Graphene epsilon-near-zero plasmonic crystalsMarios Mattheakis

mariosmat@g.harvard.eduSchool of Engineering and Applied Sciences

Harvard UniversityCambridge, MA, USA

Matthias Maiermaier@math.tamu.edu

Department of MathematicsTexas A&M UniversityCollege Station, TX, USA

Wei Xi BooUniversity of MinnesotaMinneapolis, MN, USA

Efthimios KaxirasSchool of Engineering and Applied Sciences

Department of PhysicsHarvard UniversityCambridge, MA, USA

ABSTRACTPlasmonic crystals are a class of optical metamaterials that consistof engineered structures at the sub-wavelength scale. They exhibitoptical properties that are not found under normal circumstancesin nature, such as negative-refractive-index and epsilon-near-zero(ENZ) behavior. Graphene-based plasmonic crystals present linear,elliptical, or hyperbolic dispersion relations that exhibit ENZ be-havior, normal or negative-index diffraction. The optical propertiescan be dynamically tuned by controlling the operating frequencyand the doping level of graphene. We propose a construction ap-proach to expand the frequency range of the ENZ behavior. Wedemonstrate how the combination of a host material with an opticalLorentzian response in combination with a graphene conductivitythat follows a Drude model leads to an ENZ condition spanning alarge frequency range.

KEYWORDSGraphene, metamaterials, plasmons

ACM Reference Format:Marios Mattheakis, Matthias Maier, Wei Xi Boo, and Efthimios Kaxiras.2019. Graphene epsilon-near-zero plasmonic crystals. In Proceedings ofDublin ’19: 6th ACM International Conference on Nanoscale Computing andCommunication (Dublin ’19). ACM, New York, NY, USA, 6 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn

1 INTRODUCTIONThe control of electromagnetic (EM) properties of optical materialson the nanoscale has opened the door for designing novel devicesand applications that include nano-antennas with extremely shortwavelength resonance [3], optical holography [24], and wirelessnanocommunication [6, 11]. Controlling the structure of materialsat the sub-wavelength scale enables the design of metamaterials

Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for components of this work owned by others than theauthor(s) must be honored. Abstracting with credit is permitted. To copy otherwise, orrepublish, to post on servers or to redistribute to lists, requires prior specific permissionand/or a fee. Request permissions from permissions@acm.org.Dublin ’19, September 25–27, 2019, Dublin, Ireland© 2019 Copyright held by the owner/author(s). Publication rights licensed to ACM.ACM ISBN 978-x-xxxx-xxxx-x/YY/MM. . . $15.00https://doi.org/10.1145/nnnnnnn.nnnnnnn

that possess properties that are not found under normal circum-stances in nature such as negative refraction [7] and epsilon-near-zero (ENZ) behavior [21]. One of the key ingredients for designingsuch metamaterials is the creation of EM waves with much shorterwavelengths than those of the incident light. These sub-wavelengthwaves involve electronic motion (plasmons) coupled with EM sur-face waves and are referred to as surface plasmon polaritons [5, 10].

Plasmonic crystals are a class of metamaterials that is of particu-lar interest. They consist of metallic layers arranged periodicallywith sub-wavelength distance in a dielectric host. This class of meta-materials offers fine control of EM properties and can serve as ENZ[15, 17] and negative refractive index media [23]. ENZ metamate-rials exhibit properties which cannot be obtained by traditionalphotonic systems. These features include wave propagation withno phase delay and diffraction, decoupling of spatial and temporalfield variations, propagation through very narrow channels, andultra fast phase transitions [17, 21]. Many novel optical deviceshave been proposed indicating the broad prospects of the ENZeffect [1, 9, 13]. However, novel metals have the unfortunate sideeffect that their use in plasmonic platforms restricts the frequencyrange where above exotic optical effects occur. In addition, the re-sulting plasmonic device regularly suffers from high optical lossesand construction defects yield non perfect planar layers which alsonegatively affect the optical properties. All of these issues may nolonger be critical due to the discovery of truly two-dimensional(2D) materials, such as graphene, black phosphorus, hexagonalboron nitride, and molybdenum disulfide. They promise a new eraof control of optical properties of photonic and plasmonic devices[2, 14, 18, 22, 25]. In particular, graphene supports plasmons withultra-sub-wavelength behavior, ranging from terahertz (THz) toinfrared frequencies. Due to the confining of electrons in two di-mensions, the quantum effects are very pronounced and thus 2Dmaterials pose exceptional optical properties and high quantumefficiency for light-matter interaction [5, 10].

Graphene is a quite interesting material because it allows controlof electronic and, in turn, plasmonic properties by changing thedensity of the free charge carriers through an external gate field,chemical deposition, or intercalation [5, 20, 25]. It has been shownthat graphene plasmonic crystals can serve as an ingredient in atunable metamaterial [15, 17, 23]. In particular, a plasmonic crystalconsisting of periodically stacked graphene layers embedded in a

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Dublin ’19, September 25–27, 2019, Dublin, Ireland Mattheakis et al.

dielectric host leads to negative refraction and ENZ behavior in theTHZ and infrared frequencies, and allows to tune optical propertiesby controlling the operation frequency or the doping amount [17].It has been furthermore shown that the ENZ condition coincideswith a plasmonic Dirac cone in the wavenumber space, where theplasmonic Dirac cone is formed by two asymptotically linear, in-tersecting dispersion bands [9, 13, 15, 17]. Such a graphene-basedplasmonic crystal design, however, has the major shortcoming thatENZ behavior can only be observed for a very small frequencyrange. This is due to the fact that a plasmonic crystal with a dielec-tric host that has frequency independent optical properties admitsonly a single plasmonic Dirac point which restricts the ENZ effectto a very narrow frequency range. This is disadvantageous for anumber of optical applications involving a continuous spectrum ofEMwaves. In this letter we introduce a further degree of freedom incontrolling the optical properties of graphene plasmonic crystals byusing Lorentz dispersive dielectric host materials [12]. The dielec-tric function of this family of materials depends on the operatingfrequency. We find that by using an appropriate Lorentz host we ob-tain multiple plasmonic Dirac points resulting in a wide frequencyrange exhibiting an ENZ effect. For a particular host material thatwe examine, Magnesium Oxide, we demonstrate that (a) two Diracpoints emerge in the THz band; and (b) that for a large frequency in-terval containing the two Dirac points the ENZ condition is fulfilled.As a second design example, we propose a hypothetical dielectricwith two Lorentz resonances leading to three Dirac points and aneven wider frequency spectrum of ENZ behavior. This example ismotivated by the significant progress that was made in (atomistic)material engineering recently, that has the prospect of enabling theengineering of materials with desirable electronic properties. Forinstance, Van der Waals heterostructures are comprised of differ-ent 2D materials that are stacked in a particular way to providedesirable electronic and optical features [4, 19].

The remainder of this paper is organized as follows. In Section 2we review the graphene-based plasmonic crystal by using Bloch the-ory and discuss the mathematical relation between the plasmonicDirac point and ENZ effect. In Section 3 we extend the frequencyrange of the ENZ effect by introducing Lorentz materials as theambient material of the graphene layers. The paper concludes inSection 4 with a summary of the key ideas introduced in this letterand an outlook.

2 BLOCH-WAVE THEORYWe consider a plasmonic crystal that consists of a dielectric hostsurrounding flat in which periodic 2D metallic sheets that are par-allel to the yz-plane located at x = nd , for integer n. Here, d de-notes the structural period. The metallic sheets are described by anisotropic surface conductivity σ . The dielectric host is consideredto be an anisotropic material described by a uniaxial dielectric ten-sor diag(εx , εy , εz ) with out-of-plane component εx = const andx-dependent in-plane components εy (x) = εz (x); the vacuum per-mittivity is set to unity, ε0 = 1. We examine transverse-magnetic(TM) time-harmonic waves with electric and magnetic field com-ponents E = (Ex , 0,Ez ) and H = (0,Hy , 0), respectively. Imposingthe rotational symmetry of the crystal and the TM polarizationin Maxwell equations yield fields which are translation invariant

along y direction with guided modes align along z direction [17],hence

Ez (x , z) = E(x)eikzz ,where kz is the wavenumber along z direction. The above assump-tions allow us to simplify the system of time dependent Maxwellequations to a one dimensional eigenvalue problem [15] with theHelmholtz type governing equation[

∂2

∂x2− κ(kz )εz (x)

]E(x) = 0, κ(kz ) =

k2z − k20εx

εx, (1)

where k0 = ω/c ,ω = 2π f is the angular operating frequency and fdenotes the ordinary operating frequency, c is the free space speedof light. The metallic 2D sheets carry a surface current Js = σEthat acts as a boundary. Maxwell equations dictate that on theboundaries that are defined by the metallic sheets at x = nd , thetangential electric field E must be continuous with discontinuousderivative with a jump discontinuity due to the surface currents onthe metallic plates, hence the transmission conditions are [15]:{ E+ − E− = 0,

∂

∂xE+ − ∂

∂xE− =

iσ

ωκ(kz )E+,

(2)

where (.)± denotes the limit above (+) and below (−) of a metallicboundary. We can get a closed system of equations by using theBloch-wave ansatz [15, 17] in the x direction with kx indicatingthe real Bloch number:

E(x) = eikxdE(x − d). (3)

The combination of the transmission conditions (2) and Bloch ansatz(3) yields a closed system of boundary conditions, viz.,[

E(d−)E ′(d−)

]= eikxd

[1 0

−iσ/ωκ(kz ) 1

] [E(0+)E ′(0+)

]. (4)

Here, E ′ denotes the derivative of E with respect to x . In order tosolve the differential equation (1) with boundary conditions (4) weintroduce the two fundamental solutions E(1) and E(2) of (1) thatfulfill the initial conditions:

E(1)(0) = 1, E ′(1)(0) = 0, E(2)(0) = 0, E ′

(2)(0) = 1.

By substituting a linear combination E = c1 E(1) + c2 E(2) into (4)we obtain( [

E(1)(d) E(2)(d)E ′(1)(d) E ′

(2)(d)

]− eikxd

[1 0

−i(σ/ω)κ(kz ) 1

] ) [c1c2

]= 0.

This system admits a nontrivial solution (with c1 , 0, or c2 , 0)whenever

D[k] = det

( [E(1)(d) E(2)(d)E ′(1)(d) E ′

(2)(d)

]− eikxd

[1 0

−i(σ/ω)κ(kz ) 1

] )= 0. (5)

Equation (5) now forms a dispersion relation between kx and kz . Itis convenient to work in a 2D wavenumber space k = (kx ,kz ). InRef. [15] formula (5) is used to determine the dispersion relation forseveral examples of dielectric functions, εz (x), including constant,parabolic, double-well, and nonsymmetric dielectric profiles. The

Graphene epsilon-near-zero plasmonic crystals Dublin ’19, September 25–27, 2019, Dublin, Ireland

plasmonic crystal has the character of an ENZ medium for valuesk in the vicinity of the Dirac point k∗ = (k∗x , k∗z ) = (0,±k0

√εx )

[15], that occurs at the center of the Brillouin zone [17]. It has beenshown that the occurence of the plasmonic Dirac point is a universalbehavior of 2D plasmonic crystals with any spatial dependence onthe dielectric host along the x direction [15].

At k∗ the effective in-plane permittivity component, εeffz , of themetamaterial becomes zero, hence in the neighborhood of k∗ weobserve ENZ behavior. For the purpose of examining ENZ frequencybands in the following, we make the heuristic definition of speakingof an ENZ behavior whenever

��εeffz �� < 0.25. In this letter we focus onthe frequency-dependent response of a plasmonic crystal assumingthat the host material is described by a constant in space butpossibly ω dependent permittivity. Thus, as a last preparatory stepwe solve the dispersion relation explicitly for the special case ofspatially constant εx and εz . In this case the fundamental solutionsof the differential equation (1) are given by

E(1) = cosh(√κεzx

), E(2) =

1√κεz

sinh(√κεzx

).

Substituting both solutions into dispersion relation (5) yields

cos(kxd) = cosh(√κεzd) −ξ0√κεz

2sinh(√κεzd), (6)

where

ξ0 = − iσ

ωεz(7)

is the plasmonic thickness that determines the length scale of plas-monic structures perpendicular to the 2Dmaterial sheets [15, 17, 23].For k close to k∗ the dispersion relation (6) reduces to

k2xεeffz+k2zεx= k20 .

This implies that up to first order the effective permittivity εeffz isgiven by

εeffz =

(1 − ξ0

d

)εz . (8)

It can be shown by a refined argument that equation (8) is uni-versal [16]: An EM wave traveling through the plasmonic crystalsees an effective homogeneous medium with effective permittiv-ity ε = diag

(εx , ε

effz , ε

effz

). Therefore, three distinct cases of wave

propagation are possible: elliptic propagation (ℜ εeffz > 0), hyper-bolic propagation (ℜ εeffz < 0), and ENZ behavior (ℜ εeffz ≈ 0). Theelliptical dispersion bands yield propagating waves with normaldiffraction, whereas the hyperbolic bands correspond to negativerefraction. The ENZ case is described by a linear Dirac dispersionand corresponds to propagation without dispersion and phase delay[17].

Equation (8) provides a systematic way to design plasmonic crys-tals with ENZ properties. We observe that we obtain εeffz = 0 whenξ0 = d , hence for a given d we can tune the plasmonic thickness, ξ0,to be equal to d , and vice versa. This also explains why graphene issuch an exceptional plasmonic platform. The surface conductivityof graphene depends on the operating frequency as well as thedoping, hence we can dynamically tune the surface conductivityσ to achieve ξ0 ≈ d . More precisely, the surface conductivity of

Figure 1: Dispersion relation kx (kz ) as function of frequencyfor a graphene plasmonic crystal with silica glass as dielec-tric host. The Dirac cone is marked in red and also shown ininset.

graphene is given by Kubo’s formula which includes both inter-band and intraband electronic transitions. Nevertheless, in THz andinfrared frequencies the intraband transitions dominate and thus,Kubo’s formula is simplified to Drude model [5, 10]:

σ (ω) = ie2EFπℏ2(ω + i/τ )

, (9)

where EF is the Fermi level associating with the density of electriccarriers and corresponding to the electronic doping; the EF canbe dynamically tuned by an external gate field or can be fixed bychemical deposition or intercalation [5, 20, 25]. The electron chargeis denoted by e , ℏ is the reduced Planck constant, and τ is electronicrelation time accounting for optical losses with a typically value ofτ = 0.5 ps.

As a first example we study a plasmonic crystal that consistsof doped graphene layers in a silica glass host with constant andequal permittivity components of εx = εz = 2. In Figure 1 weshow the dispersion relation kx (kz ) calculated by equation (6) fordifferent values of frequency f in the range [40, 140] THz. Thestructural period and the Fermi level are kept constant at d = 20nm and EF = 0.5 eV, respectively. We observe one plasmonic Diracpoint that appears at f = 65 THz (highlighted by red and shownin inset diagram of Figure 1), where the dispersion bands changefrom elliptical to hyperbolic.

3 LORENTZ HOST MATERIALIn the previous section we saw how a (single) plasmonic Dirac pointcan appear in the dispersion of a 2D metal-based plasmonic crystalthat then—in the vicinity of the Dirac point—behaves as an ENZmedium. The ENZ effect is, however, restricted to a narrow rangeof frequencies. This is an issue for real applications that typicallyinvolve a range of operating frequencies forming a wavepacket.We now introduce a Lorentz host material into the plasmonic crys-tal. We demonstrate how this leads to multiple Dirac points in

Dublin ’19, September 25–27, 2019, Dublin, Ireland Mattheakis et al.

the dispersion relation and consequently a significantly expandedfrequency range with ENZ behavior.

The Drude model takes only intraband electronic transition intoaccount that model free electrons in a metal response to an EMfield. Dielectric materials, however, do not have free charge carriers(electrons). Here, the optical response due to an external EM fieldis mainly caused by bound charges in the dielectric material whereradiation can be absorbed due to interband electronic transitions.The Lorentz model is a phenomenological model that takes theseinterband electronic transitions into account. Dielectrics describedby this model are called Lorentz dispersive materials [12]. The con-tribution of the interband transition leads to a frequency dependentpermittivity that reads

ε(ω) = ε∞ +(εs − ε∞)ω2

0ω20 − ω2 + iΓω

. (10)

Here, ω0 denotes the resonance (natural) angular frequency of thebound electrons, Γ is the electron damping rate, ε∞ and εs are thehigh and low frequency limits of the permittivity, respectively.

A concrete example for a plasmonic crystal with a Lorentz dis-persive host material that shows an extended ENZ band is Mag-nesium Oxide (MO). MO’s parameters for the Lorentz model read:ε∞ = 11.2, εs = 2.6, ω0 = 1 eV (f = 241.8 THz), and Γ = 0 [8].We point out that MO is considered to be a lossless dielectric andhence can be used for optical devices. We compute the dispersionrelation (6) of the plasmonic crystal for this case by setting the hostpermittivity to εz = εx = ε(ω). The resulting dispersion relationinvolving (kx ,kz ) and the operating frequency f are shown in Fig-ure 2 (upper). We indeed observe two Dirac points at frequenciesf1 = 67 THz and f2 = 99 THz. The two Dirac cones are marked inred in the diagrams and pointed out in the inset of Figure 2.Since we are interested in the ENZ range, we calculate the in-planeeffective dielectric permittivity given by equation (8). The resultsare shown in the lower panel of Figure 2. In particular, we computethe εeffz as a function of the frequency for our example of a plasmoniccrystal with MO host. For the sake of comparison we plot the εeffzfor a plasmonic crystal that was studied in previous sections, that it,with silica glass as dielectric host instead of MO. The shaded arearepresents the ENZ range (

��εeffz �� < 0.25). We observe that indeed theENZ band of the plasmonic crystal with Lorentz dispersive host issignificantly extended, about four times, compared to the previousexample of a constant permittivity host material.

We can further generalize our approach by assuming dielectricmaterials with permittivity given by a multi-oscillators Lorentzmodel [12] where many interband transitions take place and drasti-cally contribute to the dielectric function. Such a material providesmany Dirac points in the k space. Subsequently, the εeffz becomeszero many times and thus, the ENZ frequency range is significantlyexpanded. We propose this hypothetical scenario due to the greatprogress of the material engineering where new materials withdesirable electronic properties can be designed in purpose. TheLorentz model that includes multiple oscillators is

ε(ω) = ε∞ +N∑i

дi ω20i

ω20i − ω2 + iωΓi

, (11)

40 60 80 100 120 140-1

-0.5

0

0.5

1

Effe

ctiv

e pe

rmitt

ivity

Silica Glass host

Magnesium Oxide host

Figure 2: Upper: Dispersion relation kx (kz ) as a function offrequency for a graphene plasmonic crystal with Magne-sium Oxide dielectric host; the two Dirac cones are markedin red and shown in the inset. Lower: Effective dielectricpermittivity in frequency for a graphene plasmonic crystalwith Magnesium Oxide (solid) and silica glass (dashed) ashost materials; the shaded are represents the ENZ range.

where N is the number of the oscillators each one has resonancefrequencyω0i , damping factor Γi , and strengthдi . Here, we considera two-oscillator (N = 2) Lorentz material with the parameters:ε∞ = 4, (д1,д2) = (0.34, −2.4), and (ω01,ω02) = (0.59, 0.71) eV.Thence, we calculate the dispersion relation in (k ,ω) and presentthe results in Figure 3 (upper) where we observe three Dirac pointsat the frequencies (75, 114, 132) THz. In the lower panel of the samefigure we outline the effective permittivity in frequency where theENZ band has been significantly expanded in a range over than 60THz that is approximately eight times wider range compared to theexample of a constant permittivity host material.

4 CONCLUSIONMetamaterials are artificial structures that manipulate electromag-netic waves on a sub-wavelength scale and provide optical prop-erties that are not found in nature under normal conditions, such

Graphene epsilon-near-zero plasmonic crystals Dublin ’19, September 25–27, 2019, Dublin, Ireland

40 60 80 100 120 140-1

-0.5

0

0.5

1

Effe

ctiv

e pe

rmitt

ivity

Figure 3: A two-oscillator Lorentz material is used as a hostfor the graphene plasmonic crystal. Upper: Dispersion rela-tion kx (kz ) as a function of frequency; the Dirac cones aremarked in red and also shown in inset. Lower: Effective per-mittivitity; the ENZ area is shaded in grey.

as negative refraction and epsilon-near-zero behavior. ENZ meta-materials exhibit exotic properties that include wave propagationthrough very narrow channels without diffraction and phase delay.Plasmonic crystals are a particular class of metamaterials that con-sist of periodically arranged metallic layers with distances shorterthan the (free space) wavelength. Graphene-based plasmonic crys-tals are tunable metamaterials since their optical properties canbe dynamically tuned by controlling the operating frequency andthe electronic doping in graphene. Fine tuning a graphene-basedplasmonic crystal yields ENZ behavior in terahertz and infraredfrequencies. The appearance of ENZ condition coincides with a plas-monic Dirac cone in the wavenumber space and thus, Dirac pointsdetermine the neighborhood where the metamaterial behaves asan ENZ media.

Using an ordinary dielectric as the host material of the plasmoniccrystal yields a single Dirac point restricting the ENZ effect to avery narrow frequency range. This may be a critical issue for for

real applications that typically involve a range of operating frequen-cies forming a wavepacket. In this paper we proposed a modifieddesign that expands the frequency range of the ENZ behavior. Wereplaced the host dielectric in the plasmonic crystal with a Lorentzdispersive material which has a frequency dependent permittiv-ity. We demonstrated that the combination of doped graphene andLorentz materials yield multiple Dirac points and, subsequently,achieved an ENZ effect over a relatively large frequency range.

ACKNOWLEDGMENTSWe acknowledge partial support from EFRI 2-DARE NSF Grant No.1542807 (MMat, EK); and ARO MURI Award No. W911NF14-0247(MMat, MMai, EK). This project was partially supported by theUniversity of Minnesota’s Undergraduate Research OpportunitiesProgram (WXB).

REFERENCES[1] M. Z. Alam, S. A. Schulz, J. Upham, I. De Leon, and R. W. Boyd. 2018. Large

optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material.Nature Photonics 12 (2018), 79 – 83.

[2] S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam, M. Wagner, K.Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. M. Janssen, S-E. Zhu,P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov. 2015. Graphene on hexagonalboron nitride as a tunable hyperbolic metamaterial. Nat. Nano 10, 8 (2015),682–686.

[3] J. Dorfmuller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, Ch. Etrich, and K. Kern.2010. Plasmonic Nanowire Antennas: Experiment, Simulation, and Theory. NanoLetters 12 (2010), 3596–3603.

[4] A. K. Geim and I. V. Grigorieva. 2013. Van der Waals heterostructures. Nature499 (2013), 419–425.

[5] A. N. Grigorenko, M. Polini, and K. S. Novoselov. 2012. Graphene plasmonics.Nat. Photon 6 (Oct 2012), 749–758. Issue 11.

[6] N. Hassan, M. Mattheakis, and M. Ding. 2019. Sensorless Node Architecture forEvents detection in Self-Powered Nanosensor Networks. Nano CommunicationNetworks 19 (2019), 1 – 9.

[7] A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. deLeon, M. D. Lukin, and H. Park. 2015. Visible-frequency hyperbolic metasurface.Nature 522 (Jun 2015), 192–196. Issue 7555.

[8] J. Y. Horiba. 2006. Lorentz Dispersion Model. Technical notes, spectroscopicellipsometry (2006).

[9] X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan. 2011. Dirac cones inducedby accidental degeneracy in photonic crystals and zero-refractive-index materials.Nat. Mater. 10 (2011), 582–586.

[10] M. Jablan, H. Buljan, and M. Soljačić. 2009. Plasmonics in graphene at infraredfrequencies. Phys. Rev. B 80 (Dec 2009), 245435. Issue 24.

[11] J. Miquel Jornet and A. Cabellos. 2015. On the feeding mechanisms for graphene-based THz plasmonic nano-antennas. 2015 IEEE 15th International Conference onNanotechnology (IEEE-NANO) (2015), 168–171.

[12] Charles Kittel. 2005. Introduction to solid state physics (8th ed.). John Wiley andSons, Inc.

[13] Y. Li, S. Kita, P. Munoz, O. Reshef, D. I. Vulis, M. Yin, M. Loncar, and E. Mazur.2015. On-chip zero-index metamaterials. Nat. Photon 9 (Oct 2015), 738–742. Issue11.

[14] T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. Martín-Moreno, and F. Guinea.2014. Plasmons and Screening in Monolayer and Multilayer Black Phosphorus.Phys. Rev. Lett. 113 (Sep 2014), 106802. Issue 10.

[15] M. Maier, M. Mattheakis, E. Kaxiras, M. Luskin, and D. Margetis. 2018. Universalbehavior of a dispersive Dirac cone in gradient-index plasmonic metamaterials.Phys. Rev. B 97 (Jan 2018), 035307. Issue 3.

[16] M. Maier, M. Mattheakis, E. Kaxiras, M. Luskin, and D. Margetis. 2019. Homoge-nization of plasmonic crystals: Seeking the epsilon-near-zero effect. Submitted(2019). https://arxiv.org/abs/1809.08276

[17] M. Mattheakis, C. A. Valagiannopoulos, and E. Kaxiras. 2016. Epsilon-near-zero behavior from plasmonic Dirac point: Theory and realization using two-dimensional materials. Physical Review B 94 (2016), 201404(R).

[18] A. Nemilentsau, T. Low, and G. Hanson. 2016. Anisotropic 2D Materials forTunable Hyperbolic Plasmonics. Phys. Rev. Lett. 116 (Feb 2016), 066804. Issue 6.

[19] K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro Neto. 2016. 2Dmaterials and van der Waals heterostructures. Science 353, 6298 (2016).

[20] S. N. Shirodkar, M. Mattheakis, P. Cazeaux, P. Narang, M. Soljačić, and E. Kaxiras.2018. Quantum plasmons with optical-range frequencies in doped few-layer

Dublin ’19, September 25–27, 2019, Dublin, Ireland Mattheakis et al.

graphene. Phys. Rev. B 97 (May 2018), 195435. Issue 19.[21] M. Silveirinha and N. Engheta. 2006. Tunneling of Electromagnetic Energy

through Subwavelength Channels and Bends using ϵ -Near-Zero Materials. Phys.Rev. Lett. 97 (Oct 2006), 157403. Issue 15.

[22] Constantinos A Valagiannopoulos, Marios Mattheakis, Sharmila N Shirodkar,and Efthimios Kaxiras. 2017. Manipulating polarized light with a planar slab ofblack phosphorus. Journal of Physics Communications 1, 4 (nov 2017), 045003.https://doi.org/10.1088/2399-6528/aa90c8

[23] B. Wang, X. Zhang, F. J. GarcÃŋa-Vidal, X. Yuan, and J. Teng. 2012. StrongCoupling of Surface Plasmon Polaritons in Monolayer Graphene Sheet Arrays.Physical Review Letters 109 (2012), 073901.

[24] D. Wintz, P. Genevet, A. Ambrosio, A. Woolf, and F. Capasso. 2015. HolographicMetalens for Switchable Focusing of Surface Plasmons. Nano Letters 15, 5 (2015),3585–3589. PMID: 25915541.

[25] F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasubramaniam. 2014. Two-dimensional material nanophotonics. Nat. Photon 8 (2014), 899–907. Issue 12.