Topological Graphene plasmons in a plasmonic realization of the Su-Schrieffer-Heeger Model
Tatiana G. Rappoport, Yuliy V. Bludov, Frank H. L. Koppens, Nuno M. R. Peres
TTopological Graphene plasmons in a plasmonic realization of the Su-Schrieffer-HeegerModel
Tatiana G. Rappoport,
1, 2
Yuliy V. Bludov, Frank H. L. Koppens,
4, 5 and Nuno M. R. Peres
3, 6 Instituto de Telecomunicações, Instituto Superior Técnico,University of Lisbon, Avenida Rovisco Pais 1, Lisboa, 1049001 Portugal Instituto de Física, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, 21941-972 Rio de Janeiro RJ, Brazil Department and Centre of Physics, and QuantaLab,University of Minho, Campus of Gualtar, 4710-057, Braga, Portugal ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute ofScience and Technology, 08860 Castelldefels (Barcelona), Spain ICREA-Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain International Iberian Nanotechnology Laboratory (INL),Av. Mestre José Veiga, 4715-330, Braga, Portugal (Dated: February 23, 2021)Graphene hybrids, made of thin insulators, graphene, and metals can support propagating acousticplasmons (AGPs). The metal screening modifies the dispersion relation of usual graphene plasmonsleading to slowly propagating plasmons, with record confinement of electromagnetic radiation. Here,we show that a graphene monolayer, covered by a thin dielectric material and an array of metallicnanorods can be used as a robust platform to emulate the Su-Schrieffer-Heeger model. We calculatethe Zak’s phase of the different plasmonic bands to characterise their topology. The system showsbulk-edge correspondence: strongly localized interface states are generated in the domain wallsseparating arrays in different topological phases. We find signatures of the nontrivial phase which candirectly be probed by far-field mid-IR radiation, hence allowing a direct experimental confirmationof graphene topological plasmons. The robust field enhancement, highly localized nature of theinterface states, and their gate-tuned frequencies expand the capabilities of AGP-based devices.
Topology can lead to intriguing physical phenomenaand it is at the heart of modern condensed matterphysics [1–4]. It has been successfully extended to variousclassical wave systems, such as photonics [5–7], acous-tic [8] and mechanical systems [9]. It also has beenplaying an increasingly important role in nanophotonon-ics [10], offering alternative ways to design novel opticaldevices [11].In one dimension, the celebrated Su-Schrieffer-Heeger(SSH) is probably the simplest and most representativemodel with non-trivial topology [12, 13]. Originally, itdescribes electrons in a one-dimensional tight-bindingmodel with staggered hopping amplitudes, defined as in-tracell and intercell hoppings [12]. Depending of the ratiobetween the two hopping amplitudes, the chain can havetwo topologically distinct ground states. The variation ofthis ratio leads to a topological phase transition betweenthe two phases, with the band gap closing and reopening.If the intercell hopping is stronger than the intracell hop-ping, the system is in a non-trivial topological phase. Inthis case, the bulk-edge correspondence [14] predicts theexistence of end-states, and interface states when two lat-tices with different topological phases are connected [15].Photonic and plasmonic systems provide a flexibleplatform for the SSH model [16–19]. The effective in-tracell and intercell hoppings can be controlled, for ex-ample, by tuning distances via nanofabrication. Nontriv-ial topology in coupled plasmonic nanoparticle arrays hasbeen previously realized in 1D plasmonic nanoparticle ar- rays [20–24]. These systems, similar to the Su-Schrieffer-Heeger model, exhibit highly localized edge states at theirends, which are robust against perturbations [25]. How-ever, similarly to dielectric photonic crystals, it is diffi-cult to dynamically tune the 1D plasmonic nanoparticlearrays and control their edge and interface states. Toovercome these limitations, one possibility is the use ofhighly tuneable graphene plasmons [26, 27].Graphene Plasmon-polaritons (GP) are vertically lo-calized electromagnetic fields (that is, surface waves) thatcan be excited in both the mid-infrared (MIR) and theTerahertz (THz) spectral ranges. They present oscil-latory behavior at the interface between graphene anda dielectric [28]. They can exhibit high degree of spa-tial confinement when compared to a wavelength of thesame frequency in free space [28–34]. The configurationof Metal-Insulator-Graphene (MIG) for GPs [28, 35–38]involving a thin insulating layer, can hold vertically con-fined modes with much larger momentum than normalgraphene plasmons. In this limit, the dispersion rela-tion of the GP becomes linear and the mode is knownas acoustic graphene plasmon (AGP)[35, 37, 39]. Withthis hybrid system, records in the spatial confinement ofelectromagnetic radiation has been achieved [37, 40, 41].AGPs in periodic systems, e.g. involving periodicmetallic rods on graphene (separated by an insulator),form plasmonic bands and present a wealth of differentphysical effects [42]. The AGP’s lateral confinement thatoriginates from the metallic nanostructure mimics a plas- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b monic tight-binding model where graphene’s gating andthe distance between rods can control the effective hop-ping, modifying the band-structure and modulating theband gaps.Here, we propose a novel one-dimensional topologicalgraphene plasmonic crystal that consists of a monolayergraphene on top of a bulk substrate S with permittiv-ity ε S , separated from a periodic structure of silver rodswith cross-section of area W = 75 × nm by a thindielectric spacer, of thickness d = 3 nm and permittivity ε d (see Fig 1a). The main advantage of this structure isthat it is based on a recent experimental setup to creategraphene acoustic plasmons [37], and therefore is experi-mentally feasible. The extra ingredient consists in usingtwo different separations between the rods, which can beeasily fabricated with the same techniques. Furthermore,it avoids the use of metagates [26, 27].As illustrated in Fig 1b, the 1D lattice unit cell con-tains two identical silver rods separated by a distance a and symmetrically located with respect to the center ofthe unit cell. Neighboring rods from different unit cellsare separated by a distance b . The periodic structure hasa period L = a + b +2 W . As a dictates the intracell effec-tive hopping and b is linked to the intercell effective hop-ping, it is convenient to define the ratio f = ( a − b ) / ( a + b ) that controls the topology of our system. f = 0 ( Fig.1a) implies the periodic structure studied previously [42]with a single effective hopping. Positive (negative) valuesof f specifies that the intracell effective hopping of ourSSH model is larger (smaller) than the intercell one, asshown in Fig 1b.We perform full-wave finite element frequency domainsimulations[43] and semi-analytical plane-wave expan-sions to characterize our plasmonic SSH model (see Sup-plemetary Materials for the details [44]). For simplicity,graphene is simulated as a single layer with optical con-ductivity that is given by a Drude like expression σ g ( ω ) =4 σ E F / ( π ( (cid:126) γ − i (cid:126) ω )) [28], where σ = e / (cid:126) , E F is theFermi energy, γ is the relaxation rate and ω is the fre-quency of the incident light. Since we will be consideringlarge graphene Fermi energies, finite temperature play nosignificant effect in our results. The frequency-dependentrelative permittivities of Ag are taken from Ref. [45].First, we analyse the eigenfrequencies and eigenmodesof the system and distinguish their topological phasesfor different values of f . We then proceed to describethe interaction of EM radiation with our MIG structure.We consider a p -polarized monochromatic plane-wave im-pinging on the array of metallic rods at normal incidence.We calculate the absorption spectra resulting from thecoupling of AGPs with far-field radiation, which can beused to design and model experiments. In this context,we consider different setups: a periodic system, an inter-face of two semi-infinite arrays with different topologicalphases, and edge states of a semi-infinite array interfac-ing a perfect electric conductor (PEC) . Unless otherwise specified, (cid:15) S = (cid:15) d = 1 , E F = 0 . eV and γ = 3 meV.The dispersion of the plasmons in a periodic systemwith a single rod per unit cell of length L/ presentsseveral plasmonic bands [42]. If the same system is rep-resented by a unit cell of length L with two evenly locatedrods ( f = 0 ), the dispersion can be depicted in a Bril-louin zone k = [0 , π/L ] which has half of the size of theoriginal one. As a result of the band folding, the disper-sions cross each other at k = π/L , which is a point ofdegeneracy. When calculating the plasmonic band struc-ture, from the analogy with a simple one-dimensionaltight-binding SSH model, we should expect a splittingof the original bands for f (cid:54) = 0 exactly at degeneratepoints of the band folding (that is, at k = π/L ), withthe size of the new gap being proportional to | f | . Thiscan be observed in our band structure and loss functioncalculations: figure 1c shows a density plot of the lossfunction calculated with the plane-waves expansion witha superimposed band-structure for f = 1 / . Each of thetwo original lowest bands for f = 0 have one degeneratepoint at k = π/L , which leads to a band folding inducedband gap, splitting the original bands in two. The yel-low rectangle in Fig. 1c highlights one of these bandsplittings, which is very small, because of the value of f ,but illustrates nevertheless the effect of the band fold-ing. The new Bragg gaps for f (cid:54) = 0 are always locatedat k = ± π/L . Fig. 1d presents the same data but for f = 2 / and one can see that the band folding inducedgap increases for large values of f and there are fourwell separated bands labeled from 1 to 4. As expected,the band gap varies linearly with | f | for small values of f (see the supplementary material [44]), and the bandstructures for ± f are exactly the same, although theycorrespond to different topological phases. (a) (b) k (2 π/ L ) f r eq ( T H z ) (c) k (2 π/ L ) (d) FIG. 1. (a)-(b) Illustration of the one dimensional array ofrods with two rods per unit cell and f = 0 and f = 2 / re-spectively. (c)-(d) Plasmonic band structure (dotted line) andloss function superimposed with the band structure (dottedlines), calculated for f=1/15 (c) and f=2/3 (d). The yellowrectangle in (c) highlights the band splitting at k = π/L thatoccurs exactly at the degenerate point in the band folding for f = 0 . The numbers in panel (d) label the different bands The 1D array has chiral symmetry, as the unit cellconsists of two interconnected sublattices (one for eachrod) that can be interchanged without modifying thesystem properties. One-dimensional periodical systemswith chiral symmetry can be characterized by a topo-logical invariant known as Zak phase [46]. If the unitcell has an inversion symmetry, the Zak phase is quan-tized as π (non-trivial) or 0 (trivial). To evaluate theBerry phase for electromagnetic waves in the absence ofmagneto-electric coupling, either the electric or magneticfields can be considered in the calculation of the Berryconnection (cid:126) Λ n,(cid:126)k [47]. We adopted the electric field in ourcalculations, where the permittivity tensor is isotropicand given by ˆ (cid:15) ( (cid:126)r ) = (cid:15) ( (cid:126)r ) . After considering these sim-plifications, the Berry connection for an isolated band isgiven by [16, 48]: (cid:126) Λ E n,(cid:126)k = i (cid:90) u.c d(cid:126)r(cid:15) ( (cid:126)r ) (cid:126)E ∗ n,(cid:126)k ( (cid:126)r ) · ∇ (cid:126)k (cid:126)E n,(cid:126)k ( (cid:126)r ) , (1)where (cid:126)E n,(cid:126)k ( (cid:126)r ) is the periodic-in-cell part of the normal-ized Bloch electric field eigenfunction of a state on the n th band with wave-vector wavevector (cid:126)k .The periodic structure has periodicity in ˆ x and the sys-tem is a one-dimensional plasmonic lattice. In this situa-tion, the Zak phase [46] is defined as θ n = (cid:82) π/L − π/L dk Λ n,k .The integral of the Berry connection over the BZ − π/L ≤ k < π/L can be approximated as a summation of the con-tributions of small segments. If the BZ is divided into N segments where k N +1 = k , e − i θ n ( k i ) ≈ − i θ n ( k i ) =1 − iΛ n,k δk . As we are dealing with plasmons confinedin the region between the metallic rods and the graphenesheet, without loss of generality, we can calculate the Zakphase at a fixed height z located in the spacer with ho-mogenous permittivity (cid:15) S . The Zak phase of this segment θ n ( k i ) for a band n is given by e − i θ n ( k i ) = (cid:90) u.c dx (cid:126)E n,(cid:126)k i ( x, z ) (cid:126)E n,(cid:126)k i +1 ( x, z ) , (2)where θ n can be calculated in a gauge-invariant for-malism as θ n = − Im [log( (cid:81) Ni =1 e − i θ n ( k i ) )] [49].Alternatively, θ n can be obtained by inspecting theparity of the field profiles. If the symmetries of theeigenmode at k = 0 and k = ± π/L are the same (dif-ferent), the Zak phase of this band is quantized as ( π ).Following this procedure, we illustrate the differences inthe parity of the field profiles for band 2, highlighted ingreen in Fig 1d. Figure 2a, b presents the field profile E xn,k ( x, z ) k = 0 and k = π/L respectively. z is lo-cated in the middle of the spacer, between graphene andthe rods. The phase of the eigenmode is fixed in such away that (cid:126)E n,k =0 ( x, z ) is real. For k = 0 (Fig. 2a) thefield profiles for f = ± / are both even with respect tothe inversion center of the unit cell, located at x = 0 . Onthe other hand, for k = π (Fig. 2b) the profile is even for f = − / and odd for f = 2 / . When comparing Fig.
100 0 100 x(nm) E x , k ( x , z ) (a)
100 0 100 x(nm) (b) f=-2/3f=2/3 n f r e q ( T H z ) (c) FIG. 2. The periodic part of the longitudinal component ofthe electric field E xn,(cid:126)k ( x, z ) for band 2, highlighted in Figure1d for k = 0 (a) and k = π/L (b) where z is located in themiddle of the spacer 2.0 nm above graphene and − L/ ≤ x ≤ L/ . The dashed curves show the profiles for f = − / whilethe profiles of the solid curves are calculated for f = 2 / .(c) Energy spectrum of a composite system consisting of twoconnected finite arrays of 20 unit cells each with f = ± / respectively, sandwiched by PECs. The mid-gap states arelocated in gaps after an odd number of bands.
2a and b it is clear that for f < ( f > ) the symmetriesof the eigenmodes for k = 0 (a) and k = π/L (b) are thesame (opposite) so that θ n = 0( π ) , which corresponds tothe value obtained by the Zak phase calculation followingequation 2.Lets us now address the physical consequences and ex-perimental signatures of the Zak phase in the grapheneplasmonic crystal. To obtain a clear signature of thetopology, one route is the observation of interface statesfor different values of f . Figure 2c shows the energy spec-trum for a single finite system consisting of two neigh-bouring arrays of 20 unit cells each with f = ± / re-spectively. The system is sandwiched by perfect elec-tric conductors (PECs). As discussed previously, systemswith the same | f | have the same spectrum. Consequently,both arrays have the same band-structure and the spec-trum of the four lowest bands for the composite systemis similar to the unfolded version of the band structureof Fig 1d. However, there is a clear presence of mid-gap states inside the Bragg gaps. If two semi-infinitesystems with different topological phases form an inter-face, the existence of a topological interface state in agiven band gap is consistent with the bulk-edge corre-spondence. Thus, the mid-gap states of Figure 2c areassociated to the different Zak phases of each individualarray with f = ± / , corroborating the previous anal-ysis for periodic systems. The original bands for f = 0 are split in two for f (cid:54) = 0 but they do not cross any otherband, independent of the value of f . Because of the con-servation of topological numbers in band theory, the sumof the Zak phases of each pair of these bands is alwaysthe same, regardless of the sign of f , although each indi-vidual band can change its phase when inverting the signof f and the Bragg gap goes to zero. This results in theabsence of mid-gap states in gaps located after an evennumber of bands (see Fig 2c).To explore the experimental signatures of the Zakphase, we consider the coupling of the plasmonic crys-tal with far-field radiation. In this case, we have a p -polarized monochromatic plane-wave impinging on thearray of metallic rods at normal incidence. Let us firstconsider the periodic system and see if the splitting ofthe bands with different values of f can be observed infar-field experiments. At normal incidence, TM modescouple with states with k x ∼ . Because of this, far-fieldexperiments cannot directly obtain the linear dependenceof the gap with f , as the gap opening associated to theSSH model occurs at k = π/L . Still, it is possible tocapture: 1) the existence of an extra band for f (cid:54) = 0 , and2) the band separation at k x = 0 for increasing valuesof | f | . The first bands that can be seen in the far-fieldexperiments with normal incidence, are bands 3 and 4,highlighted in Fig 1 c. This is illustrated in figure 3a:the absorption spectra for f = 0 has a single peak atthis range of frequencies. For f (cid:54) = 0 , the band is split inbands 3 and 4 and produces two peaks in the absorptionspectra, where their position is dictated by the frequencyof the bandstructure at k = 0 . The peak separation isnot directly related to the band gap but, instead, to thevalues of the band structure for k = 0 .To analyse the interface states, we begin by consider-ing a finite plasmonic lattice with f = − / with PEC( perfect electric conductor) boundary conditions. Thissystem presents exactly the same far-field response of theperiodic lattice (see Fig. 3). We can now compare it withthe response for the interface considered in Figure 2c, in-volving two joined arrays with f = ± / . In this case,the interface state between different topological phases ofthe two chain leads to an extra absorption peak locatedbetween the two original peaks of the infinite system,seen in Fig. 3b. Figure 3c shows the field enhancementat the interface state at this particular frequency. Thedashed squares indicate the position of the metallic rodsand the interface between the two different lattices is lo-cated at x = 0 . One can see that in the vicinity of thedomain wall separating the two lattices, the field is en-hanced in the whole space, including the region abovethe rods. Figure 3d exhibits the profile of | (cid:126)E ( (cid:126)r ) | / | (cid:126)E | ,at z , located in the spacer between graphene and therods, where (cid:126)E is the electrical field. In this case, thefield enhancement is of the order of 120-140, which is ofthe order of the field enhancement of the main absorp-tion peaks [42]. The electric field profile in Fig. 3c hasa maxima that is strongly localized in the region of theinterface between the two arrays. This differs from theextended electric field profile normally seen in the inter-
30 35 40 freq(THz) A b s o r p t i on (a) f=0.5f=0.4f=0.3f=0.2f=0.1f=0.0
30 35 40 freq(THz) (b)
Single Domain - f =2/3Domain Wall - f = ± (c) x ( m) | E / E | (d) − − FIG. 3. (a) Absorption spectra for a periodic array and dif-ferent values of f = ( a − b ) / ( a + b ) . (b) Absorption spectra ofa finite plasmonic crystal with f = − / PEC boundary con-ditions (dashed lines) and the interface containing two con-nected arrays of 6 unit cells each, with f = ± / (solid line).(c) Electric field distribution E y ( (cid:126)r ) /E for the region of theinterface consisting of two connected arrays with f = ± / .The dashed squares denote the position of the metallic rods.(d) Electric field enhancement | (cid:126)E ( (cid:126)r ) | /E in the spacer be-tween graphene and the rods. x = 0 specifies the interfacebetween the two arrays with f = ± / . face of photonic crystals with different Zak phases, wherethe interface state has a width of several unit cells [50].The topological nature of this peak can be further con-firmed by comparing the absorption peaks of interfacesbetween two arrays with f = f and f = f where | f | (cid:54) = | f | . In this case, one can produce interfaces betweensystems in the same topological phase (sgn( f )=sgn( f ).However, only interfaces between systems with sgn( f ) (cid:54) = sgn( f ) produce interface states (see S. M). Although thetopological nature of the bands cannot be easily tuned in-situ, it is still possible to use a gate to modify the opticalconductivity in graphene. This leads to a change in thesize of the band widths and gaps and the exact frequencyof the edge state. The flexibility to modify the frequencyof the interface state can be useful for technological ap-plications.We can now consider a vacancy or a void in the unit-cellneighboring one of the PECs. This is obtained by remov-ing the two silver rods belonging to the last unit cell ofthe right, as illustrated in Fig. 4b and c. In this case, thecalculation of the eigenfrequencies of the finite system donot produce in-gap states that are related to the topol-ogy. Still, when analysing the interaction with far-fieldradiation, different responses emerge, depending on theZak phase of the plasmonic band. For positive values of f , there is an extra peak in the absorption spectra (Fig.4a) which is related to an interface state and electric fieldenhancement of the electric field at the vacancy (Fig. 4b).However, for negative f s, the absorption spectra is verysimilar to the periodic array and no interface states areobserved. Instead, a resonator is formed between the lastrod of the structure and the PEC. Depending on the size D of the resonator, its eigenfrequency can be located in-side the gap ( f > ) or in the plasmonic band f < ),which explains the absorption spectra.Although we cannot connect this response to topology,it can still be used in situations where one needs to pro-duce a field confinement in the absence of the metallicrod, such as in sensing applications.
30 35 40 freq(THz) (a) f=0.5f=-0.5 (b) x( µ m) E y / E (c) − − FIG. 4. (a) Absorption spectra for a plasmonic crystal with f = ± / with one vacancy interfacing a perfect electric con-ductor. Electric field distribution E y ( (cid:126)r ) /E for lattices with(b) f = 1 / and (c) f = − / with one vacancy interfacinga perfect electric conductor. The dashed squares denote theposition of the metallic rods. Conclusions
We proposed a simple structure to sim-ulate the Su-Schrieffer-Heeger Model for plasmons ingraphene, which avoids the use of meta-gating . Oursetup is based on Metal-Insulator-Graphene systems thathost acoustic graphene plasmons. Periodic arrays ofmetallic rods with two rods per unit cell generate plas-monic bands with topological properties that can betuned by the distances between the rods. Interface stateswith strong field enhancement can be created at the in-terface between two arrays with different topologies. Thefrequency of the localized state can be easily tuned bygating graphene, which opens new avenues in plasmonicapplications where light needs to be confined to a pre-cise location in space or where a tunable narrow bandabsorption is needed.TGR acknowledges funding from Fundação para aCiência e a Tecnologia and Instituto de Telecomunicações- grant number UID/50008/2020 in the framework ofthe project Sym-Break and Mario G. Silveirinha for use-ful discussions. N.M.R.P. and F.H.L.K. acknowledgesupport from the European Commission through theproject “Graphene-Driven Revolutions in ICT and Be-yond” (Ref. No. 881603, CORE 3). N.M.R.P. acknowl-edge COMPETE 2020, PORTUGAL 2020, FEDER andthe Portuguese Foundation for Science and Technology(FCT) through project POCI-01- 0145-FEDER-028114.F.H.L.K. acknowledges financial support from the Gov-ernment of Catalonia trough the SGR grant, and fromthe Spanish Ministry of Economy and Competitiveness,through the “Severo Ochoa” Programme for Centres of Excellence in RD (SEV-2015- 0522), support by Funda-cio Cellex Barcelona, Generalitat de Catalunya throughthe CERCA program, and the Mineco grants Ramón yCajal (RYC-2012-12281, Plan Nacional (FIS2013-47161-P and FIS2014-59639- JIN) and the Agency for Man-agement of University and Research Grants (AGAUR)2017 SGR 1656. 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