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.

The plasmas used in processing semiconductors are usually partially ionized gases, the neutral gas pressure being from about 1 mTorr to about 100 mTorr and the plasma (that is, the charged particle) number density n being in the range 1010 cm−3 ≲ n ≲ 1012 cm−3. To describe these plasmas precisely and in detail is difficult for a number of reasons. Plasma physics textbooks are largely devoted to detailed analysis of the special cases that can be described by analytic theories. Instead, it is often more useful to try to obtain approximate descriptions of the real plasma using simple physical reasoning.

In trying to understand a complex physical system it is usually useful to try several different approaches, such as studying different physical pictures, and to attempt to reconcile them with available data. In this book we will attempt to describe each situation we consider starting from “first principles” in each case. A major theme throughout will be the development of approximate quantitative models to explain experiments, based on simple reasoning. Statistical design and analysis of experiments, for example [5], provides a valuable tool for building up physical understanding.

It is tempting to say that theoretical models of processing plasmas are like houses made of cards. After a few layers (of assumptions) the whole edifice is in danger of collapsing.

Electromagnetic and Circuit Models

A great deal of insight into the behavior of the plasma can be obtained using simplified models, provided we understand when these models fail. The simplest such type of model is a circuit model, where we represent the plasma and the system enclosing it by passive circuit elements (resistors, capacitors, and inductors) and sometimes diodes. Because the plasma presents different responses depending on the set of “terminals” at which we choose to measure the impedance, we shall need several different circuit models. We also give a brief discussion of electromagnetics, which is important in its own right and also to develop the circuit models.

We begin by considering “equivalent” circuits for systems where the power supply operates at radio frequencies (rf), including ICPs and capacitive discharges. Most processing systems are driven by a radio frequency power supply, at the standard frequency f = 13.56 MHz. Some employ microwaves, at 2.8 GHz, and heat the electrons by creating Electron Cyclotron Resonance (ECR). Circuit description of ECR reactors is described next. Direct current (dc) processing systems are rare, although in some regards ECR systems behave as if they are dc. Application of a circuit model to find the voltage across an insulating surface-layer is the last topic in this chapter.

To make progress, including in developing circuit models, we now review some results from electromagnetics.

Evolution of the Trench

In this chapter we examine a number of physical processes that affect the evolution of the shape of a trench. Etching by neutrals alone is considered first, followed by charged particle effects and behavior. This leads on to charging of the trench walls and the effect of the charging on the charged particles.

The calculation of surface processes from first principles is not feasible at present. Atomistic studies of surface behavior have been widely used, employing accurate interatomic potentials [13,12,11]. Recent attempts to calculate correlation energies, with a view to improving methods such as Density Functional Theory (DFT), are described in Refs. [166]–[168]. For now, however, we stick to phenomenological approaches.

Etching by Neutrals

If the distribution of neutrals is isotropic and uniform and neutrals are responsible for the etching, then each section of the surface is etched at the same rate whatever the position or orientation of the section. If the surface is divided into sections that are all horizontal or vertical then each corner moves at 45° to the vertical. Suppose the distribution is isotropic, and that the density of neutrals varies slowly enough in space so that the sections on each side of any corner etch at roughly the same rate. This means the 45°-rule still holds approximately for each corner even though different corners may move at different rates. The best way to see how this works is to consider a few examples.

This book has suggested methods to begin to analyze the plasmas used for processing semiconductors, as well as the neutral chemistry used with the plasma, and the reactors used to create and contain the plasma. The material in this book should provide a solid base for understanding current research issues in this field and access to the relevant literature. To conclude this work, the appropriate way to describe these plasma systems will be considered, with emphasis on the tools needed to fully understand the plasma and to design a reactor.

Unlike most textbooks, there was no suggestion that we were giving a complete treatment of the problem. When textbooks give this impression, it is almost always misleading. This is particularly unfortunate, because it tends to discourage the reader from thinking creatively and independently about real problems, which cannot be handled in an oversimplified way.

To the extent that it is possible to formulate a general strategy for handling complex problems (such as processing plasmas), the strategy suggested is: Use the known experimental results and physical knowledge to set up simple models. Test the models by means of experiments. Refine the models and test them again. Repeat as often as necessary/possible.

It might seem that this process is the “hypothesis testing” that is supposed by philosophers of science to take place in all scientific endeavors. It is not the same, however. The sort of hypothesis they usually envisage is a single, straightforward idea.

In this chapter we go into more detail in describing how calculations of plasma properties may be done. Analytic models, the role of experiments in model building, and computational models are considered. We provide the basis for a mathematical description of the transport and of the density of the plasma in a plasma reactor. Next, we discuss experimental design and its role in model building. We then turn to a survey of the most widely applicable computational methods for describing plasmas.

Analytic Plasma Models

We begin this section with a review of solutions to simple diffusion equations in one and two dimensions. We turn, after that, to a one-dimensional description of the plasma. The plasma is divided into the main plasma and the sheath. The main plasma is quasineutral, whereas the sheath, which is the region of strong electric fields near the chamber wall, is positively charged since it has relatively very few electrons in it. The main plasma is also divided into a “presheath” and an interior region. The presheath is the part of the main plasma, next to the sheath (so at the outside of the main plasma) which is responsible for most of the acceleration of the ions before they enter the sheath.

In this chapter we examine long mean free path transport. This topic will involve reconsidering some of the chemistry issues from Chapter 5 and will also introduce material relevant to trench profile evolution, which is handled primarily in Chapter 7. We begin with long mean free path transport and its effect on the chemistry of neutrals in a cylindrical discharge. Ion transport and etching by ions are introduced; this is followed by a discussion of long mean free path transport in a trench that is being etched.

Chemistry at Long Mean Free Path

When the mean free path λ is large compared to the reactor dimension L (which we take to be a cylinder radius a) the particles have collisions every time they travel a distance equal to about 2a, the cylinder diameter. If λ > L, the particles predominantly have collisions with the walls. If they are moving mainly in the z direction they may go further than this, but if they are moving nearly along the surface of the wall with little motion in the z direction they will travel far less.

Plasma Models

In this chapter we begin to develop more accurate descriptions of the same plasma reactors we considered before, which go beyond the circuit models used so far. The information we can extract from circuit models is useful in some contexts but is very limited. We begin with a discussion of a parallel-plate rf discharge, which is normally operated at higher neutral pressures than the other reactors we consider. The behavior of the plasma in inductively coupled plasmas will be outlined next. Ambipolar diffusion is introduced, to help in the description of the ICP. The ECR discharge is made more difficult to describe by the presence of the magnetic field, which steers electrons almost exclusively along the magnetic field lines but which impedes ions only moderately on their way to the wall. ECR plasmas are considered next. Calculations of the electron and ion distribution functions are the last major topic in this chapter. Before discussing the rf discharge we will describe a type of experiment that can provide a very useful basis for comparison with more complicated plasmas. These experiments involve “swarms” of electrons subjected to an electric field that is constant in space and time.

Swarm Measurements

Swarm experiments [66] involve a so-called swarm of electrons with a low enough density so that their charge does not (significantly) alter the electric field. The conditions for this to occur were discussed in the section on plasma electrostatics.

This book is intended to introduce plasma processing and technology, so that the reader can readily understand the issues involved in processing and can immediately access the state-of-the-art literature. The reader is not assumed to have any knowledge of plasmas, but only some undergraduate background in basic electromagnetism. In fact, wherever possible the treatment even avoids quoting results from one part of the book for use in another part. In many cases it develops new ideas more than once, each time they are needed, to make the discussion easier to follow.

Plasma reactors and the physical and chemical processes that take place in them are discussed in considerable detail, the main emphasis being on capacitive, inductive, and electron cyclotron resonance (ECR) reactors. However, this is an area in which there are few simple (or perhaps even right) answers. It is not usually possible to give a complete description of a plasma reactor, so we are largely concerned, here, with showing how one should go about thinking about what is happening in the reactor.

The chapters of the book are organized in the order in which we need to consider material to understand the reactor. This can sometimes mean that we do not treat a single topic in one chapter separately from other topics. Instead, subject matter will often be introduced where it is needed as we build up the picture of how systems work.

Plasma Chemistry

The chemical processes that can take place in a plasma are exceptionally complicated and little understood. A treatment of all the possible chemistries of interest for semiconductor fabrication would be an enormous undertaking. Here we establish some guidelines for thinking about plasma chemistry, rather than attempting to describe all the possibilities. A number of texts on plasma chemistry are available, although the emphasis in many is more on plasma polymerization than on silicon processing [83–91].

The term plasma chemistry is usually not appropriate to describe the important effects. The hot electrons from the plasma are responsible for driving unusual chemistry – mostly neutral chemistry. (The main exception to the electrons driving the chemistry is activation of surfaces by ions.) Perhaps only the first step in a chain of reactions even involves the electrons, however. The energy an electron needs to ionize a neutral is much higher than the energy needed to dissociate most neutrals. There are usually vastly more electrons with enough energy to dissociate neutrals than there are electrons that have enough energy to ionize neutrals.

A lot of features connected with absorption and emission of light in nanocrystals can be understood in terms of the quantum confinement approach. In this approach, a nanocrystal is considered as a three-dimensional potential box in which photon absorption and emission result either in a creation or in an annihilation of some elementary excitations in an electron subsystem. These excitations are described in terms of quasiparticles known for bulk crystals, that is, electrons, holes, and excitons.

This chapter is meant to remind readers of some principal results from elementary quantum mechanics and to provide an elementary introduction to solid state physics, which is essential for the following chapters. We then depart from elementary “particle-in-a-box” problems and consider the properties of an electron in a periodic potential. In the next step, we introduce the concepts of effective mass and quasiparticles as elementary excitations of a many-body system. Finally, we give an idea of the low-dimensional structures that constitute, undoubtedly, one of the major fields of research in modern condensed-matter physics.

A few problems from elementary quantum mechanics

Particle in a potential well

To restate some basic properties of quantum particles that are necessary to consider electrons in a crystal, we start with a particle in a one-dimensional potential well (Fig. 1.1).

Semiconductor nanocrystals can be fabricated using a number of technologies, differing in the environment in which nanocrystals appear, growth conditions, size range, and size distribution, as well as physical and chemical stability and reliability. Nanocrystals can be developed in inorganic glasses and crystals, in liquid solutions and polymers, or on a crystalline surface. In this chapter we provide a brief overview of these techniques and give a synopsis of nanocrystals developed by various techniques.

Nanocrystals in inorganic matrices

Glass matrices: diffusion-controlled growth

Fabrication of nanocrystals embedded in a glass matrix by means of diffusion-controlled growth is based on commercial technologies developed for fabrication of color cut-off filters and photochromic glasses. Color cut-off filters produced by Corning (United States), Schott (Germany), Rubin (Russia), and Hoya (Japan) are just glasses containing nanometer-size crystallites of mixed II-VI compounds (CdSxSe1−x). Empirical methods of diffusion-controlled growth of semiconductor nanocrystals in a glass matrix have been known for decades or even centuries (in the case of color stained glasses). Commercial photochromic glasses developed in recent years contain nanocrystals of I-VII compounds (e.g., CuCl, CuBr, AgBr). Typically, silicate or borosilicate matrices are used with the absorption onset near 4 eV (about 300 nm), thus providing optical transmission of the semiconductor inclusions to be studied over the whole visible range.

Growth of crystallites results from the phase transition in a supersaturated viscous solution.

In quasi-zero-dimensional structures under optical excitation there are, along with reversible processes that decay over the recombination time of electron-hole pairs, processes that result in a persistent change in the optical properties. These processes are controlled by the integral dose of the absorbed radiation rather than radiation intensity. Numerous examples of similar behavior can be found in photophysics and photochemistry of molecular structures. Similar to molecular structures, semiconductor nanocrystals embedded in a matrix or precipitated in a solution exhibit a variety of guest-host effects. Some of the phenomena related to the photo-induced modifications in absorption and/or emission features will be the subject of the present chapter. The main attention will be given to laser annealing, photodarkening, persistent spectral hole-burning, and photochemical reactions resulting in permanent spectral hole-burning. Finally, we consider the intercrystallite migration of carriers and its effect on luminescence kinetics.

Laser annealing, photodarkening, and photodegradation

Semiconductor-doped glasses exposed to prolonged illumination by laser light of a wavelength corresponding to resonant absorption by nanocrystals were found to exhibit systematically a number of photo-induced modifications. These include, first of all, a sharp decrease in the intrinsic edge luminescence versus impurity and defect related emission. Second, the lifetime of electron-hole pairs decreases by several orders of magnitude and reaches 10−11 s. Finally, additional structureless absorption with a coefficient on the order of 1 cm−1 appears in a wide spectral interval. The initial properties of the samples can sometimes be restored by heating to temperatures of 400–500°C.