Introduction

Collective oscillations of conduction electrons, also known as plasmons, find a vast number of applications in nanophotonics due to their ability to enhance and confine optical fields far below the wavelength scale.1 In particular, plasmons supported by noble metal nanostructures have been extensively used for optical trapping,2 nonlinear optics,3,4,5 single-molecule detection6,7 and signal processing,8 among other feats. Recently, highly doped graphene has also been found to sustain plasmons at infrared and lower frequencies,9,10,11,12,13,14,15,16,17,18,19 with the unique advantages of possessing a large electrical tunability, long lifetimes, and strong nonlinearities.20,21,22,23 The peculiar electronic structure of this two-dimensional material24,25 allows us to substantially shift its Fermi level (EF=ħvF, where vF≈106 m s−1 is the Fermi velocity) by adding a moderate density of doping charge carriers n. Fermi energies as high as ∼1 eV have been achieved through electrostatic gating.26,27 Furthermore, strong coupling of light to localized plasmons can be realized by patterning the graphene,28,29 even reaching the limit of complete optical absorption.30 While a classical electromagnetic description of plasmons is sufficient to deal with large graphene structures compared with the Fermi wavelength ,9,16,28,31,32,33,34,35 nonlocal quantum effects play an important role in smaller islands,36 including plasmon quenching through coupling to electronic zigzag edge states,37,38 which is reminiscence of the quenching described in near-touching metallic nanoparticles.39,40

In this work, we study nonlocal quantum effects in individual and interacting graphene structures. We concentrate on nanoribbons as they are central elements in many previous theoretical28,34,37,38,41,42,43 and experimental10,13,14,18,43 graphene plasmon studies, and also stimulated by the availability of new methods of synthesis with control over the ribbon width down to the nanometer scale.44,45,46,47 Additionally, ribbons can be electrically contacted far from the region in which plasmons are exploited, thus enabling optical tunability using non-invasive electrical gating. In particular, we compare classical and quantum-mechanical models for the graphene optical response, focusing on the lowest-order dipole plasmons and examining their role in the normal-incidence extinction cross-section for light polarized perpendicular to the ribbon edges. We study ribbons of <10 nm width and different edge terminations—armchair (AC) and zigzag (ZZ)—as well as their interactions in closely spaced dimers and periodic arrays. Our results provide a roadmap of plasmons in individual and interacting narrow ribbons, reaching near-infrared energies and exhibiting important nonlocal effects that must be considered in the design of potential device applications. These results also reveal the advantage of ribbons relative to other alternative methods to exploit near-infrared plasmons, such as gratings (they should have similar periodicity as the ribbon widths here considered), tips and local emitters (they generally couple non-resonantly to propagating plasmon bands, thus resulting in broad spectral features), and prisms (the large mismatch between light and graphene-plasmon wavelengths requires the use of very high-index dielectrics that are currently unavailable in the near infrared).

Materials and methods

The extinction cross-section, which we present normalized to the graphene area, is obtained through the optical theorem from the ribbon polarizability, which we in turn calculate from the induced density using the classical28 and quantum-mechanical38 methods described elsewhere. In particular, the quantum description, which we term TB-RPA model, is based upon the random-phase approximation48 (RPA) using tight-binding (TB) electronic states as input.38 However, in the classical electromagnetic model the carbon layer is represented by a local conductivity, which is taken from the local-RPA approximation28 and used as input to solve the Maxwell equations with the boundary element method.49

In the quantum-mechanical calculations presented below for ribbon arrays, the absorbance is obtained from the effective polarizability per unit length /L for each ribbon, which we in turn obtain from the quantum TB-RPA model, conveniently modified to deal with infinite ribbon arrays (see below). This model yields the non-interacting susceptibility , which permits writing the induced density as in terms of the total potential ϕl, where l and l' run over the carbon sites of the graphene structure.41 The total potential is related to the external potential through , where vll' is the Coulomb interaction between sites l and l'. These expressions can be easily combined to find the induced density as ρ=(1−χ0⋅v)−1⋅χ0⋅ϕext, using matrix notation. For a finite number of ribbons, the one-dimensional periodicity of the atomic lattice is used to reduce the calculation to a finite number of sites (l and l' indices) within a unit cell.41 Here, for ribbon arrays, we follow this procedure to calculate χ0 for a single ribbon, but supplementing vll' with the sum of all interactions of site l with the equivalent l' sites in all other ribbons. The effective polarizability per unit length is then given by , where the sum extends over the noted unit cell of length b along the ribbon, xl is the distance across it, and Eext is the external electric field (i.e., =−xlEext). Finally, the normal-incidence reflection and transmission coefficients reduce to r=(2πik/a)(/L and t=1+r, where a is the period of the array, and from here, the absorption reduces to 1−|r|2−|1+r|2.35

For simplicity, we assume that the ribbons are in vacuum at 300 K in both the classical and quantum calculations. The addition of a substrate of permittivity ϵS can be simply introduced through an effective permittivity ϵeff=(ϵS+1)/2 of the surrounding medium. This is a rigorous correction in the local electrostatic limit,41 which essentially leads to a redshift of plasmon frequencies by a factor .

Results and discussion

Individual ribbons

We start by analyzing plasmons in individual ribbons of small width W=6 nm (Figure 1). The transverse dipole plasmon on which we concentrate here is the lowest-energy feature in the spectra. Remarkably, the classical calculations (insensitive to the edge configuration) are in excellent quantitative agreement with quantum-mechanical calculations for both AC and ZZ edges provided the plasmon energy is below the Fermi level (see EF=1 eV and 1.5 eV curves in Figure 1). If this condition is not satisfied (EF=0.4 eV), the plasmon is strongly quenched in the ZZ-edge ribbons, while it is slightly shifted and broadened in AC-edge quantum calculations in comparison with classical theory. This behavior is fully reproducible over a large range of Fermi energies and small ribbon widths, of which Figure 1 shows some representative examples. The observed plasmon quenching can be attributed to the existence of zero-energy electronic edge states, as already shown for even narrower ribbons.37 Additionally, the plasmon energy increases with EF, following the approximate behavior35 (this expression is obtained from the Drude model, while RPA corrections push the plasmon energy slightly downwards), whereas its strength grows with doping as well.

Figure 1
figure 1

Plasmons in individual graphene ribbons. We show the normal-incidence extinction cross-section of self-standing graphene nanoribbons (width W=6 nm and intrinsic damping of 0.02 eV) for several values of the Fermi energy EF, calculated using two different models: classical theory with the local-RPA conductivity;28 and quantum-mechanical TB-RPA for ribbons with ZZ and AC edges.38 We focus on the lowest-order dipolar mode (see induced charge density in inset), which is excited by light polarized across the ribbon. AC, armchair; RPA, random-phase approximation; TB, tight-binding; ZZ, zigzag.

Full size image

Ribbon dimers

Two doped coplanar parallel ribbons separated by a carbon-to-carbon edge distance d interact producing significant changes in the plasmonic spectrum of the whole structure.41 We study this configuration in Figure 2 for ribbons of width W=6 nm and Fermi energy EF=0.4 eV using both classical (upper part) and quantum TB-RPA (lower part, for AC edges) theories. The interaction is attractive and leads to redshifts in the dipole-active bonding plasmons here discussed.41 The sequence of spectra shows that the discrepancy between classical and quantum calculations increases as the ribbons approach each other. This is consistent with the intuitive idea that nonlocal interactions gain weight at shorter distances. As part of this behavior, the quantum calculations do not seem to approach the plasmon of the double-width ribbon (left dashed-curve spectrum), in contrast to the classical simulations. The evolution of the dimer plasmon energy with separation (Figure 3) clearly illustrates this effect: there is a sizeable jump in the quantum calculation of the double-width AC plasmon when a centered row of C–C bonds is removed (i.e., the TB hopping is set to zero in those bonds, represented as dashed lines in the left inset of Figure 3), thus becoming a closely spaced dimer. However, the classical calculation shows a continuous convergence of the dimer towards the double-width ribbon at small distances. Incidentally, as ribbon edges are generally passivated with hydrogens, and considering the C–H and H–H bond distances, we estimate that only carbon-to-carbon edge distances >0.3 nm are physically realistic, with well-separated and non-tunneling electronic states in each ribbon. We include smaller distances for tutorial purposes, and in particular dmin=/2 in AC ribbons in order to study the effect of removing one row of C–C bonds, as discussed above.

Figure 2
figure 2

Plasmons in graphene ribbon dimers. Normal-incidence classical (a) and quantum-mechanical TB-RPA (AC edges) (b) calculations of the extinction cross-sections of ribbon dimers (width W=6 nm, Fermi energy EF=0.4 eV and intrinsic damping of 0.02 eV) for different carbon-to-carbon edge separations d (see also Figure 3b for the relation of this parameter to the edge-to-edge distance in the classical model). Spectra for single ribbons of widths W and 2W are shown as dotted and dashed curves, respectively. The light polarization is across the ribbons. AC, armchair; RPA, random-phase approximation; TB, tight-binding.

Full size image
Figure 3
figure 3

Separation dependence of the transverse plasmon energy in ribbon dimers. (a) We represent the plasmon peak energies extracted from Figure 2 as a function of the carbon-to-carbon edge distance d using classical and quantum-mechanical (for AC ribbons) models. The classical calculations converge well to the double-width-ribbon limit for d<<W, in contrast to the quantum-mechanical model, which predicts a substantial energy gap. The left inset shows the nearest separation dmin=/2 considered for the quantum AC calculations, where a0=0.1421 nm is the C–C bond distance. (b) Scheme of the notation used for the separation distance between ribbons: we take d to be the carbon–carbon spacing throughout this paper; classical calculations are performed with a separation between electronic edges given by dcl=d−dmin. AC, armchair.

Full size image

Periodic ribbon arrays

A monolayer of extended undoped graphene exhibits a nearly constant absorption πα≈2.3%,50,51 where α≈1/137 is the fine-structure constant. This relatively small level of absorption can be enhanced through patterning and doping.30 We explore this possibility in arrays of narrow ribbons such as those of Figures 4 and 5, which show the evolution of the plasmon energies and their associated absorption. Here, we compare AC and ZZ ribbons in order to assess the strength of nonlocal effects.

Figure 4
figure 4

Absorbance of graphene ribbon arrays. We represent the absorbance of AC (a, b) and ZZ (c, d) ribbon arrays for two different Fermi energies (see upper labels) and various carbon-to-carbon edge separations d (see legend, where we use the C–C bond distance a0), as calculated from the TB-RPA model. The ribbon width is W=6 nm and the intrinsic plasmon damping is 0.02 eV in all cases. AC, armchair; RPA, random-phase approximation; TB, tight-binding; ZZ, zigzag.

Full size image
Figure 5
figure 5

Dependence of the fundamental plasmon mode on inter-ribbon distance in arrays. We show the normal-incidence dipolar plasmon frequency in AC (red curves) and ZZ (blue curves) ribbon arrays as a function of carbon-to-carbon edge separation d for various combinations of ribbon width W and Fermi energy EF (see labels). The dashed lines indicate the single-ribbon limit for each type of edge, whereas the dotted black lines show the classical single-ribbon result. AC, armchair; ZZ, zigzag.

Full size image

The resulting absorption spectra (Figure 4) exhibit again a high sensitivity to ZZ edges when the plasmon energies are above the Fermi level, leading to a significant reduction of absorption and increasing plasmon broadening (cf. Figure 4a and 4c). At higher doping (Figure 4b and 4d), plasmon broadening is limited to the intrinsic damping (0.02 eV) in both AC and ZZ ribbon arrays. Interestingly, the inter-ribbon interaction is smaller with ZZ edges, as revealed by the larger redshift observed in AC ribbons at close separations d. In all cases, these shifts are larger than in dimers (cf. for example Figures 2b and 4a). Remarkably, the maximum possible absorption of 50% is reached with ZZ edges in Figure 4d at short separations d of the order of the C–C bond distance, or equivalently, in a structure obtained by removing atomic bonds along an array of parallel rows spaced by a period of 6 nm. Full absorption can further be achieved by placing the structure approximately a quarter wavelength away from a good mirror (Salisbury screen configuration30,52,53).

The plasmon energies obtained from TB-RPA in ribbon arrays are summarized in Figure 5, where we confirm that ZZ-edge-induced plasmon quenching does not affect plasmons of energies below the Fermi level. This is clearly illustrated when moving from ribbons of width W=6 nm to W=8 nm for constant doping (EF=0.4 eV), which causes the plasmon energy to go below EF, giving rise to very similar plasmon energies in both AC and ZZ ribbons (Figure 5b). It should be stressed that in the touching limit (small d), the graphene becomes an extended sheet that presents nearly zero absorption (at photon energies below 2EF), and thus, Figure 5 shows again a discontinuity between this limit and the configurations with ribbons separated by an atomic distance, right after breaking inter-ribbon C–C bonds.

Conclusions

Nonlocal effects in the plasmons of doped graphene ribbons exhibit the following general properties: (i) plasmons are quenched by ZZ edges, unless their energies are below the Fermi level, so that they cannot decay through excitation of electronic edge states; (ii) nonlocal quantum effects (generally observed as blueshifts) increase with decreasing ribbon width, although a classical electromagnetic description produces reasonable results for individual ribbons of widths down to a few nanometers; (iii) nonlocal effects are important in interacting ribbons at short separations, leading to substantial blueshifts in the dipole-active plasmons of closely spaced ribbon dimers, in contrast to classical theory, which predicts a smooth convergence towards the double-width ribbon; (iv) remarkably, the removal of a single row of atoms produces a dramatic increase in the plasmon energy, observed both in dimers and in ribbon arrays; (v) our realistic quantum-mechanical calculations yield plasmon energies that are pushed up to the near-infrared regime for ribbon widths of a few nanometers, similar to those that can be synthesized by chemical self-assembly.46 These results provide a solid theoretical background for understanding the interaction between graphene nanoribbons, which should be relevant to the design of graphene-plasmon-based electro-optical modulators and switchers.