The enormously fast improvements achieved in solid-state-based technologies are triggered by the unusual transport properties of semiconductors. Likewise, in the “metal age” of humanity, the technological applications of semiconductors enabled us to develop tools that (mostly) increased our quality of life. Solid-crystal materials are classi”ed as metals, semiconductors, and insulators when comparing their electrical transport properties at room temperature. In general, metals are good conductors because there are many available states at the Fermi level that contribute to electronic transport [1]. In contrast, insulators have a large energy gap between their valance and conduction bands, and there are no available states at Fermi energy, EF. Semiconductors, as can be understood from their name, present a smaller band gap compared to insulators. However, they still have no states at EF under thermal and electrochemical equilibrium conditions. These classi”cations are illustrated in Figure 9.1. How do we calculate the energy band diagrams of a given material, starting from crystal (growth) parameters? It is a quite complicated procedure that is discussed in other chapters of this book. However, in this chapter, knowing only the energy gap of the given semiconductor device, we perform reasonably reliable transport calculations [2,3]. Here, we will focus on the electronic states and their in«uence on the transport properties of two-dimensional (2D) charge systems. The ”rst section is devoted to introducing the fundamental concepts of the calculation scheme to obtain the electrostatic potential, charge distribution, and energy dispersion of a semiconductor heterojunction in one dimension (1D), namely, in the growth direction z. A particular emphasis is placed on the GaAs/AlGaAs material, which we will focus on throughout this chapter, because the main objective is to discuss the intriguing physics observed in high-mobility 2D electron systems, particularly when subject to a perpendicular B ”eld. In the ”rst section, we generalize our simple 1D calculation scheme to three dimension (3D) and obtain the electrostatic properties of the crystal by solving the equation the Poisson equation (PE) numerically, utilizing iterative methods [4]. We discuss different edge pro”les of the system. Section 9.2.4 deals with the solution of the PE and the Schrödinger equation (SE) self-consistently in the growth direction to obtain the energy dispersion, where the conditions to obtain a 2D electron system (2DES) or 2D hole system will be also discussed. Next, in a short subsection, we consider multiple layers of 2D systems. In the last part of this section, we con”ne ourselves mainly to a single-layer 2D system and investigate

9.4.5 Cleaved-Edge Overgrown Samples .................................................. 247 9.4.6 Curved Crystals ................................................................................248