Modeling the Excitation of Graphene Plasmons in Periodic Grids of GrapheneRibbons: An Analytical Approach

P. A. D. Goncalves1,2,3, E. J. C. Dias1, Yu. V. Bludov1 and N. M. R. Peres1∗1Department of Physics and Center of Physics, University of Minho, P-4710–057, Braga, Portugal

2 Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark and3Center for Nanostructured Graphene, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

(Dated: June 17, 2016)

We study electromagnetic scattering and subsequent plasmonic excitations in periodic grids ofgraphene ribbons. To address this problem, we develop an analytical method to describe theplasmon-assisted absorption of electromagnetic radiation by a periodic structure of graphene ribbonsforming a diffraction grating for THz and mid-IR light. The major advantage of this method lies in itsability to accurately describe the excitation of graphene surface plasmons (GSPs) in one-dimensional(1D) graphene gratings without the use of both time-consuming, and computationally-demandingfull-wave numerical simulations. We thus provide analytical expressions for the reflectance, trans-mittance and plasmon-enhanced absorbance spectra, which can be readily evaluated in any personallaptop with little-to-none programming. We also introduce a semi-analytical method to benchmarkour previous results and further compare the theoretical data with spectra taken from experiments,to which we observe a very good agreement. These theoretical tools may therefore be applied todesign new experiments and cutting-edge nanophotonic devices based on graphene plasmonics.

I. INTRODUCTION

Nowadays, photonics — dubbed “the science of light”— is one of the branches of the physical sciences withmost impact in our daily lives. It is concerned withthe generation, manipulation and control of light (pho-tons) in a manifold of fundamental and technologicallandscapes. Recently, the “nano-revolution” under wayhas led to the miniaturization of electronics. However,in what regards electromagnetic (EM) radiation, suchminiaturization is limited by the length-scale definedby the wavelength of the employed light (known as thediffraction-limit). Naturally, using more energetic radia-tion (i.e. smaller wavelengths) is impractical due to, forinstance, the hazards of ionizing radiation or to the de-parture from technologically-relevant regions of the EMspectrum. In this context, plasmonics1–3 has been re-garded as the most promising candidate to bring EMfields to the nanoscale4–10.

Plasmonics is a branch of photonics which deals withquasiparticles known as plasmon-polaritons3,11. Sur-face plasmon-polaritons (SPPs) are electromagnetic sur-face waves coupled to collective excitations of the freeelectrons in conductors. When these hybrid excita-tions occur in conducting nanostructures — such asnanoparticles12–14 or engineered metamaterials15–17 —, the corresponding non-propagating plasmon-polaritonsare generally coined as localized surface plasmons(LSPs)3. Perhaps the most alluring property of plasmonsis that they exhibit large field-enhancements and deepsubwavelength confinement of EM fields, thereby circum-venting the diffraction limit of conventional optics5–9.For this reason, plasmonics has been considered the ulti-mate pathway to manipulate light-matter interactions atthe nanometer scale.

Very recently, graphene18–20 — a two-dimensional(2D) crystal made up of carbon atoms arranged in a hon-

eycomb lattice — has emerged as a promising plasmonicmaterial, benefiting from this material’s remarkable elec-tronic and optical properties19–22. Doped graphene is ca-pable of supporting SPPs — graphene surface plasmon-polaritons (GSPs)23–30 — in the THz and mid-IR spec-tral range. These possess tantalizing properties, outper-forming traditional noble-metal plasmonics, in that spec-tral window, in terms of mode confinement, and are pre-dicted to suffer from relatively low losses when comparedto customary three-dimensional (3D) metals24,25,28,31. Inaddition, graphene plasmons have yet another key advan-tage: the ability of being actively tunable by means ofelectrical gating or chemical doping. This feature con-stitutes a major improvement over conventional metal-based plasmonics3,11 (where tunability is usually lim-ited by the geometry and composition of the system,and therefore it is fixed), and constitutes a sought-aftercharacteristic for active nanophotonic devices and/or cir-cuitry based on graphene plasmons. Indeed, a plethora ofproof-of-concept, application-oriented experiments havealready demonstrated the capabilities of GSPs to deliverextremely sensitive biochemical sensors32–36, surface-enhanced Raman scattering (SERS)37–42, polarizers43,44,optical modulators45–48 and photodetectors49–53.

Such achievements are particularly notable in thelight that optical excitation of graphene plasmons wasonly achieved as recently as in 2011 by Ju et al., us-ing periodic arrays of graphene microribbons54. Thatfoundational publication paved the way for the emer-gence of many experimental and theoretical works thatsoon followed, thereby establishing the field of grapheneplasmonics23–30. As of today, GSPs have been realizedin a number of systems, ranging from patterned gridsof graphene ribbons32,33,44,46,54–58, disks57,59–62, andrings59,60, periodic anti-dot lattices62–64, resonators65,66,hybrid graphene/metal nano-antennas49,50,67–69, amongothers70–78.

2

In the heart of plasmonics lies the fact that freely-propagating EM radiation cannot couple directly to plas-mons owing to the momentum mismatch between plas-mons and photons of the same frequency. However, theproperty that the plasmon’s wavevector is larger than thewavevector of light of the same frequency is exactly whatenables extreme localization of light into subwavelengthvolumes. For extended graphene, these volumes can beabout α3 ≈ 10−6 times smaller (where α denotes thefine-structure constant) than the volume characterizedby the free-space light’s wavelength (i.e. λ−3

0 ). Typicalstrategies to couple light to graphene plasmons involvethe patterning of pristine graphene into gratings and re-lated nanostructures32,33,44,46,54–64, the use of dielectricgratings70,71, light scattering from a conductive tip76–78,and even non-linear three-wave mixing74,75.

In this context, the utilization of periodic grids ofgraphene ribbons — fabricated by patterning an other-wise continuous graphene sheet — has been one of themost popular setups to realize graphene plasmons withenergies from the THz up to the mid-IR regime, whichcan be tailored either by varying the size of the rib-bons or by tuning the concentration of charge-carriers ingraphene (and thus the Fermi level). Under this scheme,the array of graphene ribbons effectively acts as a diffrac-tion grating for EM radiation impinging on the system(e.g. from a laser), producing scattered waves whichcarry momenta in multiples of the reciprocal lattice vec-tor, G = 2π/L (where L is the grating period), thusovercoming the above-mentioned kinematic constraint.The reason the use of ribbon arrays to couple light toGSPs has been so predominant is essentially two-fold(apart from being easily attainable with current fabrica-tion technologies): it enable us to overcome the momen-tum mismatch between light and GSPs; and it renders astronger (composite) plasmonic response than one wouldget from a single graphene ribbon (also, in this lattercase, instead of well-defined diffracted orders, the scat-tered waves would transport a continuum of momenta).

In this work we develop an analytical framework de-scribing the interaction of EM radiation with periodicgrids of micro- and nano-sized graphene ribbons. Themain motivation driving this work was to deliver a simpleand transparent theoretical tool capable of explaining theplasmon-induced spectra measured in experiments thatdid not involve the use of computationally-heavy andtime-consuming numerical simulations. Here, we pro-vide simple closed-form expressions for the reflectance,transmittance and absorbance spectra of THz and mid-IR light through graphene patterned into ribbons. Thesespectra may then be used to design or model experi-ments with graphene plasmons in the laboratory, by sim-ply evaluating an analytical expression. The couplingbetween graphene plasmons and surface optical (SO)phonon modes of a SiO2 substrate is also considered, andwe observe a reconstruction of the polaritonic spectrumowing to the hybridization of GSPs with SO phonons ofthe underlying polar substrate. We further introduce a

semi-analytical technique developed elsewhere29,30,79–81

to benchmark our analytical theory. Finally, we comparethe outcomes of both frameworks against actual exper-imental data and demonstrate their ability to describeplasmonic excitations in periodic gratings of grapheneribbons.

II. RESULTS AND DISCUSSION

A. Analytical Method

We consider the scattering of EM radiation by a 1Dperiodic grid of graphene ribbons of width w. For thesake of simplicity, the ribbons are assumed to possessinfinite length in the longitudinal direction. In such anarrangement, the graphene grid behaves like a diffractiongrating for EM waves. The period of the grating is de-noted by L hereafter, and the system is assumed to lie inthe plane defined by z = 0, being cladded between twodielectric media with relative permittivities ε1 (for z < 0)and ε2 (for z > 0) — see Fig. 1. In what follows, we as-sume ribbons whose widths are ∼ 100 nm or larger, sothat the actual edge termination of the graphene ribbonsand finite-sized effects are not important, and thereforea classical electrodynamics framework suffices82,83.

oblique view

side view

Figure 1. Monochromatic p−polarized plane-wave impingingon a grid of graphene ribbons (not to scale) arranged in agrating-like configuration. The ribbons are sitting in the planedefined by z = 0. The structure is periodic, with period L,and the width of the graphene ribbons is defined by w. Thesystem is encapsulated between a top insulator with relativepermittivity ε1 (for z < 0), and a dielectric substrate withrelative permittivity ε2 (for z > 0).

We consider a p-polarized monochromatic plane-waveimpinging on the grid of graphene ribbons at an angle θ.For such polarization, the incident EM fields read

Bi(r, t) = Bi0 ei(k1·r−ωt) y , (1)

Ei(r, t) =(Ei0,x x + Ei0,z z

)ei(k1·r−ωt) , (2)

3

where the wavevector of the incoming wave is definedas k1 = kx x + kz z, with kx =

√ε1k0 sin θ and kz =√

ε1k0 cos θ, where k0 = ω/c. Naturally, the field am-plitudes Bi0, Ei0,x and Ei0,z are connected via Maxwell’s

equations, which establish the relations Ei0,x = c2kzωε1

Bi0

and Ei0,z = − c2kxωε1

Bi0. Furthermore, due the periodicityof the grid, one may write the reflected magnetic field inthe form of a Bloch-sum (also termed as Fourier-Floquetdecomposition),

Br(r) =

∞∑n=−∞

rn ei(qnx−κ−

z,nz) y , (3)

where an implicit time-dependence of the usual forme−iωt is assumed henceforth, and where the wavevectorsof the Bloch modes are defined as

qn = kx + nG = kx + n2π/L , (4)

where G = 2π/L is the primitive vector of the recipro-

cal lattice. In addition, note that ε1k20 = q2

n +(κ−z,n

)2as determined from Maxwell’s equations. Likewise, thefield transmitted across the graphene grating may alsobe casted as a Bloch-sum, reading

Bt(r) =

∞∑n=−∞

tn ei(qnx+κ+

z,nz) y , (5)

where(κ+z,n

)2= ε2k

20 − q2

n. As in the case of the incidentfields, we can make use of Maxwell’s curl equation ∇ ×B = −iωε/c2E to write out the corresponding reflectedand transmitted electric fields from Eqs. (3) and (5).This procedure leads to

Er(r) = − c2

ωε1

∞∑n=−∞

rn[κ−z,nx + qnz

]ei(qnx−κ

−z,nz) ,

(6)

Et(r) =c2

ωε2

∞∑n=−∞

tn[κ+z,nx− qnz

]ei(qnx+κ+

z,nz) , (7)

respectively. At this stage, the coefficients rn and tn arestill unknown. In order to determine them, one mustimpose the appropriate boundary conditions of the prob-lem. To that end, we employ the first boundary con-dition stating that the x-component of the electric fieldabove and below the graphene grid must be continuous,x ·(Ei + Er −Et

)|z=0 = 0, that is

kzBi0eikxx−

∞∑n=−∞

rnκ−z,ne

iqnx =ε1ε2

∞∑l=−∞

tlκ+z,le

iqlx . (8)

Multiplying the previous expression with a basis func-tion, e−iqmx, and integrating over the unit cell, yields

rm =kz

κ−z,mBi0δm,0 −

ε1ε2

κ+z,m

κ−z,mtm , (9)

which links the Bloch coefficients rm and tm (and Bi0 forthat matter). Moreover, according to Ohm’s law, theelectric fields produce a current given by J = σ(x)x ·Et|z=0x , which reads

Jx(x) =σ(x)c2

ωε2

∞∑n=−∞

κ+z,ntne

iqnx , (10)

where σ(x) is the position-dependent conductivity ofgraphene, which in the unit cell can be written as σ(x) =σ(ω)Θ(w/2 − |x|). In this expression, σ(ω) is the dy-namical conductivity of a graphene ribbon (here assumedto be bulk-like) and Θ(x) denotes the Heaviside stepfunction84.

We now introduce a central assumption into our an-alytic method, which is the validity of the edge condi-tion85. This condition states that the current perpendic-ular to a sharp edge — such that of a graphene ribbon —should be proportional to the square-root of the distanceto the edge, ρ, that is, Jx(ρ) ∝ √ρ. Our assumption hereis that in the regime where kw < 1 one can interpolatethe current by an expression that incorporates the edgecondition at both edges of each ribbon, e.g. x = ±w/2,simultaneously. Therefore, this ansatz allows us to writethe current within a ribbon in the unit cell as

Jx(x) = χeikxx√w2/4− x2Θ(w/2− |x|) , (11)

where χ is a coefficient to be determined. As a first steptowards the determination of the coefficient χ, we nowargue that Eqs. (10) and (11) must give rise to the sameinduced current (since they represent the same physicalquantity). Hence, one may write the following relation(in the unit cell)

χeikxx√w2/4− x2Θ(w/2−|x|) =

σ(x)c2

ωε2

∞∑n=−∞

κ+z,ntne

iqnx ,

(12)which, after multiplying by a basis function, e−iqmx, andintegrating over the unit cell, produces

χL

4mJ1(mπw/L) =

σ(ω)c2

ωε2

∞∑n=−∞

κ+z,ntn

sin ([n−m]πw/L)

[n−m]πw/L,

(13)where J1(x) is the 1st-order Bessel function of the firstkind84. This expression defines the coefficient χ in termsof the Bloch-amplitudes tm, and whose combination withEq. (9) connects the coefficients χ, rm and tm. Inorder to close the system of equations, we require an-other expression relating these quantities. Such “ex-tra” equation is the other boundary condition holding forthis system, in particular, the discontinuity of the mag-netic field across the graphene grid due to the presenceof the surface current induced by the electric field, i.e.z ×

(Bt −Br −Bi

)|z=0 = µ0Jxx which, after applying

the same operations that led to Eqs. (9) and (13), gives

Bi0δm,0 = tm − rm + µ0χw

4mJ1(mπw/L) , (14)

4

thereby closing the system. Finally, the combination ofEqs. (9) and (14) allows us to write the tm’s as

tm =ε2κ−z,m

ε1κ+z,m + ε2κ

−z,m

[2Bi0δm,0 − µ0χ

w

4mJ1(mπw/L)

],

(15)which, after using Eq. (13) endows us an expression forthe coefficient χ (from which the Bloch amplitudes tmand rm directly follow), that is

χ =2κ+

z,0κ−z,0

ε1κ+z,0 + ε2κ

−z,0

σ(ω)c2

ω

Bi0Λ(ω)

, (16)

where the quantity Λ(ω) is defined as

Λ(ω) =w

4

∞∑n=−∞

1

nJ1(nπw/L)

[1 +

σ(ω)

ωε0

κ+z,nκ

−z,n

ε1κ+z,n + ε2κ

−z,n

].

(17)It should be stressed that the sum in the previous ex-pression needs to be judiciously performed, since it is asum with alternating signs (check SI for details).

Therefore, Eqs. (9) and (15)–(17) provide us witha complete knowledge of the electromagnetic scatteringand subsequent excitation of graphene plasmons withinthe ribbons which make up the periodic system.

B. Transmittance, reflectance and absorbance fornormal incidence

Here we consider the particular case where the im-pinging radiation strikes the graphene grid at normalincidence (see SI for oblique incidence), for which wehave kx = 0 and kz =

√ε1k0, so that qn = nG.

In addition we remark that here, as in most experi-mental configurations, the z-component of the scatteredwavevectors remains real only for the zero-th mode, i.e.

κ+/−z,0 =

√ε2/1k0, while for the other diffraction orders it

is imaginary, that is κ+/−z,n = i

√q2n − ε2/1k2

0. The reason

for this is that often the period of the grating is muchsmaller than the impinging wavelength, L� λ, and thusq2n � ε2/1k

20, ∀ n 6= 0. This is no coincidence, since

our goal is to surpass the momentum imbalance betweenthe incident light and GSPs. This can only be effectivelyachieved by fabricating subwavelength gratings. There-fore, we take qn >

√max(ε1, ε2)k0 for n 6= 0 henceforth.

Consequently, only the zero-th mode reaches the far-field.

In possession of Eqs. (9) and (15)–(17), we have allthe necessary ingredients to compute the scattering effi-ciencies for EM radiation striking the array of grapheneribbons. From the aforementioned expressions, the re-flectance, transmittance and absorbance by the graphene

grid read (see SI for a detailed derivation)

R(ω) =

∣∣∣∣1− 2√ε1√

ε1 +√ε2

+ µ0χ

Bi0

πw2

8L

√ε1√

ε1 +√ε2

∣∣∣∣2 ,

(18)

T (ω) =<{1/√ε2}<{1/√ε1}

∣∣∣∣ √ε2√

ε1 +√ε2

(2− µ0

χ

Bi0

πw2

8L

)∣∣∣∣2 ,

(19)

A(ω) = 1−R(ω)− T (ω) , (20)

respectively [and recall Eqs. (16) and (17) for χ]. Fromthe inspection of the above equations it is clear that theplasmonic resonances are controlled by the poles of χ[or, similarly, −=m

{Λ−1(ω)

}]. Analyzing carefully the

structure of the quantity Λ(ω), we readily identify thatthese occur whenever the condition

1 +σ(ω)

ωε0

κ+z,nκ

−z,n

ε1κ+z,n + ε2κ

−z,n

= 0

⇔ ε1√(nG)2 − ε1k2

0

+ε2√

(nG)2 − ε2k20

+ iσ(ω)

ωε0= 0 ,

(21)

weighted by the factor J1(nπw/L)/n is met. Noticethat Eq. (21) is nothing but the implicit expression forthe dispersion relation of GSPs24–26,29,30 with wavevec-tor qn = nG. However, what particular Bragg modesconstitute the leading contributions for GSP-excitationstrongly depends on the filling ratio w/L (please refer toSI for further details).

C. Signatures of the graphene plasmon resonances

Having formulated our analytical model, we are nowable to compute the absorbance, reflectance and trans-mittance spectra of EM radiation impinging on a periodicgrid of graphene ribbons at normal incidence. Therefore,the results presented below are based on the outcome ofEqs. (18)–(20). In the following, we take the conduc-tivity of the graphene ribbons as the Drude conductivityof bulk-graphene (see Methods). This is a rather goodapproximation for doped graphene ribbons in the THzspectral range, as long as the ribbons are not too small(e.g. wider than several tens of nanometers82,83). In par-ticular, in Fig. 2 we show the absorbance spectra (leftpanel) and corresponding reflectance and transmittancespectra (right panel) for different values of the dampingparameter, Γ. The main feature figuring in the variousspectra is the presence of a well-defined peak in absorp-tion signalling the excitation of graphene plasmons. ThisGSP-assisted effect yields a dramatic enhancement in theabsorbance spectra of the grating, owing to the couplingof free-propagating THz radiation to plasmons supportedby the graphene ribbons which compose the periodic grid.Note that the aforementioned GSP-induced absoptioncomes hand in hand with a supression in transmittance

5

0 2 4 6 8 10Frequency (THz)

0

0.1

0.2

0.3

0.4

0.5A

bsor

banc

e

Γ = 3 meVΓ = 6 meVΓ = 9 meV

0 2 4 6 8 10Frequency (THz)

0

0.2

0.4

0.6

0.8

1

Tra

nsm

ittan

ce ,

Ref

lect

ance

Figure 2. Absorbance spectra (left panel), transmittance andreflectance spectra (right panel) of a p-polarized plane-waveimpinging on a periodic grid of graphene ribbons for varyingvalues of Γ = ~γ. The remaining parameters are: EF = 0.45eV, w = 2 µm, L = 4 µm, ε1 = 3, ε2 = 4, and θ = 0 (normalincidence).

and with an increase in reflectance at the GSP resonantfrequencies, which roughly correspond to the poles of χ(or, in other words, the zeros of Λ). Without surprise,smaller values of Γ render sharper resonances, with thesebecoming successively broader and less pronounced asthe electronic scattering rate increases. Also, the reso-nance frequency of the GSPs modes depends weakly onthe value of Γ, as it remains essentially unchanged despitethe different values of the damping parameter.

It should be stressed that, in principle, the interac-tion of EM radiation with the grating gives rise to mul-tiple plasmon resonances with q ≈ (2m + 1)π/w (form = 0, 1, 2, ...)30,55,86. The most prominent resonancecorresponds to the fundamental plasmon mode, whilethe higher-order resonances become increasingly weaker.Note that in the present model the latter are necessar-ily ignored, owing to the choice of ansatz for the current[cf. Eq. (11)] which can only account for the dipole-like fundamental resonance (the one that appears in Fig.2). Fortunately, this resonance carries most of the spec-tral weight (see SI) and it clearly dominates the polari-tonic spectrum29,30; in fact, the resonances that emergeat higher frequencies are often invisible (or barely visible)in many experiments, since they can only be detected forsmall values of Γ.

We now explore the dependence of the GSP-inducedabsorption spectra on the different parameters of thesystem. One of the most important parameters is theelectronic density, ne. This quantity is related with thematerial’s Fermi energy via EF = ~vF

√πne, where vF ≈

1.1× 106 m/s is the Fermi velocity of the Dirac fermionsin graphene87,88. The density of graphene charge-carrierscan be easily controlled by means of electrostatic gating.This possibility is of extreme relevance in graphene plas-monics, since it enables the excitation and control of tun-able GSPs with tailored properties at the distance of a

voltage knob. The effect of varying the electronic densitywithin the graphene ribbons which constitute the grat-ing is demonstrated in Fig. 3. From the figure, it is clearthat GSP-resonances become stronger and shift towardhigher frequencies as the density of charge-carriers in-creases. In order to quantify such behavior, in the rightpanel of Fig. 3 we have plotted the GSP-frequency (cor-responding to the fundamental mode) as a function ofthe doping level, to which we have fitted a function ofthe type fGSP(ne) ∝ nbe, having obtained b = 0.249 ' 1/4for the exponent (fitting parameter)89. This therefore

0 2 4 6 8Frequency (THz)

0

0.1

0.2

0.3

0.4

Abs

orba

nce

ne = 1.5ne = 1ne = 0.5

0 0.4 0.8 1.2 1.6 2ne (1013 cm-2)

2

3

4

5

f GSP

(TH

z)

Fit: a (ne)b

Model (analytic)

Figure 3. Dependence of the GSP-frequency on the electronicdensity of the graphene ribbons. Left panel: absorbance spec-tra for different selected values of ne; the legend gives ne inunits of 1013 cm−2. Right panel: resonant frequencies — re-trieved from our analytic theory (points) — for several valuesof ne, and corresponding fitting function to those points, thatis fGSP(ne) ∝ nb

e with b = 0.249 ' 1/4, in accordance withwhat is expected for graphene plasmons (see main text). Pa-rameters: Γ = 3.7 meV, w = 2 µm, L = 4 µm, ε1 = 3, ε2 = 4,and θ = 0 (normal incidence). We have used EF = ~vf

√πne,

with vF ≈ 1.1× 106 m/s.

demonstrates that the observed resonances scale with theelectronic density as fGSP ∝ n1/4

e , which is a specific sig-nature of graphene plasmons29,30,54,55. Contrariwise, in

typical 2DEGs a scaling with n1/2e is observed instead.

The different scaling for graphene is a direct consequenceof the linear dispersion exhibited by the Dirac particlesin this material18,19.

An alternative way to tune the GSP-resonances is topattern grids of graphene ribbons of different widths, w.For the sake of clarity, we shall keep the filling ratiow/L = 1/2 constant. Figure 4 depicts the calculated ab-sorbance spectra for periodic arrays of graphene ribbonsof different widths. Notice that structures with narrowerribbons yield GSPs with higher energies. The funda-mental plasmonic resonance of the grating resembles theexcitation of a GSP in extended graphene with q ∼ π/w.Such large wavevectors can only be attained due to thecontribution of the several Bragg diffraction orders orig-inating from the interaction of the incident light withthe grid. Within this reasoning, Fig. 4 shows that theplasmonic resonances scale as

√q, in a similar way that

plasmons in an unpatterned, continuous graphene sheetdo. We have also found that, while the filling ratio w/L

6

0 2 4 6 8 10 12Frequency (THz)

0

0.1

0.2

0.3

0.4A

bsor

banc

ew = 4 µmw = 2 µmw = 1 µmw = 0.5 µm

0 1 2 3 4 5 6 7

q (µm-1)

0

2

4

6

8

10

f GSP

(TH

z)

Fit: a qb

Model (analytic)

w = L/2 q = π/w

Figure 4. Dependence of the GSP-frequency on the period ofthe graphene grating, L. We kept the ratio w = L/2 fixed, andtherefore the GSP-fundamental mode carries a wavevector ofabout q ∼ π/w. In the right panel we show the position of theGSP-resonances rendered by the analytical method incorpo-rating the edge condition as a function of the GSP wavevector(square data points). To these data we have fitted a curvefGSP(q) ∝ qb, having obtained a exponent b = 0.502 ∼ 1/2.Parameters: EF = 0.4 eV, Γ = 3.7 meV, ε1 = 3, ε2 = 4, andθ = 0.

has a significant impact on the position of the plasmonicresonance of the graphene grid, the physics of system islargely determined by the ribbon-size, i.e. fGSP ∝ w−1/2

provided that w/L < 1/2, that is, that the interactionbetween neighboring ribbons is small.

The calculated electric field akin to the graphene plas-mons supported by the ribbons which constitute the grat-ing is depicted in Fig. 5 (depicting the unit cell of a rep-resentative array of ribbons with dimensions w = 2 µmand L = 4 µm). Such computation is straightforwardonce the resonant frequency akin to the fundamentalGSP mode is determined; that information is then fedinto the Bloch amplitudes (9) and (15) that represent thescattered field. The figure plainly demonstrates the po-tential graphene plasmons have to squeeze EM fields intodeep subwavelength dimensions. Notice that most of themodal energy is concentrated in the immediate vicinityof the graphene ribbons. In addition, the spatial distri-bution of the electric field is not uniform along the rib-bons’ transverse direction: the density of charge-carriers(and thus the electric field) is higher at both edges of theribbons. The charge-density is also antisymmetric withrespect to the ribbons’ midpoint, bearing some similar-ity to an electric dipole. Such extreme field localizationplays a pivotal role, for instance, in biosensing, allow-ing the detection of minute variations in the local di-electric environment due to the presence/adsorption of agiven target analyte. This property, together with abilityto tune the GSP-resonances, enables not only unprece-dentedly large field overlaps, but also provides a route totailor the interaction of GSPs with the vibrational reso-nances of biochemical molecules, thereby achieving hugespectral overlaps that allow specific label-free detectionof biomolecules via their vibrational fingerprints32,33.

We further note that the overall spatial configuration

of the field illustrated in Fig. 5 is qualitatively main-tained throughout a wide range of ribbon widths, fromthe micrometer to the nanometer size, provided thatwe are within the quasi-static regime (i.e. qGSP � k0)and at resonant frequencies below EF /~, beyond whichGSPs become quenched owing to the onset of interbandtransitions24,30. This scale-invariance is a property of theelectrostatic limit24,30,90. Note, however, that we includeretardation in our calculations nevertheless.

The results presented above, covering an appreciablevast parameter space, suggest that our analytical modelis able to correctly describe the fundamental plasmonicexcitations which arise in periodic grids of graphene rib-bons. We thus have built an analytic framework whichdelivered closed-form expressions for the spectra whichcan be easily evaluated, and that yield results consistentwith those found in the literature24–26,29,30,54,55.

D. THz plasmons in graphene microribbons

In order to determine to what extent our analyticalmodel is capable of explaining experimental spectra, weshall test our theory against measured data taken fromthe experiments performed by Ju et al.54. To that end,we mimic the experimental setup by feeding the reportedempirical parameters into our equations. In addition, weconcurrently employ a semi-analytical technique intro-duced elsewhere29,30,79–81 which represents the conduc-tivity of graphene (and resultant current) in terms of aFourier series (see Methods). This has the advantage oftaking into account not only the fundamental plasmonicresonance, but also the higher-order ones. On the otherhand, it requires the numerical solution of a linear algebraproblem and therefore we refer to it as a semi-analyticalmethod hereafter. We further emphasize that, to thebest of our knowledge, so far no attempt has been madeto perform a direct comparison of the outcome of thislatter method against available experimental data. Nev-ertheless, it is still far less computationally-demandingthan fully numerical simulations such as the FDTD orFEM techniques91.

In their experiments, the authors of Ref. 54 fabri-cated three different samples containing periodic grids ofgraphene ribbons with widths of 4, 2 and 1 µm, whilemaintaining the ratio L = 2w unchanged. Moreover,the authors have concluded that an effective dielectricconstant of εeff = 5 adequately accounts for the intri-cate optical constants of the cladding dielectrics (iongel and SiO2/Si), and have reported a scattering rateof γ/(2π) = 4 THz54. While at THz frequencies theconductivity of graphene can be approximated by itsDrude expression, here we model the conductivity as ob-tained using the Kubo formula at room temperature (seeMethods). This is necessary here because the experi-mental data of Ju et al.54 refers to the change in trans-mittance with respect to the same quantity measuredat the “charge neutral point” (CNP) [when the Fermi

7

- 0.4 - 0.2 0 0.2 0.4

-1 +1

Figure 5. Electric field representing graphene plasmons excited in the graphene ribbons which compose the grid (withdimensions w = 2 µm and L = 4 µm). The figures show the plasmonic fields in the system’s unit cell, and the graphene ribbonis indicated by the horizontal black line. Left panel: Vectorial representation of the electric field (in the y = 0 plane) dueto GSPs, EGSPs(x, z) = EGSPs

x (x, z)x + EGSPsz (x, z)z. The intensity plot refers to the quantity sgn(z)EGSPs

z which roughlyhighlights the charge-density within each graphene ribbon. Righ panel: normalized spatial distributions of the electric fieldcomponents EGSPs

z (top) and EGSPsx (bottom). The spatial range covered in these sub-panels is the same as in the main

panel. We note that only modes corresponding to plasmonic (evanescent) modes were included in the sums figuring in theBloch expansions, which for the parameters used here encompass all modes with the exception of the specular one (i.e. withn = 0). Parameters: EF = 0.4 eV, Γ = 3.7 meV and ε1 = ε2 = 4.

level is at the so-called Dirac point] where neither finite-temperature nor interband processes can be neglected.

The comparison between the calculated GSP-induced(normalized) change in transmittance, −∆T = TCNP−T ,and the experimental data is portrayed in Fig. 6. The ob-served agreement between theory and experiment is out-standing and constitutes compelling evidence that boththeories are capable of interpreting the measured spectra.The peaks visible in the figure originate from the exci-tation of the main GSP plasmonic resonance supportedby the graphene ribbons which form the grid. The shift-ing of the plasmon resonances towards higher frequen-cies as a consequence of the narrowing of the ribbonsexhibits the predicted fGSP ∝ w−1/2 scaling behavior.Although the degree of accordance between the data andthe full-analytical model is rather good, it seems that thismodel slightly underestimates the resonant frequency.On the other hand, the agreement among the experi-mental measurements and the spectra obtained by thesemi-analytical model is indeed quite remarkable, withthe computed (measured) GSP-resonances located at 2.9(3), 4 (4.1) and 5.6 (6) THz for arrays with ribbon widthsof 4, 2 and 1 µm, respectively. Therefore, a possible mo-tif for the small redshift visible in the spectra producedby the full-analytic theory may be concerned either withthe fact that it neglects the higher-order plasmon reso-nances and/or to the approximation made for the currentinvolving the edge condition, for instance, it may not be

exactly modeled by a square-root as in Eq. (11) [see SI].Still, and despite this fact, the fidelity of the analyticalmodel remains very good.

It should be appreciated that the use of either one ofthe above-mentioned analytical or semi-analytical tech-niques, apart from requiring less resources, provide a levelof physical insight and intuition that numerical method-ologies based on the numerical solution of Maxwell’sequations simply cannot envision.

E. Hybrid mid-IR plasmons in graphenenanoribbons: plasmon-phonon coupling

Graphene plasmonics has the potential to become aviable tool for nanophotonic devices working within abroad spectral window, from the THz/far-IR up to mid-IR frequencies. Remarkably, routes to bring grapheneplasmonics to near-IR and visible frequencies have al-ready been proposed from a theoretical perspective24.

At the time of writing, many experiments havereported GSPs at mid-IR frequencies32,33,58,60–62,65,66.Coupling light to graphene plasmons at those frequen-cies can be realized in nanostructured graphene withtypical dimensions from several tens of nanometers toa few hundreds of the nanometer. The mid-IR spectralrange is a particularly important one, as many biologicaland chemical compounds exhibit resonances in that re-

8

0.8

1

m.)

Experiment (w = 4 µm)Semi-analytical (w = 4 µFull-analytical (w = 4 µExperiment (w = 2 µm)Semi-analytical (w = 2 µ

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

-ΔT

(nor

m.)

Experiment (w = 4 µm)Semi-analytical (w = 4 µFull-analytical (w = 4 µExperiment (w = 2 µm)Semi-analytical (w = 2 µFull-analytical (w = 2 µExperiment (w = 1 µm)Semi-analytical (w = 1 µFull-analytical (w = 1 µ

0 1 2 3 4 5 6 7 8 9 10Frequency (THz)

0

0.2

0.4

0.6

-ΔT

(nor Full-analytical (w = 2 µm

Experiment (w = 1 µm)Semi-analytical (w = 1 µFull-analytical (w = 1 µm

0 1 2 3 4 5 6 7 8 9 10 11Frequency (THz)

0

0.2

0.4

0.6

0.8

1

-ΔT

(nor

m.)

Figure 6. Normalized plasmon-induced change in transmittance relative to the CNP, −∆T = TCNP − T , in periodic grids ofgraphene ribbons with different dimensions: w = 4 µm (green), w = 2 µm (red), w = 1 µm (blue), and w/L = 1/2 throughout.The dashed lines correspond to experimentally measured spectra54, while the dotted lines and solid lines correspond to thespectra obtained using our full-analytical model and the semi-analytical technique (see Methods), respectively. We have usedthe following parameters, in accordance with Ref. 54: EF = 0.497 eV, Γ = 16.5 meV and ε1 = ε2 = 5.

gion of the EM spectrum. Thus, tunable graphene plas-mons may be perceived as a fertile playground for ap-plications in biochemical sensing and spectroscopy. Fur-thermore, when graphene is deposited in a polar sub-strate — such as hBN or SiO2 — the Fuchs–KliewerSO phonons92 of the substrate can couple to plasmonsin graphene via Frohlich interaction93, leading to theemergence of new hybrid modes dubbed graphene sur-face plasmon-phonon polaritons55,66 (GSPPhs). In orderto account for the optical phonons arising in the neigh-boring polar material(s), their corresponding frequency-dependent, complex-valued dielectric function(s) can bemodeled using adequate Lorentz oscillator models94 in-corporating the phononic resonances, or more evolvedmodels, e.g. based on Gaussian functions and integrals95.The hallmark of strong coupling between graphene plas-mons and SO phonons is the complete reshaping of thetraditional fGSP ∝

√q dispersion of bare GSPs into

a set of multiple well-defined branches ascribed to hy-brid GSPPhs modes possessing mixed plasmonic andphononic character, as demonstrated in Fig. 7-c) for ex-tended graphene sitting on SiO2 (see caption for furtherdetails). In particular, notice the evident anti-crossingbehavior of the plasmon-phonon bands in the vicinity ofthe SO frequencies.

In Fig. 7 [panels a) and b)] we compare the results forarrays of graphene ribbons obtained using the analytic[a)] and semi-analytic [b)] methods (solid lines) againstthe experimental spectra (light-brown points) collectedby Luxmoore et al.58. Their data shows evidence ofstrong interaction of GSPs — excited in periodic grids ofgraphene nanoribbons — with the three SO phonons ofthe underlying SiO2 substrate. In our modeling, we have

met the experimental configuration of the fabricated de-vices, consisting in doped (EF = 0.37 eV) nanoribbonswith widths ranging from 450 nm down to 180 nm ar-ranged in a periodic array with periodicity L = 5w/258.The dynamical dielectric function of the SiO2 substratewas taken from the literature95 (check SI), and we haveemployed, as before, the optical conductivity of grapheneunder Kubo’s framework — cf. Methods.

It is worth to highlight that both the analytic and semi-analytic theories outlined above fit admirably to the ex-perimental data, whose structure is now much more intri-cate than the one seen for microribbons in the THz range.The GSPPh-induced extinction spectrum (1 − T/TCNP)of the several samples presented in Fig. 7 reveals the ex-istence of multiple peaks, which correspond to the fourpolaritonic bands visible in figure’s last panel. We stressthat, as expected, all four resonances shift toward higherfrequencies upon decreasing ribbon size. Note, however,that they disperse at different rates. This is a directconsequence of the relative plasmon-to-phonon contentwhich tends to vary depending on the distance each res-onance is from the SO flat bands: the more plasmon-likethe hybrid GSPPhs modes are (i.e. the farther they arefrom the uncoupled SO bands), the faster they disperse.Another key element, recognizable in the spectra, is theclear transfer of spectral weight from the first peak tothe other resonances ascribed to higher GSPPhs bands.Together, the above-mentioned features constitute unam-biguous manifestations of anti-crossing behavior.

From the first two panels of Fig. 7 it is apparent thatboth models closely follow the experimental data, therebyconfirming their adequacy to describe the empirical ex-tinction spectra. As in the case of the microribbons, the

9

500 1000 1500Frequency (cm-1)

0

10

20

30

40

50

60

1-T/

T CN

P(%

)

500 1000 1500Frequency (cm-1)

0

10

20

30

40

50

60Analytic Quasi-Analytic

450nm

390nm

340nm

240nm

210nm

180nm

a) b) c)c)

Figure 7. Hybrid mid-IR graphene plasmon-phonon polaritons excited in period gratings of graphene nanoribbons sitting ona polar (SiO2) substrate. Experimental data (light-brown points) and corresponding extinction spectra calculated using a) theanalytical model; and b) the semi-analytical model (solid lines). The computations were carried out in accordance with theexperimental parameters58: EF = 0.37 eV, θ = 0 (normal incidence), ε1 = 1 and the dielectric function of SiO2, ε2 ≡ εSiO2(ω),was taken from the literature95. We have used electronic scattering rates corresponding to about 25 − 30 meV dependingon the sample. c) Loss function (via =m rTM) for extended graphene deposited on SiO2, with the uncoupled GSP spectrumsuperimposed (white dashed line); we note that this serves only as an eye-guide to the interpretation of the data, since weexpect the polaritonic spectrum akin to the periodic grid of nanoribbons to be slightly different from that of unpatterned,extended graphene.

fully analytical method tends to underestimate slightlythe position of the resonances, while the semi-analyticalmethod yields an excellent agreement, particularly forthe samples with wider ribbons. The small deviationfrom the data observed in the spectra of the narrowerribbons may have multiple origins exogenous to our the-ory, for instance, an incomplete knowledge of the dielec-tric properties of the particular SiO2 substrate used inthe experiments and/or the effect of edge damage (anddefects) introduced during the etching process55,58,96 —which, naturally, should be more pronounced for smallerribbons —, and may yield ribbon edges with impairedelectrical activity (therefore rendering effective widthssmaller than the actual ribbon widths55). Here, we ne-glect the impact of the latter since it has been shownthat the damage at the edges of each ribbon is highlyheterogeneous58,96.

The theoretical results produced in this work and sub-sequent confrontation against experimental data confirmthe ability of the theoretical tools developed here to sim-ulate and interpret spectra taken from real-world ex-periments, providing excellent, reliable results almostinstantaneously97 without the need of substantial com-putational resources.

III. CONCLUSIONS

In conclusion, we have developed a novel analyti-cal approach, based on the edge condition and Bloch-expansions for the fields, to describe graphene plas-mons excited in periodic grids of graphene ribbons. Wesolved the scattering problem and provided simple closed-form expressions to compute the reflectance, absorbance,transmittance, and related extinction spectra. We thenbenchmarked the results of our analytical theory us-ing a semi-analytical model, and tested both techniquesagainst experimental data available in the literature54,58.Our results show a very good agreement between thetheoretical curves and the empirical data, which consti-tutes compelling evidence for the validity of the afore-mentioned theories. That concordance extends from theTHz, using microribbon arrays, to the mid-IR spectralregion, using nanoribbons. In the latter domain, we havealso investigated hybrid GSPPhs excitations that arisefrom the interaction of GSPs with the SO phonons of theSiO2 substrate, leading to the appearance of four com-posite modes featuring spectral weight transfer, which isindicative of anti-crossing behavior (resulting from thereconstruction of the bare GSPs spectrum owing the po-lar coupling).

The approaches developed in this work have two mainadvantages: (i) they endow us with a deeper insight andsense for the physics governing plasmonic excitations inengineered graphene structures; and, (ii) render viable

10

simulations of experimentally relevant quantities, on-de-mand and almost instantaneously, without the cost oflengthy, computationally-demanding full-wave numericalpackages, at least for patterned structures with a fair de-gree of symmetry. On the other hand, these naturallycannot compete with fully-numerical techniques such asFEM simulations in what concerns versatility to dealwith many different and complex geometries.

Our findings suggest that both the analytical andthe semi-analytical models described here could be usedto architecture new forefront nanophotonic experimentsbased on graphene plasmonics, which is emerging as apromising field to deliver cutting-edge optoelectronic de-vices with tailored light-matter interactions.

ACKNOWLEDGMENTS

The authors thank N. Asger Mortensen for insight-ful and valuable comments. PADG acknowledges finan-cial support from Fundacao para a Ciencia e a Tec-nologia (Portugal) from grant No. PD/BI/114376/2016.NMRP and YVB acknowledge financial support from theEuropean Commission through the project “Graphene-Driven Revolutions in ICT and Beyond” (Ref. No.696656). This work was partially supported by thePortuguese Foundation for Science and Technology(FCT) in the framework of the Strategic FinancingUID/FIS/04650/2013. The Center for NanostructuredGraphene is sponsored by the Danish National ResearchFoundation, Project DNRF103.

Appendix A: METHODS

Conductivity of graphene In this work we modelthe dynamical conductivity of the graphene ribbons us-ing Kubo’s formula within the local approximation atroom temperature (T = 300 K). In such framework, thematerial’s 2D conductivity reads30,98

σKubo(ω) = σintra(ω) + σinter(ω) , (A1)

σintra(ω) =σ0

π

4

Γ− i~ω

[EF +

2

βln(1 + e−βEF

)],

(A2)

σinter(ω) =σ0

π[πG(~ω/2)

+ i4~ω∫ ∞

0

dEG(E)−G(~ω/2)

(~ω)2 − 4E2

], (A3)

where β = (kBT )−1 (here kB is Boltzmann’s constant),σ0 = e2/(4~), and where the quantity G(x) is defined as

G(x) =sinh (xβ)

cosh (EFβ) + cosh (xβ). (A4)

In the THz regime, for graphene under typical dopinglevels — such that EF � kBT and 2EF > ~ω —, the

conductivity of graphene can be well approximated bythe Drude-like expression

σD(ω) ≈ σ0

π

4EFΓ− i~ω

, (A5)

provided that the conditions EF � kBT and 2EF > ~ωare met.

Semi-analytical method in a nutshell Similarly tothe fully-analytical method described in the main text,the semi-analytical method also expresses the EM fieldsin the form of Bloch-sums. Namely, the fields in themedium j may be written as (under normal incidence)

E(j)x (x, z) = Eincx eikzzδj,1 +

∑n

E(j)x,ne

inGx−ξj,n|z| , (A6)

E(j)z (x, z) = Eincz eikzzδj,1 +

∑n

E(j)z,ne

inGx−ξj,n|z| , (A7)

B(j)y (x, z) = Bincy eikzzδj,1 +

∑n

B(j)y,ne

inGx−ξj,n|z| , (A8)

where G = 2π/L, kz =√ε1k0 and ξ2

j,n = (nG)2 − εjk20.

Imposing the adequate boundary conditions, one obtainsthe following system of equations

Q0E(2)x,0 +

i

ωε0

∑l

σ−lE(2)x,l = i

2ε1kzEincx , (A9)

QnE(2)x,n +

i

ωε0

∑l

σn−lE(2)x,l = 0 . (A10)

for n = 0 and n 6= 0, respectively, and where Qn =ε1/ξ1,n + ε2/ξ2,n. In Eqs. (A9) and (A10), the quan-tities σm are the components of the Fourier series thatincorporates the system’s periodicity, that is

σ(x) =∑m

σmeimGx , (A11)

σm =1

L

∫ L/2

−L/2σ(x)e−imGxdx . (A12)

The numerical solution of the (truncated) system ofequations posed by Eqs. (A9) and (A10) for each fre-quency ω (entering as a parameter), renders the field

amplitudes, E(2)x,l , in terms of Eincx . As before, only the

mode with n = 0 is propagating, and thus only this con-tributes (i.e. reaches the far-field) to the transmittance,reflectance and absorbance, which read

T (ω) =<{√ε2}<{√ε1}

∣∣∣∣∣E(2)x,0

Eincx

∣∣∣∣∣2

, (A13)

R(ω) =

∣∣∣∣∣E(2)x,0 − Eincx

Eincx

∣∣∣∣∣2

, (A14)

A(ω) = 1− T (ω)−R(ω) . (A15)

From these expressions, the corresponding spectra akinto the semi-analytical model may be readily obtained.

11

Appendix B: Supporting Information

1. Convergence of the sum in Λ(ω)

Notice that our results for the reflection and transmis-sion amplitudes fundamentally depend on χ, which inturn strongly depends on the function Λ(ω). The latterreads [cf. Eq. (17)]

Λ(ω) =w

4

∞∑n=−∞

1

nJ1(nπw/L)

[1 +

σ(ω)

ωε0

κ+z,nκ

−z,n

ε1κ+z,n + ε2κ

−z,n

]

=πw2

8L

[1 +

σ(ω)

ωε0

κ+z,0κ

−z,0

ε1κ+z,0 + ε2κ

−z,0

]

+w

2

N∑n=1

1

nJ1(nπw/L)

[1 +

σ(ω)

ωε0

κ+z,nκ

−z,n

ε1κ+z,n + ε2κ

−z,n

],

(B1)

where in the last equality we have made explicit use ofthe fact that the summand is even with respect to n,where n ∈ integers. Note that this expression comprisesan infinite sum over n, so that we have also truncated thesum in the last step of Eq. (B1) for numerical purposes.The question that now arises is: how large should N be?And what requirements should it fulfill?

In order to answer to this question, let us plot theresults of, say, the reflectance, using several values of N .The outcome of such procedure is shown in Fig. 8. Fromthe figure, one can see a striking difference between theresults obtained using odd N -values and even N -values.In particular, note that whenever we picked N as odd wealways got the same (converged) result. This result alsocoincides with the one in the limit N → ∞ (light-greencurve). Conversely, for even N -values, one sees an erraticbehavior in the resulting spectra which indicates that theresults did not converge. Only for very large values ofN even (such has 5000), one obtains the correct result.Furthermore, notice that even when choosing a small,but odd value for N – such as N = 5 — one gets thesame (correct) result that one would obtain by choosinga large, but even N instead — such as N = 5000. Thisclearly highlights the need to choose N correctly, namelyto choose an odd-valued N .

The reason for this apparently counterintuitive behav-ior can be elucidated by plotting the values of the sum-mand in Eq. (B1) as a function of n. This is done inpanel b) of Fig. 8. Clearly, the figure shows an alternat-ing series; this kind of series usually demands proper carefor their accurate computation. In our case, one needsto have special care when choosing N , that is, to choosean odd-valued N . In that way, the total number of ele-ments of the series is N+1 (because it includes the n = 0term), which is even, and, therefore, it correctly includespairs of positive and negative values [as indicated in fig-ure’s 8 panel b)]. Naturally, as N approaches infinity,the choice between N -odd or N -even becomes unimpor-tant. However, our analysis demonstrates that by using

0 2 4 6 8Frequency (THz)

0

0.05

0.1

0.15

0.2

Ref

lect

ance

N=6N=5000N=11N=100N=5N=51N=10N=101

εtop = 3 , εbottom = 4

2 4 6 8 10 12 14

- 2

- 1

0

1

2

N should be odd, so that N+1 is even!

0 2 4 6 8Frequency (THz)

0

0.05

0.1

0.15

0.2

Ref

lect

ance

BF N=6BruteForce N=5000BF N=11BF (N=100)BF N=5AlternatingSigns (N=50)BF N=10BF N=101

εtop = 3 , εbottom = 4

N odd

N even

N -> inftya)

b)

Figure 8. a) Reflectance spectra using different values ofN , which truncates the sum entering in the function Λ(ω).Notice the difference between even and odd N -values, in theconvergence of results. b) Alternating series figuring in Λ(ω)[cf. Eq. (B1)] as a function of n.

an odd-valued N , one can obtain accurate results with(odd) N -values as small as 5.

2. Derivation of the Formulae for the Reflectanceand Transmittance Spectra

For the derivation of the reflectance and transmittancescattering probabilities, we need to introduce the Poynt-ing vector (for non-magnetic media),

S =1

µ0E×B , (B2)

which characterizes the flux of electromagnetic energyper unit area. Assuming that both E and B can be writ-ten as in terms of harmonic functions, e−iωt, one maywrite the time-averaged Poynting vector as

〈S〉 =1

2µ0<e{E×B∗} , (B3)

where the star denotes the complex-conjugate. For a TMwave polarized in the xz-plane, the time-averaged Poynt-ing vector reads

〈S〉 =1

2µ0<e{ExB∗y z− EzB∗y x} . (B4)

Then, applying the previous equation for the incidentwave, one obtains

〈Siz〉 =c2

2µ0ω<e

{kzε1

} ∣∣Bi0∣∣2 , (B5)

12

for the z-component (i.e. the component normal to thegraphene grating). Similarly, for the reflected wave onehas

〈Srz,n〉 = − c2

2µ0ω<e

{κ−z,nε1

}|rn|2 , (B6)

for the n-th diffraction order (or Bloch-mode), whereasfor the transmitted wave we obtain

〈Stz,n〉 =c2

2µ0ω<e

{κ+z,n

ε2

}|tn|2 . (B7)

At this point we should stress that for purely imagi-

nary wavevectors, κ+/−z,n → i

∣∣∣κ+/−z,n

∣∣∣, the Poynting vec-

tors associated with those Bloch modes give zero contri-

bution [since <e{i|κ+/−z,n |} = 0; cf. Eqs. (B6) and (B7)],

and, as such, they will not contribute neither to the re-flectance nor to the transmittance. This is because theyare evanescent waves, and therefore they do not carry en-ergy along the z-direction. Notice that the modes are so-called non-propagating or evanescent whenever qn > εk0,where qn = kx + nG with G = 2π/L, and k0 = ω/c, inwhich case we have

κ+/−z,n =

√ε2/1k

20 − q2

n → i√q2n − ε2/1k2

0 , (B8)

in accordance with the definitions used in the main text.Finally, the reflectance and transmittance through thestructure under oblique incidence read

R(ω, θ) =∑n

∣∣∣∣ 〈Srz,n〉〈Siz〉

∣∣∣∣ =<e{κ−z,0/ε1

}<e {kz/ε1}

∣∣∣∣ r0

Bi0

∣∣∣∣2 , (B9)

and

T (ω, θ) =∑n

∣∣∣∣ 〈Stz,n〉〈Siz〉

∣∣∣∣ =<e{κ+z,0/ε2

}<e {kz/ε1}

∣∣∣∣ t0Bi0∣∣∣∣2 , (B10)

respectively, where the sums were carried out overpropagating modes only (solely the n = 0 mode forthe parameters used in this work). The absorptionspectrum stems from these equations according toA(ω, θ) = 1−R(ω, θ)− T (ω, θ).

For normal incidence, the above formulae simplify con-siderably to:

R(ω) =

∣∣∣∣ r0

Bi0

∣∣∣∣2 , (B11)

T (ω) =<e{

1/√ε2}

<e{

1/√ε1} ∣∣∣∣ t0Bi0

∣∣∣∣2 , (B12)

A(ω) = 1−∣∣∣∣ r0

Bi0

∣∣∣∣2 − <e{

1/√ε2}

<e{

1/√ε1} ∣∣∣∣ t0Bi0

∣∣∣∣2 , (B13)

where r0 and t0 are computed using Eqs. (9) and (15)–(17). Explicitly,

t0Bi0

=

√ε2√

ε1 +√ε2

[2− µ0

χ

Bi0

πw2

8L

], (B14)

r0

Bi0= 1−

2√ε1√

ε1 +√ε2

+ µ0χ

Bi0

πw2

8L

√ε1√

ε1 +√ε2

, (B15)

with

χ

Bi0=

2κ+z,0κ

−z,0

ε1κ+z,0 + ε2κ

−z,0

σ(ω)c2

ω

1

Λ(ω), (B16)

Λ(ω) =w

4

∞∑n=−∞

1

nJ1(nπw/L)

[1 +

σ(ω)

ωε0

κ+z,nκ

−z,n

ε1κ+z,n + ε2κ

−z,n

].

(B17)

3. Modes contributing to the fundamentalresonance for different filling ratios

0 1 2 3 4 5 6 7 8 9 10Frequency (THz)

0

0.05

0.1

0.15

0.2

0.25

0.3

Abs

orba

nce

0 1 2 3 4 5 6 7 8 9 10Frequency (THz)

0

0.2

0.4

0.6

0.8

|Ex,

m(2

)/ E

xinc |2

n = 1n = -1n = 2n = -2n = 3n = -3n = 4n = -4n = 5n = -5

w = 2 µm ; L =2 w

0 1 2 3 4 5 6 7 8 9 10Frequency (THz)

0

0.05

0.1

Abs

orba

nce

0 1 2 3 4 5 6 7 8 9 10Frequency (THz)

0

0.05

0.1

0.15

0.2

|Ex,

m(2

)/ E

xinc |2

n = 1n = -1n = 2n = -2n = 3n = -3n = 4n = -4n = 5n = -5

w = 2 µm ; L =4 w

0 1 2 3 4 5 6 7 8 9 10Frequency (THz)

0

0.05

0.1

0.15

0.2

0.25

0.3

Abs

orba

nce

0 1 2 3 4 5 6 7 8 9 10Frequency (THz)

0

0.2

0.4

0.6

0.8

|Ex,

m(2

)/ E

xinc |2

n = 1n = -1n = 2n = -2n = 3n = -3n = 4n = -4n = 5n = -5

w = 2 µm ; L =2 w

0 1 2 3 4 5 6 7Frequency (THz)

0

0.05

0.1

Abs

orba

nce

0 1 2 3 4 5 6 7Frequency (THz)

0

0.05

0.1

0.15

0.2

|Ex,

m(2

)/ E

xinc |2

w = 2 µm ; L =4 w

Figure 9. Absorbance and relative modulus-squared Bloch

amplitudes |E(2)x,n|2 corresponding to several Bragg modes,

for two different configurations: one with a filling ratio ofw/L = 1/2 (left) and another with w/L = 1/4 (right). Notethat which individual Bragg modes couple the most to thefundamental GSP-resonance depends on the specific fillingratio of the system. This figure was obtained using the semi-analytical method outlined in the Methods section.

4. Spectral weight akin to the higher-ordermultipolar resonances

5. Ansatz for the current

6. Dielectric function of silicon dioxide

13

0 1 2 3 4 5 6 7 8Frequency (THz)

0

0.05

0.1

0.15R

efle

ctan

ceEF = 0.45 eVEF = 0.3 eVEF = 0.2 eV

εtop = 3 , εbottom = 4

2nd order

fundamental resonances

Figure 10. GSP-resonances in a periodic grid of grapheneribbons (w = 4 µm and L = 2w) at different carrier concen-trations. The solid lines correspond to the results obtainedusing the semi-analytical method whereas the dotted lines cor-respond to the outcome of the full-analytical approach. Notethat, as discussed in the main article, the latter does notaccount for the higher-order multipolar resonances. On theother hand, the semi-analytical method includes these reso-nances; notice, for instance, the small, weaker peaks at higherfrequencies at the right of the fundamental resonance. These,however, carry little spectral weight and can only be seen dueto the rather small damping parameter (Γ = 2.6 meV). Thisjustifies their omission when using the full-analytical tech-nique, as their contribution is rather small.

- 1.0 - 0.5 0.0 0.5 1.00

20406080

100120

x (w/2)

Cur

rent

(a.u

.)

Semi-analytical result fit

Figure 11. Current, Jx(x), within the graphene stripesobtained using the semi-analytical method (Fourier expan-sion) and corresponding fitting function of the type cte ×√w2/4− x2 to illustrate the approximation which is made

when employing the edge condition [ansatz for the current; seeEq. (11)]. The oscillations emerging in the plot correspond-ing to the semi-analytic method are a natural consequence ofthe use of a Fourier expansion to describe the current in suchgeometry — a feature that is known as Gibbs phenomenon99.These are absent in the analytic ansatz for the current.

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0 500 1000 1500

- 5

0

5

10

Frequency (cm- 1)

ϵ SiO

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Figure 12. Complex dielectric function of the silicon dioxide(SiO2) substrate in the mid-IR region of the electromagneticspectrum: real part (solid line) and imaginary part (dashedline). Notice the three phononic resonances. We have mod-eled the frequency-dependent dielectric function of the polarSiO2 substrate using Gaussian functions and integrals, follow-ing Ref. 95.

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