CHAPTER 1 Experimental and theoretical research of two-dimensional materials is highly intensifying. This is due to their fascinating physical properties which are of technological importance. We shall give a simple overview of the history and physical properties of the three most important members of the two-dimensional materials, graphene, hexagonal boron nitride and molybdenum disulfide. CHAPTER 2 Every thorough analysis of the electromagnetic properties of crystals begins with the calculation of electronic dispersions, which generally is not an easy task. Some approximations are necessary in order to find simple analytical expressions for electronic dispersions. The most important one is to focus our attention only to a few bands nearest to the Fermi level. We calculate these dispersions by using the tight binding approximation (TBA). We start from atomic orbitals or their linear combinations (hybrids), for which the hopping probability is highest. The choice of atomic orbitals is determined by crystal symmetry, which will be different for all three systems considered in this thesis: graphene, hexagonal boron nitride and molybdenum disulfide. We shall briefly introduce the hexagonal lattice with a basis, display the electronic dispersions of the valence bands in aforementioned systems for typical TBA parameters, and estimate the values of these parameters by comparison with ab initio calculations. CHAPTER 3 In this chapter we study the interaction of conduction electrons with various bosonic modes. First, we determine the electron-phonon coupling constants by looking at how lattice vibrations change the hopping matrix elements between neighbouring sites and atomic energies on those sites. We calculate these constants for acoustic and optical phonons and find their long-wavelength forms. The electron coupling to external electromagnetic fields is described by the Peierls substitution in the TBA, for a linear response regime. We shall derive the standard form of the interaction of the charge and current densities with external scalar and vector potentials and explain the role of charge, current and dipole vertex functions. These coupling constants and vertex functions are naturally defined in the direct-space representation, or in the delocalized orbital representation. Therefore, we shall use the Fourier transformations to determine the corresponding contributions to the total Hamiltonian in the Bloch representation. CHAPTER 4 Transport properties of conducting systems are usually studied in the linear response regime. This can be done using semi-classical transport equations or quantum transport equations. In both cases, the non-equilibrium distribution function has a central role. The semiclassical Boltzmann equations define it as a non-equilibrium distribution function for an electron of momentum ħk at a position r, whereas in the quantum equations and their semiclassical regimes it is related to the propagator of an electron-hole pair with momenta k + q and k. The Landau transport equations are a generalization of Boltzmann equations, where electron-electron interactions are treated in a self-consistent way and the scattering from the static disorder and from phonons is included phenomenologically. We can reach the Landau equations following the usual Landau’s procedure, or from the Heisenberg equations of motion for the electron-hole propagator. The advantage of the latter approach is that all electron scattering, from electrons, from disorder and from phonons, can be described in the same way, by studying the three contributions to the relaxation function, which will be called here the memory function. In this chapter we shall introduce the non-equilibrium distribution function n (k, r, t), derive the semiclassical equations of motion, determine the structure of the memory function for the scattering by phonons and describe the role of its real and imaginary parts in the dynamical conductivity tensor. CHAPTER 5 An alternative to the equations of motion approach from the previous chapter are the usual perturbative methods of calculating response functions. In this way various correlation functions are usually defined, named the Kubo formulae; for example, the Kubo formula for the charge-charge or current-current correlation functions, etc. There are two basic formalisms; the first one at T = 0 and the Matsubara finite-temperature formalism. The conductivity tensor, introduced in the previous chapter, represents the current-dipole correlation function, linking the current (response of the system) induced in the system by a macroscopic field (external perturbation). The external field couples to the dipole moment operator. In a similar manner, other response functions of interest can be defined. In this chapter, we shall use the T =0 formalism where response functions represent the retarded correlation functions (the response follows the external perturbation). We shall define all correlation functions related to the conductivity tensor, derive their general relations, and make sure that these relations are satisfied in the lowest order, usually called the one fermion loop approximation. We shall study the effects of phenomenologically introduced relaxation processes. We shall show that in general considerations, the continuity equation and Kramers-Kronig relations play an important role. CHAPTER 6 In this chapter, we study the DC and dynamic conductivity of graphene and doped molybdenum disulfide, using the general expressions derived in chapters 4 and 5. First, we analyze the structure of the momentum distribution function and explain its role in the calculation of the DC conductivity of undoped graphene. Afterwards, we study the intraband dynamical conductivity of doped graphene, keeping in the memory function only the scattering by phonons by disorder. These results are compared with the relaxation-time approximation results. In the end, we consider the interband conductivity of doped graphene, once again in two approximations: the relaxation-time and interband memory function approach. CHAPTER 7 In this chapter, we analyze elementary excitations in the electron subsystem. Unlike the transport equations studied in the third chapter, where long-range Coulomb interactions were taken into account implicitly through the macroscopic electric field, here we will include various electron-electron interaction contributions step by step. The primary interest here is to study the collective excitations in the electronic subsystem. In the electron-hole equations of motion for a general problem with multiple bands, we shall clearly distinguish the intraband and interband collective modes. We shall show that the "ladder" and Fock contributions cancel out in the intraband channel. What remains is the RPA contribution (being part of the macroscopic field definition) and higher order Fock contributions which lead to the electron-electron contribution to the memory functions. For the insulating case, we shall derive the equation of motion for the interband electron-hole propagator and find the energies of collective modes, excitons, which have energies similar to the 2D positronium. Energies calculated in this way do not agree well with experimental data, so it will be necessary to include electron-hole correlation effects. These effects are included in our equations by using screened Coulomb interaction instead of the bare one.