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A semi-classical electrodynamical model is derived to describe the electrical transport along graphene, based on the modified Boltzmann transport equation. The model is derived in the typical operating conditions predicted for future integrated circuits nano-interconnects,

Due to their outstanding physical properties, graphene and its allotropes (carbon nanotubes (CNTs) and graphene nanoribbons (GNRs)) are the major candidates to become the silicon of the 21st century and open the era of so-called carbon electronics [

Amongst all such applications, the low electrical resistivity, high thermal conductivity, high current carrying capability, besides other excellent mechanical properties, make graphene a serious candidate to replace conventional materials in realizing VLSI nano-interconnects [

The increasing interest in carbon nano-interconnects leads to the quest for more and more accurate and reliable models, able to include all the quantum effects arising at the nanoscale. This topic has been given great attention by the recent literature, which presented several modeling approaches, like phenomenological [

In the typical working conditions of nano-interconnects, namely a frequency up to THz and low bias conditions, such nano-structures do not exhibit tunneling transport. Thus, the electrodynamics may be studied by using a semi-classical description of the electron transport. This leads to the derivation of a constitutive relation between the electrical field and the density of the current in the form of a generalization of the classical Ohm’s law, introducing non-local interactions and dispersion. By coupling such a relation to Maxwell equations, it is possible to derive a generalized transmission line model for such nano-interconnects. This approach has been followed by the authors to model isolated metallic CNTs in [

In this paper, a comprehensive and self-consistent analysis of the electrical properties of graphene is presented, with special emphasis on its application as an innovative material for nano-interconnects. The self-consistency is guaranteed by the fact that the used transport model is derived from an electrodynamical model described in terms of physically meaningful parameters. This differs from what can be obtained by means of heuristics approaches. In particular, a self-consistent model provides a clear analytical relation between the model parameters and the physical and geometrical properties of the investigated structure. The considered frequency range of applications is limited to values up to 1 THz, and the operational conditions fall into the low bias regime (the longitudinal electric field along the interconnect should be _{z} < 0.54 V/μm). For such cases, interband transitions are absent and non-linear effects are negligible, and thus, a simple linear transport of the conducting electrons (π-electrons) may be derived based on the tight-binding model and the semi-classical Boltzmann equation.

In this chapter, we briefly summarize the main features of the band structure, first for an infinite graphene layer (a 2D structure) and, then, for the graphene nanoribbon (a 1D structure).

_{g}, is spanned by the two vectors, _{1} and _{2}, and contains two carbon atoms. The basis vectors (_{1},_{2}) have the same length, |_{1}| = |_{2}| = _{1} and _{2}, with respect to the rectangular coordinate system, (0,_{0}/ 2,_{0}/ 2) and (_{0}/ 2,−_{0}/ 2). The area of the unit cell, _{g}, is _{g}

The structure of graphene. (

In the reciprocal _{g}_{1} and _{2}, which have the same length |_{1}| = |_{2}| = _{0} = 4π_{0} and form an angle of 2π/3. The components of the vectors, _{1} and _{2}, with respect to the rectangular coordinate system (Q,_{x}_{y}_{0}_{0}_{0}_{0}_{g}_{1},_{2}) and the basis vectors of the reciprocal space (_{1},_{2}) are related by _{i}_{j}_{ij}_{g} and _{g}, are related by _{g}_{g} = (2π)^{2}.

The graphene possesses four valence electrons for each carbon atom. Three of these (the so-called σ-electrons) form tight bonds with the neighboring atoms in the plane and do not play a part in the conduction phenomenon. The fourth electron (the so-called π-electron), instead, may move freely between the positive ions of the lattice.

In the nearest-neighbors tight-binding approximation, the π-electrons energy dispersion relation is
^{(±)} denotes the energy, the + sign denotes the conduction band, the − sign denotes the valence band and γ = 2.7 eV is the carbon-carbon interaction energy. The electronic band structure is depicted in _{0} is the wavenumber at a Fermi point, ν_{F}^{6} m/s is the Fermi velocity of the π-electrons and

Graphene electronic band structure. Inset: the neighborhood of a Fermi point.

In the ground state, the valence band of the graphene is completely filled by the π-electrons. In general, at equilibrium, the energy distribution function of π-electrons is given by the Dirac–Fermi function:
_{B} is the Boltzmann constant and _{0} is the graphene absolute temperature, the electrochemical potential of the graphene being null valued. The distribution function differs from the ideal distribution function ^{(±)}] = ^{(±)}], where _{B}_{0} around the point ^{(±)} = 0; at room temperature, it results _{B}T_{0}

Let us now discuss the so-called graphene nanoribbons (GNRs),

The structure of graphene nanoribbons. (

As shown in

Given the number

Energy band structure of an armchair nanoribbon

The energy band structure of a zigzag nanoribbon

We model the propagation of an electromagnetic wave along a graphene nanoribbon in the low frequency regime, where only intraband transitions of π-electrons with unchanged transverse quasi momentum are allowed. These transitions contribute to the axial conductivity, but not to the transverse conductivity. For typical dimensions of the nano-interconnects for the 14 nm technology node and below, this assumption limits the model to a frequency up to some THz. Following the stream of what is done in [

The π-electrons in an infinite graphene layer behave as a two-dimensional electron gas moving on the surface under the action of the electric field. We disregard the Lorentz force term due to the magnetic field, because it does not contribute to the linear response of the electron gas.

The electron gases are described by the distribution functions ^{(±)} = ^{(±)}(_{BZ} and

Each of these distribution functions satisfies the quasi-classic Boltzmann kinetic equation,
_{t}_{t}^{−1}) and ^{(±)} = ^{(±)}(

The symbols, _{r} and _{k}, denote the gradients with respect to the variables,

Let us consider a time-harmonic field, in the form _{t}_{t}^{(±)} = _{0}^{(±)}(^{(±)}(^{(±)}(_{1}^{(±)}(_{1}^{(±)} is a small quantity to be found: using Equation (4) and retaining only the first order terms, we obtain:

Let us now introduce the time harmonic surface current density

Here, the first term is related to the valence bands and the second one to the conduction bands. By combining Equations (7) and (8) we get the constitutive relation of the medium:

This can be regarded as a generalized Ohm’s law, for which the conductivity is a symmetric dyad, given by:

The function:

Note that the function, Δ, behaves as an impulsive function centered at _{F}_{eff} centered at the Fermi point with electron wave vector

Along a graphene nanoribbon, in the operating conditions assumed here, the π-electrons mainly move along the longitudinal axis,

If T is the translational length of the unit cell (see _{F}

Using the above results, the generalized Ohm’s law Equation (9) may be expressed for a GNR as follows:
_{0} = 12.9 kΩ being the quantum resistance, and:

The quantity

Let us first assume the case of a graphene layer in the long wavelength limit β = 0, for which Equation (13) reduces to:

In the reference system indicated in _{xx}_{yy}_{xy}_{yx}

In the same condition, the generalized Ohm’s law for a graphene nanoribbon Equation (16) would reduce to:

Equivalent number of conducting channels for metallic and semiconducting armchair graphene nanoribbons (GNRs) at 300 K.

The conductivity, σ_{c}_{0} in Equation (23): σ_{c}_{0} by taking the limit _{0}/σ_{c}

The ratio, σ_{0}/σ_{c}

As a conclusion, in the case of a long wavelength limit, the conductivity of graphene does not show spatial dispersion, but only frequency dispersion, according to Equation (22). For a GNR (as in _{xx}

In the general case, we must take into account the dyadic nature of the conductivity. For the sake of simplicity, let us refer to the components of any vector to the parallel and orthogonal directions with respect to the vector, _{||||}(ω,_{⊥⊥}(ω,_{||⊥}(ω,_{⊥||}(ω,_{||⊥}(ω,_{⊥||}(ω,

From such components, it is possible to derive the

Real (_{||||}(ω,_{⊥⊥}(ω,_{c}_{F}

By using Equation (27), it is easy to show that there are off-diagonal terms of the dyad; hence, in general, σ_{xy}_{yx}_{||||}(ω,_{⊥⊥}(ω,_{c}_{F}_{F}

Real (_{||||}(ω,_{⊥⊥}(ω,_{c}_{F}

The relation between the dyadic nature of the conductivity and the spatial dispersion is consistent with the result obtained in [

In the operating conditions predicted for future electronic nano-interconnects, the low bias and the low frequency allows the modeling of the electrical transport with a linearized semi-classical electrodynamical model, which does not include interband transitions. This leads to a generalized non-local dispersive Ohm’s law, which is regarded as the constitutive equation for the material. In the 2D case (graphene layer), the conductivity is dyadic and shows a spatial dispersion as an effect of the coupling between the components. This coupling is absent in the long-wavelength limit. In the 1D case (graphene nanoribbon), the transverse components of the currents are negligible, and the conductivity reduces to a scalar quantity, modulated by the size and chirality, which strongly affect the number of conducting channels.

This work has been supported by the EU grant # FP7-247007, under the project CACOMEL, “Nano-CArbon based COmponents and Materials for high frequency ELectronics”.

The authors declare no conflict of interest.