In a crystalline material atoms vibrate about the rigid lattice sites and one of the most important scattering mechanisms for mobile carriers in semiconductors is due to these vibrations. In our discussions for the bandstructure we assumed that the background potential is periodic, and does not have any time dependence. In actual materials the background ions forming the crystal are not fixed rigidly but vibrate. The vibration increases as the temperature is increased. To understand the properties of electrons in a vibrating structure we use an approach shown schematically in Fig. 6.1. Scattering will occur due to the potential disturbances by the lattice vibration. Before we can answer the question regarding how lattice vibrations cause scattering, we must understand some basic properties of these vibrations. Once we understand the nature of the lattice vibrations we can begin to examine how the potential fluctuations arising from these vibrations cause scattering.

LATTICE VIBRATIONS

In Chapter 1 we have discussed how atoms are arranged in a crystalline material. The reason a particular crystal structure is chosen by a material has to do with the minimum energy of the system. As atoms are brought together to form a crystal, there is an attractive potential that tends to bring the atoms closer and a repulsive potential which tends to keep them apart. The attractive interaction is due to a variety of different causes including Van der Waals forces (resulting from the dipole moment created when an atoms' electron cloud is disturbed by the presence of another atom), ionic bonding where electrons are transferred from one atom to another and covalent bonding where electrons are shared between atoms.

In Chapter 4 we have derived a number of important mathematical relations necessary to calculate transport properties. A key ingredient of the theory is the scattering rate W(k, k′) which tells us how an electron in a state k scatters into the state k′. We will now evaluate the scattering rates for a number of important scattering mechanisms. As noted in Chapter 4, the approach used by us is semiclassical—the electron is treated as a Bloch wave while calculating the scattering rate, but is otherwise treated as a particle. The Fermi golden rule is used to calculate the scattering rate.

In Fig. 5.1 we show an overview of how one goes about a transport calculation. Once the various imperfections in a material are identified the first and most important ingredient is an understanding of the scattering potential. This may seem like a simple problem, but is, in fact, one of the most difficult parts of the problem. Once the potential is known, one evaluates the scattering matrix element between the initial and final state of the electron. This effectively amounts to taking a Fourier transform of the potential since the initial and final states are essentially plane wave states. With the matrix element known one carries out an integral over all final states into which the electron can scatter and which are consistent with energy conservation. This kind of integral provides the various scattering times.

INTRODUCTION

Interactions of electrons and photons in semiconductors form the basis of technologies such as optical communications, display, and optical memories. In this and the next chapter we will discuss how electrons in a semiconductor interact with light. To describe this interaction, light has to be treated as particles (i.e., photons). The problem is mathematically quite similar to the electron-phonon (lattice vibration) scattering problem discussed in Chapter 6. Electron-photon interactions are described via scattering theory through an absorption or emission of a photon. Both intraband and interband processes can occur as shown in Fig. 9.1. Intraband scattering in semiconductors is an important source of loss in lasers and can usually be described by a Drude-like model where a sinusoidal electric field interacts with electrons or holes. Monte Carlo methods or other transport models can account for it quite adequately. The interband scattering involving valence and conduction band states is, of course, most important for optical devices such as lasers and detectors. In addition to the band-to-band transitions, increasing interest has recently focussed on excitonic states especially in quantum well structures. The exciton-photon interaction in semiconductor structures contains important physics and is also of great technical interest for high speed modulation devices and optical switches. Excitonic effects will be discussed in the next chapter.

We will briefly review some important concepts in electromagnetic theory and then discuss the interactions between electrons and photons. We will focus on the special aspects of this interaction for semiconductor electrons, especially those relating to selection rules.

INTRODUCTION

In the previous chapters on transport we have applied Born approximation or the Fermi golden rule to describe the scattering processes in semiconductors. While the approach described in these chapters and the outcome is most relevant to modern microelectronic devices there are a number of important issues that are not described by this approach. As semiconductor devices and technology evolve, these issues are becoming increasingly important. In this chapter we will discuss some transport issues that are not described by the formalisms of the previous three chapters.

In Fig. 8.1 we show several types of structural properties of materials. In Fig. 8.1a we show a perfect crystal where there are no sources of scattering. Of course, in a real material we have phonon related fluctuations even in a perfect material. However, for short times or at very low temperatures it is possible to consider a material with no scattering. There are several types of transport that are of interest when there is no scattering: i) ballistic transport, where electrons move according to the modified Newton's equation. This kind of transport has been discussed in Section 7.3.2; and ii) Bloch oscillations, where electrons oscillate in k-space as they reach the Brillouin zone edge, as will be discussed in Section 8.2. In addition we can have tunneling type transport as well as quantum interference effects. These are discussed in Sections 8.3 and 8.4.

INTRODUCTION

The properties of electrons inside semiconductors are described by the solution of the Schrödinger equation appropriate for the crystal. The solutions provide us the bandstructure or the electronic spectrum for electrons. The problem of finding the electronic spectrum is an enormously complicated one. Solids have a large number of closely spaced atoms providing the electrons a very complex potential energy profile. Additionally electrons interact with each other and in a real solid atoms are vibrating causing time dependent variations in the potential energy. To simplify the problem the potential fluctuations created by atomic vibrations (lattice vibrations) and scattering of electrons from other electrons are removed from the problem and treated later on via perturbation theory. These perturbations cause scattering of electrons from one state to another.

The problem of bandstructure becomes greatly simplified if we are dealing with crystalline materials. An electron in a rigid crystal structure sees a periodic background potential. As a result the wavefunctions for the electron satisfy Bloch's theorem as discussed in the next section.

There are two main categories of realistic bandstructure calculation for semiconductors:

1. Methods which describe the entire valence and conduction bands.

2. Methods which describe near bandedge bandstructures.

The techniques in the second category are simpler and considerably more accurate if one is interested only in phenomena near the bandedges. Techniques such as the tight binding method, the pseudopotential method, and the orthogonalized plane wave methods fall in the first category.

The data and plots shown in this Appendix are extracted from a number of sources. A list of useful sources is given below. Note that impact ionization coefficient and Auger coefficients of many materials are not known exactly.

• S. Adachi, J. Appl. Phys., 58, R1 (1985).

• H.C. Casey, Jr. and M.B. Panish, Heterostructure Lasers, Part A, “Fundamental Principles;” Part B, “Materials and Operating Characteristics,” Academic Press, N.Y. (1978).

• Landolt-Bornstein, Numerical Data and Functional Relationship in Science and Technology, Vol. 22, Eds. O. Madelung, M. Schulz, and H. Weiss, Springer-Verlog, N.Y. (1987). Other volumes in this series are also very useful.

• S.M. Sze, Physics of Semiconductor Devices, Wiley, N.Y. (1981). This is an excellent source of a variety of useful information on semiconductors.

• “World Wide Web;” A huge collection of data can be found on the Web. Several professors and industrial scientists have placed very useful information on their websites.

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.

INTRODUCTION

In the last chapter, we examined resonator-enhanced blue-green light generation, in which a nonlinear crystal is placed inside an optical resonator so that the high circulating intensity increases the efficiency of SHG or SFG. We considered some implementations of this approach in which light from a diode-pumped solid-state laser is coupled into such a resonator, and we saw that it becomes necessary to lock the laser frequency to a resonant frequency of the enhancement cavity. Looking at such a system, we might well ask, “Since the solid-state laser itself consists of a cavity which is resonant at the infrared wavelength, why not place the nonlinear crystal inside that cavity instead of inside a separate one?” Inclusion of the nonlinear crystal within the resonator of an infrared laser is the basic idea behind intracavity SHG and SFG, which is the subject of this chapter.

Although generation of green light by intracavity frequency doubling of neodymium lasers has been pursued since the mid-1960s (Smith et al., 1965, Geusic et al., 1968), the current wave of interest in this field was ignited in the mid-1980s by the development of high-power, high-brightness diode lasers capable of efficiently pumping solid-state lasers and the demonstration that milliwatt levels of green light could be generated by placing a nonlinear crystal within the cavity of a diode-pumped Nd3+ laser (Baer and Keirstead, 1985, Fan et al., 1986).

A SHORT HISTORICAL OVERVIEW

For years after its invention in 1961, the laser was described as a remarkable tool in search of an application. However, by the late 1970s and early 1980s, a variety of applications had emerged that were limited in their implementation by lack of a suitable laser. The common thread running through these applications was the need for a powerful, compact, rugged, inexpensive source of light in the blue-green portion of the spectrum. The details varied greatly, depending on the application: some required tunability, some a fixed wavelength; some required a miniscule amount of optical power, others a great deal; some required continuous-wave (cw) oscillation, others rapid modulation.

In many of these applications, gas lasers – such as argon-ion or helium-cadmium lasers – were initially used to provide blue-green light, and in some cases were incorporated into commercial products; however, they could not satisfy the requirements of every application. The lasing wavelengths available from these lasers are fixed by the atomic transitions of the gas species, and some applications required a laser wavelength that is simply not available from an argon-ion or helium–cadmium laser. Other applications required a degree of tunability that is unavailable from a gas laser. In many of them, the limited lifetime, mechanical fragility, and relatively large size of gas lasers was a problem.

At about the same time, new options for generation of blue-green radiation began to appear, due to developments in other areas of laser science and technology.

Since the mid-1980s, the development of practical, powerful sources of coherent visible light has received intense interest and concentrated activity. This interest and activity was fueled by twin circumstances: the realization of powerful, efficient infrared laser diodes and the emergence of numerous applications that required compact visible sources. The availability of these infrared lasers affected the development of visible sources in two ways: It stimulated the investigation of techniques for efficiently converting the infrared output of these lasers to the visible portion of the spectrum and it encouraged the hope that the fabrication techniques themselves might be adapted to make similar devices working at shorter wavelengths.

Within the visible spectrum the blue-green wavelength region has demanded – and received – special attention. The demonstration of powerful red diode lasers followed relatively soon after the development of their infrared counterparts – in contrast, the extension to shorter blue-green wavelengths has required decades of wrestling with the idiosyncrasies of wide-bandgap materials systems. The first blue-green diode lasers were not successfully demonstrated until 1991, and it has only been within the past year or two that long-lived devices with output powers of tens of milliwatts have been achieved.

As this field emerged and began to grow, it quickly became evident that it would necessarily be a very multi-disciplinary one. On one hand, a variety of approaches were being pursued in order to generate blue-green light. The three main ones – nonlinear frequency conversion, upconversion lasers, blue-green semiconductor lasers – are the focus of this book.

INTRODUCTION

In the preceding chapter, we considered single-pass SHG and SFG. There, we saw that efficient frequency upconversion from infrared to blue-green wavelengths is generally possible only when the power at the fundamental wavelength is several watts. The approach to achieving such powers that we considered in Chapter 3 was a very direct and “brute force” one: build a more powerful laser. We examined several approaches that have been used for increasing the infrared power available for the nonlinear interaction, including:

• using a power amplifier to boost the output of a master oscillator;

• using high-power diode lasers that have poor spectral and spatial characteristics for pumping solid-state lasers which then act as sources for frequency-doubling;

• using pulsed, rather than cw, operation in order to achieve higher peak powers.

While these brute force approaches have the advantage of being conceptually straightforward, it has only been since about 1995 that they have succeeded in producing blue-green powers sufficient for some of the applications described in Chapter 1. In addition, these approaches suffer from a number of practical disadvantages. The powerful lasers required for efficient single-pass conversion tend to be complicated and expensive, and since they generate high powers they require substantial electrical power and thermal management. Furthermore, although pulsed configurations have succeeded in producing large average blue-green powers, the power generated by cw operation has been too low for many applications.

BACKGROUND

VCSELs have gained importance in recent years for applications where beam quality, prospects for high-density arrays, and inherent compatibility with planar processing are particularly important. In the case of resonant-cavity LEDs (RCLEDs), the quasi-beam-like directionality in the spontaneous emission and possible enhancements to the radiative recombination rates likewise have spurred active research. VCSEL technologies that rely on III–V semiconductor heterostructures have now risen to a dominant position within the semiconductor laser industry, supplying high-performance components that play an increasingly vital role in optical communications technology. Both GaAs- and phosphide-based QW VCSELs are making significant headway in penetrating into the 1.3–1.5µm wavelength region, following spectacular device successes in the roughly 650–900 nm range in the 1990s.

To date, the shortest wavelength VCSELs that have been implemented have reached the short end of the red (∼630 nm). There are a number of reasons, both fundamental and practical, that make the development of blue and green VCSELs and RCLEDs in the wide-gap semiconductors challenging. In terms of the technological approaches and prospects for short-wavelength VCSELs and RCLEDs, this chapter is speculative in tone, given the early stages of research. At this writing, it is unclear what combination of epitaxial growth and device design/processing schemes might result in a technologically viable VCSEL, for instance. On the other hand, there are ample fundamental physical reasons that suggest that microcavity emitters based on wide-gap semiconductors, and the nitrides in particular, have special properties that offer unique opportunities both in terms of the basic physics and device performance.