PlasmonâPlasmon Interactions and Radiative Damping of Graphene

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Plasmon-plasmon interactions and radiative damping of graphene plasmons Vyacheslav Semenenko, Simone Schuler, Alba Centeno, Amaia Zurutuza, Thomas Mueller, and Vasili Perebeinos ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00544 • Publication Date (Web): 09 Aug 2018 Downloaded from http://pubs.acs.org on August 9, 2018

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Plasmon-plasmon interactions and radiative damping of graphene plasmons Vyacheslav Semenenko,† Simone Schuler,‡ Alba Centeno,¶ Amaia Zurutuza,¶ Thomas Mueller,‡ and Vasili Perebeinos∗,† †Skolkovo Institute of Science and Technology, 100 Novaya Street, Skolkovo, Moscow Region 143025, Russia ‡TU Wien, Institute of Photonics, Gußhausstraße 27-29, 1040 Vienna, Austria ¶Graphenea SA, 20018 Donostia-San Sebastian, Spain E-mail: [email protected] Abstract Understanding and controlling collective oscillations of electrons in graphene have enabled new classes of devices for deep subwavelength metamaterials, extraordinarily strong light-matter interactions, and nano-optoelectronic switches. Here, we demonstrate both theoretically and experimentally that the plasmon-plasmon and plasmon-radiation interactions modify strongly the plasmon resonance energy, radiative damping, and oscillator strength in graphene nanoribbon arrays. As the graphene filling factor approaches one, plasmon resonance energy becomes zero. And even for the moderate filling factors of about 50%, plasmon radiative lifetime reduces to a ps time scale. We find scaling of plasmons with respect to the graphene

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doping level and filling factor, which both control the strength of the radiative and long range Coulomb interactions. The surprisingly large plasmon energy shift and radiative damping would significantly affect graphene based plasmonic device performance.

Keywords graphene, plasmon, infrared photonics, spectroscopy Graphene plasmons, the collective oscillations of electrons, are of particular interest due to their low damping rate and tunability of their excitation energy via an external electric field 1 . The latter opens a promising base for a variety of controllable optoelectronic applications 2,3 including bio-sensing 4–6 , photodetectors 7 , light sources 8,9 , THz spectroscopy 10–13 , metamaterials 14 , and nano-optoelectronic switches 15 . To date, plasmons in graphene nanoribbons have been well accounted for in the framework of classical quantum confined systems, where nanoribbon geometry together with the charge carrier concentration determine the optical absorption by plasmons 16–22 . However, the strength of the plasmon-plasmon interaction 23 is also determined by the doping level and the geometry. Most of the previous studies concentrated on geometries with filling factor being not very close to unity, where the interaction among plasmons is not very strong. The lifetime of the low energy plasmons is determined by electron scattering 3 and damping channels of higher energy plasmons via graphene intrinsic optical phonons 24 and surface polar phonons in polar substrates 25–27 . A typical radiative lifetime of an electronic transition is on the scale of a nanosecond, such that the role of the radiation losses on the plasmon lifetimes is considered to be negligible in graphene 26,28,29 . On the other hand, intrinsic radiative lifetime in 1D excitons can be as short as 10 ps 30 due to the large trans-


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ferred spectral weight from the free electron-hole particle continuum to the bound exciton. However, due to the thermalization an effective radiative decay rate is still on a scale of a ns at room temperatures 31,32 . This is in strong contrast to the case of nanoparticles, where large oscillator strength and coherent collective excitations lead to superradiance 33 . In the present work, we examine optical properties of graphene plasmons in highly integrated regime designable for the nanoplasmonic applications. We find surprisingly large plasmon energy red shift, radiative decay rate enhancement and oscillator strength renormalization as the spacing between the ribbons becomes much smaller than their width in nanopatterned graphene structures. As the filling factor of graphene nanoribbons is about 50%, i.e. width of a nanoribbon becomes comparable to the spacing between them, the radiative lifetime reduces to a ps time scale. Such that the radiative broadening becomes comparable to that caused by the carriers elastic scattering in typical graphene samples. A plasmon’s peak energy in this case becomes about 94% of it’s value in the absence of the plasmon-plasmon interactions, but it reduces to zero in the limit of filling factor equal to one. We identify simple analytical expressions for scaling of the energy, lifetime, and optical oscillator strength of plasmons as a function of graphene nanoribbon width, spacing between ribbons, and doping level. The observations and the in-depth understanding of the low energy plasmons scaling is the key for the design of graphene nano-photonic and optoelectronic devices. To illustrate the tunability of plasmons in graphene nanoribbon arrays (GNRs), shown in Fig. 1a, we consider the scattering problem 20 of light in normal incidence with two different polarizations of the electric field E perpendicular to the ribbon for transverse magnetic (TM) and parallel to the ribbon for the transverse electric (TE) waves. We use a Fourier expansion of the electrical current spatial distribution to solve Maxwell’s equations with the boundary conditions for electrical E- and displacement D-fields at the 3 ACS Paragon Plus Environment

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(b) Absorption (normalized to r)

a

0.3

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∞

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intraband only, L = 2000 nm full λω , L = 2000 nm full λω , L = 2 nm 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

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50

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Spacing between nanoribbons L, nm

200

Figure 1: Plasmon-plasmon and plasmon-radiation effects in GNRs. (a) Schematics of GNR on a substrate with dielectric constant ε1 and ε0 above. Electric field lines correspond to the TM polarization. (b) Typical absorption spectra normalized to the filling factor r of TM and TE polarized incident waves in suspended GNRs of a width W = 600 nm and different gaps L between them. (c) Plasmon energies as a function of inverse nanoribbon width q = π/W in the limit of isolated nanoribbon L = 2000 nm (green triangles) and the strongly interacting limit L = 2 nm (red squares). The blue circles show results with graphene conductivity without interband absorption. (d) Plasmon energies as a function of the gap L between them for a fixed width W . As the filling factor r increases the plasmon red shifts. The curves in (c) - (d) are analytical expressions using Eq. (1) and all the symbols are results of the numerical solution of the diffraction problem. The horizontal dashed lines in (d) correspond to the asymptotic values at L → ∞. In all plots we used the Fermi energy EF = 300 meV, scattering rate ~ν = 10 meV, temperature T = 300 K, and ε1 = ε0 = 1. To get the convergent results of the diffraction calculations, the highest Fourier harmonic number N that was used is 2000.


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graphene plane (see Supplemental Information). For the TM-polarized exciting wave, we also impose a boundary condition for the electrical current to be zero at the edges of the GNRs. An electrical conductivity relating the electrical current to the local electric field is the input of the problem besides the geometry of the nanoribbon array, defining the filling factor r = W/d, i.e. the ratio of the graphene nanoribbon width W to the period of the nanopatterned structure d, as shown in Fig. 1a.

Results and Discussions Theoretical predictions of the absorption spectra. The resulting absorption for the different GNRs are shown in Fig. 1b. As we fix the width of the ribbons W and reduce the distance among the ribbons L, we increase the strength of the long range Coulomb interactions. As a result of the plasmon-plasmon interaction, the peak absorption energy for the TM polarization is red shifted although the nanoribbon width W , confining collective electron motion, stays constant. For the TE polarization, i.e. E-field along the ribbon, a Drude type absorption is obtained, where the peak width is also dependent on the filling factor. As the graphene filling factor increases, the absorption per unit area and reverse process of light emission increase as well. As a result of the enhanced plasmon-radiation interaction, the plasmon peaks broaden. Our conclusion is confirmed by: a) the detailed analysis of the quasistatic approximation solution, where broadening is not observed, but only a red shift, and b) the fact that broadening in the full solution takes place also for the TE polarized exciting waves and even in the case of a plain unpatterned graphene in the absence of confined plasmons. To quantify the red shift of the plasmons, we can use a quasistatic approximation 34 , which is applicable when the wavelength of the light of the surrounding media is much 5 ACS Paragon Plus Environment

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larger than the wavelength of the plasmon. The plasmon dispersion in GNR becomes particularly simple: ω 2πq = Λ (r) , iλω κ

(1)

where ω is the plasmon frequency, λω is the graphene conductivity, which depends on the doping level, Λ (r) is a dimensionless parameter, which depends on the geometry only, q = π/W is a characteristic wavevector, and κ = (ε0 + ε1 )/2 is an average dielectric constant of the surrounding media, see Fig. 1a. Dependencies of the plasmon peak positions on the width and the gap between the ribbons are shown in Fig. 1c and 1d, correspondingly. The solid curves from Eq. (1) agree fairly well with the full solution of Maxwell’s equations 35 , shown by the symbols in Fig. 1c and 1d. As it follows from Eq. (1) and the full solution, the plasmon energy is remarkably dependent on the graphene conductivity function λω . In Fig. 1c, the curves labeled “full λω , L = 2000 nm” and “intraband only, L = 2000 nm” are both correspond to a plasmon in an isolated nanoribbon, since the interactions among plasmons are diminished at such a large gap between them. The “intraband only” model for the conductivity includes only the Drude term λω = e2 EF /π~2 (iω + ν), where ν is the carrier scattering rate and e, ~, π are the fundamental constants. The full model for λω , in addition to the Drude term, includes interband absorptions at energies above twice the Fermi energy EF . The striking difference between the two models for the high energy plasmons is due to the imaginary part of the graphene conductivity 36 , which doesn’t vanish even at low frequencies and temperatures.


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a

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test device 100 nm

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1 mm

0.4

0.2 40 nm

100 nm

600 nm

6000 nm 0.0 0.0

0.2

0.4 0.6 Filling factor r

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Figure 2: Experimental evidences for the red shift and radiative damping of plasmons in GNRs. (a) An optical image of the chip with four gratings and several test devices for electrical characterization. The inset in (a) and figures below (a) show SEM images of GNRs with a different spacing. The scale bar for 40 nm and 100 nm gratings is 100 nm and for 600 nm and 6000 nm gratings is 1200 nm. (b) Energy dependence of the measured extinction coefficient (which is proportional to the absorption): measured (circles) and simulated (solid curves) see text for details. The symbols in the inset (b) show FWHM extracted from the data as a function of the filling factor with the solid curve showing a theoretical prediction for a Fermi level EF = 300 meV and a scattering rate ~ν = 11.5 meV according to Eq. (4). (c), Circles show numerical results for the form factor Λ as a function of the filling factor r for the main plasmon (green) and the second overturn Λ2 − 2 (blue). The solid curves show analytical approximations (see text). The inset in (c) shows experimental results for the plasmon peak positions from (b) and the solid red curve is a theoretical prediction according to Eq. (1) with the shadow region reflecting uncertainties in the doping level.

Experimental signatures of the plasmon-plasmon and plasmonradiation interactions. To demonstrate the scaling of the plasmons experimentally, we fabricated several graphene gratings with a fixed nanoribbon width of W = 600 nm and different spacings L among 7 ACS Paragon Plus Environment

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them: 40, 100, 600, and 6000 nm, correspondingly, as shown in Fig. 2a. The gratings were fabricated using CVD graphene on a 285 nm thick SiO2 substrate on a highly resistive silicon wafer. The design of the GNRs is chosen such that the lowest energy SiO2 optical phonon at 60 meV lies above the energy of the plasmon to avoid an interaction between the plasmon and the surface polar phonon 26 . In order to directly compare the graphene gratings with different spacings, we fabricated them from a single layer graphene (see Supplemental Information). The extinction spectra in Fig. 2b clearly show a dramatic red shift as the filling factor varies from 0.1 to 0.9 at a constant graphene nanoribbon width. The solid curves are results of the simulations of the extinction coefficient with just two adjustable fit parameters: the Fermi level and the scattering rate, and one parameter s simul = ηext (W, L, EF , ν)·s, where W , L are nanoribbons for the overall scale, according to ηext

width and spacing between them. The parameter s = 0.8 ± 0.2 gives an overall scale of order one. This is achieved by including in simulations a second interface between 285 nm thick SiO2 with ε1 = 3.9 and an infinite thickness of an undoped Si substrate with εSi = 11.7, which introduces multiple reflections at the interface between Si and SiO2 . We should mention that, if the simulations without Si substrate are used, then parameter s would have been a factor two smaller deviating much larger from s = 1 value. The extinction coefficient η is calculated from the light transmission spectra through the gratings as (ω) ηext (ω) = 1 − ττTM , where τTM and τTE are normally incident light transmission spectra TE (ω)

with the E-field perpendicular and parallel to the nanoribbons, respectively. The global fit parameter ~ν = 11.5 meV is used in all four spectra and a Fermi level of EF = 302 ± 28 meV was used to fit the data. The Fermi levels and the scattering rates extracted from the optical measurements would correspond to a field effect transistor threshold voltage VT = EF2 /(Cox π~2 vF2 ) = 90 ± 17 V and mobility µ = evF2 /(νEF ) = 1900 ± 200 cm2 /Vs, where we used gate capacitance Cox = ε1 /(4πtox ) (SGS units) and the Fermi velocity 8 ACS Paragon Plus Environment

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in graphene vF = 108 cm/s. Optical measurements are consistent with the electrical characterization of our devices: VT = 98 ± 8 V and mobility µ = 1500 ± 400 cm2 /Vs (see Supplemental Information). From the consistency of the fit results across the different samples, we conclude that the experimental data support the theoretical predictions on the role of the plasmon-plasmon and plasmon-radiative interactions in GNRs.

Scaling of plasmons in graphene nanoribbons. Parameter Λ (r) in Eq. (1) determines the plasmon energy scaling with the filling factor r, as shown in Fig. 2c. An analytical fit to Λ (r) ≈ 0.734 × (1 − r1.75 )0.331 matches ideally the numerical results, which are indicated by the circles in Fig. 2c. Note, that in the case of an infinite graphene, the same plasmon dispersion Eq. (1) applies with Λ equals to one 16 . For the second resonances, also shown in Fig. 1b above 60 meV, a scaling law similar to Eq. (1) applies, but using the corresponding parameter Λ2 (r)−2 ≈ 0.734×(1−r12.387 )0.323 , as shown by the blue curve in Fig. 2c. In the limit r → 0, a ratio of Λ2 /Λ = 2.734/0.734 suggests a position of the second plasmon resonance to be at 1.93 times larger energy than the main plasmon absorbtion peak. The solid curve in the inset of Fig. 2c shows the theoretical prediction for the plasmon energy scaling with the filling factor using Eq. (1) and the experimental width W , which agrees fairly well with the measurement results shown by the symbols. The single particle scattering rate determines the decay rate of the plasmon collective excitation. The scattering of the low energy carriers in graphene, which determines an electrical transport, involve charge impurity and electron-phonon scattering 37–39 . The low energy plasmon lifetime can be as long as 10 ps in high mobility graphene samples encapsulated in hexagonal boron nitride 40,41 . As the plasmon energy becomes comparable to the phonon energy of either graphene or the polar substrate supporting graphene, the 9 ACS Paragon Plus Environment

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a

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b L = 2 nm L = 10 nm L = 50 nm L = 100 nm L = 200 nm L = 500 nm Theory

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Oscillation strength, meV

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Resonance Width, meV

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11 10 TM wave

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TE wave

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Figure 3: Plasmon decay and oscillator strength scalings due to the plasmonradiation interactions. (a) Resonance absorption width in GNR arrays as a function of the filling factor for a fixed separation between ribbons L: TE polarization - top, TM polarization - bottom. At the small filling factors, the canonical widths of ~ν and ~ν/2 are recovered for the TE and TM waves, correspondingly. The solid curve in the top panel shows prediction according to Eq. (4). (b) Oscillator strength in the same GNR arrays as in (a): TE polarization - top, TM polarization - bottom. The dashed curves show ”naive” expectations, if the the oscillator strength were additive, and the solid curves show results according to Eq. (6). In all plots we used EF = 300 meV, ~ν = 10 meV, and ε1 = ε0 = 1. plasmon lifetime diminishes substantially due to the strong plasmon-optical phonon scattering 24,26 . Further increase of the plasmon energy opens a Landau damping mechanism, when a plasmon can decay into an electron-hole pair. We find that while this canonical picture holds for a plasmon in an isolated graphene nanoribbon, it breaks down when the filling factor increases, leading to an additional radiation damping as large as 5 meV (2


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meV) for the TE (TM) modes for a typical doping level of graphene EF = 300 meV. In Fig. 3a, we show the main plasmon resonance width as a function of the filling factor. As the filling factor increases, the GNR absorption increases proportionally and, as a result, a reverse light emission process increases. We find that the plasmon peak width scales linearly with the filling factor independently of the gap between the nanoribbons. In our model, the graphene conductivity takes into account only the intra- and interband electron transitions, such as no phonon scattering is included. The latter is important only for plasmons at higher energies. The radiative damping decay rate can be extracted from the difference between the absorption peak width and that calculated in the quasistatic approximation (see Eq. (1) and Supplemental Information). The deviations from the linear scaling for TM modes are due to the Landau damping contribution in small-width nanoribbons. The TE mode width is roughly twice as large than that of TM mode. This can be easily understood from the fact that the spectral weights, i.e. the areas under the absorption peaks, and the absorption maxima for the two polarizations are expected to be similar. Therefore, the Drude peak for the TE mode should be twice as broad, since it lacks contribution to the spectral weight from energies below the absorption maximum at zero energy. To explain the scaling laws in Fig. 3, let’s consider the light absorption in an infinite graphene ηω given by: 4π Re λω ηω = √ |1 + ρω |2 , ε0 c where ρω is the reflection coefficient: √ √ ε0 − ε1 − 4πλω /c ρω = √ . √ ε0 + ε1 + 4πλω /c


(2)

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−1

Using the Drude conductivity λω , we obtain ηω ∝ (ω 2 + ν 02 ) , where ~ν 0 = ~ν +

2α EF , n01

where α = e2 /~c ≈ 1/137 is the fine structure constant and

(3) √ √ n01 = ( ε0 + ε1 )/2. We

find that an empirical scaling of the plasmon peak width due to the radiative damping can be well accounted by the expression:

~ν 0 (r) = ~ν +

2α EF r n01

(4)

as shown in Fig. 3a by the solid curve for the TE mode. Note, that for the TM mode, the width is about twice as small. The optical absorption increases with both the filling factor and the doping level, which, in turn, leads to the larger radiative losses and, hence, the broadening. Next we turn into discussion of the spectral weight of the plasmon absorption. Assuming a Lorentzian shape of the absorption, we calculate the spectral weight as a product of the principal resonance height and its width multiplied by π or π/2 in the case of TM or TE polarization, correspondingly. Unlike, interband broadband graphene absorption of 2.3% 42 , plasmon absorption is significantly larger 43 . In fact, it can be shown that under high doping conditions and small carrier scattering rate, plasmon absorption can be close to unity. In this regime deviations of the spectral weight from the canonical value 44 D0 = 2πα · EF /n201 are expected. In the case of an unpatterned graphene, an analytical expression for the spectral weight can be obtained by integrating the absorption from Eq. (2): D = D0

π~ν . π~ν + n01 D0


(5)

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Note, that in the limit of a very large scattering rate ~ν, Eq. (5) suggests a canonical value of D = D0 . In the case of a GNR, naively one would expect the spectral weight to scale linearly with the filling factor D = D0 r as shown by the dashed lines in Fig. 3b. However, we find large deviations in the calculated spectral weight with a naive expectation at filling factors close to unity for both TE and TM modes, as shown in Fig. 3b. An empirical expression of a form similar to Eq. (5) reproduces nicely our results of the spectral weight calculations over a wide range of parameters:

D(r) = D0 r

π~ν . π~ν + n01 D0 r

(6)

for both TE and TM modes as shown in Fig. 3b by the solid curves. Deviations for the case TM mode in small-width GNRs are due to the loss of the spectral weight to the interband plateau at energies above ∼ 2EF associated with the plasmon electron-hole continuum interaction. To demonstrate the scaling of plasmon’s oscillator strength with the doping level, Eq. (6) can be rewritten as: D(r) 2πα = 2 rEF n01

π~ν rEF π~ν rEF

+

2πα n01

.

(7)

In Fig. 4 we show the oscillator strength normalized to EF r with respect to the dimensionless scattering rate ~ν/(rEF ). All the numerical results collapse on a single curve according to Eq. (7) for both TE and TM polarizations in samples with different filling factors, scattering rates, and Fermi levels. Slight deviations for the TM polarization are due to the partial spectral weight transfer to the higher energy plasmon overturns and the interband absorption. Therefore, a “naive” scaling D(r) = D0 r holds in arrays with either small filling factor or low mobility graphene, for which ~v 2αEF r/n01 . In unpatterned graphene studies 44 , a 20% mismatch between the measured spectral weight and the value 13 ACS Paragon Plus Environment

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expected from the sum rule was reported. Using typical experimental values in Ref. 44 of ~ν ∼ 20 meV and EF ∼ 300 meV a discrepancy by n01 D0 /π~ν ∼ 0.15 according to Eq. (5) could be anticipated.

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Smooth graphene TE wave, GNRs TM wave, GNRs 0.15 0.20

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Figure 4: Scaling of the oscillator strength for both TE and TM modes. Oscillator strength scaling in W = 600 nm graphene nanoribbon arrays using n01 = 1 and different ranges of values L from 100–6000 nm, ~ν from 1–15 meV, and EF from 100–400 meV. The dashed horizontal line shows a value of 2πα as expected from the “naive” model. The n201 black solid curve shows the result for a smooth graphene, i.e. r = 1, according to Eq. (7).

Conclusions We have demonstrated that plasmon-plasmon and plasmon-radiative interactions make surprisingly large contribution to the optical properties of GNRs. The found scaling laws 14 ACS Paragon Plus Environment

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of the energy, radiative broadening, and oscillator strength of the plasmon resonances can be used to design highly integrated devices 45 and would stimulate studies of the the high frequency transport in graphene and other 2D materials.

Acknowledgement We thank Gottfried Strasser for providing access to a Fourier Spectrometer and Evgeniy Korostylev for expert help with e-beam at the MIPT’s Shared Facilities Center (Grant No. RFMEFI59417X0014). Financial support by the Austrian Science Fund FWF (START Y539-N16) and the European Union (grant agreement No. 696656 Graphene Flagship) is acknowledged. Supporting Information. Experimental details on the samples fabrication, optical and electrical characterizations, and formulation of the quasistatic approximation.

References (1) Christensen, J.; Manjavacas, A.; Thongrattanasiri, S.; Koppens, F. H. L.; Garcia de Abajo, F. J. Graphene Plasmon Waveguiding and Hybridization in Individual and Paired Nanoribbons. ACS Nano 2012, 6, 431–440. (2) Grigorenko, A. N.; Polini, M.; Novoselov, K. S. Graphene plasmonics. Nat. Phot. 2012, 6, 749. (3) Garcia de Abajo, F. J. Graphene Plasmonics: Challenges and Opportunities. ACS Photonics 2014, 1, 135–152. (4) Li, Y.; Yan, H.; Farmer, D. B.; Meng, X.; Zhu, W.; Osgood, R. M.; Heinz, T. F.;


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Avouris, P. Graphene Plasmon Enhanced Vibrational Sensing of Surface-Adsorbed Layers. Nano Lett. 2014, 14, 1573–1577. (5) Rodrigo, D.; Limaj, O.; Janner, D.; Etezadi, D.; Garcia de Abajo, F. J.; Pruneri, V.; Altug, H. Mid-infrared plasmonic biosensing with graphene. Science 2015, 349, 165– 168. (6) Farmer, D. B.; Avouris, P.; Li, Y.; Heinz, T. F.; Han, S.-J. Ultrasensitive Plasmonic Detection of Molecules with Graphene. ACS Photonics 2016, 3, 553–557. (7) Koppens, F. H. L.; Mueller, T.; Avouris, P.; Ferrari, A. C.; Vitiello, M. S.; Polini, M. Photodetectors based on graphene, other two-dimensional materials and hybrid systems. Nat. Nano 2014, 9, 780. (8) Kim, Y. D. et al. Bright visible light emission from graphene. Nat. Nano 2015, 10, 676681. (9) Kim, Y. D. et al. Ultrafast Graphene Light Emitters. Nano Lett. 2018, 18, 934–940. (10) Ju, L.; Geng, B.; Horng, J.; Girit, C.; Martin, M.; Hao, Z.; Bechtel, X., H. A. Liang; Zettl, A.; Shen, Y. R.; Wang, F. Graphene plasmonics for tunable terahertz metamaterials. Nat. Nano 2011, 6, 630. (11) Otsuji, T.; Tombet, S. A. B.; Satou, A.; Fukidome, H.; Suemitsu, M.; Sano, E.; Popov, V.; Ryzhii, M.; Ryzhii, V. Graphene-based devices in terahertz science and technology. J. of Phys. D: Appl. Phys. 45, 303001. (12) Jadidi, M. M.; Sushkov, A. B.; Myers-Ward, R. L.; Boyd, A. K.; Daniels, K. M.; Gaskill, D. K.; Fuhrer, M. S.; Drew, H. D.; Murphy, T. E. Tunable Terahertz Hybrid Metal-Graphene Plasmons. Nano Lett. 2015, 15, 7099–7104. 16 ACS Paragon Plus Environment

Page 16 of 22

Page 17 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

(13) Harada, Y.; Ukhtary, M. S.; Wang, M.; Srinivasan, S. K.; Hasdeo, E. H.; Nugraha, A. R. T.; Noe, G. T.; Sakai, Y.; Vajtai, R.; Ajayan, P. M.; Saito, R.; Kono, J. Giant Terahertz-Wave Absorption by Monolayer Graphene in a Total Internal Reflection Geometry. ACS Phot. 2017, 4, 121–126. (14) Caldwell, J. D.; Kretinin, A. V.; Chen, Y.; Giannini, V.; Fogler, M. M.; Francescato, C. T., Y. amd Ellis; Tischler, J. G.; Woods, C. R.; Giles, A. J.; Hong, K., M. Watanabe; Taniguchi, T.; Maier, S. A.; Novoselov, K. S. Subdiffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride. Nat. Comm. 2014, 5, 5221. (15) Ni, G. X.; Wang, L.; Goldflam, M. D.; Wagner, M.; Fei, Z.; McLeod, A. S.; Liu, M. K.; Keilmann, F.; Ozyilmaz, B.; Castro Neto, A. H.; Hone, J.; Fogler, M. M.; Basov, D. N. Ultrafast optical switching of infrared plasmon polaritons in highmobility graphene. Nat. Phot. 2016, 10, 244. (16) Das Sarma, S.; Hwang, E. H. Collective Modes of the Massless Dirac Plasma. Phys. Rev. Lett. 2009, 102, 206412. (17) Ju, L.; Geng, B.; Horng, J.; Girit, C.; Martin, M.; Hao, Z.; Bechtel, H. A.; Liang, X.; Zettl, A.; Shen, Y. R.; Wang, F. Graphene plasmonics for tunable terahertz metamaterials. Nat. Nano. 2011, 6, 630–634. (18) Fei, Z.; Rodin, A. S.; Andreev, G. O.; Bao, W.; McLeod, A. S.; Wagner, M.; Zhang, L. M.; Zhao, Z.; Thiemens, M.; Dominguez, G.; Fogler, M. M.; CastroNeto, A. H.; Lau, C. N.; Keilmann, F.; Basov, D. N. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 2012, 487, 82.


ACS Photonics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(19) Chen, J.; Badioli, M.; Alonso-Gonzalez, P.; Thongrattanasiri, S.; Huth, F.; Osmond, J.; Spasenovic, M.; Centeno, A.; Pesquera, A.; Godignon, P.; Elorza, A. Z.; Camara, N.; Garcia de Abajo, F. J.; Hillenbrand, F. H. L., R. Koppens Optical nano-imaging of gate-tunable graphene plasmons. Nature 2012, 487, 77–81. (20) Nikitin, A. Y.; Guinea, F.; Garcia-Vidal, F. J.; Martin-Moreno, L. Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons. Phys. Rev. B 2012, 85, 081405(R). (21) Gao, W.; Shi, G.; Jin, Z.; Shu, J.; Zhang, Q.; Vajtai, R.; Ajayan, P. M.; Kono, J.; Xu, Q. Excitation and Active Control of Propagating Surface Plasmon Polaritons in Graphene. Nano Lett. 2013, 13, 3698–3702. (22) Nikitin, A. Y.; Alonso-Gonzalez, P.; Hillenbrand, R. Efficient Coupling of Light to Graphene Plasmons by Compressing Surface Polaritons with Tapered Bulk Materials. Nano Lett. 2014, 14, 2896–2901. (23) Deng, B.; Guo, Q.; Li, C.; Wang, H.; Ling, X.; Farmer, D. B.; Han, S.-J.; Kong, J.; Xia, F. Coupling-Enhanced Broadband Mid-infrared Light Absorption in Graphene Plasmonic Nanostructures. ACS Nano 2016, 10, 11172–11178. (24) Jablan, M.; Buljan, H.; Soljaˇci´c, M. Plasmonics in graphene at infrared frequencies. Phys. Rev. B 2009, 80, 245435. (25) Fei, Z. et al. Infrared Nanoscopy of Dirac Plasmons at the Graphene-SiO2 Interface. Nano Lett. 2011, 11, 4701–4705. (26) Yan, H.; Low, T.; Zhu, W.; Wu, Y.; Freitag, M.; Li, X.; Guinea, F.; Avouris, P.; Xia, F. Damping pathways of mid-infrared plasmons in graphene nanostructures. Nat. Phot. 2013, 7, 394–399. 18 ACS Paragon Plus Environment

Page 18 of 22

Page 19 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

(27) Fei, Z.; Goldflam, M. D.; Wu, J.-S.; Dai, S.; Wagner, M.; McLeod, A. S.; Liu, M. K.; Post, K. W.; Zhu, S.; Janssen, G. C. A. M.; Fogler, M. M.; Basov, D. N. Edge and Surface Plasmons in Graphene Nanoribbons. Nano Lett. 2015, 15, 8271. (28) Yan, H.; Li, Z.; Li, X.; Zhu, W.; Avouris, P.; Xia, F. Infrared Spectroscopy of Tunable Dirac Terahertz Magneto-Plasmons in Graphene. Nano Lett. 2012, 12, 3766–3771. (29) Thongrattanasiri, S.; Koppens, F. H. L.; Garcia de Abajo, F. J. Complete Optical Absorption in Periodically Patterned Graphene. Phys. Rev. Lett. 2012, 108, 047401. (30) Spataru, C. D.; Ismail-Beigi, S.; Capaz, R. B.; Louie, S. G. Theory and Ab Initio Calculation of Radiative Lifetime of Excitons in Semiconducting Carbon Nanotubes. Phys. Rev. Lett. 2005, 95, 247402. (31) Citrin, D. S. Long intrinsic radiative lifetimes of excitons in quantum wires. Phys. Rev. Lett. 1992, 69, 3393–3396. (32) Perebeinos, V.; Tersoff, J.; Avouris, P. Radiative Lifetime of Excitons in Carbon Nanotubes. Nano Lett. 2005, 5, 2495–2499. (33) Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer, 1995. (34) Arsenin, A. V.; Gladun, A. D.; Leiman, V. G.; Semenenko, V. L.; Ryzhii, V. I. Plasma oscillations of the two-dimensional electron gas in the field-effect transistor with a cylindrical gate electrode. J. Commun. Technol. Electron. 2010, 55, 1285– 1294. (35) We neglect non-local effects which become important at width ≤ 20 nm 46,47 (36) Hanson, G. W. Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene. J. Appl. Phys. 2008, 103, 064302. 19 ACS Paragon Plus Environment

ACS Photonics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(37) Chen, J.-H.; Jang, C.; Ishigami, M.; Xiao, S.; Cullen, W.; Williams, E.; Fuhrer, M. Diffusive charge transport in graphene on SiO2. Solid State Comm. 2009, 149, 1080 – 1086. (38) Zhu, W.; Perebeinos, V.; Freitag, M.; Avouris, P. Carrier scattering, mobilities, and electrostatic potential in monolayer, bilayer, and trilayer graphene. Phys. Rev. B 2009, 80, 235402. (39) Das Sarma, S.; Adam, S.; Hwang, E. H.; Rossi, E. Electronic transport in twodimensional graphene. Rev. Mod. Phys. 2011, 83, 407. (40) Dai, S. et al. Tunable Phonon Polaritons in Atomically Thin van der Waals Crystals of Boron Nitride. Science 2014, 343, 1125–1129. (41) Woessner, A.; Lundeberg, M. B.; Gao, Y.; Principi, A.; Alonso-Gonzalez, P.; Carrega, M.; Watanabe, K.; Taniguchi, T.; Vignale, G.; Polini, M.; Hone, J.; Hillenbrand, R.; Koppens, F. H. L. Highly confined low-loss plasmons in graphene-boron nitride heterostructures. Nat. Materials 2015, 14, 421. (42) Nair, R. R.; Blake, P.; Grigorenko, A. N.; Novoselov, K. S.; Booth, T. J.; Stauber, T.; Peres, N. M. R.; Geim, A. K. Fine Structure Constant Defines Visual Transparency of Graphene. Science 2008, 320, 1308. (43) Mak, K. F.; Ju, L.; Wang, F.; Heinz, T. F. Optical spectroscopy of graphene: From the far infrared to the ultraviolet. Solid State Comm. 2012, 152, 1341 – 1349. (44) Horng, J.; Chen, C.-F.; Geng, B.; Girit, C.; Zhang, Y.; Hao, Z.; Bechtel, H. A.; Martin, M.; Zettl, A.; Crommie, M. F.; Shen, Y. R.; Wang, F. Drude conductivity of Dirac fermions in graphene. Phys. Rev. B 2011, 83, 165113. 20 ACS Paragon Plus Environment

Page 20 of 22

Page 21 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

(45) Zakrajsek, L.; Einarsson, E.; Thawdar, N.; Medley, M.; Jornet, J. M. Design of graphene-based plasmonic nano-antenna arrays in the presence of mutual coupling. 2017 11th European Conference on Antennas and Propagation (EUCAP). 2017; pp 1381–1385. (46) Wang, W.; Kinaret, J. M. Plasmons in graphene nanoribbons: Interband transitions and nonlocal effects. Phys. Rev. B 2013, 87, 195424. (47) Fallahi, A.; Low, T.; Tamagnone, M.; Perruisseau-Carrier, J. Nonlocal electromagnetic response of graphene nanostructures. Phys. Rev. B 2015, 91, 121405.


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