Damping of Terahertz Plasmons in Graphene Coupled with Surface Plasmons in Heavily-Doped Substrate
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Damping of Terahertz Plasmons in GrapheneCoupled with Surface Plasmons in Heavily-Doped Substrate
A. Satou , , ∗ Y. Koseki , V. Ryzhii , , V. Vyurkov , and T. Otsuji , Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan CREST, Japan Science and Technology Agency, Tokyo 107-0075, Japan Institute of Physics and Technology, Russian Academy of Sciences, Moscow 117218, Russia
Coupling of plasmons in graphene at terahertz (THz) frequencies with surface plasmons in aheavily-doped substrate is studied theoretically. We reveal that a huge scattering rate may com-pletely damp out the plasmons, so that proper choices of material and geometrical parameters areessential to suppress the coupling effect and to obtain the minimum damping rate in graphene.Even with the doping concentration 10 − cm − and the thickness of the dielectric layer be-tween graphene and the substrate 100 nm, which are typical values in real graphene samples witha heavily-doped substrate, the increase in the damping rate is not negligible in comparison withthe acoustic-phonon-limited damping rate. Dependence of the damping rate on wavenumber, thick-nesses of graphene-to-substrate and gate-to-graphene separation, substrate doping concentration,and dielectric constants of surrounding materials are investigated. It is shown that the dampingrate can be much reduced by the gate screening, which suppresses the field spread of the grapheneplasmons into the substrate. I. INTRODUCTION
Plasmons in two-dimensional electron gases (2DEGs)can be utilized for terahertz (THz) devices. THzsources and detectors based on compound semicon-ductor heterostructures have been extensively investi-gated both experimentally and theoretically.
The two-dimensionality, which gives rise to the wavenumber-dependent frequency dispersion, and the high electronconcentration on the order of 10 cm − allow us to havetheir frequency in the THz range with submicron channellength. Most recently, a very high detector responsivityof the so-called asymmetric double-grating-gate structurebased on an InP-based high-electron-mobility transistorwas demonstrated. However, resonant detection as wellas single-frequency coherent emission have not been ac-complished so far at room temperature, mainly owningto the damping rate more than 10 s − in compoundsemiconductors.Plasmons in graphene have potential to surpass thosein the heterostructures with 2DEGs based on the stan-dard semiconductors, due to its exceptional electronicproperties. Massive experimental and theoretical workshave been done very recently on graphene plasmons inthe THz and infrared regions (see review papers Refs. 11and 12 and references therein). One of the most im-portant advantages of plasmons in graphene over thosein heterostructure 2DEGs is much weaker damping rateclose to 10 s − at room temperature in disorder-freegraphene suffered only from acoustic-phonon scatter-ing. That is very promising for the realization of theresonant THz detection and also of plasma instabili-ties, which can be utilized for the emission. In addition,interband population inversion in the THz range was pre-dicted, and it has been investigated for the utilizationnot only in THz lasers in the usual sense but also in THzactive plasmonic devices and metamaterials. Many experimental demonstrations of graphene-baseddevices have been performed on graphene samples withheavily-doped substrates, in order to tune the carrierconcentration in graphene by the substrate as a backgate. Typically, either peeling or CVD graphene trans-ferred onto a heavily-doped p + -Si substrate, with a SiO dielectric layer in between, is used (some experimentson graphene plasmons have adapted undoped Si/SiO substrates ). Graphene-on-silicon, which is epitax-ial graphene on doped Si substrates, is also used.For realization of THz plasmonic devices, properties ofplasmons in such structures must be fully understood.Although the coupling of graphene plasmons to sur-face plasmons in perfectly conducting metallic substrateswith/without dielectric layers in between have been the-oretically studied, the influence of the carrier scat-tering in a heavily-doped semiconductor substrate (withfinite complex conductivity) has not been taken into ac-count so far. Since the scattering rate in the substrate in-creases as the doping concentration increases, it is antic-ipated that the coupling of graphene plasmons to surfaceplasmons in the heavily-doped substrate causes undisiredincrease in the damping rate.The purpose of this paper is to study theoretically thecoupling between graphene plasmons and substrate sur-face plasmons in a structure with a heavily-doped sub-strate and with/without a metallic top gate. The paperis organized as follows. In the Sec. II, we derive a dis-persion equation of the coupled modes of graphene plas-mons and substrate surface plasmons. In Sec. III, westudy coupling effect in the ungated structure, especiallythe increase in the plasmon damping rate due to the cou-pling and its dependences on the doping concentration,the thickness of graphene-to-substrate separation, andthe plasmon wavenumber. In Sec. IV, we show that thecoupling in the gated structures can be less effective dueto the gate screening. We also compare the effect instructures having different dielectric layers between the FIG. 1. Schematic views of (a) an ungated graphene structurewith a heavily-doped Si substrate where the top surface isexposed on the air and (b) a gated graphene structure with aheavily-doped Si substrate and a metallic top gate. top gate, graphene layer, and substrate, and reveal theimpact of values of their dielectric constants. In Sec. V,we discuss and summarize the main results of this paper.
II. EQUATIONS OF THE MODEL
We investigate plasmons in an ungated graphene struc-ture with a heavily-doped p + -Si substrate, where thegraphene layer is exposed on the air, as well as a gatedgraphene structure with the substrate and a metallic topgate, which are schematically shown in Figs. 1(a) and (b),respectively. The thickness of the substrate is assumedto be sufficiently larger than the skin depth of the sub-strate surface plasmons. The top gate can be consideredas perfectly conducting metal, whereas the heavily-dopedSi substrate is characterized by its complex dielectric con-stant.Here, we use the hydrodynamic equations to describethe electron motion in graphene, while using the simpleDrude model for the hole motion in the substrate (dueto virtual independence of the effective mass in the sub-strate on the electron density, in contrast to graphene).In addition, these are accompanied by the self-consistent2D Poisson equation (The formulation used here almostfollows that for compound semiconductor high-electron-mobility transistors, see Ref. 25). Differences are thehydrodynamic equations accounting for the linear dis-persion of graphene and material parameters of the sub-strate and dielectric layers. In general, the existence ofboth electrons and holes in graphene results in variousmodes such as electrically passive electron-hole soundwaves in intrinsic graphene as well as in huge dampingof electrically active modes due to the electron-hole fric-tion, as discussed in Ref. 26. Here, we focus on the casewhere the electron concentration is much higher than the hole concentration and therefore the damping associatedwith the friction can be negligibly small. Besides, for thegeneralization purpose, we formulate the plasmon dis-persion equation for the gated structure; that for the un-gated structure can be readily found by taking the limit W t → ∞ (see Fig. 1).Then, assuming the solutions of the form exp( ikx − iωt ), where k = 2 π/λ and ω are the plasmon wavenum-ber and frequency ( λ denotes the wavelength), the 2DPoisson equation coupled with the linearized hydrody-namic equations can be expressed as follows: ∂ ϕ ω ∂z − k ϕ ω = − πe Σ e m e ǫ k ω + iν e ω − ( v F k ) ϕ ω δ ( z ) , (1)where ϕ ω is the ac (signal) component of the potential,Σ e , m e , and ν e are the steady-state electron concentra-tion, the hydrodynamic “fictitious mass”, and the col-lision frequency in graphene, respectively, and ǫ is thedielectric constant which is different in different layers.The electron concentration and fictitious mass are re-lated to each other through the electron Fermi level, µ e ,and electron temperature, T e :Σ e = Z ∞ επ ~ v F (cid:20) (cid:18) ε − µ e k B T e (cid:19)(cid:21) − dε, (2) m e = 1 v F Σ e Z ∞ ε π ~ v F (cid:20) ε − µ e k B T e ) (cid:21) − . (3)In the following we fix T e and treat the fictitious mass asa function of Σ e . The dielectric constant can be repre-sented as ǫ = ǫ t , < z < W t ,ǫ b , − W b < z < ,ǫ s [1 − Ω s /ω ( ω + iν s )] , z < − W b , (4)where ǫ t , ǫ b , and ǫ s are the static dielectric constants ofthe top and bottom dielectric layers and the substrate,respectively, Ω s = p πe N s /m h ǫ s is the bulk plasmafrequency in the substrate with N s and m h being thedoping concentration and hole effective mass, and ν s isthe collision frequency in the substrate, which dependson the doping concentration. The dielectric constant inthe substrate is a sum of the static dielectric constant ofSi, ǫ s = 11 . ν s ,on the doping concentration, N s , is calculated from theexperimental data for the hole mobility at room temper-ature in Ref. 27.We use the following boundary conditions: vanish-ing potential at the gate and far below the substrate, ϕ ω | z = W t = 0 and ϕ ω | z = −∞ = 0; continuity conditionsof the potential at interfaces between different layers, ϕ ω | z =+0 = ϕ ω | z = − and ϕ ω | z = − W b +0 = ϕ ω | z = − W b − ;a continuity condition of the electric flux density atthe interface between the bottom dielectric layer andthe substrate in the z -direction, ǫ b ∂ϕ ω /∂z | z = − W b +0 = ǫ s ∂ϕ ω /∂z | z = − W b − ; and a jump of the electric flux den-sity at the graphene layer, which can be derived fromEq. (1). Equation (1) together with these boundary con-ditions yield the following dispersion equation F gr ( ω ) F sub ( ω ) = A c , (5)where F gr ( ω ) = ω + iν e ω −
12 ( v F k ) − Ω gr , (6) F sub ( ω ) = ω ( ω + iν s ) − Ω sub , (7) A c = ǫ b ( H b − ǫ b H b + ǫ t H t )( ǫ s + ǫ b H b ) Ω gr Ω sub , (8)Ω gr = s πe Σ e k m e ǫ gr ( k ) , ǫ gr ( k ) = ǫ t H t + ǫ b ǫ b + ǫ s H b ǫ s + ǫ b H b , (9)Ω sub = s πe N s m h ǫ sub ( k ) , ǫ sub ( k ) = ǫ s + ǫ b ǫ b + ǫ t H t H b ǫ b H b + ǫ t H t , (10)and H b,t = coth kW b,t . In Eq. (5), the term A c on theright-hand side represents the coupling between grapheneplasmons and substrate surface plasmons. If A c werezero, the equations F gr ( ω ) = 0 and F sub ( ω ) = 0 wouldgive independent dispersion relations for the former andlatter, respectively. Qualitatively, Eq. (8) indicates thatthe coupling occurs unless kW b ≫ kW t ≪ Equation (5) yields two modes which havedominant potential distributions near the graphene chan-nel and inside the substrate, respectively. Hereafter, wefocus on the oscillating mode primarily in the graphenechannel; we call it “channel mode”, whereas we call theother mode “substrate mode”.
III. UNGATED PLASMONS
First, we study plasmons in the ungated structure.Here, the temperature, electron concentration, and col-lision frequency in graphene are fixed to T e = 300 K,Σ e = 10 cm − , and ν e = 3 × s − . With these val-ues of the temperature and concentration the fictitiousmass is equal to 0 . m , where m is the electron restmass. The value of the collision frequency is typical tothe acoustic-phonon scattering at room temperature. As for the structural parameters, we set ǫ t = 1 and W t → ∞ , and we assume an SiO bottom dielectric layerwith ǫ b = 4 .
5. Then Eq. (5) is solved numerically. (a)(b) D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -30.150.200.250.3010 F r equen cy , T H z Doping concentration, cm -3 W b = 50, 100, 200, (cid:9) (cid:9) (cid:9) (cid:9) 300, 400 nm FIG. 2. Dependences of (a) the plasmon damping rate and (b)frequency on the substrate doping concentration, N s , with theplasmon wavenumber k = 14 × cm − (the wavelength λ =4 . µ m) and with different thicknesses of the bottom dielectriclayer, W b , in the ungated graphene structure. The inset in (a)shows the damping rate in the range N s = 10 − cm − (in linear scale). Figures 2(a) and (b) show the dependences of the plas-mon damping rate and frequency on the substrate dopingconcentration with the plasmon wavenumber k = 14 × cm − (i.e., the wavelength λ = 4 . µ m) and with differ-ent thicknesses of the bottom dielectric layer, W b . Thevalue of the plasmon wavelength is chosen so that it givesthe frequency around 1 THz in the limit N s →
0. Theyclearly demonstrate that there is a huge resonant increasein the damping rate at around N s = 3 × cm − aswell as a drop of the frequency. This is the manifestationof the resonant coupling of the graphene plasmon and thesubstrate surface plasmon. The resonance corresponds tothe situation where the frequencies of graphene plasmonsand substrate surface plasmons coincide, in other words,where the exponentially decaying tail of electric field ofgraphene plasmons resonantly excite the substrate sur-face plasmons.At the resonance, the damping rate becomes largerthan 10 s − , over 10 times larger than the contribu-tion from the acoustic-phonon scattering in graphene, ν e / . × s − . For structures with W b = 50and 100 nm, even the damping rate is so large that thefrequency is dropped down to zero; this corresponds toan overdamped mode. It is seen in Figs. 2(a) and (b)that the coupling effect becomes weak as the thicknessof the bottom dielectric layer increases. The couplingstrength at the resonance is determined by the ratio ofthe electric fields at the graphene layer and at the inter-face between the bottom dielectric layer and substrate.In the case of the ungated structure with a relatively lowdoping concentration, it is roughly equal to exp( − kW b ).Since λ = 4 . µ m is much larger than the thicknesses ofthe bottom dielectric layer in the structures under consid-eration, i.e., kW b ≪
1, the damping rate and frequencyin Figs. 2(a) and (b) exhibit the rather slow dependenceson the thickness.Away from the resonance, we have several nontrivialfeatures in the concentration dependence of the damp-ing rate. On the lower side of the doping concentra-tion, the damping rate increase does not vanish until N s = 10 − cm − . This comes from the wider fieldspread of the channel mode into the substrate due tothe ineffective screening by the low-concentration holes.On the higher side, one can also see a rather broadlinewidth of the resonance with respect to the doping con-centration, owning to the large, concentration-dependentdamping rate of the substrate surface plasmons, and acontribution to the damping rate is not negligible evenwhen the doping concentration is increased two-orders-of-magnitude higher. In fact, with N s = 10 cm − ,the damping rate is still twice larger than the contribu-tion from the acoustic-phonon scattering. The inset inFig. 2(a) indicates that the doping concentration mustbe at least larger than N s = 10 cm − for the cou-pling effect to be smaller than the contribution fromthe acoustic-phonon scattering, although the latter isstill nonnegligible. It is also seen from the inset that,with very high doping concentration, the damping rateis almost insensitive to W b . This originates from thescreening by the substrate that strongly expands the fieldspread into the bottom dielectric layer.As for the dependence of the frequency, it tends to alower value in the limit N s → ∞ than that in the limit N s →
0, as seen in Figure 2(b), along with the larger de-pendence on the thickness W b . This corresponds to thetransition of the channel mode from an ungated plasmonmode to a gated plasmon mode, where the substrate ef-fectively acts as a back gate.To illustrate the coupling effect with various frequen-cies in the THz range, dependences of the plasmon damp-ing rate and frequency on the substrate doping concen-tration and plasmon wavenumber with W b = 300 nmare plotted in Figs. 3(a) and (b). In Fig. 3(a), the peakof the damping rate shifts to the higher doping concen-tration as the wavenumber increases, whereas its valuedecreases. The first feature can be understood fromthe matching condition of the wavenumber-dependentfrequency of the ungated graphene plasmons and thedoping-concentration-dependent frequency of the sub- (a)(b) s = 2 x 10 cm -3 cm -3 cm -3 FIG. 3. Dependences of (a) the plasmon damping rate and (b)frequency on the substrate doping concentration, N s , and theplasmon wavenumber, k , with different the thickness of thebottom dielectric layer W b = 300 nm in the ungated graphenestructure. The inset of (a) shows the wavenumber dependenceof the damping rate with certain doping concentrations. Theregion with the damping rate below 0 . × s − is filledwith white in (a). strate surface plasmons, i.e., Ω gr ∝ k / , roughlly speak-ing, and Ω sub ∝ N / s . The second feature originatesfrom the exponential decay factor, exp( − kW b ), of theelectric field of the channel mode at the interface betweenthe bottom dielectric layer and the substrate; since thedoping concentration is . cm − at the resonance forany wavevector in Fig. 3, the exponential decay is valid.Also, with a fixed doping concentration, say N a > cm − , the damping rate has a maximum at a certainwavenumber, resulting from the first feature (see the in-set in Fig. 3(a)). IV. GATED PLASMONS
Next, we study plasmons in the gated structures.We consider the same electron concentration, ficticiousmass, and collision frequency, Σ e = 10 cm − , m e =0 . m , and ν e = 3 × s − , as the previous sec-tion. As examples of materials for top/bottom dielectric (a)(b) D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 F r equen cy , T H z Doping concentration, cm -3 W b = 50, 100, 200, (cid:9) (cid:9) (cid:9) (cid:9) 300, 400 nm0.150.200.250.300.3510 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 F r equen cy , T H z Doping concentration, cm -3 W b = 50, 100, 200, (cid:9) (cid:9) (cid:9) (cid:9) 300, 400 nm FIG. 4. Dependences of (a) the plasmon damping rate and(b) frequency on the substrate doping concentration, N s , withthe plasmon wavelength λ = 1 . µ m (the wavenumber k =37 × cm − ), with thicknesses of the Al O top dielectriclayer W t = 20 and 40 nm (left and right panels, respectively),and with different thicknesses of the SiO bottom dielectriclayer, W b , in the gated graphene structure. The insets in (a)show the damping rate in the range N s = 10 − cm − (in linear scale).FIG. 5. Dependence of the plasmon damping rate on the sub-strate doping concentration, N s , and the plasmon wavenum-ber, k , with different the thicknesses of the Al O top dielec-tric layer W t = 20 nm and the SiO bottom dielectric layer W b = 50 nm in the gated graphene structure. The region withthe damping rate below 0 . × s − is filled with white. layers, we examine Al O /SiO and diamond-like carbon(DLC)/3C-SiC. These materials choices not only reflectthe realistic combination of dielectric materials availabletoday, but also demonstrate two distinct situations forthe coupling effect under consideration, where ǫ t > ǫ b forthe former and ǫ t < ǫ b for the latter. Figures 4(a) and (b) show the dependences of the plas-mon damping rate and frequency on the substrate dopingconcentration with the wavenumber k = 37 × cm − (the plasmon wavelength λ = 1 . µ m), with thicknessesof the Al O top dielectric layer W t = 20 and 40 nm, andwith different thicknesses of the SiO bottom dielectriclayer, W b . As seen, the resonant peaks in the dampingrate as well as the frequency drop due to the couplingeffect appear, although the peak values are substantiallysmaller than those in the ungated structure (cf. Fig. 2).The peak value decreases rapidly as the thickness of thebottom dielectric layer increases; it almost vanishes when W b ≥
300 nm. These reflect the fact that in the gatedstructure the electric field of the channel mode is confineddominantly in the top dielectric layer due to the gatescreening effect. The field only weakly spreads into thebottom dielectric layer, where its characteristic length isroughtly proportional to W t , rather than the wavelength λ as in the ungated structure. Thus, the coupling effecton the damping rate together with on the frequency van-ishes quickly as W b increases, even when the wavenumberis small and kW b ≪
1. More quantitatively, the effect isnegligible when the first factor of A c given in Eq. (8) inthe limit kW b ≪ kW t ≪ ǫ b ( H b − ǫ b H b + ǫ t H t )( ǫ s + ǫ b H b ) ≃
11 + ( W b /ǫ b ) / ( W t /ǫ t ) (11)is small, i.e., when the factor ( W b /ǫ b ) / ( W t /ǫ t ) is muchlarger than unity. A rather strong dependence of thedamping rate on W b can be also seen with high dopingconcentration, in the insets of Fig. (4)(a).Figure 5 shows the dependence of the plasmon damp-ing rate on the substrate doping concentration and plas-mon wavenumber, with dielectric layer thicknesses W t =20 and W b = 50 nm. As compared with the case ofthe ungated structure (Fig. 3(a)), the peak of the damp-ing rate exhibits a different wavenumber dependence;it shows a broad maximum at a certain wavenumber(around 150 × cm − in Fig. 5) unlike the case ofthe ungated structure, where the resonant peak decreasesmonotonically as increasing the wavenumber. This canbe explained by the screening effect of the substrateagainst that of the top gate. When the wavenumber issmall and the doping concentration corresponding to theresonance is low, the field created by the channel modeis mainly screened by the gate and the field is weaklyspread into the bottom direction. As the doping concen-tration increases (with increase in the wavevector whichgives the resonance), the substrate begins to act as a backgate and the field spreads more into the bottom dielectriclayer, so that the coupling effect becomes stronger. Whenthe wavenumber becomes so large that kW b ≪ − kW b ). D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 F r equen cy , T H z Doping concentration, cm -3 W b = 50, 100, 200, (cid:9) (cid:9) (cid:9) (cid:9) 300, 400 nm0.00.20.40.60.81.01.210 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 D a m p i ng r a t e , s - Doping concentration, cm -3 F r equen cy , T H z Doping concentration, cm -3 W b = 50, 100, 200, (cid:9) (cid:9) (cid:9) (cid:9) 300, 400 nm (a)(b) FIG. 6. The same as Figs. 4(a) and (b) but with the DLCtop and 3C-SiC bottom dielectric layer.
As illustrated in Eq. (11), the coupling effect inthe gated strcture is characterized by the factor( W b /ǫ b ) / ( W t /ǫ t ) when the conditions kW b ≪ kW t ≪ in the top dielectric layer, it results inthe more effective gate screening than in the gated struc-ture with the Al O top dielectric layer, so that the cou-pling effect can be suppressed even with the same layerthicknesses. The structure with the DLC top and 3C-SiCbottom dielectric layers (with ǫ t = 3 . and ǫ b = 9 . O top and SiO bottom dielectric layers,the damping rate as well as the frequency are more in-fluenced by the coupling effect in the entire ranges of thedoping concentration and wavevector. In particular, theincrease in the damping rate with high doping concen-tration N s = 10 − cm − and the thickness of thebottom layer W b = 50 −
100 nm, which are typical val-ues in real graphene samples, is much larger. However,this increase can be avoided by adapting thicker bottomlayer, say, W b &
200 nm or by increasing the dopingconcentration.
FIG. 7. The same as Fig. 5 but with the DLC top and 3C-SiCbottom dielectric layer.
V. CONCLUSIONS
In summary, we studied theoretically the coupling ofplasmons in graphene at THz frequencies with surfaceplasmons in a heavily-doped substrate. We demonstratedthat in the ungated graphene structure there is a hugeresonant increase in the damping rate of the “channelmode” at a certain doping concentration of the substrate( ∼ cm − ) and the increase can be more than 10 s − , due to the resonant coupling of the graphene plas-mon and the substrate surface plasmon. The depen-dences of the damping rate on the doping concentration,the thickness of the bottom dielectric layer, and the plas-mon wavenumber are associated with the field spread ofthe channel mode into the bottom dielectric layer andinto the substrate. We revealed that even with very highdoping concentration (10 − cm − ), away from theresonance, the coupling effect causes nonnegligible in-crease in the damping rate compared with the acoustic-phonon-limited damping rate. In the gated graphenestructure, the coupling effect can be much reduced com-pared with that in the ungated structure, reflecting thefact that the field is confined dominantly in the top di-electric layer due to the gate screening. However, withvery high doping concentration, it was shown that thescreening by the substrate effectively spreads the fieldinto the bottom dielectric layer and the increase in thedamping rate can be nonnegligible. These results suggestthat the structural parameters such as the thicknessesand dielectric constants of the top and bottom dielectriclayers must be properly chosen for the THz plasmonicdevices in order to reduce the coupling effect. ACKNOWLEDGMENTS
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Plasmonics: Fundamentals and Applications (Springer Science, NY, 2007). The dielectric constant of DLC varies in the range between3 . .
8, depending on its growth condition. Here, wechoose the lowest value for demonstration of the case where ǫ t ≪ ǫ b .30