Plasmons in a superlattice of fullerenes or metallic shells

A theory for the collective plasma excitations in a linear periodic array of spherical two-dimensional electron gases (S2DEGs) is presented. This is a simple model for an ultra thin and narrow microribbon of fullerenes or metallic shells. Coulomb coupling between electrons located on the same sphere and on different spheres is included in the random-phase approximation. Electron hopping between spheres is neglected in these calculations. The resulting plasmon-dispersion equation is solved numerically. Results are presented for a superlattice of single-wall S2DEGs as a function of the wave vector. The plasmon dispersions are obtained for different spherical separations. We show that the one-dimensional translational symmetry of the lattice is maintained in the plasmon spectrum. Additionally, we compare the plasmon dispersion when the superlatice direction is parallel or perpendicular to the axis of quantization. However, because of anisotropy in the Coulomb matrix elements, there is anticrossing in the plasmon dispersion only in the direction perpendicular to the quantization axis. The S2DEG may serve as a simple model for fullerenes, when their energy bands are far apart.

Figure 1

Schematic of a superlattice of spherical two-dimensional electron gases (a) perpendicular to and (b) along the axis of quantization.

Figure 2

Nonzero Coulomb interaction matrix elements for a linear chain of spherical shells along the $z$ axis when the separation between the centers of consecutive spheres is chosen as $a=3R$, and $L=L^{\prime }=1$ with $M,M^{\prime }=0,\pm 1$.

Figure 3

Plasmon dispersion relation in the first Brillouin zone for a periodic array of S2DEGs along the $z$ axis with period $a=3R$. The radius of each sphere is $R=10$ nm, and $l_F=10$ is the angular momentum quantum number for the highest occupied electron state on each sphere. The dashed horizontal line corresponds to the plasmon energy for a single shell with $L=1$.

Figure 4

Plasmon variation for fixed $q_{z}R=2$ as a function of the separation $a$ of the shells along the $z$ axis. The dashed horizontal line gives the single-shell plasmon energy for $L=1$.

Figure 5

Coulomb interaction matrix elements for a linear array of shells along the $x$ axis for the case where $L=L^{\prime }=1$ and $M,M^{\prime }=0,\pm 1$. The radius of each sphere is $R=10$ nm.

Figure 6

Plasmon dispersion relation for a linear array of spherical shells of radius $R=10$ nm separated by a distance $a=3R$ along the $x$ axis.

Figure 7

Anticrossing of the plasmon modes shown in Fig. 6 in the vicinity of the points (a) $q_{x}a/2\pi =0.23$ and (b) $q_{x}a/2\pi =0.77$.

Figure 8

Variation of plasmon excitation energies with separation between adjacent S2DEGs on a linear chain on the $x$ axis for the chosen $q_{x}R=2$. The dashed horizontal line corresponds to the plasmon mode frequency for an S2DEG in isolation.