Graphene-based plasmonic switches at near infrared frequencies
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Graphene-based plasmonic switches atnear infrared frequencies
J. S. Gómez-Díaz and J. Perruisseau-Carrier
Adaptive MicroNanoWave Systems, LEMA/NanolabÉcole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerlandjuan-sebastian.gomez@epfl.ch, julien.perruisseau-carrier@epfl.ch
Abstract:
The concept, analysis, and design of series switches forgraphene-strip plasmonic waveguides at near infrared frequencies arepresented. Switching is achieved by using graphene’s field effect toselectively enable or forbid propagation on a section of the graphenestrip waveguide, thereby allowing good transmission or high iso-lation, respectively. The electromagnetic modeling of the proposedstructure is performed using full-wave simulations and a transmissionline model combined with a matrix-transfer approach, which takesinto account the characteristics of the plasmons supported by thedifferent graphene-strip waveguide sections of the device. Theperformance of the switch is evaluated versus different parametersof the structure, including surrounding dielectric media, electrostaticgating and waveguide dimensions.
OCIS codes: (240.6680) Surface plasmons; (130.2790) Guided waves; (250.6715)Switching
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1. Introduction
The field of plasmonics represents a new exciting area for the control of electromag-netic waves at scales much smaller than the wavelength. It is based on the propagationof surface plasmon polaritons (SPPs) [1–3], which are electromagnetic waves propagat-ing along the surface interface between a metal (or a semiconductor) and a dielectric.SPPs are typically obtained in the visible range by using noble metal such as gold orsilver [2], but they are also supported at lower frequencies by composite materials [1,4].Surface plasmons have served as a basis for the development of nanophotonic devices[5], merging the fields of photonics and electronics at the nanoscale [2] and findingapplication in different areas such as imaging [6] or sensing [7].Graphene [8] provides excellent possibilities to dynamically manipulate electromag-netic waves [9–11]. Its unique electric properties, which can be controlled by simplyapplying an external magnetostatic or electrostatic field, allows the propagation of sur-face plasmons in terahertz and infra-red frequency bands [12]. Compared to conven-tional materials, such as silver or gold, SPPs on graphene present important advantages[13] including tunability, low-losses, and extreme mode confinement. Several authorshave studied the characteristics of plasmons propagating along 2D graphene sheets[14–17] and ribbons/strips [18–20], and different configurations have already been pro-posed to enhance their guiding properties [21].The ability to allow or to forbid the propagation of SPPs on these structures is a keybuilding block for future plasmonic-based devices. Graphene-based switches have al-ready been proposed in the literature at DC [22,23] and microwave frequencies [24,25],based either on graphene electric field effect or exploiting the electromechanical prop-erties of graphene. In the optical regime, [26] proposed a structure able to switch thereflectance of a plane wave incoming from free-space between two different states,namely total reflection and total absorption. This is obviously a different functionalityrom the switching of a guided plasmonic wave as concerned here. Finally, in a re-cent work [27], graphene-based longitudinally homogeneous parallel-plate waveguideswere proposed to obtain switches and phase-shifter in the low terahertz band.In this context, we propose and design series switches able to control the propa-gation of surface plasmons on finite graphene strips at near infrared frequencies. Thestructures are composed of a chemically-doped graphene strip, host waveguide of theswitch, transferred onto a dielectric and of three polysilicon gating pads beneath thestrip. The switch consists of the central section of the host waveguide, whereas theouter sections connect the switch to the input and output ports of the device. Switchingis achieved by modifying the gate voltage of the central pad, which in turns controlsthe guiding properties of the strip in the area of the switch. In the ON state, the wholehost waveguide has the same propagating characteristics and the structure behaves as asimple plasmonic transmission line (TL) propagating the incoming energy towards theoutput port. In the OFF state, the guiding properties of the central waveguide sectionare modified to provide large isolation between input and output ports. The structuresare characterized by applying a transmission line approach and by using the commer-cial full-wave software HFSS. Note that [27] recently studied graphene-based longi-tudinally inhomogeneous parallel-plate waveguide using solely TL techniques. Here,we present a rigorous comparison between the TL approach and a full-wave solverfor the case of finite graphene-strip waveguides, demonstrating that while the formerprovides extremely fast results and physical insight into the structure, its mono-modalnature may lead to inaccuracies when characterizing the OFF state of the device. Fi-nally, several devices, based on ideal 2D graphene surfaces and on realistic finite strips,are discussed and studied, evaluating their performance versus different parameters ofthe switches.
2. Implementation and modeling
This section details the concept, implementation and modeling of the proposedgraphene-based switches. We first briefly review the characteristics of surface plas-mons polaritons propagating on ideal 2D graphene surfaces [15] and on finite strips[18, 20]. Then, we describe the proposed switches, detailing their underlying operatingprinciple and discussing its potential technological implementation. Finally, we ad-dress the modeling of the different structures, using both a TL approach and full-wavecommercial software.
Graphene is a one atom thick gapless semiconductor [8,28], which can be characterizedby a complex surface conductivity s . This conductivity can be modelled using the well-known Kubo formalism [29], and mainly depends on graphene chemical potential m c ,which may be controlled by modifying the initial doping of the material or by applyingn external electrostatic field, and on the phenomenological scattering rate G : s ( w , m c , G , T ) = jq e ( w − j G ) p ¯ h (cid:20) ( w − j G ) Z ¥ e (cid:18) ¶ f d ( e ) ¶e − ¶ f d ( − e ) ¶e (cid:19) ¶e − Z ¥ f d ( − e ) − f d ( e )( w − j G ) − ( e / ¯ h ) ¶e (cid:21) , (1)where w is the radial frequency, e is energy, T is temperature, − q e is the charge ofan electron, ¯ h is reduced Planck’s constant, k B is Boltzmann’s constant, and f d is theFermi-Dirac distribution: f d ( e ) = (cid:16) e ( e −| m c | ) / k B T + (cid:17) − . (2)Note that the first and second terms of Eq. (1) are related to intraband and interbandcontributions of graphene conductivity, respectively [29]. The real part of conductivityis sensitive to G at the frequencies where intraband contributions of graphene dominate(i.e. ¯ h w / m c ≪
1) while it mainly depends on temperate in the frequency region whereinterband contributions dominate (i.e. ¯ h w / m c ≥ −
30 THz,where interband contributions of conductivity are non-negligible) and for the parame-ters employed in this work, the influence of this phenomena on graphene conductivityis small [15], and the approximate model of graphene conductivity of Eq. (1) leads toaccurate results.One of the main advantages of graphene is that its chemical potential can be tunedover a wide range (typically from − V DC ) modifies graphene carrier density ( n s ) as C ox V DC = q e n s , (3)where C ox = e r e / t is the gate capacitance, and e r and t are the permittivity constantand thickness of the gate dielectric. Note that Eq. (3) neglects graphene quantum ca-pacitance [32], which may be important in case of extremely thin dielectrics ( t ∼ nms)or very high dielectric constants. In addition, the carrier density is related with m c viathe expression n s = p ¯ h v f Z ¥ e [ f d ( e − m c ) − f d ( e + m c )] ¶e , (4)where v f is the Fermi velocity ( ∼ cm/s in graphene). Then, the desired chemicalpotential m c is accurately retrieved by numerically solving Eqs. (3) and (4). An approx-imate closed-form expression to relate m c and V DC is given by m c ≈ ¯ hv f s p C ox V DC q , (5)hich is obtained by combining Eq. (3) with the graphene carrier density at zero tem-perature [33] n s = p (cid:18) m c ¯ hv f (cid:19) . (6)The characteristics of SPPs supported by graphene depend on the conductivity ofthe material and on the type of waveguide employed. In the case of ideal 2D graphenesheets, the dispersion relation of the propagating modes can be obtained as [15] we r e q e r k − k r − we r e q e r k − k r = − s , (7)where e is the vacuum permittivity, e r and e r are the dielectric permittivities of themedia surrounding graphene, k = w / c is the free space wavenumber and k r = b − j a is the complex propagation constant of the SPP mode, being b and a the phase andatenuation constants, respectively. In addition, the characteristic impedance of the SPPmay be expressed as [34] Z C = k r we e eff , (8)where e eff is the effective permitivitty constant of the surrounding medium. Note thatthough the characteristic impedance is not often employed to model SPPs [2], thisparameter will be useful to understand and optimize the behavior of the proposedswitches.In the case of finite graphene strips, the dispersion relation of propagating surfaceplasmons cannot be derived analytically and one has to resort to purely numericallyfull-wave solvers. There are two different types of SPP propagating along the strip[18, 20]: the waveguide type, which has the field concentrated along the whole strip,and the edge type, where the field is focused on the rims of the strip. Note that graphenerelaxation time mainly control the propagation length of the modes, barely affecting totheir field confinement. In addition, the characteristics of these modes can also be tunedby modifying the chemical potential of graphene.Let us consider, for the sake of illustration, a graphene sheet transferred on a di-electric with permittivity e r =
4. The parameters of graphene are t = / ( G ) = . T =
300 K, in agreement with measured data [35]. The characteristics of a SPPpropagating on the sheet are shown in Fig. 1 for different values of graphene chem-ical potential. We find that the propagation constant and characteristic impedance ofthe SPP mode can be tuned over a large range by varying m c . Focusing for instance inthe range between 25 and 30 THz, the structure does not support the propagation ofSPP when m c = . ( s ) >
0, as demonstrated in [14]), it can supportSPP propagating with large amount of losses (for instance with m c = . m c = . Frequency (THz) R e [ k r / k ] m c =0.50eV m c =0.25eV m c =0.17eV m c =0.12eV m c =0.10eV m c =0.00eV (a)
10 20 30 40 50 60020406080100
Frequency (THz) I m [ k r / k ] (b)
10 20 30 40 50 6001234567x 10 Frequency (THz) R e [ Z C ] ( W ) (c)
10 20 30 40 50 6001000200030004000500060007000
Frequency (THz) I m [ Z C ] ( W ) (d) Fig. 1: Normalized dispersion relation (a), attenuation constant (b), and real and imag-inary components (c-d) of the characteristic impedance of a SPP wave propagating onan air-graphene-dielectric interface versus graphene chemical potential m c computedusing Eqs. (7) and (8). The dielectric permittivity is e r = . T =
300 K and t = . The proposed graphene-based switches are shown in Fig. 2 and in Fig. 3. The electricfields of the structures are computed using the commercial software HFSS, as detailedin the next section. The devices are composed of a host graphene waveguide, namely a2D sheet in Fig. 2 and a finite strip in Fig. 3, transferred on a dielectric (with e r ) and ofthree polysilicon gating pads beneath the waveguide. The permittivity of the supportingsubstrate is also set to e r . The switch is located in the inner section of the waveguide,above the central pad, and the outer sections of the waveguide connect the switch to theinput and output ports of the device. The characteristics of the SPP propagating on theswitch and on the other sections of the waveguides are controlled by the DC voltageapplied to the gating pads. Specifically, the outer pads are biased with a voltage V out ,which provides to the graphene area located above a high chemical potential, whereasthe central pad is biased with a voltage V in . The operation principle of the switch is asfollows. In the ON state [see Fig. 2(a) and Fig. 3(a)], the DC voltages V out and V in arechosen to provide the same chemical potential to the whole graphene waveguide. In thisstate, the device behaves as a simple transmission line able to propagate an incoming a) (b) Fig. 2: Proposed graphene-based 2D sheet plasmonic switch. The device comprises amonolayer graphene sheet transferred onto a dielectric ( e r ) and three polysilicon gatingpads placed at a distance t below the sheet. The permittivity of the supporting substrateis also set to e r . The guiding properties of the SPP propagating along the sheet arecontrolled via the electric field effect by the DC bias applied to the gating pads. (a)Switch ON. Simulated results showing the z component of the electric field, E z , ofa SPP wave propagating along the sheet. The central and outer pads are biased withvoltages V out and V in , chosen to provide the same chemical potential ( m c = . V in is chosen toprovide a chemical potential of m c = . e r = . L =
350 nm, ℓ in =
50 nm, t =
20 nm, T =
300 K, t = . V in is modified to provide a much lower value to the chemical potential ofthe graphene located in the switch area. In this state, an incoming wave propagating onthe waveguide finds a strong discontinuity due to the different characteristic impedanceand propagation constant of the central section of the line. Specifically, there is a wavereflected back to the input port due to the large mismatch between the different regionsof the waveguide, some energy is dissipated in the switch due to the large losses thatnow arise, and a highly attenuated wave is transmitted towards the output port. Theperformance of the switch is determined by the isolation between the input and outputports at the OFF state and by the insertion loss in the ON state. It depends on thelength of the switch ( ℓ in ) and on the range of chemical potential values achievable bygraphene, as will be discussed in detail in Section 3.We propose two different alternatives, illustrated in Fig. 4, for the technological im-plementation of the switches. In both cases, the ON state is obtained by providing highchemical potential to the whole graphene surface while in the OFF state the chem-ical potential of the central section is highly reduced. The first approach, shown inFig. 4(a), uses uniformly highly doped graphene sheets/strips. Note that recent fabrica- a) (b) Fig. 3: Proposed graphene-based strip plasmonic switch. The device is similar to theswitch shown in Fig. 2, but here the graphene sheet is replaced by a strip of width W .(a) Switch ON. Simulated results showing the z component of the electric field, E z , of aSPP wave propagating along the strip. The voltages V out and V in are chosen to providethe same chemical potential ( m c = . V in is chosen to provide a chemical potential of m c = . e r = . L =
350 nm, W =
150 nm, ℓ in =
50 nm, t =
20 nm, T =
300 K, t = . . V out = V in = V L ) to the gatingpads, which provide to the whole graphene area the required chemical potential. Onthe other hand, the OFF state is obtained by applying a negative voltage ( − V H ) to thecentral gating pad. Indeed, due to the ambipolarity property of graphene [8, 40], an ap-plied negative DC voltage decreases the chemical potential of graphene. Therefore, thecentral and outer gating pads are biased with voltages V L and − V H , respectively. Thesecond approach [see Fig. 4(b)] relays on tailoring the chemical doping of the differ-ent graphene regions. In this way, the outer graphene surfaces are highly chemicallydoped, whereas the inner surface is slightly doped. The ON state is obtained provid-ing a low DC bias voltage to the outer gating pads ( V out = V L ) and a larger bias to thecentral one ( V in = V H ), while in the OFF state the voltage applied to the central pad isreduced ( V in = V L ). Note that this second approach requires a more complicated fab-rication process due to the tailoring of the chemical doping applied to the graphenearea. The electromagnetic modeling of the proposed graphene-based switches is performedusing two different techniques, namely a TL formalism combined with an ABCD a) (b)
Fig. 4: Cross section of the proposed switch and chemical potential profile along the‘x’ axis of the graphene area for the ON and OFF states of the device. The differentcontributions to the chemical potential of graphene (solid line), namely chemical dop-ing (dotted line) and elecrostatic DC bias (dashed line), are also shown. (a) Uniformlyhighly chemically doped graphene. The OFF state is obtained by applying a negativeDC bias to the central gating pad. (b) Non-uniformly chemically doped graphene. Outerand inner surfaces of graphene are highly and slightly chemically doped, respectively.The ON state is obtained by applying a positive DC bias to the central gating pad.transfer-matrix approach [34] and a commercial full-wave software (HFSS) [41] basedon the finite element method.The proposed graphene-based waveguide switch can be modeled by cascading threetransmission lines, as shown in Fig. 5. Each TL corresponds to a section of the hostwaveguide. The outer lines, characterized by a propagation constant g out and charac-teristic impedance Z out , are related to the outer sections of the device. In addition, theswitch (inner region of the waveguide) is modeled by the central TL, which presenta propagation constant and characteristic impedance of g in and Z in , respectively. Thedifferent lines are combined using an ABCD matrix-transfer approach, and the corre-sponding scattering parameters are then recovered using standard techniques [34]. Forsimplicity, scattering parameters are refereed to the port impedance Z P which is setequal to the characteristic impedance of the outer waveguide sections, Z out . However, itis important to point out that this method is approximate. It is well-known from trans-mission line theory [34, 42] that this approach only considers the fundamental modethat propagates along the waveguide, and neglects the presence of higher order modesthat are excited at the discontinuity between different sections. Moreover, note that theexcitation and influence of these higher order modes increase when i) the guiding char-acteristics of the different waveguide sections are very different, and ii) the length ofig. 5: Equivalent transmission line model of the proposed graphene-based switchesshown in Fig. 2 and in Fig. 3.one of these sections is very small [42]. Nevertheless, this approximate approach isindeed useful: it provides extremely fast preliminary results and physical insight intothe operation principle of the proposed switches.In the case of the graphene-based 2D sheet switch shown in Fig. 2, the propagat-ing constant and characteristic impedance of the equivalent transmission lines are di-rectly obtained using Eqs. (7) and (8), respectively. In the case of graphene-based stripswitches, depicted in Fig. 3, no analytical formulas are available. Here, we employfull-wave simulations to extract the propagation constant of SPP propagating on var-ious infinitely-long graphene strips. Then, the extracted wavenumbers are included inthe TL formalism to characterize the complete switch.The second approach is based on simulating the complete switch in a full-wave sim-ulator. To this purpose, we have employed the commercial package HFSS [41], whichimplements the finite element method (FEM) [43, 44]. There, graphene surfaces aremodeled as infinitesimally thin sheets/strips, where surface impedance boundary con-ditions ( Z Su f = / s ) are imposed. We have verified that 3D numerical FEM simu-lations are able to accurately model plasmon propagation on graphene waveguides.However, we have observed some discrepancies between the plasmons characteristics-namely propagation constant and characteristic impedance- obtained by the 2D nu-merical waveport solutions and the expected ones. These discrepancies may arise dueto numerical instabilities related to the large surface reactance of the infinitesimallythin graphene. Consequently, though we are able to accurately compute the scatte-ring parameters of the structure under analysis, they are referred to an unknown portimpedance ( Z P ). In order to solve this issue, we employ a simpler structure with exactlythe same waveport configuration to rigorously extract the port impedance. This is doneby simulating a uniform graphene-based sheet/strip waveguide with known propaga-tion constant and characteristic impedance (see Eq. (7) and [18]). Then, Z P is analyt-ically retrieved from the resulting scattering parameters applying standard techniques[34, 42]. Finally, this impedance is employed to renormalize the scattering parametersof the initial complex structure to any desired impedance, Z P = Z out in our case. . Numerical results In this section, we investigate the characteristics of the proposed switches in termsof their scattering parameters. These parameters are commonly employed in the mi-crowaves and terahertz frequency ranges [34] and are perfectly suited to evaluate thebehavior of a switch. Note that an ideal switch in its ON state propagates all input en-ergy towards the output port (i.e. S ≈ S ≈ . T = ◦ K (room tempera-ture). For simplicity, we neglect the possible fluctuations of graphene relaxation timedue to optical phonons [15].In order to assess the accuracy of the numerical methods employed to studygraphene-based switches, we first consider the propagation of surface plasmons onthe structure shown in Fig. 2. The parameters of the structure are e r = L = m mand ℓ in = m m, and the chemical potential of the central and outer graphene waveg-uide sections is set to 0 . .
15 eV, respectively. The scattering parameters ofthe structure, computed using the TL approach and HFSS, are shown in Fig. 6. Verygood agreement is found between the methods, verifying that the propagation of sur-face plasmons on the graphene waveguide is correctly modeled. Note that the accurateresults provided by the TL approach are due to the negligible influence of higher ordermodes in this structure, which are barely excited in the weak discontinuity between thedifferent waveguide sections.Fig. 7 presents the scattering parameters, obtained with the commercial softwareHFSS, of the proposed graphene-based switches suspended in free-space (see Fig. 2and Fig. 3, with e r = L = . m m and ℓ in = . m m. In the ON state, a DC bias voltage is applied toall gating pads to provide a chemical potential of m c = . . S ≈ S , whichindicates that the device is very well matched, has been obtained by renormalizing thescattering parameters to the characteristic impedance of the outer waveguide sections( Z out , see Section 2.3). This state also provides some dissipation losses, about 4 dB inboth switches, which are due to the intrinsic characteristics of graphene and directlydepends on the total length of the device. In the OFF state, the graphene-based 2D Frequency (THz) S c a tt e r i ng P a r a m e t e r s ( d B ) S HFSS S TL model S HFSS S TL model S S Fig. 6: Scattering parameters of the structure shown in Fig. 2, with e r = L = m mand ℓ in = m m, computed using the transmission line approach and the commercialsoftware HFSS. The chemical potential of the outer and central graphene waveguidesections are set to m c out = . m c in = .
15 eV.
25 26 27 28 29 30−500−25 Frequency (THz) S c a tt e r i ng P a r a m e t e r s ( d B ) S11 Switch OFFS21 Switch OFFS21 Switch ONS11 Switch ON25 26 27 28 29 30−350−300−250 S c a tt e r i ng P a r a m e t e r s ( d B ) (a)
25 26 27 28 29 30−25−500 Frequency (THz) S c a tt e r i ng P a r a m e t e r s ( d B ) S21 Switch OFFS11 Switch OFFS21 Switch ONS11 Switch ON25 26 27 28 29 30−350−300−250 S c a tt e r i ng P a r a m e t e r s ( d B ) (b) Fig. 7: Simulated scattering parameters of the proposed graphene-based switches, sus-pended in free-space, at their ON and OFF states. The parameters of the device are L = . m m and ℓ in = . m m. (a) Graphene-based 2D sheet switch, see Fig. 2. (b)Graphene-based strip switch with W = . m m, see Fig. 3. Frequency (THz) P o w e r ( d B W ) Input PowerP
OFFT
25 26 27 28 29 30−350−250−150−50−30 P o w e r ( d B W ) P OFFD P ONR P OFFR P OND P ONT
Fig. 8: Power transmitted, reflected, and dissipated in the graphene-based strip plas-monic switch shown in Fig. 7(b). The superscripts ON and OFF are related to the oper-ation state of the switch, and the subscripts T , R , and D refer to the power transmittedtowards the output port, reflected into the input port, and dissipated in the structure,respectively.surface and strip switches provide large isolation in the whole frequency band, around37 and 50 dB respectively. Importantly, the use of realistic graphene strips, instead ofideal 2D sheets, leads to devices with higher isolation levels. This is due to the fieldconfinement of surface plasmons propagating on strips, which is much larger than incase of 2D sheets [18, 20].In order to investigate the different power waves flowing along the proposedswitches, we present in Fig. 8 the power transmitted, reflected and dissipated in thegraphene-based strip plasmonic switch of Fig. 7(b) when the device is fed by a 1W-power (0 dBW) input wave. These quantities can be computed using the the relation-ship between scattering parameters and power waves [34] P T = | S | , (9) P R = | S | , (10) P D = − | S | − | S | , (11)where P T , P R , and P D are related to the total power transmitted towards the output port,reflected into the input port, and dissipated in the structure, respectively. In the ONstate all energy propagates into the structure and there are not reflected waves thanksto the good matching of the device. Besides, although there are important dissipationlosses, around − −
00 200 300 400 500−60−50−40−30−20−100 Length (nm) S ( d B ) HFSSTL approach m c =0.25 eV m c =0.125 eV m c =0.175 eV m c =0.10 eV (a)
100 200 300 400 500−60−50−40−30−20−100 Length (nm) S ( d B ) HFSSTL approach m c =0.175 eV m c =0.25 eV m c =0.125 eV m c =0.10 eV (b) Fig. 9: Parametric study of the isolation ( S ) provided by the proposed graphene-basedswitches as a function of the length ( ℓ in ) and chemical potential ( m c in ) of their centralwaveguide section at the fixed frequency of 28 THz. The length of the devices ( L = . m m) is kept constant in all cases.(a) Graphene-based 2D sheet switch, see Fig. 2.(b) Graphene-based strip switch with W = . m m, see Fig. 3.identify the reflected waves which arise due to the different characteristic impedanceof the waveguide sections as the main mechanism which provides the high isolation ofthe switches.In order to further study the isolation performance of the proposed switches, Fig. 9reports a parametric study of S as a function of the length and chemical potentialof the central waveguide section, at the fixed frequency of 28 THz. The length of thedevices ( L = . m m) is kept constant in all cases. Note that the S parameter allowsto evaluate the overall performance of the switches, implicitly providing informationabout the losses and power reflected back to the input port [see Eqs. (9)-(11)]. The re-sults have been obtained using both the TL approach (dashed line) and the full-wavesolver HFSS (solid line), and are shown in Fig. 9(a) and Fig. 9(b) for ideal 2D sheet andstrip-based graphene switches, respectively. The length of the inner (switch) waveguidesection ℓ in is swept from 20 to 500 nm, avoiding values below 20 nm where quantumeffects may be non-negligible [20]. A standing wave appears within the structure and itsinterference pattern varies versus ℓ in thus explaining the oscillatory behavior of the S ℓ in .As explained in Section 2.2, when the chemical potential of the inner graphene waveg-uide is different from the one of the outer sections of the device, the energy propa-gating along the structure finds a discontinuity due to the different impedances andpropagation characteristics of the plasmon modes supported by each region. Specifi-cally, decreasing the chemical potential of the central graphene waveguide from 0 . .
175 eV leads to switches with low isolation levels. This is because the plasmonimpedance is “weakly ” affected by the chemical potential in this range (see Fig. 1).However, decreasing m c to lower values such as 0 .
125 eV or 0 . S c a tt e r i ng P a r a m e t e r s ( d B ) S11 Switch OFFS21 Switch OFFS21 Switch ONS11 Switch ON25 26 27 28 29 30−350−300−250 S c a tt e r i ng P a r a m e t e r s ( d B ) (a)
25 26 27 28 29 30−50−250 Frequency (THz) S c a tt e r i ng P a r a m e t e r s ( d B )
25 26 27 28 29 30−350−300−250 S c a tt e r i ng P a r a m e t e r s ( d B ) S11 Switch OFFS21 Switch ONS21 Switch OFFS11 Switch ON (b)
Fig. 10: Simulated scattering parameters of the proposed graphene-based switches attheir states ON and OFF. The parameters of the structure are e r = . L = . m m and ℓ in = . m m. (a) Graphene-based 2D sheet switch, see Fig. 2. (b) Graphene-based stripswitch with W = . m m, see Fig. 3.isolation levels for almost any length of the inner waveguide section. Besides, it isobserved that the isolation level converges when the chemical potential is further de-creased. This behavior suggests that in the OFF state isolation is not governed by thefundamental plasmonic mode, but by higher order evanescent modes excited at the dis-continuities between the different waveguide sections. Consequently, the transmissionline approach -which by definition only consider the fundamental mode- is not able toprovide accurate results in this state. However, these effects are obviously accountedfor by full-wave simulations Thus, in the case of high isolation, the TL modes slightlyoverestimate the performance of the switch. In general, this parametric study demon-strates the excellent capabilities of graphene as a material for developing plasmonic-based switches, allowing isolation levels better than 40 dB using graphene sections ofabout 500 nm.The last part of our study deals with realistic graphene-based switches taking intoaccount the dielectric surrounding media. To this purpose, we consider graphene trans-ferred onto a parylene of thickness t =
90 nm and dielectric constant e r = . e r = .
9. Note that the presence of the dielectric increases the localization of the sup-ported plasmon modes, i.e. decreases their propagation length and increases their modeconfinement. The lengths of the different waveguide sections are redesigned to preservesimilar insertion losses of the complete structure as in the previous example. Specifi-cally, values of L = . m m and ℓ in = . m m are employed. Figures 10(a)-10(b) reportthe scattering parameters of the proposed graphene-based 2D sheet and strip switches,respectively. The results show similar insertion losses in the ON state (around 4 dB) ascompared with the free-space suspended devices of Fig. 7. In the OFF state, the iso-lation levels are around 30 and 40 dB for the 2D sheet and strip switches. Note thatthese values are around 10 dB lower than in the previous example, which is mainly
00 200 300 400 500−90−80−70−60−50−40−30−20−100 Length (nm) S ( d B ) HFSSTL approach m c =0.175 eV m c =0.25 eV m c =0.10 eV m c =0.125 eV (a)
100 200 300 400 500−90−80−70−60−50−40−30−20−100 Length (nm) S ( d B ) HFSSTL approach m c =0.175 eV m c =0.10 eV m c =0.125 eV m c =0.25 eV (b) Fig. 11: Parametric study of the isolation ( S ) provided by the proposed graphene-based switches as a function of the length ( ℓ in ) and chemical potential ( m c in ) of theircentral waveguide section at the fixed frequency of 28 THz. The length of the devices( L = . m m) is kept constant in all cases. The dielectric permittivity is set to e r = . W = . m m, see Fig. 3.attributed to the shorter length of the central waveguide section. In addition, Fig. 11presents a study of the switches isolation at 28 THz as a function of ℓ in and m c . Re-sults demonstrate that the influence of these parameters on the switches performanceis similar as in the case of Fig. 9, where the switches are suspended in free-space. Im-portantly, the presence of the dielectric allow to increase the isolation levels of theswitches. For instance, isolation levels larger than 80 dB are obtained for a graphene-strip based switch with a central waveguide section of ℓ in =
500 nm. This is becausethe dielectric allows the propagation of extremely localized plasmons on graphene,whose characteristics present a wider tunable range than those of SPPs propagating ongraphene transferred on a dielectric with lower permittivity. Consequently, enhancedisolation levels (or reduced device dimensions) can be obtained by designing grapheneswitches on dielectrics with high permittivities.
4. Conclusions
We have proposed and designed series switches able to dynamically control the prop-agation of plasmons on graphene surfaces at near infrared frequencies. Several con-figurations, based on 2D graphene surfaces and strips, have been analyzed and theirperformance have been evaluating versus different parameters of the structures. Twodifferent techniques, namely a transmission line approach and full-wave simulations,have been employed to characterize the switches. It has been shown that the formermethod provides fast results and physical insight into the problem, but it lacks of accu-racy to characterize the OFF state of the devices.Our results have demonstrated that controlling the properties of very reducedraphene areas provides extremely large isolation levels between the input and out-put ports. For example, isolation levels larger that 80 dB have been achieved by us-ing a graphene strip of just 500 nm. In addition, it has been shown that increasingthe permittivity of the surrounding media allows to increase the isolation level of theswitches. These interesting features can be used to further develop guided graphene-based devices at near infrared frequencies, leading to functionalities similar to currentnanophotonic plasmonic-based devices at optics.