Plasmon wave function of graphene nanoribbons

We find the low-frequency optical response of highly doped individual and arrayed graphene nanoribbons to be accurately described in terms of plasmon wave functions (PWFs). More precisely, we focus on the lowest-order transverse dipolar mode, for which we define the wave function as the induced charge density associated with the plasmon. We show that a single universal wave function is capable of describing the normal-incidence interaction of paired, co-planar, and stacked arrays of ribbons down to small inter-ribbon distances. Our work provides both intuitive insight into graphene plasmon interactions and a practical way of accurately describing complex graphene geometries based on the PWFs of the individual components.

In this work, we introduce a simple analytical procedure to describe plasmon modes in graphene nanoribbons using the concept of the plasmon wave function (PWF), which we identify with the induced charge density associated with these modes. We obtain the PWF from classical electromagnetism for an individual ribbon, focusing for simplicity on the lowest-order transverse dipolar mode excited by a plane wave under normal incidence. The interaction between neighboring ribbons is then simulated with this wave function alone, conveniently normalized to represent a single plasmon. This simple PWF model allows us to accurately study the interaction between closely spaced ribbons forming dimers and periodic arrays in co-planar and stacked graphene configurations. The validity of our approach is tested by comparison with full numerical solutions of Maxwell's equations for dimers. Additionally, the PWF formalism allows us to deal with much shorter separations than a simpler dipole-dipole interaction model [38], while this limit is trivially reached at large distances. Our method is directly applicable to complex arrangements of arbitrary nanoisland morphologies, by simply using the wave functions of the plasmons encountered within the spectral range of interest.

PWF model
The plasmon wavelength in graphene is typically much smaller than the light wavelength [22], so it is safe to describe the response of a given structure in the electrostatic limit. An eigenmode decomposition of this response [41] shows that it can be expressed in terms of the contribution arising from different plasmon resonances j. We consider a spectral region dominated by a single plasmon, corresponding to the lowest-order transverse mode of a ribbon j = 1, excited under normal incidence (see figure 1(a)). Now, we define the charge density 1 ρ associated with this plasmon as the PWF. The results presented below are then based on the use of a single universal function 1 ρ , which we obtain by numerically solving the electromagnetic problem using previously discussed methods [42].
When we consider several ribbons aligned along y (figure 1(a)), the excitation of the plasmon in a given ribbon l produces an induced line dipole p l along x. The self-consistent interaction between the plasmons of different ribbons is then described by where α ω is the single-ribbon polarizability (see appendix B, equation (B.5)) and ll′  describes the coupling between ribbons l and l′ separated by a coordinate vector r ll′ . The latter is essentially given by the Coulomb interaction between their respective PWFs, which is normalized using the dipolar mode strength This constitutes a generalization with respect to a dipole-dipole model based upon the long-distance limit of equation (2) (i.e., large r ll′ compared with the ribbon width), which produces the well-known expression x x r r 3 1 ll l l ll ll and has been discussed elsewhere in the context of graphene [37]. Further details of the PWF formalism are given in the appendix A, where we present a self-consistent derivation for arbitrary arrangements of aligned ribbons. Next, we study ribbon pairs and periodic arrays in different configurations by comparing full electromagnetic solutions [42] with both the PWF and the dipole-dipole models. The graphene response is described in the local-RPA Figure 1. Plasmon wave functions (PWFs) and their interaction in graphene ribbon pairs. (a) We define the PWF as the induced charge density associated with the plasmon, which is illustrated in this sketch for two ribbons separated by a coordinate vector r ll′ and excited by a normal-incidence plane wave of transverse polarization. We concentrate on a spectral region dominated by the lowestorder transverse dipolar plasmon. (b), (c) Extinction cross-section normalized to the total graphene area of dimers formed by selfstanding graphene nanoribbons separated by an edge-to-edge distance d in co-planar (b) and vertically-aligned (c) configurations. The ribbon width, Fermi energy, and damping are W = 6 nm, E 0.4 F = eV, and 0.02 γ = eV, respectively. We compare full numerical solutions of the Maxwell equations (solid curves) with both the PWF (dashed curves) and the analytical dipole-dipole (dotted curves) models. approximation [21] in all cases. For comparison, we also offer calculations similar to those of figures 2 and 3 using the Drude response model in figures B1 and C1 of the appendix.

Ribbon dimers
We start by analyzing the plasmons sustained by two neighboring ribbons of small width W = 6 nm and Fermi energy E 0.4 F = eV, in either co-planar (figure 1(b)) or vertically stacked (figure 1(c)) configurations. We show the extinction cross-section obtained from the optical theorem. These systems display a prominent plasmon that starts at the single-ribbon energy at large separations and undergoes dramatic redshifts (blueshifts) due to interribbon attractive (repulsive) interaction in co-planar (stacked) arrangements, as shown in figure 1(b) (figure 1(c)). The positive (negative) character of this interaction is intuitively suggested from the induced charge distribution shown in figure 1(a). Remarkably, the PWF model (dashed curves) agrees well with the full numerical solution (solid curves) down to very small separations. In contrast, the dipole-dipole model only produces accurate results at large separations, and its failure at small distances is particularly severe for stacked ribbons. This behavior is clearly observed in the distance dependence of the dimer plasmon energy, which is summarized in figure 2, as obtained from the zeros of Re{ } Incidentally, using the Drude model for the conductivity (see figure B1 in the appendix B) the stacked dimer plasmon smoothly converges to roughly the frequency of a single ribbon with double the conductivity within the PWF model, which leads to 2 p p single ω ω ≈ at d = 0; this result, which essentially stems from the fact that the plasmon energy is proportional to E F in the Drude model, is however not attained in the dipole-dipole model.

Periodic ribbon arrays
A similar analysis for periodic arrays in both co-planar (figure 3(a)) and vertically stacked ( figure 3(b)) configurations reveals larger frequency shifts due to inter-ribbon interaction than in dimers. Under normal incidence, the dipoles of different ribbons are identical, so equation (1)  (see figure 3). As expected, the bands are very flat at large separations, where the plasmons are well localized at the ribbons, but they acquire significant dispersion at short distances, indicating the presence of more delocalized plasmons that can propagate along the array.
As a final illustration of the power of the PWF model, we explore in figure 4 the absorbance of two neighboring layers of graphene ribbon arrays with a lateral offset. The spectra are dominated by a single plasmon feature, which redshifts with decreasing interlayer distance due to attractive interaction similar to the dimers, as discussed above. Interestingly, there is a clear dependence of the near-touching behavior on the Fermi energy (see figures 4(b) and (c)), although nonlocal effects can be important in this limit [43], which could be dealt with by reformulating the present classical formalism in terms of an atomistic description of the PWFs.

Outlook and conclusion
We have shown that the concept of PWF (i.e., the induced charge density associated with the plasmon) captures the details of the interaction between plasmonic structures in close proximity. For simplicity, we have limited our analysis to arrangements of graphene ribbons, but this model can be easily adapted to deal with other geometries and incorporate more plasmon modes in each structure. The PWF model constitutes a natural extension of the popular dipole-dipole interaction model, combining its ability to produce analytical expressions with a better degree of accuracy. While it can in principle deal with arbitrary distributions of plasmonic systems, it is particularly suited to understand and design nanostructures based on 2D plasmonic materials such as graphene.

Appendix A. Electrostatic eigenmode expansion and PWF
We study the optical response of graphene ribbons aligned with their long axis along the y direction and oriented perpendicularly with respect to z. The ribbons are illuminated by a normal-incidence monochromatic plane wave of external field component in what follows, keeping in mind that the full time dependence of all magnitudes under consideration is obtained as the real part multiplied by 2. The width of the ribbons is small compared with the light wavelength, and therefore, we analyze their response in the electrostatic limit, inspired in an eigenmode expansion formalism discussed elsewhere [37]. The system possesses translational symmetry, so the y component of the electric field is zero. We can formulate a self-consistent equation for the remaining in-plane component, where the integral represents the induced scalar potential and the expression in the square brackets is the induced density, which is in turn written in terms of the current via the continuity equation, using the twodimensional, local, space-and frequency-dependent graphene conductivity x ( , ) σ ω . We further assume the factorization , where f ( ) θ is 1 on the graphene and 0 elsewhere. Here, we have used a dimensionless coordinate x W θ = , where W is the ribbon width. We now define The linear operator M is clearly real and symmetric, so it admits real eigenvalues 1 j η and eigenvectors j  that satisfy and form a complete and orthonormal basis set This allows us to readily solve equation (A.2) as At this point, it is convenient to introduce the PWF associated with mode j, It should be noted that j ρ is proportional to the induced charge of mode j, whereas j ζ can be regarded as the normalized dipole moment.

Appendix B. Interaction between ribbons
For simplicity, we study the response of identical ribbons within a spectral region dominated by their lowestorder transversal dipole plasmon (j = 1), so we neglect other modes in what follows. We show the density profile of this plasmon in figure A1, as obtained by numerically solving the electromagnetic problem [42]. We can decompose  as a sum of contributions from different ribbons

Appendix C. Periodic arrays
In periodic arrays, p p l ≡ is independent of l, so we can solve equation (B.4) as In particular, for an array of co-planar ribbons, the reflection coefficient derived from this expression reduces to [37] where S a c 2πω = accounts for the coupling to propagating light and a is the lattice period. Incidentally, we have introduced the S i − term in the denominator in order to describe radiative damping in the dipole-dipole limit [37]. Finally, the absorbance of the array reduces to r t 1 2 2 − | | − | | , where t r 1 = + is the transmission coefficient.
The plasmon frequency of ribbon arrays is also represented in figures 2 and B1 as calculated from the condition