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Charge carriers in modulation–doped semiconductor quantum systems are a field of enormous and still growing research interest since they allow, in specially tailored systems, the investigation of fundamental properties, such as many–particle interactions, of electrons in reduced dimensions. Over the past decades, the experimental investigation of interacting electrons in low dimensions has led to many new and sometimes unexpected insights into many–particle physics in general. Famous examples are unique electronic transport properties as the integer and fractional quantum–Hall effects in quasi two–dimensional (Q2D) systems. Quasi one–dimensional (Q1D) electron systems, realized in semiconductor quantum wires, have been the subject of intense theoretical and experimental debates concerning the character – Fermi–liquid or Luttinger–liquid – of the interacting Q1D quantum liquid. During the past few years, tunneling–coupled electronic double–layer structures have been revisited as very interesting candidates for the realization of new quantum phases in an interacting many–particle system. A new quality came into the physics of semiconductor nanostructures by the development of quantum systems, embedded in microresonators, also called microcavities. This new inventions allowed one to investigate the light–matter interaction from an advanced point of view.

The majority of experiments of inelastic light scattering on semiconductor nanostructures has been performed on III–V semiconductors, like GaAs, as the most prominent example. In this chapter, an introduction into the basic properties of these materials is given. The first section gives a summary of the crystal and electronic band structure of the bulk material. After a short survey into the properties of electrons in different dimensions in the second section, growth methods for so called vertical nanostructures, i.e., layered heterostructures consisting of two different materials, are described in the third section. In these vertical nanostructures, quasi two–dimensional (Q2D) electron systems can be realized. This section is finalized by the description of commonly used concepts for theoretical calculations of the ground state of such systems. The second last section introduces the most important methods for the preparation of lateral micro and nanostructures. In those structures, the dimensionality of charge carriers or of quasi particles is reduced further by lithography and etching processes, or by self–organized growth methods, resulting in quasi one–dimensional (Q1D) or quasi zero–dimensional (Q0D) quantum structures. The section is finalized by an overview over methods for the calculation of the electronic ground state of lateral nanostructures. Readers who are already familiar with semiconductors and the fabrication and physics of nanostructures may skip this tutorial chapter and directly continue with Chap. 3.

In inelastic light scattering experiments on semiconductor nanostructures, electronic excitations are created or annihilated in the low –dimensional electron systems under investigation. Thus, the main body of this book will deal with the physics of those electronic elementary excitations in various systems and under various conditions. Before we elaborate on the basic concepts of the inelastic light scattering processes themselves in the following chapter, the electronic elementary excitations shall be introduced and discussed here. We will do this by the – most prominent – example of the excitations of Q2D electron systems, realized in modulation–doped GaAs–AlGaAs quantum wells. These excitations can be categorized into so called spin–density excitations (SDE) and charge–density excitations(CDE), which both are collective plasma oscillations of the Q2D system, and, single–particle excitations (SPE). In particular, the observation of intersubband SPE [1] – which are thought to be electronic excitations, which are not affected by the Coulomb interaction – has posed a puzzle, and has been controversially discussed. We will come to this discussion at various places later in this book, when considering the resonant scattering in quantum wells and in quantum dots. In particular in Chap. 5 we will see that – at least for quantum dots – the SPE’s are actually excitations: SDE’s and CDE’s. However, the many–particle interaction effects partly cancel under specific conditions so that the energies are close to single–particle energies of a noninteracting system. Historically, in 1979, intersubband CDE and SDE in GaAs–AlGaAs quantum wells [2] and heterojunctions [3] were the first electronic excitations, which were observed in semiconductor nanostructures by inelastic light scattering by A. Pinczuk et al. and G. Abstreiter et al., respectively.

Inelastic light scattering – or Raman scattering – is the scattering of light by a medium, where in the scattering process excitations are created (Stokes process) or annihilated (Antistokes process) within the medium. In solids, these excitations can be various types of elementary excitations, like phonons, magnons, or – as considered in this book – electronic excitations. Historically, the scattering by optical phonons, or by internal vibrations of molecules, is called Raman scattering, and the scattering by acoustic phonons Brillouin scattering. For Raman scattering, often the term is used, so in this book. Figure 4.1 shows schematically the Stokes and Antistokes processes, where a photon with energy and momentum is scattered by the creation or annihilation of an elementary excitation with energy and momentum . The scattered photon has an energy and momentum . This means, each scattered photon in the Stokes component is associated with a gain in energy by the sample. Similarly, the sample loses energy for each scattered photon in the Antistokes component

Semiconductor quantum dots are fascinating objects, since, in some respect, they can be regarded as artificial atoms [1]. Figure 5.1 shows a very schematic comparison of a real three–dimensional atom and a disc–shaped quantum dot. The structure of real atoms is three–dimensional, while most of the artificial quantum dots can be regarded as large Q2D atoms, since the lateral dimensions are in most cases much larger than the vertical extension. Of course, a crucial difference between the two systems is the shape of the confining potentials, which, for real atoms is essentially the Coulomb potential of the nucleus, and, for quantum–dot atoms in some approximation a two–dimensional parabolic potential.

In 1989, the first inelastic light scattering experiments on electronic excitations in quantum wires were reported [1, 2]. Since then, a number of experimental papers appeared about, e.g., many–particle interactions and selection rules in those systems [3, 4, 5, 6, 7, 8, 9] and investigations with applied external magnetic field [10, 11, 12]. All these experiments were performed on lithographically–defined GaAs–AlGaAs structures. Consequently, the lateral sizes of these structures were on the order of 100 nm, or at least not much below [8, 9]. Unlike for the case of quantum dots, there is no well– established method of self–organized growth of modulation–doped quantum wires. During the past few years, Carbon nanotubes have evolved as new and alternative quantum–wire structures. So far, the main focus in the investigation of those very promising quantum structures by optical experiments has been on phonon excitations [13]. Phonon Raman spectroscopy has greatly helped in unveiling the topological structure of Carbon nanotubes [13]. An interesting further method to produce very narrow wires with atomic–layer precision is the so called cleaved–etched overgrowth (CEO) [14]. However, with CEO it is difficult to grow very large arrays of wires, which would be necessary to get enough signal strength in inelastic light scattering experiments. Hence, there are so far no reports of inelastic light scattering experiments on CEO wires, though these might be promising structures for high–sensitivity experiments. As mentioned, most of the existing experimental reports are on lithographically–defined GaAs–AlGaAs quantum wires with rather mesoscopic widths. Hence, in those experimental structures, typically several Q1D subbands are occupied with electrons. In this chapter we will discuss both, experiments and calculations on such samples. The main focus will be on the microscopic origin of confined plasmons and interesting internal interaction effects in a magnetic field. These experimental results are described well within the RPA, i.e., a Fermi–liquid theory, as we will see later.

The plasmon spectrum of spatially separated two–component plasmas without tunneling has been studied for quite some time (see, e.g., [1]). For finite Coulomb coupling between the layers, the intrasubband charge–density excitation spectrum consists of two modes: The optical plasmon (OP) where both layers oscillate in phase parallel to the layers (see Fig. 7.1a), and, the acoustic plasmon (AP) where the carriers in both layers oscillate out of phase (see Fig. 7.1b). At long wavelengths, the energy of the OP is proportional to √ and the energy of the AP goes linear in , where is the wave vector parallel to the layers. It was shown [1] that at large spatial separation of the two layers, the AP can move outside of the continua of possible intraband single–particle transitions. The first experimental observation of coupled– layer plasmons by inelastic light scattering was reported by Fasol et al. [2] on GaAs-AlGaAs samples containing five layers in parallel. In Coulomb–coupled double quantum wells, the observation of AP and OP was reported by Kainth et al. [3].

A semiconductor microcavity is an optical resonator, where the mirrors consist of alternating layers of two different semiconductors with different refractive indices, e.g., GaAs and AlAs, which have thicknesses of a quarter wavelength each. The resonator itself is called , and has a thickness of a few half wavelengths. The electric fields of the light waves, and hence the light–matter interaction can be modified significantly inside a microcavity. In the past decades, a number of sophisticated experiments have been reported that took advantage of the strongly enhanced electric field inside the spacer of a planar semiconductor microcavity. A prominent example is the enhanced exciton–photon coupling, resulting in an enlarged Rabi splitting, in planar microcavities containing quantum wells [1]. Subsequently, a wealth of theoretical and experimental work on exciton polaritons in semiconductor microcavities, e.g., about the influence of a magnetic field [2], or coupling between different microcavities [3], followed.

In this chapter, Kronecker products corresponding to dipole matrix elements of Bloch functions of the conduction–band edge and the split–off valence–band edge are calculated. The corresponding band–edge Bloch functions are listed in Table 4.29 on page 71. The calculation shall be explained by the following example:

At the fundamental bandgap we have for Q2D systems generally a mixing of heavy and light–hole states. In this case, the dipole matrix elements between a valence–band state , () and a conduction–band state , () is given by