Semiconductors for Optoelectronics: Basics and Applications [1st ed. 2021] 3319449346, 9783319449340

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Graduate Texts in Physics

Naci Balkan Ayşe Erol

Semiconductors for Optoelectronics Basics and Applications

Graduate Texts in Physics Series Editors Kurt H. Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA Jean-Marc Di Meglio, Matière et Systèmes Complexes, Bâtiment Condorcet, Université Paris Diderot, Paris, France Sadri Hassani, Department of Physics, Illinois State University, Normal, IL, USA Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo, Norway Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan Richard Needs, Cavendish Laboratory, University of Cambridge, Cambridge, UK William T. Rhodes, Department of Computer and Electrical Engineering and Computer Science, Florida Atlantic University, Boca Raton, FL, USA Susan Scott, Australian National University, Acton, Australia H. Eugene Stanley, Center for Polymer Studies, Physics Department, Boston University, Boston, MA, USA Martin Stutzmann, Walter Schottky Institute, Technical University of Munich, Garching, Germany Andreas Wipf, Institute of Theoretical Physics, Friedrich-Schiller-University Jena, Jena, Germany

Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field.

More information about this series at http://www.springer.com/series/8431

Naci Balkan · Ay¸se Erol

Semiconductors for Optoelectronics Basics and Applications

Naci Balkan (Deceased) School of Computer Science and Electronic Engineering University of Essex Colchester, Essex, UK

Ay¸se Erol Department of Physics Istanbul University ˙Istanbul, Turkey

ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-319-44934-0 ISBN 978-3-319-44936-4 (eBook) https://doi.org/10.1007/978-3-319-44936-4 Original Turkish edition published by Seçkin Yayıncılık Sanayi ve Ticaret A.S., ¸ Ankara, 2013 Translation from the Turkish language edition: Yarıiletkenler ve Optoelektronik Uygulamaları by Naci Balkan, and Ay¸se Erol, © Seçkin Yayincilik Sanayi ve Ticaret A.S., Ankara 2013. Published by Seçkin Yayıncılık Sanayi ve Ticaret A.S., ¸ Ankara. All Rights Reserved. © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

I dedicate this book to memory of Prof. Naci Balkan

Preface

Optoelectronics have become a vital part of our lives and today’s technology. Optoelectronic devices such as light emitting diodes, semiconductor lasers, photodetectors and solar cells operate using interactions between electrons and photons. Today optoelectronic devices take place in a vast range of applications from fiberoptic communication to medical applications and remote sensing systems. Optoelectronic devices are based on semiconductors. Using modern growth techniques, it is possible to discover novel semiconductors, therefore, the operation wavelengths of optoelectronic devices cover UV to MIR regions of the electromagnetic waves. The unique properties, their controllable electrical conductivity and tailoring their bandgaps, make semiconductors indispensable in optoelectronic technology. Furthermore, using advanced lithographic techniques, it is possible to decrease the size of the semiconductors to nanometer scale that leads to fabricating high capacity, fast, high efficient, compact, high sensitive optoelectronic devices. Optoelectronics have been playing an increasingly important role in our lives since the discovery of observation light emission from a GaAs-based p-n junction and today highly efficient LEDs illuminate the world as a light source of the 21st century. To understand how optoelectronic devices operate, it is necessary to have knowledge on semiconductor physics, electronic transport and optical processes in semiconductors. Therefore, first three chapters provide knowledge on the basics of these topics. All optoelectronic devices are based on a p-n junction. Chapter 4 gives a deep insight into physics of p-n junctions. The rest of the book investigates solar cells, photodetectors, LEDs and semiconductor lasers. The target of this book is physics students, electronic engineers and researchers who are interested in semiconductors physics and applications, designing optoelectronic devices. We have written the chapters to allow readers, who are new to the subjects, to comprehend the fundamental concepts in optoelectronics. The content of the book consists of courses on Semiconductors and Semiconductor Devices given by Prof. Naci Balkan at University of Essex (UK) and Physics of Semiconductors and Optoelectronics given by me at Istanbul University (TR) for years. The book was first published in Turkish in 2014 and the second edition was released in 2015. The idea of writing a book on optoelectronics in Turkish came from vii

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Preface

Prof. Naci Balkan and his aim was to contribute to increasing the number of students and researchers to be involved in optoelectronics in Turkey. After we had an offer from Springer to write a book on optoelectronics, we started to prepare the book in January 2015, but interrupted for some time after Prof. Naci Balkan died on June 25th, 2015. Prof. Naci Balkan made his career in semiconductor physics and he spent more than almost 40 years in this field until the date of his death. His contribution to semiconductor physics, especially physics and applications of hot electron transport in low dimensional semiconductor structures is invaluable. After he died, the book was completed in memory of Prof. Naci Balkan, who would have been so happy if he had seen the book published. I have to say that this book would not have seen the light of day without his great effort. I would like to thank a number of people who have helped in the various stages of the preparation of this book. First, I would like to thank Prof. Naci Balkan, whose idea was to prepare this book and he offered me to become the co-author of the book. Our paths crossed in 1994 at a workshop on low dimensional semiconductors and since then, he had motivated and encouraged me every time. I will remain indebted to him for his continuous support and invaluable contributions to my scientific life, his presence and place in my life are very important. Special thanks for everything. I am also grateful to my colleagues Dr. Furkan Kuruo˘glu and Dr. Fahrettin Sarcan for their contributions to draw some figures, plot graphs, etc. in order to improve the manuscript. Finally, I would like to thank my doctoral supervisor, Prof. Mehmet Çetin Arıkan for pointing me toward semiconductor physics/devices and giving me the opportunity to meet and study with Prof. Naci Balkan. I encourage the readers to email their suggestions/comments to ayseerol@ istanbul.edu.tr. ˙Istanbul, Turkey

Ay¸se Erol

Contents

1 Electrical Properties of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Ohm’s Law and Ohmic Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Resistance and Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Conductance and Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Classification of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Metals, Semiconductors and Insulators . . . . . . . . . . . . . . . . . 1.4.2 Electrical Conductivity, Charge Carrier Density and Carrier Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Bandstructure of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Electron Distribution Within Energy Bands at T ~0 K . . . . . 1.4.5 Valence Band, Conduction Band and Electron Delocalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.6 Electrical Conductivity of Solids at T ~0 K . . . . . . . . . . . . . . 1.4.7 Electrical Conductivity of Solids at Finite Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.8 Free Charge Carriers in Semiconductors . . . . . . . . . . . . . . . . 1.4.9 Electrons and Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.10 Bipolar Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Bandstructure of Various Semiconductors . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 2 3 3 4

27 28 28 30 31 35

2 Intrinsic and Extrinsic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Density of States Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fermi–Dirac Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Electron Distribution at Absolute Zero Temperature . . . . . . 2.2.2 Distribution of Electrons at Finite Temperatures . . . . . . . . . 2.2.3 Charge Neutrality Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Intrinsic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Free Carrier Densities in Intrinsic Semiconductors . . . . . . . 2.3.2 Temperature Dependence of Carrier Densities . . . . . . . . . . . 2.3.3 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 39 44 46 48 50 51 51 56 57

4 5 23 25 26

ix

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2.4 Extrinsic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 n- and p-Type Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 n-Type and p-Type Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Binding Energy of Hydrogenic Impurities . . . . . . . . . . . . . . 2.4.4 The Fermi Level in Extrinsic Semiconductors . . . . . . . . . . . 2.4.5 Carrier Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Temperature Dependence of Carrier Density in Extrinsic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 59 61 67 71

3 Charge Transport in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Transport in an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Particle Current and Charge Current . . . . . . . . . . . . . . . . . . . 3.1.2 Drift Velocity and Carrier Mobility . . . . . . . . . . . . . . . . . . . . 3.1.3 Matthiessen’s Rule and the Total Mobility . . . . . . . . . . . . . . 3.1.4 Temperature Dependence of Mobility . . . . . . . . . . . . . . . . . . 3.1.5 Electric Field Dependence of Mobility . . . . . . . . . . . . . . . . . 3.1.6 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.7 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8 Bipolar Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Diffusion Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Diffusion Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Drift and Diffusion Current Densities . . . . . . . . . . . . . . . . . . 3.2.4 Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Non-equilibrium Carriers in Semiconductors . . . . . . . . . . . . . . . . . . . 3.3.1 Semiconductor in Thermal Equilibrium . . . . . . . . . . . . . . . . 3.3.2 Generation and Recombination of Excess Carriers . . . . . . . 3.3.3 Continuity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Ambipolar Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Quasi-Fermi Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Excess Carrier Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Shockley–Read–Hall Recombination Theory . . . . . . . . . . . . 3.5.2 Low Injection Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 80 80 81 84 86 87 91 92 92 93 95 95 96 97 100 100 101 104 107 115 117 118 118 121 123

4 The p-n Junction Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 p-n Junction in Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Built-In Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Depletion Layer Width and the Built-In Electric Field . . . . 4.2 p-n Junction Under an External Electric Field . . . . . . . . . . . . . . . . . . . 4.2.1 Charge Injection and Current in p-n Junction . . . . . . . . . . . . 4.2.2 Minority and Majority Carriers in a p-n Junction . . . . . . . . 4.3 High Voltage Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Forward Bias: High Injection Region . . . . . . . . . . . . . . . . . . 4.3.2 Reverse Bias: Impact Ionisation . . . . . . . . . . . . . . . . . . . . . . .

125 126 129 133 136 138 142 144 145 145

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4.4 Junction Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.5 Temperature Dependence of Diode Current . . . . . . . . . . . . . . . . . . . . 151 4.6 Tunnel Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5 Solar Cells (Photovoltaic Cells) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Principles of Solar Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Material Choice for Solar Cells . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 p-n Junction Under Illumination . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Solar Cell Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Principal Considerations for Solar Cell Design . . . . . . . . . . 5.2.2 Enhancement of Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Advantages and Disadvantages of Solar Cells . . . . . . . . . . . . . . . . . . . 5.4 Thermophotovoltaic (TPV) Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Advantages and Disadvantages of TPV Cells . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 158 158 167 170 172 177 177 181 187 187 189 191

6 Photodetectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Optical Transitions in Direct Bandgap Semiconductors . . . . . . . . . . . 6.2 Choice of Material and Wavelength of Operation . . . . . . . . . . . . . . . . 6.3 Operating Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 pin Photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Quantum Efficiency and Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Rise Time and Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Avalanche Photodiodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 194 197 200 202 204 208 211 217

7 Light Emitting Diodes and Semiconductor Lasers . . . . . . . . . . . . . . . . . 7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Absorption and Emission Rates: Einstein Relations . . . . . . 7.1.2 Population Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Optical Feedback and Laser Oscillations . . . . . . . . . . . . . . . 7.1.4 Threshold Condition for Laser Oscillations . . . . . . . . . . . . . 7.2 Optical Processes in Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Light Emitted Diodes (LEDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Semiconductor Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Homojunction Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Efficiency of a Semiconductor Laser . . . . . . . . . . . . . . . . . . . 7.4.3 Gain and Threshold Current . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Temperature Dependence of Threshold Current . . . . . . . . . 7.5 Heterojunction Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Quantum Well Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Low Dimensional Semiconductors: Quantum Wells . . . . . . 7.6.2 Density of States in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Absorption in Quantum Well Systems . . . . . . . . . . . . . . . . . .

219 220 220 226 232 235 239 244 250 252 254 256 258 259 261 262 264 265

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7.6.4 Quantum Well Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Vertical Cavity Surface Emitting Laser—VCSEL . . . . . . . . . . . . . . . 7.7.1 Temperature Dependence of VCSELs . . . . . . . . . . . . . . . . . . 7.8 Distributed Feedback Lasers (DFB) . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

266 267 272 274 277

Solution for Selected Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Chapter 1

Electrical Properties of Solids

Learning Outcomes At the end of this chapter, the reader will have: 1. 2. 3. 4. 5.

understood the physical concepts related to formation of energy bands in solids gained detailed knowledge about classification of solids as metals, semiconductors and insulators an ability to relate electrical conductivity to the bandstructure of solids a clear understanding of electron and hole transport understood the concept of bipolar conductivity.

Solids are basically classified as being a metal (conductor), a semiconductor and an insulator. All these three type of solids are utilized in different areas in device technology. Metals interconnect devices and metallic bonds are necessary to measure or apply electric signals. Insulators are necessary to insulate devices from each other and they also form oxide layer in devices such as Metal Oxide Semiconductor Field Effect Transistor (MOSFET), Light Emitting Diode (LED), etc. Among these three type of solids, undoubtedly, the most important one in device technology is semiconductors. The usage of semiconductors in electronic technology has been started with transistors in 1947 [1], then, realization of light emission from a GaAs p-n junction in 1962 [2] initiated optoelectronic technology leading to fabrication of semiconductor based devices such as light emitting diodes (LEDs), semiconductor lasers, photodiodes, solar cells etc. Semiconductor are unique type of solids because their electrical properties can be controlled and optical properties can be tailored thanks to growth technology that made possible to grow novel semiconductors with desired electrical conductivity and bandgap. In order to understand operation of optoelectronics devices, it is important to have knowledge about electrical and optical properties of semiconductors. In this chapter, we classify solids as metals, semiconductors and insulators according to their bandstructure and electrical properties and reveal that why their electrical properties are different from each other and why semiconductors are superior to metals and insulators. © Springer Nature Switzerland AG 2021 N. Balkan and A. Erol, Semiconductors for Optoelectronics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-44936-4_1

1

2

1 Electrical Properties of Solids

Fig. 1.1 a When a solid with resistance R is biased with a voltage V, a current I flows through the solid. b Graphical representation of Ohm’s law

1.1 Ohm’s Law and Ohmic Conduction When a voltage V is applied across a solid, a current I passes through it (Fig. 1.1a). If the current is directly proportional to the applied voltage and inversely proportional to resistance of the solid R, the relationship between the voltage and the current is called Ohm’s law and expressed as: I =

V R

(1.1)

The units for I, V and R are Ampere (A), Volt (V) and Ohm (), respectively. The solids obey Ohm’s law is called Ohmic conductors. Figure 1.1b gives a graphic representation of Ohm’s law. The resistance is the slope of the I-V plot.

1.2 Resistance and Resistivity Consider the section of a wire illustrated in Fig. 1.2. The resistance of the wire is directly proportional to its length l and inversely proportional to cross-sectional area Fig. 1.2 A current I flows through a conducting wire with length l and a cross-sectional area of A, when it is biased with a voltage V. Here E is the electric field

1.2 Resistance and Resistivity

3

A. The resistance of the wire is therefore written as: R=ρ

l A

(1.2)

where ρ is called resistivity. The unit of resistivity is m. The resistivity is an intrinsic property of a material, therefore its value is independent of its shape or its size and quantifies how strongly a given material opposes the flow of current. Every material has its own characteristic resistivity. For example, all aluminium wires, irrespective of their shape and size, have the same resistivity, but a long, thin aluminium wire has larger resistance than a thick, short aluminium wire.

1.3 Conductance and Conductivity The conductance of the wire shown in Fig. 1.2 S is defined to be the reciprocal of its resistance R and the conductivity of the wire σ is inverse of its resistivity ρ. We have therefore: S=

1 1 and σ = R ρ

(1.3)

From Eq. 1.3, for the wire of length l and cross-sectional area A, the conductance is expressed as: S=σ

A 

(1.4)

The units of conductance and conductivity are Siemens (S) and S/m or (m)−1 , respectively. The conductivity is a property of a material just like resistivity, that is, its value does not depend on its size or shape only on the material itself.

1.4 Classification of Solids Pure silver is known as the best conductor among bulk crystal solids with its electrical conductivity of 6.25 × 107 (m)−1 at room temperature. On the other hand, sulphur, with an electrical conductivity of 10–16 (m)−1 at room temperature, is the poorest conductor among crystal solids. Electrical conductivity values of crystal solids span a range of nearly 1024 of magnitude. This is a much wider range than that of any other physical property (e.g. carrier density, thermal conductivity, elasticity etc.) of solids. This chapter describes how this enormous range of electrical conductivity values

4

1 Electrical Properties of Solids

Table 1.1 Room temperature electrical conductivity of various metals, semiconductors and insulators[3] Metal

Semiconductor

Insulator

Materials

σ (m)−1

Materials

σ (m)−1

Materials

σ (m)−1

Ag

6.25 ×

Ge

2.25

Glass

10–12

Cu

5.9 × 107

Si

4 × 10–4

Quartz

10–13

Al

3.8 ×

GaAs

2.5 ×

Sulphur

10–14

107

107

10–7

Fig. 1.3 Grouping of solids into metal, semiconductor and insulator on a logarithmic scale of conductivity

can be understood in terms of bandstructure of solids. This leads to a description of the origin of the mobile charge carriers (free carriers), which are responsible for electrical conduction in solids.

1.4.1 Metals, Semiconductors and Insulators Solids are basically classified as metals, semiconductors and insulators accordingly to their electrical conductivity values. Table 1.1 lists examples from each of these three groups of solids and Fig. 1.3 illustrates the clustering of solids into the three groups on a logarithmic conductivity scale. Although currently this is a purely empirical division of solids according to their electrical conductivity, we will see later that this classification has an elegant explanation in terms of the bandstructure of solids. In order to do this, we start with a qualitative description of the electrical conductivity mechanism in solids.

1.4.2 Electrical Conductivity, Charge Carrier Density and Carrier Mobility The value of electrical conductivity in solids is related to the existence of a certain amount of free carrier density within the solid, which is able to move or be transported

1.4 Classification of Solids

5

under the influence of an applied electric filed. These free carriers may be either electrons which carry the elementary electronic charge –e, or holes which carry a charge +e. A solid contain either only one type of carriers or both. For simplicity, let us assume that the free carriers are only electrons in a solid with a density of n. The electrical conductivity of the solid is then given by: σ = neμ

(1.5)

In Eq. 1.5, μ is a parameter called carrier mobility and is defined as the ratio of drift velocity to applied electric field. The carrier mobility characterizes how fast a charge carrier can move in a solid under an applied electric filed. The transport under an electric filed is called drift because the charge carriers, here electrons, move by applied electric field. The drift velocity is acquired by a carrier moving under the influence of an electric field. As we shall explain in Chap. 3, charge carriers are also transported in a solid via diffusion process if the carrier density changes with position in the solid. The carrier mobility is an intrinsic property of a material. For example, room temperature electron mobility in InSb is ~8 m2 /Vs [4], on the other hand, in some organic solids can be as low as ~10–5 m2 /Vs [5]. These two extreme values of carrier mobility cover a range of less than six orders of magnitude, which is approximately 1018 smaller than the aforementioned range of electrical conductivity values of solids. Therefore, from Eq. 1.5, it can be predicted that free carrier density in solids should extend across approximately 1018 of magnitude. This vast range of free carrier can be understood by considering bandstructure of solids. The bandstructure also explains why solids group into metals, semiconductors and insulators. The importance of bandstructure in fact goes much further as it plays a significant role in the description of the physical operation within semiconductor devices. Therefore, to gain a detailed insight into the semiconductor device operation, we need to understand the essential qualitative features of the bandstructure of solids.

1.4.3 Bandstructure of Solids In order to reach an analytical picture of the bandstructure of solids, we first start with a model introduced by Bohr, which describes the allowed energy levels for the electron of a free hydrogen atom. We then generalise the model to include atoms with more than one electron and consider the effect of bringing large numbers of atoms together to form a solid.

1.4.3.1

Bohr Model for Hydrogen Atom

A free hydrogen atom has a nucleus with charge +e and an electron with charge −e orbiting the nucleus as shown in Fig. 1.4. In order to define the electronic energy

6

1 Electrical Properties of Solids

Fig. 1.4 A simplified illustration of a Hydrogen atom

levels in the hydrogen atom, Bohr postulated two approaches, classical and quantum mechanical. 1.

Classical approach: An electron can orbit the nucleus in stable orbits, such that the electrostatic Coulomb force on the electron due to the positively charged nucleus just balances the centrifugal mechanical force due to circular motion of electron.

If the mass of the electron is m, its angular frequencyω, radius of the orbit is r and the dielectric constant of free space is ε0 then: Fcentrifugal = FCoulombic , mω2 r =

e2 4π εo r 2

(1.6)

In terms of angular velocity: ω= m

v r

(1.7)

v2 e2 = r 4π εo r 2

(1.8)

From Eq. 1.8, velocity of the electron is:  v=

e2 4π εo mr

1/2 (1.9)

Clearly any value of υ can be chosen by adjusting r to satisfy Eq. 1.9. 2.

Quantum mechanical approach: The angular momentum and energy of the electrons are quantised. Bohr postulated that an atom can have only a set of allowed quantised energy levels. According to this postulate, any change in the energy of the electron, which may occur as a result of the absorption or emission of a photon should justify:

1.4 Classification of Solids

7

E m − E n = ω = E f oton

(1.10)

where ω and ω are angular frequency and energy of the photon, respectively. E m and En are the energies of the m and n states and  is Planck’s constant. Bohr’s quantum mechanical approach can be easily understood by studying the de Broglie wavelength of the electron orbiting the nucleus. As shown in Fig. 1.5, for an electron orbiting the nucleus at a constant velocity, the circumference of the orbit S must satisfy the following constructive interference condition: S = 2πr = nλ (n = 1, 2. . . .)

(1.11)

where λ is the de Broglie wavelength of the electron. The orbits satisfying Eq. 1.11 are known as allowed orbits or allowed states. If the condition given in Eq. 1.11 is not satisfied then the de Broglie waves will interfere destructively and the orbits or states will not be allowed. Assuming that the orbits are circular, we can then use Eq. 1.11 together with the de Broglie wavelength of the electron (λ = h/ p) and the linear momentum ( p = mv): S = 2πr = nh/mv

(1.12)

Combining Eqs. 1.12 and 1.9 for n = 1, 2, 3, we can obtain the radius of the allowed orbits, r n : n 2 h 2 εo rn = = π me2



 h 2 εo n 2 = 0.053 n 2 (nm) π me2

(1.13)

The radius of the first allowed orbit corresponding to n = 1 is r B = 0.053 nm and is known as the Bohr radius. We have so far established that electrons can only move in a set of allowed orbits, we can now derive the expression for the allowed energies corresponding to these orbits. The total energy of the atom is the sum of its potential energy, arising from Coulomb interaction, and its kinetic energy which comes from its velocity. Fig. 1.5 Quantum model of the hydrogen atom showing the de Broglie wave of the electron orbiting the nucleus

8

1 Electrical Properties of Solids

The kinetic energy of electron: E kin =

1 2 mv 2

(1.14)

Combining this with Eq. 1.9 for an electron in an orbit with radius r: E kin (r ) =

e2 8π εo r

(1.15)

FCoulomb dr

(1.16)

e2 4π εo r

(1.17)

The potential energy of the electron is: r E pot (r ) = ∞

E pot (r ) = −

The negative sign in Eq. 1.17 arises from the fact that in bring an electron from infinity to an orbit with radius r to assemble the atom, the system has done work and lost energy. The total energy is given by: E(r ) = E kin (r ) + E pot (r ) =

e2 e2 − 8π εo r 4π εo r

(1.18)

Putting the value of r given in Eq. 1.13 in Eq. 1.18 gives the total energy of the atom as   1 me4 (n = 1, 2, 3 . . . .) (1.19) E n (r ) = − 2 2 8εo h n 2 For a hydrogen atom, me4 /8εo2 h 2 = 13.6eV , which corresponds to the energy of the first allowed state for n = 1 and known as the Rydberg energy (E Ryd ). The electron volt (eV) is the energy acquired by an electron under a potential difference of 1 V. This unit of energy is convenient in semiconductors. It is clear from Eq. 1.19 that electron can only have a set of quantised energies corresponding to the allowed states. Coulomb potential, potential and kinetic energies, Rydberg energy and allowed quantized energies of the hydrogen atom are shown in Figs. 1.6 and 1.7. Consider now the case of an atom with Z protons in its nucleus and therefore Z electrons orbiting around it. The analysis of the hydrogen atom still can be applied to this situation except that the electrostatic Coulomb force due to the nucleus on an orbiting electron will now be:

1.4 Classification of Solids

9

Fig. 1.6 Coulomb potential of the hydrogen atom. Total energy of the atom is the sum of kinetic energy of electron and potential energy due to Coulomb forces

Fig. 1.7 Quantised energy levels in atom

1 (Z e)e 4π εo r 2

(1.20)

  Z 2 me4 1 n = 1, 2, 3 . . . . . . . 8εo2 h 2 n 2

(1.21)

FCoulomb = Therefore the allowed energies are: E n (r ) = −

To summarise, in the Bohr atom model the electrons surrounding the positively charged nucleus are constrained by the quantisation rules and occupy a well-defined set of discrete energy levels (Fig. 1.7). These allowed discrete energy levels are separated from each other by a certain energy values which electrons are forbidden to be located between these levels.

10

1.4.3.2

1 Electrical Properties of Solids

Pauli Exclusion Principle

In the full quantum mechanical description of an isolated atom’s electronic structure, the nature of the calculation, which leads to the discrete allowed electron energy values, may be indicated very qualitatively as follows. In the periodic table an element is specified by two parameters. The first is the mass number, A, which is the integer closest to the atomic weight of the element, and the second, is the atomic number, Z, which represents the number of protons in the nucleus of the element. The nucleus of an atom with atomic number Z therefore carries a positive charge +Ze. This nucleus is surrounded by Z electrons to give an electrically neutral atom. The first step of the calculation is to determine the set of allowed energies for these Z electrons. This is achieved by considering the Schrödinger’s equation for the electrons in the potential energy field of the positively charged nucleus. The solutions to this equation are a set of wavefunctions each specified by a set of quantum numbers, which are conventionally labelled either as n, l, m and s or n, l, ml and ms . The labels stand for the principle, angular momentum (azimuthal), magnetic and spin quantum numbers, respectively. From these wavefunctions the set of allowed wavefunctions are generated by the quantisation rules. n = 1, 2, 3 . . . . . . . . . . . . .n l = 0, 1, 2, 3 . . . . . . . . . .(n − 1) m = 0, ±1, ±2 . . . . . . , ±l s = ±1/2 Each allowed wavefunction has a discrete electron energy value associated with it and therefore represents an allowed quantum state of the electron. The set of allowed wavefunctions gives rise to allowed discrete energy levels of electrons with each energy level being specified by the set of quantum numbers (n, l, m, and s), which determines the corresponding wavefunction. This provides the set of allowed, discrete energy values for the electrons of the atom with atomic number Z. The question now is, how are the Z electrons of the atom distributed to allowed levels? The stable ground state of any system is the state with the minimum potential energy, so on this basis we would expect all the Z electrons to occupy the allowed state with the lowest energy, namely that specified by the set of quantum numbers (1, 0, 0, ±1/2). However, according to the Pauli Exclusion Principle, in a quantum system two electrons cannot occupy exactly the same quantum state. In other words, two electrons cannot have the same quantum numbers n, l, m, and s. Therefore, in the ground state of the atom, the Z electrons occupy the set of Z lowest allowed energy states. This completes the calculation of the electronic configuration of the atom. A periodic table of elements based on their electronic structure can be constructed using Pauli Exclusion Principle in conjunction with the relationships between the various quantum numbers. Thus for n = 1, l and m must both be zero and two electronic states exist, corresponding to the two spin quantum numbers. If n = 2, l can equal 0 or 1, and for l = 1, m may have values −1, 0, +1. The various combinations

1.4 Classification of Solids

11

Table 1.2 Electronic configurations of various light elements Element

Number of electrons

Principal quantum numbers, n

Azimuthal quantum numbers, l = 0, 1, … (n−1)

Magnetic quantum numbers, m = −l,….+l

Spectroscopic representation

H

1

1

0

0

1s1

He

2

1

0

0

1s2

Li

3

2

0, 1

−1, 0, +1

1s2 2s1

Be

4

2

0, 1

−1, 0, +1

1s2 2s2

B

5

2

0, 1

−1, 0, +1

1s2 2s2 2p1

C

6

2

0, 1

−1, 0, +1

1s2 2s2 2p2

N

7

2

0,1

−1, 0, +1

1s2 2s2 2p3

O

8

2

0, 1

−1, 0, +1

1s2 2s2 2p4

F

9

2

0, 1

−1, 0, +1

1s2 2s2 2p5

Ne

10

2

0, 1

−1, 0, + 1

1s2 2s2 2p6

Na

11

3

0, 1, 2

−2, −1, 0, + 1s2 2s2 2p6 3s1 1, +2

Mg

12

3

0, 1, 2

−2, −1, 0, + 1s2 2s2 2p6 3s2 1, +2

Al

13

3

0, 1, 2

−2, −1, 0, + 1s2 2s2 2p6 3s2 3p1 1, +2

Si

14

3

0, 1, 2

−2, −1, 0, + 1s2 2s2 2p6 3s2 3p2 1, +2

P

15

3

0, 1, 2

−2, −1, 0, + 1s2 2s2 2p6 3s2 3p3 1, +2

of (n, l, m) produce (2,0,0), (2,1,−1), (2,1,0), (2,1,1) and each of these is associated with two possible spin states, making a total of eight possible states with principle quantum number n = 2. This process can be repeated for n = 3, l = 0, 1, 2 and so on, and various combinations for periodic table are shown in Table 1.2. Electrons that have the same principal quantum number n, are said to be in the same shell. It is evident from the Table 1.2 that the maximum number of electrons per shell is 2n2 . Within a shell, each state corresponding to a particular integer value of l, the azimuthal quantum number, is given a letter designation for l = 0; s, for l = 1; p, for l = 2; d, for l = 3; f . This notation arises from the early spectroscopic identification of lines corresponding to various electronic transitions, namely sharp, principal, diffuse, fundamental etc. Thus in the electronic classification of the lighter elements, given in the last column of Table 2.1, the letters correspond to the azimuthal quantum number and the indices give the number of electrons in such subshells. It should be noted that the periodic table does not progress continuously in such a numerically logical manner. For a group of heavier elements, the energy of some electrons in an outer shell is lower than that in an inner subshell and these levels are filled before the inner subshells are fully occupied.

12

1 Electrical Properties of Solids

Fig. 1.8 Si atom with four valence electrons forms covalent bonding with nearest neighbouring atoms

The elements that form the basic materials used in semiconductor devices all have covalent bonds. Such covalent crystals are characterised by their hardness and brittleness. They are brittle because adjacent atoms must remain in accurate alignment, since the bond is strongly directional and formed along a line joining the atoms. The hardness is a consequence of the great strength of the paired electron bonding. The most important semiconducting elements all appear in group IV of the periodic table. Silicon (Z = 14) and germanium (Z = 32) are elemental semiconductors and their electronic configurations are: 14

Si :

1s2 2s2 2p6 3s2 3p2

32

Ge :

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p22

It is clear from their electronic structure that both Si and Ge have four valence electrons in their outer shells. Therefore, both Ge and Si need 4 electrons to fill their outer shells. They form covalent bonds with the four nearest neighbouring atoms by sharing each of their valence electrons with these atoms as shown in Fig. 1.8. According to the Pauli Exclusion Principle the spins of the shared pairs of electrons must be different. The compounds, which are based on the group III and V elements, such as GaAs, InP, GaN, etc. form the main semiconductors used in optoelectronic device applications. In GaAs, Ga is a group III element with three valence electrons in the outer shell (4s2 4p) and As belongs to group V of the periodic table and has 5 electrons in the outer shell (4s2 4p3 ). Ga and As atoms share their valence electrons with the neighbouring atoms. Therefore the bonding is, in principle, covalent, like in Si atoms. However, because Ga and As belong to different groups in the periodic table and the numbers of their valence electrons are different, the bonding involves charge transfer and therefore it is in fact a combination of ionic and covalent bonding. In II-VI group compounds such as ZnO and ZnS, because of the larger disparity in the number of valence electrons, the bonding is more ionic than covalent.

1.4 Classification of Solids

1.4.3.3

13

Energy Bands in Solids

In the Bohr atom model, the electrons surrounding the positively charged nucleus are allowed to have only particular energies and between these allowed energies are forbidden to electrons. When atoms assemble to form a solid, there will be allowed energy bands in solids and these bands are separated from each other by a certain value of energy. This division of electron energy values in a solid into a series of alternatively allowed and forbidden bands (bandgaps), as indicated in Fig. 1.9, is referred to as the electronic bandstructure of solids. Each crystalline solid has its own bandstructure which, as discussed, plays a crucial role in determining the electrical properties of the solid. However, in order to see this connection between bandstructure and electrical conductivity, it is necessary to look closely at the relationship between allowed discrete electron energy levels in an atom and allowed electron energy bands formed when a large number of atoms are brought together in close proximity to form a solid. We shall do this by considering the effect of bringing just two atoms close to each other and then generalising for the case of a large number of atoms which constitute a solid. The transition from the allowed discrete energy values for electrons in an isolated atom to the allowed electron energy bands in a solid is indicated in Fig. 1.10. We

(a)

(b)

Fig. 1.9 a Allowed electron energy levels in an isolated atom, b Allowed and forbidden energy bands in a solid

14

1 Electrical Properties of Solids

(a)

(b)

(c)

Fig. 1.10 a Wavefunctions of two isolated atoms and associated energy levels. b Overlapping wavefunctions of two closely placed atoms and associated energy levels. c Energy band of N neighbouring atoms in the solid with strongly overlapping wavefunctions

start with a pair of identical atoms which are sufficiently far apart from each to be considered a separate isolated quantum system. For each atom, E max represents the highest occupied electron energy level. Figure 1.10 also shows a purely schematic two-dimensional representation of the positively charged nucleus of the atom and the wavefunctions associated with the electron energy level E max . The wavefunctions are functions of space and time and in the situation depicted here, the two wavefunctions have no overlap or interaction with each other. The lack of wavefunction overlap is just another way of indicating that the two atoms may be regarded as isolated systems. In this case both atoms are in ground states. Now imagine that they are brought closer together until the wavefunctions partially overlap as shown in Fig. 1.10b. The two atoms now have a quantum interaction through this wavefunction overlap and therefore they no longer represent two separate quantum systems, instead, they are considered to be a single composite quantum system. By applying the Pauli Exclusion Principle to this quantum system it follows that the two identical atomic wavefunctions, which represent identical quantum states, must be modified by their mutual interaction. This means that there are two similar but different wavefunctions which provides two distinct quantum states that can be occupied by the two electrons without violating the Pauli Exclusion Principle. In terms of electron energy level E max , as shown in Fig. 1.10a, this is split as depicted in Fig. 1.10b, into a pair of closely spaced energy levels associated with the two new wavefunctions.

1.4 Classification of Solids

15

It is possible to generalise from the two interacting atoms to the case in Fig. 1.10c by considering a very large number of atoms (N) brought together to form a solid. This results in the single energy level E max in the isolated atom being transformed into a band of N closely spaced energy levels. If this procedure is repeated for the series of allowed discrete energy levels in an atom, then we reach the electronic bandstructure of allowed and forbidden energy bands of a solid in Fig. 1.9. There are a number of theoretical models used in the studies of energy bands in solids. One of the most commonly used model is the tight binding model and in fact the previous discussions about the energy bands are the qualitative results from this model. In order to form the solid, an alternative approach may be to consider that the isolated atoms occupy the lattice points of the periodic crystalline lattice. While doing so, we may assume that each atom is made up of a positively charged ion occupying a fixed point in the crystal lattice whilst its valence electron is free to move within the crystal. As the positively charged ions are distributed periodically within the crystal lattice, then the average charge density due to these ions will be uniform throughout the crystal within the physical boundaries of the solid. Free electrons can then be considered to be trapped within the solid, within a potential well of height V, as defined by the uniformly distributed ions of the crystal lattice. Consequently, in order to determine the energies of the electrons within the potential well (quantum well), we need to solve Schrodinger’s equation for a potential well of height V. This model is known as the Free Electron Model. The free electron model is quite simple, but very successful in explaining the electronic properties of metals. However, it falls short when predicting why some solids are metals and others are semiconductors or insulators. A better approach is to consider the interaction of free electrons with the individual positive ions which is known as the Nearly Free Electron Model. Coulomb potential associated with the positively charged ions within the solid is in fact only a modified version of the potential of an isolated atom where the modification is the result of the repulsive Coulombic forces between neighbouring ions. As a result of this, the neighbouring atomic orbits are affected by the modified Coulomb potential, resulting in the spatial extension of electron wavefunctions and thus broadening the energy levels into bands. According to the nearly free electron model, the electron in the solid is considered to be under the influence of the periodic crystal potential which exists as a result of the periodic distribution of ions. In Fig. 1.11 the periodic crystal potential is shown together with the assumed quantum well potential in the free electron model. According to the representation in Fig. 1.11, the potential profile associated with the periodic distribution of the ions within the crystal is given as: x < −L/2 V0 V (x) = Vcr ystal −L/2 < x < +L/2 x > L/2 V0

(1.22)

16

1 Electrical Properties of Solids

(a)

(b)

(c) Fig. 1.11 a Potential energy of an electron in the vicinity of isolated atom, b The overlapping of potential energies for closely spaced N number of atoms. c Filled circles: ions placed at periodic lattice points; broken lines: quantum well potential in the free electron model; continuous lines: modified potential in nearly free electron model

The potential described in Eq. 1.22 is periodic: Vcr ystal (x) = Vkristal (x + a)

(1.22a)

Herea is the separation between the adjacent ions, in other words, it is the periodicity of the crystal lattice and called as lattice constant. We can now write the time-independent Schrödinger’s equation for −L/2 < x < +L/2 as:   H ψ E (x) = H0 + Vcr ystal (x) ψ E (x)   2 ∂ 2 = − + Vcr ystal (x) ψ E (x) 2m ∂ x 2 = Eψ E (x)

(1.23)

Since the crystal has translational symmetry with period a, then charge distribution will naturally have the same symmetry. Therefore, the electron wavefunction will have the same property: |ψ E (x + a)|2 = |ψ E (x)|2

(1.24)

1.4 Classification of Solids

17

This implies that for all the values of x, there is be a phase difference between the wavefunction at x and at x + a: ψ E(k) (x) = u E(k) (x)eikx

(1.25)

where u E(k) is a periodic function with the translational symmetry: u E(k) (x + a) = u E(k) (x)

(1.26)

The free electron part of the wavefunction (eikx ) is usually referred as the envelope function. It is clear from Eq. 1.25 that energy now depends on k, and because of its periodicity, u E(k) can be expressed as a Fourier series: u E(k) (x) =



Cn (k)ei G n x

(1.27)

2nπ n = ±1, ±2, , . . . a

(1.28)

n=0,±1,...

Gn =

The modified electron wavefunction in the presence of a periodic potential was first described by Bloch, as given in Eq. 1.25, and is known as the Bloch wavefunction or simply Bloch function. It has two components namely; eikx which is the free electron wavefunction, and the periodic function, u E(k) which has the lattice periodicity. In free space the √ wavevector, k, of an electron with a finite linear momentum of p is E given as k = px = 2m . In a quantum well where the potential is periodic like u E(k) ,  however, electron wavefunction is modified as described above and consequently the de Broglie wavelength and wavevector are also affected. Now Schrödinger’s equation can be re-written to include the Bloch function and solved to obtain the energy eigenvalues:   2 ∂ 2 − + Vcr ystal (x) ψ E(k) (x) = E(k)ψ E(k) (x) 2m ∂ x 2

(1.29)

and ψ E(k) (x) = u E(k) (x)eikx =



Cn (k)ei(k+G n )x

(1.30)

n=0,±1,...

The allowed values of k are restricted by the boundary conditions. If we assume a one dimensional homogeneous macroscopic crystal with a length of L (between –L/2 and +L/2) where L is equal to the integer multiple of the lattice constant, a, then the boundary condition will be the periodic boundary conditions, i.e. ψ E(k) (−L/2) = ψ E(k) (+L/2)

(1.31)

18

1 Electrical Properties of Solids

u E(k) (x = −L/2) = u E(k) (x = +L/2)

(1.32)

The modulated envelope function eikx should also have the same value at the crystal boundaries at x = −L/2 and x = +L/2: e−ik L/2 = e+ik L/2

(1.33a)

e+ik L = 1

(1.33b)

Therefore:

The allowed k values are then: k=±

2N π L

N = 1, 2, 3 . . . ..

(1.33c)

In principle, Eq. 1.33 predicts that total number of de Broglie wavelengths (2π/k = λdB ) in a crystal with length L is N. This implies that N increases with increasing L, and for large values of L, k becomes semi-continuous. In the k space, the allowed values of k between −π/a and +π/a are equal to L/a. In other words, the allowed k values are equal to the number of atoms in the crystal. The interval between − π/a and +π/a is known as the first Brillouin zone with zone boundaries at ±π/a. Consequently, the interval between ±(N−1)π/a and ±Nπ/a is the Nth Brillouin zone with zone boundaries at ±Nπ/ a. If we now consider a three dimensional lattice, the crystal potential, similar to the one dimensional case, has the same translational symmetry: − → r + R ) Vcr ystal = Vcr ystal (

(1.34)

− → R = n 1 a + n 2 b + n 3 c is the position vector, r is the translation vector, n1 , n2 ,  c are the primitive lattice vectors, which define the smallest n3 are integers and a , b, cell in the crystal. Primitive cells are the basic building blocks of the crystal and the number of atoms and amount of valence electrons within the primitive cells determine the valence electron density within the crystal. As we shall see, the density of valence electrons within the crystal determines its electrical properties. Let us now return to the one dimensional case and express the periodic potential as a Fourier series: Vcr ystal (x + a) = Vcr ystal (x) Vn ei G n x Vcr ystal = n=±1,±2,±3,...

(1.35)

1.4 Classification of Solids

19

where Vn are the Fourier coefficients. As previously discussed the periodicity of the periodic potential function is equal to the separation between the atoms and therefore to the lattice constant, a. The term Gn is the integer multiples of 2π/a, or in other words, in the reciprocal space, the reciprocal lattice vectors. The concept of reciprocal space is similar to the Fourier representation of a time dependent electrical signal with period T, ε(t + T ) = ε(t) and is given as: ε(t) =



εn ein2πt/T

n=0,±1,±2,..

We may first consider the case where the potential is zero throughout the crystal (V crystal (x) = 0). The solution of the Schrodinger’s Eq. (1.29) is quite straight forward as the wavefunction can be just be inserted into Eq. 1.29: 1 (0) = √ eikx ψ E(k) L

(1.36)

The solution gives the allowed energies of the electron: E (0) (k) =

2 k 2 2m

(1.37)

The dispersion relation, (E-k), represents the free electron energy as a function of wavevector as shown in Fig. 1.12. As stated previously, the free electron model fails to explain why some solids are metals and some are either semiconductors or insulators. In order to have a clear idea about the physical reason for the classification of solids within these three groups, we will now consider the case when there is a non-zero periodic potential within the crystal. When there is a periodic potential within the crystal, the free electron wavefunction should be replaced by the Bloch wavefunction in Schrodinger’s equation: Fig. 1.12 Dispersion curve for free electrons

20

1 Electrical Properties of Solids



 2 ∂ 2 i Gn x + Vn e − ψ E(k) (x) = E(k)ψ E(k) (x) 2m ∂ x 2 n=±1,±2,..

(1.38)

If the crystal potential is small then Schrödinger’s equation can be solved using the time independent perturbation theory. Detailed information about the time independent perturbation theory can be found in any fundamental quantum physics books, however, we shall summarise the technique here for clarity. We start with the free electron model where the potential everywhere within the crystal is zero but infinitely high at the crystal boundaries as indicated by the broken lines in Fig. 1.11. We then assume that the crystal potential is perturbed by the periodic potential as given in Eq. 1.35. The electron wavefunction will then have two components, the first component is the free electron component and the second contains the terms which arise as a result of the perturbation: ψ E(k) (x) =

1 ikx (1) e + ψ E(k) (x) L

(1.39)

(1) (x) is the first order perturbation parameter. With the help of Eq. 1.30, Here ψ E(k) we can obtain this term: 1 ik  x (1)  e ψ E(k) (x) = C E(k) (k ) (1.40) L k

Here k  = G n + k  . The time independent perturbation theory gives the first order solution for the approximation that L is very large

compared to the wavefunction (L → ∞) and for  E (0) (k) = E (0) (k ) or |k| = k  : C E(k) (k  ) =

E (0) (k)

Vn δk  ,(k+G n ) − E (0) (k  )

(1.41)

First order wavefunction: 1 ikx (1) ψ E(k) (x) = e + ψ E(k) (x) + . . . L 1 ikx Vn 1 i(k+G n )x = e + e + .... (0) (k) − E (0) (k + G ) L E L n n=±1,±2,.. (1.42a) And the energy eigenvalues are:

1.4 Classification of Solids





(0) 2

  E (0) H

E j i

E i = E i(0) + E i(0) H E i(0) + + ... E i(0) − E (0) j i= j

21



(1.42b)

For the crystal potential given in Eq. 1.35, there is no need for a first order modification for the energy eigenvalues. Indeed the Fourier series representing Eq. 1.35 does not have the n = 0 term. Therefore the effect of the perturbation can only be important for the second order terms and the energy eigenvalues can be written as: E(k) = E (0) (k) + E (1) (k) + E (2) (k) + . . . . |Vn |2 2 k 2 = + + ... 2m E (0) (k) − E (0) (k + G n ) n=1,2,3..

(1.43)

Equations 1.42 and 1.43 carry two significant information regarding the energy of the electrons within the crystal lattice: 1. 2.

In the periodic lattice the eigenstate corresponding to the eigenenergy E(k) represents a Bloch state with a wavevector of k + G n . The electrons energy is modified in comparison to the free electron case. For k values satisfying the E (0) (k) = E (0) (k + G n ) and n = ±1, ±2…., E(k) becomes degenerate. The k values satisfying this condition are at the Brillouin zone boundaries, i.e., k 2 = (k + G)2 or (k = ±G n /2 = nπ/a).

At the zone boundaries there is a modified version of the perturbation theory, which takes into account of the degeneracy used. The behaviour of the dispersion relations at the zone boundaries determines whether the solid is metal, semiconductor or insulator. Figure 1.13 shows the dispersion curves for the free electron and nearly free electron models. The nearly free electron model predicts the electron energy changes from those determined by the free electron case at the zone boundaries. However, Fig. 1.13 Broken lines: Dispersion relation in the free electron model. Solid line: The effect of periodic crystal potential on the free electron dispersion curve. Shaded areas: Allowed electron energy bands. Between shaded areas, forbidden bands (bandgap) form

22

1 Electrical Properties of Solids

in order to understand fully how the dispersion curves differ, we have to invoke Bragg’s law. According to Bragg’s law standing waves are formed at wavevectors corresponding to the zone-boundaries, i.e., k = ±G n /2 = nπ/a and the group velocity of the electron tends to become zero. Since the group velocity is derivative of dispersion curve, the curve should deviates from the free electron dispersion curve approaching a minimum or maximum value. Therefore the dispersion curve can be qualitatively plotted around zone boundaries as depicted in Fig. 1.13. The inclusion of the periodic lattice potential results in the creation of forbidden energies for the electron states as shown in Fig. 1.13, which are known as bandgaps. Solids are classified as metals, semiconductors and insulators according to the size of these bandgaps. The solid line in Fig. 1.13 is called extended zone scheme and it is easy to identify the deviations from the free electron dispersion curve (broken line in Fig. 1.13). On the hand, it is beneficial to represent the dispersion curves within the first Brillouin zone by folding all higher-order Brillouin zones into the first zone as shown in Fig. 1.14a. This is known as the reduced zone scheme and depicted in Fig. 1.14b. In most instances, for simplicity, band diagrams are preferred, refer to left figure of Fig. 1.14b, rather than the reduced zone representation. We are now in the position of being able to classify solids. We should first consider the valence electron number in each primitive cell. In the k space, we have already established that within an interval between −π/a and +π/a the number of allowed k values is the same as the number of atoms. For every allowed k value, we can place two electrons with opposing spins. Therefore if the number of atoms is N, a total number of 2N electrons can be accommodated in the first Brillouin zone. If the number of valence electrons in the primitive cell of the solid is one, then the total number of electrons in the first Brillouin zone will be N, thus the first Brillouin zone will be half full. However, if each primitive cell has two valence electrons then the first Brillouin zone will contain 2N electrons and will be full. If the primitive cell contains three valence electrons, the first Brillouin zone will be completely full and second zone will be half full etc. In summary, when the valence electron number in the primitive cell is even the uppermost energy band will be full and when it is odd the uppermost energy band will be half full. In principle solids with odd numbered valence electrons in their primitive cells are metals; those with even numbers are either semiconductors or insulators. Semiconductors are those with smaller bandgaps and insulators have larger bandgaps. Figure 1.15 shows the band diagrams of metals, semiconductors and insulators. In this representation, at absolute zero temperature, the uppermost full band is known as the valence band and the one above which is separated from the valence band by the bandgap is known as the conduction band. In Fig. 1.15, it is clear that the uppermost energy band in metals is half full, whilst the uppermost bands in semiconductors and insulators are empty at absolute zero temperature. It is important to note the difference between the bandgaps of semiconductors and insulators.

1.4 Classification of Solids

23

Fig. 1.14 a Folding the higher-order Brillouin zone into the first zone. b Reduced zone scheme representation of dispersion curves and simplified diagram of allowed bands and bandgaps obtained from the reduced zone scheme

(a)

(b)

1.4.4 Electron Distribution Within Energy Bands at T~0 K We have now established that the energy levels allowed to the electrons, of the atoms constituting a solid, form a series of allowed energy bands. Each allowed energy band consists of a very large number of closely packed discrete energy levels. The separation in energy between these discrete levels is so small (in the order of nanoelectron volts) that these discrete levels may be regarded as quasi-continuous within

24

1 Electrical Properties of Solids

(a)

(b)

(c)

Fig. 1.15 Band diagrams of a metals, b semiconductors and c insulators. E g is the bandgap and the shaded area represents the allowed energies filled by electrons

the band. These allowed energy bands are separated from each other by forbidden bands of energy which are referred to as bandgaps. And the bandgap values can change from zero to several electron volts. Another energy scale of fundamental importance is the basic unit of thermal energy associated with an electron, or any other particle, at a given temperature. The average thermal energy is given by k B T, where k B is the Boltzmann constant. At room temperature k B T is 26 meV. The first important thing to note about this value is that it makes k B T at room temperature very much greater than the separation between the individual allowed levels which form the allowed energy bands. The second significant fact is that at room temperature k B T is very smaller than the allowed energy bands and bandgap. Table 1.3 lists the bandgaps of various semiconductors at room temperature. It is clear from this table that k B T for these semiconductors is much smaller than the bandgap. As with the earlier case of an isolated atom, we may ask the question: How are the electrons in a solid distributed among the allowed energy bands? The answer is that, as in an isolated atom, the electrons occupy the lowest available energy levels, whilst obeying the Pauli Exclusion Principle. In the case of the single atom, shown in Table 1.3 Bandgaps of various semiconductors at 300 K [6, 7] Semiconductor

Bandgap (E g ) (eV)

Semiconductor

Bandgap (E g ) (eV)

Si

1.12

InSb

0.17

Ge

0.66

GaP

2.25

GaAs

1.42

AlP

2.45

AlAs

2.15

InP

1.27

InAs

0.36

AlN

6.3

GaSb

0.68

GaN

3.4

AlSb

1.63

InN

0.7

1.4 Classification of Solids

25

(a)

(b)

Fig. 1.16 The two possible electron distributions within the energy bands of solids at T ~0 K

Fig. 1.10, this resulted in E max being the highest occupied energy level. The same will be true in the case of a solid, except that in this case the existence of a bandstructure produces two significantly different possibilities; at absolute zero temperatures, the highest populated band may be either fully or only partially occupied, as indicated below in Fig. 1.16a and b, respectively. The significance of these two possibilities will soon become clear because with Fig. 1.16 we are now quite close to our goal of demonstrating the connection between the electron energy bandstructure of a solid and its electrical conductivity. To demonstrate this connection we need to introduce one more concept into our picture which is the idea of electron delocalisation.

1.4.5 Valence Band, Conduction Band and Electron Delocalisation The highest completely filled electron energy band in a solid, at zero absolute temperature, is conventionally referred to as the valence band. The next higher empty band is known as the conduction band. The forbidden bandgap between these two bands

26

1 Electrical Properties of Solids

plays a decisive role in determining the electrical and optical properties of a solid. Therefore, it is referred to as the energy bandgap of the material and its width is usually denoted by E g . For a solid such as that in Fig. 1.16a, the conduction band is completely empty, while, for the solid in Figure, 1.16b the conduction band is partially empty. The origin of the energy bands in solids is due to the interaction between the atoms in a solid, which means that the solid can be treated as one collective quantum system. A corollary of this collective behaviour is that an electron in a band can no longer be identified as being associated exclusively with a particular atom. All the electrons which are indistinguishable particles individually, in a band must be considered to be collectively associated with the atoms forming the solid. This is particularly true of any electrons present in the conduction band because as we will show soon, it is these electrons which constitute the negatively charged free carrier population in a solid. Therefore, in addition to the collective association with all atoms in the solid, electrons in the conduction band are said to be delocalised because they are capable of being transported macroscopically through a solid. This then is the first indication of the connection between bandstructure and electrical conductivity in solids, namely that it is the presence of electrons in the conduction band which gives rise to conductivity. On the other hand, electrons in the valence band are localised and do not contribute to the conductivity as they provide the bonding between atoms and so determine the mechanical properties of the solid.

1.4.6 Electrical Conductivity of Solids at T~0 K When an external emf is applied to a solid, the electrons will have a tendency to move through the solid under the influence of the electric field. An electron in the process of drifting through the solid will need to acquire, from the driving electric field, the kinetic energy associated with its drift velocity. The total energy of a drifting electron is therefore higher than its energy in the absence of the electrical field. In other words, in order to drift an electron, electron must occupy a higher energy state than it did before the external emf was applied. Such higher energy states are readily available to the electrons in the conduction band of Fig. 1.16b in the form of the unoccupied energy levels in the band. Consequently, electrons in the conduction band represent a free (mobile) charge carrier population which is capable of carrying a macroscopic current when an external emf is applied. The solid of Fig. 1.16b would be a good electrical conductor while the bandstructure shown in Fig. 1.16b is in fact the generic bandstructure of metals. Turning now to electrons in the valence band of Fig. 1.16a, it is clear that in view of the electron drift mechanism previously described, there are no unoccupied higher energy states available to these electrons. This means that they do not constitute a free carrier population and consequently make no contribution to the electrical conductivity of the solid. Since the same applies to the electrons in all the lower

1.4 Classification of Solids

27

completely filled energy bands, then it follows that the solids represented by the bandstructure of Fig. 1.16b must all be insulators at T ~0 K.

1.4.7 Electrical Conductivity of Solids at Finite Temperatures Following the description of the relative scales of various energy values associated with the electron bandstructure of solids, it is now relatively straight forward to envisage how we would expect the electrons to be distributed within the energy bands in solids at 300 K. Let us start with the electron distribution at T ~ 0 K for a typical metal shown in Fig. 1.17a. If these electrons now possess the additional thermal energy k B T (equal to 26 meV at room temperature) then we would expect their distribution within the band to be similar to Fig. 1.17b. These mobile carriers may now be considered as an electron gas at 300 K.

(a)

(b)

(c)

(d)

Fig. 1.17 Schematic representation of electron distribution in the energy bands of a metals at T = 0 K, b metals at a finite temperature, c semiconductors at a finite temperature and d insulators at a finite temperature

28

1 Electrical Properties of Solids

Turning to the case of a solid as shown in Fig. 1.17c, with a completely empty conduction band at T ~0 K, we will now show that the electron distribution at room temperature is, in this case, critically dependent on the value of E g , the bandgap. If E g in a solid were less than 26 meV, then clearly at room temperature we would expect electrons from the full valence band to be thermally excited into the allowed states available in the empty conduction band. This solid, which behave as an insulator at T ~0 K, would then have acquired electrical conductivity at room temperature. In such a solid, the two bands may be considered as one quasi-continuous band like that represented by Fig. 1.17b, which the gives rise to an electrical conductivity. Let us now consider that the case of E g is greater than k B T. In a classical system, when E g > k B T, electron distribution at a given temperature will be identical to the distribution at T ~0 K. However, in a quantum system, there exists a small but nonzero probability, of electrons being thermally excited across a bandgap E g , even when E g > k B T. The order of magnitude of this small probability is related to the e−E g /2k B T and, at a given temperature, this probability becomes smaller for solids having larger bandgap. Consequently, thermal excitation of electrons across the bandgap give rise to increase at electrical conductivity. In more quantitative terms, in real solids at room temperature, values of E g smaller than ~3 eV lead to carrier density values in the conduction band as in Fig. 1.17c. These solids are known as semiconductors. On the other hand, values of E g greater than ~3 eV imply smaller carrier densities in the conduction band as shown in Fig. 1.17d and these solids are known as insulators. This completes our description of how the empirical classification of solids into metals, semiconductors and insulators may be understood at a more fundamental level in terms of bandstructure of solids.

1.4.8 Free Charge Carriers in Semiconductors The electron distribution within the electron energy bands of a typical semiconductor at room temperature is shown schematically in Fig. 1.18. We have already seen that the electron density, n (number of electrons per unit volume), in the conduction band constitutes a free charge carrier population. This will increase the electrical conductivity, σn = neμn . The subscript n in σ n indicates that that the conductivity is due to the negatively charged electrons and the mobility, μn , is the electron mobility. The reason for labelling conductivity and mobility in this way is that there is a contribution to the electrical conductivity by also a positively charged mobile carrier population, i.e., hole density (p) and hole mobility μp , as we will now show.

1.4.9 Electrons and Holes According to the schematic representation in Fig. 1.18, there are electrons in the conduction band. These electrons are thermally excited from the initially full valence

1.4 Classification of Solids

29

Fig. 1.18 Electrons and holes in an intrinsic semiconductor at a finite temperature

band at a finite temperature as described above. Therefore, the density of the conduction band electrons must be equal to the thermally excited electron density from the valence band leaving behind positively charged holes. In the presence of an external emf holes also make a contribution to electrical conductivity. In contrast to electrons in the conduction band where the charge carriers are negatively charged with the charge −e, holes in the valence band holes are positively charged with the charge +e. Furthermore, as we shall see in Sect. 3.1.2, the mobilities of electrons and holes are different. In contrast to the delocalised electrons in the conduction band, electrons in the valence band are localised and that they form chemical bonds between neighbouring atoms as discussed in Sect. 1.4.5. The excitation of an electron from the valence band to the conduction band therefore represents the breaking of a chemical bond between two neighbouring atoms, which results in an initially localised electron becoming delocalised. The bandgap energy E g may therefore be regarded as the energy required to break the chemical bond. If an electric field E is applied to a semiconductor then the negatively charged electrons will drift in the opposite direction to the electric field. In contrast, and disregarding holes for the moment, as the valence band electrons are localised then they

30

1 Electrical Properties of Solids

(a)

(b)

Fig. 1.19 a Electrons and holes in the valence band in the absence of an external electric field. b The motion of electrons and holes in the valence band under the influence of an external electric field

will not drift under the influence of E as we have already established. However, the presence of the hole introduces a new element into the picture. Under the influence of the applied electric field, an electron in the valence band moves into the vacant position that is left behind by another electron that has been thermally excited into the conduction band, leaving behind a vacant position itself. Consequently, while the electron moves in a direction opposite to the electric field, the hole in the valence band moves along the same direction as the electric field as if it had a charge of +e, as indicated in Fig. 1.19. In principle, holes have a positive charge simply because they move in the same direction as the electric field. Their charge must be the same magnitude as electrons because, macroscopically, total negative charge drift must be equal to the positive charge drift. The density of free holes in an intrinsic semiconductor, p, is equal to the density of free electrons, n. Even though their densities are the same the contribution to the overall conductivity by holes and electrons may be different because of various scattering mechanisms which affects carrier mobilities in the semiconductor (Sect. 3.1).

1.4.10 Bipolar Conductivity As stated previously, in an intrinsic semiconductor, electrical conductivity is due to both electrons and holes. As the polarity of these two carries have opposite signs then the conductivity is known as the bipolar conductivity. We can use the expression for conductivity in Eq. 1.5 to write down the total bipolar conductivity as:

Total conductivity is:

σn = neμn

(1.44)

σ p = peμ p

(1.45)

1.4 Classification of Solids

31

σ = σn + σ p σ = e(nμn + pμ p )

(1.46)

So far in this chapter we have studied how free carrier densities arise in solids and, with reference to intrinsic (undoped) semiconductors, how the electrical conductivity is determined by both free electrons and holes with identical densities. In Chap. 2, we will investigate extrinsic (doped) semiconductors to show that how free electron and free hole densities can be controlling changed in semiconductors significantly.

1.5 Bandstructure of Various Semiconductors Semiconductors have either direct or indirect bands. Let us now look at the bandstructure of various semiconductors to understand the concepts of direct and indirect bandgaps. Most semiconductors have their valence band maxima at k = 0. The valence band consists of heavy hole and light hole bands and these overlap at k = 0. Therefore valence band is degenerate. Effective mass of electrons or holes can be obtained from the dispersion curves (E-k) using the following expression: m ∗e =

2

(1.47)

d 2 E/dk 2

According to Eq. 1.47 the curvature of band I in Fig. 1.20 is clearly less than that for band II. Therefore the effective mass in band I will be greater than that in II. The

(a)

(b)

Fig. 1.20 a Direct and b indirect bands. Here I, II and III represents heavy, light and split-off bands, respectively

32

1 Electrical Properties of Solids

band I is known as the heavy hole band. The band below both the heavy and light hole bands arises because of the coupling between the azimuthal angular momentum and spin of electron and is known as the split-off band. When the minimum of the conduction band conduction (bandedge) occurs at k = 0, semiconductors are known as direct bandgap semiconductors, for example GaAs, InN etc. When the conduction band minimum occur at a different wavevector then this is an indirect bandgap semiconductor. When the conduction bandedge E c is at k = 0, the conduction band is parabolic, similar to that of the free electron and its energy is given as E(k) = E c +

2 k 2 2m ∗e

(1.48)

The bandstructures of Si and GaAs are shown in Fig. 1.21. Si has an indirect band and the conduction band minima is at point X. The conduction band can be expressed in the vicinity of the bandedge with an energy ellipsoid (Fig. 1.21b). And the conduction band in X valley is expresses as   2 k 2y + k 2y 2 k x2 E(k) = + 2m ∗ 2m ∗t

(1.49)

This conduction band is not isotropic and therefore the effective mass has directional dependence with two components longitudinal (m ∗ = 0.98 m 0 ) and transverse (m ∗t = 0.19 m 0 ) as depicted in Fig. 1.21b. In Si, because the effective mass has directional dependence it can be represented with a tensor. The bandgap of Si at X point is 1.12 eV at 300 K. The other conduction band valley is at L. The bandstructure of the direct bandgap semiconductor GaAs is shown in Fig. 1.22. This bandstructure is isotropic and has the following parabolic dependence: E(k) =

2 k 2 2m ∗e

(1.50)

Here m ∗e = 0.067m 0 . As the E-k dependence is parabolic, the constant energy surface is spherical and the effective mass does not have a directional dependence. The bandgap of GaAs at T = 300 K is 1.424 eV. The L valley of GaAs is at a higher energy and has a larger curvature than that at the Γ point at k = 0. The valence band of GaAs contains heavy hole, light hole and split off bands. Problems 1.

The resistance between the ends of a cylindrical semiconductor sample is 1.5k . The length of the cylinder is 10−2 m and the radius is 2 × 10−3 m. Calculate the conductivity in S/m.

1.5 Bandstructure of Various Semiconductors

33

(a)

(b) Fig. 1.21 a Si bandstructure b energy surfaces of constant energy ellipsoids: along the k x directionE has a smaller slope than along the other two directions

2. 3. 4. 5.

6.

With reference to question 1, if the free electron density in the semiconductor is 1021 m−3 then calculate the mobility of electrons in m2 /Vs. The speed of an electron in free space is 106 m/s. Calculate the wavelength of the electron. In an isolated hydrogen atom calculate the energy required to excite an electron from the first to the second energy level. A piece of semiconductor has a circular cross-section with radius r and a length l. Assuming free electron density is n, please write down the expression of electron mobility. In copper, the free electron density is n = 8.2 × 1028 m−3 . The resistivity of a Cu wire, with radius 0.95 mm and length 1 m, is 0.0198/m. Calculate the conductivity and the electron mobility in copper.

34

1 Electrical Properties of Solids

(a)

(b)

Fig. 1.22 a Bandstructure of GaAs b The constant spherical energy surface of the isotropic conduction band of GaAs

7.

8.

9. 10. 11.

In a cylindrical Si sample with diameter of 5 cm, the free carrier density and the electron mobility are 1014 cm−3 and 1400 cm2 /Vs, respectively. What is the thickness of the Si sample for the resistance between front and rear surfaces to be R = 0.1  ? Electron energy versus crystal momentum is shown in Fig. 1.12 and expressed 2 2 in Eq. 1.37 as E (0) (k) = 2mk∗ . Calculate the crystal momentum and wavevector e of an electron in GaAs with effective mass of 0.067m0 (m0 free electron mass) and an energy of 0.5 eV above the conduction band minimum. Compare the crystal momentum with that for an electron in free space but with an identical energy. Explain how the conductivity of a solid depends on whether the conduction band is completely full or empty at T ~0 K. Explain why a completely full conduction band does not make any contribution to the conductivity of a solid. (a) Explain the free electron, nearly free electron and tight binding models qualitatively. (b) At which wavevectors (crystal momentum) does the nearly free electron model differ from the free electron model? What is the physical

1.5 Bandstructure of Various Semiconductors

35

reason for this to occur? (c) With reference to the nearly free electron model, explain how the conductivity in solids is related to the valence electrons in the primitive cells.

References 1. 2. 3. 4.

The Nobel Prize in Physics (1956) http://www.nobelprize.org Retrieved June 03, 2016 Lasher G (1962) Stimulated emissions of radiation from GaAs p-n junctions. Phys Rev 133:553 Griffiths D (1999) Introduction to electrodynamics, 3rd edn. New Jersey, USA Rumyantsev S, Shur M (1996) Handbook series on semiconductor parameters, vol 1. London, UK 5. Zhou J, et al (2007) Carrier density dependence of mobility in organic solids: a Monte Carlo simulation. Phys Rev B 75:153201 6. Kittel C (2005) Introduction to solid state physics, 8th edn. New Jersey, USA 7. Vurgaftman I, et al (2001) Band parameters for III–V compound semiconductors and their alloys. 89:11

Suggested Reading List 8. Singleton J (2001) Band theory and electronic properties of solids. Oxford University Press 9. Tang CL (2005) Fundamentals of quantum mechanics for solid state electronics and optics. Cambridge University Press 10. Yu PY, Cardona M (2005) Fundamentals of semiconductors, 3rd edn, Springer 11. Gasiorowicz S (1974) Quantum mechanics. Wiley 12. Callaway J (1991) Quantum theory of solid state. Academic Press 13. Ashcroft NW, Mermin ND (1976) Solid state physics. Saunders 14. Hook JR, Hall HE (1991) Solid state physics, 2nd edn, Wiley 15. Levine SN (1965) Quantum physics of electronics. MacMillan, New York 16. Schubert EF, Physical foundation of solid-state devices http://homepages.rpi.edu/~schubert/ 17. Mishra UK, Singh J (2008) Semiconductor device physics and design, Springer

Chapter 2

Intrinsic and Extrinsic Semiconductors

Learning Outcomes At the end of this chapter, the reader will have: 1. 2. 3. 4. 5.

an in-depth knowledge of the Fermi–Dirac distribution function, an ability to calculate the electron energy distribution, detailed knowledge of intrinsic semiconductors and the temperature dependence of conductivity for intrinsic semiconductors, an understanding of how extrinsic semiconductors are obtained by doping with donor and acceptor impurities, understood the temperature dependence of the Fermi level and conductivity in extrinsic semiconductors.

Pure (intrinsic) semiconductor materials have an extremely limited range of applications in the fabrication of semiconductor devices. The reason for the development of solid-state electronics based on semiconductor materials since the late 1950’s is that the conductivity of semiconductors can be tuned with introducing small quantities of selected impurity atoms into pure semiconductors in a highly controlled methods. This precisely controlled introduction of a range of useful impurity atoms into a pure semiconductor is referred to as doping of an intrinsic semiconductor to have a doped or extrinsic semiconductor. The significance of being able to dope semiconductors in this way lies in the fact that extremely small dopant density of the appropriate elements are capable of producing relatively large, but controlled changes in the electrical conductivity of semiconductor materials. A potentially infinite range of selectively doped extrinsic semiconductors forms the material basis of all semiconductor devices. As discussed in the previous chapter, the electrical conductivity of a material is determined by two material parameters, namely free charge carrier density in the material and their respective mobilities. The change in the conductivity of a semiconductor material upon doping is due to changes in free electron and hole density in the material. To understand the physical basis of semiconductor device operation, it is necessary to acquire knowledge on how free carrier density changes by the addition of dopant atoms to semiconductor materials. © Springer Nature Switzerland AG 2021 N. Balkan and A. Erol, Semiconductors for Optoelectronics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-44936-4_2

37

38

2 Intrinsic and Extrinsic Semiconductors

(a)

(b)

(c)

Fig. 2.1 a Allowed energy levels separated by E g . b Electrons to be distributed in the allowed energy levels c Distribution of electrons in energy bands at a finite temperature

A qualitative description of free electron and hole densities responsible for electrical conduction in pure semiconductors was given in Sect. 1.4.8. In this chapter, we start by giving a quantitative analysis of extrinsic semiconductors along with electron and hole densities within intrinsic semiconductors and is extended to include doped, extrinsic semiconductors. The chapter concludes with a brief explanation of the significant role played by temperature, in determining carrier density and conductivity values in intrinsic and extrinsic semiconductor materials and the operation of semiconductor devices. In the previous chapter, the qualitative description for the electronic configuration of a semiconductor at room temperature was summarised in Fig. 1.18. To make this description more quantitative, we start by noting that we need some information about (a) (b) (c)

the number of available energy states in the conduction and valence bands (Fig. 2.1a). total number of electrons to be distributed within the two bands (Fig. 2.1b). the distribution of electrons within the two bands (Fig. 2.1c).

To quantify this qualitative description, we therefore proceed as follows. We first give a quantitative, analytical expression for the number of the allowed electron energy states in the valence and conduction bands of a semiconductor in terms of the density of states function, N(E). Total number of electrons, which have to be distributed among the allowed states in the two bands has to be known. Finally, it should be known that how these electrons are distributed among the allowed energy levels in the two bands. We will answer this last question by giving a quantitative statistical description of the required electron distribution in terms of the Fermi– Dirac distribution function, f (E). We will then use the complete description to calculate the density of free electron and hole within intrinsic semiconductors.

2.1 Density of States Function

39

2.1 Density of States Function In the schematic representation of Fig. 2.1a, the allowed energy levels within the valence and conduction bands were indicated as being distributed uniformly along the energy scale. In fact, this is not the actual case. In other words, the separation in energy between successive allowed energy levels is not the same, but increases with increasing energy. This non uniform distribution of allowed energy levels within an energy band is given a quantitative description by the density of states function,N(E). This is a function of energy, E, defined by the relationship: d N = N (E)d E

(2.1)

where dN is the number of allowed electron energy levels, per unit volume of material, with energies lying within an infinitesimal small energy interval dE centered at energy E, as indicated in Fig. 2.2. From the definition of N(E) it follows that the number, N, of allowed energy levels, per unit volume of material, with energies lying between any two values E 1 and E 2 is given by integrating the density of states function N(E) between the limits E 1 and E 2 (Fig. 2.3a). E2 N=

N (E)d E E1

It is also clear, therefore, that the total number of allowed energy levels, per unit volume of material in the valence band of a semiconductor is given by integrating N(E) with respect to energy over the total width of the valence band as indicated in Fig. 2.3b. In order to describe the density of states function quantitatively, we should first recall the two important physical principles, namely, Pauli Exclusion and Heisenberg Uncertainty Principles. Fig. 2.2 Definition of density of states function, N(E). The circled area shows the magnified energy levels at around E within a range of dE

40

2 Intrinsic and Extrinsic Semiconductors

E2

E2

N = ∫ N ( E )dE

NVB = ∫ N ( E )dE

E1

E1

(a)

(b)

Fig. 2.3 An illustration of the density of states function N(E) for the calculation of the total density of states a between energies E 1 and E 2 and b within the valence band

In Sect. 1.4.3.2, we described the Pauli Exclusion Principle and in Sect. 1.4.3.1 we introduced the concept of the wave representation of electrons and de Broglie waves. Neither the momentum nor the position of an electron within the de Broglie wave can be determined simultaneously with great precision. If the uncertainty in the position is Δx, and in momentum is Δp, then the two uncertainties are found to be related in general by the inequality: ΔpΔx ≥ h/2π

(2.2)

where Δp = mΔv and h is Planck’s constant and h = 6.626 × 10−34 J.s. Equation 2.2 is one form of Heisenberg uncertainty principle. We can now proceed with the description of the density of states function. If an electron is confined within a length L, then according to the Heisenberg uncertainty principle, the uncertainty in its wavevector is: Lk ≥ 2π

(2.3)

If we assume that the electron wavevector is between k 1 and k 2 then the allowed k values in one dimension will be: Δk = k2 − kd = 2π/L

(2.4)

2.1 Density of States Function

41

where n is an integer. In the k space, for every 2π /L interval, there is only one allowed k value. This result can be generalised for a three dimensional crystal lattice by assuming that the crystal is in the shape of a cube with sides L. The allowed values of k will therefore be: 2π n L 2π ky = m L 2π kz = p L

kx =

(2.5)

Here n, m and p are integers. In k space, the allowed k values form a cubic lattice with sides 2π/L, therefore each allowed k value occupies a volume of (2π /L)3 (Fig. 2.4). We are now going to calculate the number of allowed states within dE range of an energy E. In k space, the allowed k values are within dk of wavevector k since the unit volume of the allowed k values is: k x k y k z = (2π /L)3 . The volume of the spherical shell between the two spheres with radii k + dk and k is: d 3 k = 4π k 2 dk. Therefore, the total number of allowed states in the spherical shell: dN =

4π k 2 dk V k2 d 3k = = dk 3 3 2π 2 (2π/L) (2π/L)

where V = L 3 is the volume of the cube. Fig. 2.4 Representation of allowed k values in k space and the allowed k values within a spherical shell of thickness dk

(2.6)

42

2 Intrinsic and Extrinsic Semiconductors

Since two electrons with opposite spins occupy the same state then we can write the density of states as: dN = 2 ×

V k2 dk 2π 2

(2.7)

If we now substitute: 2 k 2 E= and dk = 2m



m dE 22 E

(2.8)

into Eq. 2.7 we can obtain the density of states per unit energy range: N (E) =

V dN = (2m)3/2 E 1/2 dE 2π 2 3

(2.9)

If we require the expression for the density of states per energy range per unit volume of the crystal, n(E) = N(E)/V, then the V term can be removed from the above equation. Example Use Eq. 2.9 to calculate the Fermi energy of free electrons at T = 0 K. The Fermi energy separates the empty and filled states and all the states at T = 0 K will be full up to the Fermi energy, thus, f (E) = 1 for E < E F and: E F n=

N (E)d E = 0

EF =

V (2m E F )3/2 3π 2 3

2  2 2/3 2 k 2F 3π n = 2m 2m

Here k F is the Fermi wavenumber and all the filled states, in k space, will be within a sphere with radius k F . The sphere is known as the Fermi sphere. In real semiconductors the density of states function for the valence and conduction bands is basically parabolic in form. It is given by 1 (2m ∗e )3/ 2 (E − E C )1/ 2 for the conduction 2π 2 3

(2.10a)

1 (2m ∗h )3/ 2 (E V − E)1/ 2 for the valence band 2π 2 3

(2.10b)

NC B (E) = N V B (E) =

where h is Planck’s constant, m ∗e is the effective mass in the conduction band, m ∗h is the effective hole mass in the valence band while E C and E v stand for the conduction and valence bandedge energies, respectively. The two density of states functions are illustrated in Fig. 2.5.

2.1 Density of States Function

43

Fig. 2.5 Density of states function for conduction and valence bands in a semiconductor. N CB and N VB are symmetrical about (E C + E V )/2

The important feature to note in Fig. 2.5 is that we can take the centre of the forbidden bandgap, at (E C + E V )/2, as a reference point of energy scale and assume that the effective electron and hole masses are identical. Then the two density of states functions are disposed symmetrically on either side of this reference point. We will make use of this important symmetry in the next section where we describe how the electrons are distributed among the allowed energy states, represented by the density of states function shown in Fig. 2.5. Example A Si sample has dimensions, 100 nm × 100 nm × 10 nm. Calculate the density of states per unit volume of the material at an energy 100 meV above the conduction bandedge (E = E C + 0.1 eV). The electron effective mass in Si is me * = 1.08 m0 . Density of states per unit volume: 1 dN = (2m ∗e )3/ 2 (E − E C )1/ 2 dE 2π 2 3 1 −31 3/2 −19 = J/eV )1/2  3 (2 × 1.08 × 9.11 × 10 kg) (0.1eV × 1.6 × 10 2π 2 6.626 × 10−34 J s

NC B (E) =

= 6.08 × 1043 m−3 J−1

Therefore, total density of states per unit energy is:     NC B (E)V = 6.08 × 1043 m−3 J−1 x 10−22 m−3 = 6.08 × 1021 J−1 = 973eV−1

44

2 Intrinsic and Extrinsic Semiconductors

2.2 Fermi–Dirac Distribution Function The density of states function that we derived above gives us the number of allowed electronic energy states. The distribution of electrons in these states, however, is given by quantum statistics. Briefly, in quantum mechanics, we can classify particles according to their spin. Particles with spins equal to the integer multiples of  (0, , 2..) are known as bosons, e.g. photons, and those with spins equal to the half integer multiples of (1/2, 3/2..) are known as fermions, e.g. electrons. In thermodynamics a system of many particles can be described by macroscopic quantities, such as volume, pressure and temperature. Microscopically, in equilibrium, the system can be described by the appropriate distribution function. The distribution function gives the probability of occupancy of a given allowed energy state. The distribution function that describes the energy distribution of bosons is known as the Bose–Einstein distribution function and the energy distribution of electrons is given by the Fermi–Dirac distribution function. The Fermi–Dirac distribution function, f (E), is a function of energy, E, which gives the probabilistic, or statistical description of equilibrium electron distribution within the valence and conduction bands of a semiconductor at any given temperature, T. By definition, the value of f (E) at an energy E represents the probability of an allowed electron energy level at energy E being occupied by an electron. Since f (E) represents a probability then its value must lie in the range zero to unity. These two extreme values represent the certainty of an allowed energy level being occupied (f (E) = 1) or unoccupied (f (E) = 0) by an electron. It also follows that the probability f (E) of an allowed level at E being occupied always implies a complimentary probability equal to (1−f (E)) of the level at E being unoccupied or empty. In Sect. 1.4.7, we showed that in a typical semiconductor, the number of allowed energy levels within the conduction or valence bands is in the order of 1028 per unit volume of material. We also showed that, at all temperatures above absolute zero, the distribution of electrons, among this very large number of allowed energy levels, is activated by the average thermal excitation energy k B T where k B is the Boltzmann constant and T is absolute temperature. The important point to realise is that the equilibrium electron distribution within the two bands at all temperatures is dynamic. Dynamic equilibrium is the result of a steady state balance between the two types of electron transitions which take place in opposite directions. The first is the random thermal excitation of electrons from occupied lower energy states to allowed and empty higher energy states. The second transition, in the opposite direction, is due to electrons in the occupied higher energy states lowering their energy, to an empty, allowed state. This dynamic nature of the equilibrium electron distribution means that there is no one specific arrangement of electrons, among the allowed energy states, which uniquely describes the electron distribution at any given temperature above absolute zero. Instead, at equilibrium the instantaneous specific patterns of occupied electron states are continually changing in a random manner. This is the reason why we are obliged to describe the equilibrium electron distribution statistically using the Fermi–Dirac function.

2.2 Fermi–Dirac Distribution Function

45

The abstract, probabilistic description of the electron distribution given by the Fermi–Dirac function is translated into an intuitively understandable physical description in the following way. Suppose that according to the Fermi–Dirac function, f (E), the equilibrium electron population probability of an allowed energy state, of energy, E is 1/3. The physical meaning of this probability is that if we could monitor the occupancy of the allowed state of energy, E, over a period of time t, then on average this energy state would be found to have been occupied during (1/3 t) of this time period. The Fermi–Dirac distribution function, f (E) is given by: f (E) =

1 1 + exp[(E − E F )/k B T ]

(2.11)

where k B is the Boltzmann constant and T is temperature. The Fermi–Dirac function is shown at T = 0 K in Fig. 2.6. The parameter E F is called the Fermi energy or the chemical potential. The Fermi energy is a parameter of central importance, since the process of describing a real physical system in terms of the Fermi–Dirac distribution function consists essentially of specifying the value of E F to be used in the general expression for f (E) in Eq. (2.11). Clearly, therefore, we would expect the value assigned to E F to be expressed in terms of (one or more) energy parameters associated with the physical system being described, and in such a way that the resulting expression for f (E) is consistent with any boundary conditions associated with the physical system. In our case the distribution of electrons within the valence and conduction bands of a semiconductor, the relevant boundary conditions consists, simply, of the first two of the three pieces of information that make up the final picture. The first of these represented by Fig. 2.1a was allowed energy levels in the two bands. The second is the total number of electrons, equal to the total number of allowed levels in the valence band, which have to be distributed among all allowed levels in the two bands. Fig. 2.6 Fermi–Dirac distribution function at T = 0K

46

2 Intrinsic and Extrinsic Semiconductors

2.2.1 Electron Distribution at Absolute Zero Temperature The form of the Fermi–Dirac distribution function f (E) given by Eq. (2.11) for T = 0 K is that of a simple step-function as shown in Fig. 2.6. For this distribution, the electron occupation probability is equal to unity for all allowed energies with E < E F , while for all allowed levels of energy E > E F the electron occupation probability is zero. In other words, all allowed levels below the Fermi energy level are occupied and all energy levels higher than the Fermi level are empty. Figure 2.7 illustrates the energy band diagram of an intrinsic semiconductor together with the Fermi–Dirac distribution function. Having introduced the density of states function N(E) and the Fermi–Dirac distribution function f (E), we now show how they can be used together to calculate carrier densities in the valence and conduction bands of an intrinsic semiconductor. From the definitions given above of N(E) and f (E), it follows that the equilibrium density of electrons, that is, the total number of electrons per unit volume of material, with energies lying in the range of E 1 to E 2 is given by integrating the product of N(E) and f (E) between limits E 1 and E2. E2 n(E 1 , E 2 ) =

N (E) f (E)d E E1

Fig. 2.7 Energy band diagram is shown together with the Fermi–Dirac distribution function at T = 0 K. The valence band (VB) is completely full and the conduction band (CB) is empty

(2.12)

2.2 Fermi–Dirac Distribution Function

47

The electron density in the conduction band of an intrinsic semiconductor with the energy bandprofile as in Fig. 2.7 is therefore: ∞ nC B =

NC B (E) f (E)d E = 0

(2.13)

EC

This is because f (E) = 0 throughout the conduction band (for E > E F : f (E) = 0). On the other hand, the electron density in the valence band of an intrinsic semiconductor is: E V nV B =

E V N V B (E) f (E)d E =

−∞

N V B (E)d E

(2.14)

−∞

Since f (E) = 1 throughout the valence band (for E < E F: f (E) = 1). At T = 0 K, from the Fig. 2.7, we can see that there are no electrons present in the conduction band and the electron density in the valence band is equal to the density of allowed states in the valence band. Remember that our real aim is to calculate free carrier densities and that these consist of electrons in the conduction band and the holes, represented by unoccupied allowed electron states, in the valence band. According to Eqs. 2.13 and 2.14 above, the density of free electrons and holes are both zero in the case of intrinsic semiconductors at T = 0 K. We need to introduce just one more idea to conclude this section. As [1 − f (E)] represents the probability of an allowed state at energy E being unoccupied, we can express the free hole density in the valence band as: E V pV B =

N V B (E)[1 − f (E)]d E

(2.15)

−∞

which gives pVB = 0 in this case, again because f (E) = 1 throughout the valence band. In summary, density of free electron and hole in a semiconductor may be calculated from the density of states and Fermi–Dirac distribution functions. Here, in order to obtain carrier density, we considered a very simple case of an intrinsic semiconductor at absolute zero temperature. However, the procedure is quite general and will be applied shortly to calculate the free carrier densities for a semiconductor at a finite temperature.

48

2 Intrinsic and Extrinsic Semiconductors

2.2.2 Distribution of Electrons at Finite Temperatures We come now to the more interesting case of the electron distribution in an intrinsic semiconductor at a finite temperature. In order to proceed with this we need to know how the Fermi–Dirac distribution changes with temperature. The form of the Fermi– Dirac distribution function at T = 0 K and at a finite temperature T > 0 K, is shown in Fig. 2.8. As we discussed above, at T = 0 K, the Fermi–Dirac distribution function has a step-function form but when the temperature is increased, the step-function form is rounded off. This variation with increasing temperature is easily understandable because we would expect that the probability of allowed electron states at energies above E F being occupied with increasing temperature. This would be accompanied by a symmetrically increasing probability of allowed states below the Fermi energy being unoccupied. This symmetrical form of f (E) about the Fermi energy E F is very important and can be expressed quantitatively as follows. We have already noted that f (E) represents the probability of an allowed state at energy E being occupied by an electron, therefore the probability of the same state at E being un occupied must be [1 − f (E)]. The form of f (E) is such that the probability, f (E + δE), of an allowed electron state, at an energy δE higher than the Fermi energy, being occupied is equal to the probability, [1 − f (E−δE)], of an allowed state at an energy δE lower than the Fermi energy. This statement of symmetry of (E) about E F is easily verified by evaluating the two probabilities from Eq. 2.11: f (E F + δ E) =

1 1 = 1 + exp [(E F + δ E − E F )/k B T ] 1 + exp[(δ E)/k B T ] (2.16)

Fig. 2.8 Fermi–Dirac distribution function at T = 0 K and T > 0 K

2.2 Fermi–Dirac Distribution Function

49

Fig. 2.9 The symmetry of the Fermi–Dirac distribution function around the Fermi level

1 1 =1− 1 + exp[(E F − δ E − E F )/k B T ] 1 + exp[(−δ E)/k B T ] 1 exp(−δ E/k B T ) = (2.17) = 1 + exp(−δ E/k B T ) 1 + exp(δ E/k B T )

[1 − f (E F − δ E)] = 1 −

The Fig. 2.9 illustrates this symmetry in the form we shall use very shortly to determine the value of E F , for an electron at a finite temperature. Closely related to the above symmetry is another important feature of f (E) curves in Fig. 2.8, namely that for each curve the electron occupation probability f (E) is equal to 0.5 at E = E F . This is a perfectly general mathematical property of the Fermi–Dirac distribution function which can be readily seen by putting E equal to E F in Eq. 2.11. Consequently, the Fermi energy of a system may be defined as the energy where an allowed electron level has an occupation probability equal to 0.5. We shall return to our objective of describing the electron distribution in an intrinsic semiconductor at a finite temperature in terms of the Fermi–Dirac distribution function. We now need to determine the value of E F for this case and this can be achieved as before by applying the appropriate boundary conditions to the distribution function f (E). The first boundary condition is to know how the number of the allowed energy levels represented by the density of states function for the valence and conduction is affected by changing temperature. The N(E) curves at a finite temperature are the same as those at T = 0 K (Fig. 2.5). Let n and p respectively, stand for free electron density in the conduction band and the mobile hole density in the valence band. If there is exactly one mobile hole in the valence band, then clearly the boundary condition associated with the number of mobile carriers to be distributed within the two bands is simply: n=p

(2.18)

Since n and p are as given by Eqs. 2.13 and 2.15, then given the symmetry of the N(E) curves about 1/2(E C + E V ), and that of f (E), it follows that to be consistent

50

2 Intrinsic and Extrinsic Semiconductors

Fig. 2.10 At finite temperatures the Fermi energy is in the middle of the bandgap. Electrons are excited from the valence band into the conduction band. Therefore the distribution function is less than 1 at the top of the valence band and greater than 0 at the bottom of the conduction band

with the above boundary conditions E F must be E F = 1/2(E C + E V ) as shown in Fig. 2.10.

2.2.3 Charge Neutrality Condition The charge neutrality condition is a very simple, indeed almost obvious, concept which is extremely useful as a boundary condition for describing the electronic structure and operation of semiconductor materials and devices. We introduce the concept here in the context of intrinsic semiconductors and we will see later that it is very easily extended for applications to extrinsic semiconductors. A piece of semiconductor material is an assembly of atoms. Electrostatically an atom is a neutral entity. It follows that, taken as a whole, an isolated piece of semiconductor is also electrostatically neutral. The electrostatic neutrality of an individual atom results from the atom containing equal numbers of electrons and protons. The overall electrostatic neutrality of a piece of semiconductor material, on the other

2.2 Fermi–Dirac Distribution Function

51

hand, must be described in a slightly different way. We have seen that the electrons and holes which constitute the charge carriers in a semiconductor are delocalised, in the sense that a mobile electron or hole cannot be considered as being associated with a particular atom. The overall electrostatic neutrality of semiconductor material must therefore be considered in terms of density of delocalised (free) electron and hole. These represent density of two electrostatic charge distributed throughout the semiconductor, which are equal in magnitude but with an opposite sign. The boundary condition which ensures the overall electrostatic neutrality of a piece of semiconductor material is called the charge neutrality condition. Equation 2.18 is the charge neutrality condition for the case of an intrinsic semiconductor.

2.3 Intrinsic Semiconductors 2.3.1 Free Carrier Densities in Intrinsic Semiconductors In an intrinsic semiconductor, the number of the allowed states in conduction and valance bands will be higher than thermally excited electron and holes, respectively. Therefore, the probability to find electron or hole at the same allowed energy state is so low that we can use Maxwell–Boltzmann distribution function instead of Fermi– Dirac distribution function. In mathematically, this status corresponds to E−E F >> k B T in Eq. 2.11, therefore, the exponential term in Eq. 2.11 is much greater than 1 and f (E) can be expressed to a very good approximation by: f (E) = exp[−(E − E F )/k B T ]

(2.19)

Equation 2.19 is the Maxwell–Boltzmann distribution. Maxwell–Boltzmann and Fermi–Dirac distribution functions are shown in Fig. 2.11 for comparison. It is clear that the former version, which is a classical distribution function, is a very good approximation to the latter at energies satisfying the E−E F >> k B T condition. Fig. 2.11 Fermi–Dirac and Maxwell–Boltzmann distributions at T > 0 K

52

2 Intrinsic and Extrinsic Semiconductors

Similarly the electron distribution in the valence band of a semiconductor is given by: f (E) = exp[−(E F − E)/k B T ]

(2.20)

Using Eqs. 2.10a and 2.19 in: ∞ N (E) f (E)d E

n= EC

We can obtain the free electron density, n, in the conduction band of an intrinsic semiconductor: ∞ n= EC

1 (2m ∗e )3/ 2 (E − E C )1/2 exp[−(E − E F )/k B T ]d E 2π 2 3

(2.21)

Solution of the integral gives the intrinsic electron density at a finite temperature, T: 

2π m ∗e k B T n=2 h2

3/2

  EC − E F exp − kB T

(2.22)

Here the pre-exponential term is known as the effective density of states in the conduction band: 

2π m ∗e k B T NC = 2 h2

3/2 (2.23)

Therefore, the free electron density in the conduction band of an intrinsic semiconductor is:   EC − E F (2.24) n = NC exp − kB T The above procedure for calculating the free electron density in an intrinsic semiconductor can also be used to calculate the free hole density in the valence band. If N v is the effective density of states in the valence band: 

2π m ∗h k B T NV = 2 h2 The free hole density in the valence band is:

3/ 2 (2.25)

2.3 Intrinsic Semiconductors

53

  E F − EV p = N V exp − kB T

(2.26)

In an intrinsic semiconductor, the electron and hole densities are equal and usually represented by ni and pi , respectively. Equations 2.24 and 2.26 can be used to express the intrinsic free carrier densities in terms of the energy bandgap. If we apply the charge neutrality condition for the intrinsic semiconductors, ni = pi : n i = pi = (n i pi )1/2

(2.27)

Substituting Eqs. 2.24 and 2.26 into 2.27: n i2

      EC − E F E F − EV exp − = n i pi = (NC N V ) exp − kB T kB T E g = EC − E V n i = (NC N V )

1/2

  Eg exp − 2k B T

(2.28)

Equation 2.28 completes our description of free carrier density in intrinsic semiconductors in terms of the fundamental parameters associated with the materials electronic bandstructure. It also shows how the carrier density, and therefore the conductivity, changes with temperature. Before looking at this temperature dependence in more detail, let us summarise the description of intrinsic free electron and hole densities, as shown in Fig. 2.12. Although our discussion has so far focused on intrinsic semiconductors, the expressions 2.24 and 2.26 for electron and holes are equally valid for extrinsic semiconductors. This is because Eqs. 2.24 and 2.26 follow simply from the definitions of N(E) and f (E). The only feature of Fig. 2.12 which is specific to an intrinsic semiconductor is that the Fermi level lies at the centre of the bandgap. We shall see later that the charge neutrality condition takes a different form for extrinsic semiconductors because the Fermi level does not lie in the middle of the bandgap. The form of the expressions for electron and hole densities, however, remains the same. At a finite temperature, the Fermi level lies near the middle of the bandgap as we briefly pointed out in Sect. 2.2.2. This can be described mathematically from the charge neutrality condition. The charge neutrality condition in an intrinsic semiconductor dictates that ni = pi , therefore:     EC − E F E F − EV n i = NC exp − = pi = N V exp − kB T kB T E F = EV +

1 kB T NV Eg + ln 2 2 NC

(2.29)

54

2 Intrinsic and Extrinsic Semiconductors

(a)

(b) Fig. 2.12 a Schematic representation of charge neutrality (ni = pi ) in an intrinsic semiconductor. b The distribution of the intrinsic carriers in the valence and conduction bands

At T = 0 K, the Fermi level is in the middle of the energy bandgap so: E F = EV +

1 Eg 2

At finite temperatures, the Fermi level moves towards the conduction bandedge. However, if m ∗e = m ∗h it is clear from Eqs. 2.23 and 2.25 that N C = N V and the last term in Eq. 2.29 is zero. Therefore, the Fermi level is independent of temperature and lies in the middle of the energy bandgap. Example Calculate energy level in units of k B T in terms of Fermi level, when the Fermi–Dirac probability is 0.05? f (E) =

1 = 0.05 1 + exp[(E − E F )/k B T ]

E − E F = ln(19)k B T = 3k B T

2.3 Intrinsic Semiconductors

55

Example What is the position of the Fermi level at T = 300 K in an intrinsic GaAs semiconductor? (me * = 0.067m0 and mh * = 0.45 m0 ).  NC = 2

 2π m ∗e k B T 3/2 h2

2π × 0.067 × 9.1 × 10−31 kg × 1.38 × 10−23 J/K × 300K =2 (6.626 × 10−34 J s)2

 NV = 2

2π m ∗h k B T h2

3/2

3/2 = 4.45 × 1023 m −3

= 7.72 × 1024 m −3

Fermi level with respect to the valence bandedge is: NV 1 kB T Eg + ln 2 2 NC 26 × 10−3 eV 7.72 × 1024 m −3 1.424eV + ln = 2 2 4.45 × 1023 m −3 = E V + 0.787eV E F = EV +

Example Calculate the position of Fermi level for intrinsic Si at T = 300 K. (m ∗ = 0.98m0 , mt * = 0.19m0 , m ∗h = 0.15m0 , mhh * = 0.5m0, E g (Si) = 1.12 eV). Effective mass of electrons with contributions from the six conduction band valleys:  1/3 = (6)2/3 (0.98 × 0.19 × 0.19)1/3 m 0 = 1.08m 0 m ∗e = (6)2/3 m ∗ m ∗2 t Hole effective mass:

2/3 ∗3/2 ∗3/2 m ∗h = m h + m hh E F = EV +

m∗ 1 3 E g + k B T ln h∗ = E V + 0.5472eV 2 4 me

Example In a semiconductor, E F = 6.25 eV. Calculate the value of the Fermi–Dirac distribution function at T = 300 K and E = 6.5 eV. Do the same calculation for T = 950 K. Assume that the position of the Fermi level remains unchanged. Determine the temperature at which the probability of the energy level E = 5.95 eV being empty is 1%? AtT = 300 K: f (6.5eV ) =

1 + exp



1 (6.5−6.25)eV 0.02586eV

= 6.29 × 10−5

56

2 Intrinsic and Extrinsic Semiconductors

At T = 950 K: f (6.5eV ) =

1 + exp



1 (6.5−6.25)eV 0.0818eV

= 0.045

Since the probability of the energy level, 5.95 eV, being empty is 1% then the probability of it being full must be 99%. f (5.95eV ) = 0.99 =

1 + exp



1 (5.95−6.25)eV k B T /e

T = 484.7 K.

2.3.2 Temperature Dependence of Carrier Densities In order to make the temperature dependence fully explicit, we now use Eqs. 2.23, 2.25 and 2.28:

3/2    2π k B m ∗e m ∗h Eg 3/2 n i = pi = 2 T exp − h2 2k B T   Eg n i = pi = AT 3/2 exp − 2k B T

(2.30)

(2.31)

whereA is a material constant given by:

3/2  2π k B m ∗e m ∗h A=2 h2

(2.32)

It is clear from Eq. 2.31 that the pre-exponential term, T 3/2 is negligible compared to the exponential temperature dependence. Therefore, in an intrinsic semiconductor, a plot of the logarithm of free carrier density versus 1/T, which is known Arrhenius plot, should give a straight line and the slope can be used to obtain the bandgap of the semiconductor. In Fig. 2.13, the temperature dependence of intrinsic carrier density for Ge, Si and GaAs semiconductors are illustrated. As a result of the different bandgaps for the three semiconductors (at T = 300 K; E g (Ge) = 0.66 eV, E g (Si) = 1.12 eV and E g (GaAs) = 1.42 eV), the values of the carrier density for these semiconductors, at a given temperature will, naturally, be different. In other words, the intrinsic carrier density is a material parameter.

2.3 Intrinsic Semiconductors

57

Fig. 2.13 Temperature dependence of the carrier densities for intrinsic Ge, Si and GaAs

2.3.3 Conductivity An intrinsic semiconductor’s conductivity σ is given by:   σ = e n i μn + pi μ p

(2.33)

Because ni = pi , conductivity can be re-written as   σ = en i μn + μ p

(2.34)

The conductivity also depends on the temperature because both the carrier density and mobility are temperature dependent. To a first approximation, if we assume that the mobility is independent of temperature, we can use Eqs. 2.28 and 2.32 to express the conductivity in an intrinsic semiconductor as:       Eg σ = en i μn + μ p = Ae μn + μ p T 3/2 exp − 2k B T

(2.35)

Therefore, a complete electronic characterisation of an intrinsic semiconductor requires a knowledge of the four material parameters σ , ni , μn and μp . Of these four

58

2 Intrinsic and Extrinsic Semiconductors

Table 2.1 Material parameters of Ge, Si and GaAs at T = 300 K7−12 Parameter

Si

GaAs

Effective density of states in 6.0 × 1024 m−3 valence band, N V

Ge

1.0 × 1025 m−3

7.0 × 1024 m−3

Effective density of states in 1.0 × 1025 m−3 conduction band, N C

2.8 × 1025 m−3

4.7 × 1024 m−3

Bandgap, E g

1.12 eV

1.42 eV

Intrinsic carrier density, ni = 2.4 × 1019 m−3 pi

0.66 eV

1.4 × 1016 m−3

1.8 × 1012 m−3

Electron mobility, μn

0.39 m2 /Vs

0.150 m2 /Vs

0.85 m2 /Vs

Hole mobility, μp

0.19 m2 /Vs

0.045 m2 /Vs

−1

Conductivity, σ

2.25

Resistivity, ρ

0.44 m

m−1

4.4 ×

10− 4

2.3 ×

105

−1

m

0.042 m2 /Vs m−1

2.6 × 10−7 −1 m−1 3.9 × 108 m

Effective electron mass, me * 0.56m0

1.08m0

0.067m0

Effective heavy hole mass, mhh *

0.29m0

0.57m0

0.45m0

Effective light hole mass, mlh *

0.044m0

0.16m0

0.082m0

Split-off effective mass, mh,SO *

0.084m0

0.29m0

0.154m0

parameters, the one easiest to obtain experimentally is conductivity σ , since it only requires a basic dc electrical measurement. Carrier mobility is usually determined by Hall Effect measurement or by a combination of Hall Effect and Haynes-Shockley type measurements with the carrier density calculated using Eq. 2.34. From the semiconductor device technology perspective, the three most important semiconductor materials are Si, Ge and GaAs. The material parameters which determine the electronic conduction properties of intrinsic Si, Ge and GaAs are shown in Table 2.1. The conductivity values of these three tabulated in Table 2.1.  semiconductors,  As conductivity is given by σ = en i μn + μ p , the differences between these conductivity values must be due to differences in either ni or (μn + μp ) values or a combination of the these. According to Eq. 2.30, ni not only depends on temperature but exponentially on the bandgap E g . In conclusion, the most significant material parameter for determining the value of conductivity in an intrinsic semiconductor is the bandgap.

2.4 Extrinsic Semiconductors

59

2.4 Extrinsic Semiconductors The conductivity of intrinsic semiconductors may be changed over a large orders of magnitude, in a highly controlled way, by introducing small density of impurity atoms. As stated at the beginning of this chapter, this is referred to as doping of intrinsic semiconductor to produce a doped or extrinsic semiconductor. The impurity atoms are referred to as the dopant atoms.

2.4.1 n- and p-Type Semiconductors An extrinsic semiconductor has a higher conductivity than that of the intrinsic material which forms its basis. This higher extrinsic conductivity is due to the fact that in the doped material, either electron density n or hole density p is greater than the corresponding intrinsic carrier density. In the first case, the extrinsic conductivity is due predominantly to electrons and the material is said to be an n-type semiconductor. In an n-type semiconductor electrons are the majority carriers and holes are the minority carriers. Conversely, in the second case the material is said to be a p-type semiconductor in which holes are the majority carriers and electrons are minority carriers. For a semiconductor formed by group IV atoms, dopants which give rise to an ntype semiconductor belong to Group V in the periodic table of elements. A Group V element used as dopant is referred to as a donoratom. This is because each Group V dopant atom donates one mobile electron into the conduction band of the host semiconductor. Dopants responsible for p-type semiconductors, on the other hand, belong to Group III in the periodic table. These are referred to as acceptor atoms because each Group III atom accepts an electron from the valence band of the semiconductor and thereby creates one hole in the band.

2.4.2 n-Type and p-Type Doping We now give a description of n-type and p-type doping of intrinsic semiconductors. Although we describe the processes for Si, the underlying concepts can be applied to any intrinsic semiconductor. Consider the schematic two-dimensional representation of the crystal structure of silicon, shown in Fig. 2.14a. Silicon, being a Group IV element, has four valence electrons in the outermost shell. The four valence electrons of each atom make covalent bonds with the neighbouring atoms. Let us now consider that we replace a Si atom with a Phosphorous (P) atom which is a Group V element in the periodic table and therefore has five electrons in the outermost shell. This is indicated in Fig. 2.14b, which shows that, in addition to the

60

2 Intrinsic and Extrinsic Semiconductors

(a)

(b)

(c) Fig. 2.14 a Schematic two dimensional representation of Si crystal. b Group V element, P, is a donor atom in Si crystal donating an electron. c Group III element, B, is an acceptor atom in Si crystal, accepting an electron

original pattern of electrons forming pairs of covalent bonds between neighbouring atoms, there is now an additional electron associated with the P atom. The fifth electron can be dissociated from its parent P atom by breaking the chemical bond between them. The energy required to break this chemical bond is very much smaller than the energy needed to break a covalent bond between adjacent atoms. Therefore, the broken bond delocalises the electron, so that it represents a free charge carrier and is able to contribute to the conductivity of the Si crystal. The breaking of the bond between the fifth electron and its parent atom is said to ionise the impurity P atom, since now it carries a net positive charge +e. The P atom remains localised at the impurity atom site and does not represent a mobile charge. The energy required to ionise the impurity atom is called the ionisation energy. The ionisation energy of dopants are sufficiently small that at room temperature. We may realistically assume that all dopant atoms to be ionised by the ambient temperature,

2.4 Extrinsic Semiconductors

61

k B T = 8.62 10−5 eV × 300 K ~26 meV. In an n-type semiconductors, density of electrons are increased by dopant atoms. Each dopant atom donates one electron to host material. Here P doped Si has a large majority of electrons and is therefore n-type. The phrase n-type comes from the negative charge of the electron. The basic ideas just described for n-type doping are also applicable, in a similar way, to p-type doping. Let us consider Si semiconductor again. In this case, the dopant atom will be a Group III element in the periodic table. This is illustrated in Fig. 2.14c, taking Boron (B) as an example of an acceptor type of impurity which has only three valence electrons in the outermost electron shell. The B atom is ionised by accepting an electron from one of its neighbouring Si atoms. In this case, Si has one less electron and this status, as we have already seen, represents a hole which then contributes to the conductivity. The net negative charge, –e, at the ionised B atom site due to the fourth electron, now forms a covalent bond with the neighbouring Si atoms. As we have seen for the donor atom, the ionisation energy of acceptor atoms is also much smaller than the bandgap E g . This is because the ionisation process involves the exchange of a Si–Si covalent bond with a Si-B covalent bond only and does not break the covalent bond. The B doped Si has a large majority of hole density and is therefore p-type. The phrase p-type comes from the positive charge of the electron.

2.4.3 Binding Energy of Hydrogenic Impurities It is useful to start by referring to the Hydrogen atom model, in Sect. 1.4.3.1, to describe the ionisation energies of dopant atoms in a semiconductor crystal. For P atom, the fifth electron, although bound to its parent atom, does not form covalent bonds with the neighbouring atoms, therefore, it can be delocalised with a very small amount of thermal energy and contribute to the conductivity of the semiconductor. This implies that the binding energy of the electron must be close to the conduction band. If we represent this energy as E D and use the conduction bandedge E c as the reference point, then a thermal energy of the order E C −E D will be sufficient to ionise P atom and excite the fifth electron into the conduction band. In order to describe the ionisation energy more quantitatively, let us assume, to a first approximation, P atom is like a Hydrogen atom where the fifth (valence) electron orbits the positively charged core P ion. There will be a Coulomb force between P atom and the valence electron modified by the dielectric constant of the host semiconductor (Si). The attractive Coulombic potential can be g iven as: V =−

e2 4π ε r

(2.36)

We can determine the value of E D by solving Schrödinger’s equation by substituting the expression in Eq. 2.36 for potential:

62

2 Intrinsic and Extrinsic Semiconductors

 2 2 e2 − ∗∇ − (r ) = (E D − E C )(r ) 2m e 4π ε r

(2.37)

Obviously, we have to replace the free electron mass m0 in Schrödinger’s equation with electron effective mass, m ∗e in the semiconductor, to obtain (Fig. 2.15): e4 m ∗e 1 n = 1, 2, .. 8ε2 2 n 2

(2.38)

 ∗  εo 2 me E D = E C − 13.6 eV m0 ε

(2.39)

E D = EC − For the ground state n = 1:

The value given in Eq. 2.39 differs from the hydrogen atom by: 

 m ∗e εo 2 m0 ε

Fig. 2.15 Four of the valence electrons of the group V element bond with Si atoms, one remaining electron is weakly bound to Si and may be treated as an electron bound to the nucleus of a hydrogen atom

2.4 Extrinsic Semiconductors

63

This is as a result of the different dielectric constant and the effective electron mass in the semiconductor compared to a hydrogen atom. For a hydrogen atom, the ground state energy was obtained as −13.6 eV (Eq. 1.19), which is much greater than the bandgap of Si. Using the electron effective mass and the dielectric constant in the semiconductor however, we see that the ground state energy is reduced to only a few meV. For example, if we assume that the dielectric constant in the semiconductor as ε = 10ε0 and the effective mass of electrons me * = 0.1m0 , from Eq. 2.39, we find E D = E C − 0.01 eV. At room temperature (T = 300 K), the thermal energy k B T = 26 meV, therefore, the donor atoms will be completely ionised by donating their fifth electrons into the conduction band. The ground state wavefunction for the donor atoms is: (r ) = √

1 π a3

e−r/a

(2.40)

where the Bohr radius is: aB =

   m0 h2ε ε = 0.053 ∗ 2 π me e ε0 m ∗e

(2.41)

Using the effective mass and dielectric constant values given above we obtain the Bohr radius ~10 nm at 300 K. Considering that the interatomic distances in a solid are in the range of a few Angstroms, a Bohr radius of 10 nm contains many atom inside. Consequently, Coulombic attraction between impurity atom and the electron will be screened and the electron binding energy will also be reduced compared to the hydrogen atom in range of eV to meV. We can use the same approach for the acceptor impurities by taking account of the different hole effective mass. The ionisation energy of the acceptors, with reference to the valence bandedge, E V , is:  ∗  mh ε0 2 eV E A = E V + 13.6 m0 ε

(2.42)

The donor and acceptor energy levels are shown in the energy band diagram in Fig. 2.16. As their ionisation energies are very small compared to the bandgap, their energy levels are known as shallow levels and the impurity atoms are called hydrogenic impurities. The hydrogen model is only applicable to situations when the difference in the valence electron numbers between the host semiconductor and impurity atoms is only one. When this is more than one, then the impurity atoms occupy deep levels within the semiconductor bandgap because the bond between parent impurity atom and its unbounded electrons are stronger and more energy is necessary to break this bond. To represent this situation, localized energy level is shown further far from the conduction band than that of Hydrogenic impurities.

64

2 Intrinsic and Extrinsic Semiconductors

(a)

(b)

Fig. 2.16 Energy bandprofiles of a n-type and b p-type semiconductors

The ionisation energies together with the Bohr radii of shallow donors and acceptors are shown in Table 2.2. The Hydrogen model is a very simple model and does not consider the electronegativity of the impurity atoms. It is therefore not right to expect that every Group V element will have the same ionisation energy in Si. For the Group IV semiconductors it is relatively straight forward to assume that the Group V elements will act as donors and Group III elements as acceptors. However, in compound semiconductors, such as GaAs, an impurity may show either of the two behaviours, i.e., it can be an acceptor or donor. These type of impurities are known as amphoteric. For example, in GaAs, if we use Si as the dopant then Si may replace a Ga atom and act as a donor, or could substitute the As atom and become an acceptor. Example Use the hydrogen model to calculate the donor and acceptor ionisation energies in GaAs (ε = 13.2ε0 , me * = 0.067m0 and mh * = 0.45m0 ). Table 2.2 Ionisation energies of donor and acceptor atoms in Si, Ge and GaAs Semiconductor

Donor atom

Si

Li

33

B

45

Sb

39

Al

67

Ge

GaAs

Donor ionisation energy (meV)

Acceptor atom

Acceptor ionisation energy (meV)

P

45

Ga

72

As

54

In

160

B

10

Li

9.3

Sb

9.6

Al

10

P

12.0

Ga

11

As

13.0

In

11

Si

5.8

C

26

Ge

6.1

Be

28

S

6.0

Mg

28

Sn

6.0

Si

35

2.4 Extrinsic Semiconductors

65

E D = E C − 5.5 meV E A = E V + 36 meV Prior to doping, at a given temperature, conduction band contains intrinsic electron density ni . Therefore, after doping, the total electron density n in the conduction band is given by: n = ni + N D

(2.43)

When the minority carriers (p) are ignored, Eq. 2.43 gives neutrality condition in an n-type semiconductor. Because ni is so small, in Eq. 2.43, ni can be neglected with respect to N D to give: N D >> n i

⇒

n∼ = ND

(2.44)

Similarly, for p-type semiconductor, density of hole can be expressed ignoring minority carriers as p = pi + N A N A >> pi

⇒

p∼ = NA

(2.45)

Let us suppose that the intrinsic Si is doped with P at a density of one P atom for every 106 Si atoms. As there are 5 × 1028 atoms/m3 of a pure Si crystal, this implies a dopant density in the extrinsic material equal to N D = 5 × 1022 phosphorous atoms/m3 . This in turn implies a donated electron density in the conduction band of 5 × 1022 electrons/m3 . Si has 1.4 × 1016 electrons/cm3 , therefore, the doping has resulted in the electron density in Si increasing by a factor of approximately 3.6 × 106 . This example demonstrates how a minute atomic density of a dopant gives rise to a comparatively enormous increase in the conductivity of a semiconductor material. The explanation for this lies in the fact that every dopant atom contributes one electron into the conduction band, but in the case of intrinsic Si atoms, less than one atom in 1012 contributes an electron into the conduction band. This is because the energy, represented by the bandgap energy, required to break a covalent bond between two adjacent Si atoms, is much greater than the ionisation energy needed to delocalise the fifth electron associated with the dopant P atom. Law of Mass Action We now consider what effect the doping has on the hole density in the material. Upon doping, the increased electron density in the conduction band is accompanied by a decrease in hole density within the valence band. An intuitive appreciation of the reasons behind this may be gained as follows. Holes in the valence band

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2 Intrinsic and Extrinsic Semiconductors

are generated by the thermal excitation of electrons from the valence band to the conduction band. In the other direction, holes are annihilated by electrons from the conduction band simultaneously recombining with the holes in the valence band. The electron density in the conduction band of an n-type semiconductor is greater than the intrinsic electron density. Consequently, the probability of a hole being annihilated by recombination is increased, leading to a lower hole density in the valence band. A quantitative description of the various electron transitions involved provides a very convenient, general result. This shows that the product of density of electron and hole in an intrinsic semiconductor is independent of the type and density of the doping used. Furthermore, this gives a constant product, np which is equal to the product of density of the electron and hole in the intrinsic material, ni pi . This general result plays a central role in the calculation of mobile carrier densities in extrinsic semiconductors and is referred to as the Law of Mass Action and expressed as np = n i pi = n i2 = pi2

(2.46)

The law of mass action is a direct result of the charge neutrality condition. In general terms, if a semiconductor is n-type, the free electron density increases and hole density decreases compared to the intrinsic case. Similarly, if a semiconductor is p-type, the free hole density increases and free electron density decreases compared to the intrinsic case. Example Intrinsic Si which has 5 × 1028 atom/m3 is doped by P with one P atom for every 106 Si atoms. Is the doped semiconductor n or p-type and what are the mobile carrier densities at T = 300 K? P is a Group V element. Therefore it is a donor impurity in Si and the extrinsic Si is n-type. In intrinsic Si, at T = 300 K, the intrinsic carrier densities are ni = pi = 1.4 × 1016 carriers/m3 . The doping density N D =

5 × 1028 = 5 × 1022 atoms/m3 106

Majority electron density n = N D = 5 × 1022 electrons/m3 From the law of mass action, hole density is obtained: p=

n i2 (1.4 × 1016 )2 = = 4 × 109 holes/m3 n 5 × 1022

Example In the example above, if the dopant atoms were to be bor, would the doped semiconductor be n or p-type and what would carrier densities be at T = 300 K? B is a Group III element in the periodic table. Therefore it is an acceptor impurity and the doped Si is p-type with a hole density of

2.4 Extrinsic Semiconductors

67

p = N A = 5 × 1022 holes/m3 From the law of mass action: n=

n i2 (1.4 × 1016 )2 = 4 × 109 electrons/m3 = p 5 × 1022

It is clear from these two examples that the majority carrier densities are orders of magnitude higher than the minority carrier densities. Therefore, the contribution to conductivity from minority carriers in extrinsic semiconductors may be neglected.

2.4.4 The Fermi Level in Extrinsic Semiconductors In Sect. 2.3, the Fermi level, in intrinsic semiconductors, was found by applying the charge neutrality condition. In that case, density of electron and hole were equal to each other and both types of carriers were distributed symmetrically along the energy scale, with respect to the centre of the forbidden energy gap. This resulted in the Fermi level lying in the middle of the bandgap. Turning now to the extrinsic case and taking the n-type Si as an example, we have an asymmetrical situation here as electron density in the conduction band is very much greater than hole density in the valence band. Now, given the symmetry of the Fermi–Dirac function f (E) around the Fermi level E F , we would expect that for this asymmetrical distribution of free electrons and holes, the Fermi level would be displaced upwards from the middle of the bandgap towards to the conduction band. The displacement of the Fermi level from the centre of the bandgap can be easily calculated as follows. From Eq. 2.24:  EC − E F = ND n = NC exp − kB T   ND E F = E C + k B T ln NC 

(2.47)

As an example, take an n-type GaAs with donor density of N D =1 × 1017 cm−3 . At T = 300 K, n = N D . Therefore, from Eq. 2.47, E F = E C – 0.039 eV. Considering the fact that bandgap of GaAs is E g = 1.424 eV at T = 300 K, the calculated value of the Fermi level represents an upward shift from the centre of the bandgap by 1.385 eV.

68

2 Intrinsic and Extrinsic Semiconductors

Similarly, in p-type semiconductors, since the hole density in the valence band is much higher than free electron density in the conduction band, the Fermi level is displaced downwards from the centre of the bandgap towards the valence band. We can calculate the position of the Fermi level in a p-type semiconductor using Eq. 2.26:   E F − EV = NA p = N V exp − kB T   NA E F = E V − k B T ln NV

(2.48)

Figure 2.17 shows the energy bandprofiles, density of states, free carrier densities together with the Fermi–Dirac distribution functions for intrinsic, n-type and p-type semiconductors. It is clear from Eqs. 2.47 and 2.48 that as the doping densities increase then the Fermi level moves towards the conduction band (n-type) or valence band (p-type). At very high doping levels, E F shifts in conduction band for n-type and in valance band for p-type sample then the semiconductor material is said to be degenerate and the carrier density can no longer be described accurately using the theoretical framework developed above. Throughout the above description of free carrier density in semiconductors, we have used the Fermi–Dirac distribution function f (E) and in Sect. 2.3 showed that this can be approximated to a Maxwell–Boltzmann distribution. This approximation is valid to a satisfactory level of accuracy provided that the Fermi level remains within the bandgap and does not approach bandedges more than 2-3kB T. Clearly, free carrier densities in a degenerate extrinsic semiconductor cannot be described in terms of the framework based on the Maxwell- Boltzmann form of f (E), because in this case, the probability to find two carriers at the same energy status is higher and Fermi–Dirac distribution should be used. The more general form of Eq. 2.13 is:   E F − EC 2 n = NC √ F1/2 kB T π

(2.49)

Here F 1/2 is the Fermi–Dirac integral and its value depends on the variables η ≡ (E − E C )/k B T and η F ≡ (E F − E C )/k B T  F1/2

E F − EC kB T

∞

 ≡ F1/2 (η F ) =

EC



1/2 ∞ (E − E C )/k B T dE η1/2 dη = 1 + exp[(E − E F )/k B T ] k B T 1 + exp(η − η F ) 0

(2.50) In Fig. 2.18, the Fermi integral is shown as a function of the Fermi level. Also shown in the figure is the result of the Maxwell–Boltzmann approximation. For ηF < 0, the Fermi integral is well represented by the simple exponential dependence

2.4 Extrinsic Semiconductors

69

Fig. 2.17 Energy bandprofiles, density of states and Fermi–Dirac distribution function for a intrinsic, b n-type  and c p-type semiconductors. In all three types of semiconductors, the law of mass action np = n i2 is valid

of the Maxwell–Boltzmann distribution. For ηF = 0, Fermi level coincides with the conduction bandedge E C and has a value of ~0.6. Therefore, free carrier density from Eq. 2.49 is, n ~ 0.7N C . In non-degenerate semiconductors, the doping density is much lower than the effective density of states N C and the Fermi level is naturally a few k B T below the conduction bandedge, E g . In this case, the Fermi integral given in Eq. 2.50 becomes:  F1/2

E F − EC kB T



√   EC − E F π = exp − 2 kB T

We then obtain the free carrier density given by Eq. 2.47.

(2.51)

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2 Intrinsic and Extrinsic Semiconductors

Fig. 2.18 Fermi–Dirac integral as a function of Fermi level. The broken lines represent the Maxwell–Boltzmann distribution approximation

Similarly, in p-type semiconductors, the general expression for the free hole density is:   E F − EV 2 p = N V √ F1/2 − kB T π

(2.52)

This expression reduces to Eq. 2.48, for ηF N A

(2.58)

An inequality of doping density in the other direction can be treated similarly with the effective acceptor density being given by: N Ae = N A − N D N A > N D

(2.59)

Example (a) Using values of the conductivity parameters in Table 2.1, calculate the intrinsic conductivity in Si. (b) A wafer of intrinsic Si is doped with one B atom for every 108 Si atoms. What are the majority and minority carrier densities in the doped material? Calculate the conductivity of the resulting extrinsic semiconductor. Is the material n-type or p-type? (c) The above wafer of extrinsic Si is now further doped with 2 × 1023 atoms per m3 of material. What will be the conductivity of Si after this second doping and are holes now the majority or minority carriers in this material? (a)

For the intrinsic Si at T = 300 K σ = en i (μn + μ p ) = (1.6 × 10−19 )(1.4 × 1016 )(0.15 + 0.045) = 4.4 × 10−4 (m)−1

(b)

B is in Group III of the periodic table, therefore it is an acceptor atom in Si and the doped semiconductor will be p-type. The acceptor density is: NA =

5 × 1028 = 5 × 1020 atoms/m3 108

At T = 300 K, all acceptor atoms will be ionised and hole density:

2.4 Extrinsic Semiconductors

73

p = N A = 5 × 1020 holes/m3 The minority carrier density can be found from the law of mass action: n=

n i2 (1.4 × 1016 )2 = = 3.92 × 1011 electrons/m3 p 5 × 1020

In order to find the conductivity, we ignore the minority carrier contribution because p >> n: σ = epμ p = (1.6 × 10−19 C)(5 × 1020 m −3 )(0.045m 2 /V s) = 3.6(−1 m −1 ) (c)

P is a donor atom in Si N D = 2 × 1023 atom/m3

After partial compensation of donor atoms, the net effective donor density will be: N De = N D − N A ≈ N D N D >> N A So the resulting material is n-type with holes as the minority carriers. We can calculate the conductivity using the majority carrier density: σ = enμn = eN D μn = (1.6 × 10−19 C)(2 × 1023 m −3 )(0.15m 2 /V s) = 4.8(−1 m −1 )

2.5 Temperature Dependence of Carrier Density in Extrinsic Semiconductors In order to quantitatively describe the temperature dependence of carrier density in extrinsic semiconductors, we need to consider three factors. (i) (ii) (iii)

The temperature dependence of the ionisation of impurity (donor or acceptor) atoms. Temperature dependence of the extrinsic carrier density. Temperature dependence of the intrinsic carrier density.

The Fermi–Dirac distribution function describes the temperature distribution in energy of both intrinsic and extrinsic semiconductors. However, for extrinsic semiconductors, the Fermi distribution function has to be a modified version of that given in Eq. 2.11. The reason for this modification is because the energy level for the

74

2 Intrinsic and Extrinsic Semiconductors

ionised donor atom (for an n-type semiconductor) can accommodate a spin up and spin down electron. An empty donor energy level implies that it has ionised twice leading to a different ionisation energy compared to an energy level with only single occupancy. The Fermi–Dirac distribution function for an n-type semiconductor is then given by: f D (E D ) =

1 1 + 21 e(E D −E F )/k B T

(2.60)

The same argument is also valid for acceptor impurities. A neutral acceptor atom does not have any extra electrons but when it is ionised it may accommodate a spin up or spin down electron. As a result the pre-exponential term in Eq. 2.60 should be replaced by 2. Furthermore, the valence band in semiconductors are doubly degenerate, thus the pre-exponential term should also be multiplied by 2. Therefore, the Fermi–Dirac distribution function for a p-type semiconductor is: f A (E A ) =

1 1+

4e(E A −E F )/k B T

(2.61)

When a donor atom has one excess electron compared to the atoms of the host intrinsic semiconductor, then the donor atom is considered neutral. If the excess electron is excited into the conduction band by thermal vibrations then the donor atom is positively ionised. The total donor density is then: N D = N D0 + N D+

(2.62)

The probability that an impurity level, donor and acceptor, will be occupied is given by Eqs. 2.60 and 2.61, respectively. Therefore, the density of neutral donors is: N D0 = N D f D (E D )

(2.63)

The ionised donor density from Eqs. 2.60 and 2.62 gives:  N D+

= N D [1 − f D (E D )] = N D   EF − ED = N D 1 + 2 exp kB T

 −1   ED − EF 1 1 − 1 + exp 2 kB T  −1 (2.64)

In Eq. 2.62, the pre-exponential factor 2 represents the degeneracy of the ground state of the hydrogen-like donor atoms, corresponding to its occupancy by spin up and spin down electrons. To elucidate further, let us consider the example of a slice of n-type Si with an impurity density of 1015 donor atoms/cm3 . At absolute zero temperature, the intrinsic

2.5 Temperature Dependence of Carrier Density …

75

carrier density will be zero and no donor atoms will be ionised. The material, like all semiconductors at 0 K, would be an insulator. At very low temperatures (k B T < E D ), the intrinsic carrier density is still negligibly small and only a few of the donor atoms are ionised. Therefore free electron density in the conduction band, donated by the donor atoms will be very low. We can use the Maxwell–Boltzmann distribution for the free carrier density in the conduction band: 

EC − E F n = NC exp − kB T



= N D+

(2.65)

Using Eq. 2.64 for N D + in Eq. 2.65: n2 −

    1 1 ED ED N D NC exp − + n NC exp − =0 2 kB T 2 kB T

(2.66)

At the low temperatures, because not all donor atoms are ionised, the free electron density in the conduction band will be smaller than the donor density (n 0 K. Prove that the probability of finding an electron at Fermi level for T > 0 K is always f (E) = 0.5. The Maxwell–Boltzmann distribution function gives the probability of an energy level E at T = 10 K as f (E) = 1.0 × 10–12 . What is the probability of the same energy level being full at T = 100 K? Show that in an intrinsic semiconductor, when the Fermi energy is measured from the conduction bandedge (E C = 0), (a)

The free carrier density can be expressed as:  n = NC exp

(b)

EF kB T



Show that the Fermi energy is:  ∗ 3 EV me − k B T ln EF = 2 4 m ∗h

(c)

Discuss the meaning of parameter X in the expression below.      EF EV − E F X = e NC exp × μe + Nr exp × μh kB T kB T

8.

In a semiconductor, the density of states per unit energy per unit volume in the conduction band is:  ∗ 3/2 2m e N (E) = (E − E C )1/2 2π 2 3 (a) (b)

9. 10. 11.

Plot the product function N(E)f (E)versus energy at T = 0 K. What is the physical meaning of the area under the N(E)f (E)versus E plot?

Calculate the total density of states between energies E C and E C + k B T in Si at T = 300 K. Germanium (Ge) is doped with donor impurities with a doping density of 3ni /2. Calculate the free electron and hole densities. In Si the intrinsic carrier density is 1010 cm−3 . If Si is doped with N A = 1016 cm−3 then how would this effect the majority and minority carrier densities?

78

2 Intrinsic and Extrinsic Semiconductors

12.

13.

An Arsenic (As) doped (N D = 1017 cm−3 ) Si sample is cut into the shape of a cuboid with length 100 μm, width 10 μm and thickness 1 μm. Calculate the resistance of the Si sample. InP semiconductor sample is doped with Si (N A (Si) = 1 × 1016 cm−3 ) and C (N D (C) = 4 × 1016 cm−3 ). Calculate: (a) (b)

14.

The electron and hole densities. The position of the Fermi level at T = 300 K.

The empirical expression for the temperature dependence of the energy bandgap of a semiconductor is: E g (T ) = E g (0K ) −

15.

16.

αT 2 T +β

where E g (0 K) is the bandgap at T = 0 K. For Si, E g (0 K) = 1.170 eV, α = 4.73 × 10–4 eV/K and β = 636 K. What is the bandgap of Si at T = 500 K? An n-type Si device is operated at T = 500 K. What is the minimum donor density in Si so that, at the operating temperature, the intrinsic electron density is only 5% of the majority carrier density? (Use your result in Problem 14 for the bandgap of Si at T = 500 K.) The Fermi level in a p-doped Si is 0.27 eV above the valence band. At T = 300 K, N V (Si) = 1.04 × 1019 cm−3 , calculate the hole density at T = 300 K and T = 400 K.

Suggested Reading List 1. 2. 3. 4. 5. 6.

Pathria RK (1996) Statistical mechanics, 2nd edn, Elsevier Reif F (1968) Fundamentals of statistical and thermal physics. McGrow-Hill Huan A, Statistical mechanics http://www.spms.ntu.edu.sg/PAP/courseware/statmech.pdf Joyce WB, Dixon RW (1997) Appl Phys Lett 31:354 Blakemore JS (1988) Semiconductor statistics. Dover Singh J (1994) Semiconductor devices. Mc-Graw Hill

Chapter 3

Charge Transport in Solids

Learning Outcomes At the end of this chapter, the reader will be competent in the full theory of transport in semiconductors and will be able to tackle any problems in the following topics: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

drift and diffusion conductivity, scattering processes, temperature dependence of mobility and Hall measurements, Matthiessen’s rule, Einstein’s relations, non-Equilibrium transport, transport at high electric fields, continuity equations, quasi-Fermi levels, excess carrier lifetime, ambipolar transport, photoconductivity.

The operation of electronic devices is based on transport of electrons and holes within the solid from which the device is fabricated. Transport occurs as a result of the application of an external electric field (drift) or by the formation of an inhomogeneous charge distribution (diffusion). It should be emphasised that for the carriers to accelerate between the contacts of the electronic device and to make a contribution to the current flow, as described in the previous chapter, they should be delocalised (free). Localised carriers cannot contribute to the conductivity. However, localised carriers are important in determining the dielectric properties of solids, thus the operation bandwidths, wavelength and polarisation characteristics of electronic and optoelectronic devices. In semiconductor devices such as diodes, transistors, photodetectors, photodiodes and lasers, both electrons and holes play an equally important role in device operation. As described in Chap. 2, in semiconductors and insulators, the free carriers are © Springer Nature Switzerland AG 2021 N. Balkan and A. Erol, Semiconductors for Optoelectronics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-44936-4_3

79

80

3 Charge Transport in Solids

electrons in the conduction band and holes in the valence band. In metals however, free carriers are only the electrons in conduction band. The motion of free carriers under the influence of an external field, is described with drift conductivity and motion resulting from an inhomogeneous distribution of free carriers is described with diffusion conductivity. In metals, i.e. passive circuit components like metal resistors and conducting paths, the conductivity is due to drift only. In semiconductors however, conductivity is determined by the sum of drift and diffusion conductivities. In this chapter we aim to study in detail these two types of conductivity mechanisms.

3.1 Transport in an Electric Field The presence of a finite density of free carriers (either electrons, holes or both) is essential for drift conductivity. Electrons have negative charges (–e) and holes have positive charges (+e). In Fig. 3.1, we show the motion of electrons and holes in a semiconductor. When a potential difference V is applied between the ends of the device, an electric field, E = V /l develops along the device originating from the cathode and terminating at the anode. Free electrons and holes drift in opposite directions and a drift current I, flows through the device and the external circuit, the value of which is the sum of electron and hole drift currents.

3.1.1 Particle Current and Charge Current It is important to distinguish the difference between direction of the charged particles’ motion and the current flow direction. Going back to Fig. 3.1, it is clear that when the applied electric field is along the x direction, electrons will move in the –x direction, while the holes move along the same direction as the electric field, x. These two directions define the motion of the two particles, negatively charged electrons and

Fig. 3.1 Representation of the drift of electrons and holes in semiconductor device with length l and cross-sectional area A, when a potential difference of V is applied between the ends of the device

3.1 Transport in an Electric Field

81

Fig. 3.2 Particle and charge currents in an electric field

positively charged holes, thus the particle current. However, conventionally, we define the direction of the charge current flow as the direction of the positive charge. Therefore, the charge current is along the same direction as the electric field for both electrons and holes, as shown in Fig. 3.2.

3.1.2 Drift Velocity and Carrier Mobility For simplicity, let us assume that in a semiconductor, the charge carriers are only electrons. The electron density (number of electrons per unit volume) in this semiconductor at room temperature is high enough for the electrons to be considered as a classical gas which is called the electron gas. At finite temperatures, because of their thermal energies, the electrons in the semiconductor have a similar, random motion as the atoms and molecules in a classical gas. They move rapidly, constantly colliding with each other and with the atoms in the crystal. Unless an external electric field is applied, the random thermal motion remains and there is no net current flow in the semiconductor. When an electric field is applied, however, the random motion is replaced by drift motion along the electric field. As shown in Fig. 3.1, in an electric field E, the electrons experience a force in the opposite direction to the electric field, F = −eE. According to Newton’s second law of motion, they will be accelerated during a period of time τ, which is defined as the average scattering time of electrons by the ionised dopant atoms, impurities, dislocations and by other scattering centres or mechanisms as described later. Therefore the velocity of the electrons cannot increase indefinitely but reaches an average value as determined by the average scattering time. Electrons drift along the field while being scattered from their original path and their motion under an electric field is depicted in Fig. 3.3. Let us now derive the appropriate expression for the average velocity, commonly known as the drift velocity. Under an applied force, the acceleration is: a=

F m ∗e

During its average scattering time, electron will gain the velocity v,

(3.1)

82

3 Charge Transport in Solids

(a)

(b)

(c) Fig. 3.3 a The motion of an electron in a solid when an external electric field E is applied, b the path of the electron versus time and c the acceleration of the electron and the time dependence of its velocity

v = aτ

(3.2)

Therefore, v=

F eEτ τ =− ∗ m ∗e me

(3.3)

Here the electron mobility is defined as:    eτ   μn = − ∗  me

(3.4)

And the electron drift velocity is: v = μn E

(3.5)

3.1 Transport in an Electric Field

83

We called the scattering time τ average scattering time, because the time difference between different scattering events will vary. We shall now explain how we acquire a quantitative expression for the average scattering time. Let us start with the plausible assumption that the motion of an electron between two scattering events is completely arbitrary. The change in the mobility of the electron during its acceleration by the electric field within a time interval Δt is:  dp  dt electric

t = m ∗e f ield

dv t = m ∗e at = Ft = −eEt dt

(3.6)

Here a is the acceleration of electron as shown in Fig. 3.3. We need to use a quantum mechanical approach to determine the change in momentum of electrons once they have been scattered. Let us consider a group of N 0 particles which undergo a quantum mechanical process such as scattering. In quantum mechanics, the number of particles remaining un-scattered after a time t, is given as N (t) = N0 e(−t/τ ) . Also in the steady state the total number of particles at a given state is constant. In other words, the number of particles entering the state is equal to the number of particles leaving the state. The number of particles leaving the state can be found from the slope of the curve shown in Fig. 3.4 and is equal to N0 /τ . The change in the momentum of the particles leaving the state (or the particles scattered) during a time interval Δt, is:  N0 d N (t) dp  t = p t t = p dt scattering dt τ

(3.7)

Fig. 3.4 Representation of the exponential time dependence of the number of electrons: N(t) = N 0 exp(−t/τ )

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3 Charge Transport in Solids

The change in the momentum of a single particle during a time interval Δt can be found by dividing both sides of Eq. 3.7 by N 0 . In the steady state, the change in momentum due to electric field and scattering must be equal:  dp  dt electric

 dp  t = t dt scattering f ield

(3.8)

Therefore: −eE = p/τ

(3.9)

Since the momentum is related to the drift velocity via: p = m ∗e v

(3.10)

The drift velocity can be obtained as: v=−

eτ E m ∗e

(3.11)

From Eqs. 3.5 and 3.10 the electron mobility is: μn =

eτ m ∗e

(3.12)

μp =

eτ m ∗h

(3.13)

Similarly the hole mobility is:

Carrier mobility, which is proportional to the scattering time and inversely proportional to the effective mass of the carriers, is a very important factor in determining the maximum operation speed of electronic and optoelectronic devices. For example, in intrinsic Si, electron mobility is 1500 cm2 /Vs and in GaAs, it is 8500 cm2 /Vs. This big difference in electron mobilities is the reason why the GaAs based devices are faster than those based on Si. Table 3.1 shows electron and hole mobilities in some common semiconductors.

3.1.3 Matthiessen’s Rule and the Total Mobility The scattering time in Eqs. 3.12 and 3.13 may be used for different scattering mechanisms. For example, if we assume that the electrons are scattered by both ionised

3.1 Transport in an Electric Field

85

Table 3.1 Electron and hole mobilities in various intrinsic semiconductors at 300 K Mobility [1–5] (cm2 /Vs)

Effective mass [6] (m0 )

μe

μp

me *

mhh *

GaAs

8500

400

0.067

0.45

GaN

1000

350

0.20

0.50

GaP

250

150

0.13

0.60

InP

5400

200

0.079

0.47

40,000

500

0.026

0.64

InAs

impurity atoms with an average scattering time, τ I , and by the lattice with a scattering time, τ L , we can calculate the average of the two scattering times by using Matthiessen’s rule, i.e. τor t = 

1 1 τL

+

1 τI

=

τL τI τL + τI

(3.14)

We can now use Eqs. 3.14 and 3.12 to find the total mobility as determined by the two scattering mechanisms as μnT =

  τL τI e m ∗e τ L + τ I

(3.15)

Naturally, if there is no lattice scattering then average scattering time for lattice scattering becomes τ L → ∞ and the total average scattering time represents solely the ionised impurity scattering, the mobility of which is: μnI =

eτ I m ∗e

(3.16)

On the other hand, if there is no impurity scattering then average scattering time for impurity scattering becomes τ I → ∞. The total average scattering time is equal to the average lattice scattering time and the mobility determined by lattice scattering is: μnL =

eτ L m ∗e

(3.17)

When we substitute Eqs. 3.16 and 3.17 into Eq. 3.15, we obtain the total mobility, μnT : 1 1 1 = L + I μnT μn μn

(3.18)

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3 Charge Transport in Solids

Equation 3.18 gives the total mobility associated with different and independent scattering mechanisms and is known as the Matthiessen’s rule. Here we considered only two different scattering mechanisms but in reality, there may be other mechanisms; in that case the general expression for Matthiessen’s rule is:  1 1 = i μi μ

(3.19)

Here (i = 1….n) represents the n independent scattering mechanisms.

3.1.4 Temperature Dependence of Mobility Let us consider the two scattering mechanisms once more in order to explain the temperature dependence of carrier mobility. For simplicity, we shall consider electrons only as the free carriers, however, the argument presented here is equally applicable to the other carrier type, holes. When the lattice temperature increases, crystal atoms vibrate increasingly more violently around their equilibrium positions. Also, with increasing temperature the thermal velocity of free electrons increases. Consequently, scattering of electrons by the crystal atoms will increase as the cross-sectional area of the vibrating atoms, as seen by the electron, will go up with temperature. There will be more scattering events per unit time and the average lattice scattering time will decrease with increasing temperature. This temperature dependence is given as: τ L ∝ T −3/2

(3.20)

However, the temperature dependence of impurity scattering will have an opposite dependence on temperature. This is because at low temperatures electrons with small thermal velocities spend a longer time in the vicinity of the ionised impurities. With increasing temperature and increasing thermal velocity, however, electrons will gradually spend shorter times near the ionised impurities. They will not feel the Coulombic attraction of the ionised impurities to the same degree and will be scattered less, as depicted in Fig. 3.5. The temperature dependence of ionised impurity scattering time is given as:

Fig. 3.5 Scattering of electrons by ionised impurities a at low temperatures b at high temperatures. I and II represent the scattering of electrons close to and at a longer distance from the impurity centre, respectively

3.1 Transport in an Electric Field

87 5200

Mobility (cm2/Vs)

5100 5000 4900 4800 4700 4600

n-type GaAs

4500

ND = 1x1017cm-3

4400 4300 50

100

150

200

250

300

Temperature (K)

(a)

(b)

Fig. 3.6: a An illustration of temperature dependence of the electron mobility as determined by ionised impurity and lattice scattering mechanisms and b Measured temperature dependence of n-type GaAs with N D = 1 × 1017 cm−3 .

τ I ∝ T 3/2

(3.21)

If we now consider the temperature dependence of the total mobility, then it can be seen that at low temperatures the dominant mechanism is ionised impurity scattering and at high temperatures it is lattice scattering. Figure 3.6a shows the overall temperature dependence of the total mobility. It is worth noting here that, in doped semiconductors, mobility decreases with increasing doping density, because higher doping density implies more scattering centres in the form of ionised impurities as seen in Fig. 3.6b for an n-type GaAs doped with Si (N D = 1 × 1017 cm−3 ). The electron mobility for the n-type GaAs at 300 K is smaller (4350 cm2 /Vs) than the value for intrinsic GaAs (8500cm2 /Vs) given in Table 2.1.

3.1.5 Electric Field Dependence of Mobility So far, we have not taken into account the effect of applied electric field on the electron mobility, which is a reasonable assumption when the thermal velocity of electrons is higher than their drift velocity. We shall now consider the case when drift velocity exceeds thermal velocity and when the applied electric field is high enough and cannot to be neglected. According to the kinetic theory of a classical gas, the molecular kinetic energy is related to thermal velocity (vt ) via: E kinetic =

1 2 3 mv = k B T 2 t 2

(3.22)

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3 Charge Transport in Solids

In this context, the classical gas is the electron gas therefore we may use Eq. 3.22 also to describe the thermal velocity of electrons. Let us start with an example. In GaAs the electron effective mass is about m* = 0.067 m0 , where m0 is the free electron mass (m0 = 9.11 × 10–31 kg). The thermal velocity of electrons at room temperature from Eq. 3.22 is then vt = 4.5 × 105 m/s. If we apply a small electric field of E = 100 V/m and assume that the electron mobility in GaAs at room temperature is 8500 cm2 /Vs, from Eq. 3.5, the drift velocity turns out to be vd = 8.5 × 103 m/s. This is considerably smaller than the thermal velocity therefore, at such low electric fields the effect of the applied electric field on electron mobility is negligible and the scattering times are solely determined by the thermal velocity. Let us now increase the applied electric field until the drift (vd ) and thermal velocities (vt ) are equal: vt = vd eτ μn E = ∗ E = me 3m ∗e k B T τ= eE

 3

kB T m ∗e (3.23)

It is clear from Eq. 3.23 that the average scattering time decreases with increasing electric field. The mean free path between the scattering events, λ, is defined as the product of the drift velocity and the average scattering time is: λ=E

q 2 τ m ∗e

(3.24a)

Scattering time in terms of the mean free path is:  τ=

λm ∗e eE

 (3.24b)

the mobility: eτ μn = ∗ = me



eλ Em ∗e

 (3.24c)

And the drift velocity is: eτ vd = ∗ = me



eEλ m ∗e

 (3.24d)

3.1 Transport in an Electric Field

89

The mobility varies as E −1/2 . This is an interesting result because so far we assumed that the mobility is constant and is equal to the ratio between the drift velocity and electric field, and is the well-known Ohm’s law. It can be seen that Ohm’s law breaks down at high electric fields and the drift velocity has a non-linear dependence on the applied electric field. The electric field at which this dependence starts can be found from Eq. 3.23: 1 E= μn



3k B T m ∗e

 (3.25)

At higher electric fields the drift velocity saturates when electrons transfer their kinetic energy, which they gain from the electric field, to the crystal lattice via the lattice scattering. Debye developed a model to explain the heat capacity of solids where he analysed the energy transfer from electrons to lattice atoms. We shall give a brief summary here. The critical energy, E c , at which the energy transfer occurs, is: Ec = k B θ D

(3.26)

where θ D is the Debye temperature. The drift velocity saturates when the kinetic energy of electrons is equal to the critical energy. Considering that kinetic energy of electrons = k B θ D : 1 ∗ m (μn E c f )2 = k B θ D 2 e

(3.27)

 where E cf is the critical electric field at which the drift velocity vd = μn E c f saturates. The electric field dependence of drift velocity is shown in Fig. 3.7. It is clear that at low fields Ohm’s law is valid where the drift velocity varies linearly with electric field. What happens at low fields is as follows. Electrons are accelerated and gain energy between two scattering events, when they are scattered, their drift velocity reduces to the thermal velocity value, and therefore, they lose the energy that they gained since the previous scattering event. At higher electric fields drift velocity varies sub-linearly with field, according to Eq. 3.24d, and when the electric field exceeds the critical field the drift velocity saturates at vd . With the recent advances in semiconductor growth and fabrication techniques, it is now possible to fabricate electronic devices with very small dimensions. For example, the channel lengths in Complementary Metal Oxide Semiconductor (CMOS) devices are currently about 20 nm. When a small voltage is applied between the electrodes of such small devices, the electric field along the channel can be extremely high. At such high electric fields, as seen already, Ohm’s law is no longer valid and electrons

90

3 Charge Transport in Solids

Fig. 3.7 a Drift velocity versus electric field in Si at room temperature, b The electic field dependence of drift velocity in various semiconductors c In the presence of NDR regime, expression of mobility in conduction band valleys.

(a)

108

Drift velocity (cm/s)

Electron Hole

InP

Ga0.47In0.53As GaAs

107

Ge

Ge

Ga0.47In0.53As

6

10

Si

105 102

103

104

Electric Field (V/cm)

(b)

(c)

105

3.1 Transport in an Electric Field

91

are not in thermal equilibrium with the crystal lattice. The average kinetic energy of the electron gas is higher than the surrounding lattice and their distribution (Fermi– Dirac) is represented by a statistical temperature, known as the electron temperature, T e which is greater than the lattice temperature, T L (T e > T L ). These electrons are known as hot electrons. Hot electrons encounter more lattice scattering (phonon scattering) with increasing electric field, therefore, their mobility is no longer constant but decreases rapidly. In the hot electron regime the mobility is defined as the differential mobility. Differential mobility is a decreasing function of electric field and eventually the drift velocity saturates and becomes independent of the electric field. Although we have chosen a nano-dimensional CMOS device as an example, it should be worth mentioning that the drift velocity—electric field relationship described, is in fact valid in devices with lengths longer that the mean free path of the electrons. In devices smaller than the mean free path, the electrons move between the electrodes of the device without experiencing any scattering. This type of transport without scattering is known as ballistic transport.

3.1.6 Conductivity In Chap. 1, the expression for conductivity was given without paying any attention to the microscopic basis of the terms. We shall now return to the concept of conductivity and explain the underlying microscopic physical phenomena. Let us start with a solid cylindrical conductor with length l and cross-sectional area A. When a potential difference V is applied between the ends of the conductor, some charge will flow through the cross-sectional area and move as a length of w per unit time (Fig. 3.8). The total charge flow per unit time (current) is then: I =

d dQ = (en Aw) dt dt

(3.28a)

dw = en Avd dt

(3.28b)

I = en A

Fig. 3.8 Current flow through volume of Aw in a cylindrical conductor

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3 Charge Transport in Solids

I = enμE A

(3.28c)

Here n, E and vd are the electron density (m−3 ), electric field (V/m) and drift velocity (m/s), respectively. I = enμV R=

A l

(3.29)

V 1 l = I enμ A

(3.30)

1 1 = enμ σ

(3.31)

ρ= And the conductivity is:

σ = enμ

(3.32)

3.1.7 Current Density Current density is used more often than current in characterising device operation and is defined as the current flow per cross-sectional area. This is related to the microscopic properties of a solid and given as: I A

(3.33a)

J = σE

(3.33b)

J = enμE

(3.33c)

J= From Eqs. 3.28a—3.28c and 3.32:

Current density changes along a conductor with non-uniform cross-sectional area but total current, however, remains constant as depicted in Fig. 3.9.

3.1.8 Bipolar Conductivity If there are more than one type of charge carriers, i.e., electrons and holes, the total conductivity is the combination of conductivities for both carriers and the total

3.1 Transport in an Electric Field

93

Fig. 3.9 Current and current density in a conductor with a non-uniform cross-sectional area

conductivity is known as bipolar conductivity. σ = e(nμn + pμ p )

(3.34)

The indices n and p represent electrons and holes respectively. The current density can be obtained by combining Eqs. 3.33c and 3.34: J = e(nμn + pμ p )E

(3.35)

In metals, electrons are the sole charge carriers, therefore the conductivity is unipolar. In contrast, the conductivity in semiconductors is bipolar.

3.2 Diffusion Conductivity It is useful to consider a simple experiment as a starting point for the study of diffusion conductivity. If we drop a single drop of ink in a bowl of water then we notice that initially the molecules of the ink will concentrate in a small volume of water about the same size of the ink drop. With time, however, the ink molecules will spread out from their initial positions and will cover larger surface area. In other words, the ink molecules move from where their density is high to positions where their density is low. The process continues until the ink molecules have evenly spread out throughout the surface of water and its density has become uniform. The density of the ink molecules [n(x)] is shown as a function of position at different timest = 0, t = t 1 and t = t 2 , (0 < t 1 < t 2 ) in Fig. 3.10. During the diffusion process the motion of the molecules is determined by their individual kinetic energies and is completely arbitrary. Let us now consider a gas with molecular density n(x) in an enclosed volume as shown in Fig. 3.11. Assume the density is higher on one side of the interface separating the volume into two sections, with a finite density gradient of dn(x)/dx at the interface. It is quite clear that on average there will be more gas molecules moving from the left hand side of the interface (higher density) to the right hand side (low density) than in the other direction. This implies that there will be a net flow of gas molecules from the right to the left. This simple qualitative description of the motion of gas molecules is the basis of Fick’s law. In the motion of gas molecules, the diffusion current is only the diffusing particle current. If we consider the diffusion

94

3 Charge Transport in Solids

(a)

(b) Fig. 3.10 a Diffusion of an ink drop on the water surface at t. b Density profiles of the ink molecules during the diffusion process where r is the radius of the ink drop at t = 0 and R is the radius of the container

Fig. 3.11 Molecular diffusion across an interface between two sides of closed volume of gas molecules. The molecular density is higher on the left hand side of the interface. The net diffusion of gas molecules per unit time is φ = φ1 − φ2

3.2 Diffusion Conductivity

95

of electrons in an electron gas, however, as well as the diffusion of particles, there is an accompanying diffusion of charge. Therefore, the diffusion of electrons will give rise not only to particle diffusion current but a charge diffusion current.

3.2.1 Fick’s Law With reference to Fig. 3.11, let us assume that the net particle flow along the x direction is φ. We can express Fick’s law as: φ = −D

dn(x) dx

(3.36)

where D is the diffusion coefficient in cm2 /s. The negative sign indicates that the diffusion current flows in a direction opposite the positive density gradient, dn(x)/dx. In other words, it flows from the higher density to the lower density.

3.2.2 Diffusion Current Density We can now write down the charge diffusion current density simply by replacing the particle density by the charge density in Eq. 3.36 for electrons: Jn (di f ) = −eφn = eDn

dn dx

(3.37)

J p (di f ) = eφ p = −eD p

dp dx

(3.38)

And for holes:

Here n, p, Dn and Dp are the electron and hole densities, and electron and hole diffusion coefficients, respectively. Diffusion coefficients for common semiconductors Si, Ge and GaAs are given in Table 3.2. When there is bipolar conductivity, involving both electrons and holes, we need to know the electron and hole densities and mobilities in order to work out the drift Table 3.2 Electron and hole diffusion coefficients for Si, Ge and GaAs at room temperature

Diffusion coefficient (cm2 /s)

Si14

Ge15

GaAs16

Dn

36

100

200

Dp

13

50

10

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3 Charge Transport in Solids

current density as given in Eq. 3.35. For the bipolar diffusion current density both the electron and hole density gradients and their diffusion coefficients are required. It will shortly be demonstrated that the carrier mobilities and diffusion coefficients are not independent of each other.

3.2.3 Drift and Diffusion Current Densities In a semiconductor, in the existence of both drift and diffusion current, the total current is expressed as the sum of these currents: Jdri f t + Jdi f f usion = Jtotal

(3.39)

We can use Eqs. 3.35, 3.37–3.38 to write down the total current explicitly: enμn E + eDn

dn dp + epμ p E − eD p = Jtotal dx dx

(3.40)

Here the electric field is chosen to be in the + x direction. Equation 3.40 is the general expression for total current in a semiconductor and may be amended for specific conditions. For example, if there is drift current only then the terms representing the diffusion current can be removed from the equation or vice versa. As mentioned previously, the carrier mobilities and diffusion coefficients are not independent of each other but related by the Einstein relations: μp e μn = = Dn Dp kB T

(3.41)

μn Dn = μp Dp

(3.42)

Einstein relations are very useful for calculating the electron and hole diffusion coefficients from the measured values of electron and hole mobilities. The carrier mobilities can be determined experimentally as a function of temperature, by using a simple experiment, known as the Hall Effect. Example In an n-type Si with N D = 5 × 1016 cm−3 , hole mobility at 300 K is 317 cm2 /Vs. If the hole density changes linearly between x = 0 and x = 1 μm from 1014 to 1013 cm−3 , calculate diffusion current density. Dp = μp

kB T 0.026eV = 317cm2 /V s × = 8.2cm2 /s e e

J p (di f ) = −eD p

dp p = −eD p = −1.6 × 10−19 C × 8.2cm2 /s dx dx

3.2 Diffusion Conductivity

97

14  10 − 1013 cm−3 × = −1.18A/cm 2 10−4 cm

3.2.4 Hall Effect The Hall Effect is a standard electrical characterisation technique employed in the experimental determination of carrier type, density and mobility in semiconductors. The principle of Hall Effect is based on the Lorentz force experienced by moving charges under electric and magnetic fields. Let us consider a semiconductor sample in the shape of a cuboid which has n type conductivity as shown in Fig. 3.12. A constant current I is applied along the x direction and a constant magnetic field B is applied along the –z direction. The Lorentz force experienced by an electrons is: − → − → − → F = −e( E + v × B )

(3.43)

− → − → where e is the electronic charge, E is the electric field, B is the magnetic field − → − → and v is the drift velocity of electrons. F , B and v are defined in the right-handed Cartesian coordinates. − → The magnetic force evx Bz will be perpendicular to both B z and vx . Consequently, electrons will experience a magnetic force in the –y direction and an electrical force in the + x direction. As a result of the Lorentz force the electrons accumulate on the

Fig. 3.12 A cuboid shaped n-type semiconductor used to measure the Hall Effect

98

3 Charge Transport in Solids

far side of the cuboid, and the near side will be depleted of electrons. The potential difference, developed between the two side surfaces is known as the Hall voltage, V H . The electric field in the + y direction, associated with this voltage is known as the Hall electric field and expressed as − → − → − → E H = |R H | J × B where RH is Hall coefficient. Let us now calculate the Lorentz force using the three dimensional vector components of all the parameters in Lorentz force given in Eq. 3.43. − → E = (E x , E y , 0)    i j k   

− → − → − → F = −e E − e( v × B ) = −e(E x , E y , 0) − e −vx 0 0   0 0 −B  z − →    F = −eE x i − e(E y − vx Bz ) j + 0k

v = (−vx , 0, 0)

− → B = (0, 0, −Bz )

(3.44)

Here the two components of the force are: Fx = −eE x Fy = −e(E y − vx Bz )

(3.45)

E y − v x Bz = 0

(3.46)

In equilibrium:

The Hall electric field E y can be obtained from the Hall voltage (V H ): Ey =

VH w

(3.47)

Here w is the width of the cuboid as indicated in Fig. 3.12. The relationship between the current density J x and drift velocity vx has been given as Jx = envx so the Hall voltage is related to electron density by: E y = v x Bz Jx = envx ⇒ vx = Ey =

Jx Bz ; en VH =

Jx en

Jx = I x Bz end

Ix wd (3.48)

3.2 Diffusion Conductivity

99

In situations where the thickness of the semiconductor (d) is not known or if the semiconductor contains more than one conducting layer, such as quantum wells, then it is more accurate to define a two dimensional electron density, which is also known as sheet electron density, n 2D = n × d n 2D =

I x Bz eVH

(3.49)

The Hall coefficient (RH ) is defined as: RH = −

rH en

RH = +

for electrons and

(3.50a)

for holes

(3.50b)

rH en

Here the Hall factor r H, varies between 1 and 1.5 but for simplicity it can be taken as close to r H = 1 for low magnetic fields. Then the Hall voltage is: V H = R H I x Bz

(3.51)

The sign of the Hall coefficient shows the type of free carriers (negative sign for electrons and positive sign for holes). In a Hall Effect experiment, I, B and the sample dimensions are known parameters. Therefore, the electron density can be obtained by inserting the measured Hall voltage into Eq. 4.49. Furthermore, the conductivity along the x direction can be obtained from the experimental results using: Jx = envx =

Ix wd

= σx E x

(3.52)

During this measurement, magnetic field should be kept zero, because the resistance of the sample can be affected by magnetic field. The Hall mobility of electrons can be calculated from: μ = |R H |σx

(3.52a)

Hall mobility differs from the drift mobility of free carriers by the Hall factor r H (μ = r H μn ). The Hall factor is related to the effective scattering mechanism and depends on the energy of the carriers. The linear relationship between the Hall voltage and the magnetic field in Eq. 3.48 is valid only at low magnetic fields. At higher magnetic fields, as the Hall factor changes with magnetic field, the linear relationship in Eq. 3.48 has to be corrected for the non-constant Hall factor.

100

3 Charge Transport in Solids

3.3 Non-equilibrium Carriers in Semiconductors In semiconductors, any departure from the equilibrium condition changes the electron and hole densities. For example, an impulse of photons with energies greater than the bandgap will increase the electron and hole densities and they will return to equilibrium with time known as the energy relaxation time. We shall first look at the case when excess electron and hole generation occurs via transitions from the valence band to the conduction band and then when excess carriers generated via de-trapping from the trap levels within the bandgap. We will consider both band to band recombination and trapping of the excess carriers. Let us start with investigating the situation when a semiconductor is in thermal equilibrium.

3.3.1 Semiconductor in Thermal Equilibrium The term carrier generation refers to the creation of excess electron and hole pairs and recombination refers to the annihilation of the excess electron and holes. In Chap. 2, expressions were derived for the equilibrium electron and hole densities in a semiconductor. As long as the thermal equilibrium is maintained then these densities are independent of time. However, because the thermal excitations due to the statistical nature of the process are completely arbitrary, electrons are continuously excited from the valence band into the conduction band. Naturally, the excited electrons in the conduction band recombine with the holes in the valence band (electron – hole recombination) and an equilibrium between thermal excitations and recombination is maintained. The net carrier density remains constant. Figure 3.13 depicts the generation and recombination processes in a semiconductor in thermal equilibrium. Let us represent the generation rate for electrons per unit volume with Gn0 and the thermal generation rate for holes per unit volume with Gp0 . There are holes left behind in the valence band for electrons thermally excited into the conduction band. Therefore: G n0 = G p0 Fig. 3.13 Generation and recombination in a semiconductor at thermal equilibrium

(3.53)

3.3 Non-equilibrium Carriers in Semiconductors

101

As for the recombination across the bandgap, we can write down for the electron (Rn0 ) and hole (Rp0 ) recombination rates per unit volume: Rn0 = R p0

(3.54)

In thermal equilibrium, the generation and recombination rates are equal: G n0 = G p0 = Rn0 = R p0

(3.55)

3.3.2 Generation and Recombination of Excess Carriers If the semiconductor is initially in thermal equilibrium and carrier generation occurs by some external stimuli then the generated carriers are known as excess carriers (excess electrons and excess holes). Let us assume that the external stimulus is photons with energies greater than the bandgap of the semiconductor. Photons will give up their energies to electrons in the valence band to excite them into the conduction band (photo-generated excess electrons), leaving behind photo-generated excess holes in the valence band. Now let us represent the non-equilibrium generation rates per unit volume with G n for the excess electrons and with G p for the excess holes. Since we are considering transitions from valence to conduction band then for every electron generated, there will be one hole generated. Therefore: G n = G p

(3.56)

The electron density in the conduction band is: n(x, t) = n 0 + δn(t)

(3.57a)

And the hole density in the valence band is: p(x, t) = p0 + δp(t)

(3.57b)

Here p0 and n0 are the carrier densities in thermal equilibrium and are independent of time and position. δn and δp are the excess, photo-generated carrier densities. The symbols used in this section are tabulated in Table 3.3. Figure 3.14 shows the excess carrier generation process where the equilibrium and photo-generated carrier densities are clearly indicated. When the semiconductor is not in thermal equilibrium, from Eqs. 3.57a and 3.57b: np = n 0 p0 = n i2

(3.58)

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3 Charge Transport in Solids

Table 3.3 Symbols used in this section Symbol

Description

n 0 , p0

Electron and hole densities in thermal equilibrium (independent of time and position)

n, p

Total electron and hole densities (time and position dependent)

δn = n − n 0

Excess carrier densities (time and position dependent)

δp = p − p0 G n0 , G p0

Generation rates for electrons and holes in thermal equilibrium

G n ,G p

Generation rates for excess electron and holes

Gn , G p

Total generation rates for electrons and holes

Rn0 , R p0

Recombination rates for electrons and holes in thermal equilibrium

Rn ,

Recombination rates for excess electrons and holes

R p

Rn , R p

Total recombination rates for electrons and holes

τn0 = τ p0

Lifetimes of electrons and holes in thermal equilibrium

τnt = τ pt

Total lifetime of minority carriers

τn = τ p

lifetime of excess carriers

Fig. 3.14 Photogeneration of excess electron and hole pairs (δn and δp) and the thermal equilibrium carriers (n0 and p0 )

As in the case of thermal equilibrium, during the excess carrier generation, the electron and hole pairs generated will recombine across the bandgap (Fig. 3.15). Their recombination rates are therefore equal: Rn = R p

(3.59)

Electron recombination rate needs to be a function of the hole density and conversely the hole recombination rate must be a function of the electron density. This is because for an electron (hole) to recombine there has to be an available hole (electron). The net rate of change in the electron density is then:

dn(t) = αr n i2 − n(t) p(t) dt

(3.60)

3.3 Non-equilibrium Carriers in Semiconductors

103

Fig. 3.15 Recombination of excess electron hole pairs

Here αr n i2 is the generation rate in thermal equilibrium. Since the electron and hole pairs are generated and recombine simultaneously then the excess carrier densities will satisfy the condition: δn(t) = δp(t) Considering that the electron and hole densities in thermal equilibrium are independent of time, Eq. 3.60 can be re-written explicitly as:

d(δn(t)) = αr n i2 − (n 0 + δn(t)) = −αr δn(t)[(n 0 + p0 ) + δn(t)]( p0 + δp(t)) dt (3.61) If the excess carrier densities are lower than the equilibrium carrier densities (low injection limit) we can easily solve Eq. 3.61. For a p-type semiconductor, where p0 >> n0 , for the low injection condition, (δn(t) = δp(t) p0 ) Eq. 3.61 becomes: d(δn(t)) = −αr p0 δn(t) dt

(3.62)

The time dependence of the excess electron density is then: δn(t) = δn(0)e−αr p0 t = δn(0)e−t/τn0

(3.63)

Here δn(0) is the initial (t = 0) excess electron density and τn0 is the time constant for the low injection condition. It is clear that the excess minority carrier density (electrons) in a p-type semiconductor decreases exponentially with the time constant τn0 and this is known as the excess minority carrier lifetime for electrons and is equal to: τn0 =

1 αr p0

(3.64)

The recombination rate of the minority electrons can now be found from Eq. 3.62 as:

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3 Charge Transport in Solids

Rn = −

d(δn(t)) δn(t) = +αr p0 δn(t) = dt τn0

(3.65)

In the band-to-band recombination, the recombination rate for the majority holes will be equal to that of the minority electron recombination rate: Rn = R p =

δn(t) τn0

(3.66)

Similarly, if we now consider the case when the semiconductor is n-type where the majority carriers are electrons and minority carriers are holes (n0 >> p0 ) then in the low injection limit (δp(t) n 0 ), the excess hole lifetime is: τ p0 =

1 αr n 0

(3.67)

The majority carrier (electron) recombination rate is equal to the minority carrier (hole) recombination rate. Therefore: Rn = R p =

δn(t) τ p0

(3.68)

Once the generation and recombination rates of excess carriers have been established, then it is possible to investigate the time and position dependence of such excess carriers in the presence of an electric field and in the presence of a finite carrier density gradient. These are known as the continuity and ambipolar transport equations.

3.3.3 Continuity Equations In a semiconductor there are four mechanisms that change the carrier density at a given position. These are generation, recombination, drift under an electric field and diffusion. In the previous sections, the time dependence of the carrier density as a result of generation and recombination were described in detail. We shall consider the influence of the last two mechanisms, namely drift and diffusion on the position and time dependences of carrier densities. We will study these concepts under the heading of continuity equations. With reference to Fig. 3.16, let us assume that holes enter into a differential volume of dV = dxdydz at x and leave the volume at x + dx. The flux of holes in one dimension (number of holes flowing through a unit area perpendicular to the x direction, per unit time) in one dimension is φ + px where:

3.3 Non-equilibrium Carriers in Semiconductors

105

Fig. 3.16 One dimensional flux of holes enclosed within a differential volume of (dxdydz)

+ φ+ px (x + d x) = φ px (x) +

∂φ + px ∂x

dx

(3.69)

The net change in the number of holes within the differential volume dV is: ∂φ +

∂p px + (x) − φ (x + d x) dydz = − d xd yd x = φ + d xd yd x px px ∂t ∂x

(3.70)

+ If φ + px (x) > φ px (x + dx), then the number of holes within the differential volume will increase. We can now re-write Eq. 3.70 for the three dimensional case, that is when the flux is all three directions. The right hand side of Eq. 3.70 can be replaced by + −∇.φ + p d xd ydz, where −∇.φ p is the divergence of the flux vector. Let us now assume that we have both generation and recombination of holes within the differential volume, dxdydz. The net change in the number of holes within the differential volume per unit time becomes:

∂φ + p ∂p px d xd yd x = − d xd yd x + G p d xd ydz − d xd ydz ∂t ∂x τ pt

(3.71)

where p is the hole density. The first term on the right hand side of Eq. 3.71 represents the change in the number of holes per unit time as a result of flux. The second term is the increase in the hole number per unit time as a result of generation and the third term is the decrease in the number of holes per unit time as a result of recombination. The recombination rate for the holes is p/τ pt where τ pt includes both the carrier lifetime in thermal equilibrium and the excess carrier lifetime. We can find the net increase in the number of holes per unit time per unit volume by dividing both sides of Eq. 3.71 by dxdydz: ∂φ + p ∂p px =− + Gp − ∂t ∂x τ pt

(3.72)

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3 Charge Transport in Solids

Equation 3.72 is known as the continuity equation for holes. φ + px is the number of holes per unit area and is known as the particle flux or particle flow for holes. Similarly, the continuity equation for electrons can be given as: ∂n n ∂φ − = − nx + G n − ∂t ∂x τnt

(3.73)

− is the number of electrons per unit area per unit time and is known as Here φnx particle flux or particle flow for electrons. In the preceding sections, the diffusion and drift currents in one dimension were obtained as:

J p = eμ p pE − eD p

∂p ∂x

(3.74)

Jn = eμn n E + eDn

∂n ∂x

(3.75)

Now, if we divide the hole density by (+e) and the electron density by (−e), we obtain the flux for each type of carriers: Jp ∂p = φ+ p = μ p pE − D p e ∂x

(3.76)

Jn ∂n = φn− = −μn n E − eDn (−e) ∂x

(3.77)

The first derivative of Eqs. 3.76 and 3.77 can be inserted into the continuity equations (Eqs. 3.72 and 3.73) to express them in the following form: ∂p ∂( pE) ∂2 p p = −μ p + Dp 2 + G p − ∂t ∂x ∂x τ pt

(3.78)

∂(n E) ∂ 2n ∂n n = μn + Dn 2 + G n − ∂t ∂x ∂x τnt

(3.79)

The terms on the right hand side of the equations can be expanded to include the partial derivatives of carrier densities and electric field: ∂p ∂E ∂( pE) =E +p ∂x ∂x ∂x

(3.80)

∂n ∂E ∂(n E) =E +n ∂x ∂x ∂x

(3.81)

When Eqs. 3.80–3.81 are inserted into Eqs. 3.78–3.79 we obtain:

3.3 Non-equilibrium Carriers in Semiconductors

  ∂p ∂E ∂p ∂2 p p = Dp 2 − μp E +p + Gp − ∂t ∂x ∂x ∂x τ pt   ∂n ∂ 2n ∂n n ∂E + Gn − = Dn 2 + μn E +n ∂t ∂x ∂x ∂x τnt

107

(3.82)

(3.83)

Equations 3.82 and 3.83 are the time-dependent continuity equations for holes and electrons, respectively. The electron and hole densities in the equations contain, naturally, the excess carrier densities. Therefore, the equations also represent the time dependence of the excess carriers. The carriers densities in thermal equilibrium are obviously time independent, and if the semiconductor is uniformly doped, as we assume here, n0 and p0 are also independent of position within the semiconductor. The Eqs. 3.82 and 3.83 can be expressed to represent the time and position dependence of the excess carriers:   ∂δp ∂(δp) ∂E ∂ 2 (δp) p E = Dp + p + Gp − − μ (3.84) p ∂t ∂x2 ∂x ∂x τ pt   ∂E ∂δn ∂ 2 (δn) n ∂(δn) = Dn +n + μn E (3.85) + Gn − 2 ∂t ∂x ∂x ∂x τnt

3.3.4 Ambipolar Transport In the derivation of Eqs. 3.74 and 3.75, the electric field term is assumed to be an externally applied field. The electric field term that appears in Eqs. 3.84 and 3.85 needs to be explored much more closely. Let us start by assuming that a pulsed electric field is applied locally to a specific point within the semiconductor to create excess carriers at that point. These ‘created’ electrons and holes will move in opposite directions; the holes in the same direction as the external field and the electrons in the opposite direction. However, because the electron hole pairs move in opposite directions so an internal field develops between them. This internal field will induce the electron and hole pairs to drift towards each other as depicted in Fig. 3.17. Fig. 3.17 The development of internal electric field when excess electrons and holes are created under the influence of a pulsed external field

108

3 Charge Transport in Solids

The electric field term in Eqs. 3.84 includes both the external electric field and the induced internal electric field. E = E applied + E internal

(3.86)

The existence of the internal electric field induces an attractive force for the excess electron and hole pairs keeping them together. We can therefore, define an effective mobility for the drift and an effective diffusion coefficient for the diffusion of the excess electron–hole pairs together. In summary, the excess electron hole pairs contribute jointly to transport within the semiconductor and this is known as ambipolar transport. As previously stated, Eqs. 3.84 and 3.85 give the time and position dependence of the excess carriers. However, these equations need to contain the induced electric field to accurately represent the ambipolar transport. Poisson’s equation can be used to include the induced, internal electric field: ∇ · E inter nal =

e(δ p − δn ) ∂ E inter nal = ε ∂x

(3.87)

Here ε is the permittivity of the semiconductor. We may make some assumptions to solve Eqs. 3.84, 3.85 and 3.87 as follows: (i)

(ii)

In order for the excess electrons and holes to stay together there does no need to be excessively high electric fields. It may be assumed, therefore, the induced, internal; electric field is negligibly small compared to the external field. However, we cannot neglect its gradient, the ∇ · E inter nal term. The charge neutrality condition can be used so that the excess electron and excess hole densities at any point in the semiconductor are equal.

In reality the second of these simplifying assumptions is not strictly true because, if the excess electron and hole densities are equal at any point in space then there cannot be an induced electric field. However, a very small difference in their densities will be sufficient for diffusion and to induce an internal electric field. This is necessary to keep the excess carrier pairs together to maintain the ambipolar drift. So we can justify this assumption for the sake of simplification. We first use Eqs. 3.84 and 3.85 to eliminate the ∇ · E term. We have already established previously that the generation and recombination rates for the excess carriers are: G p = Gn = G Rn =

n p = Rp = =R τnt τ pt

From the charge neutrality we know that δn ≈ δp. Therefore we can re-arrange Eqs. 3.84 and 3.85:

3.3 Non-equilibrium Carriers in Semiconductors

  ∂(δp) ∂E ∂δp ∂ 2 (δp) = Dp +p +G−R − μp E ∂t ∂x2 ∂x ∂x   ∂δn ∂(δn) ∂ 2 (δn) ∂E +G−R E + μ = Dn + n n ∂t ∂x2 ∂x ∂x

109

(3.88)

(3.89)

We now take the product of Eq. 3.88 with μn n and Eq. 3.89 with μn n and take the sum of the two products to obtain: ∂ 2 (δn) ∂(δn) + (μn μ p )( p − n)E 2 ∂x ∂x ∂(δn) + (μn n + μ p p)(G − R) = (μn n + μ p p) ∂t

(μn n D p + μ p p Dn )

(3.90)

If we now divide Eq. 3.90 by (μn n + μ p p): D

∂(δn) ∂ 2 (δn) ∂(δn) +G−R = + μ E ∂x2 ∂x ∂t

(3.91)

Here D  is the ambipolar diffusion coefficient and µ is the ambipolar mobility and they are given as: D =

μn n D p + μ p p Dn μn n + μ p p

(3.92)

μn μ p ( p − n) μn n + μ p p

(3.93)

μ =

Equation 3.91 is known as the ambipolar transport equation and describes the time and position dependence of non-equilibrium carrier density. We can now use   μp μn e Einstein’s relations D p = D p = k B T in Eq. 3.92 to obtain ambipolar diffusion coefficient: D =

Dn D p (n + p) Dn n + D p p

(3.94)

It is clear that the ambipolar diffusion coefficient is a function of ambipolar mobility and carrier densities. Here carrier densities include the sum of equilibrium and non-equilibrium carrier densities. Therefore, the ambipolar diffusion coefficient is not constant and when the non-equilibrium carrier density is high, Eq. 3.94 is not a linear equation. At low injection levels we can simplify and linearise Eq. 3.94 as follows: D =

Dn D p [(n 0 + δn) + ( p0 + δp)] Dn (n 0 + δn) + D p ( p0 + δp)

(3.95)

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3 Charge Transport in Solids

Therefore, we can determine the excess carrier effective diffusion coefficient end effective mobility. If we consider a p-type semiconductor (p0 >> n0 ) then for the low injection levels, that is when the excess carrier density is smaller than the thermal equilibrium carrier density (δn 0 then determine the excess carrier density as a function of time. Since the generated excess carrier density is uniform, external electric field is zero and the semiconductor is homogenous, we can write: ∂(δp) ∂ 2 (δp) =0 = 0 and ∂x2 ∂x From Eq. 3.106: G −

δp d(δp) = τ p0 dt

(3.110)

The solution of the differential equation: δp(t) = G  τ p0 (1 − e−t

 τ p0

)

(3.111)

Example In a homogenous p-type semiconductor, excess carriers are generated at x = 0 and diffuse in both x and –x directions. Work out the position dependence of the excess carrier density in the steady state. The excess minority carriers are electrons, therefore, Eq. 3.105 will be used to solve this problem. There is no applied electric field and the excess carrier generation is also zero for x = 0. = 0. In the steady state, ∂(δn) ∂t From Eq. 3.105: Dn

∂ 2 (δn) δn − =0 2 ∂x τn0

d 2 (∂n) ∂n d 2 (∂n) ∂n − = − 2 =0 dx2 Dn τn0 dx2 Ln

(3.112)

(3.113)

L n is the diffusion length of the minority electrons and expressed as Ln =



Dn τn0

(3.114)

D p τ p0

(3.115)

The hole diffusion length is: Lp =



114

3 Charge Transport in Solids

Fig. 3.18 Electrons are injected at the position of x = 0 of a semiconductor. The excess carriers generated at x = 0 position decays exponentially to its equilibrium value

The solution of Eq. 3.113 leads to: δn(x) = Ae−x/L n + Be x/L n

(3.116)

As the excess minority electrons diffuse away from x = 0, they recombine with the majority holes and their density will decrease exponentially with position as shown in Fig. 3.18. If we use the boundary conditions: At x = + ∞, δn = 0 and B = 0 and at x = −∞, δn = 0 and A = 0. Therefore the solution is: δn(x) = δn(0)e−x/L n

x ≥0

(3.117)

δn(x) = δn(0)e+x/L n

x ≤0

(3.118)

and

Example In an n-type semiconductor, electron and hole pairs are photo-generated at t = 0 and in the middle of the semiconductor at position x = 0. The photo-generation is terminated at t > 0 and an electric field E 0 is applied along the + x direction. Find the position and the time dependence of the excess carrier density. From Eq. 3.106: Dp

δp ∂ 2 (δp) ∂(δp) ∂(δp) − + μ p E0 = 2 ∂x ∂x τ p0 ∂t

(3.119)

The solution of this partial differential equation is: δp(x) = p  (x, t)e−t/τ p0

(3.120)

We can now substitute Eq. 3.120 into 3.119 and use Laplace transformation to obtain:   −(x − μ p E 0 t)2 1 (3.121) exp p  (x, t) = (4π D p t)1/2 4D p t

3.3 Non-equilibrium Carriers in Semiconductors

115

(a)

(b)

Fig. 3.19 a The position and time dependence of the excess holes (t 1 > t 2 > t 3 ) and b the change of excess hole density in presence of electric field

From Eqs. 3.119 and 3.120, we can find the position and the time dependence of the excess carrier density:    −(x − μ p E 0 t)2 e−t τ p0 exp δp(x, t) = (4π D p t)1/2 4D p t

(3.122)

The change of density of generated excess carriers is shown in Fig. 3.19 under zero electric field and applied electric field E.

3.3.5 Photoconductivity Photoconductivity in a semiconductor can be described simply as the increase in the conductivity when photons with energies greater than the bandgap are incident on the semiconductor. The absorption of photons create electron and hole pairs and when a constant external bias is applied to the semiconductor and a finite current, photocurrent, flows through the external circuit. The continuity equations can easily be applied to study photoconductivity in semiconductors. Upon the illumination of semiconductor with photons with appropriate energies, the change with time in the carrier density in the semiconductor is: 1 − ∂n → = G n − Rn + ∇. J n ∂t e ∂p 1 − → = G p − R p − ∇. J p ∂t e

(3.123)

Here the recombination rates are: Rn =

δn τn

Rp =

δp τp

(3.124a)

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3 Charge Transport in Solids

If the generation of the excess carriers are uniform throughout the semiconductor, that is when the whole semiconductor is illuminated uniformly, then the generation rates can be written as Gn = Gp = G and the carrier density becomes: p = p0 + δp n = n 0 + δn Since the applied bias is constant (time independent) and there is no diffusion as − → − → the excess carrier distribution is uniform the current density terms ∇. J n and ∇. J p in Eq. 3.123 are zero. Also, because illumination is uniform and its intensity is time independent, the generation rates, Gn = Gp = G will also be independent of time. Equations 3.124a and 3.124b then becomes: δn ∂n =G− =0 ∂t τn δn = Gτn

(3.124b)

From the charge neutrality condition: δn = δp = Gτn

(3.125)

Therefore, in thermal equilibrium, the total conductivity: σ0 = e(n 0 μe + p0 μh )

(3.126a)

Here the subscript “0” represents the thermal equilibrium condition. The contribution to the total conductivity of the photo-excited carriers is: δσ = e(δnμe + δpμ p )

(3.126b)

Here is has been assumed that the mobility of carriers is unchanged, upon illumination, from the dark values. Using Eq. 3.125 together with Eq. 3.126b: δσ = eGτn (μe + μ p )

(3.127)

The photocurrent density is: J = eE(nμn + pμ p ) = δσ E = δσ

V l

(3.128)

Here V is the applied voltage and l is the length of the semiconductor. The photocurrent is then:

3.4 Quasi-Fermi Levels

117

I = J A = δσ

A A V = e(μn + μ p )Gτn V l l

(3.129)

where A (= wd) is the cross-sectional area of the semiconductor. If the mobility of one type of carrier is higher than the other, photocurrent expression for one type of carrier can be further simplified as: I = eμn Gτn

A V l

(3.130)

3.4 Quasi-Fermi Levels The thermal equilibrium condition in a semiconductor is disturbed whenever excess electrons or holes are injected into the semiconductor by external stimuli, like photoexcitation during the photoconductivity process. In thermal equilibrium, the distribution of electrons and holes in a semiconductor is determined by the Fermi–Dirac distribution function. We shall now try to understand how the distribution function is modified under non-equilibrium conditions. In Sect. 3.2, we derived the expressions for the thermal equilibrium electron and hole densities as:     E F − EC E F − E Fi = n i exp (3.131) n 0 = NC exp kB T kB T     Ev − E F E Fi − E F p0 = N V exp = n i exp (3.132) kB T kB T Here E F is the Fermi level in the doped semiconductor and E Fi is the Fermi level in the intrinsic semiconductor. In an n-type semiconductor, E F > E Fi and n0 > ni and p0 < ni . Under non-equilibrium conditions, when excess electrons, δn and holes, δp are generated there will be quasi-Fermi levels to represent their distribution. Therefore, for the excess carrier densities, we can modify the equations for thermal equilibrium in (3.132) as:  E Fn − E Fi n 0 + δn = n i exp kB T   E Fi − E F p p0 + δp = n i exp kB T 

(3.133) (3.134)

Here E Fn and E Fp are the quasi-Fermi levels for excess electrons and holes, respectively.

118

3 Charge Transport in Solids

3.5 Excess Carrier Lifetime So far transport in an ideal semiconductor, where there are no imperfections, has been considered and along with the assumption that the recombination of excess carriers occurs only across the bandgap. In a real semiconductor, however, there are always imperfections, dislocations, mismatch, alloy fluctuations and unwanted impurities which disturb the perfect periodicity of the ideal semiconductor. These imperfections give rise to localised energy levels within the forbidden bandgap which may act as recombination centres and are also known as trap levels. A photo-excited electron in the conduction band may, for example, fall into one of these traps instead of recombining with a hole in the valance band. Obviously for an electron to be trapped, rather than to recombine with a hole in the valance band, the trapping probability must be higher than the recombination probability. We classify trap levels as shallow and deep traps. Shallow traps are usually only a few meV below or above the conduction or valence bandedges, respectively. The trapped carriers at the shallow levels can easily become delocalised even at very low temperatures. Once the electron is trapped in a deep level, however, it becomes localised and can no longer contribute to the conductivity. As previously discussed, in an ideal semiconductor the band–to-band recombination rate of excess carriers is inversely proportional to the excess carrier lifetime. In a real semiconductor, the presence of the bandgap states introduces a finite probability for the excess carrier to fall into these traps and therefore, reduces the excess carrier life time. We will now derive the appropriate expressions for the excess carrier lifetime and band-to band recombination rates in a real semiconductor using the Shockley–Read–Hall theory.

3.5.1 Shockley–Read–Hall Recombination Theory A localised energy level within the bandgap can be considered as a recombination centre with equal probability of capturing electrons and holes. In other words, the capture cross-sections of electrons and holes by the recombination centre are equal. In the Shockley–Read–Hall recombination theory it is assumed that there is a recombination centre at an energy E T within the bandgap. If this centre is due to an acceptor type imperfection then it is negatively charged when it has captured an electron and neutral when it has not otherwise. Let us now look at the possible transitions to and from the recombination centre with the aid of Fig. 3.20. a. b.

Capture of an electron from the conduction band by the initially neutral centre (Fig. 3.20a). Transition of the trapped electron into the conduction band from the negatively charged centre (Fig. 3.20b).

3.5 Excess Carrier Lifetime

119

Fig. 3.20 Probable generation and recombination processes involving trap levels

c.

d.

Capture of a hole from the valance band by the negatively charged centre (This process may also be regarded as the transition of a captured electron into the valance band) (Fig. 3.20c). Transition of a hole from a neutral trap into the valance band (This process may also be regarded as the capture of an electron from the valance band) (Fig. 3.20d).

The trapping probability of a free electron in the conduction band to be captured by a trap is proportional to both the free electron density and the trap density. The electron capture rate is then: Rcn = cn N T (1 − f (E T ))n

(3.135)

Here Rcn is the capture rate (1/cm3 s), cn is a constant related to the electron capture cross-section. N T is the density of traps, n is the free electron density and f (E T ) is the probability that the trap level at E T is occupied. The Fermi–Dirac distribution function giving the probability of the trap level to be occupied is: f (E T ) =

1   F 1 + exp E Tk −E BT

(3.136)

The (1 − f (E T ) will be probability of the trap level to be empty. The emission rate Ren (1/cm3 s) for an electron from the trap into the conduction band will depend on the probability of the trap level being full: Ren = en N T f (E T )

(3.137)

In thermal equilibrium, capture and emission rates must be equal: Rcn = Ren

(3.138)

120

3 Charge Transport in Solids

cn N T (1 − f 0 (E T ))n 0 = en N T f 0 (E T )

(3.139)

We can replace Fermi–Dirac distribution function with Maxwell–Boltzmann distribution function to obtain en as: en = n  cn

(3.140)

  (E c − E T ) n  = Nc exp − kB T

(3.141)

Here n  is:

n  is the electron density when the Fermi and trap levels coincide. At non-equilibrium in the presence of excess carriers, the capture rate of an electron from the conduction band and its emission rate from the trap will not be the same the net capture rate: Rn = Rcn − Ren

(3.142)

Using Eqs. 3.135–3.137 and 3.142, the net capture rate becomes: Rn = [cn N T (1 − f (E T ))n] − [en N T f (E T )]

(3.143)

Here n is the sum of the equilibrium and excess electron densities and the Fermi– Dirac distribution function is replaced with the quasi-Fermi–Dirac distribution. We can use Eq. 3.140 to simplify Eq. 3.143:

Rn = cn N T n(1 − f (E T )) − n  f (E T )

(3.144)

We can now use a similar approach to evaluate the capture and emission rates for holes to obtain the net capture rate as:

R p = c p N T p f (E T ) − p  (1 − f (E T ))

(3.145)

  (E T − E v ) p = Nv exp − kB T

(3.146)

where p  is: 

If the trap density is not excessively high, then the net capture rates of electrons (Eq. 3.144) and holes (Eq. 3.145) is: f (E T ) =

cn n + c p p  cn (n + n  ) + c p ( p + p  )

(3.147)

3.5 Excess Carrier Lifetime

121

We can now use the law of mass action n  p  = n i2 to obtain the electron and hole capture rates in a simple form: Rp = Rp =

cn c p N T (np − n i2 ) ≡R cn (n + n  ) + c p ( p + p  )

(3.148)

This expression is easily reduced to thermal equilibrium conditions by taking np = n0 p0 = n i2 so that Rp = Rn = 0 and: R=

δn τ

(3.149)

3.5.2 Low Injection Limit Equation 3.148 can be further simplified for the low injection limit, for example, in n-type semiconductor in the low injection limit when n0 >> p0 , n0 >> δp, n0 >> n and n0 >> p : R = c p N T δp

(3.150)

It is clear that, in n-type semiconductor, the recombination rate of excess carriers depends on the coefficient cp , which in turn depends on the capture cross section of holes. Consequently, the recombination rate is a function of minority carrier parameters. The average carrier lifetime in the low injection limit can be found by combining Eqs. 3.149 and 3.150 and written as: R=

δp δn = c p N T δp = τ τ p0

(3.151)

Here τ p0 is the lifetime of excess minority carriers. τ p0 =

1 c p NT

(3.152)

This implies that the excess minority carrier life time decreases with increasing trap density as expected. Conversely, in a p-type semiconductor, at the low injection limit, lifetime of electrons is expressed as τn0 =

1 cn N T

(3.153)

122

3 Charge Transport in Solids

Problems 1.

In an intrincis semiconductor, at 300 K, electron mobility is 3000 cm2 /Vs, hole mobility is 800 cm2 /Vs, intrincis carrier density is 1.5 × 1012 cm−3 , effective electon mass is 0.05m0 and effective hole mass is 0.5m0 . The lenght of this semiconductor is 0.1 mm and 1 V is applied to this semiconduct or. (a) (b) (c)

2. 3. 4. 5.

6.

Electron are injected into the p-type GaAs. If the minority carrier mobility is 4000 cm2 /Vs, calculate diffusion constant of electrons. In a semiconductor, the ratio between diffusion constants of electrons and holes is given as Dp = Dn /5. Using Einstein’s relation calculate μn /μp . In a semiconductor, μn /μp = 9. (a) Using Einstein’s relation, calculate Dp /Dn . (b) Explain the reason of the difference in carrier mobilities. In a p-type Si, lifetime and mobility of electrons are 10 μs and μn = 1000 cm2 /Vs, respectively. Calculate diffusion constant and diffusion lenght of electrons. In a semiconductor, free electron density is n and external electric field applied to the semiconductor is E. An excess carrir density gradient (dn/dx) is generated as an illumination of the semiconductor. Which equation express the current density in this semiconductor? (a) (b) (c) (d)

7.

What are the scattering times of electrons and holes? Calculate drift velocity of electron and holes. What is the contribution ratio of electrons and holes to the electrical conductivity?

J J J J

= n E − eDn ddnx Dn 1 dn = μn n dx = neμn − Dn ddnx = neμn E − eDn ddnx

In a p-type silicon with x 0 = 0.5  in lenght, at 300 K, acceptor density  μm is given as N A (x) = N A 0 exp − xx0 . The hole diffusion constant and hole mobility are Dp = 10 cm2 s−1 and μp = 400 cm2 /Vs, respectively. (a) (b)

8.

9.

Find an expression for built-in electric field and discuss if it depends on the position. Calculate electric field and hole current density.

In an n-type semiconductor with l = 20 mm, w = 4 mm and d = 1 mm, donor densities is 1017 cm−3 . When a current of 100 mA is passed through + x direction, and 0.1 T magnetic field is applied at + y direction, find direction of Hall electric field and calculate Hall voltage. In n-type GaAs, at 300 K, electron density changes linearly from 5 × 1018 to 5 × 1017 cm−3 along 0.05 cm in lenght. If Dn = 25 cm2 /s, calculate diffusion current density.

3.5 Excess Carrier Lifetime

10.

11.

123

An n-type Ge doped with N D = 1017 cm−3 is illuminated with photons having higher energy than the bandgap of Ge. If the generation rate is constant and uniform with a value of 1012 cm−3 s−1 , for τn = τ p = 2 ms, calculate excess carier densities. When the light source is turned off at t = t 0 , determine the expression of hole density as a function of and plot the excess minority carier density as function of time. Consider generation rate of electrons and holes in a Si sample is 1019 cm−3 s−1 . If thermal-equilibrium electron density (n) is 1014 cm−3 and τn = τp = 2 μs, ni = 1.5 × 1014 cm−3 ; (a) calculate excess electrons or holes density and (b) quasi-Fermi levels of electrons and holes.

References 1. Stillman G et al (1970) Hall coefficient factor for polar mode scattering in n-type GaAs. J Phys Chem Solids 31:1199–1204 2. Chow T et al (1996) SiC power devices in III-Nitride, SiC, and diamond materials for electronic devices. Mater Res Soc Symp Proc Pittsburgh 423:69–73 3. Kao YC et al (1983) Electron and hole carrier mobilities for Liquid Phase Epitaxially Grown GaP in the temperature range 200–550K. J Appl Phys 54:2468–2471 4. Razeghi M (1988) Very high purity InP epilayer grown by metalorganic chemical vapour deposition. Appl Phys Lett. 52:117–119 5. Rode DL (1975) Semiconductors and semimetals. Academic Press, New York, USA 6. Vurgaftman I et al (2001) Band parameters for III–V compound semiconductors and their alloys 89:11

Suggested Reading List 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Smith RA (ed) (1978) Semiconductors, 2ndedn. Cambridge University Press Mishra UK, Singh J (2003) Semiconductor device physics and design. Springer Neaman DA (2003) Semiconductor physics and devices. McGraw-Hill Schubert EF Physical foundation of solid-state devices. http://homepages.rpi.edu/~schubert/ Balkan N (ed) (1998) Hot electrons in semiconductors physics and devices. Clarendon Press, Oxford Gunn JB (1964) Instabilities of current in III-V semiconductors. IBM J Res 8:141–159 Ridley BK (1963) Specific negative resistance in solids. Proc Phys Soc 82:954–966 Ridley BK, Watkins TB (1961) The possibility of negative resistance effects in semiconductors. Proc Phys Soc 78:294–304 Patane A, Balkan N (ed) (2012) Chapter 12, A. Erol and M.Ç. Arikan, Semiconductor research experimental techniques. Springer Bube RH (1960) Photoconductivity of solids. Wiley

Chapter 4

The p-n Junction Diode

Learning Outcomes The reader on the completion of this chapter will: 1. 2.

3.

know how a p-n junction is formed and be able to derive the equations for the built-in potential, built-in electric field and the depletion width, have understood the forward and reverse biasing of a p-n junction and be able to produce the forward and reverse current–voltage characteristics of a p-n junction, have detailed knowledge of junction capacitance and the temperature dependence of diode operation.

Diodes are the backbone of today’s electronic and optoelectronics technology with a wide range of applications in amplifier circuits, digital electronics, solar cells, photodetectors, light emitting diodes and semiconductor lasers. A diode is a two terminal semiconductor device. The ideal diode would be a device exhibiting perfect rectifying action. To clarify, it would conduct in the forward direction without any potential drop across its terminals, while completely blocking conduction in the reverse direction. A real p-n junction diode, on the other hand, has the typical characteristics shown in Fig. 4.1a. For forward bias conduction, within the rated range, they have a forward voltage drop of approximately 0.3 V and 0.7 V for Ge and Si diodes, respectively. When biased in the reverse direction they allow a very small leakage current to flow, whose magnitude is approximately independent of the reverse bias voltage, V r , until V r reaches some threshold value V B , the breakdown voltage where the diode is said to breakdown. It can then carry relatively large reverse currents for very little increase in V r beyond the threshold value V B . The circuit symbol for a diode is shown in Fig. 4.1b. A diode is said to be forward biased when the external voltage is applied with the polarity depicted in the figure. A p-n junction diode is shown in Fig. 4.2. Basically it consists of a slice of semiconductor which is partly n-type material with the remainder p-type material. The transition from n-side to p-side region is assumed to abrupt. Ohmic contacts are made to the two ends of the semiconductor slice and these lead to the device terminals. We © Springer Nature Switzerland AG 2021 N. Balkan and A. Erol, Semiconductors for Optoelectronics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-44936-4_4

125

126

4 The p-n Junction Diode

Fig. 4.1 a Current–voltage characteristic of a diode and b Circuit symbol of diode

Fig. 4.2 Contact configuration and the circuit symbol of a diode

may think of the ohmic contact as simply the interface between the device and its outside terminals which takes no other action in the device operation. There are a number of fabrication techniques available for the formation of pn junctions including epitaxial growth, ion implantation and diffusion. These p-n junctions can be formed between the n and p-type same semiconductor or with nand p-type two semiconductors with different bandgaps. In this chapter we shall describe the basis of operation for single material p-n junctions, which are called homojunctions.

4.1 p-n Junction in Equilibrium In order to describe the operation of a p-n junction, we use the following two important concepts: (i) (ii)

The distribution of electrons and holes. Electron and hole currents due to diffusion and drift processes.

We have already shown in detail, in Chap. 2, that electrons and holes are the majority carriers in n-type and p-type semiconductors, respectively. Figure 4.3a shows the energy bandprofiles for both types of semiconductors. Here electron affinity (eχ ) is defined as the energy difference between the conduction bandedge

4.1 p-n Junction in Equilibrium

127

Fig. 4.3 Formation of a p-n junction a energy bandprofiles of the p- and n-type semiconductor before p-n junction formation. Here eχ is the electron affinity, eφsp and eφsn are the work functions and E Fp and E Fn are the Fermi levels. b Bandprofile of the p-n junction in equilibrium. W n is the depletion width on the n-side and W p is the depletion width on the p-side. I s and I d are the drift and diffusion currents, respectively. V bi is the built-in potential and E is the built-in electric field

and the vacuum level. The work function (φ sp and φ sn ) is the energy required for an electron to reach the vacuum level and leave the semiconductor crystal, and is equal to the difference between the Fermi energy and vacuum level. Although we cannot form a p-n junction by bringing a piece of p-type and n-type material together, an instructive way to describe the operation of a p-n junction is to visualise what would happen if we could form a junction in this way, as shown in

128

4 The p-n Junction Diode

Fig. 4.3b. The majority holes and minority electrons in the p-type semiconductor have densities pp and np , respectively. In the n-type material, nn and pn are the densities of the majority and minority electrons and holes, respectively. When the junction is formed the difference between the electron and hole densities at the junction (nn np for electrons and pp -np for holes) causes diffusion currents to flow across the junction. Electrons diffuses from the n side to p side and holes are injected in the opposite direction. When an electron crosses the junction, it leaves behind positively charged ionised donor atom so that the n-side of the junction is no longer electrically neutral but has a net positive charge of +e. Upon arriving at the p-side, this free electron recombines with one free hole, which results in one negatively charged ionised acceptor atom. The ionised acceptor and ionised donor atoms are part of the crystal lattice, therefore they are localized and separated from each other in space. These two type charges will give rise to an electric field and an electrostatic potential difference between them. The next electron to cross the junction will have to overcome this potential barrier in moving from the n-side to the p-side and when it has recombined with a mobile hole on the p-side the height of the potential barrier will have increased since the fixed net charges on the n- and p- sides giving rise to the barrier are now + 2e and −2e, respectively. As the potential barrier increases the minority electrons on the p-side and the minority holes on the n-side will drift under the influence of the built-in electric field. This drift current occurs in the opposite direction to the diffusion current, to their majority regions, therefore minority electrons go from p-side to the n-side and the minority holes move from the n-side to p-side. In equilibrium, drift and diffusion currents will balance each other out so Fermi energies align on both sides of the junction as shown in Fig. 4.3b. The p-n junction in equilibrium in Fig. 4.3b can be studied by dividing it into three regions: 1.

2.

3.

Neutral p-type region: This region is away from the junction, where the electrostatic potential is flat, Fermi energy lies close to the valence band and the majority carriers are holes and their density is equal to the acceptor doping density, pp = N A . The minority electron density, np , is obtained from the law of mass action: n n pn = n i2 Neutral n-type region: This region is away from the junction, where the electrostatic potential is flat, Fermi energy lies close to the conduction band and the majority carriers are electrons and their density is equal to the donor doping density, nn = N D . The minority hole density, pn , is obtained from the law of mass action: n n pn = n i2 (Fig. 4.4) Junction region: On both sides of the junction within close proximity of the junction plane, the electrostatic potential has a quadratic dependence on position. There exists a built-in electric filed which has a maximum value on the junction and reduces linearly with position away from the junction and becomes zero when the potential is flat in the n-type and p-type regions. There is no free charge as any free charge in this region will be depleted under the influence of the built-in electric field. This region is known as the depletion region or space

4.1 p-n Junction in Equilibrium

129

Fig. 4.4 The distribution of dopant atoms in thermal equilibrium. a In the depletion approximation (abrupt junction) b For a real distribution

charge region or intrinsic region. The only charges in the depletion region are the ionised donors on the n-side of the junction plane within a length of W n , and the ionised acceptors on the p-side of the junction plane within a length of W p . As we described previously the ionised atoms are not mobile as they are localized within the crystal and are called as space charge. We should, therefore, expect the resistance of the depletion region to be very high compared to the rest of the junction. The charge neutrality condition requires: AWn N D = AW p N A where A is the cross-sectional area of the diode.

4.1.1 Built-In Potential Figure 4.5 shows the formation of the p-n junction until equilibrium is established. At equilibrium the total hole and electron current flows across the junction plane will be zero. This implies that the drift and diffusion components of the current will

130

4 The p-n Junction Diode

Fig. 4.5 Schematic representation of the formation of a p-n junction a at t = 0, the majority carriers diffuse into the opposite regions of the junction. b At t = t 1 , the depletion region and the potential barrier start forming. The diffusion of the majority carriers continues. Minority carriers in each region drift in the opposite direction to the diffusion. c At t = t 2 > t 1 the potential barrier and the depletion region are fully formed. The diffusion of majority carriers is completely balanced by the drift of the minority carriers and the total current is zero, which means that the p-n junction is in thermal equilibrium

balance each other out. Therefore, for the hole current:   dp(x) =0 J p (x) = e μ p p(x)E(x) − D p dx

(4.1a)

or drift current is equal to diffusion current: μ p p(x)E(x) = D p

dp(x) dx

(4.1b)

4.1 p-n Junction in Equilibrium

131

where E is the built-in electric field and can be high enough for the mobile carriers to reach saturation velocities (vs ), refer to Sect. 3.1.5. Therefore the drift current in the depletion region can be expressed as ep(x)vs (x), which is independent of the electric field. We can now re-write Eq. 4.1b as: μp 1 dp(x) E(x) = Dp p(x) d x

(4.2)

From Einstein’s relations (Sect. 3.2.3): μp e = Dp kB T

(4.3)

Electric field in terms of potential is E = −d V /d x This can be used in Eq. 4.2 to obtain: −

1 dp(x) e d V (x) = kB T d x p(x) d x

(4.4)

The constant potential values on the p and n-sides of the junction area are V p and V n , respectively. When we integrate Eq. 4.4 from V p to V n : e − kB T

Vn

pn dV =

dp p

(4.5)

pp

Vp

The second integration gives: −

e pn (Vn − V p ) = ln kB T pp

(4.6)

The built-in potential (contact potential) from Fig. 4.6c is defined as Vbi = Vn − V p Using Eq. 4.6, built-in potential is obtained Vbi =

kB T pp ln e pn

Similarly for electrons the built-in potential is:

(4.7a)

132

4 The p-n Junction Diode

(a)

(b) Vn Vbi Vp (c) Fig. 4.6 a structure b energy bandprofile and c potential profile of a p-n junction. V bi is the built-in potential

Vbi =

k B T nn ln e np

(4.7b)

Equations 4.7a and 4.7b can be further simplified using the following expressions and the law of mass action: p p = N A− = N A , n n = N D+ = N D n n pn = n i2 ; p p n p = n i2 k B T p p nn kB T NA ND ln 2 = ln Vbi = e e ni n i2

(4.8)

Build-in potential can also be obtained from the energy bandprofile in Fig. 4.6b and c:

4.1 p-n Junction in Equilibrium

133

eVbi = E g − (E c − E F )n − (E F − E v ) p

(4.9)

Here the indices n and p in the above equation represent the n and p-sides of the diode, respectively and:  NC (E c − E F )n = k B T ln nn   NV (E F − E V ) p = k B T ln pp 

(4.10) (4.11)

We can now substitute Eqs. 4.10 and 4.11 into Eq. 4.9 to obtain the built-in potential: 

nn p p eVbi = E g + k B T ln NC N V





ND NA = E g + k B T ln NC N V

 (4.12)

Here:   pp nn eVbi = = exp pn kB T np

(4.13)

4.1.2 Depletion Layer Width and the Built-In Electric Field We can obtain the width of depletion region by solving the Poisson’s equation together with the charge neutrality condition as we described above. AW p N A = AWn N D

(4.14)

Poisson’s equation: −

e dE d 2 V (x) = [ p(x) − n(x) + N D (x) − N A (x)] = dx2 dx ε

(4.15)

We can divide the p-n junction into four regions to write down Poisson’s equation. (i)

In the neutral p-region: d 2 V (x) = 0 − ∞ < x < −W p dx2 d V (x) = −E(x) = 0 ⇒ V (x) = V p dx

(4.16)

134

(ii)

4 The p-n Junction Diode

Electrostatic potential is constant in this region as expected. On the p-side of the depletion region: Here the charge is due to negatively charged ionised acceptor atoms. eN A d 2 V (x) = 2 dx ε

(iii)

− Wp < x < 0

On the n-side of the depletion region: Here the charge is due to positively ionised donor atoms. eN D d 2 V (x) 0 < x < Wn =− 2 dx ε

(iv)

(4.17)

(4.18)

In the neutral n-region: d 2 V (x) = 0 Wn < x < +∞ dx2 d V (x) = −E(x) = 0 dx

⇒ V (x) = Vn

(4.19)

Electrostatic potential is constant in this region as expected. Now we can calculate the electric field and the potential in the p-side of the depletion region by integrating Eq. 4.17 with the boundary condition that E(−W p ) = 0: E(x) = −

eN A W p eN A x dV =− − dx ε ε

− Wp < x < 0

(4.20)

It is clear that the electric field is maximum at x = 0 and decreases linearly to zero when x = −W p . The integration of Eq. 4.20 with the boundary condition that, V (−W p ) = V p gives the electrostatic potential: V (x) =

eN A W p x eN A x 2 + +C 2ε ε

− Wp < x < 0

(4.21)

Here C is the integral constant and can be obtained from the boundary condition: V (x) =

eN A W p2 eN A W p x eN A x 2 + + + V p −W p < x < 0 2ε ε 2ε

(4.22)

Similarly, we can obtain the electric field and electrostatic potential on the n-side of the depletion region: E(x) =

eN D Wn eN D x − 0 < x < Wn ε ε

(4.23)

4.1 p-n Junction in Equilibrium

V (x) = −

135

eN D Wn x eN D Wn2 eN D x 2 + − + Vn 0 < x < Wn 2ε ε 2ε

(4.24)

We can now calculate the potential difference between x = −W p andx = 0 from Eq. 4.22. V (0) − V (−W p ) =

eN A W p2 2ε

(4.25)

The potential difference between x = W n and x = 0 from Eq. 4.24: V (Wn ) − V (0) =

eN D Wn2 2ε

(4.26)

eN A W p2 eN D Wn2 + 2ε 2ε

(4.27)

Therefore, the built-in potential is: V (Wn ) − V (−W p ) = Vbi =

We can now refer back to the charge neutrality condition to find the width of depletion layer, i.e., we substitute N D Wn = N A W p

(4.28)

into Eq. 4.27 to obtain:  1/2 ND 2εVbi e N A (N A + N D )  1/2  NA 2εVbi Wn (Vbi ) = e N D (N A + N D )    2εVbi N A + N D 1/2 W (Vbi ) = W p (Vbi ) + Wn (Vbi ) = e NA ND 

W p (Vbi ) =

(4.29)

(4.30)

(4.31)

Using Eqs. 4.17, 4.18 and 4.32, we can obtain the expression for the built-in potential in terms of electric field:  V (x) = E m

x2 x− 2W

 (4.32)

The maximum potential: Vbi =

1 1 E m W ≡ E m (xn + x p ) 2 2

(4.33)

136

4 The p-n Junction Diode

Fig. 4.7 a Charge distribution and b electric field in the depletion region. Electric field is the highest at the junction plane

Here W is the total depletion width and for N A >> N D it can be approximated to (Fig. 4.7):  W =

2εVbi eN D

1/2 (4.34)

and if N D > > N A , it can be approximated to:  W =

2εVbi eN A

1/2

4.2 p-n Junction Under an External Electric Field As we have seen previously, in the formation of the p-n junction, currents exist across the junction even at equilibrium, in the form of diffusion and drift currents. However, these two currents balance each other out and the net current becomes zero. When we apply an external voltage, hence an external electric field, this balance is disturbed and a finite current flows through the diode. In this section we are going to investigate the behaviour of p-n junctions in the presence of an applied electric field and to simplify the following assumptions can be made: 1.

We assume that the application of an external field has negligible effect on the equilibrium free carrier densities and thus the diode is not far from equilibrium. When the field is applied the Fermi energy splits into quasi-Fermi levels and is no longer the same on the n and p-sides of the junction, refer to Fig. 4.8c. The

4.2 p-n Junction Under an External Electric Field

137

Fig. 4.8 a Forward and reverse biasing and the changes in the b energy bandprofiles and c quasiFermi levels with biasing conditions

2.

distribution of free carriers are described with Maxwell Boltzmann distribution function. In equilibrium since there are no free carriers in the depletion region, this region will naturally have much higher resistance than the rest of the diode. Therefore most of the applied voltage drops across the depletion region.

In principle, what determines the current flow across the junction, either in the forward or reverse biasing configurations, is the change in the height of the built-in potential barrier by the application of the external bias as shown in Fig. 4.8. Under forward bias of V = V f , the potential difference between the p and n-sides of the junction is: Vtotal = Vbi − V = Vbi − V f

(4.35)

Under reverse bias of, V = −V r, V r > 0, the potential difference between the p and the n-sides is: Vtotal = Vbi − V = Vbi + Vr

(4.36)

138

4 The p-n Junction Diode

With the assumptions listed above, we can re-write the depletion layer widths under forward and reverse bias by replacing V bi in Eqs. 4.31 and 4.33 with V total . It is clear that when the diode is forward biased the built-in potential will be reduced by the same amount as the applied voltage (V bi → V bi −V f ). Due to the high resistance in the depletion region most of the applied voltage drops across this section. Consequently, the built-in potential barrier which curtails the diffusion current in equilibrium is now reduced and the current due to both hole diffusion into the n-side and electron diffusion into the p-side increases. When the diode is reverse biased the built-in potential increases by the same amount as the applied reverse voltage, (V bi → V bi + V r ). The negative electrode of the external voltage source which is connected to the p-type side of the junction causes holes to move away from the vicinity of the depletion region further out towards to the p-type region. Therefore, the depletion region on the p-side of the junction widens. Similarly, the positive electrode which is connected to the n-type side attracts the electrons from the vicinity of the depletion region further out towards the n-type region, causing the depletion region on the n-side of the junction to also widen.

4.2.1 Charge Injection and Current in p-n Junction When diode is forward biased, the potential barrier is reduced and the diffusion injection of carriers across the junction increases. The drift current, however, is not influenced by the reduction of the built-in potential. This is because the built-in electric field within the depletion region is high enough to cause the carriers to drift velocities at, or close to the saturation values. Consequently, with increasing forward bias, the diffusion current exceeds the drift current. Under reverse bias, however, the barrier height increases by as much as the applied reserve voltage, reducing injection of the carriers. Therefore, with increasing reverse bias the drift current exceeds the diffusion current. Figure 4.9 shows the distribution of majority and minority carriers when the diode is forward and reverse biased. We shall now derive the expression for the current flow under both reverse and forward bias. We start off with Eq. 4.7a, where the ratio of the majority and minority holes at –W p and W n , respectively is given in equilibrium: pp = eeVbi /k B T pn

(4.37)

Since it is assumed that under external bias, the equilibrium is little disturbed, we can use Eq. 4.37 to obtain the ratio or the majority holes and minority holes at –W p and W n when a bias voltage V is applied: p(−W p ) pp = ee(Vbi −V )/k B T ∼ = p(Wn ) p(Wn )

(4.38)

4.2 p-n Junction Under an External Electric Field

139

Fig. 4.9 a Majority and minority carrier carriers on the n and p-side of the junction. The injection of minority carriers by diffusion is controlled by the applied bias as depicted in b and c

140

4 The p-n Junction Diode

From Eqs. 4.37 and 4.38: p(Wn ) = eeV /k B T pn

(4.49)

This implies that the minority carrier density at the edge of the depletion region (W n ) can be very high when V is positive (forward bias). Here we assume that there is no electron hole recombination in the depletion region and all the holes injected move into the n region. If V is negative (reverse bias), however, the minority carrier injection will be greatly reduced as we explained above. Similarly, for the density of minority electrons injected by diffusion, at the edge of p-side of the depletion region we can obtain: n p (−W p ) = eeV /k B T np

(4.40)

Here, for forward bias, V is positive and for reverse bias, it is negative as above. The net density of the excess carriers injected across the depletion region upon the application of the bias voltage V, are then: pn = p(Wn ) − pn = pn (eeV /k B T − 1)

(4.41)

n p = n p (−W p ) − n p = n p (eeV /k B T − 1)

(4.42)

Here pn and np are the injected carrier densities in thermal equilibrium. The excess minority carriers will decay via recombination with the majority carriers within their respective diffusion lengths (L p for minority holes and L e for minority electrons). Diffusion length is the distance within which the minority carriers can travel without recombining. Therefore, using the derivations presented in Sect. 3.3, the carrier densities of the minority carriers outside the depletion region are:

δp(x) = pn (e−(x−Wn )/L p ) = pn eeV /k B T − 1 e−(x−Wn )/L p

(4.43)



δn(x) = n p e(x+W p )/L n = n p eeV /k B T − 1 e(x+W p )/L n

(4.44)

We are now in the position to write down the diode current as a function of the applied bias voltage. As explained previously, the drift current is not affected by the application of a moderately high external bias voltage, because the built-in electric field in the depletion region is already very high and the carriers drift with saturation velocities. Therefore, we can calculate the total current by using only the diffused minority carrier densities which depend exponentially on the external bias as given in Eqs. 4.44 and 4.45. The non-equilibrium hole current due to the diffusion of minority holes into the n-region is then:

4.2 p-n Junction Under an External Electric Field

I p (x) = −e AD p

141

Dp dδp(x) = eA δp(x) x > Wn dx Lp

(4.45)

Combining Eqs. 4.44 and 4.46: I p (Wn ) = e A

Dp pn (eeV /k B T − 1) Lp

(4.46)

Similarly, we can obtain the current due to the diffusion injection of nonequilibrium electrons into the p-side of the junction: In (−W p ) = e A

Dn n p (eeV /k B T − 1) Ln

(4.47)

The total current for an ideal diode is the sum of electron and hole diffusion currents: I (V ) = I p (Wn ) + In (−W p )   Dp Dn pn + n p (eeV /k B T − 1) = eA Lp Ln I (V ) = I0 (eeV /k B T − 1)

(4.48)

Here the leakage current or saturation current is given as: 

D p pn Dn n p + I0 = e A Lp Ln

 (4.49)

Equation 4.49 is known as the diode equation. It is clear from Eq. 4.49 that the diode current increases exponentially with forward bias (V = + V f ). Under reverse bias, however, if we substitute V = −V r into Eq. 4.49 for a reverse bias voltage satisfying: |eVr |  k B T, exp(−eVr /k B T )  1 and I = I0 . This is the leakage current and in an ideal diode has a very small value which can be neglected when compared with the forward current. It is for this reason that the diode only conducts in one direction (forward bias) and is known to act as a rectifier. Example A Si p-n junction is doped with N A = 1016 cm-3 and N D = 5 × 1016 cm−3 , respectively. (a) Calculate the built-in potential and depletion width of the junction in equilibrium. (b) Calculate the depletion width and the maximum electric field in the depletion region when the junction is biased with V = 0.05 V and V =−2.5V. (c) What is the potential drop on the n-side of the depletion region for these biasing conditions?

142

4 The p-n Junction Diode

k B T p p nn 0.026 eV 1 × 1016 cm −3 × 5 × 1016 cm −3 ln 2 = ln = 0.76 V e e (1010 cm −3 )2 ni    2εVbi N A + N D 1/2 W (Vbi ) = = 0.315 µm e NA ND    2ε(Vbi − 0.5) N A + N D 1/2 = 0.143 µm W (Vbi − 0.5V ) = e NA ND W (Vbi + 2.5V ) = 0.703 µm 1 2Vbi = 40 kV/cm Vbi = E m W ⇒ E m = 2 W 2(Vbi − 0.5) = 18 kV/cm E m (Vbi − 0.5) = W 2(Vbi + 2.5) = 89 kV/cm E m (Vbi + 2.5) = W eN D Wn2 V (Wn ) − V (0) = Vn = 2ε NA (Vbi − V )N A Wn = W Vn = NA + ND NA + ND Vn = 0.105 V, Vn (0.5 V ) = 0.0216 V, Vn (−2.5 V ) = 0.522 V Vbi =

4.2.2 Minority and Majority Carriers in a p-n Junction As already shown, a p-n junction diode is a device in which both electrons and holes contribute to the current. We have derived the diode equation by taking into account the sum of the hole and hole injections across the depletion region. In doing so, we used the non-equilibrium injected carrier densities at the edges (x = W n and x = −W p ) of the depletion region before they recombine with the majority carries. Therefore, the current in Eq. 4.49 is the maximum current. Once the injected minority electrons enter into the p-region and minority holes into the n-region, they recombine with the majority carriers and the injection current decreases. For example, as the excess minority holes recombine with the electrons in the n-region, an equal number of electrons are injected into this region. These electrons provide a drift current in the n-side to balance the hole current that is lost through recombination. The drift part of the minority carrier current is, however, negligible because of the relatively low carrier density and very small electric field. The hole diffusion current from Eqs. 4.44 and 4.46 is: I p (x) = e A



Dp pn e−((x−Wn )/L p ) eeV /k B T − 1 x > Wn Lp

(4.50)

4.2 p-n Junction Under an External Electric Field

143

From Eq. 4.49, the total current is: 

Dp Dn pn + n p eeV /k B T − 1 = I0 (eeV /k B T − 1) I = eA Lp Ln 

(4.51)

The electron drift current in the n-region is then: In (x) = I − I p (x) x > Wn  

Dp Dn = eA pn0 (1 − e−(x−Wn )/L p ) pn + n p eeV /k B T − 1 Lp Ln

(4.52)

Here Dp and Dn are the diffusion constants and L p and L n are the diffusion lengths for holes and electrons, respectively. pn0 and pn are the hole densities in the n-region in equilibrium and under bias, respectively. np is the minority electron density on the p-side, and W p and W n are the depletion layer widths on the p and n sides of the diode, respectively. It is evident from Eq. 4.53 that while the hole current is reduced from its value at W n , as we move into the n-region, the electron injection increases to keep the total current flow constant. Similarly, the electron current is reduced from its value at –W p , as we move into the p-region, but the hole injection increases to compensate the loss in the electron current. The behaviour of the electron and hole current in an ideal diode is shown in Fig. 4.10. The rectifying characteristic of a diode is depicted in Fig. 4.11. Under reverse bias, the diode can be assumed to be non-conducting because of the negligibly small value of the leakage current. Under forward bias, however, current increases exponentially and becomes highly conductive at bias voltages of approximately 80% of the semiconductor bandgap used in the fabrication of the diode. The diode resistance under forward bias can be obtained from Eq. 4.49 by taking the derivative dV/dI. This is known as the dynamic resistance of the diode. The dynamic resistance of a diode is inversely proportional to the forward current. rd =

  dV kB T eV = exp − dI Io e kB T

At T = 300 K (k B T = 26 meV) and dynamic resistance is rd =

26  I f (m A)

144

4 The p-n Junction Diode

Fig. 4.10 a Schematic representation of p-n junction under forward bias. b Distribution of minority carriers. c The minority and majority carrier currents

4.3 High Voltage Effects When we derived the diode equation, we made two assumptions: (1) (2)

The injected carrier density is significantly lower than the majority carrier density. The drift current under reverse bias is saturated and independent of the applied bias.

These assumptions, although perfectly plausible at moderate applied voltages, are not valid for high bias voltages as we shall see next.

4.3 High Voltage Effects

145

Fig. 4.11 I–V characteristic of a diode

4.3.1 Forward Bias: High Injection Region We have so far assumed that the injected minority carrier density is very low compared to the majority carrier densities in the n and p-n regions and that, because of its high resistivity, most of the applied voltage drops within the depletion region. When the forward biasing is increased, the injected carrier density may become comparable to the majority carrier density, thus some of the applied voltage drops across the undepleted regions, with the result that the diode current saturates. Further increasing the applied voltage will cause the diode to act as a passive resistance and if the forward current is high, the diode may burn as a result of excessive Joule heating (I 2 R).

4.3.2 Reverse Bias: Impact Ionisation When the electric field is increased above a certain value, the carriers gain enough kinetic energy to transfer electrons from the valence to conduction band, leaving behind holes in the valence band and creating electron–hole pairs. This process is known as impact ionisation. In a reverse biased diode, the electric field in the depletion region increases with increasing built-in potential. The electrons and holes which make up the drift current in the depletion region may gain sufficient kinetic energies to initiate the impact

146

4 The p-n Junction Diode

ionisation process. The current increases as: I0 = M(V )I0

(4.53)

Here I 0 is the leakage (saturation) current prior to impact ionisation and M is the impact ionisation factor which depends on the ionisation rate.

4.3.2.1

Avalanche Breakdown

The avalanche process is initiated by impact ionisation. An electron in the depletion region of a reverse biased diode gains enough kinetic energy from the electric field. If the field is sufficiently high the electron may have enough kinetic energy and collide with the lattice atoms breaking the bonds, thus delocalising the initially localised electrons in the valence band. These free electrons are transferred into the conduction band creating a free hole in the valence band. The newly created electron–hole pair will experience the same high electric field in the depletion region and create additional electron–hole pairs, and these will in turn create further electron–hole pairs. The process is called avalanche multiplication. The drift velocity, thus the kinetic energy of the impact ionised carriers, increases with distance therefore the avalanche multiplication and the current will increase with distance. This implies that the avalanche process is more likely to occur in low and moderately n and p-doped p-n junctions with wide depletion widths. If the reverse bias is increased slightly above the value that is required for the impact ionisation to start, the diode current increases rapidly as a result of the avalanche multiplication. This may result in the irreversible breakdown of the diode at a bias voltage known as the breakdown voltage as shown in Fig. 4.12.

4.3.2.2

Reverse Bias: Zener Breakdown

Zener breakdown occurs in p-n junctions fabricated using small bandgap semiconductors with very high n and p–doping. The underlying physical mechanism for Zener breakdown is quantum mechanical tunnelling. In order to understand the tunnelling process, we may consider the energy bandprofile of a highly doped p-n junction constituted of a small semiconductor bandgap shown in Fig. 4.13a. As the diode is heavily doped on both n- and p-sides (highly degenerate), then the Fermi levels are within the conduction band on the n-side and in the valance band on the p-side. As a result of heavy doping, the depletion with is very small. Upon the application of a small reverse bias, an electron in the conduction band can tunnel through the thin potential barrier to an empty (hole) state in the valance band as indicated in Fig. 4.13. The tunnelling probability of an electron from –x 2 in the conduction band a hole state at –x 1 in the valance band depends on the potential barrier, E g , and electric field, E, via;

4.3 High Voltage Effects

147

Fig. 4.12 Breakdown voltage



√ 3/2 4 2m ∗ E g T ≈ exp − 3eE

(4.54)

The tunnelling probability also increases with increasing electric field, E, which is proportional to the applied voltage and inversely proportional to the distance between the initial and final states (x 1 − x 2 ). In a Si diode with doping densities of N A = 1019 cm−3 and N D = 1016 cm−3 and with a bandgap E g = 1.1 eV, Zener breakdown occurs at a reverse voltage of approximately 50 V. In a similarly doped diamond diode with E g = 5.4 eV, the Zener breakdown occurs at around 3 kV. If the reverse bias increases beyond the Zener breakdown voltage then irreversible destruction of the diode occurs. The main difference between the avalanche and Zener processes is that in the latter the doping levels are very high, consequently the depletion width is very narrow and the breakdown occurs even before an increase in the free carrier density. Furthermore, the breakdown voltage or Zener voltage has a negative temperature dependence. Although the mechanisms are different, diodes with operation in the breakdown region are commonly known as Zener diodes.

4.3.2.3

Punch Through

The mechanism for the punch through is directly related to the geometry of the diode. As we have discussed already, the injected non-equilibrium minority carriers recombine with the majority carriers once they enter into the opposite majority regions at the edges of the depletion layer. In a narrow diode, if the depletion width, W,

148

4 The p-n Junction Diode

Fig. 4.13 a Tunnelling of an electron from conduction band into valence band under reverse bias. b The potential barrier as experienced by the tunnelling electron

is much wider than the n- and p-regions then when the diode is reverse biased the depletion width may extend throughout the device. The carriers injected from one contact will travel through the depletion region, without any recombination, to reach the other contact. As a result, no voltage drops in the depletion region and the current increases. The device current is controlled by the external circuitry. If there is no overheating there is no permanent breakdown. As an example let us consider a p+ -n diode where N A >> N D . In this case the depletion width will spread into the n-region (see Eq. 4.34). If the width of the n region is ln , the critical depletion with for the punch through is:  W = ln =

2εVr eN D

1/2 (4.55)

Here V r is the reverse bias voltage and V r >> V bi. From Eq. 4.56 the punch through occurs at a voltage, V pt :

4.3 High Voltage Effects

149

Vpt =

eN D ln2 2ε

(4.56)

4.4 Junction Capacitance In principle, a diode may be considered to be a parallel plate capacitor with the n and p regions acting as the oppositely charged capacitor plates and the depletion region as the dielectric filling the space between the plates. The capacitance is:

dQ

C =

dV

(4.57)

The charge in the depletion region is due to the space charge: |Q| = e AWn N D = e AW p N A

(4.58)

The total depletion width under external bias V (+ V for forward bias and –V for reverse bias) can be found by combining Eqs. 4.29–4.31 to give:  W =

  2ε(Vbi − V ) N A + N D 1/2 e NA ND

(4.59)

The depletion layer widths on either side of the junction are: Wn =

NA ND W ; Wp = W NA + ND NA + ND

(4.60)

We now substitute Eq. 4.61 into Eq. 4.59 to find the charge in the depletion layer width: |Q| =

  e AN A N D N D N A 1/2 W = A 2eε(Vbi − V ) NA + ND ND + NA

(4.61)

Therefore the junction capacitance is:



dQ N A N D

1/2 Aε

= A 2eε C j =

= dV 2 (Vbi − V ) N A + N D W

(4.62)

Here A is the cross-sectional area of the diode. It is clear from Eq. 4.63 that the junction capacitance varies with the reverse bias voltage (−V ) and is known as a varactor which is regularly used in electronic circuits as a variable capacitance.

150

4 The p-n Junction Diode

We may express the junction capacitance in a simpler form: 1 2(Vbi − V ) = 2 A eεNe f f ective C 2j

Ne f f ective =

NA ND NA + ND

(4.63)

We may plot the measured values of 1/C 2j versus V to obtain the built-in potential, depletion width and the doping densities from the graph. As an example, let us consider a typical 1/C 2j versus V plot for a Si diode with A = 10–3 cm2 as shown in Fig. 4.14. The line intercepts the voltage axis at 0.68 V and this is the built-in potential (Eq. 4.64). The slope

d 1/C 2 2 = 2.1 × 1023 F −2 V −1 = 2 dV A eεNe f f ective can be used to calculate N effective as 5.64 × 1013 m−3 . From:   NA ND kB T ln Vbi = e n i2 N A N D = 5.1 × 1031 cm−6 and since N effective = 5.64 × 1013 m−3 , we obtain N D = 5.5 × 1013 cm−3 and N A = 9.04 × 1017 cm−3 .

Fig. 4.14 1/C 2j versus voltage for a Si diode

4.4 Junction Capacitance

151

Example In a Si p+ -n diode the donor density is N D = 4 × 1021 m−3 and the dielectric constant of Si is ε = 11.8ε0 . If the cross-sectional area of the diode is A = 4 × 10−7 m2 , what is the junction capacitance when the diode is reverse biased at 4 V? Since the p side is heavily doped, N A >> N D . Therefore, Eq. 4.63 becomes:



N A N D

1/2 A

2eε A

2eε N D

1/2 Cj = = 2 (Vbi − V ) N A + N D 2 (Vbi − V ) If V = −4V, junction capacitance is found as C j = 33.4 pF.

4.5 Temperature Dependence of Diode Current The diode current, both in the forward and reverse bias, depends on the temperature. Here we shall derive the appropriate expressions to show this dependence. We shall start with the leakage (saturation) current: 

D p pn Dn n p + I0 = Ae Lp Ln



We can re-write this equation using the law of mass action, i.e. (n i2 = pn n n = pn N D ; n i2 = p p n p = n p N A )  Is = Ae

D p n i2 Dn n i2 + L p ND Ln NA



 = Aen i2

Dp Dn + L p ND Ln NA



  E Here since Io ∝ n i2 , and n i2 ∝ T 3 exp − k B gT then:   Eg I0 ∝ T 3 exp − kB T

(4.64)

This equation is valid only for semiconductors with bandgaps smaller than 0.8 eV, like Ge. For semiconductors with wider bandgaps, such as Si and GaAs, the temperature dependence of the leakage current is:   Eg I0 ∝ n i ∝ T 3/2 exp − 2k B T

(4.65)

The current in a forward biased diode, on the other hand, has an exponential dependence on temperature: 

eV I f = I0 exp kB T

 (4.66)

152

4 The p-n Junction Diode

In summary, diodes fabricated with small bandgap semiconductors have a stronger temperature dependence than those with wide bandgap semiconductors.

4.6 Tunnel Diode A tunnel diode or Esaki diode is a type of semiconductor that is capable of very fast operation which extends far into the microwave frequency region and the principle of the operation is based on quantum mechanical tunnelling. In heavily doped p-n junctions, the depletion region is narrow enough for the carriers to tunnel through the depletion region rather than surmounting the built-in potential barrier. Figure 4.15a shows the energy bandprofile of a heavily doped p-n junction in thermal equilibrium. In thermal equilibrium, the final states within the valance band are already occupied, therefore, there is no option for electron tunnelling. Under a small forward bias, however, as voltage is increased, electrons tunnel can through the narrow potential barrier because occupied electron states in the conduction band on the n-side become aligned with empty valence band hole states on the p-side of the junction. If the voltage is increased further, the initial and final states become misaligned and the current drops. This region where the current drops with increasing voltage is known as the negative differential resistance (NDR) region (refer to Sect. 3.1.5). The diode, when biased in the NDR region, can be used in electronic circuits as an oscillator, amplifier and in switching circuits. Beyond the NDR the diode operates as a normal diode, where minority carriers are injected across the depletion region, through the potential barrier by diffusion and no longer by tunnelling (Fig. 4.15d). It is clear from Fig. 4.15b and c that the tunnelling occurs in both forward and reverse bias configurations. Problems 1.

2.

3.

4.

A Si p-n junction is doped with N D = 1016 cm−3 and N A = 1018 cm−3 . Calculate (a) position of Fermi level in the n- and p-regions, (b) built-in potential and (c) width of depletion layer (At T = 300 K, N C = 2.8 × 1019 cm−3 , N V = 1 × 1019 cm−3 ). For a Si p-n junction with N A = 1018 cm−3 and N D = 5 × 1015 cm−3 , calculate (a) position of Fermi levels in the n- and p-regions, (b) draw the energy bandprofile of the p-n junction in equilibrium and use the figure to estimate the built-in potential (At T = 300 K, NC = 2.8×1019 cm−3 , N V = 1×1019 cm−3 ). The cross-sectional area of a Si diode with N D = 1016 cm−3 and N A = 4 × 1018 cm−3 is A = 2 × 10–3 cm2 . Calculate the built-in potential and the width of depletion of the diode at T = 300 K. The position dependence of the built-in potential in the n-side of the depletion region of a p-n junction is:

4.6 Tunnel Diode

153

Fig. 4.15 Bandprofiles of a tunnel diode a in equilibrium and when the tunnel diode is b reverse c forward and d is heavily forward biased. e Current–voltage characteristics for all these conditions

154

4 The p-n Junction Diode

Vn =

5.

6.

7.

8.

9.

10.

  eN D x2 x Wn − 0 < x < Wn ε 2

Show that the potential energy is independent of position beyond the depletion region, x>Wn, hence show that the electric field is zero in the same region. (a) Write down the expression for charge neutrality in the depletion region of a p-n junction. (b) Use Poisson’s equation to work out the charge density in the depletion region. In a p-n junction with N A = 1016 cm−3 , N D = 1015 cm−3 , ε = 11.7ε0 and with a built-in potential 0.635 V (a) what is the depletion width and the maximum electric field in the depletion region? (b) Calculate the width of depletion layer when a reverse bias of V r = 5 V is applied to the diode. Consider a Si p-n diode with N A = 1018 cm−3 . When a reverse bias of V r = 25 V is applied the maximum electric field in the diode is 3 × 105 V/cm. What is the doping density in the n-side of the junction, N D ? An ideal Si diode with a cross-sectional area A = 1.0 × 10–3 cm2 is doped with N A = 1.0 × 1018 cm−3 and N D = 1.0 × 1016 cm−3 . The transport parameters at T = 300 K are: On the n-side: Hole diffusion constant 7.8 cm2 /s and hole diffusion length 2.7 × 10–3 cm. On the p- side: Electron diffusion constant 7.3 cm2 /s and electron diffusion length 2.7 × 10–3 cm. (a) Calculate the diode current when it is forward biased with 0.2 V. (b) What is the bias voltage if the diode current density is J = 1 × 103 A/cm2 ? Assume a Si p-n junction with N D = 1017 cm−3 and N A = 5 × 1015 cm−3 . At T = 300 K, the bandgap of Si is 1.12 eV and the intrinsic carrier density is ni = 1.5 × 1010 cm−3 . (a) Plot the energy band diagram of the p-n junction in thermal equilibrium and calculate the Fermi energies in both the n and p-sides of the junction. (b) When a reverse bias voltage of 0.5 V is applied the diode current is −4.43 × 10–15 A. What would the diode current be if a + 0.5 V forward bias voltage is applied? Junction capacitance of a p-n junction diode with cross-sectional area of A = 1 × 10–3 cm2 and ε = 10ε0 is measured as a function of applied reverse bias and tabulated in the table. Plot the 1/C 2 versus V r graph and calculate the donor density (N D ) and build-in potential (V bi ).

V r (V)

0.1

0.5

1.0

1.5

2.1

2.5

C (pF)

20.0

14.5

11.8

10.2

9.0

8.4

4.6 Tunnel Diode

11.

A Si p-n junction doped with N D = 1016 cm−3 and N A = 5 × 1016 cm−3 . A forward bias voltage of V f = 0.61 V is applied to this diode. Taking diffusion constants of minority carriers Dn = 23 and Dp = 8 cm2 /s and τn = 5 × 10−8 s and τ p = 1 × 10−8 s, calculate the excess hole density as a function of distance from the junction plane for x > 0.

Suggested Reading List 1. 2. 3. 4. 5. 6. 7.

155

Bhattacharya P (1994) Semiconductor optoelectronic devices. Prentice-Hall Singh J (1994) Semiconductor devices. McGraw Hill Piprek J (2003) Semiconductor optoelectronic devices. Academic Press Sze S (1981) Physics of semiconductor devices. Wiley Streetman BG, Banerjee S (2000) Solid state electronic devices. Prentice Hall Neamen D (2003) Semiconductor physics and devices. McGraw Hill Irene EA (2005) Electronic materials science. Wiley

Chapter 5

Solar Cells (Photovoltaic Cells)

Learning Outcomes On completion of this chapter, the reader will be able to: 1. 2. 3. 4. 5.

demonstrate an understanding of the underlying physical processes in solar cell operation, design a simple solar cell and describe the operation, describe parameters defining efficiency and cost of solar cells, suggest ways of improving design for increasing efficiency and lower cost, explain various applications of solar cells.

The sun releases a vast amount of energy into the solar system. The temperature at the surface of the sun is approximately 5800 K. At a point just outside the earth’s atmosphere the solar flux is about 1353 W/m2 . Almost all renewable energy sources with exception to radiative and nuclear energy sources, have their energy from the sun. Despite the vast amount of radiant energy from the sun fossil fuels and nuclear power, which are depleting sources, still remain the main sources of primary energy in the world at 78% of the total energy demand. The renewable energy sources have a disproportionately low contribution of 22% to the total world energy use. Photovoltaic energy still takes a miniscule fraction of the renewable energy source equivalent to only 0.01% of total world energy use. With a projected world population of 12 billion by 2050 and an energy consumption estimated to be 28 TW which is double the current demand of 14 TW, the CO2 emission will be more than twice the pre-industrial level. Continuing with the current trend of energy consumption means that in less than 350 years all the depleting sources will have run out. Photovoltaics has great potential to become the most potent energy source for future generations. Besides being virtually inexhaustible, it does not have any combustible by-products and it’s CO2 ‘foot print’ is very low. The only CO2 emission is during the manufacture and this is between 5 and 13% of the contribution by fossil fuels. However, there are still challenges facing photovoltaic technologies with the major ones being the low efficiency and high installation costs. There are © Springer Nature Switzerland AG 2021 N. Balkan and A. Erol, Semiconductors for Optoelectronics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-44936-4_5

157

158

5 Solar Cells (Photovoltaic Cells)

two possible ways of curbing these problems. One alternative is to reduce the cost of production by going for relatively cheaper materials like amorphous, polycrystalline and thin films at the expense of efficiency, and the other is to go for higher efficiency at the expense of cost. Both ways are able to generate an overall profit to cover the costs for using either increased areas or higher costs in the latter case. Solar cells (or photovoltaic cells) convert the energy from the sun light directly into electrical energy. In the production of solar cells both organic and inorganic semiconductors are used and the principle of the operation is based on the current generation in an unbiased p-n junction. This chapter aims to provide an in-depth analysis of photovoltaic cells used for power generation. It starts with an introduction to the fundamental concepts in key physics and technologies, including the solar radiation spectrum, p-n junctions in equilibrium and under illumination. This is followed by photovoltaic cell design considerations, testing and efficiency measurements. The chapter finishes with a summary of novel applications of photovoltaic cells, cost and efficiency studies. We shall be using the following definitions throughout this chapter: Solar Radiation is the electromagnetic radiation emitted by the sun. Solar Spectrum is the electromagnetic spectral distribution emitted by the sun. Spectral photon flux is number of photons with energy in the range E to E + dE emitted through unit area per unit solid angle per unit time. Irradiance is the rate at which radiant energy arrives at a specific area of surface during a specific time interval. This is known as radiant flux density. A typical unit is W/m2 . Spectral Irradiance is the spectral irradiance as a function of photon wavelength (or energy). It gives the power density at a particular wavelength. The unit of spectral irradiance are in W/m2 μm1 . Spectral energy distribution is defined as the energy per unit volume per unit wavelength of frequency (W/m3 μm). Spectral energy density is the energy per unit volume per unit frequency interval (W/m3 Hz). Total power density is the integration of the spectral irradiance over all wavelengths or energies. The unit of total power density is W/m2 .

5.1 Principles of Solar Cells 5.1.1 Solar Radiation The solar spectrum can be represented by the radiation spectrum of a blackbody with temperature of T = 5800 K. Therefore this section begins by investigating the blackbody radiation spectrum.

5.1 Principles of Solar Cells

5.1.1.1

159

Bose–Einstein Distribution Law and Blackbody Radiation

As far as their energy distributions are concerned there are two types of particles, these are fermions and bosons. Fermions, for example electrons, have intrinsic angular momentum (spin) in odd half integral multiples of è, obey the Pauli Exclusion Principle and their energy distribution is represented by the Fermi–Dirac statistics (Sect. 2.2). Bosons, for example photons and phonons, have zero or integral multiples of è spins, do not obey the Pauli Exclusion Principle and are represented by Bose–Einstein statistics. According to the Bose–Einstein statistics, the probability of finding a boson at an energy E is: f B−E (E) =

1 e(E−μ)/k B T − 1

(5.1)

Here k B is the Boltzmann constant and μ is the chemical potential. When μ = 0, bosons are not conserved and the emission and absorption co-exists like in blackbody cavity walls. Blackbody is a cavity whose walls are at temperature T. The walls of the cavity are composed of atoms which emit electromagnetic radiation but at the same time they absorb radiation emitted by the other atoms in the walls (Fig. 5.1). The electromagnetic radiation occupies the whole cavity and the radiation trapped within the cavity reaches equilibrium with the atoms of the walls. The amount of energy emitted by the atoms per unit time is equal to that absorbed by them. Therefore, when the radiation in the cavity is in equilibrium with the walls, the energy density of the electromagnetic field is constant, the distribution of which is defined with the Planck function. The Planck function represents the spectral energy density of electromagnetic radiation (in Joules/ m3 Hz) at frequency ν inside a closed cavity whose walls are at a uniform temperature T. If the blackbody is larger than the average wavelength of the emitted electromagnetic spectrum, the separation of the allowed energy levels will be very small and the quantised energy spectrum may be approximated to a continuous one. The number of available states within dE interval of a given energy E, is dn. dn is the product of the density of states g(E)dE and the Bose–Einstein distribution function f B−E (E). Fig. 5.1 Schematic representation of a blackbody

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5 Solar Cells (Photovoltaic Cells)

dn = N (E) f B−E (E)d E

(5.2)

Energy of a photon with frequency v: E = hv

(5.3)

Therefore, the total density of states within the dE energy interval will be the same as the total density of states within the frequency interval dv: N (E)d E = N (v)dv

(5.4)

We can use Eq. 2.6 together with k = 2π/λ and λ = c/v to obtain the density of states in the frequency domain. N (v)dv =

4π V 2 v dv c3

(5.5)

Here V is the volume and c is the speed of light. We need to multiply the density of states in Eq. 5.5, by 2 for the two independent directions of polarization since electromagnetic waves are transverse to give: N (v)dv =

8π V 2 v dv c3

(5.6)

From Eqs. 5.2, 5.1 and 5.6: dn =

8π V 2 1 dν ν (hν/k T ) 3 B c e −1

(5.7)

Total energy corresponding to dn photons in the frequency range dν is: (dn) × (hν)

(5.8)

and the total energy per frequency range is: dn × hv dv

(5.9)

Energy per frequency range per unit volume (Spectral energy density) is then: E(ν) =

hν dn V dν

(5.10)

If we substitute Eq. 5.7, into Eq. 5.10, we obtain the spectral energy density:

5.1 Principles of Solar Cells

161

Fig. 5.2 Spectral energy density function versus photon frequency at three different temperatures. The maximum of each curve shifts to higher photon frequencies (or energies) with increasing blackbody temperature

E(ν) =

8π hν 3 c3 (ehν/k B T − 1)

(5.11)

Figure 5.2 shows the spectral energy density function for different temperatures. There are two interesting results of the Planck function: 1. Wien’s Displacement Law: The peak wavelength of the spectral energy density (λmax ) can be found first derivative of spectral energy density with respect to wavelength, i.e. d E(λ) =0 dλ Wavelength dependent density of states is obtained by placing ν = c/λ in Eq. 5.5 as N (λ)dλ =

8π V dλ λ4

(5.12)

From Eqs. 5.1, 5.2 and 5.12 the energy per unit volume and per unit wavelength is: E(λ) =

1 8π hc 5 hc/λk B T − 1) λ (e

(5.13)

The first derivation of Eq. 5.13 with respect to the wavelength for the maximum wavelength λmax , we obtain Wien’s displacement law: hc = 4.965 ⇒ λmax T = 2898 μm · K k B T λmax

(5.14)

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5 Solar Cells (Photovoltaic Cells)

This shows that temperature T and λmax are inversely proportional. Therefore, high temperature electromagnetic radiation sources emit most of their power at high frequencies or short wavelengths. At long wavelengths or low frequencies (hv > E g , electron–hole pairs will be generated at energies much higher than the conduction and valance bandedges and their excess energy will be lost as heat to the lattice via the emission of phonons. This loss of energy is an important factor in reducing the solar cell efficiency.

5.1.3 p-n Junction Under Illumination In Sect. 4.1, we showed that for a p-n junctions in equilibrium, there are no free carriers in the depletion region and the built-in electric field outside the depletion region is zero. The approximation of depletion layer fails when the junction is either forward or reverse biased and current flows through the junction (Sect. 4.2 and Fig. 4.11). Here we shall consider the case when the junction is under illumination with photons whose energy is greater than the bandgap of the semiconductor. Consider a long p-n diode in which excess carriers are generated uniformly at a rate of GL when it is illuminated. The electron hole pairs generated in the depletion region are swept out rapidly by the electric field existing in the region. Thus, electrons are swept into the n- region and holes into the p-region. The photocurrent is: x  I L1 = Ae

G L d x = AeG L W

(5.28)

0

Here A, GL and W are the cross-sectional area, generation rate and the width of the active (depletion) region where a high built-in electric field exists. Since electrons and holes contributing to IL1 move under high electric fields, the response is very fast and this component of the current is called the prompt photocurrent. In Fig. 5.9, np and pn represent the equilibrium minority charges in the p- and n-sides, respectively. n(x) and p(x) are the excess carriers generated in the neutral nand p- regions of the diode. Electrons and holes within the diffusion length of the depletion region L n and L p will be able to enter the depletion region where they will be swept by the field. Thus the photocurrent constitute of all the three regions, L n , W and L p as I L = I Ln + I L p + W = eG L (L p + L n + W )A This photocurrent flows in the reverse current direction of the diode.

(5.29)

5.1 Principles of Solar Cells

171

Fig. 5.9 The spatial distribution of minority carriers in a p-n junction diode. Under illumination (----) and at dark (—). In the depletion region all the minority carriers are swept out under the influence of the built-in electric field

In reality, GL is not constant as the intensity of light drops as I = I0 exp(−αx) with penetration depth. Thus, an average value for GL must be used. If there is an applied voltage to the diode, total current flow is expressed as:   e(V +Rs I ) I = I L + I0 1 − e k B T

(5.30)

Here Rs is the series diode resistance, n is the ideality factor and V is the voltage across the diode. The photodiode can be used in two modes as shown with their equivalent circuits in Fig. 5.10: 1.

Photovoltaic mode (solar cell): No bias is applied and diode is in-serial with an external high resistance (RL ). When the diode is illuminated, photocurrent passes through the external load resistance to generate power.

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5 Solar Cells (Photovoltaic Cells)

Fig. 5.10 Equivalent circuit of a photodiode. The device can be represented by a current source (photocurrent), I L and it consists of a shunt resistor (Rshunt ) and a capacitor (C D ). Rs is the series resistance of the diode. In the photovoltaic mode (solar cell), the diode is connected only to a load resistance (RL ). In the photoconductive mode (photodetector), it is connected to a power supply for forward biasing and to a load resistance (RL )

2.

Photoconductive Mode: In this mode, the diode is connected to an external load (RL ) and is operated under reverse bias. When it is illuminated, photocurrent passes through the circuit.

5.1.4 Solar Cell Parameters With reference to Fig. 5.10, if the circuit is used in the open circuit mode the current is zero:  e(V OC )  (5.31) I = I L + I0 1 − e k B T = 0 and, for the open circuit voltage, we get: VOC =

  IL kB T ln 1 + e I0

(5.32)

At high optical intensities, the open circuit voltage can approach the bandgap of semiconductor. If the output is short circuited then R = 0 and V = 0. From Eq. 5.30, the short circuit current is: I = I SC = I L

(5.33)

In a solar cell, under illumination, diode current varies with applied voltage as depicted in Fig. 5.11 and current flows in a direction opposite the diode current. Figure 5.12 shows the forward and reverse currents at different light intensities.

5.1 Principles of Solar Cells

173

Fig. 5.11 Current–voltage characteristic of a solar cell under illumination

Fig. 5.12 Light intensity dependence of the reverse and forward currents

It is clear from Fig. 5.13 that the short circuit current increases linearly with light intensity, whilst the open circuit voltage tends to saturate at high light intensities. Solar cells are used in harsh environments such as space and deserts with extreme temperatures from −270 °C to 60 °C. Therefore, it is important to know their temperature characteristics. Figure. 5.14 shows such characteristics of a typical GaAs solar

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Fig. 5.13 Light intensity dependence of the short circuit current and open circuit voltage

Fig. 5.14 Temperature dependence of the current–voltage characteristic of a GaAs solar cell

cell. It can be seen that the open circuit voltage decreases with increasing temperature. In order to understand the reason for this let us recall the expression for leakage current, I 0 , given in Chapter 4 in Eq. 5.4.40. I0 = e A

Dn n p D p pn + Ln Lp



= Aen i2

Dp Dn + Ln NA L p ND



In this expression apart from the constants A and e, every parameter has some degree of temperature dependence. However, the intrinsic carrier density ni depends

5.1 Principles of Solar Cells

175

on the bandgap and the temperature exponentially (Eq. 5.2.28), therefore, this exponential dependence dominates the leakage current. To determine the temperature dependence of the open circuit voltage quantitatively, let us consider for simplicity, that the leakage current is only due to drift of one type of carrier: I0 = Aen i2

Dp L p ND

(5.34)

If we substitute the intrinsic carrier density expression from Eq. 2.28 in Sect. 2, into Eq. 5.34: I0 = Ae

Dp Eg Eg ) ≈ B  T γ exp(− ) BT 3 exp(− L p ND kB T kB T

(5.35)

Here B is a material constant and B is a constant, which is independent of temperature. We have replaced the superscript 3 with γ to take into account of the temperature dependence of all other parameters (Dp , L p ) in the expression. Open circuit voltage can be obtained by substituting I 0 into: VOC

  kB T I SC = ln e I0

(5.36)

where E g is replaced by = eV g : VOC =

  eVg kB T ln I SC − ln B  − γ ln T + e kB T

(5.37)

The approximate temperature dependence of the open circuit voltage is: VOC − Vg kB d VOC = −γ dT T T

(5.38)

The reason for the photocurrent to increase with increasing temperature can be explained in terms of the shrinkage of the semiconductor bandgap with increasing temperature. This, in turn, enables the absorption of longer wavelength photons and creates more electron and hole pairs. With reference to Figs. 5.10 and 5.11, we can obtain the power consumed at the load:  P = V × I = I L V − I0 eeV /k B T − 1 × V

(5.39)

The maximum power corresponds to the maximum values of the current and the voltage (I m and V m ) in Fig. 5.11 and can be obtained by taking the first derivative of the power expression in Eq. 5.39 with respect to voltage:

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5 Solar Cells (Photovoltaic Cells)

dP =0 dV

(5.40)

P = Vm × Im

(5.41)

The power efficiency of a solar cell is defined as the ratio of the output electrical power to the input optical power. The maximum efficiency is therefore: η=

Im Vm Pm × 100 = × 100 Pin Pin

(5.42)

Another important parameter for solar cell performance is the filling factor (FF) and this is defined as the ratio of the maximum power output to the product of the open circuit voltage and the short circuit current: FF =

Im Vm Isc VOC

(5.43)

This corresponds to the ratio of the two areas, A and B in Fig. 5.11, i.e. FF =

T he ar ea A T he ar ea B

(5.44)

Pictorially, we can visualise the FF as the measure of the proximity of the area under the I-V curve to a square. The maximum value for the filling factor for a solar cell is obtained by taking the first derivative of the electrical power with respect to voltage. d(I V ) =0 dV

(5.45)

Typical solar cells, available commercially, have filling factors of FF > 70%. Solar cells with high FF have small series resistances, thus low heat losses. There are various methods to enhance the power output from a solar cell and these are: 1. Short circuit current must be high. Therefore, concentrators should be used to increase the light intensity. 2. Leakage current (I o ) must be low to increase the open circuit voltage (V OC ). 3. Series resistance should be kept very small to increase the FF. The last two conditions can be met readily by using high quality semiconductor materials with small density of defects and recombination centers. Example A solar cell generates maximum power output at I m = 2.7A and V m = 0.48 V. What is the load resistance to be used to achieve the maximum power? How many solar cells would be required to charge up a 12 V car battery?

5.1 Principles of Solar Cells

177

Load resistance: RL =

Vm 0.48 = 0.18 = Im 2.7

Number of solar cells required N=

12V = 25 0.48V

Example A Si solar cell is uniformly illuminated by sun light with a power density of 1mW/cm2 and at an ambient temperature of T = 300 K. Calculate (a) the leakage current (I 0 ), (b) open circuit voltage (c) output power when the filling factor is 0.8 and (d) the efficiency of the solar cell. Area of the solar cell (A) = 1.0 cm2 . Donor density N D = 1 × 1017 cm−3 , Acceptor density N A = 5 × 1017 cm−3 , Electron diffusion coefficient (Dn ) = 20cm2 /s, Hole diffusion coefficient (Dp ) = 10cm2 /s. Electron recombination time (τ e ) = 3 × 10–7 s. Hole recombination time (τ h ) = 1 × 10–7 s. Photocurrent (I L ) = 25 mA. (a)

(b) (c) (d)

Electron and hole diffusion lengths: 1/2  = 10.0μm L n = (Dn τn )1/2 = 24.5μm L p = D p τ p Leakagecurrent:    D n D p D I0 = e A Ln n p + Lp p n = Aen i2 L nDNn A + L p Np D I0 = 3.66 × 10−11 A Voc = k BeT ln(1 + IIL0 ) = 0.53V P = (F F)I SC VOC = 1.06mW 1.0mW η = PPino = 10mW cm −2 ×1.0cm 2 = 0.1 = 10%

5.2 Design Considerations 5.2.1 Principal Considerations for Solar Cell Design The structure of a typical solar cell is shown in Fig. 5.15. In the design of such a structure, we should firstly ensure that all the sunlight enters the solar cell without being reflected at the surface. Therefore the top surface of the solar cell is covered with an anti-reflection coating. Anti-reflection coatings help reduce the reflection of desirable wavelengths from the cell, allowing more light to penetrate the semiconductor, therefore, increasing solar cell efficiency. Anti-reflection coatings on solar cells consist of a thin layer of dielectric material, with a specially chosen thickness

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5 Solar Cells (Photovoltaic Cells)

Fig. 5.15 a Structure of a basic solar cell b A comparison of antireflective and reflective coatings c A photo of illuminated Si solar panel, showing generated photo-voltage

(a)

(b)

(c)

5.2 Design Considerations

179

so that interference effects in the coating cause the wave reflected from the antireflective coating top surface to be out of phase with the wave reflected from the semiconductor surfaces. These out-of-phase reflected waves destructively interfere with one another, resulting in zero net reflected energy. The thickness of the antireflective coating is chosen so that the wavelength in the dielectric material is one quarter the wavelength of the incoming wave. For a quarter wavelength anti-reflection coating of a transparent material with a refractive index n1 and light incident on the coating with a free-space wavelength λ0 , the thickness d 1 which causes minimum reflection is calculated by:

d1 = λ0 4n1 The p-n junction doping profile is selected so that the n-side of the depletion region is much shorter that the p-side of the depletion region. This, in turn, implies that n-type region can be kept very short by ensuring that the doping density is much higher than the p-type doping density. Due to the thin n-layer, the light now can propagate into the depletion without undergoing significant absorption. Carriers, photo-generated in the depletion region and within the diffusion lengths from edges of the depletion region (active region), are readily swept through their majority regions, then, electrodes under the influence of the built-in electric field, creating the external current in the circuit. There are four important factors that influence the solar cell efficiency: • • • •

Contacts Junction depth Choice of material Doping We shall now consider them individually.

5.2.1.1

Contacts

In order to connect the solar cells to the external circuit, contacts for both the n and p-type carriers are required. The patterns of the top contact must be carefully considered as the light falling on the contacts will be reflected by contacts and lost unless a transparent contact such as such as Indium Tin Oxide (ITO) is used. Therefore the contacts should take up a small an area as possible. On the other hand if the contact area is too small, it will result in a high series resistance for the device which will reduce the filling factor of the cell. Therefore, an optimisation of the contact design is essential. In summary: • Contacts must be ohmic. • Finger or comb-tooth contacts are used to maintain a uniform potential throughout the surface and to minimise the series diode resistance. • Shadowing from contacts will reduce efficiency unless they are transparent.

180

5.2.1.2

5 Solar Cells (Photovoltaic Cells)

Junction Depth

As we have stated above, the intensity of the absorbed electromagnetic radiation propagating thorough solar cell is:   I = I 0 1 − e−α(λ)x

(5.46)

where α(λ) is the absorption coefficient and it depends on the wavelength of the incident radiation (Fig. 5.7) With reference to Fig. 5.16 and Eq. 5.46, as we would like most photons to be absorbed within the electron and hole diffusion length from the junction, the junction depth and absorption coefficient should be considered together, i.e., if the absorption coefficient, α(λ), is large then d must be small. In fact, at short wavelengths, for high α(λ) values, most photons will be absorbed at the surface and this will limit the efficiency of the solar cell (short wavelength cut-off). At long wavelengths, when α(λ) is low, there will not be enough recombination at the junction and this will also limit the efficiency (Fig. 5.17). Fig. 5.16 Layer structure of a basic solar cells

Fig. 5.17 Absorption of the incoming photons by the active region. Only photons with energies greater than the bandgap are absorbed and very high energy photons are absorbed at the surface before reaching the active region

5.2 Design Considerations

5.2.1.3

181

Material Choice

In order to produce current a solar cell must first convert the incoming photons to free carriers by the absorption and generation process. In this process, a photon passing through the semiconductor is absorbed and in doing so gives its energy to an electron in the valance band. If the energy given to the electron is equal or more than that of the direct bandgap of the material, or if phonons are available to give the electron the momentum to make an indirect transition then the electron is promoted to the conduction band. The free electron will live for a finite amount of time before it recombines, either radiatively or non-radiatively and this is known as the carrier lifetime. If a photon has energy greater than the conduction band it will be absorbed creating a mobile carrier above the conduction band. However, the excess energy will be lost in a picosecond timescale through the emission of phonons until the carrier reaches the bottom of the conduction band where it will experience the same lifetime as an electron generated there. A photon which has energy less than the bandgap of material, to a first approximation, cannot be absorbed and will be transmitted through the material. Therefore, in designing a solar cell, bandgap of the semiconductor compared to the solar spectrum must be considered. This is because in order for the incident photons to be absorbed, the semiconductor material where the p-n junction is formed must have a bandgap smaller than the incident photon energy. If hν < E g no photons will be absorbed, but if hν >> E g then energy greater than the bandgap of the material will be transferred to the lattice as heat (See Fig. 5.7).

5.2.1.4

Doping

In order to have a small series resistance, Rs , the doping densities of the p and n regions should be high. However, the diffusion lengths should also be taken into account. This implies that the doping cannot be excessively high. For reasonably long diffusion lengths, a doping density of around 1 × 1016 cm−3 would be sufficient.

5.2.2 Enhancement of Efficiency For any ideal heat engine that converts heat into mechanical energy by transferring energy between two heat reservoirs, from a hot region to a cool region of space, cannot be more efficient than a Carnot engine and this maximum efficiency, η, is defined as: η=

TC W =1− QH TH

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5 Solar Cells (Photovoltaic Cells)

where W is the work done, QH is the heat added to the system, T C is the absolute temperature of the cold reservoir and T H is the temperature of the hot reservoir. If we now assume that the hot reservoir is the sun, the blackbody with temperature T sun = 5800 K and the cold reservoir is the solar cell on earth with temperature, T solar cell = 300 K, the efficiency is: η =1−

Tsolar cell = 94% Tsun

This is the upper limit for efficiency in an ideal system. For solar cells, the Shockley Queisser (SQ) limit refers to the maximum theoretical efficiency of a solar cell using a single p-n junction. It was first calculated by William Shockley and Hans Queisser in 1961 [1]. The limit gives the maximum solar conversion efficiency to be around 33.7% for a p-n junction with a bandgap of 1.34 eV using an AM 1.5 solar spectrum. The best known solar cell material, silicon with a bandgap of 1.1 eV, can have a maximum efficiency of 29% according to SQ limit. Commonly used commercially available mono-crystalline Si solar cells produce about 22% power efficiency. The assumptions made in the determination of the SQ limit are: • The solar cell is made of one type of semiconductor. • In the solar cell, there is only a single p-n junction. • No light concentrator is used and the limit is calculated for “1 Sun” condition (1 Sun ~ 1000 W/m2 ). • All the incident photons with energies greater than the semiconductor bandgap create electron and hole pairs well above the bandedges, which lose their excess energies in the form of heat without contributing to the solar cell efficiency. The 65% loss in the SQ limit is fundamentally due to loss of photon energy as heat (47%), non-absorption of the sun spectrum at wavelengths greater than the cut-off wavelength of the semiconductor (18%) and the recombination of the photogenerated electron hole pairs. The theoretical efficiency limit for Si is 29%. However, even the Si solar cells produced using the most sophisticated growth and fabrication techniques do not have efficiencies better than 22%. The 7% loss is arises from the effects given below. • Since the anti-reflective coating cannot stop the reflection of the full band of wavelengths of the solar spectrum, there is always reflection loss. • In the case of the presence impurities in Si, there will be localized levels in the bandgap, which act as traps for centres and reduces the free carrier density, thus the photocurrent. • In the external circuitry of the solar cell, there are always contact and lead resistances which act as a source for the loss of efficiency. There are a number of strategies to exceed the theoretical SQ efficiency limit. These can be summarised as follows:

5.2 Design Considerations

183

Fig. 5.18 A three junction solar cell with three bandgaps and the schematic representation of their absorption of corresponding regions of the AM 1.5 spectrum

• Antireflective coating, which is more effective at the wavelength region of that solar cell can absorb. • Surface passivation or window. • Surface texturing (like introducing micro grooves) and concentrators to maximize light absorption. However, if the density is too high (focused light), this might increase the temperature of the device. As the bandgap of the semiconductor decreases with increasing temperature, this, in turn, means that the output voltage decreases. • Doping or bandgap gradient to accelerate the carriers to the junction. • Use of multiple junction solar cells, for the absorption of the full solar spectrum, where each cell is made of a different bandgap material as explained below (Fig. 5.18). 5.2.2.1

Multi-Junction Solar cells

The efficiency of a solar cell made from just one direct bandgap material is limited to approximately 33% due to high and low energy cut-offs. To overcome this limit, the response of a cell needs to extend to as long a wavelength as possible as well as overcoming the losses associated with the thermalisation of carriers. This has been best achieved by using multiple p-n junctions made from different materials to absorb different wavelengths of light. Using this method the maximum difference between the materials bandgap and the photons energy is reduced. There are various configurations which can achieve this. One is to use a tandem cell where junctions with different bandgaps are stacked in series. In these devices each junction has a

184

5 Solar Cells (Photovoltaic Cells)

different bandgap so it will absorb a different wavelength range. For the tandem device the top cell will have the highest bandgap, so lower energy photons will pass through it and are then absorbed by the lower cells, as shown in Fig. 5.18. Such cells have so far achieved the highest efficiencies of any cell and as a result found usage for space applications and in the future may find wide spread terrestrial usage in concentrator systems. Another design is to somehow separate the incoming photons by wavelengths and direct them onto different cells. A system using dichroic filters to achieve this is illustrated in Fig. 5.19a. In Fig. 5.19b-c, the layer structure of the design abovementioned (tandem solar cell) is given. In these devices, each junction has a different bandgap so it will absorb a different wavelength range. The main problem in such configurations is the electrical connections between the serial cells and is usually overcome by using tunnel junctions (Fig. 5.19c). The material choice for the individual junctions in these cells is also very important. Firstly, they would ideally have the same lattice constants; a difference in lattice constant will result in dislocations at the tunnel junctions causing carriers to recombine there. This could be overcome by a stepping layer but this will increase the series resistance and consume more material. Secondly, if the junctions are connected in series by tunnel junctions they should all be producing the same amount of current as the total current of series connected solar cells will be equal to the current of the lowest current producing cell. Additionally the efficiency of each cell will be dependent on its high and low wavelength cut-offs as for a single bandgap cell. The total efficiency of a two junction tandem solar cell is modelled for the 1 sun AM1.5G spectrum for infinitely thick junctions (Fig. 5.20a), and for a solar cell with the top cell thickness optimized to allow current matching of the junction (Fig. 5.20b). Such cells have so far achieved the highest efficiencies of any cell and as a result found usage for space applications and in the future may find wide spread terrestrial usage in concentrator systems.

5.2.2.2

Light Concentrators

An alternative strategy to increase the theoretical SQ efficiency limit is to use light concentrators or concentrated photovoltaic systems (CPV). Usually lenses and less commonly curved mirrors are used to focus sunlight onto already high efficiency multi-junction solar cells, thus increasing the power efficiency per unit area of the solar cell. They are more expensive than conventional PV systems but have advantage that they possess the highest efficiency of all existing PV technologies and a smaller photovoltaic array. CPV systems may be used in conjunction with solar trackers and sometimes with a cooling system to further increase their efficiency.

5.2 Design Considerations Fig. 5.19 Connection of two different solar cells a optical connection, b electrical connection c structure of a multi-junction solar cell

185 Dichroic Filter

Small bandgap solar cell

Wide bandgap solar cell

(a)

Wide bandgap solar cell

Small bandgap solar cell

(b)

(c)

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5 Solar Cells (Photovoltaic Cells)

(a)

(b) Fig. 5.20 The optimum efficiency of two junction series connected solar cells for the AM1.5G spectrum as a function of top and bottom cell bandgap for a an infinitely thick top cell b a top cell of optimized thickness

5.3 Advantages and Disadvantages of Solar Cells

187

5.3 Advantages and Disadvantages of Solar Cells Advantages of solar cells can be summarized as: • • • • •

Low running costs and high reliability as they have no moving parts. They are competitive for applications where grid power is not viable. They can be used in consumer products such as calculators and watches. Solar cells cost much less than batteries in the long run. Solar cells can provide village power in the third world countries in useful applications such as water pumping, refrigeration, domestic uses, irrigation and communication. In all these applications solar cells are much less expensive than diesel oil. • Solar cells costs more than the grid power, but the profit through avoiding transmission losses is considerable. The cost of grid connection may be too expensive. Disadvantages of solar cells are: • High capital costs (see Fig. 5.21 for production steps). • Small efficiency, therefore large land use. Approximately, 50 m2 land area is required for the generation of approximately 1 kW. This is excessive compared to the used land per kW power in hydroelectric power stations! • Storage problems: The sun does not shine everytime therefore storage of the energy collected from solar radiation should be stored. Storage options involve battery, connection to the grid and electrolysis of water to use H2 as fuel in fuel cells. Battery is a bulky, costly and inefficient option. Transmission of the energy can causes excessive loss for connection to the grid. Electrolysis of water to use H2 as fuel in fuel cells may a viable option for clean renewable power generation.

5.4 Thermophotovoltaic (TPV) Cells TPV is the use of PV cells to convert the radiation from heat sources at lower temperatures than the solar radiation. Solar radiation corresponds to blackbody radiation spectrum corresponding to a temperature of ~5800 °C. However, most of our nonrenewable energy sources, for example coal, oil, gas, gasoline and nuclear fission, involve the burning of some form of fuel at temperatures in the range of 1000– 3000 °C. Conventional power sources use some fraction of this heat energy to boil water and drive turbines to generate electricity. However, much radiant energy is lost during the process and this lost energy will in general approximate to a blackbody radiation at the appropriate temperature. The aim of TPV cells is to harvest this waste energy using low bandgap PV cells. The ideal spectra of two of the most promising rare earth emitters are ytterbia (Yb2 O3 ) and erbia (Er2 O3 ). The peak wavelengths of the PL spectra of Er2 O3 and Yb2 O3 are 0.827 eV and 1.29 eV, respectively (Fig. 5.22). The ideal ytterbia spectrum is well

188

5 Solar Cells (Photovoltaic Cells)

Fig. 5.21 Fabrication process for a Si Solar cell from the natural compound SiO2

Fig. 5.22 Ytterbia (Yb2 O3 ) and erbia (Er2 O3 ) luminescence spectra together with the blackbody radiation spectrum at T = 2000 K

5.4 Thermophotovoltaic (TPV) Cells

189

Fig. 5.23 Components of a TVP cell unit

matched to the Si band-gap, it can therefore, be grown onto Si. The erbia spectrum is matched to GaSb and InGaAs bandgaps and these cells are lattice-matched and can be grown onto InP substrates. A typical TVP cell stack is shown in Fig. 5.23. In this system, the emitter is the solar cell which cannot absorb the long wavelength photons from the sun and lets them through. Also heat is generated in the solar cell due to phonon emission by high energy photo-generated carriers. Therefore the excess long wavelength photons and heat are selectively filtered by the filter to match the bandgap of the TVP to optimise absorption. Potential applications of TVP cells are: • • • • • • • •

Domestic boilers. Small scale co-generation of electricity for off grid houses. Military applications in low power portable technology. Replacement for small diesel generators. Recovery of industrial waste heat. Hybrid vehicles. Deep space with nuclear sources (additional satellite power). Stored PV.

5.4.1 Advantages and Disadvantages of TPV Cells Advantages: • Silent • Potentially lighter than a diesel-based equipment. • They have no moving parts, therefore, they have high reliability.

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5 Solar Cells (Photovoltaic Cells)

• They can be used in lighting security, water pumping. Disadvantages: • Low efficiency (~10%) • Cost per Watt is high (~$20/Watt) • Conversion efficiency depends strongly on: (a) (b) (c)

Making an efficient cell. Matching the radiator spectrum to the TPV response (selective emitters must be used). Collecting the radiation in the TPV instead of heating the environment or the cell (To avoid this problem back surface reflectors and monolithically integrated modules can be used).

Problems 1.

2.

Use Stephan–Boltzmann law to calculate the power density of sun’s light just outside the earth’s atmosphere (The radius of the sun is 6.96 × 108 m, distance between earth and the sun: 150 × 109 m). Which of the following does not affect the output power of solar cells? (a) (b) (c) (d)

Increase in the short cicuit current. Increase in the open circuit voltage. Increase in the filling factor. Increasing the temperature of the solar cell.

3.

The saturation current in a solar cell is given as:  eDn n p eD p I0 = A L n + Lpp n Which of the following is identical to this expression?   Dn (a) Aen i2 LDp + L p ND  n NA  D (b) Aen i L nDNn A + L p Np D   D (c) Aen i2 L nDNn D + L p Np A   D (d) Aen i2 L nDNn A + L p Np D

4.

Which of the following expressions is correct for the open circuit voltage in a solar cell? (a) (b) (c) (d)

5.

VOC VOC VOC VOC

= = = =

kB T In(1 + e kB T In(1 + e e In(1 + kB T ek B In(1 + T

I0 ) IL IL ) I0 IL ) I0 IL ) I0

When light with an intensity of 600 W/m2 is incident on a solar cell, the short circuit current is I SC = 15 mA and the open circuit voltage is V OC = 0.5 V. How

5.4 Thermophotovoltaic (TPV) Cells

6.

7.

would the short circuit current and the open circuit voltage change if the light intensity is doubled? For a p-n junction diode (a) write down the charge neutrality condition for the depletion region. (b) When the p-n juncion is used as a solar cell, the current is given with the following   general   expression. eV I (V ) = I L − I0 exp k B T − 1 Here I L and I 0 are the photocurrent and the saturation current, respectively. Derive the open circuit voltage and the short circuit from this expression to show that:   VOC = k BeT In 1 + IIL0 and I SC = I L (c)Plot the current–voltage characteristics of the solar cell under illumination and find the point corresponding to the maximum power generation in your figure. Consider the wavelength dependences of the absorption coeficients in Si and Ge to answer the following questions. (a) (b) (c)

8.

191

Why is a Si solar cell said to be better than a Ge one? Explain whether it is possible to increase the efficiency of a Si solar cell by increasing the thickness of the active region? A Ge solar cell is placed below a Si window. How would the efficiency change compared to the case when the Si window is removed?

The parameters of a Si solar cell, measured at ambient temperature of T = 30 °C, are given below:Leakage current: I 0 = 3.0nA. Area: 1.7cm2 . Internal resistance = 0.8 . Photocurrent: I ph = 36 mA. Calculate the following parameters: (a) (b) (c) (d)

Calculate the open circuit voltage. Derive the relationship between the current and the load resistance. Calculate the maximum power extracted from the cell. Calculate the filling factor.

Reference 1. Rühle S (2016) Tabulated values of the Shockley-Queisser limit for single junction. Sol Energy 130:139–147

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5 Solar Cells (Photovoltaic Cells)

Suggested Reading List 2. Wurfel P (2009) Physics of solar cells, from principles to new concepts. Wiley 3. Stix M (2002) The Sun: an introduction. Springer 4. Shockley W, Queisser HJ (1961) Detailed balance limit of efficiency of p–n junction solar cells. J Appl Phys 32:510–519 5. Nelson J (2003) The physics of solar cells. Imperial College Press 6. Singh J (1995) Semiconductor optoelectronics: physics and technology. McGraw-Hill 7. Dwivedi BN, Narain U (2006) Physics of the sun and its atmosphere. World Scientific Publishing Co 8. ASTM Standard G 173-03 (2008) Standard tables for reference solar spectral irradiances: direct normal and Hemispherical on 37° tilted surface, ASTM International, West Conshohocken, PA, https://doi.org/10.1520/G0173-03R08, www.astm.org 9. Royall B (2011) GaInNAs/GaAs Multiple Quantum Well and n-i-p-i solar cells. PhD thesis, Essex University, UK 10. Kasap SO (2000) Optoelectronic devices and photonics. Pearson 11. Moller HJ (1993) Semiconductors for solar cells. Artech House 12. Fonash SJ (2010) Solar cell device physics. Elsevier

Chapter 6

Photodetectors

Learning Outcomes On completion of this chapter, readers will be able to: 1. 2. 3. 4. 5. 6.

have a general understanding of various types of photodetectors, demonstrate an understanding of the underlying physical processes in p-n junctions and pin photodetectors, design a simple photodetector and describe the operation wavelength describe parameters defining the detector efficiency, understand the principles of avalanche photodetectors and the physics of the amplifying mechanism, suggest design improvements to increase device efficiency.

The most important part of information processing is the detection of received information. In electronics this is done using Field Effect Transistors (FET), Bipolar Junction Transistors (BJT) and diodes. These devices all have high gain, low noise, tunability and act as powerful detectors for information. However, these devices are not suitable for detecting signals in optical frequencies of ν > 1014 Hz. The detection of optical information requires conversion of an optical signal to an electrical signal, which can then be processed by electronic devices. Photodetectors are the devices that receive optical signals and convert them into electrical signals and can be classified into two broad categories; thermal photodetectors and photon detectors, as listed in Table 6.1. Thermal detectors: When light is absorbed the temperature of the device increases and its conductivity, which is temperature dependent, changes. The electrical output is proportional to the change of temperature, therefore, on the absorbed optical power, so is usually independent of the incident light wavelength. Photon detectors: These detectors are wavelength selective and absorption occurs at and below a threshold wavelength, which is determined by the detector material used during fabrication. Photodetectors are very versatile devices and by using different types of materials it is possible to design detectors to operate between the ultraviolet (UV) and Infrared (IR) region of the electromagnetic spectrum. However, © Springer Nature Switzerland AG 2021 N. Balkan and A. Erol, Semiconductors for Optoelectronics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-44936-4_6

193

194

6 Photodetectors

Table 6.1 Classification of detectors Photon detectors

Thermal detectors

Photoconductive Developed based on intrinsic or extrinsic semiconductors

Bolometric Thermistor, metal, superconducting

Photovoltaic p-n, pin, Schottky junctions and heterojunctions

Pyroelectric Operated on the basis of temperature induced polarisation

Photoemissive Photoelectric effect, metal/metal and metal/semiconductor junctions

at IR wavelengths, when the energy of the absorbed photons is small enough to be compared with the thermal energy, thermal noise interferes with the operation of device. Therefore, IR detectors need to be cooled down for efficient noise- free performance. Optical detectors based on photon effects are mainly preferred in technological applications. In the following sections, there will be a brief review of the optical processes in semiconductors, which will then concentrate on p-n junction photodiodes with or without internal gain.

6.1 Optical Transitions in Direct Bandgap Semiconductors The operation of photodetectors utilising photon effects is based on the absorption of photons. In photon absorption, like with any other physical processes, energy and momentum must be conserved. In Sect. 1.5, we defined the direct and indirect bandgap semiconductors. In Fig. 6.1, we show schematically the dispersion curve of a direct bandgap semiconductor giving Fig. 6.1 Photon absorption process in a direct bandgap semiconductor

6.1 Optical Transitions in Direct Bandgap Semiconductors

195

the energy-momentum relationship. When  a photon with energy greater than the  bandgap of the semiconductor ω ≥ E g is incident on the semiconductor then it is absorbed to transfer an electron from the valance band to the conduction band, leaving behind a hole in the valance band. The conservation of energy requires that if a photon is absorbed with energy èω, then a hole and electron pair is created at energies at E 1 and E 2 , respectively so photon energy is: ω = E 2 − E 1

(6.1)

The conservation of momentum requires that: k2 − k1 = k photon

(6.2)

Here k photon is the photon momentum, k1 and k2 are the electron momentum at the initial and final states, respectively. Most semiconductors have bandgaps between a fraction of an eV and a few eV. For example, GaAs has a bandgap of 1.42 eV. In order for a photon to be absorbed by GaAs, its energy should be equal or greater than the bandgap of GaAs: E photon = ω = h

c ≥ Eg λ

(6.3)

Therefore, in terms of wavelength, this corresponds to λ = 0.87 μm. The wavevector of the photon is then: k photon =

2π λ

(6.4)

k photon = 7.22 × 106 m −1 The corresponding photon momentum is given by: k photon = 47.80 × 10−28 kg m/s. In Fig. 6.1, k1 and k2 are inversely proportional to the lattice constant which is ≈ 1010 m−1 . only a few Angstrom and have values in the order of k1 , k2 = 2π a As a result, during the absorption process the photon momentum can be neglected. Consequently the electron momentum remains unchanged and the optical transitions for absorption (k1 → k2 ) and emission (k2 → k1 ) are vertical in momentum space. Photodetectors and solar cells use the absorption process whilst light emitting diodes and lasers require the emission process. Semiconductors such as GaAs and InAs are direct bandgap materials. For optical transitions, within indirect bandgap semiconductors, the momentum is only conserved by significantly changing the electron wavevector. This is achieved by the emission or absorption of phonons that have small energies but a large wavevector. Therefore, optical transitions for indirect

196

6 Photodetectors

bandgap semiconductors are indirect, including both photon and phonon interactions of electrons. Semiconductors such as Ge, Si and AlAs are examples of indirect bandgap materials. In the following section, there will be a detailed investigation of the absorption process in direct bandgap semiconductors. The bandstructure of the direct bandgap semiconductor in Fig. 6.1 is assumed to be parabolic around k ~ 0, therefore, the relationship between the energy and momentum is: E∝

2 k 2 2m ∗e

This expression is only valid for the small values of k. At larger k values, the dispersion curve deviates from parabolicity. However, we may assume that absorption occurs in the vicinity of the conduction and valance bandedges, so therefore the theory that the dispersion relation is parabolic is still plausible. With reference to Fig. 6.1, we may now express the energies E 1 and E 2 in terms of the conduction (E C ) and valance bandedge (E V ) energies, respectively: E2 − EC =

2 k22 2m ∗e

(6.5)

E V − E1 =

2 k12 2m ∗h

(6.6)

k2 − k1 = k photon ≈ 0 and E 2 − E 1 = ω

(6.7)

and  k2 =

  2m r ω − E g 2

(6.8)

Here mr is the reduced mass: mr =

m ∗e m ∗h m ∗e + m ∗h

(6.9)

In absorption and emission processes, the transition rates from energy states E 1 to E 2 or E 2 to E 1 depends on the probability of transition (A), the joint density of states (B) and the relative occupancy of the states (C) namely: Transition rate = A × B × C A is related to the transition matrix element via Fermi’s golden rule and to a first approximation, we may assume that this is independent of energy and can be called Γ.

6.1 Optical Transitions in Direct Bandgap Semiconductors

197

The density of states expressions were given in Sect. 2.1 with Eq. 2.10. In order to work out the joint density of states, we replace E with ω and substitute the reduced mass into Eq. 2.1 to obtain the joint density of states as N (E)d E =

 1/2 (2m r )3/2 ω − E g d(ω) 2π 2 3

(6.10)

During the absorption process, an electron absorbing a photon transfers from E 1 to E 2 leaving behind a hole. Therefore, E 1 level should be full and E 2 level should be empty. According to Fermi–Dirac statistics, the probability that the level E 1 is full is given by (f 1 ) and the probability that E 2 is empty is given by (1−f 2 ). Therefore, the absorption rate at photon energy ω is defined as. Absorption rate = A × B × ( f 1 ) × (1 − f 2 ). In order for a photon to be absorbed to transfer an electron from E 1 to E 2 , the necessary condition is: f 1 = 1 and 1 − f 2 = 0. We obtain the absorption coefficient: 1/2  α(ω) ∝  ω − E g

(6.11)

Equation 6.11 is valid for absorption involving energies near the direct bandgap.

6.2 Choice of Material and Wavelength of Operation As the definition of absorption coefficient has been established then we may now consider the choice of semiconductor material for a device that will operate at a desired wavelength range. These investigations are based on a p-n or pin junction detectors. In such devices, the operation is due to the interband (valance to conduction band) absorption of light in the of the p-n junction, where the photo-generated electron and hole pairs are spatially separated under the influence of the built-in electric field before they recombine. Let us assume that Fig. 6.2 represents the conduction and valance band energies separated by the bandgap E g . When a photon with energy ω > E g is absorbed, it transfers an electron from the valance band into the conduction band, leaving a hole in the valance band as described previously. When an external reverse bias voltage is applied to the electrodes of detector, the built-in voltage will increase the photo-generated electron hole pairs and they will move towards their respective electrodes (electrons to the anode and holes to the cathode). Therefore, when the anode and cathode are connected to an external circuit there will be a current flow. The amount of current that flows in the external circuit depends on various factors such as photon intensity, photon to charge carrier conversion factor (quantum efficiency) and wavelength of the incident photons.

198

6 Photodetectors

Fig. 6.2 a Absorption of photons by a semiconductor and the motion of the photo-generated electron hole pair in an electric field. b The flux of photons as they propagate within the detector

Cut-off wavelength: This is the wavelength of photons above which they cannot be absorbed by the semiconductor. It is determined by the bandgap energy of the semiconductor from Eq. 6.3, as the cut-off wavelength: λc =

hc Eg

(6.12)

Photon flux: As previously discussed in Chapter 5, the photon flux is defined as the number of photons incident on the detector per unit area per second (0 ): 0 =

I hν

(6.13)

Here I is the optical energy incident on the detector per unit area per second and hν is the energy of each photon. When the photons propagate through the detector, their flux will be reduced as a result of the absorption, as shown in Fig. 6.2b.

6.2 Choice of Material and Wavelength of Operation

199

If we assume that there is no reflection at the surface and all the incident photons are absorbed in the semiconductor, the flux at a depthx will be: (x) = 0 exp(−αx)

(6.14)

Here α is the absorption coefficient, and from Eq. 6.1 when hν < E g or λ > λC , there are no photons are absorbed by the semiconductor then it becomes α = 0. In Chap. 5, we investigated the wavelength dependence of the absorption coefficient for various semiconductors and showed that when the wavelength exceeds the cut-off wavelength then the absorption coefficient goes to zero. If each absorbed photon creates an electron–hole pair in equilibrium, we can write the generation rate G(x) for each carrier type as: G(x) + G(x) = −

d =0 dx

d = α 0 exp (−α x) = α (λ)(x) dx

(6.15)

Since we would wish all incident photons to be absorbed within a short depth of the semiconductor then the absorption coefficient is desired to be high at the operating wavelengths. We may, therefore, choose a semiconductor for   which the cut-off wavelength is much longer that the photon wavelength λop < λC . Therefore, the absorption coefficient: α(hv)op >> 0 This will enable all the incident photons to be absorbed by the detector. Also, because the absorption coefficient is high then most photons will be absorbed close to the surface of the photodetector. According to Eq. 6.15 most carriers will be generated at a short depth from the surface therefore, the photodetector will not be too thick and unnecessarily bulky. As long as the photo-generated carriers are not created too close to the surface where they can re-combine with the surface sates, resulting in a loss of mobile carriers, then reduced thickness is an important requirement for photodetectors. This is associated with the RC time constant of the detector. If the detector is long, its resistance will be high and therefore the RC time constant (response time) will be high. A long RC time constant is not desirable and can be detrimental for optical communication systems, especially where high speed operation is essential to receive digital signals arriving at frequencies greater than the GHz range.

200

6 Photodetectors

6.3 Operating Temperature In an ideal photodetector, the photo-generated carrier density must be much higher than the thermally generated free carrier density. At a finite temperature, under illumination, there are two components in the total detector current density: J = Jt + J f

(6.16)

Here J t is current density due to the thermally generated carriers and J f is the current density due to the photo-generated carriers. In order to have a sensitive photodetector the photo-generated current density should be much higher than that of the thermally generated carriers (J t < < J f ). However, for a given semiconductor this is not always an easy condition to satisfy, especially when the detector is designed for operation at long photon wavelengths, hence using small bandgap semiconductors. We can write down the thermally generated current density as:   Jt = e n t μn + pt μ p E

(6.17)

Here nt , pt , μn , μp and E are the thermally generated electron and hole densities, electron and hole mobilities and the electric field, respectively. For simplicity, if we concentrate on the electrons only and write down the thermally generated electron density from Eq. 2.28 in Sect. 2.3.1 then:   Eg n t = n i = (NC N V )1/2 exp − 2k B T

(6.18)

In order to preserve the charge neutrality condition free the hole density will be equal to free electron density. In Eq. 6.18 N C and N V are the effective density of states. According to Eqs. 6.16–6.18, at temperatures above the room temperature, there will be a large density of thermally generated carriers across the bandgap, giving rise to a large dark current (or the leakage current as defined on the p-n junction analysis). For low photon intensities, the thermal current and the photo-generated current become comparable. In detectors for long wavelength operation, this may be a more serious problem and dark current may actually exceed the photocurrent. One solution is to cool the detector and reduce the thermal current. According to Eq. 6.18, when the detector is cooled the thermal current will be reduced. However, there is another problem caused by the increasing bandgap of semiconductors with decreasing temperature as given by the semi-empirical Varshni Equation: E g (T ) = E g (0) −

αT 2 β+T

(6.19)

6.3 Operating Temperature Table 6.2 Varshni parameters and bandgaps of various semiconductors T = 0K [1]

201 Semiconductor

E g (0) eV

α (eV/K) × 10–4

β (K)

GaAs

1.519

5.04

204

InP

1.422

3.6

162

Ge

0.8893

6.842

398

GaP

2.88

5.77

372

GaSb

0.812

4.7

94

InAs

0.417

2.76

93

InSb

0.235

3.2

170

Here E g (T ) and E g (0) are the bandgaps of the semiconductor at a finite temperature T and T ~ 0 K. The Varshni parameters, α and β, are related to the thermal expansion and Debye temperature of the semiconductor, respectively and their values are listed for various semiconductors in Table 6.2. When the detector is cooled, the bandgap increases and the cut-off wavelength for absorption decreases so the condition in Eq. 6.12 will no longer be satisfied. As a result, one has to compromise between the detector sensitivity and operating wavelength and this is clearly demonstrated in the example below. Table 6.3 lists the cut-off wavelengths and the operating temperatures of semiconductor materials for detectors to be operated between IR and UV regions of the electromagnetic spectrum. Cooling the detector decreases the thermally excited carrier density, thus the thermal component of the current. This is fine because cooling the semiconductor will also increase its bandgap but the detector will no longer absorb the long wavelength photons as originally designed. So we have to compromise between the dark current and operating temperature to optimise the device operation. Table 6.3 Semiconductor materials, their cut-off wavelengths and operating temperatures [2]

Material

λc (μm)

Operating temperature (K)

ZnS

0.35

300

GaP

0.55

300

GaAs

0.86

300

Si

1.2

300

GaInAs

1.3

300

GaInAsP

1.5

300

Ge

1.8

> N D , this can be given as.   2εVbi N A + N D 1/2 e NA ND 

2εVbi 1/2 W ≈ Wn = eN D

W = Wn + W p =

(6.32)

210

6 Photodetectors

If d is replaced with W in Eq. 6.31, then junction capacitance per unit area of the p-n junction can be obtained using Eq. 6.32 as C = A



eε N D 2Vbi

1/2 (6.33)

If the donor density is much higher than the acceptor density (N D >> N A ), the depletion layer width is:

W ≈ Wp =

2εVbi eN A

1/2 (6.34)

And the junction capacitance per unit area is: C = A



eε N A 2Vbi

1/2 (6.35)

Obviously, under operating conditions of the photodetector, when the diode is reverse biased, V bi should be replaced with V bi + V r . If the incident light on the detector is pulsed, refer to Fig. 6.9, the detector will generate a time-dependent output voltage, V D , on the load resistor as VD = Vo (1 − e−t/τ )

(6.36)

Here the time constant τ is the sum of the RC time constant (τ RC = RC) and the transit time of the photo-generated carriers through the depletion layer (τtr = W/ νs ): τ = τ RC + τtr

(6.37)

The rise time or the response time of the photodetector is: τ R = (ln 9)(τ RC + τtr )

Fig. 6.9 Response of a photodetector to a pulsed incident light

(6.38)

6.6 Rise Time and Bandwidth

211

And the bandwidth is: f =

1 2π (τ RC + τtr )

(6.39)

In a well-designed photodetector, τ RC >> τtr , therefore,. τ RC ≈ τ . In high speed applications, it is important to employ photodiodes with small junction capacitance. For example, in optical communication systems, a photodiode with a long RC time constant will not be able to receive the high speed optical pulses resulting in the loss of information. In pin diodes, because the depletion layer is wider, the junction capacitance is appreciably smaller that the p-n counterparts, therefore they are more suitable for high speed applications. However, if the depletion region (absorption region) is wide, although the quantum efficiency of the photodetector will be higher, device resistance and transit time of the carriers will also increases. In the design of a photodetector, both factors should be taken into account for the best possible performance of the device for any particular application.

6.7 Avalanche Photodiodes Avalanche photodiodes (APD) have internal gain and are used in the detection of weak signals. They are more heavily reverse biased than the standard p-n and pin photodiodes. They have internal gain due to the carrier multiplication process with a typical current gain of M ~ 10 and 100. Therefore, they have a higher sensitivity, but more noise due to the random nature of the avalanche process. A special Si avalanche photodiode, designed for one type of carrier multiplication is shown in Fig. 6.10. The high field region is at the p-n+ junction where, because of the high density of positively charged ionised donor atoms, the electric field is very high (~106 V/cm). The carrier multiplication process takes place in this very high field region. The absorption of incident photons takes place in the intrinsic region where there is an almost constant electric field (~104 V/cm). The photo-generated electrons in the intrinsic region are accelerated towards the high field region and the holes are accelerated towards the p+ region. Electrons gain enough energy from the very high electric field and trigger the impact ionisation and avalanche process. The secondary electrons generated by the impact ionisation create further electron hole pairs and move towards the anode. The secondary holes generated by impact ionisation move in the opposite direction, through the intrinsic region to the cathode. Therefore in this type of APD, the avalanche multiplication is due to only one type of carriers (electrons) and has much better noise figures. As discussed previously, impact ionisation is the process in which high energy electrons (Fig. 6.11a) and holes Fig. (6.11b) lose their energy to create new electron hole pairs by transferring an electron from the valance band into the conduction

212

6 Photodetectors

Fig. 6.10 The structure of a Si avalanche photodiode. The absorption region is the intrinsic region between p+ and p and the high field region is at the p-n+ junction

Fig. 6.11 Impact ionisation process in a direct bandgap semiconductor a with electrons and b with holes

6.7 Avalanche Photodiodes

213

band. The requirement for the conservation of energy and momentum in the impact ionisation process imposes a minimum kinetic energy for the electron and hole impact ionisations as   m∗ e (6.40a) = Eg 1 + ∗ e ∗ E min me + mh   m∗ h E min (6.40b) = Eg 1 + ∗ h ∗ me + mh Impact ionisation rates are described by the impact ionisation coefficients, α e and αh . Electron impact ionisation coefficient, α e (1/cm) or α h (1/cm), is defined as the number of electron–hole pairs per length. Impact ionisation coefficients depend on the energy of the carriers, therefore, in general α e = α h . Figure 6.12 shows the impact ionisation leading to the carrier multiplication via the avalanche process. The gain in the carrier numbers, thus in the photocurrent, depends on the reverse bias voltage Vr: M=

1−

1 n Vr VB

(6.41)

Here n > 1 is an experimentally obtained parameter, V B is the diode breakdown voltage, the value of which may vary depending on the diode structure between 20 and 500 V. The photocurrent after the gain M is:

Fig. 6.12 Carrier multiplication through avalanche process in a reverse biased pin structure

214

6 Photodetectors

Table 6.4 Photodetector characteristics [2] Photodetector

Structure

Response time (ns)

Operation wavelength (nm)

Dark current (nA) Gain

Si

p-n junction

0.5

200–1100

0.01–0.1

0 K and N 2 < N 1 . In thermal equilibrium N2 = e−(E2 −E1 )/ k B T = e−hν/k B T N1

(7.27)

Therefore, in thermal equilibrium, there will be no gain and the electromagnetic wave will be absorbed as it propagates through the medium. Both absorption and gain conditions are shown in Fig. 7.3. Example Calculate the gain coefficient of a ruby laser with the specifications for its central frequency line using the parameters given below. N2 − N1 = 5.0 × 1017 cm −3 . ∼ 1 = 2.0 × 1011 H z At T = 300K ν = g(ν0 )

228

7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.3 Propagation of an EM wave in a medium when (a) it is amplified as a result of population inversion in the medium and (b) it is absorbed ( “•” indicates atoms in the excited state and “o” are atoms in the ground state)

τ21 = 3 ms ν = 4.326 × 1014 H z. c/n = 1.7 × 1010 cm/s We can use Eq. 7.26 to obtain the gain coefficient: γ (ν) ≈ 5 × 10−2 cm −1 Therefore the intensity of the wavelength corresponding to the central frequency line is amplified by 5% per cm when it is propagating in ruby laser rod. As previously stated, the coefficient γ given by Eq. 7.26 is positive only when there is a population inversion in the medium. A population inversion can be achieved by either electrical pumping via injecting non-equilibrium carriers by driving a current through the system or by optical pumping, for example using flash lamps. The pumping process is shown schematically in Fig. 7.4 for a two level system. Let us now investigate the rate equations for absorption and emission for the two level system. Assume that at any given time the total number of atoms in the system Fig. 7.4 Pumping in a two-level system and the spontaneous and stimulated emissions processes

7.1 Definitions

229

is N = N1 + N2 and the degeneracy of both levels is the same. Rate equitations are written as d N1 i (N2 − N1 ) = −W p (N1 − N2 ) + W21 N2 + W21 dt

(7.28a)

d N2 i = W p (N1 − N2 ) − W21 N2 − W21 (N2 − N1 ) dt

(7.28b)

N 1 and N 2 are the populations in levels 1 and 2, respectively and W p is the pumping rate. In the steady state: d N1 d N2 = =0 dt dt

(7.29)

i W p (N1 − N2 ) − W21 N2 − W21 (N2 − N1 ) = 0

(7.30)

From Eq. 7.28b:

and  N1 = N2 1 +

W21 i W p + W21

 (7.31)

In thermal equilibrium, N 2 < N 1 , and when W21 → 0 the maximum value of the population of level 2 is (N2 )max = N1 . Therefore, a population inversion cannot be achieved optical gain in two-level system. Population inversion can occur in a three-level atomic system. Indeed the first ever laser demonstrated, the ruby laser, was a three-level system. Figure 7.5 illustrates a three level atomic system which can be used to demonstrate population inversion. In such a system there is a metastable (semi-stable) intermediate level (level 2) where atoms spend longer time in comparison to levels 1 and 3. Stimulated emission occurs via transition from the metastable level to the ground level. Pumping excites the electrons from ground level to the E 3 level and the transition rates of electrons from this level to E 2 (non-radiatively), or to E 1 level (radiatively) are very fast. So, at any given time, there is always a high density of available Fig. 7.5 Energy levels and transition rates for a 3-level system

230

7 Light Emitting Diodes and Semiconductor Lasers

(empty) states in level E 3 . As the lifetime of atoms at level E 2 is much higher than that at level E 3 , then atoms will accumulate at level E 2 and consequently population inversion is achieved, which means that the E 2 population exceeds that at the ground level. Stimulated emission will occur as a result of transitions from E 2 to E 1 . i is the stimulated emission In Fig. 7.5, W p is the pumping rate (absorption rate), W21 rate, W 31 and W 21 are the 3 → 1 and 2 → 1 spontaneous emission rates, respectively. Assume that the total number of atoms in the system is N = N1 + N2 + N3 . The rate of change of the atomic density at level 1 is given by: d N1 i = W21 N2 + W21 (N2 − N1 ) + W31 N3 − W p (N1 − N3 ) dt

(7.32)

In this expression the first term on the right hand side is the total spontaneous emission per unit time. The second term is the total stimulated emission rate corresponding to the 2 → 1 transition and the third term is the total spontaneous emission rate for the 3 → 1 transition. The last term is the total absorption rate relating to the 1 → 3 transition. The rate of change of the atomic density for level 2 is then: d N2 i = W32 N3 − W21 N2 − W21 (N2 − N1 ) dt

(7.33)

Here the first term is the total non-radiative transition rate for the 3 → 2 transition. Similarly, the rate of change of the atomic density for level 3 is: d N3 = W p (N1 − N3 ) − N3 γ32 − γ31 N3 dt

(7.34)

In the steady state: d Ni = 0 (i = 1, 2, 3) dt

(7.35)

We can now re-arrange Eq. 7.32 to obtain: i W21 N2 + W21 (N2 − N1 ) + W31 N3 − W p (N1 − N3 ) = 0

(7.36)

Then Eq. 7.33 becomes: i (N2 − N1 ) = 0 W32 N3 − W21 N2 − W21

(7.37)

And Eq. 7.34 is now: W p (N1 − N3 ) − N3 γ32 − γ31 N3 = 0 We are now in the position to substitute:

(7.38)

7.1 Definitions

231

N3 = N − (N1 + N2 )

(7.39)

into Eq. 7.38 to obtain:





 N1 2W p + W31 + W32 + N2 W p + W31 + W32 − N W p + W31 + W32 = 0 (7.40) If we substitute Eq. 7.39 into Eq. 7.37 this gives:

i 

i  − W32 − N2 W21 + W21 + W32 = −N W32 N1 W21

(7.41)

N 2 - N 1 can be solved from Eqs. 7.40 and 7.41:

(W32 − W21 ) W p − W21 (W31 + W32 )  i 

 N2 − N1 = N

i W p + W31 + W32 2W21 + W21 + W p W21 + W21 + W32 (7.42) In conclusion, in order for a three level system to work as a laser the right hand side of Eq. 7.42 must be positive. Consequently, the rate of increase in the population of level 2 must be greater than the rate of decrease in its population (W32  W21 ). Furthermore, the population of level 3 must be negligibly small at any given time i . (N 3 → 0). This implies W32  W p , W21 The next stage, as shown in Sect. 7.1.3, is to place the laser in an optical cavity (resonator) to provide the optical feedback. Photons in the cavity are subject to losses such as absorption, reflection and scattering. The value N2 − N1 in Eq. 7.42 must be above a critical value to overcome the losses. Let us call this the critical value, or the threshold value Nt . Nt = N2 − N1

(7.43)

In a 3-level system (N3 → 0): N1 + N2 = N

(7.44)

Consequently, from Eqs. 7.43 and 7.44, we can obtain: N1 =

Nt N − 2 2

(7.45a)

N2 =

N Nt + 2 2

(7.45b)

and

232

7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.6 The four energy levels for the He and Ne atoms in a He–Ne gas laser. Lasing occurs as result of transitions between E 3 and E 2 levels

Therefore, in a 3-level system, the lasing threshold for the population of the excited level N 2 is given by Eq. 7.45b. It is clear that if Nt is very small then the excited level population density must at least be half the total population density. The transition energies in a He–Ne laser are shown in Fig. 7.6. The He–Ne laser is a 4-level system and requires small pumping rates for laser action. Here atoms are pumped from the ground level to the highest excited levels. The high energy He atoms collide with Ne atoms to excite the Ne atoms into the metastable E 3 level. Since the populations at levels E 4 and E 2 remain more or less unchanged, there will be a population inversion at level E 3 and the lasing action will occur as a result of transitions from E 3 to E 2 .

7.1.3 Optical Feedback and Laser Oscillations The two conditions for lasing are firstly the generation of photons with stimulated emission, which are in-phase with each other, and secondly the multiplication of the photon density within the optical cavity. An optical cavity, resonant cavity or optical resonator is an arrangement of mirrors surrounding the gain medium to provide optical feedback. The two highly reflecting mirrors are placed on either side of the cavity where the incident photons are reflected back to stimulate the generation of more photons. This feedback mechanism continues many times until there is enough gain to overcome the losses, thus a high enough density of coherent photons to be emitted through the less reflecting mirror. The optical cavity acts as a Fabry-Pérot resonator (Fig. 7.7) where the mirrors have to be perpendicular to the active region and must be absolutely parallel to each other. Stable lasing is achieved only when the optical gain within the cavity exceeds the losses such as scattering, absorption and mirror losses. Lasing is not perfectly

7.1 Definitions

233

Fig. 7.7 Fabry-Pérot cavity with plane mirrors

monochromatic but occurs within a small frequency band. The dominant laser frequency is determined by the average energy of the levels between which the stimulated emission transition occurs. Let us now derive the expressions for the available frequencies within the emission spectrum for a Fabry-Pérot cavity. Light confined in the cavity is reflected multiple times producing standing waves for certain resonance frequencies. Standing waves form when the population inversion is achieved within the gain medium. The standing wave patterns produced are called modes; longitudinal modes differ only in frequency while transverse modes change for different frequencies and have different intensity patterns across the cross section of the beam. The resonance condition occurs when the cavity length is equal to multiple integers of the half wavelength of the electromagnetic wave, so if we consider an optical cavity with length L, then the resonance condition is: L=

λq 2n

(7.46)

Here λ is the wavelength, n is the refractive index of the gain medium and q is an integer. Since λ = c/ν, ν=

cq 2n L

(7.47)

where c is the speed of light. The difference between the frequencies of consecutive longitudinal modes can be obtained via: cq c(q + 1) and νq+1 = 2n L 2n L cq c c(q + 1) − = νq+1 − νq = ν = 2n L 2n L 2n L c q ν = 2n L

νq =

(7.48)

The wavelength separation of the modes is: λ =

dλ λ2 dq = q dq 2Ln

(7.49)

234

7 Light Emitting Diodes and Semiconductor Lasers

It is clear from Eqs. 7.48 and 7.49 that for longer cavity lengths there will be many modes within the cavity. Example The cavity length of a He–Ne laser is 30 cm. What is the wavelength separation between the modes? The emission wavelength of a He–Ne laser is 632.8 nm. We can use Eq. 7.49 to calculate the mode separation: λ =

(632.8 × 10−9 m)2 λ2 = = 6.7 × 10−13 m 2Ln 2 × 1 × 0.3m

Example A ruby laser consists of a 4 cm ruby crystal with a refractive index of 1.78. The emission wavelength is 550 nm. Calculate the number of longitudinal modes and their frequency separation within the cavity. The number of modes can be calculated using Eq. 7.46: q=

2 × 1.78 × 0.04m 2n L = = 2.6 × 105 λ 550 × 10−9 m

and from Eq. 7.48 the frequency separation of the modes is: ν =

c 3 × 108 m/s = = 2.1 × 109 H z = 2.1 G H z 2n L 2 × 1.78 × 0.04m

It is clear from this example that there are a large number of modes within the cavity. However, not all of these modes will be present in the emission. Lasers will have only those modes which are within its spectral gain curve as depicted in Fig. 7.8. Laser oscillations also exist in a direction perpendicular to the cavity axis and these are known as transverse modes. Whilst the longitudinal modes appear as single spots in the emission, the transverse modes may form a few different spots. The transverse mode behaviour for a laser are synonymous with the transverse modes in waveguides, and are determined by the laser intensity distribution along its cross-sectional area. The allowed transverse modes in the cavity are represented by the acronym, TEMmn . Here T, E and M stand for transverse, electric and magnetic (fields) respectively where n and m are both integers. The lowest order mode is TEM00 with a Gaussian distribution of intensity along the cross section. The m and n numbers represent the number of zeros (or minima) in orthogonal directions within the laser light, as depicted in Fig. 7.9. For example TEM01 has two bright spots separated by a single minimum. TEM11 has two minima on either of the two orthogonal axes, dividing the light into four spots. TEM00 is the ideal mode for a laser because the Gaussian profile remains unchanged as the electromagnetic wave propagates along the emission axis. In order to design a laser with a single TEM00 mode, a restricting window with a diameter much smaller than the spot size of the TEM00 needs to be placed on the cavity axis. Another approach for a single transverse mode is to design the laser cavity so that

7.1 Definitions

235

Fig. 7.8 a Longitudinal modes in the cavity and b lasing modes within the gain spectrum

its diameter is smaller than the TEM00 spot size. Most laser cavities are designed for TEM00 mode only. However in very high power applications they may have higher order transverse modes. In summary, the number of transverse modes increases with the cavity diameter. The number of longitudinal modes and their frequency separation depends on the cavity length. A high number of transverse modes results in the poor divergence characteristic of the laser light, so in order to design a single longitudinal mode laser, the cavity length has to be kept very small, within the same order of magnitude as the lasing wavelength. However, for such small cavities the optical gain will also be reduced therefore, small cavity lasers need to have very highly reflecting mirrors to ensure multiple reflections with minimal mirror losses and this is achieved using semiconductor lasers.

7.1.4 Threshold Condition for Laser Oscillations Population inversion alone is not sufficient to achieve lasing. In order for stable laser oscillations to be maintained the gain must at least be equal to the losses in the medium. This is known as the lasing threshold condition. We will now derive the

236

7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.9 The transverse modes in laser light

appropriate equations for the threshold condition by studying the intensity of the electromagnetic wave propagating within the gain medium. In order to calculate the photon losses with the cavity, let us consider the multiple reflections as depicted in Fig. 7.10. The active region within the cavity is where the photon multiplication occurs via stimulated emission. Assume that an electromagnetic wave is incident on one of the cavity walls with amplitude E 0 and propagates

7.1 Definitions

237

Fig. 7.10 a Schematic representation of a Fabry-Pérot cavity showing the reflection and transmission of the electromagnetic wave where r 1 and r 2 are the reflection coefficients and t 1 and t 2 are the transmission coefficients at the semiconductor air interfaces, b the amplitude of the propogating electromagnetic wave within the cavity

along the cavity. With reference to Fig. 7.10 the transmitted, E t and reflected, E r wave amplitudes are given as: E t = E 4 + E 10 + . . . . =

t1 t2 A E0 1 − A2 r12

  r 1 t1 t2 A 2 E0 Er = E 2 + E 7 + . . . . . . .. = r2 + 1 − A2 r12

(7.50)

(7.51)

Here r 1 = r 2 are the reflection coefficients, t 1 = t 2 are the transmission coefficients at the semiconductor-air interfaces, respectively and A is the amplitude gain. In Eqs. 7.50 and 7.51, the electromagnetic wave amplitudes are: E 1 = t2 E 0 ,

E 2 = r2 E 0 ,

E 3 = AE 1

E 4 = t1 E 3 ,

E 5 = r1 E 3 ,

E 6 = AE 5

(7.52)

The amplitude gain of the electromagnetic wave during one transit along the cavity of length L is given as: A = exp

 γ

t

2

 + ik L

(7.53)

Here γt (net gain) includes the gain within the cavity (γ ) and the loss (α) and is given by: γt = γ − α

(7.54)

238

7 Light Emitting Diodes and Semiconductor Lasers

Laser oscillations starts when the denominator in Eq. 7.50 is zero and E t /E i tends to infinity: A2 r12 = 1

(7.55)

This corresponds to the threshold gain γth where γt = γth . We can now use Eq. 7.53 together with the threshold condition to define the threshold gain: exp 2

 γ

 1 + ik L = 2 2 r1 t

(7.56)

From the real part of Eq. 7.56: 2

γ t

2

L = ln

1 r12

γt = γth − α =

1 ln r1−2 L

We can substitute R = r12 γth = α −

1 ln R L

(7.57)

This expression is known as the amplitude condition for lasing and is achieved when the gain per unit cavity length is equal to the absorption and reflection losses. We can now use the imaginary part of Eq. 7.56 to obtain the phase condition for lasing: e2ik L = 1 and 2k L = 2π m

(7.58)

and k=

2π π = m λ L

⇒

L=

λ m 2

(7.59)

where m is an integer. This implies that the phase condition for lasing is achieved when the photons are in-phase at the end of each transit of the electromagnetic wave within the cavity. We can now substitute Eq. 7.26 into 7.57 to obtain the population inversion condition at the lasing threshold:

7.1 Definitions

239

  1 8π n 2 τ21 α − ln R Nt = (N2 − N1 )t = g(ν)λ2 L

(7.60)

Here the subscriptt stands for threshold.

7.2 Optical Processes in Semiconductors So far the optical processes and lasing in atomic systems have been considered where transitions occur between discrete energy levels. It is now possible to move on to similar processes in semiconductors where the discrete energy levels are replaced with energy bands. In this case the population inversion condition is similar in principle to atomic systems but requires a different approach. The choice of semiconductor material is a fundamental design consideration for LEDs and lasers. As far as the energy bands are concerned there are two types of semiconductors, direct bad gap and indirect bandgap. With reference to the dispersion curve (energy–momentum) diagrams, a semiconductor is known to be direct bandgap if the conduction band minimum and the valance band maximum are at the same momentum (k) value. As energy and momentum are conserved in all the optical transitions then the photon momentum is negligibly small compared to the crystal momentum. The transitions involving absorption or emission of photons therefore, are vertical in the E-k diagram. This requirement is easily met with direct band-gap semiconductors. For indirect band semiconductors, the electrons and holes have different k values at the conduction band minima and valance band maxima. In order to conserve momentum in optical transitions, the absorption or emission of long wavelength phonons will be necessary as shown in Fig. 7.11. Emission or absorption of long wavelength phonons (acoustic phonons) is a very slow process. Therefore, for the indirect bandgap semiconductors, radiative recombination time (10–2 –10−4 s) is orders of magnitude higher than that in direct bandgap semiconductors (10–8 –10−9 s). Therefore an electron in an indirect bandgap semiconductor may easily recombine non-radiatively with impurities or traps before it emits a phonon to reach the same momentum value as the hole with which it can recombine radiatively. Consequently, radiative efficiency for indirect bandgap semiconductors is very small and they are not suitable for LEDs or lasers. Table 7.1 lists the recombination coefficients of various direct and indirect bandgap semiconductors where, smaller recombination coefficients imply less radiative efficiency. Spontaneous emission occurs via the recombination of non-equilibrium electrons in the conduction band with holes in valence band. The excitation can be due to optical pumping when photons with energies equal to or greater than the bandgap are absorbed to create the non-equilibrium electron hole pairs. If the energy of the 

absorbed photons is much greater than the bandgap of the semiconductor hν0 > E g , electrons will be excited to high energy states in the conduction band (Fig. 7.12a). High energy electrons will lose their excess energy by emitting short wavelength

240

7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.11 E-k curves (bandstructure) for a direct and b indirect bandgap semiconductors

Table 7.1 Recombination coefficients for direct and indirect bandgap semiconductors [1]

Semiconductor material

Bandgap (eV)

Recombination coefficient (cm−3 s−1 )

GaAs

Direct: 1.43

7.21 × 10–10

GaSb

Direct: 0.73

2.39 × 10–10

InAs

Direct: 0.35

8.5 × 10–11

InSb

Direct: 0.18

4.58 × 10–11

Si

Indirect: 1.12

1.79 × 10–15

Ge

Indirect 0.67

5.25 × 10–14

GaP

Indirect 2.26

5.37 × 10–14

optical phonons to relax to the bottom of the conduction band (Fig. 7.10). The optical phonon emission occurs on a time scale (~10−12 s) much shorter than the e–h recombination time (~10−9 s). Therefore high energy electrons relax in energy before radiatively recombining with the holes. High energy non-equilibrium holes also undergo the same process. Before their recombination, electrons and holes are distributed in their respective bands as shown in Fig. 7.12b. The energy of the emitted photon is:  hν =

2 k 2 EC + 2m ∗e

= Eg +

2 k 2 2m r∗





2 k 2 − EV − 2m ∗h



(7.61)

7.2 Optical Processes in Semiconductors

241

Fig. 7.12 a Absorption and spontaneous emission processes. b Electron and hole distributions within the conduction and valence bands prior to recombination

where m r∗ is the reduced electron–hole mass. Equation 7.61 is also known as the joint dispersion relation and is the result of the joint density of states: N j (E) =

(2m r∗ )3/2 (E − E g )1/2 2π 2 3

(7.62)

If the density of electron and holes pairs is not too high, their distribution can be approximated to the Maxwell–Boltzmann distribution:   E f (E) = exp − kB T

(7.63)

The spontaneous emission rate is proportional to the product of Eqs. 7.62 and 7.63 and can be written as:   hν − E g (7.64) I (E = hν) ∝ (E − E g )1/2 exp − kB T Spontaneous emission starts at hν = E g and the maximum intensity occurs at an energy E g + 21 k B T (Fig. 7.13). Let us now consider the population inversion for stimulated emission at T = 0 K and T > 0 K as depicted in Fig. 7.14. Since the injected electron–hole pairs are non-equilibrium carriers, the single Fermi level in thermal equilibrium is replaced

242

7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.13 Theoretically expected spontaneous emission spectrum

Fig. 7.14 Density of states (- - -) and the energy dependence of electron density as determined by Fermi–Dirac distribution function. a Thermal equilibrium at T = 0 K. b Population inversion at T = 0 K. c At T > 0 K

with two quasi-Fermi levels for each type of carrier as described in Sect. 3.4. We can represent the non-equilibrium condition with the quasi-Fermi levels, E Fn and E F p . In other words, the conduction band is filled with electrons up to the level E Fn and the valance band is empty down to the level E F p . If we compare the situation here with the atomic systems, the energy

levels E 1 and E 2 in the atomic system are now broadened to (E C → E Fn ) and E F p → E V . Now N2 is the total electron number in the conduction band and N1 is the total number of holes in the valence band. At finite temperatures for T > 0 K, the distribution of the carriers spreads up to higher energies because of excess thermal energy. The probability of a state in the conduction band, or valence band, to be full is given by the quasi-Fermi- Dirac distribution function:

7.2 Optical Processes in Semiconductors

243



f C (E) = exp



f V (E) = exp

1 E−E Fn kB T

1 E−E F p kB T





+1

+1

(7.65a)

(7.65b)

We shall now derive the expression for the emission rate of photons with energies hν, associated with the radiative recombination of electrons at level E in the conduction band with holes at level (E − hν) in the valence band. In our calculation of emission rate, we need to know the density of (initial) occupied states at E in the conduction band ( f C NC ) and the density of (final) empty states at (E − hν) in the valence band (1 − f V )N V . However, in order to calculate the absorption rates between the two levels we also need to know the density of (initial) occupied states at (E − hν) in the valence band ( f V N V ) and density of (final) empty states at E in the conduction band (1 − f C )NC . Transition rates for absorption (W12 ) spontaneous emission (W21 ) and stimulated i ) over all energies are: emission (W21  W12 = B12

(1 − f C ) f V NC N V N f d E

(7.66)

 W21 = A21

f C (1 − f V ) NC N V d E

(7.67)

FC (1 − f V ) NC N V N f d E

(7.68)

 i = B21 W21

Here N f is the density of photons. We can make the plausible assumption that, for laser operation, the spontaneous emission can be neglected when B12 = B21 , the net optical gain becomes:  W12 −

i W21

= B21

N f ( f C − f V )NC N V d E

(7.69)

It is clear that for the optical gain to be positive the conditions that f C > f V and E Fn > E F p must be satisfied. These are the two conditions for population inversion in semiconductors. The thermal equilibrium conditions of E Fn = E F p and np = n i2 will be now have to be replaced with population inversion condition np > n i2 . This situation corresponds to the population inversion condition in an atomic system when N2 > N1 . Furthermore, from Eq. 7.65a and 7.65b and (E = hν) implies that another condition for laser emission should be: E g < hν < E Fn − E F p

(7.70)

244

7 Light Emitting Diodes and Semiconductor Lasers

Equation 7.70 states that the energy of the photons emitted by stimulated emission will be smaller than the separation of the quasi-Fermi levels. It should however, be greater than the bandgap because when hν < E g optical gain will be zero. If hν = E Fn − E F p then absorption will dominate and the gain will also be zero. This is illustrated in Fig. 7.14c.

7.3 Light Emitted Diodes (LEDs) Light emitting diodes (LEDs) are forward biased p-n junctions and depending on the semiconductor material, can emit light across the whole range of the visible spectrum. When the junction is forward biased at an appropriate voltage, electrons and holes are injected into the active region of the junction (depletion region) where they recombine radiatively to emit photons spontaneously. This is called electroluminescence and the energy of the emitted photon is determined by the bandgap of the semiconductor. The first GaAs LED was demonstrated in 1962[2] and emitted low intensity infrared light. Infrared LEDs are still frequently used as transmitting elements in remote control circuits. Modern LEDs are available across whole range of visible, NIR and UV wavelengths with high intensity emission. In recent times, LEDs have undergone a vast improvement and are now used in a wide range of applications including aviation lighting, displays, sensors, automotive headlamps, advertising, general lighting, traffic signals, optical communication technologies and room lighting. They have much lower energy consumption, longer lifetime, better physical robustness, smaller sizes and faster switching compared to incandescent light sources. LEDs emitted light at different wavelengths, from the infrared to the green. However, emitting blue light proved to be a difficult task, which took three more decades to achieve. In 1995–1996, a blue emitting high quality GaN-based LED with a 10% efficiency was fabricated [3, 4] and the invention of efficient blue LEDs has led to white light sources for illumination. Therefore, LEDs have led to a revolution of illumination technology in twenty-first century. Isamu Akasaki, Hiroshi Amano and Shuji Nakamura produced bright blue light beams from in the early 1990s and initiated a transformation of lighting technology. The Nobel Prize in Physics 2014 was awarded to Isamu Akasaki, Hiroshi Amano and Shuji Nakamura “for the invention of efficient blue light-emitting diodes which has enabled bright and energy-saving white light sources” [5]. Table 7.2 lists the currently available LEDs together with their semiconductor materials and emission wavelengths. The general lighting application for LEDs requires white light. One approach for producing a white LED is to mix the light from several coloured LEDs (Fig. 7.15) which creates a spectral emission that appears white. Another method is using phosphors together with a short-wavelength LED. For example, when one phosphor material used in LEDs, which is illuminated by a blue emitting LED, it emits yellow light having a fairly broad spectral power distribution. By incorporating the phosphor in the body of a blue LED with a peak wavelength around 450 to 470 nm, some of

7.3 Light Emitted Diodes (LEDs) Table 7.2 Semiconductors used for LEDs and their emission wavelengths[6]

245

Colour

Wavelength (nm)

Semiconductor material

IR

> 760

(GaAs) (AlGaAs)

Red

610 < λ < 760

(AlGaAs) (GaAsP) (AlGaInP) (GaP)

Orange

590 < λ < 610

(GaAsP) (AlGaInP) (GaP)

Yellow

570 < λ < 590

(GaAsP) (AlGaInP) (GaP)

Green

500 < λ < 570

(InGaN/GaN) (GaP) (AlGaInP) (AlGaP)

Blue

450 < λ < 500

(InGaN)

UV

< 400

(AlN) (AlGaN)

Fig. 7.15 a Mixing of prime colours. b Production of different colour light using LEDs emitting at wavelengths corresponding to prime colours

the blue light will be converted to yellow light by the phosphor. The remaining blue light, when mixed with the yellow results in white light. Another wide spread application of LEDs is in the field of optical communication. When compared with lasers in optical communication, LEDs have less temperature sensitivity, simpler control systems but their emission spectra are much wider, intensities are lower and switching speeds are much slower than lasers. Therefore, they can only be used as low speed (20 GHz. They can be integrated with electronic devices, such as field effect transistors, microwave oscillators, bipolar transistors and group III-V semiconductor optical components. Emission wavelengths can be tailor-made using appropriate semiconductor compounds. The spot size of the emission can be made small enough for efficient coupling to optical fibres. Therefore, they can be used in low loss and small dispersion optical communication technologies. They are low cost compared to other types of lasers. They can be mass produced and tested easily.

In order for a semiconductor laser to find applications within current technology, it should meet the following general criteria: 1. 2. 3.

The semiconductor must enable p–n junction formation with efficient carrier injection. It must have high electroluminescence efficiency. It should have the appropriate emission wavelength for a specific technological application, for example, for optical fibre communications, the emission wavelength must be either within the 1.55 or 1.3 μm communication windows.

One important advantage of semiconductor lasers is that there is no need for a large optical cavity. In gas and solid state lasers, mirrors are used for the optical feedback, in semiconductor lasers there are already high reflectivity natural mirrors at the cleaved semiconductor-air interfaces. Epitaxial growth of semiconductor lasers enables the formation of very high reflective Distributed Bragg Reflectors (DBRs) on either sides of the gain region which, even for extremely small cavity lengths, can provide very high gain. The following sections provide a detailed analysis of homojunction, heterojunction, quantum well and vertical cavity surface emitting lasers and show the chronological order of their development.

252

7 Light Emitting Diodes and Semiconductor Lasers

7.4.1 Homojunction Lasers The simplest structure for a semiconductor laser is the homojunction. It is, like an LED, a forward biased p–n junction containing one type of semiconductor only. What makes the homojunction laser different from its LED counterpart is that the p and n regions are very heavily doped (degenerate) and therefore, the Fermi level is in the conduction band in the n-side and in the valence band in the p-side as illustrated in Fig. 7.19. The forward bias is much higher than that in LEDs for increased injection of non-equilibrium carriers in order to reach the population inversion condition. Optical feedback is provided by the polished semiconductor-air interfaces on either side of the gain region as shown in Fig. 7.20. Therefore, the optical cavity in a homojunction laser is in principle identical (except for the dimensions) to the Fabry-Pérot cavity as shown in Sect. 7.1.3. As discussed in Sect. 7.1.1, in order to produce coherent and high intensity photon emission the population of the higher energy level (N 2 ) has to be much higher than that in the lower energy level (N 1 ). In the homojunction laser under forward bias this condition (population inversion) is achieved within the depletion region as shown in Fig. 7.19b. In Fig. 7.19a, the Fermi energy in the homojunction laser is shown at thermal equilibrium. When the junction is heavily forward biased, high density non-equilibrium electron and hole pairs are injected into the n and p regions, respectively, and the quasi-Fermi levels are formed at higher energies than the equilibrium value. Increasing the majority carrier densities will reduce the built-in potential and the depletion width (see Sect. 4.2). In the depletion region (active region), there is now a large number of non-equilibrium electrons and holes, thus the necessary condition Fig. 7.19 Heavily doped p-n junction a in thermal equilibrium and b under forward bias, where population inversion is achieved in the depletion region

7.4 Semiconductor Lasers

253

Fig. 7.20 a Structure of a typical homojunction laser. b Refractive index profile and the confinement of the optical wave in a heterojunction laser



for spontaneous emission of photons with energies E g < hν < E Fn − E F p is established. To achieve lasing there should be enough gain to overcome the absorption scattering and reflection losses. The gain is provided by the optical feedback from the mirrors (polished semiconductor surfaces) at both sides of the cavity and can be increased having highly polished mirrors for high reflectivity and increasing the injected carrier density. The reflectivity at the semiconductor-air interface is given by Eq. 7.75 and for a GaAs homojunction laser where the refractive index of GaAs is n2 = 3.6 and air is n1 = 1, the reflectivity is R = 33%. Figure 7.20 shows the schematic structure of a homojunction laser. Here the ends of the cavity, length L, are cleaved to provide high reflectivity. Population inversion occurs within the depletion region, width d, which is of the same order as the diffusion lengths of the electrons and holes. The spread of the optical mode (d m ) is longer than the active region so d m > d so photon confinement in a homojunction laser is rather poor. Photons which are outside the active region will be lost by absorption thus

254

7 Light Emitting Diodes and Semiconductor Lasers

delaying the condition for laser action, therefore resulting in a very high threshold current for homojunction laser operation. Good optical confinement can be achieved in heterojunction lasers as the active region is a small bandgap semiconductor and has a larger refractive index than the wide bandgap n- and p-type regions, therefore providing photon confinement (Fig. 7.20b). We shall discuss heterojunction lasers in detail in Sect. 7.5.

7.4.2 Efficiency of a Semiconductor Laser The efficiency of a semiconductor laser can be defined with differential quantum efficiency (ηd ). Differential quantum efficiency is ratio of the increase in number of photons to the increase in number of electrons above lasing threshold. ηd =

e d Pe d Pe / hν ≈ d I /e Eg d I

(7.80)

Here Pe is the emitted optical power and hν is photon energy. The energy bandgap in the Eq. 7.80 has the unit of eV. The differential quantum efficiency, which is also known as slope efficiency, can be obtained from the slope of the output power versus injected current characteristic of the laser as shown in Fig. 7.21. For semiconductor lasers operated in the cw mode, with dc current injection, the differential quantum efficiency can be between 40 to 60%. The internal quantum efficiency of a semiconductor laser, ηi is defined as the ratio of photon number produced with the cavity to the injected electron number. Its value depends on the structure of the laser but varies between 50 and 100%. The relationship between the internal and differential quantum efficiencies is given as: Fig. 7.21 Output power versus injected current in a semiconductor laser

7.4 Semiconductor Lasers

255

 ηd = ηi

1 1 + [2αL/ ln(1/r1r2 )]

 (7.81)

Here α is the loss coefficient in the laser cavity, L is cavity length, r 1 and r 2 are the reflectivity coefficients of the cleaved edges of the semiconductor. Another important parameter describing the laser efficiency is the external quantum efficiency, η0 . This is defined as the ratio of the number of output photons to the number of injected electrons. η0 =

e Pe Pe / hν = I /e I Eg

(7.82)

At injection currents well above the threshold (I > > I th ), the emitted power varies linearly with the injection current, as shown in Fig. 7.21, and the external quantum efficiency can be obtained from:   Ith ηo = η d 1 − I

(7.83)

At very high injection currents, for example I = 5I th , η0 ≈ ηd . The external power efficiency of a semiconductor laser is the ratio of the emitted optical power to the input power: ηP =

Pe Pe × 100 = × 100 P IV

(7.84)

From Eq. 7.82 we can also express the external power efficiency as:  η P = η0

Eg eV

 × 100

(7.85)

Example A GaAs laser has an external quantum efficiency of 18% and a dc voltage of V = 2.5 V is applied to the device. The bandgap of GaAs is 1.43 eV. Calculate the external power efficiency of the laser. The answer is quite straight forward because from Eq. 7.85, the external power efficiency is:   1.43eV × 100 = %10 η P = 0.18 2.5eV

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7 Light Emitting Diodes and Semiconductor Lasers

7.4.3 Gain and Threshold Current We have already studied the optical gain within the Fabry-Pérot cavity. In order to derive the correct expression for the laser threshold current, the cavity gain should be modified by the optical confinement factor G to yield the material gain, ( γ (ω)). The optical confinement factor is given as: 

=

|E(z)|2 dz

active r egion



|E(z)|2 dz

(7.86)

The higher the optical confinement factor the higher the photon density within the cavity. Therefore, in order to have high efficiency, it is vital that the optical confinement factor in the laser structure should be as close to 1 as possible. We can now derive the threshold current as a function of gain and confinement factor starting from Eq. 7.57 given in Sect. 7.1.7:

γth = α +

  1 1 ln 2L r1 r2

(7.87)

  gth J0 d 1+ ηi g0

(7.88)

The threshold current is: Jth (A/cm 2 ) =

Here d is the width of the active layer. We can combine Eqs. 7.87 and 7.88 to obtain:    1 J0 d J0 d 1 Jth (A/cm ) = + α+ ln ηi g0 ηi 2L r1 r2 2

(7.89)

Here Γ , g0 , α, r 1 and r 2 are optical confinement factor, gain, loss per unit length, reflectivity coefficients of the cleaved surfaces, respectively. While L, J 0 , d and ηi are cavity length, transparency current density, width of the active region and internal quantum efficiency, respectively. The transparency current density corresponds to the injected current density at zero material gain. A low threshold current is one of the main aims in laser design. It is clear from Eq. 7.89 that the threshold current can be reduced if the quantum efficiency and reflectivity coefficients are high and the width of the active region is small. However, although the reduction in active region width initially decreases the threshold current, when the optical confinement factor is reduced significantly with decreasing cavity width then threshold current will start increasing. Let us now derive the threshold current in terms of more easily accessible parameters of the laser. Injected electrons (and holes) arrive the active region at a rate:

7.4 Semiconductor Lasers

257

JA e

(7.90)

The recombination rate of the injected electron–hole pairs is: n Ad τr (J )

(7.91)

Here τ r (J) is the current density and is dependent on radiative recombination lifetime and A is the cross-sectional area of the p–n junction. If we take the radiative recombination efficiency as unity, the two equations above reduce to: n=

J τr (J ) ed

(7.92)

Jth τr (Jth ) ed

(7.93)

Under threshold conditions: n th =

Current density increases and the radiative lifetime decreases with increasing photon density in the dominant mode. As a result, although the injected current density increases the carrier density in the active region saturates at a value close to its threshold value. The threshold current: Ith =

n th Ad I = e τr

(7.94)

Equation 7.94 implies that for low threshold current the active region should be narrow but for high output power it should be also wide. Therefore, the length of the active region can be tailored to suit the demanded application for the laser. Example The optical confinement factor in a GaAs laser is 1. The width of the active region is 2 μm and the radiative recombination rate is 2.4 ns. Electron density at the transparency condition (when gain equals to zero) is 1.0 × 1018 cm−3 and the threshold electron density is 20% higher than that at transparency. Calculate the threshold current density. Since n = 1 × 1018 cm −3 ,n th = 1.2 × 1018 cm −3 Threshold current density is then: Jth =

en th d 1.6 × 10−19 × 1.2 × 1018 × 2 × 10−4 = = 1.6 × 104 A/cm 2 τr 2.4 × 10−9

Example In a GaAs/AlGaAs semiconductor laser, the radiative recombination lifetime is 4 ns, the cavity length is 500 μm and the width of the active region is 0.08 μm.

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7 Light Emitting Diodes and Semiconductor Lasers

The reflectivity at the semiconductor-air interface is R = 0.32. Calculate the material gain. If the injected electron density at threshold is 1.7 × 1018 cm−3 , calculate the threshold current density.

γth = α −

1 ln R L

The largest loss is: −

1 ln R = 23.4cm −1 L

If we assume α = 10 cm−1 and the optical confinement factor is equal to 1. The gain at the threshold must be equal to 33.4 cm−1 . Thus the current density at threshold is: Jth =

en th d ≈ 544 A/cm 2 τr

7.4.4 Temperature Dependence of Threshold Current When the temperature of the diode increases, threshold current density increases and emitted photon density decreases. This is because: 1.

2.

3.

The increase in temperature changes the quasi-Fermi functions so the quasiFermi levels move up to higher energies within the bands. Therefore, the population inversion condition can only be achieved at higher injection levels, increasing the threshold current density. The increase in the temperature causes the electron and hole populations to occupy states at higher energies. These carriers can go through the active region without recombining so this may be overcome by designing the laser with a longer active region. Increasing temperature, therefore the average energy of electron–hole pairs, may trigger Auger recombination which is a non-radiative recombination mechanism. This is particularly important in long wavelength lasers with narrowbandgap semiconductor materials.

As a result of these three effects, we can represent the temperature dependence of the threshold current density Jth as: Jth (T ) = Jtho exp(T /T0 )

(7.95)

7.4 Semiconductor Lasers

259

Fig. 7.22 Gain spectrum, loss and cavity modes in a semiconductor laser

Here J th o and T 0 are two empirical parameters which depend on the laser design characteristics. T 0 is known as the threshold temperature coefficient and defines the quality of laser material. In order for the laser to have a weak temperature dependent threshold current, the value for T 0 needs to be as high as possible. In a typical GaAs laser, this value is T 0 ~ 120 K and in a longer wavelength laser, it is in the range of about T 0 ~ 50 K. Another important factor is the frequency stability of the laser over temperature. There are two important factors affecting the laser frequency. Firstly, the bandgap of the laser material will decrease with increasing temperature. As a result, the peak of the gain spectrum will shift to lower energies, therefore the most dominant mode will change (Fig. 7.22). Secondly, increasing temperature causes the thermal expansion of the laser cavity which changes the refractive index of the cavity. Consequently, both the resonance modes and mode separation will change.

7.5 Heterojunction Lasers The main problems with the homojunction laser are the lack of carrier and optical confinement. This means that cavity losses are high and excessively large threshold currents are required for lasing, as shown in Eq. 7.89, where threshold current density is proportional to both d and α. In homojunction lasers, device operation is not possible in cw mode because there is a large threshold current and Joule heating is very high. The requirement for low threshold current operation can be realised with heterojunction lasers.

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7 Light Emitting Diodes and Semiconductor Lasers

Typical structures for single and double heterojunction lasers, together with their energy band diagram and refractive index profiles as well as optical confinements are shown in Fig. 7.23. In the single heterojunction laser, electrons that are injected from the n+ GaAs into the active p+ GaAs layer cannot overcome the potential barrier at the AlGaAs interface. The active region d is equal to the width of the p+ GaAs layer and a small p+ GaAs layer thickness results in lower threshold current densities. Therefore, in a single heterojunction laser there is good electron confinement. However, better optical confinement is achieved using double heterostructure lasers. In the double heterojunction laser, the n+ AlGaAs together with the p type AlGaAs have layers on either side of the small gap p+ GaAs active layer which provides the optical confinement because of the contrasting refractive indices. The wide gap material AlGaAs acts as both a carrier and photon confining layer. The power efficiency in a double heterostructure laser is enhanced and the threshold current density reduced compared to the single heterojunction laser. Alx Ga1-x As/GaAs has been most widely researched material system for a heterojunction laser. The bandgap of Alx Ga1-x As/GaAs depends

Fig. 7.23 The layer structure, energy bandprofiles, refractive index profiles and optical modes for a single heterojunction laser and b double heterojunction laser

7.5 Heterojunction Lasers Table 7.3 Bandgaps of various semiconductor compounds [8]

261 Alloys

Direct bandgap (eV)

Alx In1-x P

1.351 + 2.23x

Alx In1-x As

0.360 + 2.012x + 0.698x 2

Gax In1-x P

1.351 + 0.643x + 0.786x 2

Gax In1-x As

0.360 + 1.064x

InPx As1-x

0.360 + 0.891x + 0.101x 2

on Al density, increasing with higher Al as: E g (x) = 1.424 + 1.247x (eV )

(7.96)

The refractive index also depends on the Al content as: n r (x) = 3.590 − 0.710x + 0.091x 2

(7.97)

where Al density is 0 < x < 0.45.When the Al density exceeds x = 0.45, Alx Ga1-x As/GaAs becomes an indirect bandgap semiconductor and is no longer a suitable laser material. Alx Ga1-x As /GaAs heterojunction laser has an emission wavelength of 0.75 μm < λ < 0.88 μm depending on the composition and is regularly used in short haul (< 2 km) optical communication systems. Another semiconductor material system commonly used in lasers is Ga1-x Inx As1-y Py /InP. The wavelength of this laser ranges from 1.1 μm < λ < 1.6 μm and varies with x and y compositions. The composition dependence for various energy bandgaps are given in Table 7.3.

7.6 Quantum Well Lasers The significant progress in semiconductor lasers coincided with the advent of quantum well structures. The growth of quantum well layers with well widths comparable to the de Broglie wavelength (λ = h/p) became possible with advances in the epitaxial semiconductor growth techniques such as Molecular Beam Epitaxy (MBE) and Metal–Organic Vapour Phase Epitaxy (MOVPE). Quantum well lasers enabled lower threshold current densities, higher emission powers and higher speeds. This section reviews low dimensional semiconductor structures as well as considering the main properties of quantum well lasers.

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7 Light Emitting Diodes and Semiconductor Lasers

7.6.1 Low Dimensional Semiconductors: Quantum Wells Quantum size effects become significant when one or two dimensions of a 3D semiconductor are comparable with the de Broglie wavelength of an electron. One important example for a low dimensional semiconductor is the quantum well structure. It is formed by sandwiching a thin layer small bandgap semiconductor between two layers of a wide gap semiconductor as depicted in Fig. 7.24. If the thickness of the narrow bandgap semiconductor is sufficiently thin and of similar magnitude as the de Broglie wavelength, carriers become quantised in the growth direction (z), whilst they move freely in the x–y plane, perpendicular to the growth direction. In the growth direction, electrons have quantised energy levels and changes in the energy spectrum of the semiconductor material is a direct result of reduced dimensionality and such systems are known as quasi-two dimensional (2D)or quantum well systems. The dimension of the system is described by the number of directions, which energy of electron is given by free electron dispersion

Fig. 7.24 a Schematic representation of a quantum well formed with two semiconductors with bandgaps E g1 < E g2 . Here L z is the quantum well width and L B is the barrier width. ΔE C and ΔE V are the conduction and valence band discontinuities. E e1 , E e2 and E e3 are the quantised energy levels for electrons and E hh1 , E lh1 are quantised the energy levels for heavy and light holes.  1 is the wavefunction for the first electron subband b Subbands corresponding to the quantised electron and hole energy levels in the quantum well

7.6 Quantum Well Lasers

263

relation, not quantized. For example, if we consider a quantum wire with a length is much larger than the thickness and width of the wire, this system is called quasi-onedimensional system (1D), because electrons are just free to move along the length of the wire, that is, their energy is not quantised. As for quantum dots, energy of the electrons is quantized at all three direction and known as quasi-zero dimensional system (0D) as an atom. With reference to Fig. 7.24, quantum wells are formed by the growth of a thin small bandgap material, with thickness comparable to the de Broglie wavelength, between two large bandgap semiconductors. We can use, to a first approximation the infinitely high barrier approach to calculate the quantised energy levels formed in the quantum well. These levels E N are given as: EN =

π 2 2 N 2 2m ∗e L 2z

N = 1, 2, 3, . . . .

(7.98)

where E N is measured from the bottom of the quantum well and L Z is the quantum well width. The infinite barrier approach can also be used to obtain the quantised energy levels in the valance band for both heavy holes with effective mass, m hh and light holes with effective mass m lh . It should be made clear that the motion of the carriers in the x–y plane is not affected by energy quantisation and they move as free carriers in a continuous parabolic band as in the bulk semiconductors. In the quantum well the total energy is the sum of the quantised energy which is associated with the motion of carriers along the growth direction (z) and continuous energy which is associated with the free motion of carriers along the x–y plane. E = Ex + E y + Ez Ez = E N = E = EN +

2 (k x2 + k 2y ) 2m ∗e

π 2 2 N 2 2m ∗e L 2z

(7.99)

(7.100)

Here k x and k y are the wavevectors along the x and y directions. In Eq. 7.100, the energy levels corresponding to E N levels are known as subbands. These have the dispersion relations shown in Fig. 7.25b where parabolic subbands start from each of the quantised energy levels E e1 , E e2, etc. The valence bands are also divided into similar subbands each starting from the quantised levels E hh1 , E lh1 etc. Low dimensional semiconductors have various structures such as Single Quantum Well (SQW), Multiple Quantum Well (MQW) and superlattices (SL) as shown in Fig. 7.25. So far we have been considering SQWs. If a SQW structure is grown consecutively multiples of times then it becomes MQWs. If the barrier widths in MQWs are thin enough and form minibands along the growth direction then these structures are known as superlattices.

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7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.25 Schematic diagrams for a multiple quantum wells and b superlattice structures

7.6.2 Density of States in 2D In a three dimensional (bulk) semiconductor, the density of states varies as the square root of energy as shown in Sect. 2.1:  N3D =

2(m ∗e )3 E π 2 3

(7.101)

For a 2-dimensional semiconductor, however, the density of states for each subband is independent of energy and when E 1 < E < E 2 (Fig. 7.26, this is: N2D =

m ∗e π2

(7.102)

Electrons with energies exceeding the first subband energy populate the higher subbands. Each of the subbands has the same density of states as given in Eq. 7.102.

Fig. 7.26 a A comparison of 2D and 3D density of states. b Density of electrons in 2D and 3D semiconductors

7.6 Quantum Well Lasers

265

Consequently, the density of states function in a 2D semiconductor has a step-like formation where the edge of each step coincides with the subband energy EN and with a width: m ∗e /π 2 Equation 7.102 can be expressed in a more general form as: N2D =

m ∗e  (E − E N ) π 2 N

(7.103)

In this equation, (x) is a step function and is unity for x > 0, but otherwise is zero. In Fig. 7.26, the density of states and carrier density as function of energy are shown for 2D and 3D (bulk) systems. It is obvious that size quantisation decreases the density of states. Equation 7.103 can be also used to obtain density of states for heavy and light-hole states using their respective effective masses. The step-like characteristic of the density of states has an important role for laser operation because in 2D structures, it is easier to populate large number of carriers at a given energy, compared to the 3D devices. The most important factor influencing laser properties is the change of the density of states due to the quantum confinement effect. In a bulk semiconductor, the density of states around the bandgap energy is very low but for a 2D semiconductor, the density of states is at the bandedge and is high. Therefore, in a 2D system it is much easier to achieve population inversion than for a 3D structure. The step-like energy dependence of density of states does not only decrease the threshold current, but also significantly reduces temperature dependence of threshold current. The weaker temperature dependence of threshold current enables cw operation of the laser even at room temperature.

7.6.3 Absorption in Quantum Well Systems In a quantum well system, the electronic states are quantised in the growth direction and continuous in the plane perpendicular to the growth direction. The optical transitions take place between confined electronic states in the quantum well, therefore the threshold energy for the absorption process is equal to E = E g + E e1 + E hh1 as shown in Fig. 7.27. This indicates that the bandgap of the quantum well system shifts toward higher energies by the sum of the confined electron and hole energies. When the photon energy is equal to or exceeds the threshold energy it is absorbed by the QW. Asssume that an infinite quantum well system. The absorption of the photons in this system should follow the selection rule given by Δn = 0. Therefore, absorption process takes place between allowed energy levels, which obey selection rule such

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7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.27 a The allowed transitions in quantum wells. The “dashed line” corresponds forbidden transition, E hh1 to E e3 b Absorption spectrum in a GaAs/AlGaAs QW structure. The “arrow” on the absorption spectrum at the high energy region represent the transition from E hh1 to E e3

as E hh1 → E e1 , E lh1 → E e1 , E hh2 → E e2 , etc. as depicted in Fig. 7.27a. On the other hand, because the barrier of the quantum well system is finite, selection rules are broken because the tail of the wavefunction penetrates into the barrier layer, making forbidden transition Δn = 2 allowed. The absorption spectrum of an AlGaAs/GaAs quantum system is shown in Fig. 7.27b. The “arrow” on the absorption spectrum at the high energy region represent the transition from E hh1 to E e3 .

7.6.4 Quantum Well Lasers The active region in conventional heterostructure lasers may be kept narrow (100 nm300 nm) for better carrier and optical confinement, but the electronic and optical properties of heterostructure lasers are typical of a bulk semiconductor. This is because the

7.6 Quantum Well Lasers

267

active layer width is still much longer than the de Broglie wavelength and therefore too large for any quantum size effects. If the thickness of active region is comparable with the de Broglie wavelength however, the carrier confinement increases and quantisation effects are seen as previously discussed. The optical confinement decreases because the optical wavelengths of interest are longer than the de Broglie wavelength. The density of states in bulk structures changes with E 1/2 . Fermi–Dirac statistics dictate that the probability of occupation for these states is highest at the bottom of the bands. At a finite temperature the carriers are distributed over a wide range of energies in the bands with a relatively small carrier density at the bandedges, because the available density of states is low. Therefore, in bulk lasers the population inversion condition for lasing action is delayed and higher pumping speeds are required. In order to restrict the occupancy of states within a small energy region, it is desirable to have a large density of states around the valence and conduction bandedges. The 2D density of states in a quantum well laser has a step-like characteristic and therefore, there is always a large density of available states and carrier densities at the bandedges. Therefore quantum well lasers have lower pumping speeds thus, smaller threshold current densities are sufficient for lasing action. The lack of optical confinement in QW lasers can be addressed by including MQWs in the gain region. Furthermore, a separate confinement heterostructure can also be incorporated into the structure.

7.7 Vertical Cavity Surface Emitting Laser—VCSEL Vertical Cavity Surface Emitting Lasers (VCSELs) differ from conventional edge emitting semiconductor lasers because its cavity design is formed in a direction normal to the plane of the wafer and therefore emits from the surface. The thickness of an epitaxially grown vertical active region is limited, so the length of a VCSEL cavity is in the order of 10 μm. The short cavity length reduces the number of modes that are permitted, allowing for single mode operation at the expense of decreasing the round trip gain. This problem can be minimized by having high reflectivity mirrors. As mentioned earlier, for an edge-emitting laser, when reflectivity of the mirrors is about 33% the absorption loss is more than 103 cm−1 and results in higher threshold currents. When the reflectivity of the mirrors can be increased to 99%, absorption loss will be approximately 10−1 cm−1 thus much lower threshold current densities can be achieved. The optical cavity of a VCSEL is similar to a Fabry-Pérot cavity in conventional lasers, except that it is now cladded on either side by Distributed Bragg Reflectors (DBR layers). In principle, DBRs are quarter wave plates formed by the growth of a multi-layered, periodic structure that has contrasting layers of material with different refractive indices and produces a reflectivity higher than 99.9%. Consider a DBR with a periodically grown structure with alternating N layers, which has two types of semiconductors (e.g. GaAs/AlGaAs for a GaAs VCSEL structure) with layer thicknesses d 1 and d 2 and refractive indices of nr1 and nr2, respectively. The periodicity of the DBR structure has been selected in order to satisfy the condition:

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7 Light Emitting Diodes and Semiconductor Lasers

n 1 d1 + n 2 d2 = λ/2

(7.104)

Here λ = λnr0 corresponds to the wavelength that fulfils the Bragg condition. λ is the wavelength that coincides with the maximum wavelength of the gain curve and λo is the wavelength in vacuum. nr is effective refractive index. The thicknesses of alternating DBR layers, d 1 and d 2 , are designed to equal the quarter wavelength in order to provide constructive interference. A large contrast between the layer refractive indices increases the reflectivity of the DBR mirror. When the Bragg condition, given by Eq. 7.104, is satisfied the reflectivity at the resonant wavelength is given by: ⎛ ⎜ R=⎝

1−

ns n0

1+

ns no

 2N ⎞2 n1 n2

⎟  2N ⎠

(7.105)

n1 n2

Here ns is the refractive index of the substrate, n0 is the refractive index of air, nr1 and nr2 are the refractive indices of DBR layers and N is the number of pairs. With an appropriate selection of nr1 /nr2 , a reflectivity of 99% can be easily obtained. As an example, in Fig. 7.28, reflectivity spectrum for a GaAs/AlAs (nGaAs = 3.501, nAlAs = 2.999) DBR stack with different number of pairs are illustrated. As more layers are added the region of high reflectivity known as the stop-band develops. The stop-band is limited in extent and falls abruptly on either side to a low oscillatory value. The addition of extra layers does not alter the width of the stop band significantly, but does increase reflectivity. A typical structure of a VCSEL is shown in Fig. 7.29. Here L C and L e are the cavity and effective cavity lengths, respectively. The cavity length is the distance between the mirrors in a Fabry-Pérot (F-P) laser. However, In a VCSEL structure, because the optical field increases as result of the increased reflectivity, the effective cavity length is used instead of the F-P cavity length. The cavity in a VCSEL is perpendicular to that of an edge-emitting laser and the small active region is a single quantum well (in a SQW VCSEL) or MQW (in a MQW VCSEL) as depicted in Fig. 7.29. The cavity is therefore known as a vertical Fabry-Pérot cavity. In Fig. 7.30, the reflectivity spectrum of a typical GaAs/AlGaAs VCSEL is shown. The active region consists of an AlGaAs p–n junction with a 13 nm quantum well on the n-side of the depletion layer. The lower DBR comprises of 27 periods and the upper DBR has 17 periods of AlAs/AlGaAs layers with a reflectivity better than 99%. The high reflectivity is in the 800-870 nm wavelength region (stop-band), the reflectivity dip is located at approximately 824 nm and is the cavity resonance wavelength. The room temperature emission spectrum for this device is shown in Fig. 7.31 with the emission wavelength at the cavity resonance. VCSELs offer several advantages over Edge Emitting Lasers (EELs):1.

An all vertical construction enables the use of traditional semiconductor epitaxial processes so fabrication costs are reduced.

7.7 Vertical Cavity Surface Emitting Laser—VCSEL

269

Fig. 7.28 Reflectivity spectra for a GaAs /AlAs DBR mirror with periods of a 10 and b 25

2. 3.

Semiconductor manufacturing and wafer integration of VCSELs is compatible with detectors and other circuitry. VCSELs can be implemented in arrays and have very long lifetime and are not prone to catastrophic optical damage.

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7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.29 Schematic representation of a typical VCSEL structure with DBR mirrors at the top and bottom of the vertical cavity

Fig. 7.30 Reflectivity spectrum of a GaAs/AlGaAs quantum well VCSEL with DBR mirrors with 17 periods at the top and 27 periods at the bottom of the active layer

4. 5. 6. 7. 8. 9. 10.

VCSELS can be fully tested while still in wafer form which increases manufacturing yield and lowers costs. VCSELs are easily fabricated into one or two dimensional arrays to scale power output in order to match specific application requirements. VCSELs allow the use of traditional low cost LED packaging and can replace LEDs in existing applications. Chip-on-board technology and custom packaging are available to simplify system integration. High power efficiency extends battery life and reduces thermal design constrains in larger systems. The emission divergence from VCSELs is highly superior to EELs, so they can be coupled efficiently to optical fibers. The lasing is in single mode, therefore they are suitable for single mode fibers in optical communication systems.

7.7 Vertical Cavity Surface Emitting Laser—VCSEL

271

Fig. 7.31 Emission spectrum from a VCSEL

11. 12.

Two dimensional array of VCSELs can be fabricated and tested very easily. The small volume of the active region in VCSELs enables reduced threshold currents.

The cavity length varies between 100 to 500 μm in an edge emitting laser whereas the effective cavity length in a VCSEL is a few μm. The mode spacing for a FP cavity is, therefore, approximately: λ =

λ2 2n L

(7.106)

where n is refractive index of the semiconductor, L is the cavity length and λ is emission wavelength. In an EEL, the cavity length is approximately ~200 μm, this implies a mode spacing of about 0.56 nm. This means that more than one longitudinal mode are present within the gain spectrum. Typically for VCSELs, the effective cavity length is approximately, L = 1 μm, the mode spacing is around ~112 nm. Consequently, there is only one longitudinal mode in the gain spectrum. Although a VCSEL have several advantages compared to conventional edgeemitting lasers, the design parameters and growth conditions are crucial in determining VCSEL performance. A slight difference in the thickness of the DBR layers and/or quantum well significantly affects VCSEL operation. Therefore, the performance of the VCSEL depends strongly on the growth of a high quality active region and DBR layers. Furthermore, the resistance of the mirrors should be kept as small as possible to keep the injection current low to avoid Joule heating losses. To date, across all the VCSEL devices, the most intense research has been focused on GaAs/Alx Ga1-x As and Inx Ga1-x As/GaAs quantum well structures. GaAs/Alx Ga1-x As quantum well devices operate at 800 and 880 nm wavelengths

272

7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.32 The structure of an In1-x Gax As /InP VCSEL with Si/SiO2 DBRs

while Inx Ga1-x As/GaAs based VCSELs have wavelengths between 930 and 1100 nm. The VCSELs emitting at 1.3 or 1.55 μm are commonly used in low cost, high performance optic fiber communication systems. These two wavelengths coincide with the minima (optical windows) in the spectral fibre attenuation characteristics for optic fiber communications and the two suitable VCSELs are In1-x Gax As /InP (λ = 1.5 μm) and In1-x Gax AsyP1-y /InP (λ = 1.3 μm). Figure 7.32 shows the structure of a typical In1-x Gax As /InP with Si/SiO2 DBR stack.

7.7.1 Temperature Dependence of VCSELs Single mode VCSEL operation has an important consequence as far as the temperature dependence of the threshold current is concerned. Increasing temperature in an EEL increases the threshold current and laser, so broadens the gain spectrum, thus increasing both optical and electrical losses. For VCSELs, there is an ideal operation temperature at which the maximum of the gain spectrum and cavity resonance coincide, leading to low threshold current densities. In a conventional edge emitting laser, the mode spacing is small so there are always several lasing modes in the gain spectrum. When the cavity length or gain spectrum changes with temperature the laser mode hops to the gain peak. For an edge-emitting laser, the laser emission depends strongly on the position of the maximum gain

7.7 Vertical Cavity Surface Emitting Laser—VCSEL

273

spectrum and therefore temperature. For a VCSEL structure, due to its short cavity length, the spacing of the adjacent modes is greater than the spectral width of the gain spectrum and has only one single mode within the gain spectrum. Therefore, when the temperature changes, there is no hopping from one mode to another (Fig. 7.33). The shift in the emission spectrum in a VCSEL is caused by the temperature dependence of the bandgap alone, while the shift in the cavity resonance is largely governed by the change in the refractive index within the layers forming the DBRs. An increase in

Fig. 7.33 a The effect of temperature on the gain spectrum and the lasing modes in EEL. Mode jumping occurs with changing temperature. b In a VCSEL, since there is only one mode in the gain spectrum, the change in the temperature does not change the mode, but because of the shift in the gain spectrum the threshold current changes

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7 Light Emitting Diodes and Semiconductor Lasers

Fig. 7.34 Temperature dependence of the lasing wavelength of a VCSEL

temperature will cause both the emission spectrum and cavity resonance to red-shift. In Fig. 7.34, temperature dependence of the lasing wavelength of a VCSEL is shown.

7.8 Distributed Feedback Lasers (DFB) The Distributed Feedback (DFB) laser is a wavelength selective device with a periodic grating structure fabricated in the optical cavity. The grating is formed by a corrugated pattern which is etched in the waveguide section of the large optical cavity and acts as a Bragg diffraction grating. The periodic change of refractive index for the corrugated interface along the direction of wave propagation provides constructive interference in the cavity for a given wavelength as shown in Fig. 7.35. Optical feedback is obtained by coherent interference from the grating rather than by conventional cleaved mirrors, so the corrugated grating structure determines the lasing wavelength. When the period of the corrugation is equal to: 2 = m

λB n

(7.107)

Only the mode near the Bragg wavelength (λB) is diffracted constructively and the lasing wavelength (λ0) is expressed by:  λm = λ B ±

λ2B (m + 1) 2n L

 (7.108)

7.8 Distributed Feedback Lasers (DFB)

275

Fig. 7.35 a DBR laser structure b Reflected waves at corrugated grating structure interferes constructively when they obey Bragg’s law

Here λB is the Bragg wavelength, n is the refractive index of the waveguide, m is an integer and L is the cavity length. The DFB laser operates at the lowest threshold current for the two longitudinal modes (m = 0) closest to the Bragg wavelength and single mode operation is obtained with one of these modes as shown in Fig. 7.36. The DFB laser can be designed to have a single mode output due to its superior wavelength selectivity. The most obvious advantage of a DFB laser in comparison to conventional semiconductor lasers is single mode operation which makes them preferable for long haul optical fiber communications. This is because it avoids pulse broadening of the signal during data transmission within the fiber. InP-based DFB lasers operate at 1.3 and 1.55 μm and are suitable for transferring data in optical fibre communication systems for long distances over 50 km with data rates of > 1Gbit/s. Another important feature of DFB lasers is that temperature only affects the refractive index and it is not sensitive to bandgap changes, unlike conventional semiconductor lasers. The temperature dependence of the refractive index is less than a

Fig. 7.36 a Simplified DFB laser structure, b Ideal laser emission, c typical spectrum of a DFB laser

276

7 Light Emitting Diodes and Semiconductor Lasers

Fabry-Pérot cavity, therefore the threshold current density of DFB laser is considerably lower. Disadvantages of the DFP laser are the complex growth and fabrication processes which increase production costs. Problems 1.

2.

Electron and holes are injected into a GaAs semiconductor with a density of 1015 cm−3 . The recombination time of electrons and holes is 0.6 ns. Calculate carrier lifetime for this injection level. The physical parameters of a GaAs p-n juction LED at 300 K are given below Area of diode: 1 mm2 Electron diffusion constant: 30 cm2 /Vs Hole diffusion constant: 15 cm2 /Vs Acceptor density in p-side: 5 × 1016 cm−3 Donor density in n-side: 5 × 1017 cm−3 Electron minority carrier lifetime: 1 μs Hole minority carrier lifetime : 0.1 μs ni (GaAs): 2 × 106 cm−3 Assuming that there is no defects in GaAs: (a) (b)

3.

4.

5.

6.

7.

8.

Calculate hole and electron density, diffusion lengths and injection efficiency in n-side and p-side, respectively. When this diode is forward biased by 1 V, radiative recombination efficiency is 0.6. Calculate emitted photon number per second and optical power of the LED.

A tungsten lamp operating at 1300 K emits photons at an average frequency of 5 × 1014 Hz. Calculate the ration of spontaneous emission to stimulated emission. An InP laser emits at 0.94 μm. The modes of the laser are separated from each other with δν = 300 GHz. The refractive index of InP is n = 3.3. Determine (a) length of the cavity, (b) number of the modes. Photons with hν12 = E 2 − E 1 excite two level laser system. In thermal equilibrium, assuming upward and downward transition rates are equal, show B12 = B21 at high temperatures and find A21 /B12 . The cavity lengths of a GaAs laser operating at a current higher than threshold current is 250 μm, cavity loss is 10 cm−1 and internal quantum efficiency is 1.0. The reflectivity of the edges is 32%, determine the efficiency of the laser. The lengths of a GaAs laser cavity is 200 μm and reflectivity of the cleaved facets is 33%. Absorption loss in cavity is 10 cm−1 . Determine the photon lifetime in the cavity. A p–n junction semiconductor laser has a lengths of 200 μm and width of 10 μm. The refractive index is 3.5 and emission wavelength is 1000 nm of this laser. Calculate the wavelength separation of the longitudinal and transverse modes of this laser. If FWHM of the gain spectrum is 7 nm, find how many longitudinal and transverse modes are included in the laser spectrum. What sort of change in laser design you suggest to make single mode laser? Why?

7.8 Distributed Feedback Lasers (DFB)

9.

10.

11.

277

Draw band diagram of an AlAs/Alx Ga1-x As (x = 0.3) quantum well semiconductor structure and calculate the photon energy of the absorption onset for a quantum well with a 100 Å width (At 300 K; E g (AlAs)= 2.2 eV, E g (Alx Ga1-x As) = 1.424 + 1.247x, me (Alx Ga1-x As) = (0.067 + 0.083x)m0, mhh (Alx Ga1-x As) = (0.45 + 0.30x)m0 , me (AlAs) = 0.1m0 , mhh (AlAs) = 0.8m0 ). Long haul fibre optic communication systems uses light sources operating at 1.3 μm or 1.55 μm. Gax In1-x As/GaAs quantum well system can be used a light source for 1.3 μm when x = 0.47. (a) Sketch the band diagram of the quantum well system and calculate the bandgap of Ga0.47 In0.53 As and effective bandgap of the quantum well system considering that it emits at 1.3 μm. (b) Calculate the quantum well width of this quantum well system to be used as a light source operating at 1.3 μm (At 300 K; E g (GaAs) = 1.42 eV, E g (Gax In1-x As) = 0.36 + 1.064x, me (Gax In1-x As) = (0.023 + 0.037x + 0.03x 2 )m0 , mhh (Gax In1-x As) = (0.41 + 0.1x)m0 ). In an DFB laser corrugation period is  = 0.22 μm, d the length of the diffraction grating is 400 μm, and effective dielectric constant of the active region is 3.5. Considering that there is only first order diffraction, calculate Bragg wavelength, lasing wavelength and separation of the modes.

References 1. Varshni YP (1967) Band-to-Band Radiative Recombination in Groups IV-VI and III-V semiconductors (I). Phys. Stat. Sol. 19:459 2. Lasher G (1962) Stimulated Emissions of Radiation from GaAs p-n Junctions. Phys Rev 133:553 3. Akasaki I et al (1995) Breakthroughs in improving crystal quality of GaN and Invention of the p-n junction blue-light-emitting-diode Jpn. J Appl Phys 34:L1517 4. Nakamura S et al (1996) InGaN-based multi-quantum well-structure laser diode Jpn. J Appl Phys 35:L74–L76 5. The Nobel Prize in Physics 2014 https://www.nobelprize.org Retrieved June 03, 2016. 6. Kasap S O (2013) Optoelectronics and Photonics Principles and Practices (2nd ed.) Boston Pearson 7. Maiman T H (1960) Stimulated Optical Radiation in Ruby Nature 187: 493-494. 8. Singh J (1994) Semiconductor Devices: An Introduction International Editions, Singapore.

Suggested Reading 9. 10. 11. 12. 13. 14.

B.G. Streetman, S. Banerjee, Solid State Electronic Devices, Prentice-Hall, 2000. J. Singh, Semiconductor Devices, McGraw-Hill, 1994. P. Bhattacharya, Semiconductor Optoelectronic Devices, Prentice-Hall, 1994. S.S. Li, Semiconductor Physical Electronics, Plenum Press, 1993. S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Pearson, 2013 W.L. Leigh, Devices for Optoelectronics, Marcel Dekker, 1996.

278 15. 16. 17. 18.

7 Light Emitting Diodes and Semiconductor Lasers

M. Fox, Optical Properties of Solids, Oxford University Press, 2001. J. M. Senior, Optical Fiber Communications Principles and Practice, Pearson Education, 2009 T.E. Sale, Vertical Cavity Surface Emitting Lasers, Research Studies Press, 1995 S.F. Yu, Analysis and Design of Vertical Cavity Surface Emitting Lasers, John Wiley & Sons, 2003 19. A. Kitai, Principles of solar cells, LEDs, and diodes, Wiley, 2011

Solution for Selected Problems

Chapter 1: Electrical Properties of Solids 1. 2. 3. 4. 6. 7. 8.

2.12 S/m 0.013 m2 /V s 7.29 × 10–10 m 9.75 eV σ = 1.78 × 107 (m)−1 , μ = 1.35 × 10–3 m2 /V s 0.44 mm –25 −1 k = 9.41 × 108 m−1 ; crystal √ momentum èk = 9.88 × 10 kg m s ; momentum in free space p = mv = 2m E = 3, 82 x 10−25 kg m s−1

Chapter 2: Intrinsic and Extrinsic Semiconductors 1. 2. 3.

d 3 kB T  2/3 2 3π 2 n = 4.32 eV E F = 2m Fermi wavevector: EF =

1/3  1/3  2 k 2F ⇒ k F = 3π 2 n = 3 × π 2 × 4.7 × 1028 2m = 1.12 × 1010 m−1

Fermi velocity of an electron: vF =

1.05 × 10−34 k F = × 1.12 × 1010 = 1.29 × 106 m/s m 0.91 × 10−30

Fermi temperature: TE = k B /E E = 4.32 × 1.6 × 10−19 /1.38 × 10−23 = 5 × 104 K 4.

d

© Springer Nature Switzerland AG 2021 N. Balkan and A. Erol, Semiconductors for Optoelectronics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-319-44936-4

279

280

5.

Solution for Selected Problems

f (E) =

 1 exp

E−E F kT

 +1



E − EF E = E F ⇒ exp kB T

 =1

f (E) = 1/2 Probability of being occupied of Fermi level is 50%.

7.

(a)

n = Nc exp Ec = 0 n = Nc exp (b)

−(E c − E F ) kB T 

EF kT



n=p     EF EV − E F NC exp = N V exp kB T kB T kB T NC Ev − ln 2 2 NV  ∗ 3/2  ∗ kB T me 3k B T me Ev Ev − ln − ln EF = → EF = ∗ 2 2 mh 2 4 m ∗h EF =

(c)

(a)

N(E)F(E)

8.

    F × μp] X = e [Nc exp kEB FT × μe + Nv exp E Vk −E BT X = enμn + epμ p → X corresponds to the conductivity.

EF

E

Solution for Selected Problems

(b) 9. 10.

281

It gives electron density between E C = 0 and E = E F at unit volume

2.12 × 1019 cm−3  + − 2   N + −N − ( N D −N A ) + n 2 = n 3 + 9 + 1 = 2n n = ( D 2 A) + i i i 2 4 16 Using law of mass action p=

11. 12. 13.

n i2 ni = n 2

p = 1016 cm−3 and n = 104 cm−3 R = 8.6 k Charge neutrality: n + NA = p + ND

   2 N − N (N D − N A )  D A + n= + n i2 2 2

14.

15.

The majority carrier is semiconductor is electrons. At 300 K, in Si ni = 1.4 × 1010 cm−3 therefore. n = 1.5 × 1016 cm−3 ve p = ni 2 /n = 1.3 × 103 cm−3 . −4 2 (500)2 E g (500K ) = E g (0K ) − TαT+β = 1.170 − 4.73×10 = 1.148eV  500+636  Eg n i2 = NC N V exp − kB T         500 3 1.148 300 19 19 = 2.8 × 10 1.04 × 10 exp − 300 0.026 500 = 4.2 × 1027 cm−3 n i = 1.05Nd ND + n= 2



ND 2

2

+ n i2 (500K ) = 1.05N D ⇒ N D = 1.18 × 1014 cm−3 ND + 1.05ND = 2

16.



ND 2

2 +

  3/2 N V = 1.04 × 1019 400 = 1.60 × 1019 cm−3 300  k B T = (0.026)

400 300

 = 0.0345eV

282

Solution for Selected Problems



 −(E F − E V ) p = N V exp kB T     −0.27 19 = 1.60 × 10 exp 0.0345 = 6.43 × 1015 cm−3 Chapter 3: Charge Transport in Solids 1. 2. 3. 5. 6. 8.

(a) τ n = 0.85 ns and τ p = 23 ns (b) νn = 3 × 105 cm/s and νn = 8 × 104 cm/s (c) σ n /σ p = μ n /μ p = 3.75 Dn = 104 cm2 /s 5 Dn = 26 cm2 /s and L n = 16 μm d Hall electric field is at –z direction, therefore, Hall voltage is between top and bottom surfaces. VH = 15mV

9.

= 360A/cm2 Jn (di f ) = eDn ddnx ≈ eDn n x

Chapter 4: The P–N Junction Diode 1. 3. 4.

(a) E Fn = E C – 0.206 eV ve E Fp = E V + 0.06 eV (b) eV bi = 0.834 eV W p (V bi ) = 32 Å, W n (V bi ) = 0.32 μm V bi = 0.85 V, W = 0.334 μm, W n = 0.333 μm, W p = 8.3 Å x = W n,   eN D x2 x Wn − Vn = 0 < x < Wn εε0 2 eN D E = −d Vn /d x = − (Wn − x) εε0 eN D E = −d Vn /d x = − (Wn + Wn ) = 0 εε0

5. 6. 7.

(a)charge neutrality in depletion layer:eN D Wn = eN A W p (b) ρ = e(N D − N A ) W (V bi ) = 0.951 μm ve E m = -1.34 × 104 V/cm W (V bi + Vr ) = 2.83 μm Using Eqs. 4.29 and 4.30, the maximum electric field given in Eq. 4.32 can be obtained in case of V bi