Although Brockhouse (1961) originally invented the triple-axis spectrometer for the study of inelastic processes, it has also proven to be an exceptionally useful instrument for measuring elastic scattering in cases where the intensity is inherently low, such as in small single crystals and powders. The three-axis instrument has played an invaluable role in studies of magnetic and structural phase transitions where new superlattice Bragg peaks appear below some critical temperature. By using an analyzer, a better signal-to-noise ratio can be obtained than with the more traditional two-axis instrument with no analyzer. The intrinsic flexibility of a three-axis instrument allows one to vary the incoming and outgoing energy, and this is invaluable in detecting and eliminating the ever-present double scattering occurring within a crystal. The triple-axis instrument has developed into a versatile powder spectrometer because of the ability to significantly change the instrumental resolution by the choice of different analyzer reflections, neutron energies, and collimations.

Three-axis vs. two-axis instruments

Using a triple-axis instrument for measuring elastic scattering can significantly improve the signal-to-noise ratio when compared to the more conventional two-axis instrument used in single-crystal diffractometers or powder spectrometers. The analyzer selects only the scattering that is elastic (within the resolution of the instrument) and the inelastic scattering from the phonons, which occurs at finite energy transfers, is automatically excluded. In a two-axis instrument this phonon scattering always appears as background, and is commonly referred to as thermal diffuse scattering (TDS). Also, since analyzer crystals such as pyrolytic graphite have reflectivities of greater than 50%, there is little intensity lost.

In this introductory chapter we discuss some of the properties of the neutron and how it interacts with matter. We compare steady-state reactors to pulsed spallation neutron sources. After a review of the scattering geometry of an experiment we discuss the various instruments used for inelastic scattering.

Properties of thermal neutrons

The neutron is an ideal probe with which to study condensed matter. It was first discovered in 1932, and four years later it was demonstrated that neutrons could be Bragg diffracted by solids (see Bacon, 1986). The early experimenters struggled with the low flux from Ra–Be sources but, nevertheless, established the fundamentals of neutron diffraction. The future was assured by the construction of the first “atomic pile” by Fermi and his co-workers in 1942. Subsequent reactors produced more and more neutrons and the latest generation of high-flux reactors built in the 1960s and early 1970s produce a copious number of neutrons, making inelastic scattering studies practical. It is now widely accepted that neutron scattering is one of the most important and versatile techniques for probing condensed matter. This fact was recognized by the awarding of the Nobel Prize in Physics in 1994 to Profs. C. Shull of Massachusetts Institute of Technology (MIT) and B. Brockhouse of McMaster University (Canada) for their seminal contributions to the fields of elastic and inelastic neutron scattering. Arguably the most important instrument used in neutron spectroscopy is the triple-axis spectrometer (invented by Brockhouse, 1961) since it allows for a controlled measurement of the scattering function S(Q, ω) at essentially any point in momentum (ħQ) and energy (ħω) space.

Introductory remarks

The wide array of optical techniques and methods which are used for studying the electrodynamic properties of solids in the different spectral ranges of interest for condensed matter physics is covered by a large number of books and articles which focus on different aspects of this vast field of condensed matter physics. Here we take a broader view, but at the same time limit ourselves to the various principles of optical measurements and compromise on the details. Not only conventional optical methods are summarized here but also techniques which are employed below the traditional optical range of infrared, visible, and ultraviolet light. These techniques have become increasingly popular as attention has shifted from singleparticle to collective properties of the electron states of solids where the relevant energies are usually significantly smaller than the single-particle energies of metals and semiconductors.

We start with the definition of propagation and scattering of electromagnetic waves, the principles of propagation in the various spectral ranges, and summarize the main ideas behind the resonant and non-resonant structures which are utilized. This is followed by the summary of spectroscopic principles – frequency and time domain as well as Fourier transform spectroscopy. We conclude with the description of measurement configurations, single path, interferometric, and resonant methods where we also address the relative advantages and disadvantages of the various measurement configurations.

Optical experiments on semiconductors have led to some of the most powerful confirmations of the one-electron theory of solids; these experiments provide ample evidence for direct and indirect gaps, and in addition for excitonic states. Optical studies have also contributed much to our current understanding of doping semiconductors, including the existence and properties of impurity states and the nature of metal–insulator transitions which occur by increasing the dopant concentration. Experiments on amorphous semiconductors highlight the essential differences between the crystalline and the amorphous solid state, and the effects associated with the loss of lattice periodicity. We first focus on experiments performed on pure band semiconductors for which the one-electron theory applies, where direct and indirect transitions and also forbidden transitions are observed; in these materials the subtleties of band structure have also been explored by experimentation. This is followed by examples of the optical effects associated with exciton and impurity states. Subsequently we consider the effects of electron–electron and electron–lattice interactions, and finally we discuss optical experiments on amorphous semiconductors, i.e. on materials for which band theory obviously does not apply.

Band semiconductors

The term band semiconductor refers to materials where the non-conducting state is brought about by the interaction of electrons with the periodic underlying lattice. Single-particle effects – accounted for by band structure calculations – are responsible for the optical properties under such circumstances, these properties reflecting interband transitions.

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.