Plasmonic modes of polygonal rods calculated using a quantum hydrodynamics method

Plasmonic resonances of nanoparticles have drawn lots of attention due to their interesting and useful properties such as strong field enhancements. The self-consistent hydrodynamics model has the advantage that it can incorporate the quantum effect of the electron gas into classical electrodynamics in a consistent way. We use the method to study the plasmonic response of polygonal rods under the influence of an external electromagnetic wave, and we pay particular attention to the size and shape of the particle and the effect of charging. We find that the particles support edge modes, face modes, and hybrid modes. The charges induced by the external field in the edge (face) modes mainly localize at the edges (faces), while the induced charges in the hybrid modes are distributed nearly evenly in both the edges and faces. The edge modes are less sensitive to particle size than the face modes but are sensitive to the corner angles of the edges. When the number of sides of regular polygons increases, the edge and face modes gradually change into the classical dipole plasmonic mode of a cylinder. The hybrid modes are found to be the precursor of the Bennett mode, which cannot be found in classical electrodynamics.

Figure 1

(a) Schematic picture of the plasmonic system under consideration, in which yellow region stands for the jellium background with circumradius $r_a,{\mathrm{\Omega }}_G$ is the ground state calculation domain with radius $r_g$, and ${\mathrm{\Omega }}_D$ is the excited state calculation domain with radius $r_d$. The right inset shows different regular polygons with the same circumradius $r_a$. The green (red) arrow denotes the edge (face) direction of the regular polygons. Calculated electron distribution $\left(n_0-n_j\right)/n_{\mathrm{ion}}$ and effective potential $V_{\mathrm{eff}}$ of the ground state for a sodium triangle with radius $r_a=2\mathrm{nm}$ is shown in (b) and (c), respectively. The white dashed line in (b) highlights the jellium boundaries. The VW parameter ${\lambda }_{\omega }$ (see text) is chosen as 0.12, the radius of excited state calculation domain is $r_d=2.7\mathrm{nm}$, and that for ground state calculation domain is $r_g=3.5\mathrm{nm}$.

Figure 2

(a) Modulations of local work function as a function of polar angle $\theta /\pi$ for a triangle, square, pentagon, hexagon, 20-sided polygon, and cylinder are plotted by gray, orange, red, blue, green, and magenta lines, respectively. (b) Local work functions in the E (edge) direction and F (face) direction as a function of interior angles $\alpha$ are plotted by open squares and open triangles, respectively. The circumradii of all the regular polygons are 2.0 nm.

Figure 3

Absorption spectra of (a) a triangle and (b) a square with circumradius $r_a=2\mathrm{nm}$ under plane wave incidence with incident angle ${\theta }_{\mathrm{i}}=0$ are plotted by blue and red solid lines, respectively. The loss parameter $\gamma$ is 0.17 eV throughout the domain ${\mathrm{\Omega }}_D$. The absorption spectrum of triangle for plane wave with incident angle ${\theta }_{\mathrm{i}}=\pi /2$ is plotted in (a) by gray dots, and that of square for plane wave with incident angle ${\theta }_{\mathrm{i}}=\pi /4$ is plotted in (b) by green dots. The gray lines show the absorption spectra calculated by the classical model with the rounded corners (see text). The resonance modes are labeled according to their induced charge profiles (see text). The superscript denotes the shapes (T stands for triangle) and the subscript denotes the mode number.

Figure 4

Plot of the induced electron density ${\rho }_1/E_0$ (color/intensity) and the current density ${\mathbf{J}}_1$ (vector field) at some particular time for the ${\mathrm{Ed}}_1^{\mathrm{T}}$ mode shown in Fig. 3 for a plane wave incident at an angle (a) 0 and (b) $\pi /2$, respectively. The induced electron density and current density for the ${\mathrm{Ed}}_1^{\mathrm{S}}$ mode are shown in (c) and (d) for plane wave incident angle 0 and $\pi /4$, respectively.

Figure 5

Plot of the induced electron density ${\rho }_1/E_0$ (color/intensity) and the current density ${\mathbf{J}}_1$ (vector field) at some particular time for the ${\mathrm{Fa}}_1^{\mathrm{T}}$ mode claimed in Fig. 3 are shown in (a) and (b) for plane wave incident at angles of 0 and $\pi /2$, respectively. The induced electron density and current density for the ${\mathrm{Fa}}_1^{\mathrm{S}}$ mode are shown in (c) and (d) for plane wave incident at angles 0 and $\pi /4$, respectively.

Figure 6

(a) Absorption spectra of regular polygons with increasing side number $N$ are plotted by the solid lines. All regular polygons have the same circumradius $r_a=2.0\mathrm{nm}$, and loss parameter $\gamma$ is 0.17 eV throughout the domain ${\mathrm{\Omega }}_D$. (b) Absorption spectra of polygons with increasing side number $N$ for fixed total number of electrons. (c) Induced electron density ${\rho }_1/E_0$ (color/intensity) and current density ${\mathbf{J}}_1$ (vector field) at some particular time for the Bennett mode and its precursors [the M and Hb modes shown in (a)].

Figure 7

(a) The resonance frequencies of the edge modes as a function of the circumradius is plotted by red circles $({\mathrm{Ed}}_1^{\mathrm{T}}$ mode) and blue circles $({\mathrm{Ed}}_1^{\mathrm{S}}$ mode), respectively. The open circles are obtained by SC-HDM, and the circles with a cross inside are obtained using classical model (LRA). (b) The resonance frequencies of the face modes as a function of the radius are plotted by red stars $({\mathrm{Fa}}_1^{\mathrm{T}}$ mode) and blue stars $({\mathrm{Fa}}_1^{\mathrm{S}}$ mode), respectively. The resonance frequencies of the dipole mode (D mode) for cylindrical particles are also plotted by dark green triangles. (c) The resonance frequencies of the Bennett mode (M mode) for cylindrical particles are plotted by dark green squares. (d) The calculated resonance frequencies of the $\mathrm{E}{\mathrm{d}}_1$ mode for SC-HDM as a function of interior angles $\alpha$ are plotted by solid stars. The solid line shows the analytical dispersions of edge modes for a single corner obtained under quasistatic and small wave number limitations. The blue circles show the numerical results obtained by BEM with quantum corrections under quasistatic limits. (e) The difference between $d_{\perp }$ and $d_{\parallel }$ of a single interface for the charge neutral case, $p$ charging $\left(d_{\parallel }=-0.001\mathrm{nm}\right)$, and $n$ charging $\left(d_{\parallel }=0.001\mathrm{nm}\right)$ are plotted by red, blue, and green lines, respectively.

Figure 8

(a) Absorption spectra of a charged cylinder are shown by solid red lines for different additional amounts of charges. The open cyan triangle dots and open green square dots show the resonance frequencies as a function of $\mathrm{\Delta }S$ for dipole mode (D) and Bennett mode (M), respectively. The vertical axis is the absorption cross section ${\sigma }_{\mathrm{abs}}$ divided by geometric cross section ${\sigma }_{\mathrm{geo}}=2r_a$. (b) Absorption spectra of a charged triangle are shown by solid red lines for different additional amounts of charges. The dependence of resonance frequencies on $\mathrm{\Delta }S$ are shown by open blue circles, open orange square dots, and open blue star dots for the ${\mathrm{Ed}}_1^{\mathrm{T}}$ mode, the ${\mathrm{Ed}}_2^{\mathrm{T}}$ mode, and the ${\mathrm{Fa}}_1^{\mathrm{T}}$ mode, respectively. (c) Absorption spectra of the charged square are shown by solid red lines for different additional amounts of charges. The dependence of resonance frequencies on the amount of $\mathrm{\Delta }S$ are shown by open blue circles and open blue star dots for the ${\mathrm{Ed}}_1^{\mathrm{S}}$ mode and the ${\mathrm{Fa}}_1^{\mathrm{S}}$ mode, respectively. All the polygons have the same circumradius $r_a=2.0\mathrm{nm}$, and loss parameter $\gamma$ is 0.17 eV throughout the domain ${\mathrm{\Omega }}_D$.

Figure 9

Electric field enhancement factors of polygonal rods under plane wave illumination $\left({\theta }_{\mathrm{i}}=0\right)$ as a function of interior angles $\alpha$ for edge modes (blue circles), face modes (red triangles), and Bennett mode (green squares) calculated using SC-HDM. These plasmonic modes correspond to those shown in Fig. 6.

Figure 10

Contour plot of the energy dissipation $\langle {\mathbf{J}}_1·{\mathbf{E}}_1\rangle /I_0$ in units of $\mathrm{n}{\mathrm{m}}^{-1}$ within the domain ${\mathrm{\Omega }}_D$ for (a) dipole mode (D), (b) Bennett mode (M), (c) $\mathrm{Ed}{}_1^{\mathrm{T}}$ mode, (d) $\mathrm{Fa}{}_1^{\mathrm{T}}$ mode, (e) $\mathrm{Ed}{}_1^{\mathrm{S}}$ mode, and (f) $\mathrm{Fa}{}_1^{\mathrm{S}}$ mode. These plasmonic modes correspond to those in Fig. 6 with $r_a=2.0\mathrm{nm}$.