1	Carbon Nanotubes From Electrodynamics to Signal Propagation Models Antonio Maffucci, Sergey A. Maksimenko, Giovanni Miano, and G.Ya. Slepyan

CONTENTS

1.1    Introduction

1.2    Electrodynamics of Carbon Nanotubes

1.2.1    Band Structure of a Single CNT Shell

1.2.2    Transport Equation for a Single CNT Shell

1.2.3    Transport Equation for Two CNTs with Intershell Tunneling

1.3    Transmission Line Models for Carbon Nanotube Interconnects

1.3.1    CNTs as Future Material for Nano-Interconnects

1.3.2    Multiconductor Transmission Line Model for CNT Interconnects without Tunneling Effect

1.3.3    Transmission Line Model for CNT Interconnects with Tunneling Effect

1.4    Study of Signal Propagation Properties along CNT Interconnects

1.4.1    Performance of a CNT On-Chip Interconnect

1.5    Conclusions

References

1.1    INTRODUCTION

Carbon nanotubes (CNTs) are recently discovered materials [1] made by rolled sheets of graphene of diameters of the order of nanometers and lengths up to millimeters. Because of their outstanding electrical, thermal, and mechanical properties [2,3], CNTs are proposed as emerging materials in nanoelectronics [4,5], for fabricating nano-interconnects [6,7,8], nanopackages [9], nano-transistors [10], nano-passives [11], and nano-antennas [12,13].

Recently, the theoretical predictions have been confirmed by the first real-world CNT-based electronic devices, like the CNT bumps for nanopackaging applications presented in [14] or the CNT wiring of a prototype of a digital integrated circuit, one of the first examples of successful CNT-CMOS integration [15,16].

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 [17,18]. This requires, in principle, a quantum mechanical approach, as done in [19], where the model is derived by using numerical simulations based on first principles. Alternatively, phenomenological approaches are possible like those proposed in [20], based on the Luttinger liquid theory. Another possible way is given by semiclassical approaches, based on simplified models that yield approximated but analytically tractable results. An example is given by the imposing boundary value problems for Maxwell equations transformed into integral equations [12,18,21], which are numerically solved. A second possibility consists in modeling the transport in the frame of the transmission line (TL) theory: Examples are given by [21], where the CNT is modeled as an electron waveguide, or by [22,23,24,25,26].

Section 1.2 presents the transport equation for carbon nanotubes, first considering isolated CNTs and then considering the case of interacting CNT shells, when the tunneling effect is considered. The transport equations are derived from an electrodynamical model that describes the behavior of the conduction electrons (the so-called π-electrons and by the Maxwell equations). An isolated carbon nanotube shell is described as a quantum wire, that is, a structure with transverse dimensions comparable to the characteristic de-Broglie wavelength of the π-electrons. In the low-frequency regime only intraband transitions of π-electrons with unchanged transverse quasi-momentum are allowed. The intraband transition contributes to the axial conductivity, but not to the transverse conductivity, and excites an azimuthally symmetric electric current density, which leads to an experimentally observed conductivity peak in the terahertz frequency range [27]. In this hypothesis, a simple but physically meaningful transport model may be obtained by describing the electron cloud as if it were a fluid moving under the action of an external field, assuming that the electric fields, due to their collective motion and to the external sources, are smaller than the atomic crystal field and also slowly varying on atomic length and timescales. In these conditions the π-electrons 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 [21,28].

Under the same conditions, the transport model may be obtained by solving the Boltzmann transport equation in quasi-classical limit, which is the approach reported in Section 1.2 and described in detail in [29]. A similar approach has been adopted, for instance, in [30]. A major feature of this model is the evaluation of the longitudinal dynamic conductivity of the CNT, expressed in terms of the number of effective conducting channels, which depends on CNT chirality, size, and temperature [29,30,31]. This number estimates the subbands passing near or through the nanotube Fermi level, which effectively contribute to the conduction.

The above model is useful to describe single-wall carbon nanotubes (SWCNTs), multiwall carbon nanotubes (MWCNTs), and bundles made by them, in the assumptions that the tunneling currents between CNTs are negligible. This is the case for the MWCNT models and the CNT bundle models presented, for instance, in [11,32,33].

When operating frequencies reach the terahertz range, the tunneling effect may be no longer negligible. The transport model is then modified in Section 1.2 to account for this phenomenon, following the stream of what was done in [34], based on the density matrix formalism and Liouville’s equation [35].

In Section 1.3 the transport models are used to derive circuital representation for the propagation of an electrical signal along CNT interconnects, in the frame of the TL theory, in the transverse electromagnetic (TEM) hypothesis of propagation, assuming low-frequency and low-voltage bias conditions.

The TL model for CNT interconnects is formally given by a lossy resistance-inductance-capacitance (RLC) model, where the per unit length (p.u.l.) inductance and capacitance exhibit two new terms, besides the classical electromagnetic ones: a kinetic inductance and a quantum capacitance. The kinetic inductance takes into account the effects of the π-electron inertia, whereas the quantum capacitance describes the quantum pressure arising from the zero-point energy of the π-electrons (e.g., [23,24,25,26]). When the intershell tunneling is considered, the TL equations exhibit spatial dispersion, which may significantly affect the quality of the signals [34].

Finally, Section 1.4 shows some examples of CNT interconnects. Realistic scenarios of on-chip interconnects are considered, and performance analysis is carried out, comparing conventional copper realization with innovative SWCNT or MWCNT ones.

1.2    ELECTRODYNAMICS OF CARBON NANOTUBES

1.2.1    BAND STRUCTURE OF A SINGLE CNT SHELL

A carbon nanotube is made by rolled-up sheets of a monoatomic layer of graphite (graphene), whose lattice is depicted in Figure 1.1a. In the direct space the nanotube unit cell is the cylindrical surface generated by:

1.  The chiral vector C = na₁ + ma₂, where n and m are integers, a₁ and a₂ are the basis vectors of the graphene lattice, of length $\left|{\mathbf{a}}_{\mathbf{1}}\right|=\left|{\mathbf{a}}_{\mathbf{2}}\right|=a_0=\sqrt{3}{b}_0,$ with b₀ = 0.142.nm being the interatomic distance.

2.  The translational vector T = t₁a₁ + t₂a₂, of length T, with $t_1=(2m+n)/d_R$, $t_2=-(2n+m)/d_R$ and $d_R=\text{gcd}[(2m+n),(2n+m)]$ (gcd is the greatest common divisor). In the Cartesian system (x, y) with the origin at the center of a graphene hexagon and the x axis oriented along the hexagon side, the coordinates of a₁ and a₂ are given by $a_{\mathbf{1}}=\left(\sqrt{3a_0}/2,a_0/2\right)$ and $a_2=\left(\sqrt{3a_0}/2,-a_0/2\right)$.

A CNT shell is obtained by rolling up the graphene sheet in such a way that the circumference of the tube is given by the chiral vector C, perpendicular to the axis of the tube. The carbon nanotube radius r_c is therefore given by

$r_c=\frac{a_0}{2\pi }\sqrt{n^2+nm+m^2}$

(1.1)

Nanotubes with n = 0 (or m = 0) are called zigzag CNTs, those with n = m are the armchair CNTs, and those with 0 < n ≠ m are the chiral CNTs.

FIGURE 1.1 (a) The unrolled lattice of a carbon nanotube: lattice basis vectors of graphene, the unit cell of graphene, and the chiral vector of the tube graphene lattice. (b) A SWCNT. (c) A MWCNT.

A single-wall carbon nanotube (SWCNT) is made by a single shell (Figure 1.1b), usually with a radius of the order of fractions or of few nanometers. Instead, a multiwall carbon nanotube (MWCNT) is made by several nested shells (Figure 1.1c), with a radius ranging from tens to hundreds of nanometers, separated by the van der Waals distance δ = 0.34 nm.

The graphene layer is a zero-gap semiconductor; thus the number density of the conduction electrons is equal to zero when the absolute temperature is zero. Nevertheless, when a graphene layer is rolled up, it may become either metallic or semiconducting, depending on its geometry (e.g., [2,3]). The general condition to obtain a metallic CNT is $|n-m|=3q$, where q = 0, 1, 2, …; therefore armchair CNTs are always metallic, whereas zigzag CNTs are metallic only if m = 3q with q = 1, 2, …. In all the other cases the CNT behaves as a semiconductor. Statistically, all the chiralities have the same probability, and thus in a population of CNT shells one-third of the total number are metallic and two-thirds are semiconducting. Finally, we have to note that for m, n → ∞ (that is, for $r_c\to \infty$) any carbon nanotube shell tends to the graphene sheet.

To study the band structure of a generic CNT shell, let us analyze the graphene reciprocal lattice (see Figure 1.2). In this space we consider the Cartesian coordinate system (k_x, k_y) having the origin at the center of a hexagon, the k_y axis oriented along the hexagon side, and the k_x axis orthogonal to k_y. The basis vectors for the reciprocal lattice are given by $b_1\left(2\pi /\sqrt{3}{a}_0,2\pi /a_0\right)$ and $b_2\left(2\pi /\sqrt{3}{a}_0,-2\pi /a_0\right)$.

The first Brillouin zone of a CNT shell is the set $S=\{s_1,s_2,\dots ,s_N\}$ of N parallel segments generated by the orthogonal basis vectors ${\mathbf{K}}_1=(-t_2{\mathbf{b}}_1+t_1{\mathbf{b}}_2)/N$ and ${\mathbf{K}}_2=m{\mathbf{b}}_1+n{\mathbf{b}}_2/N$ (Figure 1.4). The K_μ points of the segment s_μ are given by

$K_{\text{μ}}(k)=k\frac{K_2}{\left|K_2\right|}+\text{μ}{K}_1\text{for}\mathrm{-}\frac{\pi }{T}<k\le \frac{\pi }{T}\text{and}\text{μ}=0,1,\dots ,N-1$

(1.2)

The vectors K₁ and K₂ are related to the direct lattice basis vectors C and T through: C · K₁ = 2π, T · K₁, C · K₂ = 0, and T · K₂ = 2π; therefore $\left|K_1\right|=1/r_c$ and $\left|K_2\right|=2\pi /L$. The first Brillouin zone is equivalent to that of N one-dimensional systems with the same length L. The position of the middle point of s_μ is given by μK₁ and the distance between two adjacent segments is $\Delta k_{\perp }=1/r_c$. The longitudinal wave vector k is almost continuous assuming the CNT shell length to be large compared with the length of the unit cell; on the contrary, the transverse wave vector $k_{\perp }$ is quantized: $\text{μΔ}{k}_{\perp }$ with μ = 0,1,…, N − 1.

FIGURE 1.2 The reciprocal graphene lattice and the first Brillouin zone referred to a CNT shell.

In the zone-folding approximation the dispersion relation for the SWCNT consists of 2N one-dimensional energy subbands given by

$E_{\text{μ}}^{(\pm )}(k)=E_g^{(\pm )}\left(k\frac{K_2}{\left|K_2\right|}+\text{μ}{K}_1\right)\text{for }\frac{\text{π}}{L}<k\le \frac{\text{π}}{L}\text{and μ}=0,1,\dots ,\text{}N-1$

(1.3)

where the function $E_g^{(\pm )}(\cdot )$ is the dispersion relation of the graphene layer, which is given, in the nearest-neighbor tight-binding approximation, by the following expression:

$E_g^{(\pm )}(k)=-\text{γ}{\left[1+4\text{ cos}\left(\frac{\sqrt{3}{k}_x{a}_0}{2}\right)\text{ cos }\left(\frac{k_y{a}_0}{2}\right)+4{\text{cos}}^2\left(\frac{k_y{a}_0}{2}\right)\right]}^{1/2}$

(1.4)

The conduction and the valence energy band of the π-electrons are obtained by putting + and – in (1.4), respectively. Here γ = 2.7 eV is the carbon-carbon interaction energy. The valence and conduction bands of the graphene touch at the graphene Fermi points. Only the energy subbands that pass through or are close to the Fermi level contribute significantly to the nanotube axial electric current, as will be pointed out in the next section.

1.2.2    TRANSPORT EQUATION FOR A SINGLE CNT SHELL

The dynamics of the π-electrons in the μth subband are described by the distribution function $f_{\text{μ}}^{(\pm )}=f_{\text{μ}}^{(\pm )}(z,k,t)$, which satisfies the quasi-classical Boltzmann equation [34]:

$\frac{\partial f_{\text{μ}}^{(\pm )}}{\partial t}+{\nu }_{\text{μ}}^{(\pm )}\frac{\partial f_{\text{μ}}^{(\pm )}}{\partial z}+\frac{e}{\hslash }{E}_z\frac{\partial f_{\text{μ}}^{(\pm )}}{\partial k}=-\nu \left(f_{\text{μ}}^{(\pm )}-f_{0,\text{μ}}^{(\pm )}\right)$

(1.5)

where e is the electron charge, ℏ is the Planck constant, $E_z=E_z(z,t)$ is the component of the electric field at the nanotube surface, ${\nu }_{\text{μ}}^{(\pm )}(k)=d{E}_{\text{μ}}^{(\pm )}/d(\hslash k)$ longitudinal velocity, ν is the relaxation frequency, and $f_{0,\text{μ}}^{(\pm )}(k)=F\left[E_{\text{μ}}^{(\pm )}(k)\right]/2{\pi }^2{r}_c$is the equilibrium distribution function, where F[E] is the Dirac-Fermi distribution function with electrochemical potential equal to zero:

$F[B]=\frac{1}{e^{E/k_{B}T}+1}$

(1.6)

in which k_B is the Boltzmann constant and T is the nanotube absolute temperature.

Assuming a time-harmonic regime for the electric field and the surface current density

$E_z(z,t)=\text{ Re }\left\{{\widehat{E}}_z{e}^{i(\text{ω}t-\beta z}\right\}, J_z(z,t)=\text{ Re }\left\{{\widehat{J}}_z{e}^{i\left(\text{ω}t-\beta z\right)}\right\}$

(1.7)

the constitutive equation for the CNT shell may be written as

${\widehat{\sigma }}_{zz}\left(\text{β},\text{ω}\right){\widehat{J}}_z={\widehat{E}}_z$

(1.8)

having introduced the CNT longitudinal conductivity ${\widehat{\sigma }}_{zz}(\text{β},\text{ω})$ in the wavenumber and frequency domain. In order to evaluate this parameter, let us assume small perturbations of the distribution functions around the equilibrium values, i.e., $f_{\text{μ}}^{(\pm )}=f_{0\text{μ}}^{(\pm )}+\text{ Re }\left\{\text{δ}{f}_{1\text{μ}}^{(\pm )}\text{ exp }[i\left(\text{ω}t-\text{β}z\right)\right\}$. From (1.5) we obtain

$\text{δ}{f}_{\text{μ}}^{(\pm )}(k,\text{ω})=\frac{i}{\hslash }\frac{\partial f_0{}_{\text{μ}}^{(\pm )}}{\partial k}\frac{e{\widehat{E}}_z}{\text{ω}-{\text{ν}}_{\text{μ}}^{(\pm )}\text{β}-i\text{ν}}$

(1.9)

hence ${\widehat{\text{σ}}}_{zz}(\text{β},\text{ω})$ is given by

${\widehat{\text{σ}}}_{\mathit{zz}}(\text{β},\text{ω})=\frac{i{e}^2}{\hslash }{\displaystyle \sum _{\pm }{\displaystyle \sum }_{\text{μ}=0}^{N-1}}\underset{-\pi /L}{\overset{\pi /L}{{\displaystyle \int }}}\frac{\partial f_{0\text{μ}}^{(\pm )}}{\partial k}\frac{{\text{ν}}_{\mu }^{(\pm )}}{\text{ω}-{\text{ν}}_{\text{μ}}^{(\pm )}\text{β}-i\text{ν}}dk$

(1.10)

For all the subbands that give a meaningful contribution to the conductivity we may put ${\text{ν}}_{\text{μ}}^{(+)}\cong {\text{ν}}_F$ in the kernel of (1.10). This assumption is well founded for the Fermi level subbands of metallic shells, whereas for the other subbands it slightly overestimates the effects of the spatial dispersion.

Starting from (1.8) and (1.10), we obtain the constitutive equation in the spatial and frequency domain,

$\left(\frac{i\text{ω}}{\text{ν}}+1\right)J_z=\frac{1}{\text{ν}\left(\frac{\text{ν}}{i\text{ω}}+1\right)}{\text{ν}}_F^2\frac{\partial {\text{ρ}}_s}{\partial z}+{\sigma }_c{E}_z$

(1.11)

where ${\text{ρ}}_s(z,\text{ω})$ is the surface charge density and

${\text{σ}}_c=\frac{{\text{ν}}_F}{\pi r_c\text{ν}{R}_0}M$

(1.12)

is the long wavelength static limit for the axial conductivity. In (1.12) we have introduced the quantum resistance $R_0=\pi \hslash /e^2\cong 12.9\text{k}\Omega$, the Fermi velocity ${\text{ν}}_F=0.87\cdot {10}^6\text{m}/\text{s}$, the relaxation frequency ν, and the equivalent number of conducting channels, defined as [26]

$M=\frac{2\hslash }{{\nu }_F}{\displaystyle \sum _{\text{μ}=0}^{N-1}{\displaystyle \underset{0}{\overset{\text{π}/L}{\int }}{({\nu }_{\text{μ}}^{+})}^2\frac{dF}{dE}{|}_{E=E_{\text{μ}}^{+}}dk}}$

(1.13)

This parameter may be interpreted as the average number of subbands around the Fermi level of the CNT shell. It depends on the number of segments s_μ passing through the two circles of radius k_eff and centered at the two inequivalent Fermi points of graphene. Figure 1.3 shows the typical configurations obtained for metallic and semiconducting carbon nanotubes.

This number increases as the CNT radius increases, since $\text{Δ}{k}_{\perp }=1/r_c$. In addition, since the radius k_eff is a function of the absolute temperature T, M depends on temperature too; the behavior of M with temperature and CNT diameter is plotted in Figure 1.4 for metallic and semiconducting CNTs.

FIGURE 1.3 The segments s_μ crossing the circumference c_k (k_eff) for (a) a metallic and (b) a semiconducting CNT shell.

FIGURE 1.4 Equivalent number of conducting channels versus CNT shell diameter, computed at T = 273 K and T = 373 K.

The constitutive relation (1.11), which can be regarded as a nonlocal Ohm’s law, may be rewritten as follows, by applying the charge conservation law and assuming the current to be uniformly distributed along the CNT contour, as shown in [21]:

$i\text{ω}{L}_{K}I(z,\text{ω})=\frac{1}{1+\text{ν}/i\text{ω}}\frac{1}{i\text{ω}{C}_q}\frac{{\partial }^{2}I\left(z,\text{ω}\right)}{\partial z^2}+E_z(z,\text{ω})-RI(z,\text{ω})$

(1.14)

Here we have introduced the per unit length (p.u.l.) kinetic inductance L_K, the p.u.l. quantum capacitance C_q, and the p.u.l. resistance R, defined as

$L_k=\frac{R_0}{2{\text{ν}}_{F}M}, C_q=\frac{2M}{{\text{ν}}_F{R}_0}\left(1+\frac{\text{ν}}{i\text{ω}}\right) , R=\text{ν}{L}_K=\frac{\text{ν}{R}_0}{2{\text{ν}}_{F}M}$

(1.15)

Equation (1.14) may be regarded as a balance of the momentum of the π-electrons and represents their transport equation: The term on the lefthand side represents the electron inertia. The first term on the righthand side represents the quantum pressure arising from the zero-point energy of the π-electrons, the second term describes the action of the collective electric field, and the third one is a relaxation term due to the collisions. Note that the relaxation frequency may be expressed as

$\text{ν}=\frac{{\text{ν}}_F}{l_{mfp}}$

(1.16)

where l_mfp is the mean free path of the electrons. In conventional conductors, l_mfp is of the order of some nanometers, and therefore it is ν → ∞; hence (1.11) becomes a local relation. The mean free path of CNTs, instead, may extend up to the order of micrometers; therefore the range of nonlocality is relatively large. An approximated expression for the mean free path is provided in Section 1.3.

The expressions of parameters (1.15) generalize those currently used in literature. We have introduced a correction factor $\left(1+\text{ν}/i\text{ω}\right)$ in C_Q, taking into account the dispersive effect introduced by the losses, and the possibility to account for the proper number of channels. Assuming metallic CNT shells with small radius, it is M = 2, and so neglecting the above correction factor, from (1.15) we derive the expressions commonly adopted in literature for metallic SWCNTs (e.g., [23,24]):

$L_k=\frac{R_0}{4{\text{ν}}_F},C_Q=\frac{4}{{\text{ν}}_F{R}_0}$

(1.17)

1.2.3    TRANSPORT EQUATION FOR TWO CNTS WITH INTERSHELL TUNNELING

When considering a collection of CNT shells (for instance, several SWCNTs in a bundle or the nested shells of MWCNTs) operating at frequencies in the terahertz band, but below interband optical transitions, we must take into account the effect of intershell tunneling conductance. To include this effect, let us analyze the simple case of the double-wall carbon nanotube (DWCNT) depicted in Figure 1.5. The approach may be generalized to any MWCNT.

We assume the worst case, i.e., the case where this effect is maximized. Since the tunnel coupling is fast diminishing quantity with the increase of the difference between the shell circumferences C₁ and C₂, the worst case is given when the combinations of chiral indexes (m₁, n₁) and (m₂, n₂) minimize such a difference mentioned. In view of this, we assume both shells to be conducting and achiral.

In the considered frequency range, we account for two types of the electron motion: intraband motion within a shell and tunneling transitions between shells. It should be noted that tunneling transitions are only of importance in the regions of the degeneracy of energy spectra of different shells, which are in the neighborhood of the Fermi points.

Let us start with the boundary conditions for the electric field on the shell surfaces, expressed in the wavenumber domain as [34,35]

${\widehat{J}}_{z1}={\widehat{\text{σ}}}_{11}\left(\text{β,ω}\right){\widehat{E}}_{z1}+{\widehat{\text{σ}}}_{12}\left(\text{β,ω}\right){\widehat{E}}_{z2}{\widehat{J}}_{z2}={\widehat{\text{σ}}}_{21}\left(\text{β,ω}\right){\widehat{E}}_{z1}+{\widehat{\text{σ}}}_{22}\left(\text{β,ω}\right){\widehat{E}}_{z2}$

(1.18)

FIGURE 1.5 DWCNT above a ground plane.

where

${\widehat{\text{σ}}}_{11}\left(\text{β,ω}\right)=\frac{i{e}^2\left(\text{ω}\text{-}i\text{ν}\right)}{2\text{π}\hslash {\text{C}}_1{\text{β}}^2}\left[g\left(\text{β,ω}\right)+\text{φ}\left(\text{β,ω}\right)\right]$

(1.19)

${\widehat{\text{σ}}}_{12}\left(\text{β,ω}\right)=\frac{i{e}^2\left(\text{ω}\text{-}i\text{ν}\right)}{2\text{π}\hslash {\text{C}}_1{\text{β}}^2}\left[g\left(\text{β,ω}\right)+\text{φ}\left(\text{β,ω}\right)\right]$

(1.20)

are, respectively, the self and intershell conductivities referred to conductor 1. Expressions for ${\widehat{\text{σ}}}_{21}\left(\text{β,ω}\right)$ and ${\widehat{\text{σ}}}_{22}\left(\text{β,ω}\right)$ follow from (1.19) and (1.20) by substitutions: C₁ → C₂, m₁ → m₂, n₁ → n_2.

The functions g(β, ω) and φ(β, ω) are evaluated from the Liouville equation solution and are given by [35]

$g=\hslash {\displaystyle \sum }_{l,l^{\prime },\text{μ}}\underset{-\text{π}/T}{\overset{\text{π}/T}{{\displaystyle \int }}}\frac{F\left[E_{\text{μ}}^{\left(l\right)}\left(k+\text{β}\right)+\hslash {\text{ω}}_t^{\left(l^{\prime }\right)}\right]-F\left[E_{\text{μ}}^{\left(l\right)}\left(k\right)+\hslash {\text{ω}}_t^{\left(l^{\prime }\right)}\right]}{E_{\text{μ}}^{\left(l\right)}\left(k+\text{β}\right)-E_{\text{μ}}^{\left(1\right)}\left(\text{β}\right)-\hslash \text{ω}+i\hslash \text{ω}}dk$

(1.21)

$\text{φ}=\hslash {\displaystyle \sum _{l,l^{\prime },\text{μ}}{\displaystyle \underset{‒\text{π}/T}{\overset{\text{π}/T}{\int }}\frac{F\left[E_{\text{μ}}^{\left(l\right)}\left(k+\text{β}\right)+\hslash {\text{ω}}_t^{\left(l^{\prime }\right)}\right]-F\left[E_{\text{μ}}^{\left(l\right)}\left(k\right)+\hslash {\text{ω}}_t^{\left(l^{\prime }\right)}\right]}{E_{\text{μ}}^{\left(l\right)}\left(k+\text{β}\right)-E_{\text{μ}}^{\left(l\right)}\left(\text{β}\right)+2\hslash {\text{ω}}_t^{\left(l^{\prime }\right)}-\hslash \text{ω+i}\hslash \text{ω}}}}dk$

(1.22)

In these formulae the relaxation frequency ν is assumed to be the same for both shells, the parameters l, l ′ belong to the finite sets of symbols {−, +}, and ${\text{ω}}_t^{(\pm )}$ = ±ω_t, where ω_t is the frequency of the intershell tunneling. The summation over µ takes into account the contribution of all the subbands passing near or through the Fermi level.

Approximated expressions for (1.19) and (1.20) are obtained by using a two-term expansion of the function $E_{\text{μ}}^{\left(\text{λ}\right)}$ (k + β), and using the Dirac-like dispersion law for the accountable subbands:

${\widehat{\text{σ}}}_{11}(\text{β},\text{ω})=-i{\text{σ}}_{C1}\frac{\text{ν}}{{\text{ω}}^{\prime }(1-\widehat{y})}\left(1-\frac{{\text{ω}}_t}{2}\frac{\widehat{\text{γ}}}{\widehat{y}}\right),{\widehat{\text{σ}}}_{11}(\text{β},\text{ω})=-i{\text{σ}}_{C1}\frac{\text{ν}}{{\text{ω}}^{\prime }(1-\widehat{y})}\frac{{\text{ω}}_t}{2}\frac{\widehat{\text{γ}}}{\widehat{y}}$

(1.23)

where

$\widehat{\text{γ}}\left(\text{β,ω}\right)=\frac{2{\text{ω}}_t-{\text{ω}}^{\prime }\left(1+\widehat{y}\right)}{{\left({\text{2ω}}_t-{\text{ω}}^{\prime }\right)}^2-{\text{β}}^2{\text{ν}}_F^2}+\frac{2{\text{ω}}_t+{\text{ω}}^{\prime }\left(1+\widehat{y}\right)}{{\left(2{\text{ω}}_t+{\text{ω}}^{\prime }\right)}^2-{\text{β}}^2{\text{ν}}_F^2}$

(1.24)

$\widehat{y}\left(\text{β,ω}\right)=\frac{{\text{β}}^2{\text{ν}}_F^2}{{{\text{ω}}^{\prime }}^2},{\text{ω}}^{\prime }\text{=}\text{ω}-i\text{ν}$

(1.25)

and the coefficient σ_C1 is given by (1.12) with M = M₁ and C = C₁. For ω_t → 0 it is ${\widehat{\text{σ}}}_{12}\to 0$, whereas ${\widehat{\text{σ}}}_{11}$ reduces to (1.11).

To move to spatial domain, let us arrange (1.17) and (1.18) and (1.23),(1.24),(1.25), to obtain

$(R+i\text{ω}{L}_k)I(z,\text{ω})=\frac{1}{1+\text{ν}/i\text{ω}}\frac{1}{i\text{ω}{C}_q}\frac{{\partial }^{2}I(z,\text{ω})}{\partial z^2}+\left(1-\frac{1}{2}{\text{ω}}_t\Theta y^{-1}\gamma \right)E_z(z,\text{ω})$

(1.26)

which generalizes (1.14). Here L_k, R, and C_q are diagonal 2 × 2 matrices character-izing the shells of the DWCNT, whose nth entry is given by (1.15), using M = M_n.

The quantity Θ is the 2 × 2 matrix whose entries are given by $\Theta ={\left(-1\right)}^{i+j}$, whereas y and γ are the operator functions obtained by the substitution β → −i∂/∂z in (1.24) and (1.25). The transport of the π-electrons in the two shells is governed by Equation (1.26) coupled to the nondiagonal part of the matrix Θ.

To have an idea of the coupling, we may introduce in the wavenumber-frequency domain the ratio ζ between the mutual conductivity ${\widehat{\text{σ}}}_{12}$ and the conductivity of the inner shell obtained in the absence of the tunneling current: $\text{ς}\text{=}{\text{ω}}_t\widehat{\text{γ}}/2\widehat{y}$. This dimensionless quantity only depends on the dimensionless groups $\text{ω}/{\text{ω}}_t,{\text{λ}/\text{λ}}_t$, and ν/ω_t, where $\text{λ}=2\text{π}/\text{β}$ and ${\text{λ}}_t=\left(2\text{π}/\text{ω}\right){\text{ν}}_F$. Typical values of the parameters are $\text{ν}\approx {10}^{12}{s}^{-1},{\text{ν}}_F\approx 0.87\cdot {10}^6\text{m/s}$, and ${\text{ω}}_t\approx {10}^{13}\text{rad}/\text{s}$ [34]; therefore it is ${\text{λ}}_t\approx 600\text{nm}$ and $\text{ν}/{\text{ω}}_t\approx 0.1$

Figure 1.6 shows the module of ς versus ω/ω_t for different values of λ/λ_t and ν/ω_t = 0.1. For characteristic wavelengths around λ_t/2 the coupling between the shells due to the tunneling is important at all frequencies. For characteristic wavelengths shorter than λ_t/2 the values of $\left|\text{ζ}\right|$ drop at low frequencies and the graphs show a peak that moves toward high frequencies in the short wavelength. Also, for characteristic wavelengths higher than λ_t/2 this value drops at low frequencies, but the curves increase with a dip at $\text{ω}/{\text{ω}}_t$. In the frequency range up to terahertz the effects of the intershell tunneling currents are important when the characteristic wavelength of the electric signals is comparable to λ_t/2 or greater than λ_t. The sharp peak at λ_t/2 has an amplitude equal to 0.25 and a full width at half maximum of roughly 0.04 λ_t.

FIGURE 1.6 Intershell coupling: absolute value of the coupling ratio ς versus $\text{ω}/{\text{ω}}_t$ for $\text{ν}/{\text{ω}}_t=0.1$, with λ/λ_t ranging from (a) 0.2 to 0.5 and (b) 0.5 to 0.8.

1.3    TRANSMISSION LINE MODELS FOR CARBON NANOTUBE INTERCONNECTS

1.3.1    CNTS AS FUTURE MATERIAL FOR NANO-INTERCONNECTS

The state of the art of the literature in terms of simulations and experimental results coming from the first measurements led to convergent conclusions:

1.  Bundles of SWCNTs or MWCNTs may be effectively used to replace copper in nano-interconnect materials, as recently shown in the first examples of CNT-CMOS integration [14,15,16].

2.  Good quality CNT bundle on-chip interconnects made by this material outperform copper in terms of electrical, thermal, and mechanical performance, at least at the intermediate and global levels, whereas at the local level the behavior is comparable [11,32,33,36].

3.  For vertical vias or interconnects for packaging applications, very high-density CNT bundles must be obtained [37].

The use of CNT interconnects is therefore strongly related to the possibility of achieving a high-quality fabrication process, which must provide low-contact resistance, good direction control, and compatibility with CMOS technology.

Parallel to this effort in improving the fabrication techniques, attention has been paid in literature to deriving more and more refined models of such interconnects, able to account in a circuital environment for their peculiar behavior.

A simple model for an interconnect made by a bundle of CNTs may be obtained by coupling Maxwell equations to the CNT constitutive relation and modeling the propagation in the frame of the multiconductor transmission line (MTL) theory, as in the conceptual scheme in Figure 1.7.

FIGURE 1.7 A bundle of CNTs modeled as a multiconductor interconnect, where a perfect electric conductor (PEC) ground is the reference for the voltages: (a) longitudinal view; transverse section of (b) a SWCNTs’ bundle and (c) a MWCNTs’ bundle.

1.3.2    MULTICONDUCTOR TRANSMISSION LINE MODEL FOR CNT INTERCONNECTS WITHOUT TUNNELING EFFECT

Let us refer to Figure 1.7 and introduce the vectors of the N voltages, currents, p.u.l. electrical charge, and magnetic flux at given z, in the frequency domain: $\mathbf{V}(\text{ω},z)={[V_1(\text{ω},z)\dots .V_N(\text{ω},z)]}^T,\mathbf{I}(\text{ω},z)={[I_1(\text{ω},z)\dots I_N(\text{ω},z)]}^T,Q(\text{ω},z)={[Q_1(\text{ω},z)\dots .Q_N(\text{ω},z)]}^T,\text{and}\Phi (\text{ω},z)={\left[{\Phi }_1\left(\text{ω},z\right)\dots .{\Phi }_N\left(\text{ω},z\right)\right]}^T$.

In the quasi-TEM assumption, the propagation is governed by the TL equations:

$i\text{ω}\Phi (\text{ω},z)+\frac{\partial V(\text{ω},z)}{\partial z}=-E(\text{ω},z) , i\text{ω}Q(\text{ω},z)+\frac{\partial I(\text{ω},z)}{\partial z}=0$

(1.27)

which may be rewritten in terms of currents and voltages by imposing

$\Phi (\text{ω},z)=L_{m}I(\text{ω},z) , Q(\text{ω},z)=C_{e}V(\text{ω},z)$

(1.28)

where C_e and L_m are, respectively, the classical p.u.l. electric capacitance and magnetic inductance matrices.

Let us assume that the tunneling between the shells is negligible: This means that the low-energy band structure of each shell is the same as if it were isolated, and thus it is possible to write (1.14) for the generic nth shell:

$i\text{ω}{L}_{K,n}{I}_n(z,\text{ω})=\frac{1}{1+{\text{ν}}_n/}\frac{1}{i{\text{ωC}}_{Q,n}}\frac{{\partial }^2{I}_n(z,\text{ω})}{\partial z^2}+E_{z,n}(z,\text{ω})-R_n{I}_n(z,\text{ω})$

(1.29)

Using (1.29) to derive the longitudinal field E_z,n, (z,ω) and replacing it in (1.27) and (1.28), we get the MTL equations:

$-\frac{dV(z,\text{ω})}{dz}=\left(R+i\text{ω}L\right)I\left(z,\text{ω}\right),-\frac{dI(z,\text{ω})}{dz}=i\text{ω}CV(z,\text{ω})$

(1.30)

with the p.u.l. parameter matrices given by

$\begin{array}{cccc}L={\text{α}}_C^{-1}(L_m+L_k),& C=C_e,& R=\text{ν}{L}_k{\text{α}}_C^{-1},& {\text{α}}_c=(I+C_e{C}_q^{-1})\end{array}$

(1.31)

where I is the identity matrix, and the other matrices are given by

$\text{ν}=diag({\text{ν}}_k) , L_k=diag(L_{kk}) , C_q=diag(C_{qk})$

(1.32)

In (1.32) each entry is computed by using (1.15).

Equation (1.30) describes a lossy TL where the quantum effects are combined to the classical electrical and magnetic ones in the definition of the p.u.l. parameters.

Note that up to hundreds of gigahertz, and for typical values of the quantum capacitance for interconnects, it is usually α_C ≈ I.

Although the CNT bundle is modeled as a MTL, in practical applications any CNT bundle is used to carry a single signal; i.e., all the CNTs are fed in parallel. Therefore a CNT bundle above a ground may be described by an equivalent single TL, as in Figure 1.8.

The parameters of this equivalent single TL may be rigorously derived from the MTL model in (1.30)–(1.32), imposing the parallel condition. Alternatively, assuming the typical arrangements proposed for practical applications, we may derive an approximated expression for these parameters (e.g., [17]). First, we assume in (1.31) that a_C ≈ I. Next, since the CNT bundles intended for practical use are very dense, assuming all the CNTs in parallel, the p.u.l. capacitance of a CNT bundle C_b of external diameter D with respect to a ground plane located at a distance h from the bundle center may be approximated by the p.u.l. capacitance to the ground of a solid wire of diameter D:

$C_b=\frac{2\text{πε}}{\text{ ln }\left(2h/D\right)}$

(1.33)

As for the p.u.l. inductance of the equivalent single TL, the magnetic component may be computed from the vacuum space electrostatic capacitance, since it is $L_m={\text{μ}}_0{\text{ε}}_0{C}_{e0}^{-1}$, where C_e0 may be computed exactly or may be approximated by (1.33). The kinetic inductance L_kb of a bundle of N CNTs may be simply given by the parallel of the N kinetic inductance L_kn associated to any single CNT. Recalling (1.15), it is

$L_{kb}={\left[{\displaystyle \sum }_{n=1}^N\frac{1}{L_{kn}}\right]}^{-1}={\left[{\displaystyle \sum }_{n=1}^N\frac{2{\text{ν}}_F{M}_n}{R_0}\right]}^{-1}$

(1.34)

where M_n is the number of equivalent channels of the nth CNT (see Figure 1.4).

Assuming, for instance, the bundle to be made by N SWCNTs, with one-third metallic, it is

$L_{kb}=\frac{1}{N}\frac{3R_0}{4{\text{ν}}_F}$

(1.35)

Finally, the p.u.l. resistance of a single CNT may be found using (1.15) and (1.16):

$R_n=\text{ν}{L}_{Kn}=\frac{\text{ν}{R}_0}{2v_F{M}_n}=\frac{R_0}{2l_{mfp}{M}_n}$

(1.36)

A complete model for l_mfp must include all the scattering mechanisms (defect, acoustic, optical, and zone boundary phonons). However, for temperatures 300 K < T < 600 K and longitudinal electric field E_z < 0.54 V/µm, an accurate fitting of experimental data provides [30]

$\frac{1}{l_{mfp}}\approx \frac{2}{D}\left[k_1+k_{2}T+k_3{T}^2\right]$

(1.37)

where T and D are the CNT temperature and diameter, respectively, and the fitting coefficients k₁ = 3.005 ⋅ 10⁻³, k₂ = −2.122 ⋅ 10⁻⁵K⁻¹, and k₃ = 4.701 ⋅ 10⁻⁸K⁻²

The resistance of a single CNT shell for varying temperature values derived from the above model is compared in Figure 1.9 to available experimental data.

FIGURE 1.8 (a) A CNT interconnect above a PEC ground. (b) Equivalent circuit: elementary cell (inset) and lumped contact resistances.

FIGURE 1.9 Resistance of a CNT shell versus temperature: fitting model compared to experimental data in [38] and [39].

The resistance of N carbon nanotube shells is simply given by the parallel of N resistances (36).

Note that beside the p.u.l. resistance, any CNT shell introduces a lumped contact resistance R_p (see Figure 1.8b). This resistance is given by the parallel of the contact resistance of each single conducting channel R₀ + R_par depends on the quality of contacts and tends to zero for ideal contacts. Even in this case, the contact resistance would not vanish, because of the presence of the quantistic term R₀.

1.3.3    TRANSMISSION LINE MODEL FOR CNT INTERCONNECTS WITH TUNNELING EFFECT

The transmission line model (30) is modified if we have to take into account the intershell tunneling currents. Let us refer to the double-wall CNT in Figure 1.5 and the results presented in Section 1.2 for this structure. By using the modified transport equation (1.26) instead of (1.14), we obtain a MTL model, which is formally described by the equations in (1.27), provided that a new definition for the p.u.l. parameters is introduced [34]:

$L=L_m+L_k+L_{tun}, C^{-1}=C_e^{-1}+C_{tun}^{-1}, R=v(L_k+L_{tun})$

(1.38)

having assumed again a_C ≈ I. The inductance and capacitance matrices are corrected by two new terms, which are the following operators:

$L=L_m+L_k+L_{tun}$

$L_{tun}=\frac{{\text{ω}}_t{\text{ω}}^{\prime }}{2v_F^3\text{ω}}\Theta C_e^{-1}\underset{-\infty }{\overset{+\infty }{{\displaystyle \int }}}{K}_1(z-z^{\prime };{\text{ω}}^{\prime })\{.\}d{z}^{\prime }$

(1.39)

$C_{tun}^{-1}=-\frac{{\text{ω}}_t{\text{ω}}^{\prime }}{2v_F}\Theta C_e^{-1}\underset{-\infty }{\overset{+\infty }{{\displaystyle \int }}}{K}_2(z-z^{\prime };{\text{ω}}^{\prime })\{.\}d{z}^{\prime }$

(1.40)

The kernels of these operators are given by

$K_1(z;{\text{ω}}^{\prime })=\text{ sin }\left(\frac{2{\text{ω}}_t}{v_F}\left|z\right|\right)\text{ exp }\left(-i\frac{{\text{ω}}^{\prime }}{v_F}\left|z\right|\right)$

(1.41)

$K_2(z;{\text{ω}}^{\prime })=\frac{1}{2i}\text{ exp }\left(-i\frac{{\text{ω}}^{\prime }}{v_F}\left|z\right|\right)\times \left[\frac{1}{2{\text{ω}}_t-{\text{ω}}^{\prime }}\text{ exp }\left(i\frac{2{\text{ω}}_t}{v_F}\left|z\right|\right)+\frac{1}{2{\text{ω}}_t+{\text{ω}}^{\prime }}\text{ exp\hspace{0.17em}}\left(-i\frac{2{\text{ω}}_t}{v_F}\left|z\right|\right)\right]$

(1.42)

The presence of these operators changes the mathematical nature of the model, which moves from a differential model to an integro-differential one. From a physical point of view, the operators L_tun and C_tun introduce a spatial dispersion. It is possible to show that in the terahertz range the norm of L_tun may rise up to 60% of the total norm of L for CNT lengths greater than 0.5 µm [34]. It is also possible to show that in the same conditions the effect of C_tun may be neglected, and hence we can again assume C = C_e in (1.38).

1.4    STUDY OF SIGNAL PROPAGATION PROPERTIES ALONG CNT INTERCONNECTS

A realistic application for CNT interconnects is given by the so-called on-chip interconnects, i.e., the electrical routes inside the different layers that compose a chip. These interconnects route the signal from the inner layers at contact with the transistor (local level) to the package pins that are connected to the outside (global level). In the next year the typical dimensions of the transistor gate will decrease to few tens of nanometers [4]: For such dimensions it is not possible to scale down the conventional interconnects, since materials like copper will exhibit dramatic loss of performance, mainly due to the steep increase of resistivity. This is the reason why carbon nanotubes are considered “the ideal interconnect technology for next-generation ICs” [36]. In this section we analyze typical arrangements for on-chip interconnects, modeled with the TL model presented in Section 1.3.

1.4.1    PERFORMANCE OF A CNT ON-CHIP INTERCONNECT

Let us consider the on-chip interconnect in Figure 1.10, which shows a stripline configuration. The signal trace is assumed to be constituted by a solid Cu conductor, by a SWCNT, or by a MWCNT bundle. The driver is modeled by a voltage source with a series resistance R_dr and a parallel capacitance C_out. The terminal receiver is a load capacitor C_L.

The electrical and geometrical parameters are given in Table 1.1 and refer to a global level interconnect for the 22 nm technology node [4].

The copper realization has been studied by assuming for copper resistivity the value reported in Table 1.1, which is slightly higher than the usual value for bulk copper, due to electromigration phenomena [40]. As for the SWCNT realization, we assume that the line section is filled by SWCNTs of diameters of 1 and 2 nm, with a filling factor of 80%. To simulate realistic conditions, only one-third of the CNT population is metallic. In addition, a parasitic lumped resistance of R_par = 50 kΩ for each CNT channel.

FIGURE 1.10 On-chip interconnect: (a) cross and (b) circuit.

TABLE 1.1
ITRS Values of the Parameters for a Global Level On-Chip Interconnect at the 22 nm Technology Node

Source:  ITRS, International Technology Roadmap for Semiconductors, 2009, http://public.itrs.net.

FIGURE 1.11 Per unit length resistance of a global level interconnect, normalized to copper value, versus line length: Bundles of MWCNTs and SWCNTs with different diameters are compared.

Finally, for the MWCNT realization we analyze two possible arrangements, corresponding to the values of external diameter D_out = [20; 40] nm, assuming that the inner diameter is D_in = 0.5D_out and that the shells are separated by the van der Waals distance of 0.34 nm.

Let us first assume the line temperature to be constant at a fixed room temperature T = T₀ = 293 K.

Figure 1.11 shows the computed line per unit length resistance, normalized to the value obtained with the copper realization. Due to the influence of the contact resistance, all the CNT realizations show worse performance with respect to the Cu one for shorter lengths. For lengths greater than 10 µm, the MWCNT solutions are always better than the Cu ones, and the performance is even better as length increases. The SWCNT solution may be better or worse than Cu, depending on the chosen CNT diameter (which affects, of course, the total number of the CNTs in the bundle). In this case, an optimal solution would be the use of 1 nm diameter SWCNTs. Note that typical lengths for global level interconnects are of the order of 0.1 mm.

FIGURE 1.12 Resistance of a global level interconnect of length 0.1 mm versus temperature: Bundles of MWCNTs and SWCNTs with different diameters are compared.

Let us now analyze the sensibility of the resistance to the temperature variation, which is an important issue for on-chip interconnects, given their major problem of Joule heating. A simple temperature-dependent model for copper resistivity is given by

$\text{ρ}\left(T\right)={\text{ρ}}_0\left[1+{\text{α}}_0\left(T-T_0\right)\right]$

(1.43)

where ρ₀ is the value given in Table 1.1 and the coefficient α₀ = 2.65 ⋅ 10⁻³.

As for the CNT realization, the increase of temperature affects the number of conducting channels and the mean free path, as pointed out in Section 1.2. Figure 1.12 shows the behavior of the resistance versus temperature, assuming a line length of 0.1 mm. The SWCNT solutions are the most sensitive to temperature increase, whereas the MWCNT ones are the more stable, the larger the CNT diameter used.

Let us now investigate the so-called signal integrity (SI) performance of the considered channel (Figure 1.10b), assuming for the chip interconnect an operating temperature of 400 K and a length of 0.1 mm. Let us consider the MWCNT case with a 40 nm diameter and the SWCNT one with a 1 nm diameter. The computed values of the line parameters are reported in Table 1.2.

Figures 1.13 and 1.14 show the eye diagram obtained by considering a transmission of a digital signal at a rate of 50 and 100 Gbaud/s. The three channels show similar performances at 20 Gbaud/s: Each channel shows an excellent response to this high-frequency digital signal. The MWCNT realization exhibits a sort of ringing due to the presence of a higher inductance (mainly related to the kinetic term). The three channels may also be used at 50 Gbaud/s, without significant degradation of the performance, unless there is an increase of the jitter for the MWCNT case.

TABLE 1.2
Computed RLC Parameters for the Global Level On-Chip Interconnect with Length 0.1 mm, for Cu, SWCNT, and MWCNT Realizations

	R [kΩ]	C [fF]	L [pH]
Cu	0.24	0.32	0.76
SWCNT	0.25	0.32	0.83
MWCNT	0.09	0.32	1.47

FIGURE 1.13 Eye diagram for the transmission of a digital signal at 20 Gbaud/s rate: (a) Cu, (b) SWCNT, and (c) MWCNT realization.

FIGURE 1.14 Eye diagram for the transmission of a digital signal at 50 Gbaud/s rate: (a) Cu, (b) SWCNT, and (c) MWCNT realization.

1.5    CONCLUSIONS

In this chapter the electrodynamics of the conduction electrons of carbon nanotubes (CNTs) have been described, and models are derived to study the propagation of signals along CNT interconnects. The conduction electrons are seen as a fluid moving under the influence of the collective field and of the interaction with the fixed ion lattice. The transport equation is derived in the quasi-classical limit, allowing the formulation of a frequency domain constitutive equation for the CNTs, in terms of a nonlocal Ohm’s law. The CNT conductivity is strongly influenced by two parameters, which account for the electron inertia and for the quantum pressure. The conductivity may be expressed in terms of the effective number of conducting channels, a parameter that counts the number of subbands significantly involved in the conduction.

The model is then extended to the case of interacting CNT shells, where the tunneling effect is considered. Self and mutual conductivities are then defined to account for this phenomenon.

By coupling the above CNT constitutive equations to Maxwell equations, a transmission line model is derived, able to describe the propagation of signals along CNT bundles, in terms of simple RLC distributed lines. The intershell tunneling occurring at terahertz is then accounted for, with a suitable recasting of the transmission line equations, which now include spatial dispersion terms.

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