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The stability of metal clusters exhibits shell effects similar to that of nuclei. This chapter reviews how this feature is treated in the jellium model. The main focus is on optical properties described by the dielectric function, which is analyzed in greater detail, first for the Drude-Lorentz model then for a fully quantal treatment. With increasing volume of the clusters, only bulk properties typical for a metal are important. For smaller systems, quantum size effects come into play. This effect is studied, reporting on microscopic calculations within the jellium model. Of special interest is the damping width, for which finite values are obtained even at small frequencies if the quantal electronic states are treated as being quasi-continuous. This mechanism is often associated with Landau damping known to conserve entropy. The problem related to this fact is examined, together with the analogous one of wall friction in finite nuclei.

Lattice vibrations which involve a fluctuating electric dipole have quite different properties from non-polar vibrations. In particular, they couple directly via the dipole moment to the radiation field in the crystal to form mixed phonon-photon modes which have a characteristic dispersion relation and are known as polaritons. Their frequencies generally depend on wavevector and on their polarization, and can be readily explored by direct absorption of infrared radiation or by the inelastic scattering of light. In this chapter, soft modes are viewed as damped harmonic oscillators with possibly temperature-dependent descriptive parameters. The effects of electronic contributions to electric dipole moment are discussed and represented by a single damped harmonic oscillator describing electron-shell vibrations about the ionic cores. The chapter also looks at the dielectric function and linear response, infrared spectra, Raman spectroscopy, some experimental studies, and Rayleigh scattering and critical opalescence.

This chapter summarizes the fundamental facts and concepts of classical electromagnetism, applied in successive chapters to the consideration of the electronic plasma excitations supported by various metal and metal–semiconductor nanostructures. The chapter outlines the basic principles of electrodynamics, the Drude theory describing dielectric function of free electrons, propagation of surface plasmon polaritons at a plane interface between a metal and an insulator, and excitation of localized surface plasmons in metal particles. The quasi-static approximation is represented for the general case of ellipsoidal particles, whereas the exact Mie theory is employed to define plasmonic excitations in conducting spheres. Thus, the goal of the chapter is to give a basic overview of the plasmonic effects and to introduce notations, conventions, and units of electrodynamics that are consistently used throughout the rest of the book.

This chapter deals with the electron dynamics in extended systems; more precisely, three-dimensional periodic solids. It begins with a brief review of electronic band-structure theory, explaining the difference between metals and insulators. It then considers linear-response theory in periodic systems, with particular emphasis on the dielectric tensor. The macroscopic dielectric function is the quantity one is usually interested in. The next section discusses the spectroscopic properties of charge-density fluctuations in metallic systems. Plasmons are introduced for homogeneous systems and real metals. Time-dependent density-functional theory gives very accurate results for plasmon dispersions. The third section focuses on optical absorption and excitons in semiconductors and insulators. It begins with a simple introduction of excitons, and then shows that time-dependent density-functional theory requires long-ranged exchange-correlation kernels to produce excitonic binding. A simple two-band model is discussed in some detail. Finally, time-dependent current-density-functional theory for periodic systems is reviewed.

This chapter covers the optical and electronic properties of available conducting materials capable of supporting plasmonic excitations. Described here are noble metals and polyvalent metals (gold, silver, indium, and aluminium), degenerate semiconductors, doped oxides and semimetals, as well as graphene. The area of applicability of the considered materials for plasmonics is determined on the basis of their complex dielectric functions defined by both interband and intraband transitions. The properties of metal–dielectric composites are treated in the framework of the generalized Maxwell–Garnett approximation. The influence on fundamental parameters of semiconductor compounds of their deviation from stoichiometry is discussed in terms of the empirical tight binding theory. Where possible, the chapter provides explicit expressions and plots defining the spectra of complex dielectric functions and other important characteristics.

This chapter focuses on the properties associated with linear response. Reversibility holds in linear transformations. Schrödinger and Maxwell equations are linear, yet the world is irreversible, with time marching forward and dissipation quite ubiquitous. The connections between the quantum and microscopic scale, which are reversible and non-deterministic, to the macroscale, where irreversibility and determinism abounds, arise through interactions where both linear and nonlinear responses can appear. Causality’s implication in linear response is illustrated through a toy example and a quantum-statistical view of response. Linear response theory—using Green’s functions—is applied to develop dispersion relationships and dielectric function. The tie-in between real and imaginary parts is illustrated as one example of the Kramers-Kronig relationship, and the linear response of a damped oscillator and the Lorentz model, together with the oscillating electron model, employed to illustrate the dielectric function implications.

This chapter develops an approach that includes electron–electron interaction, which is analogous to what one does in electromagnetic theory where the effects of the environment are incorporated through a dielectric function (or in the homogeneous static limit, a dielectric constant). It discusses the model of an interacting electron liquid. In order to preserve charge neutrality it is assumed that there is a compensating, uniform, non-deformable, positive background. The first deals with the self-consistent potential method for a uniform electron system. The second section presents applications of the dielectric function formalism, covering static screening, plasma oscillations, zero sound, and the Kohn effect. An appendix extends the periodicity associated with a crystalline solid to cover the periodic case. Sample problems are also provided at the end of the chapter.

Electrons in a semiconductor undergo polar scattering due to charged impurities, piezoelectric modes, holes, other electrons, and optical phonons — all of which are susceptible to electrical screening by the mobile electron gas. Because the frequencies of acoustic phonons which can interact with electrons are quite low, the screening of the piezoelectric interaction can be adequately described by static screening. When the density of the electron gas is high, dynamic screening and plasma effects become inextricably mixed and must be treated together. This chapter focuses on the dynamic screening of electrons in semiconductors, first by considering polar optical modes, plasma modes, and coupled modes. It then discusses the Lindhard dielectric function, fluctuations, and four regimes of dynamic screening effects.

The linearised nonlocal kinetic equation is solved analytically for impurity scattering. The resulting response function provides the conductivity, plasma oscillation and Fermi momentum. It is found that virial corrections nearly compensate the wave-function renormalizations rendering the conductivity and plasma mode unchanged. Due to the appearance of the correlated density, the Luttinger theorem does not hold and the screening length is influenced. Explicit results are given for a typical semiconductor. Elastic scattering of electrons by impurities is the simplest but still very interesting dissipative mechanism in semiconductors. Its simplicity follows from the absence of the impurity dynamics, so that individual collisions are described by the motion of an electron in a fixed potential.

Chapter 12 introduces Graphene, which is a two-dimensional “Dirac-like” material in the sense that its energy spectrum resembles that of a relativistic electron/positron (hole) described by the Dirac equation (having zero mass in this case). Its device-friendly properties of high electron mobility and excellent sensitivity as a sensor have attracted a huge world-wide research effort since its discovery about ten years ago. Here, the associated retarded Graphene Green’s function is treated and the dynamic, non-local dielectric function is discussed in the degenerate limit. The effects of a quantizing magnetic field on the Green’s function of a Graphene sheet and on its energy spectrum are derived in detail: Also the magnetic-field Green’s function and energy spectrum of a Graphene sheet with a quantum dot (modelled by a 2D Dirac delta-function potential) are thoroughly examined. Furthermore, Chapter 12 similarly addresses the problem of a Graphene anti-dot lattice in a magnetic field, discussing the Green’s function for propagation along the lattice axis, with a formulation of the associated eigen-energy dispersion relation. Finally, magnetic Landau quantization effects on the statistical thermodynamics of Graphene, including its Free Energy and magnetic moment, are also treated in Chapter 12 and are seen to exhibit magnetic oscillatory features.