See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/261318586 Transverse-resonance analysis of dominant-mode propagation in graphene nano-waveguides Conference Paper · September 2012 DOI: 10.1109/EMCEurope.2012.6396763 CITATIONS READS 2 70 1 author: Giampiero Lovat Sapienza University of Rome 166 PUBLICATIONS 2,082 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Propagation in graphene nanoribbons View project Dielectric Leaky Wave Antenna View project All content following this page was uploaded by Giampiero Lovat on 17 December 2014. The user has requested enhancement of the downloaded file. Transverse-Resonance Analysis of Dominant-Mode Propagation in Graphene Nano-Waveguides Giampiero Lovat Department of Astronautical, Electrical, and Energetic Engineering, “Sapienza” University of Rome Via Eudossiana 18, 00184 Rome, Italy [email protected] Abstract—Dispersion properties of the dominant modes supported by two-dimensional graphene-based nano-waveguides are studied by means of an exact approach based on the transverseresonance technique and using the equivalent circuit representation of graphene sheets. In particular, in the absence of magnetic bias, the graphene is characterized by a scalar conductivity obtained based on semi-classical and quantum-mechanical arguments. Such a description takes into account the possible presence of static electric biasing fields and/or chemical doping. The modal features of graphene nano-waveguides are investigated in detail thus showing their potential in future nano-electromagnetic and nano-photonic applications. z y I. I NTRODUCTION Graphene, an atomically thick layer of carbon atoms arranged in a honeycomb structure (see Fig. 1), is increasingly gaining attention all over the world as potential material for nanointerconnects from microwave to terahertz frequencies and beyond [1]–[3]. This is due to its excellent electronic, mechanical, optical, and thermal properties together with its unique feature of being only one-atom thick. From an electromagnetic point of view, graphene can be described as an infinitesimally thin medium characterized by a surface conductivity [4], [5]; in particular, a semiclassical and a quantum-dynamical approach can be used to derive the conductivity expression which also takes into account the presence of electrostatic and/or magnetostatic bias [5]. Recently, applications of graphene structures have attracted the attention of many researchers in the EMC community since graphenebased nanostructures (like graphite nanoplatelets (GNP), fewlayer graphene, and carbon nanowalls) could be investigated for their potential as nanofillers in nanocomposites or as basic building blocks in novel RF nanodevices [6]. In this contribution, dispersion properties of dominant modes supported by different two-dimensional (2-D) graphene-based nano-waveguides are studied with an exact approach based on the transverse-resonance technique and using the transverse-equivalent-network (TEN) representation of a graphene sheet [7]. In particular, the considered nanowaveguides are: i) a graphened dielectric slab, ii) a parallelplate waveguide with two graphene walls (graphene/graphene PPW), and iii) a parallel-plate waveguide with one graphene wall and one perfectly-conducting plate (graphene/PEC PPW). Such an investigation expands the preliminary studies in [8], [9] by means of a different and simpler approach. 978-1-4673-0717-8/12/$31.00 ©2012 IEEE x Fig. 1. Graphene sheet. The electromagnetic problem under analysis is thus sketched in Fig. 2. In general, it consists of an infinite graphene sheet deposited on a dielectric substrate (typically SiO2 ) of thickness h and relative dielectric permittivity εr (graphened slab). A bottom plane may also be present at z = 0, which can be a perfectly-conducting (PEC) ground plane (graphene/PEC PPW) or another graphene sheet (graphene/graphene PPW). z ε0 h ε0 εr 0 ε0 graphene plate Fig. 2. PEC plate Examples of two-dimensional (2-D) graphene nano-waveguides. II. G RAPHENE C ONDUCTIVITY In general, a graphene sheet can be modeled as a conductive sheet with a dyadic conductivity. This dyadic form can be due to the application of a magnetostatic bias together with a nonzero chemical potential μc or to spatial-dispersion effects [10]: however, the latter can usually be neglected below the THz regime and in the following analysis we will not consider the presence of a possible magnetic bias. Therefore graphene will be characterized by a scalar conductivity σ, which however depends on different parameters, e.g., frequency f = ω/(2π), temperature T , a phenomenological scattering rate Γ = 1/(2τ ) (where τ is the relaxation time depending on a variety of factors and determined experimentally) [5], and the chemical potential μc (which can be controlled either by doping or by an applied bias electric field orthogonal to the graphene plate [10]). In particular, it results q 2 (ω − j2Γ) σ=j e π +∞ 2 ∂nF (−) − Δ2 ∂nF () 1 − d · 2 ∂ ∂ (ω − j2Γ) Δ +∞ 2 + Δ2 nF () − nF (−) − 2 2 d 2 (ω − j2Γ) − 4 (/) Δ (1) where −qe is the electron charge, vF 106 m/s is the Fermi velocity in graphene, is the reduced Planck’s constant, and Δ is an excitonic energy gap with no effects at room temperature (so it will be assumed Δ = 0 in what follows). Finally, nF is the Fermi-Dirac distribution given by nF () = 1 , (2) 1+ where is the energy and kB is the Boltzmann’s constant. Moreover, when electronically controlled, the relation between the chemical potential μc and the electrostatic bias field Ebias is given by [10] +∞ ε0 π2 vF2 Ebias = [nF () − nF ( + 2μc )] d , (3) qe 0 e(−μc )/(kB T ) which must be solved numerically for the chemical potential μc (Ebias ). Also the value of the conductivity σ in (1) can easily be computed through standard adaptive integration routines. Alternatively, as mentioned above, the chemical potential is determined by the extra-carrier density ns as +∞ 2 ns = [nF () − nF ( + 2μc )] d . (4) π2 vF2 0 Finally, for Δ = 0, (1) can be approximately evaluated in a closed form as qe2 kB T μc − kμcT B σ −j 2 + 2 ln 1 + e , (5) π (ω − j2Γ) kB T thus showing a Drude-like behavior. As it will be shown later, (5) is very accurate at room temperatures and below the THz regime. 978-1-4673-0717-8/12/$31.00 ©2012 IEEE z graphened slab ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Y0 , kz 0 h Yσ Ys , kzs 0 BC at z =0 Yσ graphene/graphene PPW graphene/PEC PPW Y0 , kz 0 Fig. 3. Transverse-equivalent network (TEN) for the problem of modal propagation along the waveguides in Fig. 2. III. T RANSVERSE E QUIVALENT N ETWORK As is well known, fields in a 2-D isotropic waveguide can be decomposed into their TMz and TEz parts [11]. Both TMz and TEz fields inside the slab as well as in free space can be modeled using equivalent transmission lines (TLs) along the vertical z axis, starting from the spectral form of Maxwell’s equations. In general, this TEN model is valid for plane-wave incidence as well as for the analysis of modal propagation. As shown in [7], the TEN representation for a graphene sheet operating in general conditions is represented by a fourport network (which in general couples TMz and TEz fields), obtained by enforcing the correct boundary conditions in the Maxwell equations; however, in the absence of both electric and magnetic bias and neglecting spatial-dispersion effects, the four-port network reduces to a simple shunt admittance Yσ (or impedance Zσ = 1/Yσ ) with the same value for TMz and TEz fields. The complete TEN for the considered waveguides is then reported in Fig. 3, where the network parameters for the two polarizations in the air and slab regions (subscripts 0 and s, respectively) are kz0 = k0 k̂z0 = k0 kzs = k0 k̂zs = k0 1 − k̂ρ2 = k0 εr − k̂ρ2 = k0 1− kρ k0 kρ k0 εr − 2 , 2 (6) and Y0TE = Y0TM = k̂z0 , η0 1 η0 k̂z0 YsTE = , YsTM k̂zs η0 = εr η0 k̂zs (7) , √ where k0 = ω μ0 ε0 and η0 = μ0 /ε0 are the free-space wavenumber and impedance, respectively. The TEN in Fig. 2 can thus be used to determine the (generally complex) radial wavenumber kρ = β − jα of a mode traveling along the radial direction ρ of the waveguide as e−jkρ ρ . Y up + Y down = 0 or Z up + Z down = 0 . (8) Proper modes (which are of interest for this study) are found by choosing the proper determination for the vertical wavenumbers kz0 = k02 − kρ2 , i.e., the one with a negative imaginary part, corresponding to fields that decay exponentially at infinity in the transverse z direction. For all the structures considered below, the transverse resonance condition is applied to determine the dispersion equation of the modes supported by each nano-waveguide. Such a dispersion equation will be expressed in terms of the admittances of the individual TENs, which in turn depend on the transverse wavenumbers kz0 and kzs via (7); finally, these transverse wavenumbers are explicit functions of the radial wavenumber kρ through (6), which is the actual unknown of the dispersion equations. A. Graphened slab The TEN for the analysis of the graphened slab is reported in Fig. 4(a). It clearly results Y up = Y0 + Yσ Y0 cos (kzs h) + jYs sin (kzs h) Y down = Ys Ys cos (kzs h) + jY0 sin (kzs h) (9) so that the transverse resonance condition reads (Y0 + Yσ ) [Ys cos (kzs h) + jY0 sin (kzs h)] + Ys [Y0 cos (kzs h) + jYs sin (kzs h)] = 0 . (10) It can be noted that in the absence of a dielectric support (h = 0 or Ys = Y0 ) (10) enormously simplifies to 2Y0 + Yσ = 0 . (11) Since Yσ = σ [7], from (7) it results k0 η0 σ 2 2k0 1 =− η0 σ kz0 = − for TE modes kz0 for TM modes, 978-1-4673-0717-8/12/$31.00 ©2012 IEEE h Yσ Y up Y Yσ h Y down Ys , kzs 0 Y up down Ys , kzs 0 Y0 (b) (a) Fig. 4. TENs for dispersion analysis of the graphened-slab waveguide (a) and of the graphene/PEC PPW (b). B. Graphene/PEC PPW The TEN for the analysis of the PPW with a graphene wall and a PEC plate is reported in Fig. 4(b). It results Y up = Y0 + Yσ (13) Y down = −jYs cot (kzs h) so that in this case the transverse resonance condition reads (Y0 + Yσ ) sin (kzs h) − jYs cos (kzs h) = 0 . (14) C. Graphene/graphene PPW As it can easily be seen, the graphene/graphene PPW is simmetric with respect to the z = h/2 plane. Because of such a simmetry, the tangential component of either electric or magnetic field must vanish: in the former case, the z = h/2 plane can be substituted by a PEC plane, while in the latter by a perfectly magnetic conductor (PMC) plane. From a TEN point of view, the TEN can be simplified and the transmission line can be closed at z = h/2 either by a short ciruit or by an open ciruit, as shown in Fig. 5. It can easily be seen that the former gives rise to odd modes while the latter to even modes. For odd modes we thus have h Y up = −jYs cot kzs 2 (15) down Y = Y0 + Yσ so that the transverse resonance condition reads h h − jYs cos kzs (Y0 + Yσ ) sin kzs 2 2 . (16) For even modes we instead have Y up = jYs tan kzs h 2 (17) Y down = Y0 + Yσ (12) as also derived in [8]. Y0 Y0 IV. D ISPERSION A NALYSIS VIA T RANSVERSE R ESONANCE The semi-infinite TLs in the air region of Fig. 3 can be replaced by their characteristic admittances and the modal dispersion equation can be found by enforcing the transverse resonance condition (e.g., at z = h) of the resulting network [12]: this means that the admittance Y up (or impedance Z up ) seen looking into the TL for z > h must be equal to the admittance Y down (or impedance Z down ) seen looking into the TL for z < h, i.e., z z and the transverse resonance condition reads h h + jYs sin kzs (Y0 + Yσ ) cos kzs 2 2 . (18) z Odd modes h /2 z Yσ h Yσ 0 Y up Yσ 0 σR [Eq. (5)] Y up σJ [Eq. (5)] 0.01 Y down 0 Y0 Y down Ys , kzs σJ 0.02 Ys , kzs Y0 σR 0.03 −1 [Ω ] -0.01 z -0.02 h /2 0 200 400 Ys , kzs Y0 Yσ 0 Y up 800 1000 Fig. 6. Graphene conductivity σ = σR + jσJ as a function of frequency. Parameters: T = 300 K, τ = 0.5 ps, and μc = 0.5 eV. Y down 10 Y0 Even modes Fig. 5. 600 f [GHz] 3 Graphened slab Graphene/PEC PPW Graphene PPW (odd mode) Graphene PPW (even mode) β/k0 10 2 10 1 10 0 TENs for dispersion analysis of the graphene/graphene PPW. V. N UMERICAL R ESULTS The variation of the isotropic complex graphene conductivity σ = σR + jσJ with frequency is reported in Fig. 6, where it has been assumed T = 300 K, τ = 0.5 ps, and μc = 0.5 eV (i.e., from (4), a doping with ns 2 × 1013 cm−2 ). It can be seen that significant changes of the conductivity arise for frequencies larger than 10 GHz up to the THz regime. The behavior of the complex conductivity as a function of frequency simply follows a Drude model: such a behavior reflects the conventional dispersive conductivity arising from a Drude theory of metals [13], valid provided that spatialdispersion effects can be neglected, as for the frequency range considered here. It can also be observed that the approximate expression in (5) is very accurate in the whole considered frequency range. The following results for the dispersion curves of the dominant modes of the structured described above have been obtained with the graphene conductivity in Fig. 6 and with a dielectric board having relative dielectric permittivity εr = 3.9 (SiO2 ) and thickness h = 100 nm. The normalized wavenumber k̂ρ = β̂ − j α̂ (with β̂ = β/k0 normalized radial phase constant and α̂ = α/k0 normalized attenuation constant) have been obtained by solving numerically (10), (14), (16), and (18) using Müller’s method. The normalized phase and attenuation constants are thus reported in Figs. 7 and 8, respectively, as functions of frequency. As already shown in [9] in connection with isolated graphene sheets, the structures analyzed here do not support proper TE modes below the THz regime (more specifically, as 978-1-4673-0717-8/12/$31.00 ©2012 IEEE 10 -1 0 200 400 600 800 1000 f [GHz] Fig. 7. Normalized phase constant β/k0 as a function of frequency for the considered graphene-based nano-waveguides with graphene conductivity as in Fig. 6. Other parameters: εr = 3.9 and h = 100 nm. long as the imaginary part of the graphene conductivity σJ is negative); therefore all the modal solutions reported in Figs. 7-8 are proper TM modes. The graphened slab (black solid line) supports a TM mode which is quite fast (0.98 < β̂ < 1.14) and very slowly attenuating (α̂ 1); this entails that the mode is not well confined close to the waveguide; it should be noted that such a solution is superimposed to the TM mode supported by the isolated graphene sheet (see (12), although the relevant curve is not reported in the figures), thus showing that the effects of the dielectric nano-board are completely negligible. On the other hand, a PPW with a PEC and a graphene wall (gray solid line) supports a TM mode which is surprisingly slow (23 < β̂ < 130) and highly attenuating (α̂ > 3 in the whole frequency range), which is thus very well confined inside the nano-waveguide. Although the normalized attenuation constant is very large, the mode is still capable of carrying energy along a circuit of nanometric dimensions. 3 α/k0 10 Graphened slab Graphene/PEC PPW Graphene PPW (odd mode) Graphene PPW (even mode) 2 10 1 10 0 10 -1 10 -2 10 -3 10 0 200 400 f [GHz] 600 800 1000 Fig. 8. Normalized attenuation constant α/k0 as a function of frequency for the considered graphene-based nano-waveguides with graphene conductivity as in Fig. 6. Other parameters: εr = 3.9 and h = 100 nm. Finally, when considering a PPW with two graphene walls with equal conductivity, two fundamental TM modes exist, associated with an odd (black dashed line) and an even (gray dashed line) field distribution, respectively. It turns out that the odd mode is the perturbation of the TEM mode of the PPW with two PEC walls (and therefore its dispersion properties are also similar to those of the PEC-graphene PPW), whereas the even mode is the perturbation of the TM mode supported by the graphened slab (and therefore its propagation constant is also very similar to that of the graphened slab). VI. C ONCLUSION The modal features of dominant surface waves for different two-dimensional graphene-based nano-waveguides (i.e., a graphened dielectric nano-slab, a nano-PPW with two graphene walls, and a nano-PPW with one graphene wall and a PEC plate) have been studied by means of an exact and simple approach based on the transverse-resonance technique and a transverse-equivalent-circuit representation of graphene sheets which takes into account the possible presence of static electric biasing fields and/or chemical doping. Dispersion properties in terms of radial phase and attenuation constants are derived, thus allowing for gaining an insight into the potentialities of graphene nano-waveguides for future applications in nanoelectromagnetics. R EFERENCES [1] Y.-M. Lin, A. Valdes-Garcia, S.-J. Han, D. B. Farmer, I. Meric, Y. Sun, Y. Wu, C. Dimitrakopoulos, A. Grill, P. Avouris, and K. A. Jenkins, “Wafer-scale graphene integrated circuits,” Science, vol. 332, no. 10, pp. 1294–1297, 2011. [2] J.-S. Moon and D. K. Gaskill, “Graphene: Its fundamentals to future applications,” IEEE Trans. Microwave Theory Tech., vol. 59, no. 10, pp. 2702–2708, Oct. 2011. [3] C. Xu, H. Li, and K. Banerjee, “Modeling, analysis, and design of graphene nano-ribbon interconnects,” IEEE Trans. Electron Devices, vol. 56, no. 8, pp. 1567–1578, Aug. 2009. [4] L. A. Falkovsky, “Optical properties of graphene and IV-VI semiconductors,” Phys. Usp., vol. 51, no. 9, pp. 887–897, 2008. 978-1-4673-0717-8/12/$31.00 ©2012 IEEE View publication stats [5] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys.: Cond. Matter, vol. 19, no. 2, p. 026222, 2007. [6] M. D’Amore, M. S. Sarto, G. W. Hanson, A. Naeemi, and B. K. Tay, Guest Editors, “Special Issue on Applications of Nanotechnology in Electromagnetic Compatibility (nano-EMC),” IEEE Trans. Electromagn. Compat., vol. 54, no. 1, Feb. 2012. [7] G. Lovat, “Equivalent circuit for electromagnetic interaction and transmission through graphene sheets,” IEEE Trans. Electromagn. Compat., vol. 54, no. 1, pp. 101-109, Feb. 2012. [8] G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys., vol. 103, pp. 064302, 2008. [9] G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys., vol. 104, pp. 084314, 2008. [10] G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 747–757, Mar. 2008. [11] R. E. Collin, Field Theory of Guided Waves. 2nd ed. Piscataway, NJ: IEEE Press, 1991. [12] T. Rozzi and M. Mongiardo, Open Dielectric Waveguides. London, UK: IEE, 1997. [13] M. Dressel and G. Grüner, Electrodynamics of Solids. Cambridge, UK: Cambridge Univ. Press, 2002.