its real part, but here we focus on its imaginary part, Im{ω}, since this quantity is intrinsically related with the plasmonic losses and plasmon life-time τp, in particular, τ−1 p=−Im{ω}/2. In Fig. 6 we have plotted −Im{ω}as function of the real part of the polariton frequency, Re{ω}, both within the local and nonlocal response formalism (for the metal; graphene is modeled as nonlocal in all cases). We have considered two cases, one with a relatively high (for graphene) scattering rate of Γ = 16 meV, and another illustrating high-quality, low-loss graphene, in which Γ = 1 meV. We note that the change of graphene’s electronic scattering rate does not significantly alter the dispersion relation presented in Fig. 2 (in fact, there is no visible difference to the eye). However, it significantly impacts the losses affecting graphene plasmons sustained in the heterostructure. As Fig. 6 shows, the differences between the calculation using a local and a nonlocal metal is modest. This suggests that the losses that the plasmons incur originate from the graphene, which is natural since the metal only participates indirectly—via screening— and the mode’s spectral weight is essentially attributed to graphene’s.Furthermore, Fig. 6(b) shows that this difference becomes more evident when the spacer width is reduced, since the losses, when calculated locally, strongly increase for small spacers, whereas nonlocally their increase is very small. This means that, by reducing the spacer size, an increasingly higher confinement can be achieve without a strong increasing in the losses, what is a very important result. Let us now understand the sudden decrease of the losses seen, in Fig. 6 [panel (a)] around the hBN phonon frequencies, when large damping in graphene (16 meV) is considered. To that end we recall that the damping of the phonon modes in hBN are 2.4 meV for the in-plane phonon41 and 1.9 meV for the out-of-plane phonon36. At the frequency of the phonons the polariton spectrum has, essentially, a phononic nature. Therefore, the losses are essentially controlled by those due to phonons. Since these are much smaller than 16 meV, we see a sudden decrease in the losses. On the other hand, for a damping in graphene of 1 meV, the phonon damping is comparable to the damping in graphene. As consequence the curves of the losses in Fig. 6 [panel (a)] do not present the sudden drop at the phonon frequencies. In conclusion, away from the phonon frequencies, the losses are essentially due to graphene, whereas near the phonon frequencies of hBN the losses are essentially controlled by the behavior of the imaginary part of the dielectric function of the hBN spacer. 9 500 1000 1500 2000 0 20 40 60 80 500 1000 1500 2000 (a) 1 10 100 0 20 40 60 80 100 (b) FIG. 6. (a) Imaginary part of the frequency of the SPPs in an air/Ti/spacer/graphene/air configuration, with the spacer being hBN (left panel) or air (right panel), calculated using local (dashed) and nonlocal (continuous) models, for two different values of the graphene’s electric relaxation energy Γ, and for s= 1 nm (dot-dashed on the bottom panel). (b) Variation of the imaginary part of the SPPs, in a configuration with a hBN spacer, with the spacer width s, for a fixed frequency ω= 2000 cm−1(dot-dashed on the top panel). The remaining parameters are: h= 10 nm and EF= 0.5 eV. IV. PROBING NONLOCALITY IN METALS USING A GRAPHENE NANORIBBON GRATING A common approach to experimentally access the plasmonic properties of a system consists on performing reflectance and/or transmittance measurements of the farfield spectra, upon illuminating the sample.In this context, plasmon excitations appear as resonances in the spectra (as peaks or dips). Therefore, one may investigate the influence of nonlocal effects by studying the resulting spectra and compare it to experimental data. In the extended, continuous layered system studied in the previous section, we have seen that the metal’s nonlocal response affects the plasmon properties of the modes supported in the system. However, the wave vector mismatch between the graphene plasmons in Fig. 1 structure and the one of a photon in free-space differs by more than two orders of magnitude, and thus it has been primarily investigated by near-field techniques which are able to overcome this kinematic limitation9,52. In what follows we consider a different configuration. It is similar to the one considered in the previous section, but we now assume that the graphene monolayer has been patterned into a periodic array of graphene nanoribbons—see Fig. 7. These effectively act as a diffraction grating, whose Fourier components provide the necessary in-plane momentum to excite plasmons in the system3,38. Indeed, this mechanism not only allows the excitation of graphene plasmons by free-space photons but it also serves as a platform susceptible to enhance nonlocal effects, since the higher diffraction order can carry substantial momentum and thus promote nonlocal effects. Hence, in order to illustrate a case where nonlocality plays a major role on the optical properties of the system, we will replace the graphene sheet considered in the previous section by a graphene diffraction grating with a period d, and ribbon width w(see Fig. 7). We wd graphene hBN metal SiO2 Si air ε1 ε2 ε3 ε4 ε5 I II III IV V b s h k B Eθ FIG. 7. Schematic representation of the layered system considered in this section. It is assumed to be periodic in the x-direction and uniform in the z-direction. All the different materials, dimensions and regions considered are marked in the figure. Graphene is considered as being nonlocal throughout and the metal is assumed to be either local or nonlocal. calculate the optical properties (namely, the reflectance R, transmittance Tand absorbance A) of such structure when it is illuminated by a p–polarized plane-wave coming from region V, with monochromatic frequency ωand incident angle θ, as depicted in Fig. 7 Apart from the introduction of the diffraction grating, we will also be considering henceforth a system where the ribbon array is encapsulated between the spacer (hBN, as before) and a thick (285 nm) layer of silicon dioxide (SiO2). The latter’s permittivity is taken from Ref. 53. On top of the SiO2, we further consider a semi-infinite layer of silicon (Si) described by a isotropic dielectric constant of εSi = 11.6654. 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