This is the author accepted manuscript. The final version is available from American Chemical Society via the DOI in this record.We study plasmonic resonances in electrostatically gated graphene nanoribbons on silicon dioxide substrates. Absorption spectra are measured in the mid-far infrared and reveal multiple peaks, with width-dependent resonant frequencies. We calculate the dielectric function within the random phase approximation and show that the observed spectra can be explained by surface-plasmon-phonon-polariton modes, which arise from coupling of the graphene plasmon to three surface optical phonon modes in the silicon dioxide.This research was supported by the UK Engineering and Physical Sciences Research Council, via the award of a Fellowship in Frontier Manufacturing (EP/J018651/1) to G.N., and the European Union under the FET-open grant GOSFEL

Faist, J

Gan, CH

Li, P

Liu, PQ

Luxmoore, IJ

Nash, GR

Valmorra, F

ACS Photonics

Open Research Exeter

1  Strong coupling in the far-infrared between graphene plasmons and the surface optical phonons of silicon dioxide  I. J. Luxmoore1,a), C. H. Gan1, P. Q. Liu2, F. Valmorra2, P. Li1, J. Faist2, and G. R. Nash1 1College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF, United Kingdom. 2Institute for Quantum Electronics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland.   Abstract We study plasmonic resonances in electrostatically gated graphene nanoribbons on silicon dioxide substrates. Absorption spectra are measured in the mid-far infrared and reveal multiple peaks, with width-dependent resonant frequencies. We calculate the dielectric function within the random phase approximation and show that the observed spectra can be explained by surface-plasmon-phonon-polariton modes, which arise from coupling of the graphene plasmon to three surface optical phonon modes in the silicon dioxide.  a) E-mail:  i.j.luxmoore@exeter.ac.uk  2  Graphene has been identified as a potential plasmonic material with resonances in the mid-IR to THz region of the electromagnetic spectrum1–6, which provides a wealth of opportunity for technological exploitation in free space communication, security, bio-sensing and trace gas detection2. The ability to electrostatically gate the charge density in graphene to in excess of 1×1013cm-2 predicates tunable and switchable devices1,4,7,8 and, coupled with an effective mass which is small compared to that of two dimensional electron gases in conventional semiconductors, results in significantly enhanced light-plasmon coupling and the observation of plasmons at room temperature1.  To excite graphene plasmons using light it is necessary to overcome the momentum mismatch between the surface-plasmon and free-space photons. This can be achieved using near-field excitation, where graphene plasmons have been launched and imaged and shown to have a high degree of electromagnetic confinement9,10. Plasmons can also be observed in optical experiments by patterning the graphene into structures with dimensions from 10s of nm to several microns1,4,5,11. Etching graphene into arrays of ribbons allows electromagnetic radiation to excite plasmons with             (where w is the width of the ribbon and          ), as the incident electric field induces oscillations in the free charge, resulting in the appearance of absorption features in transmission spectra. In micron scale ribbons, the absorption spectrum is determined purely by the plasmon dispersion, with the resonance frequency,        , lying in the terahertz region1. When the ribbon width is reduced to the order of several hundred nanometers the absorption spectra become more complicated, with multiple resonances in the mid-IR, arising from coupling between the plasmon and optical phonons in the underlying substrate5,12.  3  In this work we investigate the spectrum of graphene plasmons, confined to sub-micron widths, in the mid, but also the far IR. We study the transmission of electrically contacted nanoribbon arrays, which allow control of the graphene Fermi level via an applied back gate voltage. We observe four distinct peaks in the spectra and show using a calculation of the dielectric function within the random phase approximation (RPA) that the behavior can be attributed to the coupled modes, including one previously unobserved, of the graphene plasmon and surface optical phonons of the underlying SiO2 substrate. We also show that the calculations are consistent with a large range of ribbon widths corresponding to the frequency range from the THz to mid-infrared  The devices, pictured in Fig. 1(a) are fabricated from pre-transferred chemical vapor deposited (CVD) monolayer graphene (Graphene Square) on 300nm thick SiO2. The underlying Si substrate is used as a back gate and has a sheet resistance of ~10Ωcm. Electron beam lithography and reactive ion etching are used to pattern the graphene into 300µm  300µm nanoribbon arrays with widths between 150 and 500nm, where the width is ~40% of the ribbon repeat distance. Cr/Au source and drain contacts are deposited onto the graphene at either end of the nanoribbon array to allow electrical measurement and biasing. The nanoribbons are connected by 500nm wide perpendicular bridges every 10µm to ensure that a small break in an individual nanoribbon does not result in its electrical isolation. Fig. 1(b) shows a scanning electron microscope (SEM) image of a typical graphene nanoribbon device.   Un-patterned graphene devices are also fabricated on the same chip to allow estimation of the mobility and doping of the graphene. Fig. 2(a) shows the field effect characteristic of an un-patterned graphene device, where the resistance of the graphene is 4  measured as a function of the voltage applied to the back gate, VG. This shows behavior typical of monolayer graphene with a peak in resistance at VG≈88V, corresponding to the charge neutral point (VCNP), where the carrier density is minimized. Fitting the experimental data in the voltage range close to VCNP using a phenomenological model13 allows us to estimate the  mobility, μ to be ~600cm2V-1s-1. The large positive value of VCNP indicates that the fabricated samples have a significant intrinsic hole doping, which is also the case for the nanoribbon devices. Fig. 2(b) shows the field effect characteristic for a ribbon array with w=180nm, where VCNP≈80V. The large intrinsic doping allows wide tuning of the Fermi level, which can be approximated with a simple capacitor model11 for 300nm thick SiO2 as |  |       |√       |. The resistance is ~5X larger than the unpatterned graphene, suggesting that the mobility is reduced, however, it is difficult to confirm whether this is the case as the narrow width and relatively long length between bridges means that many ribbons have breaks at some point along their length, thus contributing to an increased resistance.  Infra-red spectral transmission measurements were performed in air at room temperature using a home-built infrared microscope coupled to a Fourier Transform Infrared Spectrometer (FTIR) with a measurement range of ~250-6000cm-1. Collimated light from the FTIR is focused onto the sample using a 36X reflective objective. The transmitted light is collected with a second reflective objective and focused onto a pyro-electric detector using an off-axis parabolic mirror. The incident light is linearly polarized using a broadband wire grid polarizer.   5   Fig. 1 (a) Optical and (b) SEM image of graphene nanoribbon array device. In (b) w indicates the graphene ribbon width and Eper (Epar) represents the linear polarization of the incident electromagnetic radiation, where the electric field component is perpendicular (parallel) to the nanoribbon.  In order to investigate absorption resonances arising from plasmons in the graphene nanoribbons, transmission spectra are measured for a given Fermi level. In the following, we present results for EF=-0.37eV. The extinction is defined as         , where T is the transmission spectrum at EF=-0.37eV and TCNP is the transmission spectrum at the charge neutrality point,        . Such an extinction spectrum is shown in Fig. 2(c) for a nanoribbon with w=180nm. When the incident light is polarized parallel to the nanoribbons (Epar in Fig.1 (b)) the absorption is not strongly affected by the graphene Fermi level. The only feature in the spectrum is a small increase in the absorption at low frequencies due to the Drude absorption1. When the incident light is polarized perpendicular to the nanoribbons (Eper in Fig.1 (b)), the absorption is very different, with four sharp absorption resonances revealed, labelled P1-P4 in Fig. 2(c). These absorption resonances can be attributed to the interaction of 6  graphene plasmons with surface optical (SO) phonons14 in the underlying SiO2 substrate. Their coupling via the long range Fröhlich interaction15 results in the excitation of surface-plasmon-phonon-polariton (SP3) modes5,11,12. In previous work, peaks P2-P4 were observed5,12 due to SO phonon modes at 806 and 1168cm-1. Here, we report the observation of strong coupling between the SO phonon mode at 485cm-1 and the graphene plasmon mode, which results in the additional resonance, P1.   Fig. 2 Field effect characteristic of (a) unpatterned graphene device and (b) graphene nanoribbon array with ribbon width of 180nm. The red line in (a) is a fit to the experimental data following ref. 13. (c) Extinction spectra of 180nm wide graphene nanoribbon array on SiO2 for incident light polarized parallel and perpendicular to the ribbons. When the light is polarized perpendicular to the nanoribbon, four peaks labelled P1-P4, are clearly identified.  To confirm the origin of P1, we investigate the absorption spectrum of a range of different nanoribbon widths between 150 and 500nm. If P1 does originate from the coupling of the plasmon resonance and a substrate SO phonon, then an anti-crossing with P2 should be 7  observed as the plasmon wavevector,            is tuned by the width of the ribbon, where w0 is an experimentally determined parameter which accounts for damage at the edge of the ribbons caused by the reactive ion etch5,16,17. Fig. 3(a) plots the extinction spectra for several devices with different ribbon widths between 180 and 450nm. As the ribbon width is increased, all four peaks shift to lower frequency but at different rates, with P1 and P2 shifting the most. The spectra are fit with a Fano model5,18 and the extracted peak positions are plotted in Fig. 3(b) as a function of  . We take      as the edge damage has been shown to vary considerably5,16,17 and we do not determine it experimentally for our devices. The data suggests that there is an anti-crossing of P1 and P2 at ~485cm-1 and to confirm that this arises from the plasmon-phonon coupling we calculate the dispersion and loss function of the SP3 modes for graphene on a SiO2 substrate.  The SO phonon frequencies are first obtained by solving the dispersion relation          , where          ∑                    (1) is the ionic dielectric function of an insulator exhibiting N SO phonon modes19, ω is the frequency,      is the frequency of the nth transverse optical (TO) phonon mode and it is assumed that            . In Eq. (1),                is the oscillator strength of the nth mode, such that ∑             , with         being the static dielectric constant and         the high frequency dielectric constant. It is taken that   =3.9 and   =2.4 for the SiO2 substrate. The values of      that appear in the oscillator strength fn were approximated 8  as19                      , and the dispersion calculated from Eq. (1) for three SO modes (N=3). In the calculations, the transverse optical (TO) and longitudinal optical (LO) phonon frequencies are taken to be    =[448, 791.7, 1128.1] cm-1 and    =[498.6, 811.5, 1317] cm-1, following values obtained from the literature19,20, and the SO frequencies          √                              were found to be [484.8, 805.9, 1229.0] cm-1. The SO mode energies are plotted as dotted lines in Fig. 3(b) and agree well with previous measurements5,19 and our experimental data. The corresponding ratios of the oscillator coupling strength f2/f1=1/5 and f3/f1=4/5 are also in close agreement to our experimental data and previously reported values5,19.  To obtain the dispersion of the hybrid SP3 modes, we consider the coupling between a two dimensional electron gas (the graphene plasmons) and dispersionless SO phonons at zero temperature21. Zero-temperature calculations are expected to yield quantitatively accurate results for our measurements taken at room temperature (T~300K, KBT~26meV), since EF>>KBT and the lowest SO phonon mode energy considered is           60meV. Within the random phase approximation, the total dielectric function is5,21                       (2) where             is the 2D Fourier transform of the Coulomb potential, Π is the polarizability of the uncoupled graphene sheet as calculated in ref. 22, and  9     [       ∑                   ]   (3) where the weighted coupling coefficient                         of each mode is proportional to the Fröhlich coupling strength21 and d is the distance between the graphene and the substrate, taken to be 3.5Å in the calculations. The loss function is obtained by taking the imaginary part of the inverse dielectric function, ie.           (refs. 15,21). The RPA loss function is plotted, along with the uncoupled graphene dispersion and the peak positions from the experimental data, in Fig. 3(b). We see good agreement between the experimental data and the calculations, confirming that the measured spectra arise from the SP3 modes originating from the coupling of graphene plasmon with three SO phonon modes in the SiO2 substrate. We additionally plot in Fig. 3(b) the peak positions from micro-ribbon arrays with widths of 1 and 2µm, fabricated using the same procedure, to show the good agreement between calculations and experiment over a wide frequency range from the THz to mid-infrared.       10   Figure 3. (a) Extinction spectra measured for graphene nanoribbons with a range of widths. (b) Calculated loss plot with extracted peak frequencies overlaid. The red dashed line shows the calculated dispersion of the uncoupled graphene plasmon. The grey, light grey and white dashed lines show the calculated frequency of the three surface optical phonons, ωSO1=1229cm-1, ωSO2=806cm-1, ωSO3=485cm-1. The grey symbols are peak positions extracted from micro-ribbon arrays and show that the calculations are consistent with a large range of ribbon widths which span a frequency range from the THz to mid-infrared.   In the local limit (      and      , with a phenomenological relaxation time for electrons), one may adopt the Drude model to describe the conductivity of monolayer graphene as                      , which leads to a simplified form of the polarizability2,6                               (4) 11  In general, the dynamical conductivity of graphene has two contributions, intra-band (Drude) and inter-band (see, e.g., equations (4) - (7)  in ref. 23). Including the inter-band contribution is essential when       . In our case       , therefore the inter-band conductivity is not taken into account. Based on calculations of the dispersion of the SP3 modes with        in Eq. 2, the Drude approximation was found to be highly accurate for    350cm-1 and nanoribbon widths w ≥ 100nm.  The lifetime of the SP3 modes can be extracted from the spectral linewidth and we measure values of <100fs for all four resonances, comparable to previous measurements5. For P4, the lifetime shows a monotonic increase from ~40fs to ~80fs as the nanoribbon width is reduced from 480nm to 180nm, indicating that the effect of edge scattering is insignificant in this regime. The increase in lifetime in this range can be explained by the convergence of the resonant frequency of the SP3 mode and the SO phonon frequency at 485cm-1, whereby the SP3 mode becomes mode phonon like5. The energy is far below the graphene optical phonon energy (~0.2eV), yet despite the lack of edge effects the lifetime is still short compared to the SO phonon lifetime of ~1ps. This suggests that the lifetime is dominated by damping mechanisms intrinsic to the graphene, such as impurity24 and defect scattering25, which is consistent with the low mobility, typical of CVD graphene and high doping shown in Fig. 2(b).  In conclusion, we have experimentally observed plasmonic resonances in graphene nanoribbons. The broad measurement range allows the simultaneous observation of four absorption peaks corresponding to the surface-plasmon-phonon-polariton modes arising from the Fröhlich coupling of graphene plasmons and the three SO phonon modes with frequencies 12  of ~485, 806 and 1230cm-1. Coupling to the SO phonon at 485cm-1 pushes graphene plasmonics into the technologically relevant far infrared regime and with improvements in material quality, graphene plasmonics can play an important role in future nano-photonic devices.  Acknowledgements This research was supported by the UK Engineering and Physical Sciences Research Council and the European Union under the FET-open grant GOSFEL. 13    13  References 1L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen and F. Wang, Nat. Nanotechnol. 6, 630 (2011). 2 T. Low and P. 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Strong coupling in the far-infrared between graphene plasmons and the surface optical phonons of silicon dioxide

Strong coupling in the far-infrared between graphene plasmons and the surface optical phonons of silicon dioxide

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