Renormalization of nanoparticle polarizability in the vicinity of a graphene-covered interface

We study the electromagnetic properties of a metamaterial consisting of polarizable (nano)particles and a single graphene sheet placed at the interface between two dielectrics. We show that the particle's polarizability is renormalized because of the electromagnetic coupling to surface plasmons supported by graphene, which results in a dispersive behavior, different for the polarizability components corresponding to the induced dipole moment, parallel and perpendicular to the graphene sheet. In particular, this effect is predicted to take place for a metallic particle whose bare polarizability in the terahertz (THz) region is practically equal to the cube of its radius (times $4\pi {\varepsilon }_0$). This opens the possibility to excite surface plasmons in graphene and enhance its absorption in the THz range by simply using a monolayer of metallic particles randomly deposited on top of it, as we show by explicit calculations.

Figure 1

Schematics of the system consisting of a dipole (polarized NP) located at a point $(0,0,h)$ in the vicinity of a graphene-covered interface between two dielectrics. An image dipole located at $(0,0,-h)$ is also shown.

Figure 2

Frequency dependence of the real part of the renormalized polarizability components (divided by the bare one, ${\alpha }_0$) for a spherical Au particle of radius $R=10\mu \mathrm{m}$ located at a distance $h=10\mu \mathrm{m}$ from a suspended graphene sheet (${\varepsilon }_2={\varepsilon }_1=1$), for two values of the Fermi level as indicated. The inset shows the peak frequency dependence on the inverse of the particle's distance from the graphene sheet. Gold parameters are taken from Ref. [8].

Figure 3

Dependence of the real and imaginary parts of the renormalized polarizability components on the frequency (a) and the Fermi level (b), calculated for a spherical CdSe particle of radius $R=1\mu \mathrm{m}$ located at a distance $h=1.1\mu \mathrm{m}$ from a suspended graphene sheet (${\varepsilon }_2={\varepsilon }_1=1$). The graphene Fermi level is $E_F=0.4$ eV for (a) and the field frequency is $\hslash {\omega }_0=25.15$ meV for (b). Panel (a) also shows the frequency dependence of the bare polarizability, ${\alpha }_0$. Thin horizontal lines in panel (b) indicate the values of the real and imaginary parts of ${\alpha }_0$ for $\hslash {\omega }_0=25.15$ meV. Phonon parameters of CdSe were taken from Ref. [47].

Figure 4

Dependence of the characteristic functions $g_0$ (left column) and $g_1$ (right column) that determine the polarization charge density [Eq. (29)] upon height above the interface and the distance from the dipole projection in the graphene plane. Color code plots show the real [(a) and (d)] and imaginary [(b) and (e)] parts of $g_0$ and $g_1$ as functions of $h$ and $\rho$. Plots (c) and (f) are for $h=10\mu \mathrm{m}$. The parameters are ${\varepsilon }_2={\varepsilon }_1=1$, $R=5\mu \mathrm{m}$, $E_F=0.5\mathrm{eV}$, ${\tau }_F={10}^{-13}\mathrm{s}$, $\hslash \omega =10$meV.

Figure 5

Reflectance, transmittance (a), and absorbance (b) spectra of a monolayer of Au NPs randomly deposited on top of a free-standing graphene sheet. The corresponding spectra of graphene itself are also shown (dashed lines) for comparison. The inset shows the schematics of the considered system. The parameters are the following: $R=5\mu$m, $h=5\mu$m, $E_F=0.5$eV, $\nu =8\times {10}^{-4}\mu {\mathrm{m}}^{-2}$.