Model of plasmon excitations in a bundle and a two-dimensional array of nanotubes

We calculate the plasma excitations in a bundle as well as a two-dimensional (2D) periodic array of aligned parallel multishell nanotubes on a substrate. The carbon nanotubes are oriented perpendicular to the substrate. The model we use for the system is an electron gas confined to the surface of an infinitely long cylinder embedded in a background dielectric medium. Electron tunneling between individual tubules is neglected. We include the Coulomb interaction between electrons on the same tubule and on different tubules for the same nanotube and neighboring nanotubes. We present a self-consistent field theory for the dispersion equation for intrasubband and intersubband plasmon excitations. For both the bundle and 2D array of aligned parallel nanotubes, the dispersion relation of the collective modes is determined by a three-dimensional wave vector with components in the direction of the nanotube axes and in the transverse directions. The dispersion equation is solved numerically for a single-wall nanotube 2D array as well as a bundle, and the plasmon excitation energies are obtained as a function of wave vector. The intertube Coulomb interaction couples plasmons with different angular momenta $m$ in individual nanotubes, lifting the $\pm m$ degeneracy of the single-nanotube modes. This effect is analyzed numerically as a function of the separation between the tubules. We show that the translational symmetry of the lattice is maintained in the plasmon spectrum for the periodic array, and the plasmon energies have a periodic dependence on the transverse wave vector ${\mathbf{q}}_{\perp }$. For the bundle, the Coulomb interaction between nanotubes gives rise to optical plasmon excitations.

Figure 1

(Color online) Array of nanotubes whose axes are parallel and taken to be aligned in the $z$-direction. In this paper, we consider a subset forming a bundle, and an infinite periodic array in both $x$ and $y$-directions.

Figure 2

Undamped intrasubband $\left(m=0\right)$ plasmon excitation energy, in units of the Fermi energy $E_F$, as a function of $q_z/k_F$. Here, $k_F$ is the Fermi wave number in the ground $\left(l=0\right)$ subband obtained. The solutions were obtained by solving Eq. (25) for $q_x=\pi /a_x$ and $q_y=\pi /a_y$. The parameters used in the calculation are $m^{\ast }=0.25m_e$, where $m_e$ is the bare electron mass and $a_x=a_y=35.0\text{Å}$, $R=11.0\text{Å}$, and $E_F=0.6\text{eV}$.

Figure 3

Undamped intrasubband $\left(m=0\right)$ plasmon excitation energy as a function of the transverse wave vector $q_x$ (in units of $2\pi /a_x$). The solutions are based on Eq. (25) for $q_y=0$ and $q_z=0$. All other parameters employed in obtaining Fig. 2 are the same.

Figure 4

The plasmon excitation energy, in units of the Fermi energy $E_F$, as a function of $q_z/k_F$, for a 2D array of coaxial tubules of outer and inner radii $R_1=14\text{Å}$ and $R_2=11\text{Å}$, respectively. The lattice constants in both $x$ and $y$ directions were chosen as $35\text{Å}$. The solutions were obtained from Eq. (25), when both $a_x,a_y\to \infty$. The parameters used in the calculation are the same as in Fig. 2.

Figure 5

For a 1D periodic array, the plasmon excitation energy, in units of the Fermi energy $E_F$, as a function of $q_x$, in units of $2\pi /a_x$. Here $q_z=0$, the radius of each tubule is $11\text{Å}$, and the period is $a_x=35\text{Å}$. All other parameters are the same as in Fig. 2.

Figure 6

For a 2D periodic array, the plasmon excitation energy, in units of the Fermi energy $E_F$, as a function of $q_x$, in units of $2\pi /a_x$. Here $q_y=q_z=0$; all other parameters are the same as in Fig. 2.

Figure 7

The intrasubband plasmon excitation energy, in units of the Fermi energy $E_F$, as a function of $q_z/k_F$, for the three single wall nanotubes described in Sec. 2. The radius of each tubule was chosen as $11\text{Å}$ and the centers of the tubules are at $a_x=25\text{Å}$ and $a_y=25\text{Å}$ in (a). In (b), there are only two nanotubes with one fixed at the origin while the position of the second nanotube on the $x$ axis is at $a_x=25\text{Å}$. All other parameters are the same as in Fig. 2.

Figure 8

The intrasubband plasmon excitation energy, in units of the Fermi energy $E_F$, as a function of $q_z/k_F$, for the three single wall nanotubes described in Sec. 2. The radius of each tubule was chosen as $11\text{Å}$ and the separation between the center of the tubules on the $y$ axis is $a_y=25\text{Å}$ in (a) while the coordinate of the third tubule on the $x$ axis is varied. In (b), there are two nanotubes with one fixed at the origin while the center of the second one on the $x$ axis is varied. All other parameters are the same as in Fig. 2.