Wong H.-S.P., Akinwande D. Carbon Nanotube and Graphene Device Physics

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6.9 Kinetic inductance of metallic CNTs 149

inductance, it is enlightening to recall what the classical magnetic inductance

represents. From basic electromagnetics we understand that the motion of

charges or current produces a magnetic ﬁeld which has the ability to do work. The

magnetic energy E

of the magnetic ﬁeld serves to deﬁne a magnetic inductance

according to



allspace

dV =

, (6.51)

where H is the magnetic ﬁeld intensity, µ

is the material permeability, and dV is

the differential volume element. The magnetic inductance reﬂects the geometry of

the conductor and can be seen as a conductor property that models the magnetic

energy, a particularly useful property in circuits. However, the magnetic energy

is not the only energy that arises from the motion of charges or current. Another

manifest energy is the kinetic energy E

of the electrons, which is essentially where

the kinetic inductance originates according to the deﬁnition

, (6.52)

where the kinetic energy can be computed either from Newtonian classical mechan-

ics or quantum mechanics. Indeed, the kinetic inductance can be seen in the earliest

modern theory of conduction in metals by Drude,

where the complex impedance

has an imaginary term proportional to frequency, reﬂecting an inductance from the

kinetic energy.The total energy arising from the motion of charges is the sum of the

magnetic and kinetic energies, which means that in circuits the kinetic inductance

adds in series to the magnetic inductance.

In normal conductors, electrons scatter repeatedly, resulting in dissipation of the

kinetic energy. Invariably, whatever kinetic inductance that exists is vanishingly

small. In normal metals, therefore, we can conclude that charge transport is of an

ohmic nature with the resistance typically dominating the reactance up to optical

frequencies. This explains why the kinetic inductance is often not considered in

normal electronics. In superconductors, electrons can travel with no loss of their

kinetic energy, leading to an appreciable kinetic inductance which is employed in a

variety of superconducting devices.

In this respect, CNTs are interesting because,

while they are not superconductors, they do, however, possess an appreciable

kinetic inductance compared with normal bulk metals due to the longer electron

For the purpose of refreshing our understanding of L

, we have recalled the magnetic energy

deﬁnition of a homogeneous conductor whose material properties are independent of the ﬁeld or

current. The idea of L

applies to complex conductors as well.

An excellent discussion of Drude’s theory of conduction can be seen in, N. W. Ashcroft and N. D.

Mermin Solid State Physics (Brooks/Cole, 1976) Chapter 1. The original theory was published (in

German) by P. Drude in 1900.

For example, a search of kinetic inductance device on www.google.com reveals thousands of hits

where L

is utilized in devices ranging from thermometers to resonators to sensors for dark matter.

150 Chapter 6 Ideal quantum electrical properties

∆k

∆µ

Fig. 6.8 The coupling of energy (via an applied voltage) into a nanotube is akin to the promotion

of left-moving carriers into right-moving carriers (gray segment indicated by arrow). The

energy stored µ by the net carriers (occupying states within k) directly manifests as

thekineticinductance.AdaptedfromM.Bockrath.

mean free path. As a result, it is expected that the kinetic inductance in CNTs will

be accessible at more moderate frequencies of the order of ∼100 GHz, which is

potentially useful for high-frequency electronics.

The kinetic inductance of a ballistic metallic CNT can be derived from the net

kinetic energy of thecarriers that are responsiblefor the current ﬂow (seeFigure 6.8

for a simpliﬁed illustration). Assuming the states at the source and drain reservoirs

are populated up to µ

and µ

respectively, the net total kinetic energy is



µ

EdE =

µ

, (6.53)

where g

/2 represents the carriers that provide current and µ, as mentioned

earlier, is the difference in chemical potentials of the source and drain reservoirs,

reﬂecting the energy coupled into the system by the external potential.

The kinetic inductance is determined by equatingthe kinetic energy to the energy

stored in an inductor:

µ

≡

. (6.54)

Substituting the current given by Eq. (6.9) and replacing the DOS with the velocity

from Eq. (6.7) leads to

µ



µ



. (6.55)

It follows that the kinetic inductance is

. (6.56)

M. W. Bockrath, Carbon nanotubes: electrons in one dimension. Ph.D. dissertation, University of

California, Berkeley, CA, 2002.

6.10 An energy-based derivation of conductance 151

With v

∼ 10

cm s

−1

, the kinetic inductance is L

∼ 3.2 nH µm

−1

(includ-

ing spin and ﬁrst subband degeneracy), corroborating initial calculations by

Bockrath.

This value of the kinetic inductance of metallic CNTs is notably very

signiﬁcant when compared with the magnetic inductance of a metallic wire over

a ground plane, which is typically orders of magnitude lower.

This implies that

the kinetic inductance is potentially useful for high-frequency nanotube circuits.

The quantitative (voltage-dependent) value of the kinetic inductance of semicon-

ducting CNTs remains an open question of intellectual and practical interest. In

CNT transistor modeling and analysis, L

is often taken to be of the same order as

the kinetic inductance of metallic nanotubes in order to obtain a rough estimate of

the impact of the kinetic inductance on charge transport in the nanotube transistor.

InChapter7,whichfocusonmetallicinterconnects,wewillexploretheimpact

of the kinetic inductance in the transmission line model of CNTs. Additionally,

the symmetry that exists between the kinetic inductance and quantum capaci-

tance, such as their ratio and their products, which relate to the characteristic

impedance and group velocity in the transmission line model, will also be explored

inChapter 7.

6.10 From Planck to quantum conductance: an energy-based

derivation of conductance

The derivation of the quantum conductance in Section 6.2 is based on the DOS,

but, quite remarkably, the quantum conductance does not depend on the DOS at

all. This lack of dependence on the DOS suggests that there might exist a more

fundamental method to derive the quantum conductance from basic principles

which does not consider the DOS a priori. One line of approach is to start from

the most basic postulate in quantum mechanics proposed by Max Planck:

E = hf , (6.57)

where f is the particle-wave frequency. Since we are more concerned with the ﬂow

of electrons per unit period, it is more attractive to rewrite the postulate in terms

of electron ﬂux:

E = hf =

, (6.58)

where τ is the electron period and I

is the current of electrons (including spin) that

have an energy E. This current does not take into account the energy degeneracy

Max Planck originally proposed the postulate to explain blackbody radiation. Albert Einstein later

extended it to explain the photoelectric effect, and Louis de Broglie subsequently generalized the

postulate as the basis of his theory of matter waves. All the three ideas based on this single

postulate were priceless and, as such, all three pioneers were awarded Nobel Prizes seperately.

152 Chapter 6 Ideal quantum electrical properties

that exists at energy E; as such, it is the current per mode:

E. (6.59)

The net current in a 1D channel connected to reservoir contacts is determined by

the difference in the chemical potentials of the reservoirs. Assuming all the states

are populated up to µ

at the source reservoir and µ

at the drain reservoir, with

− µ

= eV . The net current I in the channel becomes

I =

(µ

− µ

) =

V (6.60)

and the quantum conductance per mode falls out naturally as

. (6.61)

Similarly, the quantum capacitance of metallic CNTs can also be derived in a

straightforward manner from quantum conductance. Let us proceed as follows:

the time-varying current in a capacitor is given by the usual displacement current

relation:

I(t) =

dQ(t)

= v

dQ(t)

, (6.62)

where Q is the total charge in the capacitor and dQ/dl is simply the charge den-

sity equal to the product of the capacitance per unit length and the time-varying

potential difference V (t):

I(t) = v

V (t). (6.63)

Therefore, the quantum capacitance per unit length per mode is

I(t)

V (t)

, (6.64)

which has the same form as Eq. (6.41). It is noteworthy that the quantum capaci-

tance so derived for the metallic nanotube can be extended to the semiconducting

nanotube by replacing v

with the thermally averaged particle velocity. This is a

worthwhile exercise for the interested reader seeking to gain more depth.

Likewise, the kinetic inductance of metallic CNT can be derived in a straight-

forward manner using the quantum conductance:

V (t) =

dφ(t)

= v

dφ(t)

, (6.65)

6.11 Summary 153

where dφ/dl is the equivalent ﬂux density due to the kinetic inductance and can

be expressed as the product of the current and the kinetic inductance per unit

length:

V (t) = v

I(t). (6.66)

Therefore, the kinetic inductance per unit length per mode is

V (t)

I(t)

. (6.67)

The successful derivation of the quantum capacitance and kinetic inductance of

CNTs from circuit I−V relations and the quantum conductance suggests that G

not only applies to static current voltage relations, but also applies to alternating

current voltage relations.

Noteworthy is that Planck’s postulate does not exclusively apply to electrical

conductance. Indeed, it can be shown to be useful in deriving both the thermal

and spin quantum conductances as well.

Moreover, we will employ Planck’s

postulate (6.58) to derive the ballistic current voltage relationship of a CNT ﬁeld-

effecttr ansistor inChapter 8.

6.11 Summary

This chapter has explored the electrical properties of quantum mechanical effects

in CNTs and graphene. These effects manifest as quantum conductance, quantum

capacitance, and the kinetic inductance. We have derived analytical expressions

for these electrical properties in the quasi-static limit. Moreover, these electri-

cal models for the quantum mechanical effects utilize the independent-electron

or one-electron approximations for understanding the properties of electrons in

crystalline solid matter. At low temperatures and under other certain conditions

the consequences of electron–electron interactions can become substantial, natu-

rally leading to signiﬁcant departures from the current electrical models. To ﬁrst

order, these effects can be captured in terms of a renormalized velocity.

This

The experimental measurement of the quantum thermal conductance was reported by K. Schwab,

E. A. Heriksen, J. M. Worlock and M. L. Roukes, Measurement of the quantum of thermal

conductance. Nature, 404 (2000) 974–7. For theoretical prediction of the spin quantum

conductance, see F. Meier and D. Loss, Magnetization transport and quantized spin conductance.

Phys. Rev. Lett., 90 (2003) 16724.

For example, see I. Saﬁ, Conductance of a quantum wire: Landauer’s approach versus the Kubo

formula. Phys. Rev. B, 55, (1997) R7331–4; G. Cuniberti, M. Sassetti and B. Kramer, ac

conductance of a quantum wire with electron-electron interactions. Phys. Rev. B, 57 (1998)

1515–26; and V. A. Sablikov and B. S. Shchamkhalova, Electron transport in a quantum wire with

realistic Coulomb interaction. Phys. Rev. B, 58 (1998) 13847–55.

154 Chapter 6 Ideal quantum electrical properties

renormalized velocity can be used in place of the Fermi velocity, though exper-

imental corroborations of the renormalization idea are matters of contemporary

research.

Similarly, our treatment of the MWCNTs implicitly assumes that the shells are

non-interacting; that is, electrons in each shell can be treated as independent elec-

trons oblivious of the presence of electrons in the surrounding shells. While this

assumption is expected to hold in the majority of cases, it is always possible to

devise conditions that will violate this assumption and might lead to interesting

physics.

6.12 Problem set

All the problems are intended to exercise and reﬁne analytical techniques while

providing important insights. If a particular problem is not clear, the reader is

encouraged to re-study the appropriate sections. Invariably, some problems will

involve making reasonable approximations or assumptions beyond what is already

stated in the speciﬁc question in order to obtain a ﬁnal answer. The reader should

not consider this frustrating, because this is obviously how problems are solved in

the real world.

6.1. Number of channels for metallic single-wall nanotubes.

The current and conductance of a ballistic single-wall nanotube was derived

in the text assuming low-energy transport where only the lowest subbands

contribute appreciable to current. Let us go one step further and actually derive

the quantum conductanceincluding the contribution of higher subbands while

still retaining the low-energy approximation.

(a) Derive the number of channels and quantum conductance for an armchair

single-wall nanotube including the contributions of higher subbands.

(b) For an armchair nanotube with diameters of 1 and 2 nm, how much

larger is the quantum conductance from part (a) normalized to 2G

?Is

the contribution of the higher subbands appreciable or negligible at room

temperature?

(d) Likewise, derive the quantum capacitance of metallic single-wall nan-

otubes including the contribution of higher subbands.

6.2. Voltage dependence of MWCNT conductance.

The number of channels for MWCNTs was derived in the text in the limit of

small voltages (V → 0V). This was convenient and resulted in an algebraic

expression for the number of channels for a given multi-wall nanotube that

can be used to estimate the CNT conductance. However, also of interest is the

6.12 Problem set 155

voltage dependence of the conductance. Note that the voltage dependence is

what is important here, not the ﬁeld dependence, since the nanotube is in the

ballistic mode. This exercise will explore this theme in order to obtain some

basic insights.

(a) For a model multi-wall nanotube containing all armchair shells and an

inner and outer diameter of 5 nm and 50 nm respectively, plot the number

of channels as a function of voltage from 0 V to 5 V at 77 K and at

300 K.

(b) Repeat part (a) for a multi-wall nanotube with an inner and outer diameter

of 5 nm and 100 nm respectively.

ing the voltage dependence of the conductance of ballistic multi-wall

nanotubes?

6.3. Temperature dependence of MWCNT conductance.

In the limit of small voltages, determine the temperature dependence of the

quantum conductance of an ideal multi-wall nanotube from 4 K to 500 K. Plot

the temperature dependence for a nanotube with an inner and outer diameter

of 10 nm and 50 nm respectively, for the two following conditions:

(a) All the shells of the multi-wall nanotubes make transparent contacts to

the electrodes.

(b) Only the outermost shells make transparent contacts to the electrodes.The

remaining shells make no electrical contact at all with the electrodes.

conditions.

6.4. Non-degenerate quantum capacitance of CNTs.

(a) Show that the non-degenerate quantum capacitance is given by Eq. (6.49).

(b) Plot the error of the non-degenerate quantum capacitance compared

with the two-subband analytical quantum capacitance for E

= 0to

= E

/2.

diameter, and explain this scaling trend.

6.5. Quantum capacitance of graphene.

(a) Compare the quantum capacitance of graphene (at room temperature)

with its electrostatic parallel-plate capacitance for insulator thickness

ranging from 5 nm to 1 µm, and a dielectric constant of 4. This comparison

can be obtained by superimposing the min/max electrostatic capacitances

on the C

versus µ plot. Comment on this comparison.

156 Chapter 6 Ideal quantum electrical properties

(b) Also compare the quantum capacitance of graphene with that of an array

of identical metallic single-wall nanotubes. The array can be modeled to

consist of 1 nm diameter CNTs with a density of 200 nanotubes/µm. We

want to determine which carbon nanomaterial offers the highest intrinsic

quantum capacitance. In other words, does the quantum capacitance of

an array of parallel single-wall nanotubes approach that of graphene, in

the limit of a sheet of CNTs?

7 Carbon nanotube interconnects

If you build a ‘better’ mousetrap, you’d better know what the existing

mousetrap can do.

7.1 Introduction

This chapter explores electron transport in metallic nanotubes as it relates to inter-

connect applications. Both single-wall and multi-wall CNTs are considered. The

r eader willﬁnditbeneﬁcialtobefamiliar withChapter 4,whichdiscussesthe

structureofnanotubes,andChapter6,whichexploresidealnanotubeelectrical

properties, such as the quantum conductance, quantum capacitance, and the kinetic

inductance. Employing CNTs as metallic wires to route direct-current (DC) and

high-speed signals in an integrated circuit was one of the earliest application ideas

promoting nanotubes because of their high current-carrying capability and bal-

listic transport over relatively long lengths. In this light, we will examine both

the low-frequency (lossy) and high-frequency (lossless) transmission line models

for single-wall and multi-wall nanotubes in order to elucidate their interconnect

properties. These models include bias or ﬁeld-dependent electron scattering in the

nanotube vis-á-vis the mean free path. As such, electron scattering and mean free

path will be discussed, although at a somewhat elementary level.

Additionally, the

temperature and diameter dependence of the electron mean free path and resistance

will be highlighted.

At the end of the day, future nanomaterials suchas CNTshave to be benchmarked

against existing materials to quantify any performance beneﬁts over conventional

approaches. For this purpose, we will discuss the performance of CNTs compared

with copper, which is the standardmetal used in nanoscale integratedcircuits today.

From the comparison it will be clear that an individual CNT is not competitive

against copper. Instead, a dense or tightly packed array of nanotubes is essential for

practical interconnect applications. We will conclude the discussion in this chapter

by summarizing some of the practical challenges that must be resolved for CNT

interconnect wires to be realized in an integrated circuit.

An advanced treatment of electron scattering in solids can be found in M. Lundstrom,

Fundamentals of Carrier Transport, 2nd edn. (Cambridge University Press, 2000).

158 Chapter 7 Carbon nanotube interconnects

7.2 Electron scattering and lattice vibrations

Electron transport in a conductor can be categorized into two differentregimes. One

is the ballistic regime, discussed in Chapter 6, and the other is the diffusive regime,

where electrons experience repeated scattering during transport. In describing the

transport of electrons in a metal wire or interconnect, one has to consider both

the ballistic and diffusive regimes in order to obtain a comprehensive picture of

interconnect resistance over the typical length scales involved. This section will

elucidate on the sources of scattering in CNTs, and Section 7.3 will quantify the

scattering in terms of the widely used metric known as the mean free path. In a

sense, scattering provides an opportunity for electrons that have gained energyfrom

an applied ﬁeld to lose that energy and return to thermalequilibrium with the lattice.

Just as there are several methods by which a moving vehicle loses speed on a

highway (road bumps, surface roughness, pot holes, stop signs, crowded lanes,

stray animals, etc.), there are also several means by which an electron can experi-

ence a scattering event (and lose its speed) in a CNT including the following:

(i) Lattice vibrations: The atoms in a solid are constantly vibrating about their

mean position with increased displacement as a function of temperature.

These vibrations are quantized waves and are otherwise known as phonons.

Interaction of the mobile electrons and phonons is a major source of scattering.

(ii) Lattice defects: Defects or disorder in the regularity of the carbon atoms can

lead to scattering of a mobile electron. Examples of typical defects include

missing carbon atoms in the nanotube (known as vacancies) and the presence

of extra atoms within the nanotube (interstitials).

(iii) Electron–electron interactions: The Coulombic repulsion between electrons

results in deﬂection of the electrons. We will not consider this form of

scattering in this text because at moderate temperatures (including room

temperature) transport in CNTs has been described successfully within the

independent electron formalism.

(iv) Substrate roughness: Nanotubes supported on a substrate can be disturbed by

the roughness of the substrate surface, potentially resulting in the scattering

of mobile electrons.

(v) Electron–substrate interactions: Even on a substrate with an atomically

smooth surface, electrons traveling in the nanotube can be deﬂected by the

electric ﬁeld originating from exposed charges present on the surface. This is

particularly pertinent on polar substrates, such as silicon dioxide (SiO

(vi) Electron–superstrate interactions: Similarly, exposed or unshielded charges

in the superstrate

can deﬂect mobile electrons. Even in the absence of a

superstrate, polar molecules in the surrounding ambient (e.g., H

O) may

adsorb onto the nanotube and lead to scattering of electrons.

A superstrate is the counterpart of the substrate, and can represent an insulating ﬁlm covering the

nanotube.