Semiconductors and devices based on them are ubiquitous in every aspect of modern life. From “gameboys” to personal computers, from the brains behind “nintendo” to world wide satellite phones—semiconductors contribute to life perhaps like no other manmade material. Silicon and semiconductor have entered the vocabulary of newscasters and stockbrokers. Parents driving their kids cross-country are grudgingly grateful to the “baby-sitting service” provided by ever more complex “gameboys.” Cell phones and pagers have suddenly brought modernity to remote villages. “How exciting,” some say. “When will it all end?” say others.

The ever expanding world of semiconductors brings new challenges and opportunities to the student of semiconductor physics and devices. Every year brings new materials and structures into the fold of what we call semiconductors. New physical phenomena need to be grasped as structures become ever smaller.

SURVEY OF ADVANCES IN SEMICONDUCTOR PHYSICS

In Fig. I.1 we show an overview of progress in semiconductor physics and devices, since the initial understanding of the band theory in the 1930s. In this text we explore the physics behind all of the features listed in this figure. Let us take a brief look at the topics illustrated.

In the previous chapter we have seen how the intrinsic properties of a semiconductor as reflected by its chemical composition and crystalline structure lead to the unique electronic properties of the material. Can the bandstructure of a material be changed? The answer is yes, and the ability to tailor the bandstructure is a powerful tool. Novel devices can be conceived and designed for superior and tailorable performance. Also new physical effects can be observed. In this chapter we will establish the physical concepts which are responsible for bandstructure modifications. There are three widely used approaches for band tailoring (or engineering). These three approaches are shown in Fig. 3.1 and are:

1. Alloying of two or more semiconductors;

2. Use of heterostructures to cause quantum confinement; and

3. Use of built-in strain via lattice mismatched epitaxy.

These three concepts are increasingly being used for improved performance in electronic and optical devices.

BANDSTRUCTURE OF SEMICONDUCTOR ALLOYS

The easiest way to alter the electronic properties or to produce a material with new properties is based on making an alloy. Alloying of two materials is one of the oldest techniques to modify properties of materials, not only in semiconductors, but in metals and insulators as well.

INTRODUCTION

The bandstructure and optical properties of semiconductors we have discussed so far are based on the assumption that the valence band is filled with electrons and the conduciton band is empty. The effect of electrons in the conduction band and holes in the valence band is only manifested through the occupation probabilities without altering the bandstructure. In reality, of course, there is a Coulombic interaction between an electron and another electron or hole. Some very important properties are modified by such interactions. The full theory of the electron-electron interaction depends upon many body theory, which is beyond the scope of this text. However, there is one important problem, that of excitonic effects in semiconductors, that can be addressed by simpler theoretical techniques.

In Fig. 10.1 we show how exciton effects arise. On the left-hand side, we show the bandstructure of a semiconductor with a full valence band and an empty conduction band. There are no allowed states in the bandgap. Now consider the case where there is one electron in the conduction band and one hole in the valence band. In this new configuration, the Hamiltonian describing the electronic system has changed. We now have an additional Coulombic interaction between the electron and the hole. The electronic bandstructure should thus be modified to reflect this change. The electron-hole system, coupled through the Coulombic interaction, is called the exciton and will be the subject of this chapter.

The general field of how semiconductor properties are modified in the presence of a magnetic field is a very wide one. To do justice to the field, one would need to devote several chapters to this area as we have done for electric field effects. However, it can be argued that from a technology point of view the response of electrons in semiconductors to electric fields and optical fields is more important. Magnetic field effects are primarily used for material characterization, although there is growing interest in magnetic semiconductors for device applications. In view of this fact we will provide an overview of how electrons in semiconductors respond to magnetic fields in just one chapter. In addition to a magnetic field, many important characterization techniques are carried out in the presence of an electric field or an optical field. We will therefore also discuss magneto-transport and magneto-optic properties. The general category of problems we will examine are sketched in Fig. 11.1.

In Fig. 11.1 we broadly differentiate between the “free” or Bloch states in semiconductors and the electron-hole coupled states like excitons or bound states. The magnetic field greatly alters the nature of the electronic states which then manifests itself in magneto-optic or magneto-transport phenomenon. It is important to realize that in many cases the physical phenomenon can qualitatively alter, depending upon the strength of the magnetic field. We will address the problem of electrons in the presence of a magnetic field in two steps.

We begin with a brief overview of magnetic behavior in different types of solids. We first define the terms used to describe the various types of magnetic behavior. A system is called paramagnetic if it has no inherent magnetization, but when subjected to an external field it develops magnetization which is aligned with the field; this corresponds to situations where the microscopic magnetic moments (like the spins) tend to be oriented in the same direction as the external magnetic field. A system is called diamagnetic if it has no inherent magnetization, but when subjected to an external field it develops magnetization that is opposite to the field; this corresponds to situations where the induced microscopic magnetic moments tend to shield the external magnetic field. Finally, a system may exhibit magnetic order even in the absence of an external field. If the microscopic magnetic moments tend to be oriented in the same direction, the system is described as ferromagnetic. A variation on this theme is the situation in which microscopic magnetic moments tend to have parallel orientation but they are not necessarily equal at neighboring sites, which is described as ferrimagnetic behavior. If magnetic moments at neighboring sites tend to point in opposite directions, the system is described as antiferromagnetic. In the latter case there is inherent magnetic order due to the orientation of the microscopic magnetic moments, but the net macroscopic magnetization is zero.

• 1954 Nobel prize for Chemistry: Linus Carl Pauling, for his research into the nature of the chemical bond and its application to the elucidation of the structure of complex substances.

• 1956 Nobel prize for Physics: William Shockley, John Bardeen and Walter Houser Brattain, for their research in semiconductors and their discovery of the transistor effect.

• 1962 Nobel prize for Medicine: Francis Harry Compton Crick, James Dewey Watson and Maurice Hugh Frederick Wilkins, for their discoveries concerning the molecular structure of nucleic acids and its significance for information transfer in living material.

• 1968 Nobel prize for Medicine: Robert W. Holley, Hav Gobind Khorana and Marshall W. Nirenberg, for their interpretation of the genetic code and its function in protein synthesis.

• 1972 Nobel prize for Physics: John Bardeen, Leon N. Cooper and J. Robert Schrieffer, for their jointly developed theory of superconductivity, usually called the BCS theory.

• 1977 Nobel prize for Physics: Philip W. Anderson, Sir Nevill F. Mott and John H. van Vleck, for their fundamental theoretical investigations of the electronic structure of magnetic and disordered systems.

• 1985 Nobel prize for Chemistry: Herbert A. Hauptman and Jerome Karle, for their outstanding achievements in the development of direct methods for the determination of crystal structures.

• 1985 Nobel prize for Physics: Klaus von Klitzing, for the discovery of the quantized Hall effect.

• 1986 Nobel prize for Physics: Gerd Binning and Heinrich Rohrer, for their design of the scanning tunneling microscope; Ernst Ruska, for his fundamental work in electron optics, and for the design of the first electron microscope.

• […]

Solids are the physical objects with which we come into contact continuously in our everyday life. Solids are composed of atoms. This was first postulated by the ancient Greek philosopher Demokritos, but was established scientifically in the 20th century. The atoms (ατομα = indivisible units) that Demokritos conceived bear no resemblance to what we know today to be the basic units from which all solids and molecules are built. Nevertheless, this postulate is one of the greatest feats of the human intellect, especially since it was not motivated by direct experimental evidence but was the result of pure logical deduction.

There is an amazing degree of regularity in the structure of solids. Many solids are crystalline in nature, that is, the atoms are arranged in a regular three-dimensional periodic pattern. There is a wide variety of crystal structures formed by different elements and by different combinations of elements.

In the previous two chapters we examined in detail the effects of crystal periodicity and crystal symmetry on the eigenvalues and wavefunctions of the single-particle equations. The models we used to illustrate these effects were artificial free-electron models, where the only effect of the presence of the lattice is to impose the symmetry restrictions on the eigenvalues and eigenfunctions. We also saw how a weak periodic potential can split the degeneracies of certain eigenvalues at the Bragg planes (the BZ edges). In realistic situations the potential is certainly not zero, as in the free-electron model, nor is it necessarily weak. Our task here is to develop methods for determining the solutions to the single-particle equations for realistic systems. We will do this by discussing first the so called tight-binding approximation, which takes us in the most natural way from electronic states that are characteristic of atoms (atomic orbitals) to states that correspond to crystalline solids. We will then discuss briefly more general methods for obtaining the band structure of solids, whose application typically involves a large computational effort. Finally, we will conclude the chapter by discussing the electronic structure of several representative crystals, as obtained by elaborate computational methods; we will also attempt to interpret these results in the context of the tight-binding approximation.

The tight-binding approximation

The simplest method for calculating band structures, both conceptually and computationally, is the so called Tight-Binding Approximation (TBA), also referred to as Linear Combination of Atomic Orbitals (LCAO). The latter term is actually used in a wider sense, as we will explain below.

Polarons in ionic crystals

The behavior of an electron in polarizable medium has a long history of study with versatile developments in condensed matter physics. In 1933 Landau conceived of the self-trapping of an electron in a potential field of polarization induced by itself (digging its own hole, so to speak) as a possible origin of the color center in alkali halides (see Sections 4.7 and 4.8), recognized by their strong absorption bands in the visible region. Although the color center itself turned out later to be an electron at a lattice defect, his idea was further developed by Pekar, Fröhlich, Lee and Pines, Feynman and others, partly stimulated by the field-theoretical approaches to an electron in a self-induced electromagnetic field in a vacuum, and was applied to similar or related problems in condensed matter physics. We shall describe some of these developments and applications in this chapter, with the use of the dielectric theory developed in Chapters 5 and 6.

Let us consider the fluctuating electric field in an isotropic insulator caused by lattice vibrations. The fluctuation–dissipation theorem allows us to express it in terms of the dielectric dispersion as follows. In the absence of an external charge, eq. (1.1.3) with D = ∈eE + P gives ∈ek · Ek = –k · Pk. The polarization density P due to lattice vibrations causes electrostatic potential ø(r) through E = – ∇ø(r).

Frenkel excitons

When we considered low-energy elementary excitations such as optical phonons (Section 5.3) and the low-energy binding of donor electrons (Section 7.4), we described the background polarization effect of the electrons tightly bound to host atoms phenomenologically in terms of a constant susceptibility ∈e – ∈0 assuming that the polarizations have characteristic frequencies high enough to follow instantaneously the aforementioned low-frequency motions. In this chapter we will be concerned with the high-energy elementary excitations of tightly bound electrons contributing to susceptibility ∈e – ∈0 of the background polarization, in particular with excitons, which is a first step to incorporating configuration interactions among one-electron excitations.

As the simplest model, consider an isolated atom (or ion) with one electron in the 1s state with energy ε1s. An applied electric field gives rise to electronic polarization of the atom which quantum mechanically is due to the mixing of wave functions of odd parity np(n = 2, 3, …) states with energies εnp to that of the even-parity 1s state. After the field is removed, the mixing coefficients contributing to the polarization oscillate with angular frequencies ωn = (εnp – ε1s)/ħ. Namely, the elementary excitations responsible for the polarization of an atom are excitations of its electron to higher states with different parity.

If identical atoms (or molecules) form an assemblage, the excitation of an atom resonates with those of neighboring atoms and, via the interatomic interactions, the excitation energy will propagate from atom to atom like a wave if the atoms are arrayed on a crystal lattice.

Superconductivity was discovered in 1911 by Kamerling Onnes and remains one of the most actively studied aspects of the behavior of solids. It is a truly remarkable phenomenon of purely quantum mechanical nature; it is also an essentially many-body phenomenon which cannot be described within the single-particle picture. Because of its fascinating nature and of its many applications, superconductivity has been the focus of intense theoretical and experimental investigations ever since its discovery. Studies of superconductivity have gained new vigor since the discovery of high-temperature superconductors in 1987.

Overview of superconducting behavior

Superconductivity is mostly characterized by a vanishing electrical resistance below a certain temperature Tc, called the critical temperature. Below Tc there is no measurable DC resistance in a superconductor and, if a current is set up in it, it will flow without dissipation practically forever: experiments trying to detect changes in the magnetic field associated with current in a superconductor give estimates that it is constant for 106 – 109 years! Thus, the superconducting state is not a state of merely very low resistance, but one with a truly zero resistance. This is different than the case of very good conductors. In fact, materials which are very good conductors in their normal state typically do not exhibit superconductivity. The reason is that in very good conductors there is little coupling between phonons and electrons, since it is scattering by phonons which gives rise to resistance in a conductor, whereas electron-phonon coupling is crucial for superconductivity.

Introduction

This chapter is of a different nature from the preceding chapters where more or less well-established principles of the spectroscopic studies of microscopic motions in condensed matter have been described, as is usual in texbooks and monographs. In this last chapter, however, the author is going to present his personal viewpoint on the role of solar radiation in creating life out of matter, sustaining living activities and driving the evolution of the living world – topics which have generally been considered to be beyond the scope of the physical sciences.

Obviously, life science is a vast interdisciplinary regime: the elucidation of life will need the cooperation of all areas of natural science. The majority of scientists working in any discipline will supposedly have a serious interest in the problem of life even when they do not mention that explicitly. The particular reason why the present author feels it necessary and useful to make remarks on this problem is that the various photochemical processes described in the preceding few chapters and the behaviors of charged particles in dielectrics described in other chapters have something to do with, and to shed light on, the role of solar radiation as an energy source and the role of water as a catalyst in living activities. Further investigation of these roles might contribute to constructing a bridge from our small corner of physical science, among many other such bridges under construction from different areas of science, towards elucidating the physical origin and evolution of life itself.

What the author can do as a theoretical physicist is present speculations based on the various empirical and experimental facts available.

Simple crystal lattices

Atoms in molecules and solids are bound together by the valence electrons which have been supplied by them and are shared among them. In this chapter we will consider the motion of atoms under given interatomic forces without asking about their nature – the electronic origin of the bonds. We will confine ourselves to solids and study some simple characteristics of the atomic motions within them – lattice vibrations – brought about by the periodic structure of the crystal lattice. The lattice vibrations will turn out to be a prototype of elementary excitations in a solid, other examples of which will be given in later sections.

Consider an assemblage of a great number (N) of identical atoms with interatomic potential ν(r) which typically is attractive except for the hard-core repulsion, and let them be arrayed in one-dimensional space with positions x1, x2, … in increasing order. In the approximation that only the nearest neighbor potentials are considered, as is allowed when ν(r) is short ranged, the total potential energy is given by U ≃ ν(x1 – x2) + ν(x2 – x3) + …, and the most stable arrangement of the atoms is a periodic array with equal distance a at which ν(r) is a minimum.