The main effect of nonlocality in graphene is to oppose the formation of a singularity by increasing the conductivity probed by large-momentum Fourier components. In Figure 3, we plot the transmission spectra under plane wave illumination at normal incidence (*k*=0) for different modulation strengths Δ=−log_{10}(1−*ζ*_{1}), corresponding to the number of orders of magnitude by which the conductivity is suppressed at a singular point. We assume an average Fermi level *E*_{F}=0.4 eV, a conductivity grating period *L*=5 μm, and a mobility μ_{m}=10^{4} cm^{2}/(V s) resulting in an electron scattering time $\tau ={\mu}_{\text{m}}{E}_{F}/\mathrm{(}{v}_{F}^{2}e\mathrm{)}\approx 0.44\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{ps},$ where the Fermi velocity *v*_{F}=9.5×10^{5} m/s [41] is assumed. Our results are obtained via the nonlocal mode-matching method outlined above; these were benchmarked, in the local-response limit, against finite-element method (FEM) numerical calculations using a commercially available package (COMSOL Multiphysics). For weak conductivity modulation, i.e. far from the singular limit (Figure 3A), the local and nonlocal spectra are effectively equivalent. In this limit, only momentum states well below the Landau damping regime *k*≈*ω*/*v*_{F} are populated, so that the metasurface can be accurately described via a local Drude-type conductivity model ${\sigma}_{D}\mathrm{(}\omega \mathrm{)}=\frac{{e}^{2}}{\pi {\hslash}^{2}}\frac{{E}_{F}}{\mathrm{(}\gamma -i\omega \mathrm{)}},$ where *γ*=*τ*^{−1}. As we increase the modulation strength to 99.9% of the average value (Figure 3B, Δ=2), the local and nonlocal spectra start deviating, the latter exhibiting a clear blueshift, which is a consequence of nonlocality (see, e.g. Ref. [20]), as plasmon resonance frequencies *ω*∝*σ* [see dispersion relation, Eq. (5)], and nonlocal effects lead to an increase in conductivity. Finally, for Δ=3 (Figure 3), nonlocality becomes a dominant effect, which effectively saturates the plasmonic spectrum, opposing any further merging of the plasmon resonances.

Figure 3: Nonlocality leads to saturation of the density of states in a singular graphene metasurface.

Local (red) and nonlocal (blue) transmittance spectra for plane wave illumination through the graphene metasurface at normal incidence, obtained with the mode-matching (continuous lines) and finite-element method (dots) for three increasingly singular metasurfaces corresponding to Δ=1 (A), Δ=2 (B), and Δ=3 (C), respectively. The nonlocal contribution, which is negligible away from the singular regime, becomes dominant as the singular limit is approached, opposing the merging of surface plasmon modes.

For completeness we show in Figure 4 the plasmonic band structure of our metasurfaces over the rest of the Brillouin zone, where a few additional effects are present. In order to visualize the bands, we plot in log-scale the absolute value of the reflection coefficient, which was color saturated in order to allow both propagating and evanescent modes to be identifiable. In the non-singular regime (Figure 4A), plasmonic band gaps resulting from the periodic modulation are clearly visible at *k*=*π*/*L*, whereas the bands are degenerate at *k*=0 due to the inversion symmetry of the modulation. By contrast, as the modulation strength is increased (Figure 4B), the bands flatten as a result of the stronger Bragg scattering, so that in the singular limit, they become effectively indistinguishable. In this regime, plasmons are dramatically slowed down. However as in the previous case, the introduction of nonlocality saturates the merging of the plasmon spectrum, opposing the flattening of the bands. However, in this case, we clearly see how nonlocality has the additional effect of broadening the reflection spectra dramatically, as a result of the losses introduced by Landau damping.

Figure 4: (Left-to-right) Plasmonic band structures for non-singular, singular (local) and singular (nonlocal) graphene metasurfaces.

Plasmonic band structure of the non-singular (Δ=1) (A) and singular (Δ=3) (B, C) metasurfaces, visualized by plotting the logarithm of the absolute value of the reflection coefficient. Note that to the right of the light line, the incoming waves are evanescent. Local (A, B) and nonlocal (C) spectra differ significantly for the singular case Δ=3 only. (A) In the non-singular regime, band gaps are clearly visible at the edge of the Brillouin zone, with the respective upper bands showing significantly larger broadening compared to the lower ones. (B) In the singular limit, local calculations predict that the band above of each band gap becomes indistinguishable from the lower band of the gap above, effectively realizing a series of flat bands, with extremely low group velocities. However, the onset of Landau damping in the nonlocal case (C) greatly broadens these bands, in addition to saturating their compression as in Figure 3. In addition, note that the bands are effectively degenerate at *k*=0, due to the inversion symmetry of the modulation.

The account of nonlocality can be somewhat demanding in the modeling of more complex experimental setups. Consequently, local-analogue models, which are able to incorporate the effects of nonlocality in a local simulation are valuable tools for the theoretical modeling of plasmonic systems. Here, we propose a simple local-analogue model, which can accurately reproduce the results of the fully nonlocal calculation carried out above. Local-analogue models were originally proposed for metallic plasmonic systems [32] in order to capture nonlocal effects under the framework of the hydrodynamic model of the free-electron gas at the interface between nearly touching metallic structures. In that context, the effect of nonlocality is the inward shift of the induced charges, i.e. away from the metallic surface and into the bulk, thereby effectively widening the gap between the components of the dimer (e.g. metallic cylinders or spheres). Consequently, the substitution of a thin metallic layer by *an effective* dielectric one was able to accurately reproduce the optical response of such nearly touching metallic structures.

Conversely, the type of singular structure described in this work entails the inverse effect: since the conductivity is strongly enhanced as *k*→*ω*/*v*_{F}, the effect of nonlocality is to smear out the singularity by effectively saturating the local conductivity to a minimum level *σ*_{s} dictated, qualitatively, by the condition *k*(*σ*_{s})≈*ω*/*v*_{F}, i.e. when the plasmon wavelength *λ*_{p}→*λ*_{F}, and Landau damping opposes any further confinement of the plasmonic field. The quasi-static dispersion relation of graphene plasmons is read as [20]:

$${\epsilon}_{1}+{\epsilon}_{2}+i\frac{\sigma}{{\epsilon}_{0}\omega}k=\mathrm{0,}$$(5)

where *σ*≡*σ*(*k*, *ω*) and *σ*≡*σ*(*ω*)=*σ*(*k*→0, *ω*) in the nonlocal and local cases, respectively. Herein, we set *ε*_{1,2}=1 (for simplicity alone). Moreover, we can then substitute the wavevector *k*=*βω*/*v*_{F}, where *β* is a phenomenological factor of order ~1, which quantifies the fraction of electron momentum to which the plasmon can couple before saturating (which is exactly one if momentum saturation occurs exactly at the electron momentum). In this fashion, we thus obtain the saturation value for the conductivity, *σ*_{s}=2*iε*_{0}*v*_{F}/*β*. In Figure 5, we add a positive surface conductivity offset

Figure 5: Local (red), nonlocal (blue line), and local-analogue (green triangles for *β*=1.29 and gray dashed line for *β*=1) transmittance spectra of the singular (Δ=3) graphene conductivity grating.

The inset shows how a local-analogue metasurface can be obtained by saturating the conductivity of graphene near the value, which causes the local plasmon dispersion to cross the electron dispersion *ω*=*v*_{F}k, a regime dominated by Landau damping.

$${\Delta}_{\sigma}\mathrm{(}\omega \mathrm{)}=i\Im \mathrm{[}{\sigma}_{s}-\mathrm{(}1-{\zeta}_{1}\mathrm{)}{\sigma}_{D}\mathrm{(}\omega \mathrm{)}][1-i/\mathrm{(}\omega \tau \mathrm{)}]$$(6)

in a local FEM calculation, where the factor in the first square bracket is responsible for the smearing of the imaginary part of the surface conductivity, whereas the second ensures that the loss tangent $\Re \mathrm{[}\sigma ]/\Im \mathrm{[}\sigma \mathrm{]}$ is preserved upon the conductivity offset. It is worth remarking that, as our model hinges on the relation between the plasmonic and electronic momentum, the linear dependence of the latter on frequency implies that the conductivity offset is frequency dependent.

For *β*=1, the agreement between the previous nonlocal result (Figure 3C) and the spectrum obtained using the local-analogue model is only qualitative. However, as the figure plainly shows, by choosing *β*≃1.29, this simple model is able to reproduce the entire transmission spectrum with remarkable accuracy, hereby validating the physical assumptions behind our local-analogue model. While insightful, our semiclassical theory does not provide a quantitative evaluation of the saturation parameter *β*. However, it can shine further insight into our nonlocal description: in fact, given that *β*>1, meaning that the saturation momentum surpasses the electron momentum, this result is worthy of a closer inspection. In the most singular regime, the assumption leading to the first-order Fourier expansion in Eq. (2) may lose accuracy for larger wavevectors, resulting in an underestimation of the extent of nonlocal effects in this extreme regime. In this sense, comparisons with future, fully quantum-mechanical investigations would prove extremely useful in providing an exact evaluation of the saturation momentum. Nevertheless, our local-analogue model offers a useful and intuitive method for the incorporation of nonlocal effects in the future modeling of complex metasurfaces based on 2D materials.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.