The transport of matter within materials can occur either by diffusion or by convective flow. Diffusion can occur in both solids and fluids while convective flow is found only in fluids. This chapter provides a brief overview of these processes. It also offers a summary of the mechanisms involved, for diffusion in solids, liquids and gases. While these are presented in a highly simplified fashion they do offer sufficient insight to enable many mass transport problems of practical interest to be solved.

Mass transport processes

When a drop of dye is added to a beaker of still water the highly concentrated dye spreads throughout the liquid until a uniform pale colour results. There are two processes which can contribute to this. The first is called diffusion. This process is driven by differences in the concentration of a substance (in this case the molecules that make up the dye) from one region to another. Diffusion occurs until the concentration becomes uniform, i.e. the concentration gradient goes to zero everywhere. The same process happens in the solid state when two soluble substances are mixed together.

Blends of polyolefins

Model predictions in PE blends

The interest in multicomponent materials, in the past, has led to many attempts to relate their mechanical behaviour to that of the constituent phases (Hull, 1981). Several theoretical developments have concentrated on the study of the elastic moduli of two-component systems (Arridge, 1975; Peterlin, 1973). Specifically, the application of composite theories to relationships between elastic modulus and microstructure applies for semicrystalline polymers exhibiting distinct crystalline and amorphous phases (Andrews, 1974). Furthermore, as discussed in Chapter 4, the elastic modulus has been shown to be correlated to microhardness for lamellar PE. In addition, H has been shown to be a property that describes a semicrystalline polymer as a composite material consisting of stiff (crystals) and soft, compliant elements. Application of this concept to lamellar PE involves, however, certain difficulties. This material has a microstructure that requires specific methods of analysis involving the calculation of the volume fraction of crystallized material, crystal shape and dimensions, etc. (Baltá Calleja et al., 1981).

For this reason, it is of interest to investigate the H of a model system composed of varying mixtures of two types of PE with well-differentiated microhardness values in such a way that the experimental microhardness data derived can be compared with predictions for the various component arrangements. In addition, the measurement of H of these blends at high temperature can provide direct information on the microhardness of the disordered phase.

The operation of semiconductor devices is controlled by how electrons and holes respond to applied, built-in, and scattering potentials. Electrical engineers are used to treating transport phenomenologically – carriers drift in electric fields and diffuse in concentration gradients. For much of the past 40 years during which semiconductor technology advanced from point-contact transistors to megabit memories, drift–diffusion equations have served as the backbone of device analysis. As devices continue to decrease in size and increase in sophistication, however, this simple picture of carrier transport is beginning to lose validity. Device engineers now need a clear understanding of the physics of carrier transport in a variety of semiconductors as well as an understanding of the nature of transport in modern, small devices. The focus of this book is on carrier transport fundamentals beginning at the microscopic level and progressing to the macroscopic effects relevant to devices. The reader should acquire an understanding of the general features of low- and high-field transport in common semiconductors as well as of the characteristics of transport in small devices. He or she should learn how to evaluate scattering rates and mobilities from the semiconductor's material properties and should understand the various approaches commonly used to analyze and simulate devices.

The book is directed at electrical engineering graduate students or practicing device engineers who typically possess a mature understanding of semiconductor fundamentals and devices but only an acquaintance with the basics of quantum mechanics and solid-state physics.

The search for quantitative structure-property relationships for the control and prediction of the mechanical behaviour of polymers has occupied a central role in the development of polymer science and engineering. Mechanical performance factors such as creep resistance, fatigue life, toughness and the stability of properties with time, stress and temperature have become subjects of major activity. Within this context microhardness emerges as a property which is sensitive to structural changes.

The microindentation hardness technique has been used for many years for the characterization of such ‘classical’ materials as metals, alloys, inorganic glasses, etc. Its application to polymeric materials was developed in the 1960s. The potential of this method for structural characterization of polymers was developed and highlighted to a large extent by the studies carried out in the Instituto de Estructura de la Materia, CSIC, Madrid.

Nowadays, the microhardness technique, being an elegant, non-destructive sensitive and relatively simple method, enjoys wide application, as can be concluded from the publications on the topic that have appeared during just the last five years – they number more than 100, as is shown by a routine computer-aided literature search. In addition to some methodological contributions to the technique, the microhardness method has also been successfully used to gain a deeper understanding of the microhardness-structure correlation of polymers, copolymers, polymer blends and composites.

Introduction to glassy polymers

The glass transition

When a polymer is cooled down from the liquid or rubbery state, it becomes much stiffer as it goes through a certain temperature range. This stiffening is the result of one of two possible events: crystallization or glass transition. For crystallization to occur, the polymer molecules must be sufficiently regular along their length to allow the formation of a crystalline lattice and the cooling rate must be slow enough for the crystallization process to take place before the molecular motions become too sluggish. When the polymer fails to crystallize for either reason, the amorphous, liquid-like structure of the polymer is retained, but the molecular motion becomes frozen-in and the material turns into a glass. Such a glass transition occurs over a finite temperature interval, but is still realized abruptly enough to merit the term ‘transition’. The glass transition can be recognized by the change in many properties of the material, the most important one, from a practical point of view, being the increase in the modulus of the material by several orders of magnitude.

Glass formation can be achieved with many low-molecular-weight materials and with certain metallic alloys by special preparation techniques, such as rapid quenching. With polymers, the opportunities for irregularity along the chain are numerous and the crystallization rate is inherently slow. As a result, the formation of the glassy state is a more common occurrence.

The microhardness of a polymeric material – resistance to local deformation – is a complex property related to mechanical properties such as modulus, strength, elasticity and plasticity. This relationship to mechanical properties is not usually straightforward, though there is a tendency for high modulus and strength values to correlate with higher degrees of microhardness within classes of materials. Microhardness has no simple, unambiguous definition; it can be measured and expressed only by carefully standardized tests (see Section 1.1).

Scratch tests have been used for microhardness measurements of polymeric materials (Bierbaum Scratch Hardness Test (ASTM D 1526)). These tests are related to cuts and scratches, and, to some extent, to the wear resistance of materials. Scratch tests are not always related to the resistance to local deformation and they are now being replaced by the preferred indentation test.

In the indentation test, a specified probe or indenter is pressed into the material under specified conditions, the depth of penetration being a measure of the microhardness according to the test method used. The duration of a microhardness measurement must be specified because polymeric materials differ in their susceptibility to plastic and viscoelastic deformation. An indenter will penetrate at a decreasing rate during application of the force, and also, the material will recover at a decreasing rate, reducing the depth of penetration, when the force is removed. Therefore, length of time that the force is applied must be specified. For most elastomers, the indentation will disappear when the force is removed.

The first edition of this book was written in a period when the drift–diffusion-based description of semiconductor devices was beginning to lose validity and many kinds of interesting transport effects (e.g. velocity overshoot, ballistic transport, real-space transfer, etc.) and their implications for devices were being explored. Since that time, semiconductor devices have continued to shrink in size, so that engineers and device researchers now face these issues daily. When the first edition was written, quantum transport in mesoscopic structures was also an active research field, with many uncertainties being debated. In the intervening years, this field has matured; the general principles are now understood and are becoming relevant to semiconductor technologists as devices continue their relentless march to microscopic dimensions.

The goals of the second edition are much like those of the first. The book is an attempt to help students with little formal training in quantum mechanics or solid state physics (i.e., the typical graduate of an undergraduate electrical engineering program) understand the fundamental concepts of carrier transport in semiconductors. Writing the second edition was an opportunity to update and clarify material in the first edition and to treat new topics. The most significant change in the second edition is the addition of Chapter 9 on transport in mesoscopic structures, a topic that device engineers now deal with.

Two classes of graduate students worked through early versions of this text and helped me to clarify the presentation and reduce the number of typos and errors.

In Chapter 3 we introduced the Boltzmann Transport Equation (BTE) as an alternative to calculating the position and momentum versus time for each carrier within a device. The BTE is usually very difficult to solve; it is much easier to simulate the trajectories of individual carriers as they move through a device under the influence of electric fields and random scattering forces. Since each path is determined by choosing random numbers (properly distributed to reflect the probabilities of the various scattering events) the technique is a game of chance which has become known as Monte Carlo simulation. If the number of simulated trajectories is large enough, the average results are a good approximation to the average behavior of the carriers within a real device. In many cases, Monte Carlo simulation is the most accurate technique available for simulating transport in devices; it is frequently the standard against which the validity of simpler approaches is gauged.

Much of our understanding of high-field transport in bulk semiconductors and in devices has been obtained through Monte Carlo simulation, so it is important to understand the basics of the method. Because it directly mimics the physics, an understanding of the technique is also useful for the insight it affords. This chapter's emphasis is on the underlying principles of the Monte Carlo technique and on how the results of a Monte Carlo simulation are interpreted.