Modeling the excitation of graphene plasmons in periodic grids of graphene ribbons: An analytical approach

We study electromagnetic scattering and subsequent plasmonic excitations in periodic grids of graphene ribbons. To address this problem, we develop an analytical method to describe the plasmon-assisted absorption of electromagnetic radiation by a periodic structure of graphene ribbons forming a diffraction grating for THz and mid-IR light. The major advantage of this method lies in its ability to accurately describe the excitation of graphene surface plasmons (GSPs) in one-dimensional (1D) graphene gratings without the use of both time-consuming, and computationally demanding full-wave numerical simulations. We thus provide analytical expressions for the reflectance, transmittance, and plasmon-enhanced absorbance spectra, which can be readily evaluated using any personal computer with little to no programming. We also introduce a semianalytical method to benchmark our previous results and further compare the theoretical data with spectra taken from experiments, for which we observe a very good agreement. These theoretical tools may therefore be applied to design new experiments and cutting-edge nanophotonic devices based on graphene plasmonics.

Figure 1

Monochromatic $p$-polarized plane-wave impinging on a grid of graphene ribbons (not to scale) arranged in a grating-like configuration. The ribbons are sitting in the plane defined by $z=0$. The structure is periodic, with period $L$, and the width of the graphene ribbons is defined by $w$. The system is encapsulated between a top insulator with relative permittivity ${\epsilon }_1$ (for $z<0$) and a dielectric substrate with relative permittivity ${\epsilon }_2$ (for $z>0$).

Figure 2

Absorbance spectra (left panel) as well as transmittance and reflectance spectra (dashed and solid lines in right panel, respectively) of a $p$-polarized plane wave impinging on a periodic grid of graphene ribbons for varying values of $\mathrm{\Gamma }=\hslash \gamma$. The remaining parameters are $E_F=0.45$ eV, $w=2\mu \mathrm{m}, L=4\mu \mathrm{m}, {\epsilon }_1=3, {\epsilon }_2=4$, and $\theta =0$ (normal incidence).

Figure 3

Dependence of the GSP frequency on the electronic density of the graphene ribbons. Left panel: absorbance spectra for different selected values of $n_e$; the legend gives $n_e$ in units of ${10}^{13}{\mathrm{cm}}^{-2}$. Right panel: resonant frequencies—retrieved from our analytic theory (points)—for several values of $n_e$, and corresponding fitting function to those points; that is, $f_{\mathrm{GSP}}\left(n_e\right)\propto n_e^b$ with $b=0.249\simeq 1/4$, in accordance with what is expected for graphene plasmons (see main text). Parameters: $\mathrm{\Gamma }=3.7$ meV, $w=2\mu \mathrm{m}, L=4\mu \mathrm{m}, {\epsilon }_1=3, {\epsilon }_2=4$, and $\theta =0$ (normal incidence). We have used $E_F=\hslash v_F\sqrt{\pi n_e}$, with $v_F\approx 1.1\times {10}^6$ m/s.

Figure 4

Dependence of the GSP frequency on the period of the graphene grating, $L$. We kept the ratio $w=L/2$ fixed, and therefore the fundamental GSP mode carries a wave vector of about $q\sim \pi /w$. In the right panel we show the position of the GSP resonances rendered by the analytical method incorporating the edge condition as a function of the GSP wave vector (square data points). To these data we have fitted a curve $f_{\mathrm{GSP}}\left(q\right)\propto q^b$, having obtained a exponent $b=0.502\sim 1/2$. Parameters: $E_F=0.4$ eV, $\mathrm{\Gamma }=3.7$ meV, ${\epsilon }_1=3, {\epsilon }_2=4$, and $\theta =0$.

Figure 5

Electric field representing graphene plasmons excited in the graphene ribbons which compose the grid (with dimensions $w=2\mu \mathrm{m}$ and $L=4\mu \mathrm{m}$). The figures show the plasmonic fields in the system's unit cell, and the graphene ribbon is indicated by the horizontal black line. Left panel: vectorial representation of the electric field (in the $y=0$ plane) due to GSPs, ${\mathbf{E}}^{\mathrm{GSPs}}(x,z)=E_x^{\mathrm{GSPs}}(x,z)\widehat{\mathbf{x}}+E_z^{\mathrm{GSPs}}(x,z)\widehat{\mathbf{z}}$. The intensity plot refers to the quantity $sgn\left(z\right)E_z^{\mathrm{GSPs}}$ which roughly highlights the charge density within each graphene ribbon. Righ panel: normalized spatial distributions of the electric field components $E_z^{\mathrm{GSPs}}$ (top) and $E_x^{\mathrm{GSPs}}$ (bottom). The spatial range covered in these subpanels is the same as in the main panel. We note that only modes corresponding to plasmonic (evanescent) modes were included in the sums figuring in the Bloch expansions, which for the parameters used here encompass all modes with the exception of the specular one (i.e., with $n=0$). Parameters: $E_F=0.4$ eV, $\mathrm{\Gamma }=3.7$ meV, and ${\epsilon }_1={\epsilon }_2=4$.

Figure 6

Normalized plasmon-induced change in transmittance relative to the CNP, $-\mathrm{\Delta }T=T_{\mathrm{CNP}}-T$, in periodic grids of graphene ribbons with different dimensions: $w=4\mu \mathrm{m}$ (green), $w=2\mu \mathrm{m}$ (red), $w=1\mu \mathrm{m}$ (blue), and $w/L=1/2$ throughout. The dashed lines correspond to experimentally measured spectra [48], while the dotted lines and solid lines correspond to the spectra obtained using our full-analytical model and the semianalytical technique (see Appendix pp3), respectively. We have used the following parameters, in accordance with Ref. [48]: $E_F=0.497$ eV, $\mathrm{\Gamma }=16.5$ meV, and ${\epsilon }_1={\epsilon }_2=5$.

Figure 7

Hybrid mid-IR graphene plasmon-phonon polaritons excited in period gratings of graphene nanoribbons sitting on a polar (${\mathrm{SiO}}_2$) substrate. Experimental data (light-brown points) and corresponding extinction spectra calculated using (a) the analytical model and (b) the semi-analytical model (solid lines). The computations were carried out in accordance with the experimental parameters [52] $E_F=0.37$ eV, $\theta =0$ (normal incidence), and ${\epsilon }_1=1$; the dielectric function of ${\mathrm{SiO}}_2, {\epsilon }_2\equiv {\epsilon }_{{\mathrm{SiO}}_2}\left(\omega \right)$, was taken from the literature [90]. We have used electronic scattering rates corresponding to about 25–30 meV depending on the sample. (c) Loss function (via $\mathrm{Im}{r}_{\mathrm{TM}}$) for extended graphene deposited on ${\mathrm{SiO}}_2$, with the uncoupled GSP spectrum superimposed (white dashed line); we note that this serves only as an eye guide to the interpretation of the data, since we expect the polaritonic spectrum akin to the periodic grid of nanoribbons to be slightly different from that of unpatterned, extended graphene.

Figure 8

(a) Reflectance spectra using different values of $N$, which truncates the sum entering in the function $\mathrm{\Lambda }\left(\omega \right)$. Notice the difference between even and odd $N$ values, in the convergence of results. (b) Alternating series figuring in $\mathrm{\Lambda }\left(\omega \right)$ [cf. Eq. (A1)] as a function of $n$.

Figure 9

Absorbance and relative modulus-squared Bloch amplitudes $|E_{x,n}^{\left(2\right)}{|}^2$ corresponding to several Bragg modes, for two different configurations: one with a filling ratio of $w/L=1/2$ (top) and another with $w/L=1/4$ (bottom).

Figure 10

GSP resonances in a periodic grid of graphene ribbons ($w=4\mu \mathrm{m}$ and $L=2w$) at different carrier concentrations. The solid lines correspond to the results obtained using the semianalytical method whereas the dotted lines correspond to the outcome of the full-analytical approach.

Figure 11

Current, $J_x\left(x\right)$, within the graphene stripes obtained using the semianalytical method (Fourier expansion) and corresponding fitting function of the type $c{t}^e\times \sqrt{w^2/4-x^2}$ to illustrate the approximation which is made when employing the edge condition [ansatz for the current; see Eq. (11)].