## Abstract

Plasmons in patterned graphene have attracted much interest because of possible applications in sensing, nanophotonics, and optoelectronics. We perform mid and far-infrared optical studies of electrically doped graphene nanoribbon arrays as a function of the filling factor and compare results with the unpatterned graphene. We demonstrate that an increase in both the filling factor of nanoribbon arrays and the free carrier concentration intensifies the plasmon-plasmon and plasmon-radiation interactions. As a result, the free-carrier dynamics manifested itself in the strong plasmon redshift and increased radiative damping compared to non-interacting models for the transverse magnetic polarization. Similarly, signatures of interactions are identified for plasmons in transverse electric polarization. The obtained experimental and theoretical results provide the basis for better understanding and controlling graphene-based structures’ spectral properties, thus facilitating applications’ development.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The graphene's unique optical and electronic properties are strongly related to its 2D nature and honeycomb lattice providing a gapless electronic spectrum with linear dispersion law at the Fermi energy [1–3]. Electrons in graphene act as massless Dirac fermions manifesting phenomena as universal optical absorption [4,5]. Unusual high-frequency properties and collective behavior of the 2D electron and hole systems in graphene-based heterostructures open a variety of possible applications in photodetection [6], bio- [7] and gas [8] sensing, optoelectronics [9–12], and photonics [13–17]. Recently, plasmons in graphene nanoribbons were considered for quantum qubit applications [18] due to their tunability and long plasmon lifetimes.

One of the most straightforward patterned graphene structures, graphene nanoribbon arrays (GNRs), have attracted increased attention in sensing applications [19–21]. The confined plasmonic modes in such structures have the local density of optical states more than 10^{6} larger than in free space [22]. Therefore, it leads to enormously increased light−matter interactions in the infrared spectral range, where it is mostly required to detect vibrational fingerprint modes of molecules [23,24].

In unpatterned graphene, the extinction spectrum exhibits a peak centered at zero frequency [25,26] or a Drude peak. Similar spectra are observed in a GNR with the transverse electric (TE) light polarization when the incident radiation electric field is directed along with the ribbons. In these cases, the absorption spectra can well be described by a Lorentzian representing a Drude-type frequency dependence of conductivity ${\sigma _{AC}}(\omega )= {{i{D_0}} / {\pi ({\omega + i\nu } )}}$, where *ω* is the light frequency, *ν* is the carrier scattering rate, and *D*_{0} is the Drude weight. The latter in canonical approach is based on a semiclassical Boltzmann transport theory: ${D_0} = {{{e^2}{E_F}} / {{\hbar ^2}}} = ({{{{v_F}{e^2}} / \hbar }} )\sqrt {\pi |n |} $, where *E _{F}* is the Fermi energy,

*v*is the Fermi velocity,

_{F}*n*is the carrier concentration,

*e*is the electron charge [25,27,28]. In the transverse magnetic (TM) mode, when the incident electric field is perpendicular to the ribbon's direction, the collective free carrier density oscillations, called plasmons, are excited by the light at finite frequency [29]. A damped oscillator model can accurately describe the plasmon extinction spectrum's shape as ${\mathop{\rm Im}\nolimits} ({{{ - \omega } / {{\omega^2} - \omega_p^2 + i\omega {\Gamma _p}}}} )$, where Г

*is the plasmon width, and*

_{р}*ω*is the plasmon frequency [30].

_{p}That canonical picture of graphene plasmons in GNRs has been qualitatively confirmed by many spectral measurements of the Drude-like and plasmon extinction in the infrared spectral range [14,25,31,32]. However, the electron-electron interactions [33] have been shown to cause plasmon energy shift not accounted for by the canonical dispersion relation. Moreover, the strong plasmon-phonon interactions lead to plasmon broadening [34]. Recently, plasmon-radiation interaction has been proposed to lead to additional spectra broadening [35]. Interactions among plasmon modes in neighboring strips have been shown to cause charge density redistribution across ribbons leading to a redshift of the plasmon frequency with respect to the canonical model predictions [36]. Numerical simulations using the finite-difference time-domain (FDTD) method have revealed enhancement of coupling strength between the plasmon modes due to the filling factor increasing [37]. Also, they have shown how one can control the plasmon coupling strength by severally varying the Fermi level of neighboring ribbons. The plasmon-plasmon renormalization effect is the most pronounced when the filling factor is approaching unity, as it was shown in Ref. [35] by solving Maxwell’s equations in a Fourier expansion of the electrical current, electrical E-ﬁelds and displacement D-ﬁelds at the graphene plane. The proposed various mechanisms of interactions in GNR arrays describe well deviations of both plasmon and Drude-like extinction from its canonical picture. However, it still lacks experimental verification from the point of view of functional dependencies on the GNR doping level and geometrical parameters. In this work, we perform extensive spectral studies of unpatterned graphene and graphene nanoribbon arrays of various filling factors, electrical doping levels for both TE and TM modes in mid- and far-infrared ranges, and compare the results with the interacting theory [35].

## 2. Materials and methods

The samples are fabricated on a 285 nm thick SiO_{2} substrate on a highly resistive silicon wafer (> 5000 Ω·cm), which is polished on the backside to avoid scattering due to surface roughness. The graphene ribbons’ design is chosen to ensure that the lowest energy SiO_{2} optical phonon at 60 meV lies above the plasmon's energy to avoid interaction between the plasmon and the surface polar phonon [34]. CVD-grown monolayer graphene films with a size of around 10×10 mm^{2} are transferred using a wet transfer technique. The graphene was patterned using electron-beam lithography and reactive ion etching (RIE) to define grating and test device areas, as shown in Figs. 1(a), (b). Contacts are deposited using electron-beam lithography and metallization of titanium (5 nm) and gold (60 nm). Finally, four different gratings with *L *= 0.25, 1, 1.5, and 4 µm gap widths are patterned on 2×2 mm^{2} spot size using electron-beam lithography and RIE. The ribbon width was kept the same for all gratings at *W* = 1 µm, such that the resulting filling factors have values *r* = *W/*(*W + L*) = 0.8, 0.5, 0.4, and 0.2. The samples are annealed at 300°C for around 20 hours to ensure that all residuals due to the fabrication are removed.

All GNR samples and the unpatterned graphene are electrically characterized using a constant source-drain current 10 µA. Figure 1(c) shows a typical dependence of the source-drain resistances, *R _{sd}*, as a function of the applied gate-source voltage,

*V*, in the sample with the filling factor of 0.8. The maximum value of

_{g}*R*corresponds to the charge neutrality point (CNP). The metal-graphene contact resistivity was estimated to be

_{sd}*ρ*≈ 15 kΩ·µm based on transport measurements of test devices with various channel length [38]. Therefore, the contact resistance in 2 mm wide samples can be neglected when analyzing transport data. To estimate mobility, we fit the obtained

_{c}*R*(

_{sd}*V*) dependencies using the following expression:

_{g}*V*and

_{CNP}*n*are the gate voltage and free carrier concentration at the CNP,

_{CNP}*µ*is the free carrier mobility (here we consider the same mobility for electrons and holes),

*L*and

_{GNR}*W*are the length and the width of the graphene channel, and

_{GNR}*C*= 1.211×10

_{ox}^{−8}F/cm

^{2}is the SiO

_{2}layer capacitance per unit area. The resulting mobilities varied in the range of 2200 ± 900 cm

^{2}/Vs.

The transmission spectra are acquired using a Fourier-Transform Infrared (FTIR) spectrometer Bruker IFS 125 HR with a Si bolometer as a detector in the far-infrared (FIR) spectral range and with a pyroelectric DLaTGS detector in the mid-infrared (MIR) range. The extinction spectra *Ext*.(*ω*,Δ*V _{g}*) of the samples are obtained as follows:

*Tr*(

*ω*,Δ

*V*) is a transmission spectrum at the gate-source voltage, Δ

_{g}*V*=

_{g}*V*–

_{g}*V*,

_{CNP}*Tr*(

*ω*,

*V*) is the transmission spectrum at the gate-source voltage corresponding to the charge neutral point,

_{CNP}*V*, defined by the maximum of the source-drain resistance, and

_{CNP}*ω*is the radiation frequency. The gate-source voltage swing values are Δ

*V*= -50, -30, -10, 10, 30, 50 V.

_{g}*Tr*(

*ω*,Δ

*V*) and

_{g}*Tr*(

*ω*,

*V*) spectra are measured subsequently with no manipulations in the optical path and no sample chamber venting. The corresponding

_{CNP}*Tr*(

*ω*,

*V*) spectrum is measured for each

_{CNP}*Tr*(

*ω*,Δ

*V*) spectrum. This procedure allowed us to avoid spectra distortions caused by a change in the optical path.

_{g}## 3. Results

Since the interband transitions in graphene are blocked up to twice the Fermi energy [39], i.e. 2|*E _{F}*|, by the Pauli principle, we can obtain

*E*values from the MIR spectra analysis [40]. All the measured MIR transmission spectra (Fig. 2) have a step-like shape that can be described by

_{F}*A*and

*B*represent the height and the background level of the step, respectively. The width of the slope is given by

*C*, which is of an order of a few kT. We fit the MIR spectra by (3) to obtain the

*E*values that we will use below.

_{F}The extinction spectra of the sample with the filling factor *r* = 0.5 are shown in Fig. 3(a) for TE polarization. The black lines show the fits of the spectra to the following Lorentzian curve centered at zero frequency:

*y*

_{0}is the background level, which is small in our case, being less than 10

^{−2}% of the peak maximum.

The plasmon extinction spectra for TM polarization for the same sample are shown in Fig. 3(b). A damped oscillator model fitted the plasmon line shape:

*D*is the plasmon oscillator strength.

_{p}In what follows, we analyze both TE and TM FIR extinction spectra parameters for each filling factor as functions of the Fermi energy that we extracted from the MIR spectra and compare those dependencies with the theoretical predictions, which consider the influence of the plasmon-radiation and long-range plasmon-plasmon Coulomb interactions.

## 4. Discussion

We obtained the Drude widths (Fig. 4(a)) and oscillator strengths (Fig. 4(b)) from the extinction spectra in the TE mode using (4). In the case of extinction of GNRs placed on the SiO_{2} substrate, a canonical value of the Drude oscillator strength can be written ${D_{can}} = {{r2\pi \alpha {E_F}} / {{n_{01}}}}$. Here, *α* is the fine structure constant, and *n*_{01} = (*n*_{0} + *n*_{1})/2 = 1.5 is the averaged refractive index with *n*_{0} = 1 and *n*_{1} = 2 being the refractive indexes of the air and that of SiO_{2} layer, consequently. The peak width in the non-interacting picture stays independent of both the filling factor and the Fermi energy: ${\Gamma _{can}} = \hbar \nu $.

To correctly interpret experimental data and determine the influence of plasmon-plasmon and plasmon-radiative interactions, we analyze the integral of extinction (see Eq. (2)):

_{2}substrate in the presence of graphene with Fermi energy

*E*and scattering rate ν, and of its absence, correspondingly. A similar integral calculated from absorption in graphene instead of extinction gives a physical value called spectral weight. This value characterizes conductivity in graphene and appears in such a fundamental relation as the sum rule. However, only extinction is available from the experiments. Analogously to Ref. [35], where the spectral weight was analytically calculated for plain graphene with conductivity given by the Drude model [41], we can get obtain a corresponding analytical expression for Eq. (6). It was shown in Ref. [35] that in the case of plain graphene, substituting

_{F}*E*with $r \cdot {E_F}$ in analytical expression for plain graphene reproduces the spectral weight of numerically calculated absorption spectra of GNRs for both TE and TM polarizations. To make sure that the same is true for the integral of extinction, we obtain an analogous phenomenological expression for Eq. (6):

_{F}For each sample, we fit the experimentally obtained values of the Drude peak widths as a function of the Fermi energy by linear dependencies. Then, we compare the obtained slopes with the theoretical values, which would be zero in the canonical non-interacting picture and ${{2r\alpha } / {{n_{01}}}}$ in the interacting one (Fig. 5(a)). We find that within the error bars, the experimental slopes follow the interacting theory predictions. Next, for each *r* we adjust ℏ*ν* such that (8) would fit the experimental data and use the resulting values of ℏ*ν* in (7) to fit the Drude oscillator strengths as functions of Fermi energies.

Since the GNR transmission spectrum at the CNP is influenced by the electron and hole puddles [42], the measured values of the Drude oscillator strength, Δ*D*, differs from one that to be compared with theoretical values *D*, by a constant value *D _{C}*. The latter is the Drude oscillator strength at the CNP, and it remains to be the same for all spectra in each sample,

*D*requires a pure SiO

_{C}_{2}substrate transmission spectrum to be used as the background. That measurement would require a movement of the sample between the background and the sample measurements. That would lead to the change in the optical path and, hence, to the spectral distortions.

On the other hand, the partial derivative of the Drude oscillator strength with respect to the Fermi energy isn't affected by *D _{C}*, since

*D*is a constant for a given sample. So, it can be determined experimentally without involving the direct measurement of the

_{C}*D*. In the canonical picture, the derivative depends on the filling factor only:

_{C}*E*takes a more complicated form:

_{F}*r*= 0.8. For other samples the maximal deviations are 0.15% (

*r*= 1), 0.28% (

*r*= 0.5), 0.11% (

*r*= 0.4) and 0.21% (

*r*= 0.2). Based on this, we estimate the Drude oscillator strength derivative's experimental values with respect to the Fermi energy for each filling factor as the slope of the linear fit of the measured Δ

*D*(

*E*) dependence.

_{F}Analysis of the derivative ${{\partial ({\Delta D} )} / {\partial {E_F}}}$ as a function of the filling factor demonstrates that the difference between the predictions of the canonical and the interacting theories is not so strict as in the case of ${{\partial \Gamma } / {\partial {E_F}}}$. However, at large filling factors, the interacting theory predicts lower values of ${{\partial ({\Delta D} )} / {\partial {E_F}}}$ than the canonical one, which is seen in the experiment. Moreover, for the unpatterned graphene, the lowering of ${{\partial ({\Delta D} )} / {\partial {E_F}}}$ is even more pronounced than it is expected due to the interactions.

In the interacting theory, plasmon dispersion law has the form [35]:

where*q*is the plasmon wavevector,

*κ*= (

*ε*

_{0}+

*ε*

_{1})/2 is the averaged dielectric constant with

*ε*

_{0}= 1 and

*ε*

_{1}= 4 being the dielectric constants of the air and the SiO

_{2}layer, respectively. A dimensionless parameter $\mathrm{\Lambda }(r )$ is associated with the Coulomb interaction of charge density within a given graphene stripe and between different stripes in the GNRs array. It was shown in Ref. [35] that a phenomenological expression $\Lambda (r )= 0.734 \cdot {({1 - {r^{1.75}}} )^{0.331}}$ perfectly reproduces the main peak positions of numerical absorption and extinction TM spectra. Note that Eq. (12) also follows from the quasistatic approximation in the absence of plasmon-radiation interactions. For the studied samples here, Λ(

*r*) varies from 0.505 (

*r*= 0.8) to 0.719 (

*r*= 0.2). Using q = π/W and $\sigma = {e^2}{E_F}/\pi {\hbar ^2}({i{\omega_p} + \nu } )$ one can reduce Eq. (12) to:

*r*) = 1 for all filling factors.

Figure 6 shows the measured plasmon frequency values and their fits by Eq. (13) with various Λ(*r*) values as Fermi energy functions. One can see that the plasmon energies’ experimental values are better accounted for by the interacting theory than the canonical theory.

## 5. Conclusions

We performed extensive spectral studies of graphene plasmons in nanoribbon arrays for TE and TM polarizations and compared the results against the canonical non-interacting theory and the interacting theory accounting for the plasmon-plasmon and plasmon-radiative interactions. In particular, we find the plasmon energies’ redshifts, flattering of the Drude and plasmon weights, broadening plasmon and Drude peaks with increasing free carrier concentration. The interacting theory much better accounts for the measured dependencies. The demonstrated qualitative agreement with the theoretical predictions emphasizes the role of interactions, which is essential for quantitative predictions and designs of the graphene-based plasmonic devices.

## Funding

Office of the Vice President for Research and Economic Development, University at Buffalo (75023).

## Acknowledgments

The authors would like to acknowledge the LPI Shared Facilities Center for providing research equipment for FTIR measurements [43]. V.P. acknowledges support from the Vice President for Research and Economic Development (VPRED) and the Center for Computational Research at the University at Buffalo [44]. S.S. and T.M. acknowledge funding by the Graphene Flagship.

## Disclosures

The authors declare no conflicts of interest.

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