Antonio Maffucci

A transmission line model for metallic carbon nanotube interconnects

A transmission line (TL) model describing the propagation of electric signals along metallic single wall carbon nanotube (CNT) interconnects is derived in a simple and self-consistent way within the framework of the classical electrodynamics. The conduction electrons of metallic CNTs are modelled as an infinitesimally thin cylindrical layer of a compressible charged fluid with friction, moving in a uniform neutralizing background. The dynamic of the electron fluid is studied by means of the linearized hydrodynamic equations with the pressure assumed to be that of a degenerate spin-½ ideal Fermi gas. Transport effects due to the electron inertia, quantum fluid pressure and electron scattering with the ion lattice significantly influence the propagation features of the TL. The simplicity and robustness of the fluid model make the derivation of the TL equations more straightforward than other derivations recently proposed in the literature and provide simple and clear definitions of the per unit length (p.u.l.) TL parameters. In particular, this approach has provided a new circuit model that can be used effectively in the analysis of networks composed of CNT transmission lines and lumped elements. The differences and similarities between the proposed model and those given in the literature are highlighted. Copyright

Signal Propagation in Carbon Nanotubes of Arbitrary Chirality

In carbon nanotubes (CNTs) with large radii, either metallic or semiconducting, several subbands contribute to the electrical conduction, while in metallic nonarmchair nanotubes with small radii the wall curvature induces a large energy gap. In this paper, we propose a model for the signal propagation along single wall CNTs (SWCNTs) of arbitrary chirality, at microwave through terahertz frequencies, which takes into account both these characteristics in a self-consistent way. We first study an SWCNT, disregarding the wall curvature, in the frame of a semiclassical treatment based on the Boltzmann equation in the momentum-independent relaxation time approximation. It allows expressing the longitudinal dynamic conductivity in terms of the number of effective conducting channels. Next, we study the behavior of this number as the nanotube radius varies and its relation with the kinetic inductance and quantum capacitance. Furthermore, we show that the effects of the spatial dispersion are negligible in the collision dominated regimes, whereas they may be important in the collisionless regimes, giving rise to sound waves propagating with the Fermi velocity. Then, we study the effects on the electron transport of the terahertz quantum transition induced by the wall curvature by using a quantum kinetic approach. The nanotube curvature modifies the kinetic inductance and gives arise to an additional RLC branch in the equivalent circuit, related to the terahertz quantum transition. The proposed model can be used effectively for analyzing the signal propagation in complex structures composed of SWCNTs with different chirality, such as bundles of SWCNTs and multiwall CNTs, providing that the tunneling between adjacent shells may be disregarded.

Performance Comparison Between Metallic Carbon Nanotube and Copper Nano-Interconnects

This paper addresses the problem of scaling interconnects to nanometric dimensions in future very-large-scale integration applications. Traditional copper interconnects are compared to innovative interconnects made by bundles of metallic carbon nanotubes. A new model is presented to describe the propagation of electric signals along carbon nanotube (CNT) bundles, in the framework of the classical transmission line theory. A possible implementation of a future scaled microstrip based on CNT bundle is analyzed and compared to a conventional microstrip.

A New Circuit Model for Carbon Nanotube Interconnects With Diameter-Dependent Parameters

In this paper, a new circuit model for the propagation of electric signals along carbon nanotube interconnects is derived from a fluid model description of the nanotube electrodynamics. The conduction electrons are regarded as a 2-D charged fluid, interacting with the electromagnetic field produced by the ion lattice, the conduction electron themselves, and the external sources. This interaction may be assumed to be governed by a linearized Eulers equation, which provides the nanotube constitutive equation to be coupled to Maxwell equations. A derivation of a circuit model is then possible within the frame of the classical multiconductor transmission-line (TL) theory. The elementary cell of this TL model differs from those proposed in literature, due to the definition of the circuit variable corresponding to the voltage. When considering small nanotube radius, we obtain values for the kinetic inductance and quantum capacitance that are consistent with literature. These values are corrected here to take into account the influence of larger values of radius properly. Conversely, the value of the per unit length resistance is roughly half of the value usually adopted in literature. The multiconductor TL model is used to study the scaling law of the parameters with the number of carbon nanotubes in a bundle.

Circuit Models of Carbon-Based Interconnects for Nanopackaging

This paper proposes an equivalent circuital model to describe the electrical propagation along nanoscale interconnects, made either by carbon nanotubes or graphene nanoribbons. The circuital models are derived from an electrodynamical model for the transport of conduction electrons, and are expressed in the frame of the classical transmission line theory. The per-unit-length parameters, despite their simple expressions, retain the main phenomena occurring at nanoscale, such as the kinetic and quantum effects. In addition, the circuit parameters are expressed as functions of the temperature and the transverse size of the interconnect, thus allowing a qualitative and quantitative analysis of their impact in the electrical performance of the interconnects. The models are used to study some challenging problems in nanopackaging, such as the degradation of electrical performance due to self-heating and the high-frequency current crowding problem because of the skin-effect. Interconnects and vias are analyzed, referring to the 14-nm technology node.

Transmission-Line Model for Multiwall Carbon Nanotubes With Intershell Tunneling

The electromagnetic behavior of multiwall carbon nanotubes (MWCNTs), in the frequency range where only intraband transitions are allowed, depends on the combinations of different aspects: the number of effective conducting channels of each shell, the electron tunneling between adjacent shells, and the electromagnetic interaction between shells and the environment. This paper proposes a general transmission-line (TL) model for describing the propagation of electric signals along MWCNTs at microwave through terahertz frequencies that takes into account all these aspects. The dependence of the number of conducting channels of the single shell on the shell chirality and radius is described in the framework of the quasi-classical transport theory. The description of the intershell tunneling effects on the longitudinal transport of the π-electrons is carried on the basis of the density matrix formalism and Liouvilles equation. The electromagnetic coupling between the shells and ground plane is described in the frame of the classical TL theory. The intershell tunneling qualitatively changes the form of the TL equations through the tunneling inductance and capacitance operators, which have to be added, respectively, in series to the (kinetic and magnetic) inductance matrix and in parallel to the (quantum and electrical) capacitance matrix. For carbon nanotube (CNT) lengths greater than 500 nm, the norm of the tunneling inductance operator is greater than 60% of the norm of the total inductance in the frequency range from gigahertz to terahertz. The tunneling inductance is responsible for a considerable coupling between the shells and gives rise to strong spatial dispersion. The model has been used to analyze the eigenmodes of a double-wall CNT above a ground plane. The intershell tunneling gives arise to strong anomalous dispersion in antisymmetrical modes.

Electrical Modeling of Carbon Nanotube Vias

This paper investigates the electrical behavior of vias made by bundles of either single-walled or multiwalled carbon nanotubes (CNTs). The electronic transport in the CNTs is modeled through the kinetic inductance, the quantum capacitance, and the electrical resistance, which depend on the equivalent number of the CNT conducting channels. The dependence of such a number on the CNT radius, chirality, and temperature is described by using the quasi-classical transport theory. Since for the common mode the effects of the intershell tunneling are negligible, the interaction between different shells is described by using the classical electromagnetic theory. A simple but accurate equivalent lumped model for vias made by CNT bundles is proposed. Vias of interest in nanoelectronic applications are here analyzed, with particular focus on the behavior of electrical parameters versus temperature and frequency.

Carbon nanotubes in nanopackaging applications

CNTs are recently discovered materials made by rolled-up sheets of graphene. They may be made either by a single shell [single-walled CNT (SWCNT) with radius ranging from 0.7 to 3-4 nm] or several nested shells [multiwalled CNT (MWCNT) with outer radius typically of the order of some tens of nanometers]. The length of a nanotube may reach the order of millimeters, and hence, this nanostructured material exhibits an excellent form factor, able to comply with the ultrafine pitches required in nanopackaging. CNT (carbon nanotubes) to be used for vertical vias or interconnects for packaging applications, very high-density SWCNT bundles must be demonstrated. When using MWCNT bundles instead, the achieved density comes from a compromise between the CNT radius and its shell number. The fabrication process must provide low contact resistance, good direction control, and compatibility with CMOS technology. The road for CNTs to replace copper in chip packaging is still long, but the gap between theoretical predictions and practical applications is reducing faster and faster.

An enhanced transmission line model for conductors with arbitrary cross sections

An enhanced transmission line model (ETL) has been recently proposed to describe the propagation along two parallel wires with circular cross sections up to wavelengths comparable to the distance between the wires. In this paper, a general ETL model is proposed to describe the propagation along interconnects consisting of wires with arbitrary cross sections. Since the ETL model has the same simplicity of the standard transmission line model, it allows investigating high-frequency effects, like radiation and dispersion, with a computational cost which is sensibly lower than that required by a full-wave numerical simulation. The ETL model is obtained, with suitable approximations, starting from a full-wave analysis of the propagation problem and using an integral formulation based on the electromagnetic potentials satisfying the Lorentz gauge. Some case studies are carried out and discussed, including a benchmark test with existing literature, performed to check the validity and accuracy of the proposed model.

On the Evaluation of the Number of Conducting Channels in Multiwall Carbon Nanotubes

The correct evaluation of the number of conducting channels in multiwall carbon nanotubes (MWCNTs) is essential in view of the derivation of accurate equivalent circuit models for electrical nanointerconnects made by this innovative material. The authors have recently proposed an approach to evaluate this number for a single CNT shell that takes properly into account its chirality, diameter, and temperature. In this letter, we use this approach to evaluate the number of channels in an MWCNT and compare its results to the existing literature. It is found that so far, the number of conducting channels has been sensibly overestimated by 25%-40% at room temperature.