The Lanczos tridiagonalization method orthogonally transforms a real symmetric matrix A to symmetric tridiagonal form. Traditionally, this very simple algorithm is suitable when one needs only a few of the lower eigenvalues and the corresponding eigenvectors of very large Hermitian matrices, whose full diagonalization is technically impossible. We introduce here the basic ingredients of the recursion method based on the Lanczos tridiagonalization, and explain how calculation of the DOS as well as the dynamics of wavepackets (and related conductivity) can be performed efficiently.

Lanczos method for the density of states

The Lanczos method is a highly efficient recursive approach for calculation of the electronic structure (Lanczos, 1950). This method, first developed by Haydock, Heine, and Kelly (Haydock, Heine & Kelly, 1972, 1975), is based on an eigenvalue approach due to Lanczos. It relies on computation of Green functions matrix elements by continued fraction expansion, which can be implemented either in real or reciprocal space. These techniques are particularly well suited for treating disorder and defect-related problems, and were successfully implemented to tackle impurity-level calculations in semiconductors using a tight-binding approximation (Lohrmann, 1989), and for electronic structure investigations for amorphous semiconductors, transition metals, and metallic glasses based on linear-muffin-tin orbitals (Bose, Winer & Andersen, 1988). Recent developments include the exploration of a degenerated orbital extended Hubbard Hamiltonian of system size up to ten millions atoms, with the Krylov subspace method (Takayama, Hoshi & Fujiwara, 2004, Hoshi et al., 2012).

This section presents a brief overview of the most promising graphene applications in information and communication technologies, reflecting current activities of the scientific community and the authors' own views.

Introduction

The industrial impact of carbon nanotubes is still under debate. Carbon nanotubes exist in two complementary flavors, i.e metallic conductors and semiconductors with tunable band gap (scaled with tube diameter), both exhibiting ballistic transport. This appears ideal at first sight for creating electronic circuits, in which semiconducting nanotubes (with diameter around 1–2 nm) could be used as field effect transistors, whereas metallic single-wall tubes (or large-diameter multiwalled nanotubes), with thermal conductivity similar to diamond and superior current-carrying capacity to copper and gold, would offer ideal interconnects between active devices in microchip (Avouris, Chen & Perebeinos, 2007). Nanotube-based interconnects have been physically studied over almost a decade, with companies such as Samsung, Fujitsu, STMicroelectronics, or Intel acting significantly or encouraging academic research (Coiffic et al., 2007). The current-carrying capability of bundles of multiwalled nanotubes has been practically demonstrated to fulfill the requirements for technology and thus could replace metals (Esconjauregui et al., 2010), although a disruptive technology step remains to be achieved to integrate chemical vapor deposition (CVD) growth at the wafer-scale, a step of no defined timeline.

In this chapter we start with a presentation of the so-called Klein tunneling mechanism, which is one of the most striking properties of graphene. Later we give an overview of ballistic transport both in graphene and related materials (carbon nanotubes and graphene nanoribbons). After presenting a simple real-space mode-decomposition scheme, which can be exploited to obtain analytical results or to boost numerical calculations, we discuss Fabry-Pérot interference, contact effects, and the minimum conductivity in the 2D limit.

The Klein tunneling mechanism

The Klein tunneling mechanism was first reported in the context of quantum electrodynamics. In 1929, physicist Oskar Klein (Klein, 1929) found a surprising result when solving the propagation of Dirac electrons through a single potential barrier. In non-relativistic quantum mechanics, incident electrons tunnel a short distance through the barrier as evanescent waves, with exponential damping with the barrier depth. In sharp contrast, if the potential barrier is of the order of the electron mass, eV ~ mc2, electrons propagate as antiparticles whose inverted energy–momentum dispersion relation allows them to move freely through the barrier. This unimpeded penetration of relativistic particles through high and wide potential barriers has been one of the most counterintuitive consequences of quantum electrodynamics, but despite its interest for particle, nuclear, and astro-physics, a direct test of the Klein tunnel effect using relativistic particles still remains out of reach for high-energy physics experiments.

Carbon is a truly unique chemical element. It can form a broad variety of architectures in all dimensions, both at the macroscopic and nanoscopic scales. During the last 20+ years, brave new forms of carbon have been unveiled. The family of carbon-based materials now extends from C60 to carbon nanotubes, and from old diamond and graphite to graphene. The properties of the new members of this carbon family are so impressive that they may even redefine our era. This chapter provides a brief overview of these carbon structures.

Carbon structures and hybridizations

Carbon is one of the most versatile elements in the periodic table in terms of the number of compounds it may create, mainly due to the types of bonds it may form (single, double, and triple bonds) and the number of different atoms it can join in bonding. When we look at its ground state (lowest energy) electronic configuration, 1s22s22p2, carbon is found to possess two core electrons (1s) that are not available for chemical bonding and four valence electrons (2s and 2p) that can participate in bond formation (Fig. 1.1(a)). Since two unpaired 2p electrons are present, carbon should normally form only two bonds from its ground state.

However, carbon should maximize the number of bonds formed, since chemical bond formation will induce a decrease of the system energy. Consequently, carbon will re-arrange the configuration of these valence electrons.

In this chapter we give a flavor of quantum transport beyond DC conditions, when time-dependent potentials are applied to a device. Our main focus is on Floquet theory, one of the most useful approaches for driven systems. Section 6.4 is devoted to an overview of some of the most recent advances on driven transport in graphene-related materials, while Section 6.5 presents an illustrative application to laser-illuminated graphene.

Introduction: why AC fields?

Though less explored, quantum transport beyond the DC conditions considered in previous sections also offers fascinating opportunities. Alternating current (AC) fields such as alternating gate voltages, alternating bias voltages or illumination with a laser can be used to achieve control of the electrical response (current and noise), thereby providing a novel road for applications. Furthermore, there are many novel phenomena unique to the presence of AC fields such as quantum charge pumping (Thouless, 1983, Altshuler & Glazman, 1999, Büttiker & Moskalets, 2006, Switkes et al., 1999), i.e. the generation of a DC current even in the absence of a bias voltage due to quantum inter-ference, coherent destruction of tunneling (Grossmann et al., 1991) or laser-induced topological insulators (Lindner, Refael & Galitski, 2011, Kitagawa et al., 2011).

The activity in this area has grown rapidly in the arena of nanoscale systems (Platero & Aguado, 2004, Kohler, Lehmann & Hänggi, 2005). Notwithstanding, it was not until the last few years that advances in the applications to graphene-related systems started to lourish (see the overview in Section 6.4).

Introduction

In Appendix A, a detailed description of the electronic structure calculation techniques based on the so-called density functional theory (DFT) was presented. As mentioned and illustrated in that section, DFT is widely used to investigate the electronic properties of materials, their defects, interfaces, etc. Unfortunately, the semi-local approximations of DFT, such as the local density approximation (LDA) and gradient generalized approximation (GGA), suffer from a well-known substantial underestimation of the band gap. This may be interpreted as a result of the fact that DFT does not properly describe excited states of a system. This failure of DFT may also induce a wrong estimation of the position of the electronic defect/dopant levels in the band gap.

Some empirical solutions exist to overcome the problem of DFT band gap underestimation. For example, the “scissor” technique consists in correcting the LDA/GGA gap error by shifting the conduction band up so as to match the gap relative to the experiment. However, such a method is not accurate enough for defining the accurate position of defect/dopant levels occurring in the band gap.

Another solution to the underestimation of the band gap in DFT consists in using the so-called hybrid functionals which have recently become very popular. Indeed, these functionals incorporate a fraction of Hartree–Fock (HF) exchange, which leads to improvement of the band gap compared to LDA/GGA (Curtiss et al., 1998, Muscat, Wander & Harrison, 2001, Paier et al., 2006).

This chapter illustrates the several possible computational approaches that can be used towards a more realistic modeling of disorder effects on electronic and transport properties of carbon-based nanostructures. Multiscale approaches are first presented, combining ab initio calculations on small supercells with tight-binding models developed from either a fitting of ab initio band structures, or a matching between conductance profiles with a single defect/impurity. Chemical doping with boron and nitrogen of carbon nan-otubes and graphene nanoribbons is discussed in detail, as well as adsorbed oxygen and hydrogen impurities for two-dimensional graphene, being both of current fundamental interest. Finally, fully ab initio transport calculations (within the Landauer–Büttiker conductance framework) are discussed for nanotubes and graphene nanoribbons, allowing for even more realism, albeit with limited system sizes, in description of complex forms of edge disorder, cluster functionalization or nanotube interconnection.

Introduction

In the following sections, disordered and chemically doped carbon nanotubes and graphene nanoribbons are explored. The main scientific goal consists in illustrating how defects and impurities introduce resonant quasi-localized states at the origin of electron–hole transport asymmetry fingerprints, with the possibility of engineering transport (or mobility) gaps. Several multiscale approaches are described to develop various tight-binding models from first-principles calculations. A first technical strategy (illustrated on boron-doped nanotubes, Section 7.2.2) consists in designing a tight-binding model by fitting the ab initio band structures. Such an approach is used to describe doped metallic nanotubes, but actually ceases to be accurate for graphene nanoribbons, owing to complex screening effects introduced by edges.