Investigation of Hyperbolic Metamaterials

applied sciences

Review Investigation of Hyperbolic Metamaterials

Tatjana Gric 1,2,* and Ortwin Hess 3 1 Department of Electronic Systems, Vilnius Gediminas Technical University, Vilnius LT-10223, Lithuania 2 Semiconductor Physics Institute, Center for Physical Sciences and Technology, Vilnius LT-02300, Lithuania 3 The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, UK; [email protected] * Correspondence: [email protected]; Tel.: +370-6167-8292

Received: 21 June 2018; Accepted: 12 July 2018; Published: 25 July 2018

Featured Application: It should be mentioned that the exceptional properties of SPPs, such as, subwavelength confinement and strong field enhancement give rise to a wide range of innovative applications. For example, owing to the excitation of SPPs, resolution of the so-called “superlens” has reached as high as 30 nm. The former opens the wide avenues for optical microscopy in terms of overcoming the diffraction limit. Alternatively, the resolution can be even higher in plasmonics-assisted spectroscopy, such as surface-enhanced Raman spectroscopy (SERS). The Raman signal can be considerably amplified by SERS approach. Doing so, the vibrational information of single molecules can be probed and recorded by employing the strong local field enhancement of plasmonic composites and the chemical mechanism including charge transfer between the species and the metal surface. Tip-enhanced Raman spectroscopy (TERS) may stand for the further resolution improvement. Based on the study on TERS-based spectral imaging with extraordinary sub-molecular spatial resolution, one can definitely resolve the inner structure and surface configuration of a single molecule.

Abstract: Composites designed by employing metal/dielectric composites coupled to the components of the incident electromagnetic (EM) fields are named metamaterials (MMs), and they display features not observed in nature. This type of artificial media has attracted great interest, resulting in groundbreaking theory that bridges the gap between EM and photonic phenomena. Practical applications of MMs have been delayed due to the high losses related to the use of metallic composites, on top of the challenges in manufacturing nanoscale, three-dimensional structures. Novel materials—for instance, graphene or transparent-conducting oxides (TCOs), employed for the production of multilayered MMs—can significantly suppress undesirable losses. It is worthwhile noting that three-layered nanocomposites enable an increase in the frequency range of the surface wave. This work analyzes recent progress in the physics of multilayered MMs. We deliver an outline of key notions, such as effective medium approximation, and present multilayered MMs based on the three-layered structure. An overview of graphene multilayered MMs reveals their ability to support Ferrell–Berreman (FB) modes. We also describe the tunable properties of the multilayered MMs.

Keywords: hyperbolic metamaterial; transparent conducting oxides; graphene; surface plasmon polaritons

1. Introduction At the turn of the century, a new ﬁeld of research emerged that was a hybrid of physics, electrical engineering, and materials science. The study of metamaterials (MMs) addresses the rational design and arrangement of a material’s building blocks to attain physical properties that may go signiﬁcantly beyond those of its original constituent materials. Most often, the concept of MMs is associated with

Appl. Sci. 2018, 8, 1222; doi:10.3390/app8081222 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, x FOR PEER REVIEW 2 of 17 Appl. Sci. 2018, 8, 1222 2 of 16

electromagnetic (EM) wave propagation, where the engineering of resonant subwavelength electromagneticstructures allows (EM) for the wave precise propagation, control of where effectiv thee engineering wave properties. of resonant It is this subwavelength advanced functionality structures allowsthat enables for the the precise realization control ofof effectiveunique wave properties.phenomena, It issuch this advancedas the focusing functionality of light that below enables the thediffraction realization limit of uniqueor EM wave cloaking phenomena, of objects—co such asncepts the focusing that ofhave light captured below the the diffraction imagination limit orof EMresearchers cloaking and of objects—concepts the general public that alike. have Importantly, captured the the imagination element of functionality of researchers is andan integral the general part publicof the MM alike. concept. Importantly, the element of functionality is an integral part of the MM concept. Recently,Recently, propagationpropagation ofof EMEM waveswaves onon two-dimensionaltwo-dimensional interfacesinterfaces hashas stimulatedstimulated tremendoustremendous interestinterest [[1,2].1,2]. Surface Surface plasmonic plasmonic polariton polariton (SPP) (SPP) [3–6] [3–6] attracts lots of attentionattention inin thethe scientificscientific community.community. SPPsSPPs havehave providedprovided aa fertilefertile groundground forfor newnew applicationsapplications inin manymany fieldsfields [[7–14].7–14]. DuringDuring lastlast severalseveral decades,decades, itit hashas beenbeen demonstrateddemonstrated thatthat itit isis possiblepossible toto controlcontrol SPPsSPPs inin anan unprecedentedunprecedented wayway byby nanostructuringnanostructuring thethe metalmetal surface.surface. HyperbolicHyperbolic MMsMMs [[15]15] havehave stimulatedstimulated increasingincreasing interestinterest becausebecause ofof thethe capabilitycapability ofof supportingsupporting largelarge wavenumbers.wavenumbers. TheirTheir unusualunusual propertiesproperties openedopened widewide avenuesavenues forfor manymany fascinatingfascinating applications,applications, suchsuch asas hyperlenseshyperlenses forfor biosensingbiosensing [9[9].]. TheThe potentialpotential biosensingbiosensing applicationsapplications areare achievableachievable withwith thethe novelnovel high-frequencyhigh-frequency cut-offcut-off SPPsSPPs inin hyperbolichyperbolic graphene-dielectricgraphene-dielectric multilayeredmultilayered metamaterials.metamaterials. For infraredinfrared frequencies,frequencies, infraredinfrared plasmonicplasmonic MMsMMs werewere designeddesigned by by C. C. Wu Wu et et al. al. Moreover, Moreover, their their use use has has been been suggested suggested as a as biosensing a biosensing platform platform [16]. [16]. It is worthwhileIt is worthwhile to note to that note hyperbolic that hyperbolic MMs have MMs already have beenalready treated been from treated the perspective from the perspective of supporting of photonicsupporting modes photonic with modes unbounded with k-vectorsunbounded in [k-vectors17]. Herein, in [17]. we take Herein, a step we forward take a bystep investigating forward by variousinvestigating types ofvarious modes types supported of modes by multilayered supported structures,by multilayered for example, structures, Ferrell–Berreman for example, (FB) Ferrell– and others.Berreman The (FB) former and isothers. achievable The former by employing is achievable the tunability by employing properties the tunability of the graphene properties used of as the a buildinggraphene block used of as the a building multilayered block composite. of the multilayered The experimental composite. realization The experimental of a multilayered realization structure of a ofmultilayered alternating structure graphene of and alternating Al2O3 layers graphene has been and reported Al2O3 layers in [18 has]. However, been reported the former in [18]. study However, does notthe offerformer an study overview does of not the offer tuning an over possibilities.view of the tuning possibilities. OurOur presentpresent discussiondiscussion ofof multilayeredmultilayered MMsMMs isis divideddivided intointo threethree parts.parts. In the firstfirst part, fundamentalfundamental physicalphysical notionsnotions thatthat underlieunderlie graphene-dielectricgraphene-dielectric multilayered MMs are discussed.discussed. Moreover,Moreover, thethe realizationrealization ofof thethe dispersiondispersion characteristicscharacteristics ofof thethe SPPs,SPPs, theirtheir theoreticaltheoretical implications,implications, andand theirtheir specificspecific characteristicscharacteristics areare reviewed.reviewed. InIn addition,addition, thethe conceptconcept ofof FBFB modesmodes isis introduced.introduced. InIn SectionSection4 4,, the the extension extension of of the the frequency frequency region region of of the the SPPs SPPs is is discussed discussed in in terms terms of of the the inclusion inclusion of of a a thirdthird additionaladditional layer.layer. ParticularParticular emphasisemphasis isis placedplaced onon thethe topologicaltopological structures.structures. Finally,Finally, SectionSection5 5 discussesdiscusses thethe possibility possibility of theof inclusionthe inclusion of transparent-conducting of transparent-conducting oxide (TCO) oxide layers (TCO) in multilayered layers in MMs,multilayered specifically MMs, paying specifically attention paying to the attention new kinds to the of thenew SPPs. kinds of the SPPs.

2.2. MultilayeredMultilayered MM-BasedMM-Based EffectiveEffective MediumMedium ApproachApproach Herein,Herein, itit isis aimedaimed toto derivederive thethe effectiveeffective mediummedium permittivitiespermittivities forfor anan anisotropicanisotropic multilayeredmultilayered MM.MM. TheThe techniquetechnique isis basedbased onon thethe generalizedgeneralized MaxwellMaxwell GarnettGarnett methodmethod [[19],19], whichwhich seeksseeks toto derivederive analyticalanalytical formulasformulas describingdescribing thethe materialmaterial inin different directions,directions, i.e.i.e. parallel (εk∥)) andand perpendicularperpendicular ((εε⊥⊥).). Based on the mentionedmentioned methodology, aa sufﬁcientlysufficiently dilutedilute materialmaterial isis consideredconsidered suchsuch thatthat aa hierarchyhierarchy ofof macroscopicmacroscopic andand microscopicmicroscopic ﬁeldsfields cancan bebe assumed.assumed. TheThe schematicschematic viewview ofof aa structurestructure composedcomposed ofof metallicmetallic andand dielectricdielectric layerslayers makingmaking aa multilayeredmultilayeredMM MMis isoutlined outlinedin inFigure Figure1 .1.

FigureFigure 1.1. AA viewview ofof aa multilayeredmultilayered metamaterialmetamaterial (MM).(MM). Appl. Sci. 2018, 8, 1222 3 of 16

The permittivities of the metallic and dielectric layers are εm and εd, correspondingly. Additionally, the ﬁll ratio of the total width of metal in the composite to the overall width of the MM can be deﬁned as follows: d ρ = m (1) dm + dd where dm is summation of the widths of metallic sheets in the composite and dd is summation of the widths of the dielectric sheets.

2.1. Effective Parallel Permittivity Herein, an analytical formula expressing the parallel permittivity component characterizing the structure under consideration is derived. It is worthwhile mentioning that:

→ → D = εe f f E (2) where εeff is the effective permittivity. It is a well-known fact that the tangential component of the electric ﬁeld should be continuous across a boundary of two different media. Thus, it follows:

|| || || Em = Ed = E (3)

|| || where Em is assigned as the electric field in the metallic layers, Ed is the electric field in the dielectric sheets, and E|| is the electric field of the subwavelength MM. The overall displacement might be found by averaging the displacement field components arising from the metallic and dielectric building blocks: || || || D = ρDm + (1 − ρ)Dd (4) Rearranging the above equation, the following equation is obtained:

|| || || || εe f f E = ρεmE + (1 − ρ)εdE (5)

One can conclude with the ﬁnal equation, as follows:

ε|| = ρεm + (1 − ρ)εd (6)

2.2. Effective Perpendicular Permittivity It is speciﬁcally known that the normal component of the electric displacement vector at a boundary should be continuous. The former is outlined by the following expression:

⊥ ⊥ ⊥ Dm = Dd = D (7)

It is also known that a superposition of the electric field components generated by the sheets dramatically affects the total magnitude of the electric field. Consequently, it can be defined by:

⊥ ⊥ ⊥ E = ρEm + (1 − ρ)Ed (8)

⊥ ⊥ where Em is the perpendicular component of the electric field in the metallic area, Ed is the perpendicular component of the electric field in the dielectric area, and E⊥ is the total electric field in the MM. The boundary condition from Equation (7) and Maxwell’s equation [20] can be used. Moreover, it is of particular importance to substitute them into Equation (8). If the common electric Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 17 Appl. Sci. 2018, 8, 1222 4 of 16 field components are cancelled out, and everything is solved for ε⊥, it is possible to derive the analytic formulafield components for the perpendicular are cancelled permittivity out, and everything of the multilayered is solved for MM:ε⊥, it is possible to derive the analytic formula for the perpendicular permittivity of the multilayeredεε MM: ε = md ⊥ ρε +−()ρ ε (9) dmε1mεd ε⊥ = (9) ρεd + (1 − ρ)εm

3.3. Two-Layered Two-Layered MMs MMs for for Surface Surface Plasmon Plasmon Polariton Polariton Guiding Guiding FigureFigure 22 isis a a schematic schematic view view of aof hyperbolic a hyperbolic multilayered multilayered MM consisting MM consisting of periodically of periodically changing changinggraphene graphene and dielectric and dielectric layers. layers.

FigureFigure 2. 2. AA schematic schematic diagram diagram of of the the boundary boundary surface surface separating separating two two different different infinite inﬁnite multilayered multilayered MMsMMs designed designed by by a astack stack of of periodically periodically changing changing gr grapheneaphene and and dielectric dielectric layers. layers. Here, Here, dielectric dielectric and and graphenegraphene sheets sheets are are denoted denoted by by indexes indexes “1, “1, 2”, 2”, correspondingly. correspondingly.

TheThe effective-medium effective-medium theory [[21]21] can bebe appliedapplied ifif the the width width of of any any sheet sheet is is much much smaller smaller than than the thewavelength wavelength ofradiation. of radiation. This This approach approach allows allows us to describeus to describe the optical the optical response response in such ain structure. such a structure.Herein, we present the dispersion branches calculated after carrying out the homogenization of twoHerein, MMs. The we dispersionpresent the patterndispersion for thebranches case ε 1calc= εulated3, ε2 = afterε4 and carryingd1 6= d out2 6= thed3 6=homogenizationd4 is presented of in εε= εε= dd≠≠≠ dd twoFigure MMs.3. Figure The dispersion3 demonstrates pattern that for the the SPPs case branch 13 at the, interface24 and separating 1234 MMs with isd 1presented= 0.35 nm, ind 2Figure= 10 nm, 3. Figured3 = 0.25 3 demonstrat nm, and d4es= that 10.1 the nm. SPPs Moreover, branch theat th dispersione interface diagrams separating of MMs SPPs, with characterized d1 = 0.35 nm,by thed2 effective= 10 nm, properties d3 = 0.25outlined nm, and in d [214 =], 10.1 are displayed nm. Moreover, in Figure the3. dispersion diagrams of SPPs, characterizedThe tunability by the featureeffective of properties the frequency outlined ranges in of [21], SPPs are is displayed the main outcome in Figure of 3. changing the Fermi energy.The tunability It is worthwhile feature of noting the frequency that the decreaseranges of ofSPPs the is Fermi the main energy outcomeµ causes of changing movement the Fermi of the energy.dispersion It is diagramsworthwhile to lowernoting frequency that the ranges.decrease On of thethe contrary,Fermi energy the increase μ causes of themovement Fermi energy of theµ dispersionwill stimulate diagrams their movement to lower frequency to higher frequencies.ranges. On the As illustratedcontrary, the in Figureincrease3, the of the smallest Fermi asymptotic energy μ willfrequency stimulate is achievable their movement in the case to ofhigher the smallest frequencies. Fermi energy.As illustrated The presented in Figure controllability 3, the smallest feature asymptoticprovides possibilities frequency tois engineer achievable the SPPsin the by case the Fermi of the energy smallest of the Fermi graphene energy. layers. The Consequently, presented controllabilityspecial phase-matching feature provides tools, possibilities such as a grating to engineer or prism the SPPs coupling by the [Fermi22], near-field energy of excitation the graphene [23], layers.and end-fire Consequently, coupling special [24], are phase-matching needed for their tools, excitation such as via a three-dimensional grating or prism coupling beams [3 ].[22], near- field excitationBecause of [23], the and graphene end-fire being coupling treated [24], as lossless,are neededβ is for complex their excitation resulting via in athree-dimensional finite propagation beamslength, [3]. presented in Figure3b. The next case to deal with, i.e. ε1 = ε3, ε2 6= ε4 and d1 6= d2 6= d3 6= d4, is studied by sketching four modes in Figure4 at the interface of the MMs with ε4 = 2.25. As it follows from Figure4, the case µ = 0.5 eV is characterized by the smallest asymptotic frequency. It is worthwhile noting, that Re(β) 6= 0 allows for the quasi-bound [21], leaky region of the dispersion equation. Appl. Sci.Appl.2018 Sci., 20188, 1222, 8, x FOR PEER REVIEW 5 of 175 of 16

15 x 10 2 1000

(b) (a) μ=0.5eV μ=0.5eV 800 1.5 μ=1eV μ=1eV μ μ=1.5eV =1.5eV 600 μ=2eV μ=2eV (numerical) 1 λ , Hz light line Lp/ ω 400

0.5 200

0 0 0 1 2 3 0 0.5 1 1.5 2 β 8 15 8 , 1/m x 10 ω, Hz x 10 x 10 3 (c) μ=0.5eV

) 2 μ β =1eV μ=1.5eV Im( 1 μ=2eV 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ω, Hz 15 x 10

FigureFigure 3. The 3. The dispersion dispersion of of SPPs SPPs (a ();a); propagation propagation lengths (b (b) )and and absorption absorption (c) (atc )different at different Fermi Fermi εε= εε= energy,energy, where where number number of graphene of graphene layers, i.e.layers,N = 1,i.e.d dN= 10= nm,1, ddε 1== 10ε3 ,nm,ε2 = ε134 and d, 1 6=24d2 6=andd3 6= d4 ≠≠≠ and thedd1234 blue line dd in (a) anddenotes the blue the line free in space (a) denotes light line.the free space light line. Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 17 Because of the graphene being treated as lossless, β is complex resulting in a finite propagation length, presented in Figure 3b. εε= εε≠ ≠≠≠ The next case to deal with, i.e. 13, 24 and dd1234 dd, is studied by sketching four modes in Figure 4 at the interface of the MMs with ε4 = 2.25. As it follows from Figure 4, the case μ = 0.5 eV is characterized by the smallest asymptotic frequency. It is worthwhile noting, that Re(β) ≠ 0 allows for the quasi-bound [21], leaky region of the dispersion equation.

FigureFigure 4. The 4. The dispersion dispersion of of surface surface plasmonic plasmonic polaritons (SPPs) (SPPs) (a); (a );propagation propagation lengths lengths (b) and (b) and

absorptionabsorption (c) at(c) different at different Fermi Fermi energies, energies, where where thethe number of of graphene graphene layers layers N =N 1,= dd 1, = d10d =nm, 10 nm, εε= , εε≠ , and dddd≠≠≠, and the blue line in (a) denotes the free space light line. ε1 = ε313, ε2 6= ε424, and d1 6= d12342 6= d3 6= d4, and the blue line in (a) denotes the free space light line.

Following the previous instances, the transverse magnetic (TM) mode dispersion plots are used εε= εε≠ = = to deal with the case of 13, 24 and dd13, dd24. Four various modes are presented in Figure 5. The outcome of tuning the fill ratios is examined as displayed in Figure 5. Firstly, it is worth discussing how the dielectric thickness dd impacts the dispersion branch (see Figure 5a). It is found that an increase of dd causes a shift of the upper limit to higher frequencies. Moreover, one can manipulate the frequency range for SPPs by tuning the number of graphene sheets, as shown in Figure 5c. It should be mentioned that, following [25], it is of particular importance to omit negative values of the electrical length and absorption in Figures 3b,c, 4b,c and 5b,d–f. The effect of absorption control was considered in [26]. It was proposed to tune and enhance the absorption of thin dye-doped polymeric films by metallic and lamellar metal-dielectric hyperbolic metamaterial substrates. Herein, by employing graphene as a building block of the hyperbolic metamaterial, we achieve a higher level of tunability by changing the Fermi energy. Additionally, the sketched graphs represent clear evidence of the existing SPPs propagating for a long distance. The former is possible because the real part of β is very low within this spectral range. It is of particular importance to mention that the same case appears when dealing with the boundary separating the MM and isotropic medium [27]. Appl. Sci. 2018, 8, 1222 6 of 16

Following the previous instances, the transverse magnetic (TM) mode dispersion plots are used to deal with the case of ε1 = ε3, ε2 6= ε4 and d1 = d3, d2 = d4. Four various modes are presented in Figure5. The outcome of tuning the ﬁll ratios is examined as displayed in Figure5. Firstly, it is worth discussing how the dielectric thickness dd impacts the dispersion branch (see Figure5a). It is found that an increase of dd causes a shift of the upper limit to higher frequencies. Moreover, one can manipulate the frequency range for SPPs by tuning the number of graphene sheets, as shown in Figure5c. It should be mentioned that, following [25], it is of particular importance to omit negative values of the electrical length and absorption in Figure3b,c, Figure4b,c and Figure5b,d–f. The effect of absorption control was considered in [26]. It was proposed to tune and enhance the absorption of thin dye-doped polymeric ﬁlms by metallic and lamellar metal-dielectric hyperbolic metamaterial substrates. Herein, by employing graphene as a building block of the hyperbolic metamaterial, we achieve a higher level of tunability by changing the Fermi energy. Additionally, the sketched graphs represent clear evidence of the existing SPPs propagating for a long distance. The former is possible because the real part of β is very low within this spectral range. It is of particular importance to mention that the same case appears when dealing with the boundary separating the MM and isotropic mediumAppl. Sci. 2018 [27, ].8, x FOR PEER REVIEW 7 of 17

Figure 5. The dependences of dispersion of SPPs, propagation lengths, and absorption on (a,,b,,e)) thethe thicknessthickness ofof dielectric dielectricd ddandd and (c ,(dc,,fd), numberf) number of grapheneof graphene layers. layers. Herein, Herein, the number the number of graphene of graphene layers, i.e.,layers,N = i.e., 1, µ N= = 0.5 1, μ eV = in0.5 ( aeV,b ,ine) and(a,b,ded) and= 10 d nm,d = 10µ nm,= 0.5 μeV = 0.5 in (eVc,d in,f). (c,d,f).

The imaginary parts parts of of the the wavevector wavevector (i.e., (i.e., absorption) absorption) are are demonstrated demonstrated in Figures in Figures 3c, 34cc, 4andc and5e,f.5 e,f.The Theabsorption absorption resonances resonances allow allow for for the the ri richch phenomenon, phenomenon, i.e., i.e., behavior behavior of thethe simplesimple multilayered MMs as directive and monochromatic thermal sources [28] without possessing any periodic corrugation. We used a modal analysis to demonstrate that the bulk absorption in MMs arises because of the excitation of leaky bulk polaritons named FB modes [29–31]. It is worthwhile noting that the interaction of the bulk plasmon with free space light at frequencies close to the metal epsilon-near- zero (ENZ) [32] is possible. This method has been addressed by Ferrell for metallic foils in [33], and by Berreman for polar dielectric films in [34]. Radiative excitations called FB modes are possible due to the application of the multilayered MMs with graphene layers. It should be noted that these FB modes differ from SPPs, possibly due to the usage of metal foils. The former occurs because of energy propagating along the surfaces of the metal in cases of SPP modes. With respect to Figures 3a and 4a,c, particular attention should be paid to FB modes existing at energies close to the ENZ of the hyperbolic MM to the left of the light line. A specific example of a three-layered MM and its dispersion engineering, allowing the development of new devices, are studied in detail in the next section.

4. Three-Layered MMs for Surface Plasmon Polariton Guiding Figure 6 illustrates a diagram of dielectric/graphene/dielectric multilayered MM. Appl. Sci. 2018, 8, 1222 7 of 16 multilayered MMs as directive and monochromatic thermal sources [28] without possessing any periodic corrugation. We used a modal analysis to demonstrate that the bulk absorption in MMs arises because of the excitation of leaky bulk polaritons named FB modes [29–31]. It is worthwhile noting that the interaction of the bulk plasmon with free space light at frequencies close to the metal epsilon-near-zero (ENZ) [32] is possible. This method has been addressed by Ferrell for metallic foils in [33], and by Berreman for polar dielectric ﬁlms in [34]. Radiative excitations called FB modes are possible due to the application of the multilayered MMs with graphene layers. It should be noted that these FB modes differ from SPPs, possibly due to the usage of metal foils. The former occurs because of energy propagating along the surfaces of the metal in cases of SPP modes. With respect to Figures3a and 4a,c, particular attention should be paid to FB modes existing at energies close to the ENZ of the hyperbolic MM to the left of the light line. A speciﬁc example of a three-layered MM and its dispersion engineering, allowing the development of new devices, are studied in detail in the next section.

4. Three-Layered MMs for Surface Plasmon Polariton Guiding Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 17 Figure6 illustrates a diagram of dielectric/graphene/dielectric multilayered MM.

Figure 6.6. Schematic of graphene-dielectric multilayeredmultilayered MM.MM.

ItIt shouldshould be mentioned mentioned that that the the MM MM under under study study is is described described by by effective effective permittivity permittivity tensor tensor as follows: as follows:   ε 0 0 ||ε 00  ||  ε = ε0 0 ε|| 0 (10) εε= 00 ε  0||0 0 ε (10) ε⊥ 00 ⊥ in the principal-axis coordinate system. The principal values of the tensor are obtained in [35,36] by meansin the principal-axis of the theoretical coordinate approach system. provided The in principa Sectionl2 values: of the tensor are obtained in [35,36] by means of the theoretical approach provided in Section 2: ε|| = fgεg + f1ε1 + 1 − fg − f1 ε2 (11) εεε=++−−() ε ||ffgg 1 11 ff g 1 2 (11) ε⊥ = εgε1ε2/ fgε1ε2 + f1εgε2 + 1 − fg − f1 εgε1 (12) where fg = dg/L and f1 = d1/L, L = d1 + d2 + dg represent the graphene and first dielectric filling ε⊥ =++−− εεε/1()ff εε εε() ff εε ratios, respectively, dg is the thicknessgg12 of the 1 graphene 2 1 g 2 layer, ggd1 is 1 the thickness 1 of the first dielectric(12) sheet, d2 is the thickness of the second dielectric layer, ε1 is the permittivity of the first dielectric, and ε2 iswhere the permittivity fdL= / of theand second fdL= dielectric./ , Ld=++ d drepresent the graphene and first dielectric gg 11 12 g fillingFor ratios, the different respectively, cases presenteddg is the thickness in Figure 7of, dispersionthe graphene relations layer, are d1 obtainedis the thickness as follows of the first dielectric sheet, d2 vis the thickness of the second dielectric layer, ε1 is the permittivity of the first u dielectric, and ε2 is uthe permittivityε1ε of2ε gtheε1 second− ε2 − εdielectric.1 f1 + ε2 f1 + ε2 fg − εg fg β = k (13) t 2 2 2 2 2 2 2 2 For the different casesε1εg − presentedε2εg − ε1 εing f Figure1 + ε2ε g7,f 1dispersion+ ε1ε2 fg − relationsε1εg fg − areε2ε gobtainedfg + ε2εg asfg follows

εεε ε−− ε ε + ε + ε − ε 12gggg() 1 2 11fff 21 2 f β = k (13) εε222222−− εε εε + εε + εε − εε − εε 22 + εε 121121121g g gfff g g gg ff 2 gg 2 gg f

εεε ε−− ε ε + ε 12gggg() 11fff 21 2 f β = k (14) ε2222 ε−+− εε εε ε ε ffff11gg 12 gg 1 12 g

εε β = k 1 g εε+ (15) 1 g

εε β = k 2 g εε+ (16) 2 g Appl. Sci. 2018, 8, 1222 8 of 16

v u u ε1ε2εg ε1 f1 − ε2 f1 − ε2 fg + εg fg β = k (14) t 2 2 2 2 f1ε1εg − fgε1ε2 + fgε1εg − f1ε2εg s ε1εg β = k (15) ε1 + εg s ε2εg β = k (16) + Appl. Sci. 2018, 8, x FOR PEER REVIEW ε2 εg 9 of 17

Figure 7. Schematic of graphene-dielectric multilayered MM. The ﬁguresfigures correspond to (a) EquationEquation (13),(13), ((b)) EquationEquation (14),(14), ((cc)) EquationEquation (15),(15), ((dd)) EquationEquation (16).(16).

In consequence, a surprising result isis obtained: the thicknesses of the layers (Figure7 7c,d)c,d) dodo notnot affect the dispersion of a (single) boundary mode, and it agrees with the dispersion of a conventional SPP at a metal–dielectric boundary. It can be concludedconcluded that the systems displayed in Figure7 7c,dc,d cancan support topological properties. Herein, we present dispersion diagrams calculatedcalculated after carrying out the homogenization of two MMs. Thus, Figure 88 showsshows thethe dispersiondispersion branchesbranches ofof differentdifferent casescases ofof thethe multilayeredmultilayered MM.MM. TheThe dispersion branches of spoof SPPs at the MM interface with dg = 0.35 nm are shown in Figure 8. It dispersion branches of spoof SPPs at the MM interface with dg = 0.35 nm are shown in Figure8. It should be noted that the case displayed in Figure 8a (ε1 = 18.8) corresponds to Equation (15). Thus, it should be noted that the case displayed in Figure8a ( ε1 = 18.8) corresponds to Equation (15). Thus, itcan can be be concluded concluded that that the the usage usage of of the the three-layered three-layered MM MM allows allows for for the the rich rich phenomenon, i.e., broadening of the frequency range of the SPP’s existenceexistence by approximately 800800 THz.THz. The dispersion curve computedcomputed forfor thethe three-layered three-layered MM MM case case has has special special features features deserving deserving of attention,of attention, besides besides the factthe fact that that the graphthe graph resembles resemble behaviors behavior of a of traditional a traditional metal metal SPP SPP and and multilayered multilayered MMs MMs [21]. [21]. For SPPs supported by the multilayered MM under study, thethe horizontalhorizontal asymptoteasymptote appears when β approachesapproaches infinity. inﬁnity. Based Based on on the the dispersion dispersion equation equation [21], [21 ],the the corresponding corresponding condition condition is as is asfollows: follows: 2 L L R ε||ε⊥ − ε 2 = 0 (17) εεLL−= ε R || ⊥ () 0 (17) In other words, the position of horizontal asymptotes is determined by Equation (17). It is also L L R concludedIn other that words,ε|| must the shareposition the of same horizontal sign with asymptε⊥. Itotes is worthwhile is determined noting by Equation that ε will (17). drastically It is also affect the curve type.L ε ε L ε R concluded that || must share the sameL signL with ⊥ . It is worthwhile R2 noting that will Equation (17) holds only when both ε|| and ε⊥ are negative when ε is larger than limω→∞ε||ε⊥. Thedrastically former affect scenario the matchescurve type. band 2 in Figure9. Consequently, the dispersion branch possesses only 2 ε L ε L ()ε R one horizontalEquation asymptote(17) holds resemblingonly when the both metal || SPPand case. ⊥ The are dispersion negative branchwhen under such is larger constraints than bearslim aεε close resemblance to a metal and multilayered MM SPP [21] (see Figure8). ω →∞|| ⊥ . The former scenario matches band 2 in Figure 9. Consequently, the dispersion branch Similar to ωsp for metal, the upper bound for SPP is marked by the location of the lowest possesses only one horizontal asymptote resembling the metal SPP case. The dispersion branch under horizontal asymptote. ωupper denotes the upper bound frequency, aiming to separate the upper bound such constraints bears a close resemblance to a metal and multilayered MM SPP [21] (see Figure 8). of multilayered SPPs and that of metal SPPs (i.e., ωsp). One can easily prove that it is possible to excite Similar to ωsp for metal, the upper bound for SPP is marked by the location of the lowest multilayered SPPs at any frequency lower than ωupper in band 2. horizontal asymptote. ωupper denotes the upper bound frequency, aiming to separate the upper bound of multilayered SPPs and that of metal SPPs (i.e., ωsp). One can easily prove that it is possible to excite multilayered SPPs at any frequency lower than ωupper in band 2. The dispersion curve (Figure 8) is obtained following the hypothesis of the EM field decaying exponentially in the z direction. It should be noted that not all modes with a certain frequency below ωupper is SPP in this study. The former takes place even though the dispersion curve displayed in L Figure 10 resembles the behavior of conventional metal SPP. One should be assured that both k z R and k z are imaginary when aiming to obtain the multilayered SPPs. It can be concluded, based on

Figure 10, that the two cut-off points, ωlower and ωupper cut the frequency range into three regions. One may define ωupper as the frequency of the lowest horizontal asymptote in the dispersion branch. It is Appl. Sci. 2018, 8, 1222 9 of 16

The dispersion curve (Figure8) is obtained following the hypothesis of the EM field decaying exponentially in the z direction. It should be noted that not all modes with a certain frequency below ωupper is SPP in this study. The former takes place even though the dispersion curve displayed in L Figure 10 resembles the behavior of conventional metal SPP. One should be assured that both kz R and kz are imaginary when aiming to obtain the multilayered SPPs. It can be concluded, based on Figure 10, that the two cut-off points, ωlower and ωupper cut the frequency range into three regions. One may define ωupper as the frequency of the lowest horizontal asymptote in the dispersion branch. L R It is worthwhile noting that ωlower, which is lower than ωupper, is the frequency where kz and kz are converted from real to imaginary. L The imaginary part of kz in a multilayered MM is negligible, while the real part exists as indicated in Figure 10, in region 1. The former proves the possibilities of the EM wave propagating into the material, resulting in a little damping. Region 1 is ended by frequency ωlower. The same frequency L begins region 2. kz possesses a negligible real part and relatively large imaginary part in region 2. The L former describes the evanescence of the field is in this region. The imaginary part of kz disappears and L the real part of kz occurs if ωupper is lower than the frequency (in region 3). The wavevector component R kz in the dielectric is shown in the right figure in Figure 10. In the same way, the wave can propagate into the dielectric in areas 1 and 3, however, it is bounded to the boundary in area 2. The SPPs can be controlled by tuning the Fermi energy. The former is in line with the dependence of ε⊥ on the Fermi energy [21]. It is worthwhile noting that the decreases in the Fermi energy µ will cause a shift of the dispersion curves to higher wavelengths or lower frequencies. On the contrary, they will be shifted to higher frequencies because of the increases in the Fermi energy µ. The smallest asymptotic frequency is attained in the case of the smallest Fermi energy, as indicated in Figure9. The former tunability property enables the SPP to be modified by the Fermi energy of the graphene layers. Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 17

FigureFigure 8. The 8. The impact impact of of (a) (aε1) ,(ε1b, ()b Fermi) Fermi energy energyµ μ,(, c()c) graphene graphene ﬁllingfilling ratio,ratio, ffg,, and and (d (d) )dielectric dielectric filling ﬁlling ratioratiof 1 on f1 on the the dispersion dispersion diagrams diagrams of of the the SPPs. SPPs.f g f=g = 0.1, 0.1,f afa= = 0.50.5 inin ((aa).). ffg = 0.1, 0.1, ffa a== 0.1 0.1 in in (b (b).). μµ = =1.5 1.5 eV eV in (ain,c ,(da,).c,d).

Figure 9. Real parts of ε and ε as a function of frequency. ε1 = 18.8 in (a), ε1 = 2.25 in (b), ε1 = 1 ⊥ || in (c). Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 17

Appl. Sci. 2018, 8, 1222 10 of 16

Considering the next case, i.e., ε1 6= const, in Figure8a, three modes are represented at the interface of MM with ε2 = 18.8. As displayed in Figure8a, the smallest asymptotic frequency matches the case ε1 = 18.8. Also, Re(β) 6= 0 allows for the presence of the quasi-bound [37], leaky range of the dispersion equation. Consequently, β does not approach infinity as the surface plasmon frequency is attained. The dispersion curves of the TM modes are computed while aiming to address the case f g 6= const and f 1 6= const. In Figure8c,d, eight different branches are presented. One should consider the impact of the fill ratios of the dielectric and graphene layer, as sketched in Figure8c,d. First, the impact of the width of the graphene f g on the dispersion branch (see Figure8c) should be discussed. It is found that Figure 8. The impact of (a) ε1, (b) Fermi energy μ, (c) graphene filling ratio, fg, and (d) dielectric filling the decrease of f causes a shift of the upper limit to higher frequencies. Additionally, one can control ratio f1 on theg dispersion diagrams of the SPPs. fg = 0.1, fa = 0.5 in (a). fg = 0.1, fa = 0.1 in (b). μ = 1.5 eV the frequencyin (a,c,d). range of SPPs by changing the dielectric filling ratio, as demonstrated in Figure8d.

ε ε FigureFigure 9. Real9. Real parts parts of ofε⊥ and⊥ andε|| as a|| functionas a function of frequency. of frequency.ε1 = 18.8ε1 = in18.8 (a ),inε (1a=), 2.25ε1 = 2.25 in (b in), ε(b1 ),= ε 11 in= 1 ( c). Appl. Sci. 2018, 8, x FOR PEER REVIEW 12 of 17 in (c).

FigureFigure 10. 10. EachEach component component of ofwavevector, wavevector, when when εR =ε R18.8.= 18.8. The Thecomponents components of β ( ofa) βand(a) k and in bothk in both multilayeredmultilayered MMs MMs (b ()b and) and free free space space (c). (c ).

We will address the tunable hyperbolic MM made of TCO-dielectric multilayered MMs in the We will address the tunable hyperbolic MM made of TCO-dielectric multilayered MMs in the next section, aiming to obtain a new type of SPP. next section, aiming to obtain a new type of SPP. 5. Tunable Hyperbolic MM Made of TCO-Dielectric Multilayered MMs

The proposed MM is displayed in Figure 11. Herein, dTCO and dd are the widths of the TCO and dielectric sheets, correspondingly. A TCO/PbS MM is considered, aiming to study the dispersive properties of SPPs. It is assumed that εd = 18.8 for PbS sheets, and εTCO is computed by means of the parameters outlined in [38].

d TCO d d

dielectric/TCO

L R

TCO PbS Interface z

Figure 11. Interface of multilayered MM made of periodically changing transparent-conducting oxide ε = ε = (TCO) and dielectric layers. The MM/dielectric ( R 1 or R 2.25 ) and MM/TCO boundaries are investigated.

For the instance of the anisotropic material, the EM wave dispersions can be sketched for MM/dielectric and the MM/TCO instances. Let us concentrate on a semi-infinite AZO (Aluminum Zinc Oxide)/PbS MM example while tackling the numerical calculations. Moreover, a homogenization of the MM matching the MM/dielectric and MM/TCO boundaries is considered. ε ε ε Sketching the branches of the permittivities ( ⊥ and || ) and the dielectric constant ( R ) allows us to consider the optical properties. Moreover, we consider different frequency ranges, distinguishing ε ε ε them based on the properties of and relation among ⊥ , || , and R [39].

εεε>> ε < 0 (1) ⊥ R || and || (cyan); εεε>> (2) ⊥ R || (green); εε>> ε ε < (3) || R ⊥ and ⊥ 0 (gray); εεε>> ε < (4) R || ⊥ and ⊥ 0 (magenta) and εε<<0 (5) ⊥ || (orange). Appl. Sci. 2018, 8, x FOR PEER REVIEW 12 of 17

Figure 10. Each component of wavevector, when εR = 18.8. The components of β (a) and k in both multilayered MMs (b) and free space (c).

Appl. Sci.We2018 will, 8 ,address 1222 the tunable hyperbolic MM made of TCO-dielectric multilayered MMs 11in ofthe 16 next section, aiming to obtain a new type of SPP. 5. Tunable Hyperbolic MM Made of TCO-Dielectric Multilayered MMs 5. Tunable Hyperbolic MM Made of TCO-Dielectric Multilayered MMs The proposed MM is displayed in Figure 11. Herein, dTCO and dd are the widths of the TCO and The proposed MM is displayed in Figure 11. Herein, dTCO and dd are the widths of the TCO and dielectric sheets, correspondingly. A TCO/PbS MM is considered, aiming to study the dispersive dielectric sheets, correspondingly. A TCO/PbS MM is considered, aiming to study the dispersive properties of SPPs. It is assumed that εd = 18.8 for PbS sheets, and εTCO is computed by means of the properties of SPPs. It is assumed that εd = 18.8 for PbS sheets, and εTCO is computed by means of the parameters outlined in [38]. parameters outlined in [38].

d TCO d d

dielectric/TCO

L R

TCO PbS Interface z

Figure 11. Interface of multilayered MM made of periodically changing transparent-conducting oxide Figure 11. Interface of multilayered MM made of periodically changing transparent-conducting oxide (TCO) and dielectric layers. The MM/dielectric (εR = 1 or εR = 2.25) and MM/TCO boundaries (TCO) and dielectric layers. The MM/dielectric ( ε =1 or ε = 2.25 ) and MM/TCO boundaries are are investigated. R R investigated. For the instance of the anisotropic material, the EM wave dispersions can be sketched for For the instance of the anisotropic material, the EM wave dispersions can be sketched for MM/dielectric and the MM/TCO instances. Let us concentrate on a semi-infinite AZO (Aluminum MM/dielectric and the MM/TCO instances. Let us concentrate on a semi-infinite AZO (Aluminum Zinc Oxide)/PbS MM example while tackling the numerical calculations. Moreover, a homogenization Zinc Oxide)/PbS MM example while tackling the numerical calculations. Moreover, a of the MM matching the MM/dielectric and MM/TCO boundaries is considered. Sketching the homogenization of the MM matching the MM/dielectric and MM/TCO boundaries is considered. branches of the permittivities (ε⊥ and ε||) and the dielectric constant (εR) allows us to consider the Sketching the branches of the permittivities ( ε ⊥ and ε ) and the dielectric constant ( ε ) allows us optical properties. Moreover, we consider different frequency|| ranges, distinguishing themR based on to consider the optical properties. Moreover, we consider different frequency ranges, distinguishing the properties of and relation among ε⊥, ε||, and εR [39]. ε ε ε them based on the properties of and relation among ⊥ , || , and R [39]. (1) ε > ε > ε and ε < 0 (cyan); εεε⊥ >>R || ε <|| ⊥ R || || 0 (2)(1) ε⊥ > εR > εand|| (green); (cyan); εεε⊥ >> (3)(2) ε|| > εRR >|| ε(green);⊥ and ε⊥ < 0 (gray); εε>> ε ε < (4)(3) ε||R > Rε|| >⊥ εand⊥ and ⊥ε⊥ <0 0(gray); (magenta) and εεε>> ε < (5)(4) ε⊥R < ||ε|| <⊥ 0and (orange). ⊥ 0 (magenta) and εε<<0 (5) It⊥ is worthwhile|| (orange). mentioning that the effective parameters will dramatically influence the SPPs, as outlined in [39]. The TCO filling ratio, f TCO, is used here as the main mechanism controlling the effective optical parameters. Figures 12 and 13 display characteristic dispersion branches, considering different TCO filling factors. Different branch colors are applied, aiming to distinguish various frequency ranges. One can distinguish three SPPs fitting three different ranges in the case of fTCO = 0.3 and MM/dielectric boundary. Moreover, two kinds of SPPs are observed in the instance of the MM/TCO boundary. For the MM/TCO case, the upper waves are in the gray region. The fTCO = 0.5, 0.7, 0.9 cases are associated with three exceptional cases. In the case of fTCO = 0.5, two types of SPPs appear in two color ranges, i.e., cyan and gray for the MM/dielectric and MM/TCO boundaries. The upper short branches in the case of fTCO = 0.7, lying in the gray range, are presented in Figures 12 and 13. The other SPPs belong to the orange range. The supplementary SSPs belonging to the magenta range are observed with respect to a MM/dielectric boundary. A dark-magenta color displays an extension approach for the magenta region in the case of εR = 2.25. One may observe two kinds of SPPs belonging to the orange and cyan ranges for the MM/TCO instance and three types of SPPs lying in the cyan, orange, and gray ranges for the MM/dielectric case with a filling factor fTCO = 0.9. Appl. Sci. 2018, 8, x FOR PEER REVIEW 13 of 17

It is worthwhile mentioning that the effective parameters will dramatically influence the SPPs, as outlined in [39]. The TCO filling ratio, fTCO, is used here as the main mechanism controlling the effective optical parameters. Figures 12 and 13 display characteristic dispersion branches, considering different TCO filling factors. Different branch colors are applied, aiming to distinguish various frequency ranges. One can distinguish three SPPs fitting three different ranges in the case of fTCO = 0.3 and MM/dielectric boundary. Moreover, two kinds of SPPs are observed in the instance of the MM/TCO boundary. For the MM/TCO case, the upper waves are in the gray region. The fTCO = 0.5, 0.7, 0.9 cases are associated with three exceptional cases. In the case of fTCO = 0.5, two types of SPPs appear in two color ranges, i.e., cyan and gray for the MM/dielectric and MM/TCO boundaries. The upper short branches in the case of fTCO = 0.7, lying in the gray range, are presented in Figures 12 and 13. The other SPPs belong to the orange range. The supplementary SSPs belonging to the magenta range are observed with respect to a MM/dielectric boundary. A dark-magenta color displays an ε = extension approach for the magenta region in the case of R 2.25 . One may observe two kinds of SPPs belonging to the orange and cyan ranges for the MM/TCO instance and three types of SPPs lying in the cyan, orange, and gray ranges for the MM/dielectric case with a filling factor fTCO = 0.9. It should be underlined that one SPP occurs in the cyan range for different TCO filling factors in the MM/dielectric case. The former is outlined in Figure 12. It is of importance that the case εε= RITO opens the gateway for possible applications, as the SPP always occurs in the cyan range for different filling factors (Figure 13). Appl. Sci.Following2018, 8, 1222 the essential condition for the presence of the Dyakonov-like SPP (DSW) [35], the12 SPP of 16 in the green resembles the behavior of a DSW. Thus, it is important to address DSW [40], or the Dyakonov defined in [35]. Furthermore, the frequency region of the DSW can be extended by using It shouldε = be underlined that one SPP occurs in the cyan range for different TCO filling factors in the the R 2.25 material at the right-hand side of the boundary (Figure 12). The dark-green color [39] MM/dielectric case. The former is outlined in Figure 12. It is of importance that the case εR = ε ITO is applied, seeking to outline the former extension. Particularly, the extension of the gray region [39] opens the gateway for possible applications, as the SPP always occurs in the cyan range for different is probable because of the usage of ITO (Indium Tin Oxide) instead of GZO (ZnO:Ga) at the right- filling factors (Figure 13). hand side of the boundary [39]. The SPP in the orange range is similar to the traditional SPP. The Following the essential condition for the presence of the Dyakonov-like SPP (DSW) [35], the SPP former happens because of the negative principal values of the effective permittivity in the orange in the green resembles the behavior of a DSW. Thus, it is important to address DSW [40], or the range. Thus, this SPP should be associated with an SPP of a traditional type. The others are novel Dyakonov defined in [35]. Furthermore, the frequency region of the DSW can be extended by using kinds of SPPs. Therefore, all the SPPs belong to the five different types, located in the five color ranges the εR = 2.25 material at the right-hand side of the boundary (Figure 12). The dark-green color [39] is in Figure 12, correspondingly. In the instance of the AZO/PbS MM and air/dielectric boundary, five applied, seeking to outline the former extension. Particularly, the extension of the gray region [39] is types of SPPs appear, and only three are found for the AZO/PbS MM and TCO interface. probable because of the usage of ITO (Indium Tin Oxide) instead of GZO (ZnO:Ga) at the right-hand Unavoidably, a traditional-like SPP is characterized by an ordinary dispersion for the side of the boundary [39]. The SPP in the orange range is similar to the traditional SPP. The former MM/dielectric boundary, occurring to the right of the light line. Altogether, the degradation of the happens because of the negative principal values of the effective permittivity in the orange range. dispersion in the orange range might be observed in the instance of the MM/TCO boundary. One can Thus, this SPP should be associated with an SPP of a traditional type. The others are novel kinds manipulate the dispersion features controlling the TCO filling factor in the MM realization. It should of SPPs. Therefore, all the SPPs belong to the five different types, located in the five color ranges in be noted that for a low TCO filling factor, in the case of the MM/TCO boundary, two kinds of the Figure 12, correspondingly. In the instance of the AZO/PbS MM and air/dielectric boundary, five modes always exist, while for higher TCO filling factors, modes in the gray range disappear (Figure types of SPPs appear, and only three are found for the AZO/PbS MM and TCO interface. 13d).

Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 17

== == Figure 12.12. SPP dispersion for different TCO filling ﬁlling factors, i.e., i.e., ffTCOTCO 0.30.3 inin ( (aa),), ffTCOTCO 0.50.5 in ( b), fTCO = 0.7 in (c), and fTCO == 0.9 in (d), matching the MM/dielectric boundary. fTCO 0.7 in (c), and fTCO 0.9 in (d), matching the MM/dielectric boundary. Unavoidably, a traditional-like SPP is characterized by an ordinary dispersion for the MM/dielectric boundary, occurring to the right of the light line. Altogether, the degradation of the dispersion in the orange range might be observed in the instance of the MM/TCO boundary. One

= = Figure 13. SPP dispersion for different TCO filling factors, i.e., fTCO 0.3 in (a), fTCO 0.5 in (b), = = fTCO 0.7 in (c), and fTCO 0.9 in (d), matching the MM/TCO boundary.

It should be mentioned that the complex mode diagrams (Figures 12 and 13), which match either the MM/dielectric or MM/TCO boundary, appear due to the confinement of SPPs in the direction Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 17

Appl. Sci. 2018, 8, 1222 13 of 16 can manipulate the dispersion features controlling the TCO filling factor in the MM realization. It should be noted that for a low TCO filling factor, in the case of the MM/TCO boundary, two kinds Figure 12. SPP dispersion for different TCO filling factors, i.e., f = 0.3 in (a), f = 0.5 in (b), of the modes always exist, while for higher TCO filling factors,TCO modes in the grayTCO range disappear f = 0.7 in (c), and f = 0.9 in (d), matching the MM/dielectric boundary. (FigureTCO 13d). TCO

Figure 13. SPP dispersion for different TCO ﬁlling factors, i.e., f == 0.3 in (a), f = 0.5 in (b), Figure 13. SPP dispersion for different TCO filling factors, i.e., fTCOTCO 0.3 in (a), TCOfTCO 0.5 in (b), fTCO = 0.7 in (c), and fTCO ==0.9 in (d), matching the MM/TCO boundary. fTCO 0.7 in (c), and fTCO 0.9 in (d), matching the MM/TCO boundary.

ItIt shouldshould bebe mentionedmentioned thatthat thethe complexcomplex modemode diagramsdiagrams (Figures(Figures 1212 andand 1313),), which match either thethe MM/dielectricMM/dielectric or or MM/TCO boundary, boundary, appear appear du duee to to the confinement conﬁnement of SPPs in thethe directiondirection perpendicular to the wave propagation. The origin of these EM SPPs is caused by the coupling of the EM ﬁelds to the oscillations of the conductor’s electron plasma.

6. Conclusions

6.1. Two-Layered MMs for Surface Plasmon Polariton Guiding To conclude, we have presented a research approach to describe the dispersion of hyperbolic MMs. The aim of these studies is to mitigate the problem under consideration, offering tuning possibilities of the hyperbolic features. The former is tremendously needed from the perspective of device applications. In this regard, we presented a multilayered MM. In the light of the present work, we have outlined the tunable features of the effective permittivity of the graphene-based MMs. Three signiﬁcant outcomes should be outlined, in view of the obtained results. Firstly, compared with the Appl. Sci. 2018, 8, 1222 14 of 16

dispersion equation in the instance of ε1 = ε3, ε2 = ε4, and d1 6= d2 6= d3 6= d4, the quasibound, leaky part of the dispersion equation is narrowed for the case ε1 = ε3, ε2 6= ε4 and d1 6= d2 6= d3 6= d4. Secondly, one may control the perpendicular component of effective permittivity by manipulating the Fermi energy, the width of dielectric, and the number of graphene sheets in the MMs. Thirdly, one can engineer the frequency range of the SPP by changing the Fermi energy of graphene and the ﬁll ratio of the dielectric or graphene layers in the unit cell of the MMs.

6.2. Tunable Hyperbolic MM Made of TCO-Dielectric Multilayered MMs SPPs in a TCO-dielectric MM have been studied. Like graphene-dielectric MM [21], a TCO-dielectric MM support traditional-like SPPs possessing various diagrams matching two various boundaries. Based on theoretical considerations [40], the dispersion equations of SPPs were obtained. Five types of SPPs, among which three types are novel SPP types, one is DSW, and another is the traditional-like SPP, are foreseen. The existence of these SPPs is impacted by the features of the principal values of the effective permittivity. It should be noted that the presented (new) types of SPPs appear due to the principal values of the effective pemittivity being functions of frequency. Additionally, SPPs with a dispersion coinciding with the dispersion of a surface plasmon at the interface of two isotropic media are foreseen to match the MM boundary if the material at the right-hand side is the same as that used in the MM.

6.3. Three-Layered MMs for Surface Plasmon Polariton Guiding SPP supported by graphene/dielectric multilayered MMs was studied theoretically. It is concluded that SPP can be supported by a three-layered MM. Also, one can easily manipulate the characteristics of the multilayered SPPs by manipulating the structural parameters. Thus, the active materials can be included into the MM design, which might open wide avenues for future devices. The dielectric permittivity should be smaller than the absolute value of Re(εm), aiming to fulfill the EM boundary condition and statements of continuity [41] in the case of a traditional metal SPP. If not, the SPP mode would be cut off. The property of anisotropic material is represented by multilayered MMs if the case is considered from the effective medium perspective. Further, it is shown in Figure9 that one may deal with multilayered MMs as with anisotropic metal near the upper bound of its SPP frequency region (in band 2). The Re(ε⊥) is far from zero, while Re ε|| is much closer to zero within band 2 of Figure9, compared with Re(εm) of pure metal. The effective permittivity is caused by the interaction of the EM field with the free electrons in graphene sheets and the SPP mode coupling and resonance in multilayered MM. Consequently, the SPP frequency region is extended due to the SPP mode coupling and resonance in the multilayered MMs, characterized by anisotropic permittivity. It is worthwhile noting that the multilayered SPPs could also be excited by means of the evanescent EM field generated by a prism or waveguide. Extended frequency range, lower propagation loss, and controllable features are possible due to the changes of the graphene/dielectric ratio, and these properties might open a gateway for possible applications in biosensor and active SPP devices. Herein, we have theoretically considered the SPP guided by dielectric/graphene/dielectric multilayered MMs. One may conclude that an SPP can propagate when the MM is at the frequency where a conventional metal SPP cannot occur, as in the case of high permittivity of dielectric on metal. In addition to the variance of a high frequency cut-off point in comparison with metal SPPs, it was observed that a low cut-off frequency, below which SPP could not be supported by the MM, exists. These outstanding properties open wide avenues for the usage of an SPP mode for some potential applications, which are impossible for metal SPPs. In addition to the characteristics of the multilayered SPPs being easily controlled by tuning the composite parameters, the active materials can also be incorporated into the MM. This described feature might also create new possibilities for future devices.

Funding: This research received no external funding. Appl. Sci. 2018, 8, 1222 15 of 16

Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

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