Quantum hydrodynamic theory for plasmonics: Impact of the electron density tail

Multiscale plasmonic systems (e.g., extended metallic nanostructures with subnanometer inter-distances) play a key role in the development of next-generation nanophotonic devices. An accurate modeling of the optical interactions in these systems requires an accurate description of both quantum effects and far-field properties. Classical electromagnetism can only describe the latter, while time-dependent density functional theory (TD-DFT) can provide a full first-principles quantum treatment. However, TD-DFT becomes computationally prohibitive for sizes that exceed few nanometers, which are instead very important for most applications. In this article, we introduce a method based on the quantum hydrodynamic theory (QHT) that includes nonlocal contributions of the kinetic energy and the correct asymptotic description of the electron density. We show that our QHT method can predict both plasmon energy and spill-out effects in metal nanoparticles in excellent agreement with TD-DFT predictions, thus allowing reliable and efficient calculations of both quantum and far-field properties in multiscale plasmonic systems.

Figure 1

Jellium nanosphere $\left(r_s=4\right)$ with $N_e=508$ electrons ($R=31.9$ a.u.) as obtained from the TF-HT (a)–(d) and the KS/QHT1 (e)–(h) approaches. Ground-state density $n_0\left(r\right)$ (a) and (e); real and imaginary part of the induced charge density $n_1\left(\mathbf{r}\right)$ at the plasmon resonance (b) and (f); imaginary part of $n_1\left(\mathbf{r}\right)$ at the cross-section plane (c) and (g); norm of the induced electric field at the cross-section plane (d) and (h).

Figure 2

Eigenvalues versus the inverse of the jellium nanosphere $(r_s=4$ a.u.) diameter; chemical potential $\left({\mu }^{\mathrm{OF}9}\right)$ of orbital-free DFT calculations with $\eta =9$ (purple squares); chemical potential $\left({\mu }^{\mathrm{OF}1}\right)$ of orbital-free DFT calculations with $\eta =1$ (orange circles); HOMO eigenvalues $\left({\epsilon }^{\mathrm{HOMO}}\right)$ of KS-DFT calculations (blue diamonds); effective eigenvalue $\left({\mu }^{\mathrm{eff}}\right)$ from the KS-DFT electronic density decay (green triangles); see text for details.

Figure 3

Ground-state electronic density (in a log scale) for a jellium nanosphere $\left(r_s=4\right)$ with 338 electrons ($R$=27.8 bohr) computed by KS-DFT, OF-DFT with ${\eta }_g=1$ (OF1) and with ${\eta }_g=9$ (OF9), and the model density. The inset shows the ground-state electronic density in a linear scale inside the nanosphere.

Figure 4

(a) KS Ground-state electronic density $n_0\left(\mathbf{r}\right)$ for a jellium sphere with $N_e=338$ electrons ($R=27.86$ bohr, indicated by the solid-blue vertical line); (b) plot of $-\left(E_h{a}_0^2\right)\frac{1}{8}{\left(\frac{dln\left(r^2{n}_0\left(r\right)\right)}{dr}\right)}^2$, which represents the “local” ${\mu }^{\text{eff}}$ in the case of real density. Only in the far asymptotic region (e.g., for $r>150$ bohr) ${\mu }^{\text{eff}}$ approaches ${\epsilon }^{\mathrm{HOMO}}$ indicated by a horizontal green dashed line. In the simulation domain, indicated by the vertical red line, ${\epsilon }^{\text{eff}}$ is significantly larger than ${\epsilon }^{\mathrm{HOMO}}$.

Figure 5

Graphical representation of the solutions in Eq. (21); see text for details.

Figure 6

Absorption cross section $\left(\sigma \right)$ normalized to the geometrical area $\left({\sigma }_0\right)$ for a jellium nanosphere $(r_s=$ 4 a.u.) with $N_e=338$ electrons as obtained form TD-DFT, OF9/QHT9, KS/QHT1, and Mod/QHT1. All the spectra have been obtained using an empirical broadening of 0.066 eV.

Figure 7

Induced density (complex modulus of $n_1$) at the plasmon energy for a jellium nanosphere ($r_s=4$ a.u.) with $N_e=338$ electrons ($R=27.86$ bohr), as computed from TD-DFT, OF9/QHT9, KS/QHT1, and Mod/QHT1.

Figure 8

Plasmon resonance for jellium nanospheres ($r_s=4$ a.u.) as a function of the inverse of the sphere diameter, as computed by different approaches. The behavior for large particles is shown in the inset.

Figure 9

Mod/QHT1 spectra for jellium nanospheres ranging from $R=2$ to 25 nm (red curves). The black (blue) curve shows the Mie (TD-DFT) resonance trajectory.

Figure 10

Induced polarization charge density (normalized by $R^3$) at the plasmonic resonance ${\omega }_0$ for different jellium nanospheres.

Figure 11

Dimer of Na spheres constituted by $N_e=398$ electrons each, and separated by a distance $g=1$ nm. The dimer is excited by a plane wave oscillating with energy $\hslash \omega =-2.8$ eV; $\gamma ={\gamma }_0+v_F/R$ has been used with $\hslash {\gamma }_0=0.066$ eV. (a) Geometry and incident field. (b) Real part of the induced charge density normalized with respect to the bulk positive charge density. (c) Electric field norm distribution.