Probing dynamical properties at equilibrium

By definition, the macroscopic properties of a system at equilibrium do not change with time. This does not mean, however, that the system is dynamically inert. On the contrary, the equilibrium state is associated with permanent motion at the molecular level. Although this motion is sometimes described as random ‘thermal noise’, it is in fact quite well organized, and reflects the microscopic processes that govern the dynamics of the system. As these same processes will also determine the system response to an external perturbation, understanding their organization and time evolution is of primary importance for the determination of the material response, and of its relationship to the microscopic structure.

As a simple illustration of how seemingly random dynamical processes are organized in a coherent fashion, one may consider the case of vibrations in a crystalline solid. Each atom oscillates around its equilibrium position, in a way which is apparently very random. The well known harmonic analysis, however, shows that this motion is really caused by the superposition of well defined sound waves, the phonons, with different phases and directions of propagation. This organization of the atomic motions into excitations of well defined spatial and temporal structure determines many thermodynamic and transport properties of the crystal.

The harmonic crystal is of course particularly simple, since it is possible in that case to deduce analytically from the interaction potential the structure of the coherent excitations. In more complex, disordered systems, an analytical treatment is usually out of reach. Nevertheless, the information on the way atomic motions are organized is encoded in the correlation functions of atomic positions.

While the solid state of matter is mostly associated with the mineral world and much of modern technology, liquids are more closely related to living matter and biological processes. In fact life is generally believed to have emerged in the primordial ocean which was formed when the right temperature conditions came to prevail on the young Earth, providing a striking illustration of the ‘marginal’ character of the liquid state, compared to the solid and gaseous phases of the same substances, which exist over much wider ranges of temperature and pressure. The liquid state arises from a delicate balance between ‘packing’ of molecules and cohesive forces or, more formally, between entropy and energy, which renders a statistical description very difficult, due to the absence of any obvious ‘small parameter’. This may explain that, while the gaseous and crystalline states of matter were well understood by the 1950s, significant theoretical progress on the liquid state only just started around that time, and was then speeded up by early neutron scattering and computer simulation data.

From the start, the exploration of the liquid state was interdisciplinary par excellence, thanks to the combined efforts of physicists, physical chemists and chemical engineers. However, early theoretical work evolved along two lines of thought. ‘Simple’ liquids were studied on a molecular scale, using statistical mechanics and computer simulations as basic tools, while ‘complex’ fluids (sometimes referred to as ‘soft matter’ following P.G. de Gennes) were mostly examined on a more coarse-grained level, epitomized in the scaling approach to the theory of polymer solutions.

Lattice models and mean field treatment

Changes in temperature, pressure or chemical composition can lead to continuous or discontinuous phase changes, as illustrated schematically in figure 1.1. Phase transitions are ubiquitous in simple and complex fluids, and the thermodynamic conditions for phase equilibria have been spelled out in section 2.3. The most common phase transitions, like vapour condensation, freezing or the isotropic to nematic transition in liquid crystals, are first-order transitions, characterized by discontinuities of first derivatives of the free energy, like the entropy or the molar volume. Two (or more phases) can coexist over a range of temperatures or pressures and metastable thermodynamic states of one phase can exist for appreciable times under conditions where another phase has a lower free energy. Most first-order transitions can be reasonably well described within mean field theory, which is the leitmotiv of the present chapter. A key quantity in the description of first-order transitions is the order parameter, which generally characterizes some broken symmetry, taking a finite value in the phase of lower symmetry, and vanishing discontinuously at the transition towards the phase of higher symmetry. In many cases the order parameter can be associated with some microscopic variable, like the orientation vector of mesogenic molecules, but in the case of a transition between two isotropic fluids, with full rotational and translational invariance, like a liquid and its coexisting vapour, or two liquid mixtures of different compositions, the order parameter is a macroscopic characteristic, conveniently chosen to be the difference in density or concentration between the two phases.

Thermodynamics is the branch of physical sciences developed in the 19th century to provide a systematic theoretical framework for the characterization of physical and chemical transformations of substances, involving exchange of heat, work and matter. It is an essentially macroscopic and phenomenological theory, which ignores the molecular nature of matter; in fact at the time the laws of thermodynamics were formulated, atomistic theories were far from being universally accepted. The link between macroscopic thermodynamics and the modern microscopic description of matter was provided later by statistical mechanics, which is the central theoretical tool for the study of complex fluids; this link will be briefly recalled in section 2.2.

A brief summary of macroscopic thermodynamics is provided in section 2.1. This reminder, which emphasizes key concepts based on the two fundamental laws of thermodynamics, more than specific thermodynamic relations, will serve as a constant reference throughout the book. In particular the summary will provide the basis for the phenomenology of phase transitions and interfacial phenomena (section 2.3), as well as for the macroscopic theory of fluctuations in chapter 3, which in turn can be generalized to mesoscopic and microscopic scales (chapter 3, sections 3.4 to 3.5).

State variables and thermodynamic equilibrium

Numerous measurements carried out on a broad variety of gaseous, liquid or solid substances, clearly show that the macroscopic equilibrium states of any substance can be fully characterized by specifying the values of a small number of state variables. In other words all measurable, macroscopic properties of a given substance, say its specific heat, its thermal expansivity or its viscosity, are well defined functions of these state variables.

INTRODUCTION

Semiconductors form the basis of most modern information processing devices. Electronic devices such as diodes, bipolar junction transistors, and field effect transistors drive modern electronic technology. Optoelectronic devices such as laser diodes, modulators, and detectors drive the optical networks. In addition to devices, semiconductor structures have provided the stages for exploring questions of fundamental physics. Quantum Hall effect and other phenomena associated with many-body effects and low dimensions have been studied in semiconductor structures.

It is important to recognize that the ability to examine fundamental physics issues and to use semiconductors in state of the art device technologies depends critically on the purity and perfection of the semiconductor crystal. Semiconductors are often associated with clean rooms and workers clad in “bunny suits” lest the tiniest stray particle get loose and latch onto the wafer being processed. Indeed, semiconductor structures can operate at their potential only if they can be grown with a high degree of crystallinity and if impurities and defects can be controlled. For high structural quality it is essential that a high quality substrate be available. This requires growth of bulk crystals which are then sliced and polished to allow epitaxial growth of thin semiconductor regions including heterostructures.

In this chapter we start with a brief discussion of the important bulk and epitaxial crystal growth techniques. We then discuss the important semiconductor crystal structures. We also discuss strained lattice structures and the strain tensor for such crystals.

Semiconductor-based technologies continue to evolve and astound us. New materials, new structures, and new manufacturing tools have allowed novel high performance electronic and optoelectronic devices. To understand modern semiconductor devices and to design future devices, it is important that one know the underlying physical phenomena that are exploited for devices. This includes the properties of electrons in semiconductors and their heterostructures and how these electrons respond to the outside world. This book is written for a reader who is interested in not only the physics of semiconductors, but also in how this physics can be exploited for devices.

The text addresses the following areas of semiconductor physics: i) electronic properties of semiconductors including bandstructures, effective mass concept, donors, acceptors, excitons, etc.; ii) techniques that allow modifications of electronic properties; use of alloys, quantum wells, strain and polar charge are discussed; iii) electron (hole) transport and optical properties of semiconductors and their heterostructures; and iv) behavior of electrons in small and disordered structures. As much as possible I have attempted to relate semiconductor physics to modern device developments.

There are a number of books on solid state and semiconductor physics that can be used as textbooks. There are also a number of good monographs that discuss special topics, such as mesoscopic transport, Coulomb blockade, resonant tunneling effects, etc. However, there are few single-source texts containing “old” and “new” semiconductor physics topics.

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.