The current–voltage characteristics of modern electronic devices consisting of semiconductor heterostructures, such as resonant tunneling diodes, quantum cascade lasers, and tandem solar cells, are determined by the dynamics of electrons propagating through quantum-engineered 1D potential landscapes. In this chapter, we will develop a general formalism with which to describe transmission probabilities for electron waves propagating through arbitrary potentials, which can be used for analyzing electron motion in semiconductor devices. Furthermore, we will extend our formalism to 1D electrons moving in a general spatially periodic potential, based on which we will describe the basic concepts of the band theory of solids. The central theorem in band theory is the Bloch theorem, which we will derive and then use for discussing the dynamics of electrons in crystalline solids (or Bloch electrons).

The field of quantum computing is developing at a rapid pace, and one can expect paradigm-shifting advances in coming years. The goal of this chapter is for the reader to understand fully the language and basic concepts of quantum information science needed to engage in research and development in this very exciting field in the future. We will apply the mathematical machinery we have acquired so far to develop the quantum counterparts to the classical notions of bits, logic gates, circuits, and algorithms. We will also review some of the promising examples of quantum hardware for physically realizing quantum information processing.

A regrettable amount of mathematical machinery goes into a good understanding of quantum mechanics. This could be avoided if a good intuitive understanding of many quantum systems was possible, but as intuition is generally derived from daily experience (which is governed by classical laws), we cannot expect this to be the case in general. Here, we present an in-depth introduction to the mathematical foundations of quantum mechanics, accompanied, wherever appropriate, by detailed explanations of relevant quantum concepts such as superposition, wavefunction collapse, and the uncertainty principle. As an additional benefit, the language developed in this chapter will be especially useful for describing quantum information science in .

This chapter describes specialised equipment and techniques used to perform LEED experiments and to measure intensities of diffracted LEED beams. An overview of the most common setups for experiments will be given. The diffraction geometry is important for the comparison of experimental LEED data with theory and will, thus, be covered in some detail. For the measurement of LEED intensities, close attention will be paid in particular to the preparation of the sample, the accurate alignment of the sample and the physical properties of the detectors, such as the frequently used video cameras. The instrumental response function is one aspect of detectors that can affect the measured intensities, most notably spot profiles used to measure lateral dimensions such as island sizes and disorder. Among various LEED systems that are available on the market, two types will be addressed in relatively more detail, as they provide higher resolution (i.e., are able to detect structural correlations over larger distances along the surface): spot profile analysis LEED (SPA-LEED) and low-energy electron microscope (LEEM). Finally, instrumentation will be described that has been developed for more targeted applications, such as electron-beam sensitive surfaces, and surfaces with micro- or nanoscale structures.

X-ray diffraction is the main tool used to obtain the atomic structure of 3-D crystals. The relatively weak interaction of the X-ray beam with matter and the resulting large penetration depth makes it insensitive to structural details in a small surface area: the surface is therefore usually neglected in structure determinations of 3-D crystals. This has changed with the development of synchrotron radiation as an X-ray source which provided new applications in X-ray crystallography. The very high intensity and angular resolution of the synchrotron beam allow the study of numerous effects which had been considered too weak to detect with laboratory X-ray sources. It has been shown, however, that with intensive X-ray sources that are available now, the structure analysis at surfaces is also possible in the laboratory.

The description of crystal surfaces requires some basic knowledge of crystallography. Therefore, this chapter presents a short overview of crystal lattices and their classification due to symmetry. This knowledge is required to understand the substrate structure and the orientation of the surface. However, the 3-D point groups, space groups and the mathematical description of symmetry operations in three dimensions are not described here: for a more detailed explanation the reader is referred to the International Tables of Crystallography [2.1], which is the standard reference book, or a number of textbooks on crystallography published by the International Union of Crystallography [2.2–2.5]. The 2-D space groups and symmetry operations are explained with somewhat more detail here because these are frequently used in surface structure determination. A very detailed description of the geometry of crystal surfaces is given in a recent book by K. Hermann [2.6]. A short introduction into the kinematic theory of diffraction and into diffraction at 2-D periodic lattices is also included here.

We discuss here the methods of quantitative LEED I(V) analysis and their application to relatively complex types of surface structures: quasicrystalline and modulated surfaces.