Semiconductor Transport

David Ferry

• Publisher: CRC Press
• Year: 2016
• ISBN:
  9781351973373
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Condensed Matter

The information revolution would have been radically different, or impossible, without the use of the materials known generically as semiconductors. The properties of these materials, particularly the potential for doping with impurities to create transistors and diodes and controlling the local potential by gates, are essential for microelectronics.
Semiconductor Transport is an introductory text on electron transport in semiconductor materials and is written for advanced undergraduates and graduate students. The book provides a thorough treatment of modern approaches to the transport properties of semiconductors and their calculation. It also introduces those aspects of solid state physics, which are vitally important for understanding transport in them.

Chapter 1
Introduction

As we approach the end of the twentieth century, the information revolution has significantly changed our lives. It has been more than two hundred years since the beginnings of the industrial revolution, a period characterized by the introduction of machines as an adjunct to multiply human muscle power. Over the last few decades, microelectronics has been used to multiply our computing and reasoning powers, and we speak of this using the term “information revolution.” And yet, the beginnings of this latter revolution occurred relatively early in the last century. To understand this, it is necessary to understand that the modern microelectronics revolution would not have occurred without the materials known as semiconductors. Indeed, the properties of these materials, particularly the ability to create transistors and diodes through doping with impurities and control of the local potential by gates, are essential to microelectronics. These effects could not be achieved with metals!

The first apparent measurements, and recognition of the properties, of semiconductors were made by Michael Faraday (1833, 1834; Martin, 1932). It was the property of the conductivity increasing with temperature in AgS, rather than decreasing as in a metal, that identified the material as having new properties. Today, we know that this behavior is only characteristic in the so-called intrinsic regime, and can be modified by the inclusion of impurities. Again, it is these modifications to the properties that have proven to be so useful to modern high technology. The next important breakthrough was the observation of rectification in a metal–semiconductor contact by Braun (1874), who joined an iron pyrite to PbS. The useful property of photoconductivity, also not observed in metals, was found at about the same time by Smith (1873). Only a few years later, the observation of a transverse voltage developed across a semiconductor in which a current was flowing in the presence of a magnetic field was made by Hall (1879). The Hall effect remains one of the principal characterization tools for semiconductors even today. The rectification discovered by Braun proved useful in the new radio electronics, where it could be used for direct detection of the waves (Bose, 1904; Pierce, 1907). Yet, it was not until 1938 that Schottky provided the theory of the Schottky diode to explain the action of the metal–semiconductor contact (Schottky, 1938), and by this time several suggestions for surface-field-controlled semiconductor devices had been proposed (Lilienfeld, 1930; Heil, 1935). While it may seem that little was done between 1833 and, say, the Second World War, much work on the properties of semiconductors was carried out, and this led to the discovery of the germanium transistor by Bardeen, Brattain, and Shockley in 1947 (Bardeen and Brattain, 1948; Shockley, 1949). Finally, it was the discovery of the integrated circuit in the late 1950s by Kilby (1976) and the metal-oxide-semiconductor transistor (Shockley and Pearson, 1948; Moll, 1948; Pfann and Garrett, 1959; Khang and Atalla, 1960) that provided the rapid growth in microelectronics of the past few decades. There is no obvious end to the growth or extent of the inroads that microelectronics will make in our lives. Indeed, even in everyday objects such as our automobiles, we find multiple microprocessors being used to control the engine, emission controls, radio, temperature and climate of the passenger compartment, and even the ride, through an “active” suspension system.

The growth of microelectronics has been driven, and is in turn calibrated, by growth of the density of transistors, or gates, on an individual integrated circuit. Considering that the first transistor was invented in 1947, it is indeed phenomenal that we can now routinely place more than 100 million transistors in a single integrated circuit the size of only a few square centimeters. The cornerstone of this technology is silicon, a simple semiconductor material whose properties can be modified almost at will by proper processing technology, and which has a remarkably stable insulating oxide, SiO₂. But Si is just the dominant material currently (and probably for the foreseeable future). The history of the semiconductor electronics community has seen the importance of a wide variety of materials. Indeed, arguments are currently raging over the role of GaAs circuits, particularly for microwave applications, such as cellular phones, and high-speed data processing. Ge-Si heterostructure circuits, special oxide ceramics that compose high-temperature superconducting materials, and polymeric conductors are all under investigation for possible new application technologies. We use light-emitting diodes and laser diodes made of GaAs, AlGaAs, GaP, and other related compound semiconductors composed of the group III and group V elements. Far-infrared detectors depend on the properties of HgCdTe, a compound semiconductor composed of elements from groups II and VI. We not only use these materials in their bulk and relatively well-known forms, but also create artificial superlattices and heterostructures, which mix various compounds to produce structures in which the primary property, the band gap, has been engineered to have specific properties. What makes this all possible is that semiconductors generally have very similar properties which behave in like manner across a wide range of possible materials. This follows from the fact that all of the useful materials mentioned above have a single crystal structure, the zinc-blende lattice, or its more common diamond simplification. Thus, although the wide range of properties is obtained by small changes in the basic properties of the individual atoms, the overriding observation is that these materials are characterized by their similarities.

Today’s properties that arise in small semiconductor devices require knowledge gained only recently from the study of far-from-equilibrium systems, knowledge acquired in studying the properties of nonlinear transport at high electric fields and in physically small device geometries. Most notably, the latter has led to the study of so-called mesoscopic devices where the characteristic lengths are smaller than, or comparable to, the mean free path for scattering. In understanding how devices perform under a wide range of bias voltages and at the expected very small device geometries, we require a full understanding of the transport properties of the carriers within the device that provide for the current itself. Until a few years ago, the study of transport could be covered in reasonably complete detail simply by understanding the mobility and the diffusion coefficient for the electrons and holes. Then, simple drift and diffusion processes contributed the details of the current, and these were determined by the aforementioned mobility and diffusion coefficient, respectively. This is no longer the case, and a great deal of effort has been expended in attempting to understand just when these simple concepts begin to fail and what must be done to replace them. This field has been termed the study of hot carriers in semiconductors, but is really the study of all non-equilibrium properties that are reflected in the physics of transport in semiconductors. In essence, it is also this non-equilibrium behavior that sets semiconductors apart from metals, as the distribution function of the important carriers can be far different from the Fermi–Dirac function found in metals and semiconductors at equilibrium.

It is apparent from the above that we are now faced with trying both to understand and to predict the transport properties of a great many types of small and/or heterostructure devices, in which the properties of the host semiconductor, or semiconductors in heterostructures, are modified by local configurations that may involve superlattices, strain, high fields and voltages, and illumination, and for which these variations can occur over distances that are small compared to mean free paths or wavelengths of the electrons (or holes). To understand and to predict these properties, it has become essential to understand thoroughly the manner in which the properties of an individual semiconductor depend on the subtle differences between different semiconductors.

1.1 High-fields and non-equilibrium effects in devices

The properties of semiconductor devices that are necessary are those related to the transport of non-equilibrium carriers, and the control of this transport by various biases. Normally, in determining the transport properties, particularly from theoretical considerations, one wants to deal with an “average” electron (or hole), if possible, and write some sort of equivalent single-particle equation for its motion, thereby determining the motion of the ensemble. However, when ensemble equations are formulated, there quite often arise terms that cannot be identified with single-particle effects. For example, the diffusion equation implicitly incorporates effects arising from fluctuating forces as well as the deterministic terms. In the case of a linear or weakly nonlinear system, the fluctuating forces disturb the system to give rise to a component of entropy production that subsequently will decay toward an equilibrium with a characteristic decay time usually greater than the characteristic scattering time of the system. It is therefore apparent that one must deal with a multitude of characteristic times for the semiconductor system.

The importance of the various time scales is obvious in the analysis of non-equilibrium systems, since these systems present considerably more difficulty than that of equilibrium systems, due to the necessity of evaluating the time dependence of the various measurable properties. This will become apparent in later chapters. Indeed, these properties must be determined from equations describing them as evolving ensemble averages. The latter statement follows from the observation that the non-equilibrium distribution function is itself evolving (and is therefore non-stationary) in the nonlinear case over the time scale for variations of the measurable quantity. This implies that the system is non-ergodic (by which is meant simply that time averages do not equate to ensemble, or distribution, averages, the latter of which are the important averages) over this time scale. In fact, it is important to evaluate carefully the various collection of time scales that are important. In a semiconductor, numerous collisions occur and it is these collisions that provide the mechanism of exchange of energy and momentum, and relax these quantities toward their equilibrium values. There are collisions between the carriers which randomize the energy and momentum within an ensemble but do not relax...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Contents
• List of illustrations
• Preface
• 1 Introduction
• 2 Semiconductors are not metals
• 3 The Boltzmann equation and the relaxation time approximation
• 4 Inelastic scattering and non-equilibrium transport
• 5 Topics in high field transport
• 6 Optical properties
• 7 The electron–electron interaction
• Index