Edge-plasmon assisted electro-optical modulator
Ivan A. Pshenichnyuk, Gleb I. Nazarikov, Sergey S. Kosolobov, Andrei I. Maimistov, Vladimir P. Drachev
EEdge-plasmon assisted electro-optical modulator
Ivan A. Pshenichnyuk, ∗ Gleb I. Nazarikov, Sergey S. Kosolobov, Andrei I. Maimistov, and Vladimir P. Drachev
1, 3 Skolkovo Institute of Science and Technology, Moscow 121205, Russian Federation National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow 115409, Russian Federation University of North Texas, Denton, Texas 76203, USA (Dated: January 13, 2020)An efficient electro-optical modulation has been demonstrated here by using an edge plasmonmode specific for the hybrid plasmonic waveguide. Our approach addresses a major obstacle ofthe integrated microwave photonics caused by the polarization constraints of both active and pas-sive components. In addition to sub-wavelength confinement, typical for surface plasmon polari-tons, the edge plasmon modes enable exact matching of the polarization requirements for siliconbased input/output grating couplers, waveguides and electro-optical modulators. A concept ofthe hybrid waveguide, implemented in a sandwich-like structure, implies a coupling of propagatingplasmon modes with a waveguide mode. The vertically arranged sandwich includes a thin layerof epsilon-near-zero material (indium tin oxide) providing an efficient modulation at small lengthscales. Employed edge plasmons possess a mixed polarization state and can be excited with horizon-tally polarized waveguide modes. It allows the resulting modulator to work directly with efficientgrating couplers and avoid using bulky and lossy polarization converters. A 3D optical modelbased on Maxwell equations combined with drift-diffusion semiconductor equations is developed.Numerically heavy computations involving the optimization of materials and geometry have beenperformed. Effective modes, stationary state field distribution, an extinction coefficient, opticallosses and charge transport properties are computed and analyzed. In addition to the polarizationmatching, the advantages of the proposed model include the compact planar geometry of the sil-icon waveguide, reduced active electric resistance R and a relatively simple design, attractive forexperimental realization. I. INTRODUCTION
New problems of great practical significance ariseas electronic devices like transistors get downscaled toatomic dimensions [1]. As we seek the ability to engineermaterials and devices on an atomic scale, a predictionfor the structural, electrical and mechanical propertiesof new materials, and the rates of chemical reactions be-come crucial for the transition from nano to molecularelectronics [2, 3]. Attempts to use light to interconnectelectronic units at small scales face the mismatch be-tween typical sizes in electronics (tens of nanometers)and communication wavelengths (thousands of nanome-ters) [4]. A weak light-matter interaction makes the effi-ciency of classic opto-electronic devices poor [5]. Hybridquasiparticles appear as an elegant approach, which canbe used to overcome the existing issues and bring theelectronics to the next level. For example, exotic funda-mental properties of exciton-polaritons allow to exceedlimitations of classical photonics [6]. As it was shown,the exciton-polaritons may form condensates at roomtemperatures and demonstrate nonlinear coherent behav-ior typical for quantum fluids [7–10]. Some applicationsof their properties in polariton simulators and potentialexciton-polariton integrated circuits have been discussedrecently [11–13].The technology of subwavelength optics based on sur-face plasmon polaritons (SPP) is used in numerous ap- ∗ correspondence address: [email protected] plications [14, 15]. It is known that SPP modes maycombine strong optical confinement and efficient light-matter interaction [4]. Both properties might improvethe performance of active devices for integrated electro-optical systems. Here we suggest a model of efficientand compact electro-optical modulator based on a hy-brid plasmonic waveguide (HPWG) concept employingan epsilon-near-zero (ENZ) effect.While ordinary plasmonic waveguides allow to reach ahigh degree of optical confinement, they also introduceunavoidable losses, caused by the presence of metalliccomponents. On the other hand, losses in ordinary dielec-tric waveguides can be made negligible, but the opticalmode confinement is obviously much worse. An idea ofHPWG is to combine two types of waveguides into a sin-gle structure, where a hybrid mode can be formed to com-promise losses and confinement [16, 17]. The strength ofthe coupling can be controlled by geometrical parame-ters, such as the distance between the metal surface andthe dielectric waveguide.The ENZ effect, useful for optical switching, appearsat infrared frequencies in a certain classes of materi-als, including transparent conductive oxides (TCO) [18].Applying an external electric field an increased concen-tration of electrons can be achieved, for instance, at aboundary with a dielectric [19]. When the critical con-centration is reached, the real part of the dielectric per-mittivity tends to zero in a good correspondence with theDrude theory [20]. The ENZ effect was recently demon-strated experimentally in indium-tin-oxide (ITO) [21].When the voltage is applied and the real part of permit-tivity turns into zero, field intensity inside ITO accumu- a r X i v : . [ phy s i c s . a pp - ph ] J a n lation layer grows in a resonant manner. The imaginarypart of the dielectric constant also grows significantly inthe ENZ regime, causing a strong dissipation of localizedelectromagnetic energy in the ITO accumulation layer.The described concepts in different manifestationswere used in many theoretical and experimental researchworks. The concept of plasmostor is introduced in theexperiments of Dionne et al. [22]. A silicon based plas-monic waveguide with a thin dielectric layer and silvercladdings is used. The applied voltage influences avail-able modes, allowing them to interact constructively ordestructively at the output and providing the modulationeffect. Doped silicon acts as an active material. Anothermodel based on a plasmonic slot waveguide is presentedin the work by Melikyan et al. [23]. A dielectric corewith a thin layer of ITO is used. The conducting oxideaccumulates electrons under the influence of applied volt-age, which changes its permittivity and allows to absorba propagating plasmonic mode. Charge induced changesof the refractive index are more pronounced in ITO thanin Si. A numerical model of ultrasmall fully plasmonicabsorption modulator is proposed in Krasavin and Zay-ats [24]. It is suggested to use a thin plasmonic nanowirewith rectangular cross-section separated from a metal-lic surface by a 5 nm thick sandwiched ITO/insulatorspacer. A compact geometry (25 × ×
100 nm ) and theimplementation of high quality dielectric HfO allows tomake the switching voltage as low as 1 V. Different gatemetals are compared and a better performance of goldis reported. Further development of the plasmonic slotwaveguide modulator appears in experiments of Lee et al. [25]. In their model ITO fills the space between goldencladdings with an additional thin layer of insulator. Insuch a geometry ENZ regime can be reached at relativelylow voltages ( ∼ et al. [26]. Modulating sandwich,which consists of a thin layers of ITO and an insulator,covered by a golden electrode, is placed on top of a waveg-uide, forming HPWG. More works developing this ideaare available [27, 28].There is a family of modulators where ENZ effect isexploited without the implementation of hybrid waveg-uides. In calculations of Lu et al. [29] it is suggested toincorporate thin layers of an active material (aluminumzinc oxide) and insulator into the silicon waveguide, with-out metallic layers and corresponding SPP modes. Sim-ilar approach is suggested in theoretical works Sinatkas et al. [30] and Qiu et al. [31]. ITO based capacitor eithersurrounds the waveguide or partially penetrates inside it.High quality dielectric HfO is used in these works, whichallows to reach larger charge concentrations at lower volt-ages. In general ENZ modulators with no plasmonic waveguides implemented are noticeably larger. On theother hand, fully plasmonic solutions with many metal-lic elements [32] introduce significant losses. HPWG withaccurately chosen geometry may provide a useful compro-mise between two approaches. Advanced ways to couplewaveguide modes with plasmonic modes may also help tofight plasmonic losses and bring HPWG idea to the nextlevel [33]. More details about plasmonic modulators areavailable in corresponding reviews [4, 34, 35]In current work we remove one of the main obsta-cles of integrated photonic devices based on HPWG -the polarization constraint. Indeed, plasmons can be ex-cited only by light with the polarization perpendicularto the surface of the metal. Thus, a planar device canonly operate with the vertically polarized modes [26]. Itmakes the design incompatible with modern grating cou-plers, which can provide almost 100 percent efficiency[36]. To match the polarization requirements for gratingcouplers and the modulator one should develop and usepolarization converters [37–41]. Obviously, a significantpart of the light intensity is lost after the double conver-sion, before and after the modulation. Besides, it resultsin additional complexity at the manufacturing side. Inthis paper we propose a model of a vertically assembledplasmonic modulator employing edge plasmons. Our de-vice works with the same polarization as the couplersand does not require the usage of converters. In con-trast with SPP, edge modes have a mixed polarizationstate, which allows them to serve as coupling providersbetween a horizontally polarized optical waveguide modeand a metallic electrode. Fundamental properties anddispersion relations of edge plasmons at different bound-aries are studied in order to deduce an efficient devisestructure. Our modulator has a simple geometry de-signed for the subsequent experimental realization. Theproposed planar structure possess many practical ben-efits discussed in the text. Technically, to achieve thegoal, we propose an advanced HPWG that mixes twoedge plasmons with a waveguide mode. The correspond-ing model is essentially 3D and requires numerically de-manding computations. The designed structure providesa comfortable waveguide-plasmon-waveguide conversionlength (tunable via geometrical parameters) and allowsto reach a required modulation depth.The paper is organized as following. In Sec. II we givea short review of a theoretical framework used to per-form numerical computations. In Sec. III edge plasmonproperties are investigated. The dispersion relation iscomputed numerically and compared with the SPP dis-persion. A polarization state of edge plasmons and theirstability with respect to the edge roughness are discussed.Sec. IV presents the details of the modulator geometry.3D computations of the electric field distribution insidethe device are presented and discussed. Optical losses areevaluated. High frequency charge transport properties ofthe modulator are discussed in Sec. V. The response ofthe electrons distribution in the active material on theapplied voltage is calculated. Optimal parameters forthe modulator electrical capacity and related bandwidthare discussed. ENZ wavelength in ITO, as well as thedetailed properties of the accumulation layer are investi-gated. In the last section (Sec. VI) the on- to off-statetransition of the modulator is presented and the extinc-tion coefficient is evaluated. The influence of the modu-lating sandwich length on the performance of the deviceis discussed. II. THEORY
Optical model of the modulator is based on Maxwellequations in frequency domain [42] ∇ × E ( r ) = + iωµ H ( r ) , (1) ∇ × H ( r ) = − iωε E ( r ) , (2)where ε = ε r ε is the total permittivity (defined asa product of relative and vacuum permittivities) and µ is a permeability of the medium. The time depen-dence of electric and magnetic fields is assumed to beharmonic, i.e. E ( r , t ) = E ( r ) exp( − iωt ) and H ( r , t ) = H ( r ) exp( − iωt ). For the convenience of the numeri-cal treatment two equations are combined into a singlesecond-order vector wave equation with respect to theelectric field ∇ × ∇ × E ( r ) − k n E ( r ) = 0 . (3)Here we introduce a vacuum wave vector k = ω/c ,a speed of light c = 1 / √ ε µ and a refractive index n = √ ε r ( µ = 1 is assumed in this work). Eq. 3 issolved numerically in 3D for a modulator geometry spec-ified in Sec. IV. This task is numerically expensive andrequires HPC (high performance computing) cluster tobe performed. To build the numerical model complex re-fractive indices (or permittivities) should be fixed for allthe materials used in the simulation for a specific wave-length ( λ = 1550 nm is assumed in the paper). Opticalparameters for metallic and semiconducting materials arepresented in the Tab. I.Along with full 3D numerical computations we use themode analysis technique. Considering a wave propagat-ing in x -direction and confined in y and z directions, onemay write E ( r ) = E ( y, z ) e iβx , (4)where the propagation constant β ≡ n eff k , expressedthrough effective mode indices n eff , is introduced. Sucha substitution (Eq. 4) makes Eq. 3 effectively 2D and,thus, allows to perform fast calculations and obtain use-ful results without the necessity to use supercomputers.Effective mode indices n eff and corresponding field distri-butions E ( y, z ) are computed numerically using ordinaryPC. In our model (see Sec. IV) the cross-section of themodulator has a rather complicated structure containing TABLE I: Optical model parameters Au ITO n-Si n c , cm − · · ω p , s − . · · · γ , s − . · . · . · ε ∞ . . ε ( ω ) − . . . ε ( ω ) 3 . .
008 5 · − Re n ( ω ) 0 . . . n ( ω ) 10 . .
002 7 · − References [43] [30, 44, 45] [30, 46, 47] many areas and the permittivity distribution ε = n ( y, z )should be considered as a function of y and z . Therefore,Eq. 3 under condition 4 in a scalar form prepared for thenumerical treatment reads: ∂ E x ∂y + ∂ E x ∂z + k n E x = iβ ∂E y ∂y + iβ ∂E z ∂z , (5) ∂ E y ∂z + ( k n − β ) E y = iβ ∂E x ∂y + ∂ E z ∂y∂z , (6) ∂ E z ∂y + ( k n − β ) E z = iβ ∂E x ∂z + ∂ E y ∂z∂y . (7)The propagation constant β should be considered as adiscrete eigenvalue here.To investigate charge transport in the modulator, semi-conductor drift-diffusion system of equations for the elec-tron density n = n ( r , t ) and potential ϕ = ϕ ( r , t ) in theform [48, 49] ∇ · ( ε ¯ ε ∇ ϕ ) = e ( n − N d ) , (8) ∂n∂t = ∇ · ( D n ∇ n − nµ n ∇ ( ϕ + χ )) , (9)is solved numerically. Here ¯ ε is a static permittivity [50], N d - doping level, µ n - mobility of electrons, χ - electronaffinity and D n - diffusion coefficient. Since both ITOand silicon are n-doped, only donor type conductance isconsidered. Model parameters for silicon and ITO arecollected in Tab. II.In certain regimes, when the Maxwell-Boltzmannstatistics is applicable, diffusion coefficient can be ex-pressed simply as D n = µ n k b T /e . In our computationsthe complete Fermi-Dirac statistics is taken into account,and the diffusion coefficient reads [30] D n = µ n k b Te F / ( η ) F − / ( η ) , (10)TABLE II: Drift-diffusion equations parameters ITO n-Si m ∗ /m e .
35 0 . µ n , cm /(V · s) 30 500 χ , V 4 . . ε . . N d , cm − . · References [30, 44, 45] [30, 46, 47] with η = F − / ( n/N c ) , (11)where F ± / and F − / are direct and inverse Fermi-Diracintegrals of the order ± /
2. Effective density of states inthe conduction band is expressed as [50] N c = 2 (cid:18) m ∗ k b T π (cid:126) (cid:19) / . (12)Reduced masses of electrons m ∗ for ITO and n-Si areshown in Tab. II. The room temperature T = 300 K isassumed in computations.The Drude theory links the charge density evolutionunder the applied voltage with the optical model. Drudebased frequency domain permittivity [51, 52] ε ( ω ) = ε ∞ − ω p ( ω + γ ) + i γω p ω ( ω + γ ) , (13)describes well optical properties of ITO and n-Si inC+L telecom range (1530 − ω p = (cid:112) n c e / ( ε m ∗ ) is the plasmonic frequency and γ = e/ ( µ n m ∗ ) is the relaxation coefficient. Asymptotic mate-rial permittivity at large frequencies is denoted as ε ∞ and n c stands for the charge carriers concentration. Inhomo-geneous concentration profile n ( r ) obtained from Eqs. 8and 9 can be substituted here to obtain the distribu-tion of permittivity. In particular, the structure of theaccumulation layer at the boundary between ITO anddielectric can be computed with high precision and usedin the optical model, as shown in Sec. VI. Drude basedpermittivities and refractive indices, used in optical com-putations are collected in Tab. I. They are computed atthe frequency ω = 1 . · s − (corresponding tothe wavelength λ = 1550 nm). Refractive indices of di-electrics, namely HfO and SiO , are assumed to be con-stant and real in C+L range with n = 2 . n = 1 . III. EDGE PLASMONS
Before the introduction of the modulator and its pa-rameters (see the next section) we review some importantfundamental properties of edge plasmons. As it is men-tioned in the introduction, these modes provide a usefulalternative to SPP in the case of plasmonic modulators.In contrast to SPP they have a symmetric mixed polar-ization state, which allows to interact with both polariza-tions of the waveguide mode. A field distribution of theedge mode at the ITO/Au boundary is shown in Fig. 1(c). The plasmon propagates in the x -direction (perpen-dicular to the surface of the picture). Projections E y and E z of the field on the transverse directions, revealing thepolarization state, are shown separately (Fig. 1 (d)-(e)).Dispersion relations of edge plasmons are shown inFig. 1 (a) and (b). Instead of usual dependence of angu-lar frequency on a wave vector ω ( k ), real and imaginaryparts of n eff ( λ ) are plotted (as defined in the previoussection). To go back to the usual notations one may use k = n eff ω/c and ω = 2 πc/λ , where c is a speed of lightand λ is a wavelength. While the real part of n eff is as-sociated with the mode propagation, the imaginary partreflects its decay. A family of curves describing plasmonsexcited at the golden edge surrounded by SiO or ITO(see below) is presented. The solid black line shows thedispersion relation of SPP given by the analytical formula[51] k = ωc (cid:115) ε ( ω ) ε ε ( ω ) + ε , (14)where ε = 3 . and ε ( ω ) isassociated with gold and expressed by the Drude formula(Eq. 13). C+L telecom range (1530 − boundary [26] can be understood with a help of the dis-persion relations in Fig 1 (a)-(b) (orange and blue linesrespectively). Au/ITO edge plasmons are slightly morelossy than Au/SiO plasmons at λ = 1550 nm (Fig. 1(b)). That is a consequence of the fact that ITO per-mittivity has a nonzero imaginary part. However, gen-tly doped ITO films remain mostly transparent at near-infrared and electro-magnetic losses are low [56, 57]. It isevident from the picture that the edge plasmons in ITOdecay fast at larger wavelengths, interacting strongerwith the medium. One can see very different asymp-totics of the orange line and the blue line. Both curvesdemonstrate growth at small wavelengths. An interplayof two trends produces a point with minimal losses forAu/ITO plasmons which appears at 1250 nm. Lossesin C+L range, which is slightly shifted with respect to (a) (b) (c)|E| (d) (e)|E y | |E z |Au ITO FIG. 1: Properties of edge plasmons. Dispersion relations are plotted as a dependence of the effective mode indiceson wavelength n eff ( λ ). Both real part (a) and imaginary part (b) are shown. A family of curves is plotted tocompare ordinary SPP, edge plasmons at the gold/silica and gold/ITO interfaces. Smooth edge (radius 300 nm) vssharp edge plasmons comparison is shown. Vertical dashed lines highlight C+L range. The absolute value of theelectric field distribution is shown in (c), while two transverse projection | E y | and | E z | are shown in (d) and (e)respectively.the minimum, are still quite low. Additional tuning canbe performed by changing the doping level in ITO withlimitations discussed in Sec. V. Real parts of ITO per-mittivity and SiO permittivity have the values close toeach other, which results in a family of similar curves inFig. 1(a). The SPP case is shown for comparison (blacksolid curve).In the experimental implementation metallic edgescannot be perfectly sharp. To investigate the stability ofedge modes with respect to the edge roughness we per-form the comparison between sharp edge solutions androunded edge solutions (the radius of the rounded corneris 300 nm), which is shown in Fig.1 (a) and (b) with or-ange and green curves respectively. For long-wavelengthplasmons the difference is negligible, since the plasmoncharacteristic sizes are much larger than the curvatureradius. Solutions are also similar around λ = 1550 nm.At small wavelengths the difference can be significant.When the size of the plasmon becomes smaller than thecurvature radius it perceives the edge rather like a surfaceand its dispersion curve moves closer to SPP (see how thegreen line approaches the black SPP line at small λ inFig. 1 (a)). It is interesting to note, that, due to thiseffect, rounded edge plasmon losses in ITO can be evensmaller than losses of edge plasmons in SiO (compareblue and green curves in Fig. 1 (b) below 1000 nm). IV. MODULATOR DESIGN AND FIELDDISTRIBUTION
The proposed modulator design is presented in Fig. 2.Single mode silicon waveguide (600 nm x 200 nm) is cov-ered with the modulating sandwich consisting of few lay-ers: high quality dielectric HfO (10 nm), ITO (15 nm) (a)(b) (c) xyz Aun-SiAuAir HfO (I h ) ITO (A h ) W w R h C h L yz n-Si W h R w SiO z z y x x FIG. 2: Structure and geometry of the modulator. (a)cross section of the modulator (b) 3D model used innumerical computations (c) the part, where thewaveguide enters the modulator. ITO and HfO areshown with violet and yellow colors respectively. Crosssections used throughout the paper for the visualizationof the field distribution are shown with thin red lines.The following geometrical parameters were implementedin the computations: W w = 600 nm, W h = 200 nm, I h = 10 nm, A h = 15 nm, R w = 80 nm, R h = 55 nm, C h = 155 nm, L = 6845 nm. (a)(b) V/m “Top” view, | E (x, y, z=z )|, scale x2.8“Bottom” view, | E (x, y, z=z )|, scale x1.4“Side” view, | E (x, y=y , z)| (c)(e) (f)(d) | E (x=x , y, z)| | E (x=x , y, z)| FIG. 3: Distribution of the absolute value of the electricfield | E ( x, y, z ) | inside the modulator when the voltageis not applied and the transmission is maximal(’on-state’). Five different cross-sections of 3D modelare shown: (a) horizontal plane which shows the fieldinside the sandwich, (b) horizontal cross-section whichpasses through the center of the waveguide, (c) verticalcross-section also shows the field inside the waveguide,(e) transverse cross-section which show the distributionat the entrance of the modulator, (f) the fielddistribution in the central part of the modulator. (d)the waveguide fundamental mode coupled to two edgeplasmons. It is evident how the waveguide modetransforms into the plasmonic mode and back.and golden contact (155 nm). Additional structural el-ement, which we call the plasmonic rail (or just rail),is placed on top of the ITO layer (see Fig. 2 (a)). Theresulting geometry contains two Au edges, which sup-port plasmons, as described in the previous section. Suchmodes can be coupled to the y -polarized waveguide mode.On the contrary, a flat structure without the rail supportsonly z -polarized SPP modes which do not interact withthe waveguide mode. It is relatively easy to fabricate sucha geometry which makes the resulting modulator designpromising for applications. Certain deviations from thespecified geometry of the rail are acceptable. The pre-cision of vertical lines, for example, does not have to bevery strict, since edge plasmons are quite stable to suchvariations. The edges can also be smoothed, as it wasshown in the previous section. Since the rail brakes thesymmetry in y -direction, 2D computations would not besufficient for such a device and full 3D modeling is re-quired.The size of the plasmonic rail (80 nm ×
55 nm inour model) is chosen to maximize the modulator per-formance, which means at least several things. HPWG should provide an effective conversion of the waveguidemode to the plasmon (which can be efficiently modu-lated) and back. In the on-state, when the voltage is notapplied, conversion losses should be minimal. Accordingto our computations, conversion between modes occursin a manner of beats. Such a behavior is typically ex-pected when two or more interacting modes exist in thesystem. Even in a simplest case HPWG contains at leasttwo modes, which appear as a result of hybridization ofone waveguide mode and one plasmonic mode [26]. Theanalysis for our edge modulator reveals three modes (twoedge plasmons hybridized with one waveguide mode), butjust two of them are excited. Strong interaction betweenmodes, caused by small distances and large confinement,results in a strong mixing. In contrast with directionalcouplers (and similar devices), where the coupled modetheory can be used to describe weak interaction betweenmodes, there is no simple analytical theory describingHPWG [16]. For this reason different geometries of therail, along with other parameters of the modulator, aretested numerically in order to obtain the configurationwith suitable insertion losses and beats period. Anotherfactor, which defines the performance, is related to theoff-state. When the voltage is applied the signal attenu-ation has to be maximal. It should be also taken into ac-count that the accumulation layer in ITO, being formed,changes the character of interaction between modes. Thefinal geometry of the modulator is defined by both on-and off-states. The latter one is discussed in Sec. VI.The absolute value of the electric field | E ( x, y, z ) | in-side the modulator is shown in Fig. 3. To visualize a 3Ddistribution, five different cross-sections are plotted. Theposition and orientation of the selected cross-sections isdenoted with thin redlines in Fig. 2. In Fig. 3 (a) ’hori-zontal’ cross-section | E ( x, y, z = z ) | is fixed at the centerof the gap between the silicon waveguide and the elec-trode. In Fig. 3 (b) the cross-section | E ( x, y, z = z ) | shows the field in the center of the waveguide. It is evi-dent from the pictures that the waveguide mode, passingthrough the sandwich is smoothly transformed into theplasmon and back. The ’side’ view | E ( x, y = y , z ) | inFig. 3 (c) illustrates the same process. Since the modesinside the modulator are mixed by strong interaction, werecognize a plasmon by the high value of the field in-tensity near the golden surface. Note the scaling factor,which is different in Fig. 3 (a) and (b), and was intro-duced for the better visualization. Fig. 3 (e) and (f)show the transverse cross-section that illustrate the dis-tribution of the field at the entrance of the modulator,and in the center, where the plasmonic state is most pop-ulated. One can see that the metal edges act as couplingproviders (see Fig. 3 (d)) for the y -polarized waveguidemode and allow to concentrate the field in the plasmonicgap (Fig. 3 (f)), where it can be efficiently modulated.Geometrical parameters of the plasmonic rail, in par-ticular its width R w , can be used to tune the interactionbetween modes, which influences the period of beats. Inthe proposed model (Fig. 2) the period is equal to 6 . µ mand the length L of the modulating sandwich is chosenaccordingly. If these lengths are not synchronised, lossesat the output of the modulator, related to the mode con-version (namely, the conversion of the hybrid mode intothe waveguide mode), are generally larger. It is suggestedthat the sandwich length must be equal to an integernumber of beats periods. An example of the devise witha double length is presented in the last section. How-ever, multiple conversions of the signal to the plasmonand back are also connected with additional losses. Ac-cording to our computations, 6 . µ m long sandwich, con-taining just one period, allows to reach an acceptablemodulation depth as it is discussed in Sec. VI. For theselected geometry the transmission coefficient (the ratioof the output and input intensities I out /I in ) T = 0 . κ ol = 10 log I in I out = 1 .
27 dB . (15)The second important geometrical parameter of the plas-monic rail is its height R h . If it is too large, an additionalgap mode appears which interacts with existing modesand makes the picture more complicated. In our geome-try (55 nm height of the rail) this mode is attenuated. V. SWITCHING
To understand the state switching in the modulatorand design the optimal structure of the sandwich, thedetailed drift-diffusion analysis based on Eq. 8 and Eq. 9is performed. Main results are summarized in Fig. 4.Charge density evolution in the modulator is symmetricin the x -direction. Despite the absence of the symme-try in the y -direction, caused by the plasmonic rail, onedimensional charge transport models provide a precisedescription of the accumulation layer at the ITO/HfO boundary. For drift-diffusion computations the rail hasbeen ignored and one dimensional models used to obtainthe charge concentration profiles. We also perform 2Dcomputations in some regimes for the comparison, wherethe rail was modeled explicitly (see the inset in Fig. 4(b)).In order to develop an essential understanding of themodulating sandwich one should think about it as anano-capacitor. In our model the capacitor is producedby the combination of a doped silicon, HfO and ITO(see the inset in Fig. 4 (a)). The Au contact is attachedto ITO. It is not explicitly modelled in the drift-diffusionapproach, but enters the equations through the boundaryconditions. Silicon is assumed to be grounded. The thinlayer of hafnium oxide acts as an insulator. Electron den-sities in silicon and ITO are 5 · cm − and 10 cm − respectively, which allows them to conduct current. Notethat ITO films can be doped up to the level 10 cm − [44]. If the voltage is applied, the charge is accumulatedat the capacitor. When the critical charge concentrationis reached, ENZ effect is initiated in ITO leading to the n-Si H f O I T O (a)(b) n-SiITOAu FIG. 4: Charge distribution in the modulator (a) whenthe external voltage is switched off and theconcentration profiles are determined by initial dopinglevel and electron affinities of materials (b) whenvoltage is applied (three different voltages are shown).Mind 100 times scale difference between (a) and (b).Solid vertical lines show the boundaries betweenmaterials. Horizontal dashed line marks the value ofresonant concentration n enzc . The insets show thestructure of the sandwich and 2D charge distribution,where the accumulation layer is shown with red.strong attenuation of the optical signal. It is easy to eval-uate the critical concentration from Eq. 13, assuming thereal part is equal to zero: n enzc = m ∗ e ε ∞ ε ( ω + γ ) = 6 . · cm − (16)( λ = 1550 nm is assumed). Having in mind the classicformula for a capacitor Q = CU , where C is a capaci-tance, U - voltage and Q - accumulated charge, one mayconclude that since Q is fixed by Eq. 16, and we want U to be small (for practical applications), the capaci-tance, on the contrary, should be large. To increase C one may decrease the thickness of the insulating layer,which is practically difficult and potentially allows elec-trons to tunnel through the capacitor as, for example, inthe floating gate transistor, producing undesired effects.Another way to increase C is to use high a quality dielec-tric with a large static permittivity, like HfO with ¯ ε = 25[55] [54]. In our model, 10 nm thick layer of the hafniumoxide allows to keep the voltage below 10 V. Based onthis analysis one should avoid also structures with mul-tiple layers of insulator, since the capacitance of suchjunction is decreased. Of course, if C is increased, thecharging time also becomes larger and makes the mod-ulation process slower. Thereby, there is a certain tradebetween the modulation speed and power consumption.Having in mind a sandwich with a single insulatinglayer, an active material (ITO) and a metallic contact,two possible configurations can be considered. Insteadof Si/HfO /ITO/Au structure used in our model, alter-native Si/ITO/HfO /Au sandwich can be used. In sucha configuration ITO is in a direct contact with silicon,the capacitor is formed between ITO and Au and plas-mons are excited at the boundary between Au and HfO .Since Si and ITO have different electron affinities, thereis a contact potential difference at the Si/ITO boundaryand the charge density distribution function has an ex-tra peak there. It produces undesirable refractive indexgradients in the modulator. It is also hard to model thispeak, since its amplitude depends on the details of thefabrication. On the other hand, the described effect is no-ticeable only at low voltages and becomes small at highvoltages (like the modulator switching voltage, around 8V). From this point of view both types of sandwiches canbe used. In our model we prefer to use Si/HfO /ITO/Austructure, minimizing uncertainty in parameters and pos-sible parasitic effects.The charge density distribution in the modulator(along z -axis) at the voltage switched off is plotted inFig. 4 (a). Boundaries of materials are shown with verti-cal black lines. The structure of the sandwich is demon-strated in the inset. It is clear from the picture, thatthere is a small charge at the capacitor even at zero volt-age. It is a consequence of difference between electronaffinities of ITO and Si which results in a small potentialdifference (around 0 .
75 V). Asymptotical values of thecharge distribution far from the capacitor correspond tothe doping level of materials. To switch the state of themodulator the voltage is applied to the golden contact.Electron density profiles are presented for voltages − − −
10 V in Fig. 4 (b) (note the scale differencewith Fig. 4 (a)). The horizontal dashed line shows theENZ value of the concentration (Eq. 16) that should bereached for the efficient optical modulation. For a givencapacitance C , the critical value is reached starting fromvoltage − C as C = eS mod U (cid:90) z ∈ IT O ( n Uc ( z ) − n c ( z )) dz = 0 . , (17)where S mod = 6 · nm is a transverse area of themodulating sandwich and the integral is taken through the ITO lead of the capacitor, to evaluate the accumu-lated charge in the active material. Further, n Uc is thecharge density profile at the voltage U and n c is the pro-file for U = 0 (blue line in Fig. 4). The classic formula fora parallel-plate capacitor applied to our nano-capacitorgives very close values C ≈ ε ¯ ε S mod d , (18)where d is a thickness of HfO . It is clear from this for-mula that by changing the insulating material from HfO to SiO the capacitance becomes approximately 6 timessmaller and requires 6 times larger voltage to switch themodulator. The lack of capacitance can be compensatedby decreasing the thickness of the insulator from 10 nmto 1 . remains the main candidate for the insulating ma-terial in the proposed device.One important characteristic number obtained fromthe drift-diffusion model is the thickness of the accumula-tion layer in ITO. According to our computations (Fig. 2(b)) this layer is quite thin. The specific number can bedefined using the equation1 t z + t (cid:90) z n Uc ( z ) dz = n enzc , (19)where z is a coordinate of HfO /ITO boundary and t isa thickness of the active layer. The numerical evaluationprocedure returns the value t ∼ ∂n/∂t = 0). The forma-tion of the accumulation layer, depicted in Fig. 4, is alsostudied dynamically. According to the computations, theformation time of the layer is ∆ t ∼ f ∼ t ∼ RC . To evalu-ate the active resistance of the leads R one can use theformula R = lσ − S − mod , where σ = n c µ n e is the con-ductivity and l is the conductor length (in z -direction).The active resistance of our modulator is defined by theresistance of the silicon waveguide, mainly because ofits transverse size (the resistance of ’Si lead of the ca-pacitor’ is roughly one order of magnitude larger, than (a) “Top” view, | E (x, y, z=z )|, scale x0.75“Bottom” view, | E (x, y, z=z )|“Side” view, | E (x, y=y , z)| V/m 06 (b)(c)(d) (e)
FIG. 5: Distribution of the electric field absolute value | E ( x, y, z ) | inside the modulator when the voltage U = − . R fordoped silicon to evaluate the charging time gives thevalue RC ∼ VI. MODULATOR OFF-STATE
The accumulation layer with special ENZ properties isformed at the ITO/HfO boundary as the voltage is ap-plied. The layer does not support its own optical modesdue to the small thickness. Despite the fact, that ITOin ENZ regime becomes ’more metallic’ in a sense thatits charge density becomes larger, it cannot support sub-wavelength SPP modes at the boundary with a dielec-tric either. To provide appropriate conditions for plas- mon excitation the ITO layer should have a large nega-tive real part of the permittivity, which is not the casehere. The interaction of ENZ layer with the existingmodes is, thus, dictated by boundary conditions. Sincethe component of the displacement field vector normal tothe surface of ITO should be continuous, the condition E = E ε / ( ε (cid:48) + iε (cid:48)(cid:48) ) is fulfilled [31], where E and E are electric fields outside and inside ENZ layer. If thereal part of ITO permittivity ε (cid:48) passes through zero, aresonance in the local field intensity takes place, with awidth defined by the imaginary part of permittivity ε (cid:48)(cid:48) .At the same time ε (cid:48)(cid:48) defines optical losses in ITO andthey grow notably, approximately 65 times, inside theaccumulation layer at the ENZ regime. Therefore, an ef-ficient absorption occurs, when the significant part of thefield is concentrated inside ITO accumulation layer. Anopposite effect appears when we change the polarity ofthe voltage. In the resulting depletion layer ε (cid:48) may belarger than ε . In this case ITO pushes the field out ofthe active layer.The distribution of the electric field inside the modu-lator under the applied voltage is shown in Fig. 5 (a)-(c). The output in this case is significantly weaker thanthe input. The value of the transmission coefficient is T off = 0 . T on = 0 . κ ext = 10 log I on I off = 15 .
95 dB . (20)An oscillatory mode behavior, typical for the on-state(Fig. 3), cannot be observed for the off-state in Fig. 5.The beats do not appear also for the elongated model asit will be shown below. It suggests that the presence ofthe accumulation layer changes the character of interac-tion between modes in the modulator and may affect theperiod of beats.A small, about 1 nm, width of the accumulation layercreates a resolution challenge in numerical computations.It is complicated to resolve density and, thus, permittiv-ity variation given in Fig. 4 (b) in 3D computations, sincethe profile varies significantly at a small scale. To buildan efficient numerical model we used the concept of aneffective accumulation layer with constant ENZ parame-ters placed in a contact with ordinary unperturbed ITO.For a verification of the selected approach we performed2D mode analysis, where two cases were compared. First,the hybrid plasmonic mode was computed when the elec-trons density distribution (and corresponding permittiv-ity distribution in ITO) given in Fig. 4 (b) was used ex-plicitly. Second, we computed the same mode, but usingan effective layer with a fixed width and constant ENZparameters instead of a distribution. Computations wererepeated with varying effective layer width until the effec-tive index of the mode and the field distribution becamevery close to the original computation with continuouspermittivity distribution. This procedure can be consid-ered as an alternative way to determine the width of the0 (b)(a) FIG. 6: Dependence of the modulator transmissioncoefficient on the applied voltage (a) and on thewavelength of light for both on-state and off-state (b).Vertical solid black line corresponds to the wavelength1550 nm. C+L telecom band is shown using verticaldashed black lines. The optical bandwidth of themodulator, defined in the text, is shown with verticalgray dash-dotted lines.accumulation layer. It gives 1 nm thickness, which co-incides with the evaluation in Sec. V. The effective layerconcept is used then to build a 3D numerical model. Thefield distribution in the hybrid mode is shown in Fig. 5 (e)(note the intensity maximum inside the thin accumula-tion layer at ITO/HfO boundary). Taking into account,that only a small fraction of ITO can be switched to ENZregime, we obtain the modulator characteristics that areslightly less impressive than those where the whole vol-ume of ITO is considered as active. Nevertheless, ourevaluation is realistic and obtained numbers are sufficientfor potential applications.Despite the simplification provided by the effectivelayer model, it is still a challenge to resolve a 1 nmthick material in 3D. To compute the field numerically,strongly inhomogeneous mesh is used (see Fig. 5 (d)).First, the planar triangular mesh is built at the bound- ary between the insulator and ITO with a variable ele-ment size of 10 - 20 nm. Then the plane is copied inthe directions of HfO and ITO ( z -direction) with steps3 nm and 0 . . − . /ITO boundary instead of the ac-cumulation layer. Nevertheless, according to Drude the-ory (see Eq. 13), ENZ effect is not possible in this regimeand corresponding variation of the refractive index ismuch less pronounced. For this reason, strong modula-tion regime does not exist at positive voltages and we donot consider them. Optical bandwidth of the modulatorcan be defined using the data in Fig. 6b, where the wave-length dependence of the transmission in both on-stateand off-state is shown. We define the band, using Eq. 20,as an interval around the central wavelength (1550 nm)where the extinction drop is less that 5 dB. The obtained421 nm wide band is shown in Fig. 6b using vertical graydash-dotted lines (at 1385 nm and 1806 nm). It is ob-viously much larger than the C+L telecom band shownusing vertical black dashed lines.The performance and main characteristics of plasmonicelectro-optical modulators depend on the length of themodulating sandwich in a nontrivial way. As discussedin Sec. IV, the period of a population exchange betweenthe waveguide mode and the plasmonic mode should beconsistent with the length of the modulator. Opticalcomputations for the model with 13 . µ m long modu-lating sandwich, i.e. twice the oscillation period, areshown in Fig. 7. Both on- and off-state are demonstratedwith corresponding optical losses and the extinction co-efficients. The transmission coefficient for the doubledmodel ( T = 0 . . . µ m long model ( T = 0 . . µ m long1 (a)(b) Elongated model, on-state. T=0.583. Optical losses 2.34 dB. x10 V/m1234506
Elongated model, o ff -state. T = 0.000676. Extinction coe ffi cient 29.36 dB. FIG. 7: Electric field distribution in the enlarged model (length L of the modulating sandwich is 13 . µ m) of theelectro-optical modulator. On-state (a) and off-state (b) are demonstrated. Obtained values of optical lossescoefficient and extinction coefficient are 2 .
34 dB and 29 .
36 dB respectively.model), which can be an advantage for a certain types ofapplications. Another advantage of the long model is thedecreased active resistance. Since the area of the sand-wich S mod and, consequently, the electric contact areain the xy -plane is twice larger, the resistance R is twicesmaller, which decreases the RC time of the capacitorand makes the modulator bandwidth larger. It is remark-able that 13 . µ m long waveguide based electro-opticalmodulator is still much more compact than many so-lutions without HPWG. At the same time, low opticallosses and predicted THz bandwidth limit of the shortmodel (Fig. 5) also look promising for applications.If the geometry of the sandwich is modified, the char-acter of the interaction between modes changes as well,which influences the period of beats in HPWG. There-fore, the length of the modulator should be optimizedfor each geometry of the sandwich (taking into accountboth on- and off-states). Since the numerical optimiza-tion is computationally expensive and time consuming,the development of a reasonable analytic theory of themodes interplay in HPWG would be a nice task for thefuture. CONCLUSION
The new approach utilizing edge plasmons in optoelec-tronics is developed. It allows to couple a horizontallypolarized waveguide mode to the plasmonic mode via ap-propriately designed HPWG. The described idea helps todesign compact and efficient electro-optical modulators.The following advantages of the proposed design shouldbe emphasized: (a) the possibility to remove the po-larization constraint, thus matching the modulator with recently proposed highly efficient grating couplers [36]without the need to use lossy polarization converters; (b)steady planar geometry following from the fact that sili-con waveguides for horizontally polarized modes have thewidth lager than height; (c) consequently, the possibilityto keep the active electric resistance R lower; (d) thehorizontal polarization makes it possible to put an elec-trode at the bottom of the waveguide and avoid parasiticplasmon modes excitation. The proposed design impliesthe usage of a plasmonic rail with two golden ribs atthe Au/ITO boundary supporting edge plasmon modes.The design is relatively simple, which is crucial for po-tential experimental realizations and future applications.To optimize the geometry and structure of the device,numerically heavy 3D optical model based on Maxwellequations is developed. The details of charge densitybehavior are obtained from the drift-diffusion system ofequations. The electron distribution in the accumulationlayer at ITO/HfO boundary is studied and implementedin the optical computations. The most important charac-teristics of the device, such as optical losses, the extinc-tion coefficient and the bandwidth are computed. Theobtained numbers make the device attractive for poten-tial applications. ACKNOWLEDGEMENT
This work was financially supported by the Ministry ofScience and Higher Education of the Russian Federation,project No.RFMEFI58117X0026.I.A.P. thanks Victor Vysotskiy for the support of Par-dus cluster and fruitful discussions. [1] S. Datta,
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