Robust Subnanometric Plasmon Ruler by Rescaling of the Nonlocal Optical Response

We present the optical response of two interacting metallic nanowires calculated for separation distances down to angstrom range. State-of-the-art local and nonlocal approaches are compared with full quantum time-dependent density functional theory calculations that give an exact account of nonlocal and tunneling effects. We find that the quantum results are equivalent to those from classical approaches when the nanoparticle separation is defined as the separation between centroids of the screening charges. This establishes a universal plasmon ruler for subnanometric distances. Such a ruler not only impacts the basis of many applications of plasmonics, but also provides a robust rule for subnanometric metrology.

Figure 1

Extinction cross section per length for (a) isolated jellium nanowire of diameter $D=9.8\mathrm{nm}$. Results are shown as a function of the frequency $\omega$ of the incident radiation. The incoming field is an $x$-polarized plane wave as depicted in the inset. The results from TDDFT calculations are compared with those from classical electromagnetic calculations using local (Drude) and nonlocal hydrodynamic response (NLHD). (b) Same as in (a) but for a jellium nanowire dimer with junction size $S=7.4\AA$.

Figure 2

Extinction cross section $\sigma$ per length calculated with different approaches for the $D=9.8\mathrm{nm}$ Na nanowire dimer in vacuum. The incoming plane wave is polarized along the dimer axis $x$. Waterfall plots show the results as a function of the frequency of the incoming radiation for different junction widths $S$ varied in steps of $1a_0=0.53\AA$. Red curves (gray) correspond to $S=-5.3$, $-2.65$, 0, 2.65, 5.3, 7.95, and 10.6 Å. For further details, see the text.

Figure 3

(a) Energy of the dipole plasmon resonance as a function of the junction width $S$ for a $D=9.8\mathrm{nm}$ nanowire dimer. Dots: TDDFT results; dashed and solid black lines: results of classical Drude calculations performed for the junction widths $S$ and $S-2\Delta$, respectively; dashed and dotted blue lines: results of NLHD calculations performed for the junction widths $S$ and $S-2\Delta -2\delta$, respectively. The inset gives schematic representation of the location of plasmon-induced screening charges in various approaches. (b) Energy of the dipole plasmon resonance for $D=6.2\mathrm{nm}$ and $D=9.8\mathrm{nm}$ nanowire dimers as a function of the scaled separation $S/R_{\mathrm{cl}}$. Solid lines: classical Drude calculations performed for the junction width $S-2\Delta$.