arXiv:1301.1337v2 [cond-mat.mes-hall] 2 Sep 2013 wteln.Emi:juan.mosig@epfl.ch E-mail: Switzerland. in plasmonics sion, ieMcoaoaeSses LEMA/Nanolab, Systems, MicroNanoWave tive interface the at plasmons groups propagating of waves research propagation electromagnetic the (i.e., Many reconfig- studied [11]. novel theoretically have [5], to leading possibilities field, electrostatic uration a bias of magnetostatic application or is the graphene via addition, In tunable, [10]. contrast [9], inherently range in visible propaga- the plasmonic [8] in allows tion [7], only which [6], metals noble at [3], with devices frequencies plasmonic low novel relatively of graphene development Specifically, the [5]. [3], enables infrared frequencies optical, THz low at and propagation possi- wave interesting new surface provides its for [4], bilities to properties thanks optical at- Graphene, and significant electrical [3]. [2], attracted [1], recently tention have sheets graphene (LEMA), E-mai Switzerland. julien.perruisseau-carrier@e Lausanne, sebastian.gomez@epfl.ch, 1015 Lausanne. F´ed´erale de ttelwTzband. THz devices low plasmonic of the spatial development at assessing the correctly in for conductivity. effects need graphene transitions the show to intraband results contribution These where dominant frequencies the are at increasing even and plasmons, confinement can losses, mode surface dispersion their of modifying spatial propagation notably the that affect obtained demonstrates dramatically then Application method are equation. the resonance modes of transverse TM a and applying TE by relation supported Dispersion sheet. the the for along traveling waves th constant the propagation of surfaces, the on selective depend admittances frequency analytical graphene to of model Similar non-local han- conductivity. a to using able dispersion, graphene spatial of dle circuit ad equivalent the an derives of analysis includ- mittances proposed sheet, The graphene effects. substrate dispersive ing spatially a along waves .S oe-izadJ erisa-are r ihteAda the with are Perruisseau-Carrier J. and Gomez-Diaz S. J. .Msgi ihteEetoantc n cutc Laborat and Electromagnetics the with is Mosig J. h rpgto feetoantcwvsalong waves electromagnetic of propagation The ne Terms Index Abstract feto pta iprino ufc Waves Surface on Dispersion Spatial of Effect cl oyehiu ´deaed asne 05Lausanne 1015 F´ed´erale Lausanne. Polytechnique de Ecole ´ W netgt h rpgto fsurface of propagation the investigate —We Sraewvs rpee pta disper- spatial graphene, waves, —Surface rpgtn ln rpeeSheets Graphene Along Propagating .I I. NTRODUCTION .S oe-iz .R oi,adJ Perruisseau-Carrier J. and Mosig, R. J. Gomez-Diaz, S. J. cl Polytechnique Ecole ´ pfl.ch :juan- l: ory p- e s - , , h aeo eaieylwvle ftepoaaincon- propagation the of values stant low relatively of case the ausof values n ssterlxto-ie(T,se[1)adteso- the and [21]) see (RTA, called Boltzmann relaxation-time the dispersion, semi-classical electron uses linear and the a model assuming from equation This transport derived potential. been chemical for of graphene has and of absence dominate, value where the graphene any frequencies in of at contributions valid field, intraband [20], graphene bias [19], of magnetostatic external [2], model in dispersive using presented characterized spatially is non-local applying sheet by the obtained in The conductivity tensorial even range. graphene spatial waves the THz these that low of show behavior the we the affects and sheet, dispersion effects, graphene dispersive substrate spatially including a along waves face potential. chemical zero uniform with graphene cas of particular the on graphene-based focusing in waveguides, effects parallel-plate in dispersion spatial employed study recently to also THz [18] was high model to This con- (moderate frequencies). interband dominate when graphene occurs of of this tributions case space, In free waves. in become surface graphene slow would extremely dispersion for spatial significant that model, predicted this also on [2] Based of proposed. contributions valid was intraband dominate, conductivity, where graphene range frequency graphene the non- the in a of [2], In model frequencies. local THz to low related at works propagation in wave neglected spatial been [17], though usually dispersive has However, spatially dispersion be [16]. to stack known sheets is a graphene graphene have through monolayer studies transmission of recent enhanced Furthermore, demonstrated [15]. [14 [12], waveguides pairs plate or of parallel characteristics have including the propagation, configurations improve this different to proposed Also, been [13]. recently [12], [1], [3], graphene along [2], dielectric) a and conductor a between xrsin o h rpgto osato spatially- of sheets, graphene constant on propagating waves propagation surface dispersive the closed-form obtain for to allows expressions it because [17] in described mode conductivity graphene spatially-dispersive general nti ok eivsiaetepoaaino sur- of propagation the investigate we work, this In k ρ k hl rvdsapoiaerslsfrmoderate for results approximate provides while , ρ lwapoiain.Teeoe ti cuaein accurate is it Therefore, approximations. -low k ρ eew mlyti oe nta fthe of instead model this employ we Here . e 1 ] l 2 thereby providing physical insight into their properties. The operator components of the conductivity tensor are then mapped onto the admittances of a rigorous Green’s function-based equivalent circuit of graphene [22]. These equivalent admittances, which are similar to those found in the analysis of frequency selective surfaces [23], [24], analytically show that the influence of spatial dispersion directly depends on the square of the wave propagation constant (kρ). Then, a transverse resonance equation (TRE) [25] is imposed to compute the dispersion relation (a) of surface waves along spatially dispersive graphene. Similar to the case of non spatially dispersive graphene sheets, transverse electric (TE) and transverse magnetic (TM) modes are supported. Analytical dispersion re- lations are provided for TM and TE surface waves propagating along a graphene sheet embedded into an homogeneous medium, while approximate expressions are given for the case of graphene surrounded by two different dielectrics. The derived dispersion relations analytically show that the of the surround- ing media, operation frequency and spatial dispersion similarly contribute to determine the characteristics of (b) kρ, suggesting that the influence of graphene spatial dispersion in the propagating waves may be strongly Fig. 1: Spatially dispersive graphene sheet on a dielectric affected by the environment of the sheet. Numerical substrate under arbitrary plane-wave incidence (a) and its results confirm that spatial dispersion is an important equivalent transmission line model [22] (b). mechanism for wave propagation along graphene sheets, leading to surface modes that can significantly differ from those found neglecting spatial dispersion. These dispersive (non-local) model of graphene [2] in the ab- ~ features, which include variations in the mode confine- sence of external magnetostatic biasing fields (B0 = 0). ment and higher losses, should be rigorously taken into This anisotropic conductivity reads account in the development of novel plasmonic devices σx′x′ σx′y′ σ(ω,µc,τ,T )= , (1) at the low THz band. σy′x′ σy′y′ The paper is organized as follows. Section II derives   where T is temperature, τ is the electron relaxation the analytical relations between the admittances of a time, µ is the chemical potential and ω is the angular Green’s function-based equivalent circuit of graphene c frequency. Due to the spatial dispersion of graphene, the and the components of the spatially dispersive conductiv- conductivity components become operators [2], [26] ity tensor. Then, Section III computes the dispersion re- 2 2 lation of surface waves propagating along graphene, pro- ∂ ∂ σ ′ ′ = σ + α + β , (2) viding analytical expressions for the case of a graphene x x lo sd ∂x′2 sd ∂y′2 sheet embedded into an homogeneous media. Section ∂2 ∂2 IV discusses the characteristics of surface waves along σy′y′ = σ + β + α , (3) lo sd ∂x′2 sd ∂y′2 spatially dispersive graphene, taking into account the 2 surrounding media. Finally, conclusions and remarks are ∂ σ ′ ′ = σ ′ ′ = 2β , (4) provided in Section V. x y y x sd ∂x′∂y′ where II. EQUIVALENT CIRCUIT OF A jq2k T µ SPATIALLY DISPERSIVE GRAPHENE SHEET σ = − e B ln 2 1 + cosh c , lo π~2(ω jτ −1) k T Let us consider an infinitesimally thin graphene sheet B −    (5) in the plane z = 0 and separating two media, as 3v2 σ α illustrated in Fig. 1a. The sheet is characterized by the α = − F lo , β = sd , (6) sd 4(ω jτ −1)2 sd 3 conductivity tensor σ, obtained by applying a spatially − 3

kB is the Boltzmann constant, qe is the electron charge, ky = k1 sin(φ) sin(θ), kz = k1 cos(θ), eˆx, eˆy, eˆz are unit 6 v is the Fermi velocity ( 10 m/s in graphene), and vectors, and k0 = 2π/λ0 and k1 = √ε 1k0 are the F ≈ r the subscripts “lo” and “sd” have been included to denote wavenumbers of free space and medium 1, respectively. local and spatially-dispersive (non-local) terms. Note the Imposing boundary conditions in the graphene plane and presence of a minus sign in αsd, which was not included following the approach described in [22], [27] these in [2] due to a typo [26]. It is worth mentioning that this admittances can be written as model is insensitive to the orientation of the graphene TE 2 lattice, and that anisotropy arises as the response of the Yσ (kρ)= σlo + kρ[αsd + βsd], (7) TM 2 material to the excitation, as usually occurs with spatial Yσ (kρ)= σlo + kρ[αsd + βsd], (8) dispersion [2], [26]. Consequently, the coordinate system TE/TM TM/TE Y (kρ)= Y (kρ) = 0, (9) employed in the anisotropic conductivity [x′ y′, see σ σ

− ′ ′ 2 2 2 Eqs. (2)-(4)] is related to the excitation, i.e. σx x and where kρ = kx + ky is the propagation constant of the σy′y′ provides the response parallel to the Ex′ and Ey′ wave traveling along the graphene sheet. ′ ′ fields, being x and y arbitrary perpendicular directions The importance of Eqs. (7)-(9) is two fold. First, in the infinite sheet. Importantly, Eqs. (2)-(6) are only they analytically indicate that the influence of spatial 2 strictly valid at frequencies where the intraband contri- dispersion is directly proportional to kρ. Consequently, butions of graphene conductivity dominates, usually the this phenomenon will be more important for very slow low THz regime. Intraband contributions correspond to waves (kρ k0), which usually appear at moderately electron transitions between different energy levels in high THz frequencies≫ (where interband contributions of the same band (valence or conduction), while interband graphene dominate and Eqs. (2)-(4) are not strictly valid contributions are related to electron transitions between [2]). However, note that slow waves can also be obtained different bands. The latter phenomenon becomes signif- in graphene sheets at low THz frequencies by using icant as frequency increases due to the higher photon substrates with high permittivity constant [3]. Second, energy [21]. the equivalent admittances do not depend on the direction A rigorous equivalent circuit of an anisotropic of propagation of the wave along the sheet (φ) and there graphene sheet sandwiched between two media is shown is no coupling between the TM and TE modes. This in Fig. 1b [22]. This four-port circuit relates input and is in agreement with the non-local model of graphene output TE and TM waves through equivalent shunt ad- employed in this development [2], [26] which assumes TE TM an isotropic infinite surface where spatial dispersion mittances (Yσ and Yσ , respectively), and the cross- coupling between the two polarizations through voltage- arises as a response to a given excitation. TE/TM controlled current generators, with coefficients Yσ TM/TE and Yσ for TE-TM and TM-TE coupling, respec- III. DISPERSION RELATION FOR A tively. The relation between the anisotropic conductivity SPATIALLY DISPERSIVE GRAPHENE SHEET of graphene and these admittances was provided in The dispersion relation of surface waves on a spatially [22] for the case of a local model of graphene, i.e. dispersive graphene sheet sandwiched between two dif- when anisotropy is due to an external magnetostatic bias ferent media can be obtained by imposing a transverse field. However, spatial dispersion effects have not been resonance equation [25] to the equivalent circuit shown considered so far. in Fig. 1b. The solution of the TRE provides the propa- Here, we analytically obtain the admittances of the gation constant kρ of the surface wave propagating along equivalent circuit shown in Fig. 1b using a non-local the sheet. Following the approach described in [13], the (spatially dispersive) model of graphene. For this pur- desired dispersion relation can be obtained as

pose, we include the operator components of the ten- TE TE TE TM TM TM sor conductivity, Eqs. (2)-(4), into the transmission-line Y1 + Y2 + Yσ Y1 + Y2 + Yσ = 0, (10) network formalism for computing dyadic Green’s func-   tions in stratified media [22], [27]. For the admittances where derivation, we consider a uniform plane wave impinging TE kz1 TE kz2 TM ωεr1 ε0 on the graphene sheet from an arbitrary direction (θ, Y1 = , Y2 = , Y1 = , ωµ0 ωµ0 k φ) of medium 1 (see Fig. 1a), and then we use the z1 TM ωεr2 ε0 2 2 2 2 corresponding auxiliary coordinate system for the con- Y2 = , kz1 = k1 kρ, kz2 = k2 kρ, kz2 ± − ± − ductivity tensor. The incoming wave has a wavenumber q q k1 = √ε 1 k0, and k2 = √ε 2 k0 (11) ~k = kxeˆx + kyeˆy + kzeˆz, where kx = k1 cos(φ) sin(θ), r r 4 are the TE and TM admittances, transverse propagation are fully interchangeable since graphene conductivity constant and wavenumber of media 1 and 2, respectively. itself is frequency-dependent. In the asymptotic limit, Eq. (10) does not generally admit any analytical solu- ωε0(εr1 + εr2) >> kρσlo, Eq. (14) can be solved tion and must be solved using numerical methods [28]. analytically and yields Importantly, the mathematical solutions of this equation must be carefully checked to verify that they correspond 3 (εr1 + εr2) to physical modes. Specifically, physical modes must kρ jωε0 . (15) ≈ s αsd + βsd fulfill the law of energy conservation, i.e. surface waves cannot be amplified while propagating along the struc- This equation shows that in the limit of very high ture, and the Sommerfeld boundary radiation condition frequencies, or a large permittivity of the surrounding [25]. Also, note that the solution of Eq. (10) leads to media, spatial dispersion will be the main phenomenon the propagation constant of a transverse electric (TE) governing wave propagation. However, note that in- or a transverse magnetic (TM) mode, as in the case of terband contributions of graphene conductivity, which isotropic graphene [1]. These cases are examined below. are dominant at such high frequencies, have not been considered in this analysis. A. TM modes In the absence of spatial dispersion (αsd = βsd = 0), Eq. (14) simplifies to the usual expression [3] The dispersion relation for TM modes propagating along a spatially dispersive graphene sheet can be ob- (εr1 + εr2) TM TM TM k jωε0 . (16) tained by solving Y1 + Y2 + Y = 0 [see Eq. ρ σ ≈ σlo (10)]. This equation may be expressed as The cut-off frequency of the propagating TM modes ωεr1ε0 ωεr2ε0 2 + = σlo + kρ(αsd + βsd) can. be obtained by numerically finding the lowest fre- 2 2 2 2 − εr1k0 kρ εr2k0 kρ − −   quencies which fulfill Eq. (12). A very good approxi- q q (12) mation is obtained by identifying the frequency range If the graphene sheet is embedded into an homoge- of Im(σlo) < 0, which is the condition for TM mode propagation along a non-spatially dispersive graphene neous host medium, i.e. εr1 = εr2 = εr, the dispersion relation can be simplified to sheet [1]. Note that the cut-off frequency of the supported TM modes mainly depends on the characteristics of the 6 4 2σlo 2 local conductivity of graphene, which can be externally kρ + kρ εrk0 αsd + βsd − − controlled by tuning the chemical potential of graphene  2  2 2 2 2 2 2 2 2εrσlok0(αsd + βsd) σlo 4ω εrε0 εrk0σlo via the field effect. kρ 2 − + − 2 = 0, (αsd + βsd) (αsd + βsd)   (13) B. TE modes which has 6 complex roots but can be solved analytically using standard techniques [29]. Also, considering two The dispersion relation for TE modes propagating different media and assuming the usual non-retarded along a spatially dispersive graphene sheet can be ex- regime (k k0), Eq. (12) reduces to pressed as ρ ≫ 3 σ ε0(ε 1 + ε 2) k + k lo jω r r = 0, (14) ρ ρ 2 2 2 2 αsd + βsd − αsd + βsd ε 1k0 k ε 2k0 k r − ρ r − ρ 2 which is a cubic equation with three complex roots. Note + = σlo + kρ(αsd + βsd) . q ωµ0 q ωµ0 − that the operation frequency (ω) and the permittivity  (18)  of the surrounding media (εr1 and εr2) only appear in the free term of Eq. (14), where they multiply each In case the graphene sheet is embedded into an homo- other. Therefore, the behavior of surface waves prop- geneous medium, with εr1 = εr2 = εr, this equation agating along a graphene sheet is mainly determined admits the analytical solution shown in Eq.(17). by this product, suggesting that similar responses will Similarly as in the case of TM modes, the cut off be obtained for larger frequencies if the permittivity is frequency of TE modes mainly depends on the local simultaneously reduced, or vice-versa [Similar conclu- conductivity of graphene, σlo. Specifically, an accurate sions are reached by closely examining Eq. (13)]. How- condition for the propagation of these modes along a ever, this does not mean that frequency and permittivity graphene sheet is Im(σlo) > 0 [1]. 5

2 2 2 2 2 σloω µ0(αsd + βsd) + 2(1 1+ ω µ0(αsd + βsd)[σd + (αsd + βsd)εrk0] kρ = − ± 2 2 2 (17) ±s ωpµ0(αsd + βsd)

300 100

σ=σ 250 intrabands 80 σ=σ +σ intrabands interbands 200 σ=σ +SD mode 1 intrabands ) ) 60 0 σ=σ +SD mode 2 0 /k intrabands /k ρ 150 ρ Im(k Re(k 40 100

20 50

0 0 0 1 2 3 4 5 0 1 2 3 4 5 Frequency (THz) Frequency (THz) (a) (b)

300 250

250 200

200 ) ) 150 0 0 /k /k ρ 150 ρ Im(k Re(k 100 100

50 50

0 0 0 1 2 3 4 5 0 1 2 3 4 5 Frequency (THz) Frequency (THz) (c) (d) Fig. 2: Characteristics of surface waves propagating along a spatially dispersive (SD) graphene sheet versus frequency computed using Eq. (13). Reference results, related to surface waves propagating along non spatially dispersive graphene [1] computed taking into account intraband and intraband+interband contributions of conduc- tivity, are included for comparison purposes. (a) and (b) show the normalized propagation constant and losses of surface waves along a free-space standing graphene sheet. (c) and (d) show the normalized propagation constant and losses of surface waves along a graphene sheet embedded into an homogeneous media with εr = 11.9. Graphene ◦ parameters are µc = 0.05 eV, τ = 0.135 ps and T = 300 K.

IV. NUMERICAL RESULTS numerical study, we focus our results on TM waves, which are known to be of interest for plasmonic devices In this section, we investigate the influence of spatial [3], [6]. Though TE surface waves are also supported by dispersion in the characteristics of surface waves prop- graphene sheets, they present similar characteristics to agating along a graphene sheet, taking into account the waves propagating in free space (i.e. kρ k0) [1] and surrounding media. Specifically, we study the normalized thus have less practical interest. We compute≈ reference propagation constant [mode confinement, Re(kρ/k0), results for non spatially dispersive graphene sheet con- and losses, Im(kρ/k0)] of TM waves along spatially sidering intraband and intraband+interband contributions dispersive graphene, and compare the results with the of conductivity. The aim of this comparison is to iden- ones obtained neglecting spatial dispersion [1]. In the 6

300 300

250 250 µ =0.15eV τ=0.3ps c 200 200 τ

) ) =0.05ps 0 µ =0.0eV 0 /k c /k ρ 150 ρ 150 Re(k Re(k 100 100 σ=σ +SD mode 1 σ=σ +SD mode 1 intraband intraband σ=σ +SD mode 2 50 σ=σ +SD mode 2 50 intraband intraband σ=σ σ=σ intraband intraband 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Frequency (THz) Frequency (THz) (a) (b) Fig. 3: Normalized phase constant of surface waves propagating on a spatially dispersive graphene sheet versus frequency, computed for different values of (a) chemical potential [using τ = 0.135 ps] and (b) relaxation time [using µc = 0.05 eV]. Reference results, related to surface waves on non spatially dispersive graphene sheet computed only taking into account intraband contributions of conductivity [1], are included for comparison purposes. The ◦ permittivity of the surrounding media is εr = 11.9 and temperature is T = 300 K. tify which phenomenon (spatial dispersion or interband this particular case the influence of spatial dispersion contributions) becomes dominant as frequency increases. is negligible in the low THz range. In [2] similar Furthermore, we will also investigate the influence of conclusions were obtained analyzing, in the transformed spatial dispersion in the propagating surface waves as Fourier domain, the ratio between graphene conductivity a function of different parameters of graphene. Our and the terms related to spatial dispersion. Interestingly, numerical simulations consider graphene at T = 300◦ K here we also observe that a second mode, denoted as “SD and a relaxation time of 0.135 ps, in agreement with - mode 2”, is supported due to the presence of spatial measured values [3], [4]. The results shown here have dispersion. It is observed that this mode is extremely been obtained numerically [solving Eq. (12)] or analyt- lossy in the band of interest, which greatly limits its ically [from Eq. (13)], depending on the surrounding possible use in practical applications. media of graphene. Besides, note that the analytical We present in Figs. 2c-2d a similar study, but con- solution of Eq. (14) leads to results with differences sidering now a graphene sheet embedded into an homo- smaller than 0.1% with respect to those numerically geneous material with ε = 11.9. Results demonstrate obtained from Eq. (12), further confirming the validity r that spatial dispersion becomes the dominant mechanism of this expression. of wave propagation in this case, even at relatively low First, we consider the case of a graphene sheet with frequencies, drastically changing the behavior of the chemical potential µc = 0.05 eV surrounded by air. Figs. two modes supported by the graphene sheet. At low 2a-2b show the normalized propagation constant and frequencies, the first mode (“SD - mode 1”) presents losses of surface waves propagating along the spatially very similar characteristics as compared to a TM sur- dispersive sheet. Specifically, two modes are supported face mode on along a graphene sheet neglecting spatial by the structure, which are the physical solutions of dispersion. However, the confinement and losses of the Eq. (13). The first mode (denoted as “SD - mode 1”) mode increases with frequency. At around 2 THz, losses presents extremely similar characteristics as compared exponentially decreases thus leading to a non-physical to a TM surface mode supported by a non spatially improper mode at frequencies above 2.5 THz. The dispersive graphene sheet, computed considering only behavior of the second mode (“SD - mode 2”) is also intraband contributions of graphene. Figs. 2a-2b also affected by spatial dispersion. As frequency increases, include similar computations but considering interband the confinement and losses of the mode decreases. Above contributions as well. The effect of interband contribu- 2 THz, the confinement of the mode remains relatively tion become visible at high frequencies, increasing the constant and losses slightly increase. It should be noted losses of the mode. Thus, we can conclude that for that although the phase constant of modes 1 and 2 7

300 350

250 300 σ=σ intrabands σ σ σ = + 250 200 intrabands interbands σ=σ +SD mode 1 ) intrabands ) 0 0

/k 200 σ σ /k

ρ = +SD mode 2 150 intrabands ρ Im(k

Re(k 150 100 100

50 50

0 0 2 4 6 8 10 12 2 4 6 8 10 12 Permittivity Permittivity (a) (b)

300 160 250 140

200 120 ) ) 0 0 100 /k /k ρ 150 ρ 80 Im(k Re(k 100 60

40 50 20

0 0 2 4 6 8 10 12 2 4 6 8 10 12 Permittivity Permittivity (c) (d) Fig. 4: Characteristics of surface waves propagating along a spatially dispersive (SD) graphene sheet as a function of surrounding media permittivity, computed using Eq. (13). Reference results, related to surface waves propagating along non spatially dispersive graphene [1] computed taking into account intraband and intraband+interband contributions of conductivity, are included for comparison purposes. (a) and (b) show the normalized propagation constant and losses of the surface waves at 1 THz. (c) and (d) show the normalized propagation constant and losses ◦ of the surface waves at 3 THz. Graphene parameters are µc = 0.05 eV, τ = 0.135 ps and T = 300 K.

intersect at around 2 THz, their attenuation constants in Fig. 2c) as a function of the chemical potential µc are different at that frequency thus allowing to clearly and the relaxation time τ. We observe in Fig. 3a that identify the modes. Besides, note that the frequency the chemical potential controls the frequency where the where the phase constant of the modes intersect defines phase constant of the two modes intersect, which is the frequency region where the influence of spatial the frequency where spatial dispersion starts to have a dispersion starts to be dominant. Furthermore, mode dominant effect on the behavior of the modes. Note that coupling is occurring between the two modes supported an increase of µc up-shifts this frequency. In addition, by the graphene sheet. This coupling is governed by the Fig. 3b shows that the relaxation time of graphene occurrence of a complex-frequency-plane branch point determines the phase constant of the surface waves in the that migrates across the real frequency axis [30], [31]. frequency region where spatial dispersion is dominant. Importantly, the influence of spatial dispersion on the Increasing τ modifies the phase constant of the modes, characteristics of the supported modes strongly depends which tend to have a similar behavior versus frequency, on the features of graphene. As an illustration, Fig. 3 and reduces their attenuation losses. presents the phase constant of the two modes propagating For the sake of completeness, we present in Fig. 4 along a graphene sheet (with the parameters employed a study of surface waves propagating along spatially 8

300 σ=σ 200 intraband σ=σ +σ intraband interband 250 σ=σ +SD mode 1 intraband 150 σ=σ +SD mode 2 200 intraband ) ) 0 0 /k /k ρ ρ 150 100 Im(k Re(k 100 50 50

0 0 0 1 2 3 4 5 0 1 2 3 4 5 Frequency (THz) Frequency (THz) (a) (b) Fig. 5: Characteristics of surface waves propagating along a spatially dispersive (SD) graphene sheet deposited on a silicon substrate (εr = 11.9) versus frequency, computed using Eq. (12). Reference results, related to surface waves propagating along non spatially dispersive graphene [1] computed taking into account intraband and intraband+interband contributions of conductivity, are included for comparison purposes. (a) and (b) show the normalized propagation constant and losses of the surface waves, respectively. Graphene parameters are similar to those of Fig. 2. dispersive graphene, but considering now a constant of a spatially dispersive graphene sheet deposited on a frequency and varying the permittivity of the surround- silicon substrate (εr = 11.9), as illustrated in Fig. 1a. ing media. It is worth mentioning that in practice the Very similar behavior as compared to the previous exam- material permittivity might affect graphene relaxation ples is obtained. Furthermore, these results demonstrate time [21], but this effect is neglected here for conve- that the variation of the surrounding media directly nience. Figs. 4a-4b present the normalized phase and controls the frequency range where spatial dispersion is attenuation constants of surface waves propagating along noticeable. Analyzing these variations, we can conclude graphene for a fixed frequency of 1 THz. The first that increasing the permittivity of the media surrounding mode is extremely similar to a TM surface mode along graphene down shift the frequencies where the spatial graphene neglecting spatial dispersion. As expected from dispersion phenomenon dominates wave propagation. Eq. (14), mode confinement and losses increases when the dielectric constant is very high. Interesting, there are V. CONCLUSIONS many similarities in the behavior of the different modes The propagation of surface waves along a spatially in this situation and in the previous example, where the dispersive graphene sheet has been addressed. The anal- surrounding media was constant (with a low permittivity ysis, that was based on the transverse resonance method value of ε = 1) and frequency was increased. Moreover, r extended to handle graphene spatial dispersion, allowed we present in Figs. 4c-4d the same study but at the to obtain the desired dispersion relation for surface higher frequency of 3 THz. It is observed that spatial waves propagating along a graphene sheet. Our results dispersion becomes the dominant phenomenon for wave have demonstrated that spatial dispersion is an important propagation. In fact, it leads to propagating modes with mechanism which contributes to the propagation of sur- similar behavior as in the previous example (see Figs. 2c- face waves along graphene sheets and that its influence 2d), where we used a fixed large value of permittivity cannot be systematically neglected. In fact, the presence (ε = 11.9) and varied frequency. This example confirms r of spatial dispersion at frequencies where intraband that, neglecting the frequency dependence of graphene contributions of graphene dominate lead to surface waves conductivity, permittivity and frequency play a similar with very different characteristics from those propagating role in the characteristics of surface waves propagating along graphene sheets neglecting spatial dispersion. In along spatially dispersive graphene. addition, the frequency region where spatial dispersion Finally, we investigate in Fig. 5 the propagation of is noticeable depend on the surrounding media and the surface waves in a more realistic scenario, which consists chemical potential of graphene, while the features of the 9 propagating surfaces waves are mainly determined by [11] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto- the relaxation time. These results demonstrate that the optical conductivity in graphene,” Journal of : Condensed Matter, vol. 19, no. 2, p. 026222, 2007. [Online]. influence of spatial dispersion on surface waves propa- Available: http://stacks.iop.org/0953-8984/19/i=2/a=026222 gating on graphene sheets should be rigorously taken into [12] E. H. Hwang and J. D. Sarma, “Dielectric function, screening, account for the development of novel plasmonic devices and plasmons in two-dimensional graphene,” Phys. Rev. in the low THz range. B, vol. 75, p. 205418, May 2007. [Online]. Available: The study presented here has been based on a the- http://link.aps.org/doi/10.1103/PhysRevB.75.205418 [13] J. S. Gomez-Diaz and J. Perruisseau-Carrier, “Propagation oretical model of graphene which characterizes it as of hybrid TM-TE plasmons on magnetically-biased graphene an infinitesimally thin layer with an associated tensor sheets,” Journal of Applied Physics, vol. 112, p. 124906, 2012. conductivity. This model uses the relaxation time approx- [14] G. W. Hanson, “Quasi-transverse electromagnetic modes sup- ported by a graphene parallel-plate waveguide,” Journal of imation, only takes into account intraband contributions Applied Physics, vol. 104, p. 084314, 2008. of graphene and is valid in the absence of external [15] J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. magnetostatic biasing fields. Further work is needed in Koppens, and F. J. G. de Abajo, “Graphene plasmon waveg- this area to analyze the behavior of surface waves along uiding and hybridization in individual and paired nanoribbons,” ACS Nano, vol. 6, p. 431440, 2011. spatially dispersive graphene when these assumptions are [16] C. S. R. Kaipa, A. B. Yakovlev, G. W. Hanson, Y. R. not satisfied. Padooru, F. Medina, and F. Mesa, “Enhanced transmission with a graphene-dielectric microstructure at low-terahertz frequen- ACKNOWLEDGEMENT cies,” Phys. Rev. B, vol. 85, p. 245407, 2012. [17] L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of This work was supported by the Swiss National Sci- graphene conductivity,” European Physical Journal B, vol. 56, ence Foundation (SNSF) under grant 133583 and by pp. 281–284, 2007. the EU FP7 Marie-Curie IEF grant “Marconi”, with ref. [18] G. Lovat, P. Burghignoli, and R. Araneo, “Low-frequency dominant-mode propagation in spatially dispersive graphene 300966. The authors wish to thank Prof. G. W. Hanson nanowaveguides,” IEEE Transactions on Electromagnetic Com- (University of Wisconsin-Milwaukee, USA), Dr. Garcia- patibility, in press, 2013. Vigueras and Dr. E. Sorolla-Rosario (Ecole´ Polytech- [19] G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, nique F´ed´erale de Lausanne, Switzerland) for fruitful “Comparison of spatially dispersive models for dyadic interband conductivity of graphene,” in European Conference on Antennas discussions. and Propagation, Gothenburg, Sweden, April 2013. [20] G. Lovat, G. W. Hanson, R. Araneoz, and P. Burghignoli, REFERENCES “Spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” submitted, 2013. [1] G. W. Hanson, “Dyadic green’s functions and guided surface [21] M. Dressel and G. Gruner, Electrodynamics of Solids. Cam- waves for a surface conductivity of graphene,” Journal of bridge, U.K.: Cambridge Univ. Press, 2002. Applied Physics, vol. 103, p. 064302, 2008. [22] G. 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