Optical investigations have contributed much to our current understanding of the electronic state of conductors. Early studies have focused on the behavior of simple metals, on the single-particle and collective responses of the free-electron gas, and on Fermi-surface phenomena. Here the relevant energy scales are the single-particle bandwidth W, the plasma frequency ωp, and the single-particle scattering rate 1/τ, all lying in the spectral range of conventional optics. Consequently, when simple metals are investigated standard optical studies are of primary importance. Recent focus areas include the influence of electron–electron and electron–phonon interactions on the electron states, the possibility of non-Fermi-liquid states, the highly anisotropic, in particular two-dimensional, electron gas, together with disorder driven metal–insulator transition. Here, because of renormalization effects and low carrier density, and also often because of close proximity to a phase transition, the energy scales are – as a rule – significantly smaller than the single-particle energies. Consequently the exploration of low energy electrodynamics, i.e. the response in the millimeter wave spectral range or below, is of central importance.

Simple metals

In a broad range of metals – most notably alkaline metals, but also metals like aluminum – the kinetic energy of the electrons is large, significantly larger then the potential energy created by the periodic underlying lattice. Also, because of screening the strength of electron–electron and electron–phonon interactions is small; they can for all practical purposes be neglected.

Introductory remarks

In this part we develop the formalisms which describe the interaction of light (and sometimes also of a test charge) with the electronic states of solids. We follow usual conventions, and the transverse and longitudinal responses are treated hand in hand. Throughout the book we use simplifying assumptions: we treat only homogeneous media, also with cubic symmetry, and assume that linear response theory is valid. In discussing various models of the electron states we limit ourselves to local response theory – except in the case of metals where non-local effects are also introduced. Only simple metals and semiconductors are treated; and we offer the simple description of (weak coupling) broken symmetry – superconducting and density wave – states, all more or less finished chapters of condensed matter physics. Current topics of the electrodynamics of the electron states of solids are treated together with the experimental state of affairs in Part 3. We make extensive use of computer generated figures to visualize the results.

After some necessary preliminaries on the propagation and scattering of electromagnetic radiation, we define the optical constants, including those which are utilized at the low energy end of the electrodynamic spectrum, and summarize the so-called Kramers–Kronig relations together with the sum rules. The response to transverse and longitudinal fields is described in terms of correlation and response functions. These are then utilized under simplified assumptions such as the Drude model for metals or simple band-to-band transitions in the case of semiconductors.

In Chapter 2 we described the propagation of electromagnetic radiation in free space and in a homogeneous medium, together with the changes in the amplitude and phase of the fields which occur at the interface between two media. Our next objective is to discuss some general properties of what we call the response of the medium to electromagnetic fields, properties which are independent of the particular description of solids; i.e. properties which are valid for basically all materials. The difference between longitudinal and transverse responses will be discussed first, followed by the derivation of the Kramers–Kronig relations and their consequences, the so-called sum rules. These relations and sum rules are derived on general theoretical grounds; they are extremely useful and widely utilized in the analysis of experimental results.

Longitudinal and transverse responses

General considerations

The electric field strength of the propagating electromagnetic radiation can be split into a longitudinal component EL=(nq·E)nq and a transverse component ET=(nq×E)×nq, with E=EL+ET, where nq=q/|q| indicates the unit vector along the direction of propagation q. While EL∥q, the transverse part ET lies in the plane perpendicular to the direction q in which the electromagnetic radiation propagates; it can be further decomposed into two polarizations which are usually chosen to be normal to each other.

The purpose of spectroscopy as applied to solid state physics is the investigation of the (complex) response as a function of wavevector and energy; here, in the spirit of optical spectroscopy, however, we limit ourselves to the response sampled at the zero wavevector, q=0 limit. Any spectroscopic system contains four major components: a radiation source, the sample or device under test, a detector, and some mechanism to select, to change, and to measure the frequency of the applied electromagnetic radiation. First we deal with the various energy scales of interest. Then we comment on the complex response and the requirements placed on the measured optical parameters. In the following sections we discuss how electromagnetic radiation can be generated, detected, and characterized; finally we give an overview of the experimental principles.

Energy scales

Charge excitations which are examined by optical methods span an enormous spectral range in solids. The single-particle energy scales of common metals such as aluminum – the bandwidth W, the Fermi energy εF, together with the plasma frequency ħωp – all fall into the 1–10 eV energy range, corresponding to the visible and ultraviolet parts of the spectrum of electromagnetic radiation. In band semiconductors like germanium, the bandwidth and the plasma frequency are similar to values which are found in simple metals; the single-particle bandgap εg ranges from 10−1 eV to 5 eV as we go from small bandgap semiconductors, such as InSb, to insulators, such as diamond.

The exploration of the electrodynamic response has played an important role in establishing the fundamental properties of both the superconducting state and the density wave states. The implications of the BCS theory (and related theories for density waves) – the gap in the single-particle excitation spectrum, the phase coherence in the ground state built up of electron–electron (or electron–hole) pairs, and the pairing correlations – have fundamental implications which have been examined by theory and by experiment, the two progressing hand in hand. The ground state couples directly to the electromagnetic fields with the phase of the order parameter being of crucial importance, while single-particle excitations lead to absorption of electromagnetic radiation – both features are thoroughly documented in the various broken symmetry states. Such experiments have also provided important early evidence supporting the BCS theory of superconductivity.

There is, by now, a considerable number of superconductors for which the weak coupling theory or the assumption of the gap having an s-wave symmetry do not apply. In several materials the superconducting state is accounted for by assuming strong electron–phonon coupling, and in this case the spectral characteristics of the coupling can be extracted from experiments. Strong electron–electron interactions also have important consequences on superconductivity, not merely through renormalization effects but also leading possibly to new types of broken symmetry. In another group of materials, such as the so–called high temperature superconductors, the symmetry of the ground state is predominantly d-wave, as established by a variety of studies.

In the expressions (2.4.15) and (2.4.21) we arrived at the power ratio reflected by or transmitted through the surface of an infinitely thick medium, which is characterized by the optical constants n and k. For a material of finite thickness d, the situation becomes more complicated because the electromagnetic radiation which is transmitted through the first interface does not entirely pass through the second interface; part of it is reflected from the back of the material. This portion eventually hits the surface, where again part of it is transmitted and contributes to the backgoing signal, while the remaining portion is reflected again and stays inside the material. This multireflection continues infinitely with decreasing intensity as depicted in Fig. B.1.

In this appendix we discuss some of the optical effects related to multireflection which becomes particularly important in media with a thickness smaller than the skin depth but (significantly) larger than half the wavelength. Note, the skin depth does not define a sharp boundary but serves as a characteristic length scale which indicates that, for materials which are considerably thicker than δ0, most of the radiation is absorbed before it reaches the rear side. First we introduce the notion of film impedance before the concept of impedance mismatch is applied to a multilayer system. We finally derive expressions for the reflection and transmission factors of various multilayer systems.

Optics, as defined in this book, is concerned with the interaction of electromagnetic radiation with matter. The theoretical description of the phenomena and the analysis of the experimental results are based on Maxwell's equations and on their solution for time-varying electric and magnetic fields. The optical properties of solids have been the subject of extensive treatises [Sok67, Ste63, Woo72]; most of these focus on the parameters which are accessible with conventional optical methods using light in the infrared, visible, and ultraviolet spectral range. The approach taken here is more general and includes the discussion of those aspects of the interaction of electromagnetic waves with matter which are particularly relevant to experiments conducted at lower frequencies, typically in the millimeter wave and microwave spectral range, but also for radio frequencies.

After introducing Maxwell's equations, we present the time dependent solution of the equations leading to wave propagation. In order to describe modifications of the fields in the presence of matter, the material parameters which characterize the medium have to be introduced: the conductivity and the dielectric constant. In the following step, we define the optical constants which characterize the propagation and dissipation of electromagnetic waves in the medium: the refractive index and the impedance. Next, phenomena which occur at the interface of free space and matter (or in general between two media with different optical constants) are described. This discussion eventually leads to the introduction of the optical parameters which are accessible to experiment: the optical reflectivity and transmission.

In the previous chapters the response of the medium to the electromagnetic waves was described in a phenomenological manner in terms of the frequency and wavevector dependent complex dielectric constant and conductivity. Our task at hand now is to relate these parameters to the changes in the electronic states of solids, brought about by the electromagnetic fields or by external potentials. Several routes can be chosen to achieve this goal. First we derive the celebrated Kubo formula: the conductivity given in terms of current–current correlation functions. The expression is general and not limited to electrical transport; it can be used in the context of different correlation functions, and has been useful in a variety of transport problems in condensed matter. We use it in the subsequent chapters to discuss the complex, frequency dependent conductivity. This is followed by the description of the response to a scalar field given in terms of the density–density correlations. Although this formalism has few limitations, in the following discussion we restrict ourselves to electronic states which have well defined momenta. In Section 4.2 formulas for the so-called semiclassical approximation are given; it is utilized in later chapters when the electrodynamics of the various broken symmetry states is discussed. Next, the response to longitudinal and transverse electromagnetic fields is treated in terms of the Bloch wavefunctions, and we derive the well known Lindhard dielectric function: the expression is used for longitudinal excitations of the electron gas; the response to transverse electromagnetic fields is accounted for in terms of the conductivity.

Introductory remarks

The theoretical concepts of metals, semiconductors, and the various broken symmetry states were developed in Part 1. Our objective in this part is to subject these theories to test by looking at some examples, and thus to check the validity of the assumptions which lie behind the theories and to extract some important parameters which can be compared with results obtained by utilizing other methods.

We first focus our attention on simple metals and simple semiconductors, on which experiments have been conducted since the early days of solid state physics. Perhaps not too surprisingly the comparison between theory and experiment is satisfactory, with the differences attributed to complexities associated with the electron states of solids which, although important, will not be treated here. We also discuss topics of current interest, materials where electron–electron, electron–phonon interactions and/or disorder are important. These interactions fundamentally change the character of the electronic states – and consequently the optical properties. These topics also indicate some general trends of condensed matter physics.

The discussion of metals and semiconductors is followed by the review of optical experiments on various broken symmetry ground states. Examples involving the BCS superconducting state are followed by observations on materials where the conditions of the weak coupling BCS approach are not adequate, and we also describe the current experimental state of affairs on materials with charge or spin density wave ground states.

This book has its origins in a set of lecture notes, assembled at UCLA for a graduate course on the optical studies of solids. In preparing the course it soon became apparent that a modern, up to date summary of the field is not available. More than a quarter of a century has elapsed since the book by Wooten: Optical Properties of Solids – and also several monographs – appeared in print. The progress in optical studies of materials, in methodology, experiments and theory has been substantial, and optical studies (often in combination with other methods) have made definite contributions to and their marks in several areas of solid state physics. There appeared to be a clear need for a summary of the state of affairs – even if with a somewhat limited scope.

Our intention was to summarize those aspects of the optical studies which have by now earned their well deserved place in various fields of condensed matter physics, and, at the same time, to bring forth those areas of research which are at the focus of current attention, where unresolved issues abound. Prepared by experimentalists, the rigors of formalism are avoided. Instead, the aim was to reflect upon the fact that the subject matter is much like other fields of solid state physics where progress is made by consulting both theory and experiment, and invariably by choosing the technique which is most appropriate.

The focus of this chapter is on the optical properties of band semiconductors and insulators. The central feature of these materials is the appearance of a single-particle gap, separating the valence band from the conduction band. The former is full and the latter is empty at zero temperature. The Fermi energy lies between these bands, leading to zero dc conduction at T=0, and to a finite static dielectric constant. In contrast to metals, interband transitions from the valence band to the conduction band are of superior importance, and these excitations are responsible for the main features of the electrodynamic properties. Many of the phenomena discussed in this chapter also become relevant for higher energy excitations in metals when the transition between bands becomes appreciable for the optical absorption.

Following the outline of Chapter 5, we first introduce the Lorentz model, a phenomenological description which, while obviously not the appropriate description of the state of affairs, reproduces many of the optical characteristics of semiconductors. The transverse conductivity of a semiconductor is then described, utilizing the formalisms which we have developed in Chapter 4, and the absorption near the bandgap is discussed in detail, followed by a summary of band structure effects. After discussing longitudinal excitations and the q dependent optical response, we briefly mention indirect transitions and finite temperature and impurity effects; some of the discussion of these phenomena, however, is relegated to Chapter 13.