Hot Carriers in Semiconductor Nanostructures: Physics and Applications

By Jagdeep Shah

Nonequilibrium hot charge carriers play a crucial role in the physics and technology of semiconductor nanostructure devices. This book, one of the first on the topic, discusses fundamental aspects of hot carriers in quasi-two-dimensional systems and the impact of these carriers on semiconductor devices. The work will provide scientists and device engineers with an authoritative review of the most exciting recent developments in this rapidly moving field. It should be read by all those who wish to learn the fundamentals of contemporary ultra-small, ultra-fast semiconductor devices.

• Topics covered include
• Reduced dimensionality and quantum wells
• Carrier-phonon interactions and hot phonons
• Femtosecond optical studies of hot carrier
• Ballistic transport
• Submicron and resonant tunneling devices

• Language: English
• Publisher: Academic Press
• Release date: Dec 2, 2012
• ISBN: 9780080925707

Book preview

Hot Carriers in Semiconductor Nanostructures

Physics and Applications

First Edition

Jagdeep Shah

AT&T Bell Laboratories, Holmdel, New Jersey

Academic Press, Inc.

Harcourt Brace Jovanovich, Publishers

Boston  San Diego  New York

London  Sydney  Tokyo  Toronto

Table of Contents

Cover image

Title page

Copyright page

Dedication

Preface

Contributors

Part I: Overview

I.1: Overview

1 INTRODUCTION

2 FUNDAMENTAL ASPECTS OF QUASI-2D SYSTEMS

3 MONTE CARLO SIMULATIONS

4 OPTICAL STUDIES OF HOT CARRIERS IN SEMICONDUCTOR NANOSTRUCTURES

5 TRANSPORT STUDIES AND DEVICES

6 SUMMARY

Part II: Fundamental Theory

II.1: Electron–Phonon Interactions in 2D Systems

1 INTRODUCTION

2 QUANTUM CONFINEMENT

3 THE ELECTRON–PHONON SCATTERING RATE

4 MODEL RATES FOR THE FRÖHLICH INTERACTION

5 SCATTERING BY ACOUSTIC PHONONS

6 CONCLUDING REMARKS

II.2: Quantum Many-Body Aspects of HoT-Carrier Relaxation in Semiconductor Microstructures

1 INTRODUCTION

2 ENERGY RELAXATION OF AN EXCITED ELECTRON GAS

3 SINGLE-PARTICLE INELASTIC LIFETIME

4 CONCLUSION

ACKNOWLEDGMENTS

II.3: Cooling of Highly Photoexcited Electron–Hole Plasma in Polar Semiconductors and Semiconductor Quantum Wells: A Balance-Equation Approach

1 CARRIER COOLING IN BULK POLAR SEMICONDUCTORS

2 CARRIER COOLING IN QUANTUM-WELL STRUCTURES

3 SUMMARY AND CONCLUSIONS

II.4: Tunneling Times in Semiconductor Heterostructures: A Critical Review

1 Introduction

2 PHASE TIME, DWELL TIME, BÜTTIKER–LANDAUER TIME, LARMOR TIMES, AND COMPLEX TIMES

3 ANALYSIS AND DOMAIN OF VALIDITY OF THE PROPOSED TUNNELING TIMES

4 EXPERIMENTAL METHODS FOR DETERMINING TUNNELING TIMES

ACKNOWLEDGMENTS

II. 5: Quantum Transport

1 INTRODUCTION

2 THE GENERAL PROBLEM AND THE VARIOUS APPROACHES

3 APPLICATIONS

4 CONCLUSIONS

Part III: Monte Carlo Simulations

III.1: Hot-Carrier Relaxation in Quasi-2D Systems

1 Introduction

2 Scattering in Quasi-2D Systems

3 Monte Carlo Simulation

4 Analysis of Experimental Results

5 Summary and Conclusions

Acknowledgments

III. 2: Monte Carlo Simulation of GaAs–AlxGa1 − x as Field-Effect Transistors

1 INTRODUCTION

2 ENSEMBLE MONTE CARLO DEVICE MODEL

3 NONSTATIONARY TRANSPORT AND SCALING OF MODFETS

4 PHYSICS OF REAL-SPACE TRANSFER TRANSISTORS

5 EXTENDED DRIFT-DIFFUSION FORMALISM

6 CONCLUSIONS

ACKNOWLEDGMENTS

Part IV: Optical Studies

IV.1: Ultrafast Luminescence Studies of Carrier Relaxation and Tunneling in Semiconductor Nanostructures

1 INTRODUCTION

2 ULTRAFAST LUMINESCENCE STUDIES OF CARRIER RELAXATION

3 ULTRAFAST LUMINESCENCE STUDIES OF TUNNELING IN SEMICONDUCTOR NANOSTRUCTURES

4 SUMMARY

ACKNOWLEDGMENTS

IV.2: Optical Studies of Femtosecond Carrier Thermalization in GaAs

1 INTRODUCTION

2 EXPERIMENTAL METHODS

3 EXPERIMENTAL RESULTS

4 THEORETICAL APPROACHES

5 CONCLUSION

ACKNOWLEDGMENTS

IV.3: Time-Resolved Raman Measurements of Electron–Phonon Interactions in Quantum Wells and Superlattices

1 INTRODUCTION

2 EXPERIMENTAL CONSIDERATIONS

3 RAMAN MEASUREMENTS OF INTERSUBBAND RELAXATION IN QUANTUM WELLS

4 LO-PHONON EMISSION IN INTRASUBBAND RELAXATION

5 PHONON-ASSISTED CHARGE TRANSFER IN TYPE II GAAS–ALAS SUPERLATTICES

6 SUMMARY

ACKNOWLEDGMENTS

IV.4: Electron–Hole Scattering In Quantum Wells

1 INTRODUCTION

2 EXPERIMENTAL TECHNIQUES

3 QUANTITATIVE RESULTS ON MOMENTUM AND ENERGY RELAXATION

4 PHOTOCONDUCTIVITY EXPERIMENTS

5 OUTLOOK

ACKNOWLEDGMENTS

Part V: Transport Studies

V-1: Ballistic Transport in a Two-Dimensional Electron Gas

1 HOT-ELECTRON TRANSPORT

2 HOT BALLISTIC TRANSPORT

3 ENERGY DEPENDENCE OF HOT-ELECTRON TRANSPORT

4 BALLISTIC TRANSPORT IN UPPER SUBBANDS

5 ANGULAR DISTRIBUTION AND ELECTRON-BEAM STEERING

6 ELECTROSTATIC FOCUSING OF BALLISTIC ELECTRONS

7 A BALLISTIC HOT-ELECTRON DEVICE

8 SUMMARY

ACKNOWLEDGMENTS

NOTE ADDED IN PROOF

V.2: Resonant-Tunneling Hot-Electron Transistors

1 Introduction

2 Modeling of RHET Operation [7]

3 Experimental Analyses of RHET Microwave Performance

4 RHET Performance Improved by a New Collector Structure

5 RHET Logic Family [19]

6 Summary

Acknowledgments

V.3: Resonant Tunneling in High-Speed Double Barrier Diodes

1 Introduction

2 Principles of Resonant Tunneling

3 Resonant-Tunneling Device Physics

4 Experimental Results

5 Summary

Acknowledgments

Index

Copyright

copyright© 1992 by american telephone and telegraph company

all rights reserved.

no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

ACADEMIC PRESS, INC.

1250 Sixth Avenue, San Diego, CA 92101

United. Kingdom Edition published by

ACADEMIC PRESS LIMITED

24–28 Oval Road, London NW1 7DX

Library of Congress Cataloging-in-Publication Data:

Hot carriers in semiconductor nanostructures: physics and applications Jagdeep Shah [editor].

p. cm.

Includes bibliographical references and index.

ISBN 0-12-638140-2 (acid-free paper)

1. Hot carriers—Congresses. 2. Semiconductors—Congresses.

I. Shah, J. (Jagdeep) II. Title: Nanostructures.

QC611.6.H67H66  1992

621.381′52—dc20     91-15684

CIP

printed in the united states of america

92 93 94 95    9 8 7 6 5 4 3 2 1

Dedication

To the memory of my parents

Preface

Jagdeep Shah, Holmdel, New Jersey

The success of various epitaxial growth techniques such as molecularbeam epitaxy, vapor-phase epitaxy and chemical vapor deposition techniques has made it possible to grow a large class of high-quality semiconductor structures where the composition and doping can be controlled down to a single monolayer (≈3 Å). Furthermore, remarkable advances in semiconductor processing technology have allowed fabrication of structures with lateral dimensions of tens of nanometers. It is no exaggeration to state that these nanostructures have revolutionized the world of semiconductor physics and devices, by leading to novel physical phenomena and to smaller and faster devices.

The field of hot carriers in semiconductors occupies a pivotal position in semiconductor science. Investigation of hot carriers provides important information about many fundamental scattering processes that determine high-field transport in semiconductors, and such knowledge is invaluable in understanding high-speed electronic and optoelectronic devices operating at high electric fields.

There are several excellent books covering hot-carrier effects in bulk semiconductors, e.g., by Conwell, by Nag and by Reggiani. Various aspects of growth and fabrication of semiconductor nanostructures, physics of semiconductor heterostructures and devices made from such structures have also been covered in several excellent books, e.g., by Dingle, by Capasso and Margaritondo and by Capasso. References to these books are provided in the Overview (Chapter I). There are, however, no books dealing with hot carriers in semiconductor nanostructures. This book attempts to fill this gap and reviews the most exciting recent developments in the field of hot carriers in semiconductor nanostructures, a field that is important from fundamental as well as device points of view.

It is hoped that this book will be useful to a wide range of researchers: to specialists as a source of references and of information on subfields related to their interests, to nonspecialists as an overview of the field, to researchers interested in the basic physics of semiconductor nanostructures as a source of information about scattering processes in quasi-2D systems, and to researchers interested in nanostructures devices as an overview of some of these devices and as a source of information about the basic physics governing them. It is indeed fortunate that each chapter is written by an internationally recognized expert or group of experts who have played leading roles in the advancement of their fields.

There are some topics that logically should be a part of a book of this kind but are omitted either because the subject has just been reviewed or because it is not yet ripe for a review. The books on heterostructures mentioned above include excellent reviews on resonant tunneling bipolar and unipolar transistors, ballistic transport in vertical structures, transport in quasi-ID mosfets, and modulation-doped field-effect transistors. The recent work on coherent spectroscopy of free carriers in semiconductors and their nanostructures, and on one- and zero-dimensional nanostructures, might well form the subject matter of future books.

I am grateful to AT&T Bell Laboratories for permission to publish this book and for providing an intellectually stimulating environment conducive to successful and productive research in a rapidly developing field. I would like to thank many colleagues, both within and outside AT&T Bell Laboratories, with whom I have collaborated and interacted in the course of research on hot-carrier relaxations in semiconductor and their nanostructures, the colleagues who have contributed to this volume, the colleagues who provided valuable feedback on the scope and the content of this volume, and Mr. Robert Kaplan of Academic Press for encouragement to undertake this project and for providing a smooth interface with the publishers. Last but not least, I wish to express my appreciation to my wife and children for their encouragement, understanding and support.

Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

E.R. Brown     (469), Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA 02173-9108

Rossella Brunetti     (153), Dipartimento di Fisica, Università di Modena, Via Campi 213/A, 41100 Modena, Italy

S. Das Sarma     (53), Department of Physics, University of Maryland, College Park, MD 20742-4111

Stephen M. Goodnick     (191), Department of Electrical and Computer Engineering, Oregon State University, Corvallis, OR 97331

M. Heiblum     (411), Weizmann Institute of Sciences, Rehovot, Israel 76100

Karl Hess     (235), Beckmann Institute, Department of Electrical and Computer Engineering, University of Illinois, Urbana, IL 61801

Ralph A. Höpfel     (379), Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria

Kenichi Imamura     (443), Fujitsu Limited, Fujitsu Laboratories Ltd., 10-1 Morinosato-Wakamiya, Atsugi 243-01, Japan

Carlo Jacoboni     (153), Dipartimento di Fisica, Universitá di Modena, Via Campi 213/A, 41100 Modena, Italy

A.P. Jauho     (121), Physics Laboratory, H.C. Ørsted Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark

Simon Juen     (379), Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria

Isik C. Kizilyalli     (235), AT&T Bell Laboratories, Allentown, PA 18103

Wayne H. Knox     (313), AT&T Bell Laboratories, Holmdel, NJ 07733

P. Kocevar     (87), Institut für Theoretische Physik, Universität Graz, Universitätsplatz 5, A-8010 Graz, Austria

Paolo Lugli     (191), Dipartimento di Ingegneria Meccanica, II Università di Roma, Via O. Raimondo, 00173 Roma, Italy

Toshihiko Mori     (443), Fujitsu Limited, Fujitsu Laboratories Ltd., 10-1 Morinosato-Wakamiya, Atsugi 243-01, Japan

Shunichi Muto     (443), Fujitsu Limited, Fujitsu Laboratories Ltd., 10-1 Morinosato-Wakamiya, Atsugi 243-01, Japan

Hiroaki Ohnishi     (443), Fujitsu Limited, Fujitsu Laboratories Ltd., 10-1 Morinosato-Wakamiya, Atsugi 243-01, Japan

W. Pötz     (87), Department of Physics, University of Illinois at Chicago, Chicago,IL 60680

B.K. Ridley     (17), Department of Physics, University of Essex, Colchester, United Kingdom

Fausto Rossi     (153), Dipartimento di Fisica, Università di Modena, Via Campi 213/A, 41100 Modena, Italy

J.F. Ryan     (345), Clarendon Laboratory, University of Oxford, Oxford, England

Jagdeep Shah     (3, 279, 379), AT&T Bell Laboratories, Holmdel, NJ 07733

Akihiro Shibatomi     (443), Fujitsu Limited, Fujitsu Laboratories Ltd., 10-1 Morinosato-Wakamiya, Atsugi 243-01, Japan

U. Sivan     (411), IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598

Motomu Takatsu     (443), Fujitsu Limited, Fujitsu Laboratories Ltd., 10-1 Morinosato-Wakamiya, Atsugi 243-01, Japan

M.C. Tatham     (345), Clarendon Laboratory, University of Oxford, Oxford, England

Naoki Yokoyama     (443), Fujitsu Limited, Fujitsu Laboratories Ltd., 10-1 Morinosato-Wakamiya, Atsugi 243-01, Japan

Part I

Overview

I.1

Overview

Jagdeep Shah    AT&T Bell Laboratories Holmdel, New Jersey

1. Introduction   3

2. Fundamental Aspects of Quasi-2D Systems   5

2.1. Electron-Phonon Interaction in Quasi-2D Systems   6

2.2. Many-Body Effects   7

2.3. Hot-Phonon Effects   7

2.4. Scattering Processes Specific to Quasi-2D Systems   8

2.5. Tunneling Times   8

2.6. Quantum Transport   8

3. Monte Carlo Simulations   9

3.1. Monte Carlo Simulations of Ultrafast Optical Studies   9

3.2. Monte Carlo Simulations of Submicron Devices   10

4. Optical Studies of Hot Carriers in Semiconductor Nanostructures   10

4.1. Ultrafast Luminescence Studies of Carrier Relaxation and Tunneling   10

4.2. Femtosecond Pump-and-Probe Transmission Studies   11

4.3. Ultrafast Pump-and-Probe Raman Scattering Studies   11

4.4. Electron–Hole Scattering   12

5. Transport Studies and Devices   12

5.1. Ballistic Transport in Nanostructures   12

5.2. Resonant Tunneling Hot-electron Transistors   13

5.3. Resonant Tunneling Diodes   13

6. Summary   13

References   14

1 INTRODUCTION

In thermal equilibrium, all elementary excitations in a semiconductor (e.g., electrons, holes, phonons) can be characterized by a temperature that is the same as the lattice temperature. Under the influence of an external perturbation such as an electric field or optical excitation, the distribution functions of these elementary excitations deviate from those in thermal equilibrium. In general, the nonequilibrium distribution functions are nonthermal (i.e. cannot be characterized by a temperature). But, under special conditions, they can be characterized by a temperature that may be different for each elementary excitation and different from the lattice temperature. The term hot carriers is often used to describe both these nonequilibrium situations.

Investigation of hot-carrier effects plays a central role in modern semiconductor science. Properties of hot carriers are determined by various interactions between carriers and other elementary excitations in the semiconductor. Therefore, investigations of hot-carrier properties provide information about scattering processes that are of fundamental interest in the physics of semiconductors. Furthermore, these processes determine high-field transport phenomena in semiconductors and thus form the basis of many ultrafast electronic and optoelectronic devices. The field of hot carriers in semiconductors thus provides a link between fundamental semiconductor physics and high-speed devices.

Although some theoretical work on high-field transport in semiconductors dates from 1930s, experimental investigations started in 1951 with the high-field experiments of Ryder and Shockley (the early work is referenced by Conwell [1]). These and other investigations that followed in the next quarter of a century concentrated on bulk semiconductors and semiconductor devices, and provided quantitative understanding of many phenomena and new insights into the high-field transport processes in semiconductors. This work is extensively covered in excellent books by Conwell [1], Nag [2,3], and Reggiani [4]. The topic has also been the subject of NATO Advanced Study Institutes [5,6].

The direction of the field changed considerably in 1970s and 1980s because of several developments. The quasi-two-dimensional nature of carriers in the conducting channels in Si mosfets brought into play new physical phenomena [7]. The mid 1970s brought the first high-quality quantum-well heterostructures, consisting of thin layers of semiconductors with different bandgaps and grown using the techniques of molecular-beam epitaxy (for a recent review, see, for example, Madhukar in [8]). Semiconductor nanostructures have led to many exciting developments in the physics of semiconductors [8–10]. Furthermore, the ability to grow and fabricate semiconductor structures on nanometer scales has led to the development of many new devices, such as modulation-doped field-effect transistors and resonant tunneling diodes. Nonequilibrium transport of carriers is a common thread in these ultrasmall, ultrafast devices operating at high electric fields. Ballistic transport in nanonstructures provided another focal point of interest. These developments have led to considerable interest in the investigation of hot-carrier effects in semiconductor nanostructures.

An important milestone in the field of hot carriers in semiconductors was the demonstration in late 1960s that optical excitation can create hot carriers and optical spectroscopy can provide information about the distribution function of hot carriers. Although transport measurements provide considerable information about various scattering processes in semiconductors, they are averaged over the carrier distribution functions. In contrast, optical techniques, by providing the best means of determining the carrier distribution functions, allow one to investigate the microscopic scattering processes. Another development that has significantly altered the course of this field is the recent availability of ultrafast lasers with pulsewidths as short as 6 fs (for a recent review of the field of ultrafast lasers and their applications to physics, chemistry and biology, see [11]). These lasers allowed the investigation of the time evolution of the carrier distribution functions on ultrashort time scales. Since different scattering processes occur on different time scales, it became possible to isolate various scattering processes by appropriate choices of time windows.

The availability of high-speed computers has made it possible to carry out ensemble Monte Carlo simulations of submicron devices and ultrafast carrier relaxation in semiconductors. Detailed comparison of these simulations with the device performance or with experimental observations of carrier relaxations obtained with ultrafast lasers has provided valuable new information.

Finally, the ability to grow nanostructures has led to interesting new transport phenomena such as ballistic transport of electrons and led to devices based on nonequilibrium transport through such nanostructures. Examples of the devices are resonant tunneling diodes, resonant tunneling hot-electron transistors and modulation-doped field-effect transistors.

As one can see from this brief historical survey, the field of hot carriers in semiconductors and their nanostructures has been a dynamic field with many important developments in the past decade. The purpose of this book is to review the most exciting of these developments in the four areas discussed above. The book is divided into four parts, with several chapters in each part. Part II deals with the fundamental aspects of hot-carrier physics in quasi-2D systems. Part III deals with Monte Carlo simulations of ultrafast optical experiments in quasi-2D systems and of submicron devices. Part IV discusses optical studies of hot carriers in quasi-2D systems, and Part V deals with ballistic transport, resonant tunneling transistors and diodes. In the remainder of this chapter, I will present an overview of these developments.

2 FUNDAMENTAL ASPECTS OF QUASI-2D SYSTEMS

Hot-carrier effects are determined by many different scattering processes, such as carrier–carrier scattering, carrier–phonon scattering, intervalley scattering, and intersubband scattering. An understanding of these processes is essential for an understanding of hot-carrier phenomena and devices. These fundamental processes are reviewed in Part II.

2.1 Electron–Phonon Interaction in Quasi-2D Systems

Electronic states in a quantum confined system are different from those in a bulk semiconductor. The conduction and valence bands break up into various subbands as a result of confinement. The wavefunctions of the confined states penetrate into the barrier for finite barrier heights but vanish at the boundary for infinitely high barriers. For thick barriers, each well in a multiple quantum-well structure can be treated as independent of the other wells. With decreasing barrier thickness, the wavefunctions in the adjacent wells overlap with each other and lead to the phenomenon of minibands, with some interesting transport consequences [12]. These modifications of the electronic states in quasi-2D systems are well known and have been discussed in many reviews (see, for example, Weisbuch in [8]). A brief discussion is given by Ridley in Chapter II.1.

While the electrons are simple, the holes in quasi-2D systems are extremely complicated. The valence bands in bulk III–V semiconductors are nonparabolic and anisotropic. Inclusion of quantization effects leads to very complicated band structure for the heavy- and light-hole valence subbands. It is only very recently that it has become possible to map out the dispersion relations of the valence bands in quantum wells experimentally. Detailed understanding of hot-hole phenomena will clearly require a better understanding of these complicated bands.

For this reason, most of the work on carrier–phonon interactions in quasi-2D systems deals with electron–phonon interactions. This is reviewed in detail by Ridley in Chapter II.1. Early work in this field considered confined carrier states but bulk phonon modes. However, phonon modes in quasi-2D systems have been investigated extensively by light-scattering techniques [13]. Many interesting features of phonon modes must be considered in carrier–phonon scattering. One subject of considerable current interest is whether slab modes, confined modes or those calculated on the basis of microscopic theories provide the correct description of phonons and carrier–phonon interactions. A discussion of theoretical aspects of electron–phonon interactions in quasi-2D systems is given by Ridley in Chapter II.1. Direct information on the dynamics of quasi-2D phonon modes can be obtained by time-resolved light-scattering experiments. This subject is reviewed by Ryan and Tatham in Chapter IV.3.

Carrier–phonon interaction in a semiconductor can be modified by a number of effects. The two most important are many-body effects arising from a large density of carriers and nonequilibrium phonon effects (commonly referred to as hot-phonon effects) that occur at high electric fields or high excitation densities.

2.2 Many-Body Effects

When the carrier density is high, one cannot treat plasma oscillations and lattice vibrations as independent phenomena; one must consider a unified picture of coupled phonon–plasma modes. In general, one must consider a frequency- and wavevector-dependent dielectric constant. However, researchers have quite often made simplifying assumptions that do not correctly represent the physical situation, and this has led to considerable confusion in the literature. For example, the use of static screening approximations may considerably overestimate the effects of screening and therefore underestimate carrier–phonon interactions. Another problem that must be considered properly is what happens following the excitation of a coupled phonon–plasmon mode by a high-energy electron. What fraction of the energy comes back to the electronic system, and what fraction goes to the lattice? These considerations have important effects in bulk as well as quasi-2D systems. Das Sarma (Chapter II.2) reviews how many-body effects influence carrier–phonon interactions and carrier relaxation processes in quasi-2D systems.

2.3 Hot-Phonon Effects

Another phenomenon influencing how effectively hot carriers lose energy to the lattice is the hot-phonon effect. This refers to the creation of a large nonequilibrium population of phonons in the semiconductor so that a free carrier has increased probability of absorbing a phonon. Such a process leads to a reduction in the net probability of emission of a phonon and therefore reduces the energy loss rate from carriers to the lattice. This effect is present in bulk as well as quasi-2D systems. Most treatments of this effect in quasi-2D systems assume bulklike phonon modes, an assumption that is just beginning to be examined critically. Pötz and Kocevar (Chapter II.3) review the results obtained by a balance-equation approach to the problem; Goodnick and Lugli discuss Monte Carlo simulations of this problem (Chapter III.1), and Ryan and Tatham (Chapter IV.3) discuss how time-resolved Raman scattering experiments can provide information about the dynamics of hot phonons in quasi-2D systems.

2.4 Scattering Processes Specific to Quasi-2D Systems

It is clear that additional scattering processes become available in quasi-2D systems as a result of the spatially inhomogeneous potential and modified energy-band structure. One such process is the intersubband scattering of electrons and holes. This has been investigated theoretically and experimentally and is reviewed by Ridley (Chapter II.1) and Ryan and Tatham (Chapter IV.3) respectively. Another process that becomes important is the transfer of carriers from wells to barriers as the carrier kinetic energy increases. Such real-space transfer effects form the basis of some hot-electron devices (for a recent review, see [14]). Finally, the inverse of this process, namely capture of electron and holes from the barriers into the quantum wells, is also important. Some aspects of this problem are reviewed by Shah (Chapter IV.1).

2.5 Tunneling Times

Tunneling is a quantum-mechanical phenomenon that is of considerable fundamental interest, and there are many devices based on tunneling of carriers through nonostructures. Tunneling also plays an important role in the perpendicular transport of carriers in multiple quantum-well structures. An important aspect of this problem is the question of tunneling times. There are many different approaches to this problem and some confusion on this subject in the literature. Jauho (Chapter II.4) reviews this fundamental topic. Further theoretical work is to be expected in this area. Although tunneling is not a traditional hot-carrier phenomenon, it is included in this book because it plays an important role in nonequilibrium transport of carriers through nanostructures. Optical studies of tunneling are discussed by Shah (Chapter IV.1), and devices based on tunneling are discussed by Yokoyama et al. (Chapter V.2) and Brown (Chapter V.3).

2.6 Quantum Transport

A discussion of fundamental aspects of transport in semiconductor nanostructures would be incomplete without considering the question of quantum transport. When the length scales in transport become comparable to the carrier de Broglie wavelength or when the time scales become comparable to the duration of collisions, many of the usual assumptions in classical transport break down. This is an emerging field, but certainly very important in the context of this book, because technology continues to provide smaller structures and faster devices, and faster lasers to investigate them. This topic is reviewed by Rossi et al. (Chapter II.5) in the final chapter of this part.

3 MONTE CARLO SIMULATIONS

Classical transport is determined by the Boltzmann transport equation, which cannot be solved analytically in any realistic physical situation of interest. For this reason, a number of techniques have been developed for solving this equation numerically. The Monte Carlo method has been widely used in hot-electron studies and has been reviewed in earlier books [4,15]. The method simulates the motion of either a single particle or an ensemble of particles in a crystal under the influence of external perturbations subject to various scattering processes present in the crystal. Quite often, the Monte Carlo simulation goes beyond the standard Boltzmann equation and includes numerous quantum effects. The method has been applied with considerable success to studies of submicron devices and photoexcited semiconductors.

3.1 Monte Carlo Simulations of Ultrafast Optical Studies

As discussed by Shah in Chapter IV.1, optically excited carriers are subject to a number of different scattering processes in semiconductors. Although an experimentalist strives to devise experiments that isolate a specific process, such attempts are usually not entirely successful. One strength of Monte Carlo simulations in this respect is the ease with which a specific scattering process can be turned on or off. Thus, comparisons of simulations and experiments allows one to obtain insight into the physics and quantitative information about scattering rates. Goodnick and Lugli (Chapter III.1) review this field and illustrate it with various examples.

In the past ten years, Monte Carlo simulations of optical experiments have made progress in several directions. Techniques have been developed to simulate an ensemble of particles so that carrier–carrier scattering can be included in the simulations. The complicated structure of valence bands in bulk semiconductors has been included, and efforts are under way to do the same for quantum wells. Various forms of screening of carrier–phonon and carrier–carrier interactions are included, and efforts are under way to include the phonon modes calculated on the basis of micoroscopic models and to include the full dynamic screening by using a molecular-dynamics approach. Effects of carrier densities have been taken into account in a self-consistent manner. Thus, Monte Carlo simulations have become extremely valuable in interpreting ultrafast optical experiments. There are, however, limitations to this approach. One limitation that requires further attention is the influence of many-body exchange and correlation effects in the simulation of optical experiments, a topic that will be further discussed in Chapters IV.1 and IV.2.

3.2 Monte Carlo Simulations of Submicron Devices

Monte Carlo simulations of submicron devices have provided new insights into the physics of such devices by comparison of the predicted performance of the device with its measured performance. As discussed by Kizilyalli and Hess (Chapter III.2), the Monte Carlo method provides several advantages: it applies from diffusive to ballistic transport regimes, it can incorporate band structure effects, and it can incorporate complicated boundary conditions. Kizilyalli and Hess review this technique and apply it to field-effect transistors in the GaAs system.

4 OPTICAL STUDIES OF HOT CARRIERS IN SEMICONDUCTOR NANOSTRUCTURES

It is now well established that optical spectroscopy provides valuable insight into processes related to hot-carrier effects and nonequilibrium transport in semiconductors. This subject has been reviewed in earlier books [16], so that the emphasis in this book is on recent developments related to quasi-2D systems. Ultrafast optical spectroscopy allows an investigation of the dynamics of nonequilibrium processes and has dominated the field in the past ten years. A general introduction to various relaxation processes in photoexcited semiconductors is presented by Shah in Chapter IV.1. We present here an overview of Part IV, which deals with various aspects of optical studies of hot-carrier and nonequilibrium transport effects in semiconductor nanostructures.

4.1 Ultrafast Luminescence Studies of Carrier Relaxation and Tunneling

Luminescence spectroscopy provides a powerful technique for studying carrier distribution functions by offering excellent time resolution (≥ 50 fs), background-free detection with large dynamic range and an internal probe of the system. These characteristics make this the preferred technique for investigating the cooling of thermalized carriers to the lattice temperature. In addition, luminescence spectroscopy provides an excellent means for investigating perpendicular transport in semiconductors using the technique of optical markers, which provide a unique spectral signature to specific spatial regions. Both topics are reviewed by Shah (Chapter IV.1). The discussion of carrier cooling concentrates on the influence of dimensionality, screening and hot- phonon effects on cooling rates in quasi-2D systems. The discussion of transport phenomena focuses on the luminescence studies of tunneling in double-barrier structures and double-quantum-well structures. The tunneling studies are relevant to the discussion by Jauho (Chapter II.4) and Brown (Chapter V.3).

4.2 Femtosecond Pump-and-Probe Transmission Studies

Femtosecond pump-and-probe transmission studies provide several complementary strengths. These include possibility of better time resolution (6 fs has been demonstrated), the ability to probe resonantly as well as over a wide spectral range, and the possibility of studies at lower densities. Femtosecond pump-and-probe transmission studies have been performed with excitation far above the bandgap (providing the possibility of exploring processes such as intervalley scattering, intersubband scattering, and real-space transfer) and with excitation near the bandgap (providing information about excitonic effects and carrier–carrier scattering). A number of studies have been performed on quasi-2D systems in the past decade and are reviewed by Knox (Chapter IV.2). While these studies have provided considerable empirical understanding of processes such as electron–electron and electron–hole scattering, a comprehensive theoretical understanding is not yet available, as discussed by Knox.

4.3 Ultrafast Pump-and-Probe Raman Scattering Studies

Raman scattering provides information that is complementary to that obtained by the two techniques discussed above. In particular, ultrafast pump-and-probe Raman scattering studies can investigate the dynamics of phonons (albeit only at the wavevectors accessible to Raman scattering) and the dynamics of a single carrier type, i.e., electron or hole, through studies of intersubband transitions. Although the time resolution of Raman studies has not approached that of pump-and-probe transmission or luminescence studies, they have provided valuable information, which is reviewed by Ryan and Tatham (Chapter IV.3).

4.4 Electron–Hole Scattering

Optical excitation creates both electrons and holes, so that momentum and energy exchange between electrons and holes may play an important role in optical studies. These effects are also important for devices, e.g., for electrons traveling in the p-type base region of a bipolar transistor. In spite of their importance, electron–hole scattering has received serious attention only recently. One way to investigate this interaction is a combination of transport and optical studies in quantum wells. Such studies have not only determined the rate of energy exchange between electrons and holes, but also demonstrated the absolute negative mobilities of carriers resulting from electron–hole momentum scattering. These studies have led to a determination of electron–hole scattering rates in quantum wells and are reviewed by Höpfel et al. (Chapter IV.4).

5 TRANSPORT STUDIES AND DEVICES

Part V of this book deals with nonequilibrium transport in nanostructures and devices made out of nanostructures. At least a part of the current excitement in the field of hot carriers in semiconductor nanostructures arises from the fact that devices using some of the unique properties of nanostructures lead to improved performance in many instances. The topics selected for this part are investigations of ballistic transport in lateral nanostructures, resonant hot-electron transistors and resonant tunneling diodes.

5.1 Ballistic Transport in Nanostructures

With reduction in dimensions, the possibility of ballistic transport of carriers increases. This is an exciting aspect of nonequilibrium transport in nanostructures and was first explored for electrons in vertical nano-structures, which also allowed energy analysis of the transported electrons. These studies have been reviewed earlier (Heiblum and Fischetti in [10]). The most recent development in this field is ballistic transport in lateral nanonstructures defined by lithography. The results in this field are reviewed by Heiblum and Sivan (Chapter V.1).

5.2 Resonant Tunneling Hot-electron Transistors

Several configurations can be used to inject hot electrons in a transistor structure. Yokoyama et al. have used resonant tunneling through a double-barrier structure as a hot-electron injector and made transistors using such injectors. Resonant tunneling hot-electron transistors of InGaAs-based material have shown significantly improved performance in recent years. Small-scale integrated circuits based on these transistors are currently under serious development. Progress in this area is reviewed by Yokoyama et al. in Chapter V.2.

5.3 Resonant Tunneling Diodes

The high-frequency potential of resonant tunneling diodes made out of double-barrier structures was responsible for much of the early interest in this field (Sollner et al. in [10]). There have been some important breakthroughs in this field in recent years. In particular, the use of material systems other than GaAs in these diodes is promising. Devices based on these new material systems have shown oscillations at frequencies approaching the terahertz region. Brown (Chapter V.3) reviews these recent developments as well as the fundamental physics of resonant tunneling devices. Although considerable progress has been made in understanding tunneling in double-barrier diodes, microscopic theories for certain aspects of the problem are still lacking.

6 SUMMARY

This book presents reviews of the most exciting recent developments in the field of hot carriers in semiconductor nanostructures. It covers basic physics and device physics as well as applications. This overview has discussed interrelations between these topics and has attempted to put various developments in perspective.

REFERENCES

1. Conwell EM. High Field Transport in Semiconductors. In: Seitz F, Turnbull D, Ehrenreich H, eds. Solid State Physics, Supplement 9. New York: Academic; 1967.

2. Nag BR. Theory of Electrical Transport in Semiconductors. Pergamon; 1972.

3. Nag BR. Electron Transport in Compound Semiconductors. Berlin: Springer; 1980.

4. Reggiani L, ed. Hot Electron Transport in Semiconductors. Berlin: Springer; 1985.

5. Ferry DK, Barker JR, Jacoboni C, eds. Physics of Nonlinear Transport. New York: Plenum; 1980.

6. Grubin HL, Ferry DK, Jacoboni C, eds. The Physics of Submicron Semiconductor Devices. New York: Plenum; 1988.

7. Ando T, Fowler AB, Stern F. Rev. Mod. Phys. 1982;54:437–672.

8. Applications of Multiquantum Wells, Selective Doping, and Superlattices. In: Dingle R, ed. New York: Academic; . Semiconductors and Semimetals. 1987;24.

9. Capasso F, Margaritondo G, eds. Heterojunction Band Structure Discontinuities: Physics and Device Applications. Amsterdam: North-Holland; 1987.

10. Capasso F, ed. Physics of Quantum Electron Devices. Berlin: Springer; 1990.

11. (eds.) Harris CB, Ippen EP, Mourou GA, Zewail AH. Ultrafast Phenomena VII: Proceedings of the 7th International Conference. Berlin: Springer; 1990.

12. Esaki L, Tsu R. IBM J. Res. Dev. 1970;14:61.

13. Cardona M, Güntherodt G, eds. Light Scattering in Solids V. Berlin: Springer; 1989.

14. Sze SM, ed. High Speed Semiconductor Devices. New York: Wiley; 1990.

15. Jacoboni C, Lugli P. The Monte Carlo Method for Semiconductor Device Simulation. Berlin: Springer; 1989.

16. Shah Jagdeep, Leheny RF. In: Alfano RR, ed. Semiconductors Probed by Ultrafast Laser Spectroscopy. New York: Academic; 1984:45–75.

Part II

Fundamental Theory

II.1

Electron–Phonon Interactions in 2D Systems

B.K. Ridley    Department of Physics University of Essex Colchester, United Kingdom

1. Introduction   17

2. Quantum Confinement   20

3. The Electron–Phonon Scattering Rate   30

4. Model Rates for the Fröhlich Interation   34

5. Scattering by Acoustic Phonons   46

6. Concluding Remarks   49

References   50

1 INTRODUCTION

The electron–phonon interaction is a central topic in solid state physics, notably as the process that determines electrical resistance, superconductivity, and the equilibrium dynamics of hot electrons. Theories of the way electrons interact with lattice vibrations in bulk crystalline material are well developed, and their predictions are borne out in reasonable detail by a wealth of experiment work. In recent years interest has turned to the problem of describing the interaction in layered semiconductor structures in which both electrons and phonons exhibit quantum confinement. In such circumstances the basic question we would like to answer is: how does the confinement of electrons and phonons affect their interaction?

A major feature of experimental work in this area is the prevalence of hot carriers. This is nothing new as regards devices such as field-effect transistors (FETs) and avalanche photodiodes (APDs), but it is more unusual as regards optical devices such as quantum-well lasers and far-infrared detectors, and quite new as regards superlattice and tunnelling structures. In all of these, the performance crucially depends on the rôle of phonons in relaxing the energy of energetic carriers induced by high electron fields or by optical injections, and it is important to understand how that rôle is modified by quantum confinement. Recent developments of ultrafast spectroscopy involving picosecond and femtosecond laser pulses have opened up the possibility of directly observing the rate of scattering of carriers by phonons under favourable conditions. One striking observation has been the appearance of so-called hot phonons in the presence of high carrier concentrations, and these have the effect of markedly slowing down the energy relaxation rate. These topics will receive individual attention in the rest of this book; the aim in this chapter is to lay a foundation on which an understanding of the part played by phonons in layered structures can be built, and only the simplest of models of the electron–phonon interaction will be described.

Growth of layered material has been mostly of III–V semiconductors, and we will have this category in mind throughout. The basic electron–phonon interactions are:

1. Polar interaction with long-wavelength longitudinally polarized optical (LO) phonons.

2. Deformation-potential interaction with LO and TO (transversely polarized) phonons.

3. Piezoelectric interaction with acoustic modes.

4. Deformation-potential interaction with acoustic modes.

Table 1 summarizes the selection rules for intravalley processes near the valley minimum that involve only long-wavelength modes, these being the only ones that can satisfy the conservation of energy and momentum. Intervalley processes are mediated by short-wavelength LA and LO modes via a deformation-potential interaction. By far the most important intravalley process for Γ valley electrons at room temperature is the polar interaction with LO modes. Since many of the technologically important materials such as GaAs, InGaAs, InP are direct-gap semiconductors with Γ valley electrons, the polar LO interaction is of particular interest, and we will be concentrating on this, though we will not forget the others.

Table 1

Selection rules

A word is in order here on what will not be dealt with. The optical-phonon processes mentioned above are zero-order processes in the sense that the interaction depends directly on the amplitude of the ionic displacements. Acoustic-phonon processes are therefore first-order in that the interaction here depends upon differential displacement i.e. strain. Where an optical- or intervalley-phonon zero-order process is forbidden by symmetry, a first-order process may be allowed. However, we do not intend to treat first-order processes of this type here. Nor will we concern ourselves with two-phonon and many-phonon processes. It is not that these processes are of no importance, but rather that little work has been done on them in the context of low-dimensional structures (LDSs).

Another type of excitation involving ionic vibrations is the polariton, which is a transversely polarized electromagnetic wave coupled with the optical modes of the solid. We will not be dealing with the electron–polariton interaction, but only mention polaritons here because as Fuchs–Kliewer slab modes and interface modes they have frequently been given the properties of LO modes, in that a scalar potential field has been assigned to them and this has allowed a strong coupling to the electron to be deduced. However, transversely polarized electromagnetic waves do not exhibit a scalar potential—the interaction with the electron is magnetic via the usual A ⋅ p term and hence weaker. An ascription to polaritons of a scalar potential needs considerable justification, but the latter is so far lacking.

One of the major facets of the LDS field is the largeness of the carrier density in the vast majority of experiments. The shrinking of dimensions has almost inevitably entailed an increase of carrier density in order to overcome extraneous effects in the substrate or cladding material. As a consequence, the effects of quantum confinement have often been diluted by the sheer complexity introduced by having large number of electrons. In the present context this means that the bare electron–phonon interaction to be discussed in this chapter will require modification to take into account the coupling between phonons and plasmons. In the case of acoustic modes this can be done comparatively simply via a static screening approximation. In the case of optical modes the situation is more complicated, since the screening is dynamic, giving antiscreening as well as screening, and producing frequency shifts. Such coupled-mode effects are discussed in Chapter II.2. Another complication of high densities is the production of a nonthermal distribution of phonons—hot phonons. This has the effect of increasing the probability of the electrons reabsorbing phonons, which slows down the process of energy relaxation and tends to enhance momentum relaxation. This topic will be discussed in Chapter III. 1.

Application in practice commonly requires the evaluation of energy relaxation rates and momentum relaxation rates for the ensemble in question. Bare scattering rates have to be weighted by energy or momentum and by the occupation probabilities of the initial and final states, which in turn requires knowledge of the distribution function. In this chapter we will limit our attention to the bare scattering processes.

We begin by discussing the quantum confinement of electrons and phonons in Section 2. The confinement of phonons is still a topic of active investigation, and we spend some time in discussing the main issues. General aspects of the electron–phonon scattering rate are discussed in Section 3, and in Section 4 we concentrate on the rates for the Fröhlich interaction with LO modes as described by the dispersive continuum model. We conclude with an account of acoustic-phonon scattering.

2 QUANTUM CONFINEMENT

Wherever dissimilar semiconductors adjoin there will be, in general, discontinuities of the conduction-band and valence-band edges, and discontinuities of dielectric and elastic properties. These discontinuities are the source of electron and phonon confinement. To be specific, let us consider the best-investigated case—that of the lattice-matched GaAs/AlxGa1 − xAs system. AlGaAs has a larger band gap than that in GaAs, the difference being distributed between conduction-band and valence-band offsets in a ratio that is difficult to determine accurately but is about 70:30. Consequently both electrons and holes can be confined in a GaAs layer sandwiched between two layers of AlGaAs. If the layer is thin enough, that is, thin relative to the energy relaxation length for a carrier, the wavefunction maintains coherence across the layer, and as a result the confinement is quantized. If, moreover, the chance of an elastic collision is negligible, the carrier wavefunction can be taken to be an unmodified Bloch function provided that the layer is thick enough to contain across its width several unit cells.

The conditions of wavefunction and current continuity which must be satisfied as the interfaces are well known [1,2], and so are the difficulties that can arise when the cell-periodic parts of the Bloch functions on either side of the interface do not coincide [3–5]. If none of these difficulties arise, the boundary conditions refer solely to envelope functions, and so for k|| common, where k|| is the wavevector in the plane,

   (1)

which are satisfied only for certain values of kz, where mw* and mB* refer the effective masses in the well and barrier respectively. The scheme can be extended easily to the case of a superlattice. Figure 1 shows the subband structure for electrons in a GaAs–AlGaAs superlattice.

Figure 1 (a) Electron band structure in a Al 0.3 Ga 0.7 As–GaAs superlattice (100-Å barriers), (b) ground-state wavefunctions, (c) excited-state wavefunctions, L  = 40 Å.

Describing the confinement of holes is less simple because of the mixing of heavy-hole (HH) and light-hole (LH) states [6]. The explicit conditions of Eq. (1) work only for k|| = 0, but for k|| ≠ 0 this mixing means that k|| does not remain fixed for a given subband, and its variation with k|| contributes to the zone-centre effective mass. The latter turns out to be much smaller than the bulk heavy-hole mass. Thus, holes are very seriously affected by confinement, far more than electrons are, and the variation of kz with k|| makes working out the rates for inelastic scattering more complicated, especially