Influence of the electron spill-out and nonlocality on gap plasmons in the limit of vanishing gaps

We study the effect of electron spill-out and of nonlocality on the propagation of light inside a gap between two semi-infinite metallic regions. We compare the predictions of a local response model taking into account only the spill-out, to the predictions of a quantum hydrodynamic model able to take both phenomena into account. We show that only the latter is able to correctly retrieve the correct limit when the gap closes, while the local model suffers from undesirable features (divergence of the fields, overestimation of the losses). Finally, we show that, to a certain extent, the correct results can be retrieved using a simple local approach without spill-out or a conventional Thomas-Fermi approximation, but considering an effective gap width.

Figure 1

Visual representation of overlapping of the equilibrium charge densities $n_0\left(y\right)$ due to electron spill-out in the dielectric gap $g$ between two semi-infinite bulk metals. Deep inside the bulk metal, $n_0$ is equal to the constant density of the bulk metal $n_b$. The magnetic mode profile $H_x\left(y\right)$ of a $p$-polarized light propagating along the $z$-direction is also shown. The structure is invariant along the $x$- and $z$-directions.

Figure 2

A comparison between the local response approximation (LRA) with and without electron spill-out from the metal surfaces. (a) Equilibrium electron density $n_0$ normalized with the bulk density $n_{\mathrm{b}}$ for $g=1\mathrm{nm}$. The (b) real and (c) imaginary parts of the effective refractive index $n_{\mathrm{eff}}$ plotted at $E=1\mathrm{eV}$. The inset in (b) depicts a zoom-in of the region where the spill-out has a significant impact on the mode index. The horizontal dashed gray line marks the refractive index of the bulk metal, $n_{\mathrm{bulk}}$. The dotted red lines represent the region where it becomes hard to find a proper solution due to numerical artifacts. Parts (d)–(f) present a comparison between the electric and magnetic field components considering $g=1\mathrm{nm}$ and $E=1\mathrm{nm}$. The plots in the right column are normalized with the field value at the center of the gap, except in (e) where the field is zero at the center of the gap and the maximum value of the field is considered for normalization.

Figure 3

Equilibrium charge density $n_0$ calculated self-consistently for an air gap sandwiched between the Na metals. (a) For a different gap size ($g$). The vertical dashed lines represent the metal boundaries for the corresponding gaps. (b) A comparison between the LRA, TFHT, and QHT for the spill-out parameter ${\lambda }_{\mathrm{vW}}=1/9$ and 1, considering $g=1$ nm and (c) the decay of the charge density at ${\lambda }_{\mathrm{vW}}=1/9$ and 1. The results are normalized with the bulk density $n_b$, and the shaded gray area represents the metal region.

Figure 4

Effective refractive index $n_{\mathrm{eff}}$ of Na-air-Na configuration as a function of gap distance $g$. A comparison between the LRA, TFHT, and QHT is given. (a) Real and (b) imaginary parts of $n_{\mathrm{eff}}$, plotted at $E=1$ eV. The horizontal dashed gray lines represent the refractive index of the bulk metal.

Figure 5

Normalized (a) electric $|E_y|$ and (b) magnetic $|H_x|$ field components plotted along the $y$-direction, i.e., in the normal direction to the interfaces, plotted under different approaches. The inset of (b) is a closeup view of the $|H_x|$ at the interfaces. The plots are normalized by the field value at the center of the gap. Parts (c) and (d) show decay of the fields at longer distances inside the metal. In these plots, we have considered $g=1$ nm, $E=1$ eV, and ${\lambda }_{\mathrm{vW}}=1/9$ in the QHT case.

Figure 6

Normalized equilibrium charge density $n_0$ and induced charge density $n_1$ computed using the QHT with ${\lambda }_{\mathrm{vW}}=1/9$. The other parameters are the same as those used in Fig. 5. The shaded gray area represents the metal regions.

Figure 7

(a) Real and (b) imaginary parts of $n_{\mathrm{eff}}$ within the LRA with electron spill-out and full self-consistent QHT for ${\lambda }_{\mathrm{vW}}=1/9$. The inset shows a zoom-in of the effective mode index in the tunneling regime. The dotted red lines represent the gap sizes, where it becomes hard to find a proper solution due to numerical artifacts, as was discussed in the caption of Figs. 2 and 2.

Figure 8

Effective refractive index for Ag-air-Ag configuration as a function of gap distance. A comparison between the LRA, TFHT, and QHT is given. The upper and lower panels show the real and imaginary parts of the effective index, respectively. The results are plotted at $E=1$ eV. The dashed gray line represents the refractive index of the bulk metal.

Figure 9

Effective refractive index as a function of energy (in eV) at $g=1\mathrm{nm}$. A comparison between the LRA and QHT is given. The upper and lower panels shows the real and imaginary parts of the effective index. The results are normalized with the screened plasma frequency, ${\omega }_p^{\prime }={\omega }_p/\sqrt{{\varepsilon }_{\infty }}$.

Figure 10

The effective mode index for Ag-air-Ag waveguide (a) real and (b) imaginary part as a function of gap distance $g$ between the metal edges plotted at $E=1\mathrm{eV}$. The LRA and TFHT curves are shifted by ${\delta }_g$ to cope with the QHT results, which allows us to introduce the concept of an effective gap that can be defined as $g_e=g-{\delta }_g$. The $g_e$ seems to appear smaller than the mechanical gap $g$ by an amount of ${\delta }_g$. In particular, using ${\delta }_g=0.15$ nm in the LRA and ${\delta }_g=0.22$ nm in the TF approximation gives a good fit with QHT. The insets plot $n_{\mathrm{eff}}$ as a function of the actual gap.

Figure 11

Comparison between the finite-element method (FEM) and the finite-difference method (FDM). (a) Real and (b) imaginary parts of the effective refractive index $n_{\mathrm{eff}}$. The horizontal dashed gray line represents the refractive index of the bulk Na.