In this chapter, we begin our study of crystalline solids by concentrating on multielectron systems. The first topic we consider is called the Born–Oppenheimer or adiabatic approximation. This approximation enables us to treat the electronic and the ionic components of a solid as two separate, distinct subsystems coupled only through the electron–lattice scatterings. The next topic is that of multielectron systems. The nature of the exchange interaction is examined, and the energies or multielectron systems are studied. Finally, we examine crystalline symmetries and how these properties facilitate our understanding of crystalline solids.

The Born-Oppenheimer Approximation

The basic assumption implicit to the theory of crystalline solids is called the Born–Oppenheimer or the adiabatic approximation. Simply put, the Born–Oppenheimer approximation assumes that a solid can be treated as being composed of two separate subsystems, the electronic and the lattice systems. The elections are considered as moving in a stationary lattice, and the lattice system is treated as being embedded within a uniform electron gas. The dynamics of each subsystem can then be treated independently of the dynamics of the other system, which holds to zeroth order. Coupling between the two subsystems occurs through electron–lattice scatterings. Therefore the Born–Oppenheimer approximation enables us to treat a solid symbolically in the same way as the two separate subsystems shown in Figure 5.3.1 when energy exchange is allowed between them.

The use of the adiabatic approximation is justified in the following way. The masses of the electrons and the ions differ by several orders of magnitude.

As discussed in Chapter 10, a direct-gap semiconductor is a far more efficient emitter and detector of optical radiation since radiative transitions in these materials proceed to first order. In indirect-gap semiconductors, such as silicon and germanium, radiative transitions cannot proceed to first order but require a second process, phonon absorption or emission, etc., to occur. One of the most important characteristics of the compound semiconductors is that many of them are direct-gap semiconductors. For this reason, many of the compound semiconductors are used for light emitters such as light-emitting diodes (LEDs) and semiconductor diode lasers. In this chapter, we discuss optoelectronic semiconductor devices that emit photons.

Light-Emitting Diodes

We begin our discussion of optoelectronic emitters. Perhaps the simplest semi-conductor light emitter is the LED. LEDs have become pervasive because of their low cost, high efficiency, wide spectral capabilities, relatively simple drive circuitry, high reliability, and very long lifetime. LEDs are most familiar as displays and indicator lamps. However, the relatively low cost, high efficiency, long lifetime, and reliability make them attractive candidates to replace incandescent bulbs in many applications, that is, particularly in those applications in which it is difficult to replace the lamp (like automobile dashboards). Future lighting systems in automobiles, traffic lights, outdoor lighting fixtures, etc., may also utilize LEDs for these reasons.

As discussed in Chapter 10, there are two basic types of radiative transitions, spontaneous and stimulated. Stimulated emission requires the presence of an electromagnetic field to induce a transition.

In the previous chapters, we have solved a series of problems that all have an exact solution. However, there are only a select few problems in quantum mechanics that provide exact analytical solutions. Those problems that can be solved exactly are the free particle, the one-dimensional barrier potential, the finite and the infinite square wells, the infinite triangular well, the harmonic oscillator, and the hydrogen atom, including both its nonrelativistic and relativistic solutions. However, there are many problems different from the above-mentioned few as well as problems that cannot be approximated by these solutions. What happens then if we want to solve a different problem, such as the helium atom or the hydrogen atom in an applied electric field? To solve these problems it is necessary to use approximation techniques. In this chapter, we consider some of these techniques, the most important being the time-independent and the time-dependent perturbation theories. We begin with a discussion of time-independent disturbances in a nondegenerate system.

Time-independent Perturbation Theory: The Nondegenerate Case

We seek to determine the solution to a problem in which a small disturbance is applied to an otherwise determinable system. Perturbation theory is used most often in cases for which the solution to the undisturbed (unperturbed) system is known and the perturbation is small. For example, perturbation theory is useful in solving for the energy levels in a hydrogen atom in the presence of a small external electric field.

In this chapter, we consider what happens when the lattice is no longer in equilibrium. In Chapter 8 we considered the motion of electrons in a crystal assuming that the ions were at rest for all times about their equilibrium positions. The presence of the ionic potential on the electrons results in energy-band formation: allowed ranges of energies for the electrons. However, this is not a physically realistic situation, since at nonzero lattice temperatures, the ions undergo some oscillation about their equilibrium positions. This additional motion, called lattice vibration, has important consequences on the behavior of the electrons. In Section 9.1 we consider lattice vibrations and the quanta of lattice vibrations, phonons. In the subsequent sections, we then examine how phonons influence the electronic properties of a crystal.

Lattice Vibrations and Phonons

From the discussion in Chapter 7, we found that the equilibrium configuration of the ions results in the minimum potential energy of the crystal. As the ions become more compressed, a repulsive force due to the nuclear-nuclear interaction acts to restore the system back to equilibrium. Similarly, as the lattice is stretched, the attractive force from the molecular bonds formed between adjacent ions acts again to restore equilibrium. Hence a lattice can be thought of as a collection of mass centers situated at definite positions and held together by the action of an elastic restoring force. The system has a definite minimum potential energy.

The maturation of epitaxial crystal growth capabilities, such as molecularbeam epitaxy and chemical beam epitaxy, has enabled the realization of a host of new ultrasmall semiconductor devices. Aside from the feature size reduction of conventional semiconductor devices, particularly transistors, a totally new class of semiconductor devices has been invented. These structures, called superlattices/multiple-quantum-well devices, consist of alternating layers of different semiconductor materials, often measuring only a few atomic layers thick. These new semiconductor devices operate well within the range in which quantum-mechanical phenomena become prevalent. As a consequence, most new semiconductor devices behave according to quantum-mechanical effects rather than classical effects. Therefore the understanding of these new device types requires a firm grounding in the basics of quantum mechanics. It is the purpose of this book to introduce the engineering student, particularly those interested in studying solid-state devices, to the principles of quantum mechanics, statistical mechanics, and solid-state physics. Following this introduction, the physics of semiconductors and various device structures is examined.

The book contains fourteen chapters in total. The first four chapters are concerned with the standard principles of quantum mechanics for a one-particle system. I have attempted to condense the vast literature on this subject into just four chapters that will present the salient features of quantum mechanics. I have included a few topics, most notably a short presentation on relativistic quantum mechanics, for completeness. The instructor may elect to skip different sections as he or she sees fit.

One of the most important physical mechanisms of importance to semiconductor devices is the field effect. Several important devices exploit this effect in their operation, such as metal-oxide–semiconductor field-effect transistors (MOSFETs), metal–semiconductor field-effect transistors (MESFETs), and junction field-effect transistors (JFETs). In fact, the field effect transistor (FET) is arguably the most important innovation that has fueled the computer and information revolution. In this chapter, the fundamentals of the FET operation are presented; the reader is directed to the references for a more comprehensive study.

The Field Effect

The field effect can be simply defined as the modulation of the conductivity of an underlying semiconductor layer by the application of an electric field to a gate electrode on the surface. As we learned in Chapter 11, the application of a bias to a MIS structure results in a modulation in the carrier concentration within the underlying semiconductor layer. If the semiconductor is naturally n type and a positive gate bias is applied, electrons accumulate at the semiconductor-insulator interface. Conversely, if a negative gate bias is applied to the same structure, the electrons are repelled from the interface and, depending on the magnitude of the bias, the underlying semiconductor layer is either depleted or inverted. If the semiconductor becomes inverted, the carrier type changes.

Quantum mechanics forms the basis of modern physics. In a sense it is the parent theory about which we construct our view of the physical world. Briefly, quantum mechanics is the theory by which we describe the behaviors of subatomic and atomic particles, such as electrons, of which the macroscopic world is made. Although it is not necessary to treat macroscopic objects by use of quantum mechanics, the laws of quantum mechanics and their implications are completely consistent with Newton's Laws of Motion, which we know are applicable to most macroscopic objects. As we will see below, Newton's Laws of Motion are a special subset of quantum mechanics; quantum mechanics reduces to Newton's Laws at macroscopic dimensions.

Before we begin our study of quantum mechanics, it is of interest to explain why quantum mechanics is of importance in the study of modern electrical engineering. Many new areas of electrical engineering are based on developments that can be understood only through the use of quantum mechanics. Among these are the broad areas of

1. semiconductors and solid-state electronic devices,

2. electro-optics and lasers,

3. superconductors.

It would be fair to say that in the study of each of the above areas some knowledge of quantum mechanics is essential. In this book, some basic concepts in quantum mechanics are presented that are necessary in the study of the above-mentioned disciplines.

Introduction

The concept most basic toward the understanding of quantum mechanics is the concept of measurement.

In this chapter, we consider generation and recombination processes in semiconductors. In a semiconductor, electrons and holes can be either generated or recombined within a given volume, thereby changing the local carrier concentrations. In a sense, there are sources and sinks of particles within the semiconductor itself. Although generation and recombination events change the local carrier concentrations, the entire semiconductor must always remain space-charge neutral. This requirement leads to the injection or extraction of charge at the contacts. In this chapter, we examine the different types of generation and recombination processes and outline their behaviors.

Basic Generation-Recombination Mechanisms

There are in general three basic generation–recombination channels available in semiconductors. These are

1. Auger,

2. radiative,

3. thermal.

These mechanisms are defined as follows.

An Auger process is defined as an electron–hole pair (EHP) recombination followed by a transfer of energy from the recombined EHP to a free carrier, which is then excited to high energy within the band. The inverse Auger effect, in which an EHP is produced, is called impact ionization. In this case, a high-energy free carrier collides with the lattice and transfers its excess kinetic energy to an electron in the valence band, promoting it to the conduction band. Hence an EHP is produced after the event.

In a radiative recombination event, an EHP recombines with the emission of a photon. The electron recombines from the conduction band with a hole in the valence band.

In this chapter, we discuss the physics of any two junctions formed between two crystalline solids both in equilibrium and nonequilibrium. In general there are many types of junctions that can be formed between two different crystalline materials. Specifically, the junctions we are most concerned with are p–n homojunctions, p–n or n–n heterojunctions, metal–semiconductor junctions, and metal-insulator–semiconductor (MIS) junctions. The p–n homojunction consists of n- and p-type layers made from the same material type, a common silicon p–n junction diode, for example. A heterojunction is formed from two dissimilar material types that are often doped differently as well. For example, a common heterojunction of great use in modern semiconductor devices is that formed from n-type AlGaAs on either intrinsic GaAs or p-type GaAs.

In addition to semiconductor–semiconductor junctions, metal–semiconductor and MIS junctions can be formed as well. The two most important types of metal-semiconductor junctions are Schottky barriers, which have diodelike, rectifying current-voltage characteristics, or ohmic contacts, which have linear current-voltage characteristics.

Knowledge of the equilibrium and the nonequilibrium properties of these junction types, along with the earlier topics covered in this book, will provide us with sufficient background to study advanced semiconductor devices in the next chapters. First we consider the equilibrium properties of each junction type. Next we consider the nonequilibrium current-flow processes in each junction. Our method is to treat these different junction types, when possible, by using a unified approach.

A preface is supposed to alert potential readers to the contents and style of what lies within so they can decide whether to proceed further. Superconductivity is now a vast subject extending from the esoteric to the very practical; the people who study or work on it have different preparations, goals, and talents. No treatment can or should address all these dimensions.

Part I is devoted to the phenomenological aspects of superconductivity, e.g., London's and Pippard's electrodynamics, the Ginzburg–Landau theory and the Landau Fermi liquid theory. These theories allow a discussion of the effects of magnetic fields, interfaces and boundaries, fluctuations, and collective response (which may all be thought of as different manifestations of inhomogeneities).

Since there is currently much interest in unconventional (non-s-wave) superconductivity, we have included a discussion of the associated Ginzburg–Landau theory (which then has a multidimensional, complex order parameter). He is the only established example of an unconventional superfluid (triplet p-wave) and therefore our discussions of this substance are somewhat longer.

Part II is devoted to the microscopic theory of uniform superconductors: the theory of Bardeen, Cooper, and Schrieffer (BCS) and the Bogoliubov–Valatin canonical transformation, where the latter so greatly simplifies the discussion of excited states and finite temperature effects. Although not strictly a uniform-superconductor phenomenon, the theory of tunneling and the accompanying Josephson effects are also discussed in Part II.