Electrical Propagation on Carbon Nanotubes: from Electrodynamics to Circuit Models

Carbon nanotubes (CNTs) are recently discovered materials made by rolled sheets of graphene of diameters of the order of nm and lengths up to mm. CNTs have been proposed as emerging materials able to give solutions to many of the problems given by the tight requirements of the technologies nodes below 22 nm. Presently, CNTs are considered for a large variety of micro and nano-electronics applications, like nano-interconnects, nano-packages, nano-transistors, nano-passives, and nano-antennas. Given these perspectives, many efforts have been made in literature to derive models able to describe the electrical propagation along carbon nanotubes. The electromagnetic response of carbon nanotubes has been widely examined in frequency ranges from microwave to the visible, taking properly into account the graphene crystalline. For each carbon atom in the graphene only one out of the four valence electrons (the electron) contributes to the conduction phenomenon, hence in order to model the electromagnetic response of carbon nanotubes there is the need to describe the interaction of the electrons with the electromagnetic fields produced by the electrons themselves and by the external sources, under the action of the electric field generated by the fixed positive ions of the lattice. This requires, in principle, a quantum mechanical approach, since the electrical behavior of the electrons depends strongly on the interaction with the positive ion lattice. A quantum-mechanical approach has been for instance used in, where the model is derived by using numerical simulations based on first principles. Alternatively, phenomenological approaches are possible like those based on the Luttinger liquid theory. Another possible way is given by semi-classical approaches, based on simplified models that yield approximated but analytically tractable results. Among these models, the fluid ones play a central role in CNTs modeling; in fact, despite their simplicity and immediate physically intuition, they are able to describe the main physical processes arising on characteristic lengths involving many unit cells, such as the collective effects. These models assume that the electric fields due to the collective motion of the electrons themselves and to the external sources are smaller than the atomic crystal field and also slowly varying on atomic length and time scales. In these conditions the lectrons behave as “quasi-classical particles” and the equations governing their dynamics are the classical equations of motion, provided that the electron mass is replaced by an “effective mass”, which endows the interaction with the positive ion lattice. Section 2 presents an electrodynamical model of the propagation along carbon nanotubes, derived by using the above semi-classical fluid description. The model is here presented to any type of CNTs, both metallic and semiconducting, with any chirality. The model introduces the concept of “equivalent number of conducting channels”, which represents a measure of the number of subbands in the neighbors of the nanotube Fermi level that effectively contribute to the electric conduction. This number depends on the chirality, the radius and the temperature of the CNT. Section 3 provides an example of applications of CNTs as electromagnetic material. The problem of the evaluation of the scattering characteristics of a CNT antenna is presented. An electromagnetic model is derived and discussed, based on the description provided in Section 2. Finally, in Section 4 a circuit model is derived, able to describe the behavior of CNT interconnects in the frame of the classical transmission line theory. The model describes either single or bundled CNTs. An application of CNTs as materials for innovative nanopackaging interconnect is discussed and comparisons with conventional copper technology is provided.