Plasmon polaritons in nanoparticle supercrystals: Microscopic quantum theory beyond the dipole approximation

Crystals of plasmonic metal nanoparticles have intriguing optical properties. They reach the regimes of ultrastrong and deep strong light-matter coupling, where the photonic states need to be included in the simulation of material properties. We propose a quantum description of the plasmon polaritons in supercrystals that starts from the dipole and quadrupole excitations of the nanoparticle building blocks and their coupling to photons. Our model excellently reproduces the results of finite difference time domain simulations. It provides detailed insight into the emergence of the polariton states. Using the example of a face centered cubic crystal we show that the dipole and quadrupole states mix in many high-symmetry directions of the Brillouin zone. A proper description of the plasmon and plasmon-polariton band structure is only possible when including the quadrupole-derived states. Our model leads to an expression of the reduced coupling strength in nanoparticle supercrystals that we show to enter the deep strong-coupling regime if the metal nanoparticles take up $>60\%$ of the total volume. In addition to the plasmon-polariton energies, we analyze the relative contributions of the dipole, quadrupole, and photonic states to their eigenfunctions and are able to demonstrate the decoupling of light in the deep strong-coupling regime. Our results pave the way for a better understanding of the quantum properties of metallic nanoparticle supercrystals in the ultrastrong- and deep strong-coupling regime.

Figure 1

Polariton band structure of an fcc crystal of spherical nanoparticles along the (a) $\mathrm{\Gamma }L$ and (b) $\mathrm{\Gamma }K$ high-symmetry directions. Solid lines were calculated with the microscopic quantum model. The color map shows the magnitude (in log scale) of the normalized electric field obtained from FDTD simulations as a function of energy and momentum. $\hslash {\omega }_p=9$ eV, ${\epsilon }_m={\epsilon }_d=1, a=65$ nm, and $f=0.46$.

Figure 2

Collective plasmon dispersion along the high-symmetry directions of an fcc lattice for fill factors (a) $f=0.06$, (b) 0.31, and (c) 0.90. The blue (red) lines are induced by the dipole (quadrupole) plasmons of the nanoparticles neglecting dipole-quadrupole interactions. The black lines are a full calculation including dipole and quadrupole modes and their interaction. The insets show a sketch of the supercrystal structure for each configuration. $\hslash {\omega }_p=9$ eV, ${\epsilon }_m={\epsilon }_d=1$, and $a=62$ nm.

Figure 3

Plasmon-polariton dispersion along the high-symmetry directions of an fcc crystal with metal fill factors (a) $f=0.06$, (b) 0.31, and (c) 0.90. Black lines are calculated including all terms and interactions, while green dashed lines show the dispersion of the bare plasmons. Yellow dashed lines show the bare photon dispersion. The insets show a sketch of the supercrystal structure for each configuration. $\hslash {\omega }_p=9$ eV, ${\epsilon }_m={\epsilon }_d=1$, and $a=62$ nm.

Figure 4

Energies of (a) the bare plasmon and photon states and of (b) the plasmon-polariton states for $q=0.17\mathrm{\Gamma }K$. The coloring indicates the dipole, quadrupole, and photon contribution to each state—see the inset in (a). (c) and (d) show zoomed images of the rectangular areas in (a) and (b), respectively.

Figure 5

(a) Energies of the upper and lower dipole (blue symbols) and quadrupole (red symbols) plasmon bands for different values of $f$ of an fcc crystal. The dipole (quadrupole) energies are evaluated at the $\mathrm{\Gamma }$ ($X$) point. Red and blue lines are fits to the data points, using Eq. (45) (see Table 1). (b) Normalized coupling strength ${\eta }_t$ of the transverse dipole-derived plasmon band for the $\mathrm{\Gamma }L$ (squares), $\mathrm{\Gamma }X$ (triangles), and $\mathrm{\Gamma }K$ (dots) high-symmetry directions. Blue and red lines are coupling strengths predicted with Eq. (46) for the dipole and quadrupole modes, respectively.

Figure 6

Decomposition of the plasmon-polariton bands into their dipole (blue), quadrupole (red), and photon (yellow) contributions for fill factors (a) $f=0.06$, (b) 0.31, and (c) 0.90. The size of the data points shows the magnitude of the relative contributions of each of these bare excitations to the plasmon-polariton bands.