Nonlocal and nonlinear plasmonics in atomically thin heterostructures

Abstract

Plasmons in atomically thin materials offer a compelling route to trigger nonlinear light-matter interactions through extreme optical confinement in the two-dimensional (2D) limit. However, optical nonlocality in plasmons is typically associated with losses in the linear response regime. Here, we show that nonlocal effects mediate strong plasmon-assisted optical nonlinearity in electrically reconfigurable 2D heterostructures. Using atomistic simulations that capture quantum finite-size and nonlocal effects in the nonlinear plasmonic response of graphene and phosphorene nanoribbon dimers, we reveal how symmetry and inter-ribbon coupling shape harmonic generation processes in perturbative and high-harmonic regimes. Independent tuning of geometry and carrier density in nanoribbon heterostructures is shown to induce inter-ribbon plasmon hybridization, impacting inversion symmetry governing even-ordered nonlinear processes like second-harmonic generation. These results reveal design principles for active and passive tuning of nonlinear plasmonic effects and enable selective enhancement of specific harmonic processes, establishing 2D heterostructures as a versatile platform for nonlinear nanophotonics.

Figures (4)

• Figure 1: Active tuning of linear and second-order optical processes in mesoscale co-planar graphene nanoribbon dimers. (a) Illustration of second harmonic generation in a co-planar graphene nanoribbon (GNR) dimer comprised of ribbons with unequal widths and independent doping charge carrier concentrations (indicated by the Fermi energies $E_{\rm{F,a}}$ and $E_{\rm{F,b}}$). (b) The extinction cross-section $\sigma^{\rm ext}$, calculated from the linear optical response to normally impinging plane-wave illumination polarized along the ribbon confinement direction, of a dimer formed by co-planar armchair-edge terminated GNRs of width $W_{\rm a}\approx10$ nm and $W_{\rm b}\approx5$ nm with a 2 nm edge-to-edge separation. The first ribbon has a fixed electron doping density $n_{\rm a}=1.2\times10^{14}$ cm$^2$, corresponding to a plasmon resonance energy $\hbar\omega=0.75$ eV for the isolated ribbon (indicated by the horizontal dashed line), while the doping density $n_{\rm b}$ of the second ribbon varies continuously from hole doping ($n_{\rm b}<0$) to electron doping ($n_{\rm b}>0$), with its plasmon resonance (in isolation) indicated by the dashed curves. Panels (c) and (d) show the nonlinear optical susceptibilities $\chi_{2\omega}^{(2)}$ and $\chi_0^{(2)}$ associated with second-harmonic generation and optical rectification, respectively, for the system considered in (a). Panels (e-h) show analogous results as panels (a-d) when the dimer is comprised of identical armchair edge-terminated GNRs of width $W_{\rm a}=W_\mathrm{b}\approx10$ nm and 2 nm edge-to-edge separation. All results are obtained by assuming a phenomenological inelastic scattering rate $\gamma$ corresponding to $\hbar\gamma=50$ meV.
• Figure 2: Linear and nonlinear near-fields produced by hybridized plasmons in graphene nanoribbon dimers. (a) Extinction cross-section of a dimer formed by co-planar armchair edge-terminated graphene nanoribbons (GNRs) with 2 nm end-to-end separation, where one ribbon has width $W_{\rm a}\approx10$ nm and a fixed doping density $n_{\rm a}=1.2\times10^{14}$ cm$^{-2}$, and the other has width $W_{\rm b}\approx5$ nm and tunable doping density $n_{\rm b}$. The horizontal dashed lines indicate the first three plasmon modes of the static GNR and the dashed curve shows the lowest-order dipolar mode in the tunable GNR. (b) The induced near-fields ${\bf E}^{(1)}$ (to linear order) of hybridized plasmon modes, i.e., at the doping densities $n_{\rm b}$ and photon energies $\hbar\omega$ indicated by the numbered points in (a), are calculated from the induced charge density $\rho_{i,l}^{\rm ind}$, the imaginary parts of which are represented by color-coordinated open circles (see first colorbar) at the atomic positions $x_{i,l}$ (marked by filled black circles). Likewise, the imaginary part of the induced field is plotted as quivers superimposing contour plots of the total field magnitude (see second colorbar). (c) The second-harmonic susceptibility for the same GNR configuration in panels (a) and (b), with dashed curves indicating where the generated frequency $2\omega$ coincides with the plasmon modes highlighted in (a) and solid curves indicating the frequency of the incoming radiation $\omega$ matching these modes. (d) The magnitude of the induced near-fields ${\bf E}^{(2)}$ (to second order) at the parameters corresponding to the numbered points in (c) with atomic positions marked by filled circles. Panels (e-h) show analogous results to panels (a-d) for a stacked configuration of the same GNRs with 2 nm vertical separation and one aligned edge.
• Figure 3: Plasmon tunability and hybridization effects in the linear and third order optical response of graphene nanoribbon dimers. (a) Illustration of harmonic generation in co-planar graphene nanoribbon (GNR) dimers comprised of ribbons with equal width but independent doping charge carrier concentrations, indicated by the Fermi energies $E_{\rm{F,a}}$ and $E_{\rm{ F,b}}$. (b) Spectral dependence of the linear optical response for a dimer of $W\approx10$ nm wide GNRs with edge-to-edge separation 2 nm and armchair edge terminations, characterized by the optical extinction cross section $\sigma^{\rm ext}$ spectra per unit of the total graphene area, for a fixed carrier density $n_{\rm a}=1.2\times10^{14}$ cm$^2$ in the first GNR and a variable density $n_{\rm b}$ in the second GNR. The horizontal dashed lines indicate the fixed plasmon resonance energy $\hbar\omega=0.75$ eV of the first GNR in isolation while the dashed curves correspond to that of the second GNR. Panels (c-e) correspondingly show the nonlinear optical response associated with (c) third-harmonic generation (quantified by the nonlinear susceptibility $\chi_{3\omega}^{(3)}$) (d) the nonlinear refractive index $n_2$, and (e) the two-photon absorption coefficient $\beta$. Panels (f-j) show analogous results to panels (a-e) but for stacked GNR dimers separated by 2 nm.
• Figure 4: High-harmonic generation in nanoribbon dimers of graphene and phosphorene. (a) Extinction cross section (top) and harmonic spectra (bottom) for $W_\mathrm{a}=W_\mathrm{b}\approx10$ nm GNRs. Results are shown for a single ribbon and for two co-planar zigzag-terminated ribbons with an edge-separation of 2 nm. The isolated or first ribbon is doped at $n_{\rm a}=1.2\times10^{14}$ cm$^{-2}$, while the second ribbon is doped at $n_{\rm b}=-1.2\times10^{14}$ cm$^{-2}$, breaking inversion symmetry and aligning the plasmonic resonances of the two GNRs. (b) Comparison of optical rectification and the first six harmonic orders, obtained when the driving pulse frequency is resonant with the plasmon (dashed vertical lines in (a)), quantified by the squared modulus of the dipole acceleration per unit area, for a single ribbon, a co-planar dimer, and a stacked dimer with 2 nm vertical spacing (dashed lines). (c-d) Results for $W_\mathrm{a}=W_\mathrm{b}\approx5$ nm phosphorene nanoribbons (PNRs) with zigzag edge terminations. Doping densities are $n_{\rm a}=-1.5\times10^{14}$ cm$^{-2}$ and $n_{\rm b}=2.1\times10^{14}$ cm$^{-2}$, chosen to align the two plasmon resonance frequencies. (e) Results for $W_\mathrm{a}=W_\mathrm{b}\approx5$ nm PNR-GNR hetero-dimers in co-planar (left panel) or stacked (right panel) geometries. The PNR is doped at $n_{\rm a}=-1.5\times10^{14}$ cm$^{-2}$ and the GNR at $n_{\rm b}=-1.0\times10^{14}$ cm$^{-2}$, both corresponding to $\hbar\omega_p=0.78$ eV. (f) Same as (b) and (d), but for the graphene-phosphorene dimers, with comparison to a single zigzag-terminated GNR or PNR (dashed lines).