Polaritons in layered two-dimensional materialsTony Low1*, Andrey Chaves2,3, Joshua D. Caldwell4, Anshuman Kumar1,5, Nicholas X. Fang5,Phaedon Avouris6, Tony F. Heinz7, Francisco Guinea8,9, Luis Martin-Moreno10 and Frank Koppens11,12
In recent years, enhanced light–matter interactions through a plethora of dipole-type polaritonic excitations have beenobserved in two-dimensional (2D) layered materials. In graphene, electrically tunable and highly confined plasmon-polaritonswere predicted and observed, opening up opportunities for optoelectronics, bio-sensing and other mid-infrared applications.In hexagonal boron nitride, low-loss infrared-active phonon-polaritons exhibit hyperbolic behaviour for some frequencies,allowing for ray-like propagation exhibiting high quality factors and hyperlensing e�ects. In transition metal dichalcogenides,reduced screening in the 2D limit leads to optically prominent excitons with large binding energy, with these polaritonic modeshaving been recently observed with scanning near-field optical microscopy. Here, we review recent progress in state-of-the-art experiments, and survey the vast library of polaritonic modes in 2D materials, their optical spectral properties, figures ofmerit and application space. Taken together, the emerging field of 2D material polaritonics and their hybrids provide enticingavenues for manipulating light–matter interactions across the visible, infrared to terahertz spectral ranges, with new opticalcontrol beyond what can be achieved using traditional bulk materials.
In many materials, electric dipoles (for example, infrared-activeoptical phonons, excitons in semiconductors and plasmons indoped materials) can be excited when illuminated1,2, producing
hybrid quasiparticles with photons called polaritons. Thesepolaritons can be sustained as electromagnetic modes at theinterface between a positive (for example, normal dielectric) andnegative permittivity material, leading to propagating polaritons. Inthe case of the plasmon-polaritons (PPs), the negative permittivityis provided by the coherent oscillations of the free carriers, andcan be described by the Drude model. For exciton-polaritons (EPs)and phonon-polaritons (PhPs), it is associated with their resonantoptical absorption, resulting from a highly dispersive permittivity.These optical resonances can also result in a negative permittivity,albeit over a narrow spectral window. To provide a simple, yetcomprehensive overview of these different optical modes, we haveprovided a summary in Box 1.
These polariton modes are characterized by two related lengthscales: the polariton wavelength along the interface and theextension of the evanescent field in the perpendicular direction,both of which are smaller than the free-space wavelength. Theassociated reduced modal volume presents an extremely largelocal density of electromagnetic states at the interface, leadingto strong light–matter interactions. Hence, polaritonics providea way to confine, harness and manipulate light at dimensionssmaller than the diffraction limit3. The emergence of so-called2D or van der Waals materials as hosts for these polaritonicmodes has enabled the first realization and imaging of polaritonswithin atomically thin materials. Due to the inherent materialanisotropy and the exceptional variation of material types (forexample, metallic, dielectric, semiconductor) available within theknown library of 2Dmaterials, a large breadth of different flavours ofpolaritonic modes have been realized and envisioned. This includes
tunable graphene PPs4,5, chiral6,7, anisotropic and hyperbolicPPs8,9; long-lived hyperbolic PhPs10–12 with slow light13,14 andhyperlensing behaviour15,16; EPs with strong binding energies17–21,their complexes20,22–24, and anisotropic excitons25–28. This review isdevoted to the emerging but rapidly developing field of polaritonicsand nanophotonics in the family of 2D materials, with an emphasison the materials and the polaritons physics.
Frommetal to graphene plasmon-polaritonsThe most well-known physical realization of polaritons consistsof electromagnetic modes bound to a flat interface between ametal and a dielectric, called surface plasmon-polaritons (SPPs)3.The field of metal plasmonics has developed tremendously overthe last few decades, with a number of interesting effects such asextraordinary transmission through nanoholes in metals29, single-molecule detection30, compact nanophotonic components31, andnovel optical phenomena with metamaterials32 and metasurfaces33.However, metal plasmonics suffers from the problem of absorptionlosses34, which traditionally has limited the range of possiblematerials to metals such as silver and gold, and constrains theoperating frequencies to the near-infrared, visible and ultraviolet. InBox 1, we illustrate how one can viewPPs in graphene as the extremecase of the so-called short-ranged SPPs in metal film, as the metalfilm thickness approaches atomic level.
State-of-the-art graphene plasmonicsGraphene plasmonics35–39 presents several advantages as comparedwith metal plasmonics. First, the carrier density, which determinesits plasmonic Drude weight (as defined in Box 2), can beelectrically40, chemically41 or optically42 tuned. This is due to thefact that graphene is a semimetal with a small density of states,where typical carrier concentrations are less than ∼0.01 free
1Department of Electrical & Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA. 2Universidade Federal do Ceará,Departamento de Física, Caixa Postal 6030, 60455-760 Fortaleza, Ceará, Brazil. 3Department of Chemistry, Columbia University, New York,New York 10027, USA. 4US Naval Research Laboratory, 4555 Overlook Avenue SW, Washington DC 20375, USA. 5Mechanical Engineering Department,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 6IBM T.J. Watson Research Center, 1101 Kitchawan Road, YorktownHeights, New York 10598, USA. 7Department of Applied Physics, Stanford University, Stanford, California 94305, USA. 8IMDEA Nanociencia, Calle deFaraday 9, E-28049 Madrid, Spain. 9Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK. 10Institutode Ciencia de Materiales de Aragon and Departamento de Fisica de la Materia Condensada, CSIC-Universidad de Zaragoza, E-50012 Zaragoza, Spain.11ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain. 12ICREA InstitucióCatalana de Recerça i Estudis Avancats, 08010 Barcelona, Spain. *e-mail: [email protected]
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Themost well-known physical realization of polaritons consists ofelectromagnetic modes bound to a flat interface between a metaland a dielectric, called surface plasmon polaritons (SPPs). Thephysical characteristics of SPPs in metals are fixed by the materialproperties but can be changed by changing the geometry. Forinstance, let us consider a thin metal film, where the SPPs at bothinterfaces couple in symmetric and antisymmetric combinations.The symmetric mode, also often called the ‘short-range SPP’,has the electric field |E| concentrated in the metal film asillustrated for the case where the metal film thickness is smallerthan its skin depth. This mode is more strongly confined andpropagates a smaller distance than the isolated SPP. Grapheneplasmons-polaritons (PP) can be understood as the extreme case
of ‘short-range SPPs’ as the thickness approaches the atomic level,and light is being confined to dimensions 2–3 orders smallerthan that of the free-space wavelength. To allow for boundelectromagnetic modes, the real part of the permittivity has tobe negative. In the case of PP, this is provided by the intrabandmotion of the free carriers, commonly described by the Druderesponse. However, polar dielectrics (for example, silicon carbide,but not silicon) and semiconductors supporting excitons mightalso exhibit negative permittivity, enabling phonon-polaritons(PhPs) and exciton-polaritons (EPs). See the illustration ofthe dielectric function Kramers–Kronig pairs (ε1 + iε2) andthe electric field line distributions for these polaritons in 2Dmaterial systems.
conduction electrons per atom, in contrast to the case of ∼1 ingold. The consequence of this lower carrier concentrationis that graphene PPs can be stimulated in the terahertz tomid-infrared frequencies4,5,39–41, and in some cases even intothe short-wave infrared43, while metal SPPs are found inthe ultraviolet, visible to the near-infrared. Second, the lowelectronic density of states and relatively weak electron–phononcoupling endows graphene with very high intrinsic carriermobilities, attainable via graphene encapsulated within hexagonalboron nitride (hBN) multilayers made with the van der Waalsassembly technique44.
The first experiments4,5 on propagating PPs in graphenewere observed using scattering-type scanning near-field optical
microscopy (SNOM). Recently, similar measurements have beenperformed on high-mobility encapsulated graphene45, with aschematic of the set-up presented in Fig. 1a. In this experiment,the metal-coated atomic force microscope tip scatters incidentfree-space light into graphene PPs, where the sharpness of thetip provides the needed momentum to overcome the momentummismatch between free-space photons and the confined PPs. ThePP propagates away from the tip as circular waves with complexwavevector q. The PP propagates towards the edge of the grapheneflake, and provided this edge is within the polariton propagationlength, it will be reflected back towards the tip and is detected asout-scattered light. Spatial scanning of the tip near the grapheneedge shows characteristic fringes due to interference between the
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Figure 1 | State-of-the-art graphene plasmonics. a, Schematic of the SNOM measurement, where the probe tip is excited with a laser source, launchingplasmons radially from the tip, with the scattered plasmons also collected by the tip. b, Measured optical signal from a 2D scan of the SNOM tip near thegraphene edge (dashed line) at room temperature. Plasmons are reflected o� the graphene edge, and appear as interference fringes. c, Calculatedgraphene plasmon inverse damping ratios, γ−1, due to graphene acoustic phonons (blue dashed line), substrate phonons of hBN (yellow dashed line), andthe combination of these mechanisms (red line). γ−1 due to charge impurities at concentration 1.9× 1011 cm−2 (green dashed–dotted line) are alsodisplayed. Experimentally measured γ−1 are shown in solid symbols with error bars representing the 95% confidence intervals. d, Experimental SNOMimage of a convex Au antenna extremity due to laser excitation at 11.06 µm, demonstrating the possibility of plasmon launching and wavefront engineeringin graphene. e, A similar SNOM image of refraction of graphene plasmons launched from a Au antenna due to a graphene bilayer prism as indicated.f,g, Calculated confinement factor, β (f), and inverse damping ratio, γ−1 (g), for various 2D materials such as graphene, bilayer graphene, blackphosphorus (BP), and MoS2. Result for Au monolayer is shown for comparison. BP exhibits highly anisotropic in-plane electronic dispersion, with e�ectivemasses along the two crystal axes di�ering by a factor of∼10. We display results for both the high (x)- and low (y)-mass directions. Adapted from ref. 45,Nature Publishing Group (a–c) and ref. 47, AAAS (d,e).
Box 2 | 2D plasmonic materials and optical conductivity.
Isotropic plasmon(graphene)
Hyperbolic plasmon(anisotropic materials)
Chiral plasmon(gapped Dirac materials)
Plasmon-polaritons (PPs) can be launched with a nano-antenna,such as a gold disc, as illustrated. The propagating PPs wavefrontcan then be spatially mapped with the SNOM47. Graphene PPshave a circular wavefront, since its optical conductivity tensor isisotropic, that is, σxx=σyy≡σ0, with σ0= iD/π(ω+ i/τ), whereD= e2µ/}2 is known as the Drude weight, µ is the chemicalpotential in graphene and τ is the electron’s lifetime. Anisotropic2D materials (for example, black phosphorus8,56) imply σxx 6=σyy , and could potentially host hyperbolic PPs. In hyperbolicmedia, the permittivities along the Cartesian axes are opposite insigns (that is, Im[σxx ·σyy]<0), which can fundamentally changehow light interacts with matter. The mechanism relies on theinterplay between anisotropic intraband and interband motions9,which can render capacitive and inductive optical responsesalong the two axes. In SNOM experiments, these effects couldbe observed as PPs rays launched with an optical antenna, asillustrated above. Gapped Dirac materials (for example, transitionmetal dichalcogenides54,55) are a unique class of materials wherethe two electronic valleys at the Fermi energy exhibit circular
dichroism. The electronic wavefunctions in the two valleys ‘twist’differently (that is, Berry phase), hence imparting a sense ofchirality, and result in a different response to circular polarizedlight54,55. Pumping with circularly polarized light effectively leadsto population imbalance in the two valleys, which are relatedto each other by time-reversal symmetry. This results in a finitetransverse conductivity (that is, σxy 6= 0) and the appearance ofnon-reciprocal chiral edge modes6,7 as illustrated. It should bepossible to detect these effects with ultrafast pump–probe SNOMwith a small radius of curvature tip of∼10 nm.
reflected and incident PPs (see Fig. 1b) from which the complexq can be extracted. In such experiments, spacing between fringesis the PP half-wavelength, that is, π/Re[q], while the fringes decayexponentially as exp(−Im[q]x).
To directly compare and quantify the behaviour of variouspolaritonic materials, a series of figures of merit have beenestablished. Commonly used figures of merit for PPs are:γ ≡ Im[q]/Re[q], where γ −1/2π gives the number of cyclesthe PP can propagate before its amplitude decays by 1/e, and thelight confinement factor β , which is obtained by normalizing Re[q]to the free-space wavevector. Experiments with hBN-encapsulatedgraphene have achieved β∼150 and γ −1>25 (ref. 45), see Fig. 1c.Hence, the current state of the art in graphene PP has alreadysurpassed the performance of SPPs along air/silver interfaces(γ −1≈10 across the visible spectral range, where β >10 (ref. 36)).In these graphene-based devices, the inverse damping γ −1 wasfound to be dominated by phonons in graphene and hBN46,
rather than those of extrinsic ionized impurities as in earlierexperiments on SiO2 substrates4,5 where γ −1∼ 5. Steps have alsobeen undertaken to develop resonant optical gold rod antennas and2D spatial conductivity patterns for direct launching and control ofpropagating PPs47 as illustrated in Fig. 1d,e respectively.
Plasmon-polaritons beyond grapheneGraphene has paved the way for the discovery and exploration ofother atomic 2D materials with new physical properties48, amongthemnew types of PP. For example, Bernal-stacked bilayer graphenehas recently received attention as a new plasmonic material,both theoretically49 and experimentally50,51. It allows for efficientswitching of the plasmonic Drude weight with the applicationof a vertical electric field50, due to the opening of an electronicbandgap52. It also accommodates a mid-infrared-active opticalphonon mode53, which can hybridize with the PPs to feature noveleffects such as electromagnetically induced transparency and slow
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light49–51. This, in addition to other new emerging 2D materialssuch as transition metal dichalcogenides (TMDs)54,55 and blackphosphorus (BP)56, presents exciting opportunities for exploringnew plasmonic effects6–9 (Box 2).
As gold has perhaps been the most utilized plasmonic material,it is instructive to compare the associated figures of merit for thecorresponding SPPs with those of 2Dmaterials. In themid-infrared,due to the large negative values associated with the real part ofthe dielectric function, SPPs in gold exhibit very poor β . However,β can be increased by considering metal films, and utilizing the‘short-range SPP’ mode (Box 1). To facilitate comparison, weconsider a very thin gold film (most optimal in terms of β),assuming a thickness of 1 nm. Bulk dielectric constants for goldare used57. Optical constants for graphene and its bilayer, MoS2and BP have been obtained from the well-known Kubo formula inconjunction with their low-energy Hamiltonians7,8,49. In all cases,we have assumed a temperature of 300K and an n-doped materialwith a typical doping of 5× 1012 cm−2, which can be obtainedwith standard electrical gating48. We obtained the free carrierscattering time τ from current state-of-the-art experiments45,58,59.For graphene, its bilayer, and MoS2, τ is approaching the intrinsiclimit determined by their thermal and optical phonons, that is,τ ∼ 1 ps (ref. 46) and ∼ 0.1 ps (ref. 60) respectively. For BP, τ canapproach ∼1 ps (ref. 61); however, the best experiments to dateachieved only∼0.1 ps (ref. 58), so for our purposes the latter valuewas used. All calculations are for a configuration where the 2Dmaterial is placed on a substrate with dielectric constant of 2.25,which corresponds to that of SiO2 in the mid-infrared.
Figure 1f,g shows the calculated field confinement β anddamping ratio γ −1 across the far- and mid-infrared frequencies for2D plasmonic materials and the fictional atomic gold film. In allcases, β is observed to increase with increasing frequency; however,it is only of order 1 for the gold film even on approaching thenear-infrared. On the other hand, 2Dmaterials present much largerβ ∼ 100–1000. This implies that 2D materials are clearly betterthan metals for field enhancement and localization in the mid-infrared. From these plots, it is clear that not one material providesthe optimal behaviour at all frequencies. For instance, while thehighestβ values are observed in BP, these are observed along a singlecrystallographic axis (x) and only over a relatively narrow spectralwindow. In contrast, graphene andMoS2 both provide high β over abroad spectral range extending from the terahertz out into the mid-and near-infrared, respectively, with MoS2 offering higher β valuesoverall. In the case of graphene, the operational range is limitedon the high-frequency end to about 3,500 cm−1 due to interbandlosses. These occur when the plasmon energy is roughly twice theFermi energy62. A very interesting behaviour is observed in bilayergraphene. Within the mid-infrared, β is found to degrade rapidlywith increasing frequency. However, after an extended spectral gapbetween roughly 1,450–3,250 cm−1, the β recovers. This behaviouris related to the two nested conduction bands in the bilayer, whichdue to interlayer coupling can transfer the energy to another higherenergy plasmonic mode49. Thus, bilayer graphene can be used toextend polaritonics into the near-infrared.
Semiconducting TMDs, especially monolayer MoS2, haveattracted significant attention due to their novel optical dichroic andcoupled spin–valley physics55. Despite the considerable bandgapof ∼2 eV, a metallic state can still be induced with a verticalelectric field if gap trap-states63 can be sufficiently suppressed. Thedevelopment of van der Waals heterostructure device platformswhere MoS2 layers are fully encapsulated within hBN representsimportant progress in this direction, with a record-high Hallmobility reaching 34,000 cm2 V−1 s−1 for six-layer MoS2 at lowtemperature59. Figure 1f shows that TMDs can have a very largeconfinement of β∼ 103, an order larger than graphene, albeit itsγ −1 is smaller over most of the spectral range where graphene
plasmons can be supported due to its smaller carrier lifetime.A new class of anisotropic 2D materials56,64,65 has also receivedconsiderable attention recently, particularly BP, which has a bulkgap of 0.3 eV and decent carrier mobility of∼1,000 cm2 V−1 s−1 forBP thin film56. In BP, the in-plane electronic mass anisotropy canbe as large as 10, which offers the promise of anisotropic PPs8 asreflected in their β and γ −1 presented in Fig. 1f,g.
Hyperbolic phonon-polaritons in hBNWhile the in-plane anisotropy of BP provides the potential to realizehyperbolic PP behaviour within the plane of a 2D material (forexample, Box 2), the structural anisotropy in all van der Waalscrystals results in a strong optical birefringence. Hyperbolicityis defined as an extreme type of birefringence, whereby thepermittivities along orthogonal crystal axes are not just different,but opposite in sign66. In 2Dmaterials, the Cartesian axes exhibitingthe sign inversion of the permittivity tensors tend to be betweenthe in- and out-of-plane directions, with hBN offering a primeexample. Bulk hBN features two sets of infrared-active opticalphononmodes, which results in two spectrally distinct Reststrahlenbandswhere negative permittivity can be observed. These two bandsare now referred to as the lower (‘LR’; ∼760–825 cm−1) and upper(‘UR’;∼1,360–1,620 cm−1) Reststrahlen bands. As shown in Fig. 2a,its in-plane permittivity is positive (negative) within the LR (UR),while opposite in sign along the out-of-plane c axis.
The inversion of signs of the permittivity components betweenthese two bands gives rise to a type I and II hyperbolic behaviourwithin the LR and UR respectively66, which impacts the resultantdispersion relationships of the in-plane transverse (kt ) and out-of-plane (kz ) hyperbolic modes. These hyperbolic PhPs in hBNwere first explored within the UR using SNOM10 through asimilar interferometric study to that previously described forPP in graphene. This initial experiment recorded β ∼ 25 andγ −1 ∼ 20, which further improved to γ −1 ∼ 35 in a subsequentexperiment16. Such high γ −1 is not surprising since PhPs do notsuffer from electronic losses, instead being dictated by the opticphonon scattering lifetimes, which are intricately tied to the crystalstructural quality and should improve with better growth.
Unlike traditional polaritonic materials, hyperbolic PhP modesexhibit multiple branches within the dispersion relationship asshown in Fig. 2b. Later work demonstrated that these multiplebranches are directly tied to the thickness of the hyperbolicmaterial layer thickness16.While the initial work10 observed only thefundamental, lowest order hyperbolic modes, work using conical-shaped hBN nanostructures experimentally observed up to 7 and4 branches within the UR and LR, respectively, by varying theaspect ratio of the structures11 (Fig. 2c). Unlike the case of SNOM,where the higher order branches exhibit higher in-planewavevector,kt , in the resonators, the modes increased in kz . As the aspectratio for these nanostructures is directly proportional to the out-of-plane kz , this gives rise to the apparent inversion of the dispersionrelationships between Fig. 2b (SNOM) and 2c (nanostructures).Subsequent work using SNOMwas able to further map out the firstfew orders of the kt dispersion as well16.
Slowing light and imaging within hBNThe PhP dispersions reported in refs 11,16 suggest negative(positive) group velocity for the kz (kt ) modes within the UR, andthe opposite sign for the LR. Interestingly, recent time-resolvedSNOM measurements provided unambiguous evidence that it isinstead the phase velocity that exhibits negative values13,14. Figure 2ddepicts the launching of PhP from the edge of a gold pad, andvisualizing its propagation in the time domain for PhP in the UR(Fig. 2e) and LR bands (Fig. 2f). By tracing its wave envelope,one can discern that the PhP moves with a positive group velocityregardless of the Reststrahlen band. However, by monitoring the
Figure 2 | Hyperbolic phonon-polaritons in hBN. a, Real parts of the in-plane (εt) and out-of-plane (εz) permittivity tensor components of hBN. Type Ilower and II upper Reststrahlen bands are shaded. A schematic of the hBN crystal structure is presented in the inset. b, Phonon-polariton dispersion withinReststrahlen bands experimentally obtained from the SNOM images near edges of a 105-nm-thick hBN, versus the in-plane momenta kt (filled symbols).Solid lines are theory results. c, Hyperbolic phonon-polariton dispersion of the out-of-plane wavevector kz as determined by the aspect ratio dependence ofthe resonance frequencies for di�erent conical-shaped nanostructures. This is plotted for both the upper (top) and lower (lower) Reststrahlen band. Thesolid lines are analytical calculations for ellipsoidal particles, while the various symbols indicate the resonant frequencies for experimental conicalnanostructures. d, Schematic of time-domain SNOM measurement of phonon-polaritons in hBN. Mid-infrared light incident on the Au antenna launcheshyperbolic polaritons in hBN, which propagate away from the Au edge and decay exponentially in amplitude and are finally collected by the nanotip.e,f, Line scans of the SNOM amplitude taken as a function of the delay time between the incident (on Au) and detected fields (by tip). The polaritons’group velocity (measured for frequency within the type II and I Reststrahlen bands respectively) can be extracted from the rate at which the ‘envelope’ ofthe fields propagates away from the Au edge, while both sign and magnitude of the phase velocity can be determined from the direction and speed of thered/blue fringes with respect to the envelope. g, Schematic showing the launching of hBN hyperbolic phonon-polaritons from the edges of the Au disc,when it is illuminated with mid-infrared light. h, Atomic force micrograph of Au discs defined lithographically on SiO2/Si substrate before the hBNtransfer. i, SNOM image of a 395-nm-thick hBN at laser frequency ω= 1,515 cm−1, where the observed ‘rings’ produced by the hyperbolic polaritons areconcentric with the Au discs. Adapted from ref. 11, Nature Publishing Group (a,c); ref. 16, Nature Publishing Group (b,g–i); and ref. 13, Nature PublishingGroup (d–f).
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Figure 3 | Hyperbolic polaritons beyond hBN. Chart showing the type I and II hyperbolic frequency ranges for various naturally occurring hyperbolic layeredmaterials, that is, cuprates70 (BSCCO (Bi2Sr2Can−1CunO2n+4+x), GBCO (GdBa2Cu3O7−x), LASCO (La1.92Sr0.08CuO4)), ruthenates70(SRU (Sr2RuO4),SR3U (Sr3Ru2O7)), hBN12,13, TMDs74, tetradymites (Bi2Se3)75 and graphite70,73. The colour map depicts the calculated figure of merit (FOM)Re[q]/Im[q].
fringe velocity, the sign of the phase velocity can be identifiedas being negative in the LR, while positive in the UR. Such anegative phase velocity necessarily implies that kt is negative.Presumably, for modes where kz is changing (for example, 3Dconfined resonators), the opposite behaviour would be anticipated.These experiments also allowed for direct measurement of thehyperbolic PhP modal lifetimes, recorded to be 1.8 and 0.8 ps inthe LR and UR, respectively, demonstrating that these values areon the order of the lifetimes of the optical phonons themselves.With lifetimes for PPs in metals being on the order of 10s fs,such long lifetimes would naturally be anticipated to result incorrespondingly long polariton propagation lengths. However, aspreviously discussed12, the group velocity at which the polaritonpropagates can be exceptionally slow, therefore limiting the ultimatepropagation length. For hBN, this was certainly the case with thegroup velocity being demonstrated to be on the order of 0.027and 0.002 times the speed of light in vacuum for the UR andLR, respectively.
Another promising application for hyperbolic media thathas received a lot of attention is the direct imaging of deeplysubdiffractional sized objects via the so-called ‘hyperlens’67.Originally demonstrated using hyperbolic metamaterials68 in thevisible, the high optical losses associated with the metal plasmonicconstituents and the fabrication complexities have limited itsimaging capabilities and resolution to the smallest meta-atom. Onthe contrary, hyperbolic materials based on hBN are homogeneous,ideally infinite, and low-loss by the nature of the PhP. Proof-of-principle hyperlens experiments were recently demonstrated15,16,as illustrated in Fig. 2g–i. In both works, subdiffractional metalobjects were fabricated on a substrate surface using electron beamlithography (Fig. 2h) and covered with a flat slab of hBN (onthe order of 100 nm thick) and imaged using a SNOM (Fig. 2g).Multiple rings result from the PhP modes launched from the edges
of the metal nanoparticles at angles dictated by the hyperbolicnature of the PhPs (Fig. 2i). This propagation angle (with respectto the surface normal) results from the fact that while hyperbolicmaterials can support very high k modes, they can propagate onlyat an angle given by tan(θ)=
√Re[εt]/Re[εz]/i. For both the
UR and LR of hBN, either εt or εz will be negative and dispersivewhile the other is positive and nominally constant; this angleof propagation is therefore a function of frequency. Because ofthis, the image that results from the corresponding hyperlens canvary from a direct replica (θ = 0◦) to a highly magnified object,which can be user-controlled. It is important to note that thepropagation angle θ is fixed with respect to the crystal axes ofhBN. As a result, in a non-planar geometry of a truncated conegeometry (as illustrated in the inset of Fig. 2c), non-specularreflections can occur off the sidewall surfaces as recently confirmedwith SNOM69.
Natural hyperbolic layered 2D materials beyond hBNResearch in this area has focused on the natural hyperbolicproperties of hBN. However, the natural optical anisotropyassociated with van der Waals crystals, and the polar natureof many should in principle offer a broad range of naturallyoccurring hyperbolic materials covering a very broad spectralrange70–72. Strong anisotropy in electron motion along the in-plane(metallic) and out-of-plane (insulating) layeredmaterials can lead tohyperbolicity for specific frequency bands, for example, in graphiteand magnesium diboride70,73. Ruthenates have different Drudeweights70 along the in- and out-of-plane axes, and are hyperbolicbetween the two plasma frequencies. High-quality semiconductingTMDs, such as MoS2, also accommodate far-infrared anisotropicpolar optical phonons, and should result in hyperbolic bandsas in hBN in the Reststrahlen frequencies74. Furthermore, thetetradymites, such as Bi2Se3 and Bi2Te3, should also host such
hyperbolic polaritons, albeit in the terahertz regime75, while theywere also observed to support hyperbolicmodes in the near-infraredto visible due to different highly resonant interband transitionenergies along the in-plane and out-of-plane axes76. Lastly, thehigh-Tc superconducting cuprates are highly metallic in-plane, butwith an out-of-plane dielectric response typical of an insulatorcharacterized by several Lorentzian-type resonances70. As such,one would naturally infer that hyperbolic behaviour would also bereadily observed.
Figure 3 presents representative materials from the above-mentioned layered 2D systems. Their hyperbolicity spectral range,type, and inverse damping ratio γ −1 are plotted. Here, we haveused the quasistatic approximation and considered only the lowestorder mode. This allows us to characterize a broad class ofmaterials on equal footing. Among the listed materials, onlyhBN and Bi2Se3 whose hyperbolicity is phonon in origin, showγ −1>10, indicating that while a wealth of hyperbolic 2D materialsawait further study, optical losses may still limit advancements inthe near-term.
Strong excitons in 2D semiconductorsOver the past few years, a plethora of photoluminescence (PL),absorption and reflectance experiments have been reported in theliterature for several 2D materials, such as TMD monolayers17–21(including the distorted 1T phase ReS2 (ref. 77)), BP mono- andmultilayers25–28, and, more recently, monolayer organic–inorganicperovskite (OIP) crystals78, namely (C4H9NH3)2PbI4. By probingthe formation (absorption and reflectance) and recombination(PL) of electron–hole pairs, these experiments have provided usinformation not only about their optical gaps, which are found tobe significantly far from the theoretical quasiparticle bandgaps, butalso about their excitonic Rydberg series. Figure 4a summarizes theexperimentally observed optical gaps (filled symbols), which covera wide range of wavelengths from blue, for monolayer OIPs, all theway to the near-infrared, for BPmultilayers. Recent calculations alsosuggests possible extension into the mid-infrared using arsenene, astable analogue of BP but with arsenic atoms79.
Binding energies of excitonic states can be obtained byEn=EXn−EQP, where EXn and EQP are the nth exciton state energyand quasiparticle bandgap, respectively, where the former are shownin Fig. 4b. Unlike bulk semiconductors, excitons in 2Dmaterials arestrongly confined to a plane and experience a reduced screeningfrom their surrounding dielectric environment80, which modifiesthe character of the Coulomb interaction potential, leading to non-hydrogenic Rydberg series of exciton states81. Excited exciton stateswithin the Rydberg series (that is, 2s, 3s, 4s... states) have beenobserved in OIPs78 and TMDs81–84, using different methods such asreflectance and absorption spectroscopy, as summarized by circlesin Fig. 4b. Triangles in Fig. 4b are excited states with p symme-try, observed by two-photon luminescence experiments in WS2and WSe2(refs 82–84). As a consequence of the non-Coulombicelectron–hole interaction potential, the degeneracy between s and pstates is lifted, so that two-photon experiments exhibit peaks in thePL spectra that do not match those of the Rydberg series states83.In general, binding energies in 2D materials are about an order ofmagnitude higher than those of bulk semiconductors85, such as Si,Ge and III–V or II–VI alloys, being comparable only to excitonbinding energies observed in carbon nanotubes86 and conjugatedpolymer chains87.
Such strong electron–hole interaction makes these materialsa playground for investigating excitons and their complexes(trions and biexcitons), which are usually harder to observe inconventional bulk semiconductors. In Fig. 4b, we summarize therecently experimentally observed binding energies of trions andbiexcitons in 2D materials, which range from tens to a hundredmillielectronvolts (refs 20,22–24,88–91). All of these high binding
energies for excitons and their complexes as observed are consistentwith theory within the generalized Wannier–Mott model describedelsewhere92. In tightly bound excitons in layered 2D materials,there is a large spatial overlap of the respective electron and holewavefunctions, with the corresponding Bohr radius being on theorder of only 1 nm (refs 81,83). This leads to a particularly strongcoupling of excitons to photons93, resulting in both large absorptioncoefficients and efficient emission in this class of materials. Theabsolute absorption values at the excitonic transition peak areas high as 10–20% (ref. 18) for single layers with subnanometrethickness, as shown in Fig. 4c. The corresponding area underthe resonance (shaded), proportional to the strength of the light–matter coupling94, is orders of magnitude larger than the respectivevalues in more conventional inorganic semiconductors, such asGaAs (ref. 95).
Exciton-polaritonsEP was first observed in 2D materials within optical microcavitieswith TMDs96,97. Experimentally observed EP branches97 inmonolayer MoS2 follow anti-crossing paths over the excitonand cavity mode energy lines, with a Rabi splitting as large as∼46meV. The strong coupling regime, typically defined by the rateof the exciton–phonon scattering being larger than the competingdephasing processes of the two particles, was already shown tobe attainable even at room temperature97. However, real-spaceobservation of EP with SNOM was observed only very recently98in an exfoliated 260 nm WSe2 thin flake. The field of EPs in2D materials is still at the nascent stage, and we expect excitingfuture developments.
Hybrid polaritonsLooking forward, perhaps one of the most intriguing prospectsassociated with 2D van der Waals crystals is the ability to cleaveand combine layers of different 2D materials to realize heterostruc-tures of different constituents and thicknesses99, and engineer newhybrid polaritons with novel physics100–103. Two recent examples aredepicted in Fig. 5. By combining graphene with hBN, one canmarrythe advantage of tunable PP in graphene with high quality, low-loss, PhP in hyperbolic hBN104. A recent SNOM measurement per-formed on a graphene–hBN heterostructure104 (Fig. 5a) revealed acoupled PP and PhP (Fig. 5b). This hybrid polaritonmode, termed a‘plasmon–phonon polariton’, exhibited a broadband dispersion, ex-tending beyond the Reststrahlen band of hBN, and offered electro-static gate tuning of the hyperbolic mode primarily confined withinthe hBN. In addition to these so-called electromagnetic hybridswhere combinations of PPs and PhPs are induced, another newtype of atomic-scale heterostructure consisting of polar dielectric2Dmaterials can also lead to the creation of a newmaterial resultingfrom the hybridization of optic phonons at the heterostructureinterfaces, which can provide direct control of the dielectric functionwithin the spectral regime of the Reststrahlen bands103.
On the EP front, heterostructures of TMDs where electronsand holes are confined to different layers, as illustrated inFig. 5c, would allow for the formation of ‘indirect excitons’at an energy lower than that of its single-layer constituents.Many pairs of 2D materials are known to be compatible withthe type-II band alignment105 required for such situations.Having a low oscillator strength, but also lower energy,this charge-transfer exciton manifests itself as a clear low-energy peak in PL experiments that is absent in absorptionexperiments102, as shown in Fig. 5d for a WSe2/MoS2 hybridheterostructure100.
OutlookAs indicated by the breadth of work highlighted above, there is agreat deal of promise for nanophotonics based on 2D materials and
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Figure 4 | Excitons in TMDs and beyond. a, Survey of experimentally observed optical bandgaps (full symbols) in di�erent 2D materials: OI in both phases(I and II)78, WS2, WSe2 (refs 81–84), MoS2 (refs 17–19), MoSe2 (refs 20,21), ReS2 (refs 64,77), ReSe2 (ref. 118), and n-layer black phosphorus (n-BP)25–28.We note that for anisotropic materials such as ReS2, ReSe2 and n-BP, the optical spectra are polarization sensitive, and the optical gap is defined by thelowest energy resonance. Open symbols are theoretical calculations based on the Wannier–Mott model. Theoretical predictions of quasiparticle (opencircles)119 and optical (stars) gaps for n-As are shown, extending the frequency window into the mid-infrared. b, Binding energy of exciton states in theupper panel, in their Rydberg sequence for s states (filled circles) and p states (filled triangles), as well as trions and biexcitons in the lower panel. Sketchesof the electron–hole structure of these excitonic complexes are illustrated on the right. The binding energy can be obtained from the observed opticalbandgap and the quasiparticle gaps, and the results for excitonic complexes in black phosphorus91, WS2 (ref. 89), MoS2 (refs 22,88,90), MoSe2 (ref. 20)and WSe2 (refs 23,24) are presented. c, Measured absorption spectrum of a single WS2 layer, as sketched in the inset. Adapted from ref. 120, APS.
their heterostructures. While still in the earliest stages of work, herewe offer our perspective on how we anticipate the development ofthe field, highlighting the key physics and the potential applicationspace. The latter is summarized in Fig. 5e. Graphene plasmonics
has led the way in terms of demonstrating proof-of-principle deviceconcepts such as mid-infrared optoelectronics106, bio-sensing107and fingerprinting108. We anticipate that these applications shouldcontinue to develop over the next few years, and also envision
Optical components such as filters, phase shifters
Far- Mid- Near-
Solar cells
Microwave
Proof-of-principleexperiment demonstrated
No experiment yet
10 cm−1
1,000 μm100 cm−1
100 μm200 cm−1
50 μm3,333 cm−1
3 μm13,000 cm−1
0.78 μm
Figure 5 | Hybrid polaritonic and application space of polaritons. a, Schematic of the SNOM experiment performed on a graphene–hBN heterostructure.b, Experimental dispersions of hybrid plasmon-phonon-polaritons (blue circles) obtained from the graphene–hBN heterostructure, compared againstphonon-polariton (red triangles) from hBN. Theoretical results are shown as white lines and a colour map. c, Sketch of a charge-transfer (indirect) excitonin a van der Waals heterostructure with type-II band alignment. d, Photoluminescence and absorption spectra of WSe2 and MoS2 monolayers and theirhybrid heterostructure. e, Summary of the possible application space of 2D materials polaritons (excitons, plasmons, phonons) across frequency ranges(see main text). Adapted from ref. 104, Nature Publishing Group (a,b) and ref. 100, National Academy of Sciences (d).
ventures into free-space beam shaping and steering with graphenemetasurfaces109. Going forward, these high-quality graphene PPscan also provide an excellent platform for realizing tunable 2Dmid-infrared nanophotonics circuits with novel functionalities notpreviously attainable110.
The discovery of new plasmonic materials beyond graphenemight potentially offer newmeans of light–matter interactions, suchas hyperbolic9 and chiral6,7 plasmons in anisotropic and gappedDirac materials respectively. Plasmonic loss remains a fundamentalissue that should be addressed, and non-equilibrium processes orgain media111 could offer enticing routes, particularly in gapped
materials for example, bilayer graphene. However, while there isa strong desire to extend the operating spectral range into thevisible, this would also necessarily require the realization of newmaterials, for instance intercalated graphene112 or the recentlyobserved hyperbolic polaritons in tetradymites76.
Because of the optical phonon origin of PhPs, these modesare observed in the terahertz to mid-infrared spectral ranges12,as summarized in Fig. 3. In 2D materials, the natural hyperboliccharacter of these phononic modes makes them preferredcandidates for subdiffraction imaging (such as hyperlensingdiscussed earlier), and super-Planckian thermal emission113.
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However, while the slow-moving PhP modes are not ideal for mostwaveguide applications, the long residence times associated withthese long modal lifetimes and slow velocities can be extremelybeneficial for enhancements of local emitters, molecular vibrationalmodes and applications where strong light–matter interactionsare desired. Indeed, these slow-light modes in natural hyperbolicmaterials would also manifest a van Hove singularity in thenear-field local density of optical states114, offering control ofspontaneous emission.
Long-lived excitons are of great importance for example,for possibly allowing future observation of their superfluidityand exciton condensation at relatively high temperatures101.Such unprecedented tightly bound exciton complexes alsobring the exciting prospect of realizing efficient energy transferby driving charged excitons through applied electric fields.This could be interesting for future applications in solar cellsand photodetectors115, where suppression of electron–holerecombination is also desirable. The coupling of excitons withplasmons to achieve sustained propagating EPs98 is also interesting.Last but not least, the emergence of topological materials withnon-trivial ‘twisted’ electronic wavefunctions has also motivatedstudies on imbuing topological property to excitons116, plasmons6,7and phonons117 in solids. This could potentially open up newavenues for the observation of topological polariton physics.
In summary, the use of 2Dpolaritons has enabled the engineeringof light–matter interactions beyond the diffraction limit across theterahertz to visible spectrum. The ability to manipulate polaritonswithin the vast library of van der Waals 2D materials, in additionto nano- and heterostructuring, promises the on-demand designof new optical properties that are not possible with traditionalplasmonic materials.
Received 28 May 2016; accepted 5 October 2016;published online 28 November 2016
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AcknowledgementsT.L. acknowledges financial support by DARPA grant award FA8650-16-2-7640. A.C.acknowledges support by CNPq, through the PRONEX/FUNCAP and Science WithoutBorders programs. J.D.C. acknowledges financial support from the Office of NavalResearch that was administered by the NRL Nanoscience Institute. A.K. and N.X.F.acknowledge the financial support by AFOSR MURI (Award No. FA9550-12-1-0488).L.M.M. acknowledges the Spanish Ministry of Economy and Competitiveness underproject MAT2014-53432-C5-1-R. F.K. acknowledges financial support from the SpanishMinistry of Economy and Competitiveness, through the ‘Severo Ochoa’ Programme forCentres of Excellence in R&D (SEV-2015-0522), support by Fundacio Cellex Barcelona,the European Union H2020 Programme under grant agreement no 604391 Graphene
Flagship’, the ERC starting grant (307806, CarbonLight), and project GRASP(FP7-ICT-2013-613024-GRASP). We also acknowledge useful discussion withA. Chernikov.
Additional informationReprints and permissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to T.L.
Competing financial interestsThe authors declare no competing financial interests.
