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.

The Landauer-Büttiker (LB) formalism is widely used to simulate transport properties at equilibrium. The applications range from 1D conductors such as nanowires, nanotubes, nanoribbons, to 3D conductors such as molecular junctions with two or more contacts. At the ab initio level, this LB formalism is quite practical thanks to the Fisher–Lee relation, which connects the Landauer expression to the Green's function formalism. The transport properties of a given material can be simulated by finding the Green's function of the system within DFT (or even MBPT).

In this appendix the Green's function formalism is briefly reviewed. Section C.1 provides an introduction with a derivation of the trace formula starting from the Lippmann–Schwinger equations, then Section C.2 discusses recursive Green's function techniques, while Dyson's equation is introduced and applied to the case of a disordered system in Section C.3. Finally, Section C.4 is devoted to the implementation of LB formalism in conventional ab initio codes in order to investigate coherent electronic transport in nanoscale devices.

Phase-coherent quantum transport and the Green's function formalism

Green's functions are one of the most useful tools (Economou, 2006) for calculation of different physical quantities of interest such as the density of states or the quantum conductance and conductivity. In the context of phase-coherent quantum transport, they play a crucial role because their relation with the scattering matrix can be exploited to compute the quantum transmission probabilities as needed within the Landauer–Büttiker formalism presented in Section 3.3.

Once deemed impossible to exist in nature, graphene, the first truly two-dimensional nanomaterial ever discovered, has rocketed to stardom since being first isolated in 2004 by Nobel Laureates Konstantin Novoselov and Andre K. Geim of the University of Manchester. Graphene is a single layer of carbon atoms arranged in a flat honeycomb lattice. Researchers in high energy physics, condensed matter physics, chemistry, biology, and engineering, together with funding agencies, and companies from diverse industrial sectors, have all been captivated by graphene and related carbon-based materials such as carbon nanotubes and graphene nanoribbons, owing to their fascinating physical properties, potential applications and market perspectives.

But what makes graphene so interesting? Basically, graphene has redefined the limits of what a material can do: it boasts record thermal conductivity and the highest current density at room temperature ever measured (a million times that of copper!); it is the strongest material known (a hundred times stronger than steel!) yet is highly mechanically flexible; it is the least permeable material known (not even helium atoms can pass through it!); the best transparent conductive film; the thinnest material known; and the list goes on …

A sheet of graphene can be quickly obtained by exfoliating graphite (the material that the tip of your pencil is made of) using sticky tape. Graphene can readily be observed and characterized using standard laboratory methods, and can be mass-produced either by chemical vapor deposition (CVD) or by epitaxy on silicon carbide substrates.

Introduction

As described in Chapter 1, the sp2 carbon-based family exhibits a great variety of allotropes, from the low-dimensional fullerenes, nanotubes and graphene ribbons, to two-dimensional monolayer graphene, or stacked graphene multilayers. Two-dimensional monolayer graphene stands as the building block, since all the other forms can be derived from it. Graphene nanoribbons can be seen as quasi-one-dimensional structures, with one lateral dimension short enough to trigger quantum confinement effects. Carbon nanotubes can be geometrically constructed by folding graphene nanoribbons into cylinders, and graphite results from the stacking of a very large number of weakly bonded graphene monolayers.

The isolation of a single graphene monolayer by mechanical exfoliation (repeated peeling or micromechanical cleavage) starting from bulk graphite has been actually quite a surprise, since it was previously believed to be thermodynamically unstable (Novoselov et al., 2004, Novoselov et al., 2005b). At the same time, the route for controlling the growth of graphene multilayers on top of silicon carbide by thermal decomposition was reported, and eventually led to fabrication of single graphene monolayers of varying quality depending on the surface termination (silicon or carbon termination) (Berger et al., 2006). Basic electronic properties of graphene were actually well-known since the seminal work by Wallace in the late forties (Wallace, 1947), such as the electron–hole symmetry of the band structure and the specific linear electronic band dispersion near the Brillouin zone corners (Dirac point), but it was after the discovery of carbon nanotubes by Iijima from NEC (Iijima, 1991) that the exploration of electronic properties of graphene-based materials was revisited (for a review see Charlier, Blase & Roche 2007).

Introduction

Over the last few decades, there has been a significant increase in the use of computational simulation within the scientific community. Through a combination of the phenomenal boost in computational processing power and continuing algorithm development, atomistic scale modeling has become a valuable asset, providing a useful insight into the properties of atoms, molecules, and solids on a scale “often inaccessible” to traditional experimental investigation.

Atomistic simulations can be divided into two main categories, quantum mechanical calculations and classical calculations based on empirical parameters. Quantum mechanical simulations (often referred to as ab initio or first-principles) aims at solving the many-body Schrödinger equation (Schrödinger, 1926). The original reformulation of the Schrödinger equation offered by the DFT provides valuable information on the electronic structure of the system studied.

The very essence of DFT is to deal with noncorrelated single-particle wavefunctions. Many of the chemical and electronic properties of molecules and solids are determined by electrons interacting with each other and with the atomic nuclei. In DFT, the knowledge of the average electron density of the electrons at all points in space is enough to determine the total energy from which other properties of the system can also be deduced. DFT is based on the one-electron theory and shares many similarities with the Hartree–Fock method. DFT is presently the most successful and promising (also the most widely used) approach to computing the electronic structure of matter. In this appendix, the basics of DFT modeling techniques are explained.

This chapter is devoted to the properties of granular media immersed in a liquid. Mixtures of grains and fluids are used in many industrial processes, for example in civil engineering projects with concrete. In environmental problems, the coupling between a granular soil and water controls the soil stability and is important in the understanding of many natural disasters such as landslides and mud flows. The physics of two-phase flows involving grains and liquid is a vast area of research. In this book dedicated to granular media, we will restrict ourselves to the high-concentration regime, for which the grains are in contact and interact primarily through contact interactions. Our goal is to illustrate through examples how the concepts developed in the previous chapters on dry granular materials are modified in the presence of an interstitial fluid. In the first part (Section 7.1) we introduce two-phase-flow equations, the relevant theoretical framework for studying immersed granular media. In this approach, the granular medium and the liquid are described as two interpenetrating continuum media, which interact. In the second part, the use of two-phase-flow equations is illustrated in simple examples for which the granular skeleton is static (Section 7.2) and for which it is slightly deformed (Section 7.3). Finally, the influence of the interstitial fluid on the rheology of sheared granular media is discussed in Section 7.4.

A granular medium without external perturbation can be considered primarily as a solid. A pile of sand, the soil on which a house is built and a silo filled with grains are examples of situations in which the grains do not move. The material supports external forces without flowing, just like a solid. This chapter is dedicated to the statics of granular media: how are grains organized in a packing? Howare the forces distributed among the particles to ensure the mechanical balance of the pile? Is it possible to describe the granular medium as a continuum and to define stresses? The chapter starts with the description of the geometrical properties of packings by introducing the concepts of volume fraction and compaction of a granular medium (Section 3.1). Then the problem of the mechanical equilibrium of a sand pile is addressed, and the statistical properties of the inter-particle force distribution are presented (Section 3.2). Following the analysis at the microscopic level, the possibility of a continuum description is discussed in Section 3.3. The concept of stresses in granular media and the relation between inter-particle forces and macroscopic stresses are presented. Simple cases for which the stress distribution can be calculated are studied (Section 3.4). Finally, the issue of elasticity and sound propagation in a granular packing is discussed in Section 3.5.

Granular packings

Packings of grains have been studied since antiquity. Mathematicians, physicists and engineers, in a quest to optimize the storage of granular matter, have been interested in these issues. This section is an introduction to the concepts which are useful for characterizing packings of grains. For more details, the reader is referred to more specialized works, such as the book by Cumberland and Crawford (1987).

Sand dunes, screes, river deltas … Most geological structures at the Earth's surface involve granular materials. This final chapter proposes an application of the various properties of granular media developed throughout this book to geomorphology, i.e. to the study of the nature and origin of landforms, particularly regarding the formative processes of weathering and erosion that occur in the atmosphere and hydrosphere. These processes continually shape the Earth's surface, and generate the sediments that circulate in the rock cycle. Landforms are the result of the interactions among the geosphere, atmosphere and hydrosphere.

On the one hand, we will propose a description of sedimentary landscape elements that is based on natural history. On the other hand, we will analyse various sedimentary structures in terms of physical mechanisms. When possible, we will detail the associated scaling laws, which allow one to reproduce geophysical phenomena in the lab, at small scale, in a controlled way. In particular, we have established in the previous chapter a description of erosion and sediment transport in a homogeneous flow. We will now use these results by considering the coupling between relief and transport. We will pay special attention to linear instabilities, which can explain the emergence and organization of geological objects. We will study successively gravity-driven flows (Section 9.1), ripples and dunes (Section 9.2), coastal instability (Section 9.3) and, finally, rivers (Section 9.4).

The behaviour of a granular material is closely related to the nature of the interactions between grains. In this chapter, we focus on these forces at the grain level. We first discuss solid contact, which is dominant in the case of dry granular media made of macroscopic particles (Section 2.1). The basics of Hertz elastic contact, solid friction and the rules of inelastic collisions between solid particles are given. We then discuss other kinds of interaction between grains such as electrostatic and adhesive forces, capillary cohesion and solid bridges (Section 2.2). The last part of the chapter gives a brief overview of the hydrodynamic forces produced on a particle immersed in a fluid (Section 2.3). Our aim in this chapter is to provide some background in contact physics and hydrodynamics that will be useful for our study of granular media. More detailed treatments can be found in the classical books given in the text.

Solid contact forces

The contact force between two dry grains is usually split into a normal force and a tangential force. The physical origin of these forces at the microscopic level involves many phenomena, such as surface roughness, local mechanical properties (elasticity, plasticity, viscoelasticity) and physical and chemical properties (the presence of electrical charge, oxidation, temperature, the presence of lubricant film). In the following, we will not consider these microscopic features in detail but rather focus on the macroscopic laws of solid contact. At the macroscopic level, these laws are dominated by elastic repulsion (Hertz contact) and solid friction (Coulomb’s law).

Sand, gravel, rice, sugar … Granular matter is familiar and abounds around us. However, the physics of granular media is still poorly understood and continues to fascinate scientists and other people, more than three centuries after the work of Coulomb on slope stability. A pile of grains actually exhibits a great variety of behaviours with unique properties. Strong enough to support the weight of a building, grains can also easily flow like water in an hourglass or be transported by wind to sculpt dunes and deserts. For a long time, the study of granular materials has remained the preserve of engineers and geologists. Therefore, important concepts arose from the need to build structures on solid ground, store grains in a silo or predict the history of a sediment. More recently, the study of granular media has entered the field of physics, at the crossroads of statistical physics, mechanics and soft-matter physics. The combination of results from laboratory experiments on model materials, discrete numerical simulations and theoretical approaches from other fields has enriched and renewed our understanding of granular materials.

This book has been written in this context. Our goal is to provide an introduction to the physics of granular media that takes into account recent advances in this field, while describing the basic concepts and tools useful in many industrial and geophysical applications. This book is intended primarily for students, researchers and engineers willing to become familiar with the fundamental properties of granular matter.

The previous chapters show that a granular medium can behave as a solid. In the opposite limit, when grains are strongly shaken in a box, particles are agitated and interact mainly by binary collisions. The medium is then more similar to a gas. This chapter is devoted to this ‘gaseous’ regime of granular matter. The analogy between agitated grains and molecules in a gas was the basis for the development of kinetic theories of granular media that provide constitutive equations for rapid and diluted granular flows. In this chapter, we first introduce the notion of granular temperature and briefly discuss the analogies and differences between a granular gas and a classical molecular gas (Section 5.1). We then present a first approach to the kinetic theory, which gives insight into the physical origin of the transport coefficients (Section 5.2). A more formal presentation of the kinetic theory that is based on the Boltzmann equation for inelastic gases is given in the next section (Section 5.3). We then apply the hydrodynamic equations of the kinetic theory to various situations that highlight the role of inelastic collisions in the behaviour of granular gases (Section 5.4). Finally, some limits of the kinetic theory are discussed, in particular concerning dense media (Section 5.5).

Analogies and differences with a molecular gas

Figure 5.1 gives two examples of granular materials in a ‘gaseous’ state. The first one is obtained by shaking vertically a box containing beads. The second example shows steel beads flowing down a steeply inclined plane under the action of gravity. In both cases, the medium looks like a gas. Particles are strongly agitated and move independently, except when collisions occur.

Most granular flows encountered in nature and industry lie between the quasistatic and gaseous regimes seen in the previous chapters. In this intermediate ‘liquid’ regime, particles remain closely packed and interact both by collision and through long-lived contacts. Understanding and modelling the flow of dense granular media is challenging and many questions remain to be answered, despite important advances having been made during the last decade. In this chapter, we first present the basic features of dense granular flows (Section 6.1), before focusing on the rheology of this peculiar liquid (Section 6.2). A phenomenological constitutive law that is based on dimensional analysis is presented, in which the medium is described as a viscoplastic fluid with a frictional behaviour. The success and limitations of this approach are then discussed, in particular close to the solid–liquid transition where complex collective behaviours are observed. The second part of the chapter presents a hydrodynamic description of dense flows that is valid for a shallow layer flowing under gravity (the Saint-Venant equations) (Section 6.3). This depth-averaged approach enables one to gather the complex rheology into a single basal friction term and is commonly used in geophysics to describe rock avalanches and landslides. We close the chapter with a presentation of the phenomenon of size segregation, which occurs when the medium is composed of particles of different sizes. The consequences of segregation for polydisperse granular flows in various configurations are presented (Section 6.4).