Carbon-Based Nanoelectromagnetics

By Antonio Maffucci

Carbon-Based Nanoelectromagnetics provides detailed insights into the electromagnetic interactions of carbon-based nanostructured materials such as graphene and carbon nanotubes. Chapters within the book offer a comprehensive overview on this discipline, starting with an introduction to the field-matter interaction, its features, and finally, its applications in microwave, THz and optical frequency ranges. Electromagnetics at the nanoscale level has become a major research area in recent years as the synthesis of a variety of carbon-based nanostructures has progressed dramatically, thus opening the era of nanoelectronics and nanophotonics.

To meet the challenges of these new fields, a thorough knowledge is required of the peculiar properties of the electromagnetic field. The novel behavior of the electromagnetic fields interacting with nano-sized elements and nano-structured has motivated the birth of this new research discipline, ‘Nanoelectromagnetics’.

• Presents a one-stop resource that explores the emerging field of nanoelectromagnetics
• Focuses on modeling, simulation, analysis, design and characterization, with an emphasis on applications of nanoelectromagnetics
• Explores the optical properties and applications of a range of carbon-based nanomaterials

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 8, 2019
• ISBN: 9780081023945

Book preview

Finland

Preface

Antonio Maffucci, Department of Electrical and Information Engineering, University of Cassino and Southern Lazio, Cassino, Italy

Sergey Maksimenko, Institute for Nuclear Problems, Belarusian State University, Minsk, Belarus

Yuri Svirko, Institute of Photonics, University of Eastern Finland, Joensuu, Finland

Nanoelectromagnetism has recently emerged as the branch of applied sciences that studies electromagnetic interactions in nanostructures of reduced dimensionality. The ever-growing interest to nanoelectromagnetism is fueled by the astonishing progress of nanotechnology in the last decades that nowadays allows us to employ semiconductor heterostructures, quantum dots, nanostructured carbon, noble metal nanowires, and organic macromolecules in electronic and photonic devices.

In addition, the challenges imposed by the technology scaling from micro- to nanoscale inevitably require the development of new materials that enable exploring and exploiting the novel physical phenomena at nanoscale (such as spatial confinement of the charge carrier motion and sequential quantum mechanical effects). Among the most promising nanostructures, carbon allotropes (e.g., graphene, carbon nanotubes, and fullerenes) have sparked a lot of interest due to their outstanding physical properties. There is a common belief that the advent of carbon is indeed opening a new era in electronics and photonics.

This book provides a comprehensive overview of carbon-based nanoelectromagnetics, including the fundamentals of the interactions between electromagnetic field and carbon nanostructures; the features of the electromagnetic fields at nanoscale; and the latest applications of carbon nanomaterials in the microwave, terahertz, and optical frequency ranges. The book covers theoretical modeling and experimental characterization, as well as design and synthesis of materials and devices.

This interdisciplinary book addresses specific know-how and expertise in the fields of fundamental and applied electromagnetics, chemistry and technology of nanostructures and nanocomposites, and physics of nanostructured systems. The authors who contributed chapters to this book are worldwide recognized experts in such fields, and therefore, we are extremely thankful to them for accepting our invitation.

We hope that the readers will find this book useful in their pursuits of carbon nanotechnology and nanoelectromagnetism.

The Editors

Chapter 1

Electrodynamics of carbon nanotubes

Sergey Maksimenko⁎; Mikhail V. Shuba⁎; Gregory Y. Slepyan†    ⁎ Institute for Nuclear Problems, Belarusian State University, Minsk, Belarus

† Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel

Abstract

A theoretical concept on electrodynamics of carbon nanotubes (CNTs) based on the effective two-side impedance boundary conditions for electromagnetic field is presented. The effective tensor of sheet impedance is introduced as a phenomenological parameter for single- and double-walled CNTs basing on the quantum models of charge carrier transport. Two types of boundary value problems are formulated for CNT electrodynamics: (i) eigenmodes of CNTs and (ii) CNT excitation by the external field. The second type of the problems takes into account the finite-length effects basing on the integral equation technique. These problems have been applied to the surface plasmon propagation and antenna resonances. In particular, it gives an interpretation of the terahertz conductivity peak, which physical origin has been debated by different research groups for a long time. The generalization to the quantum optical problems based on the concept of the surface noise current is given. It is applied to analysis of Purcell effect and thermal radiation in CNT.

Keywords

Carbon nanotube; Electromagnetic waves; Terahertz frequency range; Effective boundary conditions; Surface plasmon; Integral equation technique; Geometric resonance; Surface noise current; Purcell effect; Thermal radiation

1.1 Introduction

Over the past decades, there has been significant progress in the synthesis and fabrication of different nanosized objects and nanostructured artificial materials with peculiar mechanical, electronic, and optical properties irreducible to properties of bulk media. Among others, different forms of nanocarbon—graphene, carbon nanotubes, fullerenes, carbon black, etc.—are of special interest as a basis for future progress in solving of two engineering and technological tasks:

−Miniaturization of electric circuit elements and interelement distances in integrated circuits to values of 5–10 nm, which allows achieving the integration level of the order of tens of billions elements on a chip

−Active development and exploitation of the terahertz frequency range (300 GHz–3 THz) aiming with the orders of magnitude more capacity and data transfer rate in wireless communication technology, design of security control systems for airports and mass events, progress in spectroscopic method of the study of complex organic molecules, and biological tissues.

Owing to their fascinating physical properties [1,2], carbon nanotubes (CNTs) and graphene are considered both as building blocks of nanoelectronics (nanosized diodes and transistors [3–8], filters [9], transmission lines and antennae [10–11], amplifiers and generators of electromagnetic signals [12–17], etc.) and as source components for the design of macroscopic devices and functional materials for the control of electromagnetic fields in upper giga- and terahertz frequency ranges [18–21] and in optics [22].

On the way of wide application of carbon nanostructures in electronics, there are a variety of problems related to the quality of synthesized particles, reproducibility of processes and tenability properties, etc. Different methods are in quest to synthesize high-quality CNTs in bulk; different routes have been developed and established over the last two decades. Arc discharge, laser ablation, and catalyst-enhanced chemical vapor deposition are the main three types of technology for CNT synthesis suitable for different applications in nanoelectronics and nanophotonics. Each of these techniques has some advantages and disadvantages, which are discussed in the large number of recent reviews (e.g., [23–26]). However, under optimized conditions, these processes produce the reproducible CNTs with the highest achievable crystallinity for the parallel integration of CNTs into microelectronic systems.

There are numerous potential applications of the different types of carbon-based materials, where lightweight, flexible, and stretchable conductors are needed. The high potential of carbon-based metamaterials is illustrated by the example given in [27]. Normal metal- and gold-polydimethylsiloxane (PDMS) interconnects fracture due to the deformation of the substrate by a few percent [28] with stand unidirectional elongation of 10%–15% but fail to be elongated in an arbitrary direction. In contrast, CNT-based stretchable interconnects can be elongated by 100% in an arbitrary direction.

The quality of the various carbon-based materials for different applications in the high order of degree is indicated by their conductivity. The maximum conductivity should be achieved by means of maximization of the number of conducting channels [29] per unit cross-sectional area. It is worth looking at the specific resistivity of SWCNT and MWCNTs. The resistivity of the SWCNTs is defined by the energy spectrum and density of electronic states (DOS) in CNT. These physical characteristics were analyzed on the basis of the tight-binding approximation [30–32]. It was shown that the value of DOS in CNTs is four orders of magnitude smaller than that of metals. This would normally result in rather poor conductivity. However, the one-dimensional origin of transport in CNTs leads to a giant mean free path of the order of 1–10 μm of the conductive electrons, in contrast to that of metals (e.g., it is around 40 nm for copper). Additional states have to be available for the scattered carriers, to where charge state can be scattered. However, additional states for small-angle scattering events are energetically far away due to the azimuthal confinement and quantization of transverse component of the electron moment. As a result, the only allowed states are in the direction along the CNT.

Promising potential of carbon nanostructured elements for transmission and processing of high-frequency signals motivates active studying their electromagnetic response properties; see Refs. [33–50] as examples. Two recent reviews [18,21] should also be mentioned as very important for the understanding of the role of carbon nanostructures in high-frequency technique and devices. In this regard, researchers face a lot of new problems, while the traditional electromagnetic methods gain new life in their application to the new objects. In this chapter, we give an overview of peculiar electromagnetic problems on the nanoscale focusing mainly on specific electromagnetic effects in CNTs and in graphene. CNTs could even improve the nanoelectronic and nanophotonic devices due to the following key advantages. First, the CNT acts as an electromagnetic waveguide for the surface plasmon with the reachable roughly 50–100 of delay coefficient in the THz frequency range [33]. Second, there are special types of geometric resonances in CNTs and graphene with the resonant wavelengths large compared with the typical lengths of CNTs [38,44,50]. Third, the photonic density of states is controllable by the radius and chiral angle of CNT [44]. Forth, as was mentioned above, the existence of highly delayed eigenwave opens the way for the implementation of synchronous interaction of electron beam and free-electron lasing with CNT and graphene. The basic theoretical approach allowing description of a family of electromagnetic effects in nanocarbons, mainly in CNTs, and physical origin of these effects will be considered in the given chapter.

1.2 Electron properties of hexagonal lattice

1.2.1 Tight-binding approximation. Fermi-velocity and overlap integral

General properties of carbon nanostructures have been described in many classical books and papers (see, e.g., [1, 2] for carbon nanotubes and graphene, respectively). We also attract your attention to paper [51] where different forms of carbon are classified and considered as a unique model material for condensed matter physics and engineering science. Here, we restrict ourselves to a short introduction facilitating further reading.

Graphene is a plane honeycomb lattice of carbon atoms as depicted in Fig. 1.1 with interatomic distance b = 1.42 Å. A single-walled CNT is graphene rolled up into a cylinder. Let Rc be the relative position vector (chiral vector) in the graphene plane drawn between two sites on its honeycomb lattice, as it is shown in Fig. 1.1. In terms of the lattice basic vectors a1 and a2, Rc = ma1 + na2, where m and n are integers. Thus, in the conventional graphene parametrization [1], the geometric configuration of CNTs can be classified by a dual index (m,n) that has the form (m,0) for zigzag CNTs, (m,m) for armchair CNTs, and any other combinations of m and n . Typically, CNTs are 0.1–10 μm in length; their cross-sectional radius varies within the range 1–10 nm, while 0 ≤ θcn ≤ 30°.

Fig. 1.1 Configuration of the graphene crystalline lattice, R c  =  m a 1  +  n a 2 . White and black circles mark two sublattices, whose combination forms the graphene structure, and τ 1,2,3 are the position vectors connecting a chosen atom with its proximate neighbors.

Physical mechanisms of electron transport—tunneling, avalanche, and resistive instabilities and interaction between charged quasiparticles and periodic potential—are well known in solid-state physics. However, in nanotubes, as different from bulk media, these mechanisms express themselves in a significantly different manner because of the quasi-one-dimensional conductivity of CNTs. Therefore, the classical results for three-dimensional macroscopic bodies cannot be extended to nanotubes, which necessitate new fundamental investigations of electronic processes in CNTs.

Properties of electrons in nanotubes and electron transfer processes in them have been studied in detail both theoretically and experimentally. The theoretical analysis is usually confined to dynamics of π-electrons within the tight-binding approximation [31,32], which allows for interaction between only three adjacent atoms of the hexagonal structure. In the framework of this model, electron properties of graphene are described by the well-known dispersion law first time proposed by Wallace [30]. In the tight-binding approximation, the Hamiltonian matrix for π-electrons in the plane monoatomic carbon crystalline structure is the following:

   (1.1)

where the asterisk means the complex conjugate and

   (1.2)

Here, γ0 ≈ 2.7 − 3.0eV is overlapping integral, and px, y are the projections of the electron quasimomentum p on the corresponding axes; ℏ is the Planck constant. The eigenvalues of the matrix (1.1) are the energy values of π-electrons:

   (1.3)

where signs + and − correspond to conduction and valence band, respectively. The definition range for the quasimomentum p (the first Brillouin zone) is the hexagon; see Fig. 1.2A and B. The vertices are the Fermi points where the energy gap for the π-electrons is absent.

Fig. 1.2 Configuration of the first Brillouin zone for zigzag (A) metallic and (B) semiconducting CNT. Schemes of the dispersion characteristics for (C) metallic and (D) semiconducting CNTs.

1.2.2 Quantum confinement

Dispersion properties of nanotubes essentially differ from dispersion properties of graphene because of the difference in their topology. Due to the cylindrical geometry of carbon nanotube, the periodical boundary conditions for electrons in the circumference direction lead to the quantization of the electron transverse quasimomentum:

   (1.4)

where s is an integer. The axial projection pz of the quasimomentum is continuous. For zigzag CNT, inserting px → pz and py → pφ into Eq. (1.3) and taking into account Eq. (1.4) yields

To find dispersion relations for chiral CNTs, the analogous procedure is specified by px → pz cos θcn + pφ sin θcn and py → pz sin θcn − pφ cos θcn.

Due to the quantization, the first Brillouin zone in CNT is transformed from a hexagon to a set of one-dimensional subzones, which are defined by segments of straight parallel lines confined to the interior of the hexagon. Dependent on whether or not these lines pass through the hexagon vertices (Fermi points), the bandgap in the electron spectrum either disappears or appears. Unlike graphene, the density of states at the Fermi level in one-dimensional zones is nonzero. Accordingly, the nanotube is either metal or semiconductor. The armchair nanotubes exhibit metallic conductivity at any m, whereas the zigzag nanotubes behave as a metal only at m = 3q, where q is an integer; see Fig. 1.2A and B. Comparison of the dispersion characteristics for metallic and semiconducting CNTs is shown in Fig. 1.2C and D.

1.3 Linear ac-conductivity of carbon nanotubes

1.3.1 Kinetic equation. Semi-classical conductivity of CNTs

For a metallic CNT of a small radius and for a CNT of a very large radius (m → ∞), the approximate dispersion law for π -electrons

   (1.5)

has been proposed [52], where pF is the constant electron quasimomentum corresponding to the Fermi level and υF = 3γ0b/2ℏ is the velocity of π-electrons on this level. In both cases, the foregoing approximate dispersion law is applicable, because the regions near the Fermi points give the maximum contribution to the conductivity.

We apply semiclassical approximation to describe the motion of π-electrons exposed to axial polar-symmetrical component of the electric field Ez =  Re [Ez⁰ exp (− iωt)] with ω as the angular frequency. Electron transport is characterized by distribution function f(p, z, t), which satisfies the Boltzmann kinetic equation in the relaxation time approximation [53]:

   (1.6)

where e is the electron's charge, υz = ∂ E(p)/∂ pz is the electron's velocity, p is the electron momentum, ν[F(p) − f(p, z, t)] is the collision integral, ν is the relaxation frequency (ν = τ− 1, where τ is the relaxation time), and F is the Fermi-Dirac distribution function:

   (1.7)

Here, EF is the Fermi energy, kB is Boltzmann's constant, and T is temperature. We restrict the analysis to the linear approximation with respect to Ez⁰. Setting f = F +  Re [δf exp (− iωt)]EF with δf as a small unknown value and keeping only terms linear in Ez⁰, we arrive from Eq. (1.6) at

   (1.8)

The surface density of the axial polar-symmetrical current can be found from the expression jz =  Re [jz⁰ exp (− iωt)] where

   (1.9)

Writing jz⁰ = σ(ω)Ez⁰, from Eqs. (1.8), (1.9), we obtain the formula for the axial surface conductivity of a carbon nanotube

   (1.10)

For small radius CNTs (< 2 nm), Eq. (1.10) can be approximated by

   (1.11)

Frequency dependence of the conductivity σ is shown in Fig. 1.3.

Fig. 1.3 Frequency dependence of the surface conductivity of (12,0) zigzag metallic CNT.

1.3.2 Quasiclassical conductivity of the multi-wall carbon nanotubes: Inter-tube tunneling

Let us, for simplicity, consider the electron transport in double-walled CNT (DWCNT) above a ground plane (see Fig. 1.4). This system is embedded in a homogenous dielectric with permittivity ɛ. The shells of DWCNT with indexes (m1, n1) and (m2, n2) are coupled by intershell electron tunneling, which is characterized by the intershell tunneling frequency ωt, so that ωtℏ is the binding energy due to electronic delocalization of π-electrons. Let the shells be the commensurate, that is, the translation symmetry along the axis of DWCNT exists. In this case, the binding energy is maximal, and the typical value of ωtℏ is < 35 meV [54]. For incommensurate shells, ωt → 0.

Fig. 1.4 DWCNT above a ground plane.

Let us refer to the hth shell, being h = 1 and 2. Let Ih = Ih(z) be the axial electric current intensity, Eh = Eh(z) be the axial component of the electric field on the shell, Vh = Vh(zbe shell circumferences.

We focus on the frequency range below interband optical transitions and take into account two types of the electron motion: intraband motion within a shell and tunneling transitions between shells. Note that tunneling transitions occurs in the neighborhood of the Fermi points where the degeneracy of energy spectra of different shells occurs. The general formalism developed in [29,45] leads to the following transport equations for the π-electrons in the wave number and frequency domain (β,ω):

   (1.12)

where the quantities σ11 and σ22 are the self-conductivities of the shells while σ12 and σ21 are the mutual intershell conductivities due to the tunneling effect:

   (1.13)

where

   (1.14)

ω′ = ω − iν. The coefficient σC1 is the static and long-wavelength limit for the axial conductivity

   (1.15)

where R0 = πℏ/e² is the quantum resistance, lmfp = υF/ν is the mean free path of the π-electrons, and

   (1.16)

is the number of effective conducting channels, supposed to be the same for both shells. In this equation, T and N are, respectively, the length and the number of graphene hexagons of the shell unit cell and υμ = dEμ/d(ℏk) and Eμ are the longitudinal electron velocity and electron energy in the conduction μ-subband, respectively. Expressions for σ21(β, ω) and σ22(β, ω) follow from Eq. (1.13) by replacing C1 and σC1 with C2 and σC2 and by the substitutions m1 → m2 and n1 → n2.

The governing equations of the transmission line in Fig. 1.4 are obtained from Maxwell's equations:

   (1.17)

is a diagonal 2 × 2 matrix of the classical p.u.l. electric capacitance of two cylindrical shells above the ground plane, V = (V1 V2)T and I = (I1 I2)Tare given by

   (1.18)

are the matrix integral operators

   (1.19)

in the Hilbert space of functions defined on the real axis − ∞  < z <  + ∞, and

   (1.20)

are given by