Mechanism for graphene-based optoelectronic switches by tuning surface plasmon-polaritons in monolayer graphene

0.43 MB·English

by  Yu. V. Bludov

epl draft
Mechanism for graphene-based optoelectronic switches by
tuning surface plasmon-polaritons in monolayer graphene
YU. V.BLUDOV,M.I.VASILEVSKIY,ANDN.M.R.PERES
1 CentrodeF´ısicaeDepartamentodeF´ısica,UniversidadedoMinho,CampusdeGualtar,Braga4710-057,Portugal
1
0
2
PACS 81.05.ue–Graphene
n
a PACS 72.80.Vp–Electronictransportingraphene
J PACS 78.67.Wj–Opticalpropertiesofgraphene

5
Abstract.- Itisshownthatonecanexploretheopticalconductivityofgraphene,togetherwiththeability

ofcontrollingitselectronicdensitybyanappliedgatevoltage,inordertoachieveresonantcouplingbetween
]
l anexternalelectromagneticradiationandsurfaceplasmon-polaritonsinthegraphenelayer. Thisopensthe
l
a possibilityofelectricalcontroloftheintensityoflightreflectedinsideaprismplacedontopofthegraphene
h layer,byswitchingbetweentheregimesoftotalreflectionandtotalabsorption. Thepredictedeffectcanbe
-
usedtobuildgraphene-basedopto-electronicswitches.
s
e
m
.
t
a
m
Amongthemanypromisedgraphenedreams[1–3],thepos- the effect we discuss below, and therefore the calculations we
-
sibility of exploring the electronic, thermal, and mechanical presentbelowusetheexperimentallymeasuredconductivityof
d
n properties of graphene, having in view a new generation of graphene.Theexactformofσ(ω)belowthe2µthresholdises-
o optoelectronic devices, is one of the most exciting of those sentialfortheparticulartypeofinteractionoftheelectronsin
c dreams. Understanding the fundamental physics of the inter- graphenewithanelectromagneticfieldleadingtotheformation
[
actionofelectrons(ingraphene)withanelectromagneticfield ofsurfaceplasmon-polaritons[12].

2 isakeysteptowardtherealizationofsuchdevices. Surface plasmon-polariton (SPP) is an evanescent electro-
v
Theopticalresponseofgraphenehasbeenanactivefieldof magneticwaveinducedbythecouplingoftheelectromagnetic
9
research, both experimental [4–7] and theoretical [8–10], and fieldtotheelectronsnearthesurfaceofametalorasemicon-
3
7 muchisalreadyunderstood.Fromthetheoreticalpointofview, ductor. Itsamplitudedecaysexponentiallyatbothsidesofthe
1 the independent electron approximation predicts that the real interface. TheSPPpropertiesaredeterminedbythedielectric
9. part of optical conductivity of graphene, at zero temperature, function of the conductor (which is related to the optical con-
0 has the form σ(cid:48) = σ θ((cid:126)ω −2µ), where σ = πe2/(2(cid:126)) is ductivity)andthedielectricconstantsofthesurroundingmedia
0 0
0 the AC universal conductivity of graphene, (cid:126)ω is the photon [12,13]. Clearly, graphene is the ultimate thin surface layer.
1 energy, µ is the chemical potential, and θ(x) is the Heaviside This fact, together with the ability of tuning the value of the
:
v step function. The imaginary part of the conductivity is finite chemicalpotentialbyanexternalgate,makesthissystempar-
i everywhere, as long as µ is also finite [11]. For zero chemi- ticularlyinteresting,sincetheformationofthistypeofelectro-
X
calpotential,theexperiments[4,5,7]confirmtheindependent magneticsurfacewavescanbycontrolledexternally.
r
a electronmodelpredictions. Ontheotherhand,forfinitechem- From the point of view of applications, surface plasmon-
ical potential and ω < 2µ, the experiments show a substan- polaritons can be explored in plasmon sensors [14] and high-
tial absorption in this energy range, at odds with the theoreti- resolution imaging [15], as well as for the miniaturization of
calprediction. Insimpleterms, therealpartoftheconductiv- photonicscomponents[16–18]. InthisLetter, weshowthatit
ity, as measured experimentally, follows roughly the formula isfeasibletoachieveasharpresonanceoftheattenuationofan
σ(cid:48) (cid:39)σ θ((cid:126)ω−2µ)+0.4σ θ(2µ−(cid:126)ω)θ((cid:126)ω−E ),whereE electromagnetic wave by transferring its energy into the exci-
0 0 D D
istheenergyatwhichtheDrudepeakstartsdeveloping. Below tation of surface plasmon-polaritons in graphene. Indeed, our
E ,theopticalresponseincreasesdramatically.Thedifference results demonstrate that, by adjusing the external gate voltage
D
betweenexperimentandtheorycanbeexplainedbybothinter- of an attenuated total internal reflection (ATR) structure with
bandandintra-bandscattering,duetoimpuritiesandelectron- graphenelayer,itispossibletotune,reversibly,thereflectance
electron interactions. In what concerns our present study, the of the incident electromagnetic wave from total absorption to
deviations seen in the experimental data are actually vital for totalreflection. Thestructureincludingagraphenelayer,con-
p-1
Yu. V.Bludov,M.I.Vasilevskiy,andN.M.R.Peres
(cid:113)
InEq. (1)q = (k/κ)2−(cid:15) (n=1,2,3)with
n n
√
k =κ (cid:15) sinθ, (4)
3
denoting the in-plane component of the photon wavevector,
κ = ω/c, and c is the speed of light in vacuum. In what fol-
lowswewilldiscussthephysicaleffectsimpliedbyEq.(1)and
howtheycanbeusedtobuildagraphene-basedoptoelectronic
switch.
AtresonancebetweentheincidentwaveandSPP,theampli-
tudeofthereflectedwaveisminimaland,ifthedielectriccon-
stants (cid:15) are all real, the reflectance vanishes, that is R = 0.
m
At the same time, the SPP amplitude, determined by the field
inthemedium2,isgivenby
Fig.1: Schematicrepresentationoftheexperimentalsetupneededto
excite surface plasmon-polaritons in graphene. The graphene layer e(2) ≡ Ex(2)(0,0) = 2(cid:15)2D−1(ω) (5)
issandwichedbetweentwodielectricmediaofrelativepermitivity(cid:15)1 x Ex(i) q2cosh(κq2d)
(consideredsemi-infinite)and(cid:15) (ofwidthd).Ontopofthedielectric
2
layerwithpermitivity(cid:15)2 thereisaprismofrelativepermitivity(cid:15)3 > withD(ω)denotingthedenominatorofEq. (1). TheSPP-light
(cid:15)2(whichisnecessaryfortotalinternalreflection).Theincidentangle couplingingraphenecanalsobeunderstoodfromthefollow-
oftheincomminglightisθ, andtheelectricandmagneticfieldsare ing simplified picture. In the case of d → ∞, the condition
E =(E ,0,E )andH =(0,H ,0),respectively.
x z y D(ω)=0reducesto
(cid:15) (cid:15) 4πσ(ω)
sideredinthecalculationspresentedbelowandsuitableforthe 1 + 2 = , (6)
q q ic
SPPexcitationingrapheneisshowninFig. 1wheretheneces- 1 2
sarydefinitionsaregiven. which gives the dispersion relation for the SPP excitations in
ThegeometryoftheexperimentalsetupdepictedinFig. 1is graphene surrounded by two semi-infinite media ((cid:15) and (cid:15) ).
1 2
knownasATRconfiguration. Thekeypointistoshinelighton The dispersion curve ω(k), determined by Eq. (6), allows for
theprismatananglelargerthanthecriticalangle(θ )fortotal aqualitativeanalysisoftheSPPpropertiesandtheconditions
c
internal reflection, characteristic of the interface between the fortheirexcitation.WenotethatinEq.(6)thewavevectorcom-
prismandthedielectricunderneath(whichcanbeanairgapor ponent along the x direction, k = k(cid:48) +ik(cid:48)(cid:48), is complex and
adielectriclayerdepositedontopofgraphene). FromSnell’s its imaginary part k(cid:48)(cid:48) describes the decay of the electromag-
law, we have sinθ = max((cid:15) ,(cid:15) )/(cid:15) ; choosing large (cid:15) , θ neticwavealongthegraphenesheet. Iftherewerenodissipa-
c 1 2 3 3 c
can be made small thus allowing for a broad range of angles, tion in the graphene layer (i.e. if σ(cid:48)(ω) ≡ 0), SPP would be
θ ≤ θ < π/2, to be scanned. Since graphene is at a finite a nondecaying wave with a purely real wavevector (k(cid:48)(cid:48) ≡ 0),
c
distancedfromthereflectinginterface,itispossibletotransfer propagatingalongxwithaphasevelocityv ,smallerthanthe
f
theenergyoftheincominglighttotheSPPexcitationviafrus- speed of light in either of the surrounding dielectrics, that is
(cid:112)
trated(orattenuated)totalinternalreflection. Tocomputethe v < c/ max(ε ,ε ). Inreality,SPPisdecayingbecauseof
f 1 2
electromagnetic response of such a system, we need to solve the energy dissipation in graphene (σ(cid:48) (cid:54)= 0) and the product
Maxwellequationsinthepresenceofthethreedielectrics,with k(cid:48)k(cid:48)(cid:48) must be positive. It follows from Eq. (6) that the con-
grapheneactingasaconductivesurface.Themathematicalpro- dition k(cid:48)k(cid:48)(cid:48) > 0 is satisfied only when σ(cid:48)(cid:48)(ω) > 0. This is a
cedureiswellknown[13]butthecalculationsarelengthy(we qualitative criterium for the existence of SPPs in the consid-
shall outline the derivation procedure below). The central re- eredstructure. OurnumericalresultspresentedinFigs. 2(a)to
sultofthisLetteristheequationdefiningthereflectanceofan 2(d)confirmthatforSPPsingraphenetheabovecriteriumin-
incomingelectromagneticwaveusingtheATReffect,withthis deedworks. Fromthesefigureswecanseethatasω increases
quantitydefinedasR=|r|2andrbeingthecomplexreflection the real part of the wavevector (k(cid:48)) initially increases (hence,
coefficientgivenby SPPs possess a positive group velocity) and then it reaches a
maximumandstartsdecreasing(herethegroupvelocityisneg-
ative). Finallyitcomestozeroatacriticalfrequencyω∗where
r = Ex(r) = [(cid:15)1/q1+4πiσ/c]χ−1 +(cid:15)2χ−2/q2 , (1) the imaginary part of the conductivity vanishes, σ(cid:48)(cid:48)(ω11∗) = 0.
Ex(i) [(cid:15)1/q1+4πiσ/c]χ+1 +(cid:15)2χ+2/q2 Notice that there is a second SPP band at ω > ω2∗ (where ω2∗
is another critical frequency, σ(cid:48)(cid:48)(ω∗) = 0), separated by a fi-
2
where nite gap from the lower band [Figs. 2(a) to 2(d)]. The gap
q (cid:15) correspondstothefrequencyrangeω∗ <ω <ω∗anditispre-
χ± =tanh(κq d)± 3 2 , (2) 1 2
1 2 q (cid:15) ciselywheretheimaginarypartoftheconductivityofgraphene
2 3
q (cid:15) is negative and SPPs cannot exist, according to the above cri-
χ±2 =1± q3(cid:15)2 tanh(κq2d). (3) terium.Theimaginarypartofthewavevectork(cid:48)(cid:48)increaseswith
2 3
p-2
Mechanismforgraphene-basedoptoelectronicswitches
Fig. 2: (color on line) Real (solid lines) and imaginary k(cid:48)(cid:48) (dashed
lines)partsoftheSPPwavevectorversusfrequency,calculatedfora
singlegraphenelayerwiththefollowingparameters: ε = 1, ε =
2 1
3.9(panelsa,b),ε = 1(panelsc,d),µ (cid:39) 0.22eV(panelsa,c),µ (cid:39)
1
0.16eV (panels b,d). Chemical potentials, corresponding to panels
(a)and(b),canbeachievedbyapplyingthegatevoltageV = 50V
orV = 25V,correspondingly, tographenelayerdepositedonSiO
2
substrate with thickness 300nm [6]. In all panels the ATR scanline
forangleofincidenceΘ = 60◦ andprismwithε = 16isdepicted
3
bythedash-and-dotline. Themeaningofcriticalfrequenciesω∗ is
1,2
designatedbyverticallinesinpanel(c).
the frequency in each of the SPP bands and has its maximum
in the vicinity of ω∗. By decreasing the chemical potential µ
1
themaximumvaluesofk(cid:48)andk(cid:48)(cid:48)alsodecrease[compareFigs.
2(a) and 2(b), as well as Figs. 2(c) and 2(d)]. The same re-
sult can be achieved by decreasing the dielectric constants of
thesurroundingmedia. Forinstance,whenthegraphenelayer
is deposited on the SiO substrate with ε = 3.9 for the fre-
2 1
quency range relevant to our situation [see Figs.2(a),2(b)] the Fig.3:(coloronline)Panels(a,b):reflectivityR[panel(a)]andSPP
maxima of k(cid:48) and k(cid:48)(cid:48) are higher than in the case of vacuum squareamplitude|e(2)|2 [panel(b)]versusangleofincidenceθ,and
x
[Figs.2(c),2(d)]. frequencyωfortheATRstructurewithµ (cid:39) 0.16eV(corresponding
Passingtothedescriptionofexcitationproblem,thiscanbe toV =25Vofgatevoltage);Panel(c):reflectivityRversusangleof
seen as a coupling of the exponential tails of the electromag- incidenceθ,correspondingtotheminimalvalueofR(ω,θ),depicted
netic wave in the prism and SPPs in graphene, similar to the inpanel(a);Panel(d):thefrequencyωatwhichforgivenΘthemin-
imumofRinpanel(a)occurs. Panel(e): angleofincidence,θ,and
tunneling effect. We notice that now the in-plane wavevector
frequency,ω,correspondingtozeroreflectivityoftheATRstructure
of the radiation in the prism is a real value defined by its fre-
versusgatevoltageV. InallpanelstheparametersoftheATRstruc-
quency,theangleofincidenceandthedielectricconstantofthe
ture(Fig.1)areε =3.9,ε =1,ε =16,d=40µm.
1 2 3
prismmaterial,asexpressedbyrelation(4)andknownasATR
scan line. The resonant coupling occurs when the ATR scan-
line crosses the SPP dispersion curve ω(k) [see inset in Fig. beborneinmind,however,thatthetruepositionandthewidth
2(a)].Thevalueofk(cid:48)(cid:48)atthecrossingpointdeterminesthequal- oftheresonancecanbefoundonlyconsideringthewholesys-
ity factor Q of the resonant excitation and, hence, the relative tem shown in Fig. 1 and including the real part of the optical
amplitudeoftheexcitedSPP,e(2). Consequently,theefficient conductivityofgraphene.
x
SPPexcitationisonlypossibleatsmallvaluesofω,becausethe Figure 3 shows a particular example of ATR via coupling
large k(cid:48)(cid:48) hampers the excitation in the vicinity of ω∗ . Since to the gated graphene layer at work. In panels (a)–(d) of this
1,2
the imaginary part of the optical conductivity of graphene de- figure,thegatevoltageV ofthecapacitormadeofagraphene
pends on the gate-controlled chemical potential (for example, layer as one plane of the plates of the field effect transistor is
atlowfrequencies,ω << |µ|,σ(cid:48)(cid:48) ≈ 4σ |µ|/(π(cid:126)ω)[11]),itis kept constant. Panel (a) shows the minimum value of R at a
0
clear from Eq. (6) that the resonance can be tuned by chang- particular value of the pair of variables θ and ω that can be
ing the gate voltage applied to the graphene layer. It should controlled independently. Comparison of panels (a) and (b)
p-3
Yu. V.Bludov,M.I.Vasilevskiy,andN.M.R.Peres
demonstrates that resonance minimum of reflectance R corre-
sponds to the maximal amplitude of the excited SPP. In other
words,thesereflectanceminimaoccurasaresultofthetrans-
formationoftheincidentwaveenergyintotheenergyofSPPs,
excitedingraphene. ItisclearthatthevalueofRcanbemade
verysmall andeven zeroby anappropriate choiceof θ and ω
asdepictedinpanels(b)and(c). Aswemovealongthecurve
ofpanel(c),bychangingtheangleθ,thevalueofRatthatan-
gledependsonthefrequencyoftheincominglight. Thevalue
ofω, correspondingtotheminimumofthereflectivityR, can
be read off from panel (d) by drawing a straight vertical line
crossing both (c) and (d) panels; an example of such a line is
giveninFig. (3). Inpanel(e)wesetR = 0andobservehow
the values of θ (left axis) and ω (right axis), change as func-
tions of the gate voltage. For a given gate voltage, a vertical
lineisdrawnintersectingbothθ(V)andω(V)curvesandthe
pair of these two variables giving R = 0 is read off from the
corresponding vertical axis. From this panel, we see that it is
alwayspossibletofindapairofθ(V)andω(V)satisfyingthe
conditionfortotalabsorption. Ofcourse,itispossibletodraw
the corresponding curves for any given value of R. Thus, we
can see that indeed it is possible to tune externally the reso-
nanceabsorptionconditionbychangingtheexternalgatevolt-
age. We would like to stress at this point that it is feasible Fig.4: Panels(a): shapeoftheacgatevoltageV(t)withamplitude
to implement the predicted resonance absorption effect using 50Vandfrequency10kHz.Panels(b-f):reflectanceR(Θ,ω)forthe
large area graphene samples grown by chemical vapor depo- attenuatedtotalreflectionstructureattimemoments,correspondingto
sition (CVD) [19,20], since also in this case the optical con- pointsb–fatpanel(a),isdepictedinrespectivepanels.Theparameters
ductivity of graphene does not change substantially from that oftheATRstructurearethesameasinFig.3.
ofexfoliatedproducedsamples. SincetheCVDmethodallows
graphenetobetransferredtodifferentsubstrates,itisalsofea-
as the derivative g = dR/dV at the level R = 0.5. As one
sibletorealizetheATRstructuredepictedinFig. 1.
can see from Fig. 5(b), g shows an increase with V and then
The possibility of changing the chemical potential in
saturatesatV ≈100V.
graphene by applying a gate voltage makes it possible to tune
We shall now present the derivation of our main result.
theATRconditionsbychangingthislatterquantity, similarto The solution of Maxwell equations rotE(cid:126)(m) = iκH(cid:126)(m),
the tuning of the plasmon spectrum as proposed in Ref. [21]. rotH(cid:126)(m) =−iκε E(cid:126)(m) forthep-polarizedwavecanbewrit-
m
Fig.4 demonstrates the variation in time of the reflectivity of
tenintheform:
theATRstructureasthefunctionoffrequencyandangleofin-
(cid:104) (cid:105)
cidence. The gate voltage is changed adiabatically according E(m)(x,z) = A(m)exp(κq z)+A(m)exp(−κq z) eikx,
x + m − m
totheharmoniclawV(t) = 50cos(2πft),withthefrequency
κε iε (cid:104)
f = 10kHz [Fig.4(a)]. As it can be seen from Fig.4, a de- H(m)(x,z) = − mE(m)(x,z)= m A(m)exp(κq z)
y k z q + m
crease of the absolute value of gate voltage corresponds to a m
(cid:105)
shift of the polaritonic resonance to the low-frequency region −A(m)exp(−κq z) eikx.
− m
[e.g., compare Fig.4(b) and 4(c), as well as Fig.4(d) and 4(e)]
andabroadeningoftheresonance. The meaning of the coefficients A(m) is different for the
±
Now a natural question arises: how sensitive is the re-
different media. Since in the medium 3 the electromag-
flectanceoftheATRstructuretothevariationofgatevoltage,
netic wave is propagating in z-direction, q is imaginary
3
when frequency ω and angle of incidence Θ are fixed? The [Im(q ) < 0], and the coefficients A(3) = E(r)exp(κq d),
answer follows from Fig. 5(a), which demonstrates the pos- 3 + x 3
A(3) = E(i)exp(−κq d) are proportional to the amplitudes
sibility of changing the ATR structure’s reflectivity from zero − x 3
to almost unity (full reflectance) by varying the gate voltage (at z = −d) of the reflected Ex(r) and incident Ex(i) waves,
within a range of δV ∼ 10V. As a result, the ATR structure respectively. In the media 2 and 1 the electromagnetic waves
withgraphenelayercanoperateasaswitchofelectromagnetic are exponentially decaying along the z-axis, and qm are real
radiationintheterahertzdomain,wheretheintensityofthere- (q , q > 0). In the medium 2, the coefficients A(2) =
2 1 ±
(cid:104) (cid:105)
flectedelectromagneticwavecanbetunedbythegatevoltage, E(2)(0,0)±(q /iε )H(2)(0,0) /2aresuperpositionsofthe
x 2 2 x
applied to graphene. An important characteristic of the THz
switchisthesharpnessoftheresonance,g,whichisanalogous electricEx(2)(0,0)andmagneticHy(2)(0,0)fieldsatboundary
tothecross-conductivityofafield-effecttransistor, calculated z = 0. In the medium 1, the coefficient A(1) = 0, while
+
p-4
Mechanismforgraphene-basedoptoelectronicswitches
[4] KUZMENKO A. B. and et al., Phys. Rev. Lett., 100 (2008)
117401.
[5] KINFAIMAKandetal.,Phys.Rev.Lett.,101(2008)196405.
[6] LIZ.Q.andetal.,NaturePhys.,4(2008)532.
[7] NAIRR.R.andetal.,Science,320(2008)1308.
[8] PERESN.M.R.andetal.,Phys.Rev.B,73(2006)125411.
[9] FALKOVSKY L. A. and PERSHOGUBA S. S., Phys. Rev. B, 76
(2007)153410.
[10] GUSYNINV.P.andetal.,NewJ.Phys.,11(2009)095013.
[11] STAUBERT.andetal.,Phys.Rev.B,78(2008)085432.
Fig.5: (coloronline)Panel(a): reflectivityRversusgatevoltageV
[12] H.Ra¨ther,SurfacePlasmonsonSmoothandRoughSurfacesand
forATRstructurewithΘ=34.40◦,ω=2.894meV[dashedline]or
onGratings,(Springer-Verlag:Berlin,1988).
Θ = 34.35◦,ω = 3.044meV[solidline];Panel(b): theabruptness
[13] PITARKE J. M. and SILKIN V. M. and CHULKOV E. V. and
g=dR/dV asfunctionofgatevoltageV ofthecharacteristicR(V),
ECHENIQUEP.M.,Rep.Prog.Phys.,70(2007)1.
calculatedonthelevelR = 0.5forΘ,ω,correspondingtothosein
[14] HOMOLAJ.andetal.,SensorsandActuatorsB,54(1999)3.
Fig.3(e).OtherparametersoftheATRstructurearethesameasinFig.
[15] KAWATAS.andetal.,NaturePhotonics,3(2009)388.
3.
[16] VASAP.andetal.,Laser&Photon.Rev.,3(2009)483.
[17] ZAYATSA.V.andetal.,PhysicsReports,408(2005)131.
A(1) = E(1)(0,0) stands for the electric field amplitude at [18] BARNESW.L.andetal.,Nature,424(2003)824.
− x [19] KEUNSOOKIMandetal.,Nature,457(2009)706.
z =0. Boundaryconditionsatz =−dimplythecontinuityof
[20] XUESONGLIandetal.,Science,324(1312).
the tangential components of the electric and magnetic fields,
[21] RYZHII V. and SATOU A. and OTSUJI T., J. Appl. Phys., 101
[Ex(3)(x,−d) = Ex(2)(x,−d), Hy(3)(x,−d) = Hy(2)(x,−d)]. (2007)024509.
Atz = 0thetangentialcomponentoftheelectricfieldiscon- [22] PERESN.M.R.,Rev.Mod.Phys,82(2010)2673.
tinuous E(1)(x,0) = E(2)(x,0), while the discontinuity of
x x
the tangential component of the magnetic field, H(1)(x,0)−
y
H(2)(x,0) = −(4π/c)j = −(4π/c)σ(ω)E (x,0), stems
y x x
fromthepresenceofsurfacecurrents(causedbytheSPPelec-
tricfield)inthegraphenelayer. Matchingtheseboundarycon-
ditions,weobtainasystemoflinearequationsforthefieldam-
plitudesinthematrixform,
 Ch iεq22Sh −1  e(x2)   1 
 εε23qq32Sh iεq33Ch 1  h(y2) = 1 . (7)
iε1 − 4πσ 1 0 r 0
q1 c
where C = cosh(κq d), S = sinh(κq d), and h(2) =
h 2 h 2 y
H(2)(0,0)/E(i). Solution of (7) yields the reflection coeffi-
y x
cient[Eq. (1)]andthefieldamplitudeinthegap[Eq. (5)].
To conclude, we have demonstrated that the ATR structure
incorporatingamonolayergraphenesheetcanoperateasTHz
switch where the reflectance of an electromagnetic wave can
be switched from nearly unity to nearly zero by applying an
external gate voltage to the graphene layer. Since the typical
frequenciesare∼5meV(or∼1.2THz,whichcorrespondsto
∼ 0.25mmofwavelengthinvacuum),thisstructurecanoper-
ateinthesubmillimeterrange. ThefrequencyofexcitedSPPs
canbeincreasedbyusingaprismmaterialwithahigherdielec-
tricpermeability(cid:15) . Theproposeddevicecanalsobeusedfor
3
spectroscopyofthegrapheneopticalconductivity[22]through
measuringthecharacteristicsoftheATR-excitedSPPs.
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p-5