Fundamentals of Solid State Engineering

In this chapter the electronic structure of single atoms will be discussed. A few quantum concepts will be introduced, as they are necessary for the understanding of many aspects in solid-state physics and device applications.

In this chapter we review the basic properties of the element carbon as an atom and list the basic properties such as energy levels and electronic structure. We then look at the various ways carbon bonds to form chemical complexes and allotropes emphasizing the amazing versatility of the carbon bond. The most important class of carbon allotropes are enumerated and the structures shown. Carbon isotopes are also examined. The remaining material in this chapter is dedicated to the question of carbon bonding and energy bands in solids. The theory of energy bands in solids in the tight binding methods (TBM) is developed in a separate appendix.

This chapter gives a brief introduction to crystallography, which is the science that studies the structure and properties of the crystalline state of matter. We will first discuss the arrangements of atoms in various solids, distinguishing between single crystals and other forms of solids. We will then describe the properties that result from the periodicity in crystal lattices. A few important crystallography terms most often found in solid state devices will be defined and illustrated in crystals having basic structures. These definitions will then allow us to refer to certain planes and directions within a lattice of arbitrary structure.

In Chapter 0 we saw that classical mechanics was incapable of explaining the optical spectra emitted by atoms or even the existence of atoms. Bohr developed a model for the atom of hydrogen by assuming the quantization of the angular momentum, which was an introduction to wave or quantum mechanics. Quantum mechanics is a more precise approach to describe nearly all physical phenomena which reduces to classical mechanics in the limit where the masses and energies of the particles are large or macroscopic.

In Chap. 4, we introduced quantum mechanics as the proper alternative to classical mechanics to describe physical phenomena, especially when the dimensions of the systems considered approach the atomic scale. The concepts we learned will now be applied to describe the physical properties of electrons in a crystal. During this process, we will make use of the simple quantum mechanical systems which were mathematically treated in the previous chapter. This will lead us to the description of a very important concept in solid-state physics, namely, that of the “energy band structures.”

In the previous chapters, we have considered the electrons in a crystal that consisted of a rigid lattice of atoms. This represented a good approximation because the mass of an atom is more than 2000 of the mass of an electron. However, such assumptions founder when considering specific heat, thermal expansion, the temperature dependence of electron relaxation time, and thermal conductivity. In order to interpret these phenomena involving electrons and atoms, a more refined model needs to be considered, in which the atoms are allowed to move and vibrate around their equilibrium positions in the lattice. In this chapter, we will present a simple yet relatively accurate mathematical model to describe the mechanical vibrations of atoms in a crystal. We will first cover one-dimensional monatomic and diatomic crystals followed by three-dimensional crystals. We will then consider the collective movement or excitations of the atoms in a crystal, the so-called phonons, and conclude with a section on the velocity of sound in a medium.

In Chap. 4, we discussed the quantum mechanical states of electrons in a periodic crystal potential and the resulting formation of energy bands. We also introduced the concept of effective mass, that of holes, and the Fermi energy which provides an easy way to differentiate a semiconductor from a metal.

In the previous chapter, we established the basic relations and formalism for the distribution of electrons in the conduction band and holes in the valence band at thermal equilibrium.

Until now, our discussion was based solely on homogeneous semiconductors whose properties are uniform in space. Although a few devices can be made from such semiconductors, the majority of devices and the most important ones utilize nonhomogeneous semiconductor structures. Most of them involve semiconductor p-n junctions, in which a p-type doped region and an n-type doped region are brought into contact. Such a junction actually forms an electrical diode. This is why it is usual to talk about a p-n junction as a diode. Another important structure involves a semiconductor in intimate contact with a metal, leading to what is called a metal-semiconductor junction. Under certain circumstances, this configuration can also lead to an electrical diode.

In previous chapters, we introduced the reader to the fundamental concepts of quantum mechanics, band structure, and semiconductor physics. In this chapter we have the opportunity to apply this acquired knowledge of the electronic structure of solids to understand the optical properties. We do this by modeling the optical response properties, in particular the permittivity of the solid. We present the formalism which allows one to calculate the permittivity and then study how this permittivity affects the light penetrating the solid. We shall demonstrate how band structure and free electrons determine the permittivity, and therefore the way light propagates in a solid, and how much of this light gets absorbed. We shall investigate under what circumstances the lattice can couple to photons and how this coupling can affect the velocity of light in a medium. But we shall see in the next chapters that band structure depends on the dimensionality of the system, and we have already seen in Chaps. 8 and 9 that carriers can be added or neutralized in semiconductors. So it turns out that just in the same way that the energy bands can be engineered, so can the optical properties. Atom by atom growth and miniaturization are modern key engineering tools, but so is the application of external electric and magnetic fields. In the last sections of this chapter, we therefore investigate how an electric or a magnetic field modifies the band structure and how this reflects on the optical properties. The fundamental concepts developed in this chapter are a necessary prerequisite to understand the way optical methods can be used to characterize the electronic structure of semiconductors as is described in Chap. 15.

In this chapter we focus on the problem of energy harvesting using hot sources. The phenomenon known as thermoelectric power harvesting rests on the fact that current can be generated by using hot sources linked to cold sources, and vice versa, currents can transport heat from cold sources to hot sinks making coolers. The mechanism is reviewed and explained. The efficiency involves electrical and thermal transport properties of materials. In this short chapter, the emphasis is on harvesting not cooling. The theory of electrical conductivity, Seebeck coefficients, Peltier coefficients, and thermal conductivity is considered, and because a low thermal conductivity raises the thermo-harvesting efficiency, disorder can actually play a very useful role in this context. Another question examined here is as to the role of many body interactions. It will be shown that the so-called Kondo insulators, where localized spins bind conduction electrons, exhibit very high Seebeck coefficients (energy transported per carrier), but this exciting phenomenon can at present only be exploited at low temperatures. Photovoltaic cells constitute well-established ways of harvesting solar power, so we also consider the problem of how to efficiently harvest the solar heat, i.e., the energy in the wavelength region not normally addressed by conventional solar technology. This is called “thermophotonic energy harvesting,” and it turns out that some interesting new science has been developed in recent years which has helped to address this question and to raise the efficiency. Thermophotonics is beyond the scope of this book and we refer the reader to the original literature. But one of the basic ideas in this technology is to make devices which strongly absorb radiation in a broadband, the body heats up, and the hot body then emits black body radiation. The short-wavelength components of the reemitted radiation are filtered out by conventional devices leaving the long-wavelength part of the hot body spectrum. The long-wavelength part of the radiation is reflected back into the absorber and recycled back again into the heat using “photonic crystal” mirror technology. If little radiation is allowed to leak out, then the heat absorbed and reabsorbed raises the temperature of the body. In this way the radiation reflected will eventually be radiated out in the allowed higher energy photonic window where photovoltaic cells collect it.

The other and more common strategy for harvesting energy from heat and light is to use PVC devices (see the previous chapter on light harvesting). The PVC devices are very well documented and constitute a mature technology that utilizes solar cells which the reader can access in the literature and books and buy in shops. The problem with current PVC technology is that it is mostly geared to the harvesting of shorter-wavelength region of the sun’s spectrum; see Fig. 12.​32. Silicon is currently still the typical and best material class; the lowest-energy photons collected have energies around 1 eV or 1.2 μm (Figs. 13.1 and 13.2, 13.3).

In this chapter we will investigate how the presence of other charges and dipoles influences the charge-charge interaction.

In Chap. 4, we have introduced the basic concepts and formalism of quantum mechanics. In Chap. 0, we have determined the energy spectrum, or energy-momentum or E–k relations, for electrons in a crystal which governs their interaction with external forces and fields. Moreover, we saw that the quantum behavior of particles is best observed in small, typically nanometer scale (one billionth of a meter or 10⁻⁹ m) dimension structures, as illustrated in the example of a particle in a 1D box.

We have seen in Sect. 16.2.1 how we could define current in classical Drude theory in terms of electrons or charges obeying Newton’s law with frictional forces giving rise to resistance. In Chap. 4 we had introduced the methodology of quantum mechanics and argued that classical physics was not really the right way of looking at dynamics on a microscopic scale. In practice it turns out that the classical theory of transport is very useful indeed, and one can go a long way in understanding transport phenomena in solid-state physics and engineering using the classical method. But there comes a point beyond which the classical description does not work well anymore, and we have to consider the quantum mechanical aspects. This happens on many occasion most of which we cannot discuss here, but we can consider a very simple and common situation where quantum mechanics is needed. Consider a beam of electrons injected, for example, in the conduction band of a semiconductor via an electrode and traveling to the other electrode. Now we can ask what is the current? In classical physics, the answer is obvious if we know the velocity of the carriers. Now we can insert a potential barrier on the way, for example, a material with a higher bandgap as in Fig. 16.1, and ask: what is the resistance produced by the potential barrier on the electrons impinging on it? A classical Drude approach would obviously give us a totally oversimplified and misleading answer to this question. It would require the definition of “frictional force” but which acts only in the form of one obstacle and would not give a satisfactory picture of this well-defined and concrete transport problem. So the right starting point in this case is the quantum mechanical definition of the current (quantum current). To do this, and for simplicity, we consider a one-dimensional situation and write down the continuity equation:

A key component in semiconductor microtechnology is the production and quality control of the basic semiconductor materials from which devices and integrated circuits are made. These semiconductor materials are usually composed of single crystals of high perfection and high purity.

Semiconductor characterization techniques are used in order to gain knowledge on the physical properties of a semiconductor crystal. The process is similar to decoding the DNA sequence of a living organism as it involves understanding the nanoscale structure of the crystal, i.e., its atoms, electrons, structures, and interactions with the surrounding environment. The knowledge gained from the characterization process is essential in determining whether the semiconductor crystal probed is suitable for a particular device component with certain functionalities.

An ideal crystalline solid has a periodic structure that is based on the chemical properties of its constituent atoms (see Chap. 3). However, real crystals are not perfect. They always have imperfections such as extra/missing atoms or impurities, which are called defects.