Starting from a detailed explanation of Klein paradox of relativistic quantum mechanics, we consider a motion of massless Dirac fermions through potential barriers. It is shown that chiral properties of these particles guarantee a penetration through arbitrarily high and broad potential barriers. The role of this phenomenon (chiral tunneling) for graphene physics and technology is explained. We discuss analogy between electronic optics of graphene and optical properties of metamaterials, especially, Veselago lensing effect for massless Dirac fermions. Chiral tunneling in bilayer graphene is discussed.

Using graphene on hexagonal boron nitride (hBN) as an example, we introduce the concept of van der Waals heterostructures. First, we explain extraordinary high quality of graphene on hBN. Then we discuss the physics of formation of moiré patterns and a general problem of commensurability and incommensurability. We also discuss the basic consequences for electronic structure and electron transport properties, including a conductivity along zero-mass lines, formation of additional Dirac points and recently experimentally discovered new types of magneto-oscillation effects in graphene superlattuces.

We discuss the physics of pseudomagnetic field,s which can be induced in graphene by applying strains, and show how they can be used to manipulate electronic transport through graphene heterostructures (strain engineering). We consider strain-induced pseudo-Landau levels, which were observed in graphene, and discuss the related valley quantum Hall effect. At the end of this chapter we demonstrate that a combination of strain and electric gating can open energy gap in electron spectrum of graphene which can be potentially interesting for applications.

Exact solution of two-dimensional Dirac equation for Coulomb potential (Dirac–Kepler problem) is presented. Linear and nonlinear screening of the Coulomb potential is discused. The main focus is on the phenomenon of relativistic collapse of supercritical charges, which was discussed for many years in high-energy physics (this is the process that determines the end of the periodic table) and was at last discovered in graphene. We introduce Hartree–Fock theory for massless Dirac electrons and show that their Coulomb interaction essentially renormalizes Fermi velocity in such a way that Dirac cone is, strictly speaking, no more cone.

We continue the discussion of Van der Waals heterostructures for the case of twisted blayer graphene. After a general consideration, we discuss a special case of graphene dodecagonal quasicrystal for misorientation angle 30⁰. We also discuss a formation of flat bands for small misorientation angles and give a brief introduction to the physics of flat electron bands.

The chapter explains a physics of minimal conductivity in graphene. It is shown that a new type of electronic transport arises in graphene, namely, electron tunneling via zero modes of Dirac operator. The relation to Zitterbewegung concept of relativistic quantum mechanics is demonstrated. We calculate the value of minuimal conductivity and shot noise in the neutrality point, and consider Aharonov–Bohm effect in undoped graphene rings.

Optical properties of massless Dirac electrons are considered. In particular, it is shown that they provide a universal, frequency-independent adsorption coefficient determined by fine structure constant. The possible effect of interelectron interaction on this property is discussed. Using a perturbation theory for density matrix, we derive Kubo formula for various response functions and use it to consider optics, magnetooptics, charge screening and diamagnetism of massless Dirac electrons. Graphene plasmonics is briefly reviewed.

The chapter starts with the derivation of effective Hamiltonian for band electrons in magnetic field and continues with discussion of energy levels in magnetic field for massless Dirac fermions. The case of bilayer graphene is also considered. Special attention is paid to a formation of topologically protected zero-energy modes. Using band electrons in magnetic field as an example, a general concept of Berry phase is introduced. Magneto-oscillation effects for two-dimensional Dirac fermions are considered. Quantum Hall effect is discussed, via topological approach by Thouless and coworkers. At the end, we discuss electronic structure in the presence of crossed electric and magnetic fields and the effects of a smooth disorder on Landau levels for massless Dirac electrons.

We discuss scattering theory for massless Dirac fermions and for a new type of wave equation describing low-energy electrons in bilayer graphene. After that, we present a general theory of defects in solids via Green's function formalism. We apply it to consider mid-gap states due to vacancies or adatoms in graphene and calculate interaction energy between these adatoms. The basic physics of scanning tunneling spectroscopy as an experimental tool to study defect states in solids is presented.

After general discussion of itinerant-electron magetism, Hubbard model and Lieb theorem, we discuss magnetic moments at different types of defects in graphene and supposed ferromagnetism at zigzag edges. We consider various mechanisms for determining spin-orbit coupling, with especial emphasis on the importance of full band structure, and the effect of spin-orbit interaction on electronic structure. In this respect, we briefly discuss the difference between graphene, silicene, and germanene, and Kane–Mele model, which initiated development of the field of topological insulators. At the end, we consider the effect of magnetic edges on spin relaxation in graphene nanoribbons.

In this chapter, we discuss how to build effective many-body models starting from first principles electronic structure calculations and apply this general approach to graphene. We present quantitative results for the Fermi velocity renormalization, which were preliminary announced in Chapter 8. After that, we discuss many-body effects in graphene electron spectrum, static screening, and optical conductivity based on the results of lattice quantum Monte Carlo simulations. At the end, we consider many-body renormalization of minimal conductivity in graphene within the concept of environment-induced suppression of quantum tunneling.

Boundary conditions for electron wave functions in graphene are discussed, both in Dirac approximation and for the honeycomb lattice. We start with the model of "neutrino billiard." Then, we discuss typicality of zigzag boundary conditions for the terminated honeycomb lattice, existence of zero-energy edge mode for these conditions, electronic states and conductance quantization in graphene nanoribbons, level statistics for graphene quantum dots, explanation of quantum Hall effects in terms of topologically protected edge modes, and Aharonov–Bohm effect in multiconnected graphene flakes. The latter case is used as an example of the general topological concept of spectral flow.

After discussion of basic concepts of the covalent chemical bond with applications to carbon, the chapter presents tight-binding description of electronic structure of single-layer and multilayer graphene, with a special emphasis on emergence of massless Dirac fermions in honeycomb lattice, effects of trigonal warping, and symmetry protection of conical points in band structure.