Formulation in terms of normalized propagators of a charge-dipole model enabling the calculation of the polarization properties of fullerenes and carbon nanotubes

We present a model for the calculation of the polarization properties of fullerenes and carbon nanotubes. This model describes each atom by both a net electric charge and a dipole. Compared to dipole-only models, the consideration of electric charges enables one to account for the displacement of free electrons in structures subject to an external field. It also enables one to account for the accumulation of additional charges. By expressing the electrostatic interactions in terms of normalized propagators, the model achieves a better consistency as well as an improved stability. In its most elementary form, the model depends on a single adjustable parameter and provides an excellent agreement with other experimental and theoretical data. The technique is applied to a ${\mathrm{C}}_{720}$ fullerene and to open and closed (5,5) nanotubes. The simulations demonstrate the improved stability of our algorithm. In addition, they quantify the role of free charges in the polarization of these structures. The paper finally investigates the field-enhancement properties of open and closed (5,5) nanotubes.

Figure 1

(Color online) Axial polarizability of a (5,5) nanotube, as obtained analytically with a cylinder (solid) or using the Q+P iso $\left[R\right]$ (dashed), the Q+P iso $[R,{\alpha }_{\mathrm{iso}}]$ (dot-dashed), and the Q+P aniso $[R,{\alpha }_{\mathrm{par}}$, ${\alpha }_{\mathrm{perp}}]$ (dotted) models. The lower curve (dashed) stands for results obtained using an anisotropic dipole-only model.

Figure 2

(Color online) Electric charges and dipoles of a ${\mathrm{C}}_{720}$ fullerene, as induced by a uniform external field of $1\mathrm{V}∕\mathrm{nm}$ applied horizontally to the left of the figure. The atoms are represented with a brightness that is proportional to the amplitude of their charge. Positive charges (in red) characterize the left part of the figure, while negative charges (in blue) characterize the right part. Top: the Q+P aniso model of Ref. 28 is used. The minimal and maximal values of the atomic charges are $-9.51\times {10}^{-3}e$ and $9.50\times {10}^{-3}e$, respectively (with $e$ the absolute value of the electronic charge). Bottom: the Q+P aniso [$R,{\alpha }_{\mathrm{par}}$, ${\alpha }_{\mathrm{perp}}$] model of this paper is used. The minimal and maximal values of the atomic charges are $-7.75\times {10}^{-3}e$ and $7.73\times {10}^{-3}e$, respectively.

Figure 3

Polarizability of a ${\mathrm{C}}_{720}$ fullerene, as a function of the dilatation of this molecule (measured by the factor by which the interatomic distances are multiplied). Top: the Q+P aniso model of Ref. 28 is used; Bottom: the Q+P aniso [$R,{\alpha }_{\mathrm{par}}$, ${\alpha }_{\mathrm{perp}}$] model of this paper is used. The solid curve indicates the contribution of charges and the dashed one the contribution of dipoles.

Figure 4

Potential energy of a closed (5,5) carbon nanotube subject to a uniform $0.1\mathrm{V}∕\mathrm{nm}$ external field applied downward. The nanotube consists of 4030 atoms and stands vertically on a metallic substrate. The potential energy is calculated using Eq. (10) in the text.

Figure 5

Net electric charge $q_i$ characterizing the atoms of a closed (5,5) nanotube placed vertically on a metallic substrate and subject to a uniform external field of $0.1\mathrm{V}∕\mathrm{nm}$ applied downward.

Figure 6

(Color online) Electric charges and local fields of a closed (5,5) nanotube, as induced by a uniform external field of $0.1\mathrm{V}∕\mathrm{nm}$ applied downward. The nanotube stands vertically on a metallic substrate. The atoms are represented with a brightness that is proportional to the amplitude of their charge. The minimal and maximal values of the atomic charges are $-0.177e$ and $-1.25\times {10}^{-4}e$, respectively (with $e$ the absolute value of the electronic charge). This maximal value is not represented on this figure.

Figure 7

Local fields characterizing the atoms of a closed (5,5) nanotube placed vertically on a metallic substrate and subject to a uniform external field of $0.1\mathrm{V}∕\mathrm{nm}$ applied downward.

Figure 8

Field-enhancement factor of open (solid) and closed (dotted) (5,5) carbon nanotubes. These tubes are placed vertically on a metallic substrate. The elementary unit of each nanotube consists of 20 atoms and has a length of $0.24595\mathrm{nm}$. The analytical expressions given by Eqs. (14, 15) are included as well (they are hardly distinguishable from the numerical results).