An Introduction to Quantum Transport in Semiconductors

David K. Ferry

• Publisher: Jenny Stanford Publishing
• Year: 2017
• Print ISBN: 9789814745864
• eBook ISBN: 9781351796378
• Topic: Physical Sciences
• Subtopic: Biology

Throughout their college career, most engineering students have done problems and studies that are basically situated in the classical world. Some may have taken quantum mechanics as their chosen field of study. This book moves beyond the basics to highlight the full quantum mechanical nature of the transport of carriers through nanoelectronic structures. The book is unique in that addresses quantum transport only in the materials that are of interest to microelectronics—semiconductors, with their variable densities and effective masses.

The author develops Green's functions starting from equilibrium Green's functions and going through modern time-dependent approaches to non-equilibrium Green's functions, introduces relativistic bands for graphene and topological insulators and discusses the quantum transport changes that these bands induce, and discusses applications such as weak localization and phase breaking processes, resonant tunneling diodes, single-electron tunneling, and entanglement. Furthermore, he also explains modern ensemble Monte Carlo approaches to simulation of various approaches to quantum transport and the hydrodynamic approaches to quantum transport. All in all, the book describes all approaches to quantum transport in semiconductors, thus becoming an essential textbook for advanced graduate students in electrical engineering or physics.

Chapter 1

Introduction

The transport of carriers, electrons and holes, in semiconductors has been of interest for quite some time. It certainly became a subject of central interest when the inventors of the transistor were trying to understand the properties of the carriers in this new device [1]. But almost immediately, there was interest in the behavior of the carriers at high electric fields, in efforts to understand the breakdown of the oxides in use at that time [2]. Of course, there was increased interest in the materials important to the new semiconductor devices, such as silicon [3]. By understanding the transport properties of the carriers, one could certainly understand more about the physics governing the interactions between the carriers and their environment—the surfaces, the phonons, and so on. Over the decades since, we have found that the careful modeling of transport and the semiconductor devices has contributed to the ability to push the technology to ever smaller physical sizes. Today, the critical length in a modern trigate transistor is approaching the distance between the individual atoms of the underlying semiconductor. Indeed, we have seen the fabrication of a device in which the active region consists of a single phosphorus atom [4]! If the atoms of the semiconductor are held together by quantum mechanical forces, then it is quite likely that we will need to describe the transport in such small transistors via a fully quantum mechanical approach (and, indeed, this was done to gain understanding of the physics within the single-atom transistor).

Thus, it is clear that more detailed modeling of the quantum contributions in modern semiconductor devices is required. These contributions appear in many forms: (1) changes in the statistical thermodynamics within the devices themselves as well as in its connection and interaction with the external world, (2) new critical length scales, (3) an enlarged role for ballistic transport and quantum interference, and (4) new sources of fluctuations, which will affect device performance. Indeed, many of these effects have already been studied at low temperatures where the quantum effects appear more readily in such devices [5].

A fair question to ask at this point is why are not quantum effects seen in today’s very small devices? In fact, quantum effects are an integral part of the design of today’s devices, but they are not seen in the observed output characteristics for one good reason. Most of the important quantum effects are in a direction normal to that in which the current flows. But this does not diminish their importance. For example, strain is a common part of every device in a modern microprocessor. This strain is used to distort the band structure and improve the mobility of the electrons and holes. So controlled introduction of quantum modifications has been a part of the fabrication of devices for more than a decade. And there has been an ongoing effort to design and create simulation tools for the semiconductor world, which incorporate the quantum effects in the very base of the physics included within the tool. On the other hand, many people have studied quantum transport (and written books on the subject) in metals for quite a long time. But semiconductors are not metals. The differences are large and significant. So while one would like to extrapolate from what is known in metals, this can be taken only so far. What we would like to do here is to examine what approaches work for semiconductors and to try to learn from the many places where studies have been done for these materials and the resulting devices. In the following sections, we will try to describe what the key features are that differentiate quantum transport from the classical transport world that has been used so successfully in semiconductors and semiconductor devices.

1.1 Life Off the Shell

One of the hallmarks of classical physics is the s...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1. Introduction
• 2. Approaches to Quantum Transport
• 3. Equilibrium Green’s Functions
• 4. Interaction Representation
• 5. Role of Temperature
• 6. Quantum Devices
• 7. Density Matrix
• 8. Wigner Function
• 9. Real-Time Green’s Functions I
• 10. Real-Time Green’s Functions II
• 11. Relativistic Quantum Transport
• Index