Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay

We propose a concept that allows for efficient excitation of surface plasmon spolaritons (SPPs) on a thin graphene sheet located on a substrate by an incident electromagnetic field. Elastic vibrations of the sheet, which are generated by a flexural wave, act as a grating that enables the electromagnetic field to couple to propagating graphene SPPs. This scheme permits fast on-off switching of the SPPs and dynamic tuning of their excitation frequency by adjusting the vibration frequency (grating period). Potential applications include single molecule detection and enhanced control of SPP trajectories via surface wave patterning of graphene metasurfaces. Analytical calculations and numerical experiments demonstrate the practical applicability of the proposed concept.

Figure 1

Geometry under investigation: a thin graphene sheet on top of a substrate is simultaneously excited by a biharmonic plane wave with wave number ${\beta }_b$ and an electromagnetic field with wave number ${\beta }_0$. The inset shows the graphene patterned as a sinusoidal grating. The thick red arrow indicates propagating GSPPs with wave number ${\beta }_{\mathrm{SPP}}\gg {\beta }_0$.

Figure 2

(a) Dispersion relation of the flexural mode. The horizontal axis is the projection of the Bloch vector on the boundary of the first irreducible Brillouin zone $\Gamma XM$. The vertical axis is the square of the flexural wave number ${\beta }_b^2={\omega }_b\sqrt{\rho \delta /D}$ with $\delta =0.3\mathrm{nm}$, $\rho =2300\mathrm{kg}/{\mathrm{m}}^3$, and $D=2.34\times {10}^{-18}\mathrm{Pa}{\mathrm{m}}^3$. The inset shows a snapshot of the vertical displacement field $W$ in the unit cell and the arrows represent the energy flow of elastic power. (b) Dispersion relation of the GSPP with $E_F=0.7\mathrm{eV}$ and dispersion relation of an electromagnetic wave in surrounding space. The inset shows the geometry of the thin graphene sheet lying on top of a dielectric substrate (silicon) of permittivity ${\epsilon }_d=2.25$.

Figure 3

(a) Absorption, transmission, and reflection spectra of the graphene sheet with $\delta =0.3\mathrm{nm}$ and $E_F=1\mathrm{eV}$, which is vibrated at flexural frequency ${\omega }_b/(2\pi )=175\mathrm{MHz}$ (${\Lambda }_b=250\mathrm{nm}$). The inset shows the direction of electromagnetic excitation ($\theta =0$) and the resulting reflected waves (thin arrows). Note that only the specular reflection was taken into account since the period of the grating ${\Lambda }_b=250\mathrm{nm}$ is very small compared to wavelength of the incident electromagnetic field ${\lambda }_0=7\mu \mathrm{m}$. (b) Electric and magnetic field norms in the unit cell for the $N=2$ mode, where the induced GSPP’s wave number is $4\pi /{\Lambda }_b$. Different components of (c) the electric field and (d) the magnetic field demonstrate the resonances associated with the $N=1$ mode, where the induced GSPP’s wave number is $2\pi /{\Lambda }_b$. The periodicity in the profile of the electric field norm is plotted along the grating’s upper boundary and it shows the order of each corresponding grating mode.

Figure 4

Absorption spectra of the vibrating graphene sheet when ${\Lambda }_b=250\mathrm{nm}$ and $\theta =0$ for four values of $E_F$ (a) for the $N=1$ mode and (b) for the $N=2$ mode. Absorption amplitude vs $\theta$ when $E_F=1\mathrm{eV}$ and ${\Lambda }_b=250\mathrm{nm}$ (c) at frequency 51 THz for the $N=1$ mode and (d) at frequency 72.1 THz for the $N=2$ mode. GSPP resonance frequency ${\omega }_{\mathrm{res}}/(2\pi )$ vs flexural frequency ${\omega }_b/(2\pi )$ when $E_F=1\mathrm{eV}$ and $\theta =0$ (e) for the $N=1$ mode and (f) for the $N=2$ mode. The continuous curves are obtained by solving Eq. (2) for ${\omega }_{\mathrm{res}}$ with given ${\omega }_b$ or ${\Lambda }_b$. Dots represent frequencies ${\omega }_{\mathrm{res}}$ obtained via electromagnetic simulations. Note the good agreement.