Plasmon-plasmon interactions and radiative damping of graphene

Letter Cite This: ACS Photonics 2018, 5, 3459−3465

pubs.acs.org/journal/apchd5

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 Downloaded via KAOHSIUNG MEDICAL UNIV on September 24, 2018 at 08:07:38 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

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 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

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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 temperature.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 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%, that is, the 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 its 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,

raphene 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 applications2,3 including biosensing,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 interaction23 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 scattering3 and damping channels of higher energy plasmons via graphene intrinsic optical phonons24 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 ps30 due to the large transferred spectral weight from the © 2018 American Chemical Society

Received: April 25, 2018 Published: August 9, 2018 3459

DOI: 10.1021/acsphotonics.8b00544 ACS Photonics 2018, 5, 3459−3465

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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) and (d) are analytical expressions using eq 1 and all the symbols are the result 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 , F = 300 meV, scattering rate ℏν = 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.

graphene nanoribbon width W to the period of the nanopatterned structure d, as shown in Figure 1a.

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 nanophotonic and optoelectronic devices. To illustrate the tunability of plasmons in graphene nanoribbon arrays (GNRs), shown in Figure 1a, we consider the scattering problem20 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-fields and displacement D-fields at the graphene plane (see Supporting 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, that is, the ratio of the

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RESULTS AND DISCUSSIONS Theoretical Predictions of the Absorption Spectra. The resulting absorption for the different GNRs are shown in Figure 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, that is, 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 the reverse process of light emission increases 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 3460

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Figure 2. Experimental evidence 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 and 100 nm gratings is 100 nm and for 600 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 , F = 300 meV and a scattering rate ℏν = 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.

them. The “intraband only” model for the conductivity includes only the Drude term λω = e 2 , F/π ℏ2(iω + ν), where ν is the carrier scattering rate and e, ℏ, and π are fundamental constants. The full model for λω, in addition to the Drude term, includes interband absorptions at energies above twice the Fermi energy , F. The striking difference between the two models for the high energy plasmons is due to the imaginary part of the graphene conductivity,36 which does not vanish even at low frequencies and temperatures. Experimental Signatures of the Plasmon−Plasmon and Plasmon−Radiation 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 them: 40, 100, 600, and 6000 nm, correspondingly, as shown in Figure 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 Supporting Information). The extinction spectra in Figure 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:

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

(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 Figure 1a. Dependencies of the plasmon peak positions on the width and the gap between the ribbons are shown in Figure 1c,d, correspondingly. The solid curves from eq 1 agree fairly well with the full solution of Maxwell’s equations,35 shown by the symbols in Figure 1c,d. As it follows from eq 1 and the full solution, the plasmon energy is remarkably dependent on the graphene conductivity function λω. In Figure 1c, the curves labeled “full λω, L = 2000 nm” and “intraband only, L = 2000 nm” both correspond to a plasmon in an isolated nanoribbon, since the interactions among plasmons are diminished at such a large gap between 3461

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Figure 3. Plasmon decay and oscillator strength scalings due to the plasmon−radiation 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 small filling factors, the canonical widths of ℏν and ℏν/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 “naı̈ve” expectations, if the oscillator strength were additive, and the solid curves show results according to eq 6. In all plots we used , F = 300 meV, ℏν = 10 meV, and ε1 = ε0 = 1.

ization of our devices: VT = 98 ± 8 V and mobility μ = 1500 ± 400 cm2/(V s) (see Supporting 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 Figure 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 Figure 2c. Note that in the case of an infinite graphene, the same plasmon dispersion eq 1 applies with Λ equal to 1.16 For the second resonances, also shown in Figure 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 Figure 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 1.93 times larger energy than the main plasmon absorption peak. The solid curve in the inset of Figure 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

the Fermi level and the scattering rate, and one parameter s for simul = ηext(W , L , , F, ν) ·s , the overall scale, according to ηext where W and L are nanoribbons 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 ηext(ω) = 1 −

τTM(ω) , τTE(ω)

where τTM and τTE are normally incident

light transmission spectra with the E-field perpendicular and parallel to the nanoribbons, respectively. The global fit parameter ℏν = 11.5 meV is used in all four spectra and a Fermi level of , F = 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 = , 2F/(Coxπ ℏ2vF2) = 90 ± 17 V and mobility μ = evF2 /(ν , F) = 1900 ± 200 cm2/(V s), where we used gate capacitance Cox = ε1/(4πtox) (SGS units) and the Fermi velocity in graphene vF = 108 cm/s. Optical measurements are consistent with the electrical character3462

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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 value44 2 D0 = 2πα ·, F/n01 are expected. In the case of unpatterned graphene, an analytical expression for the spectral weight can be obtained by integrating the absorption from eq 2:

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 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 meV) for the TE (TM) modes for a typical doping level of graphene , F = 300 meV. In Figure 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 Supporting Information). The deviations from the linear scaling for TM modes are due to the Landau damping contribution in smallwidth nanoribbons. The TE mode width is roughly twice as large than that of the TM mode. This can be easily understood from the fact that the spectral weights, that is, 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 Figure 3, let us consider the light absorption in an infinite graphene ηω given by ηω =

Reλω 4π |1 + ρω |2 ε0 c

D = D0

ρω =

ε1 − 4πλω /c

ε0 +

ε1 + 4πλω /c

D(r ) = D0r

2α ,F n01

D(r ) 2πα = 2 r ,F n01

(2)

(3)

where α = e /ℏc ≈ 1/137 is the fine structure constant and 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: 2α , Fr n01

(6)

π ℏν r ,F π ℏν r ,F

+

2πα n01

(7)

In Figure 4 we show the oscillator strength normalized to , Fr with respect to the dimensionless scattering rate ℏν /(r , F). 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 “naı̈ve” scaling D(r) = D0r holds in arrays with either small filling factor or low mobility graphene, for which ℏv ≫ 2α , Fr /n01. In unpatterned graphene studies,44 a 20% mismatch between the measured spectral weight and the value expected from the sum rule was reported. Using typical experimental values in ref 44 of ℏν ∼ 20 meV and , F ∼ 300 meV a discrepancy by n01D0/ πℏν ∼ 0.15 according to eq 5 could be anticipated.

2

ℏν′(r ) = ℏν +

π ℏν π ℏν + n01D0r

for both TE and TM modes as shown in Figure 3b by the solid curves. Deviations for the TM mode in small-width GNRs are due to the loss of the spectral weight to the interband plateau at energies above ∼2, F 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

Using the Drude conductivity λω, we obtain ηω ∝ (ω2 + ν′2)−1, where ℏν′ = ℏν +

(5)

Note that in the limit of a very large scattering rate ℏν, eq 5 suggests a canonical value of D = D0. In the case of a GNR, naı̈vely one would expect the spectral weight to scale linearly with the filling factor D = D0r, as shown by the dashed lines in Figure 3b. However, we find large deviations in the calculated spectral weight with a naı̈ve expectation at filling factors close to unity for both TE and TM modes, as shown in Figure 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:

where ρω is the reflection coefficient: ε0 −

π ℏν π ℏν + n01D0

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(4)

CONCLUSIONS We have demonstrated that plasmon−plasmon and plasmon− radiative interactions make surprisingly large contribution to

as shown in Figure 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 3463

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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, ℏν from 1−15 meV, and , F from 100−400 meV. The dashed horizontal line shows a value of 2πα , as expected 2 n01

from the “naı̈ve” model. The black solid curve shows the result for a smooth graphene, that is, r = 1, according to eq 7.

the optical properties of GNRs. The found scaling laws of the energy, radiative broadening, and oscillator strength of the plasmon resonances can be used to design highly integrated devices45 and would stimulate studies of the high frequency transport in graphene and other 2D materials.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.8b00544. Experimental details on the samples fabrication, optical and electrical characterizations, and formulation of the diffraction problem and its quasistatic approximation (PDF).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Vasili Perebeinos: 0000-0002-0234-8188 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Gottfried Strasser for providing access to a Fourier Spectrometer and Evgeniy Korostylev for expert help with ebeam 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.

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REFERENCES

(1) Christensen, J.; Manjavacas, A.; Thongrattanasiri, S.; Koppens, F. H. L.; Garcia de Abajo, F. J. Graphene Plasmon Waveguiding and 3464

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DOI: 10.1021/acsphotonics.8b00544 ACS Photonics 2018, 5, 3459−3465