Ab initio approach for gap plasmonics

Gap plasmonics deals with the properties of surface plasmons in the narrow region between two metallic nanoparticles forming the gap. For subnanometer gap distances, electrons can tunnel between the nanoparticles, leading to the emergence of novel charge-transfer plasmons. These are conveniently described within the quantum corrected model by introducing an artificial material with a tunnel conductivity inside the gap region. Here we develop a methodology for computing such tunnel conductivities within the first-principles framework of density functional theory and apply our approach to a jellium model representative for sodium. We show that the frequency dependence of the tunnel conductivity at infrared and optical frequencies can be significantly more complicated than previously thought.

Figure 1

Gap plasmonics for (a) two coupled metallic nanoparticles is modeled by approximating the gap region by (b) the interspace between two metallic slabs and computing the tunnel conductivity ${\sigma }_t(\omega ,\ell )$ within the framework of density functional theory (DFT) for various gap distances $\ell$. In a consecutive step, ${\sigma }_t(\omega ,\ell )$ is submitted to the quantum corrected model [3, 4, 19]. Panel (b) reports results for a jellium model representative for sodium. For a homogeneous external excitation one can use periodic boundary conditions (PBCs), as discussed in the text.

Figure 2

(a) Density $n\left(x\right)$ (green shaded area), sum of exchange-correlation potential, and external potential for the corrected jellium model [37] as computed within the local (LDA, dashed line) and weighted (WDA, solid line) density approximations, and image charge potential (circles, image plane position of 0.5 atomic units [34]) for two jellium slabs representative for sodium separated by 1 nm. $E_F$ denotes the Fermi energy position. (b) Energy splitting of WDA Kohn-Sham energies for selected gap distances and for slab widths of 20 nm (filled circles) and 10 nm (open circles).

Figure 3

(a) Tunnel conductivity ${\sigma }_t$ as computed from Eq. (6) within the LDA (circles) and WDA (square) approximations, and TDDFT and QCM results of Esteban et al. [3, 4] (to facilitate the comparison we show ${\sigma }_t$ in atomic units). For the QCM expression we use ${\omega }_p=5.16$ eV, ${\gamma }_p=0.218$ eV, and ${\ell }_c=0.075$ nm. (b) ${\sigma }_t(\omega ,\ell )$ as a function of photon energy $\hslash \omega$ and for selected gap separations $\ell$.

Figure 4

Time dependence of (a) paramagnetic and (b) diamagnetic current for a photon energy of 1.34 eV [see arrow in Fig. 3] and for an external excitation that starts at time zero. The currents in the gap regions are magnified for clarity by a factor of 20. (c) Amplitude (square modulus of ${\sigma }_t$) and (d) phase of paramagnetic (circles) and diamagnetic (solid lines) current as a function of gap distance.

Figure 5

Optical extinction spectra for two coupled nanospheres (50 nm diameter) using the dielectric function for a jellium model representative for sodium, and for electrodynamic simulations (a) without and (b) with consideration of quantum tunneling. Spectra for different gap separations given on the right-hand side of each panel are offset for better visibility. The tunnel results are obtained within the boundary QCM approach [19] using the tunnel conductivities of Esteban et al. [3, 4] (red lines) and of Eq. (6). Circles indicate the position of the dipole peak, for sodium the charge-transfer peak is at photon energies below 0.5 eV.