Matthias and co-workers, in a series of electron spin resonance (ESR) and nuclear magnetic resonance (NMR) experiments on non-magnetic metals – metals with no permanent magnetic moment – observed surprising evidence for long-lived local spin packets in the ESR lineshape (M1960). These data indicated the persistence of local magnetic moments. The magnetic moment was quickly traced to the presence of small amounts of magnetic impurities. While various systems were studied, such as Mn, Fe, and other iron group impurities in host materials such as Cu, Ag, and Au, the common ingredient shared by all the impurity ions is that they possessed one or more vacant inner-shell orbitals. In addition, the experiments demonstrated that varying the kind and amount of the magnetic impurities did not always result in the formation of local magnetic moments in non-magnetic metals. This finding added to the intrigue and established the question of the formation of local magnetic moments as central to understanding magnetism and transport in solids. In this chapter, we describe the origin of local moments, focusing primarily on Anderson's model (A1961), the model that rose to the fore as the standard microscopic view of local magnetic moment formation in metals.

Local moments: phenomenology

An impurity in a non-magnetic metal can give rise to a local moment if an electronic state on the impurity is singly occupied, at least on the time scale of the experiment. Friedel (F1958) was the first to introduce a phenomenological model to explain the onset of local moments.

When an electron gas is confined to move at the interface between two semiconductors and a magnetic field is applied perpendicular to the plane, a new state of matter (TSG1982) arises at sufficiently low temperatures. This state of matter is unique in condensed matter physics in that it has a gap to all excitations and exhibits fractional statistics. It is generally referred to as an incompressible quantum liquid or as a Laughlin liquid (L1983), in reference to the architect of this state. While the Laughlin state is mediated by the mutual repulsions among the electrons, it is the presence of the large perpendicular magnetic field that leads to the incompressible nature of this new many-body state. The precursor to this state is the integer quantum Hall state. In this state, disorder and the magnetic field conspire to limit the relevant charge transport to a narrow strip around the rim of the sample. The novel feature of this rim or edge current is that it is quantized in integer multiples of e2/h (KDP1980). The equivalent current in the Laughlin state is still quantized but rather in fractional multiples of e2/h. We present in this chapter the phenomenology and the mathematical description needed to understand the essential physics of both of these effects.

As we will see, topology is an integral part of the quantum Hall effect. Regardless of the geometry or smooth changes in the Hamiltonian, the quantization of the conductance depends solely on the existence of edge states which have a well-defined chirality.

In the previous chapter we developed a mean-field criterion for local magnetic moment formation in a metal. As mean-field theory is valid typically at high temperatures, we anticipate that at low temperatures, significant departures from this treatment occur. The questions we focus on in this chapter are: (1) how does the presence of local magnetic moments affect the low-temperature transport and magnetic properties of the host metal, and (2) what is the fate of local magnetic moments at low temperatures in a metal? These questions are of extreme experimental importance because it has been known since the early 1930s that the resistivity of a host metal such as Cu with trace amounts of magnetic impurities, typically Fe, reaches a minimum and then increases as – ln T as the temperature subsequently decreases.

At the conceptual level, the quantization procedure replaces the pure particle description of classical mechanics by a particle-wave treatment satisfying the requirements of the fundamental wave–particle dualism. Technically, one can always start from a generalized wave equation and use it as a basis to discuss wave-front propagation in the form of rays following classical particle trajectories, see Sections 2.2.1 and 3.1. Consequently, any wave theory inherently contains that aspect of the wave–particle dualism.

Especially, Maxwell's equations of electrodynamic theory include both wave and particle aspects of electromagnetic fields. Hence, it could be that there are no additional “quantum,” i.e., wave–particle dualistic aspects. However, as we discuss in this chapter, Maxwell's equations can be expressed in the framework of Hamilton's formulation of particle mechanics. This then suggests that in addition to the approximate ray-like behavior, electromagnetic fields do have a supplementary level of particle-like properties.

Due to the fundamental axiom of wave–particle dualism, the additional particle-like aspects of electromagnetic fields must then also have a wave counterpart that can be axiomatically introduced via the canonical quantization scheme described in Section 3.2.3. Since the spatial dependency in Maxwell's equation contains wave–particle dualism already at the classical level, it is clear that the new level of quantization cannot be a simple real-space feature but has to involve some other space describing the structure of the electromagnetic fields. In fact, this quantization yields several very concrete implications such as quantization of the light energy, elementary fluctuations in the light-field amplitude and intensity, and more.

The optically generated excitations in semiconductors constitute a genuine many-body system. To describe its quantum-optical features, we have to expand significantly the theoretical models used so far. However, the important insights of Chapters 16–23 are already presented in a form in which most of them can directly be used and generalized to analyze central properties of the optical excitations in solids. As for atoms, the optical transitions in semiconductors are induced via dipole interaction between photons and electrons. We can thus efficiently construct a systematic quantum-optical theory for semiconductors by following the cluster-expansion approach.

One of the main differences from atoms is that the electronic excitations in semiconductors form a strongly interacting many-body system. Thus, we must systematically treat the arising Coulomb-induced hierarchy problem together with the quantum-optical one. Moreover, the coupling of electrons to lattice vibrations, i.e., the phonons, produces yet another hierarchy problem. In addition, in solid-state spectroscopy one often uses multimode light fields such that one cannot rely on the single-mode simplifications to study semiconductor quantum optics.

As shown in Chapter 15, the Coulomb-, phonon-, and photon-induced hierarchy problems have formally an identical structure. Thus, we start the analysis by investigating how semiconductor quantum optics emerges from the dynamics of correlated clusters. We first focus on the basic properties of the optical transitions in the classical regime. This means investigating the fundamental optical phenomena resulting from the singlets. The full singlet–doublet approach is presented in Chapters 28–30.