Handbook of Nitride Semiconductors and Devices, Materials Properties, Physics and Growth (Handbook of Nitride Semiconductors and Devices (VCH)) (Volume 1)

Hadis Morkoc¸ Handbook of Nitride Semiconductors and Devices

Related Titles Piprek, J. (ed.)

Nitride Semiconductor Devices: Principles and Simulation 2007 ISBN: 978-3-527-40667-8

Adachi, S.

Properties of Group-IV, III-V and II-VI Semiconductors 2005 ISBN: 978-0-470-09032-9

Ruterana, P., Albrecht, M., Neugebauer, J. (eds.)

Nitride Semiconductors Handbook on Materials and Devices 2003 ISBN: 978-3-527-40387-5

Ng, K. K.

Complete Guide to Semiconductor Devices 2002 ISBN: 978-0-471-20240-0

Hadis Morkoç

Handbook of Nitride Semiconductors and Devices Vol. 1: Materials Properties, Physics and Growth

The Author Prof. Dr. Hadis Morkoç Virginia Commonwealth University Dept. of Chemical Enigineering Richmond, VA USA Cover SPIESZDESIGN GbR, Neu-Ulm, Germany

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at . # 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Typesetting Thomson Digital, Noida, India Printing betz-druck GmbH, Darmstadt Binding Litges & Dopf GmbH, Heppenheim Printed in the Federal Republic of Germany Printed on acid-free paper ISBN: 978-3-527-40837-5

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Contents Preface

XIII

Color Tables XXI

1 1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5

General Properties of Nitrides 1 Introduction 1 Crystal Structure of Nitrides 1 Gallium Nitride 30 Chemical Properties of GaN 35 Mechanical Properties of GaN 35 Thermal Properties of GaN 47 Aluminum Nitride 62 Mechanical Properties of AlN 62 Thermal and Chemical Properties of AlN Electrical Properties of AlN 69 Brief Optical Properties of AlN 71 Indium Nitride 75 Crystal Structure of InN 77 Mechanical Properties of InN 77 Thermal Properties of InN 79 Brief Electrical Properties of InN 81 Brief Optical Properties of InN 84 Ternary and Quaternary Alloys 89 AlGaN Alloy 90 InGaN Alloy 92 InAlN Alloy 97 InAlGaN Quaternary Alloy 99 Dilute GaAs(N) 105 References 110

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morkoç Copyright # 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.8.1 2.8.2 2.9 2.9.1 2.9.2 2.10 2.10.1 2.10.2 2.10.3 2.10.4 2.10.5 2.10.6 2.10.7 2.10.8 2.11 2.11.1 2.11.2 2.11.3 2.12 2.12.1 2.12.2 2.12.3 2.12.3.1 2.12.3.2 2.12.3.3 2.12.4 2.12.4.1 2.12.4.2 2.12.5

Electronic Band Structure and Polarization Effects 131 Introduction 131 Band Structure Calculations 132 Plane Wave Expansion Method 134 Orthogonalized Plane Wave (OPW) Method 134 Pseudopotential Method 135 Augmented Plane Wave (APW) Method 135 Other Methods and a Review Pertinent to GaN 136 General Strain Considerations 154 Effect of Strain on the Band Structure of GaN 159 k
p Theory and the Quasi-Cubic Model 160 Quasi-Cubic Approximation 167 Temperature Dependence of Wurtzite GaN Bandgap 169 Sphalerite (Zinc blende) GaN 172 AlN 176 Wurtzite AlN 177 Zinc Blende AlN 183 InN 184 Wurtzitic InN 185 Zinc Blende InN 200 Band Parameters for Dilute Nitrides 202 GaAsN 205 InAsN 208 InPN 209 InSbN 209 GaPN 210 GaInAsN 210 GaInPN 212 GaAsSbN 212 Confined States 212 Conduction Band 216 Valence Band 224 Exciton Binding Energy in Quantum Wells 227 Polarization Effects 230 Piezoelectric Polarization 236 Spontaneous Polarization 241 Nonlinearity of Polarization 245 Origin of the Nonlinearity 250 Nonlinearities in Spontaneous Polarization 253 Nonlinearities in Piezoelectric Polarization 256 Polarization in Heterostructures 264 Ga-Polarity Single AlGaN/GaN Interface 272 Ga-Polarity Single AlxIn1xN/GaN Interface 276 Polarization in Quantum Wells 278

Contents

2.12.5.1 2.12.5.2 2.12.6 2.12.7

Nonlinear Polarization in Quantum Wells 280 InGaN/GaN Quantum Wells 286 Effect of Dislocations on Piezoelectric Polarization Thermal Mismatch Induced Strain 290 References 299

3

Growth and Growth Methods for Nitride Semiconductors 323 Introduction 323 Substrates for Nitride Epitaxy 324 Conventional Substrates 326 Compliant Substrates 327 van der Waals Substrates 328 A Primer on Conventional Substrates and their Preparation for Growth 329 GaAs 329 A Primer on GaAs 330 Surface Preparation of GaAs for Epitaxy 331 Si 332 A Primer on Si 332 Surface Preparation of Si for Epitaxy 333 SiC 334 A Primer on SiC 334 Surface Preparation of SiC for Epitaxy 338 Sapphire 342 A Primer on Sapphire 343 Surface Preparation of Sapphire for Epitaxy 346 ZnO 350 A Primer on ZnO 351 Substrate Preparation for Epitaxy 353 LiGaO2 and LiAlO2 355 LiGaO2 Substrates 355 LiAlO2 Substrates 358 AlN and GaN 359 Seedless Growth of GaN 362 Seedless Growth of GaN by High Nitrogen Pressure Solution Growth (HNPSG) for Substrates 362 Seeded Growth of GaN by HNPSG Method for Substrates 363 Pertinent Surfaces of GaN 365 GaN Surface Preparation for Epitaxy 369 Other Substrates 371 GaN Epitaxial Relationship to Substrates 372 Epitaxial Relationship of GaN and AlN with Sapphire 372 Epitaxial Relationship of GaN and AlN with SiC 381 Epitaxial Relationship of GaN and AlN with Si 381 Epitaxial Relationship of GaN with ZnO 381

3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.1.1 3.2.1.2 3.2.2 3.2.2.1 3.2.2.2 3.2.3 3.2.3.1 3.2.3.2 3.2.4 3.2.4.1 3.2.4.2 3.2.5 3.2.5.1 3.2.5.2 3.2.6 3.2.6.1 3.2.6.2 3.2.7 3.2.7.1 3.2.7.1.1 3.2.7.1.2 3.2.7.2 3.2.7.3 3.2.8 3.3 3.3.1 3.3.2 3.3.3 3.3.4

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Contents

3.3.5 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.1.3 3.4.1.3.1 3.4.1.3.2 3.4.1.3.3 3.4.1.3.4 3.4.1.3.5 3.4.1.3.6 3.4.2 3.4.2.1 3.4.2.2 3.4.2.3 3.4.2.4 3.4.2.5 3.4.2.6 3.4.2.7 3.4.2.8 3.4.2.9 3.4.2.9.1 3.4.2.9.2 3.4.2.9.3 3.4.2.9.4 3.5 3.5.1 3.5.1.1 3.5.1.2 3.5.2 3.5.3 3.5.3.1 3.5.3.2 3.5.3.2.1 3.5.3.2.2 3.5.3.2.3 3.5.3.2.4 3.5.3.2.5 3.5.3.3 3.5.3.4

Epitaxial Relationship of GaN with LiGaO2 and LiAlO2 and Perovskites 382 Nitride Growth Techniques 384 Vapor Phase Epitaxy 385 Hydride Vapor Phase Epitaxy 385 Organometalic Vapor Phase Epitaxy 393 Modeling of OMVPE Growth 398 Thermal Decomposition of GaN as it Relates to Growth 398 Ga and N Precursor Adsorption and Decomposition 400 Ga and N2 Desorption from the Surface 401 Ga and N Surface Diffusion 403 Kinetic Model: Balance Between Adsorption and Desorption 405 Comparison of Model with Growth Conditions for Surface Morphology 406 Molecular Beam Epitaxy 409 Adsorption 411 Desorption 412 Surface Diffusion 413 Incorporation 415 Decomposition 416 Reflection High-Energy Electron Diffraction 417 Plasma-Assisted MBE (PAMBE) or RF MBE, Primarily N Source 428 Reactive Ion MBE 435 Principles of RMBE and PAMBE Growth 436 Growth by RMBE 437 Growth by PAMBE 446 Which Species of N is Desirable? 451 The Effect of III/V Ratio and Substrate Temperature on Surface Morphology 455 The Art and Technology of Growth of Nitrides 462 Sources 467 HVPE Buffer Layers and Laser Liftoff 468 Benchmark HVPE Layers/Templates 472 Growth on GaAs Substrates 477 Growth on SiC: Nucleation Layers and GaN 479 Stacking and Interfacial Relationship 480 Nucleation Layers on SiC 482 High-Temperature AlN Nucleation Layers on SiC 486 Low-Temperature GaN Nucleation Layers on SiC 488 High-Temperature GaN Nucleation Layers on SiC 489 Alloy and Multiple Layer Nucleation Layers on SiC 491 Nucleation Layers on SiC by MBE 492 Substrate Misorientation and Domain Boundaries 495 Polarity of GaN on SiC 498

Contents

3.5.3.4.1 3.5.3.5 3.5.3.6 3.5.4 3.5.5 3.5.5.1 3.5.5.1.1 3.5.5.1.2 3.5.5.1.3 3.5.5.2 3.5.5.2.1 3.5.5.2.2 3.5.5.2.3 3.5.5.2.4 3.5.5.2.5 3.5.5.2.6 3.5.5.3 3.5.5.3.1 3.5.5.4 3.5.5.5 3.5.6 3.5.6.1 3.5.7 3.5.8 3.5.9 3.5.10 3.5.11 3.5.11.1 3.5.11.2 3.5.11.3 3.5.12 3.5.13 3.5.14 3.5.14.1 3.5.15 3.5.15.1 3.5.15.1.1 3.5.15.1.2 3.5.15.2 3.5.15.2.1 3.5.15.2.2 3.5.15.2.3 3.5.15.3

GaN Growth on SiC 499 Growth on Porous SiC 503 Zinc Blende Phase Growth 507 Growth on Si 507 Growth on Sapphire 512 OMVPE Low-Temperature Nucleation Buffer Layers 513 The Effect of V/III Ratio on Nucleation Buffer Layer 523 Effect of Epitaxial Growth Temperature 525 Effect of Process Pressure 525 Epitaxial Lateral Overgrowth 528 Selective Epitaxial Growth and Lateral Epitaxial Overgrowth with HVPE 537 Lateral Epitaxial Overgrowth on Si 539 Pendeo-Epitaxy 540 Pendeo-Epitaxy on SiC Substrates 542 Pendeo-Epitaxy on Silicon Substrates 544 Point Defect Distribution in ELO Grown GaN 558 Nanoheteroepitaxy and Nano-ELO 564 SiN and TiN Nanonets 569 Selective Growth Using W Masks 583 Low-Temperature Buffer Interlayer 584 Polarity and Surface Structure of GaN Layers, Particularly on Sapphire 586 MBE Buffer Layers 597 Growth on ZnO Substrates 598 Growth on LiGaO2 and LiAlO2 Substrates 603 Growth on GaN Templates 605 Growth on Spinel (MgAl2O4) 611 Growth on Non c-Plane Substrates 611 The a-Plane GaN Growth 613 Epitaxial Lateral Overgrowth of a-plane GaN 616  00) m-Plane GaN Growth 623 The (11 Growth of p-Type GaN 627 Growth of InN 629 Growth of AlN 638 Surface Reconstruction of AlN 642 Growth of Ternary and Quaternary Alloys 652 Growth of AlGaN 653 Growth of p-Type AlGaN 666 Ordering in AlGaN 668 Growth of InGaN 671 Doping of InGaN 678 Phase Separation in InGaN 679 Surface Reconstruction of InGaN 689 Growth of AlInN 695

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Contents

3.5.15.3.1 3.5.15.4 3.5.16 3.5.16.1 3.5.16.2 3.5.16.3 3.5.16.4 3.5.16.4.1 3.5.16.4.2 3.5.16.5 3.5.16.5.1 3.5.16.5.2 3.6

Miscibility Gap in InAlN 697 InGaAlN Quaternary Alloy 699 Growth of Quantum Dots 706 Quantum Dots by MBE 711 Quantum Dots by OMVPE 719 Quantum Dots by Other Techniques 723 Preparation and Properties of Nanostructures 725 Approaches for Synthesis 726 Vapor Phase Growth 726 Nanowires and Longitudinal Heterostructures 737 Coaxial Heterostructures 755 Nanotubes 756 Concluding Remarks 759 References 760

4

Extended and Point Defects, Doping, and Magnetism 817 Introduction 817 A Primer on Extended Defects 818 Dislocations 819 Misfit Dislocations 822 Threading Dislocations 822 Edge Dislocations 824 Screw Dislocations 828 Mixed Dislocations 836 Nanopipes or Hollow Pipes 840 Planar Defects: Domain Boundaries 844 Stacking Faults 851 Grain Boundaries 862 Electronic Structure of Extended Defects 863 Open Core Versus Close Core in Screw Dislocations 865 Edge and Mixed Dislocations 866 Simple Stacking Faults: Electrical Nature 884 TEM Analysis of High Nitrogen Pressure (HNP) Solution Growth (HNPSG) and HVPE-Grown GaN 886 Extended Defect Characterization 887 Pyramidal Defects 894 V-Shaped Defects (Pits) in InGaN Multiple Quantum Wells (MQWs) 905 Structural Defect Analysis by Chemical Etch Delineation 905 Structural Defect Observations with Surface Probes 910 Point Defects and Autodoping 917 Theoretical Studies of Point Defects in GaN 919 Hydrogen and Impurity Trapping at Extended Defects 924 Vacancies, Antisites, Interstitials, and Complexes 928 Vacancies 929

4.1 4.1.1 4.1.1.1 4.1.1.2 4.1.1.2.1 4.1.1.2.2 4.1.1.2.3 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.1.6.1 4.1.6.2 4.1.6.3 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.3 4.3.1 4.3.1.1 4.3.1.2 4.3.1.2.1

Contents

4.3.1.2.2 4.3.2 4.3.2.1 4.3.2.2 4.3.2.3 4.3.2.4 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.6 4.6.1 4.6.2 4.7 4.8 4.9 4.9.1.1 4.9.1.2 4.9.1.3 4.9.2 4.9.2.1 4.9.2.2 4.9.2.3 4.9.2.4 4.9.3 4.9.3.1 4.9.3.1.1 4.9.3.1.2 4.9.3.1.3 4.9.3.2 4.9.3.3 4.9.3.4 4.9.3.5 4.9.3.6 4.9.3.7 4.9.3.8 4.9.4 4.9.4.1 4.9.4.2 4.9.5 4.9.6

Interstitials and Antisite Defects 934 Complexes 935 Shallow Donor – Gallium Vacancy Complexes 936 Shallow Acceptor – Nitrogen Vacancy Complexes 936 Hydrogen-Related Complexes 937 Other Complexes 938 Defect Analysis by Deep-Level Transient Spectroscopy 938 Basics of DLTS 939 Applications of DLTS to GaN 948 Dispersion in DLTS of GaN 970 Applications of DLTS to AlGaN, In-Doped AlGaN, and InAlN 977 Minority Carrier Lifetime 977 Positron Annihilation 982 Vacancy Defects and Doping in Epitaxial GaN 983 Growth Kinetics and Thermal Behavior of Vacancy Defects in GaN 993 Fourier Transform Infrared (FTIR), Electron Paramagnetic Resonance, and Optical Detection of Magnetic Resonance 998 Role of Hydrogen 1004 Intentional Doping 1006 Shallow Donors 1007 Substitutional Acceptors 1007 Isoelectronic Impurities 1009 n-Type Doping with Silicon, Germanium, Selenium, and Oxygen 1010 Si Doping 1010 Ge Doping 1011 Se Doping 1012 p-Type Doping 1013 p-Type Doping and Codoping with Donors and Acceptors 1014 Magnesium Doping 1014 Codoping for Improving p-Type Conductivity 1019 Use of Superlattices for Improving p-Type Conductivity 1032 Role of Hydrogen and Defects in Mg-Doped GaN 1034 Beryllium Doping 1038 Mercury Doping 1040 Carbon Doping 1040 Zinc Doping 1042 Calcium Doping 1042 Cadmium Doping 1043 Other Acceptors in GaN 1043 Doping with Isoelectronic Impurities 1044 Arsenic Doping 1044 Phosphorus Doping 1045 Doping with Rare Earths 1045 Doping with Transition Metals and Rare Earths 1046

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Contents

4.9.6.1 4.9.6.2 4.9.6.3 4.9.6.4 4.9.6.5 4.9.6.5.1 4.9.6.5.2 4.9.6.5.3 4.9.6.6 4.9.6.6.1 4.9.6.7 4.9.6.7.1 4.9.6.7.2 4.9.7 4.9.7.1 4.9.7.2 4.9.7.3 4.9.8 4.9.9 4.9.10 4.9.11 4.9.12 4.10 4.11

Manganese Doping for Electronic Properties 1046 Other TM Doping for Electronic Properties 1060 General Remarks About Dilute Magnetic Semiconductors 1063 General Remarks About Spintronics 1075 Theoretical Aspects of Dilute Magnetic Semiconductor 1082 Carrier – Single Magnetic Ion Interaction 1084 Interaction Between Magnetic Ions 1086 Zener, Mean Field, RKKY, and Ab Initio Treatments 1089 A Primer to Magnetotransport Measurements 1103 Faraday Rotation, Kerr Effect, and Magnetic Circular Dichroism (MCD) 1104 II–VI and GaAs-Based Dilute Magnetic Semiconductors 1123 II–VI-Based Dilute Magnetic Semiconductors 1124 III–V-Based DMS: (GaMn)As 1133 Experimental Results of TM-Doped GaN 1141 Magnetotransport Properties TM-Doped GaN 1141 Magnetic Properties of Mn-Doped GaN 1143 Magneto-Optical Measurements in TM-Doped GaN 1146 Magnetic, Structural, Optical, and Electrical Properties of Cr-Doped GaN 1156 Other TM and Rare Earth Doped GaN:(Co, Fe, V, Gd, and so on) 1163 TM-Doped Nanostructures 1166 Applications of Ferromagnetism and Representative Devices 1167 Summarizing Comments on Ferromagnetism 1186 Ion Implantation and Diffusion for Doping 1188 Summary 1190 References 1191 Index

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Preface This three-volume handbook represents the only comprehensive treatise on semiconductor and device fundamentals and technology under the overall umbrella of wide bandgap nitride semiconductors with comparison to GaAs when applicable. As it stands, the book is a reference book, a handbook, and a graduate text book all in one and would be beneficial to second-year graduate students majoring in semiconductor materials and devices, graduate research assistants conducting research in wide bandgap semiconductors, researchers and technologists, faculty members, program monitors, and managers. The philosophy of this endeavor is to present the material as much clearly and comprehensively as possible, so that there is very little need to refer to other sources to get a grasp of the subjects covered. Extreme effort has been expended to ensure that concepts and problems are treated starting with their fundamental basis so that the reader is not left hanging in thin air. Even though the treatise deals with GaN and related materials, the concepts and methods discussed are applicable to any semiconductor. The philosophy behind Nitride Semiconductors and Devices was to provide an adequate treatment of nitride semiconductors and devices as of 1997 to be quickly followed by a more complete treatment. As such, Nitride Semiconductors and Devices did not provide much of the background material for the reader and left many issues unanswered in part because they were not yet clear to the research community at that time. Since then, tremendous progress both in the science and engineering of nitrides and devices based on them has been made. While LEDs and lasers were progressing well even during the period when Nitride Semiconductors and Devices was written, tremendous progress has been made in FETs and detectors in addition to LEDs and lasers since then. LEDs went from display devices to illuminants for lighting of all kinds. Lasers are being implemented in the third generation of DVDs. The power amplifiers are producing several hundred watts of RF power per chip and the detectors and detector arrays operative in the solar-blind region of the spectrum have shown detectivities rivaling photomultiplier tubes. The bandgap of InN has been clarified which now stands near 0.7 eV. Nanostructures, which did not exist

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morkoç Copyright # 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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Preface

during the period covered by Nitride Semiconductors and Devices, have since become available. The technological breakthroughs such as epitaxial lateral overgrowth, laser liftoff, and freestanding GaN were either not fully developed or did not exist, neither did the highly improved quantum structures and devices based on them. In the interim period since then, the surfaces of nitrides and substrate materials, point defects and doping, magnetic ion doping, processing, current conduction mechanisms, and optical processes in bulk and quantum structures have been more clearly understood and many misconceptions (particularly, those dealing with polarization) identified, removed and/or elucidated. The handbook takes advantage of the fundamental and technological developments for a thorough treatment of all aspects of nitride semiconductors. In addition, the fundamentals of materials physics and device physics that are provided are applicable to other semiconductors, particularly, wurtzitic direct bandgap semiconductors. The handbook presents a thorough treatment of the science, fundamentals, and technology of nitride semiconductors and devices in such a width and depth that the reader would seldom need to engage in time-consuming exploration of the literature to fill in gaps. Last but not the least, the handbook contains seamless treatments of fundamentals needed or relied on throughout the entire book. The following is a succinct odyssey through the content of the three-volume handbook. Volume 1, Chapter 1 discusses the properties of nitride-based semiconductors with plenty of tables for reference. Volume 1, Chapter 2 treats the band structure of III–V nitrides, theories applied to determining the band structure, features of each theory with a succinct discussion of each, band structure of dilute III–V semiconductors doped with N, strain and stress, deformation potentials, and in-depth discussion of piezo and spontaneous polarization with illustrative and instructive artwork. Volume 1, Chapter 3 encompasses substrates that have been and are used for growth of nitride semiconductors, mainly, structural and mechanical (thermal) properties of those substrates, surface structure of planes used for growth, and substrate preparation for growth. Orientation and properties of GaN grown on those substrates are discussed along with commonly used surface orientations of GaN. The discussion is laced with highly illustrative and illuminating images showing orientations of GaN resulting through growth on c-plane, a-plane, m-plane, and r-plane substrates whichever applicable and the properties of resulting layers provided. The treatment segues into the discussion of various growth methods used for nitrides taking into account the fundamentals of growth including the applicable surface-oriented processes, kinetics, and so on, involved. A good deal of growth details for both OMVPE and MBE, particularly, the latter including the fundamentals of in situ process monitoring instrumentation such as RHEED, and dynamics of growth processes occurring at the surface of the growing layer are given. Of paramount interest is the epitaxial lateral overgrowth (ELO) for defect reduction. In addition to standard single multistep ELO, highly attractive nanonetwork meshes used for ELO are also discussed. Specifics in terms of growth of binary, ternary, and quaternaries of nitride semiconductors are discussed. Finally, the methods used to grow nanoscale structures are treated in sufficient detail.

Preface

Volume 1, Chapter 4 focuses on defects, both extended and point, doping for conductivity modulation and also for rendering the semiconductor potentially ferromagnetic segueing into electrical, optical, and magnetic properties resulting in films, with sufficient background physics provided to grasp the material. A clear discussion of extended defects, including line defects, are discussed with a plethora of illustrative schematics and TEM images for an easy comprehension by anyone with solid-state physics background. An in-depth and comprehensive treatment of the electrical nature of extended defects is provided for a full understanding of the scope and effect of extended defects in nitride semiconductors, the basics of which can be applicable to other hexagonal materials. The point defects such as vacancies, antisites, and complexes are then discussed along with a discussion of the effect of H. This gives way to the methods used to analyze point defects such as deep level transient spectroscopy, carrier lifetime as pertained to defects, positron annihilation, Fourier transform IR, electron paramagnetic resonance, and optical detection of magnetic resonance and their application to nitride semiconductors. This is followed by an extensive discussion of n-type and p-type doping in GaN and related materials and developments chronicled when applicable. An in-depth treatment of triumphs and challenges along with codoping and other methods employed for achieving enhanced doping and the applicable theory has been provided. In addition, localization effects caused by heavy p-type doping are discussed. This gives way to doping of, mainly, GaN with transition elements with a good deal of optical properties encompassing internal transition energies related to ion and perturbations caused by crystal field in wurtzitic symmetry. To get the reader conditioned for ferromagnetism, a sufficient discussion of magnetism, ferromagnetism, and measurement techniques (magnetic, magneto transport, magneto optics with underlying theory) applied to discern such properties are given. This is followed by an indepth and often critical discussion of magnetic ion and rare earth-doped GaN, as well as of spintronics, often accompanied by examples for materials properties and devices from well-established ferromagnetic semiconductors such as Mn-doped GaN and Cr-doped ZnTe. Volume 2, Chapter 1 treats metal semiconductor structures and fabrication methods used for nitride-based devices. Following a comprehensive discussion of current conduction mechanisms in metal semiconductor contacts, which are applicable to any metal semiconductor system, specific applications to metal-GaN contacts are treated along with the theoretical analysis. This gives way to a discussion of ohmic contacts, their technology, and their characterization. In particular, an ample discussion of the determination of ohmic contact resistivity is provided. Then etching methods, both dry (plasma) and wet, photochemical, process damage, and implant isolation are discussed. Volume 2, Chapter 2 deals with determination of impurity and carrier concentrations and mobility mainly by temperature-dependent electrical measurements, such as Hall measurements. Charge balance equations, capacitance voltage measurements, and their intricacies are treated and used for nitride semiconductors, as well as a good deal of discussion of often brushed off degeneracy factors.

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Volume 2, Chapter 3 is perhaps one of the most comprehensive discussions of carrier transport in semiconductors with applications to GaN. After a discussion of scattering processes in physical and associated mathematical terms, the methods discussed are applied to GaN and other related binaries and ternaries with useful ranges of doping levels, compositions, and lattice temperatures. Comparisons with other semiconductors are also provided when applicable. This discussion segues into the discussion of carrier transport at high electric fields applicable to field-effect transistors, avalanche and pin (biased) photodiodes. This is followed by the measurement of mobility and associated details, which are often neglected in text and reference books. The discussion then flows into magnetotransport beyond that present in Hall measurements. Low, medium, and high magnetic field cases, albeit only normal to the surface of the epitaxial layers, determination of which is affected by the value of the mobility and various cases are treated. The treatise also includes cases where the relaxation time, if applicable, is energy-dependent and somewhat energy-independent. The discussion of the magnetotransport paves the way for a fundamental and reasonably extensive discussion of the Hall factor for each of the scattering mechanisms that often are not treated properly or are treated only in a cursory manner in many texts leading to confusion. After providing the necessary fundamental knowledge, the transport properties of GaN are discussed. This gives way to the discussion of various scattering mechanisms in two-dimensional systems that are relied on in high-performance FETs. For determining the mobility of each layer (in the case of multiple layer conduction), quantitative mobility spectrum analysis including both fundamentals and experimental data obtained in nitride semiconductors is discussed. The quantum Hall effect and fractional quantum Hall effect in general and as germane to GaN are discussed along with parameters such as the effective mass determined from such measurements. Volume 2, Chapter 4 is devoted to p–n junctions, beginning with the discussion of band lineups, particularly, in the binary pairs from the point of view of theoretically and experimentally measured values. Current conduction mechanisms, such as diffusion, generation-recombination, surface recombination, Poole–Frenkel, and hopping conductivity are discussed with sufficient detail. Avalanche multiplication, pertinent to the high-field region of FETs, and avalanche photodiodes, are discussedfollowed by discussions of the various homojunction and heterojunction diodes based on nitrides. Volume 2, Chapter 5 is perhaps the most comprehensive discussion of optical processes that can occur in a direct bandgap semiconductor and, in particular, in nitride-based semiconductors and heterostructures inclusive of 3, 2-, and 0-dimensional structures as well as optical nonlinearities. Following a treatment of photoluminescence basics, the discussion is opened up to the treatment of excitons, exciton polaritons, selection rules, and magneto-optical measurements followed by extrinsic transitions because of dopants/impurities and/or defects with energies ranging from the yellow and to the blue wavelength of the visible spectrum. Optical transitions in rare earth-doped GaN, optical properties of alloys, and quantum wells are then discussed with a good deal of depth, including localization effects and their possible sources particularly media containing InN. The discussion then leads to the

Preface

treatment of optical properties of quantum dots, intersubband transitions in GaNbased heterostructures, and, finally, the nonlinear optical properties in terms of second and third harmonic generation with illuminating graphics. Volume 3, Chapter 1 is devoted, in part, to the fundamentals of light emitting diodes, the perception of vision and color by human eye, methodologies used in conjunction with the chromaticity diagram and associated international standards in terms of color temperatures and color rendering index. Specific performances of various types of LEDs including UV varieties, current spreading or the lack of related specifics, analysis of the origin of transitions, and any effect of localization are discussed. A good deal of white light and lighting-related standards along with approaches employed by LED manufacturers to achieve white light for lighting applications is provided. The pertinent photon conversion schemes with sufficient specificity are also provided. Finally, the organic LEDs, as potential competitors for some applications of GaN-based LEDs are discussed in terms of fundamental processes that are in play and various approaches that are being explored for increased efficiency and operational lifetime. Volume 3, Chapter 2 focuses on lasers along with sufficient theory behind laser operation given. Following the primer to lasers along with an ample treatment based on Einstein’s A and B coefficients and lasing condition, an analytical treatment of waveguiding followed by specifics for the GaN system and numerical simulations for determining the field distribution, loss, and gain cavity modes pertaining to semiconductor lasers are given. An ample fundamental treatment of spontaneous emission, stimulated emission, and absorptions and their interrelations in terms of Einstein’s coefficients and occupation probabilities are given. This treatment segues into the extension of the gain discussion to a more realistic semiconductor with a complex valence band such as that of GaN. The results from numerical simulations of gain in GaN quantum wells are discussed, as well as various pathways for lasing such as electron-hole plasma and exciton-based pathways. Localization, which is very pervasive in semiconductors that are yet to be fully perfected, is discussed in the light of laser operation. Turning to experimental measurements, the method for gain measurement, use of various laser properties such as the delay on the onset of lasing with respect to the electrical pulse, dependence of laser threshold on cavity length to extract important parameters such as efficiency are discussed. The aforementioned discussions culminate in the treatment of performance of GaN-based lasers in the violet down to the UV region of the optical spectrum and applications of GaN-based lasers to DVDs along with a discussion of pertinent issues related to the density of storage. Volume 3, Chapter 3 treats field effect transistor fundamentals that are applicable to any semiconductor materials with points specific to GaN. The discussion primarily focuses on 2DEG channels formed at heterointerfaces and their use for FETs, including polarization effects. A succinct analytical model is provided for calculating the carrier densities at the interfaces for various scenarios and current voltage characteristics of FETs with several examples. Experimental performance of GaNbased FETs and amplifiers is then discussed followed by an in-depth analysis of anomalies in the current voltage characteristics owing to bulk and barrier states,

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including experimental methods and probes used for cataloging these anomalies. This is followed by the employment of field spreading gate plates and associated performance improvements. This segues into the discussion of noise both at the low-frequency end and high-frequency end with sufficient physics and practical approaches employed. The combined treatment of various low-frequency noise contributions as well as those at high frequencies along with their physical origin makes this treatment unique and provides an opportunity for those who are not specialists in noise to actually grasp the fundamentals and implications of low- and high-frequency noise. Discussion of high-power FETs would not be complete without a good discussion of heat dissipation and its physical pathways, which is made available. Unique to GaN is the awareness of the shortfall in the measured electron velocity as compared to the Monte Carlo simulation. Hot phonon effects responsible for this shortfall are uniquely discussed with sufficient theory and experimental data. A section devoted to reliability with specifics to GaN based high power HFETs is also provided. Finally, although GaN-based bipolar transistors are not all that attractive at this time, for completeness and the benefit of graduate students and others who are interested in such devices, the theory, mainly analytical, of the operation of heterojunction bipolar transistors is discussed along with available GaN based HBT data. Volume 3, Chapter 4 discusses optical detectors with special orientation toward UV and solar-blind detectors. Following a discussion of the fundamentals of photoconductive and photovoltaic detectors in terms of their photo response properties, a detailed discussion of the current voltage characteristic of the same, including all the possible current conduction mechanisms, is provided. Because noise and detectors are synonymous with each other, sources of the noise are discussed, followed by a discussion of quantum efficiency in photoconductors and p–n junction detectors. This is then followed by the discussion of vital characteristics such as responsivity and detectivity with an all too important treatment of the cases where the detectivity is limited by thermal noise, shot current noise, generation-recombination current noise, and background radiation limited noise (this is practically nonexistent in the solar-blind region except the man-made noise sources). A unique treatment of particulars associated with the detection in the UV and solar-blind region and requirements that must be satisfied by UV and solar-blind detectors, particularly, for the latter, is then provided. This leads the discussion to various UV detectors based on the GaN system, including the Si- and SiC-based ones for comparison. Among the nitride-based photodetectors, photoconductive variety as well as the metal-semiconductor, Schottky barrier, and homo- and heterojunction photodetectors are discussed along with their noise performance. Nearly solar-blind and truly solar-blind detectors including their design and performance are then discussed, which paves the way for the discussion of avalanche photodiodes based on GaN. Finally, the UV imagers using photodetectors arrays are treated. It is fair to state that I owe so much to so many, including my family members, friends, coworkers, colleagues, and those who contributed to the field of semiconductors in general and nitride semiconductors in particular, in my efforts to bring this manuscript to the service of readers. To this end, I thank my wife, Amy, and son,

Preface

Erol, for at least their understanding why I was not really there for them fully during the preparation of this manuscript, which took longer than most could ever realize. Also, without the support of VCU, with our Dean R. J. Mattauch, Assistant Dean Susan Younce, Department Chair A. Iyer, and my coworkers and students, it would not have been possible to pursue this endeavor. Special recognitions also go to Dr N. Izyumskaya for reading the entire manuscript for consistency in terms of figures, references, and so on, which had to have taken perseverance beyond that many could muster; Dr Ü. Özgür for being the bouncing board and proofing many parts of the book, particularly chapters dealing with optical processes, lasers and magnetism; my colleague P. Jena for reading and contributing to the band structure section; my coworker Professor M. Reshchikov for his contributions to the point defects and doping sections; Professor A. Baski for her expert assistance in obtaining microprobe images; Dr D. Huang for his many contributions to the quantum dots section; Dr Y-T Moon for his assistance in current crowding; C. Liu for her assistance with ferromagnetism; Prof. A. Teke for reading the chapter on detectors; Dr. R. Shimada for her contributions to the surface emitting laser section; Dr. J.-S. Lee for his help in updating the LED chapter; Dr Q. Wang for her help in generating the accurate ball and stick diagrams in Volume 1, Chapter 1; Dr V. Litvinov for calculating the energy levels in quantum wells; students Y. Fu, Q. Fan, X. Ni, and S. Chevtchenko for their contributions to various sections of the book with proofing equations, redoing calculations, and so on; and to J. Leach who took it upon himself to be the local expert in the latest in semiconductor and organic LEDs and helped with the chapter on LEDs and read the chapter on transport as well as proofread some of the other chapters and create the figures; Ms G. Esposito for reading a large portion of the text for English. Undergraduate students K. Ngandu, D. Lewis, B. D. Edmonds, and M. Mikkelson helped in reading various parts of the manuscript as well as helping with the artwork. Unbeknown to them, many graduate students who took classes from me helped in many immeasurable ways. Special thanks go to Professors R. M. Feenstra, A. Matulionis, A. Blumenau, P. Ruterana, G. P. Dimitrakopulos, P. Handel, K. T. Tsen, T. Yao, P. I. Cohen, S. Porowski, B. Monemar, B. Gil, P. Le Febvre, S. Chichibu, F. Tuomisto, C. Van de Walle, M. Schubert, F. Schubert, H. Temkin, S. Nikishin, L. Chernyak, J. Edgar, T. Myers, K. S. A. Butcher, O. Ambacher, V. Fiorentini, A. di Carlo, F. Bernardini, V. Fiorentini, M. Stutzmann, F. Pollak, C. Nguyen, S. Bedair, N. El-Masry, S. Fritsch, M. Grundman, J. Neugebauer, M. S. Shur, J. Bowers, J. C. Campbell, M. Razhegi, A. Nurmikko, M. A. Khan, J. Speck, S. Denbaars, R. J. Trew, A. Christon, G. Bilbro, H. Ohno, A. Hoffmann, B. Meyer, B. Wessels, N. Grandjean, and D. L. Rode; and Drs Z. Liliental-Weber, P. Klein, S. Binari, D. Koleske, J. Freitas, D, Johnstone, D. C. Look, Z.-Q. Fang, M. MacCartney, I. Grzegory, M. Reine, C. W. Litton, P. J. Schreiber, W. Walukiewicz, M. Manfra, O. Mitrofanov, J. Jasinski, V. Litvinov, Jan-Martin Wagner, K. Ando, H. Saito, C. Bundesmann, D. Florescu, H. O. Everitt, H. M. Ng, I. Vurgaftman, J. R. Meyer, J. D. Albrecht, C. A. Tran, S.-H. Wei, G. Dalpian, N. Onojima, A. Wickenden, B. Daudin, R. Korotkov, P. Parikh, D. Green, A. Hansen, P. Gibart, F. Omnes, M. G. Graford, M. Krames, R. Butte, and M. G. Ganchenkova for either reading sections of the book, providing unpublished data, or providing

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suggestions. Many more deserve a great deal of gratitude for willingly spending considerable time and effort to provide me with digital copies of figures and highquality images, but the available space does not allow for individual recognition. They are acknowledged in conjunction with the figures. In a broader sense, I benefited greatly from the counsel and support of Professor T. A. Tombrello of Caltech. I also would like to use this opportunity to recognize a few of the unsung heroes, namely, Dr Paul Maruska and Professor Marc Ilegems who truly started the epitaxy of nitrides with the hydride VPE technique independently, and Dr S. Yoshida and Professor T. Matsuoka for their pioneering work in AlGaN and InGaN, respectively. Richmond, VA January 2008

Hadis Morkoç

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Figure 1.4 A stick-and-ball stacking model of crystals with (a, both top and bottom) 2H wurtzitic and (b, both top and bottom) 3C zinc blende polytypes. The bonds in an A-plane (1 1 2 0) are indicated with heavier lines to accentuate the stacking sequence. The figures on top depict the

three-dimensional view. The figures at the bottom indicate the projections on the (0 0 0 1) and (1 1 1) planes for wurtzitic and cubic phases, respectively. Note the rotation in the zinc blende case along the h1 1 1i direction. (This figure also appears on page 6.)

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morkoc Copyright
2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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Figure 1.9 The bandgaps of nitrides, substrates commonly used for nitrides, and other conventional semiconductors versus their lattice constants. (This figure also appears on page 12.)

Figure 1.23 An artist’s view of the scanning thermal microscope. Patterned after D.I. Florescu and F.H. Pollak. (This figure also appears on page 57.)

Color Tables

50

40

q D = 800 K

–1

–1

Molar specific heat, Cp (cal mol K )

q D = 500 K

30

20

Cp data, GaN q D = 500 K q D = 600 K

10

q D = 700 K q D = 800 K

0 0

200

400

600

800

1000

Temperature (K) Figure 1.26 Molar specific heat at constant pressure, Cp (cal mol1 K1), of GaN versus temperature. Open circles represent the experimental data. The solid lines are calculation based on the Debye model for Debye temperatures, yD, of 500, 600, 700, and 800 K. Unfortunately, it is difficult to discern a Debye

temperature that is effective over a wide temperature range because a large concentration of defects and impurities is present in GaN. However, a value of 600 K estimated by Slack is used commonly. The data are taken from Refs [215,216], as compiled in Ref. [88]. (This figure also appears on page 61.)

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50

40 Specific heat, Cp (J mol–1 K–1)

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30

20 Specific heat AlN (J mol–1 K–1)

800 K 850 K 900 K 950 K 1000 K 1050 K 1100 K

10

0

0

200

400

600

800

1000

Temperature (K) Figure 1.29 Molar specific heat at constant pressure, Cp (J mol1 K1, 1 cal ¼ 4.186 J), of AlN versus temperature. Open circles represent the experimental data. The solid lines are calculation based on the Debye model for Debye temperatures, yD, in the range of 800–1100 K

with 50 K increments. The data can be fit with Debye expression for yD ¼ 1000 K, which compares with 950 K reported by Slack et al. The data are taken from Ref. [88]. (This figure also appears on page 68.)

Color Tables

Figure 1.41 Bandgap versus composition for quaternary AlxInyGa1xyN (assumed InN bandgap ¼ 0.8 eV). (This figure also appears on page 101.)

Figure 1.42 Bandgap versus composition for quaternary AlxInyGa1xyN (assumed InN bandgap ¼ 1.9 eV). (This figure also appears on page 101.)

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0.02

Spontaneous polarization (C m–2)

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+0.333 InxAl1–xN

InxGa1–xN

0.00

+0.193

–0.02

InxGa1–xN

–0.04

GaN

+0.095 –0.06

InxAl1–xN

InN AIxGa1–xN

+0.037 –0.08

Random alloy CH-like CP-like

AIN

AIN

–0.10 0

0.2

0.4

0.6

1

Molar fraction, x Figure 2.37 Spontaneous polarization versus the molar fraction in all three ternary nitride alloys. Circles, squares, and triangles represent random alloy, CH-/LZ-, and CP-like structures, respectively. The dashed/dotted lines (blue) with solid triangles are for the CP-like alloys, the dashed lines (green) with solid squares are for CH-like alloys, and solid lines (black) with filled

circles are for random alloys. The black dashed lines represent the data calculated using Vegard’s law. Numbers indicated in the figure are for CP and CH-/LZ-like ordered alloy bowing parameters in terms of C m2. Courtesy of F. Bernaridini and V. Fiorentini. (This figure also appears on page 249.)

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25

AlN Si 6HSiC 4HSiC 3CSiC LiGaO2 Al2O3 MgO GaAs ZnO MgAl2O4 LiAlO2 ScMgAlO4 NdGaO3

Curvature (1 m–1)

20

15

10

5

0.0

–5 0.0

20

40

60

Thickness (μm) Figure 2.59 A compilation of the variation of thermal curvature, a measure of strain, in epitaxial GaN layers grown on different substrates with respect to layer thickness [492]. (This figure also appears on page 294.)

80

100

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Figure 2.59 (Continued )

Color Tables

5108

Stress on various substrates (Pa)

0

–5108

–1109 AlN Si 6HSiC 4HSiC 3CSiC LiGaO2 Al2O3 MgO GaAs ZnO MgAl2O4 LiAlO2 ScMgAlO4 NdGaO3

–1.5109

–2109

–2.5109 –3109 0100

210–5

410–5

610–5

810–5

Thickness (m) Figure 2.60 A compilation of residual thermal stresses in epitaxial GaN layer on different substrates with respect to layer thickness [492]. (This figure also appears on page 296.)

110–4

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Stress versus thickness of GaN/potential substrates 5.00E+08

0.00E+00 0

0.00002

0.00004

0.00006

0.00008

0.0001

AlN Si MgO

–5.00E+08

3C-SiC 6H-SiC 4H-SiC

Stress (Pa)

–1.00E+09

ZnO Al2O3 LiGaO2

–1.50E+09

MgAl2O4 GaAs NdGaO3** ScAlMgO**

–2.00E+09

LiAlO2 LSAT –2.50E+09

–3.00E+09

Thickness (m) Figure 2.60 (Continued )

Color Tables

Figure 3.1 The perspective view of the GaAs crystal (a) along [1 0 0] (1 · 1 · 1 unit), (b) [1 1 0] (2 · 2 · 2 units), and (c) [1 1 1] (2 · 2 · 2 units) directions [13]. (This figure also appears on page 330.)

Figure 3.2 The perspective view along (a) the [0 0 1], (b) [0 1 1], and (c) [1 1 1] directions of a Si cell. (This figure also appears on page 333.)

Figure 3.3 Tetragonal bonding of a carbon atom with its four nearest silicon neighbors. The bond lengths depicted with a and C–Si (the nearest neighbor distance) are approximately 3.08 and 1.89 Å, respectively. The right side is the three-dimensional structure of 2H-SiC structure. (This figure also appears on page 334.)

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Figure 3.9 (a) Top view of the oxide structure on SiC ð0 0 0 1Þ. The Si2O3 silicate adlayer consisting of a honeycomb structure with SiOSi bonds. At the center of the hexagons, one carbon atom of the topmost substrate bilayer is visible [the dark shaded area indicates the (1 1; 1) unit cell and light shaded the

pffiffiffi pffiffiffi ð 3  3ÞR30  -unit cell]; (b) side view of the oxide structure on the SiC (0 0 0 1) in ð0 1 1 0Þ SiC projection. Linear SiOSi bonds connect the silicate layer and the underlying SiC substrate. Courtesy of N. Onojima (patterned after Ref. [40]). (This figure also appears on page 342.)

Figure 3.10 (a) Top view (projection on the Si-plane of the basal plane of SiC) and (b) side view of SiC after an in situ Ga exposure indicating of the lack of silicate adlayer. Courtesy of N. Onojima. (This figure also appears on page 342.)

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Figure 3.12 The unit cell of sapphire: (a) rhombohedral unit cell; (b) hexagonal unit cell. Smaller spheres are for O and large ones are for Al [13]. (This figure also appears on page 344.)

Figure 3.13 Perspective views in (2 · 2 · 1) unit cells: (a) along the [0 0 0 1] direction in a rhombohedral unit cell; (b) along the (0 0 0 1) direction in hexagonal unit cell [13]. (This figure also appears on page 344.)

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Side view of sapphire r-plane {1 0 1 2}

Oxygen

(a)

Al

Al2O3 [1 0 1 1]

Color Tables

a

2

m

C

b

m

LiGaO2

Lower O

Li

Upper O

Ga

Figure 3.24 Example of the exact fit of GaN atoms over the LiGaO2 lattice if there is no distortion. Courtesy H. Paul Maruska. (This figure also appears on page 356.)

3

Figure 3.20 (a) Sapphire r-plane stacking sequence showing O atoms in larger clear circles and Al atoms in smaller, filled circles. The salient feature is that each Al layer has an O layer above and below it. (b) The atomic arrangement on three layers (the uppermost one is O,

immediately below is Al and third layer down is another O layer) on the r-plane of sapphire. The lines are there just guides to eye and do not represent bonds. (This figure also appears on page 352.)

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LGO: orthorhombic a = 5.402 Å b = 6.372 Å c = 5.007 Å

c b

a

(a)

LGO: orthorhombic a = 5.402 Å b = 6.372 Å c = 5.007 Å

c b

a

(b)

Projection on c-plane ∆b = –0.19 % aG

aN

=3

.18

9Å

∆a = +1.1%

bLGO = 6.372 Å

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∆a = +1.9 % ∆b = –1.1% (c)

aLGO = 5.402 Å

Figure 3.25 Structure of (a) orthorhombic LiGaO2 (LGO), (b) GaN, and (c) a detailed view of the relative orientation of GaN with respect to LGO. Courtesy of H. Paul Maruska. (This figure also appears on page 357.)

Color Tables

Figure 3.43 A schematic representation of a vertical OMVPE system employed at Virginia Commonwealth University along with a picture of the deposition chamber (a); a photograph of the reactor chamber of the same (b). (This figure also appears on page 394.)

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Figure 3.82 Cartoons illustrating the laser liftoff process sequence of 20 GaN membranes. (a) Laser lift-off of the GaN film coated with silicone elastomer and affixed onto a support template; (b) removal of sapphire following laser scanning; (c) deposition of an approximately

3 mm thick thermoplastic adhesive layer at 120  C; (d) peeling-off of the silicone elastomer. In the last step the GaN film is fully removed by dissolving the thermoplastic adhesive in acetone. Courtesy of M. Stutzmann, Ref. [382]. (This figure also appears on page 471.)

Figure 3.128 SEM and CL wavelength image of a cross section of HVPE ELO sample. Courtesy of J. Christen and A.G. Hoffmann, Ref. [695]. (This figure also appears on page 560.)

Color Tables

Figure 3.129 SEM (a) and CL wavelength (b) images of two different regions, coherently grown above the openings and overgrown above the SiO2 stripes. The growth in the windows (between the SiO2 stripes) and the wing

(coalesced regions over the SiO2 stripes) regions indicated in the schematic drawing (c) are clearly visible in the CL wavelength image. Courtesy of J. Christen and A.G. Hoffmann, Ref. [697]. (This figure also appears on page 561.)

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Figure 3.130 A m-Raman scans along c-axis of overgrowth GaN (in blue) and coherently grown GaN (in red): (a) free carrier density and (b) biaxial compressive stress. Courtesy of J. Christen and A. G. Hoffmann, Ref. [697]. (This figure also appears on page 562.)

Color Tables

Figure 3.160 Schematic representation of m-, a-, and c-planes of GaN. (This figure also appears on page 613.)

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Figure 3.165 Cartoon of epitaxial lateral overgrowth (ELO) on a-plane GaN with growth along the [0 0 0 1] direction, representing the Gapolar growth front and along the ½0 0 0 1 direction, representing the N-polar growth front: (a) top view and (b) side view. The growth along the Ga-polar front is about a factor of g ¼ 3 times faster. Courtesy of VCU students Vishal Kasliwal

and Xianfeng Ni. (c) Left 30 mm · 30 mm AFM image for sample B. Right 4 mm · 4 mm AFM image near the window and N-polar wing boundary of sample B, showing different surface pit densities for the window and the wing. Courtesy of VCU students Vishal Kasliwal and Xianfeng Ni. (This figure also appears on page 618.)

Color Tables

Figure 3.168 (a) AFM and (b) NSOM scans from a 40 mm · 40 mm area of a-GaN ELO sample B [875]. (This figure also appears on page 620.)

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Figure 3.249 (a) I–V characteristics of a core multishell (CMS) nanowire device, the top view of which is shown in the inset in the form of a field emission scanning electron microscopy image. Scale bar is 2 mm. (b) Optical microscopy images collected from around theopaque p-contact ofcore multishell nanowire LEDs with increasing In

concentration in the shell quantum well and in forward bias, showing purple, blue, cyan, green, and near yellow emission, respectively. (c) Normalizedelectroluminescencespectraobtained from five representative multicolor CMS nanowire LEDs. Courtesy of C.M. Lieber. (This figure also appears on page 747.)

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Figure 3.249 (Continued )

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(c)

1.0

0.8 Normalized intensity (au)

XLVI

0.6

0.4

0.2

0.0 300

400

500 Wavelength (nm)

Figure 3.249 (Continued )

600

700

Color Tables

Figure 4.8 Side view (projection onto the ð1 1  2 0Þ plane) of a relaxed and neutral screw dislocation: (a) full-core screw dislocation; (b) Ga-filled screw dislocation. Core of a full-core screw dislocation (discussed in greater detail in Figure 4.10 and the associated text) showing the

double helix of Ga bonds. The supercell is repeated twice in the [0 0 0 1] for clarity. Note that the bonds at the core are heavily distorted. Courtesy of Blumenau et al. (patterned after Ref. [29]). (This figure also appears on page 829.)

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Figure 4.36 Potential profile across the dislocation in n-type GaN deduced from the holographic phase map compared with theoretical profile (a). Courtesy of D. Cherns. False colored map showing phase shifts produced by edge dislocations viewed end-on in nominally undoped GaN. Contour lines emphasize dipole-like phase shifts near

dislocation cores (b). Line profile through indicated dislocation in (b) allows quantification of nominally undoped GaN of electric fields, yielding an estimated bound surface charge of 4 · 1011 e cm2 on either side of the defect (c). Courtesy of M. McCartney. (This figure also appears on page 874.)

Color Tables

2 1.5

Phase (rad)

1 0.5 0 –0.5 –1 –1.5 –2 –150

–100

(c)

–50 0 Distance (nm)

50

100

Figure 4.36 (Continued )

Figure 4.42 Contour plots of dislocation-induced electronic gap states for three edge dislocation configurations, namely, (a) fourcore, (b) full-core, and (c) open-core structures. The plots are obtained by calculating atomic geometries with DFT theory used as input to image simulations. Large (small) balls correspond to Ga (N) atoms [125]. (This figure also appears on page 883.)

150

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Mean implantation depth (µm)

0

0.09

0.26

0.50

0.79

1.13

HVPE GaN

0.50

10–14 µm

Mg-doped reference

0.49

S parameter

L

1 µm

36–39 µm

5 µm

49–68 µm

VGa

0.48

0.47

0.46 Defect free

0.45 0

5

10

15

20

Positron energy (keV) Figure 4.97 Ga vacancies and 1, 5, 10–14, 36–39, 49–68 mm thick HVPE GaN layers indicating increased S parameter, thus increased Ga vacancy concentration toward the GaN/Al2O3 interface in each of the films. A Mg-doped p-type sample with very low, if any, Ga vacancy is shown as the reference. Courtesy of K. Saarinen. (This figure also appears on page 986.)

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Figure 4.118 Hole concentration versus ND where the acceptor–donor–acceptor complex model of Ref. [495] is shown with thin and bold lines for NA ¼ 1 · 1019 cm3 (i) and 1 · 1020 cm3 (ii), respectively. The optimum hole concentration where NA ¼ 2ND, as expected from the complex formation, is shown with diamonds. For comparative purpose, the simple compensation model which assumes a single

donor and (unpaired) acceptor is depicted with thin and thick lines for NA ¼ 1 · 1019 cm3 (iii) and 1 · 1020 (iv), respectively. The random pair model is also plotted with thin and thick lines for NA ¼ 1 · 1019 cm3 (v) and 1 · 1020 cm3 (vi), respectively. Discussions with Dr R. Korotkov are acknowledged. (This figure also appears on page 1024.)

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Figure 4.126 Proposed bond center and antibonding site incorporation of H in GaN and its passivation of Mg during growth (the Mg atom is directly below the H atom). In part courtesy of C. Van de Walle. (This figure also appears on page 1037.)

GaN: Mn Mnd projected PDOS (/eV cell)

LII

t+

e-

t+

e+ t-

GaP: Mn

e-

t+ e+

t+

ee+

t-

t+

tt+

GaSb: Mn

e-

e+ t–3

t-

t+

t-

GaAs: Mn

–4

t-

–2

t-

t+ –1

–F Energy (eV)

1

2

3

Figure 4.150 In Mn d projected partial density of states for a single Mn in GaN, GaP, GaAs, and GaSb, where the symmetry (t2 and e) as well as the spin (þ and ) have been indicated. The shaded region represents the t2þ states (after Ref. [699]). (This figure also appears on page 1097.)

Color Tables

3d ion d n-1 t-(d )

e-(d )

t+(d)

Mn on Ga site t-CFR

Anion dangling bonds V Ga3-

CFR

e-

t+CFR t+(p)

e+(d) VBM

CFR

e+

DBH

t-

t-(p)

DBH

t+

Figure 4.151 A schematic energy-level diagram for the levels (central panel) generated from the interaction between the crystal field and exchange-split levels on the 3d transition metal ion (left panel) with the anion dangling bond levels (right panel), when the TM d levels are energetically shallower than the dangling bond levels (after Ref. [700]). (This figure also appears on page 1098.)

LIII

LIV

Color Tables Figure 4.155 Interband transitions in GaAs selected because of its well-known band structure and also its well-established and wellcharacterized properties in terms of magnetic ion doped diluted magnetic semiconductors: (arrows indicate emission but the concept is just as applicable to transitions from the valence band subband to the conduction band as in absorption). (a) Schematic band structure of GaAs near the G point, the center of the Brillouin zone. As for the terms, Eg is the bandgap and DSO the spin–orbit splitting; CB, conduction band; HH, valence heavy hole; LH, light hole; SO, spin–orbit split-off subbands; G6,7,8 are the corresponding symmetries at the k ¼ 0 point representing conduction, HH, LH, spin–orbit (SO) bands, or, more precisely, the irreducible representations of the tetrahedron group Td (see, e.g., Ref. [721]). The terms s1/2 and p3/2 and p1/2 represent the conduction band (s-like) and valence band (p-like) type of orbitals. (b) Selection rules for interband transitions between the Jz projection of the angular momentum along z-direction, sublevels for circularly polarized light sþ(right-hand circular polarization or positive helicity that results from transitions between the Jz ¼ 1/2 conduction band states and Jz ¼ 3/2 heavy-hole states, and Jz ¼ þ1/2 conduction band and Jz ¼ 1/2 light-hole states), and s (left-hand circular polarization or negative helicity, which results from transitions between the Jz ¼ þ1/2 conduction band states and Jz ¼ þ3/2 heavy-hole states, and Jz ¼ 1/2 conduction band and Jz ¼ þ1/2 light-hole states). The numbers by each transition indicate the relative transition intensities, with respect to the light-hole subband to the conduction band (absorption or excitation of carriers to higher bands), or the conduction band to the light-hole subband transition (emission), which apply to both excitation and radiative recombination (depicted by the arrows). The circular polarization (s polarization) for light energies that would not excite the spin split-off band is ideally 50%, which becomes 0 if the spin–orbit split-off band is also excited. For completeness, the transitions between the Jz ¼ 1/2 conduction band states and Jz ¼ 1/2 light-hole states, and Jz ¼ þ1/2 conduction band states and Jz ¼ þ1/2 light-hole states, which are linearly polarized (p polarization), are also shown as depicted by two-way arrows in the figure. The

" transition probability or the emission intensity normalized to the Jz ¼ 1/2 conduction state to the Jz ¼1/2 state transitions (indicated with 1) are also indicated in numbers for GaAs. The circular polarization resulting from the conduction band to the heavy-hole states are three times more intense than the circular polarization resulting from the conduction band states to the light-hole valence band states. The linearly polarized transitions are twice as intense as the circular polarization involving light-hole states. (c) Removal of the valence band heavyand light-hole degeneracy by, for example, strain inducing either by lattice mismatch or by confinement in a quantum well, which increases the electron polarization to nearly 100%. Note that heavy- and light-hole states are no longer degenerate. Both the tensile (left) and compressive (right) in-plane biaxial strain cases are shown. The respective ratios of various transitions (oscillator strengths) have been assumed to be the same as in the relaxed case. Note that spin is indifferent to strain, which means that spin-up and spin-down states are moved in the same direction by strain, but not to magnetic field, as spin-up and spin-down states in a given band are split and moved in opposite directions as shown in (d). In part courtesy of W. Chen, Linko1ping University. (d) Removal of the valence band heavy- and light-hole degeneracy as well as splitting the spin-up and spin-down states by application of magnetic field. The total splitting is enhanced due to sp-d interaction in DMS materials in the form of xN0a<Sz> for the conduction band states, xN0b<Sz> for the HH and LH valence band states, and (1/3)xN0a<Sz> for the spin–orbit split-off band. Here, N0, x, a, b, <Sz> represent the number of cations per unit volume, mole fraction of magnetic ions, the product of Bohr magneton and the g factor for the respective bands, and average spin for each magnetic ion site, respectively. Note that magnetic field/magnetization causes Zeeman splitting, and direction of splitting either up or down in energy is spin dependent. If the semiconductor is ferromagnetic as is the case of GaMnAs, one can either couple polarized light to the symmetry/splitting allowed bands or cause polarized light emission by tuning the wavelength. (This figure also appears on page 1112.)

Color Tables

LV

LVI

Color Tables

Figure 4.155 (Continued )

Color Tables

(a) x = 0.053 B ⊥ plane

2K 55 K 25 K

100 K 125 K 300 K

0.00

0.5

Rsheet(kΩ)

R Hall (kΩ )

0.03

-5

25 K

2K 0.3 -5

-0.03

100 K 125 K 55 K

0.4

300 K 0

5

B (T)

0 B (T )

5

0.08

1/χ Hall(au)

(RHall R ) sheet s

(b) 0.06 0.04 0.02 0.00 0

100

200

300

T (K) 120 (c)

T c (K)

80

40

0 0.00

0.04 x

Figure 4.170 (a, top) Temperature dependence of the Hall resistance RHall for a 200 nm thick Ga0.947Mn0.053As sample for which direct magnetization measurements have been performed but not shown. The inset shows the temperature dependence of the sheet resistance Rsheet. (b, center) Temperature dependence of the saturation magnetization [RHall/Rsheet]S obtained using Arrott plots (solid circles) and inverse susceptibility 1/wHall (open circles), both

0.08 deduced from the transport data shown in (a). Solid lines depict [RHall/Rsheet]S and (c, bottom) 1/wHall (bottom, c) calculated using the mean field Brillouin theory with S ¼ 5/2 for the Mn spin and the Curie–Weiss law, respectively. The dependence of magnetic transition temperature TC on Mn composition as determined from the transport data. Courtesy of Ohno and Matsukura [777]. (This figure also appears on page 1140.)

LVII

LVIII

Color Tables

+1/2 CB

Jz = −1/2

σ+

σ+

σ− π

σ−

π

σ+ σ−

+3/2 HH

−3/2 −1/2

+1/2 LH +1/2 CR

−1/2 VB;HH Figure 4.172 The GCB , G7VB;LH 7 conduction band and G9 VB;CR (spin–orbit split-off band), and G7 (crystal field split off band) valance bands in wurtzitic GaN at the G point along with polarization (sþ right-hand and s left-hand circular polarizations) of various transitions between the conduction and valence band states in the presence of a magnetic field. (This figure also appears on page 1147.)

0.15

Cr: 1%

5K

0.05

Cr: 3%

0.05

Magnetization (emu g–1)

Magnetization (emu g–1)

0.1

0.2 μ B /Cr

Cr: 5%

Cr: 0.5%

0

–0.05 –0.1 –0.15 –10 000

Cr : 5% Cr :1%

0

H c =100 Oe

–0.05 –1000

0

1000

Magnetic Field (Oe)

–5000

0

5000

10 000

Magnetic Field (Oe) Figure 4.177 Magnetization curves for Cr-doped GaN in atomic concentrations of 0.5, 1, 3, and 5% up to a magnetic field normal to the surface of 10 000 Oe (1 T). Note that the film containing 5% Cr does not show any saturation magnetization in the range measured and

appears to be paramagnetic. The blow-up version near the origin indicates hysteresis for 1% Cr sample and a coercive field of 100 Oe. Courtesy of F. Hasegawa. (This figure also appears on page 1157.)

Color Tables

0.04

Magnetization (emu g–1)

H = 200(Oe)

Ferromagnetic

Cr: 1%

0.02

Paramagnetic + ferromagnetic

Cr: 3 %

Paramagnetic

Cr : 5%

0 0

50

100

150 200 250 Temperature (K)

300

Figure 4.178 Temperature dependence of magnetization for 1, 3, and 5% Cr-containing GaN. As indicated, the film with 1% Cr is consistent with ferromagnetic behavior. The films with 3 and 5% Cr exhibit a combination of ferromagnetic and paramagnetic behavior, and paramagnetic behavior, respectively. Courtesy of F. Hasegawa. (This figure also appears on page 1158.)

350

LIX

LX

Color Tables

VG < 0 R (kΩ) Hall

0.04

1

1.5 K 5K 10 K

0

20 K

0.02

RHall (kΩ)

–1 –0.5

0.0 B(T)

VG > 0

0.5

0.00

22.5 K

VG = 0V

–0.02

+125 V –125 V 0V –0.04 –1.0

–0.5

0.0

0.5

1.0

B (mT) Figure 4.186 Hall resistance RHall of an insulated gate (In,Mn)As field effect transistor at 22.5 K as a function of the magnetic field for three different gate voltages. RHall is proportional to the magnetization of the (In,Mn)As channel. Upper right inset shows the temperature dependence of

RHall. Left inset shows schematically the gate voltage control of the hole concentration and the corresponding change of the magnetic phase. Courtesy of Ohno et al. [855]. (This figure also appears on page 1172.)

Color Tables

Figure 4.187 Injection of spin-polarized holes into a light-emitting p–n diode using a ferromagnetic semiconductor (Ga,Mn)As. (a) Sample structure. Spin-polarized holes hþ travel through the nonmagnetic GaAs and recombine with spin-unpolarized electrons in the (In,Ga)As quantum well. I represents the current, and sþ represents circularly polarized light emitted from the edge of the quantum well. (b) Dependence of the polarization DP of the emitted light on the magnetic field B at temperatures of 6, 31, 52 K, the latter above the Curie value. The solid and hollow symbols represent the degree of polarization when the magnetic field is swept in the positive and negative directions, respectively.

The magnetic field was applied parallel to the surface along the easy axis of magnetization of the (Ga,Mn)As. The temperature dependence of the residual magnetization M in (Ga,Mn)As, where the degree of polarization of the zero magnetic field seen in the emitted light exhibits the same temperature dependence as the magnetization (not shown). Dependence on temperature for B ¼ 0 of the change in the relative remanent polarization, DP, (hollow circles) and magnetic moment measured by a SQUID magnetometer (solid circles). Courtesy of Ohno and coworkers [770]. (This figure also appears on page 1174.)

LXI

Color Tables

∆P 1.00

T=6K

Magnetization 6

16 K 0.75

4 0.50

31 K

2

0.25

52 K 0.00 0

20

40

60

Temperature (K) (c) Figure 4.187 (Continued )

80

100

Magnetization (10-5emu)

Relative polarization,∆P (%)

LXII

Color Tables

High resistance state

Low resistance state

(b) Figure 4.190 (a) Schematic representation of a spin valve, a normal metal straddled by two ferromagnetic metals. When the spins in ferromagnetic metals on either end are aligned parallel to each other, the system is in the lowresistance state top. When, for example, the spin of the FM metal on the right is flipped by a magnetic field, making the spins of the

ferromagnetic metals antiparallel, a high resistance state is attained. (b) Schematic representation of transport that is parallel to the plane of a layered magnetic metal sandwich structure for antialigned (upper figure – high resistance) and aligned (lower figure – low resistance) orientations. (This figure also appears on page 1180.)

LXIII

j1

1 General Properties of Nitrides Introduction

GaN as a representative of its binary cousins, InN and AlN, and their ternaries along with the quaternary, is considered one of the most important semiconductors after Si. It is no wonder that it finds ample applications in lighting and displays of all kinds, lasers, detectors, and high-power amplifiers. These applications stem from the excellent optical and electrical properties of nitride semiconductors. The parameters are imperative in determining the utility and applicability of this class of materials to devices, as will be made evident in this chapter and throughout the book. In this chapter, the structural, mechanical, thermal, chemical, electrical, and optical properties of GaN and its binary cousins as well as the substrates commonly used for nitride epitaxy are treated in a general sense for quick reference. The detailed properties associated with electrical and optical parameters and properties are discussed in chapters dealing with transport and optical processes in GaN and related alloys. Because GaN is used in the form of a thin film deposited on foreign substrates, meaning templates other than GaN, a discussion of heteroepitaxial thin films is of paramount importance. Consequently, the properties of nitride films intricately depend on substrates, inclusive of the inherent properties such as lattice constants and thermal expansion coefficients, and on the process-induced characteristics such as surface preparation and chemical and physical interactions at the surface. These too are discussed in the book.

1.1 Crystal Structure of Nitrides

Group III nitrides can be of crystalline structures: the wurtzite (Wz), zinc blende (ZB), and rock salt. Under ambient conditions, the thermodynamically stable structure is wurtzite for bulk AlN, GaN, and InN. The zinc blende structure for GaN and InN has been stabilized by epitaxial growth of thin films on {0 1 1} crystal planes of cubic substrates such as Si [1], SiC [2], MgO [3], and GaAs [4]. In these cases, the intrinsic tendency to form the Wz structure is overcome by the topological compatibility.

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morko
c Copyright
2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

j 1 General Properties of Nitrides

2

However, Wz structure could very likely be present at the extended defect sites. The rock salt form is possible only under high pressures and, therefore, is laboratory form of exercise. Let us now discuss the space groups for the various forms of nitrides. The rock salt, or NaCl, structure (with space group Fm3m in the Hermann–Mauguin notation and O5h in the Schoenflies notation) can be induced in AlN, GaN, and InN under very high pressures. The reason for this is that the reduction of the lattice dimensions causes the interionic Coulomb interaction to favor the ionicity over the covalent nature. The structural phase transition to rock salt structure was experimentally observed at the following estimated pressure values: 22.9 GPa (17 GPa in other estimates) for AlN [5], 52.2 GPa for GaN [6], and 12.1 GPa for InN [7]. Rock salt III nitrides cannot be produced by any epitaxial growth. The space grouping for the zinc blende structure is F43m in the Hermann– Mauguin notation and T 2d in the Schoenflies notation. The zinc blende structure has a cubic unit cell, containing four group III elements and four nitrogen elements. (Although the term zinc blende originated in compounds such as ZnS, which could be in cubic or hexagonal phase, it has been used ubiquitously for compound semiconductors with cubic symmetry. The correct term that should be used for the cubic phase of GaN is actually sphalerite. However, to be consistent with the usage throughout the literature, even at the expense of accuracy, the term zinc blende has been used in this book). The position of the atoms within the unit cell is identical to the diamond crystal structure. Both structures consist of two interpenetrating facecentered cubic sublattices, offset by one quarter of the distance along a body diagonal. Each atom in the structure may be viewed as positioned at the center of a tetrahedron, with its four nearest neighbors defining the four corners of the tetrahedron. The stacking sequence for the (1 1 1) close-packed planes in this structure is AaBbCc. Lowercase and uppercase letters stand for the two different kinds of constituents. The wurtzite structure has a hexagonal unit cell and thus two lattice constants, c and a. It contains six atoms of each type. The space grouping for the wurtzite structure is P63mc in the Hermann–Mauguin notation and C46v in the Schoenflies notation. The point group symmetry is 6 mm in the Hermann–Mauguin notation and C6v in the Schoenflies notation. The Wz structure consists of two interpenetrating hexagonal close-packed (hcp) sublattices, each with one type of atom, offset along the c-axis by 5/8 of the cell height (5c/8). The wurtzite and zinc blende structures are somewhat similar and yet different. In both cases, each group III atom is coordinated by four nitrogen atoms. Conversely, each nitrogen atom is coordinated by four group III atoms. The main difference between these two structures lies in the stacking sequence of closest packed diatomic planes. The Wz structure consists of alternating biatomic close-packed (0 0 0 1) planes of Ga and N pairs, thus the stacking sequence of the (0 0 0 1) plane is AaBbAa in the (0 0 0 1) direction. Although the main interest is in Wz GaN as opposed to zinc blende GaN, a description of stacking sequence of both GaN polytypes with the accepted Ramsdel notation is warranted, so is the stacking order of SiC polytypes that are relevant to GaN because they are used for substrates in GaN epitaxy. Therefore, a generic description of stacking in Wz semiconductors is given below. A comprehensive description of the tetrahedrally coordinated structures is imperative for a clear picture

1.1 Crystal Structure of Nitrides

j3

of nitride semiconductors, particularly the extended defects that are discussed in detail in Chapter 4. The bonds describe a tetrahedron denoted by T, which has one atom species at each of the three corners and the other atom species in its center [8,9]. The basal plane of this structure is defined by one face of the tetrahedron and the bond perpendicular to this plane defines the c-axis. A rotation of 180
around the caxis produces a twin variant denoted by T0 as shown in Figure 1.1a (left).

(a)

Atom a Atom b [0001]

T

T'

〈1120〉

{1100} In-plane bonds Out-of-plane bonds

(b)

A

T 3´

b

B

T1

a

A b

B

Figure 1.1 Representation of the tetrahedrally coordinated materials in the Ramsdel notation. (a) The two possible 0 0 tetrahedra. (b) The T1, T3, T1, T3, tetrahedral stacking composing the 2H sequence. Courtesy of Pierre Ruterana [9].

T 3´

T1

j 1 General Properties of Nitrides

4

The two variants (twins T and T0 ) are related to one another by mirror symmetry about one of the {1 1 0 0} m-planes. A tetrahedron can occupy one of the three possible positions in the basal plane. The representation of the tetrahedrally coordinated materials in the Ramsdel notation is shown in Figure 1.1a for two possible tetrahedra, one is the mirror image twin of the other with respect to the (1 1 0 0) m-plane. The single bonds are on the (1 1 2 0) plane, called the a-plane. The 0 0 0 layers of the tetrahedra can then be denoted by T1, T2, T3, and by T1 ; T2 ; T3 for its 0 0 twin. An example of T1 ; T ; T1 ; T3 stacking order representing 2H ordering as in wurtzitic GaN is shown in Figure 1.1b. The structure of nitride semiconductors and most relevant polytypes of SiC can be completely described by a combinatorial stacking of the aforementioned six tetrahedra layers. Naturally, not all the stacking sequences must obey the following two rules to keep a corner sharing structure, as such not all stacking orders are allowed: (i) A tetrahedron T can be followed by another one of the same kind with the 0 0 0 following subscript: T1T2T3, and inversely for the twin variant: T3 T2 T1 . (ii) A tetrahedron T1 must be followed by the twin variant of the preceding subscript: 0 0 T1 T3 , and inversely for its twin variant: T1 T2 . In the Ramsdel notation, the stacking order for the wurtzite structure corresponding to various polytypes can be denoted as . . . .

0

0

0

T1 T3 or T2 T1 or T3 T2 for the 2H polytype, which is also applicable to Wz nitride semiconductors; 0 0 T1T2T1 T3 for the 4H polytype; 0 0 0 T1T2T3T2 T1 T3 for the 6H polytype; 0 0 0 T1T2T3 or T3 T2 T1 for the 3C polytype.

The 3C, 4H, and 6H stacking sequences as well as 2H sequence on 6H sequence are discussed in Chapter 3. Recall that GaN crystallizes in the cubic structure (zinc blende or sphalerite, the latter being the correct term and the former being the one used universally) or in the more stable hexagonal structure (wurtzite). The anions (N3) form an hcp structure in which the cations (Ga3þ) occupy half of the tetrahedral sites. The structure of a unit cell of GaN projected along [0 0 0 1] is depicted schematically in Figure 1.2. The open symbols represent g sites that are occupied by nitrogen atoms; the Ga atoms are in the tetrahedral sites, b. These latter sites can either be at heights (3/8)c above (b1) or below (b2) N site, depending on the crystal polarity. A stick-and-ball representation of Ga-polarity and N-polarity Wz structure is depicted in Figure 1.3. The Wz and zinc blende structures differ only in the bond angle of the second nearest neighbor (Figure 1.4). As clearly shown, the stacking order of the Wz along the [0 0 0 1] c-direction is AaBb, meaning a mirror image but no in-plane rotation with the bond angles. In the zinc blende structure along the [1 1 1] direction, there is a 60
rotation that causes a stacking order of AaBbCc. The point with regard to rotation is illustrated in Figure 1.4b. The nomenclature for various commonly used planes of hexagonal semiconductors in two- and three-dimensional versions is presented in Figures 1.5 and 1.6. The Wz group III nitrides lack an

1.1 Crystal Structure of Nitrides

a γ1

β1

γ1

γ1

* γ1

β2

c

β1

γ2

β2 β1

γ1

u

β2

*

β2 β1

β2 β1

γ1

γ1

γ1

Figure 1.2 Schematic diagram showing the b1 and b2 tetrahedral sites of GaN unit cell. Starting with the assumption that N occupies the g sites, only one family of b sites can be simultaneously occupied by Ga atoms. Courtesy of Pierre Ruterana [9].

inversion plane perpendicular to the c-axis; thus, nitride surfaces have either a group III element (Al, Ga, or In) polarity (referred to as Ga-polarity) with a designation of (0 0 0 1) or (0 0 0 1)A plane or a N-polarity with a designation of (0 0 0 1) or (0 0 0 1)B plane. We will use the former notations for each. The distinction between these two directions is essential in nitrides because of their implications for the polarity of the polarization charge. Three surfaces and directions are of special importance in

Ga-polarity

N-polarity

[0001]

[0001]

Ga

N

Ga

N

Ga

N

Ga N

Ga N

Ga N

Ga N

Ga

N

Ga N

Ga

Ga

Ga

N

N

N

Ga

Ga

Ga

N

N

N

Ga

N

Ga

Ga

N

N

N

N

N

Ga

Ga

Ga

N

N

Ga

Ga

N Ga

Ga

N

N

N

N

Ga

Ga

Ga

N Ga

Figure 1.3 A stick-and-ball diagram of a hexagonal structure.

N Ga

N

N

Ga

Ga

j5

j 1 General Properties of Nitrides

6

C View normal to [0001] and [111]

B

B

A

A

Ga N

Wurtzitic

Zinc blende

View along [0001] and [111]

(a) Figure 1.4 A stick-and-ball stacking model of crystals with (a, both top and bottom) 2H wurtzitic and (b, both top and bottom) 3C zinc blende polytypes. The bonds in an A-plane (1 1 2 0) are indicated with heavier lines to accentuate the stacking sequence. The figures on top depict the three-dimensional view. The

(b) figures at the bottom indicate the projections on the (0 0 0 1) and (1 1 1) planes for wurtzitic and cubic phases, respectively. Note the rotation in the zinc blende case along the h1 1 1i direction. (Please find a color version of this figure on the color tables.)

nitrides, which are (0 0 0 1) c-, (1 1 2 0) a-, and (1 1 0 0) m-planes and the directions associated with them, h0 0 0 1i, h1120i, and h1100i as shown in Figure 1.7. The (0 0 0 1), or the basal plane, is the most commonly used surface for growth. The other two are important in that they represent the primary directions employed in reflection high-energy electron diffraction (RHEED) observations in molecular beam epitaxial growth, apart from being perpendicular to one another. They also represent the direction of stripes employed in the epitaxial lateral overgrowth (ELO), details of which are discussed in Section 3.5.5.2. The cohesive energy per bond in the wurtzite form is 2.88 eV (63.5 kcal mol1), 2.2 eV (48.5 kcal mol1), and 1.93 eV (42.5 kcal mol1) for AlN, GaN, and InN, respectively [10]. The calculated energy difference DEW-ZB between wurtzite and zinc blende lattice is small [11]: DEW-ZB ¼ 18.41 meV/atom for AlN, DEW-ZB ¼ 11.44 meV/atom for InN, and DEW-ZB ¼ 9.88 meV/atom for GaN. Wurtzite form is energetically preferable for all three nitrides compared to zinc blende, although the energy difference is small. The Wz structure can be represented by lattice parameters a in the basal plane and c in the perpendicular direction, and the internal parameter u, as shown in Figure 1.8.

1.1 Crystal Structure of Nitrides 〈1100〉

〈1120〉

(m)

(a )

o

1010

30

(m )

o

30

r

n n

n n

c

1100

0110

r

r n

2110

1120

n 0111 s (c)

1210 a 57

o

0114 d 1213 n

1012 r

2113 n

1101 s

1104 d 0001 c

1210 1213 n

o

.6

61

1123 n

c

(n) r

(r) r

n

n

1102 r

32.4

n

o

1100 m

a

a

a

1014 d

2113 n

0112 r 0110

1123 n

1011 s

(m) 2110

1120 1010

n

n

r

n

Common crystallographic planes in sapphire d Plane Miller name index spacing a m c r n s

(1120) (1010) (0001) (1102) (1123) (1011)

2.379 Å 1.375 Å 2.165 Å 1.740 Å 1.147 Å 1.961 Å

Angles between common planes (0001) ^ (1102) (0001) ^ (1123) (0001) ^ (1011) (0001) ^ (1121) (0001) ^ (1120) (0001) ^ (1010) (1120) ^ (1010)

c^r c^n c^s c^ c^a c^m a^m

57º 35' 61º 11' 72º 23' 79º 37' 90º 00' 90º 00' 30 00'

Figure 1.5 Labeling of planes in hexagonal symmetry ( for sapphire).

The u parameter is defined as the anion–cation bond length (also the nearest neighbor distance) divided by the c lattice parameter. The c parameter depicts the unit cell height. The wurtzite structure is a hexagonal close-packed lattice, comprising vertically oriented X–N units at the lattice sites. The basal plane lattice parameter (the edge length of the basal plane hexagon) is universally depicted by a and the axial lattice parameter, perpendicular to the basal plane, is universally described by c. The interatomic distance in the basic unit is described by the internal parameter u. In an ideal wurtzite structure represented by four touching hard spheres, pffiffiffiffiffiffiffiffi the values of the axial ratio and the internal parameter are c=a ¼ 8=3 ¼ 1:633 and u ¼ 3/8 ¼ 0.375, The crystallographic ! pffiffiffi respectively. pffiffiffi ! vectors of wurtzite are a ¼ að1=2; 3=2; 0Þ, b ¼ að1=2;  3=2; 0Þ, and ! c ¼ að0; the basis atoms are (0, 0, 0), (0, 0, uc), pffiffiffi0; c=aÞ. In Cartesian coordinates, pffiffiffi a(1=2; 3=6; c=2a), and a(1=2; 3=6; ½u þ 1=2c=a). Table 1.1 tabulates the calculated structural parameters a, c/a, and e1 ¼ u  uideal for the III–V nitrides by three different groups [12–14]. In the case of Bernardini et al. [12], they optimized the structure within both the generalized gradient

j7

j 1 General Properties of Nitrides

8

(tuvw) coordinate system v

1010 2110

1120

0110

0111 s

1123 n

1012 r 2113 n

0114 d

1104 d 1210

t

0001 c 0112 r

1102 r 2113 n

1100 m

1100

1213 n

1213 n

1210 a

1101s

1014 d

1123 n

0110

1011 s 1120

2110

u

1010

Figure 1.6 A magnified view of labeling of planes in hexagonal symmetry in the (tuvw) coordinate system with w representing the unit vector in the c-direction. The lines are simply to show the symmetryonly. If the lines connecting m-points among each other and a-points among each other were to be interpreted as the projection of those

planes on the c-plane, the roles would be switched in that the lines connecting the m-points would actually represent the a-planes and lines connecting the a-points would actually represent the m-planes that are normal to the plane of the page.

(1120) a-plane [1120] v

(1100) m-plane [0110]

[1010]

[1210]

[1100]

[2110]

t

[2110]

[1010]

[1100]

Ga N

[0110] u [1210]

m-planes a-planes

[1120]

Figure 1.7 The orientations which are commonly used in nitrides, namely the (1 1 2 0) and (1 1 0 0) planes and associated directions are shown as projections on the (0 0 0 1) basal plane.

1.1 Crystal Structure of Nitrides

a

M

M

α c N M

M

b=u x c M

M

b1

N

[0001]

N

b '2

M

N

b 3' N

β b'

N

1

N

M M

M

Figure 1.8 Schematic representation of a wurtzitic metal nitride structure with lattice constants a in the basal plane and c in the basal direction, u parameter, which is expressed as the bond length or the nearest neighbor distance (b) divided by c (0.375 in ideal crystal), a and b (109.47 in ideal crystal) are the bond 0 0 0 angles, and b 1, b 2, and b 3, represent the three types of second nearest neighbor distances.

approximation (GGA) and local density approximation (LDA). The experimental data are from Leszczynski et al. [15]. In all Wz III nitrides, experimentally observed c/a ratios are smaller than ideal and it has been postulated that not being so would lead to the zinc blende phase [16]. There are two avenues that can lead to a deviation from ideal: changing the c/a ratio or changing the u value. It should be pointed out that a strong correlation exists between the c/a ratio and the u parameter so that when c/a decreases, the u parameter increases in a manner to keep the four tetrahedral distances nearly constant through a distortion of tetrahedral angles. For equal bond length to prevail, the following relation must hold: uð1=3Þða2 =c 2 Þ þ 1=4:

ð1:1Þ

Table 1.1 Structural parameters for GaN reported by Bechstedt,

Großner, and Furthm€ uller (BGF) [13] and by Wei and Zunger (WZ) [14] using the local density approximation (LDA).

BGF WZ BFV (LDA) BFV (GGA) Experimental data

a (Å)

c/a

e1 (103c/a)

3.150 3.189 3.131 3.197 3.1890

1.6310 1.6259 1.6301 1.6297 1.6263

6.5 1.8 1.6 1.9 2.0

However, Bernardini, Fiorentini, and Vanderbilt [12] employed both the LDA and GGA methods. Lattice constant a is given in Å and e1 in 103c/a.

j9

j 1 General Properties of Nitrides

10

The nearest neighbor bond length along the c-direction (expressed as b in Figure 1.8) and off c-axis (expressed as b1 in Figure 1.8) can be calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 1 a þ  u c2 : ð1:2Þ b ¼ cu and b1 ¼ 3 2 In addition to the nearest neighbors, there are three types of second nearest 0 0 neighbors designated in Figure 1.8 as b 1 (one along the c-direction), b 2 (six of them), 0 and b 3 (three of them), which are given as [17] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 1 0 0 0 2 a þ c2 u : b 1 ¼ cð1  uÞ; b 2 ¼ a2 þ ðucÞ ; and b 3 ¼ 3 2 ð1:3Þ The bond angles, a and b, are given by [17] "qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1 a ¼ p=2 þ arccos 1 þ 3ðc=aÞ2 ð  u þ 1=2Þ2 ; "qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b ¼ 2 arcsin

2

4=3 þ 4ðc=aÞ ð  u þ 1=2Þ

2

1

ð1:4Þ

# :

Table 1.2 tabulates the calculated as well as experimentally observed structural parameters discussed above, inclusive of the lattice parameters, the nearest and second nearest neighbor distances, and the bond angles for three end binaries, GaN, AlN, and InN. The distances are in terms of Å. The lattice parameters are commonly measured at room temperature by X-ray diffraction (XRD), which happens to be the most accurate one, using the Bragg law. In ternary compounds, the technique is also used for determining the composition; however, strain and relevant issues must be accounted for as the samples are in the Table 1.2 Calculated (for ideal crystal) and experimentally

observed structural parameters for GaN, AlN, and InN [17]. GaN

u a (Å) c/a b (Å) b1 (Å) 0 Â b 1 ðeÞ 0 Â b 2 ðeÞ 0 Â b 3 ðeÞ a b

AlN

InN

Ideal

Exp.

Ideal

Exp.

Ideal

Exp.

0.375 3.199 1.633 1.959 1.959 3.265 3.751 3.751 109.47 109.47

0.377 3.199 1.634 1.971 1.955 3.255 3.757 3.749 109.17 109.78

0.375 3.110 1.633 1.904 1.904 3.174 3.646 3.646 109.47 109.47

0.382 3.110 1.606 1.907 1.890 3.087 3.648 3.648 108.19 110.73

0.375 3.585 1.633 2.195 2.195 3.659 4.204 4.204 109.47 109.47

0.379 3.585 1.618 2.200 2.185 3.600 4.206 4.198 108.69 110.24

1.1 Crystal Structure of Nitrides

form of epitaxial layers on foreign substrates. The accuracy of X-ray diffraction and less than accurate knowledge of the elastic parameters together allow determination of the composition to only within about 1% molar fraction. In addition to composition, the lattice parameter can be affected by free charge, impurities, stress (strain), and temperature [18]. Because the c/a ratio correlates with the difference of the electronegativities of the two constituents, components with the greatest differences show the largest departure from the ideal c/a ratio [19]. For GaN, the c/a ratio and the value of u are measured as 1.627 (1.634 in Ref. [17]) and 0.377, respectively, which are close to the ideal values [20]. AlN deviates significantly from the ideal parameters: c/a ¼ 1.601 (1.606 in Ref. [17]) and u ¼ 0.382. Although the data for InN are not as reliable, values of u ¼ 0.379 and c/a ¼ 1.601 have been reported [17]. Inhomogeneities, strain, partial relaxation of strain, and high concentration of structural defects may distort the lattice constants from their intrinsic values and cause a wide dispersion among the reported values. Table 1.3 lists a comparison of measured and calculated lattice parameters reported for AlN, GaN, and InN crystallized in the wurtzite structure in more detail in terms of the specifics of the sample used for measurements and complements. The dispersion is even a greater concern in ternary and quaternaries, as compositional inhomogeneities, in addition to the aforementioned issues, cause an additional dispersion. The particulars of the ternaries are discussed in Section 1.5. The wurtzite polytypes of GaN, AlN, and InN form a continuous alloy system whose direct bandgaps range, according to data that adorned the literature for years, from 1.9 eV for InN, to 3.42 eV for GaN, and to 6.2 eV for AlN. A revisit of the InN bandgap indicates it to be about 0.78 eV [30] and the same for AlN is about 6 eV in which case the energy range covered would be about 0.7–6 eV. Thus, the III–V nitrides could potentially be fabricated into optical devices, which are active at wavelengths ranging from the red well to the ultraviolet. The bandgaps of nitrides, Table 1.3 Measured and calculated lattice constants of wurtzite AlN, GaN, and InN.

Compound

Sample

a (Å)

c (Å)

AlN

Bulk crystal [21] Powder [22] Epitaxial layer on SiC [23] Pseudopotential LDA [24] FP-LMTO LDA [25] Bulk crystal [18] Relaxed layer on sapphire [26] Powder [29] Relaxed layer on sapphire [27] GaN substrate – LEO [28] Pseudopotential LDA [24] FP-LMTO LDA [25] Powder [29] Pseudopotential LDA [24] FP-LMTO LDA [25]

3.1106 3.1130 3.110 3.06 3.084 3.189 3.1892 3.1893 3.1878 3.1896 3.162 3.17 3.538 3.501 3.53

4.9795 4.9816 4.980 4.91 4.948 5.1864 5.1850 5.1851 5.1854 5.1855 5.142 5.13 5.703 5.669 5.54

GaN

InN

LDA: local density approximation; FP-LMTO: full-potential linear muffin–tin orbital.

j11

j 1 General Properties of Nitrides

12

Figure 1.9 The bandgaps of nitrides, substrates commonly used for nitrides, and other conventional semiconductors versus their lattice constants. (Please find a color version of this figure on the color tables.)

substrates commonly used for nitrides, and other conventional semiconductors are shown in Figure 1.9 with respect to their lattice constants. All III nitrides have partially covalent and partially ionic bonds. The concept of fractional ionic character (FIC) is useful in interpreting many physical phenomena in the crystals [31,32]. The FIC may be defined for a binary compound AB as FIC ¼ jQ A  Q B j=jQ A þ Q B j, where Q A and Q B are effective charges on atoms A and B. The FIC values range from zero for a covalent compound (each atom has four electrons) to 1 for an ionic compound (all eight electrons belong to the anion). Figure 1.10 displays the charge distribution along the AB bond for all three compounds. The arrow along the bond charge indicates the atomic boundaries in the crystals that are not always at the minimum of the line charge along the bond AB. This should be expected taking into account the partial covalent bond of the compounds, because only in the ionic crystals, the atomic boundary is clearly defined. Table 1.4 lists the calculated effective radii, rIII and rN, the effective charges, and FIC for AlN, GaN, and InN. The ionicity of AlN is high. This may explain the difficulties with AlN doping. It is well known that only covalent semiconductors or semiconductors with a large covalent component can form hydrogen-like shallow levels in the bandgap by substitution of a host atom with a neighbor with one more or one less electron. GaN and InN have a smaller than AlN but nearly equal ionicity. GaN was doped both p- and n-type. Thus, one can expect that InN can also be doped n- and p-type. To date, only n-type InN has been obtained because of high volatility of nitrogen and easiness of nitrogen vacancy formation that acts as a donor in this compound.

1.1 Crystal Structure of Nitrides 2.0

AlN

1.5 1.0 0.5 0.0 0.0

0.5

1.0 r(Al-N)(λ)

1.5

2.0

1.5

2.0

2.0

ρ(r) (electrons/a.u.3)

GaN 1.5 1.0 0.5 0.0 0.0

0.5

1.0 r(Ga-N)(λ)

2.0 InN 1.5 1.0 0.5 0.0 0.0

0.4

0.8

1.2

1.6

2.0

r(In-N)(λ) Figure 1.10 Charge density along the III–N bond in III nitride semiconductors.

The III-nitrides are commonly grown on mismatched substrates because of the lack of suitable native substrates. Thus, the epitaxial layers are strained during cool down, if they are sufficiently thick for them to relax at the growth temperature. The mechanical forces related to strain dramatically change the band structure of the epitaxial layers. The pressure dependence of the bandgap energy Eg can be expressed as Eg ¼ Eg(0) gP þ dP2, where Eg(0) is the bandgap of stress-free semiconductor, g and d are the Table 1.4 Calculated ionic radii (Å), effective charges (electrons), and the fractional ionic character (FIC) for III nitrides [32].

Compound

rIII (Å)

rN (Å)

QIII (e)

QN (e)

FIC

AlN GaN InN

0.8505 0.9340 1.0673

1.0523 1.0119 1.0673

1.12 1.98 1.83

6.88 6.02 6.17

0.72 0.51 0.54

j13

j 1 General Properties of Nitrides

14

3.8

Energy gap (eV)

3.7

3.6

3.5

3.4

3.3

-2

0

2

4

6

8

10

Pressure (GPa) Figure 1.11 Pressure dependence of the GaN energy gap, showing the typical sublinear character. Solid line represents the calculations of Christiansen and Gorczyca [35], which have been rigidly upshifted by 0.82 eV for a better fit with experiments, and the squares represent experimental results [6].

pressure coefficients, and P is the pressure. For GaN, g and d parameters are 4.2 · 103 and 1.8 · 105, respectively [33,34]. The bandgap is in terms of eVand the pressure is in terms of kbar. The pressure dependence has, in general, a sublinear character. The variation of the GaN energy gap with pressure, both theoretical [35] and experimental [6], is shown in Figure 1.11. The calculated pressure coefficients for III nitrides are given in Table 1.5. Parameters associated with mechanical properties of GaN in wurtzitic phase are tabulated in Tables 1.6 and 1.7, the latter dealing with the sound wave velocity. The same parameters for the zinc blende phase of GaN are tabulated in Tables 1.8 and 1.9.

Table 1.5 Calculated pressure coefficients for III nitrides including wurtzitic, zinc blende, and rock salt phases (g in units of meV GPa1 and d in units of meV GPa2) [35].

Zinc blende polytype

Wurtzite polytype

Rock salt polytype

Compound

G

d

g

d

g

d

AlN GaN InN

40 39 33

0.32 0.32 0.55

42 40 16

0.34 0.38 0.02

43 39 41

0.18 0.32 0.08

It should be noted that rock salt phases cannot be synthesized and exist only under high pressures beyond the phase transition point. g and d parameters with values of Eg ¼ Eg(0) þ gP þ dP2.

1.1 Crystal Structure of Nitrides Table 1.6 Parameters related to mechanical properties of wurtzitic GaN (in part after Ref. [36]).

Wurtzite polytype GaN

Parameter value/comments

Group of symmetry Molar volume, Vc (cm3 mol1) Molecular mass (g mol1) Density (g cm3)

C 46v (P63mc) 13.61 83.7267 6.11 or 6.15

Number of atoms in 1 cm3 Lattice constants

8.9 · 1022 a ¼ 3.1893 Å for powder, c ¼ 5.1851 Å for powdera 210 [38]b or 20.4 · 1011 dyn cm2 (204 GPa) 4 150 0.23  0.06 (0.198–0.37) from C parameters 15.5 1200–1700

Bulk modulus B (GPa) (compressibility1) dB/dP Young’s modulus (GPa) Poisson’s ratio, n or s0 (n ¼ C13/(C11 þ C12)) Knoop’s hardness (GPa) Surface microhardness (kg mm2) Nanoindentation hardness (GPa) Yield strength (GPa) Deformation potential, Eds C11 (GPa) C12 (GPa) C13 (GPa)

C33 (GPa) C44 (GPa)

10.8 at 300 K 0.1 at 1000 K 8.54 eV unscreened, 12 eV screened 390  15, 29.6 · 1011 dyn cm2 (296 GPa) 145  20, 13.0 · 1011 dyn cm2 (130 GPa) 106  20, 15.8 · 1011 dyn cm2 (158 GPa)

Comments/ references

The latter by Bougrov et al. [37]

The latter by Bougrov et al. [37]

[39] At 300 K 300 K, using Knoop’s pyramid test [36,40,41]

[38,42] The second set is from Ref. [43] The second set is calculated from the mean square displacement of the lattice atoms measured by X-ray diffraction

398  20, 26.7 · 1011 dyn cm2 (267 GPa) 105  10, 2.41 · 1011 dyn cm2 (241 GPa)

nh0 0 0 1i ¼ (Da/arelaz)/(Dc/crelax) or nh0 0 0 1i ¼ (Da/a0)/(Dc/c0) with Da ¼ ameas  arelax and Dcmeas  crelax. Conversion: 1 dyn cm2 ¼ 0.1 Pa (i.e., 1 GPa ¼ 1010 dyn cm2). For details of elastic constants and piezoelectric constants, see Tables 2.27 and 2.28, and at 300 K Bs ¼ 210  10 GPa, Bs ¼ [C33(C11 þ C12)  2(C13)2]/[C11 þ C12 þ 2C33  4C13]. a See Section 1.2.2 for details and lattice parameter for GaN on different substrates b Average of Voigt and Reuss bulk modulus.

j15

j 1 General Properties of Nitrides

16

Table 1.7 Wave propagation properties in wurtzitic GaN [36].

Wave propagation direction [1 0 0]

[0 0 1]

Wave character VL (longitudinal) VT (transverse, polarization along [0 0 1]) VT (transverse, polarization along [0 1 0]) VL (longitudinal) VT (transverse)

Expression for wave velocity

Wave velocity (in units of 105 cm s1)

(C11/r)1/2 (C44/r)1/2

7.96 4.13

(C11  C12)/2r)1/2

6.31

(C33/r)1/2 (C44/r)1/2

8.04 4.13

Parameters associated with thermal properties of GaN in wurtzitic and zinc blende phases (expected to be identical or nearly identical – treated to be identical here) are tabulated in Table 1.10. The parameters associated with electrical and optical properties of wurtzitic GaN are tabulated in Table 1.11. The same parameters associated with the zinc blende phase of GaN are tabulated in Table 1.12. Table 1.8 Parameters related to mechanical properties of zinc blende GaN (in part after Ref. [36]).

Zinc blende polytype GaN

Parameter value/comments

Group of symmetry Molar volume, Vc, na, or O (cm3 mol1) Molecular mass (g mol1) Density (g cm3) Number of atoms in 1 cm3 Lattice constant (Å) Bulk modulus, B (GPa)

2 Tp d ðF43mÞ ffiffiffi ð 3a2 cÞ=4 ¼ 2:28310  23 cm3

dB/dP Young’s modulus (GPa)

1.936 · 1023 6.15 8.9 · 1022 a ¼ 4.511  4.52 Bs ¼ 204 [36], 201 (theory) [45], 237 [46], 200 [47] 3.9, 4.3 181 [36]

Shear modulus, C 0 (GPa) Poisson’s ratio, n or s0

67 [36] 0.352 [36]

Knoop’s hardness Surface microhardness Nanoindentation hardness Yield strength Deformation potential, Eds C11 (GPa) C12 (GPa) C44 (GPa)

293 159 155

Bs ¼ [C33(C11 þ C12)  2(C13)2]/[C11 þ C12 þ 2C33  4C13] or Bs ¼

Comments/references

2

C12 Þ  2ðC 13 Þ Bs ¼ CC3311ðCþ11C12þ þ 2C 33  4C13

Y0 ¼ (C11 þ 2C12) (C11  C12)/(C11 þ C12) C0 ¼ (C11  C12)/2 n or s0 ¼ C13/ (C11 þ C12)

[42]

C 33 ðC 11 þ C 12 Þ  2ðC13 Þ2 . C11 þ C 12 þ 2C33  4C 13

1.1 Crystal Structure of Nitrides Table 1.9 Wave propagation properties in zinc blende GaN (after Ref. [36]).

Wave propagation direction

Wave character

Expression for wave velocity

Wave velocity (in units of 105 cm s1)

[1 0 0]

VL (longitudinal) VT (transverse)

(C11/r)1/2 (C44/r)1/2

6.9 5.02

[1 1 0]

VL (longitudinal) Vt//(transverse) V t? (transverse)

[(C11þCl2þ2C44)/2r]1/2 Vt// ¼ VT ¼ (C44/r)1/2 [(C11  C12)/2r]1/2

7.87 5.02 3.3

[1 1 1]

V l 0 V l

[(C11 þ 2C12 þ 4C44)/3r]1/2 [(C11  C12 þ C44)/3r]1/2

8.17 3.96

0

For the crystallographic directions, see Ref. [44].

Table 1.10 Parameters related to thermal properties of GaN, wurtzitic, and zinc blende phases are expected to be the in this respect with the exception of the first two parameters, which are for the wurtzitic phase (in part Ref. [36]).

GaN

Parameter value/comments

Comments/references

Temperature coefficient (eV K1) Thermal expansion (K1)

dEg/dT ¼ 6.0 · 104

Wurtzite structure only

Da/a ¼ 5.59 · 106, a|| ¼ aa ¼ 5.59 · 106 (wurtzite structure) [48]

Dc/c ¼ 3.17 · 106; for a plot versus temperature, see Ref. [49] (wurtzite structure only) For low dissociation material (106 cm2)

Thermal conductivity k (W cm1 K1) Debye temperature (K) Melting point (
C) Specific heat (J g1
C1) Thermal diffusivity (cm2 s1) Heat of formation, DH298 (kcal mol1) Heat of atomization, DH298 (kcal mol1) Heat of sublimation (kcal mol1) Heat capacity (J mol1 K1) Specific heat (J mol1 K1) (298 K < T < 1773 K) Enthalpy, DH0 (kcal mol1) Standard entropy of formation, DS0 (cal mol1 K1)

11.9 at 77 K, 2.3 at 300 K, 1.5 at 400 K 600 >1700 (at 2 kbar), 2500 (at tens of kbar) 0.49 0.43 26.4

[50] [37] [37]

203 72.4  0.5 35.4 at 300 K Cp ¼ 38.1 þ 8.96 · 103T

[51]

37.7 32.43

The specific heat Cp of Wz GaN at constant pressure for 298 K < T < 1773 K is Cp ¼ 38.1 þ 8.96 · 103T (J mol1 K1) [51].

j17

j 1 General Properties of Nitrides

18

Table 1.11 Parameters related to electrical and optical properties

of Wz GaN (in part after Refs [36,44]).

Wurtzite polytype GaN

Parameter value/comments

Bandgap energy, Eg (eV), direct Breakdown field (cm1) Electron affinity (eV) Energy separation between G and M–L valleys (eV)

3.42 at 300 K, 3.505 at 1.6 K

Energy separation between M–L valleys degeneracy (eV) Energy separation between G and A valleys (eV) Energy separation between A valley degeneracy (eV) Index of refraction

Dielectric constants (static) Dielectric constants (high frequency)

Optical LO phonon energy (meV) A1-LO, nA1(LO) (cm1)

Comments/ references

3–5 · 106 at 300 K 4.1 1.9 at 300 K

[53] [37] [37]

1 at 300 K 0.6 at 300 K

[52] [37]

0.6 at 300 K 1.3–2.1 at 300 K

[52] [37]

2 at 300 K 1 at 300 K

[52] [37]

0.2 at 300 K n (1 eV) ¼ 2.35 or 2.3 2.29, n (3.42 eV) ¼ 2.85 at 300 K (extrapolated to 0 eV), E?c interference method (the value for E||c is 1.5(2)% lower at 500 nm); also see energy dependence and long-wavelength value [54] 10.4 (E||c) 9.5 (E?c) 8.9 in c-direction (E||c) at 300 K 5.35 5.8 (E||c) at 300 K 5.35 (E?c) at 300 K 5.47 (E||c) 91.2

[52]

[55] [55] [37] [37] [56] [55] [37]

710–735

[57]

744

A1-TO, nA1(TO||) (cm1) E1-LO, nE1(LO? ) (cm  1)

533–534 741–742

[34] [58]

533 746

E1-TO, nE1(TO? ) (cm  1) E2 (low) (cm1) E2 (high) (cm1)

556–559 143–146 560–579

[59]

559

Reflectivity [55] Raman [56] Reflectivity [55] Raman [60]

1.1 Crystal Structure of Nitrides Table 1.11 (Continued)

Wurtzite polytype GaN

Parameter value/comments

Energy of spin–orbital splitting, Eso (meV)

11 (þ5, 2) at 300 K calculated from the values of energy gap Eg,dir (given in this table) 40 at 300 K

Energy of crystal-field splitting, Ecr (meV)

Effective electron mass, == me or me

Effective electron mass, me? or m? e Effective hole mass Effective hole masses (heavy), mhh

Effective hole masses (light)

Effective hole masses (split-off band), ms

Effective mass of density of state, mv Effective conduction band density of states (cm3) Effective valence band density of states (cm3) Electron mobility (cm2 V1 s1) Hole mobility (cm2 V1 s1) n-doping range (cm3)

22 (2), calculated from the values of energy gap Eg,dir (given in this table) 0.20m0 at 300 K

Comments/ references [61]

[37] [61] [37]

0.20m0 0.27m0 by Faraday rotation 0.138–0.2 0.20m0, 300 K; fit of reflectance spectrum

[62] [52]

0.15–0.23m0 0.8m0 at 300 K mhh ¼ 1.4m0 at 300 K

[52] [64] Calculated

==

mhhz ¼ mhh ¼ 1:1m0 at 300 K mhh? ¼ m? hh ¼ 1:6m0 at 300 K == mhh ¼ 1:1  2:007m0 m? hh ¼ 1:61  2:255m0 mlh ¼ 0.3m0 at 300 K ==

mlhz ¼ mlh ¼ 1:1m0 at 300 K m? lh ¼ mlh? ¼ 0:15m0 at 300 K == mlh ¼ 1:1  2:007m0 mlh? ¼ 0:14  0:261m0 msh ¼ 0.6m0 at 300 K ==

mshz ¼ mch ¼ 0:15m0 at 300 K msh? ¼ m? ch ¼ 1:1m0 at 300 K == msh? ¼ mch ¼ 0:12  0:16m0 ¼ 0:252  1:96m0 m? ch 1.4m0

[63]

[15] [70] [52] [52] Calculated [15] [70] [52] [52] Calculated [36] [70] [52] [52] [37]

2.3 · 1018 at 300 K

4.6 · 1019 at 300 K 1400 experimental at 300 K <20

50 000 at 20 K [65] At 300 K

1016 cm3–high 1019

(Continued )

j19

j 1 General Properties of Nitrides

20

Table 1.11 (Continued)

Wurtzite polytype GaN

Parameter value/comments

Comments/ references

p-doping range (cm3) Diffusion coefficient for electrons (cm2 s1) Diffusion coefficient for holes (cm2 s1)

1016 cm3–mid 1018 25

[36]

5, 26, 94

[36,53]

The details of the energies of high symmetry points compiled by Fritsch et al. [52] are given in Table 2.1. Dependence of the bandgap on hydrostatic pressure: Eg ¼ Eg(0) þ gP þ dP2, where Eg(0) is the bandgap of stress-free GaN. The values for g parameter are 39–42 meV GPa1 and the same for d parameter are 0.18 to 0.32 meV GPa2. For others, see Table 1.5.

The phonon energies are discussed later on in Table 1.27 in detail and effective masses are discussed in Chapter 2. More details of effective masses can be found in Table 2.9. More should be said about the dielectric constants. Electromagnetic theory indicates that for any longitudinal electromagnetic wave to propagate, the dielectric function e(o) must vanish. Doing so leads to [66] eðwÞ w2LO  w2 ; ¼ eð¥Þ w2TO  w2

ð1:5Þ

where oLO and oTO represent the transverse optical (TO) and longitudinal optical (LO) phonon frequencies and e(o) and e(1) represent the low and high (optical) frequency dielectric constants. The phonon branches associated with a wurtzitic symmetry are discussed in Section 1.2.2 dealing with the mechanical properties of GaN. When o ¼ oLO, the dielectric function vanishes, e(oLO) ¼ 0. Equation 1.5 can be expanded to the directional dependence of the dielectric function in nitrides in general and GaN in particular. In the direction parallel to the c-axis or the z-direction, from the G point to the A point, in the k-space, (with x, y representing the in-plane coordinates), the lowand high-frequency dielectric functions are related each with the help of A1(LO) and E1(TO) phonons through [67] e== ðwÞ ¼ e¥?

w2  w2== ðLOÞ w2  w2== ðTOÞ

:

ð1:6Þ

Likewise, Equation 1.5 can be expanded in the direction perpendicular to the c-axis or in the basal plane or the (x, y) plane, the z-direction (in k-space between the G point and M (1/2, 0, 0) or K (1/3, 1/3, 0) points), the low- and high-frequency dielectric functions are related each with the help of A1(TO) and E1(LO) phonons through e? ðwÞ ¼ e¥?

w2  w2? ðLOÞ ; w2  w2? ðTOÞ

ð1:7Þ

where ? and // indicate in the basal plane and along the c-direction, respectively.

1.1 Crystal Structure of Nitrides Table 1.12 Parameters related to electrical and optical properties

of zinc blende GaN (in part after Refs [36,44]). Zinc blende polytype GaN

Parameter/comments

Comments/references

Bandgap energy (eV) Breakdown field (V cm1) Index of refraction Dielectric constant (static)

3.2–3.28 at 300 K 5 · 106 n (at 3 eV) ¼ 2.9, 2.3 9.7 at 300 K == 9.2 by ð2e? 0 þ e0 Þ=3 of wurtzitic form 5.3 at 300 K

3.302 at low temperature [36]

Dielectric constant (high frequency) Energy separation between G and X valleys, EG (eV) Energy separation between G and L valleys, EL (eV) Spin–orbit splitting in valence band, Dso or Eso (eV) Effective electron mass, me Effective hole masses (heavy)

1.4 1.1 1.6–1.9 2 0.02 0.017 0.13m0 0.14m0 mhh ¼ 1.3m0, ½110 mhh ¼ 1:52m0 m[100] ¼ 0.8m0,

[37] [52] Using Lyddane–Sachs–Teller relation (e0/ehigh ¼ o2LO/ o2TO) [36] [52] [36] [52] At 300 K [36] [69] At 300 K [37] [52] At 300 K [36,70]

½100

mhh ¼ 0:84m0 m[111] ¼ 1.7m0, ½111

mhh ¼ 2:07m0 Effective hole masses (light)

Effective hole masses (split-off band), ms, mch, or mso

Effective conduction band density of states Effective valence band density of states Electron mobility (cm2 V1 s1) Hole mobility (cm2 V1 s1) Diffusion coefficient for electrons (cm2 s1) Diffusion coefficient for holes (cm2 s1) Electron affinity Optical LO phonon energy (meV)

Second set of figures are from Ref. [52], which are deemed more reliable

mlh ¼ 0.19m0, ½110 mlh ¼ 0:20m0 m[100] ¼ 0.21m0, ½100 mlh ¼ 0:22m0 m[111] ¼ 0.18m0, ½111 mlh ¼ 0:19m0 msh ¼ 0.33m0, mso ¼ 0.35m0 m[100] ¼ 0.33m0 m[111] ¼ 0.33m0 1.2 · 1018 cm3

At 300 K [37]

4.1 · 1019 cm3

At 300 K [37]

1000 at 300 K 350 at 300 K 25

[36] [36] [36]

9, 9.5, 32

[36]

4.1 eV 87.3

[37] At 300 K

The details of the energies of high symmetry points are given in Table 2.1.

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j 1 General Properties of Nitrides

22

For wurtzitic GaN, the various directional components of phonon frequencies are o?(LO) ! E1(LO) ¼ 91.8 meV, oz(LO) ¼ o//(LO) ! A1(LO) ¼ 91 meV, o?(TO) ! E1(TO) ¼ 69.3 meV, and oz(TO) ¼ o//(TO) ! A1(TO) ¼ 66 meV. In the z-direction (along the c-direction) and perpendicular to the z-direction (in basal plane), LO and TO phonons are not mixed. For any direction other than the in-plane and out-of-plane configurations, the LO and TO phonons mix and hybridize. For a given propagation direction with an angle y relative to the c-axis (0z), one finds three phonon branches. One is an ordinary TO phonon mode with atomic displacement in the (0xy) plane. The other two branches have a mixed TO and LO character and their dielectric functions are given by the solutions of [68] e== cos2 q þ e? sin2 q ¼ 0:

ð1:8Þ

Using the above relationship, the phonon energy as a function of the angle can easily be calculated. Doing so leads to the conclusion that the upper branch (LO-like) remains between A1(LO) and E1(LO) energies, whereas the lower branch (TO-like) remains between A1(TO) and E1(TO) energies. Therefore, the dispersion remains small compared to the LO–TO separation, owing to the relatively small cell asymmetry and the large ionicity of atomic bonds. A more important consequence of LO–TO mixing is that the TO-like mode becomes coupled to carriers whereas in the c-direction A1(LO) mode and in the basal plane E1(LO) phonons couple to the carriers. For the special case o ¼ 0 (or very small frequencies compared to the LO and TO phonon frequencies), the relationship between the optical and static dielectric constants reduces to the well-known Lyddane–Sach–Teller relationship eðwÞ w2LO ; ¼ eð¥Þ w2TO

ð1:9Þ

which will be used to determine the optical frequency dielectric constant from the knowledge of A1(LO) and A1(TO) phonon frequencies along the c-direction and E1(LO) and E1(TO) in the basal plane. This relationship is used very often. Parameters related to the energy bandgap, carrier mass, and mechanical properties of AlN have been determined [71–76]. Extensive data on all the binary and ternary band structure parameters can be found in Chapter 2. For example, additional parameters on the critical point energies for Wz GaN, AlN, and InN are given in Tables 2.1–2.3, respectively. Tables 2.4–2.6 list the critical point energies for ZB GaN, AlN, and InN, respectively. Effective masses and other band parameters for Wz GaN are listed in Table 2.9. Table 2.10 tabulates the Luttinger band parameters for ZB GaN. Table 2.14 lists the effective band parameters for Wz AlN, whereas the effective masses and band parameters for Wz AlN are tabulated in Table 2.15. Luttinger parameters for ZB AlN are listed in Table 2.16. The band parameters and effective masses for Wz InN are tabulated in Tables 2.19 and 1.20, respectively. Returning to the content of this chapter, parameters associated with the mechanical properties of AlN in wurtzitic and zinc blende phases are tabulated in Tables 1.13 and 1.14, respectively. The parameters related to the sound wave velocity in wurtzitic AlN are listed in Table 1.15.

1.1 Crystal Structure of Nitrides Table 1.13 Parameters related to mechanical properties of wurtzitic AlN (in part after Ref. [36]).

Wurtzite AlN

Parameter/comments

C46v (P63 mc) 9.58 · 1022 12.47 40.9882 3.28 3.255 g cm3 by X-ray 3.23 g cm3 by X-ray Lattice constants a ¼ 3.112 Å, c ¼ 4.979–4.982 Å Bulk modulus, B (GPa) 159.9–210.1, 21 · 1011 dyn cm2 (210 GPa) (Bs ¼ 210) dB/dP 5.2–6.3 374, 308 Young’s modulus, E or Y0 (GPa) 0.18–0.21 Poisson’s ratio, n or s0 {0001}, c-plane Poisson’s ratio s0 along the different crystallographic f1 1 2 0g, a-plane ðl ¼ h0001i; m ¼ h1 1 0 0iÞ directions f1 1 2 0g, a-plane ðl ¼ h1 1 0 0i; m ¼ h0001iÞ Knoop’s hardness (GPa) 10–14 at 300 K Nanoindentation hardness (GPa) 18 Yield strength (GPa) 0.3 at 1000
C Surface microhardness on basal 800 kg mm2 by 300 K, using Knoop’s pyramid test plane (0 0 0 1) 410  10 C11 (GPa)a 149  10 C12 (GPa)a 99  4 C13 (GPa)a 389  10 C33 (GPa)a 125  5 C44 (GPa)a Velocity of the longitudinal sound 10 127 m s1 waves, vl 6333 m s1 Velocity of the shear waves, vs

Comments/references

Group of symmetry Number of atoms in 1 cm3 Molar volume, Vc (cm3 mol1) Molecular mass (g mol1) Density (g cm3)

Longitudinal elastic modulus, Cl Shear elastic modulus, Cs

[77] [78] [78]

The latter from Ref. [79] 0.287 Ref. [80] 0 Ref. [80] 0.216 Ref. [80] [81] [82] [36,40,41] [42,83] Refer to Table 1.29 as well Refer to Table 1.29 as well Refer to Table 1.29 as well Refer to Table 1.29 as well [79] The sound velocities and related elastic module (experimental data)

334 GPa 131 GPa

Conversion: 1 dyn cm 2 ¼ 0.1 Pa (i.e., 1 GPa ¼ 1010 dyn  cm2). See Table 1.29 for more details. For details of elastic constants and piezoelectric constants, see Tables 2.27 and 2.28. This expression is given already in conjunction with Table 1.6 and it is also Bs ¼ [C33(C11 þ C12)  2 (C13)2]/[C11 þ C12 þ 2C33  4C13]. Temperature derivatives of the elastic module: dln Cl/ dT ¼ 0.37 · 104 K1; dln Cs/dT ¼ 0.57 · 104 K1; dln Bs/dT ¼ 0.43 · 104 K1. a See Table 1.29 for a more in-depth treatment of these parameters.

Parameters associated with thermal properties of wurtzitic AlN are tabulated in Table 1.16. Parameters associated with electrical and optical properties of wurtzitic AlN are tabulated in Table 1.17. The same range of parameters associated with the zinc blende

j23

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24

Table 1.14 Parameters related to mechanical properties of zinc blende AlN.

Not much is known about the zinc blende phase

Zinc blende AlN Lattice constant (Å) Bandgap (eV) E Xg ðeVÞ E Lg ðeVÞ Bulk modulus, B (GPa) Young’s modulus (GPa)

a ¼ 4.38 5.4, indirect 4.9 9.3 228

Shear modulus (GPa) Poisson’s ratio, n or s0 C11 (GPa)

348

304

C12 (GPa) C44 (GPa) me ðGÞ ml ðXÞ mt ðXÞ

168 135 0.25 0.53 0.31

160 193

Comments/references

[84] [84] [84] Bs ¼ (C11þ2C12)/3 Y0 ¼ (C11 þ 2C12) · (C11  C12)/(C11 þ C12) C’ ¼ (C11  C12)/2 n or s0 ¼ C12/(C11þC12) The latter figures are from Ref. [84]

[84] [84] [84]

See Table 2.18 for more details.

phase of GaN is tabulated in Table 1.18. Mechanical, phonon, properties of epitaxial AlN (deposited on silicon and sapphire substrates at 325 K by ion beam assisted deposition (IBAD)) have been investigated by Ribeiro et al. Raman scattering measurements revealed interesting features related to the atomic composition and structure of the films [94]. Vibrational modes corresponding to 2TA(L) at 230 cm1, 2TA(X) at 304 cm1, 2TA(S) at 435 cm1, TO(G) at 520 cm1, TA(S) þ TO(S) at 615 cm1, accidental critical points at 670 and 825 cm1, 2TO(D) at 950 cm1, 2TO(L) at 980 cm1, 2TO(G) at 1085 cm1, 2TA(X) þ 2TO(G) at 1300 cm1, and 3TO(G) at 1450 cm1 have been observed. While identifying the vibrational modes, one should be wary of the peak at 2330 cm1 caused by the molecular nitrogen on Table 1.15 Acoustic wave propagation properties in wurtzite AlN [36].

Wave propagation direction [1 0 0]

[0 0 1]

Wave character VL (longitudinal) VT (transverse, polarization along [0 0 1]) VT (transverse, polarization along [0 1 0]) VL (longitudinal) VT (transverse)

For the crystallographic directions, see Ref. [44].

Expression for wave velocity (C11/r)1/2 (C44/r)1/2 (C11-C12)/2r)1/2 (C33/r)1/2 (C44/r)1/2

Wave velocity (in units of 105 cm s1) 11.27 6.22 6.36 10.97 6.22

1.1 Crystal Structure of Nitrides Table 1.16 Parameters related to thermal properties of wurtzitic AlN (in part after Ref. [36]).

Wurtzite polytype AlN

Value

Comments/references

Thermal expansion (K1)

Da/a ¼ a|| ¼ aa ¼ 4.2 · 106, Dc/c ¼ aort ¼ ac ¼ 5.3 · 106 Da/a ¼ 2.9 · 106, Dc/c ¼ 3.4 · 106 aort ¼ ac ¼ 5.27 · 106, a|| ¼ aa ¼ 4.15 · 106

[48,85–87]

Thermal conductivity (W cm1 K1) Thermal diffusivity (W cm1
C1) Debye temperature (K) Melting point (K)

Specific heat (J g1
C1) Thermal diffusivity (cm2 s1) Heat of formation, DH298 (kcal mol1) Heat of atomization, DH298 (kcal mol1) Free energy, DG298 (kcal mol1)

k ¼ 2.85–3.2 2.85 at 300 K 950, 1150 3273 3023 (between 100 and 500 atm of nitrogen) 3487 (2400
C at 30 bar) 0.6

[88] T ¼ 20–800
C. X-ray, epitaxial layers, by Sirota and Golodushko [89], also see Ref. [48] [90]; later results by Slack et al. [91] [90]

[92] [78]

See Figure 1.29 and the expressions below this table

1.47 64 209.7 68.15

For 293 < T < 1700 K, Da/a300 ¼ 8.679 · 102 þ 1.929 · 104T þ 3.400 · 107T2 7.969 · 1011T3. For 293 < T < 1700 K, Dc/c300 ¼ 7.006 · 102 þ 1.583 · 104T þ 2.719 · 107T2 5.834 · 1011T3. The specific heat Cp of AlN for constant pressure: for 300 < T < 1800 K, Cp ¼ 45.94 þ 3.347 · 103T  14.98 · 105T2 (J mol1 K1); for 1800 < T < 2700 K, Cp ¼ 37.34 þ 7.866 · 103T (J mol1 K1). After Ref. [93]. Optical emission measurements indicate the FXA transition at 6.023 with an associated binding energy of 63 meV, which sets the bandgap of Wz AlN at 6.086.

the surface of c-Si [95]. It is worth noting that, owing to the extremely weak Raman signal usually presented by AlN films, it is not uncommon to ascribe some of the features erroneously to AlN [96]. Conduction band first- and second-order pressure derivatives [36]: Eg ¼ Eg(0) þ 3.6 · 103P  1.7 · 106P2 (eV) [121] EM ¼ EM(0) þ 7.5 · 104P þ 1.0 · 106P2 (eV) EL ¼ EL(0) þ 8.0 · 104P þ 6.9 · 107P2 (eV) Ek ¼ Ek(0) þ 6.3 · 104P þ 1.7 · 106P2 (eV) where P is pressure in kbar.

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j 1 General Properties of Nitrides

26

Table 1.17 Parameters related to optical and electrical properties

of wurtzitic AlN [97–102] (in part from Ref. [36]). Wurtzite polytype AlN

Parameter

Comments/references

Bandgap energy (eV) From the dichroism of the absorption edge, it follows that the G1 0 state lies slightly above the G6 state (transition E||c (G1 0 v  Glc) at lower energy than transition E ? c (G6v  Glc)), both states being split by crystal field interaction [105]

6.026 at 300 K 6.2 eV at 300 K

[103,104] Excitonic contribution near direct edge [105]

6.23 at 77 K

Excitonic contribution near direct edge [105] Excitonic edge assuming exciton binding energy of 75 meV [106] With a free exciton binding energy of 63 meV [107,108]

6.28 at 300 K

6.086 at 5 K

Breakdown field (V cm1) dEg/dP (eV bar1) Conduction band energy separation between G and M–L valleys (eV) Conduction band energy separation between G and M–L valleys Conduction band energy separation between M–L valleys degeneracy (eV) Conduction band energy separation between G and K valleys (eV) Conduction band K valley degeneracy (eV)

Valence band energy of spin–orbital splitting, Eso (eV)

6.0 at 300 K 6.1 at 5 K 1.2–1.8 · 106 3.6 · 103 0.7 1

[36] [109,110] [78] [52]

0.6

[78]

0.2

[52]

1.0

[78]

0.7 2

[52] [78]; empirical pseudopotential calculations of Fritsch et al. [52] do not show degeneracy at this critical point

0.019 at 300 K 0.036

[108]

1.1 Crystal Structure of Nitrides Table 1.17 (Continued)

Wurtzite polytype AlN

Parameter

Comments/references

Valence band energy of crystal field splitting, Ecr (eV), G7 on top of G9 Effective conduction band density of states (cm3) Effective valence band density of states (cm3) Index of refraction Dielectric constant (static)

0.225

[108]

6.3 · 1018

[78]

4.8 · 1020

[78]

Dielectric constant (high frequency)

4.68 4.77 4.84 at 300 K 4.35 for E//c (modeling) 4.16 for E?c (experiment) 2.1–2.2 at 300 K

Infrared refractive index

Effective electron mass, me

Effective hole masses (heavy) == For kz direction mhz or mhh For kx direction mhx or m? hh

Effective hole masses (light) == For kz direction mlz or mlh For kx direction mlx or m? lh Effective hole masses (splitoff band) ==

For kz direction msoz or mch For kx direction msox or

n (3 eV) ¼ 2.15  0.05 9.14 at 300 K 7.34 8.5  0.2 at 300 K 9.32 for E//c (modeling) 7.76 for E?c (experiment) 4.6 at 300 K

m? ch

1.9–2.1 at 300 K 1.8–1.9 at 300 K 3 for E//c (modeling) 2.8 for E?c (experiment) 0.27 and 0.35m0 0.25–0.39m0 0.4m0 at 300 K == me ¼ 0:231  0:35m0 m? e ¼ 0:242  0:25m0 == mhh ¼ 3:53m0 at 300 K 2.02–3.13m0 at 300 K

By reflectivity [111] [78,112] [113] [113] [111] [78] Reflectivity [113] [113] Epitaxial films and monocrystal Polycrystalline films Amorphous films [87] [113] [113] [14,114,115] [107] [52]

m? hh ¼ 10:42m0 at 300 K == mhh ¼ 1:869  4:41m0 m? hh ¼ 2:18  11:14m0 3.53m0, 0.24m0 == mlh ¼ 1:869  4:41m0 m? lh ¼ 0:24  0:350m0 0.25m0 at 300 K

[116] [107]; from Mg binding energy [116] [52] [52] At 300 K [116] [52] [52] [116]

3.81m0 at 300 K == mch ¼ 0:209  0:27m0

[52]

m? ch

[52]

¼ 1:204  4:41m0

(Continued )

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j 1 General Properties of Nitrides

28

Table 1.17 (Continued)

Wurtzite polytype AlN

Parameter

Comments/references

Effective mass of density of state, mv Optical phonon energy (meV) nTO(E1) phonon wave number (cm1)b

7.26m0 at 300 K

[116]

nLO(E1) phonon wave number (cm1) nTO(A1) phonon wave number (cm1) nLO(A1) phonon wave number (cm1) n(E2) phonon wave number (cm1) nTO(E1) phonon wave number (cm1) nTO(A1) phonon wave number (cm1) nLO(E1) phonon wave number (cm1) nLO(A1) phonon wave number (cm1) n(1)(E2) phonon wave number (cm1) n(2)(E2) phonon wave number (cm1)

99.2 895

614

608

671.6

821

888.9

888

514

667.2

659.3

663

909

303a

426

657–673

First column [117]; second column [118]; third column [113] See Table 1.30 for more details

[87,111,119,120]

607–614 or 659–667 895–924 888–910 241–252 655–660

The details of the energies of high symmetry points are given in Table 2.2. See Table 1.30 for additional details for phonon wave numbers. More details of effective masses can be found in Table 3.15. Temperature dependence of energy gap: Eg ¼ Eg(0) – 1.799 · 103T2/(T þ 1462) (eV) by Guo and Yoshida [103]. a Room-temperature Raman, tentative. b For more details regarding vibrational modes, refer to Section 1.3.1.

Phase transition from the wurtzite phase to the rock salt structure (space group O5h ; lattice parameter 4.04 Å) takes place at the pressure of 17 GPa (173 kbar) [109,110]. Parameters associated with the electrical and optical properties of zinc blende AlN are listed in Table 1.18. For details regarding the Luttinger parameters for the valence band in zinc blende AlN, refer to Table 2.16. Parameters associated with the mechanical properties of wurtzitic InN are tabulated in Table 1.19. For wurtzite crystal structure, the surfaces of equal energy in G valley should be ellipsoids, but effective masses in the z-direction and perpendicular directions are estimated to be approximately the same.

1.1 Crystal Structure of Nitrides Table 1.18 Parameters related to optical and electrical properties of zinc blende AlN.

Zinc blende polytype of AlN

Value

Comments/references

Bandgap energy (eV)

All at 300 K [52]

Dielectric constant (static) Dielectric constant (high frequency) Energy separation between G and X valleys EG (eV) Energy separation between G and L valleys EL (eV) Spin–orbit splitting in valence band, Dso or Eso (eV) Deformation potential (eV) Effective electron mass, me

4.2; 6.0 5.8 All (theory) 9.56 4.46 0.7 0.5 2.3 3.9 0.019

[52] [84] [52] [84] [84]

9 0.23m0

[84] [52]

Effective hole masses (heavy)

mhh ¼ 1:02m0

Effective hole masses (light)

Effective hole masses (split-off band), ms, mch, or mso Luttinger parameter c1 Luttinger parameter c2 Luttinger parameter c3 a

½100

½111 mhh

¼ 2:64m0

½110 mhh

¼ 1:89m0

mlh

½100

¼ 0:37m0

½111 mlh

¼ 0:30m0

½110 mlh

¼ 0:32m0

a a

[52]

[52]

0.54m0

[52]

1.85 0.43 0.74

[52]

C. Persson, and A. Ferreira da Silva, “Linear optical response of zinc-blende and wurtztie III-N (III ¼ B, AI, Ga, and In)”, Journal of Crystal Growth 305 pp. 408–413 (2007)

The parameters associated with thermal properties of wurtzitic InN are tabulated in Table 1.20. The specific heat Cp of InN at constant pressure for 298 K < T < 1273 K [51] is Cp ¼ 38.1 þ 1.21 · 102T (J mol1 K1). Refer to Table 1.31 for a detailed treatment of mechanical properties of InN. The parameters associated with electrical and optical properties of wurtzitic InN are tabulated in Table 1.21. Available parameters associated with the mechanical properties of zinc blende InN, primarily calculated, are tabulated in Table 1.22. Other parameters dealing with electrical and optical properties of zinc blende InN, primarily calculated, are listed in Table 1.23.

j29

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30

Table 1.19 Parameters related to mechanical properties of wurtzitic InN (in part after Ref. [36]).

Wurtzite InN

Value

Group of symmetry

C 46v (P63 mc)

Molar volume (cm3 mol1)

18.49

Molar mass (g mol1)

128.827

Density (g cm3)

6.89 6.98 6.81 a ¼ 3.548 a ¼ 3.5446 a ¼ 3.533 c ¼ 5.760 c ¼ 5.7034 c ¼ 5.693 165 140

Lattice constants (Å)

Bulk modulus B (GPa) dB/dP

3.8

Nanoindentation hardness(GPa)

11.2

Comments/references

Measured by displacement X-ray, 298.15 K [122] Epitaxial layers, X-ray [123]; 300 K [124] Epitaxial layers, X-ray [123]; 300 K [124]

[124]

[125]

Young’s modulus (GPa) Poisson’s ratio, n or s0 Knoop’s hardness (GPa) Deformation potential, Eds C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa)

Can be calculated using S parameters and Equation 2.10 0.82, 0.68 7.10 eV 223 115 92 224 48

C31 ¼ 70 205

Estimate [42] [42] [42] [42] [42]

Eg ¼ Eg(0) þ 3.3 · 102P (eV), where P is pressure in GPa [35,126]. For details of elastic constants and piezoelectric constants, see Table 2.19. Bs ¼ [C33(C11 þ C12)  2(C13)2]/ [C11 þ C12 þ 2C33  4C13].

1.2 Gallium Nitride

Despite the fact that GaN has been studied far more extensively than the other group III nitrides, further investigations are still needed to approach the level of understanding of technologically important materials such as Si and GaAs. GaN growth often suffers from large background n-type carrier concentrations because of native defects and, possibly, impurities. The lack of commercially available native substrates

1.2 Gallium Nitride Table 1.20 Parameters related to thermal properties of wurtzitic InN (in part after Ref. [36]).

Wurtzite polytype InN

Value

Temperature coefficient Thermal expansion

dEg/dT ¼ 1.8 · 104 eV K1 Da/a ¼ 2.70 · 106 K1; Dc/c ¼ 3.40 · 106 K1 Da/a ¼ 2.85 · 106 K1; Dc/c ¼ 3.75 · 106 K1 Da/a ¼ 3.15 · 106 K1; Dc/c ¼ 4.20 · 106 K1 Da/a ¼ 3.45 · 106 K1; Dc/c ¼ 4.80 · 106 K1 Da/a ¼ 3.70 · 106 K1; Dc/c ¼ 5.70 · 106 K1 aa ¼ 3.8 · 106 K1; ac ¼ 2.9 · 106 K1 0.8  0.2 W cm1 K1 0.45 W cm1
C1 1.76 W cm1
C1, 300 K (estimate for ideal InN) 4.6

Thermal conductivity

Heat of formation, DH298 (kcal mol1) (Wz) Heat of atomization, DH298 (kcal mol1) Melting point

Debye temperature Specific heat (J mol1 K1) Heat capacity, Cp (cal mol1 K1) Entropy, S0 (cal mol1 K1) TSFCw at formation, DH0f (kcal mol  1) TSFCa at formation, DS0f (kcal mol  1 K  1) TSFCw at formation, DG0f (kcal mol  1) TSFCw at fusion, DHm (kcal mol1) TSFCw at fusion, DSm (cal mol1 K1) N2 equilibrium vapor pressure

Comments/references

At 190 K At 260 K At 360 K At 460 K At 560 K [124] Estimate [127]

175 1373 K 2146 K, vapor pressure 105 bar at 1100–1200
C 660 K at 300 K 370 K at 0 K Cp ¼ 38.1 þ 1.21 · 102T 9.1 þ 2.9 · 103T 10.4 34.3, 30.5

[92]

25.3

Experimental 298.15 K

22.96

Experimental 298.15 K

14.0 10.19 cal mol1 K1 1 atm 105 atm

Theoretical Theoretical 800 K 1100 K

[124] [128] [51] 298–1273 K 298.15 K Experimental 298.15 K

a

TSFC: thermodynamic state function changes.

exacerbates the situation. These, together with the difficulties in obtaining p-type doping, and the arcane fabrication processes caused the early bottlenecks stymieing progress. Information available in the literature on many of the physical properties of GaN is in some cases still in the process of evolution, and naturally controversial. This

j31

j 1 General Properties of Nitrides

32

Table 1.21 Parameters related to electrical and optical properties

of wurtzitic InN (in part after Ref. [36]). Wurtzitic InN

Value

Comments/references

Bandgap energy, Eg (300 K)

1.89 eV, 1.5 eV, 0.78 eV

See Section 1.3.1 for an expan ded discussion

15.3 e0,ort ¼ 13.1

300 K [124] 300 K [128]

e0, || ¼ 14.4,

300 K [128]

8.4

300 K, using the Lyddane– Sachs–Teller relation (e0 =ehigh ¼ w2LO =w2TO ) [129,130] Heavily doped film, infrared reflectivity [131] [132] At 300 K [124] At 300 K and l ¼ 1.0 mm, interference method; n ¼ 3–1020 cm3 [131] At l ¼ 0.82 mm [131] At l ¼ 0.66 mm [131]

Electron affinity Dielectric constant (static) Dielectric constant (static, ordinary direction) Dielectric constant (static, extraordinary direction) Dielectric constant (high frequency)

9.3

Infrared refractive index

Energy separation between G and M–L valleys (eV) Energy separation between M–L valleys degeneracy Energy separation between G and A valleys (eV) Energy separation between A valley degeneracy Energy separation between G and G1 valleys (eV) Energy separation between G1 valley degeneracy (eV) Effective conduction band density of states Effective valence band density of states Valence band crystal field splitting, Ecr

5.8 2.9 2.56

2.93 3.12 Reported range: 2.80–3.05 2.9–3.9

300 K [124]

4.8 0.6

[52] 300 K [124]

0.7 0.7–2.7

[52] 300 K [124]

4.5 1

[52] 300 K [124]

0.6 1.1–2.6

[52] 300 K [124]

1

300 K [124]

9 · 1017 cm

300 K [124]

5.3 · 1019 cm3

300 K [124]

0.017 eV

300 K [124]

1.2 Gallium Nitride Table 1.21 (Continued)

Wurtzitic InN

Value

Comments/references

Valence band spin– orbital splitting, Eso Index of refraction

0.003 eV

300 K [124]

2.9 at 300 K 2.56 at 300 K (interference method; n ¼ 3–1020 cm3, l ¼ 1.0 mm) 2.93 3.12 0.11m0 == me ¼ 0:1  0:138m0 m? e ¼ 0:1  0:141m0 1.63m0 at 300 K 0.5m0 at 300 K == mhh ¼ 1:350  2:493m0 m? hh ¼ 1:410  2:661 m0 0.27m0 at 300 K == mlh ¼ 1:350  2:493m0 m? lh ¼ 0:11  0:196m0 0.65m0 at 300 K == mch ¼ 0:092  0:14m0 m? ch ¼ 0:202  3:422 1.65m0 at 300 K

[124] [131]

[131] [131] [133] [52] [52] [32,134,135] [136] [52] [52] [32,134,135] [52] [52] [32,134,135] [52] [52] [134,135]

73 at 300 K

[124]

Effective electron mass, me

Effective hole masses (heavy), mh

Effective hole masses (light), mlp Effective hole masses (split-off band), ms Effective mass of density of state, mv Optical LO phonon energy (meV)

The details of the energies of high symmetry points are given in Table 2.3. More details of effective masses can be found in Table 2.19.

Table 1.22 Available parameters for mechanical for zinc blende InN.

Zinc blende InN

Value

Lattice constant Density (g cm3) Bulk modulus (GPa) dB/dP Young’s modulus (GPa), Y0 or E

a ¼ 4.98 Å 6.97 138–155, 145.6 [42]

Shear modulus (GPa) Poisson’s ratio, n or s0 C11 (GPa) C12 (GPa) C44 (GPa)

187 125 86

Comments/references

Derived from X-ray data Bs ¼ (C11 þ 2C12)/3 3.9–4.0 Y0 ¼ (C11 þ 2C12) (C11  C12)/(C11 þ C12) C’ ¼ (C11  C12)/2 n or s0 ¼ C12/(C11 þ C12) See Ref. [42] See Table 1.31 for details

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j 1 General Properties of Nitrides

34

Table 1.23 Available electrical and optical properties of zinc blende InN, primarily calculated.

Zinc blende InN

Value

Comments/references

Bandgap energy, Eg (300 K)

2.2 eV

In the absence of any reliable data, the bandgap to a first extent can be assumed to be similar to that for Wz InN. See Ref. [137] for a detailed treatment

Dielectric constant

1.5–2.1 (theory) 0.2 eV below the Wz polytype 8.4

2.88  0.30 2.90  0.30 2.90 3.05  0.30 2.65 6.97 3

LWL == ð2e? 0 þ e0 Þ of wurtzitic By 3 form (the spur) Using Lyddane–Sachs–Teller relation (e0 =ehigh ¼ w2LO =w2TO ) Theory — Transmission interference Transmission interference NIRSR Derived from X-ray data [52]

2.6

[52]

0.006

[84]

0.13m0

[52]

12.45 Dielectric constant (high frequency) Refractive index at LWL at 600–800 nm at 900–1200 nm at 900–1200 nm at 620 nm Density (g cm3) Energy separation between G and X c1 valleys EG (eV) Energy separation between G and L valleys EL (eV) Spin–orbit splitting in valence band, Dso or Eso (eV) Effective electron mass, me Effective hole masses (heavy)

Effective hole masses (light)

Effective hole masses (split-off band), ms or mch or mso

[138]

½110 mhh ½100 mhh ½111 mhh ½110 mlh ½100 mlh ½111 mlh

¼ 2:12 m0

[52]

¼ 1:18 m0 ¼ 2:89 m0 ¼ 0:20 m0

[52]

¼ 0:21 m0

¼ 0:19 m0 0.36m0

[52]

LWL: long-wavelength limit; NIRSR: normal incidence reflectance of synchrotron radiation.

is in part a consequence of measurements being made on samples of widely varying quality. For this book, when possible we have disregarded the spurious determination. However, measurements are too few to yield a consensus, in which case the available data are simply reported.

1.2 Gallium Nitride

The burgeoning interest in nitrides has led to substantial improvements in the crystal growth and processing technologies, thus overcoming many difficulties encountered earlier. Consequently, a number of laboratories consistently obtained highquality GaN with room-temperature background electron concentrations as low as 5 · 1016 cm3. The successful development of approaches leading to p-type GaN has led to the demonstration of excellent p–n junction LEDs in the UV, violet, blue, green, and even yellow bands of the visible spectrum with brightness suitable for outdoor displays, CW lasers, and UV detectors, including the ones for the solar blind region. Moreover, power modulation doped field effect transistors (MODFETs) also generically referred to as heterojunction field effect transistors (HFETs) have been developed. What follows reports on the state of knowledge regarding the physical properties of GaN. 1.2.1 Chemical Properties of GaN

Since Johnson et al. [139] first synthesized GaN in 1932, a large body of information has repeatedly indicated that GaN is an exceedingly stable compound exhibiting significant hardness. It is this chemical stability at elevated temperatures combined with its hardness that has made GaN an attractive material for protective coatings. Moreover, owing to its wide energy bandgap, it is also an excellent candidate for device operation at high temperatures and caustic environments. Although the hardness may have initiated the interest in GaN, it is the excellent semiconducting features that have piqued the attention of researchers. While the thermal stability of GaN allows freedom of high-temperature processing, the chemical stability of GaN presents a technological challenge. Conventional wet etching techniques used in semiconductor processing have not been as successful for GaN device fabrication. For example, Maruska and Tietjen [140] reported that GaN is insoluble in H2O, acids, or bases at room temperature, but does dissolve in hot alkali solutions at very slow rates. Pankove [141] noted that GaN reacts with NaOH forming a GaOH layer on the surface and prohibiting wet etching of GaN. To circumvent this difficulty, he developed an electrolytic etching technique for GaN. Low-quality GaN has been etched at reasonably high rates in NaOH [142,143], H2SO4 [144], and H3PO4 [145–147]. Although these etches are extremely useful for identifying defects and estimating their densities in GaN films, they are not as useful for the fabrication of devices [148]. Well-established chemical etching processes do help for the device technology development, and the status of these processes in the case of GaN can be found in Volume 2, Chapter 1. Various dry etching processes reviewed by Mohammad et al. [149] and Pearton et al. [150] are promising possibilities and are discussed in Volume 2, Chapter 1. 1.2.2 Mechanical Properties of GaN

GaN has a molecular weight of 83.7267 g mol1 in the hexagonal wurtzite structure. The lattice constant of early samples of GaN showed a dependence on growth conditions, impurity concentration, and film stoichiometry [151]. These observations were

j35

j 1 General Properties of Nitrides

36

attributed to a high concentration of interstitial and bulk extended defects. A case in point is that the lattice constants of GaN grown with higher growth rates were found to be larger. When doped heavily with Zn [152] and Mg [153], a lattice expansion occurs because at high concentrations the group II element begins to occupy the lattice sites of the much smaller nitrogen atom. At room temperature, the lattice parameters of GaN platelets [18] prepared under high pressure at high temperatures with an electron concentration of 5 · 1019 cm3 are a ¼ 3.1890  0.0003 Å and c ¼ 5.1864  0.0001 Å. The freestanding GaN with electron concentration of about 1016 cm3, originally grown on sapphire (0 0 0 1) by hydride vapor phase epitaxy (HVPE) followed by liftoff, has lattice constants of a ¼ 3.2056  0.0002 Å and c ¼ 5.1949  0.0002 Å . For GaN powder, a and c values are in the range of 3.1893–3.190 and 5.1851–5.190 Å, respectively. Experimentally observed c/a ratio for GaN is 1.627, which compares well with 1.633 for the ideal case, and the u parameter calculated using Equation 1.1 is 0.367, which is very close to the ideal value of 0.375. For more established semiconductors with the extended defect concentration from low to very low, such as Si, GaAs, and so on, the effect of doping and free electrons on the lattice parameter has been investigated rather thoroughly. In bulk GaN grown by the high-pressure technique, the lattice expansion by donors with their associated free electrons has been investigated [18]. However, large concentration of defects and strain, which could be inhomogeneous, rendered the studies of this kind less reliable in GaN layers. In spite of this, the effect of Mg doping on the lattice parameter in thin films of GaN has been investigated. Lattice parameters as large as 3.220–5.200 Å for a and c values, respectively, albeit not in all samples with similar hole concentrations, have been reported [154]. For GaN bulk crystals grown with high-pressure techniques and heavily doped (a small percentage) with Mg, the a and c lattice parameters were measured to be 3.2822–5.3602 Å [155]. Suggestions have been made that the c parameter of implanted GaN layers increases after implantation and languishes after annealing [156]. However, the a parameter could not be precisely measured because sharp off-normal diffraction peaks are needed to determine this parameter accurately. For the zinc blende polytype, the calculated lattice constant, based on the measured GaN bond distance in Wz GaN, is a ¼ 4.503 Å. The measured value for this polytype varies between 4.49 and 4.55 Å, while that in Ref. [18] is 4.511 Å, indicating that the calculated result lies within the acceptable limits [157]. A high-pressure phase transition from the Wz to the rock salt structure has been predicted and observed experimentally. The transition point is 50 GPa and the experimental lattice constant in the rock salt phase is a0 ¼ 4.22 Å. This is slightly different from the theoretical result of a0 ¼ 4.098 Å obtained from first-principles nonlocal pseudopotential calculations [158]. Tables 1.6 and 1.10 compile some of the known properties of Wz GaN. Parameters associated with electrical and optical properties of Wz GaN are tabulated in Table 1.11. The same parameters associated with the zinc blende phase of GaN are tabulated in Table 1.12. The bulk modulus of Wz GaN, which is the inverse of compressibility, is an important material parameter. Various forms of X-ray diffraction with the sample being under pressure can be used to determine the lattice parameters. Once the

1.2 Gallium Nitride

lattice parameters are determined as a function of pressure, the pressure dependence of the unit cell volume can be obtained and fitted with an equation of state (EOS), such as the Murnaghan’s EOS [159], and based on the assumption that the bulk modulus has a linear dependence on the pressure: 0 !  1=B BP 1þ ; B0 0

V ¼ V0

ð1:10Þ

where B0 and V0 represent the bulk modulus and the unit volume at ambient pressure, respectively, and B0 the derivative of B0 versus pressure. X-ray diffraction leads to the determination of the isothermal bulk modulus, whereas the Brillouin scattering leads to the adiabatic one. Nevertheless, in solids other than molecular solids, there is no measurable difference between the two thermodynamic quantities [160]. The bulk modulus (B) of Wz GaN has been calculated from first principles [161] and the first-principle orthogonalized linear combination of atomic orbitals (LCAO) method [158], leading to the values of 195 and 203 GPa, respectively. Another estimate for B is 190 GPa [158]. These figures compare well with the value of 194.6 GPa estimated from the elastic stiffness coefficient [79] and a measured value for 245 GPa [6]. The bulk modulus is related to the elastic constants through B¼

ðC 11 þ C12 ÞC33  2C 213 C11 þ C12 þ 2C33  4C13

ð1:11Þ

and the range of bulk modulus values so determined is from about 173 to 245 GPa [160]. Using the room-temperature elastic constants of single-crystal GaN calculated by Polian et al. [38] yields an adiabatic bulk modulus, both Voigt and Reuss averages, of 210 GPa [91]. Earlier experimental investigations of the elastic constants of Wz GaN were carried out by Savastenko and Sheleg [162] using X-ray diffraction in powdered GaN crystals. The estimates of the Poisson’s ratio from the early elastic coefficients (n ¼ C13/C11 þ C12) [162] and its measured [26] values of 0.372 (for nh0 0 0 1i ¼ (Da/arelax)/(Dc/crelax)) and 0.378 (for nh0 0 0 1i ¼ (Da/a0)/(Dc/c0)), respectively, are in good agreement (to avoid confusion the R value is defined as R ¼ 2C31/C33). The experiments were performed on GaN layers on sapphire substrates because of X-ray diffraction. However, the results obtained later point to a Poisson’s ratio of more near 0.2 as tabulated in Table 1.6 and depend on crystalline direction. The Poisson’s ratio for the ZB case can be calculated from the elastic coefficients for that polytype as n or s0 ¼ (C12/C11 þ C12) leading to values of about 0.352 as tabulated in Table 1.8. The Poisson’s ratio varies along different crystalline directions as tabulated in Table 1.13 for AlN. It should be noted that there is still some spread in the reported values of elastic stiffness coefficients, as discussed in detail in the polarization sections of Section 2.12. More importantly, Kisielowski et al. [39] pointed out that expression ðn  1Þh0 0 0 1i ¼ ðDa=arelaz Þ=ðDc=c relax Þ;

ð1:12Þ

j37

j 1 General Properties of Nitrides

38

where Da ¼ ameas  arelax and Dc ¼ cmeas  crelax, should be used to calculated the Poisson’s ratio, n. Doing so leads to a Poisson’s coefficient of nGaN ¼ 0.2–0.3. Chetverikova et al. [163] measured the Young’s modulus and Poisson’s ratio of their GaN films. From the elastic stiffness coefficients, Young’s modulus Eh0001i is estimated to be 150 GPa [157,162]. Sherwin and Drummond [164] predicted the elastic properties of ZB GaN on grounds of values for those Wz GaN samples reported by Savastenko and Sheleg [162]. The elastic stiffness coefficients and the bulk modulus are compiled in Table 1.24. Considering the wide spread in the reported data more commonly used figures are also shown. Wagner and Bechstedt [178] calculated the elastic coefficients of Wz GaN using a pseudopotential plane wave method and pointed out the discrepancies among the results from different calculations and measurements tabulated in Table 1.24. It is argued that reliable values produce 2C13/C33 ¼ 0.50–0.56 and n ¼ 0.20–0.21 [178]. The agreement between ab initio calculations [42,178] and some measure-

Table 1.24 Experimental and calculated elastic coefficients (Cii), bulk modulus (B) and its pressure derivative (dB/dP), and Young’s modulus (E or Y0) and (in GPa) of Wz GaN and ZB GaN (in part from Ref. [160]).

Technique

C11

C12

C13

C33

C44

B0

X-ray [162] XAS EDX ADX Brillouin [38] Brillouin Brillouin [166] Brillouin [28] Brillouin [169] Brillouin [170] Ultrasonic [171] Ultrasonic [165] Single crystal X-ray Most commonly used values PWPP [42] FP-LMTO [165] Kim [168] PWPP (Wagner) ZB GaN

296

130

158

267

24.1

195 245 188 237 210 180 204 192 175 192 173 208 207

390 374 365 373 315 373 377 370

145 106 135 141 118 141 160 145

380

110

367 396 431 C11 253–264

135 144 109 515–C11 153–165

106 70 114 80.4 96 80 114 110

398 379 381 387 324 387 209 390

105 101 109 94 88 94 81.4 90

B0

E 150

4 3.2 4.3 356 329 362 281 362 161 343

105 103 100 64 104

405 392 476 414

95 91 126 60–68

202 207 201 207 200–237

363 355 461 373 3.9–4.3

The room-temperature elastic constants of single-crystal GaN have been determined by Polian et al. [38] yielding an adiabatic bulk modulus, both Voigt and Reuss, averages, of 210 GPa. The term B0 ¼ dB/dP represents the derivative of B0 versus pressure. EDX: energy dispersive X-ray; ADX: angular dispersive X-ray diffraction; XAS: X-ray absorption spectroscopy; PWPP: plane wave pseudopotential; FP-LMTO: full-potential linear muffin–tin orbital.

1.2 Gallium Nitride

ments [38,165,166] is satisfactory. However, several calculations [167,168] and measurements [28,169–171] suffer from deviations in one or more of the values of elastic constants. The results from Savastenko and Sheleg [162] show excessive deviation for all the elastic constants and, therefore, should be avoided completely. The results from surface acoustic wave measurements of Deger et al. [165] on epitaxial epilayers have been corrected for piezoelectric stiffening and, therefore, are among the most reliable. The vibrational properties of nitrides can best be described within the realm of mechanical properties. These vibrations actually serve to polarize the unit cell [172]. Phonons can be discussed under mechanical and optical properties. Here an arbitrary decision has been made to lump them with the mechanical properties of the crystal. Using GaN as the default, a succinct discussion of vibrational modes, some of which are active Raman modes, some are active in infrared (IR) measurements, and some are optically inactive called the silent modes, is provided [173]. Vibrational modes, which go to the heart of the mechanical properties, are very sensitive to crystalline defects, strain, and dopant in that the phonon mode frequencies and their frequency broadening can be used to glean very crucial information about the semiconductor. The method can also be applied to heterostructures and strained systems. Electronic Raman measurements can be performed to study processes such as electron–phonon interaction in the CW or time-resolved schemes. Time-resolved Raman measurements as applied to hot electron and phonon processes under high electric fields have important implication regarding carrier velocities. A case in point regarding GaN is treated in this context in Volume 3, Chapter 3. The wurtzite crystal structure has the C 46v symmetry and the group theory predicts the existence of the zone center optical modes A1, 2B1, E1, and 2E2. In a more simplified manner, one can consider that the stacking order of the Wz polytype is AaBb while that for the ZB variety is AaBbCc. In addition, the unit cell length of the cubic structure along [1 1 1] is equal to the width of one unit bilayer, whereas that for the hexagonal structure along [0 0 0 1] is twice that amount. Consequently, the phonon dispersion of the hexagonal structure along [0 0 0 1] (G ! A in the Brillouin zone) is approximated by folding the phonon dispersion for the ZB structure along the [1 1 1] (G ! L) direction [174], as shown in Figure 1.12. Doing so reduces the TO phonon mode at the L point of the Brillouin zone in the zinc blende structure to the E2 mode at the G point of the Brillouin zone in the hexagonal structure. This vibrational mode is denoted as EH 2 with superscript H depicting the higher frequency branch of the E2 phonon mode. As indicated in the figure there is another E2 mode at a lower frequency labeled as EL2 . This has its genesis in zone folding of the transverse acoustic (TA) mode in the zinc blende structure. It should be noted that in the hexagonal structure there is anisotropy in the macroscopic electric field induced by polar phonons. As a result, both the TO and LO modes split into the axial (or A1) and planar (or E1) modes where atomic displacement occurs along the c-axis or perpendicular to the c-axis, respectively. This splitting is not shown in Figure 1.12 as it is very small, just a few meV, near zone center; phonon dispersion curves for GaN including the splitting of the A1 and E1 modes can be found in Volume 3, Figure 3.84.

j39

j 1 General Properties of Nitrides

40

Zinc blende

Wurtzitic A1

LO

H

B1

H

TO

E1 A

[0001]

Phonon frequency

E2

Γ

L

B1

LA

L

E2

TA

Γ

A

[111]

L

Wave vector Figure 1.12 Schematic depiction of the phonon dispersion curves for ZB and Wz structures. Also shown are the G and A points of the zone in relation to the real space hexagonal structure. Phonon branches along the [1 1 1] direction in the ZB structure are folded to approximate those

of the wurtzite structure along the [0 0 0 1] direction, because the unit cell length of the cubic structure along the [1 1 1] direction is equal to the width of one unit bilayer, while that for the hexagonal structure along the [0 0 0 1] directions is twice that amount. Patterned after Ref. [174].

As discussed below, in the context of hexagonal structures, group theory predicts eight sets of phonon normal modes at the G point, namely 2A1 þ 2E1 þ 2B1 þ 2E2. Among them, one set of A1 and E1 modes are acoustic, while the remaining six modes, namely A1 þ E1 þ 2B1 þ 2E2, are optical modes. As shown in Figure 1.12, one A1 and one B1 mode (BH 1 ) derive from a singly degenerate LO phonon branch of the zinc blende system by zone folding, whereas one E1 and one E2 mode (EH 2 ) derive from a doubly degenerate TO mode in the cubic system. The first-order phonon Raman scattering is due to phonons near the G point zone center, that is, with wave vector k 0, because of the momentum conservation rule in the light scattering process. Raman measurements typically are employed to probe the vibrational properties of semiconductors. When performed along the direction perpendicular to the c-axis or the (0 0 0 1) plane, the nomenclature used to describe this configuration is depicted as ZðXY; XYÞZ. Here, following Porto’s notation [175] A(B, C)D is used to describe the Raman geometry and polarization, where A and D

1.2 Gallium Nitride

represent the wave vector direction of the incoming and scattered light, respectively, whereas B and C represent the polarization of the incoming and scattered light. In Raman scattering, all the above-mentioned modes, with the exception of B1 modes, are optically active. Because of their polar nature, the A1 and E1 modes split into longitudinal optical (A1-LO and E1-LO) meaning beating along the c-axis, and transverse optical (A1-TO and E1-TO), meaning beating along the basal plane. To reiterate, the A1 and B1 modes give atomic displacements along the c-axis, while the others, E1 and E2, give atomic displacements perpendicular to the c-axis, meaning on the basal plane. Here, the A1 and E1 modes are both Raman and IR active whereas the two E2 modes are only Raman active and the two B1 modes are neither Raman nor IR active, meaning silent modes. In the ZðX Y; X YÞZ configuration, only the E12 (or EL2 or E2 low), E22 (or EH 2 or E2 high), and A1(LO) modes should be observable. In particular, in ZðX ; XÞZ and ZðY; YÞZ geometries, all three modes are observable, while in ZðX; YÞZ or ZðY; X ÞZ geometries only E2 modes are detected [175]. The details of the mode–Raman configuration relationship are tabulated in Table 1.25. Shown in Figure 1.13 are the modes in the Raman backscattered geometries in relation to hexagonal crystalline orientation that can be used to sense the various phonon modes indicated. The acoustic modes, which are simple translational modes, and the optical modes for wurtzite symmetry are shown in Figure 1.14. The calculated phonon dispersion curves [57] for GaN are shown in Figure 1.15. There is another way to describe the number of vibrational modes in zinc blende and wurtzitic structures, which is again based on symmetry arguments. In the wurtzite case [66], the number of atoms per unit cell s ¼ 4, and there are total of 12 modes, the details of which are tabulated in Table 1.26. This table also holds for the zinc blende polytypes with s ¼ 2. This implies a total of six modes in zinc blende as opposed to 12 in wurtzite, three of which are acoustical (1 LA and 2 TA) and the other three are optical (1 LO and 2 TO) branches. These phonon modes for a wurtzite symmetry, specifically the values for wurtzite GaN, are listed in Table 1.27 obtained from Refs [56,157,176,177] along with those obtained from first-principles pseudopotential calculations [161,178]. Also listed are TO and LO optical phonon wave numbers of ZB GaN [25,179].

Table 1.25 Raman measurement configuration needed to observe

the phonon modes in hexagonal nitrides. Mode

Configuration

A1(TO), E2 A1(TO) E1(TO) E1(TO), E1(LO) E2 E2 A1(LO), E2

X ðY; YÞX X ðZ; ZÞX X ðZ; YÞX X ðY; ZÞY X ðY; YÞZ ZðY; X ÞZ ZðY; YÞZ

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j 1 General Properties of Nitrides

42

Z(X,X)Z + Z(X,Y)Z L E2

H E2

A1(LO)

X(Z,Z)X

A1 (TO)

X(Z,Y)X

A1 (TO)

Quasi-E1 (LO) H

E2

Y Y

Z

X GaN Substrate

GaN Z

Substrate

c-axis

X

c-axis Figure 1.13 Schematic representation of two Raman configurations with incident and scattered light directions in the backscattering geometry for ZðX ; X ÞZ þ ZðX; YÞZ configuration to sense EL2 , EH 2 , and A1(LO) modes, X ðZ; ZÞX configuration to sense A1(TO) and quasi-E1(LO) modes, and X ðZ; YÞX configuration to sense A1(TO) and EH 2 modes.

Owing to the presence of elastic strain, thin epilayers have different phonon energies compared to the bulk samples. In the general case, strain can give rise to a shift and splitting of phonon modes. However, for uniaxial strain along the c-axis or biaxial strain in the c-plane, the crystal retains its hexagonal symmetry resulting in only a shift of the phonon frequencies. The strain effects on GaN optical phonon energies have been studied experimentally [181] and theoretically [178]. Within a perturbative approach, the change in the frequency of a given phonon l under symmetry-conserving stress can be expressed in terms of the two strain components, exx and ezz, representing perpendicular and parallel to the z-axis, respectively, as DOl ¼ 2 alexx þ blezz, where al and bl are the corresponding deformation potential constants. The derivation of strain values from the Raman measurements of phonon frequencies is straightforward, once the phonon deformation potentials are known. Davydov et al. [181] combined high-resolution X-ray diffraction (HRXRD) measurements and Raman spectroscopy results to determine the phonon deformation potentials in GaN epitaxial layers grown on 6H-SiC. The strain components were obtained by comparing the lattice constants of the epitaxial layers with those of the strain-free GaN (a0 ¼ 3.1880 Å, c0 ¼ 5.18 561 Å). The Raman spectrum of a strain-free 300 mm thick GaN layer grown on sapphire was used as a reference. Also, using their relation to the hydrostatic pressure coefficients [6] through the bulk elastic coefficients, the sets of phonon deformation potentials were derived for most of zone center optical phonons. As seen in Table 1.27, except for the A1-TO mode, the phonon energies and the deformation potentials reported by Davydov et al. [181] agree well with the pseudopotential plane wave calculations reported by Wagner and Bechstedt [178]. Wagner and Bechstedt argue that the published conversion factors

1.2 Gallium Nitride

A 1 mode

Z

Z

Z

B 1 (1) mode

N

N

N

Ga

Ga

Ga

N

N

N

Ga

Ga

Ga

X

X Z

B 1 (2) mode

Z

E 1 mode

X Z

E 2 (1) mode

N

E 2 (2) mode N

N Ga

Ga

Ga

N

N Ga

N

Ga

Ga

X

(a)

X

X

A mode 1 Z

Z

B1H

Z

L

B1

N

N

N

Ga

Ga

Ga

N

N

N

Ga

Ga

L

Z

Ga

X

X

Z

E2

H

E2

N

N

Ga

Ga

N

Ga

(b)

X

N

X

Figure 1.14 Atomic vibrations in Wz GaN. The larger atom represents Ga while the smaller one is for N. X ¼ (1 0 0), Y ¼ (0 1 0), and Z ¼ (0 0 1) represent the optical polarization directions: (a) for general wave vector and (b) for zone center phonons.

Ga

X

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j 1 General Properties of Nitrides

44

E1(LO)

100

E1 (LO)

GaN

700 B 1

B1 A1(LO)

600 E2

E2 E (TO) 1

80

A1(TO)

A1(TO)

500

60

400 B1

B1

40

300 200

Energy (meV)

Frequency (cm–1)

800

E2

E2

20

100 0

0 Γ

K

M

Γ

A

H

L

A

DOS

Reduced wave vector coordinate Figure 1.15 Calculated phonon dispersion curves and phonon density of states for hexagonal bulk GaN. The solid and dashed lines correspond to the L1(or T1) and L2 (or T2) irreducible representations (following Ref. [180]), respectively. Note how close the E1(LO) and A1(LO) modes are, making high-quality samples with sharp modes imperative for their experimental delineation [169].

Table 1.26 Acoustic and optical phonon modes in a crystal with wurtzite symmetry such as GaN, AlN, and InN, where s represents the number of atoms in the basis.

Mode type

Number of modes

LA TA Total acoustic modes LO TO All optical modes All modes

1 2 3 s1 2s  2 3s  3 3s

The s parameter for wurtzite symmetry is 4. This table is also applicable to the zinc blende case but with s ¼ 2.

Wz [157,177]

556–559 533–534 741–741 710–735 143–146 560–579

Phonon mode

E1-TO A1-TO E1-LO A1-LO EL2 EH 2 BL1 BH 1

(a)

558.5 532.5 745.0 737.0

Wz template [177] 558.8 531.8 741 734 144 567.6

[169]

Wz relaxed

568 540 757 748 142 576 337 713

Wz unstrained (calculated) [178]

bk 680  50 1290  80

80  35 920  60

ak 820  25 630  40

115  25 850  25

Raman [181]

717 640 775 664 75 742 334 661

ak

Calculated [178]

Deformation potentials – Wz

by Akasaki and Amano [157] (Wz) and Huang et al. [177] (Wz template). Also shown are the calculated values. (b) Measured phonon wave numbers (in units of cm1) for wurtzitic GaN. (c) Zinc blende phase phonon wave numbers for zinc blende GaN [179] (theory [59]).

Table 1.27 (a) Zone center optical phonon wave numbers (in units of cm  1) of GaN obtained from Raman scattering at 300 K compiled

591 695 703 695 4 715 275 941

bk

1.2 Gallium Nitride

j45

GaN on sapphire, about 50–70 mm thick, at 300 K.

LO(L) (cm1) TO(L) (cm1) LA(L) (cm1) TA(L) (cm1) TO (cm1) LO (cm1)

LO(G) (cm1) TO(G) (cm1) LO(X) (cm1) TO(X) (cm1) LA(X) (cm1) TA(X) (cm1) 748 [59] 562 [59] 639 [59] 558 [59] 286 [59] 207 [59]

Mode

Mode

(c)

E1-TO A1-TO E1-LO A1-LO EL2 EH 2 BL1 BH 1

Phonon mode

(b)

Table 1.27 (Continued)

675 [59] 554 [59] 296 [59] 144 [59] 558 [179] 730 [179]

558.8 531.8 741 741 144 567.6

560 [58] 750 [58]

Wz unstrained (measured) (collected in Refs [174,178], but based on Refs [6,169])

46

j 1 General Properties of Nitrides

1.2 Gallium Nitride

between the luminescence or Raman shifts and the corresponding biaxial stress are seldom directly measured data. They are either obtained using elastic constants or are constructed from deformation potentials, which have been obtained by means of additional hydrostatic pressure coefficients. Owing to these varying procedures and different sets of parameters used to extract the conversion coefficients from the raw experimental data, discrepancies in the experimental reports of deformation potentials are present. 1.2.3 Thermal Properties of GaN

The lattice parameter of semiconductors depends on temperature and is quantified by thermal expansion coefficient (TEC), which is defined as Da/a or aa and Dc/c or ac, for in-plane and out-of-plane configurations, respectively. It depends on stoichiometry, extended defects, and free-carrier concentration. As in the case of the lattice parameter, a scatter exists in TEC particularly for nitrides as they are grown on foreign substrates with different thermal and mechanical properties. Measurements made over the temperature range of 300–900 K indicate the mean coefficient of thermal expansion of GaN in the c-plane to be Da/a ¼ aa ¼ 5.59 · 106 K1. Similarly, measurements over the temperature ranges of 300–700 and 700–900 K, respectively, indicate the mean coefficient of thermal expansion in the c-direction to be Dc/c ¼ ac ¼ 3.17 · 106 and 7.75 · 106 K1, respectively [140]. Sheleg and Savastenko [49] reported a TEC near 600 K for perpendicular and parallel to the c-axis of 4.52  0.5 · 106 and 5.25  0.05 · 106 K1, respectively. Leszczynski and Walker [182] reported aa values of 3.1 and 6.2 106 K1 for the temperature ranges of 300–350 and 700–750 K, respectively. The ac values in the same temperature ranges, in order, were 2.8 and 6.1 106 K1. In a similar vein, GaN and other allied group III nitride semiconductors are grown at high temperatures and also subjected to increased junction temperatures during operation of devices such as amplifiers and light emitting devices. As such, the structures are subjected to thermal variations as well. In this context, it is imperative to have knowledge of the thermal expansion coefficients, which are termed as TEC. Assuming that these figure remain the same with temperature, the linear expansion coefficients for the a and c parameters are tabulated in Tables 1.10 and 1.28 for heteroepitaxial GaN. However, it is instructive to know the temperature dependence of these parameters, which is shown in Figure 1.16. Being grown on various substrates with different thermal expansion coefficients leads to different dependencies of the lattice parameter on temperature. Temperature dependence of GaN lattice parameter has been measured for a bulk crystal (grown at high pressure) with a high free-electron concentration (5 · 1019 cm3), a slightly strained homoepitaxial layer with a low free-electron concentration (about 1017 cm3), and a heteroepitaxial layer (also with a small electron concentration) on sapphire [88]. The results of such study are tabulated in Table 1.28. It can be seen that the bulk sample with a high free-electron concentration exhibits a thermal expansion that is about 3% higher as compared to the homoepitaxial layer.

j47

j 1 General Properties of Nitrides

48

7

Linear expansion coefficent, a (10–6 K–1)

GaN 6

a

5

a

4

3

2 200

100

300

400

500

600

700

800

Temperature (K) Figure 1.16 Wz GaN coefficient of linear thermal expansion versus temperature for basal plane (a||), a parameter, and out of the basal plane (a?), c parameter, directions [49].

As for the case of the heteroepitaxial layer on sapphire, the thermal expansion of the substrate affects the dependence of the lattice parameter on temperature. Various spectroscopic techniques, such as Auger electron spectroscopy, X-ray photoemission spectroscopy (XPS), and electron energy loss spectroscopy (EELS) have been very useful for the study of surface chemistry of GaN. Building on earlier investigations of the thermal stability of GaN by Johnson et al. [139] and employing Table 1.28 Lattice parameters for GaN samples at various temperatures (lattice parameters c were measured with accuracy of 0.0002 Å, lattice parameter a with accuracy of 0.0005 Å) [88].

T (K)

20 77 295 500 770

GaN bulk, n ¼ 5 · 1019 cm3

Homoepitaxial GaN on conductive GaN substrate

GaN on sapphire

c (Å)

a (Å)

c (Å)

a (Å)

c (Å)

a (Å)

5.1836 5.1838 5.1860 5.1885 5.1962

3.1867 3.1868 3.1881 3.1903 3.1945

5.1822 5.1824 5.1844 5.1870 5.1944

3.1867 3.1868 3.1881 3.1903 3.1945

5.1846 5.1865 5.1888 5.1952

3.1842 3.1859 3.1886 3.1941

1.2 Gallium Nitride

the aforementioned techniques, the thermal stability and dissociation of GaN have been examined further. As indicated earlier, the materials characteristics depend, to a large extent, on defects and impurities, which in turn depend somewhat on growth conditions. Because of this the materials, obtained from various sources, studied in various laboratories exhibit different characteristics. This led to inconsistent results from different laboratories. While some experimental studies on GaN stability conducted at high temperatures suggested that significant weight losses occur at temperatures as low as 750
C, others contradicted this proposition and suggested that no significant weight loss should occur even at a temperature of 1000
C. Sime and Margrave [183] followed the investigation by Johnson et al. [139] by studying the evaporation of GaN and Ga metal in the temperature range 900–1150
C under atmospheric pressure in N2, NH3, and H2 environments with an emphasis on the formation and decomposition equilibrium. The heat of evaporation was determined and the existence of (GaN)x polymers in the gas phase was suggested. Morimoto [184] and Furtado and Jacob [185] observed that GaN is less stable in an HCl or H2 atmosphere than in N2. Some controversy exists regarding the process steps that dictate the decomposition of GaN. Using mass spectroscopy, Gordienko et al. [186] noted that (GaN)2 dimers are the primary components of decomposition. Others [187,188] found only N2 þ and Ga þ to be the primary components in the vapor over GaN. On the basis of measurements of the apparent vapor pressure, Munir and Searcy [189] calculated the heat of sublimation of GaN to be 72.4  0.5 kcal mol  1. Thurmond and Logan [190] determined the equilibrium N2 pressure of GaN as a function of temperature by measuring the partial pressure ratios existing in a (H2CNH3) gas mixture streaming over Ga and GaN. Thermal stability of GaN was taken up later by Karpinski et al. [191] with a detailed investigation of the problem at high temperatures and under pressure up to 60 kbar by employing a tungsten carbide anvil cell activated by a gas pressure technique. The bond strength in gallium nitride is high with bonding energy of 9.12 eV/molecule [192], particularly as compared to the more conventional semiconductors such as GaAs, which has a bonding energy of 6.5 eV/atom pair. As a result, the free energy of GaN is very low in relation to the reference states of the free N and Ga atoms. However, the N2 molecule is also strongly bonded with 4.9 eV/atom. Therefore, the free energies of the constituents of GaN (Ga and N2) at their normal states are close to that of the GaN crystal as illustrated in Figure 1.17, where the free energy of GaN (1 mol) and the free energy of the sum of its constituents (Ga þ 1/2N2) are shown as a function of temperature and N2 pressure. As the temperature increases, the Gibbs free energy, G(T), of the constituents decreases faster than G (T) of the GaN crystal. More importantly, GaN becomes thermodynamically unstable at high temperatures. The crossing of G(T) curves determines the equilibrium temperature where GaN coexists with its constituents at a given N2 pressure. The application of pressure increases the free energy of the constituents more than G(T) of the GaN crystal, which causes the equilibrium point to shift to higher temperatures, increasing the range of GaN stability. The data on phase diagrams of GaN are limited and contradictory by reason of high melting temperatures (Tm) and high nitrogen dissociation pressures (Pdis N2 ). Dissociation

j49

j 1 General Properties of Nitrides

50

–600 Ga(s) + (1/2)N2(g)

p = 1 bar

Ga(l)+(1/2)N2(g)

p = 1 kbar

G(kJ mol–1)

p = 10 kbar –800 p = 20 kbar GaN(s)

–1000

–1200 0

400

800

1200

1600

2000

T(K) Figure 1.17 Gibbs free energy of GaN and its constituents as a function of temperature and pressure [192].

pressure of MN, where M stands for Al, Ga, and In, and N for nitrogen, is defined as the nitrogen pressure at the thermal equilibrium of the reaction [193]: MN(s) ¼ M(l) þ 1/2N2(g), where s, l, and g stand for solid, liquid, and gas states, respectively. Reported values for Pdis N2 for GaN [193] show large discrepancies [191,194]. Specifically, in the high-pressure range, the partial pressure, p, versus the inverse temperature, 1/T, curve of Karpinski et al. deviates markedly from the linear dependence proposed by Thurmond and Logan as shown in Figure 1.18. Despite the discrepancies, there is a good agreement in the Gibbs free energy with DG0 32.43T  3.77 · 104  700 cal mol1 for GaN synthesis between the two references. The value of enthalpy DH0 (37.7 kcal mol1) is in good agreement as well with that estimated by Madar et al. [194]. The stars in Figure 1.18 indicate the melting point of AlN at T M AlN ¼ 3487 K, GaN M at T M GaN ¼ 2791 K, and InN at T InN ¼ 2146 K. The GaN and InN melting points so indicated may underestimate the real values, as perhaps a sufficient overpressure was not maintained. Line fits correspond to 8.3  109 exp(  5.41 eV/kT), 1.5 · 1014 exp(3.28 eV/kT), and 7.9 · 1017 exp(2.78 eV/kT) bar for AlN, GaN, and InN, respectively. The data over the larger temperature range are those compiled by Ambacher [196]. The results of Madar et al. [194], Thurmond and Logan [190], and Karpinski et al. [191] are also shown in a limited temperature range. For GaN (see Figure 1.18), the nitrogen dissociation pressure equals 1 atm at approximately 850
C and 10 atm at 930
C. At 1250
C, GaN decomposed even under pressure of 10 000 bar of N2. The turning over of the partial pressure for GaN and InN at temperatures approaching the melting point may need to be reexamined. What is clear, however, is that GaN and particularly InN have very high partial pressures that make it imperative to maintain high fluxes of N during growth. It should, therefore, come as no surprise that the incorporation of nitrogen is not a trivial problem at high temperatures. For the pressures below equilibrium at a given temperature, the thermal dissociation occurs at a slow and apparently constant rate suggesting a diffusion-controlled

1.2 Gallium Nitride

Temperature (oC) 105

4700

1400

440

700

104 u Th

103

rm

InN

d an Lo

2

n ga

101

Ma

dis

d

PN (atm)

on

102

dar et a

AlN

l. Ka

100

n rpi et ski

GaN

al.

10–1 10–2 10–3

2

4

6

10 8 104 /T ( K–1)

Figure 1.18 Equilibrium N2 pressure over the MN(s) þ M(l) systems corresponding to GaN (M ¼ Al, Ga, In) reported and/or compiled by Slack and McNelly[195], Madar et al. [194], Thurmond and Logan [190], Karpinski et al. [191], and Ambacher [196]. The melting points of the three binaries are indicated by stars. The desorption activation energies, EMN, determined by straight line fits to the data points are 3.5 eV

12

14

(336 kJ mol1), 3.9 eV (379 kJ mol1), and 4.3 eV (414 kJ mol1) for InN, GaN, and AlN, respectively. Caution should be exercised as there is significant deviation from the activation line for GaN and InN. This may simply be a matter of not being able to maintain sufficient pressure on GaN and InN at very high temperatures to reach the real melting point.

process of dissociation. Expanded equilibrium vapor pressure data inclusive of GaAs and InP in addition to the three nitride binaries reported by Matsuoka [197] are shown in Figure 1.19. Melting points and other thermodynamic characteristics of III-N compounds are given in Tables 1.10, 1.16, and 1.20 as compiled by Popovici and Morko
c [198] as well as those collected from various sources as indicated in the pertinent tables. Investigations utilizing epitaxial thin films of GaN, as well as AlN and InN, have been conducted by Ambacher et al. [199], who heated the samples in vacuum and recorded the partial pressure of relevant gases with a quadrupole mass spectrometer. Desorption spectra were then analyzed [200] to find the binding energies of various desorbed species as well as the thermal stability of the sample for a given thermal treatment. As expected, the nitrogen partial pressure increases exponentially above TE ¼ 850
C for GaN underscoring the point that the decomposition temperature in vacuum is much lower than the melting point shown in Figure 1.18. The rate of

j51

j 1 General Properties of Nitrides

52

Temperature ( oC)

Equilibrium vapor pressure (atm)

2000

1000

500

100 InN 10–2

10–4

GaN

AlN GaAs

10–6

InP 10–8 0.0

0.5

1.0

1.5

2.0

1000/T (K–1) Figure 1.19 Equilibrium vapor pressure of N2 over AlN, GaN and InN, the sum of As2 and As4 over GaAs, and the sum of P2 and P4 over InP [197].

nitrogen evolution F(N) was set equal to the rate of decomposition, and the slope of ln [F(N)] versus 1/T gives the effective activation energy of the decomposition in vacuum as compared to those shown in Figure 1.18. The decomposition rate equals the desorption of one monolayer every second (FN ¼ 1.5 · 1015 cm2 s1) at 970
C, and the activation energy of the thermally induced decomposition is determined to be EMN ¼ 3.9 eV (379 kJ mol1) for GaN. Despite some disagreement, as mentioned above, investigations of the equilibrium nitrogen overpressure versus temperature, PN2 –T, for GaN [92,190,201], including a very complete and consistent set of data obtained by Karpinski et al. [191,202] have set the stage for bulk template growth as well as setting benchmarks for growth of GaN by nonequilibrium methods. Those authors employed direct synthesis and decomposition experiments and used the gas pressure technique (for pressures up to 20 kbar) and the high-pressure anvil method beyond the reach of gas pressure technique (up to 70 kbar). The results of these experiments are shown in Figure 1.20 [203,204]. The message in the form of N2 partial pressure is that one must stay below the decomposition curve. This means that the selection of GaN synthesis temperature directly depends on the pressure that the vessel can provide. For example, if a pressure of 20 kbar is all that is available, then the temperature should be kept below about 1660
C. For a review of the stability of GaN as well as the growth GaN templates, the reader is referred to Ref. [192]. As alluded to earlier, nitride semiconductors in general and GaN in particular are considered for high-power/high-temperature electronic and optoelectronic devices

1.2 Gallium Nitride

Temperature (K) 3

10

2 × 103 1.5 × 103 1.25 × 10

5

1×10 3

Pressure limit of 20 kbar

Pressure (bar)

104

103 Ga+1/2N 2

GaN

102

101

100

4

6

8

10

12

1/T (10 –4 K –1) Figure 1.20 N2 partial pressure as a function of temperature GaN. Ref. [192], originally in Refs [203,204].

where thermal dissipation is a key issue. Device applications assure that the thermal conductivity (k) is an extremely important material property. Thermal conductivity is a kinetic property determined by contributions from the vibrational, rotational, and electronic degrees of freedom, and as such it is related to the mechanical properties of the material. However, for convenience, this property is generally categorized under the thermal properties of nitrides in this book. The electronic thermal conductivity contribution is negligible for carrier concentrations 1019 cm3. The heat transport is predominantly determined by phonon–phonon Umklapp scattering, and phonon scattering by point and extended defects such as vacancies (inclusive of the lattice distortions caused by them), impurities, and isotope fluctuations (mass fluctuation) as elaborated on by Slack et al. [91]. For pure crystals, phonon–phonon scattering, which is ideally proportional to T1 above the Debye temperature, is the limiting process. The lattice contribution (phonon scattering) to the thermal conductivity, k, in a pure solid is obtained from the kinetic theory as [205] 1 k lattice ðTÞ ¼ vs C lattice ðTÞLðTÞ; 3

ð1:13Þ

where T is the temperature, vs is the velocity of sound (nearly independent of temperature), Clattice(T) is the lattice specific heat, and L(T) is the phonon mean free length. In nearly all materials, the thermal conductivity, k(T), first increases with temperature, reaches a maximum (kmax) at some characteristic temperature Tch, and

j53

j 1 General Properties of Nitrides

54

then decreases. At low temperatures, L is relatively long and is dominated by extrinsic effects such as “defects” and/or finite crystal size and Clattice(T)  (T/yD)3, where yD is the Debye temperature. As the temperature increases, Clattice(T) begins to saturate and the intrinsic temperature-dependent Umklapp processes become dominant, leading to a decrease in L. The other contribution, the electronic contribution, to the thermal conductivity is negligible for carrier concentrations 1019 cm3. It can be expressed as [206] k electr ðTÞ ¼

p2 nk2B Ttelectr ; 3mc

ð1:14Þ

where n is the carrier density, kB is the Boltzmann constant, telectr is the scattering time of the electrons, and mc is the conduction band effective mass. The first measurements of k of GaN were by Sichel and Pankove [207] on “bulk” GaN (400 mm of material grown by HVPE) as a function of temperature (25–360 K): k ffi 1.3 W cm1 K1 (along the c-axis) at 300 K. This room-temperature value measured is a little smaller than the value of 1.7 W cm1 K1 predicted in 1973 [77] and much smaller than the k  4.10 W cm1 K1 calculated by Witek [208]. Using the elastic constants reported by Polian et al. [38], Slack et al. [91] calculated a Debye temperature of 650 K, which led to a more recent thermal conductivity at 300 K for GaN of k ¼ 2.27 W cm1 K1, assuming that there is no isotope scattering in GaN. This is very close to the measured values in high-quality freestanding GaN samples, the electrical properties of which are discussed in Volume 2, Chapter 3 and optical properties of which are discussed in Volume 2, Chapter 5. Using a steady-state four-probe method and a high-quality freestanding GaN template, Slack et al. [91] measured a value for k of 2.3 W cm1 K1 at room temperature, which increased to over 10 W cm1 K1 at 77 K. The method holds for four-probe thermal measurement, where the term “four probe” is analogous to the four-probe electrical measurement method. Namely, a heater is attached on the end of the sample – sandwiched in a copper clamp so that the heat flows through the entire width of the sample, not just the surface –supplying a heat current Q (analogous to an electrical current I). Two thermocouple junctions are attached along the length of the specimen by two little copper clamps separated by a distance L, the schematic representation of which is shown in Figure 1.21. The heat current Q creates a temperature gradient of DT across the wafer. The k value is calculated using k ¼ (P/DT)(L/A), where P is the power (¼voltage · current) supplied to the heater and A represents the cross-sectional area of the sample. Although the technique sounds simple, its accuracy depends very critically on making sure that the heat conduction is through the specimen and along the direction in which the temperature gradient is measured. To make certain that heat is transferred in the said direction only, the radiation losses must be minimized as well as making sure that the electrical wires used do not remove heat. To this end, the sample is placed in a turbo pumped vacuum to eliminate conduction and convection through the surrounding medium. Heat losses via conduction through the wires are minimized using long (10 cm), thin (<100 mm) wires of low thermal conductivity, typically chromel/constantan thermocouple and heater wires. Radiation losses are

1.2 Gallium Nitride

j55

Variable temperature copper finger Chromel Thermocouple wires

Thermocouple 2

Constantan Chromel

Δ ΔT

Thermocouple 1

low t f a He Copper Heater resistor

le mp Sa

r te de un

st

Thermocouple for absolute temperature measurement

Figure 1.21 Schematic representation of the four-probe thermoelectric measurement setup used to measure the thermal conductivity of freestanding GaN [91].

minimized by surrounding the sample with a heat shield anchored thermally to the cold tip of the cryostat. By carefully designing this shield, Slack et al. [91] were able to reduce the total heat loss to the order of 1–2 mW K1 at room temperature. Because the radiation losses follow T3 dependence, they die off rather quickly below room temperature. For a sample with a thermal conductivity of 1 W cm1 K1, cross section of 1 · 3 mm2, and thermocouple probe separation of 5 mm, the thermal conductance is about 60 mW K1, so the heat losses are less than 5%. Fortunately, for wide bandgap semiconductors such as GaN the thermal conductivities are high enough so that the heat conduction is mainly through the sample, reducing the measurement error. Just as a reference point, for samples of lower thermal conductance (either lower conductivity or thinner), the heat losses can become important near room temperature. Thus, the samples for lower thermal conductivity materials (e.g., glasses or thermoelectric alloys) usually need to be short with large cross-sectional areas. The temperature-dependent thermal conductivity so measured for freestanding GaN is shown in Figure 1.22. From that temperature dependence and assuming the heat dissipation is through acoustic phonons, a Debye temperature of yD 550 K was deduced, which compares with 650 K reported by Slack et al. [91]. As can be seen in Figure 1.22, the measured thermal conductivity of GaN in the temperature range of 80–300 K has a temperature power dependence of 1.22. This slope is typical of pure adamantine crystals below the Debye temperature indicating acoustic phonon transport where the phonon–phonon scattering is a combination of

j 1 General Properties of Nitrides

56

Thermal conductivity (W cm–1 K –1)

10

2

T –1.22

101 500 μm phonon mean free path 100

Sichel and Pankove

10

–1

1

10

100

1000

Temperature (K) Figure 1.22 The thermal conductivity of 200 mm thick freestanding GaN sample (Samsung) as a function of temperature. The dashed line indicates calculation using the boundary scattering limit for a phonon mean free path of 500 mm. Also shown is the T 1.22dependence between about 80 and 300 K, and earlier results of Sichel and Pankove [207] measured using a 400 mm HVPE sample. Courtesy of Slack and Morelli [91].

acoustic–acoustic and acoustic–optic interactions. This temperature dependence strongly suggests that the thermal conductivity depends mainly on intrinsic phonon–phonon scattering and not on phonon–impurity scattering. Keep in mind that the net electron concentration in the measured film is about 1016 cm3 and the hole concentration is in the 1015 cm3 range. In addition, the dislocation density is low, about 106 cm2. It should be pointed out that the thermal conductivity degrades with increased dislocation density, particularly above 107 cm2, with a slope of 0.4 W cm1 K1 per decade dropping down to slightly above 1 W cm1 K1 for a dislocation density of mid-109 cm2. The dashed curve at low temperatures in Figure 1.22 has been calculated for boundary scattering assuming a mean free path of 500 mm [91]. The mean free path is comparable with the average sample diameter, which indicates that impurity scattering in this region is not dominant either. The thermal conductivity of GaN is a tensor quantity and has two principal values k? and k|| perpendicular and parallel to the c-axis, respectively, that is, in-plane and out-of-plane values. The anisotropy in the sound velocity, which relates to the phonon propagation velocity, has been calculated by Polian et al. [38] in the form of n? ¼ 5.56 · 105 cm s1 and n|| ¼ 5.51 · 105 cm s1 for in-plane and out-of-plane directions, respectively. These values are smaller than the measured values, reported by Deger et al. [165], of n? ¼ 8.02 · 105 cm s1 and n|| ¼ 7.79 · 105 cm s1 for in-plane

1.2 Gallium Nitride

and out-of-plane directions, respectively. The in-plane and out-of-plane sound velocities reported in Ref [36] are tabulated in Tables 1.7 and 1.9 for wurtzitic and zinc blende phases of GaN. Because the difference is negligible and the anharmonicity producing the phonon–phonon scattering is not discernible, one can conclude that in-plane thermal conductivity measurements are a good representative of k in GaN [91]. A newer method, named the scanning thermal microscopy (SThM) [209], has been developed to measure thermal conductivity and is purported to provide nondestructive, absolute measurements with a high spatial/depth resolution of about 2–3 mm. Thermal imaging is achieved using a resistive thermal element incorporated at the end of a cantilever/AFM-type feedback as shown in Figure 1.23. The resistive tip forms one element of a Wheatstone bridge as shown in Figure 1.24. The spatial/depth resolution is estimated to be 2–3 mm for GaN and AlN. Upon contact with the sample, the tip tends to cooldown due to heat conduction into the sample, which is related to its thermal conductivity, k. The bridge circuit applies a compensating voltage (Uout) to maintain its target operating temperature. The feedback signal for constant resistance is a measure of the thermal conductivity of the material with which the tip is in contact, specifically V 2out is proportional to k because power dissipation is the mechanism here. Measurements of the absolute values of k are based on a calibration procedure. This simply comprises calibrating the feedback signal, V 2out , for a constant thermal element resistance against that for samples with known conductivities such as GaSb, GaAs, InP, Si, and Al metal, as shown in Figure 1.25. The influence of the surface roughness on the effective thermal conductivity is of concern. For a perfectly flat surface,

Figure 1.23 An artist’s view of the scanning thermal microscope. Patterned after D.I. Florescu and F.H. Pollak. (Please find a color version of this figure on the color tables.)

j57

j 1 General Properties of Nitrides

58

2 Vout ~ κ

R1

Tip

R2

Vout R probe

Sample

R control

(a)

(b)

Figure 1.24 (a) Wheatstone bridge arrangement in which the tip temperature is kept constant before and after contact with the material whose thermal conductivity is being measured. The feedback signal Uout is related to thermal

conductivity, k. A calibration against known samples such as Si, GaAs, GaP, and so on, leads to absolute values of k. (b) Schematic diagram of heat dissipation into the sample from the tip. Courtesy of D.I. Florescu and F.H. Pollak.

the contact between the probe tip (radius of curvature 1 fm) and the sample surface is very small. For rough surfaces, however, the tip could impinge on a valley- or hillocklike feature with the valley/hillock leading to increased/decreased thermal signal accompanied by a corresponding change in the measured effective thermal conductivity. 0.68 Al

0.66

V 2out

0.64 0.62 0.60 Si 0.58 InP 0.56 GaAs 0.54 0.0

GaSb 0.5

1.0

1.5 2.0 – 1 –1 k (W cm K )

2.5

Figure 1.25 The feedback signal, V 2out, which is a measure of the thermal conductivity of the material under test, for a constant thermal element resistance for samples with known conductivities such as GaSb, GaAs, InP, Si, and Al metal. Courtesy of D.I. Florescu and F.H. Pollak.

1.2 Gallium Nitride

The SThM method has been applied to the measurement of the room-temperature thermal conductivity on both fully and partially coalesced epitaxial lateral overgrown GaN/sapphire (0 0 0 1) samples [209]. As expected, a correlation between low threading dislocation density and high thermal conductivity values was established. The reduction in the thermal conductivity with increased dislocation density is expected as threading dislocations degrade the sound velocity and increase the phonon scattering in the material. In fact, due to the high defect concentrations in early films, the thermal conductivity value measured was 1.3 W cm1 K1 [207]. Using this method, the highest GaN k values, 2.0–2.1 W cm1 K1, were found in the regions of the samples that were laterally grown and thus contained the lowest density of threading dislocations. This compares with a value of 2.3 W cm1 K1 in a freestanding sample measured by the steady-state four-probe method discussed earlier. Even then, it falls short of the predictions by Witek [208]. An explanation for the dramatic increase from to k  1.3 W cm1 K1 for the early samples to 2.3 W cm1 K1for the freestanding sample, as iterated above is most likely related to extended defect concentration (Dd) and the differences in background doping. The effect of dislocation density on the thermal conductivity has been calculated by Kotchetkov et al. [210]. The dislocation density in the thick film measured by Sichel and Pankove was between 109 and 1010 cm2, while the freestanding sample exhibited densities of less than 106 cm2 near the top surface (Ga-polarity) and 107 cm2 near the bottom surface (N-polarity). Kotchetkov et al. showed that k remains fairly independent of Dd up to some characteristic after which it decreases about a factor of 2 for every decade of value Dchar d increase in Dd. The thermal conductivity has also been correlated to doping levels in HVPE n-GaN/sapphire (0 0 0 1) by SThM on two sets of samples [211,212]. In both sets of data, the thermal conductivity decreased linearly with log n, n being the electron concentration, the variation being about a factor of 2 decrease in k for every decade increase in n. Significantly, it was concluded that the decrease in the lattice contribution to k, due to increased phonon scattering from impurities and free electrons, predominates the increase in the electronic contribution. Also, a correlation between the film thickness and the improved thermal conductivity was found, which is consistent with the observed general reduction of both extended (dislocations) and point defects with film thickness [212]. The k values at 300 K before and after plasma-induced effects on a series of n-GaN/ sapphire (0 0 0 1) samples fabricated by HVPE were also measured [213]. The sample thicknesses were 50  5 mm and the carrier concentrations were 8 · 1016 cm3, as determined by Hall effect measurements. The thermal conductivity before treatment was found to be in the 1.70–1.75 W cm1 K1 range, similar to that previously reported for HVPE material with this carrier concentration and thickness [211,212]. The k value was reduced, however, when the samples were processed under constant Ar gas flow and pressure for a fixed period of time (5 min). The only variable processing parameter was the DC bias voltage (125–500 V). After the initial 125 V procedure, k exhibited a linear decrease with the DC voltage in the investigated range. At 125 V, the thermal conductivity was only slightly less (k  1.65 W cm1 K1)

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j 1 General Properties of Nitrides

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than the untreated case. The values of k had dropped to 0.3 W cm1 K1 for the 500 V case. To a first extent, the temperature dependence of the specific heat of Wz GaN (Cp) at constant pressure can be expressed by phenomenological expression [51]. In this vein, the specific heat Cp of Wz GaN at constant pressure for 298 K < T < 1773 K can be expressed as CP ðTÞ ¼ 9:1 þ ð2:14  10  3 TÞðJ mol  1 K  1 Þ; Cp ¼ 38:1 þ 8:96  10  3 T ðcal mol  1 K  1 Þ ð1 cal ¼ 4:186 JÞ:

ð1:15Þ

However, this expression is very simplistic, as will be seen below. As already mentioned, free electrons (very effective at low temperatures), impurities, defects (inclusive of point defects), and lattice vibrations contribute to specific heat. If GaN with negligible free-electron concentration and defects were available, only the lattice contribution would be considered, which is also the case in texts [214]. The specific heat of Wz GaN has been studied by Koshchenko et al. [215] in the temperature range of 5–60 K and also by Demidienko et al. [216] in the temperature range of 55–300 K and discussed by Krukowski et al. [88,127]. The Debye expression for the temperature dependence of specific heat in a solid at a constant pressure (Cp) can be expressed as [214] 

T Cp ¼ 18R qD

3 xðD : 0

x 4 ex ð ex

 1Þ2

dx;

ð1:16Þ

where x D qD =T and R ¼ 8.3144 J mol1 K1 is the molar gas constant. The coefficient in front of the term R has been multiplied by 2 to take into account the two constituents making up the binary GaN. By fitting the measured temperature-dependent heat capacity to the Debye expression, one can obtain the Debye temperature yD specific to heat capacity. The experimental data of Demidienko et al. and Krukowski et al. are plotted in Figure 1.26. Also shown in the figure is the calculated specific heat using the Debye expression for Debye temperatures of 500, 600, 700, and 800 K. It is clear that the quality of the data and/or sample prevents attainment of a good fit between the experimental data and Equation 1.16. Consequently, a Debye temperature with sufficient accuracy cannot be determined. It is easier to extract a Debye temperature using data either near very low temperatures or well below the Debye temperature where the specific heat has a simple cubic dependence of temperature [214]:  3 T Cp ¼ 234R : ð1:17Þ qD Unfortunately, the GaN samples contain large densities of free carriers and defects that compromise the application of the Debye specific heat expression. Consequently, a good fit to the data is not obtained and the Debye temperature so extracted is not as dependable as desired. There is a spread in the reported Debye temperatures for GaN, yD, in the range of about 600 to 700 K. Slack [77] estimated a value of 600 K at 0 K by utilizing the more established Debye temperatures for BeO and AlN. This compares with 550 K deduced

1.2 Gallium Nitride

50

40

q D = 800 K

–1

–1

Molar specific heat, Cp (cal mol K )

q D = 500 K

30

20

Cp data, GaN q D = 500 K q D = 600 K

10

q D = 700 K q D = 800 K

0 0

200

400

600

800

1000

Temperature (K) Figure 1.26 Molar specific heat at constant pressure, Cp (cal mol1 K1), of GaN versus temperature. Open circles represent the experimental data. The solid lines are calculation based on the Debye model for Debye temperatures, yD, of 500, 600, 700, and 800 K. Unfortunately, it is difficult to discern a Debye temperature that is effective over a wide

temperature range because a large concentration of defects and impurities is present in GaN. However, a value of 600 K estimated by Slack is used commonly. The data are taken from Refs [215,216], as compiled in Ref. [88]. (Please find a color version of this figure on the color tables.)

from heat transfer due to acoustic phonons, as mentioned above. Because the samples used in these measurements contained defects and large density of free electrons, the dispersion among the data and Debye expression is attributed to defects at high temperatures and free electrons at low temperatures. Elastic properties of GaN can also be used to deduce the Debye temperature. In this vein, Raman scattering measurements yielded a Debye temperature of yD ¼ 650 K [38]. Calculations since the estimate of Slack [77] yielded a range of 620–690 K [88]. Thermodynamic properties of Wz GaN have been reported by Elwell and Elwell [217]. From the reaction GaðsÞ þ 1=2N2 ðgÞ ¼ GaNðsÞ;

ð1:18Þ

the heat of formation of Wz GaN was calculated to be DH298 K ¼ 26.4 kcal mol1 [217], or as the standard heat of formation DH ¼ 37.7 kcal mol1 [194]. The equilibrium vapor pressure of N2 over solid GaN has been found to be 10 MPa at 1368 K and 1 GPa at 1803 K [202]. A thorough description of the GaN phase diagram including the equilibrium vapor pressure of N2 over GaN as well as AlN and InN has been presented by Porowski and Grzegory [218] (1 cal ¼ 4.186 J).

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1.3 Aluminum Nitride

AlN exhibits many useful mechanical and electronic properties. For example, hardness, high thermal conductivity, resistance to high temperature and caustic chemicals combined with, in noncrystalline form, a reasonable thermal match to Si and GaAs in somewhat relaxed terms, make AlN an attractive material for electronic packaging applications. The wide bandgap is also the reason for AlN to be touted as an insulating material in semiconductor device applications. Piezoelectric properties make AlN suitable for surface acoustic wave device applications [219]. However, the majority of interest in this semiconductor in the context of electronic and optoelectronic device arena stems from its ability to form alloys with GaN producing AlGaN and allowing the fabrication of AlGaN/GaN and AlGaN/ InGaN-based electronic and optical devices, the latter of which is active from the green wavelengths well to the ultraviolet. AlN also forms a crucial component of the nitride-based AlInGaN quaternary, which makes tuning of the bandgap independent of composition to some extent. This way, lattice-matched conditions to the underlying epitaxial structure can be maintained while being able to adjust the bandgap. AlN is not a particularly easy material to investigate because of the high reactivity of aluminum with oxygen in the growth vessel. Early measurements indicated that oxygen contaminated material can lead to errors in the energy bandgap and, depending on the extent of contamination, in the lattice constant. Only recently achieved contamination-free deposition environments coupled with advanced procedures have allowed researchers to consistently grow improved-quality AlN. Consequently, many of the physical properties of AlN have been reliably measured and bulk AlN synthesized. 1.3.1 Mechanical Properties of AlN

When crystallized in the hexagonal wurtzite structure, the AlN crystal has a molar mass of 40.9882 g mol1, restated for convenience. The cubic form is hard to obtain and thus will be ignored. The point group symmetry for the wurtzite structure in the Schoenflies notation is C46v (P63mc in the Hermann–Mauguin notation), restated for convenience. Reported lattice parameters range from 3.110 to 3.113 Å for the a parameter (3.1106 Å for bulk, 3.1130 Å for powder, and 3.110 Å for AlN on SiC), and from 4.978 to 4.982 Å for the c parameter. The c/a ratio thus varies between 1.600 and 1.602. The deviation from that of the ideal wurtzite crystal (c/a ¼ 1.633) is probably because of the lattice stability and ionicity. The u parameter for AlN is 0.3821, which is larger than the calculated value of 0.380 using Equation 1.1. This means that the interatomic distance and angles differ by 0.01 Å and 3
, respectively, from the ideal [16]. Whereas the metastable zinc blende polytype AlN has a value of a ¼ 4.38 Å [220], the rock salt structure has a value of a ¼ 4.043–4.045 Å at room temperature [5,221]. Table 1.13 summarizes some of the observed structural properties of AlN.

1.3 Aluminum Nitride

Early investigations of the elastic properties of AlN were carried out on sintered polycrystalline specimens, owing to the unavailability of large single crystals. This, however, paved the way to more refined measurements as single crystalline AlN became available. The measured bulk modulus B, which is related to elastic stiffness coefficients through Equation 1.11, and Young’s modulus Y0 or E are compiled in Table 1.29 along with the entire set of elastic stiffness coefficients. The latter were obtained by fitting the results of surface acoustic wave measurements made on epitaxial AlN films and by Brillouin scattering measurements made on an AlN single crystal [160]. Wagner and Bechstedt [178] suggested that reliable values for the elastic constants should produce 2C13/C33 ¼ 0.5–0.6 and n ¼ 0.18–0.21. The ab initio calculations [42,178] and some measurements [83,165] provide similar results. However, the values from some of the calculations [165,222] and measurements [169,223] should be used with caution because of the large deviations in one or more coefficients. Surface acoustic wave measurements of Deger et al. [165] are very reliable because they include the correction for piezoelectric stiffening. Bulk modulus values range from 159.9 GPa, measured by an ultrasonic method, to 237 GPa, measured by Brillouin scattering. The range for the same from calculations is 111–239 GPa [160]. Young’s modulus is measured as 374 GPa for single-crystal AlN [82] and 295 GPa for AlN thin films [224]. The hardness of AlN has been measured to be 12 GPa on the basal plane (0 0 0 1) using a Knoop diamond indenter [225]. Some anisotropy in Knoop’s hardness has been observed for the indent direction perpendicular to the c-axis with measured values in

Table 1.29 Experimental bulk modulus and elastic coefficients (in GPa) of AlN (from Ref. [160] and references therein).

Method

C11

C12

C13

C33

C44

B

Ultrasonics [223] Ultrasonic Ultrasonic [165] ADX Brillouin [83] Brillouin [169] EDX Hardness [82] PWPP [42] FP-LMTO [165] HF [222] PWPP [178]

345

125

120

395

118

201 159.9 209 207.9 210.1 237 185 220 207 218 231 210

Zinc blende HF [222]

410

140

100

390

120

410.5 419

148.5 177

98.9 140

388.5 392

124.6 110

396 398 464 C11 FIX THIS 348

137 140 149 538 168

108 127 116 113

373 382 409 370

116 96 128

135

228

B0

E or Y0

5.2

334 308 354

6.3 354 326 5.7 374 329 322 365 322 Y0 ¼ (C11 þ 2C12) · (C11  C12)/ (C11 þ C12)

ADX: angular dispersive X-ray diffraction; EDX: energy dispersive X-ray; PWPP: plane wave pseudopotential; FP-LMTO: full-potential linear muffin–tin orbital; HF: Hartree–Fock.

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j 1 General Properties of Nitrides

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the range of 10–14 GPa [81]. More recent nanoindentation measurements on singlecrystal AlN revealed a hardness of 18 GPa [82]. The phonon structure of AlN has been the subject of numerous investigations. As in the case of GaN, the phonon dispersion spectrum of Wz AlN has 12 branches, 3 acoustic, and 9 optical ones [226]. LO and TO phonon energies have been obtained from fits to infrared reflectivity measurements, the results of which are tabulated in Table 1.30. Raman-active optical phonon modes belong to the A1, E1, and E2 group representations. Several Raman scattering studies on AlN have been conducted, and the measured phonon energies are listed in Table 1.30 along with the calculated values by Wagner and Bechstedt [178]. The shift of phonon energies with strain was studied experimentally by Gleize et al. [227] and theoretically by Wagner and Bechstedt [178]. Gleize et al. [227] investigated strained 500 nm thick AlN layers grown on 6H-SiC. Using the strain values deduced from high-resolution X-ray diffraction and phonon frequency shifts measured by micro-Raman spectroscopy, the deformation potentials were obtained for most of the zone center optical phonons of Wz AlN. The determination of the deformation potentials is based on the knowledge of the ideal equilibrium state of the material from which strains and phonon shifts are defined. Additionally, hydrostatic pressure coefficients and elastic constants of the bulk material are also needed for extracting the deformation potentials from the raw experimental data. The deviations in the published data originate from the fact that different sets of parameters are typically used for this purpose. Table 1.30 lists the phonon deformation potentials from Gleize et al. [227] along with the results from pseudopotential plane wave calculations of Wagner and Bechstedt [178], which produced slightly lower values. The frequencies or energies of vibrational modes are very sensitive to the strain state of the samples. Strain inhomogeneities and imperfections cause the linewidths in Raman observable mode to be broad reducing the accuracy of central frequency determination. Tischler and Freitas Jr [228] utilized freestanding and high-quality AlN oriented along the (0 0 0 1) plane, as characterized by X-ray with full width half maximum (FWHM) of 36–54 arcsec. Listed in Table 1.30 are also the data of Tischler and Freitas Jr, which by virtue of the high quality of sample should be used as standard. Tischler and Freitas Jr [228] also estimated the linewidths of the Raman modes by fitting the data with Lorentzian peaks from which the phonon decay times were deducedby relying on theuncertainty principle in the form ofDE= h ¼ G= h ¼ 1=t, where DE, G, and t represent the error bar for the energy and linewidths, and the phonon decay time, respectively. The phonon decay times so deduced are in the range of 0.7 ps for E1(LO) to 5.3 ps for E12. To get a flavor of the range of experiments regarding the mechanical properties, including vibrational phonon properties, of epitaxial AlN, a mention of the investigations on AlN deposited on silicon and sapphire substrates at 325 K by IBAD undertaken by Ribeiro et al. [230] is made. Raman scattering measurements revealed interesting vibrational features related to the atomic composition and structure of the films. Features related to crystalline (c-) Si and corresponding to 2TA(L) at 230 cm1, 2TA(X) at 304 cm1, 2TA(S) at 435 cm1, TO(G) at 520 cm1, TA(S) þ TO (S) at 615 cm1, accidental critical points at 670 and 825 cm1, 2TO(D) at

E1-TO A1-TO A1-LO E1-LO E12 ðE2 lowÞ E22 ðE2 highÞ

Mode

82.8 75.4 75.4 112.8 30.5 81.2

(meV) 667.2 608.5 888.9 909.6 246.1 655.1

(cm  1)

Wz (Raman) unstrained [228]

895–921

667–673 614–667

Ref. [157]

246 655

668 608 890

Ref. [229]

Wz (Raman range) (cm1)

Table 1.30 Optical phonon energies and phonon deformation potentials for AlN.

924

677 618

Wz unstrained (calculated) (cm  1) [178]

bk 901  145 904  163

ak 982  83 930  94

Raman [227]

744 394 808

835 776 867

bk

Calculated [178] ak

Deformation potentials

1.3 Aluminum Nitride

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1000

AlN

800

800

600

600

400

400

200

200

0 Γ

K

M

Γ

A

DOS

Frequency(cm –1 )

Frequency(cm–1)

1000

0

Figure 1.27 Phonon dispersion curves and phonon density of states for Wz AlN reported in Ref. [234]. The data points have their roots in Ref. [83] and are discussed and compared with the ab initio calculations in Ref. [233].

950 cm1, 2TO(L) at 980 cm1, 2TO(G) at 1085 cm1, 2TA(X) þ 2TO (G) at 1300 cm1, and 3TO(G) at 1450 cm1 have been observed. A very narrow peak seen at 2330 cm1 has been attributed to molecular nitrogen on the surface of c-Si [231]. It is worth noting that because of the extremely weak Raman signal usually presented by low quality AlN films, some of the previously reported features have been erroneously identified [232]. Misidentification of some vibration modes could lead to incorrect interpretations of the crystalline quality of AlN films. As in the case of GaN, the acoustic modes are simple translational modes, whereas the optical modes for wurtzite symmetry are more complex as shown in Figure 1.12. The calculated phonon dispersion curves [233,234] along with the phonon density of states for wurtzitic AlN are shown in Figure 1.27. 1.3.2 Thermal and Chemical Properties of AlN

Single crystalline forms of this compound either in the epitaxial form or bulk form represent the focus of this treatment. In its most commonly available form, AlN is an extremely hard ceramic material with a melting point higher than 2000
C. In single crystalline form, the melting of AlN was measured to be 2750–2850
C at nitrogen pressures of 100 and 200 atm (or bar) [235]. The melting temperatures for various nitrides were also determined by Van Vechten [236], who made use of a semiempirical theory for electronegativity and concluded that the melting point of AlN is close to 3487 K. Slack and McNelly [195] calculated the N2 equilibrium pressures over liquid Al to be 1, 10, and 100 bar at 2563, 2815, and 3117
C, respectively, as shown in Figure 1.18 in the context of the GaN discussion.

1.3 Aluminum Nitride

6.00 AlN

Δa/a

Δ (×10 3 ) Δa/a, Δc/c Δ

5.00

Δc/c

4.00

3.00

2.00

1.00

0.00 200

400

600

800 T (K)

1000

1200

1400

Figure 1.28 Variation of the thermal expansion coefficient of AlN with temperature in the c-plane and in the c-direction [86,87,225].

Using X-ray techniques across a broad temperature range (77–1269 K), it was noted by Slack and Bartram [85] that the thermal expansion of AlN is isotropic with a roomtemperature value of 2.56 · 106 K1. The thermal expansion coefficients of AlN measured by Yim and Paff [237] have mean values of Da/a ¼ 4.2 · 106 K1 and Dc/c ¼ 5.3 106 K1. The dependence of the thermal expansion coefficient on temperature in the c-plane and in the c-direction is shown in Figure 1.28, which can be fitted by the following polynomials (for 293 < T < 1700 K): Da=a0 ¼  8:679  10  2 þ 1:929  10  4 T þ 3:400  10  7 T 2  7:969  10  11 T 3

ð1:19Þ

and Dc=c 0 ¼  7:006  10  2 þ 1:583  10  4 T þ 2:719  10  7 T 2  5:834  10  11 T 3 ;

ð1:20Þ

where a0 and c0 represent the 300 K lattice parameters. For AlN powder, Krukowski et al. [88] reported the expansion coefficient for the a parameter to be 2.9 · 106 K1, and the same for the c parameter to be 3.4 · 106 K1. The specific heat of AlN has been discussed extensively. Mah et al. [238] approximated the specific heat Cp of AlN in the temperature interval 298–1800 K as Cp ¼ 45:94 þ 3:347  10  3 T  14:98  10  5 T  2 J mol  1 K  1

ð1 cal ¼ 4:186 JÞ:

ð1:21Þ

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The same for the higher temperature range of 1800–2700 K was approximated by Glushko et al. [239] as Cp ¼ 37:34 þ 7:86  10  3 T J mol  1 K  1

ð1 cal ¼ 4:186 JÞ

ð1:22Þ

relying on the following points: the specific heat Cp ¼ 51.5 J mol1 K1 at T ¼ 1800 K, and an estimated value, Cp ¼ 58.6 J mol1 K1 at T ¼ 2700 K, as outlined by Krukowski et al. [88]. Specific heat obtained from the above approximations coupled with the measured values for a constant pressure from the literature has been tabulated [88], a plot of which is shown in Figure 1.29 along with the calculated specific heat using Equation 1.16 for Debye temperature values of 800–1100 K with 50 K increments. The best fit between the data and the Debye specific heat expression for insulators

50

Specific heat, Cp (J mol–1 K–1)

40

30

20 Specific heat AlN (J mol–1 K–1)

800 K 850 K 900 K 950 K 1000 K 1050 K 1100 K

10

0

0

200

400

600

800

1000

Temperature (K) Figure 1.29 Molar specific heat at constant pressure, Cp (J mol1 K1, 1 cal ¼ 4.186 J), of AlN versus temperature. Open circles represent the experimental data. The solid lines are calculation based on the Debye model for Debye temperatures, yD, in the range of 800–1100 K

with 50 K increments. The data can be fit with Debye expression for yD ¼ 1000 K, which compares with 950 K reported by Slack et al. The data are taken from Ref. [88]. (Please find a color version of this figure on the color tables.)

1.3 Aluminum Nitride

indicates a Debye temperature of 1000 K, which is in good agreement with 950 K reported by Slack et al. [240]. Compared to the GaN figure, the Debye temperature so obtained for AlN appears more dependable owing to a much better fit. The equilibrium N2 vapor pressure above AlN is relatively low compared to that above GaN that makes AlN easier to be synthesized. The calculated temperatures at which the equilibrium N2 pressure reaches 1, 10, and 100 atm are 2836, 3088, and 3390 K, respectively [195]. Details of the thermodynamic properties of AlN can be found in Refs [87,88]. Similar to GaN, albeit to a lesser extent, AlN exhibits inertness to many chemical etches. A number of AlN etches have been reported in the literature. However, molten salts such as KOH or NaOH at elevated temperatures such as 50–100
C, lower than what is required for GaN by as much as 200
C, etch AlN at appreciable rates. The surface chemistry of AlN has been investigated by numerous techniques including Auger electron spectroscopy, XPS, ultraviolet photoemission spectroscopy (UPS), ultraviolet photoelectron spectroscopy, and electron spectroscopy. One of these investigations by Slack and McNelly [195] indicated that the AlN surface grows an oxide 50–100 Å thick when exposed to ambient air for about a day. However, this oxide layer was protective and resisted further decomposition of the AlN samples. Details can be found in Refs [87,157]. The thermal conductivity k of AlN at room temperature has been predicted as k ¼ 3.19 W cm1 K1 [77,241]. Values of k measured at 300 K are 2.5 [241] and 2.85 W cm1 K1 [85]. The predicted values are near 3.2 W cm1 K1 in an O-free simulated material but are based on measurements in AlN containing O [77]. A more recent prediction for the value in AlN is 5.4 W cm1 K1, which is much larger than any measured value [208]. The measured thermal conductivity as a function of temperature in bulk AlN containing some amount of O is plotted in Figure 1.30a and b. Also shown is a series of samples with estimated concentrations of O showing an overall reduction in the thermal conductivity with O contamination. The calculation results of Slack et al. for impurity- and defect-free AlN are shown as well. In the temperature range of interest where many of the devices would operate, the thermal conductivity in the sample containing the least amount of O assumes a T 1.25 dependence. In freestanding and 300–800 mm thick AlN samples grown by HVPE, originally on Si(1 1 1) substrates, values in the range of 3.0–3.3 W cm1 K1 were measured by the SThM method [242]. The dislocation density in these freestanding AlN templates was about 108 cm2 [243]. 1.3.3 Electrical Properties of AlN

Owing to the low intrinsic carrier concentration and the deep native defect and impurity energy levels, the electrical characterization of AlN has usually been limited to resistivity measurements. One such measurement by Kawabe and coworkers [244] on transparent AlN single crystals yielded resistivities r ¼ 1011–1013 O cm, a value consistent with other reports [245–247]. However, it was found that impure crystals,

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Thermal conductivity (W cm–1 K–1)

103

(a)

AlN

"Pure AlN"-Theor

4 x 1019 cm–3

102

5 x 1019 cm–3 2 x 10 20 cm–3 3 x 10 20 cm–3

101 100

(J/mol-1 K-1)

10 –1 10 –2 10 –3 10 –1

10 0

10 1 10 2 Temperature (K)

10 3

10 4

102 "Pure AlN" 19

Thermalconductivity(W cm–1 K–1)

4 x 10

19

5 x 10

20

2 x 10

20

3 x 10

–3

cm

–3

cm

–3

cm

101

100 10 (b)

–3

cm

100

1000

Temperature (K)

Figure 1.30 (a) Thermal conductivity of singlecrystal AlN. The solid line alone indicates the theory whereas the others represent measurements of AlN with various concentrations of O. The lower the O concentration, the higher the thermal conductivity. The theoretical data are from

Ref. [77], the data for the lowest O concentration are from Ref. [241], and the data for the other samples are from Ref. [91]. (b) The thermal conductivity of AlN in a limited temperature range of common interest that underscores the detrimental effect of O on the thermal conductivity [240].

1.3 Aluminum Nitride

which exhibited a bluish color possibly because of the presence of Al2OC, have resistivities of r ¼ 103–105 O cm, much lower than those reported by Chu et al. [248], who were purportedly able to obtain both n- and p-type AlN by introducing Hg and Se, respectively. However, they failed to determine the net carrier concentrations owing to very high resistivities. The n-AlN films grown by Rutz [249] had a quite low resistivity (r ¼ 103 O cm), which is comparable to those of Kawabe et al. [81]. Although Rutz [249] did not determine the source of the electrons, Rutz et al. [250] observed an interesting transition in their AlN films in which the resistivity abruptly decreased by two orders of magnitude with an increase in the applied bias. This observation found applications to switchable resistive memory elements that are operated at 20 MHz. The insulating nature of these early films hindered meaningful studies of their electrical transport properties. With the availability of refined growth techniques, AlN, presently grown with much improved crystal quality, shows both n- and p-type conductions. This has rejuvenated efforts to measure both electron and hole Hall mobilities. Edwards et al. [251] and Kawabe et al. [252] carried out some Hall measurements in p-type AlN producing a very rough estimate of the hole mobility mp ¼ 14 cm2 V1 s1 at 290 K. The predictions for the Hall mobility in the entire range of AlGaN alloy including the binary end points are treated in Volume 2, Chapter 3. Not all the parameters needed for the calculations are known precisely, reducing the confidence in predicted values somewhat. As in the case for GaN, the roomtemperature mobility is dominated by the polar optical phonon scattering. 1.3.4 Brief Optical Properties of AlN

Investigations of the optical absorption coefficient, a, of AlN at room temperature were reviewed as early as 1976 by Slack and McNelly [195]. Harris and Youngman [253] have reviewed photoluminescence and cathodoluminescence characteristics of AlN. Because the AlN lattice has a very large affinity to oxygen, it is almost impossible to eliminate oxygen contamination in AlN that affects observations. Commercially available AlN powder is said to contain about 1–1.5 at.% oxygen. Some oxygen is dissolved in the AlN lattice, with the remainder forming an oxide coating on the surface of each powder grain. After irradiation with ultraviolet light, AlN doped with oxygen was found to emit a series of broad luminescence bands at near-ultraviolet frequencies at room temperature no matter whether the sample was powdered, single crystal, or sintered ceramic. Pacesova and Jastrabik [254] observed two broad luminescence lines centered in the vicinity of 3.0 and 4.2 eV and more than 0.5 eV wide for samples contaminated at about 1 to 1.5 at.% oxygen. Youngman and Harris [255] and Harris et al. [256] investigated the luminescence characteristics of polycrystalline-sintered AlN samples and noted a continuous shift of the peak position in the ultraviolet luminescence line as a function of oxygen content up to a critical concentration of about 0.75 at.%. The luminescence lines beyond this limit of oxygen concentration remained stationary.

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Epitaxial layers of about 1 mm or less in thickness on sapphire substrates led to observation of donor bound excitons with phonon replicas and some deep emission [257]. Cathodoluminescence spectra obtained in reactive MBE (RMBE), using ammonia as the nitrogen source, show a sharp band edge peak, which is tentatively assigned to the optical recombination of a donor bound exciton (D0, X), sometimes accompanied by weak one and two longitudinal optical phonon replicas. Also observed is a broad band with maximum in the range 320–370 nm, although some variation of this low-energy peak has been reported. The 6 K CL spectrum of the near band edge region shows a sharp and strong peak at 2068 Å in addition to two weak peaks on its low-energy side, which are not observed for all samples. The symmetric [0 0 0 2] X-ray diffraction peak is narrow and the linewidth of the prominent near band edge CL peak is 23 meV. Bulk single crystalline AlN has also been investigated for its CL emission properties. In addition to the near band edge emission at about 6 eV, the ultraviolet oxygen luminescence peak at 4.0 eV was also observed. In the investigation of Youngman and Harris [255], 4.64 eV photons were used to specifically explore the below the band transitions such as the peak near 4 eV. That investigation showed a blue shift with increased O concentration in the peak under question. With the availability of highquality AlN substrates, presumably high-quality epitaxial AlN layers have been grown on them and characterized for their optical properties by CL at 5 K [107]. In this particular work, the CL measurements were carried out at different temperatures for a fixed electron beam energy density value. The beam current was held at 5 mA and the voltage at 10 kV. The energy density was about 700 W cm2. Some six emission lines between the energies of 5.98 and 6.03 eV have been delineated in the lowtemperature near-band edge emission spectra of those AlN films. In addition to unidentified neutral donor and possibly acceptor bound exciton lines, free exciton A and its excited state, and free exciton B were observed. The availability of A exciton ground and excited states led to a binding energy of 63 meV that gives a lowtemperature bandgap of 6.086 eV for this material. The provisionally accepted value for the bandgap of this material is 6.2 eV (which is questionable now that the lower value is supported by measurements performed in homoepitaxial layers). In terms of absorption, Yim et al. [246] characterized AlN by optical absorption determining the room-temperature bandgap to be direct with a value of 6.2 eV. It should be pointed out that this early figure is not consistent with measurements performed later on using high-quality samples as discussed in the previous paragraph. Perry and Rutz [258] performed temperature-dependent optical absorption and determined a bandgap of 6.28 eV at 5 K compared to their room-temperature value of 6.2  0.1 eV. We should point again that it has been lowered to slightly over 6 eV at low temperatures in high-quality samples. Several groups have reported comparable values whereas others have produced questionable values considerably below 6.2 eV, probably due to oxygen contamination or nonstoichiometry. In addition to the band edge absorption, a much lower energy absorption peak at 2.86 eV (although some variation in the peak position has been recorded from 2.8 to 2.9 eV) is likely owing to nitrogen vacancies or nonstoichiometry as proposed by Cox et al. [245]. Yim et al. [237] also observed a broad emission spectrum range of 2–3 eV with a peak at about 2.8 eV.

1.3 Aluminum Nitride

10

4

(0001) AlN Single crystals 300 K Nitrogen vacancy

Absorption coefficient α (cm–1)

3

10

Oxygen

Pastrnak and Roskovcova

2

10

1

10

0

10

1

2

4 3 Energy (eV)

5

6

Figure 1.31 Room-temperature absorption spectra of several AlN films of varying thicknesses whose principal absorption edge occurs at 6.2 eV. The bump near 4.5–4.8 eV in the data is attributed to the oxygen absorption bands. The one at about 2.8 eV is attributed to N vacancies [245–247,259–261]. The ones not marked as Pastrnak and Roskovcova [259] are from Slack et al. [91].

This peak does not correlatewith the presence of oxygen. The oxygen absorption region lies between 3.5 and 5.2 eV, as originally found by Pastrnak and Roskovcova [259–261]. The exact position of this particular peak appears to change with the oxygen content from 4.3 eV at low oxygen content to 4.8 eV at high oxygen content. The results of the studies on the room-temperature absorption coefficient, a, are shown in Figure 1.31 for three different crystals along with the results of Pastrnak and Roskovcova [259]. Low-temperature and room-temperature absorption data taken at 2 K in a thin film on double side polished sapphire are shown in Figure 1.32. In the only optical study of AlN impurities, Karel and coworkers [262–266] reported on the luminescence of Mg and rare earth centers in AlN. Measurements of the refractive index of AlN have been carried out in amorphous, polycrystalline, and single-crystal epitaxial thin films. The values of the refractive index, n, are in the range of 1.99–2.25 with several groups reporting n ¼ 2.15  0.05.

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2 × 10 10

RT absorption2

1.5 × 1010

1 × 1010

5 × 10

9

0 1900

1950

2000

2050

2100

2150

2100

2150

Wavelength (Å)

(a)

3.5 × 10 9 3 × 10 9

2K absorption 2

2.5 × 10 9 2 × 10 9 1.5 × 10 9 1 × 10 9 5 × 10 8 0 1900 (b)

1950

2000 2050 Wavelength (Å)

Figure 1.32 Room-temperature (a) and 2 K (b) absorption spectrum of a thin AlN film grown on double side polished c-plane sapphire by RF MBE. The data were taken at the University of Pittsburgh by Song Bay and W.J. Choyke using a sample prepared in author’s laboratory.

These values are found to increase with increasing structural order, varying between 1.8 and 1.9 for amorphous films, 1.9–2.1 for polycrystalline films, and 2.1–2.2 for single-crystal epitaxial films. The spectral and polarization dependence of the index of refraction has been measured and showed a near-constant refractive index in the wavelength range of 400–600 nm. Some of these measurements also indicate that, in the long-wavelength range, the dielectric constant of AlN (e0) lies in the range of

1.4 Indium Nitride

8.3–11.5, and that most of the values fall within e0 ¼ 8.5  0.2. Other measurements in the high-frequency range produced dielectric constants of 4.68 and e1 ¼ 4.84. AlN has also been examined for its potential for second harmonic generation. Synchrotron radiation studies of AlN single crystals up to 40 eV have been performed, which resulted in the observation of an 8 eV luminescence peak. The same peak was also found in vacuum–ultraviolet reflection measurements.

1.4 Indium Nitride

InN forms the third binary anchor of the nitride family and was first synthesized in 1938 [267]. Compared to the other two, GaN and AlN, it is very difficult to form in high quality because of an inherent reason. The disparity in size, electronegativity, and very high vapor pressure of N over In imposes intractable tasks on the crystal grower [197,268]. The very early attempts explored absorption properties of polycrystalline InN films grown by DC discharge [269], reactive cathodic sputtering [270], or RF sputtering [271], and somewhat later RF deposition [272]. Inclusive properties of InN grown by various methods have been discussed in a review [273]. Essentially, the early results suggested a bandgap Eg ¼ 1.8–2.0 eV at room temperature. Those techniques gradually gave way to more refined growth methods such as MBE and OMVPE. The stoichiometry, however, has always been a pestering issue and will always remain so, which appears to have been one of the sources of controversy as to InN’s true bandgap following the longstanding value of about 1.89 eV. An inordinate number of reports adorned many reputable journals and filled the programs in technical meetings wherein researchers in great numbers argued that the true bandgap of InN is actually 0.7–0.8 eV. In fact, lower values such as 0.65 [274] and 0.67 eV [275] have also been reported. What appeared to be experimentally impeccable results made the theorists to reexamine and redebate their band structure calculations [138,276]. The previously accepted larger value for the bandgap was explained away by O contamination and Moss–Burstein shift [277] because of high electron concentrations inherent to this semiconductor. Excellent reviews chronicling the developments in InN and controversy surrounding its bandgap are available in the literature [278,279]. Owing in part to visceral difficulties touched upon above and its bandgap, regardless of the controversy, InN has received nowhere near the attention given to GaN and AlN. Reiterating, the problems with InN are difficulties in growing highquality crystalline InN samples, poor luminescence properties of InN, and the existence of alternative well-characterized semiconductors such as AlGaAs and (Ga, Al)AsP, which have energy bandgaps close to that of the old value of InN bandgap (1.89 eV) and InGaAs close to that of the new InN bandgap. Setting the bandgap aside, which is the holy grail of optoelectronic devices, InN possesses the largest roomtemperature mobility among all the nitride-based semiconductors. Predictions point to InN being a superior channel layer with its higher mobility for field effect transistors, a topic that will receive more coverage in the pages to follow.

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Just when the data and calculations appear to indicate the bandgap of InN to be close to 0.7–0.8 eV [30], there came the reports casting doubt on the accuracy of the small bandgap. The controversy regarding the true bandgap of InN brought this semiconductor, from “obscurity to controversy,” in the words of Scott Butcher, who is one of the pioneers involved in InN studies of and instrumental in determining the 1.89 eV bandgap [278]. The controversy is actually not limited to just the bandgap. It spans the whole gambit of its properties including the lattice constant, the effective electron mass, not to mention the hole mass, which is simply left in the dark, the role of hydrogen and oxygen, nonstoichiometry-induced defects, and point and extended defects. Returning to the controversy surrounding the bandgap of InN, proponents of the smaller bandgap argue that the measured larger bandgap from absorption data is most likely skewed by Moss–Burstein shift and/orO contamination.Theyare alsoquick toargue that emission near 1.8–1.9 eV has not been observed casting doubt on its accuracy. The opponents of the larger bandgap put forward arguments ranging from Mie resonance owing to scattering or absorption of light in InN-containing clusters of metallic In [280] to In-rich nonstoichiometry-induced defects active near the 0.7–0.8 eV region [281]. In the efforts of the authors of Ref. [280], microcathodoluminescencestudies coupled with imaging of metallic In have shown that bright infrared emission at 0.7–0.8 eV arises in the close vicinity of In inclusions and is likely associated with surface states at the metal/InN interfaces. Employing thermally detected optical absorption measurements, Shubina et al. [280] suggested that a true bandgap near 1.5 eV, reserving a more accurate figure for the bandgap until after more detailed measurements are carried out. The presence of In inclusions would also make suspect the mobility and doping level data published for this material. In fact, metal inclusions placed by design inGaAs have been shown to skew the electron mobility [282]. Setting this issue aside for now and referring the reader to Chapter 2 for bandgap-related discussion and Chapter 3 for growth-related discussion, let us segue into the discussion of InN properties. As mentioned, the energy bandgap of InN corresponds to a portion of the electromagnetic spectrum in which alternative and well-developed semiconductor technologies are already available. Consequently, practical applications of InN are more or less restricted to its alloys with GaN and AlN or related heterostructures. The growth of high-quality InN and the enumeration of its fundamental physical properties remain for the present a purely scientific enterprise except of course their impact on the properties of the ternaries it makes with GaN and AlN. InN is not different from GaN and AlN in the sense that it suffers from the same lack of a suitable substrate material and, in particular, a high native defect concentration. All these have hindered its progress. In addition, because of its rather poor thermal stability InN cannot be grown at the high temperatures required by CVD growth processes. As InN rapidly dissociates at high temperatures, even as low as 600
C, an extraordinarily high nitrogen overpressure would be required to stabilize the material up to the melting point, which is practically impossible. The large disparity of the atomic radii of In and N is another factor that increases the difficulty in obtaining InN of good quality. Notwithstanding the aforementioned characterization, the seminal work of Tansley and Foley [271] first characterized many of the fundamental physical properties of InN.

1.4 Indium Nitride

Tables 1.19–1.22 list the physical properties of InN. It has proved difficult to grow high-quality single crystalline material that would enable detailed optical, structural, and electrical measurements to be performed. Given the fact that the growth of bulk single-crystal InN films using equilibrium techniques is unlikely, attention turned to the deposition of thin films using nonequilibrium techniques. All of the early data summarized below under various properties, unless otherwise specified, were obtained from highly conductive n-type polycrystalline InN grown by nonequilibrium techniques. There have been several studies that report rapid dissociation of InN at temperatures above 500
C. Because no high-quality InN has yet been grown, the resistance of the high-quality material to chemical etching is unknown. Successful etching of single crystalline InN films in a hot H3PO4 : H2SO4 solution has been measured as was surface oxidation. 1.4.1 Crystal Structure of InN

To reiterate, indium nitride normally crystallizes in the wurtzite (hexagonal) structure. The zinc blende (cubic) form has been reported to occur in films containing both polytypes. Because of the absence of good-quality single-crystal films, early studies dealing with the crystal structure of InN were limited to mainly less than ideal thin films, particularly the ordered polycrystalline films with crystallites in the thickness range of 50–500 nm. Basically, the measurements confirm that, although InN normally crystallizes in the Wz structure, it occasionally also crystallizes in the zinc blende (cubic) polytype. Thermal instability of InN forbids growth at high temperatures and a large lattice mismatch with most available substrates inevitably leads to less than perfect structural quality of the epitaxial films grown by any method although the electron mobility in MBE-grown layers have improved considerably. A study based on XRD analysis has provided important insights into the dependence of the structural properties on the degree of lattice mismatch and film thickness of InN [283] on three different kinds of substrates, namely sapphire, GaN, and AlN. Significant improvement in the structural quality of the InN films, which do suffer from the residual strain, was observed on GaN templates. Below a thickness of 1200 Å, the InN film is composed of grain islands with different crystal orientations. Above this thickness, screw dislocations are nucleated, relieving the strain and leading to a reduction of the observed mosaicity in the surface morphology as grains with the same orientation grow with film thickness exceeding 1200 Å. 1.4.2 Mechanical Properties of InN

The measured InN lattice parameters using powder technique are in the range of a ¼ 3.530–3.548 Å and c ¼ 5.704–5.960 Å with a consistent c/a ratio of about 1.615  0.008. This c/a ratio is close to the more optimistic value of 1.633 determined

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from layers especially grown under significant precautions, best possible growth conditions, and presumably with reduced nitrogen vacancies [284]. Another value of the c/a ratio of 1.612 has also been reported using powder diffractometry [29]. There is no reliable experimental u parameter for InN. An examination of the reported data indicates an unacceptably large scatter. This is possibly caused by nitrogen deficiency because nitrogen atoms are closely packed in (0 0 0 1) planes. The single reported measurement yields a lattice constant of a0 ¼ 4.98 Å in zinc blende (cubic) form InN occurring in films containing both polytypes [284]. While the cubic polytype of InN yields a molecular cell volume of 30.9 Å3, the hexagonal polytype gives a molecular cell volume of 31.2  0.2 Å3. The experimental density of InN deduced from Archimedean displacement measurements is 6.89 g cm3 at 250
C [285]. This is comparable with 6.81 g cm3 estimated from X-ray data. In a hexagonal structure, the second-order elastic moduli are C11, C12, C13, C33, and C44. The only report on InN elastic coefficients is by Savastenko and Sheleg [162], but their results are lower than the values calculated by the linear muffin–tin orbital (LMTO) and the plane wave pseudopotential (PWPP) methods and are suggested to be completely unreliable [178]. Table 1.31 summarizes the measured and the calculated elastic coefficients for both Wz and ZB InN. Because these figures depend on the lattice constants that are within some 10%, values of other nitrides can be used as a first approximation when absolutely needed [271,286]. The bulk modulus has been calculated from first principles by the local-density approximation [287] and by the LMTO method [288] suggesting bulk modulus B ¼ 165 GPa. The results of other calculations for bulk modulus are shown in Table 1.31. Most of the properties of InN are tabulated in Tables 1.19–1.22. As in the cases of Wz GaN and Wz AlN, Wz InN has 12 phonon modes at the zone center (symmetry group: C6v), 3 acoustic and 9 optical ones with the acoustic

Table 1.31 Theoretical and experimental elastic coefficients and bulk modulus (in GPa) of the various forms of InN [160].

Method

Structure

C11

C12

C13

C33

C44

B

X-ray [162] LMTO [24] PWPP [42] PWPP [42] PWPP LMTO PWPP PWPP ADX LMTO PWPP PWPP

Wz Wz Wz ZB ZB Wz Wz ZB Wz Wz Wz ZB

190 271 223 187

104 124 115 125

121 94 92

182 200 224

10 46 48 86

139 141 145.6 155 165 166 138 125.5 165 139 140

dB/dP ¼ B0

4 3.8 3.9 12.7

ADX: angular dispersive X-ray diffraction; PWPP: plane wave pseudopotential; LMTO: linear muffin–tin orbital.

1.4 Indium Nitride

branches near zero at k ¼ 0. The infrared active modes are of the E1(LO), El(TO), A1(LO), and A1(TO) type. Raman spectroscopy studies [289,290] have yielded four optical phonons characteristic for InN with wave numbers 190 cm1 (E2), 400 cm1 (A1), 490 cm1 (E1), and 590 cm1 (E2) in InN layers grown by atomic layer epitaxy (ALE). Moreover, a TO mode has been observed at 478 cm1 (59.3 meV) by reflectance and 460 cm1 (57.1 meV) by transmission measurements [286]. From other reflectance data, the existence of a TO phonon mode at 478 cm1, consistent with Ref. [286], and an LO mode at 694 cm1 was deduced [291]. 1.4.3 Thermal Properties of InN

The linear thermal expansion coefficients measured at five different temperatures between 190 and 560 K [292] indicate that both along the parallel and perpendicular directions to the c-axis of InN these coefficients increase with increasing temperature. Thermal conductivity derived from the Leibfried–Schloman scaling parameter, assuming that the thermal conductivity is limited by intrinsic phonon–phonon scattering, is about 0.80  0.20 W cm1 K1. Using InN microcrystals prepared by microwave nitrogen plasma, the specific heat of InN has been measured with differential scanning calorimeter with a precision better than 1% [88,293]. The data have been fit to the Debye equation, Equation 1.16, as shown in Figure 1.33a. An expanded view of the experimental data over a temperature range of 150–300 K is given in Figure 1.33b. Below 200 K, a Debye temperature of 600 K appears to fit the data well, tabulated in Table 1.32. However, above 200 K a Debye temperature of 700 K fits the data better. This paradox indicates the poor quality of InN and significant contribution by nonvibrational modes, as the Debye theory is developed for a perfect dielectric. Others [88] argue that the Debye temperature of yD ¼ 660 K, albeit in a relatively small range, describes InN best. As shown in Figure 1.18, the melting point of InN, T M InN , is 2146 K and the line fit to the partial pressure data for N indicates a temperature dependence of 7.9  1017 exp (  2.78 eV/kT) bar for InN. As can be seen in Figure 1.18, the nitrogen partial pressure increases exponentially above TE ¼ 630
C for InN, illustrating that the decomposition temperature in vacuum is much lower than the melting point achievable under high pressures. To determine the effective decomposition activation energy more precisely, the nitrogen flux was calculated from the measured nitrogen pressure [196,199]. The rate of nitrogen evolution F(N) is equal to the rate of decomposition and the slope of ln[F (N)] versus 1/T in Figure 1.18 gives the effective activation energy of decomposition in vacuum, EMN. The decomposition rate is that corresponding to desorption of one monolayer in 1 s, in other words, FN ¼ 1.5 · 1015 cm2 s1, at 795
C. The activation energy of the thermally induced decomposition is determined as EMN ¼ 3.5 eV (336 kJ mol1) for InN (M is for metal and N is for nitrogen). This indicates temperature limits for high-temperature or high-power devices. This together with the reported small values of InN bandgap does not bode well for power devices based on InN alone. Of course, the picture is different if this

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50

–1

Cp (J mol–1 K )

40

30 –1

–1

Cp (J mol K ) T = 600K T = 700K T = 800K

20

10

0 0

(a)

200

400 600 Temperature (K)

800

1000

–1 Specific heat, C p (J mo l K–1)

40

35

30

25

20 150 (b)

200

250 Temperature (K)

300

Figure 1.33 (a) The specific heat of InN with experimental data points, albeit in a small range, and the 600, 700, and 800 K Debye temperature fits. Data from Refs [88,127]. (b) Experimental specific heat of InN in a temperature range of 150–300 K [88,127].

material is used in conjunction with GaN if the associated technological difficulties can be overcome. While the heat capacity of InN is 9.1  2.9 · 103 cal mol1 K1) at temperatures between 298 and 1273 K, the entropy is 10.4 cal mol1 K1 at 298.15 K. The

1.4 Indium Nitride Table 1.32 Specific heat, Cp, of InN at constant pressure [88,127].

T (K)

Cp (J mol1 K1)

153 163 173 183 193 203 213 223 233 243 253 263 273 283 293

25.38 26.54 27.96 29.12 30.15 31.18 31.95 32.59 33.50 34.26 35.17 35.81 36.97 37.61 38.65

equilibrium partial pressure of N2 above InN is about 1 atm at 800 K and it increases exponentially with temperature to 105 atm at 1100 K.

1.4.4 Brief Electrical Properties of InN

It is fair to state that reliable experimental data for the true electron mobility in InN is waiting to be obtained. As mentioned repeatedly, InN suffers from the lack of a suitable substrate material and high native defect concentrations that limit its quality. In addition, a large disparity of the atomic radii of In and N makes it more difficult to obtain InN of high quality. As a result, nitrogen vacancies are thought to lead to large background electron concentrations in InN. Because of all these factors, the electron mobilities obtained from various films vary very widely, as can be inferred from Table 1.33. The electron mobility in InN can be as high as 3000 cm2 V1 s1, perhaps even much higher, at room temperature [294]. A study of the electron mobility of InN as a function of the growth temperature indicates that the mobility of ultrahighvacuum electron cyclotron resonance radio-frequency magnetron sputtering (UHV ECR-RMS) grown InN can be as much as four times the mobility of conventionally grown (vacuum deposition) InN [295]. The dependence of the electron concentration and mobility on the InN film thickness grown on AlN and GaN buffer layers by plasma-enhanced MBE is shown in Figure 1.34. Hall measurements in InN films grown on AlN buffer layers [300] that are in turn grown on sapphire indicated electron mobility to be 1310 cm2 V1 s1 at room temperature [307] for MBE-grown and 830 cm2 V1 s1 for OMVPE-grown material [304]. As discussed in Section 3.5 in conjunction with growth-related issues,

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Table 1.33 A compilation of electron mobilities obtained in InN on different substrates and for various deposition conditions in InN as compiled in part in Ref. [296].

n-type carrier concentration (cm3)

Carrier mobility (cm2 V1 s1)

5–8 · 1018

250  50

1020 3–10 · 1018

20 20–50

1–200 · 1018 2–80 · 1020 5 · 1018 6 · 1020 6 · 1016

3 35–50 20 2 2

7–70 · 1016 3 · 1016 at 150 K 1020

730–3980 5000 at 150 K 10

4.8 · 1020 1–8 · 1020 1–10 · 1020 2–3 · 1020

38 50 50 20–60

2 · 1020 1020ae 2.0 · 1020 5.98 · 1018

100 220a 100 363

Sapphire Sapphire Sapphire Sapphire, silicon, mica GaAs GaAs GaAs Glass

5.0 · 1019 3.0 · 1018

700 542

Sapphire Sapphire

8.8 · 1018 2–3 · 1018 1019

500 800 306

Sapphire Sapphire Glass, KBr

1.0 · 1019 4 · 1017

830 2100

Sapphire Sapphire

1.4 · 1018

1420

Sapphire

5 · 1016 (3 · 1016 at 150 K)b

2700 (5000 at 150 K)b

a b

Zinc blende polytype. Have not yet been reproduced by others.

Substrate

Deposition technique

Sapphire, silicon, various metals Sapphire Glass, fused quartz Fused quartz Sapphire Glass, NaCl Fused quartz Glass, silicon, 304 stainless steel Glass, silicon Glass, silicon Glass

Reactive sputtering Reactive evaporation Reactive sputtering Reactive sputtering CVD Reactive sputtering Cathodic sputtering RF ion plating Reactive sputtering Reactive sputtering Reactive DC magnetron sputtering Magnetron sputtering Plasma-assisted MOVPE MOVPE Reactive RF magnetron sputtering ECR-assisted MOMBE Plasma-assisted MBE ECR-MOMBE [297] Reactive RF magnetron sputtering [298] MOVPE [299] Migration-enhanced epitaxy [300] MOMBE [301] MBE [302] RF reactive ion sputtering [303] RF MBE [304] MBE on thick HVPE GaN [305] Plasma-assisted MBE [306] Sputtering [308]

10

21

104

10

20

103

10

19

102

10

18

10

InN on GaN

mobility electron conc.

InN on AlN

mobility electron conc.

17

10

1

10

2

Mobility (cm2 V-1 s-1)

Electron density (cm–3)

1.4 Indium Nitride

101

100 3

10 Thickness (nm)

10

4

Figure 1.34 Electron density and mobility as a function of InN thickness. Samples grown on a GaN buffer are indicated by diamond and square symbols, and on an AlN buffer are indicated by circles and inverted triangles. The diamond and inverted triangle symbols indicate the electron mobility, whereas the squares and circle symbols indicate the electron concentration, all measured at room temperature. Courtesy of W.J. Schaff.

room-temperature mobilities increased to 2000 cm2 V1 s1 in MBE-grown InN films. Some directions, such as the insertion of low-temperature intermediate layers or AlN buffer layers have been pointed out for improvement. Very high inadvertent donor concentrations, >1018 cm3, seem to be one of the major problems for further progress. ON and SiIn have been proposed to be the likely dominant defects responsible for high electron concentration based on their low formation energies. Additionally, H has been proposed [307] as the dominant impurity candidate for the state-of-the-art MBE-grown InN. Assuming that the measurements were performed flawlessly and interpreted, there clearly seems to be some way to go to reach the goal of a mobility of 2700 cm2 V1 s1 at an electron concentration of 5 · 1016 cm3 reported for RF reactive ion sputtered growth of InN nearly a few decades earlier [308]. Ensemble Monte Carlo calculations have been the popular tool to investigate the carrier velocity field characteristics theoretically. Although early application of this method to InN was done by O’Leary et al. [309], a more detailed calculation based on the full details of the conduction band structure appeared later in a paper by Bellotti et al. [310]. A peak electron drift velocity of 4.2 · 107 cm s1 has been predicted at an electric field of 65 kV cm1, substantially higher than that in GaN,

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with a noticeable anisotropy for field direction parallel or perpendicular to the basal plane. The velocity is seen to decrease to 3.4 · 107 cm s1 with an increase of field to 70 kV cm1 and reach a value of 2.0 · 107–1.0 · 107 cm s1 at the onset of impact ionization, for field directions parallel and perpendicular to the basal plane, respectively. The calculated low-field mobility is 3000 cm2 V1 s1. This study claims to report the first calculation of high-field electron transport in InN. Electron–phonon coupling would hurt the high-field velocity as it seems to be the case in GaN, see Volume 3, Chapter 3 for details. Another interesting aspect of electron transport is its transient behavior, which is relevant to short channel devices with dimensions smaller than 0.2 mm, where a significant overshoot is expected [311] to occur in the electron velocity over the steadystate drift velocity. In yet another calculation, Foutz et al. [311] found that of the three III nitride binaries, GaN, InN, and AlN, this overshoot is most pronounced in InN and occurs above a critical field of 65 kV cm1. The peak velocity at this field is 4.2 · 107 cm s1 and the velocity overshoot is retained over longer distances as compared to that for GaN and AlN. 1.4.5 Brief Optical Properties of InN

As mentioned in the opening statements for Section 1.4, while the bandgap value reported in the early stages of InN development dating back to as early as 1980s, a controversy developed as to the true value of the InN bandgap developed. This is detailed in Section 2.9.2. However, a brief treatment is provided here for completeness. Excellent reviews chronicle and detail the evolution of the controversy in InN [278,279]. A number of groups have described optical measurements performed on InN [25,313,314]. Early values of the room-temperature InN direct bandgap ranged from 1.7 to 2.07 eV. A value of 1.89 eV was measured by optical absorption by Tansley and Foley [308], who also measured the infrared absorption of InN and observed an unidentified donor level approximately 50–60 meV below the conduction band edge. Reflectance spectroscopy on single-crystal material, using synchrotron radiation over the range 2–20 eV[315], later extended to 130 eV [316], was performed to determine the optical parameters of InN. A few studies of the interband optical absorption performed on InN thin films deposited by sputtering techniques [271] and OMVPE [103], were found consistent with a fundamental energy gap of about 2 eV. However, weak photoluminescence peaks with energies ranging from 1.81 to 2.16 eV were observed for InN grown on Si substrates later on [317]. In one such case, an emission centered at 1.86 eV at a temperature below 20 K was seen, whereas the reflectance measurement showed a strong plasma reflection at 0.7 eV, corresponding to an effective mass of 0.12m0. Another branch of studies shows that in improved InN films a strong photoluminescence transition at energies around 1 eV [30,318,319] appears. Observing that the position of the photoluminescence energy agrees with the onset of strong absorption, the optical transition at about 1 eV has been attributed to the fundamental bandgap [319].

1.4 Indium Nitride

4

Assuming large gap Assuming small gap

E g (eV)

3

2

1

0 0 InN

20

40

60

80

Composition

100 In2 O3

Figure 1.35 Vegard’s law plot of InN–In2O3 pseudobinary alloy system with both 0.7 and 1.9 eV bandgaps shown (dashed lines and solid line, respectively) versus O composition reaching In2O3, which has a bandgap of 3.75 eV reported in Ref. [320]. Also shown is the calculated (using LCAO) bandgap with 10% oxygen if O-free bandgap is 0.7 eV [278,322].

Following the wide acceptance of the above-mentioned data by the community, a couple of crystal growth groups, often in collaborations with others who focused on characterization, began to question the nearly 1.9–2.0 eV bandgap, because the newer and supposedly more improved samples showed a strong emission at much smaller energies, primarily around 0.7–0.8 eV. Those latter groups argued that the early samples did not show efficient PL near the band edge and had to have been contaminated with O to support the earlier measurements. One should, however, keep in mind that In2O3 has a bandgap of 3.75 eV [320], and if the Vegard’s law is applicable it would take some 35% O in InN to boost its bandgap from say 0.8 to 1.9 eV, which is substantial, as shown in Figure 1.35. Also shown is the bandgap of InN for 10% O contamination assuming an O-free bandgap of 0.7 eV [321,322]. Not only large amounts O is needed, but also that O must form an alloy with InN. Clearly, the assignment of the 1.9 eV bandgap to O, on the premise that the bandgap for O-free InN is 0.7–0.8 eV, requires large amounts of O, which is not supported by experiments, as discussed in detail in Section 2.10.1. To account for the 1.9 eV measured bandgap, the proponents of the 0.7 eV bandgap also suggested that the former result could be accounted for by Moss–Burstein blue shift owing to high electron concentration. This effect relates to semiconductors where the electron concentration is larger than the density of states and the Fermi level is actually in the conduction band itself. The extent of the penetration naturally depends on the electron concentration. Consequently, the measured apparent optical gap is skewed upward. This possibility was suggested in 1974 by Trainor and Rose [323] and has been the topic of an extensive study using In2O3 as the model wide bandgap material [320]. What accompanies the Moss–Burstein shift is the

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2.4

E G +E F (eV)

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6

Moss–Burstein effect

0.4 10 16

10 17

10 18

10 20

10 19

a b c d e f g h i

10 21

–3

Carrier concentration (cm ) Figure 1.36 The measured bandgap of InN (combination of the bandgap and the Moss–Burstein shift) deduced from absorption measurements versus the carrier concentration reported by various groups. The solid line shows the expected blue shift in the bandgap because of Moss–Burstein shift. Solid and dashed lines

indicate nonparabolic and parabolic theories, respectively. (a) Ref. [341]; (b) Ref. [324]; (c) Refs [274,318]; (d) Ref. [331]; (e) Ref. [271]; (f) Refs [303,325]; (g) Refs [131,272,326,327]; (h) Ref. [328]; (i) Ref. [329]. Collated by Butcher and Tansley [278,279].

bandgap renormalization, which is a red shift, caused by tail states extending into the bandgap and counters the former effect to some extent. Here too there is a controversy in that the samples of older times did not indicate large Moss–Burstein effect, which has been attributed by the proponents of the 0.7 eV bandgap, as having been heavily reduced owing to band tail states. Butcher [278] collated the apparent bandgap (presumably a combination of the bandgap with its band tailing and Moss–Burstein effect) measured by optical absorption and the electron concentration data for a large group of samples as shown in Figure 1.36. Although an argument for Moss–Burstein effect can be made for samples with high carrier concentrations, the same cannot be applied to a good many samples exhibiting the large bandgap while having low carrier concentration. To be thorough, one should recognize that many of the InN samples contained in Figure 1.36 with low carrier concentrations are heavily compensated, casting some amount of uncertainty as well. Turning the argument around and assuming that the large bandgap data are more dependable, the low bandgap data could be explained with some defect level in In-rich material or by Mie resonant absorption due to In inclusions. These are discussed in Section 2.10.1. After discussing the controversy regarding the bandgap and making many references to the low bandgap measured in InN, let us briefly discuss the data obtained in layers grown by MBE under In-rich conditions, which exhibit the so-called small bandgap. The optical absorption data in InN, purportedly having 0.7–0.8 eV bandgap, show an onset at 0.78 eV not near 1.9–2.0 eV, as shown in

1.4 Indium Nitride

Figure 1.37a. The absorption coefficient increases gradually with increasing photon energy and at the photon energy of 1 eV it reaches a value of more than 104 cm1. This high value is consistent with an interband absorption in semiconductors. Moreover, the integrated PL intensity increased linearly with excitation intensity over three orders of magnitude, lending more credence to the notion that the observed nonsaturable peak relates to the fundamental interband transition. The absorption squared versus the energy plots used to obtain the apparent bandgap in a semiconductor with very high carrier concentration underestimates the bandgap owing to band tailing. Briot et al. [330,331] estimated a bandgap near 1.2 eV from the absorption data while taking the large carrier concentration of 1019 cm3 into account. This, however, does not account for the large discrepancy between the 0.7–0.8 eV group and 1.8–2.0 eV group. The free-electron concentration in this sample was measured by Hall effect to be 5 · 1018 cm3. Figure 1.37a also shows that the samples exhibit intense roomtemperature luminescence at energies close to the optical absorption edge. Additionally, the 77 K photoreflectance (PR) spectrum exhibits a transition feature at 0.8 eV with a shape characteristic for direct gap interband transitions. Consistent with the absorption data, no discernible change in the PR signal near 2 eV is seen. The simultaneous observations of the absorption edge and PL and PR features at nearly the same energy led Wu et al. [30] to argue that this energy position of 0.78 eV is the fundamental bandgap of InN. This value is very close to the fundamental gap for InN reported by the other group, Davydov et al. [319], in favor of the smaller bandgap for InN. Tsen et al. [332] studying nonequilibrium optical phonons in a high-quality single-crystal MBE-grown InN with picosecond Raman spectroscopy reached the conclusion that their results are not consistent with the large bandgap of InN. Using the possible phonon emission allowed by energy and momentum conservations for a range of excitation photon energies, they argued that their observations are consistent with the small bandgap of InN. The basis of the argument is that if the bandgap energy were 1.89 eV, no nonequilibrium phonons could be observed, contradicting the observation of Tsen et al. [332], details of which are discussed in Section 2.10.1. Figure 1.37b shows the room-temperature electron mobility, the peak energy of PL, and the transition energy determined by PR as functions of electron concentration, showing that the transition energies increase with increasing free-electron concentration. This indicates that the transitions from higher energy occupied states in the conduction band contribute significantly to the PL spectrum. The PL energy as a function of pressure shows a linear pressure coefficient of 0.6 meV kbar1, which is considerably smaller than that for other III–V compounds. For comparison, the pressure coefficient of GaN is 4.3 meV kbar1 [333], AlxGa1xN is 4.1 meV kbar1 for 0.12 < x < 0.6 [334], and GaAs is 11 meV kbar1 [335], as compiled by Wu et al. [30]. The sapphire substrate on which the InN layer is grown has a larger bulk modulus than InN, which will reduce hydrostatic pressure transmission to the InN film providing that the film remains coherently strained. Using experimental elastic constants for sapphire and theoretical elastic constants for InN, Wu et al. [30] estimated the correction factor for coherently strained InN on sapphire to be 1.45, leading to between 0.6 and 0.9 meV kbar1 for the bandgap change. More work is needed to shed light on this unusually low pressure dependence although

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8

6

PL or PR signal

ab. at 300 K 5 4 3 2

PR at 77 K

Absorption coefficient (104 cm-1)

7

PL at 300 K

1

0.5

1

1.5

2

0 2.5

Energy (eV)

(a) 0.90

103 mobility at 300 K

Energy (eV)

102

0.80

PL at 300 K PL at 12 K

Mobility (cm2 V-1s-1)

0.85

101

0.75 PR at 77 K

100

0.70 1018

(b)

1019

1020

Electron concentration, n (cm–3)

Figure 1.37 (a) Optical absorption (300 K), PL (300 K), and PR (77 K) spectra of a typical InN sample. This sample is undoped with roomtemperature electron concentration of 5.48 · 1018 cm3. The spike on the PR spectrum at 0.97 eV is an artifact due to the light source used in the PR measurement. (b) Room-

temperature mobility, PL peak energy (300 and 12 K), and the critical energy determined by PR (77 K) as a function of free-electron concentration. The sample with n ¼ 1 · 1019 cm3 (indicated by an arrow) is the Ritsumeikan sample.

1.5 Ternary and Quaternary Alloys

studies of InGaN showing smaller pressure dependence as the InN concentration is increased are consistent with data on InN [336]. Tyagai et al. [337] performed reflection and transmission measurements in InN with electron concentration larger than 1020 cm3. They were able to estimate an effective mass of me ¼ 0:11m0 and an index of refraction of n ¼ 3.05  0.05, which is in reasonable agreement with the value measured by Tsen et al. [332]. The longwavelength limit of the refractive index was reported to be 2.88  0.15. The temperature dependence of the InN bandgap indicates a bandgap blue shift of 23 meV from 300 to 77 K [291,338]. Inushima et al. [132] reported the effective mass to be 0.24m0 in InN grown by UV-assisted atomic layer epitaxy under atmospheric pressure. Using infrared spectroscopic ellipsometry, Kasic et al. [339] arrived at a value of m ¼ 0.14m0 in an MBE-grown InN layer having an electron concentration of n ¼ 2.8 · 1019 cm3. Using surface reflection of extrinsic semiconductors in the infrared region by the free-carrier plasma, used earlier for GaN [340], Wu et al. [341] determined the effective mass of the free carriers to be 0.07m0 at the bottom of the conduction band. This method hinges on the knowledge of the plasma frequency, electron concentration, and optical dielectric constant through the relation ne2 m ¼ ; ð1:23Þ ee¥ w2P where all the terms have their usual meanings and wP represents the plasma frequency. Using e¥ ¼ 6:7 (compares with figures in the range of 5.8–9.3 tabulated in Table 1.21) and utilizing infrared reflection measurements for a series of samples with different carrier concentrations, Wu et al. [341] arrived at an electron effective mass value of 0.07m0 at the bottom of the conduction band. The temperature dependence of the bandgap of InN indicates a bandgap temperature coefficient of [342] ðdE g =dTÞ ¼ 1:8  10  4 eV K  1 :

ð1:24Þ

1.5 Ternary and Quaternary Alloys

Many important GaN-based devices involve heterostructures as the primary means of achieving an improved performance. Ternary alloys of wurtzite polytypes of GaN, AlN, and InN have been obtained in the continuous alloy systems whose direct bandgap ranges from the old value for InN of 1.9 eV (the new value is approximately 0.7 eV according to Ref. [30]) for InN to 6.2 eV for AlN (the new value is approximately 6 eV). For an in-depth understanding of the physical mechanisms that underlie their operations, the properties of these alloys need to be extensively studied. Many of these properties such as the energy bandgap, effective masses of the electrons and holes, and the dielectric constant depend on the alloy composition. Although measured data for these parameters in InGaN and InAlN have been obtained, they are still not very precise (for AlGaN see Ref. [157] and for InGaN see Ref. [343]). Yamasaki et al. [344] have reported on p-InGaN. More research is necessary to confirm

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these conclusions regarding p-AlN and p-InGaN. AlN and GaN are slightly lattice mismatched (2.4%). It has been noted that, for many devices, only small amounts of AlN are needed in the GaN lattice to provide sufficient carrier and optical field confinements. 1.5.1 AlGaN Alloy

The ternary alloys of wurtzite and zinc blende polytypes of GaN with AlN form a continuous alloy system with a wide range of bandgap and a small change in the lattice constant. An accurate knowledge of the compositional dependence of the barrier as well as material is a requisite in attempts to analyze heterostructures in general and quantum wells (QWs) and superlattices in particular. The barriers can be formed of AlGaN or AlN, and while dependent on the barrier material, the wells can be formed of GaN or AlGaN layers. The compositional dependence of the lattice constant, the direct energy gap, and electrical and CL properties of the AlGaN alloys were measured by Yoshida et al. [345]. A similar investigation followed a few years later [346]. On the structural side, namely the calculated lattice parameter of this alloy, predictions indicate that Vegard’s law applies [347] (also reviewed in Ref. [17]): ˚ and aAlx Ga1  x N ¼ 3:1986  0:0891x A

˚: c Alx Ga1  x N ¼ 5:2262  0:2323xA ð1:25Þ

By bringing to bear various tools such as HRXRD, the experimental data for various AlGaN support the applicability of Vegard’s law in that the experi˚ and c Al Ga N ¼ mental data aAlx Ga1  x N ¼ ð3:189  0:002Þ  ð0:086  0:004Þx A x 1x ˚ ð5:188  0:003Þ  ð0:208  0:005Þx A are within about 2% of those predicted by linear interpolation, Vegard’s law. However, the bond lengths exhibit a nonlinear behavior, deviating from the virtual crystal approximation. Essentially, the nearest neighbor bond lengths are not as dependent on composition as might beexpected from the virtual crystal approximation. The ensuing investigations to pin down the compositional dependence of the bandgap of this important alloy continued with conflicting results. These are discussed below following the presentation of an empirical expression used to relate the bandgap to composition. The compositional dependence of the principal bandgap of AlxGa1xN can be calculated from the following empirical expression providing that the bowing parameter, b, is known accurately: E g ðxÞ ¼ xE g ðAlNÞ þ ð1  xÞE g ðGaNÞ  bxð1  xÞ;

ð1:26Þ

where Eg(GaN) ¼ 3.4 eV, Eg(Al N) ¼ 6.1 eV, x is the AlN molar fraction, and b is the bowing parameter.

1.5 Ternary and Quaternary Alloys

An earlier compilation by Amano et al. [348] already pinpointed the discrepancy in the reported bowing parameters. For example, Yoshida et al. [349] concluded that, as the AlN mole fraction increases, the energy bandgap of AlxGa1xN deviates upward, implying a negative value for the bowing parameter b. This contrasts the data of Wickenden et al. [350] that support a vanishing bowing parameter b. Koide et al. [351] observed that the bowing parameter is positive and that the bandgap of the alloy deviates downward indicating a positive value for the bowing parameter. To determine the bowing parameter accurately, as investigations expanded [352–362] so did the dispersion in the bowing parameters ranging from 0.8 eV (upward bowing) to þ2.6 eV (downward bowing), as compiled by Yun et al. [363]. Much of this spread emanates from the likely dispersion in the quality of AlxGa1xN, thus erroneous determination of its bandgap and to a lesser extent its lattice parameter. Because the genesis of PL transitions could be nonintrinsic, a technique relying on absorption or modulated photoreflectance is more accurate in the determination of bandgap energy. Using X-ray and surface analytical techniques, such as secondary ion mass spectroscopy (SIMS) and Rutherford backscattering (RBS), for determining composition, reflectance, and absorption for bandgap for AlGaN layers spanning the entire compositional range, Yun et al. [363] revisited the bowing parameter. The results of this study, shown in Figure 1.38 in the form of AlGaN bandgap versus the composition, yield a bowing parameter of b ¼ 1.0 eV for the entire range of alloy compositions. In this figure, the solid line represents a least square fit to the data, which in turn are depicted by solid circles. X-ray diffraction peaks generally tend to be wider for alloy compositions around the midway point that is the most likely source of error in determining the bowing parameter. The situation is exacerbated by the fact that the data points near the middle of the compositional range determine the bowing parameter to a much larger extent, as near each of the binary ends the compositional variation approaches the linear line. It is still possible that as the quality of the films improves smaller bowing parameters could result. A bowing parameter as low as 0.7 has been reported. Hall measurements for n-Al0.09Ga0.91N demonstrated a carrier concentration of 5 · 1018 cm3 and a mobility of 35 cm2 V1 s1 at 300 K [364]. This measurement did not reveal any temperature-dependent mobility of n-A10.09Ga0.91N. Other Hall measurements [365] on Mg-doped p-Al0.08Ga0.92N grown by MOVPE, however, addressed the temperature dependence of the mobility [365]. They indicate that the hole mobility decreases with increasing temperature, reaching a value of about 9 cm2 V1 s1 for a doping density of 1.48 · 1019 cm3. This low mobility is ascribed to a high carrier concentration and the intergrain scattering present in the samples. While the lattice constant was studied, it was observed to be almost linearly dependent on the AlN mole fraction in AlGaN. Until recently, the resistivity of unintentionally doped AlGaN was believed to increase so rapidly with increasing AlN mole fraction that AlGaN became almost insulating for AlN mole fractions exceeding 20%. As the AlN mole fraction increased from 0 to 30%, the n-type carrier concentration dropped from 1020 to 1017 cm3 and the mobility increased from 10 to 30 cm2 V1 s1. An increase in

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6.5

Bangap of AIGaN (eV)

6.0

5.5

5.0

b=0

b=1

4.5

4.0

3.5 0

0.2

0.4

0.6

0.8

1

Al composition (x) Figure 1.38 Experimental data of energy bandgap of AlGaN (0  x  1), plotted as a function of Al composition (solid circle), and the least squares fit (solid line) giving a bowing parameter of b ¼ 1.0 eV. The dashed line shows the case of zero bowing. As the quality of the near 50 : 50 alloy layers get better, giving rise to sharper X-ray and PR data, smaller bowing parameters may ensue. Bowing parameters as low as 0.7 have been reported.

the native defect ionization energies with increasing AlN may possibly be responsible for this variation. Our knowledge of the doping characteristics of AlGaN is still incomplete. For example, it is not known how the dopant atoms such as Si and Mg respond to the variation of the AlN mole fraction in AlGaN. However, it was suggested that as the AlN mole fraction increases, the dopant atom moves deeper into the forbidden energy bandgap. AlGaN with an AlN mole fraction as high as 50–60% may be doped by both n- and p-type impurity atoms. The ability to dope a high mole fraction AlGaN, especially when low-resistivity p-type material is required, is important because it may otherwise restrict the overall characteristics of devices such as laser diodes. A low AlN mole fraction in AlGaN has been considered sufficient for acceptable optical field confinement. However, this must be addressed before the potential of AlGaN with respect to the other wide bandgap semiconductors is fully realized. 1.5.2 InGaN Alloy

The growth of high-quality InN and an enumeration of its fundamental physical properties remain somewhat elusive as compared to the other alloy, AlGaN.

1.5 Ternary and Quaternary Alloys

Notwithstanding the difficulties in technology, InGaN is already an integral part of important device designs. InxGa1xN (x is the InN mole fraction) is not any less important than AlxGa1xN for the fabrication of electrical and optical devices, such as LEDs and lasers, which can emit in the violet or blue wavelength range. It can be a promising strained QW material for these devices, but added complexities such as the phase separation and other inhomogeneities make the determination of the bandgap of InGaN versus the composition a very difficult task. The first growth of single crystalline InGaN by MOVPE was realized by Nagatomo et al. in 1989 [366] and Matsuoka et al. [367], followed by Yoshimoto et al. in 1991 [368]. Since then, considerable effort has been expended worldwide on this material, as it is responsible for emission in the near-UV, violet, blue, and green colors of the optical spectrum. High-efficiency blue and green LEDs utilizing InGaN active layers are commercially available. This material, however, is not as easy to grow because of the high vapor pressure of N on In and also mismatch between the large In atom and the small N atom. To mitigate this problem, V/III ratios in excess of 20 000, increasing with InN mole fraction, as well as reduced growth temperatures are employed. Matsuoka et al. [369] discovered that lowering the growth temperature to 500
C from nominal temperatures such as 800
C increased the In content in the layers, but at the expense of reduced quality. Efforts to increase the In concentration by raising the indium precursor temperature or the carrier gas flow rate resulted in the degradation of the structural and surface morphology so much that In droplets were formed on the surface [370]. The great disparity between Ga and In could lead to issues such as phase separation and instabilities. In this vein, Ho and Stringfellow [371] investigated the temperature dependence of the binodal and spinodal boundaries in the InGaN system with a modified valence force field model. The calculation of the extent of the miscibility gap yielded an equilibrium InN mole fraction in GaN of less than 6% at 800
C [371]. In the annealing experiments in argon ambient, the phase separation in an InxGa1xN alloy with x 0.1 was observed at temperatures between 600 and 700
C [372], pointing to the large region of solid immiscibility of these alloys. However, under nonequilibrium growth conditions, InxGa1xN layers were grown in the entire range of compositions. But, the decomposition into two phases upon annealing of the InxGa1xN alloys (x ¼ 0.11 and x ¼ 0.29) at 600 and 700
C was observed pointing to the existence of the miscibility gap. For some alloys with x ¼ 0.6, the phase separation could not be observed at 600
C. Above 800
C, the alloy samples with x ¼ 0.1 actively evaporated from the substrate. These results suggest that the solid solutions are grown in metastable conditions and decomposed under annealing conditions. Koukitu et al. [373] performed a thermodynamical analysis of InGaN alloys grown by MOCVD. They found that in contrast to other III–III–V alloy systems where the solid composition is a linear function of the molar ratio of the group III metalorganic precursors at constant partial pressure of group V gas, the solid composition of InGaN deviates significantly from a linear function at high substrate temperatures. Kawaguchi et al. [374] reported a so-called InGaN composition pulling effect in which the indium fraction is smaller during the initial stages of growth but increases with increasing growth thickness. This observation was to a first extent independent

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of the underlying layer, GaN or AlGaN. The authors suggested that this effect is caused by strain caused by the lattice mismatch at the interface. They found that a larger lattice mismatch between InGaN and the bottom epitaxial layers was accompanied by a larger change in the In content. What one can glean from this is that the indium distribution mechanism in InGaN alloy is caused by the lattice deformation because of the lattice mismatch. With increasing thickness, the lattice strain is relaxed owing to the formation of structural defects and roughness, which weakens the composition pulling effect. Other substrates have also been used for InGaN growth. It has been reported that the crystalline quality of InGaN is superior when grown with the composition that lattice matches ZnO substrate to that grown directly on (0 0 0 1) sapphire substrate [368,369]. In the same investigations, it was observed that InGaN films grown on sapphire substrates using GaN as buffer layers exhibited much better optical properties than InGaN films grown directly on sapphire substrates [375]. For a given set of growth conditions, an increase of InN in InGaN can be achieved by reducing the hydrogen flow [376]. As in the case of AlGaN, the calculated lattice parameter of this alloy follows Vegard’s law [347] (also reviewed in Ref. [17]): ˚ aInx Ga1  x N ¼ 3:1986 þ 0:3862x A

and

˚ c Inx Ga1  x N ¼ 5:2262 þ 0:574x A: ð1:27Þ

By bringing to bear various tools such as HRXRD, the experimental data for various AlGaN support the applicability of Vegard’s law in that the experimental data ˚ and c Al Ga N ¼ ð5:195  0:002Þþ aAlx Ga1x N ¼ ð3:560  0:019Þþð0:449  0:019Þx A x 1x ˚ are within about 2% of that predicted by linear interpolation, ð0:512  0:006Þx A Vegard’s law. As in the case of AlGaN but to a larger extent, the bond lengths exhibit a nonlinear behavior, deviating from the virtual crystal approximation. Essentially, the nearest neighbor bond lengths are not as dependent on composition as might be expected from the virtual crystal approximation. The compositional dependence of InGaN bandgap is a crucial parameter in designs of any heterostructure utilizing it. As such, the topic has attracted a number of theoretical [377–382] and experimental (to be discussed below) investigations and reports. Similar to the case of AlGaN, the energy bandgap of InxGa1xN over 0  x  1 can be expressed by the following empirical expression: g

g

g

E Inx Ga1  x N ¼ xE InN þ ð1  xÞE GaN  bInGaN xð1  xÞ ¼ 0:7x þ 3:4ð1  xÞ  bInGaN xð1  xÞ eV; g

g

ð1:28Þ

where E GaN ¼ 3:40 eV and E InN ¼ 1:9 eV or near 0.7 eV. For the nomenclature GaxIn1xN, the terms x and 1  x in Equation 1.28 must be interchanged. Another point of caution is that the sign in front of the bowing parameter is changed to positive in some reports.When a comparison ismade, the sign of theb parametermust bechanged. An earlier investigation of InGaN bowing parameter for alloys with small concentrations of InN by Nakamura et al. [383] led to a bowing parameter of 1.0, which is in disagreement with the value of 3.2 reported by Amano et al. [348], who took into

1.5 Ternary and Quaternary Alloys

consideration the strain and piezoelectric fields as well. It should be mentioned that these reports dealt with the Ga-rich side of the alloy. To obtain a fit over a large range of compositions, a composition-dependent bowing parameter has been suggested. As the InGaN is grown on GaN, there are many complicating factors, such as the piezoelectric effect and the nonuniform strain; the impact of the former can be made negligible by growing thick films. Moreover, compositional inhomogeneities due to partial phase separation are present. If the strain caused by the lattice mismatch were uniform, it would be compressive due to the InN lattice constant being 11% larger than that of GaN with an accompanying blue shift of the band edge. Herein lies the dilemma faced by the experimentalists. Growing thick films could minimize the extent of strain and the piezoelectric effect. However, this is nearly an intractable proposition. The relaxation value of the lattice constant and the origin of the optical transitions must be known accurately to determine the bandgap versus composition dependence. Absorption and/or reflection measurements, provided that the absorption edge is sharp, are more useful in determining the bandgap but again require thick and/or high-quality films. Detailed X-ray reciprocal-space mapping undertaken by Amano et al. [348] purportedly indicated that InGaN wells and even somewhat thicker InGaN layers grown on GaN buffer layers are coherently strained; a conclusion reached by the observation that the in-plane lattice constants of GaN and InGaN match. At the same time, though, the layer thicknesses well exceeded the calculated critical values. When a bandgap of 1.9 eV for InN is assumed as the end point value for InN in regard to InGaN ternary, large and/or more than one bowing parameter is required to fit the compositional dependence of the bandgap energy. For example, a bowing parameter of 2.5 eV was obtained from optical absorption measurements and a value of 4.4 eV was obtained from the position of the emission peaks [384]. Nagatomo et al. [366] noted that the InxGa1xN lattice constant varies linearly with the In mole fraction up to at least x ¼ 0.42, but it violates the Vegard’s law for x > 0.42, which may be caused by erroneous determination of the composition and illustrates well the problem at hand. Even additional investigations did not agree on the exact value of the bowing parameter. For example, a value for bInGaN ¼ 3.9  0.5 eV was reported when 0.9 eV was used for the InN gap, but the bowing parameter had to be increased to 5.1  0.4 eV when 1.9 eV was used for the InN gap [385]. Using the bandgap determined by PL, a bowing parameter of 4.5 eV was also reported [386]. However, when reflectivity measurements together with PL data corrected for Stokes shift were used, bInGaN ¼ 2.5  0.7 eV was obtained for 0.9 eV bandgap of InN and bInGaN ¼ 3.5  0.7 eV for 1.9 eV InN bandgap. Optical transmission measurements led to a bowing parameter of 8.4 eV [387]. At least one theoretical effort resulted in a bowing parameter of 1.2 eV [377]. In fact, linear bandgap dependence on composition with a slope of 3.57 eV for up to 25% InN content has also been reported [388]. Linear dependence with a slope of 4.1 eV for InN mole fraction, x < 0.12, has been reported in another publication as well [389]. Wu et al. [390] visited the bandgap dependence of InGaN on composition by considering 0.8 eV for the bandgap of the end binaryInN. Figure1.39 shows the composition dependence of the bandgap of InGaN, determined by using photomodulated transmission [391] and optical absorption [392] measurements, as a

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Lattice constant (Å) 3 .2

Energy gap (eV)

4

3.3

3 .4

3 .5

3 .6

abs. 300 K PL PT Old data Calorimetric

GaN

3 2

InN

1 0 0.0

0.2

0.4

0.6

0.8

1.0

Composition (x) Figure 1.39 PL peak energy and bandgap of InGaN determined by optical absorption as a function of composition, as compiled in Ref. [390], including previously reported data for InN. The solid curve shows the fit to the bandgap energies (determined by absorption and phototransmission) using a bowing parameter b ¼ 1.43 eV [390].

function of GaN fraction. The data near the GaN binary end include those reported by Pereria et al. [392], Shan et al. [390], and O’Donnel et al. [393]. Care was taken by observing the dependence of squared absorption coefficient on light probe energy and seeing nearly a linear dependence to gain confidence on the measured bandgap and also confirming the values by bandgaps determined by photomodulated transmission measurements. The slight deviation from linearity near the InN end of the ternary has been attributed to the nonparabolicity of the conduction band caused by the k p interaction between the G6 symmetry conduction band and the G8 symmetry valence bands. As shown by the solid curve in Figure 1.39, the compositional dependence of the bandgap in the entire composition range can be well fit by a bowing parameter of b ¼ 1.43 eV. Shown in Figure 1.39 with dashed line is the fit to the empirical expression using energy of 1.9 eV for InN and bowing parameter of 2.63 eV to demonstrate that it does represent the Ga-rich side of the compositions well. However, the bowing parameter of 1.43 eV that is good for the entire compositional range is the one that utilizes 0.77 eV for the bandgap of InN. In an investigation with a different set of objectives, Yoshimoto et al. [368] studied the effect of growth conditions on the carrier concentration and transport properties of InxGa1xN. They observed that if the deposition temperature is increased from 500 to 900
C, InxGa1xN grown on sapphire with x 0.2 suffers from a reduction in carrier concentration from 1020 to 1018 cm3, but gains from an increase in the carrier mobility from less than 10 to 100 cm2 V1 s1. The same group later noted that this trend does not change if the films are grown on ZnO substrates instead of sapphire [369]. They could achieve good InGaN material with In mole fractions as

1.5 Ternary and Quaternary Alloys

large as 23%. Nakamura and Mukai [394] discovered that the film quality of InxGa1xN could be significantly improved if these films are grown on high-quality GaN films. Thus, from the reports cited above it may be concluded that the major challenge for obtaining high-mobility InGaN is to find a compromise in the growth temperature, because InN is unstable at typical GaN deposition temperatures. This growth temperature would undoubtedly be a function of the dopant atoms, as well as the method (MBE, OMVPE, etc.) used for the growth. This is evident from a study by Nakamura et al., who have since expanded the study of InGaN employing Si [395] and Cd [396] as dopants. A review of various transport properties of GaInN and AlInN by Bryden and Kistenmacher [296] is available but predates the bandgap reconsideration of InN; the growth and mobility of p-GaInN is discussed by Yamasaki et al. [344]. 1.5.3 InAlN Alloy

In1xAlxN is an important compound that can provide a lattice-matched barrier to GaN, low fraction AlGaN, and InGaN, and consequently lattice-matched AlInN/ AlGaN or AlInN/InGaN heterostructures. Although there was some discrepancy as to which composition really lattice-matched GaN, continued improvement in layer quality and persistence narrowed the In composition for matching. Compositions In0.29Al0.71N and In0.17Al0.83N have been reported as matching, but the value around the latter composition is gaining more acceptance [397]. The growth and electrical properties of this semiconductor have not yet been as extensively studied compared to the other two ternaries, particularly AlGaN, as the growth of this ternary is also challenging because of diverse thermal stability, lattice constant, and cohesive energy of AlN and InN. Moreover, thermal instability resulting from, for example, the spinodal phase separation phenomenon, which is more of an issue in Al1xInxN than in InyGa1yN [398], must be considered. Despite the above-mentioned difficulties, lattice matching and the lack of crack formation when AlGaN is replaced with InAlN in distributed Bragg reflectors (DBRs) and other structures requiring relatively thicker layers are more than enough to pursue this material. In fact, light emitters, field effect transistors, and DBRs, as mentioned, using InAlN barriers as opposed to AlGaN are gaining considerable momentum. As in the case of AlGaN and InGaN, the calculated lattice parameter of this alloy follows Vegard’s law [347] (also reviewed in Ref. [17]) as ˚ aAlx In1  x N ¼ 3:58480  4753x A

˚ and c Alx In1  x N ¼ 5:8002  0:8063x A: ð1:29Þ

By utilizing various tools such as HRXRD, the experimental data for various AlInN layers support the applicability of Vegard’s law in that the experimental data ˚ and c ¼ ð5:713  0:014Þ  aAlx In1  x N ¼ ð3:560  0:019Þ  ð0:449  0:019Þx A ˚ ð0:745  0:024Þx A are within about 2% of that predicted by linear interpolation, the Vegard’ law. As in the case of AlGaN and InGaN, the bond lengths exhibit a

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nonlinear behavior, deviating from the virtual crystal approximation. Essentially, the nearest neighbor bond lengths are not as dependent on composition as might be expected from the virtual crystal approximation. Early experimental results [288] for the bandgap of In0.29Al0.71N, which was thought lattice matched to GaN, indicate that this alloy has an energy gap of 3.34 eV at low temperatures (the room-temperature value is actually closer to 4.5 eV) that is even below that for GaN. The estimations by Wright and Nelson [399] that followed pointed to a bandgap of about 5 eV for the zinc blende variety. The accompanying bowing parameter reported by Wright and Nelson is 2.53 eV at the time when the larger InN bandgap was accepted. Naturally, when the bandgap is in question the bowing parameter is even more in question. As in the case of AlGaN and InGaN, the compositional dependence of the bandgap of AlInN can be expressed with the following empirical expression using a bowing parameter, bAlInN, as g

g

g

E Alx InN ¼ xE AlN þ ð1  xÞE InN  bAlInN xð1  xÞ ¼ 6:1x þ 0:7ð1  xÞ  bAlInN xð1  xÞ eV:

ð1:30Þ

In addition to the aforementioned calculations, experimental data for the bowing parameter, b, exist. Using a bandgap of 6.2 eV for AlN (the new figure is closer to 6 eV), the values that have been reported encompass b ¼ 3.1 eV deduced by fitting the bandgap of this alloy determined by PL [400], b ¼ 2.384 eV by absorption measurements but by using 2.0 eV for the bandgap of InN and 5.9 eV for AlN [401], and b ¼ 5.4 eV in a review where 1.95 eV was used for InN bandgap [17]. Scaling the bandgap of AlN to about 6.0 eV would reduce the bowing parameter a little. Despite the scattered data, reasonably useful bandgap variation of AlInN with composition can be obtained as shown in Figure 1.40. Kim et al. [404] deposited thin AlInN films with X-ray rocking curve FWHM values between 10 and 20 arcmin. They observed an increase of In content in AlInN of up to 8% by lowering the substrate temperature to 600
C. A further reduction of substrate temperature during OMVPE is not useful because of the needed efficient pyrolysis of ammonia. Yamaguchi et al. [405] also reported on OMVPE growth of AlInN on GaN templates that were in turn deposited on low-temperature AlN nucleation layers on cplane sapphire. In macroscopic sense, the alloys grown were not phase separated and the bandgap variation followed the compositional variations in the InN composition range of 19–44%. From the square of the absorption coefficient versus E–Eg, the bandgap of the alloy was determined. Starosta [406] and later Kubota et al. [407] grew InAlN alloy by radio frequency (RF) sputtering. Kistenmacher et al. [408], however, used the RF magnetron sputtering (RF MS) from a composite metal target to grow InAlN at 300
C. It was observed that the energy bandgap E of this semiconductor varies between 2.0 eV (this is supposed to represent the InN binary end point, which assumes the old and incorrect value) and 6.20 eV (this too represents the old value for the bandgap of AlN with the new figure being approximately 6 eV) for x between 0 and 1 [407]. The carrier concentration and the mobility of In1xAlxN for x ¼ 0.04 were 2 · 1020 cm3 and 35 cm2 V1 s1, respectively, and for x ¼ 0.25 were 8 · 1019 cm3 and 2 cm2 V1 s1,

1.5 Ternary and Quaternary Alloys

Lattice constant (Å) 7 a b c d e

AlN

6

Energy gap (eV)

5

f g h i j k l

4 3 2

InN

1 Eg=6(1–x)+0.7x–3.1x(1–x)

0 0.0

0.2

0.4

0.6

0.8

1.0

Composition (x) Figure 1.40 Dependence of bandgap of the InAlN alloy on composition. Unless otherwise stated, the measurement temperature is room temperature. The solid line between the 0.8 eV gap of InN and 6 eV of AlN is deemed as being reasonably accurate. (a) Absorption; (b) RT PL;

(c) RT absorption [404]; (d) Ref. [401]; (e) absorption, poly [416]; (f) absorption [402]; (g) a theory [403]; (h) RT PL and CL [400]; (i) 8 K optical reflection [400]; (j) RT absorption [405]; (k) RT PL [405]; (l) fit to Eg ¼ 6(1  x) þ 0.7 x  3.1 x (1  x). In part courtesy of Wladek Walukiewicz.

respectively [296]. Thus, the mobility decreases substantially with an increase in the Al mole fraction because the structure of the InAlN approaches the structure of the insulating AlN. 1.5.4 InAlGaN Quaternary Alloy

By alloying InN together with GaN and AlN, the bandgap of the resulting alloy(s) can be increased from 1.9 eV (or near 0.7 eV if we use the updated InN bandgap) to a value of 6.2 eV (or 6 eV if we use the updated value), which is critical for making highefficiency visible light sources and detectors. In addition, the bandgap of this quaternary can be changed while keeping the lattice constant matched to GaN [409,410]. In quaternary alloys of nitrides, the N atoms constitute anion sublattice, whereas group III elements (In, Ga, Al) constitute the cation sublattice. Use of this quaternary material allows almost independent control of the bandgap and thus the band offset in AlInGaN-based heterostructures. However, among other difficulties brought about by the four-component system, the optimal growth temperature is important to optimize and control, as aluminum-based compounds generally require higher growth temperatures and In-based ones require lower

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temperatures. Higher temperatures are also desirable for reducing the O incorporation in the growing film as oxides of Ga and In desorb from the surface. The growth temperature will therefore govern the limits of In and Al incorporation into the AlGaInN quaternary alloy [409]. The quaternary alloy (Ga1xAlx)In1yN is expected to exist in the entire composition range 0 < x < 1 and 0 < y < 1. Unfortunately, as in the case of the InGaN alloy, incorporation of indium in these quaternary alloys is not easy. To prevent InN dissociation, InGaN crystals were originally grown at low temperatures (about 500
C) [411], which also applies to InGaAlN. The use of a high nitrogen flux rate allowed the high-temperature (800
C) growth of high-quality InGaN and InGaAlN films on (0 0 0 1) sapphire substrates. Note that the incorporation of indium into InGaN film is strongly dependent on the flow rate, N/III ratio, and growth temperature in an OMVPE environment. The incorporation efficiency of indium decreases with increasing growth temperatures. Observations made in the case of InGaN should be applicable to In incorporation in quaternary nitrides. Ryu et al. [412] reported on the optical emission in this quaternary system and AlInGaN/AlInGaN multiple quantum wells grown by pulsed metalorganic chemical vapor deposition. Strong blue shift with excitation intensity was observed in both the quaternary layers and quantum wells that was attributed to localization. This would imply the inhomogeneous nature of the structures and/or presence of band tail states indicative of early stages of material development and/or serious technological problems involved. The relationships between composition and bandgap (or lattice constant) can be predicted by the equation below, which was originally developed for the InGaAsP system [413]. Qðx; y; zÞ ¼

xyT 12 ðð1  x þ yÞ=2Þ þ yzT 23 ðð1  y þ zÞ=2Þ þ zxT 31 ðð1  z þ xÞ=2Þ ; xy þ yz þ zx

T ij ðaÞ ¼ aBj þ ð1  aÞBi þ að1  aÞbij : The parameters x, y, and z represent the composition of GaN, InN, and AlN. If GaN, InN, and AlN are represented by 1, 2, and 3, T12 would represent GaxInyN. Further, the term T12 can be expressed as T 12 ðaÞ ¼ aB2 þ ð1  aÞB1 þ að1  aÞb12 , where b12 is the bowing parameter for the GaxInyN alloy and a ¼ ð1  x þ yÞ=2 or ð1  x þ yÞ=2 or ð1  z þ xÞ=2 is the effective molar fraction for GaInN, InAlN, and AlGaN, respectively, B2 the bandgap of InN, and B1 is the bandgap of GaN. Similar expressions can be constructed for T23 and T31 by appropriate permutations. An alternative approach is discussed in conjunction with Equation 1.31. The results of these calculations for the bandgap and lattice constant dependence on composition are shown in the three-dimensional diagrams of Figures 1.41–1.43. An empirical expression similar to that used for the ternaries can also be constructed for the quaternary as g

g

g

g

E Alx Iny Ga1  x  y N ¼ xE AlN þ yE InN þ ð1  x  yÞE GaN  bAlGaN xð1  xÞ  bInGaN yð1  yÞ;

ð1:31Þ

1.5 Ternary and Quaternary Alloys

Figure 1.41 Bandgap versus composition for quaternary AlxInyGa1xyN (assumed InN bandgap ¼ 0.8 eV). (Please find a color version of this figure on the color tables.)

Figure 1.42 Bandgap versus composition for quaternary AlxInyGa1xyN (assumed InN bandgap ¼ 1.9 eV). (Please find a color version of this figure on the color tables.)

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Figure 1.43 Lattice constant a versus composition for quaternary AlxInyGa1xyN.

where the first three parameters on the right-hand side of the equation are contributions by the binaries to the extent of their presence in the lattice, the third term represents the bowing contribution related to Al, and the last term depicts the bowing contribution due to In. The bowing parameters, bAlGaN and bInGaN, indicated in Equation 1.31 are the same as those discussed in conjunction with InGaN and AlInN. As such, the values are the same. The parameters x, y, and z represent the molar fraction of binaries in the quaternary. After discussing all three ternary alloys of the nitride semiconductor family, the bandgap (both in terms of energy and also corresponding air wavelength) versus the lattice parameter is shown in Figure 1.44 for convenience. The discussion of alloys individually up to this point paves the way to a collective discussion of alloys in terms of structural parameters for a rapid observation of trends. This discussion would be of special value particularly for the least discussed of alloys, InAlN. Let us first discuss the structural properties such as the lattice constants and bond lengths, and angles of nitride semiconductor alloys, following the discussion in Sections 1.5.1–1.5.3 and that surrounding Figure 1.8, Equation 1.3, and Table 1.2. Following Ref. [17], the lattice parameter calculated using Equation 1.25 (for AlGaN), Equation 1.27 (for InGaN), and Equation 1.29 (for InAlN) can be used to calculate the lattice constants for the three ternaries for all compositions and compared with experiments for AlGaN [414], InGaN [415], and InAlN [416], as shown in Figure 1.45.

1.5 Ternary and Quaternary Alloys

Figure 1.44 The bandgap versus the lattice parameter for AlGaN, InGaN, and InAlN using bowing parameters in the same order, 1, 1.43, and 3.1 eV, and bandgap values of 6 eV for AlN, 3.4 eV for GaN, and 0.8 eV for InN. The lattice constants used for the binary AlN, GaN, and InN are 3.11, 3.199, and 3.585 Å, respectively.

Following the case for the binaries tabulated in Table 1.2 and discussed from a theoretical point of view in Refs [347,417], and the experimental points of view in Refs [414] (for AlGaN), [415] (for InGaN), and [416] (for AlxIn1xN), the cell parameter, u, has been calculated for randomly distributed A0.5B0.5N (here A and B represent the metal components forming the alloy) alloys by the theoretical approach of Ref. [418], the pertinent parts of which are succinctly discussed in Section 1.1. The internal cell parameter can be approximately expressed by the quadratic equation uAx B1  x N ¼ xuAN þ ð1  xÞuBN  bAB xð1  xÞ;

ð1:32Þ

where bAB is the bowing parameter defined as bAB ¼ 2Y AN þ 2Y BN  4Y A0:5 B0:5 N :

ð1:33Þ

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3.8

(a)

Lattice constant, a (x) (Å)

3.6

Alx In1–xN Inx Ga1–xN

3.4

Alx Ga1–x N

3.2

Theory (T = 0 K) Experiment (T = 300 K) 3.0 0

0.2

0.6 0.4 Molar fraction, x

0.8

6.0

1.0

(b)

5.8

Alx In1–xN

Lattice constant c(x) (Å)

Inx Ga1–xN 5.6

5.4

Alx Ga1–x N

5.2

5.0 Theory (T = 0 K) Experiment (T = 300 K) 4.8 0

0.2

0.4

0.6

0.8

1.0

Molar fraction, x Figure 1.45 (a) The a(x) lattice parameter and (b) the c(x) lattice parameter for random ternary alloys of AlxGa1xN, InxGa1xN, and AlxIn1xN as measured by HRXRD at room temperature (solid lines) and the calculated values using Equation 1.25 (for AlGaN) and Equation 1.27 (for

InGaN) for T ¼ 0 K (dashed lines). The agreement between calculations and measured lattice constants is better than 2% over the entire range of compositions, compiled in Ref. [17] utilizing Refs [414,415]. Courtesy of O. Ambacher.

1.5 Ternary and Quaternary Alloys

The internal cell parameters for each of the three alloys then are uAlx Ga1  x N ¼ 0:3819x þ 0:3772ð1  xÞ  0:0032xð1  xÞ; uInx Ga1  x N ¼ 0:3793x þ 0:3772ð1  xÞ  0:0057xð1  xÞ; uAlx In1  x N ¼ 0:3819x þ 0:3793ð1  xÞ  0:0086xð1  xÞ:

ð1:34Þ

The structural and other polarization related parameters of ternaries do not follow a linear relationship of the composition, as discussed in detail in Section 2.7. The nonlinearity in question for an alloy, AxB1xN, where A and B represent the metal components, is approximated by quadratic equations of the form [418] Y Ax B1  x N ¼ xY AN þ ð1  xÞY BN  bAB xð1  xÞ;

ð1:35Þ

where Y represents any parameter, namely the lattice constant, the u parameter or polarization, and the bowing parameter is defined in Equation 1.33. As in the case of binaries discussed in Section 1.1, the cell parameter, u, and the c/a ratio do not follow the ideal crystal values for the three ternaries of nitride semiconductors. They are shown for the three ternaries for varying composition in Figure 1.46. Similar to the binaries, tabulated in Table 1.2 in conjunction with Figure 1.8, the aforementioned two parameters, the nearest and the second neighbor bond lengths, as well as the bond angles have been calculated for the three ternaries and those associated with 50% alloys are tabulated in Table 1.34. As displayed in Figure 1.46, the cell internal parameter increases as one goes from GaN to InN and, more significantly, to AlN. The nonlinear dependencies on the alloy composition are described by a bowing parameter, bAB, whose values are 0.0032, 0.0057, and 0.0086 for AlxGa1xN, InxGa1xN, and AlxIn1xN, respectively. The bowing parameter increases from AlxGa1xN to InxGa1xN, and continues on to AlxIn1xN. It is worth noting that the bowing parameter is negative for all the three ternaries, the average cell internal parameter of the same alloys is always above the ideal value of 0.375. If the lattice constants scale linearly with the alloy composition but the internal parameter does not, the bond angles and/or the bond lengths of the real and the virtual crystal must depend nonlinearly on the alloy composition. The average nearest neighbor bond lengths (b and b1, see Figure 1.8 for a graphical description) and bond angles (see Figure 1.8 for a graphical description) calculated by using Equations 1.2–1.4 are shown in Figure 1.47a and b and listed in Tables 1.2 and 1.34. The average cation–anion distances to the nearest and second nearest neighbors scale nearly linearly with alloy composition for AlxGa1xN, InxGa1xN, and AlxIn1xN. The average bond length along the c-axis is 0.7–0.9% longer than the nearest neighbor bonds in the direction of the basal plane (Figure 1.47a). 1.5.5 Dilute GaAs(N)

When small amounts of N and As are incorporated into GaAs and GaN lattices, respectively, a large negative bandgap bowing parameter results. Consequently, with

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very small amounts of N in the GaAs lattice, its bandgap can be made very small, to a point where 1.3 mm lasers and 1.5 mm lasers if In and Sb are also added to the lattice can all be built with GaAs technology. Anomalously large bandgap bowing parameters exhibited by GaAsN and GaNAs are caused by large chemical and size 1.64 (a)

Ideal

1.63

c(x)/a(x) ratio

1.62

Inx Ga1–xN

Alx Ga1–xN

1.61

Alx In1–xN

1.60 Theory (T = 0 K) Experiment (T = 300 K)

1.59 0

0.2

0.4

1.0

0.8

0.6

molar fraction, x 0.383 (b)

cell-internal parameter u(x)

0.381

Alx Ga1–x N

b = –0.0032

0.379

Alx In1–xN

b = 0.0086

Inx Ga1–xN

0.377

b = –0.0057

Ideal

0.375

0

0.2

0.6 0.4 molar fraction, x

0.8

1.0

1.5 Ternary and Quaternary Alloys Table 1.34 Calculated cell internal parameter, a lattice parameter,

c/a ratio, cation–anion distance between the nearest and second nearest neighbors, and bond angles (given in degrees) for the three ternary random alloys in the virtual crystal limit with a composition of 50%.

u a (Å) c/a b (Å) b1 (Å) 0 b1 (Å) 0 b2 (Å) 0 b3 (Å) a b

Al0.5Ga0.5N

In0.5Ga0.5N

In0.5Al0.5N

0.379 3.154 1.620 1.935 1.924 3.175 3.701 3.694 108.80 110.14

0.377 3.392 1.625 2.078 2.073 3.436 3.977 3.975 109.13 109.81

0.378 3.347 1.612 2.042 2.041 3.354 3.921 3.920 108.76 110.18

The distance is in Å and the angles are in degrees [17].

~

differences between As and N [419–422]. Dependence of the bandgap energy in GaAsN and InPN on nitrogen content is shown in Figure 1.48. To a first extent, the dashed lines originating from both GaN end (in which case small amounts of As are added to GaN) and GaAs end (in which case small amounts of N are added to GaAs) represent the bandgap dependence of GaNAs. However, one must keep in mind that for both GaAsN and InPN the simple treatment behind the aforementioned statement fails and that the arrows shown in the figure indicate the boundaries of the regions where the gap dependence on composition may be predicted with any accuracy. Also shown is the bandgap variation with composition for other commonly used ternaries. The thicker vertical line through GaAs represents the bandgap attainable with GaInAsN, at least in theory, while maintaining lattice matching to GaAs. The decrease in the lattice constant caused by N can be compensated with In added to the lattice. The potential of covering a large range of bandgap energies on GaAs substrates has attracted a great deal of interest in this material system. In fact, the first laser containing N was an InGaAs(N) active layer one. Owing in part to

Figure 1.46 (a) The c/a ratio for the three random ternary alloys determined by HRXRD at room temperature (solid lines) and calculated using Equation 1.24 for T ¼ 0 K (dashed lines). The measured and calculated data confirm that the c/a ratios of Wz InGaN, AlGaN, and AlInN crystals are always less than the value of 1.633 for ideal hexagonal crystal. (b) The cell internal parameter, u, for three random AlGaN, InGaN, and AlInN alloys calculated using the quadratic

Equation 1.35. The nonlinearity of the internal cell internal parameter in its compositional dependence can be described by a negative bowing parameter b. This bowing parameter is 0.0032, 0.0057, and 0.0086 for AlxGa1xN, InxGa1xN, and AlxIn1xN, respectively, as indicated in the figure as well. The u parameter of the ternaries is always larger than 0.375 that is the value for an ideal hexagonal crystal [17]. Courtesy of O. Ambacher.

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(a) M-Nc1

InN

2.2

InN

Metal-N average bond length (A)

M-Nb1

2.1 Inx Ga1–xN Alx In1–xN

2.0 Alx Ga1–xN

M-Nc1

GaN b=M-Nc1 b1= M-Nb1

1.9

0

0.2

0.4

M-Nb1 AlN 0.6

0.8

1.0

Molar fraction, x

111

T=0K

(b)

AlN

Bond angle of virtual lattice (deg)

InN

Alx In1-x N

110

InN

Alx Ga1-x N GaN

Inx Ga1-x N

ideal: α = β =109.47 ο

Inx Ga1-x N 109

β α

Alx Ga1-x N Alx In1-x N

AlN 108

0

0.2

0.8 0.6 0.4 Molar fraction, x

Figure 1.47 (a) The compositional dependence of the average nearest neighbor bond lengths, b and b1 (see Figure 1.8 for a graphical description) in the virtual crystal limit for the metal–nitrogen bonds along the c-axis (solid line) and off c-axis (dashed line). (b) The compositional dependence of the average bond angles a (dashed lines) and b (solid lines) of random

1.0

AlxGa1xN, InxGa1xN, and AlxIn1xN alloys (see Figure 1.8 for a graphical description). Clearly, the average bond angles deviate noticeably from the ideal hexagonal crystal for which a ¼ b ¼ 109.47. Moreover, the deviation increases from GaN to InN and continues onto AlN in a nonlinear fashion [17]. Courtesy of O. Ambacher.

1.5 Ternary and Quaternary Alloys

Γ valley energy gap (eV)

6

AlN

zinc blende T=0K

5 4

AlP AlAs

3

GaP

GaN

2 InN 1

AlSb

GaAs GaAsN InPN

0 4.5

5.0

5.5

InP GaSb InSb InAs 6.0

6.5

Lattice constant (A) Figure 1.48 Direct G valley energy gap as a function of lattice constant for the zinc blende form of 12 III–V binary compound semiconductors (filled circles) and some of their random ternary alloys (lines connecting the solid circles) at zero temperature. The energy gaps for certain ternaries such as AlAsP, InAsN, GaAsN, InPN, and GaPN are extended into regions where

no experimental data have been reported. For GaAsN and InPN, the arrows indicate the boundaries of the regions where the gap dependence on composition may be predicted with any accuracy, patterned after Ref. [423] with necessary changes, particularly the one reflecting the small bandgap of InN.

extreme nonequilibrium conditions employed for growth, MBE is the dominant growth approach for dilute arsenides with nitrogen. The critical issues are compositional control, incorporation of more than a small percentage of N, doping inefficiency, and layer quality. The situation is exacerbated on all fronts when the N concentration is increased for achieving 1.5 mm wavelength of emission. Postgrowth annealing is often employed to improve the crystal quality and/or to increase Si dopant incorporation, however, at the expense of blue shift in the bandgap. While GaAsN is chosen here for the present discussion, there are many other dilute nitride semiconductors as discussed in Section 2.11 in conjunction with band parameters. As alluded to earlier, the chemical and size differences between the N and As atoms are the challenges facing experimentalists. In addition, the generation of atomic nitrogen, although not that different from the technology required for hexagonal GaN growth [424], deserves some attention. While basic mismatch between N and As can be dealt with by growing the layers under nonequilibrium conditions, the issue of atomic nitrogen can be handled by compact RF sources that have seen a good degree of improvement lately. By adjusting the RF power and pressure in the cell, one can tailor the source to produce mostly the atomic species by optimizing the emission at 745 nm of wavelength. Note that the substrate and most of the structure are zinc blende and, consequently, the dilute material assimilates and assumes the same crystalline structure. The desired nitrogen concentrations are in the range of 1–10% for red shifting the transitions out to as long as 1.55 mm. Larger growth rates lead to a reduced incorporation of N in the lattice. Similarly, higher growth temperatures lead to the same. Consequently, when 1.55 mm wavelength material is desired, lower

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growth rates must be employed as well as lower growth temperatures. At substrate temperatures of 500
C or below, if very large As overpressure is employed, incorporation of N is limited because the flux of atomic nitrogen is small. However, atomic nitrogen is very reactive and, therefore, compositional control should be much better as compared to quaternaries relying on P and As (InGaAsP). As expected, owing to dissimilarities of N and As, the luminescence properties of GaInNAs degrade rapidly with increasing nitrogen concentration. Employing remedies such as postgrowth annealing enhances the luminescence efficiency of GaInNAs. However, this enhancement is accompanied by a blue shift in the transition in bulk and quantum well materials. Nitrogen and possibly In diffusion out of GaInAsN are responsible for the observed luminescence shift to shorter wavelengths. For completeness, a one-paragraph discussion of device issues will be made in conjunction with the edge emitting and vertical cavity lasers operating at 1.3 and 1.5 mm portion of the optical spectrum, although other applications such as heterojunction bipolar transistors are possible. Several groups have reported lasers operating at 1.3 mm region [425–442], where the silica-based fiber dispersion is zero, and 1.5 mm region [443–447] (albeit with addition of Sb to the lattice as the quality required for laser operation for InGaAsN layers cannot be obtained), where the loss is low, again for the silica-based fibers. Both are intended for telecommunication purposes. Even 8 W [448] and 12 W [449] CW operation has been reported. Highspeed testing of these lasers has also been performed [450] with data transmission rates as high as Gbit s1 having been achieved already [451]. For interconnects and high-speed data links, vertical cavity surface emitting lasers (VCSELs) have received a great deal of attention. Now that dilute nitrides are becoming potential candidates for long-wavelength lasers, efforts are under way to explore VCSELs in this material system as well [452,453].

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428 Yang, K., Hains, C.P. and Cheng, J.L. (2000) Efficient continuous-wave lasing operation of a narrow-stripe oxideconfined GaInNAs–GaAs multiquantumwell laser grown by MOCVD. IEEE Photonics Technology Letters, 12 (1), 7–9. 429 Kondow, M., Kitatani, T., Nakahara, K. and Tanaka, T. (1999) A 1.3-mm GaInNAs laser diode with a lifetime of over 1000 hours. Japanese Journal of Applied Physics, Part 2: Letters, 38 (12A), L1355–L1356. 430 Li, N.Y., Hains, C.P.,Yang, K., Lu, J., Cheng, J. and Li, P.W. (1999) Organometallic vapor phase epitaxy growth and optical characteristics of almost 1.2 mm GaInNAs three-quantum-well laser diodes. Applied Physics Letters, 75 (8), 1051–1053. 431 Sato, S. and Satoh, S. (1998) Roomtemperature pulsed operation of strained GaInNAs/GaAs double quantum well laser diode grown by metal organic chemical vapour deposition. Electronics Letters, 34 (15), 1495–1497. 432 Kitatani, T., Kondow, M., Nakahara, K., Larson, M.C. and Uomi, K. (1998) Temperature dependence of the threshold current and the lasing wavelength in 1.3-mm GaInNAs/GaAs single quantum well laser diode. Optical Review, 5 (2), 69–71. 433 Nakatsuka, S., Kondow, M., Kitatani, T., Yazawa, Y. and Okai, M. (1998) Index-guide GaInNAs laser diode for optical communications. Japanese Journal of Applied Physics, Part 1: Regular Papers, Short Notes & Review Papers, 37 (3B), 1380–1382. 434 Ougazzaden, A., Bouchoule, S., Mereuta, A., Rao, E.V.K. and Decobert, J. (1999) Room temperature laser operation of bulk InGaAsN/GaAs structures grown by APMOVPE using N2 as carrier gas. Electronics Letters, 35 (6), 474–475. 435 Fischer, M., Reinhardt, M. and Forchel, A. (2000) High temperature operation of GaInAsN laser diodes in the 1.3 mm regime. 58th Device Research Conference, IEEE, pp. 119–120. 436 Shimizu, H., Kumada, K., Uchiyama, S. and Kasukawa, A. (2000) High

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performance CW 1.26 mm GaInNAsSbSQW and 1.2 mm GaInAsSb-SQW ridge lasers. Electronics Letters, 36 (20), 1701–1703. Setiagung, C., Shimizu, H., Ikenaga, Y., Kumada, K. and Kasukawa, A. (2003) Very low threshold current density of 1.3mm-range GaInNAsSb–GaNAs 3 and 5 QWs lasers. IEEE Journal of Selected Topics in Quantum Electronics, 9 (5), 1209–1213. Bank, S.R., Yuen, H.B., Bae, H., Wistey, M.A. and Harris, J.S. (2006) Overannealing effects in GaInNAs(Sb) alloys and their importance to laser applications. Applied Physics Letters, 88, article 221115. Gollub, D., Moses, S., Fischer, M. and Forchel, A. (2003) 1.42 mm continuouswave operation of GaInNAs laser diodes. Electronics Letters, 39 (10), 777–778. Ikenaga, Y., Miyamoto, T., Makino, S., Kageyama, T., Arai, M., Koyama, F. and Iga, K. (2002) 1.4 mm GaInNAs/GaAs quantum well laser grown by chemical beam epitaxy. Japanese Journal of Applied Physics, 41, 664–665. Tansu, N. and Mawst, L.J. (2002) Lowthreshold strain-compensated InGaAs(N) (l ¼ 1.19–1.31 mm) quantum well lasers. IEEE Photonics Technology Letters, 14 (4), 444–446. Peng, C.S., Jouhti, T., Laukkanen, P., Pavelescu, E.-M., Konttinen, J., Li, W. and Pessa, M. (2002) 1.32-mm GaInNAs-GaAs laser with a low threshold current density. IEEE Photonics Technology Letters, 14 (3), 275–277. Yang, X., Jurkovic, M.J., Heroux, J.B. and Wang, W.I. (1999) Molecular beam epitaxial growth of InGaAsN:Sb/GaAs quantum wells for long-wavelength semiconductor lasers. Applied Physics Letters, 75 (2), 178–180. Yang, X., Heroux, J.B., Mei, L.F. and Wang, W.I. (2001) InGaAsNSb–GaAs quantum wells for 1.55 mm lasers grown by molecular-beam epitaxy. Applied Physics Letters, 78 (26), 4068–4070.

445 Vurgaftman, I., Meyer, J.R., Tansu, N. and Mawst, L.J. (2003) (In)GaAsN–GaAsSb type-II “W” quantum-well lasers for emission at l ¼ 1.55 mm. Applied Physics Letters, 83 (14), 2742–2744. 446 Bank, S.R., Wistey, M.A., Goddard, L.L., Yuen, H.B., Lordi, V. and Harris, J.S., Jr (2004) Low-threshold continuous-wave 1.5-mm GaInNAsSb lasers grown on GaAs. IEEE Journal of Quantum Electronics, 40 (6), 656–664. 447 Fischer, M., Reinhardt, M. and Forchel, A. (2000) Room-temperature operation of GaInAsN/GaAs laser diodes in the 1.5 mm range. Conference Digest, 2000 IEEE 17th International Semiconductor Laser Conference (Cat. No. 00CH37092), IEEE, Piscataway, NJ, pp. 115–116. 448 Livshits, D.A., Egorov, Yu.A. and Riechert, H. (2000) 8 W continuous wave operation of InGaAsN lasers at 1.3 mm. Electronics Letters, 36 (16), 1381–1382. 449 Bugge, F., Erbert, G., Fricke, J., Gramlich, S., Staske, R., Wenzel, H., Zeimer, U. and Weyers, M. (2001) 12 W continuous-wave diode lasers at 1120 nm with InGaAs quantum wells. Applied Physics Letters, 79 (13), 1965–1967. 450 Reinhardt, M., Fischer, M., Kamp, M. and Forchel, A. (2000) 7.8 GHz small-signal modulation bandwidth of 1.3 mm DQW GaInAsN/GaAs laser diodes. Electronics Letters, 36 (12), 1025–1026. 451 Steinle, G., Mederer, F., Kicherer, M., Michalzik, R., Kristen, G., Egorov, A.Y., Riechert, H., Wolf, H.D. and Ebeling, K.J. (2001) Data transmission up to 10 Gbit/s with1.3 mmwavelengthInGaAsNVCSELs. Electronics Letters, 37 (10), 632–634. 452 Fischer, M., Reinhardt, M. and Forchel, A. (2000) A monolithic GaInAsN verticalcavity surface-emitting laser for the 1.3-mm regime. IEEE Photonics Technology Letters, 12 (10), 1313–1315. 453 Schneider, H.C., Fischer, A.J., Chow, W.W. and Klem, J.F. (2001) Temperature dependence of laser threshold in an In GaAsN vertical-cavity surface-emitting laser. Applied Physics Letters, 78 (22), 3391–3393.

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2 Electronic Band Structure and Polarization Effects Introduction

The band structure of a given semiconductor is pivotal in determining its potential utility. Consequently, an accurate knowledge of the band structure is critical if the semiconductor in question is to be incorporated in the family of materials considered for serious investigations and device applications. The group III–V nitrides are no exception, and it is their direct bandgap nature and the size of the energy gap that spurred the interest in nitride semiconductors. Nitride semiconductors can be classified into two groups. One group pertains to stoichiometric systems where N represents 50% of the constituents while the other half is made of metal constituents. These stoichiometric nitrides come in wurtzitic and zinc blende (ZB) forms. The other class of nitrides is the dilute compound semiconductors, wherein very small amounts of N are added to the lattice for remarkably large negative bowing of the bandgap, making these dilute nitride systems compete for longer wavelength applications. For example, the bandgap of GaAs can be extended to 1.3 mm applications. Likewise, the bandgap of InGaAs coherently grown on GaAs can be extended with dilute amounts of N in the lattice to be a contender for 1.5 mm applications, which has been the domain of In0.53Ga0.47As lattice matched to InP. The impact of dilute nitrides is that, in at least the aforementioned example, what used to be the domain of InP-based technology can be met by GaAs technology with untold consequences in terms of not only technology but also the cost of that technology. A number of researchers have published band structure calculations for both wurtzite (Wz) and zinc blende GaN, AlN, and InN. To make matters more interesting, the bandgap of InN transmogrified from 1.9 eV downward to about 0.7 eV between the first edition and the current one. It is argued that the first set of bandgap measurements might have been conducted in films containing a large amount of O, which could have caused an upward shift in the measured data. The initial estimate of the 1.9 eV bandgap of InN, in addition to creating confusion concerning the nature and applications of InN, caused uncertainties in the bandgap of the InGaN ternary as well. The situation is exacerbated by inhomogeneities in composition and strain as well as poor sample quality. The situation in fact transformed into one in which

Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morkoç Copyright
2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

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multiple bowing parameters, depending on the InN composition in the alloy, were proposed. It can now be said that, with the small bandgap of about 0.8 eV for InN as agreed on, only a single bowing parameter can indeed account for the data dating back to 1980s where incompatibilities with the 1.9 eV InN bandgap were reported [1]. The first Wz GaN band structure found through a pseudopotential method led to a 3.5 eV direct bandgap. The band structure for ZB GaN has been obtained by a firstprinciples technique within the local density functional framework with a direct bandgap of 3.40 eV and a lattice constant of 4.50 Å. A treatise of the band structure in bulk and quantum wells (QWs) with and without strain for conventional nitrides, GaN, InN, and AlN, along with their alloys in the wurtzite and zinc blende form, and “dilute” nitrides (all of which are zinc blende ternaries and quaternaries, for example, GaAsN and GaInAsN) is discussed. Complete and consistent parameter sets are provided with tabulations of the direct and indirect energy gaps and spin–orbit and crystal field splitting. The alloy bowing parameters, electron and hole effective masses, deformation potentials, elastic constants, and piezoelectric (PE) and spontaneous polarization coefficients are given in many tables in Chapter 1. However, the basis of spontaneous and piezoelectric polarization effects and their practical impact on single- and dual-interface heterostructures are discussed in this chapter following the band structure discussion. The heterostructure band offsets are discussed in Volume 2, Chapter 4. The temperature dependence of the bandgap parameter of nitrides inclusive of the case of band tailing, primarily GaN, is discussed in this chapter as well as in Volume 2, Chapter 5 in conjunction with its optical properties.

2.1 Band Structure Calculations

A glossary of band structure calculations will be given before delving into the specific calculations employed to determine the band structure of nitride semiconductors. This is not an all-inclusive treatment of the field, but it is a compact treatment of the salient features of methods used to illustrate and/or calculate the band structure. The first exposure of many students to band structure calculations is that of a free-electron or nearly free electron approach in a periodic lattice [2–4]. In this one-electron model, the periodic potential can be thought of as arising from the periodic charge distribution associated with ion cores that are situated at the lattice sites. To expand the picture to include many electrons, an average constant potential contribution is added to account for all the other electrons in the system. This problem was first considered by Felix Bloch [5]. In its simplest form, the wave function representing the electrons in a periodic potential would be composed of the product of a plane wave function representing an electron in free space and a function representing the periodicity of the crystal. In the nearly free electron approach, the effect of the crystalline potential on the electronic structure is considered to be weak, and the energy levels of the electrons have little resemblance with those of the atoms. The allowed energies occur in bands of allowed states separated by forbidden energy regions (gaps). Within an allowed and

2.1 Band Structure Calculations

mainly occupied energy band, the electron motion is in many ways similar to that of free particles with an appropriate charge and effective mass. Whether the crystal is an insulator or conductor depends on whether the states within a band or set of bands are completely filled or partially empty. The crystal is considered a semiconductor if the gap between a filled band and the empty band is small, the exact value of which has changed over the years. For example, a gap of 3 eV was taken to be associated with insulators, which is smaller than the gap of GaN. The one-electron picture is an approximation and does not take into consideration processes such as electron– electron interaction, which is neglected other than what is convoluted in the average potential. To understand the basic aspects of band structure, it is instructive to consider an infinite periodic one-dimensional square well potential, which forms the basis for the Kronig–Penney approach [6]. This approximation leads to an exact solution of the Schr€odinger equation. Even though the square well potential approximation is very crude, it serves to illustrate explicitly many important characteristic features of the electron behavior in periodic lattices. In contrast to the free-electron approximation in which the potential energy of the electron is assumed to be small in comparison to its total energy, the opposite is assumed in another commonly practiced approximation, called the tight binding (TB) approximation. Specifically, it is assumed that the potential energy of the electrons accounts nearly for all of the total energy in the case of which the allowed energy bands are narrow compared to the forbidden ones. Unlike the free-electron model, the electronic wave functions are more or less localized around the atoms. Thus, the interaction between neighboring atoms is relatively weak, and the wave functions and the allowed energy levels of the crystal as a whole resemble the wave function and energy levels of isolated atoms. In a sense, an electron associated with an atom is assumed to remain in an orbit associated with that atom, and these orbitals are combined linearly in a form to be consistent with the Bloch–Floquent theorem to represent a state running throughout the crystal. Again, each orbital is localized on a particular atom. As such the results are very sensitive to overlap integrals, which in turn are sensitive to the details of the orbitals outside the cores and to the lattice spacing. Naturally, in this approach, electrons are not affected by atoms more than a single atomic spacing away. The choice of whether the free electron or the tight binding method is good depends on the particular crystal. In fact, in some crystals neither of these is good. The tight binding approach was assumed by Bloch in his original discussion of energy bands. If there is an appreciable interatomic interaction, the tight binding approach must use a linear combination of atomic orbitals (LCAO), in which a quantum mechanical variational procedure is employed to find the combination of s, p, and d orbitals that correspond to the lowest energy in the system. Not surprisingly, the tight binding methods are successful when the effect of the periodic core potential is quite large. This is true, for example, when the band is derived from the 3d states in the first series of transition elements. In these elements, the 3d states are partially filled and one or two electrons are present in 4s subshells. When atoms of these elements are brought together to make a solid, the interaction between the 4s states is very strong while the overlap between the partially filled inner shells is rather weak.

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There are other band structure methods that provide much improvement over the nearly free electron model and the tight binding model. These methods rely on choosing an appropriate basis for the electrons that represent electron behavior both inside and outside the atomic sphere. For example, while a single plane wave may be adequate to represent electron wave function in the interstitial space, to account for the rapid change in the function near the core region, a combination of a large number of plane waves would be necessary. Because the core functions are described in terms of radial functions and spherical harmonics, all unknown wave functions of crystals can be expanded in a set of known functions such as plane waves, radial functions, and spherical harmonics. The various methods for band structure calculations, therefore, differ in the initial choice of boundary conditions that the wave functions must satisfy. A brief qualitative discussion of these methods is given in the following sections. 2.1.1 Plane Wave Expansion Method

Here, the Bloch wave is expanded as a linear combination of plane waves (LCOPW), namely, ! ! ! X ! ð Þ:r Ck ðKÞei k þ k : ð2:1Þ Fk ð r Þ ¼ !

k

The combination coefficients Ck(K) are determined by solving the determinantal equation " # X ! !
h2 ðk þ KÞ2 0
E K jCk ðKÞj2 dK;K 0 þ Ck ðKÞCk ðK ÞVð k
k 0 Þ ¼ 0; 2m 0 K;K

!

! 0

ð2:2Þ

where EK is the energy eigenvalue and Vð k
k Þ is the crystalline potential. Although this is a simple method, its practical implementation is difficult, as it requires a large number of plane waves to represent the behavior of electrons near a core region. Consequently, the convergence in the eigenvalues is poor and it requires solving a large determinantal equation. 2.1.2 Orthogonalized Plane Wave (OPW) Method

The method originally proposed by Herring and Hill [7] and later discussed by Woodruff [8] considers the Bloch function to be a linear combination of OPW basis. The OPW basis consists of a plane wave orthogonalized to the atomic core functions, such that the electron behaves like a core electron while inside the core and like a plane wave while in the interstitial region. The OPW basis can be written as ! ! X 1 ! ! X k ð r Þ ¼ pffiffiffiffiffiffiffiffi ei k : r
mkj Fkj ð r Þ; ð2:3Þ NW j

2.1 Band Structure Calculations

where Fkj are Ðthe core wave functions for constant j. The term mkj is evaluated by ! ! requiring that Fkj ð r ÞX k ð r Þd3 r ¼ 0mX , where W is the volume of the Wigner–Seitz cell, and N is the number of atoms. The Bloch function is then expressed as a linear combination of the OPWs, and the energy eigenvalues are computed by solving the appropriate determinantal equation. This method has been applied to band structure calculation of metallic and nonmetallic solids. 2.1.3 Pseudopotential Method

Phillips and Kleinman [9] later demonstrated that it is possible to rewrite the crystal potential that includes contributions from the core and valence electrons in such a way that the Bloch function can be written in terms of a linear combination of planes as in the plane wave expansion method without compromising the convergence advantages of the OPW method [10,11]. The crystal potential thus obtained is called the pseudopotential. As mentioned earlier, the orbitals between the cores (outer regions) are smooth where the wave functions are somewhat like plane waves. Near the core regions, however, the wave functions are complicated by the strong and rapidly varying potential. The orthogonality requirement causes nodes (zeros) in the wave function in the core region. The weak potential experienced by the electrons in the outer region can be treated as a perturbation that mixes the plane wave components (strongly only at the Brillouin zone (BZ) boundaries). While the orbitals are not like plane waves near the cores and potentials vary rapidly, it is argued that what goes on near the core is irrelevant to the dependence of the energy on the wave vector. The energy wave vector dependence can be calculated by applying the Hamiltonian operator to an orbital at any point in space, which when applied to the outer regions, the picture would be that of nearly free electron energy. The actual potential energy in the core region can be represented by an effective potential energy called the pseudopotential that gives the same wave functions outside the core regions as the actual potential. Surprisingly, the pseudopotential is nearly zero in the core region, which is arrived at by experience with such potentials as well as theoretical considerations. Although use of these pseudopotentials may lead to incorrect wave functions, doing so can, with acceptable accuracy, indicate how the energy varies. 2.1.4 Augmented Plane Wave (APW) Method

This method proposed by Slater [12] also makes use of the fact that the wave function inside the core behaves like atomic functions and outside the core like plane waves as in the OPW method. The difference lies in how one applies the boundary conditions. Unlike the OPW method where the wave function outside and inside the core are matched by the Schmidt orthogonalization condition, in the APW scheme one expands the wave functions outside the core region (r  ri) by a set of plane waves and

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inside the core region (r  ri) by a sum of spherical waves, namely, !! X ! Fk ð r Þ ¼ a0 Qðr
r i Þei k : r þ alm Qðr i
rÞY lm ðq; fÞRl ðE; rÞ; l;m

where Y is the step function with Y(x) ¼ 1 for x  0 and Y(x) ¼ 0 for x < 0. The ! coefficients alm are chosen in such a way that the wave function Fk ð r Þ is continuous across the sphere of radius ri. The term Rl(E, r) represents the solution of the radial equation. Slater [12] proposed a reasonable approximation to the potential with a constant potential outside the cores and an ordinary atomic potential inside a sphere surrounding each ion core. The Schr€odinger equation is then solved separately for each of the two regions and solutions are matched over the spherical boundaries between them, and within each core the wave function can be expanded in spherical harmonics. The solution outside the cores is a superposition of plane waves. The combination gives this method its name, the augmented plane waves. 2.1.5 Other Methods and a Review Pertinent to GaN

Another class of calculations relies on what is called the all-electron approach. Here, one develops potentials within the sphere around the core region and uses their energy derivatives to expand the wave functions in a manner to match those outside the sphere. Among the methods relying on this premise are linear muffin-tin orbital (LMTO) and linearized augmented plane wave (LAPW) methods. Essentially, in the LMTO method, the crystal is divided into nonoverlapping muffin-tin spheres surrounding the atomic sites and an interstitial region outside of spheres. Inside the muffin-tin sphere, the potential is assumed spherically symmetric while in the interstitial region the potential is assumed constant or slowly varying. The early application of LMTO method made use of what is called the atomic sphere approximation (ASA), in which the crystal potential is treated as a superposition of weakly overlapping spherical potentials centered around lattice sites but in such a way as to fill the space and the total volume of muffin-tin sphere is the same as the atomic volume. The potential is also assumed to be spherically symmetric inside each muffin-tin sphere. Additionally, the kinetic energy of the basis function in the interstitial regions is restricted to be a constant and that constant is typically assumed to be zero in the calculations. In open structures, it is customary to include spheres centered on interstitial sites. It is worth noting that the potential so determined is very close to the true full potential and provide a much better representation of the potentials than the original muffin-tin potentials that are constant in the interstitial regions and spherical within nonoverlapping spheres. The limited precision, 0.1 eV, prevents this method from being applied satisfactorily to the very intricate portions of the band structures. For example, very small differences in the total energy between the zinc blende and wurtzitic forms of GaN are a troublesome point for these ASA-LMTO calculations. Likewise, the crystal field splitting at the valence band maxima presents similar problems.

2.1 Band Structure Calculations

The LMTO method has advantages such as using a minimal basis for computational efficiency and thus allowing large unit cell calculations. It also treats all elements in the same manner and is accurate due to an augmentation procedure, which gives the wave function a reasonably correct shape near the nuclei. The method also uses atom-centered basis functions of well-defined angular momentum. The full potential LMTO (FP-LMTO) calculations are all fully relativistic electrons with the shape approximation to the charge density or the potential. As in the LMTO, the crystal is divided into nonoverlapping muffin-tin spheres and interstitial regions outside the spheres. Then, the wave function is presented differently in those two regions in that inside the muffin-tin spheres, the basis functions are as in the LMTO-ASA method and are of Bloch sum of linear muffin-tin orbitals and the kinetic energy is not restricted to zero in the interstitial regions. All of the other above methods rely on our knowledge of the potential in which the electrons inside a crystal move. This potential results from the interaction between electrons, electrons and nuclei, and between nuclei. This presents a many body problem that cannot be solved exactly. Thus, approximations are needed. One of these techniques developed by Hohenberg and Kohn and Sham, known as the density functional theory (DFT) [13,14], has shown a great promise in treating energy bands of solids. DFT can be construed as a self-consistent field method for attaining the crystal potential. Over the years, many band structure calculations evolved to include not only the electronic properties but also binding energies, through which the lattice constant, bulk modulus, elastic constants, and vibrational properties such as phonon frequencies can be computed. The DFT method is well suited for calculating the aforementioned parameters. The basic quantity is the electronic charge density, and the total energy is expressed as functional of this density. The total energy expression as a function of density includes exchange and correlation effects in an average electron gas like manner [15], which is termed as the local density approximation (LDA). Most common form expresses the charge density in terms of occupied oneelectron wave functions, the corresponding eigenvalues of which constitute the band structure [16]. Calculations of the self-energies are best done by the DFT-LDA method within the GW approximation. The nomenclature is adopted from the original paper of Lars Hedin [17] for one-electron Green’s function and Hedin and Lundqvist [18] for screened Coulomb interaction. This approximation represents the first term in a perturbation expansion of the self-energy, which can also be viewed as the screened Hartree–Fock (H–F) theory. It should be pointed out that in semiconductors the GW method differs from the band structure determined by LDA chiefly due to a shift of the conduction band relative to the valence band. Consequently, the fundamental bandgap is underestimated. The genesis for Kohn–Sham [15] local DFT lies in lowering the conduction band in a manner that is k-value dependent, as has amply been pointed out by Sham and Schl€ uter [19] and Perdew and Levy [20] in back-to-back articles. It is for this reason that DFT-based calculations are supplemented by a shift of the calculated fundamental gap inspired by a body of experimental data. Most band structure calculations for GaN and related materials rely on the DFT-LDA approach. Computational development of the GW approximation was later accomplished by a

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series of authors, namely, Hyberstsen and Louie [21], Godby, Schl€ uter, and Sham [22], and even later on by Aryasetiawan and Gunnarsson [23,24]. Let us now discuss the band structure of some specific nitride semiconductors. This semiconductor family can exist in wurtzite and zinc blende crystal polytypes with the Wz phase being the stable and widely used form. While both the Wz and ZB polytypes of GaN have been given in quite some detail, the ZB varieties of AlN and InN are not thermodynamically stable and reports are very sketchy. In fact, there are predictions that ZB AlN has an indirect bandgap [25]. A review of the band structure calculation and the methodologies used for both polytypes of all group III nitrides, BN, AlN, GaN, and InN, has been given by Lambrecht and Segall [16]. Calculations of electronic and optical properties of Wz GaN and related structures have been undertaken over many years [16,26] with more of them emerging continually, as nitride-based devices become more popular. Methods such as the ab initio, tight binding [27], pseudopotential, ASA LMTO, a general treatment of which has been treated by Andersen [28], LCAO, LAPW, full potential linearized augmented plane wave (FP-LAPW) method such as that reported by Wimmer et al. [29] for calculating the electronic band structure of Wz semiconductors within the LDA [15,16], DFT, and GW methods have been employed to calculate the energy bands for both wurtzite and zinc blende GaN, InN, and AlN bulk materials. Christensen and Gorczyca [30,31] utilized the ASA-LMTO method for group III-nitrides mainly GaN and AlN, albeit in relation to their behavior under pressure. The ASA-LMTO method was also applied by Lambrecht and Segall [32] to contrast and compare the nature of the direct and indirect bandgap of various ZB and Wz nitrides and, in particular, in terms of the directness or indirectness of the bandgap. These methods change in their capabilities to varying degrees. Early versions of pseudopotential calculations did not include the contribution by 3d-electrons because of the difficulty of attaining convergence in-plane wave expansions due to the deep N pseudopotential [33]. A mixed basis set was later used to overcome this apparent shortcoming [34] but with the consequence that ZB GaN would be lower in energy, meaning favored over the Wz phase that is inconsistent with experiments, and also was not confirmed with well-converged plane wave calculations performed for GaN by Yeh et al. [35] and InAlN [36]. Yeh et al. [35] focused on the issue of polymorphism for a large number of semiconductors including AlN, GaN, InN, AlP, AlAs, GaP, GaAs, ZnS, ZnSe, ZnTe, CdS, C, and Si, and using the local density formalism (LDF), developed a simple scaling at T ¼ 0 that systematizes the energy difference (DE LDF W-ZB ) between the ZB and Wz forms. This energy difference was found ~ to be linearly dependent on the atomistic orbital radii coordinate RðA; BÞ that depends only on the properties of the free atoms A and B, making up the binary compound AB. Of special interest for the topic under discussion is that Yeh et al. [35] found that DE LDF W-ZB ðABÞ for GaN is
9.9 meV/atom, which is within 0.7 meV/atom of the calculations by Van Camp et al. [37] and within 0.3 meV/atom of the calculations by Min et al. [34]. The energy difference, DE LDF W-ZB ðABÞ, for all three nitride binaries are negative when this quantity is scaled with differences in tetrahedral radii and Pauling’s electronegativity, implying that the equilibrium state of all three nitride binaries is the Wurtzitic form.

2.1 Band Structure Calculations

The issue of d-electrons is an important one, as in Ga 3d, in relation to pseudopotential and all-electron calculations, specifically, to know whether these calculations can handle these bands. Because both In and Ga are heavy and have d cores, Ga 3d and In 4d states overlap with the deep N 2s states with serious implications about bonding and band structure. The way in which the d-electrons are treated as core orbitals with or without nonlinear core corrections or as valence states in pseudopotential methods caused some confusion, which has been the topic of some discussion [32,38]. In the case of Fiorentini et al. [38], the structural and electronic properties, albeit cubic GaN, were studied within the local density approximation by the full potential linear muffin-tin orbitals method, wherein the Ga 3d-electrons were treated as band states with no shape approximation to the potential and charge density. Owing to the resonance of Ga 3d-states with nitrogen 2s states, the cation d bands were found not to be inert, and features unusual for a III–V compound were found in the lower part of the valence band as well as in the valence charge density and density of states. Additional full and frozen (T ¼ 0) overlapped core calculations performed for GaN, ZnS, GaAs, and Si (all cubic) showed that an explicit description of closed-shell interaction has a noticeable effect on the cohesive properties of GaN. The resulting energy resonance causes the Ga 3d-electrons to strongly hybridize with both the upper and lower valence band s and p levels. Such hybridization is predicted to have a profound influence on the GaN properties, including quantities such as the bandgap, the lattice constant, acceptor levels, and valence band heterojunction offsets. Because Al has no 3d core states, there is no hybridization between the cation d states and the N 2s states. In short, the band structure and cohesive properties of GaN are very sensitive to the cation d bands. On the pseudopotential side, Wright and Nelson [36] provided a framework in which accurate calculations treating the Ga 3d- and In 4d-electrons explicitly as valence states were performed by extending the plane wave cutoff to 240 Ry to ensure convergence. An interesting observation is that while d-electrons are important in bonding, they appear as separate states in considering quasi-particle excitations in photoemission experiments [39]. Somewhat of a side note but with legitimate relevance, it has been predicted in the cases of ZnS and ZnSe that potential acceptors, such as Cu, whose d-electrons are resonant with the lower valence band, are repelled by the d-hybridized upper valence band, resulting in a deep level. Impurities without d-electron resonance form shallow acceptors. Mg has no d-electrons and turns out to be sufficiently shallow for roomtemperature p-type doping of GaN. On the contrary, Zn, Cd, and Hg, which all have d-electrons, form deep levels in GaN [40]. Further insight is warranted before conclusive statements can be made with certainty as, for example, photoemission data show the N 2s to be well below the Ga 3d band. Pseudopotential calculations can also be applied to defects [41] and surfaces [42], as has been done for GaN. The calculations relating to surfaces are discussed in Section 3.2.7.2 in reasonable detail. Likewise, the calculations in relation to defects are treated in Section 4.3.1. Other applications of the calculations discussed above are for the determination of dielectric properties and susceptibility [43] and vibrational properties [44].

j139

j 2 Electronic Band Structure and Polarization Effects

140

Calculations in the Hartree–Fock approximation rather than the local density approximation have also been performed for nitride semiconductors, notably GaN [45] and AlN [46,47]. While LDA calculations underestimate the bandgap, the opposite is the case for Hartree–Fock calculations. In the H–F method, exchange is treated exactly but the correlations are fully ignored. Specific to the case of GaN, its total energy as a function of unit cell volume has been calculated for the wurtzite, zinc blende, and rock salt phases by the ab initio all-electron periodic Hartree–Fock method by Pandey et al. [45]. In this case, the gallium 3d levels were treated as fully relaxed band states, and the internal parameters c/a and u in the wurtzite phase were optimized. The calculated transition pressure between the wurtzite and rock salt phases were found to be about 52 GPa at the Hartree–Fock level and about 35 GPa at the correlated level. The calculated electronic structure shows strong hybridization of Ga 3d and N 2s states with the ordering as Ga 3d–N 2s–N 2p in all the phases. The results indicate the bandgap to be direct at G in the wurtzite and zinc blende phases and indirect in the high-pressure rock salt phase where the valence band maximum is shifted away from the G point. The electronic structure of Wz AlN has been investigated by means of periodic ab initio Hartree–Fock calculations for the purpose of calculating the binding energy, lattice parameters (a, c), and the internal coordinate or parameter (u) [47]. The values of the bulk modulus, its pressure derivative, the optical phonon frequencies at the center of the Brillouin zone, and the full set of elastic constants have been calculated and compared with experimental data. When ab initio Hartree–Fock calculations were used to determine the electronic structure of AlN in high pressure, the rock salt phase resulted [46]. In this phase, the calculated lattice constant is 3.982 Å with the bulk modulus of 329 GPa. As in the case of GaN, the rock salt phase is predicted to be indirect at the X point with a gap of 8.9 eV. Moreover, the bonding is essentially ionic between Al and N. The direct gap shows a stronger linear dependence on pressure with a pressure derivative of 68 meV GPa
1 compared to that of the indirect X-valley gap with a pressure derivative of 31.7 meV GPa
1. It should be emphasized that the rock salt phase is favored to exist under high pressure, and as such throughout this book and literature, nitrides are spoken of as if they are wurtzitic with GaN being cubic also when grown away from thermodynamic equilibrium conditions on cubic substrates along h0 0 1i directions. As mentioned above, the electronic properties of nitride semiconductors can more accurately be calculated using first-principles techniques like density functional theory [25] within the Green’s function theory with the characteristic GW approximation of the exchange correlation self-energy [17]. These calculations have been applied to Wz and ZB GaN and AlN by Rubio et al. [25] and to ZB GaN by Palummo et al. [48]. The computational complexity of the full GW method is prohibitive for applications to complex systems with large number of atoms, such as surfaces, interfaces, and clusters. However, it should be mentioned that efficient simplified version of the GW method has been reported to reduce the central processing unit (CPU) time by a factor of 100 (in conjunction with semiconductors Si, GaAs, AlAs, and ZnSe) [49,50]. Unless the simplified GW method is used [48,49], the full GW method is typically limited to simple systems, for example, elemental or binary semiconductors.

2.1 Band Structure Calculations

The GW calculations are reasonably consistent with each other and also with experiments in many cases. In the calculations of Rubio et al. [25], the ab initio pseudopotential method within the local density approximation and the quasi-particle approach have been employed to determine the electronic properties of both Wz and ZB phases of AlN and GaN. The quasi-particle band structure energies were calculated using a model dielectric matrix for the evaluation of the electron self-energy. In the zinc blende structure, AlN was predicted to be indirect (G to X) with (4.9 eV) and that GaN to be direct with 3.1 eV at the G point, the latter in good agreement with absorption experiments on cubic GaN, showing the bandgap to be 3.2–3.3 eV. In the calculations of Palummo et al. [48], models of diagonal and off-diagonal screening with LDA-RPA full calculations in cubic GaN were considered. Simplified GW calculations relying on these models were also compared with full GW calculations. At the time empirical pseudopotential calculations were not available, necessitating ab initio RPA calculations to be done within the DFT-LDA approach. These calculations have already been used for obtaining the full GW band structure of GaN [51]. It should be mentioned that with respect to pure LDA results, the valence band shifts down and the conduction band shifts up, resulting in larger bandgap estimation. The amount of downward shift of the valence bands increases with the increase in energy below the valence band maximum. The N2s states are about 1.5–2 eV more than the valence band maximum. Moreover, the bottom of the N2p valence band, the character of which is of a mixture of N2p cations, shifts by an amount of about 0.5 eV more than the maximum. The absolute shift of the valence band maximum is a problem in GW theories, which is also the case with this method for very established materials such as Si. This seems to stem from the choice parameterization used for the LDA starting point of the calculations. The GW method changes the gaps of GaN and AlN by 1 and 2 eV, respectively [16]. The conduction band correction is on the order of 0.1 eV across the k-points and specific states. This figure appears to increase with the increase in energy dealing with higher conduction bands, awaiting further refinement following any comparison with experiments when accurate measurements become available. Unlike the full GW calculations, the TB approach provides an attractive possibility for an extension of the system size accessible to electronic structure calculations with atomic resolution. TB calculations have been applied to nitride-based systems [52–54]. Moreover, the Slater–Koster parameters transferable between the ZB and Wz crystal phases have been treated [27]. There has been a plethora of reports regarding band structure calculations in nitride semiconductors. The calculated band structures of Wz GaN, AlN, and InN are exhibited in Figure 2.1. For semblance of completeness, results from empirical pseudopotential method are also included here. Fritsch et al. [55] investigated the electronic band structure of both the wurtzite and zinc blende group III nitride semiconductors GaN, AlN, and InN within the empirical pseudopotential approach. Using ionic model potentials and a static dielectric screening function derived, the cationic and anionic model potential parameters were obtained from the zinc blende GaN, AlN, and InN experimental data. Using these model potentials, Fritsch et al. [55] calculated the band structure of group III nitrides in both the wurtzite and zinc blende

j141

j 2 Electronic Band Structure and Polarization Effects

142

(a)

GaN

12 10

3

1 1

3

6

Energy ( eV )

6

1

8

3 1

2

4

3

2

1 1 6 5

0 –2

2

4 3 2

3

–4

3 1

–6

1 3 1

3

–8 –10

3

–12

3

–14 –16

3

1 3

3

1 3

A

S H

R L T U

P K

M

1 M

K

M U L

T

(b) 6

3

3

10

H

P K

4 1

3

8

3 3 1

1

6

S

A

R

AlN

12

3

2

1

4 Energy (eV)

3

4

3

2 1 5

4 3 1

3

3 1

6

0 3

–2

2 3

–4

1

–6 –8

A

S

Γ

R L T U

–10 3

–12 –14 M

K

T

3

1

1

3

M U

H P K

M

L

Figure 2.1 Calculated band structures of (a) wurtzite GaN, (b) AlN, and (c) InN in the LDA within the FP-LMTO method at the experimental lattice constant and optimized u-value. The first Brillouin zone is also shown for convenience [16].

R

A

S

H P K

2.1 Band Structure Calculations

(c)

8

6

3 1 3 2

6 4

1 1

1

3 1 1

3

2 Energy (eV)

InN 4 3

1

2

10

1 6

0 2

–2

5

4

1

3 2

3 1 3

–4 –6

3

3

1 3 1

A

S R L T U

–8 3

–10

3

1 3

–12 3

–14 M

K

1 3 T

H P K

M

3 1 M U

L

R

A

S

H P K

Figure 2.1 (Continued )

form, recognizing the necessity of including the anisotropy of wurtzite crystals in the screening function. The band structures so calculated for wurtzitic GaN, AlN, and InN are shown in Figure 2.2. The same for the zinc blende variety is shown in Figure 2.3. It should be noted at the outset that all these binary materials, including alloy compounds obtained by combinations of these binaries, are wide direct bandgap semiconductors in both crystal phases, except zinc blende AlN that is expected to have an indirect gap with the conduction band minimum being at the X valley. Due to the lack of reliable experimental data, many details of these studies must be improved to provide an accurate band description. Approaches such as the kp model [56,57] including strain have been employed to calculate the valence band structure of Wz GaN [56,58]. First-principles calculations of effective mass parameters and valence band structures in bulk and confined systems with and without strain, utilizing the FP-LAPW method [59–61,66] and envelope function formalism for valence bands in wurtzite quantum wells, have been undertaken [62]. The wurtzite structure has a hexagonal unit cell and thus two lattice constants, c and a. It contains six atoms of each type. The space group for the wurtzite structure is P63mc (C46v ) [63]. The wurtzite structure consists of two interpenetrating hexagonal close-packed (HCP) sublattices, each with one type of atoms, offset along the c-axis by 5/8 of the cell height (5c/8). The zinc blende structure has a cubic unit cell, containing four group III elements and four nitrogen elements. The space group for the zinc blende structure is T 2d : F
43m. The position of the atoms within the unit cell is identical to the diamond crystal structure. Both structures consist of two

j143

j 2 Electronic Band Structure and Polarization Effects

144

interpenetrating face-centered cubic sublattices, offset by one quarter of the distance along a body diagonal. Each atom in the structure may be viewed as positioned at the center of a tetrahedron with its four nearest neighbors defining the four corners of the tetrahedron. The zinc blende and wurtzite structures are similar. In both cases, each

Energy (eV)

10

5

0

–5

A

R

L U M

Σ

Γ Δ A

S

H P K

T

Γ

(a)

15

Energy (ev)

10

5

0

–5

A

R

L U M

Σ

Γ Δ A

S

(b) Figure 2.2 Band structure of (a) wurtzitic GaN, (b) wurtzitic AlN, and (c) wurtzitic InN along high-symmetry lines in the Brillouin zone calculated within the empirical pseudopotential method (EPM), using ionic model potentials obtained experimentally from zinc blend varieties. Courtesy of Daniel Fritsch et al. [55].

H P K

T

Γ

2.1 Band Structure Calculations

j145

Energy (eV)

10

5

0

–5

A

R

L U M

Σ

Γ Δ A

S

H P K

Γ

T

(c) Figure 2.2 (Continued )

Energy (eV)

15 L3

10

K1 X3

L1

5

X1 L3

0

X5 X3

L2

–5

L

Λ

Γ

Δ

X

(a) Figure 2.3 Band structure of (a) zinc blende GaN, (b) zinc blende AlN, and (c) zinc blende InN along high-symmetry lines in the Brillouin zone calculated within the empirical pseudopotential method (EPM), using ionic model potentials obtained experimentally. Courtesy of Daniel Fritsch et al. [55].

K1

K2 K1 K1 UK

Σ

Γ

j 2 Electronic Band Structure and Polarization Effects

146

20

15 Energy (eV)

L3 10

X1

5

0

L3

–5

L2

10

K1

K2

X5

K1

X3 Λ

L

(b)

K1

X3

L1

Γ

Δ

K1 X

U,K

Σ

Γ

L3

Energy (eV)

K1 X3

5

X1

K1

L1 0

K2

L3

X5

K1

L2

–5 L

K1

X3 Λ

Γ

Δ

X

U, K

Σ

Γ

(c) Figure 2.3 (Continued )

group III atom is coordinated by four nitrogen atoms. Conversely, each nitrogen atom is coordinated by four group III atoms. The main difference between these two structures lies in the stacking sequence of closest packed diatomic planes. For the wurtzite structure, the stacking sequence of the (0 0 0 1) planes is ABABAB in the h0 0 0 1idirection. For the zinc blende structure, the stacking sequence of the (1 1 1) planes is ABCABC in the (1 1 1) direction.

2.1 Band Structure Calculations

The structure and the first Brillouin zone of a wurtzite and zinc blende crystal along with the irreducible wedges, calculated using the LDA within the FP-LMTO method at the experimental lattice constant and optimized u value, are displayed in Figure 2.4a and b, respectively. In a crystal with Wz symmetry, the conduction band kz

So

A o

R

Δo

T o

Γ

H L o

o

S’

oP

U K

o

o

ky

T’

Σ

M

kx (a)

kz

L U Q

S Z

kx

K

X

ky

W

(b) Figure 2.4 (a) Structure and the first Brillouin zone of a wurtzite crystal. Schematics of the irreducible wedges of Wz structure, indicating the high-symmetry points and lines, are also shown. The Umin point of the Wz phase is located on the M–L line at two-thirds a distance away

from the M point. (b) Structure and the first Brillouin zone of a zinc blende crystal. Schematics of the irreducible wedges of ZB structure indicating the high-symmetry points and lines are also shown.

j147

j 2 Electronic Band Structure and Polarization Effects

148

Conduction band

E gA E0

E0

C6v

or

5

9

A

6

7

7

7

Valence band

9

, J = 3/2

Valence band

B cr

15

so 15

1 7

ZB

W Crystal field cr = so

W Spin orbit

7

C

W

8 , J = 1/2

Crystal field

1

=

ZB

ZB Spin orbit cr = 0 so

=

2

=

3

Figure 2.5 Schematic representation of the splitting of the valence band in Wz crystals due to crystal field and spin–obit interaction. From left to right, the crystal field splitting is considered first. From right to left, the spin–orbit splitting is considered first. Regardless of which is considered first, the end result is the same in that there are three valence bands that are sufficiently close to one another for band mixing to be nonnegligible.

wave functions are formed of the atomic s orbitals, which transform the G point congruent with the G7 representation of the space group C 46v , The upper valence band states are constructed out of appropriate linear combinations of products of p3-like (px-, py-, and pz-like) orbitals with spin functions. Under the influence of the crystal field and spin–orbit interactions, the hallmark of the wurtzite structure, the sixfold degenerate G15 level associated with the cubic system, splits into Gv9 , upper Gv7 and lower Gv7 levels (Figure 2.5). Figure 2.6 shows the dispersion of the uppermost valence and conduction band structures in Wz GaN and ZB GaN ((a) near the G band for Wz, (b) inclusive of M, L, and A minima in Wz, and (c) inclusive of G, L, and X minima in ZB GaN). The influence of the crystal field splitting, which is present only in the wurtzite structure, transforms the semiconductor from ZB to Wz, which is represented in the section on the left-hand side in Figure 2.5. The crystal field splits the G15 band of the ZB structure into G5 and G1, states of the wurtzite structure. These two states are further split into Gv9 , upper Gv7, and lower Gv7 levels by spin–orbit interactions. Application of the spin–orbit splitting, from right to left, splits the G15 band of the ZB crystal into G8 and G7 states while the crystal possesses the zinc blende symmetry. Application of a crystal field further splits these states into Gv9 , upper Gv7, and lower Gv7 levels, and the crystal now possesses the wurtzite symmetry.

2.1 Band Structure Calculations

E(k)

c

Γ7

Γ9 :HH Δ1 Γ7:LH Γ7:ΧΗ

kz

(a)

Γ

Figure 2.6 (a) Schematic representation of the G point valence and conduction bands in crystal with wurtzite symmetry, such as GaN, where the spin–orbit splitting leads to the bands labeled as HH and LH. The one caused by splitting due to crystal field is labeled as CH [59,60]. (b) Schematic representation of the band diagram for Wurtzite GaN showing the separation between the G, A, and M–L band symmetry points at 300 K. The values with respect to the top of the valence band are EG ¼ 3.4 eV, EM–L ¼ 4.5–5.3 eV, EA ¼ 4.7–5.5 eV,

kx , ky Eso ¼ 0.008 eV, Ecr ¼ 0.04 eV [56]. The values of EG ¼ 6 eV, EM–L ¼ 7 eV, and EA ¼ 8 eV are given by Fritsch et al. [55]. (c) Schematic representation of the band diagram for zinc blende GaN showing the separation between the G, X, and L band symmetry points at 300 K. The values with respect to the top of the valence band are EG ¼ 3.2 eV, EL ¼ 4.8–5.1 eV, Ex ¼ 4.6 eV, and Eso ¼ 0.02 eV. Note that in the ZB structure, the valence band is degenerate [56]. The values of EG ¼ 3.2 eV, EL ¼ 5.1 eV, EX ¼ 4.3 eV are given by Fritsch et al. [55].

j149

j 2 Electronic Band Structure and Polarization Effects

150

Energy

A valley

M and L valleys

valley

EA

EM–L

E Ecr

HH band LH band

kz

k x,y

Split–off band (b) Energy L valleys X valley

EL

valley

EX E <1 0 0>

Eso

HH band

<111>

LH band Split–off band (c) Figure 2.6 (Continued )

Literature values of the calculated and experimental critical point transition energies for wurtzitic GaN, AlN, and InN are tabulated in Tables 2.1–2.3. Literature values of the calculated and experimental critical point transition energies for zinc blende GaN, AlN, and InN are tabulated in Tables 2.4–2.6. As shown in Figure 2.6a for wurtzitic GaN, the hole effective masses of the three uppermost valence bands Gv9 , Gv7 , and Gv7 exhibit large k-dependence. The bands are labeled as HH (heavy Gv9 ), LH (light, upper Gv7 spin–orbit split) and CH (Gv7 , crystal field split). The mass of the Gv9 band is heavy in all k-directions, whereas that of the upper Gv7 is relatively light in the x- and y-planes but heavy in the z-direction. That of the lower Gv7 is light in the x- and y-planes, but it is heavy along the z-direction

2.1 Band Structure Calculations Table 2.1 Literature values of calculated and experimental critical point transition energies for wurtzitic GaN [55].

A (eV) Parameter

Anisotropic

Isotropic

B (eV)

M v2
Mc1 Mv4
Mc1 Mv4
Mc3 Gv6
Gc1 Gv3
Gv6 Gv1
Gv6 Gv5
Gv6 Gv5
Gc3 Gv5
Gc6 Hv3
Hc3 K v3
K c2 K v2
K c2

7.67 6.07 7.68 3.47 6.97 0.043 1.00 5.96 10.74 8.06 8.54 8.68

7.67 6.07 7.68 3.47 6.94 0.023 1.00 5.96 10.74 8.07 8.55 8.68

8.26 6.61 7.69 3.50 6.80 0.021

9.0 9.43 10.10

C (eV)

3.0 7.0 0.0 1.0 5.9 11.1 8.3 7.9

D (exp) (eV) 7.05 7.0a 7.05 3.6, 3.44a, 3.50a 7.0a 0.022a 5.3 9.4 7.9a 7.65, 9.0a

The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor. A: empirical pseudopotential calculation from Ref. [55]; B: ab initio pseudopotential calculation within local density approximation from Ref. [25]; C: LCAO within local density approximation from Ref. [64]; D: unless stated otherwise, the experimental values are taken from Ref. [65]. a Experimental values taken from Ref. [66].

(c-direction). Two different definitions are prevalent in the literature. The G6, Gl pair has been used in Refs [63,71] and the G5, G1 pair in Refs [62,72,73]; for background information on group theory and symmetries in physics, see Refs [74,75]. We should mention that a carryover habit from the zinc blende nomenclature is still used for wurtzite symmetry by referring to the crystal field split-off band with the nomenclature “SO” as if it is the spin–orbit split-off band, because it happens to be the farthest from the HH band. In the zinc blende symmetry, the crystal field splitting is nonexistent, making the top of the valence band degenerate, and the spin–orbit splitting is large. Portions of this book, unfortunately, participate in the misuse of this nomenclature. Shown in Figure 2.6b are the most pertinent bands near the zone center and A, M, and L valleys. The same for zinc blende GaN is shown in Figure 2.6c. Without the spin–orbit interaction, the valence band would consist of three doubly degenerate bands: HH, LH, and CH bands. The spin–orbit interaction removes this degeneracy and yields six bands. Some band calculations based on the empirical pseudopotential method (EPM) [66] or the empirical tight binding method (ETBM) [76] have shown this splitting to be about 10 meV near the G point, which is comparable to the energy separation of the split-off band in GaN. In early attempts, the general Hamiltonian in kp theory included the spin–orbit interaction,

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j 2 Electronic Band Structure and Polarization Effects

152

Table 2.2 Literature values of calculated and experimental critical point transition energies for wurtzitic AlN [55].

A (eV) Parameter

Anisotropic

Isotropic

B (eV)

M v2
Mc1 M v4
Mc1 M v4
Mc3 Gv6
Gc1 Gv3
Gv6 Gv1
Gv6 Gv5
Gv6 Gv5
Gc3 Gv5
Gc6 Hv3
Hc3 K v3
K c2 K v2
K c2

9.56 7.88 8.81 6.11 6.44 0.13 1.04 8.95 12.99 10.10 9.43 9.59

9.54 7.87 8.83 6.11 6.41 0.16 1.03 8.94 12.97 10.91 9.43 9.57

10.0 8.3 8.5 6.0 6.7 0.2 0.9 9.4 14.0 10.5 9.6 9.7

C (exp) (eV)

6.29

8.02 14.00 10.39

The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor. A: empirical pseudopotential calculation from Ref. [55]; B: ab initio pseudopotential calculation within local density approximation from Ref. [25]; C: experimental values taken from Ref. [67].

but it took a while for it to be applied to the calculations of the band structure of wurtzite materials such as GaN. Naturally, the band structures of wurtzitic and zinc blende polytypes are very distinct due to the differences in underlying symmetries. For the zinc blende case, the three Luttinger parameters and the spin–orbit splitting provide a minimal description of the valence band structure. Moreover, the energy gap and interband coupling strength are also required for complete parameterization of both the conduction and valence bands. The split-off hole mass can be treated as an independent parameter within the commonly used eight-band kp model. The increase in the electron effective mass due to interactions with higher conduction bands can be included via the F parameter (see Section 2.2 for details) [72,77–79]. The set of band parameters needed to describe the wurtzite lattice must be augmented due to its lower symmetry. Neglecting the effect of valence band and upper conduction bands on the electron effective mass allows one to omit the interband matrix element and the F parameter. Owing to the reduced symmetry, the electron mass can display a rather weak anisotropy. In contrast, a full description of the valence band in the wurtzite polytype band structure requires both the spin–orbit splitting, Dso, and the crystal field splitting, Dcr, along with the seven so-called A parameters. Analogous to the Luttinger parameters in zinc blende materials, the latter parameterizes the hole masses along the different directions. Figure 2.6a and b highlights the differences in wurtzite and zinc blende varieties in terms of their band structure.

2.1 Band Structure Calculations Table 2.3 Literature values of calculated and experimental critical point transition energies for wurtzitic InN [55].

A (eV) Parameter

Anisotropic

Isotropic

B (eV)

d (exp) (eV)

M v2
Mc1 Mv4
Mc1 Mv4
Mc3 Gv6
Gc1 Gv3
Gv6 Gv1
Gv6 Gv5
Gv6 Gv5
Gc3 Gv5
Gc6 Hv3
Hc3 K v3
K c2 K v2
K c2

7.30 5.94 6.71 2.58 5.63 0.214 0.90 5.22 10.16 7.34 8.13 8.60

7.23 5.88 6.70 2.59 5.50 0.084 0.89 5.18 10.12 7.36 8.12 8.50

6.65 5.05 5.80 2.04 5.77 0.017 1.05 4.65 8.74 6.51 7.38 7.20

7.3, 4.95a 7.3 2.11, 2.0a;b

5.0, 5.5, 4.7a 8.8, 8.9a 5.4a 7.3, 7.2a

The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor. A: empirical pseudopotential calculation from Ref. [55]; B: empirical pseudopotential calculation from Ref. [66]. a Experimental values taken from Ref. [68]. b It should be pointed out that the values of bandgap values listed in this table compares with 1.8–2.1 eV values reported during the early stages of InN development and 0.7–0.8 eV reported later on. For an in-depth discussion of this seemingly controversial bandgap determined experimentally, the reader is referred to Section 2.9.1.

Table 2.4 High-symmetry point energies in zinc blende GaN in

reference to the top of the valence band for cases where spin–orbit effects are neglected (included) [55]. Parameter G15 c Gc1 Gv15 X c3 X c1 X v5 X v3 Lc3 Lc1 Lv3 Lv2

(Gc7 ) (Gc6 ) (Gv8 ) (X c7 ) (X c6 ) (X v7 ) (X v6 ) (Lc4;5 ) (Lc6 ) (Lv4;5 ) (Lv6 )

A (eV)

B (eV)

C (eV)

10.098 3.308 0.000 6.010 4.428
2.459
6.294 10.416 5.149
0.834
6.812

10.300 3.383 0.000 6.805 4.571
2.693
6.149 9.916 5.636
0.931
6.743

10.248 3.213 0.000 6.265 4.585
2.086
5.923 10.606 5.510
0.772
6.644

A: empirical pseudopotential calculation from Ref. [55]; B: empirical pseudopotential calculation from Ref. [69]; B: empirical pseudopotential calculation from Ref. [70].

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Table 2.5 High-symmetry point energies in zinc blende AlN in

reference to the top of the valence band for cases where spin–orbit effects are neglected (included) [55]. Parameter G15 c Gc1 Gv15 X c3 X c1 X v5 X v3 Lc3 Lc1 Lv3 Lv2

(Gc7 ) (Gc6 ) (Gv8 ) (X c7 ) (X c6 ) (X v7 ) (X v6 ) (Lc4;5 ) (Lc6 ) (Lv4;5 ) (Lv6 )

A (eV)

B (eV)

12.579 5.840 0.000 8.794 5.346
2.315
5.388 12.202 8.264
0.718
6.251

13.406 5.936 0.000 10.661 5.102
2.337 5.262 12.014 9.423
0.728
6.179

A: empirical pseudopotential calculation from Ref. [55]; B: empirical pseudopotential calculation from Ref. [69].

Table 2.6 High-symmetry point energies in zinc blende InN in

reference to the top of the valence band for cases where spin–orbit effects are neglected (included) [55]. Parameter G15 c Gc1 Gv15 X c3 X c1 X v5 X v3 Lc3 Lc1 Lv3 Lv2

(Gc7 ) (Gc6 ) (Gv8 ) (X c7 ) (X c6 ) (X v7 ) (X v6 ) (Lc4;5 ) (Lc6 ) (Lv4;5 ) (Lv6 )

A (eV)

B (eV)

9.722 2.112 0.000 6.416 5.187
1.555
4.303 10.168 4.733
0.480
4.667

10.193 1.939 0.000 7.392 2.509
1.408
4.400 8.060 5.818
0.456
5.200

A: empirical pseudopotential calculation from Ref. [55]; B: empirical pseudopotential calculation from Ref. [69].

2.2 General Strain Considerations

Strain–stress relationship or Hooke’s law can be used to describe the deformation of a crystal ekl, due to external or internal forces or stresses sij, X sij ¼ Cijkl ekl ; ð2:4Þ k;l

2.2 General Strain Considerations

where Cijkl is the fourth ranked elastic tensor and represents the elastic stiffness coefficients in different directions in the crystal, which due to the C6v symmetry can be reduced to a 6 · 6 matrix using the Voigt notation: xx ! 1, yy ! 2, zz ! 3, yz, zy ! 4, zx, xz ! 5, xy, yx ! 6. The elements of the elastic tensor can be rewritten as Cijkl ¼ Cmn with i, j, k, l ¼ x, y, z and m, n ¼ 1, . . . , 6. With this notation, Hooke’s law can be reduced to si ¼

X

Cij ej :

ð2:5Þ

j

or as treated in Ref. [80] for C6v symmetry, we have 2

3 2 sxx C11 6 syy 7 6 C12 6 7 6 6 szz 7 6 C13 6 7 6 6 sxy 7 ¼ 6 0 6 7 6 4 syz 5 4 0 szx 0

C12 C22 C13 0 0 0

C13 C13 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

32 3 0 exx 6 7 0 7 76 eyy 7 6 7 0 76 ezz 7 7; 6 7 0 7 76 exy 7 0 54 eyz 5 ezx C66

ð2:6Þ

with C66 ¼ C11
2 C12 . If the crystal is strained in the (0 0 0 1) plane, and allowed to expand and constrict in the [0 0 0 1] direction, the szz ¼ sxy ¼ syz ¼ szx ¼ 0, sxx 6¼ 0, and syy 6¼ 0, and the strain tensor has only three nonvanishing terms (with C11 ¼ C22), namely, 

sxx ; syy szz



 ¼

C11 þ C12 2C13

C 13 C 33



 exx ; eyy ; ezz

ð2:7Þ

with exx ¼ eyy ¼ a
a0a0 and ezz ¼ c
c0c0 ¼
CC1333 ðexx þ eyy Þ, the latter of which describes the Poisson effect, and a and a0 and c and c0 represent the in-plane and out-ofplane lattice constants of the epitaxial layer and the relaxed buffer (substrate), respectively. The above assumes that the in-plane strain in x- and y-directions is identical, namely, exx ¼ eyy. When the crystal is uniaxially strained in the (0 0 0 1) c-plane and free to expand and constrict in all other directions, szz is the only nonvanishing stress term, and the strain tensor is reduced to     1 eyy C 12 C33
C213 ð2:8Þ ¼ 2 exx : ezz C13
C11 C33 C 11 C13
C12 C13 If the growth is performed on the ð1
1 0 0Þm-plane, meaning the growth plane is the xz-plane with the growth direction being along the y-axis, the in-plane strain anisotropy dictates that exx 6¼ eyy . The out-of-plane stress, syy ¼ 0, which when utilized in the stress–strain relationship of Equation 2.6 leads to eyy ¼


C12 exx þ C13 ezz : C11

ð2:9Þ

If the growth is performed on the ð1 1
2 0Þa-plane, meaning the growth plane is the yz-plane with growth direction being along the x-axis, the in-plane strain anisotropy

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j 2 Electronic Band Structure and Polarization Effects

156

dictates that exx 6¼ eyy. The out-of-plane stress, sxx ¼ 0, which when utilized in the stress–strain relationship of Equation 2.6 leads to exx ¼


C12 eyy þ C13 ezz : C 11

ð2:10Þ

The values of the elastic stiffness coefficients for GaN have been measured by Sheleg and Savastenko [81] and reproduced in Ref. [59]. These data are listed in Table 1.24. The inversion of the 6 · 6 elastic constant matrix in Equation 2.6 leads to the elastic compliance constants as follows: S11 ¼

C11 C 33
C213 ; ðC 11
C12 Þ½C33 ðC 11 þ C12 Þ
2C213 

S12 ¼

C12 C 33
C213 ; ðC 11
C12 Þ½C33 ðC 11 þ C12 Þ
2C213 

S13 ¼

C13 ; ½C33 ðC11 þ C 12 Þ
2C 213 

S33 ¼

C11 þ C12 ; ½C33 ðC11 þ C 12 Þ
2C 213 

S44 ¼

1 : C44

ð2:11Þ

Through the use of the aforementioned compliance constants, a very useful figure of merit can be determined, which in turn would lead to the directional hardness and the reciprocal Young’s modulus as a function of orientation to the crystal axis. For 0 hexagonal symmetry, the reciprocal Young’s modulus S11 along an arbitrary direction at an angle y with respect to the [0 0 0 1] axis is given by [82–84] 0

S11 ¼ S11 sin4 q þ S33 cos4 q þ ðS44 þ 2S13 Þsin2 qcos2 q:

ð2:12Þ

0

In Figure 2.7a and b, polar plots of S 11 as a function of the direction in reference to the [0 0 0 1] axis and for directions along the basal plane are shown for InN, GaN, and AlN binaries. Clearly, AlN and GaN are harder than InN by more than a factor of 2. The hardness of AlN is almost isotropic, whereas that for GaN and InN exhibit some preferential “softness” along the [0 0 0 1] and ½2
1
1 0 axes. Of paramount interest here is that the hardness of all the wurtzitic binaries is isotropic in the basal plane. This takes on a special meaning as the strain, with growth along the c-axis, caused in epitaxial heterostructures by mismatch lattice and thermal mismatch is along the basal plane. Lack of any force in the growth direction and the fact that the crystal can relax freely in this direction leads to a biaxial strain e1 ¼ e2, which in turn causes stresses s1 ¼ s2

with s3 ¼ 0:

The internal strain is defined by the variation of the internal parameter under strain, (u
u0)/u0. In the limit of small deviations from the equilibrium, Hooke’s law

2.2 General Strain Considerations

[1 0 1 0] 8 S 11 [10 –12 m 2 /N–1]

(a)

6 In N 4

GaN

2 AlN –8

–6

–4

–2

0

0

[1 2 1 0] 2

4

6

8

–2 –4 –6 - –8 [0 0 0 1] 8

S 11 [10 –12 m 2 / N–1]

(b)

6 InN

4

GaN

2

AlN –8

–6

–4

–2

[2 1 1 0]

q 0

0

2

4

6

8

–2 –4 –6 - –8 Figure 2.7 (a) The reciprocal Young’s moduli, S0 11, in the basal plane of InN, GaN, and AlN indicating AlN and GaN are harder than InN to be harder by more than a factor of 2. Of paramount importance, the hardness of the wurtzitic crystals is isotropic in the basal plane. (b) The reciprocal

Young’s modulus, S0 11, along an arbitrary direction making an angle y with respect to the c-axis in the [0 0 0 1] direction. In basal plane the stiffness of AlN is isotropic whereas GaN and InN show preferential “softness” along the [0 0 0 1] and [2
1
1 0] directions [84].

gives the corresponding diagonal stress tensor s with the elements [85] sxx ¼ syy ¼ ðC11 þ C12 Þexx þ C13 ezz ; szz ¼ 2C13 exx þ C33 ezz :

ð2:13Þ

In Equation 2.13, four of the five independent stiffness constants Cij of the wurtzite crystal are involved. The modifications of Equation 2.13 by the built-in electric field due to the spontaneous and piezoelectric polarization are neglected, as the effect is small.

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j 2 Electronic Band Structure and Polarization Effects

158

In the case of uniaxial stress, for example, along the c-direction, there is an elastic relaxation of the lattice in the c-plane. The ratio of the resulting in-plane strain to the deformation along the stress direction is expressed by the Poisson’s ratio, which in general can be anisotropic. For the wurtzite lattice subjected to a uniaxial stress szz parallel to the c-axis, sxx ¼ syy ¼ 0 holds. Then Equation 2.13 gives the relation exx ¼
½C13 =ðC 11 þ C12 Þezz ¼
nezz ;

ð2:14Þ

with n ¼ C13 =ðC 11 þ C12 Þ being the Poisson’s ratio. The uniaxial stress correlates with strain along the direction of the stress by the Young’s modulus E as szz ¼ Eezz and thus E ¼ C33


2C 213 : C11 þ C 12

ð2:15Þ

A homogeneous biaxial stress in the basal plane is described by a constant force in the plane with sxx ¼ syy and vanishing force in the c-direction szz ¼ 0. The Hooke’s law of Equation 2.13 leads to a relationship between axial and basal plane strain components as ezz ¼
RBexx, which reproduces Equation 2.7 with the biaxial relaxation coefficient being RB ¼

2C13 : C33

ð2:16Þ

The biaxial relaxation coefficient is also referred to as simplylater in this chapter in the polarization section. The in-plane stress–strain relationship using the biaxial modulus is sxx ¼ Yexx, which leads to Y ¼ C11 þ C12


2C 213 : C33

ð2:17Þ

The strain–stress relationship along the c-axis is sxx ¼
ðY=RB Þezz :

ð2:18Þ

We can then relate the Young modules E (is also commonly described by nomenclature as Yo) to biaxial modulus Y as E¼

C33 Y 2n or E ¼ B Y: C11 þ C12 R

ð2:19Þ

Equation 2.13 can now be expressed as sxx ¼


E ezz : 2n

ð2:20Þ

In the case of hydrostatic pressure sxx ¼ syy ¼ szz ;

ð2:21Þ

and from the Hooke’s law ezz ¼ RH exx ;

ð2:22Þ

2.3 Effect of Strain on the Band Structure of GaN

with RH expressed as RH ¼

C11 þ C12
2C13 : C33
C 13

To calculate elastic stiffness constants, Wagner and Bechstedt [85] considered C11 þ C12 as an independent quantity and made use of the relation between the elastic constants and isothermal bulk modulus, B0 (nomenclature Bs is also used): B0 ¼

ðC11 þ C12 ÞC13
2C213 : C11 þ C12 þ 2C 33
4C13

ð2:23Þ

Equation 2.22 can be obtained from Equation 2.13 with the aid of linearized relation DP/B0 ¼
DV/V0 ¼
2(e11 þ e33) or Dp/B0 ¼
DV/V0 ¼
2(e11 þ e33), where DV is the variation of volume with pressure and V0 is the static volume. The values of the bulk modulus B0 have been obtained by fitting the Vinet equation of state to the calculated dependence of energy on volume [86,87], using     1 Y ¼ 2 þ 1=2 ð2:24Þ
4 RB B 0 ; v and C13 ¼ Y=½ð1=vÞ
RB :

ð2:25Þ

Wagner and Bechstedt [85] obtained the absolute values of the elastic stiffness coefficients. The strain- and stress-related issues represent the cornerstone of the discussion on piezoelectric polarization, and the above discussed section provides sufficient material to embark on the discussion of polarization issues in nitride semiconductor heterostructures.

2.3 Effect of Strain on the Band Structure of GaN

The strain in conventional group III–V semiconductors has been a much desired feature for its beneficial effects [88]. In the world of GaN, however, it is not necessarily a desirable commodity but could be construed as a nemesis brought upon by the lack of lattice- and thermal-matched substrates and uncomfortably large lattice and thermal mismatch with its ternaries. It is therefore imperative that strain effects be considered. Figure 2.8 exhibits the valence band structure of GaN in the x- and y-planes under biaxial compressive strain and uniaxial strain in the c-plane with the direction of strain as in Figure 2.8c. There are no major changes in the HH, LH, and CH bands, other than crystal splitting becoming larger, with the hole effective mass remaining heavy, the density of states staying high, and the crystal symmetry remaining the same, C46v . In contrast, the uniaxial strain in the c-plane causes an anisotropic energy splitting in the x- and y-planes, which leads to a symmetry lowering from C 46v to C2v . When a compressive uniaxial strain is induced along the y-direction, the HH band in the x-direction and the LH band in the y-direction move to higher energies. This causes a

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j 2 Electronic Band Structure and Polarization Effects

160

(a)

(b)

E

E

(c) Biaxial eff so

eff cr

Uniaxial z

kx

ky

kx

ky

y x

Figure 2.8 The valence band structure of GaN under (a) biaxial strain in the c-plane, (b) uniaxial strain in the c-plane, and (c) schematic of the particulars of the strain [59,89].

reduction in the density of states. A tensile uniaxial strain along the x-direction has the same effect. On the contrary, when a tensile uniaxial strain is induced along the ydirection, the HH band in the x-direction and the LH bands along the y-direction move to lower energies. This causes a reduction in the density of states. A compressive strain along the x-direction has the same effect [89].

2.4 kp Theory and the Quasi-Cubic Model

The conduction and valence bands of nitride semiconductors are comprised of s- and p-like states, respectively. Unlike the conventional ZB III-N semiconductors and the lack of a high degree of symmetry, the crystal field present removes the degeneracy at the top of the conduction band. Moreover, unlike the ZB case, the spin–orbit splitting is very small and makes all three bands in the valence band closely situated in energy. Consequently, the three valence bands and the conduction band must be considered in unison and this makes the use of an 8 · 8 kp Hamiltonian imperative. Because the bandgaps of nitrides are very large, the coupling between the conduction and valence bands can be treated as a second-order perturbation, which allows the 8 · 8 Hamiltonian to be split into one 6 · 6 Hamiltonian dealing with the valence band and another 2 · 2 dealing with the conduction band [59]. As indicated above, the conduction band is made of s-like states, which means that it can be treated as parabolic with the dispersion relation EðkÞ ¼ E c0 þ


2 k2z h

==

2mc

þ


2 ðk2x þ k2y Þ h 2m? c

== þ a? c ðexx þ eyy Þ þ ac ðezz Þ;

ð2:26Þ

==

where a? c and ac represent the in-plane and out-of-plane deformation potentials, respectively. For an isotropic parabolic conduction band, Equation 2.26 reduces to EðkÞ ¼ E c0 þ


2 k2 h þ ac e: 2mc

ð2:27Þ

2.4 kp Theory and the Quasi-Cubic Model

Ec0 is the conduction band energy at the k ¼ 0 point, e is the strain, and ac is the deformation potential for the conduction band. The other terms have their usual meanings. It should be pointed out that we are dealing with a linear system. Using the basis jY 11 ">; jY 11 #>; jY 10 ">; jY 10 #>; jY 1
1 ">; jY 1
1 #>, the 6  6 Hamiltonians can be expressed as 0 1 F 0
H 0 K 0  B 0 G D
H 0 K C B C B
H D l 0 I 0 C B C; ð2:28Þ B 0
H 0 l D I C B C @ K 0 I D G 0 A 0 K 0 I 0 F where F, G, l, D, H, I, and K are defined as (two forms are given by Ren et al. [90] and Albrecht et al. [91]) F ¼ D1 þ D2 þ l þ q; G ¼ D1
D2 þ l þ q; l ¼ A1 k2z þ A2 ðk2x þ k2y Þ þ D1 ezz þ D2 ðexx þ eyy Þ; q ¼ A3 k2z þ A4 ðk2x þ k2y Þ þ D3 ezz þ D4 ðexx þ eyy Þ; H ¼ iA6 kz ðk2x þ k2y Þ1=2
A7 ðk2x þ k2y Þ1=2 ; H ¼ ðiA6 kz
A7 Þðkx þ iky Þ þ iD6 ðexz þ ieyz Þ; I ¼ iA6 kz ðk2x þ k2y Þ1=2 þ A7 ðk2x þ k2y Þ1=2 ; I ¼ ðiA6 kz þ A7 Þðkx þ iky Þ þ iD6 ðexz þ ieyz Þ; K ¼ A5 ðk2x þ k2y Þ; K ¼p A5ffiffiffiðkx þ iky Þ2 þ D5 ðexx
eyy þ i2exy Þ; D ¼ 2D3 ;

½91 ½92 ½91 ½92 ½91 ½92

ð2:29Þ

where Ai is the valence band parameter corresponding to the Luttinger parameters in the ZB system, D1 and D2,3 are the crystal field and spin–orbit splitting energies (D2, ¼ D3 ¼ Dso), Di parameters represent the deformation potentials for the valence band, and exx, eyy, and ezz are the strain tensors (also referred to as e11, e22, and e33 in this book and many other publications). Both forms used by Albrecht et al. [91] and Ren et al. [90] for H, I, and K parameters are given. The former uses the basis and Hamiltonian in the (kx þ iky) form and contains all of the shear strain information. The latter does the phase rotation to get two 3 · 3 and the relevant matrix elements are then functions only of k-transverse and all of the directional phase information has been compiled onto the basis states by a unitary transformation. For presentation of the pseudomorphic strain, either form would be just as easy to use. Another important point is that in the case where the shear strain is nonzero, such as when uniaxial in-plane strain, the D5 and D6 terms must be included as represented by the Albrecht et al. [91] notation above. If only biaxial strain is considered, the shear terms of the strain tensor are zero and thus the D5 and D6 terms vanish. In the latter case, the representation by Ren et al. [90] and others similar to it hold. A good description of pertinent issues is discussed in Ref. [62]. Similar to the cubic system, the 6 · 6 matrix can be block diagonalized into two 3 · 3 matrices, and this can considerably simplify the band structure calculation. It should

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j 2 Electronic Band Structure and Polarization Effects

162

be mentioned that A7 can be assumed nearly zero due to symmetry considerations. Chuang and Chang [58] derived the two 3 · 3 Hamiltonians when A7 is neglected and obtained three doubly degenerate bands. Ren et al. [90] investigated the effect of A7 parameter on the valence band dispersion in wurtzitic crystals such as GaN. In fact, Ren et al. [90] argued that theories forwarded by Chuang and Chang [58] replicates that of Bir and Pikus [72] reported decades earlier. Choosing A7 ¼ 0 reduces the results to that of Chuang and Chang [58] and Sirenko et al. [62]. To underscore the effect of A7 parameter, Ren et al. [90] compiled data from empirical pseudopotential method (EPM) [67]. The same method was also applied to wurtzitic and zinc blende phases of all the three binaries of nitrides by Fritsch et al. [55]. Ren et al. [90] calculated the band structure for GaN for values of A7 ¼ 93.7 meV Å and 0 as, shown in Figure 2.9. Clearly, inclusion of the A7 parameter results in a much better fitting between the kp theory and EPM calculations. From the (a) 0.00

Wz GaN

–0.01

HH

Energy (eV)

–0.02 LH –0.03

–0.04 CH

–0.05 0.0

0.050

0.100

0.15

Wave vector, kx,( k// ) (1 Å–1) Figure 2.9 (a) Valence band structures of wurtzite GaN with the kp theory fitting including the spin–orbit interaction with A7 ¼ 93.7 meVÅ in the solid line. The dash-dotted line is the result of fitting with A7 ¼ 0. The empirical pseudopotential method (EPM) calculation data (o) are from Ref. [67] ([90]). (b) Valence band

structure of wurtzite GaN, using the parameters recommended by Ren et al. Courtesy of I. Vurgaftman and J. Meyer and Ref. [90]. (c) The same using the parameters recommended in Ref. [152]. The dashed lines represent the case for A7 ¼ 0. Courtesy of I. Vurgaftman and J. Meyer and Ref. [90].

2.4 kp Theory and the Quasi-Cubic Model

(b) 0.00

Wz GaN

HH

–0.01

Energy(eV)

LH –0.02

–0.03

–0.04 CH –0.05

–0.06 –0.15

–0.10 k z , (k )

–0.050

0.0

0.050

0.100

0.15

-1 Wave vector (1 Å ) kx, (k//)

(c) 0.00

Wz GaN (c)

HH

–0.01 LH

Energy(eV)

–0.02

–0.03

–0.04 CH –0.05

–0.06 –0.10

–0.15

k z , (k ) Figure 2.9 (Continued )

–0.050

0.0

0.050

Wave vector (1 Å-1)

kx, (k//)

0.100

0.15

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j 2 Electronic Band Structure and Polarization Effects

164

Table 2.7 Fitted splitting energies and Luttinger-like parameters

for the valence band of the wurtzite GaN: Di parameters are in unit of meV, and the Rydberg terms are in units of
h2 =2m0 except for A7, which is in unit of meV Å. D1

D2

D3

A1

A2

A3

A4

A5

A6

A7

21.1

3.61

3.61


7.21


0.440

6.68


3.46


3.40


4.9

93.7

quality of the fit, one can argue that the 93.7 value for the Luttinger-like parameter A7 is a good one. The other parameters giving this fit are listed in Table 2.7. In addition to the cycled parametric values of A parameters by kp theorists, a good extraction of A1–A7 parameters from empirical pseudopotential method band structures of AlN, GaN, and InN can be found in Ref. [55]. Vurgaftman and Meyer [152] followed a nearly identical path to that of Ren et al. [90] but calculated the band structure in both kx and ky crystal directions, as shown in Figure 2.9. The dashed lines, as in the case of the paper by Ren et al. [90], correspond to the case representing the case of A7 ¼ 0. In addition, Vurgaftman and Meyer [152] and Vurgaftman et al. [92] mentioned the calculation of the valence band structure for GaN using parameters representing the properties of GaN that they prefer, the results of which are also shown in Figure 2.9 for comparison. The effective masses are calculated using the parallel and perpendicular hole masses m//and m? that can be expressed in terms of their dependence on the Luttinger-like parameters Ai as follows: ==

m0 =mhh ¼
ðA1 þ A3 Þ; ==

m0 =mlh ¼
ðA1 þ A3 Þ; == m0 =mso

ð2:30Þ

¼
A1 :

?

Here, m represents the mass in the (kx, ky) plane, which means kz ¼ 0 and m0 =m? hh ¼
ðA2 þ A4
A5 Þ; 2 m0 =m? lh ¼
ðA2 þ A4
A5 Þ
2A7 =jD1 j ;

m0 =mh? so

¼

ð2:31Þ


A2 þ 2A27 =jD1 j;

where m//is along the kz-direction (kx ¼ ky ¼ 0) and m? in the (kx, ky) plane, which means kz ¼ 0. The effective masses are calculated using the parallel and perpendicular hole masses m//and m? together with Luttinger-like parameters using Equations 2.30 and 2.31. As indicated in the schematic of Figure 2.5, both the spin–orbit and the crystal field splitting affect the structure of the valence band in wurtzitic crystals [93]. Typically, the relevant parameters are correlated to one another as Dso ¼ 3D2 ¼ 3D3, in spite of the fact that a small D2/D3 anisotropy has sometimes been reported [94,95] and Dcr ¼ D1. Experimentally, the splitting parameters are obtained from the energy differences of the A, B, and C free excitons, which have nonlinear dependencies on the various splittings [96]. It should be pointed out that the nomenclature for the three valence

2.4 kp Theory and the Quasi-Cubic Model

bands for hexagonal system is A, B, and C for HH, LH, and SO (CH) bands when including A7 terms, because spin splitting and strain can significantly alter as to which band of eigenstates is “heavy” or “light” at various k-values, particularly in the c-plane. An early experimental undertaking by Dingle et al. [97] led to Dcr ¼ 22 meV and Dso ¼ 11 meV. An analysis by Gil et al. [98] yielded Dcr ¼ 10 meV and Dso ¼ 18 meV. Chuang and Chang [58] attempting to rederive these parameter from the same data but with what was termed as a more precise description of the effect of strain on the valence band edge energies arrived at values of Dcr ¼ 16 meV and Dso ¼ 12 meV. Reynolds et al. [99] obtained Dcr ¼ 25 meV and Dso ¼ 17 meV from a fit to exciton energies, with A and B determined by photoluminescence (PL) and C determined by reflection but with a geometry not fully ideal in terms in that some error is introduced in the value of C exciton energy. Again, using exciton energies values of Dcr ¼ 22 meV and Dso ¼ 15 meV were obtained by Shikanai et al. [100], Dcr ¼ 37.5 meV and Dso ¼ 12 meV by Chen et al. [101], Dcr ¼ 9 meV and Dso ¼ 20 meV by Korona et al. [102], and Dcr ¼ 9–13 meV and Dso ¼ 17–18 meV by Campo et al. [103] and Julier et al. [104]. The values of Dcr ¼ 10 meV and Dso ¼ 17 meV were determined by both Edwards et al. [105] and Yamaguchi et al. [96]. Noticeable is one of the smallest reported crystal field splittings to date, Dcr ¼ 9 meV, along with Dso ffi 18 meV reported by Rodina et al. [106] based on detailed experimental investigation. On the theoretical side, an ab initio calculation by Wei and Zunger [107] overestimates the crystal field splitting Dcr ¼ 42 meV, but arrives at a Dso ¼ 13 meV that agrees well with the experimental data. Suzuki et al. [56] reported Dcr ¼ 40 meV and Dso ¼ 8 meV or Dso ¼ 3D2 ¼ 3D3 1.16 mRy and Dcr ¼ D1 ¼ 5.36 mRy for these splittings. Many firstprinciples calculations focusing on the valence band splitting are available in the literature [58,90,67,154,108]. The experimental data, however, appear to converge on the splittings Dcr ffi 10 meV and Dso ffi 17 meV, as suggested by Vurgaftman and Meyer and tabulated in Table 2.8 [152]. The spin splitting of the valence band of wurtzitic GaN can be determined via the A7 parameter, as shown in Figure 2.9a, which is derived by Vurgaftman and Meyer [152] assuming the parameters of Ren et al. [90]. On the contrary, Figure 2.9b is derived by Vurgaftman and Meyer [152] using A parameters from Ren et al. [90] combined with what is believed to be the more representative spin–orbit and crystal field splittings. Modification of the A parameters alone, but with the corrected values of the splitting energies, does not allow the recovery of the band structure resembling that shown in Figure 2.9a. This simply implies that the field is not yet settled on a set of reliable parameters and more refinement is needed for the most appropriate values of the A parameters to be arrived. To be sure six distinct valence band deformation potentials, as well as the strain tensor and the overall hydrostatic deformation potential, are necessary to describe the band structure of GaN under strain. In the cubic approximation, these can be expressed in terms of the more familiar av, b, and d potentials [93]. In terms of the calculation and combined calculation and measurement efforts, Christensen and Gorczyca [31] reported a hydrostatic deformation potential a ¼
7.8 eV, which has been shown to agree well with
8.16 eV obtained by Gil et al. [98]. On the contrary, a somewhat lower value of a ¼
6.9 eV was attained through an ab initio calculation by

j165

j 2 Electronic Band Structure and Polarization Effects

166

Table 2.8 Recommended band structure parameters for wurtzitic GaN from Ref. [152].

Parameter

Value

Parameter

Value

Parameter

Value

Eg (eV, low temperature) a (meV K
1) b (K)

3.510

A1


7.21

D1 (eV)


3.7

0.909 (1 in [119]) 830 (1100 in [119]) 10 17 0.20 0.20
4.9
11.3

A2 A3


0.44 6.68

D2 (eV) D3 (eV)

A4 A5 A6 A7 (meV Å) d13 (pm V
1) d33 (pm V
1) d15 (pm V
1) Psp (C m
2)


3.46
3.40
4.90 93.7
1.6b 3.1b 3.1b
0.034

D4 (eV) D5 (eV) D6 (eV) c11 (GPa) c12 (GPa) c13 (GPa) c33 (GPa) c44 (GPa)

Dcr (meV) Dso (meV) == me =m0 m? e =m0 a1 (eV) a2 (eV)

4.5 8.2
4.1
4.0
5.5 390 145 106 398 105

367a 135a 103a 405a 95a

See Tables 2.27 and 2.28 for details related to the elastic constants, piezoelectric constants, and spontaneous polarization charge. Any dispersion among the tables is a reflection of the uncertainty in the available parameters. See Volume 2, Chapter 5 for an extended discussion of Varshni parameters. a The second column figures for the Cii parameters are from Table 2.28 where a more expanded list of elastic coefficients is given. b Table 2.28 provides additional data on d-parameters.

Kim et al. [109]. Noting that the hydrostatic potential is anisotropic, owing to the reduced symmetry of the wurtzite crystal, Wagner and Bechstedt [85] calculated values of
4.09 and
8.87 eV for the two hydrostatic interband deformation potentials. On the transport side, fits to the experimental mobility data [110,111] yielded a conduction band deformation potential approaching
9 eV, a topic discussed in some detail in Volume 2, Chapter 3. Using pressure-dependent optical transition energies with pressure, Shan et al. [112] reported a1 ffi
6.5 eV and a2 ffi
11.8 eV and uniaxial deformation potentials b1 ffi
5.3 eV and b2 ffi 2.7 eV for the two hydrostatic interband components. Employing photoreflectance measurements on compressively strained M-plane GaN films (grown along the h1 0
1 0i direction) grown on g-LiAlO2 (1 0 0), Ghosh et al. [113] reported a1 ¼
3.1 eV with a2 ¼
11.2 eV. Again, using M-planeoriented GaN layers, Gil and Alemu [114] obtained a1 ¼
5.22 eV with a2 ¼
10.8 eV. Vurgaftman and Meyer [152] recommend a set of a1 ¼
4.9 eV and a2 ¼
11.3 eV, which represents an average of all the measured values. Numerous sets of valence band deformation potentials have been derived from both first-principles calculations [58,60,61,115,116] and fits to experimental data [94,98,100,96,112–114,117]. The dispersion among the reported data is unacceptable, which calls for further work to resolve the discrepancies and converge on accurate parameters. If one were to average the deformation potentials that are most widely quoted, values of D1 ¼
3.7 eV, D2 ¼ 4.5 eV, D3 ¼ 8.2 eV, D4 ¼
4.1 eV, D5 ¼
4.0 eV, and D6 ¼
5.5 eV, which satisfy the quasi-cubic approximation that is about to be discussed [58], are obtained. The deformation potential values for GaN are

2.5 Quasi-Cubic Approximation

tabulated in Table 2.8, and their effect on optical transitions are discussed in detail in Volume 2, Chapter 5.

2.5 Quasi-Cubic Approximation

The genesis of the quasi-cubic approximation relies on the fact that the Wz and ZB structures are both tetrahedrally coordinated and hence are closely related. The nearest neighbor coordination is the same for Wz and ZB structures but differs at the next nearest neighbor positions. The basal plane (0 0 0 1) of the Wz structure corresponds to one of the (1 1 1) planes of the ZB. When the in-plane hexagons are lined up in Wz and ZB structures, the Wz [0 0 0 1], ½1 1
2 0, and ½1
1 0 0 planes are

parallel to the ZB [1 1 1], ½1 0 1, and ½1 2 1 planes, respectively. This, in turn, leads to correlations between the symmetry direction and the k-points for the two polytypes. There are, however, twice as many atoms in the Wz unit cell as there are in the ZB one. In addition to the band structure similarities between the doubled ZB and Wz structures, one can establish a correlation between the Luttinger parameters in the ZB system and parameters of interest in the Wz system by taking the z-axis along the [1 1 1] direction and the x- and y-axes along the ½1 1
2 and ½
1 1 0 directions. For details regarding the symmetry relations between the ZB and Wz polytypes, refer to Refs [32,118]. Doing so leads to D2 ¼ D3 ; A1 ¼ A2 þ 2A4 ;pffiffiffi A3 ¼
2A4 ¼ 2A6
4A5 ; A7 ¼ 0; D1 ¼ D2 þ 2D4 ;pffiffiffi D3 ¼
2D4 ¼ 2D6
4D5 :

ð2:32Þ

The A parameters can be related to the classical Luttinger parameters gi through A1 A2 A3 A4 A5 A6

¼
ðg 1 þ 4g 3 Þ; ¼
ðg 1
2g 3 Þ; ¼ 6g 3 ; ¼
2g 3 ; ¼
p ðg ffiffi2ffi þ 2g 3 Þ; ¼
2ð2g 2 þ g 3 Þ:

ð2:33Þ

The calculated values of the spin–orbit and crystal field splitting parameters with those deduced from the observation of A, B, and C excitons are listed in Tables 2.8 and 2.9, which will be presented shortly. The calculations agree well in terms of the spin–orbit splitting, but the theoretical crystal field splitting is much too large compared to experimental data. The discrepancy may be due to the unaccounted residual strain and strain inhomogeneities present in GaN films, as well as the inaccuracy of the parameter values. The debate will probably continue until strainfree or homogeneously strained films can be prepared.

j167

j 2 Electronic Band Structure and Polarization Effects

168

Table 2.9 Effective masses and band parameters for wurtzitic GaN.

Parameter Aniso ==

me m? e == mhh == mlh == mch m? hh m? lh m? ch A1 A2 A3 A4 A5 A6 A7 D1

0.138 0.151 2.000 2.000 0.130 2.255 0.191 0.567
7.692
0.575 7.192
2.855
2.986
3.360 0.160 0.043

Iso

A

B

0.138 0.151 2.007 2.007 0.130 2.249 0.261 0.317
7.698
0.600 7.200
2.816
2.971
3.312 0.171 0.023

0.20 0.18 1.10 1.10 0.15 1.65 0.15 1.10
6.56
0.91 5.65
2.83
3.13
4.86

0.20 0.18 1.76 1.76 0.16 1.61 0.14 1.04
6.27
0.96 5.70
2.84
3.18

a

0.039

C

0.14 0.15 1.479 1.479 0.130 1.592 0.299 0.252
7.706
0.597 7.030
3.076
3.045
4.000 a 0.194 0.038 0.022

D

E

0.14 0.15 1.453 1.453 0.125 1.595 0.236 0.289
7.979
0.581 7.291
3.289
3.243
4.281 0.179 0.022

0.19 0.17 1.76 1.76 0.14 1.69 0.14 1.76
7.14
0.57 6.57
3.30
3.28

F

0.19 0.17 1.96 1.96 0.14 1.87 0.14 1.96
7.24
0.51 6.73
3.36
3.35
4.72 0 0 0.021 0.021

G

H

I

0.19 0.19 0.23 0.19 1.89 2.00 1.96 1.89 2.00 1.96 0.12 0.16 0.16 2.00 2.04 1.20 0.15 0.18 0.16 0.59 1.49 1.96
7.21
6.4
6.36
0.44
0.50
0.51 6.68 5.9 5.85
3.46
2.55
2.92
3.40 2.56 2.60
4.9 3.06 3.21 0.094 0.108 0 0.021 0.036

Effective masses in units of free-electron mass m0, Luttinger-like parameters Ai (i ¼ 1, . . ., 6) in units of h2 =2m0 , and A7 in units of eV Å . The crystal field splitting energy D1 is given in units of meV. The term
aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric == tensor. Here, me and m? e represent the effective electron masses along and perpendicular to the c-axis [55]. Anisotropically screened and isotropically screened values are from Ref. [55]. A: FP-LAPW band structure calculations are from Ref. [56], and effective mass parameters are obtained through a 3D fitting procedure within cubic approximation; B: FP-LAPW band structure calculations are from Ref. [56], and effective mass parameters are obtained by direct line fit; C: Ai from Ref. [153] obtained through a Monte Carlo fitting procedure to the band structure and effective masses calculated using Equations 2.30 and 2.31; D: direct kp calculations for Ai from Ref. [153] and effective masses calculated using Equations 2.30 and 2.31; E: effective masses and Ai from Ref. [67] obtained through a line fit to the band structure; F: direct kp calculation in a 3D fit from Ref. [67]; G: Ai obtained through a direct fit from Ref. [90] and effective masses calculated using Equations 2.30 and 2.31; H: direct fit of Ai to firstprinciples band structure calculations from Ref. [154]; I: Ai and effective masses obtained in the quasicubic model from zinc blende parameters from Ref. [154]. a A7 in the range of 0.136 eV Å has been set to zero.

The parameters mentioned in bandgap-related discussion for wurtzitic GaN are tabulated in Table 2.8 [152] for the wurtzitic phase GaN. All conventional nitridesinthe wurtzite phase exhibit a directenergy gap, and the next satellite conduction valley, which is the M valley, is some 2 [31]to 5 eV [56]higher than the G valley. In addition to the one made available here, the wurtzite indirect gap related issues have been amply discussed in the literature, some of which can be found in Refs [31,53,75,120–122]. Sufficing it to state the bottom of the conduction band in GaN can be well approximated by a parabolic dispersion relation, although a slight anisotropy is expected due to the reduced lattice symmetry [56]. In many devices, the pertinent property of the band structure is the region near the bottom of the conduction band, which can be represented to a great deal with effective mass. In relatively early

2.6 Temperature Dependence of Wurtzite GaN Bandgap

experimental studies in GaN grown by hydride VPE, Barker and Ilegems [123] obtained an electron effective mass of mn ¼ 0.20m0 from reflectivity measurements. Again early on, Rheinlander and Neumann [124] inferred 0.24–0.29m0 for the effective mass from a Faraday-rotation investigation of heavily n-doped GaN. Using heavily doped samples, which was the norm then, and fits to the thermoelectric power, Sidorov et al. [125] obtainedelectroneffective masses of0.1–0.28m0, dependingon whatprimary scattering channel was assumed. For a review of early investigations of these and other properties, thereaderisreferredtoreviewsfromthe1970s,suchastheonebyPankoveetal.[126]and that by Kesamanly [127]. Congruent with the increased activity in GaN, fuelled by the device demonstrations, particularly LEDs and later on lasers, a substantial body of work has since produced more precise estimations of the electron mass. Among them are the works by Meyer et al. [128] and Witowski et al. [129] who obtained masses of 0.236m0 and 0.222m0, respectively, utilizing shallow donor transition energies; the latter is with the smallest error bars quoted in the literature (0.2%). Underscoring the importance of the polaron correction, which is about 8% in GaN and comes about because of the strong polar nature of GaN, Drechsler et al. [130] derived a bare mass of 0.20m0 from cyclotron resonance data. A similar result was obtained by Perlin et al. [131] using infrared reflectivityandHalleffectmeasurements,whichalsoledtoananisotropyoflessthan1%. For comparison, a slightly larger dressed mass of 0.23m0 has been obtained by Wang et al. [132] and Knap et al. [133]. A small downward correction may be necessary in the former,astheelectronswereconfinedataninterface.Thelatterauthors,however,appear to have corrected for that effect. Using n-type bulk GaN, which does not require an appreciable correction that is needed in confined systems and employing infrared ellipsometry measurements, Kasic et al. [134] reported slightly anisotropic electron masses of 0.237 0.006m0 and 0.228 0.008m0 along the two axes. Again, using modulation-doped structures, a series of authors, Elhamri et al. [135], Saxler et al. [136], Wong et al. [137], Wang et al. [138], and Hang et al. [139], also reported on the effective mass, the values of masses for which ranged from 0.18m0 to 0.23m0 from Shubnikov–de Haas data. Elhamri et al. [135] suggested that strain effects, which are somewhat difficult to be certain of, could have compromised somewhat the masses reported in some of these reports. A value of 0.20m0 is very commonly used for the bare electron effective mass and 0.22m0 for the experimentally relevant dressed mass. A more in-depth discussion of the cyclotron and Shubnikov–de Haas measurements can be found in Volume 2, Chapter 3. This bare mass figure of 0.20m0 agrees reasonably well with a number of estimates based on theory, as outlined in a list in Ref. [140]. Owing to the large uncertainty, no attempt is made to specify an F parameter for wurtzite GaN. However, the interband matrix element may be obtained from the relation between the electron mass and the relevant zone center energies [92].

2.6 Temperature Dependence of Wurtzite GaN Bandgap

The temperature dependence of the bandgap in semiconductors is often described by an imperial expression (assuming no localization)

j169

j 2 Electronic Band Structure and Polarization Effects

170

EðTÞ ¼ Eð0Þ
aT 2 =ðb þ TÞ:

ð2:34Þ

In the case of localization, which can also be construed as band tail effect, the temperature dependence deviates from the above equation. In the framework of the band tail model and Gaussian-like distribution of the density of states for the conduction and valence band, the temperature-dependent emission energy could be described by the following modified expression [141], which is based on a model developed for Stokes shift in GaAs/AlGaAs quantum wells [142]. EðTÞ ¼ Eð0Þ
½aT 2 =ðb þ TÞ
½s2 =ðkTÞ;

ð2:35Þ

where the last term represents the localization component with s indicating the extent of localization or band tailing, which is nearly imperative for In-containing alloys. The values of the parameters a (in units of energy over temperature) and b (in units of temperature), for wurtzitic GaN, are listed in Volume 2, Table 5.1. Although a detailed discussion of these parameters in very high-quality samples is deferred to Volume 2, Chapter 5, the evolution of them is discussed here to give the reader a flavor that when the sample quality is under question, the fits to experiments could lead to varying if not erroneous parameters. In concert with this approach, the spread in the values of a and b for A exciton is also discussed in the text surrounding Volume 2, Table 5.3. Varshni parameters have been deduced from the measured variation of the A, B, and C excitonic energies with temperature early on by Monemar [143] with Varshni parameters of a ¼ 5.08 · 10
4 meV T
1 and b ¼
996 T, the sign for the latter of which is contradictory to the agreed upon values deduced from high-quality samples. In chronological order of the reports, using optical absorption measurements on bulk single crystals and also epitaxial layers grown on sapphire, Teisseyre et al. [144] reported a ¼ 0.939–1.08 meV K
1 and b ¼ 745–772 K; note the positive sign of b. Shan et al. [145] reported a ¼ 0.832 meV K
1 and b ¼ 836 K deduced from the temperature variation of the A exciton resonance. Petalas et al. [146] determined the Varshni parameters to be a ¼ 0.858 meV K
1 and b ¼ 700 K using spectroscopic ellipsometry. Relying on PL measurements, Salvador et al. [147] obtained a ¼ 0.732 meV K
1 and b ¼ 700 K. Using absorption measurements, Manasreh [148] reported a ¼ 0.566–1.156 meV K
1 and b ¼ 738–1187 K on samples grown by MBE and OMVPE. Using a variation of electroreflectance, the contactless electroreflectance, Li et al. [149] led to a ¼ 1.28 meV K
1 and b ¼ 1190 K for the A exciton transition energy. Utilizing PL spectra of excitonic transitions, Zubrilov et al. [150] reported values of a ¼ 0.74 meV K
1 and b ¼ 600 K based on exciton luminescence spectra. PL data of free and bound excitons were fitted by Reynolds et al. [151] to a modified Varshni-like form that resulted in a ¼ 0.5 meV K
1 and b ¼ 1060. Some of the dispersion in the reported values can be attributed to the difficulty in identifying and resolving various excitonic transitions. Vurgaftman and Meyer [152], averaging what they term as more credible results, recommend a ¼ 0.909 meV K
1 and b ¼ 830 K. A detailed discussion of the temperature dependence of the bandgap of GaN along with the Varshni parameters for all three excitons (A, B, and C) are listed in Volume 2, Table 5.3, where A exciton related parameters are a ¼ 1 meV K
1 and b ¼ 1100 K.

2.6 Temperature Dependence of Wurtzite GaN Bandgap

The complete Ai parameters calculated by Fritsch et al. [55] as well as others deduced from alternative methods are tabulated in Table 2.9. In addition, a compilation of the dispersion in the effective mass for both the conduction band and various valence bands as obtained by various computational methods as well as parameters used in the description of the bandgap for wurtzitic GaN, particularly, in the context of empirical pseudopotential method, as described in Ref. [55], are also included in Table 2.9. Suzuki and Uenoyama [59] have determined the deformation potentials by the fullpotential linearized augmented plane wave (FLAPW) calculations. The values recommended for GaN by Vurgaftman and Meyer [152] are tabulated in Table 2.8. In the calculations of Suzuki and Uenoyama [59], biaxial and uniaxial strains have been introduced and reduced shifts in the G point energy to which a linear fit in terms of strain was obtained. From the linearfit,the deformation potential values for thevalence band were deduced. The figures obtained from the quasi-cubic approximation are listed too. The good agreement between the calculated values and those determined from the quasi-cubic model is strikingly obvious, which is indicative of the excellence of the quasi-cubic approximation. Assuming an approximately spherical potential in the neighborhood of the N atoms, of the two spin states, the higher energy is in the one in which the electron spin and the orbital angular momentum are parallel. This result is also anticipated on the basis of the atomic spin–orbit splitting in which the P1/2 state is known to have energy higher than the P3/2 state. The contributions of spin–orbit interaction and the crystal field perturbation to the experimentally observed splittings (E1,2 and E2,3) have been calculated with different linear combination of atomic orbitals (LCAO) approximations [155,156]. The large effective mass and the small dielectric constant of GaN, relative to more conventional group III–V semiconductors, lead to relatively large exciton binding energies and make excitons, together with large exciton recombination rates, clearly observable even at room temperature. The bottom of the conduction band of GaN is predominantly formed from the s levels of Ga, and the upper valence band states from the p levels of N. Even though sophisticated methods have been introduced and discussed on these pages, the method of Hopfield and Thomas [157], which treats the wurtzite energy levels as a perturbation to the zinc blende structure is discussed briefly, as it provides a physical picture of band splitting in the valence band. Using the quasi-cubic model of Hopfield [98], one obtains E 1 ¼ 0; v" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # u   dþD u dþD 2 2 t E2 ¼
dD ; þ 2 2 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v" # u  2 dþD u d þ D 2
dD ;
t E3 ¼ 2 2 3

ð2:36Þ

ð2:37Þ

ð2:38Þ

j171

j 2 Electronic Band Structure and Polarization Effects

172

where Dcr and Dso represent the contributions of uniaxial field and spin–orbit interactions, respectively, to the splittings E1,2 and E2,3.

2.7 Sphalerite (Zinc blende) GaN

The zinc blende GaN crystal consists of two interpenetrating face-centered cubic lattices, one having a group III element atom, for example, Ga, and the other a group V element atom, for example, N. The matrix element of the momentum operator between the conduction and valence bands has been expressed by Kane [92,158] in terms of a single parameter P whose value is termed as EP, which is an important parameter. The other parameter of importance is the F parameter, again defined by Kane, which comes about from a second-order perturbation theory and takes into account the higher order band contributions to the conduction band. Their importance aside [92] EP and F are inherently difficult to determine accurately, due to the fact that the remote band effects can be calculated but are not measurable quantities. One experimental technique relies on measuring the effective g factor, which is not as influenced by remote bands as the effective mass. A number of experimental and theoretical studies have determined energy gaps for zinc blende GaN [69,70,146,159–163]. The term “gaps” is used, as there seems to be a dispersion in the reported values. Typically, the excitonic transitions [164–166] observed in low-temperature PL is used to infer the bandgap, provided the exciton binding energy is known, which in this case is 26.5 meV. Although, low-temperature bandgaps ranging from 3.2 to 3.5 eV have been measured, most of them tend to be between 3.29 and 3.35 eV. It is therefore reasonable to use a low-temperature bandgap of 3.3 eV for zinc blende GaN. A reasonable figure for the room-temperature fundamental bandgap is 3.2 eV, although the range of 3.2–3.3 eV stated in Chapter 1 remains. As in the case of wurtzitic GaN, the temperature dependence of the energy gap was also studied for zinc blende GaN, examples of which can be found in the works of Petalas et al. [146] and Ramirez-Flores et al. [167]. Both group of authors found b ¼ 600 K (using the more reliable model 1 in Ref. [146]), but the a parameters differed somewhat. Not having any real basis for selecting one or the other, the average value of 0.593 meV K
1 is considered the default value. Although the indirect gap energies have not been measured, for a calculation by Fan et al. [69], the X-valley and L-valley minima had been put at 1.19 and 2.26 eV above the G valley, respectively. These compare with an earlier prediction by Suzuki et al. [56] of about 1.4 and 1.6 eV, respectively, as shown in Figure 2.6c. Ramirez-Flores et al. [167] measured the spin–orbit splitting in zinc blende GaN to be 17 meV. Electron spin resonance measurements on zinc blende GaN determined an electron effective mass of 0.15m0 [168], which may represent the only experimental results, and the value is similar to the G-valley masses derived from first-principles calculations by Chow et al. [169] and Fan et al. [69]. Effective masses of ml ¼ 0:5m0 and mt ¼ 0:3m0 have been calculated for the X valleys in GaN [70], which are similar to the theoretical results of Fan et al. [69].

2.7 Sphalerite (Zinc blende) GaN

0.00

HH

ZB GaN

LH

–0.01

Energy (eV)

j173

–0.02

SO

–0.03

–0.04

–0.05

–0.06 –0.15

–0.10 [1 1 1]

–0.050

0.0

0.050

Wave vector (1 Å-1)

0.100 [0 0 1]

Figure 2.10 Valence band structure of zinc blende GaN [152].

The valence band of zinc blende GaN has been the topic of various theoretical efforts, and the E–k diagram by Vurgaftman and Meyer [152] is show in Figure 2.10. Although the hole effective masses in zinc blende GaN have apparently not been measured, a number of theoretical predictions of Luttinger parameters are available in the literature [69,70,164–166,168,169,425,170,171]. Once the Luttinger parameters are known, the full picture in terms of the hole effective masses can be determined. First, it should be pointed out that in polar semiconductors such as the III–Vcompounds in general and GaN in particular, it is the nonresonant polaron [172] mass that is actually measured. The polaron mass exceeds the bare electron mass by about 1–2%, the exact value of which depends on the strength of the electron–phonon interaction. Because the band structure is governed by the bare electron mass, this is the quantity that is typically reported whenever available. At the valence band edge, the heavy hole (hh) effective masses in the different crystallographic directions are related to the free mass by the Luttinger parameters in the following manner [92]: mzhh ¼

m0 2m0 m0 ½110 ½111 ; mlh ¼ ; mlh ¼ : g 1
2g 2 2g 1
g 2
3g 3 g 1
2g 3

ð2:39Þ

0.15

j 2 Electronic Band Structure and Polarization Effects

174

Here, the z-direction is perpendicular to the growth plane of (0 0 1). These expressions described by Equation 2.39 show the relationship of the Luttinger parameters to the hh effective masses that can typically be measured in a more direct manner. The light hole (lh) and so hole effective masses are given by mzlh ¼

m0 2m0 m0 ½110 ½111 ; mlh ¼ ; mlh ¼ ; g 1 þ 2g 2 2g 1 þ g 2 þ 3g 3 g 1 þ 2g 3

m0 E P Dso ¼ g1
: mso 3E g ðE g þ Dso Þ

ð2:40Þ

ð2:41Þ

Equation 2.41, which relates the split-off hole mass to the Luttinger parameters, should in principle contain an additional parameter to account for the effects of remote bands that is analogous to the F parameter [92], but the remote bands are not necessarily the same ones that cause the largest correction to the electron mass. Due to the dominance of the wurtzitic GaN, insufficient effort and thus data exist for zinc blende GaN, which is also true for even the well-investigated III–V materials to describe the effect of the interaction with remote bands on the split-off hole mass quantitatively. To restate, although the hole effective masses in zinc blende GaN have apparently not been measured, a number of theoretical predictions of Luttinger parameters are available in the literature. The values are based on averages of the heavy-hole and lighthole masses along [0 0 1], as well as the degree of anisotropy in g3–g2. Doing so leads to the parameter set as g1 ¼ 2.70, g2 ¼ 0.76, and g3 ¼ 1.11. Similarly, averaging all the reported split-off masses [69,70,163,171,173] leads to mso ¼ 0:29m0 . In its simplest form, the Luttinger parameters can be used to quickly determine the effective masses in various valence bands both in equilibrium and also under biaxial strain. In fact, with biaxial strain, the valence band degeneracy can be removed, and most strikingly the heavy-hole in-plane mass can be made smaller by compressing strain, a notion that has been exploited in the InGaAs/GaAs system very successfully. An average of the two theoretical values for EP in zinc blende GaN [163,173] yields EP ¼ 25.0 eV, which in turn implies F ¼
0.95. Caution is advised because these values have not been verified experimentally. In conjunction with calculations of the electronic band structure of binary nitrides and specifically effective masses in the valence band, Fritsch et al. [55] also arrive at the Luttinger-like kp parameters by empirical fits for the effective masses at the G point. These Luttinger-like parameters for the valence band of zinc blende GaN are listed in Table 2.10. Those for zinc blende AlN and InN will be given in Sections 2.8.2 Sections 2.9.2. The Ai parameters transformed from the Luttinger parameters obtained with the help of the quasi-cubic approximation can be found in Tables 2.8 and 2.9. It has been argued that the Ai parameters calculated in this manner are in good agreement with the calculated values that have been used to support the value and validity of the quasi-cubic approximation, which greatly simplifies the calculations. Fritsch et al. [55] obtained the valence band effective masses for zinc blende binaries by solving the eigenvalues of the kp matrix while taking the spin–orbit interaction into account. The effective masses so calculated for the conduction and valence bands, the latter involving the light and heavy holes, as well as the spin–orbit

2.7 Sphalerite (Zinc blende) GaN Table 2.10 Luttinger parameters g1, g2, and g3 for zinc blende GaN obtained from a fit along the [1 1 0] direction along with those available in the literature, as compiled in Ref. [55].

Parameter

A

B

C

D

E

c1 c2 c3

2.89 0.85 1.20

2.96 0.90 1.20

2.70 0.76 1.07

3.07 0.86 1.26

2.67 0.75 1.10

A: empirical pseudopotential calculation by Fritsch et al. [55]; B: self-consistent FP-LAPW method within local density approximation from Ref. [171]; C: first-principles band structure calculations from Ref. [170]; D: empirical pseudopotential calculation from Ref. [174]; E: recommended values taken from Ref. [152].

split-off mass, and the anisotropy taken into account are listed in Table 2.11 for zinc blende GaN. The data contain those obtained by full potential linearized plane waves (FP-LAPW), empirical pseudopotential method (EPM) calculations and those calculated with Luttinger parameters employing ðm0 =mhh=lh Þ½100 ¼ g 1 2g 2 ; ðm0 =mhh=lh Þ½111 ¼ g 1 2g 3 ; 2g g 2 g 3 ðm0 =mhh=lh Þ½1100 ¼ 1 ; 2

ð2:42Þ

where subscripts hh and lh represent the heavy-hole and light-hole effective masses, respectively. The spin–orbit split-off-hole effective mass, mso, is isotropic in all the three directions and is given by ðm0 =mso Þ ¼ g 1 :

ð2:43Þ

Table 2.11 Effective masses for electrons (e), heavy holes (hh), light holes (lh), and spin–orbit split-off (so) holes in units of the free-electron mass m0 along the [1 0 0], [1 1 1], and [1 1 0] directions for zinc blende GaN. [100]

[100]

[111]

[111]

[110]

[110]

Reference

me

mhh

mlh

mhh

mlh

mhh

mlh

mso

A B C D E F

0.14 0.14 0.13 0.17 0.15 0.12

0.84 0.86 0.76 0.85 0.85 1.34

0.22 0.21 0.21 0.24 0.24 0.70

2.07 2.09 1.93 1.79 2.13 1.06

0.19 0.19 0.18 0.21 0.21 0.63

1.52 1.65 1.51 1.40 1.55 1.44

0.20 0.19 0.19 0.21 0.21 0.58

0.35 0.30 0.32 0.37 0.29 0.20

Compiled by Fritsch et al. [55]. A: after Ref. [55]; B: self-consistent FP-LAPW method within local density approximation from Ref. [171]; C: empirical pseudopotential calculation from Ref. [69]; D: calculated from Luttinger parameters from Ref. [170], using Equations 2.42 and 2.43; E: calculated from recommended Luttinger parameters from Ref. [152], using Equations 2.42 and 2.43; F: empirical pseudopotential calculation from Ref. [70].

j175

j 2 Electronic Band Structure and Polarization Effects

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Because various calculations place the hydrostatic deformation potential for zinc blende GaN in the range
6.4 to
8.5 eV [31,69,70,104,169,175], for the lack of a better choice, the average value of the hydrostatic deformation potential a ¼
7.4 eV is chosen. It should be mentioned that due to the nature of the atomic bonding in III–V materials, the bandgap increases for a compressive strain. Under positive hydrostatic pressure that produces negative strain because the lattice constant gets smaller, the change in the bandgap corresponding to a change in volume, DV/V ¼ (exx þ exx þ ezz), given by DEg ¼ a(exx þ exx þ ezz) is positive, which necessitates the sign of the deformation potential to be negative. The deformation potential is the sum of the conduction and valence band deformation potentials, a ¼ ac þ av. Under pressure, the conduction band edge is believed to move upward in energy while the valence band moves downward, with most of the change being in the conduction band edge. Similar to the total bandgap change, the change in valence band under pressure can be described as DEg ¼ av(exx þ exx þ ezz). Naturally, it is much easier to measure the change in the total bandgap due to strain than its component effects on the conduction and valence bands. The valence band deformation potential value reported by Wei and Zunger [176], namely, av ¼
0.69 eV, is suggested. The sign convention for av is adopted from Vurgaftman et al. [92]. To restate, the shrinking volume and negative strain cause the valence band to move down. The values reported in the literature are in the range of
0.69 to
13.6 eV. The suggested value is consistent with the expectation that most of the strain shift should occur in the conduction band. As for the shear deformation potentials b and d, the same is applied, which yields a suggestion that b ¼
2.0 eV with the full range of reported values being
1.6 to
3.6 eV. The recommendation d ¼
3.7 eV is an average of the published results from Ohtoshi et al. [175], Van de Walle and Neugebauer [177], and Binggeli et al. [178]. No experimental confirmations of any of these deformation potentials for zinc blende GaN appear to exist. Turning to elastic constants, the values of C11 ¼ 293 GPa, C12 ¼ 159 GPa, and C44 ¼ 155 GPa have been taken from the theoretical calculations of Wright [179]. Very similar sets have also been calculated by Kim et al. [109,180] and Bechstedt et al. [181]. For more details of elastic constants for all three binaries, refer to Tables 2.25–2.27 and 2.28 that will follow later on in this chapter. The parameters in conjunction with the band structure for zinc blende GaN are compiled in Table 2.12.

2.8 AlN

AlN forms the larger bandgap binary used in conjunction with GaN for increasing the bandgap for heterostructures. As in the case of GaN, AlN also has wurtzitic and zinc blende polytypes, the latter being very unstable and hard to synthesize. Owing to increasing interest in solar blind devices and expectations that larger AlGaN with large mole fractions of AlN would have large breakdown properties, this material has been steadily gaining interest. It should also be mentioned that the N overpressure on Al is the smallest among those over Ga and In, paving the way for equilibrium growth of AlN bulk crystals, albeit not without O contamination.

2.8 AlN Table 2.12 Parameters associated with the band structure for zinc blende GaN.

Parameter alc (Å) at T ¼ 300 K E Gg (eV) a (G) (meV K
1) b (G) (K) E Xg ðeVÞ a (X) (K) b (X) (meV K
1) E Lg ðeVÞ a (L) (meV K
1) b (L) (K)

Value 4.50 3.3 0.593 600 4.52 0.593 600 5.59 0.593 600

Parameter

Value

Parameter

Value

Dso (eV) me ðGÞ ml ðXÞ mt ðXÞ c1 c2 c3 mso

0.017 0.15 0.5 0.3 2.70 0.76 1.11 0.29

EP (eV) F VBO (eV) ac (eV) av (eV) b (eV) d (eV) c11 (GPa) c12 (GPa) c44 (GPa)

25.0
0.95
2.64
6.71
0.69
2.0
3.7 293 159 155

Bandgaps are for low temperature [152].

2.8.1 Wurtzite AlN

Wurtzite AlN is a direct bandgap semiconductor with a bandgap near 6.1 eV and still considered to be semiconductor. The zinc blende polytype is not stable with a predicted indirect bandgap, as will be discussed in the next section. The AlN derives its technological importance from providing the large bandgap binary component of the AlGaN alloy, which is commonly employed both in optoelectronic and electronic devices based on the GaN semiconductor system. Early on, the absorption measurements carried out by Yim et al. [182] and later by Perry and Rutz [183] indicated a large energy gap for wurtzite AlN of 6.28 eVat 5 K to 6.2 eVat room temperature. The actual figures are converging at values about 0.1 eV below those contained in those early reports. As for the dependence of the bandgap on temperature, the Varshni parameters of a ¼ 1.799 meV K
1 and b ¼ 1462 K were reported by Guo and Yoshida [184], who also found the low-temperature gap to be 6.13 eV, which is similar to that reported by Vispute et al. [185], a value closer to the values observed of late. Tang et al. [186] resolved what they believed to be the free or shallow impurity bound exciton in their cathodoluminescence (CL) data, at an energy of 6.11 eV at 300 K. Brunner et al. reported a variation from 6.19 eV at 7 K to 6.13 eV at 300 K [187]. A group of same authors Wethkamp et al. [188] used spectroscopic ellipsometry and determined that the energy gap varies from 6.20 eV at 120 K to 6.13 eV at 300 K. Kuokstis et al. [189] resolved a low-temperature free-exciton transition at 6.07 eV. Guo et al. [190] reported the temperature dependence of the reflectance spectra, while fitting it to the Bose–Einstein expression. Using the low-temperature data in conjunction with the Varshni parameters of Guo and Yoshida [184], leads to an intermediate value of 6.23 eV for the lowtemperature bandgap. The Varshni parameters reported by Brunner et al. [187] indicate no significant divergence from GaN for the entire AlGaN alloy composition range, which may bring the accuracy of the data into question. As in any semiconductor, the quality and strain nature of the films can alter the results. The availability

j177

j 2 Electronic Band Structure and Polarization Effects

178

of high-quality homoepitaxial AlN with presumably no strain has shed the much needed light onto the issues surrounding the actual bandgap of AlN [191]. However, even then O contamination could cause the near-band emission peak observed for shift, as it has done so for high-quality bulk substrates reported by Slack et al. [192]. In the experiments of Silviera et al. [191], the epitaxial layer was 0.5 mm, plenty considering the small penetration depth of 10 keV electrons used for the CL experiments, and the substrate was 287 mm thick with O concentrations of about 5 · 1019 cm
3 as measured by Neutron activation. The homoepitaxial AlN films have been grown by organometallic vapor-phase epitaxy on the single-crystal AlN substrates and efforts were undertaken to reduce the oxygen content of the film. The lowtemperature near band edge CL spectrum of the AlN film is shown in Figure 2.11. The open squares correspond to the experimental data obtained at 5 K, while the full line representing the best fit to the experimental data using Corinthian line shapes with transitions are indicated with dashed lines. The assignments shown are a result of thermal quenching behavior. The full widths at half maxima (FWHM) of the narrowest emission line at 6.023 eV is about 1.0 meV and was reported to be perhaps limited by the slit size used during the experiment. A measurement of the 253.65 nm emission line of a low-pressure Hg lamp using the same slits size set resulted in a FWHM of about 0.7 meV, as tabulated in Table 2.13. Figure 2.12 shows the temperature-dependent CL spectra for the AlN film. A rapid decrease in the intensities of the four peaks initially observed between 5.98 and 6.01 eV with the increase in temperature is evident, which is consistent with recombination processes involving excitons bound to shallow neutral centers. The peaks at 6.023 and 6.036 eV remain intense with increasing temperature, which lets them gain the free exciton A (FXA) and free exciton B (FXB) assignments, respectively,

Figure 2.11 High-resolution CL spectrum of an AlN homoepitaxial film. The full line represents our best fit using Lorentzian line shapes, and the dashed lines are the transitions composing the full line [191].

2.8 AlN Table 2.13 Energy positions, full widths at half maxima (FWHM),

and preliminary assignments associated with the transitions shown in Figure 2.11 [191]. Energy (eV)

FWHM (meV)

Assignment

5.98 6.000 6.008 6.01 6.023 6.036

49.4 11.0 1.5 44.0 1.0 8.0

A07 X A D027 X A D017 X A D017 X B FXA FXB

due to their large binding energies. The line around 6.07 eV, shown as FX2A , is some 2 orders of magnitude weaker than the most intense bands observed in the spectrum. On the basis of the similarity in the luminescence spectra of both GaN and AlN, the peak at 6.07 eV is attributed to the first excited state of the FXA. This assignment allows an estimation of the FXA binding energy as 63 meV, which is about twice the value for GaN. This leads to an estimated low-temperature bandgap of AlN of 6.086 eV (the sum of the FXA energy and its binding energy). Returning to the band structure of AlN, of considerable significance, the crystal field splitting in wurtzitic AlN is believed to be negative. The ramification of this is that the topmost valence band is the crystal hole band. Calculations have yielded a range of crystal field splittings, namely, Dcr ¼
58 meV by Suzuki et al. [56], Dcr ¼
217 meV by Wei and Zunger [107],Dcr ¼
176 meV by Shimada et al. [193], and Dcr ¼
244 meV by Wagner and Bechstedt [85]. Moreover, splittings of Dcr ¼
104 and
169 meV were obtained from first-principles and semiempirical pseudopotential calculations, respectively, by Pugh et al. [163] and Dcr ¼
215 meV by Kim et al. [154]. Averaging all of the available theoretical crystal field splittings, one obtains a value of Dcr ¼
169 meV. Silveira et al. [194], using optical reflectance data performed on a- and c-plane bulk AlN and a quasi-cubic model developed for the wurzite crystal structure, determined the crystal field splitting to be D ¼
225 meV. Note that the negative sign for the crystal

Figure 2.12 Temperature-dependent CL spectra of 0.5 mm thick AlN film deposited on 287 mm bulk AlN substrate [191].

j179

j 2 Electronic Band Structure and Polarization Effects

180

field splitting has important implications, namely, that the G7 valence band is on the top of G9 valence band, which is opposite of that in GaN. As for the spin–orbit splitting, the literature values range from 11 [163] to 20 meV [124]. Silveira et al. [194] again using the optical reflectance spectra in bulk AlN determined the spin–orbit splitting energy to be d ¼ 36 meV. In view of the experiments in high-quality bulk AlN, the value of 36 meV is recommended even though it is much larger than the calculated value of 19 meV recommended by Wei and Zunger [107]. As in the case of GaN, the region of the energy band near the bottom of the conduction band, as it manifests itself in devices, can be represented by the effective mass. The same of course applies for the top of the valence band. A number of investigators have calculated the AlN electron effective mass [154,124,163,195,196], with the prediction that it displays a greater anisotropy than that for wurtzitic == GaN [56]. The bare mass values of m? e ¼ 0:30m0 and m e ¼ 0:32m0 obtained by averaging the available theoretical masses may represent a good set of default values as this stage. It should again be underscored that further experimental studies are needed to verify the calculations. As for the valence band, a number of theoretical sets of valence band parameters are available [56,58,154]. There is an apparent disagreement in the signs for A5 and A6 among these reports, which may be irrelevant, because only absolute values of those parameters enter the Hamiltonian [154,163]. The A parameters given by Kim et al. [154] are suggested because the crystal field and spin–orbit splittings reported by these authors are closest to the ones suggested here. The hydrostatic deformation potential for wurtzite AlN has been reported to be in the range of
7.1 and
9.5 eV [31,180], which is consistent with the observation that the bandgap pressure coefficients for AlGaN alloys have little dependence on composition, as reported by Shan et al. [197]. The calculated values of a1 ¼
3.4 eV and a2 ¼
11.8 meV reported by Wagner and Bechstedt [85] are assumed to represent the material well. Theoretical values are also available for a few of the valence band deformation potentials such as D3 ¼ 9.6 eV, and D4 ¼
4.8 eV [180]. However, the complete set D1 ¼
17.1 eV, D2 ¼ 7.9 eV, D3 ¼ 8.8 eV, D4 ¼
3.9 eV, D5 ¼
3.4 eV, and D6 ¼
3.4 eV with the last value derived using the quasi-cubic approximation presented by Shimada et al. [193] can be used in the absence of any other reliable data. For mere availability reasons and few other issues, the mechanical properties of AlN have seen a good deal of experimental activity very early on, which later was followed by theory. Tsubouchi et al. [198], McNeil et al. [199], and Deger et al. [200] measured the elastic constants of wurtzitic AlN. A good many theoretical papers have also been reported [154,193,201–204]. The values suggested by Wright [179], who also provided a detailed discussion of their expected accuracy, namely, C11 ¼ 396 GPa, C12 ¼ 137 GPa, C13 ¼ 108 GPa, C33 ¼ 373 GPa, and C44 116 GPa, are recommended. Several piezoelectric coefficients [205,206] for early AlN at least in part can be found in Ref. [207]. The result for d33 ¼ 5.6 pm V
1 reported in Ref. [207] is in reasonably good agreement with the previous determinations but differs somewhat from d33 ¼ 5.1 pm V
1 measured by Lueng et al. [208]. While these experiments focused on only d33, both d33 and d13 can be determined from first-principles

2.8 AlN Table 2.14 Recommended band structure parameters for wurtzitic AlN [152].

Parameter

Value

Parameter

Value

Parameter

Value

Eg (eV, low temperature) a (meV K
1) b (K) Dcr (meV) Dso (meV) == me =m0 m? e =m0 a1 (eV) a2 (eV)

6.077 [194] 1.799, 0.9a 1462, 1000a
225 [194] 36 [194] 0.32 0.30
3.4
11.8

A1 A2 A3 A4 A5 A6 A7 (meV Å) d13 (pm V
1) d33 (pm V
1) d15 (pm V
1) Psp (C m
2)


3.86
0.25 3.58
1.32
1.47
1.64 0 (default)
2.1 5.4 3.6
0.090

D1 (eV) D2 (eV) D3 (eV) D4 (eV) D5 (eV) D6 (eV) C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa)


17.1 7.9 8.8
3.9
3.4
3.4 396 137 108 373 116

See Tables 2.27 and 2.28 for details related to the elastic constants, piezoelectric constants, and spontaneous polarization charge. Any dispersion among the tables is a reflection of the uncertainty in the available parameters. Note that the G7 valence band is above the G9 valence band, which is opposite of GaN. It is also similar to that in ZnO, which is somewhat controversial. See Zinc Oxide: Materials Preparation, Properties, and Devices, by H. Morkoç and Ü. Özg€ ur, Wiley (2008) regarding the valence band ordering in ZnO. a Obtained using the temperature dependence of the A exciton energy reported in Ref. [194].

calculation [193,209–212]. The recent theoretical values of Bernardini and Fiorentini, d33 ¼ 5.4 pm V
1 and d13 ¼
2.1 pm V
1 [210], are suggested although the elastic coefficients given in that reference are somewhat larger than the recommended values. On the basis of recent measurements [206,207] and a calculation [210] of the shear piezoelectric coefficient, Vurgaftman and Meyer [152] recommend d15 ¼ 3.6 pm V
1. The parameters concerning the bandgap-related issues for wurtzitic AlN recommended by Vurgaftman and Meyer [152] are tabulated in Table 2.14. A compilation of the dispersion in the effective mass for both the conduction band and various valence bands as obtained by various computational methods as well as parameters used in the description of the bandgap for wurtzitic InN, particularly, in the context of empirical pseudopotential method, as described in Ref. [55], are tabulated in Table 2.15. Even though a detailed discussion of polarization is reserved (Section 2.12), a succinct treatment of the topic is given here as it is relevant to the topic under discussion. The difference between the GaN and AlN spontaneous polarizations strongly causes a net polarization at the interface between the two materials that extends to the GaN/AlGaN interfaces as well. This charge, which is bound, influences the band profiles and energy levels in GaN/AlN quantum heterostructures. Although rigorous calculations [84,181,209,213] of the spontaneous polarization Psp(AlN) have produced results spanning a fairly broad range, from
0.09 to
0.12 C m
2, values for the difference Psp(AlN)
Psp(GaN) have tended to be more consistent, with most falling between 0.046 and 0.056 C m
2. Experimentally, for some time the majority of workers on the GaN/AlGaN system reported somewhat smaller Psp(AlN)
Psp(GaN). For example, Leroux et al. [214,215] derived 0.051 < Psp <
0.036 C m
2 for AlN. A study of the charging of GaN/AlGaN field effect transistors led to a similar

j181

j 2 Electronic Band Structure and Polarization Effects

182

Table 2.15 Effective masses and band parameters for wurtzitic AlN.

Parameter ==

aniso

iso

A

B

C

D

E

F

me m? e

0.231 0.242

0.232 0.242

0.33 0.25

0.33 0.25

0.24 0.25

0.24 0.25

0.35

0.33

mhh

2.370

2.382

3.68

3.53

1.949

1.869

3.53

4.41

2.370

2.382

3.68

3.53

1.949

1.869

3.53

4.41

0.209 3.058 0.285 1.204
4.789
0.550 4.368
1.511 1.734
1.816 0.134
0.128

0.209 3.040 0.287 1.157
4.794
0.571 4.374
1.484 1.726
1.788 0.153
0.160

0.25 10.42 0.24 3.81
4.06
0.26 3.78
1.86 2.02

0.229 2.584 0.350 0.709
4.367
0.518 3.854
1.549 1.680
2.103 0.204
0.093

0.212 2.421 0.252 1.484
4.711
0.476 4.176
1.816 1.879
2.355 0.096
0.093

0.26 11.14 0.33 4.05
3.86
0.25 3.58
1.32 1.47 1.64

0.27 2.18 0.29 4.41
3.74
0.23 3.51
1.76 1.52 1.83 0

==

== mlh == mch m? hh m? lh m? ch

A1 A2 A3 A4 A5 A6 A7 D1

0.25 6.33 0.25 3.68
3.95
0.27 3.68
1.84 1.95
2.91 0
0.059

0
0.059


0.215

Effective masses in units of free-electron mass m0, Luttinger-like parameters Ai (i ¼ 1, . . ., 6) in units of
h2 =2m0 , and A7 in units of eV Å. The crystal field splitting energy D1 is given in units of meV. The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor [55]. Anisotropically screened and isotropically screened values are from Ref. [55]. A: FP-LAPW band structure calculations are from Ref. [170], and effective mass parameters are obtained through a 3D fitting procedure within cubic approximation; B: FP-LAPW band structure calculations are from Ref. [170], and effective mass parameters are obtained by a direct line fit; C: Ai from Ref. [153] obtained through a Monte Carlo fitting procedure to the band structure and effective masses calculated using Equations 2.30 and 2.31; D: direct kp calculations for Ai from Ref. [153] and effective masses obtained from Ai using Equations 2.30 and 2.31; E: direct fit of Ai to first-principles band structures from Ref. [154]; F: Ai and effective masses obtained in the quasi-cubic model from zinc blende parameters from Ref. [154].

conclusion [216], and Hogg et al. [217] were able to fit their luminescence data by assuming negligible spontaneous polarization. Park and Chuang [218] required Psp ¼
0.040 C m
2 to reproduce their GaN/AlGaN quantum well data. On the contrary, Cingolani et al. [219] reported good agreement with experiment using a higher value derived from the original Bernardini et al. [209] calculations. A significant step toward resolving this discrepancy has been the realization that the AlGaN spontaneous polarization cannot be linearly interpolated between the values at the binary end points [220–222]. In combination with an improved nonlinear strain treatment of the piezoelectric effect, the discrepancy between theory and experiment for GaN/AlGaN quantum wells has been largely eliminated [85]. For additional details, see Section 3.14 and the text dealing with Tables 2.25, 2.27 and 2.28. We adopt Psp ¼
0.090 C m
2 as the recommended value for AlN, in conjunction with Psp(GaN) ¼
0.034 C m
2. The recommended band structure parameters for wurtzite AlN are compiled in Table 2.14.

2.8 AlN

2.8.2 Zinc Blende AlN

Only a handful of purportedly successful growths of zinc blende AlN on zinc blend substrates, such as GaAs and 3-C SiC, and Si substrates following low-temperature zinc blende GaN buffer layers have been reported [223–226]. Consequently, much of the discussion here relies primarily on theoretical projections and so do the parameter set recommended for this polytype. The only quantitative experimental study of the bandgap indicated a G-valley indirect gap of 5.34 eV at room temperature [224]. Assuming that the Varshni parameters for the wurtzitic AlN hold for the zinc blende polytype, the aforementioned room-temperature bandgap translates to a low-temperature gap of 5.4 eV. Vurgaftman and Meyer [152] recommend 4.9 and 9.3 eV for the X- and L-valley gaps, respectively [31,69,163]. The spin–orbit splitting is expected to be nearly the same as in wurtzite AlN at 19 meV [154,107,171,227]. Averaging the theoretical results from a number of different publications [69,154,163,170,173], one arrives at a G-valley effective mass of 0.25m0. The longitudinal and transverse masses for the X valley have been predicted to be 0.53m0 and 0.31m0, respectively [69]. If the method used previously for the GaN is applied to zinc blende AlN, one arrives at recommended Luttinger parameters of g1 ¼ 1.92, g2 ¼ 0.47, and g3 ¼ 0.85, and mso 0.47m0 [69,154,170,173]. These as well as the other literature values of the Luttinger parameters are listed in Table 2.16. Fritsch et al. [55] calculated the effective masses for conduction and valence bands, the latter involving the light and heavy holes, as well as the spin–orbit split-off mass, which with the anisotropy taken into account are listed in Table 2.17 for zinc blende AlN. The data contain those obtained by FP-LAPW, EPM calculations and those calculated with Luttinger parameters (Equations 2.42 and 2.43). When the calculated [163,173] values for the momentum matrix are averaged, a value of EP ¼ 27.1 eV (with F ¼
101) is obtained. Hydrostatic deformation potentials of
9.0 eV [31] and
9.8 eV [69] have been reported. The deformation potential values, a ¼
9.4 eV, av ¼
4.9 eV [69,107], b ¼
1.7 eV [69,177,178], and d ¼
5.5 eV [180,177,178], have been suggested [152]. The elastic constants of C11 ¼ 304 GPa, C12 ¼ 160 GPa, and C44 ¼ 193 GPa calculated by Wright [179] are Table 2.16 Luttinger parameters g1, g2, and g3 for zinc blende AlN obtained from a fit along the [1 1 0] direction along with those available in the literature, as compiled in Ref. [55].

Parameter

A

B

C

D

E

c1 c2 c3

1.85 0.43 0.74

1.54 0.42 0.64

1.50 0.39 0.62

1.91 0.48 0.74

1.92 0.47 0.85

A: empirical pseudopotential calculation by Ref. [55]; B: self-consistent FP-LAPW method within local density approximation from Ref. [171]; C: first principles band structure calculations from Ref. [170]; D: empirical pseudopotential calculation from Ref. [174]; E: recommended values taken from Ref. [152].

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Table 2.17 Effective masses for electrons (e), heavy holes (hh), light holes (lh), and spin–orbit split-off (so) holes in units of the free-electron mass m0 along the [1 0 0], [1 1 1], and [1 1 0] directions for zinc blende AlN. [100]

[100]

[111]

[111]

[110]

[110]

Reference

me

mhh

mlh

mhh

mlh

mhh

mlh

mso

A B C D E

0.23 0.28 0.21 0.30 0.25

1.02 1.44 1.05 1.39 1.02

0.37 0.42 0.35 0.44 0.35

2.64 4.24 2.73 3.85 4.55

0.30 0.36 0.30 0.36 0.28

1.89 3.03 2.16 2.67 2.44

0.32 0.37 0.31 0.38 0.29

0.54 0.63 0.51 0.67 0.47

Compiled by Fritsch et al. [55]. A: after Ref. [55]; B: self-consistent FP-LAPW method within local density approximation from Ref. [171]; C: empirical pseudopotential calculation from Ref. [69]; D: calculated from Luttinger parameters from Ref. [170], using Equations 2.42 and 2.43; E: calculated from recommended Luttinger parameters from Ref. [152], using Equations 2.42 and 2.43.

similar to the sets quoted in other theoretical works [180,181,228] and are therefore suggested. These and other band structure parameters recommended for zinc blende AlN are tabulated in Table 2.18 [152].

2.9 InN

As in the case of AlN, the interest in InN has so far been not necessary because of its properties, but because of the InGaN alloy that is used in lasers and LEDs operative in the visible and violet regions of the optical spectrum. In fact, if and when the technological issues are overcome, the InGaN channel FETs may also be superior to the GaN channel varieties [229], some details of which are discussed in Volume 3,

Table 2.18 Parameters associated with the band structure for zinc blende AlN.

Parameter

Value

Parameter

Value

Parameter

Value

alc (Å) at T ¼ 300 K E Gg ðeVÞ a (G) (meV K
1) b (G) (K) E Xg ðeVÞ a (X) (K) b (X) (meV K
1) E Lg ðeVÞ a (L) (meV K
1) b (L) (K)

4.38 5.4 0.593 600 4.9 0.593 600 9.3 0.593 600

Dso (eV) me ðGÞ ml ðXÞ mt ðXÞ c1 c2 c3 mso

0.019 0.25 0.53 0.31 1.92 0.47 0.85 0.47

EP (eV) F VBO (eV) ac (eV) av (eV) b (eV) d (eV) c11 (GPa) c12 (GPa) c44 (GPa)

27.1
1.01
3.44
4.5
4.9
1.7
5.5 304 160 193

Bandgaps are for low temperature [152].

2.9 InN

Chapter 3. The properties, particularly the fundamental parameters of InGaN for a given composition, depend very much on the InN parameters, particularly its bandgap. It is therefore important to understand the properties of bulk InN in its wurtzitic form. InN, however, is not all that easy, even considering the general difficulties encountered in the nitride semiconductor system, to synthesize. The somewhat intractable problem with InN is the enormous difference in the ionic size of its constituent atoms in that the atomic radii for In and N are largely different, which leads to highly distorted interatomic distances, interatomic bonding charges, tendency to form metallic clusters of the group III constituent, and inhomogeneous strain. All of these could, in principle, lead to pronounced anomalies in all the properties of InN, inclusive of measured bandgap and nature of defects. To make matters worse, the InN layers are grown at best on GaN epitaxial layer with large lattice mismatch (lm), aggravating many of the aforementioned problems. In spite of all these, progress is steadily made. It should be added that after having been accepted as the bandgap of InN, the 1.98 eV figure came under new scrutiny in that a plethora of reports concluded the actual bandgap to be 0.7–0.8 eV. Just at a time when a good many got convinced of the newer data, questions have been raised about the models along with heightened level of scrutiny of the new low-bandgap data. Assuming that interpretation of experiments pointing to the small bandgap figures are impeccable, theories also begin to be developed, even though long-standing understanding such as cation rule would be broken by the small bandgap figure. 2.9.1 Wurtzitic InN

The bandgap of InN has been a point of controversy dating back to the early days of InGaN development [1,230,231]. Early absorption studies on sputtered thin films concluded that the InN bandgap is in the 1.7–2.2 eV range [232–236]. However, no band-to-band PL could be observed in the samples prepared by sputtering in early developmental stages or later on in films grown by OMVPE and MBE. A review [237] of various crystal growth related issues and resultant properties as well as a proceedings [238] of a meeting devoted to debating these issues is available in the literature. In contrast to earlier reports of no near band edge emission, Davydov et al. [239–242] and others [243–245] reported near band edge emission but at much lower energies near 0.6–0.8 eV, depending on the report. Also see the comment, Ref. [246], on Ref. [239] and reply to that comment [247]. These reports relied to various extents on absorption, photoluminescence, and photoluminescence excitation experiments that showed the experimental evidence for the bandgap of InN to be overwhelmingly in the range of 0.7–0.8 eV and recommended a zero-temperature gap of 0.78 eV [248] and 692 2 meV [249], and in fact values as low as 0.65 eV [239] and 0.67 eV [250] have been reported as well. InN films of N-polarity grown by RF molecular beam epitaxy exhibited a bandgap value of about 0.7–0.75 eV as measured by optical transmission and reflection measurements [251–256]. The PL emission

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line appeared at 0.7 eV with no shift in energy between 300 and 77 K. These particular samples also exhibited room-temperature electron mobilities in the range of m ¼ 1750–2000 cm2 V
1 s
1 and an electron concentration in the range of n ¼ 2–3 · 1018 cm
3 at room temperature. Some details about the growth of these films are provided in Section 3.5.13. The polycrystalline or nanocrystalline nature of those early thin films associated with high electron densities and low mobilities led the proponents of the small bandgap for InN to suggest that those early films most likely contained a good deal of oxygen coupled with Moss–Burstein effect that could push the apparent bandgap upward. Another possible explanation for the dispersion in the reported values of the bandgap and also in support of the smaller bandgap may have to do with blue shift caused by any quantum size effects. At least the correct trend has been established by Lan et al. [257] who reported PL emission at 1.9 eV in nanorods of diameter between 30 and 50 nm (dubbed the brown InN), whereas 0.766 eV emission (measured at 20 K) was observed in rods with a diameter in the range of 50–100 nm (dubbed the black InN). Further refinement of the work led to the observation that the samples with fine (10 nm) or containing very high carrier concentrations exhibit the visible emission. In fact, both IR and visible peaks could be observed in the same sample when the samples show bimodal distribution of grain size [258]. The InN nanorods catalytically formed in the upstream portion of the substrate were of the brown type while those downstream were of the black variety. The Au catalyst that floats on the top of the InN nanorods as the growth progresses were shown to be encapsulated with In2O3, which is somewhat unexpected and may be made possible owing to close epitaxial relationship between wurtzitic InN and In2O3, indicating the participation of O in the catalytic process. The source of O, in this case, was attributed to residual O in the reactor as well as the quartz tube used. Furthermore, the black InN photon emission in a PL experiment quenched above 150 K while that from the brown InN exhibited strong emission at room temperature, albeit broad. The variation in the observed emission wavelength has been attributed to a variety of sources including O incorporation, Moss–Burstein shift due to high electron concentration, and quantum size effect. The quantum size effect would require diameters of less than 5 nm that is much larger than the 20–50 nm brown InN rods, which cannot therefore explain the bandgap shift to 1.9 eV from about 0.7 eV. It should be stated that the characteristic PL peak in O-implanted InN is different from the broad visible peak. The possibility of InN quantum dot formation in InGaN as being responsible for bandgap variations has not been of as much use in that even intentional dot formation, albeit limited to possibly only one report, did not lead to any blue shift [259]. The results are not precise but speak to the trend and demonstrate that great care must be taken to make sure that all the samples are grown under identical conditions, and doping levels are the same and not very large. Another issue of paramount importance that comes into the equation is the degree of crystallinity of the InN films. For example, Anderson et al. [260] produced polycrystalline InN with 0.8 eV luminescence present but did not identify the size of polycrystalline grains.

2.9 InN

Proponents of the earlier and larger bandgap for InN bring on the table several arguments undermining the validity of the low-bandgap figure. Among theme is that the absorption squared versus energy plots used to obtain the apparent bandgap in a semiconductor with very high carrier concentration underestimates the bandgap due to band tailing [261]. In addition, the 0.7–0.8 eV peak ascribed by some to the bandgap of InN is attributed to defects caused by nonstoichiometry of the films, which are grown under extremely In reach conditions and far away from equilibrium condition [261]. Moreover, The Moss–Burstein blue shift used by proponents of the small bandgap to account for the large bandgap reported earlier is not consistent among all the samples in that the sample with low carrier concentrations in the past confirmed the large bandgap of InN, albeit they were most likely heavily compensated. Finally, the opponents of the larger bandgap argue that the observed transition at 0.7–0.8 eV is due to Mie resonance that in turn is due to scattering or absorption of light in InNcontaining clusters of metallic In, which may have been mistaken for the low bandgap, as only the In cluster containing samples do show the 0.7–0.8 eV peak [262]. Let us now discuss the evolution of InN, and in particular, its perceived and admittedly controversial perceived bandgap. The early attempts to produce InN relied on not well-developed methods, as compared to modern crystal growth techniques, and as such produced mostly powder and nanopowders [263–265]. Naturally, the early work on thermodynamic decomposition studies and X-ray diffraction (advantageous for power diffractometry due to the nature of the films) were performed on powder films. Even in these very powder samples, there was some observed variation in the color of the material in that some have been noted to be black or blackbrown [264–266] as opposed to the deep red that one would expect for the 1.8 eV bandgap originally estimated [265]. For a witty historical account of these early efforts through the controversy, the reader is referred to Refs [267,268]. Below a discussion of the role of oxygen, possible defects that might be responsible for the 0.7 eV emission, the Moss–Burstein shift, particulars of carrier absorption, and Mie resonance is given in an effort to give an overall appreciation of issues causing the controversy in determining the bandgap parameter of InN. Focusing on the issue of O, because In2O3 has a bandgap of 3.75 eV [269], the argument goes that if InN films are heavily O contaminated, the bandgap would be pushed upward. This is one of the points used by the proponents of the small bandgap. However, as shown in Figure 1.34, to bring the bandgap from 0.7 to 1.9 eV would require some 30% O in the film assuming that the Vegard’s law holds. It should also be noted that the stated O must form an alloy for increasing the bandgap not just as surface contamination or inclusion of O at grain boundaries. One of the samples used to support the InN–In2O3 alloy formation was an RF-sputtered film with poor mobility and high carrier concentration [270]. Auger analysis used to determine the O content in the aforementioned layer is sensitive to only O and cannot determine whether it is alloyed in the semiconductor InN. In addition, any depth profiling accompanied by that technique requires argon ion etching, which is known to result in severe nitrogen loss and lead to overestimation of the atomic oxygen concentration owing to the recycling of sputtered oxygen on the film surface coupled with the strong bond between oxygen and surface indium [271]. An alternative method, Rutherford backscattering (RBS), was

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188

also used by Davydov et al. [270] with the sample mounted on glass, which in consideration of the sample thickness may not have allowed a definitive determination whether the O signal is due to O in InN or the glass substrate used. Nevertheless, the O content so measured was 20%, which is to be contrasted to 10% obtained [272] from measurements performed in similar layers using elastic recoil, which is accurate for elemental analysis. It should again be mentioned that elemental O in the film, even if present in stated quantities, is not the same as InN:In2O3 alloy, which would increase the bandgap, not to mention the fact that such alloying is not favored by temperatures employed in RF sputter deposited InN. One can surmise that amorphous InON, NO2 and surface hydroxide species are implicated as being responsible for the 1.8–2 eV bandgap reported. These species are only as surface species, and there has not as yet been any evidence reported of any other form of InN–In2O3 alloy species during the growth of InN, as can be discerned from Figure 1.34 that 10% or even 20% oxygen could not account for the bandgap of 1.9 eV if a bandgap of 0.7 eV is assumed for InN. Figure 1.34 clearly indicates that an alloy with about 37 at.% oxygen, which translates to 44% In2O3, would be needed to provide a bandgap of 2.0 eV if the bandgap of pure InN is 0.7 eV. Consequently, 10% (at.) oxygen, even if all were alloyed with InN, would account for a blue shift of about only 0.3 eV, much less than some 1 eV for consistency. If the bandgap of the alloy follows a bowing parameter, unless it is positive, which is unlikely, the blue shift caused by alloyed O would even be smaller. This view is supported by the results of Yoshimoto et al. [273] that reported a bandgap value of 1.8–2.0 eV for MBE InN grown on quartz with 3% atomic oxygen present in the film. Evidently, this level of O contamination does not prevent one from arriving at the longstanding bandgap of InN, as any blue shift caused by this level of elemental O even if all is in the alloy form does amount to much. The role of O in InN had been investigated early on in sputtered films. Among them is the work of Westra et al. [274] who produced InN with carrier concentrations between 7 · 1019 and 2 · l020 cm
3 and mobility of 4–10 cm2 V
1 s
1. Rutherford backscattering data indicated 11% atomic oxygen, and indium-to-nitrogen ratios slightly above a value of 1. No evidence of oxygen or oxynitride phases was observed in the X-ray diffraction spectra, which led those authors to propose that oxygen is in the form of an amorphous indium oxynitride, similar to the observations of Foley and Lyngdal [275]. In this case the structure would maintain the stoichiometry, while the oxygen would not be detected by X-ray diffraction. Contribution from NO2 has been observed for samples containing higher oxygen content in a polycrystalline InN matrix with no evidence for O in X-ray diffraction or infrared absorption spectra [271], wherein grain boundaries were proposed to be the host for O. Raman data for sputtered InN with 10% atomic oxygen concentration reveals only InN-related phonon peaks, implying the lack of alloy formation [272,276]. The moral of the aforementioned discussion is that any O present in at least polycrystalline InN that has been examined with the role of O in mind did not seem to be incorporated as an alloy, which explicitly leads to the conclusion that could not contribute to the bandgap. An in-depth discussion of the topic can be found in Ref. [267]. Moss–Burstein effect has been forwarded as a plausible argument for observing 1.8–2 eV absorption if the 0.7 eV bandgap is assumed. This shift occurs when carrier

2.9 InN

concentration is above the Mott critical density, meaning larger than the conduction band density, that is, the Fermi level lies in the conduction band. In such a case, electrons fill the bottom of the conduction band so that the apparent bandgap measured by optical absorption is increased by an amount of extension of the Fermi level. As a result, extra photon energy is required to excite electrons from the top of the valence band to the Fermi-level position within the conduction band suggested by Trainor and Rose [277]. But the required large electron concentrations produces bandgap renormalization, which is in competition with the Moss–Burstein effect. High doping concentrations cause band tailing effect, which acts to reduce the apparent bandgap. Layers of InN with large variation in electron concentration have been reported along with the measured bandgap, as compiled in Figure 1.35, as function of electron concentration. The measured bandgap is the convolution of Moss–Burstein shift and bandgap renormalization culminating in the measurement of EG þ EF, the EF being measured with respect to the conduction band edge. If the plot is limited to MBE films only, the data appear to support the argument for small bandgap and variation attributed to Moss–Burstein shift. However, when all the data in films prepared by any growth method in aggregate are plotted, there seems to be quite a scatter, including large bandgap associated with relatively lower impurity levels. The data basically do not show any trend in that a good many of the samples with high and low electron concentration both do exhibit the large bandgap value. The critical data having to do with the samples of relatively lower electron concentrations are associated with compensated samples. This is an important issue in that material compositional, or stain-related nonuniformities may also hamper efforts to determine the bandgap accurately. A critical view may raise question about the quality of the samples. What is clear is that the bandgap of the nondegenerate InN is still unknown and will require considerably more investigation [272]. What about some yet unknown defect being responsible for or the source of the 0.7 eV emission and absorption? Ask Butcher [267] who considered this possibility and provided arguments backed by experimental data in support of it. Butcher [267] suggested that the evidence for a 0.7 eV bandgap is also consistent with the presence of a 0.7 eV deep-level trap in that the available absorption data for the best published apparently 0.7 eV material (without the Moss–Burstein shift due to low carrier concentrations) exhibit an energy dependence consistent with a deep-level trap of |si-like orbital symmetry and was inconsistent with direct band-to-band transitions. Butcher [267] also made the case that the slope dependence of absorption coefficient plots is consistent with variations in the density of such a deep-level trap. Further, the 0.7 eV material showed that emission to be emanating from regions of indium-rich aggregates [270]. If one wants to be skeptical, one could argue that it is likely that the said emission could be a result of surface states at the metal–semiconductor interface rather than being associated with the InN band edge. It was also shown that material grown without such aggregates had an absorption edge nearer 1.4 eV [278]. The early InN data were produced by N-rich material, whereas the 0.7 eV material is grown under In-rich conditions and the same probably holds for OMVPE-grown layers. The early samples grown under N-rich conditions exhibited a higher than usual unit cell

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volume [279]. Such variation in the unit cell volume with stoichiometry would be consistent with that observed for GaN [280]. Being forced to grow InN at very low temperatures, typically below 550  C, with the caveat that the temperature measurements are not absolute, the driving force for desorption of excess In is not all that potent. The potency of In droplet formation has been noted by Yamaguchi et al. [281] who indicated that as little as a 20  C increase in temperature from optimum growth conditions can result in indium droplets. Clearly the stoichiometry, specifically In inclusion issue, is something that would need to be dealt with. With the evidence for indium-rich aggregates in the InN matrix and suggestion that they are responsible for the 0.7 eV luminescence properties of InN, the data of Wu et al. [282] has been discussed under a different light [267]. The samples investigated by Wu et al. had undergone an irradiation with 2 MeV protons to a dose of 2.23 · 1014 protons cm
2 that resulted in a factor of 2 increase in the intensity of 0.7 eV PL peak, which was interpreted as radiation hardness. This increase in PL peak intensity can be ascribed to increased density of radiative defects, as has been well documented for III–V materials by Lang [283]. This is typically accompanied by band-edge photoluminescence that is either quenched or left unperturbed by radiation damage. Another interpretation of the observations of Wu et al. [282] might therefore be an increase in radiation-induced defects although not being promoted [267]. An issue is the fact that InN is particularly susceptible to nitrogen loss when bombarded with ions [284–287], leading to possible indium-rich aggregates. The aforementioned discussion lays the groundwork for a plausible connection between the 0.7 eV peak and In-rich aggregates, but does not quite attempt to discuss the nature of that emission. Shubina et al. [288] and also Ivanov et al. [278] attempted to do just the same, meaning shed some light into the nature of the 0.7 eV emission. The efforts of those authors utilizing microcathodoluminescence studies coupled with imaging of metallic In have shown that bright infrared emission at 0.7–0.8 eV arises in the close vicinity of In inclusions and is likely associated with surface states at the metal–InN interfaces. Employing thermally detected optical absorption (TDOA) measurements, Shubina et al. [288] suggested a bandgap near 1.5 eV, reserving a more definitive judgment until after more accurate measurements could be performed. Shubina et al. [288] have actually broadened the range of samples, examining various substrates including sapphire by choosing two representative sets of InN epilayers grown by both plasma-assisted molecular beam epitaxy and organometallic VPE methods. The dominant IR emission in these samples were observed to be in the range of 0.7–0.8 eV, independent of the growth technique used to prepare them and of excitation, such as optical with different laser lines, or by an electron beam in conjunction with a CL performed at 5 K in an analytical scanning electron microscope. No correlation between the IR emission and the electron concentration, which ranged from 2.1 up to 8 : 4 · 1019 cm
3 (determined from IR ellipsometry measurements using an effective electron mass of me ¼ 0 : 11m0) [234] and measurement temperature was discernable, consistent with other reports [289]. Not all the 18 samples studied emitted light, and all the samples emitting IR PL emission did contain In-rich aggregates. Analytical microscopy sensitive to atomic weight in

2.9 InN

backscattered electrons (BSE) geometry, and energy dispersive X-ray (EDX) analysis along with CL were employed to establish a definite correlation between the In-rich aggregates and IR PL emission. Again, the bright 0.7–0.8 eV IR emission in both MBE and OMVPE sets of the samples was found to be associated with the In aggregates. Total optical extinction losses in a semiconductor matrix with metallic clusters have been established. In addition to the interband absorption in the matrix, those losses contain two additional components, namely, a bipolar absorption of radiation energy and its conversion into heat in small particles, and resonant scattering on plasmon excitations, which is important for larger particles [290]. Both characteristic components have been observed in the optical spectra obtained by Shubina et al. [288]. Those authors also observed emission in the range of 0.8–1.4 eV that was attributed to the scattered background signal, the root cause of which was most likely associated with nonchromatic spontaneous emission of the semiconductor laser used or the fluorescence of all optical components at high excitation power levels. Owing to electron beam excitation, the scattered signal was absent in the CL spectrum. Unless the aforementioned spurious signals are accounted for, erroneous conclusions could be drawn. Citing the absence of such signals, Shubina et al. [288] employed thermally detected optical absorption technique performed at 0.35 K. The method is based on the detection of a small increase in the sample temperature caused by phonons produced by nonradiative recombination processes as a result of optical absorption and bipolar absorption of light in In-rich aggregates. The TDOA spectra contained additional peaks below the principle absorption edge in films containing In-rich aggregates but not others, owing to absorption within those aggregates. The sharpness of the observed feature is related to the resonance at the extremely low measurement temperature that prevents thermal broadening induced by electron acoustic phonon scattering [288]. A natural conclusion of the aforementioned observations is that the strong IR absorbance is most likely associated with Mie [291] resonances due to scattering or absorption of light in InN-containing In aggregates or metallic In clusters. In the Mie theory [291], the extinction losses for a metallic sphere depend on the complex dielectric functions of both the matrix material, e, which is InN in this case, and metal, em, which is In in this case. Consequently, the resonance energy of the In clusters in InN matrix with a high-frequency dielectric constant of e0 ¼ 8.4 (based on stoichiometric InN, which is the case here as the InN films studied were In-rich) is considerably smaller than that in vacuum. For a treatment of the dielectric constant in InN, the reader is referred to Ref. [292]. Shubina et al. [288] argued that the In inclusions are predominantly formed either between columns or initiated at the interface with the substrate. Not knowing the shape and density of those clusters accurately, the authors employed a model developed for nonspherical metal particles in an absorbing matrix [293], which is based on the Maxwell–Garnett approximation for an effective medium [294], to demonstrate that the resonance absorption energy in the InN–In composite can shift down to the IR range depending on the In content and the shape of the In clusters. A lack of accurate knowledge of the InN complex dielectric constant and the specifics of In clusters prevented the authors to arrive at a

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shape and density of In clusters responsible for the IR absorption. Suffice it to say that given the available parametric data, it is very plausible that the observations are related to Mie resonance absorption caused by In-rich aggregates in the InN matrix. Clearly, additional experiments are imperative and will surely be available in due time. Even with the reported available data and analysis, Shubina et al. [288] argue that it is unlikely that the bandgap of InN is at 0.7–0.8 eV range. Confirming to a large extent the results obtained in InN reported in 1980s, using epitaxially grown wurtzite InN, Guo and Yoshida [184] measured low-temperature and room-temperature gaps of 1.994 and 1.97 eV, respectively, along with Varshni parameters of a ¼ 0.245 meV K
1 and b ¼ 624 K. Estimates of the crystal field splitting in wurtzite InN range from 17 to 301 meV [66,107,163]. A value of 40 meV can be adopted. On the basis of the calculation, spin–orbit splittings vary from 1 to 13 meV [107,163], but a value of Dso ¼ 5 meV is recommended by Vurgaftman and Meyer [152]. Ironically, the small bandgap of InN actually goes against the long held cation rule, in which for isovalent, meaning common-cation semiconductors, the direct gap at the G point increases as the anion atomic number decreases. This implies, for example, that the bandgap of InN should be larger than that of InP, which is 1.4 eV, which is consistent with the values of 1.5 eV and higher reported for InN. It should, however, be stated that the breakdown of the common-cation rule is not unusual in ionic semiconductors. This is articulated in an interesting observation where in Nag [246] pointed out that this gap is unusually low in the context of trends exhibited by other semiconductor materials. Fore example, the bandgap of ZnO is also smaller than that of ZnS. This unexpected behavior has been attributed to two effects. Wei et al. [295] argued that a much lower 2s atomic orbital energy of N (
18.49 eV) compared with P (
14.09 eV) and other group V elements lowers the conduction band minimum at the G point. Moreover, the smaller bandgap deformation potential of InN (
3.7 eV) in comparison to InP (
5.9 eV) weakens the atomic size effect. The atomic size effect is the one that forms the basis of the common-cation rule. Similarly, the effect of the lower orbital energy of O as compared to the other group VI elements such as S is responsible for ZnO bandgap being smaller than that of ZnS. The controversy on the theory side matches that of the experiments. It could be argued that in the end carefully thought of experimental evidence will carry the day and theories with appropriate approximations will be developed to support the general direction of experiments. Tsen et al. [296] studying nonequilibrium optical phonons in a high-quality singlecrystal MBE-grown InN with picosecond Raman spectroscopy reached the conclusion that their results are not consistent with the large bandgap of InN. The basis for this argument is as follows: above gap photons cause creation of electron and hole pairs that very quickly relax toward the bottom/top of conduction/valence band by emitting phonons. For wurtzitic InN, using me ¼ 0:14 me [297], mh ¼ 1:63 me [298],
hwL ¼ 2:34 eV, and Eg ¼ 1.89 eV gives an index of refraction n ¼ 3.0 (measured by Tsen et al. [296]) and
hwLO ¼ 73:4 meV (corresponding to A1(LO) phonon mode energy); the phonon wave vector in the experiments of Tsen et al. [296] is q ¼ 2nkL 7.08 · 107 m
1, where kL is wave vector of the excitation laser with photon energy

2.9 InN

2.34 eV. Owing to a much larger associated hole mass, the phonons emitted by holes can be ignored. In a parabolic band, excess electron energy is given by   mh DE ¼ ð
hwL
E g Þx ffi 0:41 eV; ð2:44Þ me þ mh which is about five times the LO phonon energy
hwLO ¼ 73:4 meV, which means that the energetic electrons are capable of emitting five LO phonons during their thermalization to the bottom of the conduction band. However, because of energy and momentum conservation during the electron–LO phonon interaction, there exists a range of LO phonon wave vectors that electrons can emit. As depicted in Volume 2, Figure 3.7, for an electron with wave vector ~ ke and excess energy DE, the minimum and maximum LO phonon wave vectors it can emit are given by [299] pffiffiffiffiffiffiffiffiffi 2me pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð DE
DE

hwLO Þ; kmin ðor qmin Þ ¼
h and kmax ðor qmax Þ ¼

pffiffiffiffiffiffiffiffiffi 2me pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð DE þ DE

hwLO Þ:
h

ð2:45Þ

Because of the nature of E–k relationship of the electron, the lower the electron energy, the larger kmin (also referred to as qmin) and the smaller kmax (also referred to as qmax) are. Therefore, at some electron energy, kmin of the LO phonon will be larger than the wave vector probed by the Raman scattering experiment, which is q ¼ 7.08 · 107 m
1. When this occurs, the energetic electrons can no longer emit LO phonons with wave vectors detectable in the Raman scattering experiments. These conditions corresponds to kmin ffi 1.1 · 108 m
1 and kmax ffi 2.4 · 109 m
1 in the experiments of Tsen et al. [296]. Although the effect of nonparabolicity of the conduction band for InN is currently not known, the general trend is that the effective electron mass increases with the increase in energy in the conduction band for a typical semiconductor. By taking the nonparabolicity into account, the detectable kmin can be revised as being larger than 1.1 · 108 m
1. Consequently, no nonequilibrium A1(LO) phonon population should be detected in the Raman experiments of Tsen et al. [296] when the effect of nonparabolicity of the conduction band is considered. This means that if the bandgap energy of InN were 1.89 eV, there would be no detectable nonequilibrium A1(LO) phonon population with the excitation laser photon energy of 2.34 eV. This contradicts the fact that nonequilibrium A1(LO) phonons have been observed and argues against the large InN bandgap. A similar argument can be applied to the scenario that the bandgap of InN is ffi0.8 eV, in the case of which the same laser photon energy (2.34 eV) can excite the electrons up in the conduction band so that it would take 6 A1(LO) phonons to be emitted for full thermalization. No A1(LO) phonons should be detected with 1.96 eV laser excitation. Both of the above arguments are consistent with the experimental observations. Earlier, it was stated that the proponents of the larger bandgap InN argue that the small bandgap reported for InN could be due to incorrect attribution of the deep-level emission to the band-edge emission as defects capture electrons or

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j 2 Electronic Band Structure and Polarization Effects

194

holes and emit photons (through a radiative relaxation process) of lower energy than the bandgap. This capture process can also emit phonons through a nonradiative relaxation process. Let us for a moment suppose that the 0.8 eV luminescence reported in the literature is due to such a capture process by some unknown deep-level defect(s) in InN. If so, the observations of Tsen et al. [296] of nonequilibrium A1(LO) phonons in InN with excitation laser having photon energy 2.34 eV suggests that nonradiative relaxation processes also play a role in the capture. However, due to deep-level defects having very localized wave functions, their momenta are widely spread due to the uncertainty principle. This suggests that electron–phonon interaction during the capture process does not need to conserve momentum. Therefore, phonon wave vectors of almost every magnitude can be emitted. In other words, the defect-model predicts that if one detects nonequilibrium A1(LO) phonons with 532 nm excitation then one should also detect nonequilibrium A1(LO) phonons with 634 nm excitation. This is in contradiction with the experimental observation of Tsen et al. [296]. In short, the nonequilibrium phonon experiments are consistent with the 0.8 eV bandgap. In a renewed effort to determine the fundamental gap of InN, Arnaudov et al. [249] studied the shape and energy position of near band edge photoluminescence spectra of InN epitaxial layers with different doping levels. They implied that samples with high doping concentration have been used to infer the bandgap of InN without properly accounting for the effects of band filling, band nonparabolicity, and electron–electron and electron–impurity interactions. In addition to usual difficulties associated with highly doped samples, another aggravating factor is a clear lack of convergence in the value of the electron effective mass. Properly accounting for the aforementioned effects, Arnaudov et al. suggest a bandgap of Eg ¼ 692 2 meV for an effective mass at the conduction band minimum mn0 ¼ 0.042m0. They also argue that the value of Eg ¼ 0.69 eV reported in Ref. [270] extracted from the absorption and photoluminescence spectra of samples with carrier concentration n > 6 · 1018 cm
3 within the band-to-band recombination model while taking into account the Burstein–Moss shift and bandgap renormalization due to many-body effects agrees with the data; the value used for the effective mass at the bottom of the conduction band, mn0 ¼ 0.1m0, is inconsistent with the universal Kane’s relation (mn0; Eg). This relation predicts a bandgap energy of 1.7 eV for mn0 ¼ 0.1m0, and to obtain near to 0.7 eV bandgap one must use a much smaller effective mass of mn0 ¼ 0.042m0. Interestingly, the same value of Eg ¼ 0.69 eV was obtained by fitting the absorption spectra, this time in a lightly doped sample with n ¼ 3.5 · 1017 cm
3, with a sigmoidal function that includes only the band tailing effect and does not involve any value for the effective mass [300]. Taking advantage of improved crystalline quality and utilizing samples with electron concentrations in the range of 7.7 · 1017–6 · 1018 cm
3 Arnaudov et al. [249] undertook the task of determining the bandgap on InN from optical data but with the interpretation of the emission spectra in such highly conducting layers in terms of the free-electron recombination band (FERB) model, which has been previously reported in the context of GaAs [301,302], InP [303], InSb [304], and GaN [305]. By analyzing not only the emission energy position but also the shape of the spectra simultaneously and taking into consideration the specifics associated with both high and low electron concentrations, Arnaudov

2.9 InN

E

1.0

Ec

0.8 PL intensity (au)

n = 1.7 × 1018cm–3

E F = E Fn Degenerate band tails G* = E

0.6

Ev

0.4

n = 6 × 1018 cm–3

Fp

gn ,gp

0.2

Eg -G2*Eg-G1* Eg EF2

0.0 0.5

0.7

0.6

EF 3 0.8

Energy (eV) Figure 2.13 Experimental, depicted with symbols, and calculated, depicted with solid lines, PL spectra of samples with 1.7 · 1018 and 6.0 · 1018 electron concentrations as measured by Hall effect. The inset schematically depicts the recombination mechanism between the

degenerate electrons in the conduction band DOS to the level G* in the valence band tails as relied on in modeling. The energy positions representing the best fits for Eg, EF, and the intrinsic, or the unperturbed, bottom of the conduction band Eg–G* are also indicated [249].

et al. [249] are able to determine the fundamental bandgap for the electron effective mass in InN as Eg ¼ 692 2 meV for an effective mass at the bottom of the conduction band mn0 ¼ 0.042m0, which is consistent with Kane’s relation. PL spectra of two samples with electron concentrations of 1.7 · 1018 cm
3 6 · 1018 cm
3, as determined by Hall measurements, by Arnodov et al. [249] are shown in Figure 2.13 (points)). Both samples exhibit a broad emission band with a maximum at 685 meV and 705 meV, respectively, the former for the lower doped and the latter for the higher doped sample due to a larger Burstein–Moss shift. Noteworthy, however, is that the emission band of the sample with a higher Hall concentration is broader and more asymmetric, and its PL peak is at a higher energy due to the larger Burstein–Moss shift. The low-energy side of the spectral band of the sample with higher electron concentration shifts to lower energy compared to the sample with lower Hall concentration, which tends to narrow the apparent optical bandgap, consistent with emission spectra from highly doped semiconductors. In addition, these observations are characteristic of free to bound recombination of degenerate conduction electrons with nonequilibrium valence holes in the valence band tail, as shown in the inset of Figure 2.13 [301–305]. Moreover, the shape of the emission bands follows the energy distribution of electrons in the conduction band, all the while their energy positions are determined by the interplay of the equilibrium Burstein–Moss shift (blue shift) and the effective bandgap renormalization (red shift).

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j 2 Electronic Band Structure and Polarization Effects

196

To deduce the bandgap energy Arnaudov et al. [249] fit the experimental emission spectra with that obtained from the general expression for the intensity versus the photon energy I(hn) given by (neglecting the energy dependence of the probability for radiative transitions, as was done for heavily doped GaN [305]) ð¥ ð¥ IðhnÞ g n ðE Fn Þf n ðE n
E Fn Þg p ðE p Þf p ðE p
E Fp ÞdðE n
E p
hnÞdE n dE p ; 00

ð2:46Þ where gn(En) and gp(Ep) are the density of states in the conduction and valence bands at electron and hole energies En and Ep, respectively. The terms fn and fp represent the associated nonequilibrium Fermi–Dirac functions, and EFn and EFp are the quasiFermi levels for electrons and holes. The conduction band density of states gn(En), as well as the electron effective mass mn(En), can be calculated in the framework of Kane’s two-band kp model as described in Ref. [300]. Although the bandgap information is implicit in the En
Ep term of the delta Dirac function, the bandgap term Eg is sometimes explicitly subtracted from En
Ep for emphasis, as done by Arnaudov et al. [249]. The reader is referred to Volume 3, Chapter 2 for a detailed discussion of spontaneous emission intensity calculations. Because the nonequilibrium holes are situated in a relatively narrow energy window deep in the band tails, they do not significantly affect the spectral distribution of the emitted light. Therefore, the quantity gn(En) fn(En
EFn) in Equation 2.46 roughly reproduces the shape of the FERB shown in the upper part of the inset in Figure 2.13. Likewise, the term gp(Ep) fp(Ep
EFp) in Equation 2.46 determines primarily the energy position of the emission band associated with the unperturbed fundamental bandgap Eg via Equation 2.47, as depicted in the lower part of the inset in Figure 2.13. It should be pointed out that the FERB model includes a calculation of the spectral shape as well as analytical renormalization of the bandgap due to the presence of ionized impurities. In this case, the energy positions of both the low- and high-energy slopes of the emission band are sensitive to the electron concentration induced by ionized impurities. In conjunction with the nonequilibrium Fermi–Dirac function of electrons, fn, Arnodov et al. [249] used the Fermi level EFn ffi EF corrected for the temperature of electrons y, which can differ from the lattice temperature T, electron–electron and electron–impurity interactions [305], and the nonparabolicity of the conduction band density of states (DOS), which can be determined in framework of the two-band kp model. The valence band DOS gp(Ep) is replaced by a Gaussian, determining the tails deep in the bandgap through the root mean square (rms) impurity potential G as detailed in Ref [305]: pffiffiffi 4pe2 ðN i R3s Þ1=2 ; G¼2 p eRs with Rs ffi

ð2:47Þ

aBe e h2 ðna3Be Þ
1=6 and aBe ffi ; 2 4pe2 4pmn0

where the terms e represent in order the electric permittivity, Rs the Thomas–Fermi screening length, Ni ¼ [(1 þ K)/(1
K)]n the total ionized impurity concentration,

2.9 InN

K the compensation ratio, n the extrinsic electron concentration, and aBe the effective Bohr radius of electrons. The value of Rs is smaller than aBe and thus the equilibrium and nonequilibrium degenerate electrons are free above the bottom of the conduction band [305]. The situation with the nonequilibrium holes is quite different in that in III–V materials the effective Bohr radius aBh is much smaller than Rs due to the relatively large hole effective mass mp. Moreover, holes are classically localized, at least for not extremely high impurity concentrations and high temperatures [306], at the potential minima of the valence band tails near the thermal equilibrium level pffiffiffi G ¼
E v þ 2G
kT=2. As shown for heavily doped GaN [305], the level G* plays the role of the quasi-Fermi level in the nonquasi-equilibrium recombination FERB model. Thus, we can replace the value of EFp in the Fermi–Dirac function for holes fp can be replaced by G*, meaning one case set EFp ¼ G*. To comment on the electron effective mass, the FERB emission spectra of samples with 1.7 · 1018 and 6.0 · 1018 cm
3 Hall electron concentration can be calculated with varying n, y, and K and using the value of mn0 ¼ 0.042m0 suggested in Ref. [307]. For the relative static permittivity, a value of e ¼ 14.61 was used by Arnaudov et al. [249]. Assuming a zero-compensation ratio, which is a good first-order approximation, the best fits of the spectra are obtained with bandgap values of 690 and 692 meV, for the sample with 1.7 · 1018 cm
3 electron concentration and the sample with 6.0 · 1018 cm
3 electron concentration, respectively. To reconcile the small difference in the aforementioned bandgap values, a small compensation ratio of K ¼ 0.06 for sample with 1.7 · 1018 cm
3 electron concentration and K-value of 0.01 for sample with 6.0 · 1018 cm
3 electron concentration was introduced. Doing so resulted in a bandgap valued of Eg ¼ 692 meV for both samples. The best fit values for the two samples are shown in Figure 2.13 (solid lines). It should be noted that best fit values of the electron concentration nopt are noticeably lower than those deduced by the measured Hall effect. This dispersion may perhaps be related to the inhomogeneities in the films. In spite of this both represent degenerate cases because the Mott’s transition concentration is estimated to be about nMott ¼ 5 · 1016 cm
3 for mn0 ¼ 0.042 m0. The calculated curves agree very well with the experimental spectra, with the exception for the low-energy range. In this region, an additional contribution from a deeper emission center could in principle be possible, which is not included in the model. In the high-energy portion of the spectrum, the accuracy is more reliable. To summarize the above discussion, using the shape as well as the energy position of the near band edge PL spectra of InN epitaxial layers with different doping levels, Arnaudov et al. [249] concluded that the radiative transition is between the degenerate electrons in the conduction band and nonequilibrium holes in the valence band tails and that the fundamental bandgap of InN is Eg ¼ 692 2 meV for an effective mass at the conduction band minimum of 0.042m0, which is consistent with the Kane model. The optical transmission and reflection data obtained from a high-quality InN film grown by MBE with N-polarity support the smaller bandgap figures of InN as shown in Figure 2.14. The growth details and transport properties of InN layers similar to the one that led to the data presented in Figure 2.14 are discussed in Section 3.5.13.

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j 2 Electronic Band Structure and Polarization Effects

198

70

100

80

65

60

60

40

55

20

Reflection (%)

Transmission (%)

E g = 0.7– 0.75 eV

50

0 45 1.0

1.5

2.0

2.5

3.0

3.5

Wavelength (μm) Figure 2.14 Optical reflection and transmission data obtained in an N-polarity InN film grown by MBE indicative of a bandgap between 0.7 and 0.75 eV. Courtesy of A. Yoshikawa.

As for the true value of bandgap of wurtzitic InN, although the data for small bandgap data are convincing and the arguments for the small bandgap are compelling, for some reason, some controversy remains. This controversy is expected to evaporate when and if the large bandgap observed is earlier and some InN samples are explained satisfactorily. We should reiterate that the data in high-quality samples converge on the small bandgap. However, there is still some dispersion in the exact bandgap value that seems to be between 0.65 and 0.8 eV. In this chapter, the arguments for both the large and small bandgap as well as the pitfalls for each in terms of the reliability of the data and a historical review are provided for the reader to be abreast with the conflicting issues surrounding the matter. Turning our attention momentarily to other electronic properties affected by the band structure, measurements of the electron effective mass in wurtzitic InN produced values of 0.11m0 [234], 0.12m0 [308], and 0.14m0 [297], as well as 0.24m0, for the mass perpendicular to the c-axis [309]. Kasic et al. [297] used infrared spectroscopic ellipsometry and micro-Raman scattering to study vibrational and electronic properties of wurtzitic in 0.22 mm thick InN layers grown by RF MBE, as well as Hall effect measurements, and arrived at the isotropically averaged effective electron mass of 0.14m0. The mass value of 0.14m0 closely matches at least one theoretical projection [53]. It should be mentioned that all the InN films used for these investigations featured very high electron concentrations, which are endemic to InN, in the 1018 cm
3 or higher, and causes the Fermi level to degenerate well in the conduction. Consequently, any nonparabolicity in the conduction band would affect the effective mass measurements. However, the realization that the InN bandgap is narrower than that previously thought prompted a reexamination of the effective mass issue also [310]. Accounting for the substantial nonparabolicity that can cause

2.9 InN

an overestimate of the mass because high doping leads to a band-edge effective mass of 0.07m0, which is what is recommended here as was done by Vurgaftman and Meyer [152]. Turning our attention to holes, valence band mass parameters have been calculated by Yeo et al. [67] using the empirical pseudopotential method, and also by Pugh et al. [163] and Dugdale et al. [153] using more or less the same technique. The results of the first two investigations are quite similar, Pugh et al. [163] employed three different levels of computation comprising first-principles total energy calculations, semiempirical pseudopotential calculations and kp calculations. Band structures were obtained from each method in a consistent manner and were used to provide effective masses and kp parameters. These parameters are useful in investigating the electronic structure of alloys and quantum well heterostructures. These valence band parameters are the recommended values with the caveat in that the lower InN energy gap may require a downward revision of the light-hole mass. The parameters concerning the bandgap-related issues for wurtzitic InN recommended by Vurgaftman and Meyer [152] are tabulated in Table 2.19. A compilation of the dispersion in the effective mass for both the conduction band and various valence bands as obtained by various computational methods, as well as parameters used in the description of the bandgap for wurtzitic InN, particularly, in the context of empirical pseudopotential method, as described in Ref. [55], are tabulated in Table 2.20. Christensen and Gorczyca [31] predicted a hydrostatic deformation potential of
4.1 eV for wurtzite InN, which compares to a smaller value of
2.8 eV calculated by Kim etal. [180].Inthe absence of any predilection forany of the tworeports, averaging the two leads to a ¼
3.5 eV. In the absence of any calculations of the valence band deformation potentials, appropriating the parameter set specified above for GaN could be a good default at this point. The elastic constants measured by Sheleg and Savastenko [81], early on more refined values arrivedby calculations suchasthe set reported by Wright [179], are available, which are C11 ¼ 223 GPa, C12 ¼ 115 GPa, C13 ¼ 92 GPa, Table 2.19 Recommended band structure parameters for wurtzitic InN [152].

Parameter

Value

Parameter

Value

Parameter

Value

Eg (eV, low temperature) a (meV K
1) b (K) Dcr (meV) Dso (meV) == me =m0 m? e =m0 a1 (eV) a2 (eV)

1.5–1.8 0.245 624 0.040 0.005 0.07 0.07
3.5
3.5

A1 A2 A3 A4 A5 A6 A7 (meV Å) d13 (pm V
1) d33 (pm V
1) d15 (pm V
1) Psp(C m
2)


8.21
0.68 7.57
5.23
5.11
5.96 0 (default)
3.5 7.6 5.5
0.042 (
0.041)

D1 (eV) D2 (eV) D3 (eV) D4 (eV) D5 (eV) D6 (eV) c11 (GPa) c12 (GPa) c13 (GPa) c33 (GPa) c44 (GPa)


3.7 4.5 8.2
4.1
4.0
5.5 223 115 92 224 48

See Tables 2.27 and 2.28 for details related to the elastic constants, piezoelectric constants, and spontaneous polarization charge. Any dispersion among the tables is a reflection of the uncertainty in the available parameters.

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200

Table 2.20 Effective masses and band parameters for wurtzitic InN.

Parameter ==

me m? e == mhh == mlh == mch m? hh m? lh m? ch A1 A2 A3 A4 A5 A6 A7 D1

anisotropic

isotropic

A

B

C

D

0.138 0.141 2.438 2.438 0.140 2.661 0.148 3.422
7.156
0.244 6.746 3.340
3.208
4.303 0.072 0.214

0.137 0.140 2.493 2.493 0.137 2.599 0.157 1.446
7.298
0.441 6.896 3.064
3.120
3.948 0.103 0.084

0.11 0.10 1.56 1.56 0.10 1.68 0.11 1.39
9.62
0.72 8.97 4.22
4.35

0.11 0.10 1.67 1.67 0.10 1.61 0.11 1.67
9.28
0.60 8.68 4.34
4.32
6.08 0

0.10 0.10 1.431 1.431 0.106 1.410 0.196 0.209
9.470
0.641 8.771 4.332
4.264
5.546 0.278 0.0375

0.10 0.10 1.350 1.350 0.092 1.449 0.165 0.202
10.841
0.651 10.100
4.864
4.825
6.556 0.283 0.0375

0

Effective masses in units of free-electron mass m0, Luttinger-like parameters Ai (i ¼ 1, . . ., 6) in units of
h2 =2m0 , and A7 in units of eV Å. The crystal field splitting energy D1 is given in units of meV. The term aniso represents the values derived using a band structure calculation with anisotropically screened model potentials, whereas the term iso describes a comparative band structure calculation on the basis of isotropically screened model potentials using an averaged e0 value by taking the spur of the dielectric tensor [55]. Anisotropically screened and isotropically screened values are from Ref. [55]. A: effective masses and Ai are from Ref. [67] obtained through a line fit to the band structure; B: direct kp calculation in a 3D fit from Ref. [67]; C: Ai from Ref. [153] obtained through a Monte Carlo fitting procedure to the band structure and effective masses calculated using Equations 2.30 and 2.31; D: direct kp calculations for Ai from Ref. [153] and effective masses calculated using Equations 2.30 and 2.31.

C33 ¼ 224 GPa, and C44 ¼ 48 GPa. Other sets of parameters calculated by Kim et al. [180] and Davydov [201] are also available in the literature. For a more detailed discussion of elastic and piezoelectric coefficients as well as polarization issue, refer to Section 2.12 and Tables 2.25–2.27 and 2.28. Owing to the fact that the piezoelectric coefficients in InN have apparently not been measured, for consistency the theoretical values of Bernardini and Fiorentini [210], d33 ¼ 7.6 pm V
1, d13 ¼
3.5 pm V
1, and d15 ¼ 5.5 pm V
1, are suggested. In spite of the fact that the spontaneous polarization data for GaN/GaInN structures are not as conclusive as one would like at this point, most likely owing to relatively poor material quality, the value Psp(InN) ¼
0.042 C m
2 is consistent with a thorough comparison of experiment and theory [84]. Recommended band structure parameters for wurtzite InN are compiled in Table 2.19 [152]. 2.9.2 Zinc Blende InN

Experiments on zinc blend InN are very rare although this polytype has been reported [311]. The bulk of the reports comprise theoretical estimates of its band

2.9 InN Table 2.21 Luttinger parameters g1, g2, and g3 for zinc blende InN

obtained from a fit along the [1 1 0] direction along with those available in the literature, as compiled in Ref. [55]. Parameter

A

B

c1 c2 c3

7 0.97 1.22

3.27 1.26 1.63

A: empirical pseudopotential calculation by Fritsch et al. [55]; B: recommended values taken from Ref. [152].

parameters. The zinc blende variety has been projected to have a direct band alignment, with G-, X-, and L-valley gaps of 1.94, 2.51, and 5.82 eV, respectively [70]. However, this particular calculation was performed before the controversy in the bandgap of wurtzitic InN. If the arguments available in the literature and presented in the Section 2.9.1 dealing with wurtzitic InN were to hold in favor of the large bandgap, then the aforementioned bandgap values will be more credible. For the valence band, spin–orbit splittings in the range 3–13 meV have been projected [107,171,227]. Vurgaftman and Meyer [152] recommends the 5 meV value among them. As in the case of the bandgap, the effective mass value for the wurtzitic polytype is recommended for the zinc blende variety, that is, 0.07m0. As mentioned in the previous section, this value is arrived at after the nonparabolicity effects are accounted for. To reiterate, the range of values reported for the Wz InN is 0.10–0.14m0 [70,163,173]. The longitudinal and transverse masses for the X valley have been calculated to be 0.48m0 and 0.27m0, respectively [70]. The recommended Luttinger parameter set by Vurgaftman and Meyer [152] is g1 ¼ 3.72, g2 ¼ 1.26, and g3 ¼ 1.63, which is derived from the work of Pugh et al. [163], and the split-off mass is chosen to be mso ¼ 0.3m0 [70,163]. These as well as the other literature values of the Luttinger parameters for zinc blende InN are listed in Table 2.21. Fritsch et al. [55] calculated the effective masses for conduction and valence bands, the latter involving the light and heavy holes, as well as the spin–orbit split-off mass, which along with the anisotropy taken into account are listed in Table 2.22 for zinc blende InN. The data contain those obtained by FP-LAPW, EPM calculations, and those calculated with Luttinger parameters Equations 2.42 and 2.43. The EP parameter value given by Meney et al. [173] is more likely because the alternative value by Pugh et al. [163] implies too large a value for F. The resulting parameter set is EP ¼ 17.2 eV and F ¼
4.36. For the hydrostatic deformation potential, an average value of
3.35 eV from the theoretical [31,180] candidates of
2.2 to
4.85 eV has been recommended [152]. The valence band deformation potentials listed in Table 2.19 are compiled from the calculations of Wei and Zunger [107], Kim et al. [180], Tadjer et al. [70], and Van de Walle and Neugebauer [177] are av ¼
0.7 eV, b ¼
1.2 eV, and d ¼
9.3 eV. Elastic constants of C11 ¼ 187 GPa, C12 ¼ 125 GPa, and C44 ¼ 86 GPa are assumed from the calculations of Wright [179],

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202

Table 2.22 Effective masses for electrons (e), heavy holes (hh), light holes (lh), and spin–orbit split-off holes (so) in units of the free-electron mass m0 along the [1 0 0], [1 1 1], and [1 1 0] directions for zinc blende InN. [100]

[100]

[111]

[111]

[110]

[110]

Reference

me

mhh

mlh

mhh

mlh

mhh

mlh

mso

A B C

0.13 0.12 0.10

1.18 0.83 2.18

0.21 0.16 0.89

2.89 0.83 2.29

0.19 0.16 0.93

2.12 1.55 3.10

0.20 0.15 0.79

0.36 0.30 0.30

Compiled by Fritsch et al. [55]. A: after Ref. [55]; B: calculated from recommended Luttinger parameters from Ref. [152], using Equations 2.42 and 2.43; C: empirical pseudopotential calculation from Ref. [70].

Table 2.23 Parameters associated with the band structure for zinc

blende InN bandgaps are for low temperature [152]. Parameter

Value

Parameter

Value

Parameter

Value

alc (Å) at T ¼ 300 K E Gg ðeVÞ a (G) (meV K
1) b (G) (K) E Xg ðeVÞ a (X) (K) b (X) (meV K
1) E Lg ðeVÞ a (L) (meV K
1) b (L) (K)

4.98 0.78 1.5–1.8 0.245 624 2.51 0.245 624 5.82 0.245 624

Dso (eV) me ðGÞ ml ðX Þ mt ðX Þ c1 c2 c3 mso

0.005 0.07 0.48 0.27 3.72 1.26 1.63 0.3

EP (eV) F VBO (eV) ac (eV) av (eV) b (eV) d (eV) c11 (GPa) c12 (GPa) c44 (GPa)

17.2
4.36
2.34
2.65
0.7
1.2
9.3 187 125 86

which are similar to other calculated sets [180,181]. The recommended band structure parameters for zinc blende InN are compiled in Table 2.23.

2.10 Band Parameters for Dilute Nitrides

Dilute nitrides can be described as conventional III–V compound semiconductors, wherein a small N fraction on the order of a small percentage is added. Addition of even very small amounts of N causes substantial changes to the bandgap and the lattice contact of the host compound to the point that standard bowing parameters for the bandgap variation linear interpolation of lattice constants between the host material and zinc blende GaN or GaInN do not apply. A single bowing parameter is inadequate even if the goal is only to describe the energy gap for a relatively wide range of compositions [312]. The technological driving force is really compelling in

2.10 Band Parameters for Dilute Nitrides

that wavelengths of interest for short and long haul communications systems can be obtained on GaAs technology by adding small fractions of N into the GaAs or InGaAs lattice. The motivation here is in materials incorporating only a small percentage of nitrogen [313], because it is highly questionable whether more than 10–16% N can be incorporated stably. Even in cases when one can, the quality is very inferior. Annealing techniques employed to improve layer quality end up causing N segregation, reducing the fraction in the bulk. Having mentioned this, it should also be pointed out that N isoelectronic doping of compound GaP, which has an indirect bandgap, for LEDs [314] predates the flurry of activity aimed at imparting substantial changes in the band structure, which is the topic of this section. The band structure of dilute nitride compound semiconductors has been reviewed in the literature by Vurgaftman and Meyer [152]. The treatise here follows the same philosophy in that the N-containing GaAs followed InP is treated before segueing onto other compound semiconductors such as those based on Sb anion. The properties of the conduction band in dilute nitride semiconductors can be described in terms of the band anticrossing (BAC) model [315]. It should be mentioned that a small percentage of N usually has little effect on the valence bands. This twoparameter anticrossing model can be spun formally in terms of the many-impurity Anderson model within the coherent potential approximation. It can also be thought of as the interaction between a single, spatially localized N level and the conduction band of the underlying traditional As-, P-, Sb-based compound semiconductors. Alternatively, Lindsay et al. [316] predicted the identical fundamental bandgap by assuming that the interaction involves a weighted average of perturbed upper states as opposed to a single N level. If one neglects the effect on the valence bands completely, the energy dispersion relation for the two-coupled bands within the BAC model can be expressed as   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  C E ðkÞ ¼ ð2:48Þ E ðkÞ þ E N ½E C ðkÞ þ E N 2 þ 4xV 2 : 2 Here, EC(k) is the conduction band dispersion of the nominal nonnitride semiconductor (e.g., GaAs in the case of GaAsN), EN is the position of the nitrogen isoelectronic impurity level in the nonnitride semiconductor, V is the interaction potential between the two bands, and x is the N mole fraction. The energy dispersion relation for the twocoupled conduction bands in GaAs0.99N0.01 showing the characteristic anticrossing is plotted in Figure 2.15. As can be gleaned from the band structure treatment for standard nitride semiconductors, any temperature dependence arises from the shift of the conduction band dispersion EC(k) not that of the valence band. In the dilute semiconductor case also, the bandgap is assumed to follow the Varshni formula of Equations 2.34 and 2.35, but with EN taken temperature independent. One consequence of this assumption is a considerable weakening of the variation of the fundamental energy with temperature [317]. The weak shift of EN with applied pressure necessitates a fresh look at the deformation theory. The strain dependence of the E transitions can be determined by substituting the applicable deformation potentials of the host nonnitride semiconductor to obtain EC(k), followed by deriving E (k) from Equation 2.48. The implementation of Equation 2.48 requires input of

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2.4

T = 300 K

GaAs0.99N0.01

Energy (eV)

2.2

E+

2.0

1.8 EN 1.6

EC E–

1.4

1.2 –0.15

–0.10

–0.050

0.0

0.050

0.100

0.15

Wave vector (1 Å–1) Figure 2.15 Conduction band dispersion relations for GaAs0.99N0.01 at 300 K from the band anticrossing model (BAC) [319] (solid curves). For comparison, the unperturbed GaAs conduction band and the position of the nitrogen level are shown as the dotted and dashed curves, respectively [152].

band parameters of the host semiconductor. Although, we do not reproduce the nonnitride parameters in this work to preserve space, all of the required values are tabulated in a review [92]. The more detailed kp theory can also be applied. The BAC model can be extended to treat spin-doubled conduction bands, valence band, and nitrogen impurity bands by modifying the eight-band kp theory [318–321]. Coupling of the nitrogen band to the X and L valleys have also been introduced [322,323] but at the expense of additional complexity, which may not be warranted. Unless compelled, it is preferable to stick to the simple two-parameter fit of Equation 2.48. It has been pointed out that the fixed position of the nitrogen level with respect to the vacuum level implies a tandem shift with the valence band maximum of the host nonnitride material. To account for the experimental observation of nonnegligible deviations from referencing to the valence band offset, a separate nitrogen level for each host material is typically specified. Vurgaftman and Meyer [152] suggest that the valence band offset for an unstrained dilute nitride be set equal to that of the host semiconductor. As mentioned above, within the realm of the BAC model, the primary effects of the nitrogen are on the conduction band. Further,

2.10 Band Parameters for Dilute Nitrides

even the 10-band model does not shift the valence band maximum in the absence of strain or quantum confinement, while of course influencing the hole dispersion relations. Although a finite type-I or type-II offset in strained structures have been reported on the basis of experiments, they are not sufficiently compelling to deviate from a null offset relative to the host in the absence of strain. Despite its simplicity, Equation 2.48 provides a basis for describing material properties, such as the fundamental energy gap, which are governed by the transition from E
to the top of the valence band, the temperature dependence of the gap, the electron effective mass, and the characteristics of the upper band Eþ. It must be kept in mind that within the theory of Lindsay et al. [316], there is not necessarily a single well-defined Eþ band. Somewhat in similar vein, the extent to which the BAC representation may be considered fundamentally realistic is still a matter of active discussion [315,324]. Clearly, the BAC model considers only a single nitrogen level on a substitutional lattice site or a narrow impurity band formed from such levels. In the process, it neglects mixing with the L and X valleys, and more complex nitrogen behavior in the semiconductor such as the formation of nitrogen pairs and clusters. In contrast, the more complicated pseudopotential calculations that consider some of these issues are computationally demanding [325–327]. In addition, the numerical results do not lend themselves to simple formalism such as Equation 2.48. Buttressing the simpler BAC approach is the discovery by Lindsay et al. [316] that the verifiable prediction of the dependence of the bandgap on the N content may be unaffected by generalizing to a multiplicity of higher lying states. Let us now turn our attention to specific dilute nitride semiconductor systems. 2.10.1 GaAsN

In conjunction with early investigations motivated by visible LED development, it has been known that small quantities of nitrogen in GaAs and GaP form deep-level impurities [314]. However, it has only been after the advent of nitride semiconductors that also paved the way for investigating traditional compound semiconductors, such as GaAs with N content definitely beyond the quantities ( 1% or more) used for doping experiments [328,329]. On the flip side of this, reports [330] of the incorporation of small amounts of As into GaN are neither common nor easy. The presence of As in GaN has been stated to cause modified surface reconstruction and/or act as surfactant or be a source of dopant impurities. Even though care must be exercised, PL measurements can be used to determine the bandgap, provided the relative position of the particular transition energy is known, and have been applied to GaAsN with N fractions up to 1.5% in an effort to determine the dependence of the energy gap on N composition [331]. As compared to other ternaries, a large bandgap bowing parameter of 18 eV was found, which for small compositions is equivalent to a linear model [332]. Early theoretical studies projected bandgap bowing parameters based primarily on the dilute nitride semiconductor with large N content [333–341]. Although the large bowing parameter was originally supposed to produce a semimetallic overlap at intermediate

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compositions [336], more detailed investigations that followed led to reduced bowing parameter with increasing composition [338,339], with experimental backing based on investigations of Bi and Tu [342], who studied N compositions as large as 15%. In discord with the above-mentioned linear dependence of the bandgap for small N fractions, subsequent investigations pointed to a highly nonlinear reduction in the energy gap for small N compositions [343–349]. Another noteworthy discovery was the realization of a significant weakening of the temperature and pressure dependencies of the bandgap for GaAsN and also GaInAsN with small In fractions [346,350,351]. In aggregate, it is clear that a simple bowing approximation could not adequately describe the GaAsN alloy. In this vein, Shan et al. [352] proposed the band anticrossing model and confirmed a weak pressure dependence for the nitrogen band transitions with a deduced deformation potential of
1.2 eV. To put matters in context, the density functional calculation of Jones et al. [353,354] also predicted reduced pressure dependence without invoking the BAC model. Additional report supporting the BAC model has been the finding by Skierbiszewski et al. and others pointing to a significantly heavy electron mass in GaInAsN [355–361]. On the anticorrelation side is another set of measurements by Young et al. [362] who found a reduction in the effective mass with increasing N content, in direct conflict with the BAC model that predicts an increase even at the zone center. Increasing mass with increasing N content, however, appears to hold. Consistent with the overall expected behavior of dilute nitride semiconductors based on the BAC model, the temperature dependence of the bandgap has been confirmed to be notably weaker in GaAsN than in GaAs [317,363]. In a similar vein, GaAsN film electroreflectance [364] resolved both the E
and Eþ transitions, the band description of which can be seen in Figure 2.15. Extending to a ternary, the bandgap reduction was also observed in nitrogen-implanted Al0.27Ga0.73As samples [365]. The transition between the doped and alloyed materials was studied by Zhang et al. [366,367]. Here, doped implies quantities manifesting themselves as dopants without radical changes incurring on the host material. The alloy material implies that the N concentration is sufficiently high to cause substantial changes in the host material. Zhang et al. observed evidence for impurity banding at N concentrations as small as 0.1% N. Zhang et al. also proposed an alternative method for the characterization of the bandgap energy, which is not based on the BAC model [368]. This transition point was quantified as 0.2% by Klar et al. [369]. Figure 2.16 depicts the fundamental energy bandgap that is between the valence band maximum and the E
conduction band minimum, as a function of N fraction, x, for GaAs1
xNx at 300 K. Also shown is a curve with a constant bowing parameter of 18 eV (dotted line) along with another incorporating a variable bowing parameter of (20.4–100x) eV, as suggested in a review by Vurgaftman et al. [92] (dashed line). Ostentatiously, the BAC model predicts a substantially higher energy gap beyond the N fraction of 1.5%. The available experimental data (points in Figure 2.16), compiled in Ref. [315], show much better consistency with the BAC model than with either of the two utilizing bowing parameters. It should be reiterated that GaAs1
xNx alloys with x > 5% become increasingly difficult to grow while retaining quality. As such, compositions with large N fraction

2.10 Band Parameters for Dilute Nitrides

GaAs 1.4

Bandgapenergy (eV)

GaAs1–xNx 1.2

1.0

BAC 0.8

C = 20.4 –100 xeV C = 18 eV Experimental data

0.6 0.00

0.01

0.02

0.03

0.04

0.05

N mole fraction, x Figure 2.16 Energy of the fundamental bandgap in GaAsN as a function of nitrogen concentration x (a) from the BAC model (solid curve), (b) using the variable bowing parameter from the review of Vurgaftman et al. [92] (dashed curve), and (c) using a constant bowing parameter (dotted curve). For comparison, the available experimental data as compiled in Ref. [315] are also plotted (circles) [92,152].

may not have the technological significance [313]. On the theory side, no upper limit on the N compositions beyond which the BAC model becomes invalid has been reported. Despite some spread in the reported values for the two primary parameters EN, 1.65–1.71 eV, referenced with respect to the GaAs valence band maximum, and V (2.3–2.7 eV), the values reported by Shan et al. [352], namely, EN ¼ 1.65 eV, V ¼ 2.7 eV, are suggested. In discord with the conventional BAC model, which assumes that the addition of N has little effect on the valence bands, two reports noted a larger than expected heavy-/ light-hole splitting in GaAsN containing a small percentage nitrogen [370,371]. This implies a strong bowing in the valence band shear deformation potential b, although the increase in the deformation potential is inconsistent in these two reports. Expanding their earlier work, Egorov et al. [372] again observed strain-induced splitting of light-hole and heavy-hole bands of tensile-strained GaAsN. The observed dependence of the bandgap in unstrained GaAsN on the nitrogen content differed substantially from that predicted by the theory assuming that the bandgap in GaAsN

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can be reduced to zero. In cases like this when there is discrepancy, the default position is to underscore the importance of additional investigations. An issue of importance in device applications is the nature and magnitude of the band alignment at a GaAs/GaAsN heterojunction. While the BAC model implies that the GaAsN conduction band minimum must lie below that in GaAs, it is not clear whether the valence band maximum in GaAsN should exhibit any relative shift. Various predictions disagree on this issue [334,341]. On the experimental side, X-ray photoelectron spectroscopy data suggest a type-II band alignment [373] but with quite large error bars on the valence band offset (VBO). This type of band alignment was supported by PL measurements [374]. Follow-up investigations utilizing not just optical but electrical characterizations as well have concluded a definite type-I alignment [375–379]. A point of discord may lie in the fact that the built-in strain was not completely relaxed in any of the heterostructures covered in the aforementioned investigations. Egorov et al. [377] accounted for strain effects and deduced that the band offset for unstrained GaAs0.98N0.02 with respect to GaAs. The value of the valence band discontinuity between GaAs and GaAsN0.02 at 18 K was evaluated to be 15 5 meV, taking into account the bandgap of GaAsN alloy [372]. On the contrary Egorov et al. [379] reported the band alignment of InxGa1
xAs/GaAsN heterojunctions to be type I or type II, depending on the In content x, a point that needs to be confirmed by additional measurements. The BAC model parameters recommended by Vurgaftman and Meyer [152] for GaAsN and all of the other dilute nitrides for which information is available are summarized in Table 2.24. 2.10.2 InAsN

Another binary compounded with N is InAsN, which garnered early theoretical interest [334,380], followed by tight binding calculation focused on the effects of nitrogen clustering in the alloy [381]. Experimental investigations of this dilute nitride have been reported [382–386]. Measurements of the electron effective mass in this alloy indicated a large increase [380,390], analogous to that in GaInAsN [355].

Table 2.24 Band anticrossing (BAC) model parameters for some

of the dilute nitride semiconductors [152]. Parameters

EN w.r.t. VBM (eV)

V (eV)

GaAsN InAsN Ga1
xInxAsN GaPN InPN Ga1
xInxPN InSbN

1.65 1.44 1.65(1
x) þ 1.44x
0.38x(1
x) 2.18 1.79 2.18(1
x) þ 1.79x 0.65

2.7 2.0 2.7(1
x) þ 2.0x
3.5x(1
x) 3.05 3.0 3.05(1
x) þ 3.0x
3.3x(1
x) 3.0

2.10 Band Parameters for Dilute Nitrides

While the authors [385,386] appear to argue that the BAC model could not account for any increase greater than doubling of the mass in the nitrogen-free host material, upon closer examination of the model it becomes evident that their view is contradicted, and also by the results available for GaInAsN [355], clearly displaying a similarly large increase in mass. Consequently, it is reasonable to assume that the BAC model is applicable to InAsN. At the same time, however, it should be stated that the available information is somewhat incomplete and future investigations may alter this assertion. Extracting the position of the nitrogen level with respect to the valence band maximum in the host InAs from the valence band offsets tabulated in a review article [92], a value for EN ¼ 1.44 eV is recommended. The measurements of Naoi et al. [382] are consistent with values for the potential V ranging between 1.9 and 2.3 eV. A round figure of V ¼ 2.0 eV could be a good default. 2.10.3 InPN

A few experimental studies of InP1
xNx are available in the literature [387,388]. Because the bandgaps and valence band offsets of GaAs and InP differ only slightly, it would follow that the BAC model would apply equally well to InPN. Yu et al. [388] derived band structure parameters in which the GaAs/InP VBO was assumed to be 0.35 eV. However, Vurgaftman et al. [92] recommended a value of 0.14 eV, but considering the result EN ¼ 2.0 eV by Yu et al. led them to update their recommended value to EN ¼ 1.79 eV, both being with respect to the valence band maximum of InP. [92] A reexamination of the coupling potential that is most consistent with the data of Yu et al. [388] led Vurgaftman et al. [92] to recommend V ¼ 3.0 eV. 2.10.4 InSbN

Even though InSb has the smallest bandgap for the conventional III–V binaries, addition of N has attracted some attention because of the potential that even longer wavelengths, beyond those accessible by InSb, can be accessed. Murdin et al. [389– 391] made experimental observations that the effective mass in InSb1
xNx increases despite a considerable reduction in the bandgap, which is consistent with all the other dilute nitride semiconductors discussed above. The authors derived EN ¼ 0.65 eV. This compares with EN ¼ 0.85 eV above the top of the valence band. At this stage, the experimental values are recommended as default. The same authors also reported V ¼ 2.2 eV and supplemented the minimal BAC dispersion relation of Equation 2.48 with an additional shift of the level of nitrogen position with increasing N fraction [389,390]. It should, however, be pointed out that it appears that their value for V underestimates the observed bandgap decrease [389]. This led Vurgaftman et al. [152] to recommend V ¼ 3.0 eV, which should yield better consistency with the available data, albeit little. Because the effect of nitrogen on the band structure of InSbN is significant in terms of the light-hole dispersion, the full 10-band kp model may be required for any realistic calculations.

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2.10.5 GaPN

The early LED [314] developments utilized N-doped GaP because of the unique band structure of this binary. It is an indirect gap semiconductor with both X and L valleys lower in energy than the G point. Nitrogen acts as an isoelectronic impurity in GaP and has been employed as the active material of visible LEDs [314] until the advent of other conventional direct bandgap and III–V and nitride-based varieties. Initial studies of GaP with an alloy-like concentration of N were reported by Baillargeon et al. [392,393]. Miyoshi et al. [394] investigated the transmogrification of the GaPN luminescence spectrum with increasing N content and were able to observe the emission from excitons bound to nitrogen pairs for x < 0.5%. Bi and Tu [395] reported GaP1
xNx with x as large as 16% using gas-source molecular beam epitaxy. A number of theoretical reports predicted that GaPN retains its indirect gap nature up to relatively large (beyond dilute) N concentrations [334,339,341]. Shan et al. [396,397] reexamined this view of indirect bandgap within the BAC model, and in the process found out that the anticrossing between G-valley states and the N impurity band moves E
below the X valley for arbitrarily small values of x. They derived V ¼ 3.05 eV employing the well-established value of EN ¼ 2.18 eV, relative to the valence band maximum. This places the nitrogen level slightly below the conduction X valley. As in the case of GaAsN whose band structure is less dependent on pressure and temperature than its host material GaAs, GaPN too sports a decrease in the pressure dependence of the fundamental transition with a deformation potential of
1.2 eV [396]. The temperature dependence of the fundamental gap also sees a reduction [398,399]. As in the case of all the dilute nitrides, a large increase in the electron effective mass is observed [400]. Congruous with popular LED materials for decades, strong luminescence for small N fractions [401,402], occurring despite the indirect gap of the GaP host, is another feature of this material. The evolution of transitions due to isolated N centers, N pairs, and N clusters that hasbeenobservedforx < 1%,andthemixingwithX-andL-valleystates,however,cannot be described by the simple BAC model [403–405]. For example, Buyanova et al. observed a sudden reduction in the radiative lifetime of the fundamental transition for x > 0.5%, which they attributed to an effective indirect-to-direct crossover [406]. The wealth of phenomena reported for GaPN can perhaps beexplainedwith a more flexible theoretical approach such as the supercell pseudopotential formulations of Kent and Zunger [324– 326]. Even though the accuracy of the BAC model is more limited in GaPN than in the other dilute nitride alloys due to the proximity of the X valley and the plethora of complex experimental observations for intermediate compositions, the parameter set EN ¼ 2.18 eV and V ¼ 3.05 eV of Shan et al. [398] can be used as default values. 2.10.6 GaInAsN

Let us now extend our treatment of the effect of N to ternary alloys. Having established BAC parameters for GaAs1
xNx and InAs1
xNx, they need to be determined for the

2.10 Band Parameters for Dilute Nitrides

Ga1
yInyAs1
xNx alloy whose host material is InGaAs. Owing to applications in long haul communications systems, most of the technological interest among all the dilute nitrides has so far focused on this quaternary. Additional impetus can be found in its applications to solar cells and photovoltaics that can be grown on GaAs substrates rather than InP, the former having a more established and less expensive technology base. Addition of N in InGaN narrows the bandgap for In concentrations, while at the same time providing tensile strain compensation far less than those required in conventional GaInAs quantum wells that are under compressive strain, countering the composition-induced reduction in the bandgap. Moreover, the wavelengths near 1.55 mm are not accessible with standard InGaAs-based quantum wells on GaAs substrates, which necessitates the use of InP substrates, which allow employment of larger concentrations of In. Incorporation of N in InGaAs can mitigate the situation and allow the use of GaAs substrates even for the extended wavelength for long haul fiber-based communications systems. Specific to the discussion of the bandgap, the applicability of the BAC model to the case of GaInAsN is quite well established [320,321,351,352,355,357] in that much of what is reported for the dilute nitrated binaries discussed above has also been found to be applicable to this quaternary. Investigations employing very low In fractions [351,352,355], for example, on the order of 10%, typically found no differences of significance from GaAsN except of course the decrease in the InGaAs energy gap [92]. For materials with larger In fractions, prepared on both GaAs [407,408] and InP [409] substrates, Zhukov et al. [410] proposed an alternative model. Pan et al. [359] took EN to be independent of the In concentration and employed V ¼ 2.5 eV. Although Choulis et al. [320,321,411] utilized the same assumption with respect to EN, their value for the coupling potential was considerably lower: V ¼ 1.675 eV. A similar value (V ¼ 1.7 eV) was independently deduced by Polimeni et al. [408] for In compositions ranging from 25 to 41%. In contrast, Sun et al. [412] found that, depending on the particular transition between the conduction and valence subbands, V in the range of 2.8–3.0 eV was imperative to account for the luminescence data for InGaAsN/GaAs quantum wells having an In content of 27.2%. The finding of a smaller bandgap reduction in GaInAsN than in GaAsN with N fraction is in fact expected and is due to the ordering of the nitrogen atoms in the InGaAs matrix [413,414]. There is some experimental evidence for carrier localization in the presence of both In and N in the quaternary alloy [415,416]. In one investigation [416] a series of five distinct transitions, which were attributed to five different environments for the N atom in the alloy, have been reported. Vurgaftman et al. [92] attempted to provide the best available parameters for the InGaAsN quaternary, while maintaining consistency with the parameters recommended above for GaAsN and InAsN. Accordingly, the position of the nitrogen band EN should be determined from the shift of the valence band offset in GaInAs [92]. This should include the small, yet nonnegligible, bandgap bowing with composition. This leads to a smooth variation of EN between 1.65 eV (for GaAsN) and 1.44 eV (for InAsN) and is consistent with the intuitive expectation, the basis of which is in no contradiction with any definitive experiments in that the position of the nitrogen level should not vary with respect to vacuum. Within this framework, Vurgaftman et al. [92] proposed a

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bowing of the coupling potential V in the Ga1
yInyAsN alloy: V ¼ 2.7(1
y) þ 2.0y
3.5y(1
y) eV. Admittedly, this parameterization does not fully agree with the experimental results for this quaternary [320,321,351,352,355,359,408]. But, it is nearly consistent with the median values and the recommended binary end points. As in the case of the ternaries, any strain must be added to the host semiconductor properties in the basic BAC model while employing these parameters. 2.10.7 GaInPN

It must be mentioned at the get go that there are not as much data available in this quaternary system. One report of the energy gap in bulk Ga0.46In0.54PN is available in the literature [417]. According to estimates based on the VBO dependence in GaInP, the quaternary alloy under discussion represents a special case in that one would expect a proximity of the nitrogen energy level and the host conduction band edge, perhaps within as little as 10–20 meV [92]. If one presumes, for the sake of discussion, that the BAC model remains valid in this limit, the reported energy gaps allow one to obtain a coupling potential in the 2.1–2.3 eV range. Considering that this is considerably smaller than the recommended values for both InPN and GaPN, by analogy with GaInAsN one could surmise a bowing of the Ga1
yInyPN interaction potential: V ¼ 3.05(1
x) þ 3.0x–3.3x(1
x) eV. It is noteworthy that the bowing parameters derived for the two quaternaries are quite similar. It should again be underscored that further studies are needed to confirm and/or update this value. 2.10.8 GaAsSbN

In addition to the mixed cation ternary host materials, mixed anion host materials also present opportunities. Among them is the alloy host GaAsSb that produces the quaternary alloy GaAs1
x
ySbyNx [418–420]. This quaternary has the potential for reaching long wavelengths also on GaAs substrates. Unfortunately, because of the sparse nature of the data for this material, and the total lack of any reports on GaSb1
xNx, one can only conjecture that the procedure recommended above for GaInAsN be followed. The material-specific parameters, that is, the constant V ¼ 2.7 eV at least for Sb fractions 20% should be assumed.

2.11 Confined States

If the physical size of the semiconductor in any direction is comparable to the de Broglie wavelength for electrons in electronic processes and exciton diameter in lowtemperature optical processes, the size effects become important. The constriction can form one side, two sides, and three sides in the case of which the system represents three-dimensional, two-dimensional, one-dimensional, and zero-dimensional, as

2.11 Confined States

Figure 2.17 Schematic representation of three-dimensional, twodimensional, one-dimensional, and zero-dimensional systems in real space.

shown in Figure 2.17. The two-dimensional, one-dimensional, and zero-dimensional correspond to quantum wells, wires, and dots, respectively. From the transport point of view, when the size of the well in a quantum well structure is comparable to the de Broglie wavelength in the semiconductor forming the well, the conduction and valence bands are modified noticeably in that the density of states in both the conduction and valence bands are discretized. From the excitonic optical transitions point of view, the length scale is the Bohr radius. This picture is depicted schematically in Figure 2.17a and b for a wurtzite semiconductor bulk (3D system), quantum well (2D system), quantum wire (1D system), and quantum dot (0D system). Moreover, equally important is the modification of these energies in the semiconductor caused by strain effects. In short, eigenstates of strained-layer superlattices require the consideration of both strain and quantum size effects. Eigenstates applicable to the early versions of compound semiconductor based quantum wells without strain were modeled by the envelope function formalism of Bastard [421]. If this method is to be used, one has to calculate the strain-dependent bandgap first. Bastard’s formalism can then be utilized to determine the transition energies in quantum wells, which must be added to the strain component, as shown by Marzin [422]. Marzin developed a method for calculating the bandgap of a strained cubic semiconductor that, when employed in conjunction with the envelope function approximation, leads to eigenstates in quantized structures of strained systems (Figure 2.18). In Bastard’s method revised by Marzin, an 8 · 8 Kane Hamiltonian matrix [423] is employed to give an accurate description of a strained quantum structure. The

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Figure 2.18 Schematic representation of (a) a GaN/AlGaN multiple quantum well structure and the conduction and valence band edges at the G point, and (b) valence and conduction band diagrams of the same with the confined states indicated.

size of the Hamiltonian is justified because there are three valence bands, in addition to the conduction band, with spin-up and spin-down for each band. Similar to cubic semiconductors, the wurtzite phase also has one conduction band and three valence bands, heavy- and light-hole states as well as spin–orbit bands, each with spin-up and spin-down, necessitating the use of an 8 · 8 Hamiltonian. The strain effects in the GaN system have been treated by Gil et al. [98,424] and others, as reviewed in Ref. [59]. However, until the time when uniformly strained films can be grown, correlations to experiments will remain weak and reduce the level of confidence in the predictions especially and the analysis as a whole. This is particularly true for InGaN wells, as there is also phase separation to deal with, in addition to nonuniform strain. We shall once again emphasize that there exist many discrepancies concerning the determination of some important parameters for the relevant bulk material properties such as hole masses, bandgap energy bowing parameters, shear and deformation potentials, and band offsets that are necessary for determining and understanding the properties of confined structures. Calculations, using ab initio methods, have been applied to estimate the unknown but necessary parameters for the band structure calculations. Later on, Sirenko et al. [62] have performed envelope function calculations of the valence band in wurtzite quantum wells following the formalism of Rashba– Sheka–Pikus (RSP) developed for bulk wurtzite semiconductors. Employing a 6 · 6 Luttinger–Kohn model, Ahn [425] studied the effect of a very strong spin–orbit split-off band coupling on the valence band structure of GaN-based materials. Considering that the spin–orbit band is extremely important for GaN because of its very narrow

2.11 Confined States

spin–orbit splitting (10 meV), the spin–orbit split-off coupling was taken into consideration in the calculations. In addition, it was assumed that the electrons in QW are confined by the conduction band offset (DEc) and the holes by the valence band offset (DEv), the values of which are also of some controversy. When the size of the well is comparable to the Bohr radius of an exciton in the semiconductor forming the well, the exciton transition energies are modified. The Bohr radius is given by a0 ¼

4pes
h2 ; m  q2

ð2:49Þ

where es is the dielectric constant of the semiconductor. In GaN, due primarily to its large effective mass, this radius is about 28 Å, necessitating very small physical dimensions before noticeable quantization can occur. In an experimental quantum well, the wave function is constricted along the growth direction, which we shall term as the z-direction. In the orthogonal directions, the system is free. The problem is similar to that of a vibrating string with the ends held stationary. The vibration wavelengths are given by ln ¼

2L n

with n ¼ 1; 2; 3;

ð2:50Þ

where L represents the length of the string. In a quantum well, the rapidly varying Bloch waves will be affected by the barriers in the z-direction, and the effect can be lumped into an envelope function that is slowly varying. The wave function can be expressed as [49,426] Cn ðk? ; zÞ ¼

X n

f n ðk? ; zÞu n expðjk?  r ? Þ;

ð2:51Þ

where fn(k?, z) and uv represent the envelope and Bloch functions, respectively, and n is the subband index. The summation is performed for spin-up and spin-down of the conduction band where the value of six is assumed for the three valence bands. The wave function expression must be solved for the conduction band and the valence band with the envelope function satisfying Schr€ odinger’s equation for the particular potential barrier height [59]. For the case where the conduction band is almost s-like, the G7 state suffices. However, for the valence band, due to band mixing, a 6 · 6 Hamiltonian including all the three uppermost valence bands must be used. Moreover, if strain is present, which is the case in almost all nitride-based structures, the Hamiltonian must include the effect of strain as well. To complicate matters further, the piezoelectric effect induces large electric fields at the heterointerfaces, particularly, in samples utilizing InGaN wells. The envelope function must satisfy    X
h q Hnn0 k? ; þ V n ðzÞdnn þ Hvn0 ðeÞ yv;n0 ðk? ; zÞ j qz ð2:52Þ v0 ¼ E n ðk? Þyv;n0 ðk? ; zÞ;

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j 2 Electronic Band Structure and Polarization Effects

216

where m ¼ 1, 2 for the conduction band state and n ¼ 1, . . . , 6 for the valence band state. The details of how the dispersion of the conduction and valence band states can be found in Ref. [59]. Suffice it to say that the conduction band is formed of nearly slike states and can be considered nearly parabolic, ameliorating the confined state calculations, as will be discussed in the next section. In finding the total energy between the confined conduction and valence band states, one may assume that the majority of the contribution is due to the confinement energy in the conduction band because of the large disparity between the electron and hole effective masses in favor of the conduction band. However, when gain in semiconductor lasers is considered, the dispersion of valence band states must be taken into consideration. 2.11.1 Conduction Band

If the potential barrier is infinitely high, the wave vector in the z-direction will be quantized and assumes the discrete values of   2p np ; n ¼ 1; 2; 3; ð2:53Þ ¼ kzn ¼ ln L where L is the thickness of the quantum well. Assuming a parabolic band structure that satisfactorily describes the s-like conduction band, the confinement energy can be expressed as
h2 k2zn
h2 np2 ¼ n ¼ 1; 2; 3: ð2:54Þ DE conf ¼ 2m 2m L Taking into account the energy dispersion relationship in the x- and y-directions for a parabolic conduction band, we have  
h2 np2 þ k2x þ k2y n ¼ 1; 2; 3; ð2:55Þ E ¼ Ec þ 2m L with n ¼ 1, 2, 3, and Ec representing the conduction band edge. For a true one-dimensional wire along the x-direction, discretization along the ydirection would also occur in addition to the z-direction, giving rise to confined energy states (with complete confinement) of  
h2 np2 E ¼ Ec þ þ k2x þ k2y ; n; m ¼ 1; 2; 3 . . . : ð2:56Þ 2m L If we consider a semiconductor whose constant energy surface for conduction band in k-space is a sphere, such as the case in GaN, the volume of that sphere and the number of available states in k-space are proportional to k3 in terms of momentum and E3/2 in terms of energy, as shown in Figure 2.19a. The density of states per unit energy associated with that system is proportional to E1/2, again as shown in Figure 2.19a. The area and the number of available states in k-space in an ideal system confined in one direction only (representing quantum wells), which is often the z or the growth direction, is proportional to k2 or E, as shown in Figure 2.19b. The

2.11 Confined States

density of states in this case is given by m  =p
h2 and forms a staircase as shown in Figure 2.19b. If we continue and place a confinement in the x-direction in addition to the z-direction, which represents quantum wires, the line length and the number of available states in k-space is proportional to k in terms of momentum and E1/2 in terms of energy as shown in Figure 2.19c. The corresponding density of states takes the dependence of E
1/2, again as shown in Figure 2.19c. If confinement is imposed in all three directions, which represent the pseudoatomic or quantum dot state, the energy is discretized in all directions and the resultant density of states takes a deltalike function in energy, as shown in Figure 2.19d. The case for quantum dots where

(a)

kz

dN/dE ~E 1/2 3

N ~ k (N-E3/2)

ky

E

kx

(b) kz dN/dE N ~ k 2(N-E)

~Constant

ky E kx Figure 2.19 Constant energy surfaces and density of states for 3D in (a) , 2D in (b) , 1D in (c) , and 0D in (d) systems, respectively. The constant energy surface are represented by a sphere, circle, a line, and a point in 3D, 2D, 1D, and 0D systems, respectively, in semiconductor such as GaN conduction band that has a spherical constant energy surface in 3D.

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j 2 Electronic Band Structure and Polarization Effects

218

(c)

kz Bulk dN/dE ~ E–1/2 N ~ k(N-E 1/2 )

n=4

n=3 n=2 n=1

ky

E

kx

(d)

dN/dE ~ δ (E)

kz

Bulk n=4

n=3 n=2 n=1

ky E

kx Figure 2.19 (Continued )

physical dimension in all directions are scaled to the size of the wave function or smaller is treated in Volume 2, Chapter 5. Considering quantum wells formed in growth along the z-direction, the confinement occurs in the same direction and thus the wave vector is quantized. The component of the wave vector experiencing confinement is typically referred to as the z-component, out-of-plane component, or simply kz. In-plane components of the wave vector kx and ky, however, are not quantized and the usual energy momentum dispersion (E–k) diagram would apply. A pictorial description of the E–k diagram in a two-dimensional system with confinement along the z-direction for two quantum energy levels is shown in Figure 2.20. If the barrier potential is large but not infinite, the wave function outside the well decays exponentially, which is called the evanescent wave. No analytical solution exits for the subband energies when the potential barrier is not infinitely high. Graphical solutions treated in many textbooks on quantum mechanics and numerical solutions as well do exist. The problem is made even more complicated in semiconductors in

2.11 Confined States

E

E

E2 E1 π/Lz 2π /L z

kx ,k y

kz

kx Figure 2.20 Energy momentum dispersion relation in a twodimensional system with confinement along the z-direction. Shown on the left is the three-dimensional view while that on the right represent two slices depicting the E–kin-plane relationships for quantum levels.

that not only is the barrier not infinite but also the barrier and well materials do not have the same carrier mass. In this case, the boundary condition must be changed from the continuity of the derivative of the wave function in the z-direction to the continuity of the particle flux in the z-direction, that is,     1 qyB 1 qyW ¼  ; ð2:57Þ mB qz mW qz at Z L/2, assuming the origin of the z-axis to be in the middle of the well. The terms mB and mW represent the effective masses in the barrier and well materials, respectively. Likewise, fnB and fnW depict the envelope wave functions [427] in the barrier and well materials, respectively. The solution for the subband energies can be computed numerically. The conduction band minimum for GaN as well as AlN is at the zone center and twofold degenerate. The confinement energies for the GaN/AlGaN quantum wells can reasonably be estimated by means of the envelope function approximation in the same manner as that extensively used for the GaAs/AlGaAs material system [427,428]. Following the Weisbuch and Vinter notation, the low-lying conduction electron state can be represented by [427,428] X CðrÞ ¼ ½expðik? ðrÞucj ðrÞf n ðzÞ; ð2:58Þ j¼W;B j

where W and B represent the well and barrier materials, uc ðrÞ is the conduction band zone center Bloch wave function of GaN or AlGaN, fn(z) is a slowly varying envelope function, k? is the transverse (in-plane) wave vector, is the envelop wave function j [427], and the growth direction is along the z-axis. Because uc ðrÞ is the same for GaN and Al(Ga)N, Schr€odinger’s equation reduces to  
h2 q2
þ VðzÞ f n ðzÞ ¼ E n f n ðzÞ: ð2:59Þ 2m0 qz2

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j 2 Electronic Band Structure and Polarization Effects

220

In the above equation, m*(z) is the corresponding effective mass of the conduction electron, V(z) represents the profile of the minimum of the conduction band along the growth direction, and E(z) is the confinement energy. Assuming no doping in either regions, V(z) has a rectangular well-like profile. The solution of the confinement energies is similar to a particle in a box problem. The boundary conditions are 1 df n ðzÞ that fn(z) and mðzÞ dz be continuous across the interface. The latter is necessary to ensure the conservation of the particle current.

Al x Ga1–x N/GaN, x = 0.2, L B = L W = 2 nm

0.25

Energy (eV)

0.20 0.15 0.10 0.05 0.00

–2

–1

0

1

2

3

4

Distance (nm) (a) Figure 2.21 (a) The conduction band edge potential profile for a representative Al0.2Ga0.8N/ GaN single quantum well with a barrier and well thickness of 2 nm (or 20 Å) each for a Ga polar sample with the [0 0 0 1] direction pointing to the left. Polarization charge causes deviation from the square well and no screening due to free carriers is accounted for. The scales for the vertical and horizontal axes are in terms of eV and nm, respectively. Courtesy of V. Litvinov. (b) The conduction band edge potential profile for a representative Al0.3Ga0.7N/GaN single quantum well with a barrier and well thickness of 2 and 4 nm, respectively for a Ga-polar sample with the [0 0 0 1] direction pointing to the right.

Polarization charge causes deviation from the square well and no screening due to free carriers is accounted for r. The scales for the vertical and horizontal axes are in terms of eV and nm, respectively. Courtesy of V. Litvinov. (c) The conduction band edge potential profile for a representative In0.3Ga0.7N/GaN single quantum well with a barrier and well thickness of 2 and 4 nm, respectively, for a Ga-polar sample with the [0 0 0 1] direction pointing to the right. Polarization charge causes deviation from the square well and no screening due to free carriers is accounted for. The scales for the vertical and horizontal axes are in terms of eV and nm, respectively. Courtesy of V. Litvinov.

2.11 Confined States

Al x Ga1-x N/GaN, x = 0.3, LB = 4 nm, L W = 2 nm

0.5

Energy (eV)

0.4 0.3 0.2 0.1 0.0 –4

–2

0

2

4

Distance (nm)

(b)

In x Ga

1–x

N/GaN, x = 0.3, LB = 4 nm, LW = 2 nm

1.4 1.2

Energy (eV)

1.0 0.8 0.6 0.4 0.2 0.0 –4 (c)

–2

0

2

4

Distance (nm)

Figure 2.21 (Continued )

The case of quantum wells in the nitride system requires not just confinement effects due to barriers but also polarization-induced charge. The latter tends to distort the band profile due to the induced electric field. The details of the polarization charge and its effect on the quantum wells can be found in Section 2.12.5 and Volume 2,

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j 2 Electronic Band Structure and Polarization Effects

222

Chapter 5. Essentially, the conduction and valence band edges get skewed due to the field induced by polarization, as shown in Figure 2.21a and b for representative Al0.2Ga0.8N/GaN (Ga-polar) and Al0.3Ga0.7N/GaN (Ga-polar) single quantum well structures, the former with barrier and well thicknesses of 2 nm (or 20 Å) each, with [0 0 0 1] direction pointing to the left, and the latter with barrier and well thickness of 2 and 4 nm and [0 0 0 1] direction pointing to the right. The same for a representative In0.3Ga0.7N/GaN single quantum well with a barrier and well thickness of 2 and 4 nm, respectively, for a Ga-polar sample with the [0 0 0 1] direction pointing to the right is shown in Figure 2.21c. The band profile depicts the case with no screening. Free carrier induced screening would alter the profile. The calculated eigenstates, only the ground levels exist, for Al0.1Ga0.9N/GaN and Al0.2Ga0.8N/GaN as a function of quantum well thickness for a barrier thickness of 2 nm are shown in Figure 2.22. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells discussed in Volume 3, Chapter 1. The calculated eigenstates for Al0.3Ga0.7N/GaN ground and the first excited states (only two states are available) for a barrier thickness of 4 nm as a function of quantum well thickness are shown in Figure 2.23. For increased activity in deeper UV devices, both

0.30

x = 0.1, level 1 x = 0.2, level 1 x = 0.2, level 2 LB = 2 nm

Alx Ga1–x N/GaN Dashed lines: without polarization

0.25

Energy (eV)

0.20

0.15

0.10

0.05

0.00 1

2

3

4

5

LW (nm) Figure 2.22 Calculated eigenstates, only the ground levels exist, for Al0.1Ga0.9N/GaN (lower curve) and Al0.2Ga0.8N/GaN (upper curve) for a barrier width of 2 nm as a function of quantum well thickness. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

6

7

2.11 Confined States

in terms of emitters and detectors, it is necessary to consider structures where AlxGa1
xN is used as the active part of the device in the form of a quantum well. Choosing AlN, arbitrarily, as the barrier for AlxGa1
xN, the eigenstates for two representative cases are presented. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells discussed in Volume 3, Chapter 1. Shown in Figure 2.24 are the ground and first excited states as a function of well width for an AlN/Al0.4Ga0.6N quantum having a barrier width of 2 nm. It should be noted that only the ground and first excited states are available. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells discussed in Volume 3, Chapter 1. The same structure but with AlN/Al0.5Ga0.5N, representing nearly the solar blind region of the Sun’s spectrum, is shown in Figure 2.25. Also shown in dashed lines are the energy levels without polarization, such as the case of a-plane (nonpolar surface) quantum wells discussed in Volume 3, Chapter 1. For the visible or nearly visible part of the spectrum, InGaN quantum wells typically with GaN barriers are employed. This is in part due to not only benefits gained by heterostructure devices but also technological reasons, the latter due to the decomposition of thick InGaN layers and inordinate amounts of ammonia required

0.45 0.40 0.35

AlxGa1–x N/GaN

level 1 level 2 L B = 4 nm, x = 0.3 Dashed lines: without polarization

Energy (eV)

0.30 0.25 0.20 0.15 0.10 0.05 0.00 1

2

3

4 LW ( nm )

Figure 2.23 Calculated eigenstates for Al0.3Ga0.7N/GaN for ground and the first excited states (only two states are available) for a barrier thickness of 4 nm as a function of quantum well thickness. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

5

6

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j 2 Electronic Band Structure and Polarization Effects

224

1.0

level 1 level 2 x = 0.4 L B = 2 nm

AlN/Al x Ga 1–x N Dashed lines: without polarization

Energy (eV)

0.8

0.6

0.4

0.2

0.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

L W (nm) Figure 2.24 Calculated ground and first excited states (only two states are available) as a function of well width for an AlN/ Al0.4Ga0.6N quantum having a barrier width of 2 nm. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

to grow thick layers. In this vein, the calculated ground state (for x ¼ 0.1) and ground and excited states (for x ¼ 0.2) as a function of well width for an InxGa1
xN/GaN quantum well having a barrier thickness of 2 nm are presented in Figure 2.26. 2.11.2 Valence Band

In an attempt to determine the valence band subband structure, Suzuki and Uenoyama [59] calculated the band discontinuities from first-principles calculations and found them to be 0.11 and 0.43 eV for the valence and conduction bands of GaN/ Al0.2Ga0.8N. The elastic stiffness constants, taken from prior experimental data, employed for GaN were, in units of 1011 dyn cm
2, 29.6, 13.0, 15.8, 26.7, and 2.41 for C11, C12, C13, C33, and C44, respectively (see Table 1.24). For illustrative purposes, the valence band structure in unstrained 30 and 50 Å GaN/Al0.2Ga0.8N quantum wells is exhibited in Figures 2.27a and 2.28 where the strain due to the lattice and thermal mismatch are neglected. Bandgap discontinuities of 0.11 and 0.43 eV were adopted for the valence and conduction bands, respectively. The confinement energies in deep wells are inversely proportional to the effective mass in the growth direction and

2.11 Confined States

1.4 AlN/Al x Ga1–xN 1.2

Dashed lines: without polarization

level 1 level 2 x = 0.5 LB = 2 nm

Energy (eV)

1.0

0.8

0.6

0.4

0.2

0.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

LW (nm) Figure 2.25 Calculated ground and first excited states (only two states are available) as a function of well width for an AlN/ Al0.5Ga0.5N quantum well having a barrier width of 2 nm, representing nearly the solar blind region of the Sun’s spectrum. Also shown in dashed lines are the energy levels without polarization such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

directly proportional to the square of the well length. The HH and LH bands are not coupled. Consequently, the HH band can be construed as parabolic with little, if any, change in strain. This is because the C6v crystal symmetry of the bulk remains. The upper pffiffiffi (LH) and lower (CH) bands are coupled with a constant coupling coefficient 2D3 , which means that coupling of these bands is independent of kz and strain. The effective masses of the HH and LH bands are too heavy to cause substantial confinement energy, whereas the CH band with a lighter mass causes more split due to quantization and, in a sense, makes the crystal splitting (Dcr) larger. For comparison, in ZB structures the coupling between the light hole and the spin–orbit band is dependent on kZ and thus the bands change considerably with strain, and the in-plane heavy-hole mass becomes light and the in-plane light-hole mass becomes heavy. The effect of strain on the quantum well was also considered by Suzuki and Uenoyama [60], assuming coherently strained quantum wells. Consequently, the inplane lattice constant of the layers is assumed to be that of the substrate with a substantial effect on the subband structure of the quantum wells. The beneficial effects of strain in electronic and optoelectronic devices based on ZB crystals have

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j 2 Electronic Band Structure and Polarization Effects

226

0.8 0.7

Dashed lines: without polarization

In x Ga1–x N/GaN

x = 0.1 x = 0.2, first x = 0.2, second L B = 2 nm

Energy (eV)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

2

3

4

5

6

LW (nm) Figure 2.26 Calculated ground state (for x ¼ 0.1) and ground and excited states (for x ¼ 0.2) as a function of well width for an InxGa1
xN/GaN quantum wells having a barrier width of 2 nm. For x ¼ 0.2, the ground and the first excited states are available. Also shown in dashed lines are the energy levels without polarization, such as the case on a-plane (nonpolar surface) quantum wells. Courtesy of V. Litvinov.

been documented well [88]. This and the fact that there is some lattice mismatch between GaN and its ternaries make it imperative that the effect of strain on the properties of Wz quantum wells be considered. Figures 2.27b and 2.28b exhibit the valence band subband structure for 30 and 50 Å GaN/Al0.2Ga0.8N quantum wells. The biaxial strain was assumed to be 0.5% in the (0 0 0 1) c-plane. Superimposed is the valence band structure for 30 and 50 Å Wz GaN/Al0.2Ga0.8N quantum wells with 0.5% tensile strain in the c-plane. Results indicate that the wells get deeper for compressive strain and shallower for tensile strain. The density of states at the valence band maximum gets smaller for the compressive and larger for tensile strain. However, the change is very small as the symmetry remains as in the bulk with no further removal of the degeneracy. It must be mentioned though that uniaxial strain along the x- or the y-direction only causes the HH band to move to a higher energy and leads to a reduced density of states. Consequently, the effects of this type of strain resemble those in bulk GaN. In a sense, the Q well alone does not result in any special characteristic that would lead to much improved results for lasers [429]. Valence band confinement energies for relaxed and 0.5% compressively strained GaN/Al0.2Ga0.8N quantum wells in the c-plane are depicted in Figure 2.29. The data

2.11 Confined States

50

L = 30 Å

100

No strain

50

0

–50

0 Energy (meV)

Energy (meV)

HH1 LH1 LH2 –100 HH2 –150

–200 0

–50

L = 30 Å

j227

With 0.5 % compressive strain HH1

LH1

LH2 HH2

–100 –150

20 10 15 5 kx ,ky ,Wave number (× 106 cm –1)

–200 0

(a)

20 10 15 5 kx ,ky ,Wave number (× 10 6 cm –1) (b)

Figure 2.27 Plot of the upper valence band structure (HH and LH bands) in a 30 Å GaN/Al0.2Ga0.8N quantum well with bandgap discontinuities of 0.11 and 0.43 eV for the valence and conduction bands, respectively; (a) without strain and (b) with 0.5 compressive strain in the c-plane [60].

have been deduced from the calculations of Suzuki and Uenoyama [59] for 30, 40, 50, and 60 Å well thicknesses and band discontinuities of 0.11 and 0.43 eV for the valence and conduction bands, respectively. The other parameters utilized can be found in the tables presented in Chapter 1. 2.11.3 Exciton Binding Energy in Quantum Wells

Exciton binding energies in reduced dimensional systems vary from those of the bulk [430]. In the GaN/AlGaN system, one would expect the binding energy to go up, as the confinement gets stronger. If the well thickness is continually reduced, at some point the overlap with AlGaN becomes very noticeable. Toward zero well thickness, the binding energy should approach the value of AlGaN. Considering the strong localization and possible role of excitons in optical processes even at room temperature, it is imperative that the binding energy be known. Bigenwald et al. [431] considered the very problem of exciton binding energy in a GaN/Al0.2Ga0.8N system with results leading to the obvious conclusion that the effect of confinement is large and cannot be ignored. They calculated the exciton binding energies and oscillator strengths with the formalism developed [432] prior to the aforementioned investigation by applying a two-parameter trial function. Due to the anisotropy of the structure, the dielectric constant was globalized and the particle masses were weighted with the

j 2 Electronic Band Structure and Polarization Effects

228

50

100

L = 50 Å

Strain free

L = 50 Å HH1

50

0

0.5 % Compressive strain LH1

E n e r g y (m e V )

HH1 –50 HH2 –100 LH1

0

LH2

HH2

–50 –100

HH3

LH2

–150

–150 –200 0

–200 0

5

10

15

20

5

10

15

20

k x ,k y ,Wave number (× 106 cm–1)

kx,ky,Wave number (× 106 cm–1) (a)

(b)

Figure 2.28 Plot of the upper valence band structure (HH and LH bands) in a 50 Å GaN/Al0.2Ga0.8N quantum well with bandgap discontinuities of 0.11 and 0.43 eV for the valence and conduction bands, respectively; (a) without strain and (b) with a compressive strain of 0.5 in the c-plane [60].

0

HH LH CH

100

CH 1

Strained b ul k GaN

50

LH 2 Bulk Al 0.2Ga 0.8N

C onfin em ent en ergy (m eV)

LH 1

HH 2 CH 2

( 9 )1 HH 1 ( 9 )2 HH 2

HH 1

150 HH LH CH

200 0

50 Well width (Å)

Figure 2.29 Valence band confinement energies versus the thickness of the well in GaN/Al0.2Ga0.8N quantum wells. Courtesy of Professor Bernard Gil.

100

(

) 71 1

LH 1

(

71 )2

LH 2

( (

) CH 1 72 1 ) 72 2 CH 2

2.11 Confined States

j229

40

Exciton binding energy (meV)

35 B Exciton A Exciton 30

25

C Exciton 20

15 0

20

40

60

80

GaN Q Well thickness (Å) Figure 2.30 A, B, and C exciton binding energies as a function of the well width in a GaN/ Al0.2Ga0.8N system. Note that the exciton binding energies reported for GaN range from about 20 to 30 meV. The likely value of A exciton

binding energy in GaN is 21 meV. If so, the absolute values of the binding energies shown should be treated with caution. However, the trend with quantum well thickness holds, which is the reason for inclusion of this figure [431].

probability densities. The A, B, and C exciton binding energies computed as a function of GaN well thickness are shown in Figure 2.30. Two essential points stand out. The first point is that the G9v and G17v states are confined in the well as is the electron state. The largest binding energy corresponds to a well thickness of L  15 Å, which is when the spreading of both electron and hole functions in the barrier area minimal. The second point is that the relatively light G27v hole state leaks out of the well (for L < 100 Å) so that the electron–hole pair has a small binding energy and is almost constant for 20 < L < 100 Å. Caution should be exercised in applying the calculations of Ref. [432] for well thicknesses larger than about 100 Å, three times the Bohr radius. In GaN-based systems, the terms quantum well and superlattices have been used very liberally in that structures with well thicknesses well in access of the Bohr radius are referred to as quantum wells. The term superlattice requires that barriers are penetrable by the wave function, and further, the wave functions in adjoining wells overlap and form the superlattice minibands. If these standards are strictly applied at this point in time, there may not be much to discuss. Consequently, a conscious decision was made to treat many heterostructures with reduced dimensions, in at least one direction, as “quantum-confined structures.”

100

j 2 Electronic Band Structure and Polarization Effects

230

2.12 Polarization Effects

Solids are different from vacuum in that they respond to electric fields present or applied. There are three forms of polarization that are present at the atomic level [2]. One is due to partial or complete alignment of dipole moments of polar molecules with the electric field. When atoms forming the solid are different, as in binary, ternary, and quaternary semiconductors, and they have different electronegativities, any asymmetrical molecule has a permanent dipole moment, a process referred to as dipole orientation or paraelectric response. This component cannot keep pace with varying electric field above about 1010 Hz, causing a drop in the real part of the dielectric constant and a jump in the imaginary part (the loss part – appreciable loss factor). The loss is caused by dipoles attempting to respond to the field but seriously lagging in phase. At higher frequencies, the dipoles cannot follow the field and the effect is negligible including the loss factor. Electric field paves the way for rotation of dipole to participate in the electric displacement and align the dipoles collectively, barring thermal disturbances. In a completely or partially ionic solid, dipoles can be induced by relative motion of positive and negative ions under the influence of electric field, causing what is termed as the ionic polarization. Similar to the case of dipole orientation, this process cannot respond to frequencies above 1013 Hz (Reststrahlen frequency), causing yet an additional drop in the dielectric constant and a surge in the loss factor due to the phase lag. Again, as the frequency is increased further, above the Reststrahlen frequency, ionic motion cannot respond to the field and the loss factor due to this process diminishes. The third kind, which occurs in every dielectric, is called the electronic polarization. This is caused by displacement of electrons in an atom relative to the nucleus under the influence of electric field, in a sense deforming the electron shells. The electronic polarization remains at frequencies above Reststrahlen frequency, making the real part of the dielectric constant larger than unity. This process too cannot follow the field above 1015 Hz, above which the dielectric constant of the solid becomes very close to unity. Group III–V nitride semiconductors exhibit highly pronounced polarization effects. Semiconductor nitrides lack center of inversion symmetry and exhibit piezoelectric effects [209] when strained along h0 0 0 1i. Piezoelectric coefficients in nitrides are almost an order of magnitude larger than in many of the traditional group III–V semiconductors [209,433–437]. The strain-induced piezoelectric and spontaneous polarization charges have profound effects on device structures. The piezoelectric effect has two components. One is due to lattice mismatch (misfit) strain while the other is due to thermal strain (ts) caused by the thermal expansion coefficient difference between the substrate and the epitaxial layers. The low symmetry in nitrides, specifically, the lack of center of inversion symmetry present in zinc blende structure, may be interpreted as some sort of nonideality, which is not the case. Nonvanishing spontaneous polarization is allowed in an ideal wurtzite structure [212,438]. This spontaneous polarization is noteworthy, particularly when heterointerfaces between two nitride semiconductors with varying electronegativity are involved. This manifests itself as a polarization charge at heterointerfaces.

2.12 Polarization Effects

Spontaneous polarization was only understood fully not too long ago by King-Smith and Vanderbilt [439] and Resta et al. [440]. In heterojunction devices such as modulation-doped field effect transistors (MODFETs) where strain and heterointerfaces are present, the polarization charge is present and is inextricably connected to free carriers, which are indeed present. As such, polarization charge affects device operation in all nitride-based devices, particularly HFETs, and thus must be taken into consideration in device design unless nonpolar surfaces such as the a-plane are used. The quality of films on nonpolar surface has not kept pace with those on polar basal plane, which make the topic of discussion quite relevant. As mentioned above, polarization charge arises from two sources: piezoelectric effects and the difference in spontaneous polarization between AlGaN and GaN, even in the absence of strain. These charges exit in all compound semiconductors to varying degrees unless self-cancelled by the symmetry of the particular orientation under consideration such as the nonpolar surfaces/interfaces. In relative terms, spontaneous polarization is larger than the piezoelectric polarization in AlGaN/GaN-based structures. In the case of InGaN/GaN structures, spontaneous polarization is relatively small but not as small as the earlier predictions called for, but still noteworthy, as spontaneous polarizations in GaN and InN are not as different from one another. However, the strain-induced piezoelectric polarization can be sizable. If and when defect-associated relaxation occurs reducing the strain in the films, the strength of the piezoelectric polarization is lowered. Spontaneous polarization and piezoelectric polarization affect the band diagram of heterostructures. The effects are very large and can easily obscure the engineered designs. Polarization is dependent on the polarity of the crystal, namely, whether the bonds along the c-direction are from cation sites to anion sites or visa versa. The convention is that the [0 0 0 1] axis points from the face of the N-plane to the Ga-plane and marks the positive z-direction. In other words, when the bonds along the c-direction (single bonds) are from cation (Ga) to anion (N) atoms, the polarity is said to be the Ga-polarity, and the direction of the bonds from Ga to N along the c-direction marks the [0 0 0 1] direction, which is generally taken to be the þz-direction. By a similar argument, when the bonds along the c-direction (single bonds) are from anion (N) to cation (Ga) atoms, the polarity is said to be the N-polarity, and the direction of the bonds from N to Ga along the c-direction marks the direction, which is generally taken to be the
z-direction. To shed further light, the Ga-polarity means that if one were to cut the perfect solid along the c-plane where one breaks only a single bond, one would end up with a Ga-terminated surface. The Ga- and N-polarity of a model GaN crystal is shown in Figure 1.3. A schematic representation of the spontaneous polarization in a model GaN/AlN/GaN wurtzitic crystal is shown in Figure 2.31. The spontaneous polarization Pspont (also commonly referred to as P0) in a solid has not always been well defined, although much better understanding of it has been emerging. Only those differences in P between two phases that can be linked by an adiabatic transformation that maintains the insulating nature of the system throughout are well defined. For example, one phase can be considered unstrained and the other strained. Vanderbilt proved that the polarization difference DP between the

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j 2 Electronic Band Structure and Polarization Effects

232

[0 0 0 1] Axis

P0AlN

P0GaN N

N

Ga

Ga

Al

Al

N

N

N

Al

Al

N

N

Ga

Ga

N

P0

P0 GaN

P0GaN

P0AlN

AlN

Figure 2.31 Schematic depicting the convention used for determining the polarity and crystalline direction in wurtzitic nitride films. The diagram shows the case for a Ga-polarity film with its characteristic bonds parallel to the c-axis (horizontal in the figure) going from the cation (Ga or Al) to the anion (N). The spontaneous polarization components P0Ga and P0AlN for a periodic GaN/AlN structure are also indicated with that for AlN having a larger magnitude. The

GaN

spontaneous polarization is negative and thus points in the [0 0 0
1] direction. Caution must be exercised here as there is no long-range polarization field, just that it is limited to the interface. The polarization in AlN is larger in magnitude than in GaN. There exists a difference in polarization at the interface, DP0 pointing in
direction for both GaN/AlN the [0 0 0 1] interfaces. The Born factor is defined in Equation 2.82 [433].

wurtzite and zinc blende phases could be calculated by considering an interface between the two phases and by defining Pspont to be zero in the zinc blende phase. In short, by calculating the integral of a quantum mechanical Berry phase along a line in the Brillouin zone from one end to the other in the bulk wurtzite symmetry leads to polarization P with respect to that in zinc blende (which is zero by definition because zinc blende is cubic and cannot have a spontaneous polarization in an infinite bulk periodic crystal). The Berry phase actually represents an overlap integral between the periodic part of the Bloch function at k and a neighboring k-point, k0 . Zorroddu et al. [212] and Bernardini et al. [433] showed that the charges accumulating at each interface in a self-consistent calculation can be obtained from the DP of the two bulk layers forming the heterointerface. The relation between the charge and P follows basically from Gauss’s law. The bound charge density rb ¼
ÏP. This means that across an abrupt interface with P1 on one side and P2 on the other side, one gets P P2
P 1 ¼ i rs (surface charge density at the interface with the appropriate signs). Even though it is overly simplistic, a graphical picture of polarization due to strain (piezo component) and heterointerfaces (spontaneous component), the latter is in the case of different ionicity, can be obtained, which is helpful. Shown in Figure 2.32 is a ball-and-stick diagram of a tetrahedral bond between Ga and N in Ga- and N-polarity configurations, showing the polarization vector due to the electron cloud being closer to the N atoms. Actually, the cumulative polarization due to the triply bonded atoms is along the direction of the single bond. The in-plane and vertical components of polarization due to pairs of atoms cancel one another if the tetrahedron is ideal.

2.12 Polarization Effects

Ga Polarity

[0 0 0 1]

N Polarity

N

[0 0 0 1]

P0 P0 P0

P0zr

P0

P0

Ga

Ga P0

P0zr

P0

P0

N

Ga

P0

N

j233

Ga

P0

N

Ga

Figure 2.32 Ball-and-stick configuration of an ideal GaN tetrahedron with proper c/a ratio and internal parameter u for both Ga and N polarity in a relaxed state.

However, when a Ga-polarity film is under homogeneous in-plane tensile strain, the cumulative z-component, [0 0 0 1] direction, of the polarization associated with the triple bonds decreases, causing a net polarization that would be along the ½0 0 0
1 direction, as shown Figure 2.33. In a nitrogen-polarity film, the same occurs except that the net polarization would be in the opposite, [0 0 0 1], direction. When an inplane and homogeneous compressive strain is present, the net polarization would be in the [0 0 0 1] direction in the Ga-polarity case and ½0 0 0
1 direction in the N-polarity case, as shown in Figure 2.34. Ga Polarity

[0 0 0 1]

N Polarity

N

P0

P0 P0 P0

P0zr

P0

P0 Ga

N

[0 0 0 1]

F

P0

P0

N

N

Ga

Ga P0 Ga

Figure 2.33 Ball-and-stick configuration of a GaN tetrahedron for both Ga and N polarity with a homogeneous in-plane tensile strain showing a net polarization in the [0 0 0 1] direction for Ga-polarity and [0 0 0
1] polarization for N polarity.

P0

P0

P0zr

P0

Ga

F

j 2 Electronic Band Structure and Polarization Effects

234

Ga Polarity

[0 0 0 1]

N Polarity

N

[0 0 0 1]

Ga

P0 P0 P0

P0zr

P0

Ga P0

N N

P0

P0 P0zr

P0

P0

N

Ga Ga

P0

Ga Figure 2.34 Ball-and-stick configuration of a GaN tetrahedron for both Ga and N polarity with a homogeneous in-plane compressive
direction for Ga strain showing a net polarization in the [0 0 0 1] polarity and [0 0 0 1] polarization for N polarity.

The same graphical argument can be used to attain a mental image of spontaneous polarization at heterointerfaces as well. For example, if we were to construct two tetrahedra, one representing a GaN bilayer and another on top of it representing an AlN bilayer, the top N atom shown in Figure 2.32 for the Ga-polarity configuration would make triple bonds with it. Because AlN is more electronegative than GaN, the net component of the polarization vector in the [0 0 0 1] direction in triply bonded N with Al is larger in amplitude than in the GaN tetrahedron, and there would be a net interfacial polarization in the ½0 0 0
1 direction even without strain. In short, the source of piezoelectric polarization is strain in an electronegative binary. That for spontaneous polarization is the change in electronegativity across an interface such as the AlN and GaN interface. Substrates upon which nitride films are grown lack the wurtzitic symmetry of nitrides. Consequently, the polarity of the films may not be uniform, as schematically depicted in Figure 4.21 where the section on the left is of Ga-polarity and the section on the right is of N-polarity, representing a Holt-type inversion domain. In this type of inversion domain, the wrong type of bonds, for example, Ga
Ga and N
N, are formed at the boundary projected on (1 1
2 0) plane. The structural and electrical details of the inversion domains observed and investigated in GaN can be found in Section 4.1.3. Inversion domains combined with any strain in nitride-based films lead to flipping piezoelectric fields with untold adverse effects on the characterization of nitride films in general and the polarization effect in particular, and on the exploitation of nitride semiconductors for devices. Such flipping fields would also cause much increased scattering of carriers, as they traverse in the c-plane. Having made the case, it should be mentioned that if proper measures are taken, the Gapolarity films grown by OMVPE and to a lesser extent by MBE are nearly or completely inversion domain boundary free even on sapphire substrates. However,

2.12 Polarization Effects

in the case of MBE, unless optimum AlN buffer layers are employed or N-polarity films are grown by incorporation of GaN initiation layers, inversion domain boundaries do occur, sometimes in high concentrations, see Section 3.5.6. The magnitude of the polarization charge converted into number of electrons can be in the mid-1013 cm
2 level for AlN/GaN heterointerfaces, which is huge by any standard. For comparison, the interface charge in the GaAs/AlGaAs system is used for MODFETs in less than 10% of this figure. An excellent review of the polarization effects can be found in Ref. [440]. The magnitude of the polarization charge is tabulated in Table 2.25 along with elastic coefficients taken from a series of publications by Bernardini, Fiorentini, and Vanderbilt. The data in bold are those reported in an earlier publication [433] and the remaining data points are taken from a later publication [212]. Following the initial reports of piezoelectric and spontaneous polarization [441], the authors returned to the topic [212] as the values of the initial parameters were not consistent with other reports [181]. Bernardini et al. [442] reanalyzed the polarization as obtained using the Berry phase method within two different DFT exchange correlation schemes. Specifically, the authors used the Vienna ab initio simulation package (VASP) and the pseudopotentials provided therewith, as in Ref. [180]. The newer calculations were carried out using both the generalized gradient corrected local density approximation (GGA) to density Table 2.25 Elastic constants and spontaneous polarization charge

in nitride semiconductors.


2

e33 (C m ) LDA GGA e31 (C m
2) LDA GGA p e31 LDA GGA C33 (GPa) GGA C31 (GPa) GGA P0 (C m
2) LDA GGA P0 (C m
2), ideal wurtzite structure R ¼
C31/C33 LDA GGA e31
(C13/C33)e33 (C m
2)

AlN

GaN

InN

1.46 1.8 1.5
0.60
0.64
0.53
0.74
0.62 377 94
0.081
0.10
0.090
0.032
0.578
0.499
0.86

0.73 0.86 0.67
0.49
0.44
0.34
0.47
0.37 354 68
0.029
0.032
0.034
0.018
0.40
0.384
0.68

0.97 1.09 0.81
0.57
0.52
0.41
0.56
0.45 205 70
0.032
0.041
0.042
0.017
0.755
0.783
0.90

The data in bold are associated with DFT in the generalized gradient approximation (GGA) that are more accurate than others reported prior. Moreover, the resultant predictions are in relatively better agreement with experimental data as well as the bowing parameters observed in polarization charge in alloys [212,442]. e31 and e33 are piezoelectric constants. C m
2 is Coulomb p per square meter. C31 and C33 are elastic stiffness coefficient or elastic constants. e31 is the proper piezoelectric constant. The data in bold, which are from Refs [212,442], are recommended. The remaining data are from Refs [212,433,442].

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j 2 Electronic Band Structure and Polarization Effects

236

functional theory in the Perdew–Wang PW91 version, and the LDA in the Ceperley– Alder–Perdew–Zunger form. For a glossary of these approximations within the DFT method, the reader is referred to Section 2.15. Ultrasoft potentials were used treating Ga and In d-electrons as valence at a conservative cutoff of 325 eV. Finally, the reciprocal space summation was done on a (8 8 8) Monkhorst–Pack mesh. The results of the refined GGA calculations in terms of spontaneous polarization, piezoelectric, and elastic constants so calculated are tabulated in Table 2.25 along with earlier calculations. The two sets of data, the earlier and refined, are within 10% agreement. The results of more refined LDA calculations are also provided. For deference to earlier work, the piezoelectric constants, albeit an incomplete list dealing with e14, for GaN were estimated theoretically for cubic GaN [443] and used in early investigation of piezoelectric effects in GaN [444] and deduced from the mobility data [445], which is indirect particularly in samples containing many scatterers. For comparison, the interface charge in the GaAs/AlGaAs system is used for MODFETs (or HFET) in less than 10% of this figure. An excellent review of the polarization effects can be found in Ref. [446]. 2.12.1 Piezoelectric Polarization

In a polarizable medium, the displacement vector can be expressed in terms of two components due to the dielectric nature of the medium and the polarizability nature of the medium as [447] !

!

!

!

!

!

D ¼ eE þ 4pP in cgs and D ¼ eE þ P in mks units; !

!

!

$ !

$

!

ð2:60Þ

where E and P represent the electric field and polarization vectors. Considering only the piezoelectric component, the piezoelectric polarization vector is given by [448] PPE ¼ e e ;

ð2:61Þ

where e and e are the piezoelectric and the stress tensors. To gain a quantitative understanding of the piezoelectric polarization, the piezoelectric tensor, which is defined as the derivative of the polarization with respect to strain, must be considered. In hexagonal P63mc symmetry, piezoelectric polarization is related to strain through the piezoelectric tensor (ei,j) as [449] 2 3 exx 36 eyy 7 2 3 2 7 Px 0 0 0 0 e15 0 6 6 ezz 7 5 4 Py 5 ¼ 4 0 6 7: 0 0 e24 0 0 6 ð2:62Þ eyz 7 7 Pz 0 0 6 e31 e31 e33 0 4 exz 5 exy Note that e24 ¼ e15 for hexagonal symmetry that reduces to 2 3 2 3 e15 exz Px 4 Py 5 ¼ 4 ðe24 ¼ e15 Þeyz 5; e31 ðexx þ eyy Þ þ e33 ezz Pz

ð2:63Þ

2.12 Polarization Effects

where Pi, eij, and eij represent the electric polarization, electric piezoelectric coefficient, and strain, respectively. It is clear from Equation 2.63 that the piezoelectric polarization along the ½1
100mdirection Px ¼ e15exz ¼ 0, along the ½11
20a-direction Py ¼ e15eyz ¼ 0, and along the [0 0 0 1] c-direction Pz ¼ e31(exx þ eyy) þ e33ezz. The strain components given in Equation 2.7 are repeated here for convenience, without shear and eyz ¼ ezx ¼ exy ¼ 0. If only a biaxial strain is present, the in-plain strain can be calculated by relative difference in the in-plane lattice constants of the epitaxial layer and template (buffer or substrate) through a
a0 : ð2:64Þ exx eyy e11 e22 ¼ a0 Here, abuffer a0 and aepi a represent the relaxed (equilibrium) in-plane lattice constants of the buffer layer or the substrate, depending on layers and their thicknesses, and of the epitaxial layer of interest, the strained epitaxial layer, respectively. The nomenclature as or a0 for the substrate (buffer) and ae or a for the epitaxial layer is also used. The expression for the out-of-plane strain is e33 ezz ¼

c
c0 : c0

ð2:65Þ

Similarly, c0 and c represent the relaxed and the out-of-plane lattice parameters, which would correspond to the buffer layer and epitaxial layer, respectively. In case the in-plane strain is anisotropic, e11 6¼ e22. As it may have already become obvious, the nomenclature used in the literatures for parameters surrounding strain and piezoelectric and elastic constants vary in that, for example, exx and e11 are interchangeably used. The two-way transformation of x 1, y 2, z 3 can be used to convert from one nomenclature to the other. Likewise, PPE, pz pz PPE, PPE are commonly used in literature interchangeably to depict 3 , P 3 , and P piezoelectric polarization. If the subscript 3 is also used as in P PE 3 , it specifically indicates that for biaxial in-plane strain, the only nonvanishing polarization is along the c-axis. Even in cases where the subscript is not employed, the underlying assumption is that the polarization is along the c-direction, as growth of nitride semiconductor structures is performed predominantly on the basal plane. In the same vein, P0, Psp, and Psp indicate spontaneous polarization along the c-axis. DPsp and DP0 both represent differential spontaneous polarization at a heterointerface such as an AlN/GaN interface, a topic which follows the discussion of piezoelectric polarization. The components of the piezoelectric polarization tensor given by Equation 2.62 pz can be expressed in terms of a summation, using Pi instead of PPE, as X pz Pi ¼ eij ej with i ¼ 1; 2; 3 and j ¼ 1; . . . ; 6; ð2:66Þ j pz

where P i is the ith component of the piezoelectric polarization. The wurtzite symmetry reduces the number of independent components of the elastic tensor e to three, namely, e15, e31, and e33. The third independent component of the piezoelectric tensor, e15, is related to the polarization induced by a shear strain that is not applicable to the epitaxial growth schemes employed. The index 3 corresponds

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j 2 Electronic Band Structure and Polarization Effects

238

to the direction of the c-axis. It is clear that the piezoelectric properties of the Wz structures are somewhat more complicated. If we restrict ourselves to structures with growth along the [0 0 0 1] direction or z-direction, or along the c-axis, only the e31 and e33 components need to be considered. The piezoelectric polarization in the [0 0 0 1] direction can be obtained by setting i ¼ 3. The electric polarization component in the c-direction, which is designated by z- in the above nomenclature, is given by Pz ¼ e31 exx þ e31 exx þ e33 ezz ¼ 2e31 exx þ e33 ezz :

ð2:67Þ

For isotropic basal plane strain, the strain components exx e?/2 and thus Equation 2.67 can be written as pe

Pz P 3 ¼ e31 e? þ e33 ezz :

ð2:68Þ

In hexagonal symmetry, strain in the z-direction can be expressed in terms of the basal plane strain e? through the use of Poisson’s ratio, which is expressed in terms of the elastic coefficients Cij as ezz ¼
2(C13/C33)exx ¼
(C13/C33)e? (see Equations 2.7, 2.14–2.16 for specifics). In the case of externally applied pressure in addition to mismatch strain, the out-of-plane strain can be related to the in-plane strain through ezz ¼
[(p þ 2C13exx)]/C33, where p is the magnitude of compressive pressure (in the same unit as the elastic coefficients). In terms of the nomenclature again, it should also be noted that e1 e11 exx and e3 e33 ezz in the other notation used in the literature and also in this text.   C13 Pz ¼ e31
e33 ð2:69Þ e? : C33 pz

3 The z-component of the electric polarization is also referred to as P PE 3 , P 3 , and P pz . Piezoelectric polarization is also described in terms of piezoelectric moduli in the literature, which is treated here for completeness. In terms of piezoelectric moduli, dij, which are related to the piezoelectric constants by X eij ¼ dik Ckj with i ¼ 1; 2; 3 and j ¼ 1; . . . ; 6 and k ¼ 1; . . . ; 6: ð2:70Þ k

Using Equation 2.70 in Equation 2.66 and strain–stress relationship (stress is equal to the product of elastic constant and strain in a tensor form), the piezoelectric polarization can be expressed in terms of piezoelectric moduli as X pz dij sj with i ¼ 1; 2; 3 and j ¼ 1; . . . ; 6: ð2:71Þ Pi ¼ j

Symmetry considerations lead to d31 ¼ d32, d15 ¼ d24 and all other components dij ¼ 0, and thus Equation 2.71 reduces to a set of three equations: 1 1 pz pz P1 ¼ d15 s5 ; P2 ¼ d15 s4 ; 2 2

pz

and P 3 ¼ d31 ðs1 þ s2 Þ þ d33 s3 :

ð2:72Þ

For biaxial strain, which is the case with epitaxial layers, additional conditions are imposed in that s1 ¼ s2, s3 ¼ 0. Moreover, the shear stresses are negligible, which leads to s4 ¼ s5 ¼ 0. Consequently, in cases primarily applicable to epitaxial layers grown along the c-direction, the piezoelectric polarization is left with only one

2.12 Polarization Effects

nonvanishing component, which is in the growth direction and is given by, using Equation 2.72, pz

P3 ¼ 2d31 s1 :

ð2:73Þ

Utilizing stress–strain relationship   C2 s1 ¼ e1 C 11 þ C12
2 13 ; C33

ð2:74Þ

we obtain   C2 pz P3 ¼ 2d31 e1 C 11 þ C12
2 13 ; C33

ð2:75Þ

pz

where P 3 represents the piezoelectric polarization along the c-direction and similar to Equation 2.69, which expresses the same in terms of piezoelectric constants as opposed to piezoelectric moduli. For hexagonal crystals, the relations between piezoelectric constants and piezoelectric moduli expressed in Equation 2.70 can be reduced to e31 ¼ e32 ¼ C11 d31 þ C12 d32 þ C13 d33 ¼ ðC11 þ C 12 Þd31 þ C13 d33 ; e33 ¼ 2C13 d31 þ C33 d33 ; e15 ¼ e24 ¼
C 44 d15 ; eij ¼ 0 for all other components:

ð2:76Þ

The nonvanishing component of the piezoelectric polarization of Equation 2.66 due to only the biaxial strain is for i ¼ 3 and is given by, which recovers Equation 2.67, pz

P3 ¼ e1 e31 þ e2 e32 þ e3 e33 ¼ 2e1 e31 þ e3 e33 ¼ e? e31 þ e3 e33 because e31 ¼ e32 : ð2:77Þ Steps can be taken to express Equation 2.77 in the form of Equations 2.68 and 2.69, which are straightforward and therefore not presented. We have so far focused on lattice mismatch induced strain. However, the thermal expansion coefficients of the layers used to compose many of the heterostructures are different, which upon cooling from growth temperature could lead to thermalinduced strain. A larger effect in this vein, however, is that caused by differences in the thermal expansion coefficient between the substrate and epitaxial stack used. In that case, the piezo component would have two parts, namely, the lm or misfit strain and by the thermal strain leading to Ppe ¼ Plm þ Pts. Another issue that must be considered is that the electric field induced due to strain (piezoelectric field) in adjacent layers of a heterostructure comprised of A and B (i.e., A for AlGaN, B for GaN, and A,B for AlGaN/GaN) is sp

pe

E A;B ¼ E A;B þ E A;B :

ð2:78Þ

If, for example, A is composed of a ternary, then the linear interpolation for both sp and pe polarizations can be used to a first extent. Again, the nonlinearities are discussed later on in Section 2.12.3.

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j 2 Electronic Band Structure and Polarization Effects

240

Another relevant but not discussed nearly as much topic is that the properties that cause the piezoelectric polarization can also lead to pyroelectric effects. Such phenomena can be rather important in nitride-based devices, as the junction temperature is high by the nature of the applications such as lasers and high-power amplifiers. Consequently, the thermally induced electric field, pyroelectric effect, would most likely be present [450] with consequences similar to those ascribed to polarization effects. If one considers the distortion of the u parameter as well, the piezoelectric polarization can be expanded as qP 3 ¼

qP3 qP3 qP3 ða
a0 Þ þ ðc
c 0 Þ þ ðu
u0 Þ: qa qc qu

ð2:79Þ

The internal u parameter is defined as the average value of the projection of the connecting vector of a nitrogen atom with its first neighbor in the ð0 0 0
1Þ direction along this same direction. The three parameters, a, c, and u, are not independent of each other. If the partial derivatives in Equation 2.79 are known, following Ref. [209], one can write for the two piezoelectric constants qP3 4qc 0 qu þ pffiffiffi 2 Z  ; qc qc 3a0

ð2:80Þ

e31 ¼

a0 qP 3 2q qu þ pffiffiffi Z ; 2 qa qa 3a0

ð2:81Þ

Z ¼

pffiffiffi 2 3a0 qP3 ¼ ZT3 ; 4q qu

ð2:82Þ

e33 ¼ c 0

where

is the axial component of the Born, or the transverse component of the charge tensor ZT3 . Various structural parameters that are useful in treating the polarization issue in nitride semiconductors are tabulated in Table 2.26.

Table 2.26 Structural parameters of wurtzitic AlN, GaN, and InN

reported in Ref. [209] and updated with DFT in the GGA approximation in Ref. [212]. c0/a0

a0 (Å) Ref. [209] GGA [212] Exp. AlN 5.814 GaN 6.040 InN 6.660

3.1095 3.1986 3.5848

u0

Ref. [209] GGA [212] Exp.

3.1106 1.6190 3.1890 1.6336 3.538 1.6270

1.6060 1.632 1.6180

Ref. [209] GGA [212] Exp.

1.6008 0.380 1.6263 0.376 1.6119 0.377

0.3798 0.3762 0.377

0.3821 0.377

It should be mentioned that GGA produced data for structural as well as polarization-related parameters, see Table 2.25, are in better agreement with refined experimental data.

2.12 Polarization Effects

In Equation 2.80, it is implicit that the vector connecting the cation with the anion has a modulus uc associated with the internal cell parameter and points in the direction of the c-axis. The first term in Equations 2.80 and 2.81 signifies the term called the clamped-ion term, and represents the effect of the strain on the electronic structure. The second term represents the effect of internal strain on the polarization. The derivatives of u with respect to c and a in Equations 2.80 and 2.81 are related to the strain derivatives of u through c0du/dc ¼ du/de3 and a0du/da ¼ 2du/de1. In addition to binaries, the nitride heterojunction system utilizes ternary and to a lesser extent quaternary alloys as well. Knowing the piezoelectric parameters of the end binary points is generally sufficient, to a first order, to discern parameters for more complex alloys. For example, in the case of AlxGa1
xN, the piezoelectric polarization vector expression, using linear interpolation within the framework of Vegard’s law, can be described as [448] $

$

!

Ppe ¼ ½x e AlN þ ð1
xÞ e GaN e ðxÞ:

ð2:83Þ

The same argument can be extended to piezoelectric polarization in quaternary alloys such as AlxInyGa1
x
yN in a similar fashion as $

$

$

$

!

Ppe ¼ ½x e AlN þ ð1
xÞ e InN þ ð1
x
yÞ e GaN e ðxÞ:

ð2:84Þ

The linear interpolation is very convenient and does give reasonably accurate values. However, as will be discussed later in this section, while the Vegard’s law applies to the alloys, the polarization charge itself is not a linear function of composition [220,221]. 2.12.2 Spontaneous Polarization

Spontaneous polarization calculated for the binary nitride semiconductors are tabulated in Table 2.25. For ternary and quaternary alloys, the simplest approach is to use a linear combination of the binary end points, taking into account that the mole fraction can be used under the auspices of the Vegard’s law. However, this linear interpolation falls short of agreeing with the experimental variation of spontaneous polarization with respect to the mole fraction. Consequently, nonlinear models have been developed that are discussed later in Section 2.12.3. For now, the linear interpolation is applied for simplicity in which the spontaneous polarization in quaternary alloys such as AlxInyGa1
x
yN can be expressed as
sp þ yP
sp
sp
sp ðx; yÞ ¼ x P P InN þ ð1
x
yÞP GaN : AlN

ð2:85Þ

The ternary cases can be obtained by simply setting either x or y to zero. This is again predicated on the assumption that polarization charge obeys Vegard’s law, as shown below. Later in this section, the nonlinearity involved is discussed. Using the GGA calculation results for the spontaneous polarization and linear interpolation, as in Equation 2.95, for ternaries one gets

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j 2 Electronic Band Structure and Polarization Effects

242

sp

PAlx Ga1
x N ¼
0:09x
0:034ð1
xÞ; sp PInx Ga1
x N ¼
0:042x
0:034ð1
xÞ; sp PAlx In1
x N ¼
0:09x
0:042ð1
xÞ:

ð2:86Þ

The total polarization charge inclusive of spontaneous and piezoelectric must be considered in dealing with heterojunctions such as quantum wells, quantum wires, and quantum dots. The case of the quantum wells and dots are treated in Volume 2, Chapter 5. The case of the modulation-doped structures is treated in Volume 3, Chapter 3. Therefore, the total polarization involving layers A (AlxGa1
xN) and B (GaN) in contact is Ptotal ¼ P sp þ Ppe :

ð2:87Þ 


2

Spontaneous polarization P (C m ), Born Z or the transverse component of the charge tensor ZT3, piezoelectric constants (C m
2), elastic constants (GPa), and the ratio R ¼
2C31/C33 of wurtzitic nitrides, as obtained in the LDA and GGA approximation [212] are tabulated in Table 2.27. Also, tabulated following the elastic ðpÞ constants e33 and e31 is e31 , which is the applicable piezoelectric constant in the context of experiments dealing with current flow across the sample. The constant e31 is relevant to systems in depolarizing fields such as nitride nanostructures [448]. A comprehensive table including experimental and calculated values of elastic compliance, elastic constants, and piezoelectric constants as well as the Poisson number of wurtzite binary group III nitrides at room temperature are shown in Table 2.28. Table 2.27 Spontaneous polarization, Born effective charges, Z*

(in units of e), piezoelectric constants, dynamical charges, elastic constants (GPa), and the ratio R ¼
2C31/C33 of wurtzitic nitrides, as obtained with DFT calculations in the LDA and GGA approximation.

AlN LDA LDA [179] GGA GaN LDA [179] LDA GGA InN LDA LDA [179] GGA

P (C m
2)

Z

e33 (C m
2)

e31 (C m
2)

«(p) 31 (C m
2)

C33 (GPa)


0.100

2.652

1.80


0.64


0.74

384 373

111 108


0.578
0.579


0.090

2.653

1.50


0.53


0.62

377

94


0.499


0.032

2.51

0.86


0.44


0.47

415

83


0.400


0.034

2.67

0.67


0.34


0.37

405 354

103 68


0.508
0.384


0.041

3.045

1.09


0.52


0.56

233 224

88 92


0.755
0.821


0.042

3.105

0.81


0.41


0.45

205

70


0.683

The last column reports the proper e31 piezoelectric constant [212].

C31 (GPa)

R¼
2C31/C33

0.22c

367 135 103 405 95 0.52 3.267 1.043 0.566 2.757 10.53

Theory [179]


1.253 2.291
1.579


0.34 0.67

0.38

68 354

Theory [209,212]

GaN

396 137 108 373 116 0.58 2.993 0.868 0.615 3.037 8.621

370a 145a 110a 390a 90a 0.56 3.326 1.118 0.623 2.915 11.11

0.30d

Theory [179]

Experiment


2.298 5.352
2.069


0.53 1.50

0.50

94 377

Theory [212,433]

AlN

a

For an expanded list of elastic constants, see tables in Chapter 1 under mechanical properties [84]. Ref. [200]. b Ref. [198]. c Shur, M.S., Bykhovski, A.D. and Gaska, R. (1999) MRS Internet Journal of Nitride Semiconductor Research, S41, G16. d Ref. [437].

C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa) n(0001) S11 (10
12 N m
2) S12 (10
12 N m
2) S13 (10
12 N m
2) S33 (10
12 N m
2) S44 (10
12 N m
2) e31 (C m
2) e33 (C m
2) e24 ¼ e15 (C m
2) d31 (10
12 C m
2 Pa) d33 (10
12 C m
2 Pa) d15 (10
12 C m
2 Pa)

Parameter

of wurtzite binary group III nitrides at room temperature (theory [179,212,433]).

410a 140a 100a 390a 120a 0.51 2.854 0.849 0.514 2.828 8.333 0.58b 1.55b 0.48b
2.65 5.53
4.08

Experiment 223 115 92 224 48 0.82 6.535 2.724 1.565 5.750 20.83

Theory [179]

Table 2.28 Experimental and predicted elastic compliance, elastic constants, and piezoelectric constants as well as the Poisson number


3.147 6.201
2.292

0.41 0.81

0.68

70 205

Theory [212,433]

InN

2.12 Polarization Effects

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j 2 Electronic Band Structure and Polarization Effects

244

In heterojunctions containing donors and acceptors and shallow defects, the associated free carriers within the Fermi statistics diffuse to the semiconductor with the smaller bandgap where they are confined, due to potential barriers, to potential minima. The resulting charge separation due to free carriers causes an internal electric field, screening field, which is represented by the first term in Equation 2.60. In addition, an electric field can also be induced by the application of an external voltage such as done through the use of Schottky barriers, metal oxide semiconductor structures, and p–n junctions. Spontaneous and strain-induced piezoelectric polarization can influence the final status of the interfacial free-charge density in these heterostructures. Any shallow defects (induced fields would change the ionization ratio), free carriers, and surface contacts must be included for a complete treatment. Let us now calculate the polarization charge for model heterojunctions using the linear interpolation method. For AlxGa1
xN coherently strained on a relaxed GaN substrate, the strain e? is expected to be proportional to x and given by e? ¼ 2 (aGaN
aAlGaN)/aAlGaN, which is 0.051x and is tensile. The piezoelectric polarization is then Ppiezo ¼
0.0464x, that is, pointing in the ½0 0 0
1 direction. The corresponding difference in spontaneous polarization between AlxGa1
xN and GaN is also expected to be proportional to x, the AlN mole fraction, and is given by DPspon ¼
0.056x. Consequently, the two are in the same direction for this particular orientation and are comparable in magnitude. This treatment assumes that the polarization charge scales linearly with alloy composition, which does not necessarily hold but is used for simplicity. The matter is discussed below in more detail. The total polarization for AlN–GaN interface, which is defined in this case as the sum of the piezoelectric polarization and the differential polarization charge is
0.102x. Note that these are all in C m
2 and that 1 C m
2 ¼ 0.624 · 1015 electrons cm
2. Thus, for x on the order of 0.1, we are dealing with total polarization charge of the order of mid-1012 cm
2. In case the ternary AlxIn1
xN is used for the barrier, the composition of Al0.82In0.18N can be grown lattice matched to GaN [451] and the piezoelectric polarization vanishes. For lower Al concentrations, that is, x < 0.82, the piezoelectric polarization increases due to the increase in biaxial compressive strain. For higher Al concentrations, that is, x > 0.82, the layer is under tensile strain and the piezoelectric polarization becomes negative. For a coherently strained InxGa1
xN layer on relaxed GaN, the difference in spontaneous polarization is much smaller, DPspon ¼
0.012x. Furthermore, the InxGa1
xN layer on GaN would be under compressive strain e? ¼
0.203x and Ppiezo ¼ þ0.139x. Here, the piezoelectric polarization dominates and is opposite in direction to the spontaneous polarization charge but even larger in absolute magnitude. Unlike for tensile-strained AlxGa1
xN or AlxIn1
xN (for large x-values) on GaN layer, wherein the piezoelectric and the spontaneous polarization are negative and point in the same direction, thus they add up, the spontaneous and piezoelectric polarizations oppose one another for compressively strained InxGa1
xN or AlxIn1
xN (for small x-values) layers. To calculate the differential spontaneous and piezoelectric polarization associated with alloys, one can to a first order employ a

2.12 Polarization Effects

linear interpolation for the spontaneous polarization and piezoelectric and elastic constants from the binary compounds [452]. In addition to nonlinear behavior of polarization that is discussed in Section 3.14.3, there are technological issues that must be considered. Some further words of caution about the above estimates based on linear interpolation are needed. If the AlGaN layers are not pseudomorphic but partially relaxed (by misfit dislocations for example), then the piezoelectric effect would be reduced but the spontaneous polarization would still be present. Finally, if domains with inverted polarity exist, the overall polarization effects may be washed out. Also note that in an inverted structure with nitrogen (N) polarity toward the surface, it may be possible to create a two-dimensional hole gas (2DHG) at the AlGaN/substrate GaN interface, provided that free holes are available. However, if an n-type GaN layer is placed on top, a 2DEG may form on top of the AlGaN layer. Polarization effects and devices are inextricable. In devices with a large concentration of free carriers, the polarization charge would be screened. In devices where the modulation of charge is the basis of operation such as the MODFETs, a detailed accounting of all polarization charge must be undertaken. In considering a Normal MODFET (N-MODFET) structure where the larger bandgap AlGaN donor layer is deposited on top of a GaN channel layer, both the spontaneous polarization and the piezoelectric polarization must be accounted for. For an N-MODFET structure with Ga-polarity, the potential will slope down from the surface of the AlGaN layer toward the AlGaN–GaN interface. The topic is discussed in some detail for both AlGaN/GaN and AlxIn1
xN/GaN structures in Section 2.12.4.1 and for the InGaN/GaN heterojunction system in Section 2.12.4.2. 2.12.3 Nonlinearity of Polarization

In the treatment above and in the literature for quite sometime, a linear interpolation from the binary end points were used to deal with both piezoelectric and spontaneous components of polarization. Despite the linear interpolation method being reasonably successful in getting the figures to a first extent, as done in conjunction with Equation 2.86, discrepancies with experiments were noted. Thus, efforts continued to refine the polarization figures and, while in the process, entertain whether a bowing parameter such as the case in bandgap of ternaries, while the lattice parameter obeys the Vegard’s law, that is, linear interpolation, could also be considered for polarization. This forms the basis of the treatment of the problem by Bernardini and Fiorentini [220,221,453] and others [454,455]. The nonlinearity is quite pronounced in AlInN and InGaN alloys for which the binary constituents are very largely lattice mismatched. Spontaneous polarization bowing strongly depends on the microscopic nature of the alloy. What is more is that chemical ordering in the form of short period superlattices may increase the bowing up to a factor of five. Similarly, the piezoelectric polarization is also nonlinear. However, in random alloys, this nonlinearity is entirely due to a nonlinear strain dependence on piezoelectric polarization in pure end binary compounds while the

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j 2 Electronic Band Structure and Polarization Effects

246

piezoelectric coefficients follow Vegard’s law. On the contrary, in chemically ordered InGaN and AlInN alloys piezoelectric coefficients deviate from Vegard’s law, this effect reduces the strength of the piezoelectric polarization up to 38% of its value in AlInN alloy. However, linear extrapolation is simple and gets reasonably accurate figures. The details of this nonlinearity are discussed below. To reiterate, discrepancies between the experimental data and the theory using linear interpolation led Bernardini and Fiorentini to consider a bowing parameter for polarization in conjunction with ternaries, and it follows that this applies to quaternaries as well. For freestanding alloys, meaning relaxed for this purpose, and assuming that the ternary nitride alloys have random microscopic structure with no strain, the values of the lattice constants within the realm of Vegard’s law are given by aAlx Ga1
x N ¼ xaAlN þ ð1
xÞaGaN ; c Alx Ga1
x N ¼ xc AlN þ ð1
xÞcGaN :

ð2:88Þ

The spontaneous polarization Psp of an alloy, for example, AlxGa1
xN, can be expressed in terms of the polarization values of the binary constituents, AlN and GaN, in the realm of nonlinearities (i.e., non-Vegard behavior). The polarization in an alloy can be described in a generic measurable quantity to a first approximation by a parabolic model involving a bowing parameter, similar to that used for the bandgap of alloys. In conjunction with nonlinear polarization effects in alloys, Bernardini and Fiorentini [453] considered ordered structures to get at the spontaneous polarization across the entire composition range by calculating it for compositions of 0.25, 0.5, and 0.75 in addition to the binary end points. The chalcopyrite-like (CH) structure, used for the 0.5 composition, is formed by each anion site being surrounded by two cations of one species and two of the other, with the overall condition of conforming to periodical (2 · 2 · 2) wurtzite supercell. This structure is highly symmetric, as there are only two kinds of inequivalent anion sites, differing in the orientation of the neighbors but not in their chemical identity. Among the possible ordered structures, CH is in a sense the most homogeneous for a given composition. A further useful ordered structure considered is a luzonite-like (LZ) structure, used for the 0.25 and 0.75 points, resembling zinc blende based alloys. In this structure, each nitrogen atom is surrounded by three cations of one species and one of the other: in a sense, this is the analog to the CH structure for molar fractions x ¼ 0.25 and 0.75. In the comparison between CH and random structures, the latter used for the 0.5 composition could provide insight into the effect of randomness versus ordering without the biases due to specific superlattice ordering, as in the CuPt (CP) structure used for the 0.5 composition point. Luzonite-like structure can provide values of the polarization in the intermediate molar fractions. Shown in Figure 2.35 is a comparison of calculated equilibrium basal and axial lattice parameters a and c for three binaries and their alloys with those determined in the realm of Vegard’s law (dashed lines). The agreement between the equilibrium a and c lattice parameters so calculated and those determined using Vegard’s law is quite good. The dependence of the polarization on composition is then the same as that on the lattice parameter(s), modulo a multiplicative factor.

2.12 Polarization Effects

5.9 InN

c-Lattice constant (Å)

5.7 In x G

a

1–x

N

5.5

In x

A l 1 –x

N

GaN 5.3

5.1

A lx G a

AlN

1 –x N

AlN

4.9 0

0.2

0.4

0.6

1.0

0.8

Molar fraction, x InN

a-Lattice constant (Å)

3.6 In x G

a 1 –x

N In x

3.4

A l 1 –x

N

GaN

A lx G a

3.2

1 –x

N

AlN

AlN 3.0 0

0.2

0.4

0.6

0.8

1.0

Molar fraction, x

Random alloy CH-like CP-like Figure 2.35 The basal plane lattice parameter a and axial lattice parameter c of wurtzite nitride binaries and alloys directly calculated versus those determined by Vegard’s law. The open circles denote the random alloy with 0.5 molar fraction. The dashed lines are Vegard’s law. Courtesy of F. Bernaridini and V. Fiorentini.

Shown in Figure 2.36 are again the spontaneous polarization values for the aforementioned ternaries in the freestanding strain-free form. The solid lines represent interpolations utilizing Equation 2.95 for AlInN, the binary points of which were determined using 32-supercell calculations. The dashed lines represent the simple Vegard’s law based interpolations and the numbers indicate the bowing parameters in terms of C m
2.

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j 2 Electronic Band Structure and Polarization Effects

248

–0.02 In x Ga 1–x N

GaN

Spontaneous polarization (C m–2)

+0.038 InN –0.04 +0.071

+0.019 –0.06

In x Al1–x N Alx Ga1–xN

–0.08

AlN

AlN

–0.10 0

0.2

0.4

0.6

0.8

1

Molar fraction, x Figure 2.36 Spontaneous polarization versus molar fraction in freestanding (strain free) random nitride alloys (solid circles). The solid lines represent the results of the bowing model of Equation 2.95. The dashed lines are determined by Vegard’s law. The numbers shown in the figure are the bowing parameters expressed in units of C m
2. Courtesy of F. Bernaridini and V. Fiorentini.

Depicted in Figure 2.37 are the spontaneous polarization values versus the molar fraction. The solid circles, squares, and triangles represent the values for the random alloy, CH-/LZ-, and CP-like alloys, respectively. The dashed lines represent the values calculated using Vegard’s law. The numbers shown in the figure represent the bowing parameters for the CH-/LZ- and CP-like ordered alloys. The CH and LZ calculations can be used to verify efficacy of the interpolation model inclusive of the bowing parameter. The ordered LZ structure is analogous to CH for molar fractions of 0.25 and 0.75. The extent to which the LZ values deviate from those calculated by Equation 2.95 and the CH (x ¼ 0.5) value indicates whether or not nonparabolicity occurs in the P(x) relation for CH-like order. Because the polarization of the CH structure behaves qualitatively as that of the random structure (Figure 2.37), the conclusions drawn for CH are applicable to the random phase. Fortuitously, the values of the polarization calculated for the LZ structures at molar fractions 0.25 and 0.75, shown in Figure 2.37, are very close to the parabolic curve for InGaN and AlGaN calculated using the quadratic expression depicted in Equation 2.95, paving the way for the use of the analytical (quadratic) expression for polarization calculations. However, for AlInN, the calculated values are somewhat

2.12 Polarization Effects

Spontaneous polarization (C m–2)

0.02

+0.333 InxAl1–xN

InxGa1–xN

0.00

+0.193

–0.02

InxGa1–xN

–0.04

GaN

+0.095 –0.06

InxAl1–xN

InN AIxGa1–xN

+0.037 –0.08

Random alloy CH-like CP-like

AIN

AIN

–0.10 0

0.2

0.4

0.6

1

Molar fraction, x Figure 2.37 Spontaneous polarization versus the molar fraction in all three ternary nitride alloys. Circles, squares, and triangles represent random alloy, CH-/LZ-, and CP-like structures, respectively. The dashed/dotted lines (blue) with solid triangles are for the CP-like alloys, the dashed lines (green) with solid squares are for CH-like alloys, and solid lines (black) with filled

circles are for random alloys. The black dashed lines represent the data calculated using Vegard’s law. Numbers indicated in the figure are for CP and CH-/LZ-like ordered alloy bowing parameters in terms of C m
2. Courtesy of F. Bernaridini and V. Fiorentini. (Please find a color version of this figure on the color tables.)

above the quadratic relation for x ¼ 0.25 and below it for x ¼ 0.75, indicating some nonparabolicity in polarization. Specifically, the bowing is higher for low In concentration in AlInN. This nonparabolicity is relatively modest, of order 10%, as compared to the quadratic nonlinearity Equation 2.95 for AlInN. One can then conclude that the analytical (quadratic) expression would predict polarization in AlGaN and InGaN fairly accurately and also for AlInN but with about 10% accuracy. To understand the physical origin of the spontaneous polarization bowing, Bernardini and Fiorentini [453] decomposed the spontaneous polarization into three distinct components on the basis of their genesis, namely, the internal structural and bond alternation, volume deformation, and disorder. The internal structural and bond alternation (strain) can be caused by varying cation–anion bond lengths. The volume deformation can be due to compression or dilation of the bulk binaries from their original equilibrium lattice constants to the alloy values. The disorder effect is due to the random distribution of the chemical elements on the cation sites. Bernardini and Fiorentini [453] showed that in ordered alloys the structural

j249

j 2 Electronic Band Structure and Polarization Effects

250

contribution is dominant, the volume deformation accounts for one-third of the bowing found in random alloys, and the effect of disorder appears insignificant in terms of its effect on the bowing of spontaneous polarization. 2.12.3.1 Origin of the Nonlinearity To understand the origin of the structural contribution to the nonlinear behavior, Bernardini and Fiorentini [453] considered that while the bandgap is a scalar, the polarization is a vector of defined direction. The basal plane normal in wurtzite structures is the [0 0 0 1] direction. Thus, the bond length and angle alternation will affect the polarization bowing only if it changes the projection of the bond length along the c-axis, which is the [0 0 0 1] axis. This is consistent with the notion that the polarization in pure binaries is strongly affected by the relative displacement of the cation and anion sublattice sites in the [0 0 0 1] direction [441]. Also consistent is that there is a clear correlation between the u parameter of the wurtzite structure, which is the bond length along the singular polar (or pyroelectric) axis, and the value of the polarization. Shown in Figure 2.38 is the calculated spontaneous polarization of freestanding x ¼ 0.5 alloys of AlGaN, InGaN, and AlInN versus the average internal 0.02 Random CH – like CP – like Spontaneous polarization (C m–2)

0.00 InGaN AlInN – 0.02

– 0.04

AlGaN – 0.06

0.380

0.378

0.376

0.374

Average u lattice parameter Figure 2.38 Spontaneous polarization versus the average internal parameter u in AlGaN, InGaN, and InAlN ternary alloys. Open circles, squares, and triangles, refer to random, CH-like, and CPlike structures, respectively. Courtesy of F. Bernaridini and V. Fiorentini.

0.372

0.370

2.12 Polarization Effects

parameter u. The internal u parameter is defined as the average value of the projection of the connecting vector of a nitrogen atom with its first neighbor in the ð0 0 0
1Þ direction along this same direction. This is in the realm of the wurtzite structure convention in which each anion is situated at the (0, 0, u) Cartesian point from the cations and all the vertical bonds between Ga and N point along [0 0 0 1] direction. This definition can also be used for random phase alloys in spite of the displacement from the ideal sites. Figure 2.38 shows that for a given alloy composition, the spontaneous polarization of relaxed (freestanding) nitride alloys of different microscopic structure depends linearly on the average internal u parameter of the alloy structure. This indicates that spontaneous polarization differences between alloys of the same composition are primarily due to structural and bond alternation effects; disorder appears to have a negligible influence. The structural and bond alternation effects discussed above can also shed light on the random and CP phases in AlGaN having almost the same average u, hence, nearly the same polarization. In InGaN, the random alloy has a larger u than the CH phase, while the opposite is true for AlInN and AlGaN accounting for the CH versus random bowing behavior in InGaN being opposite of that in AlInN and InGaN. Moreover, the large bowing of CP-ordered AlInN and InGaN is consistent with the very large deviation of the average u as compared to the random and CH-like structures. If the internal strain were the only source of polarization bowing, all of the points in Figure 2.38 would fall on the same straight line. This not being the case suggests that another factor related to the chemical identity of the constituents plays a role of some importance and brings to the role of volume deformation. To investigate this role, Bernardini and Fiorentini [453] set up a model based on the polarization in a constrained ideal wurtzite structure in which only the a parameter is a variable, whereas c and u are fixed atp the determined by maximal sphere packing, ffiffiffiffiffiffiffivalues ffi namely, u ¼ 0.375 and c=a ¼ 8=3. Each nitrogen atom is then surrounded by four equidistant cations, meaning all bonds have the same length for a given lattice constant a. By design, the bond alternation caused by perturbation in the internal parameter u would not play any role, and the effects of chemical identity of the constituents can be easily distinguished. To continue to tackle the aspect of volume deformation, Bernardini and Fiorentini [453] assumed that Vegard’s law holds for the lattice constant a. Naturally, this establishes a linear relationship between the composition and the lattice constant. This segues into the calculation of the polarization in each of the binary nitrides in their ideal structure as a function of the lattice constant a(x). Finally, they could express the alloy polarization as a composition-weighted Vegard-like average of the polarizations of the binary end points (through a reduction of Equation 2.85) as aðxÞ Psp ðAlx Ga1
x NÞ ¼ xPaðxÞ sp ðAlNÞ þ ð1
xÞP sp ðGaNÞ:

ð2:89Þ

In this approach, any nonlinearity must have its origin in the different response of polarization to perturbations in a(x), hence, to hydrostatic compression. To illustrate this point, shown in Figure 2.39 are calculated polarizations in the ideal wurtzite structure for the three ternary alloys, polarization of the binaries in the ideal

j251

j 2 Electronic Band Structure and Polarization Effects

252

InGaN

Spontaneous polarization (C m–2)

–0.015 GaN

InN

–0.020 AlInN –0.025

AlGaN

AlN GaN InN

–0.030 AlN –0.035 3.0

3.1

3.2

3.3

3.4

3.5

3.6

3.7

Equilibrium lattice constant (Å) Figure 2.39 Spontaneous polarization versus the lattice constant in ideal wurtzite structures. Solid circles depict the values of binary compounds and random ternary alloys. Open circles, squares, and triangles represent the polarization calculated as a function of the lattice constant in bulk AlN, GaN, and InN. Solid lines correspond to Vegard interpolations based on

the ideal binaries under hydrostatic pressure. The dashed lines represent the Vegard interpolation of the polarization using the values for the binaries at equilibrium. The data show that the polarization in the alloys is a direct result of the hydrostatic pressure and thus volume deformation. Courtesy of F. Bernaridini and V. Fiorentini.

structure, and polarization interpolated by the Vegard interpolation of Equation 2.89. It can be seen that the calculated values of polarization and those interpolated by Vegard’s prediction agree well. Because the Vegard interpolation intrinsically account for the volume deformation, the origin of the volume deformation component of the nonlinearity and its large values in In-containing alloys become clear. Essentially, this has its genesis in the fact that polarization decreases with hydrostatic pressure in AlN and GaN, while it increases in InN. Also to be noted, polarizations in the ideal structure are between 35 and 50% of their values in freestanding (relaxed) alloys, and despite the absence of bond alternation, which is designed for the purpose separating the components in effect, the bowing is still very large. The above model dealing with the effect of strain on polarization provides the basis for developing an expression of the bowing parameter bmodel for an ideal wurtzite structure alloy as a function of the polarization response to hydrostatic pressure (for the model case of AlGaN): 0 1 qP qP GaN AlN A AlGaN @
bmodel ¼ ðaGaN
aAlN Þ qa qa a¼að1=2Þ 0 1 ð2:90Þ 2 2 1 q P q P GaN AlN A 2@ þ ðaGaN
aAlN Þ
: qa2 qa2 4 a¼að1=2Þ

2.12 Polarization Effects

The agreement of the latter expression with the b resulting from a fit to the calculated values is very good (e.g., for the extreme case of AlInN, bmodel
20.0225 C m
2, while from direct calculation, we get
20.0208 C m
2). On the basis of the model, it is now understandable that the AlGaN bowing is pretty moderate because the region of interest is small (3.1–3.2 Å) and the responses to hydrostatic pressure of AlN and GaN are similar. On the contrary, in the large range 3.1–3.6 Å, AlN and InN have opposite behavior, when the huge bowing is found in AlInN alloys. The same goes, although to a lesser extent, for the InGaN alloys. 2.12.3.2 Nonlinearities in Spontaneous Polarization Any nonlinearity in the spontaneous polarization can be treated by using a bowing parameter as commonly employed in interpolating the bandgap of an alloy from the binary point with the help of a bowing parameter. In this vein, the spontaneous polarization for a ternary Psp (AxB1
xN) with A and B representing the metal components and N representing nitrogen is given by [84,220,221] sp

sp

sp

PAx B1
x N ¼ xP AN þ ð1
xÞP BN
bAB xð1
xÞ: sp

ð2:91Þ

sp

PAN and PBN are the spontaneous polarization terms for the end binaries forming the alloy. The bowing parameter is as defined bAB ¼ 2PAN þ 2P BN
4P A0:5 B0:5 N ;

ð2:92Þ

which requires only the knowledge of the polarization of the ternary alloy at the midpoint, that is, molar fraction x ¼ 0.5. Knowledge of the bowing parameter from Equation 2.97 would lead to the determination of the spontaneous polarization at any composition. For AlxGa1
xN, Equations 2.91 and 2.92 take the form sp

sp

sp

PAlx Ga1
x N ¼ xPAlN þ ð1
xÞPGaN
bAlx Ga1
x N xð1
xÞ;

ð2:93Þ

bAlx Ga1
x N ¼ 2P AlN þ 2P GaN
4P Al0:5 Ga0:5 N :

ð2:94Þ

with

The first two terms in Equation 2.93 are the usual linear interpolation terms between the binary constituents. The third term, quadratic, represents the nonlinearity. Higher order terms are neglected because their contribution is estimated to be less than 10%. Using the numerical GGA values in Table 2.25 for the spontaneous polarization in AlN and GaN and the bowing parameter for random alloy AlGaN given in Refs [220,221] leads to sp

PAlx Ga1
x N ¼
0:09x
0:034ð1
xÞ þ 0:0191xð1
xÞ; and sp

sp

sp

PInx Ga1
x N ¼ xPInN þ ð1
xÞPGaN
bInx Ga1
x N xð1
xÞ;

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j 2 Electronic Band Structure and Polarization Effects

254

with bInx Ga1
x N ¼ 2P InN þ 2PGaN
4PIn0:5 Ga0:5 N : Again, using the numerical GGA values in Table 2.25 and the bowing parameter for random alloy InxGa1
xN given in Refs [220,221] leads to sp

PInx Ga1
x N ¼
0:042x
0:034ð1
xÞ þ 0:0378xð1
xÞ; and sp

sp

sp

PAlx In1
x N ¼ xP AlN þ ð1
xÞP InN
bAlx In1
x N xð1
xÞ; with bAlx In1
x N ¼ 2PAlN þ 2PInN
4P Al0:5 In0:5 N: Using the numerical GGA values in Table 2.25 and the bowing parameter for random alloy InxAl1
xN given in Refs [220,221] leads to sp

PAlx In1
x N ¼
0:090x
0:042ð1
xÞ þ 0:0709xð1
xÞ:

ð2:95Þ

In calculating the bowing parameters for the three ternaries mentioned above, Bernardini and Fiorentini [453] used a 32-atom supercell for both alloys and binary nitrides for spontaneous polarization for the end binaries and ternaries. These calculations, while being more efficient in terms of computer time, are not as accurate as those reported in Refs [220,221,441]. In the above treatment, the bowing parameters are taken from Ref. [453], while the spontaneous polarization for the end binaries is taken from Ref. [441]. Relying simply on the 32-atom supercell calculations, the bowing parameters and related spontaneous polarization figures for ordered alloys such as CuPt-like ordered alloy (CP-like), chalcopyrite-like ordered alloy (CH-like), luzonite-like ordered alloy (LZ-like) have also been obtained by Bernardini and Fiorentini [453]. In the series of tables below, the results of such calculations for the aforementioned ordered alloys are given for completeness. The binary figures in terms of spontaneous polarization and the lattice parameter resulting from the 32-atom supercell calculations are also given for consistency. Again, more accurate binary data exist in Refs [441]. For convenience, the lattice parameter (a), spontaneous polarization (Psp), and bowing parameter (bAB) for the three ternary nitride alloys in the form of random, ordered chalcopyrite (CH-like), ordered luzonite (LZ-like), and CuPt-ordered alloy (CP-like) are tabulated in Tables 2.29–2.35, in addition to the data shown in Figures 2.35–2.37.

Table 2.29 The lattice parameter (a) and spontaneous polarization (Psp) for AlN, GaN, and InN determined by 32-atom supercell calculations by Bernardini and Fiorentini.

AlN a (Å) Psp (C m
2)

3.1058
0.0897

GaN 3.1956
0.0336

InN 3.5802
0.0434

2.12 Polarization Effects Table 2.30 The lattice parameter (a), spontaneous polarization (Psp), and the bowing parameter (bAB) for random alloy ternaries with a molar fraction of x ¼ 0.5 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453].

R 50%

Al0.5Ga0.5N

In0.5Ga0.5N

Al0.5In0.5N

a (Å) Psp (C m
2) bAB (C m
2)

3.1500
0.0569 þ0.0191

3.3872
0.0290 þ0.0378

3.3352
0.0488 þ0.0709

Table 2.31 The lattice parameter (a), spontaneous polarization (Psp), and the bowing parameter (bAB) for CuPt ordered alloy (CPlike) with a molar fraction of x ¼ 0.5 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453].

CP 50%

Al0.5Ga0.5N

In0.5Ga0.5N

Al0.5In0.5N

a (Å) Psp (C m
2) bAB (C m
2)

3.1489
0.0573 þ0.0176

3.3884 þ0.0098 þ0.1934

3.3222 þ0.0168 þ0.3336

Table 2.32 The lattice parameter (a) and spontaneous polarization (Psp) for the luzonite (LZ-like) and chalcopyrite (CH-like) alloy with a molar fraction of x ¼ 0.25 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453].

LZ 25%

Al0.25Ga0.75N

In0.25Ga0.75N

Al0.25In0.75N

a (Å) Psp (C m
2)

3.1724
0.0413

3.2920
0.0323

3.4510
0.0385

Table 2.33 The lattice parameter (a) and spontaneous polarization

(Psp) for the luzonite (LZ-like) and chalcopyrite (CH-like) alloy with a molar fraction of x ¼ 0.5 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453]. CH 50%

Al0.5Ga0.5N

In0.5Ga0.5N

Al0.5In0.5N

a (Å) Psp (C m
2) bAB (C m
2)

3.1474
0.0523 þ0.0374

3.3949
0.0328 þ0.0226

3.3369
0.0427 þ0.0952

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j 2 Electronic Band Structure and Polarization Effects

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Table 2.34 The lattice parameter (a) and spontaneous polarization

(Psp) for the luzonite (LZ-like) and chalcopyrite (CH-like) alloy with a molar fraction of x ¼ 0.75 determined by 32-atom supercell calculations by Bernardini and Fiorentini [453]. LZ 75%

Al0.75Ga0.25N

In0.75GaN0.25

Al0.75In0.25N

a (Å) Psp (C m
2)

3.1276
0.0690

3.4828
0.0366

3.2146
0.0564

Table 2.35 The bowing parameter (a) for the random alloy and luzonite (LZ-like), chalcopyrite (CH-like), and CuPt ordered alloys (CP-like) calculated by Bernardini and Fiorentini [453].

Random CH and LZ CP

AlGaN

InGaN

AlInN

þ0.019 þ0.037 þ0.018

þ0.038 þ0.023 þ0.193

þ0.071 þ0.095 þ0.333

2.12.3.3 Nonlinearities in Piezoelectric Polarization The total polarization at heterointerfaces is the sum of spontaneous and piezoelectric polarization. Having treated the nonlinearity in spontaneous polarization, attention must be turned to piezoelectric polarization, specifically, its nonlinear dependence on composition in alloys. Some if not all of the components of the heterojunctions are grown pseudomorphically and are, therefore, under strain on the (0 0 0 1) axis. The ensuing symmetry-conserving strain causes a change in polarization that amounts to a piezoelectric polarization. The aim here is to show that piezoelectricity in nitride alloys is nonlinear, and that this nonlinearity is due to a pure bulk effect with its nonlinear behavior of bulk binary piezoelectric constants versus symmetry-conserving strain. The model heterostructure considered by Bernardini and Fiorentini [453] is a coherently strained alloy grown on a relaxed binary buffer layer (bulk for this purpose) in which the in-plane lattice parameter is aalloy ¼ aGaN. The piezoelectric component is the difference between the total polarization to be obtained and the spontaneous polarization discussed above. Shown in Figure 2.40 is the piezoelectric polarization as a function of the alloy composition. Symbols, which have similar designation as in the spontaneous polarization, represent the calculated polarizations for AlGaN, InGaN, and InAlN alloys as a function of compositions. After the publication of Ref. [453], to reconcile a discrepancy with a paper by Al-Yacoub and Bellaiche [454] who showed that CuPt-like ordering in In0.5Ga0.5N wurtzite-structure alloys causes a sizable deviation of the piezoelectric constants from Vegard’s like behavior. Bernardini and Fiorentini [220,221] revisited the piezoelectric polarization in nitride alloys. In fact, Figure 2.40 contained here represents the updated calculations. The error in question stems from the fact that the polarization values for the strained and unstrained alloys were subtracted

Piezoelectric polarization (C m–2)

2.12 Polarization Effects

InN

Vegard's Random alloy CH,LZ CP

0.2

0.1

InGaN AlInN GaN

0.0 AlGaN AlN

AlN – 0.1 0

0.2

0.4

0.6

0.8

1

Alloy molar fraction Figure 2.40 Piezoelectric component of the macroscopic polarization in ternary nitride alloys epitaxially strained on a relaxed GaN layer (template). Open symbols represent the directly calculated values for random alloy (circles), CHlike and LZ-like (squares), and CP-like (triangles)

structures, respectively. Dashed lines represent the prediction of linear piezoelectricity, while the solid lines are the prediction of Equation 2.96 using the nonlinear bulk polarization as shown in Figure 2.41. Courtesy of F. Bernaridini and V. Fiorentini.

correctly, but values calculated with different k-point meshes were used instead of those for the same k-point [454,455]. It is clear that, contrary to the spontaneous component of polarization discussed above, the piezoelectric polarization component hardly depends on the microscopic structure of the alloy. One might then ask whether the piezoelectric polarization of the alloy can be reproduced by a Vegard-like model interpolated from the binaries in the form pe

pe

pe

PAlGaN ðxÞ ¼ xPAlN ½eðxÞ þ ð1
xÞP GaN ½eðxÞ; pe P AlN ½eðxÞ

ð2:96Þ

pe PGaN ½eðxÞ

and represent the strain-dependent bulk piezoelectric where polarization of the binary end points. With obvious permutations, this expression can be constructed for InGaN and InAlN ternaries as well. To a first approximation, one may calculate the piezoelectric polarization of the binary compounds for symmetry conserving in-plane and axial strains as pe

PAlN ¼ e33 e3 þ 2e31 e1 :

ð2:97Þ

The piezoelectric constants e can be calculated for the equilibrium state of the binary, AN, and as such they do not depend on strain. The dashed lines in Figure 2.40 represent the piezoelectric term as computed from the above relations using the piezoelectric constants computed for the binaries [433]. The Vegard’s law of Equation 2.96 when combined with Equation 2.97 clearly fails to reproduce the calculated polarization and misses the strong nonlinearity of the piezoelectric term evident in

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j 2 Electronic Band Structure and Polarization Effects

258

Figure 2.40. This is due to a valid nonlinearity of the bulk piezoelectricity of the binary constituents, which is of nonstructural origin. It should be stated that bowing due to the microscopic structure of the alloys is negligible. The argument forwarded by Bernardini and Fiorentini [453] is that they calculate the piezoelectric polarization as a function of the basal strain for AlN, GaN, and InN while optimizing all structural parameters. The results depicted in Figure 2.41 clearly indicate that the piezoelectric polarization of the binaries is an appreciably nonlinear function of the lattice parameter a, which is related to basal strain. Because all lattice parameters closely follow Vegard’s law, the nonlinearity cannot be related to deviations from linearity in the structure. Bernardini and Fiorentini [453] substitute the nonlinear piezoelectric polarization computed for the binaries into the Vegard interpolation, Equation 2.96 In doing so, they obtain excellent agreement with the polarization calculated directly for the alloys as shown with solid lines in Figure 2.40. This led them to conclude that the nonlinearity in bulk piezoelectricity dominates over any effects related to disorder, structure, bond length and angle alternation, and so on. Importantly, they concluded that Vegard’s law still holds in calculating the piezoelectric polarization of the III–V nitrides alloys, provided that the nonlinearity of the bulk piezoelectric of the constituents is accounted for. Serendipitously, this means that the piezoelectric polarization of any nitride alloy at any strain can be found by noting the value for x (the composition), followed by calculating the basal plane strain, e(x) from Vegard’s law, and Ppe from Equation 2.96 using the nonlinear piezoelectric polarization of the binaries (Figure 2.41). This approach is of paramount value in the modeling of nitride heterostructures, especially those with high In content, and AlInN alloys. It should be noted that nonlinearities in the calculated piezoelectric constants of AlN and GaN have also been reported by Shimada et al. [456] but in the realm of

Piezoelectric polarization (C m–2)

0.4

AlN GaN InN

0.3

AlN InN

0.2

GaN 0.1

0.0 0.00

– 0.05

– 0.10

– 0.15

Basal strain Figure 2.41 Piezoelectric polarization in binary nitrides as a function of basal strain (symbols and solid lines) compared to linear piezoelectricity prediction (dashed lines). The c- and u-lattice parameters are optimized for each strain. Courtesy of F. Bernaridini and V. Fiorentini.

2.12 Polarization Effects

volume-conserving strain. A direct quantitative comparison with the results of Shimada et al. and Bernardini and Fiorentini is not possible, because the latter seeks to optimize the volume of the cell so that the stress component along [0 0 0 1] vanishes, which is more appropriate for epitaxial structures. In spite of this substantial difference, there is a common trend between the derivative of the piezoelectric polarization shown in Figure 2.41 (i.e., an effective piezoconstant) and the values of the piezoelectric constants reported in Ref. [456]. The nonlinear piezoelectricity of the binaries can be described by the relations (in C m
2) [84] pz

PAlN ¼
1:808e þ 5:624e2 for e < 0;

pz

P AlN ¼
1:808e
7:888e2 for e>0;

pz

PGaN ¼
0:918e þ 9:541e2 ; pz

PInN ¼
1:373e þ 7:559e2 : ð2:98Þ It is important again to note that the nonlinearity in the bulk piezoelectricity exceeds any effects related to disorder or bond alternation, which have been taken into account [220,221]. The calculation of the piezoelectric polarization of an AxB1
xN alloy for any level of strain would proceed with first calculating the strain e ¼ e(x) for a given molar fraction, x, using the Vegard’s law, and the piezoelectric polarization by pz

pz

pz

PAx B1
x N ¼ xP AN ðeÞ þ ð1
xÞPBN ðeÞ; pz

ð2:99Þ

pz

where PAN ðeÞ and PBN ðeÞ are the end binary strain dependent piezoelectric polarizations that can be calculated for a given strain, eðxÞ, using Equation 2.98. Application of this process to each of the three ternaries for all the possible cases of ternaries is as follows. Using Equation 2.96 and linear interpolation for the elastic constants that are tabulated in Table 2.28 and strain determined from Equation 2.64, the piezoelectric polarization between a given ternary and binary can be calculated, as represented below (in C m
2) [84]. pz

PAlx Ga1
x N=GaN ¼
0:0525x þ 0:0282xð1
xÞ; pz P Alx Ga1
x N=AlN ¼
0:026x þ 0:0282ð1
xÞ; pz PAlx Ga1
x N=InN ¼
0:28x
0:113ð1
xÞ þ 0:042xð1
xÞ:

ð2:100Þ

pz

PInx Ga1
x N=GaN ¼
0:148x þ 0:0424xð1
xÞ; pz PInx Ga1
x N=AlN ¼
0:182x þ 0:026ð1
xÞ
0:0456xð1
xÞ; pz P Inx Ga1
x N=InN ¼ 0:113ð1
xÞ þ 0:0276xð1
xÞ:

ð2:101Þ

pz

P Alx In1
x N=GaN ¼
0:0525x þ 0:148ð1
xÞ þ 0:0938xð1
xÞ; pz PAlx In1
x N=AlN ¼
0:182ð1
xÞ þ 0:092xð1
xÞ; pz PAlx In1
x N=InN ¼
0:028x þ 0:104xð1
xÞ:

ð2:102Þ

The calculated values are a function of molar fraction for the ternaries for the epitaxial layer and template combinations represented in Figure 2.42, assuming that the template is fully relaxed and the epitaxial layer is completely coherently strained

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j 2 Electronic Band Structure and Polarization Effects

260

with strain relaxation. For partial relaxation, unless the degree to which the relaxation that occurs is known, the calculations cannot be made. The extent of relaxation just depends on whether strain-relieving defects propagate from the template to the epitaxial layers. In addition, the effect of cooldown-induced thermal mismatch strain due to cooling from the growth temperature down to the operating temperature of the structure must also be considered. As stated, the results shown are for a perfect system with fully relaxed template and fully strained epitaxial layer on top of it. We should point out that for a particular pair with a particular composition, Al0.82In0.18N/ GaN heterostructure, there is a perfect lattice match; thus, the misfit-induced piezoelectric polarization is equal to zero. It should also be pointed out that other

0.2

Al x In1–x N/AlN In x Ga1-x N/AlN

b = 0.092

Piezoelectric polarization (C m–2)

b = – 0.046

0.1

A1 x Ga 1–x N/AlN b = 0.025 0.0

–0.1 0 (a)

0.2

0.4

0.6

0.8

1.0

Molar fraction, x

Figure 2.42 (a) Piezoelectric polarization of fully and coherently strained ternary alloys on fully relaxed AlN templates in the case of which the ensuing positive piezoelectric polarization and the negative spontaneous polarization are antiparallel. The bowing parameters describing the nonlinearity in the compositional dependence of Ppz are also indicated. (b) The piezoelectric polarization of coherently strained ternary alloys on fully relaxed GaN template. Note that Al0.82In0.18N/GaN heterojunction (indicated with an arrow) is lattice matched, thus

strain and piezoelectric polarization vanishes. (c) Also note that other experiments indicate the lattice matching composition for AlInN on GaN where the piezoelectric polarization charge vanishes to be different. The piezoelectric polarization of coherently strained ternary alloys on fully relaxed InN template. For ternary alloys grown on InN, the negative piezoelectric polarization and the spontaneous polarization are parallel and oriented to along the c-axis. Courtesy of O. Ambacher.

2.12 Polarization Effects 0.2

Piezoelectric polarization (C m–2)

Al x In1–x N/AlN b = 0.094

In x Ga 1–x N/GaN b = – 0.042

0.1

0.0

A1 x Ga 1–x N/GaN b = 0.028

–0.1

0

0.2

(b)

0.8

0.6

0.4

1.0

Molar fraction, x

0.0

Al x In 1–x N/InN b = 0.104

Piezoelectric polarization (C m–2)

In x Ga 1–x N/InN b = – 0.028 –0.1

–0.2

Al x Ga1–xN/InN b = 0.042

–0.3 0

(c) Figure 2.42 (Continued )

0.2

0.4

0.6

Molar fraction, x

0.8

1.0

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j 2 Electronic Band Structure and Polarization Effects

262

Nonlinear

InN 0.4

Piezoelectric polarization, P3 (C m–2)

AlN GaN 0.2

0.0

AlN

–0.2

–0.4

–0.2

–0.1

0.0

0.1

0.2

Strain, ε pz

Figure 2.43 The piezoelectric polarization (P 3 ) of wurtzitic binary group III nitrides under biaxial tensile or compressive strain calculated using Equations 2.98 and 2.99, the premise of which relies on nonlinearity as depicted in the figure. Only the AlN case is shown for positive strain consistent with Equation 2.98. In addition, the values of polarization exceed those of the linear model expressed in Equation 2.69. Courtesy of O. Ambacher.

experiments indicate the lattice matching composition for AlInN on GaN, where the piezoelectric polarization charge vanishes, to be different. The piezoelectric polarization versus strain, ignoring nonlinearity, can be calculated using Equation 2.69 for both compressive and tensile biaxial strain for all the epitaxial layer/template configurations. Use of Equations 2.98 and 2.99, however, takes the nonlinearity discussed above into consideration, as shown in Figure 2.43. As expected, the piezoelectric polarization for a given strain increases from GaN to InN and AlN. Significantly, higher piezoelectric polarizations, especially in cases of high strains (high In concentrations), result when nonlinearities in the piezoelectric charge are considered. The piezoelectric polarization associated with coherently strained random ternary AlGaN, InGaN, and AlInN alloys grown on GaN templates is shown in Figure 2.44. To reiterate, the piezoelectric polarization is nonlinear with respect to the alloy

2.12 Polarization Effects

Piezoelectric polarization, P3 (C m–2)

0.2

Al x In1–x N

In x Ga 1–x N

0.1

0.0

Al xGa1–x N/

0

0.2

0.4

0.6

0.8

1.0

Molar fraction, x Figure 2.44 The piezoelectric polarization of random coherent ternary alloys on relaxed GaN templates. Those calculated using Equation 2.69 and linear interpolations of the physical properties such as the piezoelectric constants and elastic coefficients (exy , Cxy) for the relevant binaries are shown in dashed lines. Calculations considering the nonlinearity in the piezoelectric polarization in terms of strain reflected in

Equations 2.98 and 2.99 are shown in solid lines. For alloys under high biaxial strain such as those containing high In concentration, the piezoelectric polarization is underestimated by the linear approach. It should be pointed out that the lattice matching composition for AlInN on GaN where the piezoelectric polarization charge vanishes is converging on 16–18% In in the InxAl1
xN lattice. Courtesy of O. Ambacher.

composition, spanning a range of
0.461 to 0.315 C m
2, the end points corresponding to coherently strained AlN on InN and InN on AlN, respectively. A point of side interest, albeit not practical, is that in alloys with high indium concentration, leading to high strain, the piezoelectric polarization is larger than that of the spontaneous polarization. As mentioned earlier in conjunction with Figure 2.42 that Al0.82In0.18N can be grown on GaN in a lattice-matched form making the piezoelectric polarization for this heterostructure vanish. To underline the point, the nonlinear model described by Equations 2.98 and 2.99 leads to larger piezoelectric polarization charge than linear interpolation where strain, elastic constants, and piezoelectric constants are assumed to vary linearly with composition.

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2.12.4 Polarization in Heterostructures

Investigation of heterostructures such as quantum well and single heterointerfaces such as those present in modulation-doped structures are ideal platforms for putting to test the knowledge base of spontaneous and piezoelectric polarization. Depending on the polarity of the sample, that is, Ga or N, the order of growth, meaning GaN on AlGaN or the other way around, and the buffer layer that determines to a first extent whether the barrier, the well, or both are in strain, the piezoelectric and spontaneous polarization charge add or subtract. Regardless of this, the resultant band bending leads to a red shift in energy due to the presence of electric field, commonly referred to as quantumconfined Stark effect (QCSE), and the ensuing deformation of the wave functions being pushed to opposite ends of the interface leads to reduction of radiative efficiency, increase in lifetime, and also blue shift with injection that is screening. In parallel, if the well size is small enough, carrier confinement induces a blue shift. Consequently, the red shift induced by polarization and the blue shift induced by localization compete in determining the transition energy. The optical transitions both in CW and timeresolved forms are discussed in detail in Volume 2, Chapter 5. Here, the manifestation of polarization charge in heterostructures is treated. We should mention that the electric field resulting from the polarization in GaN would cause a large Stark shift in optical measurements and reduce the effective bandgap in, for example, quantum wells (see Volume 2, Chapter 5), which is of paramount importance in optoelectronic devices. The field induced has been calculated in the context of the GaAs-based superlattices, which show notable Stark shift when grown along the [1 1 1] direction [457]. Stark shift can be screened on a length scale of the order of the Debye length ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pThe ekT=q2 n by injecting free carriers in the GaN layer(s), as is the case in LEDs, lasers, and PL experiments with large excitation intensities. In such a case, the strain-induced field causes carrier separation that, in turn, causes a field that opposes the strain field. Strain-induced polarization in such a heterostructure can also lead to a net electric field that can be measured as a voltage drop across the sample. The impact of the Stark shift and concomitant screening has been considered for laser gain also [458]. Fiorentini et al. [448] and Bernardini and Fiorentini [459] demonstrated that for an alternating sequence of wells (w) and barriers (b), the total electric field in the well can be calculated by recognizing that the normal components of the displacement vector D of Equation 2.60 are continuous (provided that there is no interface charge). total eW E W þ 4pPtotal W ¼ DW ¼ DB ¼ eB E B þ 4pP B

ð2:103Þ

in cgs units (for MKS remove the 4p terms). Utilizing Equation 2.103 together with the periodicity-imposed equality of voltage drops being equal but with opposite sign LW E W ¼
LB E B : One arrives at

ð2:104Þ

2.12 Polarization Effects total E W ¼
4pLB ðPtotal W
P B Þ=ðLW eB þ LB eW Þ;

or

total E W ¼
4pðPtotal W
P B Þ=ðLW =LB eB þ eW Þ;

total E B ¼
4pLW ðPtotal B
P W Þ=ðLB eW þ LW eB Þ; 0 1 L B total @ or E B ¼
4pðPtotal þ eB A: B
P W Þ= LW eW

ð2:105Þ

where eW and eB are the dielectric constant of the well and barrier layers, respectively. Likewise, LW and LB represent the well and barrier thicknesses. The same notation is used for polarization also. Both W and B superscripts and subscripts relate to wells and barriers, respectively. The total polarization term can be changed to piezoelectric term and spontaneous polarization term for cases when only the former or the latter is in effect, respectively. total Explicit in Equation 2.105 is that whenever Ptotal W „P B there will be an electric field. Due to strain or screening, the electric field will be present in both the well and the barrier. If we limit ourselves to piezoelectric polarization and somehow achieve relaxed heterostructures, the piezoelectric polarization charge and therefore the field will be zero. The electric field in wells and barriers in a quantum well has two components, one from spontaneous and the other from piezoelectric polarization. If the thickness of the wells and barrier are the same, the field in wells and barriers are related to each other as sp

pe

sp

pe

E W þ E W ¼ E W ¼
E B
ðE B þ E B Þ;

ð2:106Þ

where EW and EB represent the electric field in the well and barrier material. The superscripts indicate the field due to spontaneous and the other from piezoelectric polarization. Additional comments that can be made are that if lattice-matched AlInGaN alloy is used, the piezoelectric component in that material is eliminated. However, the spontaneous component would still be present. Another point to be recognized is that the piezoelectric field induced in InGaN and AlGaN layers of the same composition of In in the former and Al in latter, if grown on relaxed GaN, is larger in the former because of the larger lattice mismatch between GaN and InN as compared to that between GaN and AlN. The spontaneous polarization in alloys can be found using Equations 2.85 and 2.86 together with values listed for the binary end points in Table 2.25. The spontaneous polarization so formulated together with piezoelectric polarization of Equation 2.77 (and also Equation 2.83) would allow the computation of total polarization charge. This must be done for both the barrier and well material. The electric field can then be calculated using Equation 2.105. Any free carrier present in the well and barrier regions as well as those injected by optical and/or electrical means tend to screen the polarization-induced field. A complete picture can be obtained by solving self-consistently a set of effective mass theory or tight binding theory and simultaneously the Schr€ odinger–Poisson equation [460], which is discussed in Section 2.1.14. Combining the ab initio calculations

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j 2 Electronic Band Structure and Polarization Effects

266

of Fiorentini and Schr€odinger–Poisson solver of Di Carlo allowed the authors to calculate the charge distribution and field profiles in quantum wells in the presence of free carriers [448,461]. Poisson’s equation is solved using the boundary condition that the electric field is zero at the ends of the simulated regions. This corresponds to LW reaching the infinity limit in Equation 2.105. The potential thus obtained across the structure is plugged into the tight binding Schr€ odinger equation, which is solved to obtain energies and wave functions [462]. The new quasi-Fermi levels are then calculated, which then lead to carrier concentrations, and the procedure is iterated until self-consistency is obtained. A case of interest for FETs and parts of quantum wells is one that features a ternary on the surface where polarization charge would exist. The same is true for the interface between an AxB1
xN and a GaN heterostructure. The characteristic of this charge is that it changes abruptly, leading to a fixed two-dimensional polarization charge density s on the surface and at the interface, which is given by sp

pz

sAx B1
x N ¼ P Ax B1
x N þ P Ax B1
x N sp

pz

on the surface and sp

pz

sAx B1
x N=GaN ¼ ðPGaN þ P GaN Þ
ðP Ax B1
x N þ P Ax B1
x N Þ at the interface; ð2:107Þ respectively. Figure 2.45 shows the polarization-induced surface and interface sheet density s/q for relaxed and coherently strained binary nitrides and ternary–binary AxB1
xN/GaN interfaces, such as AlxGa1
xN/GaN. The spontaneous polarization surface charge density on relaxed InN, GaN, and AlN layers is 2.62 · 1013 cm
2, 2.12 · 1013 cm
2, and 5.62 · 1013 cm
2, respectively (there is no piezoelectric charge in this case). If a biaxial compressive strain of e ¼
0.002 is present, the surface charge is reduced to 0.72 · 1013 cm
2 · 1013, 0.74 · 1013, and 3.22 · 1013 cm
2, as can be deduced from Figure 2.45a. For compressively strained InN, GaN, and AlN at the level of e ¼
0.025,
0.030, and
0.045, respectively, the piezoelectric polarization fully compensates the spontaneous polarization and thus no polarization-induced field should exist. As can be seen from Figure 2.45a, polarization-induced surface charge is reduced by compressive and increased by tensile strain. For fully relaxed layers grown on the c-axis on a substrate, the spontaneous polarization-induced surface charge is negative for Ga-polarity and positive for N-polarity samples. The bound surface charge can be screened by oppositely charged surface defects and charges adsorbed on the surface. If this screening were incomplete, the carrier concentration profiles in the crystals would be different. Consequently, surface band bending, as well as that of interface, would be affected by polarization, and further such band bending would be dependent of the polarity of the sample. Experimental results are not sufficiently complete to draw a definitive conclusion. For coherently strained Ga-face AlxGa1
xN/GaN (for 0 < x  1) and AlxIn1
xN/ GaN (for 0.71 < x  1 interfaces), the polarization-induced interface charge is calculated to be positive as shown in Figure 2.45b. For both ternary and binary interfaces, the bound polarization charge increases nonlinearly with composition, x,

2.12 Polarization Effects

up to 7.06 · 1013 cm
2. However, for Ga-face InxGa1
xN/GaN (for the entire compositional range, 0 < x  1) and AlxIn1
xN/GaN (or 0 < x  0.71), the polarization charge is calculated to be negative and nonlinear with respect to composition. At the limit for the former, InN/GaN structure, a bound sheet density of 14.4 · 1013 cm
2 is calculated. Upon screening in n-type heterostructure, a positive polarization sheet 4

3 AlN Bound surface density s /q (1014 cm–2)

2

1

0

AlN

–1 InN –2

–3 GaN –4 –0.2

–0.1

0.0

0.1

0.2

Strain, ε (a) Figure 2.45 (a) In-plane biaxial strain dependence of surface polarization (piezoelectric plus spontaneous) charge for wurtzitic GaN, InN, and AlN. As in the case of Figure 2.43, the AlN case is with positive strain. (b) Bound interface charge density for coherent AlxGa1
xN, InxGa1
xN, and AlxIn1
xN grown on relaxed Ga-polarity GaN. Polarization-induced

bound interface charge (positive in n-type samples and negative in p-type samples) is screened by free electrons in n-type and holes in p-type samples, respectively, leading to twodimensional gas. Note that the interface charge can be converted into carrier concentration by dividing it with the electronic charge of q ¼ 1.602 · 10
19 C. Courtesy of O. Ambacher.

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j 2 Electronic Band Structure and Polarization Effects

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2.0 Ga-Polarity on GaN

2DEG

Bound interface density s /q (1014cm–2)

1.5

1.0 AlN 0.5

N A l xG a1 –x

GaN 0.0

–0.5 In

x Ga

N

I n 1 –x

Al x

–1.0

1– x

N

InN

–1.5

InN 2DHG

–2.0 0

0.2

0.4

0.6

0.8

1.0

Molar fraction, x (b) Figure 2.45 (Continued )

charge, which is bound, leads to a 2DEG with a sheet carrier concentration close to the concentration of the bound interface density þs/e. Now that a conceptual issues related to polarization has been discussed, the attention can now be turned to calculating the sheet charge distribution and band structure at a single heterointerface typically used for MODFET. The available data indicate that GaN channel devices are the only viable ones, even though In-containing channels may someday work well as well. For the moment, the likely GaN channel MODFET structures utilize AlxGa1
xN, AlxIn1
xN, or perhaps the quaternary. The criterion is that the bandgap of AxB1
xN ternary must be larger than that of GaN. As discussed often, the AlxIn1
xN ternary lattice matches GaN; however, the reported values for lattice matching composition vary. For example, AlN molar fraction of 71% (corresponds to InN molar fraction of 29%) and 82–83% (corresponds to InN molar fraction of 18–17%, which is the more likely lattice matching composition) have been reported to lattice match GaN [451]. To treat the MODFET interface charge problem,

2.12 Polarization Effects

the bandgap of nitride semiconductor alloys, which are discussed in Section 1.5 in detail, must be known. For completeness, the expressions for the compositional dependence of the alloy bandgap are repeated below. It should be pointed out that the bandgap bowing parameter data bAx B1
x N for AlGaN are converging onto a value of nearly 1 eV even though early figures spanned from
0.8 eV (upward bending) to þ 2.6 eV (downward bending). The data for InGaN are fluidic also in part because of reasons having to do with difficulties in determining the composition and bandgap, detailed in Section 1.5.2, and also lack of high-quality samples with composition midway in the alloy range. For example, optical reflectivity measurements together with PL data corrected for Stokes shift led to a bInGaN ¼ 2.5 0.7 eV for 0.9 eV bandgap of InN and bInGaN ¼ 3.5 0.7 eV for 1.9 eV InN bandgap [463]. PL data alone in another report states the bowing parameter to be 4.5 eV [464]. The figure determined by optical transmission measurements is 8.4 eV [465], while the theory indicates 1.2 eV [466]. Other experimental values for lower InN molar fraction end of the ternary are near null, meaning linear variation of the bandgap [467,468]. The reported data in aggregate, taking into consideration the small bandgap of InN, leads to a value of 2.53, as discussed in Section 1.5.2. The value for the bowing parameter in AlInN is somewhat too controversial. For example, one particular theoretical report points the bowing parameter to be 2.53 eV [469] while a value of 3.1 eV was determined by fitting the bandgap of this alloy determined by photoreflection [470], 2.384 eV by absorption measurements but by using 2.0 eV for the bandgap of InN, and 5.9 eV for AlN [471] and 5.4 eV in a review article where 1.95 eV was used for InN bandgap [84]. However, when all the available data are considered in aggregate with 0.7 eV bandgap for InN, a bowing parameter of about 3 eV appears to be a very good value. g

g

g

E Alx Ga1
x N ¼ xE AlN þ ð1
xÞE GaN
bAlGaN xð1
xÞ ¼ 6:1x þ 3:42ð1
xÞ
xð1
xÞ eV; g E Inx Ga1
x N

g

g

¼ xE InN þ ð1
xÞE GaN
bInGaN xð1
xÞ ¼ 6:1x þ 0:7ð1
xÞ
1:43xð1
xÞ eV;

g

ð2:108Þ

g

E Alx InN ¼ xE AlN þ ð1
xÞE EInN
bAlInN xð1
xÞ ¼ 6:1x þ 0:7ð1
xÞ
bAlInN xð1
xÞ eV: The presence of displacement or polarization gradient leads to a volume charge density rv given by !

!

!

r: D ¼ r:ðe E þ P total Þ ¼
rv

!

in MKS units: For cgs; add 4p before P : ð2:109Þ

In a one-dimensional system, which is considered here along the c-axis, and utilizing E ¼
dV/dz, Equation 2.109 can be rewritten as the Poisson’s equation in the form of dD=dz ¼ d=dzð
eðzÞdV=dz þ P total ðzÞÞ ¼
qrz ¼
q½N Dþ ðzÞ þ pðzÞ
nðzÞ
N A
ðzÞ:

ð2:110Þ

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j 2 Electronic Band Structure and Polarization Effects

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The term on the right-hand side of Equation 2.110 represents the volume density of net charge, and the position-dependent quantities D, e, and V are the displacement field, dielectric constant, and potential, respectively. The term Ptotal is the position-dependent total transverse polarization (along the c-axis and perpendicular to the interfaces). N Dþ ðzÞ and N A
ðzÞ represent the ionized donor and acceptor concentrations, and p(z) and n(z) represent the hole and electron concentrations, respectively. The effects of composition, polarization, and free-carrier screening are thus fully included. With the knowledge of the band profile for both the valence band and the conduction band, one can determine the electronic states in the heterostructure by solving the Schr€odinger (Equation 2.59) equation, as a function of the spatial coordinate z. In the effective mass approximation, one needs to solve the following eigenvalues problem:  
2 d 1 dCi ðzÞ h
þ V c ðzÞCi ðzÞ ¼ E i Ci ðzÞ; 2m0 dz mz dz

ð2:111Þ

for which the appropriate choice of the effective masses in the conduction band and in the valence band is done by preliminary calculation using the tight binding approximation. The solution of Equation 2.111 determines the eigenstates Ei in the conduction band and in the valence band, and the corresponding eigenfunctions Ci. It is imperative to note that free carriers cannot eradicate the polarization charge, which is bound and invariable (unless structural changes are made), but one can screen it to a degree determined by carrier concentration. Likewise, polarization charge is bound charge and cannot by itself be the source of free carriers, but it can cause a redistribution of free carriers that would tend to screen the polarization charge. In addition to the polarization charge related parameters that must be known to solve the Schr€odinger and Poisson equations, one must also know the dielectric constants, doping levels, effective masses, and band discontinuities, in addition to compositions and the layer thicknesses used. The band discontinuity issue is discussed in Volume 2, Chapter 1, but suffice it to say that the bandgap of the ternaries is found from Equation 2.108 with the appropriate bowing parameter, and a certain fraction of the total bandgap difference is assigned to the conduction band and the rest to the valence band. To mitigate the process, the band alignment for all the nitride heterostructures is of type I, which means that the larger bandgap straddles the smaller bandgap one. Typically, 60–70% of the total band discontinuity is assigned to the conduction band. A linear interpolation for the effective masses, given in Chapter 1, in ternaries from binary end points can be used. While the tables provided in Chapter 1 list the dielectric constants for binaries, knowledge of them leads to the ternary dielectric constants via linear interpolation. Relative dielectric constants of 10.28, 10.31, and 14.61 (compares with 15.1 presented in Chapter 1) for dielectric constants in GaN, AlN, and InN, respectively, have been calculated [472]. The GaN dielectric constant is nominally larger than that of AlN, but in this particular investigation the data are as listed above. Following the so calculated figures and applying the linear interpolation leads to dielectric constant for ternaries.

2.12 Polarization Effects

eAlx Ga1
x N =e0 ¼ 10:28 þ 0:03x; eInx Ga1
x N =e0 ¼ 10:28 þ 4:33x; eAlx In1
x N =e0 ¼ 14:61
4:33x: Using the values for the binary dielectric constants tabulated in Chapter 1, the linear interpolation scheme for the ternary dielectric constants lead to eAlx Ga1
x N =e0 ¼ 10:4
1:9x; eInx Ga1
x N =e0 ¼ 10:4 þ 4:9x; eAlx In1
x N =e0 ¼ 15:3
4:9x:

ð2:112Þ

In dealing with devices such as MODFETs where a Schottky barrier is placed on top, the Poisson’s equation, through boundary conditions, is affected by the metal barrier height as well as the applied bias on the metal. The Schottky barrier height for the ternaries can also be interpolated from the binary end points. It should also be noted that the barrier height depends on the particular metal used, the details of which is discussed in Volume 2, Chapter 1. Because the Fermi level on the surface of GaN is not fully pinned in that heavier metals with large work functions lead to larger barrier heights, it is plausible and natural to assume that the barrier height would increase with an increase in AlN content and decreases with InN content. However, for InN the Fermi level is most likely in the conduction band already, as in InSb and InAs. Limited studies of metal contact potential on nitride semiconductors make it difficult to state a molar fraction dependence of this parameter. However, Ti and Ni contacts on Al0.15Ga0.85N have been studied and were compared to those on GaN. The barriers heights for Ti and Ni increased on Al0.15Ga0.85N [473]. Comparing the data deduced for Ni using I–V, C–V, and photoemission methods, barrier heights of 0.95 (0.84 eV if not corrected for the nonideal ideality factor), 0.96, and 0.91 eV, respectively, have been obtained. For Al0.15Ga0.85N, the same figures are 1.25 (1.03 eV if not corrected for the nonideal ideality factor), 1.26, and 1.28 in order, using the same methods. These figures represent an increase of about 0.3 eV in barrier height for Al0.15Ga0.85N over GaN. Assuming that Ni is used for Schottky barriers, the barrier height for GaN is 0.95 eV, and more boldly reported figures for one mole fraction do actually represent a figure consistent with a linear interpolation, one can express the molar fraction dependence of the barrier height on AlxGa1
xN as fAlx Ga1
x N ¼ 0:95 þ 2xV:

ð2:113Þ

Following Ref. [84], the linear interpolation for the other two ternaries modified for barrier height on GaN used here are fInx Ga1
x N ¼ 0:95
0:36xV; fAlx In1
x N ¼ 0:59 þ 1:36xV:

ð2:114Þ

These expressions lead to barrier heights (qf or ef) of 0.95, 2.95, and 0.59 eV on GaN, AlN, and InN, respectively. With the exception of the value for GaN, the rest is really speculative at this point.

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j 2 Electronic Band Structure and Polarization Effects

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2.12.4.1 Ga-Polarity Single AlGaN/GaN Interface Returning to single-interface structures, such as those used in MODFETs (or HFETs), the total interface charge, for example, at a gated AlxGa1
xN and GaN interface, rs, for an n-type case would be sum of total polarization charge rp and free-carrier charge ns.    er e0 E Fi
DE C ; ð2:115Þ rs ¼ rp
ns ¼ rp
V G
fB
V p2 þ qdAlGaN q

where VG is the applied gate bias in terms of V, fB is the Schottky barrier height in terms of V, on AlxGa1
xN, EFi is the Fermi level in GaN at the interface with respect to the edge of conduction at the interface, DEC is the conduction band discontinuity between the ternary AlxGa1
xN and GaN, and Vp2 is the voltage drop across the doped AlGaN, which in turn is given by V p2 ¼ qN d d2d =2er e0 where Nd is the donor concentration in AlxGa1
xN, all assumed to be ionized, dd is the thickness of the doped AlGaN, and ere0 is the dielectric constant of AlxGa1
xN. Vp2 is negative for depleting voltage in the case of which it would add to the Schottky barrier height. We should point out at this stage that the form of Equation 2.115 is good for determining the sheet carrier concentration at the interface, but when used for FETs both the polarization and the charge induced by doping must be lumped into the threshold or off voltage. For details see Volume 3, Chapter 3. For a p-type semiconductor, the semiconductor charge as well as the reference for the Fermi level and the band discontinuity should be that at the valence band edges [88,474]. The effect of any undoped AlxGa1
xN layer designated as having a thickness of di is small because its thickness is several nanometers and is neglected in Equation 2.115. This setback layer was originally employed by the author in the GaAs system to further reduce remote Coulomb scattering for increased mobility. A detailed treatment including the effect of di can be found in Ref. [474]. For an undoped AlGaN/GaN heterostructure, the Vp2 term can be set to zero. In this case the boundary conditions for potential or the Fermi level are made consistent with the Schottky barrier height on the Alx Ga1
x N surface. In the bulk of the structure, due the special nature of the quasi-triangular barrier at the interface, shown in Figure 2.46, the Fermi level is generally taken to be near the midgap of the smaller bandgap material, which in this case is GaN. As in the case of quantum wells, the one-dimensional Schr€ odinger–Poisson solver can be iteratively used to determine the band and carrier profile (simultaneous solution of Equations 2.110 and 2.111 in an iterative mode either in the tight binding realm or the effective mass approximation, but in the current example it is in the effective mass realm). The bound charge can be represented by a thin and heavily doped interfacial layer, the thickness of which is about 1 nm or less, keeping in mind that the total charge associated with such a fictitious layer must be equal to the bound charge. In one investigation, a thickness for this fictitious layer of 0.6 nm was used. To reiterate, to solve this pair of equations boundary conditions at the interface and surface as well as the structural parameters must be known. In a typical undoped AlGaN/GaN MODFET, the GaN buffer layer is unintentionally doped with a level of ND  1016 cm
3, and the thickness of this buffer layer spans 1–3 mm. The barrier thickness spans 10–20 nm. Because the structure is not intentionally doped,

2.12 Polarization Effects

Ec

AlGaN

GaN

di

qφB

Ec ΔEc

-qVG

E1 E0

EF

EFi

dd

x=0 Figure 2.46 Conduction band edge of what is generically referred to as a modulation-doped structure based on the AlGaN/GaN system. The origin of the interface charge is aggregate due to polarization and free carriers. In Ga-polarity samples, AlGaN grown on GaN produces polarization charge (due to spontaneous polarization and piezoelectric polarization

x because AlGaN is under tensile strain), causing accumulation of electrons at the interface in addition to any electron donated by the any donor impurities, intentional or unintentional, in AlGaN. The diagram is shown for a doped AlGaN, the doped portion of which is indicated by dd and the undoped part is indicated by di.

it is assumed that unintentional impurities in the form of contamination or native defects, such as O, Si, VN, are responsible for supplying the free carriers, which is measured with the accompanying assumption that it is sufficient to completely screen the positive polarization charge, þr/q, at the interface [476]. The influence of doping specifically in the case of AlGaN/GaN has been treated, for example, by Chu et al. [475]. In short, the sheet charge at the interface of such an undoped structure is dominated by the polarization-induced charge. On Ga-polarity surfaces, this charge increases with increasing AlN molar fraction in the barrier because both the piezo and spontaneous components of the polarization charge increase, assuming of course that the barrier is coherently strained. The conduction band and electron concentration profiles for an undoped Ga-face Al0.3Ga0.7N/GaN (30 nm/ 2 mm) heterostructure with an Ni Schottky contact on top are shown in Figure 2.47a. An electric field strength of about 0.4 MV cm
1 in the barrier and a sheet electron concentration of 1.2 · 1013 cm
2 are induced by polarization. The bound sheet density and 2DEG sheet carrier density induced by polarization in heterostructures (identical to that shown in Figure 2.47a with the exception that the alloy compositions in the barrier has been changed) are shown in Figure 2.47b and c with solid lines for both the linear and nonlinear polarization cases. The underlying assumptions are that the GaN buffer layer is relaxed, the barriers are coherently strained, and the physical parameters of importance (Cij, eij, and Psp) linearly scale from binaries to ternaries.

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j 2 Electronic Band Structure and Polarization Effects

274

The estimated error depicted by the gray area in Figure 2.47c is primarily due to uncertainties in the barrier thickness and the conduction band offsets. The sheet carrier concentrations of 2DEGs confined in an AlxGa1
xN/GaN heterostructure for x ¼ 0.5 have been measured by C–V profiling using Ti/Al ohmic and Ni Schottky contacts [84]. The highest measured and calculated sheet carrier concentration for 2.0

+s E = 0.41 MV cm–1

1.5

2DEG

1.0

Energy (eV)

13

Ns = 1.2 10 cm –2

eΦ

1.23 eV

0.5

CB

0

Δ

0.30 eV

–0.5 (a)

0

10

ΔE

EF

c

0.38 eV

E0 = 0.17 eV

30 40 20 Depth (nm)

50

Figure 2.47 (a) Self-consistent calculation of the Schr€ odinger–Poisson equations for the conduction band edge and the electron density profile for an undoped Ga-polarity Al0.3Ga0.7N (30 nm)/GaN (2 mm) single-interface heterostructure. Also shown are the Schottky metal on the surface and the polarizationinduced surface and interface charges. (b) The polarization-induced bound interface charge density as a function of the alloy composition in the barrier for Ga-polarity AlxGa1xN/GaN heterostructures for the case of relaxed buffer layer and coherently strained barrier layer. The upper solid line for sheet charge corresponds to the case of linear interpolation of physical parameters (Cij, eij, and Psp) from the binary compounds. The lower solid line for the sheet charge, on the contrary, corresponds to the case of nonlinear extension of polarization from binary end points. The dashed lines depict the lagging sheet density with increasing molar fraction, x, due to partial relaxation, which is

60 accounted for by the measured degree of barrier relaxation into account. (c) 2DEG sheet carrier concentrations ns in unintentionally doped Gaface AlxGa1
xN/GaN heterostructures (with 30 nm AlGaN and 2000 nm GaN – the structure referred to in (b)) as obtained by C–V profiling versus alloy composition of the barrier (open symbols), compared with the theoretical predictions for the bound interface charge r/q calculated using (i) a linear interpolation between the macroscopic polarizations of the binary compounds (upper solid line) and (ii) the nonlinear piezoelectric and spontaneous polarization (lower solid line). The sheet charge of the 2DEGs is then calculated as in (ii), considering in addition the depletion by a Ni Schottky contact (dotted line). The dashed lines account for the experimentally observed strain relaxation of the barrier for x > about 0.4. Courtesy of O. Ambacher. The related details can be found in Refs [476,84,479].

2.12 Polarization Effects

1014

Bound interface density σ/q(cm–2)

Linear interpolation

Coherently strained

1013

Nonlinear

1012

0

0.4

0.2

(b)

0.6

0.8

1.0

Molar fraction, x

3.5

Sheet carrier concentration ns (1013 cm–2)

Relaxed

Ga-face Alx Ga1–x N/GaN (30/2000 nm)

3.0

Pseudomorphic

2.5 2.0 Partly relaxed

Linear interpolation

1.5 1.0 0.5

Nonlinear

0 (c)

0

0.1

0.3 0.4 0.2 Molar fraction, x

0.5

0.6

Figure 2.47 (Continued )

AlxGa1
xN/GaN heterostructures is 2 · 1013 cm
2 for x ¼ 0.37, as higher AlN molar fractions in the alloy for 30 nm barrier causes a partial relaxation, lowering the piezo contribution [476]. The extrapolation, indicated by dashed lines in Figure 2.47c, represents the case when the mole fraction is sufficiently large to cause some degree of relaxation. As expected, the sheet density decreases nonlinearly with reducing mole fraction for the 30 nm barrier modeled. Among the two curve that bend over, the upper one is the total calculated polarization charge and the lower one is the

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j 2 Electronic Band Structure and Polarization Effects

276

calculated screening charge using the one-dimensional Schr€ odinger–Poisson equations. When full or partial relaxation occurs, the piezo component of the polarization charge is reduced while the spontaneous component remains unchanged. The experimental data along with the error bars are shown in open circles. Overall, there is then a reduction of the charge. Likewise, the screening sheet carrier concentration is also reduced. The obvious conclusion that can be made is that a much better agreement is attained between calculations and experiments when the nonlinear polarization is used. The case of the nonlinear polarization is stronger in the 2DEG case than it is in the quantum well case due to the direct nature of the measurement. Ridley et al. [477] presented an empirical expression relating the sheet carrier concentration for a nominally undoped AlGaN/GaN single-interface heterostructure having barrier thicknesses of greater than 15 nm and AlN molar fractions over 6%. ns ðxÞ ¼ ½
0:169 þ 2:61x þ 4:50x 2 1013 cm
2 :

ð2:116Þ

Naturally, the sheet charge becomes affected by parameters other than the AlN molar fraction when the barriers are made much thinner, as can be deduced from Equation 2.115. Using an expression similar to Equation 2.115, Ambacher et al. [84] plotted the dependence of the sheet density of barrier thickness for several AlN molar fractions, namely, x ¼ 0.15, 0.30, and 0.45. A priori, it is clear that beyond a certain thickness of the barrier, the density should saturate even if coherent strain prevails, as shown Figure 2.48. Ambacher et al. [84] also measured the sheet density for a set of samples with x ¼ 0.3 by C–V profiling for barrier thicknesses spanning the range of 1–50 nm, and presented the data together with those from other reports. 2.12.4.2 Ga-Polarity Single AlxIn1
xN/GaN Interface This system, if for nothing else, is of importance because the entire structure can be lattice matched, leading to vanishing piezoelectric polarization and thus allowing one to investigate and probe only the spontaneous polarization in undoped structures. Doing so would enhance our confidence in spontaneous polarization calculations, as there are fewer parameters to be determined and thus less uncertainty. For high concentrations of Al (for x > 0.6), the bandgap of AlxIn1
xN is larger than that of GaN, and if used in conjunction with Ga-polarity GaN, a 2DEG would result due to polarization or doping in AlxIn1
xN or both. For lattice-matched conditions where the x-value is about 0.82, the bandgap of the AlxIn1
xN alloy is about 4.7 eV to produce sufficiently large band discontinuity with GaN, imperative for MODFET structures. Ambacher et al. [84] prepared coherently strained AlxIn1
xN (50 nm)/ GaN (540 nm) single-interface heterostructures with Al concentrations between 0.78 and 0.88, and applied X-ray diffraction to determine the structural state of GaN, meaning relaxed or strained (GaN grown on sapphire is typically under compressive strain due to thermal mismatch). Through the in-plane and out-of-plane strain (which can both be measured using reciprocal space X-ray diffraction mapping) or the lattice constants, the mole fraction of the barrier and, through repeated attempts, the mole fraction giving rise to lattice matching conditions can be determined. Ambacher et al. [84] determined the lattice matching composition of AlxIn1
xN to

2.12 Polarization Effects

10

14

Sheet carrier concentration ns (cm–2)

x = 0.45

10

13

= 0.3

= 0.15

1012

Ga-face AlGaN/GaN

10

11

0

10

20

30

40

50

60

d AlGaN (nm) Figure 2.48 Barrier thickness, d, dependence of sheet density in nominally undoped and coherently strained AlxGa1
xN/GaN heterointerfaces for x ¼ 0.15, 0.30, and 0.45 (solid lines). Experimental data available for x ¼ 0.3 measured by C–V profiling for barrier thicknesses spanning 1 and 50 nm, representing an aggregate from several reports, are shown [84].

GaN to be x ¼ (0.83 0.01), as shown in Figure 2.49, which is compared with other experimental values of 0.82–0.83 [451]. The weak compressive strain in GaN was determined to be e ¼
1.9 · 10
3, which would result in a piezoelectric polarization of 1.5 · 10
3 Cm
2. The residual strain in GaN would lead to a bound sheet density of only
1012 cm
2, which is much smaller than the 1013 cm
2 electron sheet density. This implies explicitly that interface charge is dominated by the gradient in the spontaneous polarization across the GaN/AlxIn1
xN interface, as show in Figure 2.50, which shows the calculated bound sheet density induced by a gradient in spontaneous polarization (upper dashed line) and strain-induced piezoelectric (lower solid line) for a range of compositions near the lattice matching conditions and the resultant 2DEG sheet carrier density (upper solid line). Also shown are the experimental 2DEG densities obtained by C–V measurements. The polarization and thus the sheet density have been obtained by a linear interpolation of the physical parameters (Cij, eij, and PSP) of the binary compounds but by taking the nonlinearity of Ppz and PSP into account. The calculations confirm the obvious in that the measured high electron sheet density can be accounted for by the spontaneous polarization charge.

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j 2 Electronic Band Structure and Polarization Effects

278

6

cAlInN (x) 4 aAlInN (x)

Strain, e (10–3)

2 Tensile strain 0 Compressive strain –2

–4

Lattice matched –6 0.75

0.85 0.80 Molar fraction, x

Figure 2.49 Strain in AlxIn1
xN barrier along the c-axis and on the basal plane as determined from the measured lattice constants, cAlxIn1
xN and cAlxIn1
xN by high-resolution X-ray diffraction reciprocal mapping. AlxIn1
xN for x ¼ 0.83 can be

0.90 grown lattice matched to GaN, leading to vanishing piezoelectric polarization due to misfit. However, residual strain, due to thermal mismatch between GaN and the substrate, can still induce piezoelectric polarization [84].

Note that for Ga-polarity samples and AlN compositions less than the lattice match conditions, the AlxIn1
xN barrier layer is under compressive strain in-plane and therefore the piezo-induced polarization is opposite in sign to that for spontaneous polarization. When the AlN molar fraction exceeds that for matching condition, the in-plane strain is tensile in the case of which the two polarization charges are additive. Near the lattice matching conditions, the calculated sheet density is about 2.95 · 1013 cm
2 and falls above the measured values. The reason for this discrepancy could be related to samples themselves, either in the form of defects near the interface, surface adsorbates, and/or inhomogeneities caused by In, and underestimation of the bowing parameters associated with the polarization charge. 2.12.5 Polarization in Quantum Wells

For multiple interface heterostructures, the sheet carrier density and barrier thickness, as well as the width quantum wells, are of interest because the total potential drop across the structure is directly proportional to the product of polarization field and well width in constant field approximation if free-carrier screening is neglected.

2.12 Polarization Effects

4

s sp e (P )

2

n (x) S

1

0

–1

–2 0.75

s z (P ) E

0.80

Lattice matched

Sheet carrier concentration (1013cm–2)

3

0.85

0.90

Molar fraction, x Figure 2.50 Compositional dependence of the spontaneous polarization (dashed line) and piezoelectric polarization (lower solid line) induced charge density reduced to interface sheet charge in an AlxIn1
xN/GaN heterojunction using the nonlinear interpolation for polarization discussed in the text. For AlN molar fractions below the lattice matching

conditions, the two polarization charges oppose and above the lattice matching value, they add. The calculated sheet electron concentration (upper solid line) and the measured sheet electron density are also shown, the latter of which has been obtained by both Hall effect (solid symbols) and C–V profiling (open symbols) [84].

This issue goes to the heart of lasers, particularly in the stages as the gain is built up, in that the well widths greater than about 5 nm are not used. At higher injection levels, the polarization-induced field is screened pretty much [458]. To illustrate the point, the conduction band profile of a 10 nm GaN/In0.2Ga0.8N quantum well as calculated by Della Salla et al. [461] is shown in Figure 2.51 for several sheet densities. Even at a substantial sheet density of n2D ¼ 5 · 1012 cm
2, a nearly uniform electrostatic field of strength 2.5 MV cm
1 is still present in the well. One needs to increase n2D to 5 · 1013 cm
2 before recovering the quasi-field-free shape of the quantum well that is needed for lasers. This is achieved much earlier in thinner quantum wells. In wells, the electrons and holes are indeed spatially separated by the polarization field, but the free carrier induced field acts to cancel the polarization field, which is efficient for high sheet densities. This reestablishes the efficient electron–hole recombination. Time-resolved PL (TRPL) is a wonderful way of seeing the effect of polarization. Due to band bending induced by polarization, when the optical excitation pulse is turned off, the free-carrier density goes down and so does the screening. As a result, a

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j 2 Electronic Band Structure and Polarization Effects

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[0 0 0 1]

Conduction band edge energy (eV)

4.0

GaN/In0.2Ga0.8N

3.5 3.0 2.5 2.0

n2 D = 1013cm–2 0.5 1.0 2.0 5.0

1.5 1.0 0.5

0

10

20 30 Depth (nm)

40

50

Figure 2.51 Conduction band profile of a 10 nm GaN/In0.2Ga0.8N quantum well for various levels of free carriers ranging from 5 · 1012 cm
2 to 5 · 1013 cm
2 either present by doping or injection. The effect of polarization is nearly all but wiped out for the largest sheet electron concentration [461].

red shift accompanied with increased carrier lifetime due to lowering of the overlap between the electron and hole wave function occurs, as they are pushed to the opposing end of each well [478]. For details, the reader is referred to Volume 2, Chapter 5. 2.12.5.1 Nonlinear Polarization in Quantum Wells Asinthecaseofbulk binariesand alloys, the polarization issue inheterostructuresneeds a revisit to consider the nonlinearities discussed above in the context of binary and alloy nitride bulk layers. Assuming that the ternary nitride alloys have random microscopic structure, the spontaneous polarization of random ternary nitride alloys, in unit C m
2, has been expressed by Fiorentini et al. [479] to the second order in the composition parameter x, as expressed in Equation 2.95 but repeated here for convenience. sp

PAlx Ga1
x N ¼
0:09x
0:034ð1
xÞ þ 0:019xð1
xÞ; sp

P Inx Ga1
x N ¼
0:042x
0:034ð1
xÞ þ 0:038xð1
xÞ; sp P Alx In1
x N

ð2:117Þ

¼
0:09x
0:042ð1
xÞ þ 0:071xð1
xÞ:

The first two terms in all three equations indicated in Equation 2.117 are the usual linear interpolation between the binary compounds represented by Equation 2.86. However, the third term is the so-called bowing term encompassing the quadratic nonlinearity as in the case of the bandgap bowing parameter discussed in Section 1.5. The coefficient of the third term is the bowing parameter discussed in conjunction with Equations 2.95 and 2.92. Higher order terms are neglected, but their effect was estimated to be smaller than 10% in the worst case being the AlInN alloy [453].

2.12 Polarization Effects

For piezoelectric polarization, it was shown [453] in conjunction with Equation 2.96 that Vegard’s law holds provided that the appreciable nonlinearity of the bulk piezopolarization of the component binaries as a function of strain is accounted for. Doing so has led, in general, to a good agreement with experimental results [480]. For a model AlxGa1
xN alloy, the piezoelectric polarization can be related to the binary end points using the Vegard’s law, but recognizing that the terms for the bulk binaries must contain the nonlinear terms as indicated in Equation 2.96. Such polarizations can be expressed accurately and compactly (in units of C m
2) as pe

PAlN ¼
1:808e þ 5:624e2 pe

PAlN ¼
1:808e
7:888e2 pe PGaN ¼ pe PInN ¼

for e < 0; for e > 0;


0:918e þ 9:541e ; 2

ð2:118Þ


1:373e þ 7:559e2 ;

as a function of the basal strain of the alloy layer in question, with a(x) and asubs or a0 as the lattice constants of the unstrained alloy at composition x and of the relaxed buffer layer or the substrate. eðxÞ ¼ ½asubst
aðxÞ=aðxÞ:

ð2:119Þ

In the case of pseudomorphic growth on GaN buffer layers, basal strain e can be calculated directly from the lattice constants, which are found to follow Vegard’s law as a function of composition (depending on the lattice constants used): aAlx Ga1
x N ðxÞ ¼ aGaN
xðaGaN
aAlN Þ ¼ 0:31986
0:00891x nm; aInx Ga1
x N ðxÞ ¼ aGaN þ xðaInN
aGaN Þ ¼ 0:31986 þ 0:03862x nm; aAlx In1
x N ðxÞ ¼ aInN
xðaInN
aAlN Þ ¼ 0:35848
0:04753x nm:

ð2:120Þ

The combination of Equations 2.118–2.120 provide a convenient way of determining the polarization dependence on basal strain. The coefficients in Equation 2.118 are related (not equal) to piezoelectric constants and come about from the ab initio calculations [453]. The polarization charge values calculated using Equations 2.118–2.120 for heterostructures can be used together with a self-consistent Schr€ odinger–Poisson solver based, for example, on effective mass theory (not as accurate but efficient) or tight binding (more accurate but computation intensive) to determine field and charge distribution in the entire heterostructure as well as the effect of free carriers [448,460]. Typically, for a given structure or set of structures, two classes of observable interest can be simulated and compared with experiment. Experimental confirmation for the nonlinear theory can be garnered from quantum wells by probing the red shift for a given sized well and its dependence on excitation, and measuring the magnitude and gate voltage dependence of the interface sheet carrier concentration at an AlGaN–GaN interface with the aid of the one-dimensional self-consistent simultaneous solution of Schr€ odinger and Poisson equations in the effective mass approximation. Additional confidence can be estimated by repeating these experiments for these structures where, for example,

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j 2 Electronic Band Structure and Polarization Effects

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only the barrier mole fraction is changed [479]. Fortunately, the results are very sensitive to the values of the polarization in the different layers, and therefore the built-in field or the screening charge. The particular bowing parameters in the polarization expression yielding the best agreement with experiment can be assumed to be valid. The point should be made that the C–V measurements are probably the most direct and optical shifts the least direct. The C–V data and the simulations based on the theories described above agree very well [479]. The attempt here is to show that inclusion of the nonlinear effects in calculations lead to a better agreement between the experiments and theory in quantum wells and AlGaN/GaN heterointerfaces. In terms of the quantum confined stark effect (QCSE), the photoluminescence energy of AlxGa1
xN/GaN QWs with well thicknesses of 1,1.5, 2.5, 4, 6, and 8 nm, and with barrier alloy composition of x ¼ 0.08, 0.13, 0.17, and 0.27, respectively, reported in Ref. [481] were used to determine the polarization-induced electric field. The field determined by measuring red shift from the PL transition energies versus the mole fraction in the barrier of quantum wells as well as calculated values from the polarization charge determined using the linear and the nonlinear approach are shown in Figure 2.52. In both types of calculations, the electric field in the GaN QWs was obtained by solving self-consistently the coupled Schr€ odinger–Poisson equations including polarization-induced interface charges. In the first approach the polarization of the AlGaN barriers that are assumed under tensile coherent strain, as the bulk buffer layers were GaN, is determined by a linear interpolation between the elastic 2.5 Linear interpolation

Electric field (MV cm–1)

2.0

Nonlinear approach

1.5

1.0

Ga-polar AlxGa1–xN/GaN QWs

0.5

0 0

0.1 0.2 0.3 0.4 0.5 0.6

AIxGa1–x N molar fraction, x Figure 2.52 Polarization-induced electric fields in high resistivity and Ga-face AlxGa1
xN/GaN MQWs versus the alloy composition of the barrier. The upper line represents the field predicted by linear interpolation of binary compound polarization. The lower line represents the calculated field using the nonlinear polarization concept. Open circles

are deduced, via self-consistent effective mass calculations, from the polarization-induced Stark shift of excitonic recombination reported in Ref. [481] via a self-consistent effective mass calculation. In addition, the experimental data published by Langer et al. [482] and Kim et al. [483] are also shown in filled circles [479].

2.12 Polarization Effects

and piezoelectric constants and the spontaneous polarizations of the binary compounds AlN and GaN. In the second approach, the nonlinearity of the polarization of AlGaN as described by Equations 2.117 and 2.118 was taken into consideration. Clearly, the electric field calculated including the nonlinearity in polarization versus barrier alloy composition does much better at reproducing the experimental data taken from Ref. [481]. In addition, the experimental data published by Langer et al. [482] and Kim et al. [483] are also shown in filled circles. A more convincing arguments for nonlinearity in polarization in the context of polarization charge and resultant red shift in the spectra is made by pressuredependent measurements of the transition energies as presented by Vaschenko et al. [484]. They considered a quantum well system with background unintentional dopants and excitation-induced carriers in the case of which the field deviates from that indicated in Equation 2.105. total ðPtotal Vs W
PB Þ þ r þ LW þ LB LW eB þ LB eW      1 d
LW 1 d
LB
qN D þ ; 2 2 eW eB

E W ¼
LB

ð2:121Þ

where eW,B is the permittivity of the GaN wells and AlxGa1
xN barriers, respectively, (assumed to be independent of pressure in this work), LW,B are the cumulative thicknesses of the wells and the barriers in the MQW structure, r is the total twodimensional photogenerated charge density in the wells, Vs is the surface barrier potential determined as in Ref. [485], ND ¼ 1017 cm
3 is the assumed background doping concentration based on bulk GaN layers grown by MBE, which was used to produce the structure, and d is the distance from the barrier–buffer interface to the well where the field is calculated. The first term in Equation 2.121 is similar to Equation 2.105 but the former having an additional charge r (the total two-dimensional photogenerated charge) in the first term in addition to second and third terms. The second term represents the field due to the surface barrier potential which neglects the effect of dielectric discontinuity between AlGaN and GaN, and the third terms represents the effect of dopant on the field. If Vs, r, and ND were made to go to zero, Equation 2.121 reduces to Equation 2.105. To find PW
PB as a function of pressure, the experimentally measured PL peak energy variation, which was assumed to be representing the n ¼ 1 electron and heavyhole transitions, with well width was fitted to the calculated dependence of the same transition, The nonlinear behavior in this case is revealed by determining the PW
PB (barrier–well polarization difference) as a function of applied hydrostatic pressure. Figure 2.53 shows the fit to the measured PL peak energies in samples in which the AlN mole fraction varied as 0.2, 0.5, and 0.8 at a pressure of 5 GPa. The solid lines are calculations with the polarization-induced field, whereas the dashed ones are without such field. The experimental data are shown in open symbols. The good agreement between the calculations where PW
PB was treated as fitting parameter and the experiment underscores the crucial nature of the built-in electric field in the determination of the well width dependence of the ground-state energy.

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j 2 Electronic Band Structure and Polarization Effects

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4.2 4.0 3.8

Photon energy (eV)

3.6 3.4 3.2 3.0 2.8 2.6

x = 0.2 x = 0.5

2.4

x = 0.8

2.2 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Well width (nm) Figure 2.53 Well width dependence of the PL peak energy at 5 GPa. Open symbols correspond to the experimental points and the solid lines depict the fits to the experimental data obtained with PW
PB as an adjustable parameter. The dashed lines correspond to the e1
hh1 transitions calculated assuming zero field [484].

The polarization difference PW
PB and resulting electric field in the 2.9 nm wells for the x ¼ 0.5 MQW sample, as determined by the procedure described above, is presented in Figure 2.54. For the x ¼ 05, and the others that are not shown here, PW
PB noticeably increases with pressure, resulting in an increase of the built-in field in accordance with the mole fraction in the barrier reaching 0.76 MV cm
1 in the samples with x ¼ 0.5 at 8 GPa. Figure 2.54 also shows the calculated results of the pressure dependence of PW
PB in that the solid lines correspond to PW
PB calculated with the linear polarization [448]. This linear polarization model overestimates the values of PW
PB at atmospheric pressure and underestimates the pressure dependence as compared to experiments. The dashed-dotted lines represent calculations where only the volume-conserving strain dependence of the GaN and AlN piezoelectric coefficients is taken into account [456]. Clearly, this model agrees better with the experimental data than the linear polarization model that is consistent with the conclusions of Perlin et al. [486], where the pressure dependence of PL in GaN/Al0.13Ga0.87N QWs was found to be adequately described by the volumeconserving strain dependence of the piezoelectric coefficients. Lastly, the dashed line

2.12 Polarization Effects

0.058

3.6

0.056

0.052

3.2

0.050 3.0 0.048 2.8

0.046 0.044

Built in electric field, MV/cm

Polarization difference (Pw-PB), (cm–2)

3.4 0.054

2.6

0.042 2.4

0.040

T = 35 K x = 0.5

0.038 0

2

4

6

8

10

2.2

Pressure (GPa) Figure 2.54 Pressure dependence of PW
PB and corresponding electric field in 2.9 nm AlxGa1
xN/GaN MQWs with x ¼ 0.5. The open circles represent the calculated points obtained from the fit to the PL data as shown in Figure 2.53. The solid line corresponds to PW
PB calculated with the linear polarization that overestimates the data obtained from experiments at zero pressures and underestimates the slope of the pressure dependence of polarization. The dashed-dotted

line shows the calculations, where only the volume-conserving strain dependence of the GaN and AlN piezoelectric coefficients is considered, which overestimates the differential polarization but does very well in terms of the pressure dependence of the differential polarization. Lastly, the dashed line shows the result of calculations using the nonlinear polarization behavior with the distinctly best overall fit to the experimentally determined values. The dotted line is a guide to eye [484].

shows the result of calculations where the nonlinear behavior of both the spontaneous and piezoelectric polarizations had been taken into account using the results of Bernardini and Fiorentini [220,221]. Only the change in piezoelectric polarization due to hydrostatic compression of the ideal crystal and that due to the increase in the internal parameter u with pressure is considered here [86]. The spontaneous polarization bowing was included at p ¼ 0 [220,221]. Although this is only an approximation of the theory developed in Refs [220,221], this model also predicts the slope of the pressure dependence of PW
PB significantly larger than that of the linear model. In short, the added value of pressure dependence of differential polarization causes this parameter to be very sensitive to which polarization picture is employed for the transition energies in AlGaN/GaN MQWs to the point that one can clearly state that the nonlinear dependence of the piezoelectric polarization in GaN and AlN unequivocally predicts the experimental data best.

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j 2 Electronic Band Structure and Polarization Effects

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2.12.5.2 InGaN/GaN Quantum Wells Relatively better transport properties of InN as compared to GaN and possible translation of the same to InxGa1
xN are one of the draws for considering InxGa1
xN channels [229] for MODFETs, discussed briefly in Volume 3, Chapter 3. However, experimental results have so far been disappointing. On the contrary, InxGa1
xN proved to be the magical material for its high radiative recombination efficiency for optical emitters. As detailed in Volume 3, Chapters 1 and 2, all the highperformance optical emitters feature InxGa1
xN in one form or another in their active regions. Unlike the FETcases where GaN is used as the active layer and also the buffer layer where the buffer layer is high resistivity, the conductivity of InxGa1
xN is high, which rules out its use as the buffer layer in an InxGa1
xN channel FET. Consequently, even if InxGa1
xN were to be considered for the active layer, it must be grown on GaN buffer layers. As for the barrier layer, it can be made of GaN or some composition of AlxGa1
xN. In any case, the InxGa1
xN layer would be straddled by large bandgap material on both sides with the resultant single quantum well structure. Consequently, polarization in single-well InxGa1
xN quantum wells must be considered as discussed here. In a single quantum well structure, in addition to interface sheet charge that contains information on polarization and can be measured by electrical means, these structures also offer an additional avenue to probe the polarization-induced charge through quantum-confined Stark shift. While the optical properties of InxGa1
xN quantum wells are discussed in great detail in Volume 2, Chapter 5, including the associated QCSE and any Stokes shift, a discussion of single quantum well for the sole purpose of polarization effects is provided here for completeness. Ambacher et al. [84] reviewed the issue in conjunction with nominally undoped, n-type GaN/In0.13Ga0.87N/GaN structures with Ga-polarity, where the width of the quantum well dInx Ga1
x N varied between sp 0.9 and 54 nm. The spontaneous polarization of the InGaN layer, P In0:13 Ga0:87 N ,
2
is
0.031 Cm and points in the ½0 0 0 1 direction and the piezoelectric polarization is calculated to be 0.016 Cm
2, which is antiparallel to the spontaneous polarization because the InxGa1
xN quantum well is under compressive strain. The bound charge at the GaN–InGaN interface near the surface is positive and that at the lower interface is negative for Ga-polarity sample due the compressive strain that InxGa1
xN is under. This implies that the electron accumulation caused by screening would occur at the interface near the surface in n-type samples. If p-type samples were considered, a hole accumulation would occur at the other interface. The total polarization-induced interface sheet density is then given by sp

pz

sp

pz

ðPGaN þ P GaN Þ
ðPInx Ga1
x N þ PInx Ga1
x N Þ:

ð2:122Þ

Recognizing that GaN is relaxed in this case, the samples are grown on GaN buffer layers that are presumed to be relaxed and any residual strain is neglected. This leads pz to PGaN ¼ 0, and substituting the numerical values, one gets for the total polarization ð
0:034 þ 0Þ
ð
0:031 þ 0:016Þ Cm
2 ¼ q1:18  1013 Ccm
2 ;

or 1:18  1013 electrons cm
2 :

2.12 Polarization Effects

InGaN SQW x = 0.13

d = 54 nm

ns = 5 × 1012 cm–2

22

10

26 nm

Electron concentration (cm–3)

2.0 ×

× 10

1012

10

10 16 12 nm 2.6 × 1011

18

16

10

10

20

1018

2

20

10

10

10

22

1014

Electron concentration (cm-3)

10

10 12

4.3 nm 3.6 × 1011

+s –s

14

12

10

1

10

2

2

× 10

10

3

10 4

Depth (nm) Figure 2.55 Electron concentration profiles unintentionally doped, n-type GaN/In0.13Ga0.87N/GaN QWs having well widths of 4.3, 12, 26, and 54 nm, respectively, as deduced from C–V depth profiling [84].

The GaN/In0.13Ga0.87N/GaN the heterostructures with GaN top layer and In0.13Ga0.87N quantum well with thicknesses of 130 and 20 nm, respectively, have been examined by Ambacher et al. [84], and their electron profiles, as determined by C–V measurements, are shown in Figure 2.55. For well widths less than 4 nm, any electron accumulation was not observed, which implies failure to screen the bound charge fully. In fact, as the quantum well thickness was increased from 4.3 to 54 nm, the sheet carrier concentration increases from ns ¼ 3.6 · 1010 to 5 · 1012 cm
2. The increase in 2DEG sheet carrier concentration with well width, which follows a 2.5 power of well width, is much faster than the well width (the volume), which may be attributed to the system not being in equilibrium in terms of screening for thinner wells. Overall, the sheet density is about half of what is expected even in thick quantum wells. The discrepancy between predictions and experiments raises an interesting question if the polarization-induced charge is fully screened by electrons and ionized donors. It is clear that an electric field causes band bending, the total extent of which scales with thickness, which in turn causes a red shift in radiative recombination transition energy. Assuming that the polarization-induced charge at the free surface and the GaN–substrate interface are fully screened, an electric field in the constant

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j 2 Electronic Band Structure and Polarization Effects

288

field approximation forms, which can be expressed as E well ¼

ptotal e0 ðeInGaN r


1Þ

¼

total P total GaN
P InGaN : InGaN e0 ðer
1Þ

ð2:123Þ

The terms here have their usual meanings. The electric field has been predicted to be 2.2 MV cm
1 for In0.13Ga0.87N QWs, which causes band bending to the extent of turning otherwise square quantum well potential distribution to a triangular distribution in the constant field approximation. The resultant red shift in transition energy, Stark shift, can be related to the polarization charge through the field as  g

E InGaN
E energy ¼ qE well W þ

9phqE W pffiffiffi 8 2

2=3 

1 mnInGaN

þ

1 p

mInGaN

1=3 ; ð2:124Þ

where mnInGaN

p and mInGaN

are the electron and hole effective masses in InGaN. For an InN mole fraction of 0.13, these values will not be much different from those for GaN. W represents the InGaN well width. This would provide an additional means for determining the polarization charge with an all-optical method such as photoluminescence. Extreme care must then be exercised to be certain about the InN composition and its bandgap as well as making sure that the optical transitions observed are accurately related to the band edge, meaning their nature must be known. The experiments must also be conducted at low injection levels as to not screen the charge, and high injection levels to screen the charge. If screening is full and low injection levels is truly low, then the difference in energy between the very low and very high injection levels would represent the red shift due to the polarization charge, as was done by Ambacher et al. [84] who also determined the bandgap of InGaN by reflection measurements such as spectroscopic ellipsometry and room-temperature PL using an excitation energy of 3.41 eV, only absorbed by the InGaN well. The QW region was pumped with high-intensity light to generate a high density of electron–hole pairs to fully screen the polarization-induced charge, thus the field. For In0.13Ga0.87N QWs with widths of greater than 26 nm, a bandgap of 2.902 0.012 eV was measured, which is consistent with the value for bulk (2.946 eV). The spectroscopic ellipsometry and PL data show a monotonic increase in energy, reaching a value of 3.21 eV for a well width of 0.9 nm, as the Stark shift diminishes with decrease in well width while the quantum confinement increases, as depicted in Figure 2.56. The PL spectra measured by 3.81 eVexcitation with a pump power density of 103 Wcm
2 yield a PL peak position that is increasingly shifted to lower energies if the QW width is increased from 0.9 to 5.3 nm (Stark effect). Knowledge of this red shift in the PL peak position together with Equation 2.124 allows one to calculate the electric field strength as 0.83 MV cm
1. This field corresponds to a polarization-induced bound charge density of about 5 · 1012 cm
2 and compares well with the data of 0.62 MV cm
1 and charge of 3.7 · 1012 cm
2 for In0.12Ga0.88N/GaN MQWs by Wetzel et al. [487]. As can be deduced from Figure 2.56, the polarization-induced red shift (Stark shift) is not notable for QWs wider than 26 nm where the bandgap of InGaN and the PL

2.12 Polarization Effects

3.4

3.2 g

Energy (eV)

3.0

2.8

E InGaN

660 meV

E 0 e, 0 h

2.6 5 nm

2.4

2.2 10 –1

10 1 10 0 InGaN SQW thickness (nm)

Figure 2.56 The dependence of the effective bandgap (without confinement effects) of In0.13Ga0.87N used in QWs (solid line) and the energy of the radiative recombination in QWs assumed to be between electron ground state (E0e) and hole ground state (E0h) on well width. The effective bandgap is determined by

10 2 spectroscopic ellipsometry and RT PL with 3.41 eV excitation (high and low intensity for PL to account for the Stark shift). Also noted in the figure is that a change of 5 nm in the well thickness leads to a change of 660 meV change in the QW emission energy [84].

peak position agree within 40 meV. It also appears that polarization-induced charged may either be fully screened, negating the constant field approximation throughout the quantum well, or be the notorious In fluctuations discussed in details in Chapter 3 in terms of growth, Chapter 4 in terms of its effects on optical transitions, and Volume 3, Chapters 1 and 2 in terms of its effects on optical emitters. 2.12.6 Effect of Dislocations on Piezoelectric Polarization

To fully consider the true nature of III-Nitride epilayers, the effect of dislocations on piezoelectric polarization must be included. The case does hardly needs to be made owing to the high density of dislocation (edge, screw, and mixed) that alters the strain distribution and thus the piezoelectric polarization. In this vein, Shi et al. [488] calculated the piezoelectric polarization around c-oriented screw and edge dislocations in Wz GaN and found that polarization around screw dislocations (having Burgers vector h0 0 0 1i) has no z-component, which is similar to a magnetic field around a conducting line, so there is no charge induced either at the core or around the screw dislocation. In the case of edge dislocations (having Burgers vector 1/3h1
2 0i), in which the strain field is compressive on one side and tensile on

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j 2 Electronic Band Structure and Polarization Effects

290

(0 0 0 1) c-plane

-

Qs

+

Pe Ps bs be

+

-

Figure 2.57 Schematic diagram illustrating the dislocation geometry, associated polarizations and charge densities [488].

the other, calculations show that piezoelectric polarization has only the z-component and its divergence vanishes at the core and around the dislocation. But at the interface, it results in an effective surface charge for the difference in polarization across the interface, as shown in Figure 2.57. It was estimated that for a c-oriented edge dislocation the charge density could reach 1011 e cm
2 within 0.1 mm of the core. On the experimental side, electron holography was applied to investigate the built-in field caused by polarization [489] and charge distribution at the dislocation [490]. In undoped GaN, the holography results confirm that all dislocations are negatively charged and the line charge densities are calculated as 1 and 0.3 for screw and edge dislocations, respectively. Cai and Ponce [491] argued that screw dislocations always have relatively higher charge density. The electrical activity associated with extended defects is discussed in considerable detail in Section 4.1.6. We should be cognizant of the fact that not only extended defects in general cause local strain and therefore inhomogeneous strain but they also attract and often trap impurities, point defects, and free charge. Moreover, the strain component affects the charge through the piezoelectric component. 2.12.7 Thermal Mismatch Induced Strain

Having made the case that strain induced by both lattice mismatch and also by thermal mismatch plays a profound role in polarization, let us discuss the thermal mismatch case in some detail because the lattice mismatch component got good deal of coverage already. Within the realm of thermal mismatch, the dominant component is that introduced by the nonnative substrates used. The substrates used are many in number and kind, but the dominant ones are sapphire and SiC both of which introduce sizable thermal mismatch. An incomplete list of

2.12 Polarization Effects

substrates used, in addition to the ones cited, include other substrates, GaN, AlN, g-LiAlO2, b-LiGaO2, NdGaO3, Si, GaAs, MgO, ZnO, ScAlMgO4, MgAl2O4, and (La, Sr)(Al,Ta)O3. For a complete discussion refer to Chapter 4 and for a complete compilation refer to Ref. [492]. In a nut shell, the thermal stress is relatively small if GaN layers are grown on AlN, SiC, ScAlMgO4, Si, GaAs, and ZnO. However, it is relatively larger when all other substrates are used. The stress in GaN is compressive for all the substrates substrate except Si and SiC, which due to their small expansion coefficients give rise to tensile strain, which is notorious for layer cracking. The overall stress remains the same for nominally thick GaN layers when a sapphire substrate is used with or without an AlN buffer layer but reduces by an order when a 6H-SiC substrate is used with an AlN buffer layer. In these pages, the treatment reported in Refs [492,493], which followed the model of Olsen and Ettenberg [494], for an arbitrary stack of epitaxial layers on a substrate is followed. The three-layer heterostructure stack layer model used to study the thermal-induced strain is shown in Figure 2.58 with length L, width W, Young’s modulus Ei, layer thickness ti, moments Mi, coefficient of thermal expansion a, forces Fi, strain e, and curvature k. The index i ¼ 1 represents the substrate and i ¼ 2, 3, 4, . . . represents the epitaxial layers.

Figure 2.58 (a) A set of unstrained platelets of thicknesses ti (i ¼ 1, 2, 3, . . .) used to construct the composite layer. (b) A case depicting the films and the substrate to be in contact, representing the epitaxial growth of two different films on a substrate with a larger thermal

expansion coefficient (positive bending) as compared to the films. The dimensions are characterized by length L (into the page), width W, and thickness ti. The terms Ei, Mi, and Fi represent the Young’s modulus, moments (bending forces), and forces, respectively.

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j 2 Electronic Band Structure and Polarization Effects

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For one epitaxial layer with i ¼ 1 representing the substrate and i ¼ 2 representing the epitaxial layer on it, one can surmise that from equilibrium of force, the total force must vanish as F 1 þ F 2 ¼ 0:

ð2:125Þ

Similarly, equilibrium of moments leads to

Wk t1 t2  ¼ 0: ðE 1 t31 þ E 2 t32 Þ þ F 1 þ F 2 t1 þ 2 2 12

ð2:126Þ

The strain at the interface between the film and the substrate in terms of mechanical parameters can be expressed as e¼

F2 F1 ðt1 þ t2 Þk

: E 2 t2 L E 1 t1 L 2

ð2:127Þ

The strain is also determined from the difference between the coefficients of thermal expansion (CTE) of the substrate and the film multiplied by the difference between growth and room temperature, DT. e ¼ DTða1
a2 Þ:

ð2:128Þ

Considering a negligibly small bending stress, the one-dimensional stress in the ith epitaxial layer is then taken as constant and given by si ð1DÞ ¼

Fi ; ti W

ð2:129Þ

where si(1D) is the one-dimensional stress. By assuming a spherical bending for a square sample (meaning L W), the twodimensional stress can be deduced from the one-dimensional stress expression of Equation 2.129 as follows: si ð2DÞ ¼

si ð1DÞ

ð1
vÞ
1

;

ð2:130Þ

where si(2D) and n are the two-dimensional stress and Poisson’s ratio, respectively. For two epitaxial layers the equilibrium of forces Equation 2.125, the equilibrium of moments Equation 2.126, strain in terms of mechanical parameters Equation 2.127, strain in terms of the difference in CTE, and differential temperature Equation 2.128 take the form of F 1 þ F 2 þ F 3 ¼ 0;

ð2:131Þ



Wk t1 t2  t3  þ F 3 t1 þ t2
¼ 0; ðE 1 t31 þ E 2 t32 þ E 3 t33 Þ þ F 1 þ F 2 t1 þ 2 2 2 12 ð2:132Þ e1 ¼

F2 F1 ðt1 þ t2 Þk

¼ DTða1
a2 Þ; E 2 t2 L E 1 t1 L 2

ð2:133Þ

2.12 Polarization Effects

e2 ¼

F3 F2 ðt2 þ t3 Þk

¼ DTða2
a3 Þ; E 3 t3 L E 2 t2 L 2

ð2:134Þ

where e1 and e2 represent the strain between the first epilayer and substrate and the second and first epilayers, respectively. Then si(2D), the two-dimensional stress in the epilayer i, is given as si ð2DÞ ¼

ðF i =ti WÞ

ð1
nÞ
1

:

ð2:135Þ

To calculate the misfit strain and to some extent CTE mismatch, the epitaxial relationship between the epitaxial layer and the substrate must be known, which is known simply as epitaxial relationship. That relationship between GaN and a variety of other substrates are discussed in Section 3.3. In spite of this, a succinct description is provided here as a part of the present topic on stress and strain for continuity and to help support the discussion on polarization. For sapphire substrates, orientations of (0 0 0 1), ð0 1
1 0Þ, ð2
1
1 0Þ, and ð0 1
1 2Þ (basically c-, m-, a-, and r-planes) have been used, see Table 3.6. The largest lattice mismatch,
33%, is between (0 0 0 1) GaN and ð2
1
1 0Þ sapphire along the ½0 1
1 0==½0 1
1 0 in-plane direction. The smallest mismatch, 1.19%, is between ð2
1
1 0Þ GaN and ð0 1
1 2Þ sapphire (r-plane) along the ½0 0 0 1==½0
1 1 1 in-plane direction. This orientation produces a-plane sapphire on r-plane sapphire and is discussed at some length in Sections 3.3.1 and 3.5.11. From the view point of lattice mismatch alone, the r-plane of sapphire is predicted to be most suitable for GaN growth. A thermal strain of
0.18% exists in GaN (0 0 0 1) when grown on sapphire (0 0 0 1), which is compressive. The next common substrate used to grow GaN is various polytypes of SiC. The epitaxial relationship on SiC substrates of hexagonal symmetry is tabulated in Table 3.7. The lattice parameter misfits between GaN and 6H-SiC, 3C-SiC, and 4H-SiC are very close to each other, namely, 3.48%, 3.46%, and 3.50%, respectively, and the corresponding thermal strains are 0.01, 0.09, and 0.03, respectively. The thermal strain in GaN on SiC is tensile and notorious for causing cracks in the epitaxial layers, beginning at about 2 mm thickness. In case AlN substrates would become available, growth on that substrate is also considered. Lattice misfit between GaN and AlN is only 2.41% and the thermal strain is
0.06% and 4.08% for epitaxial relationships (0 0 0 1)//(0 0 0 1) and ð1 1
2 0Þ==ð1 1
2 0Þ, respectively. From the thermal strain point of view, growth on c-plane AlN is preferred. GaN and ZnO share the stacking order and close lattice parameter, the epitaxial relationship for which is provided in Section 3.3.4. GaN can be grown on (0 0 0 1) ZnO, with the lattice parameter mismatch being only
1.97% and the misfit strain
0.21%. Some oxides have also been explored because of the small lattice misfit with GaN they provide. The epitaxial relationship between GaN and LiAlO2 is ð1
1 0 0Þ==ð1 0 0Þ with a lattice misfit of 0.31% along the [0 0 0 1]//[0 1 0] in-plane direction and a misfit strain of
0.41%. Some details can be found in Section 3.3.5. The structure of LiGaO2 is similar to the wurtzite structure, but because Li and Ga atoms have different ionic radii, the crystal has orthorhombic structure. Figure 3.36 shows the

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j 2 Electronic Band Structure and Polarization Effects

294

transformation of the hexagonal unit cell of GaN to an orthorhombic cell, which has lattice parameters close to that of LiGaO2. Similarly, Table 3.10 shows the corresponding lattice parameters of GaN before and after the transformation to orthorhombic cell as well as that of orthorhombic LiGaO2 unit cell. The misfit along [1 0 0] is 2.13%, with a misfit strain of
0.54%. Hexagonal (0 0 0 1) GaN has been prepared on (0 0 0 1) ScAlMgO4. The lattice parameter mismatch between the ½1 0
1 0 GaN and ½1 0
1 0 ScAlMgO4 is
1.49% and the misfit strain is
0.15%. Wurtzite (0 0 0 1) GaN films have been grown on cubic (1 1 1) spinel (MgAl2O4) substrates, as detailed in Section 3.5.10. The lattice parameter mismatch between the ½1 1
2 0 GaN and ½1
1 0 MgAl2O4 is 11.55% and the misfit strain is 0.06%. Zinc blende GaN was grown on (0 0 1) MgO with an epitaxial misfit of 6.99% and a misfit strain of
1.19%. Zinc blende GaN was grown on (0 0 1) MgO with an epitaxial misfit of 6.99% and a misfit strain of
1.19%. Mixed perovskite (La,Sr)(Al,Ta)O3 (LSAT) that is grown by Czochralski method could also be a promising substrate for GaN epitaxial layers and are reported. The lattice parameter 25

AlN Si 6HSiC 4HSiC 3CSiC LiGaO2 Al2O3 MgO GaAs ZnO MgAl2O4 LiAlO2 ScMgAlO4 NdGaO3

Curvature (1 m–1)

20

15

10

5

0.0

–5 0.0

20

40

60

Thickness (μm) Figure 2.59 A compilation of the variation of thermal curvature, a measure of strain, in epitaxial GaN layers grown on different substrates with respect to layer thickness [492]. (Please find a color version of this figure on the color tables.)

80

100

2.12 Polarization Effects

Figure 2.59 (Continued )

mismatch between ½1 1
2 0 GaN//LSAT ½1
1 0 is 16.64% and the corresponding misfit strain is 0.04%. Cubic GaN films can be epitaxially grown onto (0 0 1) Si and hexagonal polytype on (1 1 1) Si with a lattice misfit of
16.93% and misfit strain of 0.19%. The epitaxial relationship between GaN andSi is discussed in Section 3.3.3 and tabulated in Table 3.8. As for the case on GaAs, GaN with (0 0 0 1) orientation can be grown on (1 1 1) GaAs with an in-plane lattice arrangement of ½1 1
2 0==½1
1 0. The corresponding lattice misfit is
20.19% and the misfit strain 0.07%. Perovskite oxide substrates have also been considered as substrates for GaN and related structures in an effort to perhaps find a better match and/or utilize the

j295

j 2 Electronic Band Structure and Polarization Effects

296

nonlinear optical properties of perovskite oxide along with what GaN has to offer. The lattice mismatch of GaN to NdGaO3 has been calculated by assuming a perovskite cell of NdGaO3 with lattice parameters a, b, and c each being equal to 3.86 Å. Next, a new unit cell is constructed with a0 and b0 , where a0 and b0 are the diagonals of the old perovskite cell, as shown in Figure 3.37. The c0 -axis of the new cell is parallel to the caxis of the perovskite cell, but its length is doubled. In a sense this transforms a perovskite unit cell to a tetragonal unit cell. Accordingly, (1 0 0) plane becomes ð1
1 0Þ plane and (0 0 1) becomes (0 0 1). The corresponding lattice misfit is 1.72% and the misfit strain is
0.66%. Owing to the difference in stress between the thin film and substrate, the composite of film and substrate bends, which is the basis for many predigital thermometers and temperature controllers with spiral elements. The strain can be deduced from bending radius, for example, by optical means even during growth at elevated temperature, to monitor the evolution of stress and in attempts to reduce stress by epitaxial heterojunction layer design as performed on SiC substrates, see Section 3.5.3. It is therefore imperative to establish the relationship between the radius of curvature and the difference in the strain between the film and substrate, as discussed in the model of Olsen and Ettenberg [494]. In the calculation of curvature and stress instead of an average CTE over the entire range of temperature, the variation CTE with temperature has been considered for accuracy [492]. The final 5108

Stress on various substrates (Pa)

0

–5108

–1109 AlN Si 6HSiC 4HSiC 3CSiC LiGaO2 Al2O3 MgO GaAs ZnO MgAl2O4 LiAlO2 ScMgAlO4 NdGaO3

–1.5109

–2109

–2.5109 –3109 0100

210–5

410–5

610–5

810–5

Thickness (m) Figure 2.60 A compilation of residual thermal stresses in epitaxial GaN layer on different substrates with respect to layer thickness [492]. (Please find a color version of this figure on the color tables.)

110–4

2.12 Polarization Effects

j297

Stress versus thickness of GaN/potential substrates 5.00E+08

0.00E+00 0

0.00002

0.00004

0.00006

0.00008

0.0001

AlN Si MgO

–5.00E+08

3C-SiC 6H-SiC 4H-SiC

Stress (Pa)

–1.00E+09

ZnO Al2O3 LiGaO2

–1.50E+09

MgAl2O4 GaAs NdGaO3** ScAlMgO**

–2.00E+09

LiAlO2 LSAT –2.50E+09

–3.00E+09

Thickness (m) Figure 2.60 (Continued )

curvature and stress is then the integrated values from growth temperature to room temperature. Figures 2.59 and 2.60 show the curvature and the residual thermal stresses in epitaxial GaN layers grown on various substrates with respect to layer thickness, respectively. The parameters associated with each of the substrate used as well as the growth temperatures are tabulated in Table 2.36. An inspection of Figure 2.60 leads to the conclusion that the thermal stress is relatively small when GaN is grown on AlN, SiC, ScAlMgO4, Si, GaAs, and ZnO and much higher when NdGaO3 and MgO substrates are used. In all other cases (Al2O3, LiAlO2, LiGaO2, MgAl2O4, LSAT) the values are intermediate. Of paramount importance, the thermal stress is tensile in GaN when grown on Si and SiC, whereas in all other cases it is

j 2 Electronic Band Structure and Polarization Effects

298

Table 2.36 Properties and residual thermal stress of 1 mm epitaxial GaN film with other III-N compounds and substrates [492].

Substrate

Melting point ( C)

GaN AlN a-Al2O3 6H-SiC 3C-SiC 4H-SiC c-LiAlO2 b-LiGaO2 Si GaAs NdGaO3 MgO ZnO ScAlMgO4 MgAl2O4

>1700 at 2 kbar 2400  C at 30 bar 2030 2700 sublimes 1825 sublimes 2797 1700 1595 1415 1238 1600 2852 1975 2130

CTE (·10
6  C) (room temperature)

Growth temperature of GaN by grown OMVPE ( C)

aa ¼ 4.997, ac ¼ 4.481, a ¼ 5.45 aa ¼ 5.411 aa ¼ 8.31, ac ¼ 8.5 aa ¼ 4.76, ac ¼ 4.46 4.5 [76] 4.75 aa ¼ 12.1 aa ¼ 10.1, ab ¼ 21.1, ac ¼ 13.6 3.9 6.7 aa ¼ 11.9, ab ¼ 6.6, ac ¼ 5.8 13.9 aa ¼ 6.9, ac ¼ 4.75 aa ¼ 6.2, ac ¼ 12.2 8.7

950–1050 450–1040 950–1100  1000 600 600–1000 600 700 810 450–800 700 650 1000

compressive in nature, which leads to serious cracking issues. This is caused by CTE in SiC and Si being much smaller than that for GaN. As a result, during cooldown GaN is not freely permitted to reduce its lattice constant to the extent GaN naturally would like to, and consequently, the film remains under tensile strain, which causes cracks when the film thickness is about 2 mm or larger. Focusing on the most commonly used substrates for GaN and related epitaxy, Figures 2.61 and 2.62 show variation of the residual thermal stress in GaN versus the 7 × 107

Stress (Pa)

6 × 107 5 × 107 4 × 107 3 × 107 2 × 107 0

20

40

60

80

Thickness (μm) Figure 2.61 Residual thermal stress in GaN for a dual layer GaN/ AlN (0.1 mm) structure on 6H-SiC with respect to GaN thickness [492].

100

References

–0.2

Stress (GPa)

–0.4 –0.6 –0.8 –1.0 –1.2 0

20

60 40 Thickness (μm)

80

100

Figure 2.62 Residual thermal stress in GaN for dual layer GaN/ AlN (0.1 mm) structure on sapphire (Al2O3) with respect to GaN thickness [492].

thickness of GaN with an AlN buffer layer of thickness of 0.1 mm for dual-layer GaN/ AlN on 6H-SiC and GaN/AlN on Al2O3 heterostructures, respectively. There is very little change in thermal stress when GaN is grown on Al2O3 with or without a buffer layer of AlN. On the contrary, the stress decreases by an order by using a buffer layer of AlN while growing GaN on 6H-SiC. In summary, GaN layers grown on substrates such as AlN, SiC, ScAlMgO4, Si, GaAs, and ZnO has a residual thermal stress that is smaller by a factor of two or more as compared to the cases when the GaN layers are grown on other substrates. Moreover and very pivotally, the thermal stress is tensile in nature when grown on Si and SiC substrates whereas in all other cases it is compressive. The tensile residual strain in GaN grown on SiC and Si is notorious for cracking, and multiheterolayer buffer structures are used to deal with the problem, but not without limitations on layers thicknesses for a given substrate temperature used.

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xInxN epitaxial films on GaN templates grown by metalorganic vapor phase epitaxy. Journal of Applied Physics, 94 (4), 2449. 471 Lukitsch, M.J., Danylyuk, Y.V., Naik, V.M., Huang, C., Auner, G.W., Rimai, L. and Naik, R. (2001) Optical and electrical properties of Al1
xInxN films grown by plasma source molecular-beam epitaxy. Applied Physics Letters, 79 (5), 632. 472 Bernardini, F., Fiorentini, V. and Vanderbilt, D. (1997) Polarization-based calculation of the dielectric tensor of polar crystals. Physical Review Letters, 79 (20), 3958–3958. 473 Yu, L.S., Qiao, D.J., Xing, Q.J., Lau, S.S., Boutros, K.S. and Redwing, J.M. (1998) Ni and Ti Schottky barriers on n-AlGaN grown on SiC substrates. Applied Physics Letters, 73 (2), 238. 474 Morkoç, H., Ünl€ u, H. and Ji, G. (1991) Principles and Technology of MODFETS, vol. II, John Wiley & Sons, Ltd, Chichester, UK, p. 317.

References 475 Chu, R.M., Zhou, Y.G., Zheng, Y.D., Han, P., Shen, B. and Gu, S.L. (2001) Influence of doping on the two-dimensional electron gas distribution in AlGaN/GaN heterostructure transistors. Applied Physics Letters, 79 (14), 2270. 476 Ambacher, O., Foutz, B., Smart, J., Shealy, J.R., Weimann, N.G., Chu, K., Murphy, M., Sierakowski, A.J., Schaff, W.J., Eastman, L.F., Dimitrov, R., Mitchell, A. and Stutzmann, M. (2000) Journal of Applied Physics, 87, 334. 477 Ridley, B.K., Ambacher, O. and Eastman, L.F. (2000) Semiconductor Science and Technology, 15, 270. 478 Reale, A., Massari, G., Di Carlo, A., Lugli, P., Vinattieri, A., Alderighi, D., Colocci, M., Semond, F., Grandjean, N. and Massies, J. (2003) Comprehensive description of the dynamical screening of the internal electric fields of AlGaN/GaN quantum wells in time-resolved photoluminescence experiments. Journal of Applied Physics, 93 (1), 400. 479 Fiorentini, V., Bernardini, F. and Ambacher, O. (2002) Evidence for nonlinear macroscopic polarization in III–V nitride alloy heterostructures. Applied Physics Letters, 80 (7), 1204. 480 G€ orgens, L., Ambacher, O., Stutzmann, M., Miskys, C., Scholz, F. and Off, J. (2000) Applied Physics Letters, 76, 577. 481 Grandjean, N., Damilano, B., Dalmasso, S., Leroux, M., La€ ugt, M. and Massies, J. (1999) Journal of Applied Physics, 86, 3714. 482 Langer, R., Simon, J., Ortiz, V., Pelekanos, N.T., Barski, A., Andre, R. and Godlewski, M. (1999) Applied Physics Letters, 74, 3827. 483 Kim, H.S., Lin, J.Y., Jiang, X.H., Chow, W.W., Botchkarev, A. and Morkoç, H. (1998) Applied Physics Letters, 73, 3426. 484 Vaschenko, G., Patel, D., Menoni, C.S., Ng, H.M. and Cho, A.Y. (2002) Nonlinear

485

486

487

488

489

490

491

492

493

494

macroscopic polarization in GaN/ AlxGa1
xN quantum wells. Applied Physics Letters, 80 (22), 4211. Simon, J., Langer, R., Barski, A., Zervos, M. and Pelekanos, N.T. (2001) Physica Status Solidi a: Applied Research, 188, 867. Perlin, P., Suski, T., Lepkowski, S., Teisseyre, H., Grandjean, N. and Massies, J. (2001) Physica Status Solidi a: Applied Research, 188, 839. Wetzel, C., Takeuchi, T., Amano, H. and Akasaki, I. (1999) Japanese Journal of Applied Physics, 38, L163. Shi, C., Asbeck, P.M. and Yu, E.T. (1999) Piezoelectric polarization associated with dislocations in wurtzite GaN. Applied Physics Letters, 74, 573. Cartney, M.R., Ponce, F.A., Cai, J. and Bour, D.P. (2000) Mapping electrostatic potential across an AlGaN/InGaN/AlGaN diode by electron holography. Applied Physics Letters, 76, 3055. Cherns, D., Jiao, C.G., Mokhtari, H., Cai, J. and Ponce, F.A. (2002) Physica Status Solidi b: Basic Research, 234, 924. Cai, J. and Ponce, F.A. (2002) Determination by electron holography of the electronic charge distribution at the threading dislocation in epitaxial GaN. Physica Status Solidi a: Applied Research, 192, 407. Barghout, K. and Chaudhuri, J. (2004) Calculation of residual thermal stress in GaN epitaxial layers grown on technologically important substrates. Journal of Materials Science, 39 (18), 5817–5823. Morkoç, H., Ünl€ u, H. and Ji, G. (1991) Fundamentals and Technology of MODFETs, vols I and II, John Wiley & Sons, Ltd, Chichester, UK. Olsen, G.H. and Ettenberg, M. (1977) Journal of Applied Physics, 48, 2543.

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3 Growth and Growth Methods for Nitride Semiconductors Introduction

Although the synthesis of GaN goes to back more than a half century, there are several pivotal developments, which, in the opinion of the author, are responsible for laying the technological framework and paving the way for the tremendous commercial and scientific interest in nitrides. They are as follows: synthesis of AlN by Tiede et al. [1], synthesis of GaN through the reaction of Ga, and ammonia by Johnson et al. [2] synthesis of InN by Juza and Hahn [3], epitaxial deposition of GaN using the hydride VPE technique by Maruska and Tietjen [4] employment of nucleation buffer layers by Amano et al. [5] and Yoshida et al. [6] and achievement of p-type GaN by Akasaki et al. [7]. A more recent development that paved the way for all the commercial activity is the preparation of high-quality InGaN by Nakamura et al. [8], which followed the synthesis of InGaN by Osamura et al. [9]. Nearly every crystal-growth technique, substrate-type, and orientation has been tried in an effort to grow high-quality group III–V nitride thin films. Increasingly, researchers have successfully taken advantage of the hydride vapor phase epitaxy (HVPE), organometallic vapor phase epitaxy (OMVPE), and molecular beam epitaxy (MBE) techniques, which have yielded greatly improved film quality. All of these epitaxial methods must contend with two main problems: the lack of native GaN substrates and difficulty with nitrogen incorporation and concomitant high ammonia flow rates needed particularly for In-containing nitride semiconductors. A major drawback of GaN is that native substrates are not yet available in large quantities. This is, in part, owing to the low solubility of nitrogen in bulk Ga and the high vapor pressure of nitrogen over GaN at the growth temperature of bulk crystals. The best alternatives now lie in development of sapphire, SiC, or AlN substrates. Interest in AlN substrates has increased recently owing to the closer lattice match over sapphire, matched stacking order, and high thermal conductivity. These factors make AlN one of the best choices for growth of detectors requiring high AlN content AlGaN and backside illumination. Early work on producing bulk AlN looked promising [10]. The problem of nitrogen is endemic in epitaxial deposition techniques as well. Regardless of the growth method employed, the major difficulty in growing group III Handbook of Nitride Semiconductors and Devices. Vol. 1. Hadis Morko
c Copyright
2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40837-5

j 3 Growth and Growth Methods for Nitride Semiconductors

324

nitrides arises from the need to incorporate stoichiometric quantities of nitrogen into the film. This is accomplished in vapor phase processes at high substrate temperatures by decomposing a nitrogen-containing molecule, such as ammonia, on the substrate surface. It should be noted that as the N vapor pressure increases, going from AlN to GaN and then to InN, the ammonia flow rate must be increased reaching tens of liters per minute for InGaN and V/III ratios of 200 000 for InN. It can also be accomplished at lower temperatures in MBE growth by increasing the reactivity of nitrogen through remote plasma excitation or ionization or using ammonia as the reactive nitrogen source. A great deal of effort has been spent in trying to overcome the problems arising from lack of GaN substrates. Some of the best device results have been achieved through use of buffer layers such as thick HVPE, including freestanding, variety, or lateral epitaxial overgrowth variety.

3.1 Substrates for Nitride Epitaxy

As there is lack of a commercial native substrate, a plethora of substrates have been employed in the growth of GaN films [11–14]. Recognizing the thermodynamic bottleneck with regard to native substrates, unconventional methods employing various implementations of compliant substrates and lateral epitaxy have been explored. The most promising results on more conventional substrates so far have been obtained on sapphire and SiC, with SiC making substantial inroads. Also coming on the scene are thick freestanding GaN templates grown by HVPE and then separated from the sapphire substrates. GaN, AlN, and InN have been grown primarily on (0 0 0 1) sapphire but also on the ð2 1  3 1Þ, ð1 1 0  1Þ, ð1  1 0 2Þ, and  ð1 1 2 0Þ surfaces. In addition, III–V nitrides have been grown on Si, NaCl, GaP, InP, SiC, W, ZnO, MgAl2O4, TiO2, and MgO. Other substrates as well have been used for nitride growth, including Hf and LiAlO2, and LiGaO2. Table 3.1 is a compilation of the lattice parameters and thermal characteristics of a number of prospective substrate materials for nitride growth. Lattice-mismatched substrates lead to a substantial density of misfit and threading dislocations (in the range of 108 and 1010 cm
2), though selective epitaxy followed by coalescence, which goes by many names such as epitaxial lateral overgrowth (ELO), lateral epitaxial overgrowth (LEO), and epitaxial lateral overgrowth (ELOG), is a promising method for reducing dislocations down to 106 cm
2. In comparison, the extended defect densities are essentially zero for silicon homoepitaxy and 102–104 cm
2 for gallium arsenide homoepitaxy [13]. Additional crystalline defects besetting the layers include inversion domain boundaries (IDBs) and stacking faults. Such defects are either directly or indirectly responsible for the creation of nonradiative recombination centers, which manifest themselves as energy states in the forbidden energy bandgap reducing the quantum efficiency as well as producing scattering centers. Other adverse effects of structural and point defects are that impurities diffuse more readily along threading dislocations, and carrier transport is

3.104 3.189 4.758

AlN (hexagonal) GaN (hexagonal) Al2O3 (sapphire) (rhombohedral) 4H-SiC (hexagonal) 6H SiC (hexagonal) ZnO (hexagonal) ScAlMgO4 (hexagonal) c-LiAlO2 (tetragonal) LiGaO2 (orthorhombic) MgAl2O4 (cubic/spinel) Si (cubic) GaAs (cubic) b-SiC (cubic) MgO (cubic/rock salt) 6.372

—

Conventional b (Å)

10.053 15.1123 5.2065 25.195 6.2679 5.407

4.966 5.175 12.991

c (Å)

3.1340 6.372

3.0817 3.2496

3.104 3.19 2.747

Matched a (Å)

1.5 0.5 4.9

4.9 0.3–0.4

3.2 2.3 0.3–0.5

Thermal conductivity, j (W cm
1 K
1)

10.5

4.2, 4.68 4.75, 2.9 6.2, 12.2 7.1, 15 a ¼ 6, b ¼ 9, 7 7.45 3.59 6

4.2, 5.3 5.59, 3.17 7.5, 8.5

Da/a, Dc/c (·10
6 K
1)

In part from Landolt-B€ ornstein, vol. 17, Springer, New York, 1982. ZnO data are from W. Harsch of Eagle Picher.

3.073 3.0817 3.2496 3.246 5.1687 5.402 8.083 5.4301 5.6533 4.36 4.216

a (Å)

Crystal

Table 3.1 Lattice parameters and thermal characteristics of a number of the prospective substrate materials for nitride growth and their lattice mismatch with GaN.

P63mc P63mc R3m P41212 Pna21 Fd3m Fd3m F 43m F 43m Fm3m

P63mc P63mc R3c

Space group


3.63%
3.36% þ1.9% þ1.8%
1.7% 0.18%


2.7% 0% 49% (13%)

Mismatch

3.1 Substrates for Nitride Epitaxy

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j 3 Growth and Growth Methods for Nitride Semiconductors

326

either impeded, as in lateral transport, or aided, as in vertical transport. The high density of defects also leads to boundary-limited transport making the important basic parameters such as diffusion constant and mobility nearly impossible to measure. The extended defects in nitrides also lead to inhomogeneities in electric potential because of the high piezoelectric constants of GaN. Electrically active defects induced either directly or indirectly by extended defects cause excess leakage that is detrimental to both optical devices (in the form of dark current in detectors and reduced quantum efficiency in emitters) and electrical devices (in the form of increased gate current leakage and output conductance in field-effect transistors). Details regarding the piezoelectric properties can be found in Chapter 2. Lattice-mismatched substrates are commonly used at present in spite of efforts to produce GaN [15,16] and AlN [10] bulk materials. Despite the lack of matched substrates, remarkable progress in the growth of high-quality epitaxial III nitride films has been achieved by a variety of methods such as hydride vapor phase epitaxy (inorganic VPE or hydride VPE also goes with the acronym HVPE) [11,17], OMVPE [18], and reactive molecular beam epitaxy (RMBE) [19,20]. Moreover, thick freestanding GaN templates for further epitaxy have been prepared by HVPE [21]. By far the most frequently used methods are the VPE methods with heterojunction capability. The most versatile among the VPE methods is the metalorganic chemical vapor deposition (OMVPE). OMVPE is the primary method employed in the investigation and production of optoelectronic devices, such as LEDs and lasers, albeit the quality of MBE films comes close to that grown by OMVPE. Electronic devices with higher quality interfaces are achieved principally with OMVPE and MBE. Inorganic VPE was the first method used to grow epitaxial III-N semiconductors but was nearly abandoned [22]. The technique, however, got revived recently by growing very high-quality and thick GaN buffer layers and templates [23] for the growth of device structures using MBE or OMVPE [11]. Efforts are underway to expand the method to the growth of AlGaN. Below, a discussion of the class of substrates that have been explored is followed by a discussion of the properties of and processing steps for the conventional substrates before growth for each of these substrates. 3.1.1 Conventional Substrates

GaN, as the most studied member of the semiconducting group III nitrides, has been grown on many substrates. Many of the major problems that have hindered the progress in GaN and related semiconductors can be traced back to the lack of a suitable substrate material that is lattice and thermally matched to GaN. Lattice mismatch is responsible for stacking faults and dislocations. Thermal mismatch causes the epilayer to crack on cooling. Specifically, the semiconductors GaN, AlN, and InN have been grown primarily on sapphire, most commonly in the c (0 0 0 1) orientation but also on the a- ð1 1 2 0Þ and R- ð1  1 0 2Þ planes [14]. Growth on a-plane produced c-plane GaN, but growth on R-plane sapphire produces a nonpolar, a-plane

3.1 Substrates for Nitride Epitaxy

GaN. In addition, the group III–V nitrides have been grown on SiC, ZnO, MgAl2O4, Si, GaAs, MgO, NaCl, W, and TiO2. As high-resistivity SiC substrates became available, it became the proffered substrate for transistor work, in part owing to its high thermal conductivity. In addition, freestanding GaN prepared by HVPE is at or nearing production capacity primarily for low threshold lasers that need low-defect material. Some of the suitable substrate materials have become commercially available only recently. Almost all the group III–V nitride semiconductors have been deposited on sapphire despite its poor structural and thermal match to the nitrides. The preference for sapphire substrates can be ascribed to its wide availability, hexagonal symmetry, and ease of handling and pregrowth cleaning. Sapphire is also stable at high temperatures (1000  C) required for epitaxial growth using the various CVD techniques commonly employed for GaN growth. Owing to thermal and lattice mismatches between sapphire and the group III–V nitrides, it is necessary to grow a thick epilayer to obtain good-quality material. 3.1.2 Compliant Substrates

When large mismatch exists between an epilayer and its substrate, the misfit is typically accommodated by the introduction of misfit dislocations at the interface, which are accompanied by threading dislocation segments in the epilayer. To overcome this problem, a compliant substrate is used in high-misfit systems. The role of the compliant substrate is to accommodate the large mismatch either by plastic deformation of the compliant substrate in a manner that avoids the formation of dislocations in the heteroepitaxial film or by homogeneous elastic strain of the threading dislocation, which also avoids formation of the threading dislocations. Both mechanisms are facilitated by a compliant substrate whose stiffness constants are well below those of the epilayer and the supporting bulk substrate. As for the former mechanism, the soft and thin nature of the compliant substrate energetically favors the capture of dislocations resulting from the mismatch by the substrate rather than by the stiffer epilayer, thus paving the way for predilection toward misfit accommodation (MA) by homogeneous elastic strain rather than misfit dislocations. An attractive approach is to insert a pillarlike interfacial layer that is capable of accommodating thermal strain, the effectiveness of which depends on the height of the pillars and the size of the wafer. An effort has been made to find a universal substrate onto which any epitaxial layer can be grown with a very low density of structural defects. Some experimental success has been achieved, but only in specific cases, since 1991 when Lo and colleagues [24,25] introduced the basic idea of a compliant substrate. Different kinds of epitaxial layers have been grown on compliant substrates of GaAs twist bonded to bulk substrates of GaAs. Among them are InGaP, In0.22Ga0.78As, GaSb, and InSb where the respective misfits with GaAs are 1,
1.5,
8, and
15%, respectively [26,27], with positive figures denoting tensile stresses in the layer on GaAs. The use of a thin compliant substrate is not limited to the epitaxial growth of III–V compounds but may also be found in SiGe grown on a thin compliant

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substrate of Si on a viscous SiO2 layer [28]. Other approaches pursue compliancy by twist bonding, that is, deposition of a low melting temperature interlayer. Twist bonding Si to SOI provides a high density of interlayer dislocations that can elastically accommodate misfit between the compliant substrate and the heteroepitaxial film. Otherwise, if the bonding between the compliant layer and substrate is strong, an array of screw dislocations form, transforming to edge/screw dislocations upon deposition of a lattice-mismatched film by elastic deformation. Various explanations have been offered to understand the mechanisms of misfit accommodation [29]. The term compliant is used liberally to describe an approach or a set of approaches to grow lattice mismatch materials where the substrate or some interface layer accommodates the mismatch by expansion/contraction or generating defects within itself as opposed to epitaxial layers. Specifically, the concept behind the latter case is to force the defects caused by mismatch to propagate into the substrate as opposed to the epilayer, requiring generation of misfit dislocations in the thin, weakly bound template layer rather than the growing epitaxial layer. In case of nitride, the initial approach proposed was that Si on an insulator be utilized for GaN growth. Here, a thin Si layer on silicon dioxide, which, in turn, is on Si would be employed. The downside is that the quality of GaN on compliant Si has been poor at best. The modified approach to overcome this barrier is to carbonize the Si to convert it to SiC. If one utilizes the (1 1 1) orientation, one would then get the wurtzitic phase of GaN. To be specific, Yang et al. [30] suggested that before the beginning of the carbonization process, SiO2, Si, and C need to be deposited successively on a Si substrate. By exposing the new composite substrate to a flux of acetylene or carbon particles at 900  C, a thin layer (less than 50 nm) of Si (on SiO2) will be partially or completelyconverted into SiC. GaNis then grownonthis SiC. Again, the problemhere, setting aside the problems associated with the growth on SiC, is that SiC so formed is not contiguous and is extremely defective both in terms of bulk and surface structural properties. In addition, air gaps form beneath the layer surface. Consequently, this technique has not yet lived up to the original proposal and expectations. Efforts still continue to exploit this approach despite the lack of progress so far. Compliance based on expansion and/or contraction is a very neat idea and may be workable for small-sized wafers. However, it is impractical for larger wafers. For example, if the lattice mismatch between the compliant substrate is 4% (epitaxial layer having the larger lattice constant) and the wafer is 50 mm, an expansion in the substrate required for producing defect-free epitaxial layer is 2 mm, which is substantial and unlikely. 3.1.3 van der Waals Substrates

To get around the lattice-mismatch problem, a new growth method called van der Waals epitaxy has been proposed [31], which delivers strain-free films. In this approach, the substrate and epitaxial film are separated by an intermediate epitaxial two-dimensional (2D) buffer material such as MoS2, WS2, or other materials such as II–VI (ZnTe) or III–VI compounds (GaSe, InSe, etc.) having weak van der Waals

3.2 A Primer on Conventional Substrates and their Preparation for Growth

bonding to the substrate and the film. Strain from lattice mismatch between the epitaxial film and the substrate is completely relieved in the region between the layer and the buffer. As in the case of the compliant substrate scheme, this approach has not been very successfully applied to nitrides.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

A substrate is like the foundation of a building. As such, substrate preparation deserves the mostattention.Though thedetailsof the proceduresemployed vary fromone growth method to the next, a chemical preparation before loading into the growth reactor is common. In the OMVPE technique, this is followed by either a simple heat treatment or a combination of heat treatment with gas-phase etching, where temperatures for heat treatment in the vicinity of 1200  C are possible. In the case of vacuum-deposition techniques where it is not always possible to achieve sufficiently high temperatures, dry processing techniques are employed. One of the dry processing techniques is utilizing ECR remote plasma etching with a mixture of hydrogen and helium, as discussed below. The purpose of the He gas is to take advantage of its energetic metastable states with long mean free paths. In addition to a clean surface, the goal is to get as flat a surface as possible, because the nitride stacking order, AaBbAaBb, is different from the stacking order found in most of the substrates under investigation. The exception is ZnO whose stacking order matches that of the nitrides. Because the atomic steps on the (0 0 0 1) surface are of the bilayer type, the surface terraces would have the same surface polarity so that stacking mismatch boundaries (SMB) can be avoided. The surfaces of the substrates used have to be prepared for epitaxial growth, a process that includes degreasing followed by chemical etching when possible. Surfaces of as-received sapphire and SiC substrates contain mechanical polishing damage that must be removed. Chemical etches are not yet available for this purpose. Consequently, a high-temperature treatment under a controlled environment is employed, as will be discussed below. The degreasing procedure, which is the first step for growth, for Si, sapphire, SiC, ZnO, LiGaO2 and LiAlO2, and GaN and AlN, whose specifics will be discussed below, is the same. The substrate is first dipped in a solution of trichloroethane (TCE) kept at 300  C, for 5 min. It is then rinsed for 3 min each in acetone and methanol. This is followed by a 3-min rinse in deionized (DI) water. The above process is repeated three times to complete the degreasing process. The substrates are then etched, which is a substrate dependent procedure. Following degreasing, a variety of substrate-specific methods are employed to get various substrates growth ready as discussed below. 3.2.1 GaAs

GaAs, as a substrate for GaN epitaxy, is justified on the premise of obtaining pure (wurtzite-free) zinc blende GaN on GaAs(1 0 0), attaining thick wurtzite GaN films on

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GaAs(1 1 1) substrates, and dilute GaAsN films for infrared applications. The bulk of the research on GaAs based nitrides is on (1 0 0) surface for cubic GaN. GaAs(1 0 0) is one of the few semiconductor substrates on which metastable zinc blende GaN epitaxial films readily form, and many researchers have investigated the best ways of avoiding any inclusion of the wurtzite polytype into these films. GaAs is much more readily wet etched than any of the other substrates used for nitride epitaxy, which also makes GaN films easier to separate from GaAs than sapphire. Thus, GaAs(1 1 1) substrates are considered a template for creating freestanding thick GaN films for subsequent epitaxy. Because the decomposition rate of GaAs in NH3 or an ultrahigh vacuum (UHV) rapidly increases at temperatures above 700  C, a deposition process including multiple temperatures is required. Unless the substrate temperature is high, the maximum growth rate (GR) attainable is limited. Moreover, even a small amount of GaAs decomposition could be detrimental, as surface roughening or faceting enhances the onset of mixed polarity growth. Because MBE is capable of depositing epitaxial GaN films at a lower temperature as compared to vapor phase methods, it has been more commonly employed in this respect. The maximum allowed temperature could be increased once the GaAs substrate is completely encased with GaN deposited at a low temperature (LT), thereby making OMVPE and HVPE more viable. 3.2.1.1 A Primer on GaAs GaAs has the zinc blende crystal structure with the symmetry group of F  43m. Figure 3.1 displays the perspective view of the GaAs crystal along [1 0 0], [1 1 0], and [1 1 1] directions. Table 3.2 lists the physical, chemical, thermal, mechanical, and optical properties of GaAs important for the GaN epitaxy. For additional details, see Liu and Edgar [13]. GaAs substrates are grown with either liquid encapsulated Czochralski (LEC) or vertical gradient freeze (VGF) methods. GaAs has seen a spectacular improvement over a period of two decades in that crystal defects, impurities, and micro-inhomogeneities have been reduced. In fact, GaAs wafers with diameters greater than

Figure 3.1 The perspective view of the GaAs crystal (a) along [1 0 0] (1 · 1 · 1 unit), (b) [1 1 0] (2 · 2 · 2 units), and (c) [1 1 1] (2 · 2 · 2 units) directions [13]. (Please find a color version of this figure on the color tables.)

3.2 A Primer on Conventional Substrates and their Preparation for Growth Table 3.2 Properties of GaAs at room temperature (partially after Ref. [13]).

Parameter

Value

Lattice constant (Å) Density (g cm
3) Melting point ( C) Heat capacity (J g
1 K
1) Thermal conductivity (W cm
1 K
1) Thermal diffusivity (cm2 s
1) Thermal expansion (linear) (·10
6 K
1) Percent change in lattice (300–1200 K) Bulk modulus (GPa) Young’s modulus (GPa) Poisson’s ratio Refractive index Relative dielectric constant Electrical resistivity (undoped)

5.6536 5.32 1240 0.327 0.45 0.26 6.03 Da/a0 ¼ 0.5876 75.0 85.5 0.31 3.66 near band edge e0 ¼ 13.1 1.0 · 104 O cm, nonstoichiometric defect compensated

150 mm and with various doping types and concentrations are commercially available. Silicon and tellurium are common n-type dopants with resultant electron concentrations in the range of 1016–1018 cm
3. On the contrary, zinc is the standard p-type dopant with resultant hole concentrations in the range of 1018–1019 cm
3. Commercial LEC GaAs has a typical etch pit density (EPD) of less than 104 cm
2 and an electron mobility greater than 4000 cm2 V
1 s
1. But, VGF grown GaAs offers a lower defect density, with EPD less than 103 cm
2. Both (1 0 0) and (1 1 1) types of GaAs with different vicinal degrees are available. 3.2.1.2 Surface Preparation of GaAs for Epitaxy The substrate preparation, following degreasing, includes etching in acid such as H2SO4 : H2O2 : H2O. After rinsing, the surface can be treated in dilute HF for H passivation of the surface. The H passivation layer can be desorbed in the deposition chamber. However, owing to the well-advanced nature of GaAs technology, epi-ready substrates with protective oxides are commercially available. Once in the deposition reactor, the oxide layer can be removed by thermal desorption. Direct nucleation of GaN on GaAs is difficult owing to great chemical and mechanical mismatch with GaN (the large lattice match of about 20% caused by 5.65 Å vs. 4.51 Å for zinc blende GaN), but it is a task mitigated somewhat with the deposition of a GaAs prelayer. Zinc blende GaN on (2 · 4) GaAs can be grown by first establishing an atomically smooth GaAs surface with minimized step density. This is accomplished by depositing a GaAs prelayer, on the order of 100 nm to as high as 1 mm, and choosing appropriate nucleation conditions. Without a GaAs prelayer, the more stable wurtzite polytype of GaN grows preferentially once (1 1 1) facets of the GaN film are generated on the rough substrates. The content of hexagonal GaN phase

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is drastically reduced with an epitaxial GaAs prelayer in MBE growth. The growth temperature for a GaAs prelayer is typically about 600  C. It is imperative that the GaAs surface is nitridated prior to GaN growth for chemical and mechanical transitioning. Nitridation of the GaAs substrate results in a thin nitride surface layer, which provides a cubic template for growth, improving the quality of the GaN layer and suppressing GaAs decomposition at higher substrate temperatures needed for GaN epitaxy. GaAs substrates are nitridated either by exposure to nitrogen plasma (MBE) or by annealing in ammonia (OMVPE and RMBE). At temperatures below 200  C, the nitridation is hindered by kinetic limitations. At temperatures above 600  C, simultaneous etching of the surface may occur along with the nitridation process. Nitridation of GaAs(0 0 1) does not take place homogeneously but proceeds along {1 1 1} facets into the underlying GaAs layer. Complete nitridation can lead to a highly facetted interface between the GaN layer and GaAs substrate, which, in turn, leads to the nucleation of the wurtzite phase and could lead to polycrystalline GaN. Using the highest growth rate possible to quickly bury the interface or maintaining an As flux during growth of the first few monolayers of GaN helps to prevent the deterioration of the GaAs surface. A buffer layer of GaN at low temperature following the initial nitridation generally improves the eventual quality. AlN is not used as buffer because its zinc blende polytype is extremely difficult to nucleate. 3.2.2 Si

Si is the most perfected and least expensive substrate that is available in sizes up to 300 mm. Unlike GaAs, ZnO, and a few others, silicon has good thermal stability under conditions used for GaN epitaxy. However, Wz GaN and AlN grown on Si(1 1 1) are highly defective. The incentives for using Si substrates remain high, however, and good progress in reducing the defect density by using epitaxial lateral overgrowth or pendeo-epitaxy has been reported. 3.2.2.1 A Primer on Si Si has a diamond-lattice structure with the space group of Fd 3m (No. 227) and can be thought of as two interpenetrating fcc sublattices with one sublattice displaced from the other by one quarter of the distance along a body diagonal of the cube (i.e., the pffiffiffi displacement of a 3=4, where a ¼ 5.43102 Å is the lattice constant). Each atom in the lattice is surrounded by four equidistant nearest neighbors that lie at the corners of a tetrahedron. Figure 3.2 illustrates the perspective view along the [0 0 1], [0 1 1], and [1 1 1] directions of a Si cell. Table 3.3 lists physical, chemical, thermal, mechanical, and optical properties of Si at room temperature. Single crystalline ingots are produced by the Czochralski (CZ) method (over 85% of silicon crystals are grown by this method) or the Float Zone (FZ) method, used mostly for purification. These ingots eventually become thin Si wafers through the processes of shaping, slicing, lapping, etching, polishing, and cleaning. Impurities

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Figure 3.2 The perspective view along (a) the [0 0 1], (b) [0 1 1], and (c) [1 1 1] directions of a Si cell. (Please find a color version of this figure on the color tables.)

can be added directly to the melt to create p-type and n-type silicon. The only discernible half-drawback of the CZ method is that oxygen (typical at a level of 1018 cm
3) and carbon (typical at a level of 1016 cm
3) can be incorporated because of the reduction of the quartz crucible and contamination by graphite fixtures. It should be noted that these impurities are not without benefits in that oxygen increases the yield strength or acts as internal getter to tie up metallic contaminants. The FZ method does not use any crucible, and thus the impurity level is markedly reduced, making it easier to grow high-resistivity material. For more details, refer to Ref. [13]. 3.2.2.2 Surface Preparation of Si for Epitaxy For wurtzitic GaN growth, (1 1 1) plane Si is used. The (0 0 1) surface is also used for cubic GaN growth, albeit in only a few cases. As-received Si surface is already

Table 3.3 Properties of Si at room temperature (partially after Ref. [13]).

Parameter

Value

Lattice constant (Å) Density (g cm
3) Melting point ( C) Heat capacity (J g
1 K
1) Thermal conductivity (W cm
1 K
1) Thermal diffusivity (cm2 s
1) Thermal expansion (linear) Percent change in lattice (298–1311 K) Shear modulus (GPa) Bulk modulus (GPa) Young’s modulus (GPa) Poisson’s ratio Refractive index Relative dielectric constant Electrical resistivity (undoped)

5.43102 2.3290 1410 0.70 1.56 0.86 2.616 · 10
6 K
1 Da/a0 ¼ 0.3995 680 97.74 165.6 0.218 3.42 e0 ¼ 11.8 Up to 50 kO cm

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excellent and removal of only a very thin surface layer, using the RCA etch followed by hydrogenation of surface dangling bonds, is sufficient. This is accomplished by immersing Si for 10 min in a 1 : 1 : 5 solution of HCl : H2O2 : H2O kept at 60  C, which grows a porous oxide, followed by a rinse in deionized water. The resulting oxide layer is then removed by dipping the substrate in a 10 : 1 solution of H2O : HF for 20 s. The hydrogenation process takes place through a short exposure of the wafer to an HF solution. 3.2.3 SiC

The cohesive bond strength of SiC is so large that it was once considered an element under the name of carborendum. Owing to its large thermal conductivity and dearth of defect causing in-plane rotation of GaN with respect to SiC lattice, and availability of high-resistivity substrates, SiC is continually gaining recognition as a very viable substrate for epitaxy for both optical and electronic devices. Much of the early drawbacks having to do with pre-epitaxy surface preparation, micropipes, size, and, to an extent, cost issues have been mitigated to the point that some commercial LEDs and almost all of high power field-effect transistors utilize nitride heterostructures on SiC. 3.2.3.1 A Primer on SiC A basic unit of crystalline SiC is a covalently bonded tetrahedron of C atoms with a Si atom at its center or vice versa, that is, either SiC4 or CSi4, as illustrated in Figure 3.3. Variation in the stacking order of SiC along the c-direction leads to more than 250

Figure 3.3 Tetragonal bonding of a carbon atom with its four nearest silicon neighbors. The bond lengths depicted with a and C–Si (the nearest neighbor distance) are approximately 3.08 and 1.89 Å, respectively. The right side is the three-dimensional structure of 2H-SiC structure. (Please find a color version of this figure on the color tables.)

3.2 A Primer on Conventional Substrates and their Preparation for Growth

A B C

Carbon Si Base

B A

C

A

B

C

A

C

A

A

C

B

B

B

A

A

B A

A 3C

A

2H

4H

6H

Figure 3.4 Stacking sequence of cubic and three polytypes of wurtzitic SiC.

polytypes, of which a few prominent ones are shown in Figure 3.4. (A basic discussion of stacking is given in Section 1.1.) By observing the SiC crystal from the side, the stacking sequence can be projected as in Figure 3.5. The distance a between neighboring silicon or carbon atoms is approximately 3.08 Å for all polytypes. The height of the unit cell c varies with the different polytypes, as tabulated in Table 3.4. Consequently, the c/a ratio varies from polytype to polytype but is always close to the ideal for a close packed structure. This ratio is approximately 1.641, 3.271, and 4.908 for the 2H-, 4H-, and 6H-SiC polytypes, respectively, whereas theffiffiffiffiffiffiffi equivapffiffiffiffiffiffiffiffi p ffi lent ideal ratios for these polytypes are 1.633, 3.266, and 4.899 ( 8=3 , 2 8=3 , and pffiffiffiffiffiffiffiffi 3 8=3), respectively [13]. Each polytype has a unique set of electronic and optical properties. The bandgaps at liquid helium temperature of the different polytypes range between 2.39 eV for 3CSiC and 3.33 eV for the 2H-SiC polytype. The two most important polytypes as substrates for GaN epitaxy, 6H-SiC and 4H-SiC, have bandgaps at liquid helium temperature of 3.02 and 3.27 eV, respectively. The hexagonal polytypes of SiC, such as 4H- and 6H-SiC, belong to the same space group, P63mc (No. 186), as wurtzite GaN. The most studied substrates for GaN epitaxy are the 3C-SiC/Si(1 0 0) and 6H-SiC, as these polytypes have been the most readily prepared or commercially available for the longest time. With 4H-SiC now commercially available, its use will become more common. Table 3.4 shows the physical, chemical, thermal, mechanical, and optical properties of SiC at room temperature. The thermal expansion coefficient of SiC in c- and a-planes as a function of temperature is shown in Figure 3.6. Bulk SiC crystals are produced by sublimation in the modified Lely process, developed by Tairov and Tsvetkov [33], which employs a SiC seed crystal for the control of polytype and orientation. Growth is achieved by the vapor transport of Si, Si2C, and SiC2 driven by a temperature difference in an argon atmosphere in a graphite, tantalum, or tantalum carbide crucible at 20–500 Torr and at about 2200  C. The 4H-SiC and 6H-SiC(0 0 0 1) varieties both on- and off-axis (typically 3.5 for 6HSiC and 8 for 4H-SiC), silicon and carbon face, are available in sizes up to 100 mm in

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Figure 3.5 Views of the ½1 1 2 0 planes for the 3C-, 2H-, 4H-, and 6H-SiC polytypes.

diameter. Screw dislocations occur in high densities and depending on the magnitude of their Burgers vector, the core of a screw dislocation can be hollow (nanopipes or micropipes) or closed and run through the entire wafer. Hollow core screw dislocations take place when the Burgers vector (b) is two or more times the c-lattice constant (c) for 6H-SiC or three times the lattice constant for 4H-SiC. Because the best micropipe density has been reduced to about 1 cm
2, research has shifted on to closed-core screw dislocations, which occur in densities of approximately 103–104 cm
2. Both 4H-SiC and 6H-SiC wafers are available in low resistivity nand p-type forms with concentrations in the range of 1015–1019 cm
3. The resistivities for n- and p-type material for the aforementioned doping range lie in the range 0.01–0.10 and 1–10 O cm, respectively. Interest in semi-insulating SiC is driven by

3.2 A Primer on Conventional Substrates and their Preparation for Growth Table 3.4 Properties of SiC at room temperature (after Ref. [13]).

Parameter

Polytype

Value

Lattice constant (Å)

3C 2H 4H 6H 3C 2H 6H 3C 6H 3C 4H 6H 3C 6H 6H 3C 3C Ceramic 3C 2H 4H 6H 3C 6H

a ¼ 4.3596 a ¼ 3.0753, c ¼ 5.0480 a ¼ 3.0730, c ¼ 10.053 a ¼ 3.0806, c ¼ 15.1173 3.166 3.214 3.211 2793 0.71 3.2 3.7 3.8 3.9 4.46 for a-axis, 4.16 for c-axis Da/a0 ¼ 0.4781, Dc/c0 ¼ 0.4976 Da/a0 ¼ 0.5140 440 0.183–0.192 2.6916 at l ¼ 498 nm 2.6686 at l ¼ 500 nm 2.6980 at l ¼ 498 nm 2.6894 at l ¼ 498 nm e(0) ¼ 9.75, e(1) ¼ 6.52 e(0) ¼ 9.66, e(1) ¼ 6.52 ? c-axis e(0) ¼ 10.3, e(1) ¼ 6.70 || c-axis 102–103, higher in V doped

Density (g cm
3)

Melting point ( C) Heat capacity (J g
1 K
1) Thermal conductivity (W cm
1 K
1) Linear thermal expansion coefficient (·10
6 K
1) Percent change in lattice (300–1400 K) Young’s modulus (GPa) Poisson’s ratio Refractive index (ordinary ray)

Dielectric constant

Electrical resistivity (undoped)

4H

Thermal expansion coefficient (%)

1.8

α−SiC

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 –0.2 0

200 400

600

800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800

Temperature (K) Figure 3.6 Thermal expansion coefficient of SiC in c and a planes as a function of temperature [32].

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FETs. The resistivities close to 1010 O cm in both V-doped and intrinsically compensated SiC are possible. 3.2.3.2 Surface Preparation of SiC for Epitaxy The surface preparation of SiC prior to deposition takes on a different meaning in MBE as opposed to OMVPE. Lacking high temperature capability and H, MBE growth relies on ex situ cleaning procedures. In contrast, SiC surfaces can be epi readied in situ during growth by vapor phase. However, one of the best, if not the best, approach involves gas phase preparation of SiC surface in which the surface is exposed to in H and/or HCl gases at high temperatures, the details of which are discussed below. Unless SiC substrates are H polished a priori or in the growth reactor, as can be done in HVPE or OMVPE methods, it is recommended that approximately 3 mm from the surface be removed in a hot KOH solution (300–350  C). If the substrate quality is not high, the etch rate in defective regions is high and smooth surfaces do not follow. Assuming that the previous step is successful, it is followed by a DI rinse for 3 min and the wafer is blow dried by N2. The SiC substrate is then subjected to a series of oxidation and passivation procedure. The substrate is immersed for 5 min in a 5 : 3 : 3 solution of HCl : H2O2 : H2O at 60  C, followed by a 30 s DI rinse. The resulting oxide layer is then removed by dipping the substrate, for 20 s, in a 10 : 1 solution of H2O : HF. This procedure is repeated several times (three to five times) after which the substrate should not be exposed to the atmosphere for longer than 30 min, otherwise another oxidation–passivation procedure would be required. Skipping the chemical etching step leaves the surface with residual damage from mechanical polishing. As mentioned above, techniques have been developed to remove the surface damage by plasma or vapor etching. One such technique is the mechanical chemical polish, which has made substantial progress [34] (commercial service is available from Novasic), and the other is etching in H and/or Cl environments at very high temperatures. The surface morphology of the as-received SiC substrate that underwent a standard mechanical chemical polish (MCP) contains much surface and subsurface damage as characterized by many grit scratches shown in Figure 3.7a. Special MCP procedures have been developed to obtain smoother surfaces (see Ref. [34]), two example of which obtained by Eagle Picher and Novasic are shown in Figure 3.7b and c. Schottky barrier diodes on as-received and Novasic MCP-treated substrates indicated as much as four orders of magnitude reduction in the reverse bias current, albeit with a considerable nonuniformity. Another method that leads to atomically smooth SiC surfaces and much improved GaN overlayers, in terms of reduced extended defect concentration as shown by plan Figure 3.7 (a) AFM image of as-received SiC surface following a standard mechanical chemical polish. Image size is 10 mm · 10 mm and vertical scale is 50 nm. Note the presence of scratches; (b) AFM image of SiC surface after a mechanical chemical polish performed at Eagle

" Picher. Note that scratches are no longer present. Image 10 mm · 10 mm, vertical 5 nm; (c) a 2 mm · 2 mm AFM image of a Cree 6H-SiC wafer MCP polished by Novasic showing a root mean square roughness of 0.134 nm.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

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view transmission electron microscopy (TEM) images, is the high temperature H annealing [35]. Those investigating various issues dealing with SiC have recently developed and exploited the in situ hydrogen etching [36]. H2, H2 þ HCl, or H2 þ C3H8 etching at temperatures between 1300 and 1550  C removes the scratches caused by mechanical polishing [37]. This is more effective than wet HF etching of SiO2 after oxidation. Other wet etching techniques in bases such as molten salts (Na2O2, NaOH, KOH, etc. at temperatures approaching or at 500  C) reveal the defect features of SiC surface and thus are not suitable for surface preparation for epitaxy. Even though nitridation has been shown to improve the smoothness of SiC substrate owing to a combination of nitrogen chemisorption and etching at 1050  C in NH3 flow for 30 min, the utility of long nitridation processes is questioned because of SixNy formation. An atomic force microscopy (AFM) image of a SiC substrate polished by a hightemperature H treatment (1500  C), similar to that reported in Ref. [38], and used for GaN growth in Ref. [35], is shown in Figure 3.8, which clearly shows well-ordered and unbroken atomic terraces indicative of superb surface quality. Similar results can be obtained by a H treatment in a typical SiC growth reactor at about 1500  C. A light follow-up etch in molten KOH solution ensures atomically smooth terraced surfaces if the H polishing steps are not ideal. As previously mentioned, vacuum deposition equipment, such as the one used in MBE, is not compatible with the high-temperature H or HCl treatment, but remote plasma etching techniques can be employed to at least remove the damaged surface layer if H-etched samples are not available. To circumvent the need for high temperatures and exotic treatments incompatible with conventional MBE setups, a preparation procedure adapted from conventional Si technology, and augmented by H plasma cleaning, has been shown to work for SiC. In the first step of this procedure reported by Lin et al. [39], the surface is hydrogen passivated using an HF dip before

Figure 3.8 AFMimageofSiCsurfaceaftera5 min1600  C hydrogen polishing step. Note that scratches are no longer present. The step height seen in a full c-direction lattice parameter for 6H-SiC.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

being introduced into vacuum. In the second step, the substrate is treated with hydrogen plasma, which reduces the CO level (oxygen–carbon bonding) to a value below the X-ray photoemission detection limit. Detailed investigation of SiC surface after H anneal have been undertaken [40] using such techniques as low-energy electron diffraction (LEED) experiments and Auger electron spectroscopy (AES) in addition to reflection high-energy electron diffraction (RHEED). The emphasis is to determine the surface chemical and structural properties taking it beyond what may be needed simply for epitaxy but looking at it with the precision needed for MOS-like structures with AlN/SiC composite. When (0 0 0 1) SiC samples subjected to a H polish at 1500  C for 5 min in flow 3000 scum are introduced into the UHV system, they exhibited a 2 ffiffi pffiffiaffi Hp ffi ( 3  3) R30 LEED pattern with bright and sharp superstructure spots. For comparison on the C-polarity surfaces, ð0 0 0 1Þ, no background was observed, which indicates a high degree of order. On the (0 0 0 1) surfaces, however, always a faint background was visible. The ratio between the average intensities of fractional and integer order beams was above 0.5 for both surfaces, indicating strong surface reconstruction in both cases. The typical Auger spectra displayed a strong OKLL peak in addition to the typical SiLVV and CKLL peaks [40]. The SiLVV signal on the ð0 0 0  1Þ surface showed both bulk-related peak at 90 eV and a feature at around 65 eV attributed to oxygen-bonded silicon. The SiLVV signal of the (0 0 0 1) surface is more complex as expected because of differently coordinated Si for both surfaces. The LEED pattern on the ð0 0 0 1Þ surface changed to (3  3) structure following a 30-min  annealing at 1050 pffiffiffi C.pIn ffiffiffi addition, both the oxygen and the Si–O AES signals vanished. The ( 3  3) R30 structure remained for the (0 0 0 1) surface after a 30-min annealing at 1000  C. However, OKLL peak disappeared and SiLVV signal returned the bulklike shape. The simultaneous structural evolution and oxygen removal from both surfaces are indicative oxide on the surface, which was pffiffiffi of pffiffisilicon ffi confirmed by LEED analysis of the ( 3  3) R30 phase of the presence of Si2O3 overlayer, referred to as the honeycomb silicate adlayer, on the SiC surface. The honeycomb silicate adlayer is formed by two Si atoms per unit cell and each oxygen atom connects two of the Si atoms completing a ring-type structure, as shown in Figure 3.9 pffiffiffi p ffiffiffi depicting the top view or the projection on the c-plane. The silicate-related ( 3  3) R30 structure on the (0 0 0 1) surface, shown as a side view, in Figure 3.9 is identically arranged with Si atoms also oriented toward the substrate. Interestingly, the silicate layer and substrate are linearly bridged by oxygen, Si–O–Si, not connected via Si–Si bonds. This simply implies that the surface as it stands is not optimum for nitride growth. It is possible that in OMVPE environment, the silicate adlayer may be removed, a statement that cannot be unequivocally made for MBE growth. In fact, the growth of AlN on such SiC surface leads to three-dimensional (3D) growth and, as discussed earlier, exposureto Gaathightemperature removes thesilicate layer [41–44]. Treating the surface of ex situ HCl-treated (1300  C) SiC with in situ Ga spray, Onojima et al. [41] obtained SiC surfaces free of the silicate adlayer, as shown schematically in Figure 3.10, and were able to obtain a 2D AlN growth. Owing pffiffiffi topGa ffiffiffi deposition on the SiC surface and subsequent flash-off, an oxygen-free ( 3  3) R30 surface structure was achieved and initial 2D growth with an evident RHEED

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[1 1 0 0]

(a)

[1 1 2 0] [0 0 0 1]

Top Si Second Si (b) Silicate adlayer

C Top O

SiC surface

Second O

Figure 3.9 (a) Top view of the oxide structure on SiC ð0 0 0 1Þ. The Si2O3 silicate adlayer consisting of a honeycomb structure with Si
O
Si bonds. At the center of the hexagons, one carbon atom of the topmost substrate bilayer is visible [the dark shaded area indicates the (1 1; 1) unit cell and light shaded the

pffiffiffi pffiffiffi ð 3  3ÞR30 -unit cell]; (b) side view of the oxide structure on the SiC (0 0 0 1) in ð0 1 1 0Þ SiC projection. Linear Si
O
Si bonds connect the silicate layer and the underlying SiC substrate. Courtesy of N. Onojima (patterned after Ref. [40]). (Please find a color version of this figure on the color tables.)

intensity oscillation was demonstrated. The initial growth mode of AlN closely correlated with the crystalline quality of AlN layer. Figure 3.11 shows the RHEED images of HCl-treated surface with a silicate adlayer, HF treated surface with has residual O, and finally in situ Ga spray treated SiC, of O on the surface for the ½1 1  2 0 and ½1  1 0 0 azimuths. Note the pffiffiffi which pffiffiffi is void  ( 3  3) R30 RHEED surface reconstruction for the HCl-treated surface, 1 · 1 pffiffiffi pffiffiffi RHEED reconstruction, and again ( 3  3) R30 RHEED surface reconstruction for the in situ Ga spray treated surface. 3.2.4 Sapphire

Owing to its relatively low cost, availability in large area, and continual improvement in its quality, both in terms of bulk and surface properties, sapphire has become the [1 1 0 0] [1 1 2 0] [0 0 0 1]

(a) Si

(b) 1/3 ML Si ad atom

ad-Si C

SiC surface Figure 3.10 (a) Top view (projection on the Si-plane of the basal plane of SiC) and (b) side view of SiC after an in situ Ga exposure indicating of the lack of silicate adlayer. Courtesy of N. Onojima. (Please find a color version of this figure on the color tables.)

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Figure 3.11 RHEED images along the ½1120 and ½1100 azimuths for HCl, HF,pand in ffiffisitu pffiffiSi-polarity ffi pffiffiffi SiC surface ffiffiffi p ffi Ga spray treated indicating ð 3  3ÞR30 , 1 · 1, and ð 3  3ÞR30 , respectively. Courtesy of N. Onojima.

dominant substrate material for epitaxy. Although there are other reasons, sapphire is transparent for most of the bandgaps of nitride alloys; thus, it affords certain benefits in detectors, for example, for back illumination and in LEDs, for lack of absorption. 3.2.4.1 A Primer on Sapphire Sapphire has the space group of R3c (No. 167), as provided in the International Tables for Crystallography, and is primarily of ionic bond nature. It can be represented by both rhombohedral unit cells, with volume 84.929 Å3, and hexagonal unit cell, with volume 254.792 Å3, which is displayed in Figure 3.12 [13]. In the rhombohedral unit cell there are 10 ions in total, 4 Al3 þ ions and 6 O2
ions. The hexagonal unit cell has 30 ions in all, 12 Al3 þ ions and 18 O2
ions. Oxygen is located at (x, y, z) ¼ (0.306, 0, 0.25). If this position is approximated to (x, y, z)  (1/3, 0, 1/4), the anion framework forms an hcp lattice with a ¼ 0.476 nm and c ¼ 1.299 nm. The unit cell described by Miller–Bravais indices consists of six close-packed (0 0 0 1) planes of O2
ions sandwiching 12 planes of Al3þ ions that occupy two thirds of the available octahedral voids created by the O2
ions. An Al3þ ion is located at (x, y, z) ¼ (0, 0, 0.352) instead of (0, 0, 1/3), thus the cations are shifted by 0.025 nm along the c-axis from the ideal octahedral sites. The oxygen ion is larger than the aluminum ion by a factor of about 3 in terms of its radius; therefore, the steps on the substrate are limited to those in the oxygen sublattice, leading to step heights in multiples of c/6 (d(0006)  0.216 nm). The (0 0 0 1) Al2O3 surfaces are oxygen terminated and present steps along f1 1 2 0g and f1 1 0 0g planes [45]. Two crystallographically equivalent surfaces are related by a symmetry operation of the space group. Along the [0 0 0 1] direction, A–A or B–B surfaces are separated by c/3, 2c/3, and c steps. Steps separating two “A” surfaces are noted as A–A, and c/3 steps of height c/6, c/2, or 5c/6

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separate the two surfaces related by a glide symmetry operator. Such steps are dubbed demi-steps and are noted as A–B, c/6 [46,47]. The unreconstructed basal c-plane perspective views for both unit cells are given in Figure 3.13 [13], where the cell boxes are polyhedra. A schematic representation of sapphire unit cell indicating the six O layers in the unit cell is shown in Figure 3.14. The oxygen ions form a pseudohexagonal lattice. The small Al ions occupy the octahedral sites. The labeling of planes and directions in the context of sapphire substrates are shown in Figure 1.5. Properties of sapphire are provided in Table 3.5 [13]. All common surfaces employed for GaN epitaxy including the (0 0 0 1) and ð1  1 0 0Þ are nonpolar. Thus, the polarity control on sapphire depends on the particulars of growth conditions employed with the ominous inversion domain formation always a possibility. Because

Figure 3.13 Perspective views in (2 · 2 · 1) unit cells: (a) along the [0 0 0 1] direction in a rhombohedral unit cell; (b) along the

3.2 A Primer on Conventional Substrates and their Preparation for Growth 0.287 nm

0.252 nm

b B

O 2–

Al3+

a A

0.052 nm

b

0.0797 nm

C

0.1358 nm

a B b

0.1441 nm

0.1661 nm [0 0 0 1]

A a

[1 0 1 0]

Figure 3.14 A schematic diagram of the Al2O3 sapphire unit cell, there are six oxygen layers in the unit cell, the distances between the various atomic layers change as shown in the figure. The oxygen ions form a pseudohexagonal lattice. The small Al ions occupy the octahedral sites. Courtesy of P. Ruterana and Ref. [47]. Table 3.5 Properties of sapphire (in part after Ref. [13] and references therein).

Parameter

Value

Condition

Lattice constant (Å) Melting point ( C) Density (g cm
3) Thermal expansion coefficient (K
1)

a ¼ 4.765, c ¼ 10.2982 2030 3.98 6.66 · 10
6 || c-axis 9.03 · 10
6 || c-axis 5.0 · 10
6 ? c-axis a/a0 ¼ 0.83, c/c0 ¼ 0.892

20  C

Percent change in lattice constants with DT Thermal conductivity (W cm
1 K
1) Heat capacity (J K
1 mol
1) Young’s modulus (GPa)

Tensile strength (MPa) Poisson’s ratio Hardness: Knoop nanoindentation (GPa) Energy band gap (eV) Resistivity (O cm)

20  C 20–50  C 20–1000  C 20–1000  C 293–1300 K

0.23 || c-axis 0.25 || a-axis 77.9 452–460 in [0 0 0 1] direction, 352–484 in ½1 1 2 0 direction 190 0.25–0.30 23.9  2.0

296 K 299 K 298 K

8.1–8.6 >1011

Experimental value 300 K

300 K 300 K 300 K

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2.4 2.2 2.0 1.8

Polycrystalline

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 –0.2

0

c-axis

Al2O 3

a-axis Si

200

400

600

800

100

1200

1400 1600

1800 2000

Temperature (K) Figure 3.15 Thermal expansion coefficient of sapphire on the cplane (along the a-axis) and along the c-axis, and that of Si as a function of temperature [48].

of heteroepitaxy of GaN on sapphire, it is useful to display the thermal expansion coefficient of sapphire as done in Figure 3.15. 3.2.4.2 Surface Preparation of Sapphire for Epitaxy As-received sapphire substrates contain scratches caused by mechanical polishing with root mean square (RMS) roughness values between 0.8 and 2.1 nm over l mm2 areas. Wet chemical etches such as phosphoric acid (H3PO4), sulfuric–phosphoric acid combination (H2SO4–H3PO4), fluorinated and chlorofluorinated hydrocarbons, tetrafluorosulfur (SF4), and sulfur hexafluoride (SF6) have been employed. None of these techniques, however, produces a surface free of damage and scratches. For MBE growth, which does not allow in situ cleaning of the surface in H at high temperatures, a 3 : 1 solution of H2SO4 : H3PO4 is used as the etchant. The substrate is dipped in this solution and kept at 300  C for 20 min. This is followed by a rinse in DI water for 3 min. Although the hot etching removes some material, the resultant surface still bears the scratches caused by mechanical polishing. However, the surface becomes flatter after etching, with the RMS roughness being reduced from 0.323 nm in image to 0.211 nm in image in one case. A high-temperature annealing technique after wet chemical etching of mechanically polished sapphire substrates has been shown to result in atomically smooth surfaces [49]. Figure 3.16 shows the AFM images of two c-plane sapphire surfaces, (a) before and (b) after the chemical etching. In an OMVPE or HVEP environment, the typical process is to simply heat the sapphire under flowing hydrogen at temperatures between 1000 and 1100  C. This process etches sapphire slightly leading to the formation of hexagonal pits if there are residual amounts of gallium left in the reactor from prior runs. The crystal quality of subsequently deposited GaN films was insensitive to the presence of these pits [13]. To eliminate surface damage altogether, a high-temperature annealing step has been employed, which gives rise to atomically smooth surfaces. A very high

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Figure 3.16 (a) AFM image of an as-received sapphire substrate. Note the scratches caused by mechanical chemical polishing; (b) AFM image of a sapphire substrate after a 180  C etch in sulfuric/ phosphoric acid. Some improvements are apparent, but the scratches remain and are accentuated to some extent. Image size 2 mm · 2 mm.

temperature annealing investigation of sapphire substrates was recently undertaken. Annealing experiments in air at 1000, 1100, 1200, 1300, and 1380  C (the ceiling of the furnace employed) for 30- and 60-min periods were conducted to determine the best conditions with the aid of AFM images of the finished surface. This was followed by observation of RHEED patterns once in an MBE system. A small, but progressive, improvement was observed in the reduction of scratches up to 1300  C. However, annealing at 1380  C for 1 h led to scratch-free and smooth surfaces to the point where the only noticeable feature in AFM images were the atomic steps about 0.15 nm in height. AFM images indicated that annealing at

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Figure 3.17 An AFM image of sapphire following a 1380  C–1 h annealing in atmosphere. Atomically flat surface is clearly visible. Atomic step heights are about 0.15 nm, which represent the only roughness in the image. The diagonal lines, from left to right, are the artifacts of AFM.

1380  C for 1 h leads to atomically smooth surfaces as shown in Figure 3.17. An atomically smooth surface is maintained after nitridation as well. RHEED images typically show extended and bright rods associated with sapphire at temperatures as low as 600  C during the ramp-up as shown in Figure 3.18.

Figure 3.18 A RHEED image at about 800  C of an annealed sapphire at 1380  C for 1 h (½1 1 2 0 azimuth). Clear streaky RHEED pattern observed at temperatures as low as 600  C indicates that the high temperature annealing step produces clean epiready surfaces. Without the annealing procedure, the RHEED images are not as clear and elongated and not reproducible.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Sapphire is nitridated by exposing it to nitrogen plasmas or thermally cracked ammonia (a practice that has been abandoned) in MBE reactors or by to ammonia/ hydrogen gas mixtures in OMVPE reactors. Sapphire substrates that have not undergone a heat treatment in O, as described above, exhibit the polishing damage in the form of random scratches when they undergo MBE-like pregrowth in situ heat treatment followed by an exposure to ammonia [50]. Considering the stacking-order mismatch between sapphire and nitrides, these features are likely to have deleterious effects on growth. A low density (108 cm
2) of surface outgrowths was observed after 30 min of nitridation. The presence of surface damage does not appear to have influenced the formation of protrusions. There is no clear correlation between the positions at which the protrusions have formed and the local surface topography. Uchida et al. [51] observed similar protrusions after 5 min of nitridation at 1050  C in an OMVPE system, but at a much higher density than that observed in the MBE process. It is likely that a combination of a higher substrate temperature and the background ammonia pressure promotes a more rapid nitridation reaction leading to a higher density of protrusions in OMVPE-grown samples. Although what is reported for these particular samples may hold, it should again be noted that atomically smooth surfaces following annealing in air, as described above, do not show discernible change after nitridation. Noting that AlOxN1
x would be unstable at the nitridation temperatures employed, the nitridation of sapphire should result in the formation of AlN. In fact, in the MBE process, this can be observed with RHEED in that the pattern associated with the ½1 1 2 0 azimuth of sapphire gives way to the pattern associated with the ½1 1 2 0 azimuth of AlN but with a 30 rotation to minimize misfit strain. Nitridation has direct consequences in the quality of the low-temperature buffer layer and final layer(s). The ultimate test whether nitridation was done properly requires going through the buffer and final layer growth. The particulars of the final layer are then used to draw conclusions about the nitridation process as was done by Wickenden et al. [52], the details of which are discussed in Section 3.5.5.1. In the study of Uchida et al. [51], nitridation was reported to occur very rapidly for times less than 3 min and then slowed considerably. For short nitridation times, of <3 min, the surface was reported to be relatively smooth, but stress-induced protrusions developed for longer nitridation times. The density of the protrusions increased with time, making the surface progressively rougher. Uchida et al. [51] posed the argument that nitridation produced an amorphous AlNxO1
x layer via the exchange of oxygen atoms from sapphire and nitrogen atoms from ammonia. However, this amorphous layer was not seen in subsequent TEM micrographs after the deposition of a 4.0 mm thick GaN layer, which they attributed to diffusion of N and O atoms into the crystalline layer. X-ray photoelectron spectroscopy (XPS) analyses were carried out to confirm the incorporation of nitrogen atoms into the sapphire substrate during the nitridation process. Sapphire substrates exposed to ammonia even for as little as 1 min did indeed show the 1s nitrogen peak; the same is expected from RF nitrogen exposure as well. This indicates that nitrogen atoms react with the surface. The downshifting of the oxygen 1s line by about 0.25 eV, relative to the spectra obtained from a bare sapphire sample, and the sample nitrided for 1 min

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suggests the generation of a significant bonding between oxygen and nitrogen atoms in the nitrided layer [50]. The RHEED observations during nitridation indicate that AlN forms on sapphire during nitridation and its orientation along the c-direction is rotated by 30 to accommodate the strain caused by lattice mismatch. The duration of nitridation is an important factor in terms of the quality of the eventual GaN layer. Increase in nitridation times from 60 to 400 s has been reported [53] to result in mobility reduction in GaN films, the overall features of which are consistent with the study of Kim et al. [50,54]. The nitridation process, both in OMVPE and MBE, has a great impact on the subsequent layers. In MBE, if nitridation is done well, which can clearly be observed by RHEED image changing from sapphire to AlN, the subsequent AlN layer is of high quality. In OMVPE, the picture is more complex in that nitridation is followed by the low-temperature buffer layer that is then annealed prior to the commencement of growth of subsequent layers. The structure of the low-temperature buffer layer and the change following annealing are such that the morphology of the subsequently deposited GaN buffer layer changes from rough to highly faceted (possibly mixed with the zinc blende structure) and to smooth films of the wurtzite structure. The details of the lowtemperature buffer and other buffer layers as well as other issues related to GaN growth in general are discussed in Section 3.5.5.1. The calculated lattice mismatch between the basal GaN before the in-plane rotation and the basal plane of sapphire is about 49%. However, the actual lattice mismatch of nitride layers with sapphire is reduced by the rotation of the nitride lattice with respect to the substrate unit cell by 30 . Consequently, the lattice mismatch is reduced to 13% to AlN, 16% to GaN, and 29% to InN. This large mismatch would cause even the very thin layers to be fully relaxed at growth temperatures. When the samples are cooled down after the growth, a residual thermal strain is created. In general, films grown on the basal plane show either very little or none of the cubic GaN phase. Among the faces of sapphire that have been utilized for nitrides are the c-, a-, and r-planes. The stacking configurations perpendicular to the c- and a-planes are displayed in Figure 3.19a and b. The atomic arrangements on the c- and a-planes of sapphire are also shown in Figure 3.19. Figure 3.20 depicts the stacking arrangement perpendicular to the r-plane (a) and the atomic arrangement on the r-plane of sapphire (b). 3.2.5 ZnO

Unlike sapphire and polytypes of SiC that are available, ZnO and GaN have a common stacking order and a small lattice mismatch as shown in Table 3.1. In addition, high-quality substrates, nearly 2 in. in diameter, are available. ZnO is up against the fact that most laboratories select sapphire substrates with SiC making inroads. The reasons are that the layers grown on sapphire and SiC have in many cases better quality, and sapphire is available up to 6 in. in diameter and is inexpensive. The temporary problems plaguing the SiC approach are high cost, fluctuating

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Basal or c-plane sapphire

[0 0 0 1]

O O O Al Al O O O Al Al O O O Al Oxygen

(a)

Al

a-plane (1 1 2 0) sapphire

[1 1 2 0]

O O Al Al O

O Oxygen Al

(b)

Figure 3.19 (a) The atomic arrangement and stacking order (left) and the top view (right) of the c-plane sapphire. (b) The atomic arrangement and stacking order (left) and the top view (right) of the a-plane sapphire.

quality, and poor surface finish. A more permanent, fundamental problem is the stacking-order mismatch on SiC and sapphire. As mentioned earlier, ZnO has the desired stacking order and a reasonably close lattice match to GaN. Moreover, as in the case of LiGaO2, it can be selectively removed from GaN followed by GaN being transferred to a template with good thermal conductivity. 3.2.5.1 A Primer on ZnO The thermal expansion coefficients and the lattice constants of ZnO are Da/a ¼ 4.8 · 10
6 K
1, Dc/c ¼ 2.9 · 10
6 K
1, a ¼ 3.2426 Å, and c ¼ 5.1948 Å, respectively. The thermal expansion coefficient of ZnO in the c-plane as a function of temperature

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Side view of sapphire r-plane {1 0 1 2}

Oxygen

(a)

Al

Al2O3 [1 0 1 1]

3.2 A Primer on Conventional Substrates and their Preparation for Growth

1.2

Thermal expansion coefficient (%)

ZnO

1.0

a-axis

0.8 Polycrystalline

0.6 0.4 c-axis

0.2 0.0 –0.1 0

200

400

600

800

1000 1200 1400 1600 1800 2000

Temperature (K) Figure 3.21 Thermal expansion coefficient of ZnO on the c-plane (along the a-xais), along the c-axis as a function of temperature; the thermal expansion coefficients of polycrystalline ZnO [48].

is depicted in Figure 3.21. Also shown are the thermal expansion coefficients of polycrystalline ZnO. High-quality ZnO substrates have recently become available in laboratory quantities. While stable in air and O environments at temperatures as high as 900  C, perhaps even higher, exposure to ammonia etches ZnO even at temperatures as low as 600  C. It is believed that atomic hydrogen reacts with O forming a volatile water vapor. The Zn metal is removed from the surface by evaporation. 3.2.5.2 Substrate Preparation for Epitaxy The chemical preparation of ZnO involves a 5 min acetone bath followed by the same procedure in methanol while using ultrasound agitation to remove particulates. The wafers are then rinsed in deionized water, followed by blow drying with filtered nitrogen prior to introduction into the growth chamber. The details of the growth processes are given in Section 3.5.7. ZnO is similar to the SiC and sapphire substrates and yet very different. It shares very similar problems, chiefly the scratches caused by mechanical polishing. 3 Figure 3.20 (a) Sapphire r-plane stacking sequence showing O atoms in larger clear circles and Al atoms in smaller, filled circles. The salient feature is that each Al layer has an O layer above and below it. (b) The atomic arrangement on three layers (the uppermost one is O,

immediately below is Al and third layer down is another O layer) on the r-plane of sapphire. The lines are there just guides to eye and do not represent bonds. (Please find a color version of this figure on the color tables.)

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Figure 3.22 An AFM image of a ZnO surface after a 3-h annealing procedure in air at Ta ¼ 900  C. The RMS is about 0.1 nm. The image is 500 nm · 500 nm.

Chemical etches have not yet been developed to deal with this problem. In fact, it may not even be possible to accomplish this task with chemical etches. However, as in the case of sapphire, annealing in oxygen appears to lead to improved surfaces over mechanically polished substrate. Annealing at 900  C for 3 h in air leads to atomic bilayer steps, which is about the best that can be achieved, as shown in Figure 3.22. Annealing in air at 850 and 950  C also led to very smooth surfaces [55]. This implies that the process is not very temperature sensitive and that reproducible results should be obtained without extreme control over the temperature employed. In an MBE system, the RHEED patterns also bear this out by showing sharp 1 · 1 diffraction rods during the ramp-up of the temperature, as shown in Figure 3.23.

Figure 3.23 RHEED image of ZnO taken at 780  C (½1 1 2 0 azimuth.

3.2 A Primer on Conventional Substrates and their Preparation for Growth

3.2.6 LiGaO2 and LiAlO2

There are other nontraditional substrates that are beginning to appear; among them are LiGaO2 (lithium gallium oxide (LGO), which is orthorhombic with space group Pna21 No. 33), and LiAlO2 (lithium aluminum oxide, which is tetragonal with space group P41212). 3.2.6.1 LiGaO2 Substrates Lithium gallate (LiGaO2) is the most closely lattice-matched substrate currently being considered for GaN heteroepitaxy, with an average lattice constant mismatch of only 0.9% in the basal plane. It is therefore expected that the quality of the very thin GaN films on LiGaO2 would be better than those on other substrates discussed so far. There is some experimental evidence to this effect. The GaN films grown on LiGaO2 are of Ga polarity. Although LiGaO2 is easily etched, which is useful for transferring the GaN film to another substrate, it is not easily etched to a smooth surface for epitaxy. As in the case of ZnO, etching for a few tens of seconds in H3PO4 removes the surface damage caused by mechanical polishing and reduces the short-range roughness. The main disadvantages of LiGaO2 are its low thermal stability under OMVPE growth conditions, low thermal conductivity, high thermal expansion coefficients, and electrical insulation. However, when GaN layers of about 300 mm are grown by HVPE followed by the removal of LiGaO2, the thermal and electrical conductivity of LiGaO2 are not relevant. The structure of LiGaO2 is similar to the wurtzitic structure, but because Li and Ga have different ionic radii, the crystal has an orthorhombic structure. The atomic arrangement in the (0 0 1) face is hexagonal, which promotes the epitaxial growth of (0 0 0 1) GaN, so that the epitaxial relationship (0 0 0 1) GaN/(0 0 1) LiGaO2 is expected. The crystal structure deviates slightly from the hexagonal symmetry because of the need to accommodate two different metallic atoms of Ga and Li. For the visualization of LiGaO2, one can think of it as the I–III–V analogue of ZnO, with one half of the Zn replaced by Li and the other half by Ga. The metal atoms are ordered, alternating in the [0 1 0] direction between Li layers and Ga layers. Unlike ZnO, LiGaO2 melts under atmospheric pressure, and hence large single crystals can be pulled from the melt using the Czochralski method. The orthorhombic lattice dimensions of LiGaO2 are a ¼ 5.402 Å, b ¼ 6.372 Å, and c ¼ 5.007 Å. The distance between the nearest cations in LiGaO2 is in the range of 3.133–3.189 Å, while the distance between nearest anions is in the range pffiffiffiof 3.021–3.251 Å. For comparison, the basal plane wurtzite lattice parameter h ¼ a= 3 ¼ b=2. The main virtue of LiGaO2 is that it has a very small lattice constant mismatch with GaN (averaging 0.9% at room temperature: 1.9% in the a-direction and
0.19% in the b-direction). The orientation relationship between GaN and LiGaO2 has been studied by several groups, yielding ½1 1 2 0 GaN||[0 1 0] LiGaO2 and [0 0 0 1] GaN||[0 0 1]LiGaO2. The atomic relationship between GaN and LiGaO2 in the ideal case is shown in Figures 3.24 and 3.25. Anion (oxygen) or cation (gallium and lithium) termination is available on the (0 0 1) substrate surfaces. As is the case in polar surfaces in all compound substrates,

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a

2

m

C

b

m

LiGaO2

Lower O

Li

Upper O

Ga

Figure 3.24 Example of the exact fit of GaN atoms over the LiGaO2 lattice if there is no distortion. Courtesy H. Paul Maruska. (Please find a color version of this figure on the color tables.)

the A-face (oxygen-terminated surface) can be etched easily, whereas the B-face (metal-terminated surface) is difficult to etch in an aqueous solution of nitric acid (H2O : HNO3 ¼ 1 : 1). This is attributed to surface oxygen atoms having a dangling bond with two electrons, whereas surface metal atoms have no dangling bonds (hard to etch). A maximum etch rate of 0.25 mm min
1 for the cation face was obtained at pH 9.2 and temperature 50  C. The etch is selective in relation to GaN, which means that the entire LGO substrate could be chemically removed without affecting the GaN layer. As-received LGO surfaces contain scratches and pits as a result of polishing damage and crystalline defects, as shown in Figure 3.26. However, following a chemical cleaning in acetone, methanol, and DI water and etching in H3PO4 for 10 s

3.2 A Primer on Conventional Substrates and their Preparation for Growth

LGO: orthorhombic a = 5.402 Å b = 6.372 Å c = 5.007 Å

c

(a)

b

a

LGO: orthorhombic a = 5.402 Å b = 6.372 Å c = 5.007 Å

c

(b)

b

a

Projection on c-plane Δb = –0.19 % a

Ga

N

=3

.18

9Å

bLGO = 6.372 Å

Δa = +1.1%

Δa = +1.9 % Δb = –1.1% (c)

aLGO = 5.402 Å

Figure 3.25 Structure of (a) orthorhombic LiGaO2 (LGO), (b) GaN, and (c) a detailed view of the relative orientation of GaN with respect to LGO. Courtesy of H. Paul Maruska. (Please find a color version of this figure on the color tables.)

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Figure 3.26 AFM images of LiGaO2 (LGO): (a) as-received, (b) after etching in H3PO4 for 10 s at 80  C, and (c) H3PO4 for 20 s at 80  C.

at 80  C, the surface is improved, with more improvement resulting when the etching time is increased to 20 s as shown in Figure 3.26b and c, with reduced pits and shortrange roughness. Compared to the as-received LGO, which exhibited a RMS roughness of 1.12 nm, a 20-s etching reduces this value to 0.78 nm. The RHEED analysis also shows a marked improvement in the sharpness and intensity of the reconstruction rods. The bandgap of LGO is 5.6 eV. The Young’s modulus of LGO is about 150 GPa and its density is 4.175 g cm
3. The hardness is 7.5–10 GPa, as measured by nanoindentation. Because LiGaO2 is asymmetric along the [1 0 0] and [0 1 0] directions, the coefficients along these two directions are different. The thermal expansion coefficients are Da/a ¼ 6 · 10
6 K
1, Db/b ¼ 9 · 10
6 K
1, and Dc/c ¼ 7 · 10
6 K
1. Optically, it is biaxial with a transparency ranging from 0.3 to 6 mm and refractive indices of na ¼ 1.7617, nb ¼ 1.7311, and nc ¼ 1.7589 at 620 nm. The crystal has no natural cleavage planes. The structure of interest is the only stable form of the compound in the range from room temperature to its melting point. Because LiGaO2 melts congruently at 1585  C, the requirements for its crystal growth are relatively modest – no high-temperature or high-pressure apparatus is necessary. Therefore, good-quality wafers are available with relatively large diameters at a price comparable to that of sapphire. Crystals up to 50 mm in diameter and 200 mm in length have been produced at pull rates of 2–5 mm h
1 by the Czochralski method from the mixture of Li2CO3 and Ga2O3 [56]. Crystals have been grown in [1 0 0], [0 1 0], and [0 0 1] orientations. 3.2.6.2 LiAlO2 Substrates The lattice constants of LiAlO2 are a ¼ 5.1687 Å and c ¼ 6.2679 Å with a density of 2.615 and hardness of 8 GPa, as determined by nanoindentation. The thermal expansion coefficients are Da ¼ 7.1 · 10
6 K
1, and will cause a compressive strain in GaN g/cm3 and Dc ¼ 15 · 10
6 K
1. Optically, it is uniaxial with transparency ranging from 0.2 to 4 mm and refractive indices of ne ¼ 1.6014 and n0 ¼ 1.6197 at

3.2 A Primer on Conventional Substrates and their Preparation for Growth

633 nm. The crystal has no natural cleavage planes requiring, as in the case of LGO, the facet development with a chemically assisted ion etching or chemical mechanical polishing. Of course, the former is better in terms of performance and cost reduction. The structure of interest is the high temperature form (or g form) of the compound. It melts congruently around 1700  C and is stable at room temperature. Single crystals can be grown by the conventional Czochralsky melt pulling method. The epitaxy relationships between GaN and LiAlO2 are expected to be ð0 1  1 0Þ GaN/(1 0 0) LiAlO2 with ½2 1 1 0 GaN/[0 0 1] LiAlO2. Unlike Al2O3 and 6H-SiC substrates with very smooth surfaces, except the scratches caused by mechanical chemical polishing, the LiAlO2 substrate exhibited a wavelike surface with equidistant grooves about 10 nm deep, which could have originated from the mechanical surface polishing. Because the preferred growth direction of the GaN epitaxial film is [0 0 0 1], the match along the a-axis is more critical for the film deposition than the c-axis. Considering the a-axis lattice constant, it is obvious that LiGaO2 is preferred for GaN and LiAlO2 for Al1
xGaxN. Another important factor is that because these two crystals have exactly the same structure as GaN, the growth orientation may not have to be limited just to the c-axis or the [0 0 0 1] direction. In fact, epitaxial growth can be achieved at any orientation. The degree of lattice matching may vary slightly depending on the structure of epitaxial nitrides in exact orientation. 3.2.7 AlN and GaN

It is well known that nitride semiconductors do not enjoy native substrates, notwithstanding considerable efforts to produce them. The main impediment is the large vapor pressure of N on AlN, GaN, and InN, in ascending order, coupled with a low solubility of N in the molten metal at reasonable temperatures and pressures. It is thus imperative to consider the phase diagrams of these binaries. Shown in Figure 1.18 are the partial pressures of N2 over AlN and Al liquid as a function of temperature, as determined by Slack and McNelly [57], on GaN by Karpinski et al. [58], and on InN by Porowski and Grzegory [59]. The calculated values for the N2 vapor pressure on AlN by Slack and McNelly are 1, 10, and 100 atm at about 2550, 2800, and 3120  C, respectively. The GaN data, however, tell a different story in that the partial pressure of N2 is very high, necessitating high-pressure experiments to collect data. Karpinski et al. employed a tungsten carbide anvil cell and pressures of up to 60 kbar to collect the data presented. The GaN data deviate from the calculations of Thurmond and Logan [60] most noticeably at high temperatures. The equilibrium N2 pressure data were calculated from the measurement of the equilibrium ammonia pressure over GaN assuming N2 to be an ideal gas and thus would predict a linear dependence in the log vs. 1/Tscale. The partial pressure data alone are indicative that AlN would be easier to synthesize than GaN. In fact, Slack and McNelly [10] obtained growth rates in the millimeters per hour range under very moderate pressures with a duration of growth determined by the reaction rate of Al with a tungsten crucible. The same partial pressure data also indicate the difficulties associated with an epitaxial

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deposition going from AlN to GaN and then to InN requiring an increasing amount of nitrogen overpressure necessary to avoid decomposition. The data of the phase diagrams of GaN, AlN, and InN are limited and contradictory by reason of the high melting temperatures (TM) and the high nitrogen dissociation pressures (P dis N2 ), particularly for InN and to a lesser extent for GaN. Dissociation pressure of MN, where M represents a metal species such as Al, Ga, or In, is defined as the nitrogen pressure at the thermal equilibrium of the reaction [61] MClðgÞ þ NH3 ðgÞ ! MNðsÞ þ HClðgÞ þ H2 ðgÞ;

ð3:1Þ

where s, l, and g indicate solid, liquid, and gas state, respectively. Reported values for P dis N2 of GaN are plotted in Figures 1.18 and 1.19. Examining the available data, Sasaki and Matsuoka [61] concluded that the data of Madar et al. [62] and Karpinski et al. [58] are most reliable. The data of Karpinski et al. are therefore shown. As can be deduced from Figure 1.18, the nitrogen dissociation pressure equals 1 atm at approximately 850  C and 10 atm at 930  C. At 1250  C, GaN is unstable and decomposes even under a pressure of 10 000 bar of N2. The case of InN is even more problematic at the decomposition temperature of about 700  C [59]. It should therefore come as no surprise that the incorporation of nitrogen at high temperatures is a nontrivial problem at best. For pressures below equilibrium at fixed temperatures, the thermal dissociation occurs at a slow and apparently constant rate, suggesting a diffusive process of dissociation. Despite the fact that equilibrium growth of InN appears nearly impossible, InGaN grows at temperatures in the range of 700–900  C with very high amounts of reactive nitrogen on the surface. The difficulties of growing bulk GaN and especially InN are alleviated somewhat with AlN synthesis. Judging from the N pressure at equilibrium, AlN is the closest to successful production of all the group III–V nitride semiconductors under discussion for bulk growth in the conventional sense. The most satisfactory method of growing highpurity AlN is the one in which AlN itself is the starting material. Slack and McNelly [57,63] employed a technique in which high-purity AlN is produced through an intermediate AlN powder formed by utilizing AlF3. Sublimation converts the AlN powder into single crystals in a closed tungsten crucible or in an open tube with a gas flow. The main problem with this growth technique is perhaps the surface oxidation of the powder owing to the strong reactivity of oxygen and aluminum. If this oxidation is minimized, then AlN could be produced with only 100 ppm of oxygen and with lower amounts of other impurities. The purest AlN prepared by Slack and McNelly employing a tungsten crucible had 350 ppm oxygen. However, when W or Re was used for the crucible, very little contamination of the AIN with metal impurities was found. Furthermore, the crystals grown in W or Re crucibles generally showed uniform amber color indicating that, indeed, both oxygen and carbon contaminations were scrupulously minimized. GaN is more difficult to grow than AlN, but not as difficult as InN. A much lower equilibrium pressure of N on AlN, on the contrary, lends itself to the growth of this alloy utilizing the sublimation technique. Samples with dimensions of some 4 mm diameter and 12 mm length have already been prepared [64] and used to determine much of the AIN thermal data discussed in this

3.2 A Primer on Conventional Substrates and their Preparation for Growth

chapter. The deposition temperature is around 2250  C. Deterioration of the W boat determines the size of the crystal. The thermodynamic properties of GaN [65], in particular its melting conditions, are so extreme that the application of the common growth methods utilizing stoichiometric solutions is technically impossible. To increase the solubility of N in Ga melt and also to reduce N desorption, nitride crystal growth can be attempted under high N2 pressures as has been done by Leszczynski et al. [66,67], Porowski et al. [68,69], and Grzegory et al. [70,71]. The GaN templates are crystallized in gas pressure vessels with volumes up to 1500 cm3 and with a workable crucible volume of 50–100 cm3. The high pressure– high temperature reactor consists of a pressure chamber and a multizone furnace. It also has features for in situ annealing in vacuum and electronics for stabilizing and programming of pressure and temperature. The walls of the vessel are water cooled considering the high temperatures involved. The pressure in the chamber is stabilized to a precision better than 10 bar. The temperature is measured by an array of thermocouples in the furnace. This allows a stabilization of temperature to 0.2 C and programmable variations of the temperature distribution in the crucible. GaN crystals presented are grown from nitrogen solutions in pure liquid gallium or in Ga alloyed with 0.2–0.5 at.% of Mg or Be at pressures in the range of 10–20 kbar and temperatures of 1400–1600  C. Magnesium and beryllium, being the most efficient acceptors in GaN, are added to the growth solutions to obtain p-type crystals. Supersaturation in the growth solution is obtained by a deliberate application of temperature gradient of 2–20  C cm
1 along the long axis of the crucible. This approach assures a continuous flow of nitrogen from the hotter part of the solution to the cooler parts. Crystallization experiments performed without intentional seeding resulted in crystals nucleating spontaneously on the internal surfaces of polycrystalline GaN crusts in the cooler zone of the solution. Typical duration of the growth process was 120–150 h. Leszczynski et al. [15], Porowski et al. [16], and Grzegory et al. [70] performed nitride crystal growth from a solution under high N2 pressure. The growth experiments under high pressure were carried out in a gas pressure chamber of 30 mm internal diameter with a furnace dimension of 14 mm (1500  C) or 10 mm (1800  C) and with a boron nitride (BN) crucible containing Al, Ga, or In. The temperature was stabilized to a precision better than 10  C. Measures were made to optimize the pressure range for growth. With this optimization, the crystals grew only at a pressure for which the nitride was stable over the entire temperature range. In the case of GaN, single crystals were grown from a solution in liquid Ga under a N2 pressure of 8–17 kbar at temperatures ranging between 1300 and 1600  C. The quasi-linear temperature gradient in the process, which spanned over 5–24 h, was 30–100  C cm
1. The nucleation and growth of single-crystal GaN took place through the process of dissociation and transport of thin polycrystalline GaN film deposited on a Ga surface into the cooler part of the crucible. The technique has also been applied to AlN and InN as well, with GaN being the most successful. At high N2 pressure, the synthesis rate of AlN is high and that of InN is extremely low. The rate of AlN growth is so high that, at a pressure lower than

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6.5 kbar, thermal explosion takes place during heating of a bulk Al sample. Owing to a low stability, the crystallization rate of AlN at 1600–1800  C is marginally low (0.5 mm h
1 in the literature). On the contrary, owing to kinetic (low-temperature) and thermodynamical (low-stability) barriers, crystal-growth experiments of InN result in very small crystallites (5–50 mm), particularly when grown by slow cooling of the system from the temperatures exceeding the stability of InN. A wide variety of GaN, AlN, and InN crystals result from changes in temperature and pressure used during growth. 3.2.7.1

Seedless Growth of GaN

3.2.7.1.1 Seedless Growth of GaN by High Nitrogen Pressure Solution Growth (HNPSG) for Substrates The GaN crystals grown by the high nitrogen pressure solution method without seeding are wurtzitic and mainly have the form of hexagonal platelets. They grow at a rate of just below 0.1 mm h
1, along the h1 0  1 0i like directions and have nearly perfect morphology indicating a stable layer-by-layer growth mode. As one can deduce from the form of the crystals, the growth is strongly anisotropic in that it is much faster (about 100 times) in the c-plane, as can be deduced from the size of crystals produced. This holds true for supersaturations corresponding to an average growth rate in h1 0 1 0i directions of 0.05–0.1 mm h
1. The templates are transparent with flat mirrorlike faces. The behavior of the crystals remains the same for solutions containing Mg or Be. The average size of crystals, grown without any intentional seeding, scales with the diameter of the high-pressure reactor. A photograph of a slice of one such sample prepared at UNIPRES in Warsaw [72] is displayed in Figure 3.27. Growth rate anisotropy in favor of enhanced rate along the c-plane is desirable for acceptable size templates. However, this at the expense of a much reduced growth rate in the c-direction leads to only template growth as opposed to a boule growth, which would allow many substrates to be sliced. The vertical growth rate in the c-direction can

Figure 3.27 GaN crystals grown in high pressure chambers of different sizes. The grid size corresponds to 1 mm · 1 mm. The schematic cross section of the hexagonal platelet is shown [71].

3.2 A Primer on Conventional Substrates and their Preparation for Growth

be increased by increasing the supersaturation of the Ga melt, but this has the undesirable effect of unstable needlelike forms. The supersaturation of the solution is determined primarily by the temperature of growth and its gradient, mass transport mechanisms in Ga, and by the local competing processes such as neighboring crystals because we are dealing with seedless, or spontaneous, growth. For large GaN templates to be obtained without accelerated growth near the edges and corners, it is imperative that the supersaturation be attained. It should be noted that for extremely high supersaturation, edge nucleation on the hexagonal faces of GaN platelets often occurs, which seeds the unstable growth on that particular face. The tendency toward unstable growth is stronger for the Ga-face for undoped templates and N-face for doped templates (with sufficient Mg to render the crystal p-type), resulting in rough surfaces leaving the unaffected face mirrorlike. However, if Mg concentration is not high, the aforementioned instability has no effect in that the N-face remains mirrorlike [73]. The instability is characterized as macrosteps, periodic inclusions of solvent or cellular growth structures. The polarity of the crystal surfaces can be identified by etching in hot alkali solutions, because the Ga-polar surface is inert to etching, whereas the N-polar one etches very rapidly. The validity of the etching method has been verified by performing convergent beam electron diffraction (CBED) [74] and XPS [75] measurements in the same samples. The topic is further discussed in Section 4.2, in conjunction with extended defects. The low doping level, with no bearing on stability, may be related to the position of the Fermi level that may influence the microscopic nature of growing surfaces, which is consistent with ab initio calculations and underscores the importance of both native and impurity-related point defect formation in GaN [76,77]. This would presuppose that the instabilities mentioned above may have their genesis in point defect formation for which there is no direct evidence as yet. Summarizing, the seedless growth method leads to rates of 0.05–0.1 mm h
1 along the h1 0 1 0i directions and the crystal is morphologically stable. However, the crystal is morphologically stable along the h1 0 2 0i directions even for higher growth rates. This explicates that the size of the stable platelets would depend on the volume of the solution and growth time. One can argue that increasing the volume of the crucible and growth time might lead to scaling up the size of the crystal. 3.2.7.1.2 Seeded Growth of GaN by HNPSG Method for Substrates To circumvent at least some of the scalability-related issues, seeded growth can be attempted using the templates prepared by the seedless method. However, seeding growth along the c-axis seems to be much more challenging because the observed growth rates are small and the growth tends toward instability [71]. Parameters such as seed preparation and the configuration of the experiment in the context of N flow could be optimized to enhance the in-plane or out-of-plane growth rates. For example, the macrosteps at the edges result from exposure to higher nitrogen fluxes, because the supply is from the hotter part of the solution. This can be alleviated by configuring the seed to melt relation, which can also lead to much more uniform supersaturation across the growth front. Other complications result from constitutional supercooling of the solution, caused by a very low temperature gradient and high thermal and low

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solute diffusivity [78]. This leads to the creation of a depleted zone at the growth front, which cannot be remedied because the concentration cannot be recovered by diffusion owing to low solute diffusivity. N-type GaN platelets have been used for seeded crystallization to suppress the cellular growth on the Ga-polar (0 0 0 1) surface. The N-polar ð0 0 0  1Þ surfaces have also been used for comparison. Large positive temperature gradients of the order of 100  C cm
1 have been applied at a nominal average crystallization temperature of about 1500  C. After 20–50 h growth, the substrates with the new crystals were removed from the solution and investigated [71]. Essentially, the new growth, which was a hillock 6 mm in diameter, was transparent, colorless, and void of periodic and cellular structures. The dominant feature was the propagation of the macrosteps from the hillock center. Growth on the N-polar surface was mainly by the propagation of the macrosteps [71]. Several growth centers were also noted for similar experimental conditions. It is still not clear if this is related to differences in surface nucleation mechanisms for the different polarities or due to inadequate surface preparation. The average growth rates in these experiments were between 4 and 8 mm h
1 for both polarities, which depends on supersaturation at the growth front, which, in turn, is a function of temperature gradient and the amount of liquid Ga layer on the substrate. Microscopic observation of cross sections of the samples indicates a stable growth in terms of continuity of the newly grown material, that is, inclusions of the solvent and/or voids were not observed. The interface between template (80–100 mm thick) and the newly grown crystals was not noticeable, indicating that the surface preparation and the wetting procedures were adequate. The X-ray diffraction (XRD) analysis indicated similar quality for the template and overgrowth. Future efforts should include crystallization experiments for finding the optimum configuration for a stable growth along the h0 0 0 1i directions. Of the particular issues remaining to be addressed adequately are step bunching and its reduction, and uniform supersaturation across growing surface. The uniform supersaturation can be achieved by reducing the radial temperature gradient and/or decreasing the width of the Ga layer over the substrate. Summarizing, an examination of the crystal morphology indicates that the crystal shape and size depend on the pressure, the temperature range, and the supersaturation during growth. For pressures and temperatures lying deeply in the GaN stability field (e.g., higher pressure and lower temperature), the crystals are hexagonal prisms elongated in the c-direction. Under conditions close to the equilibrium curve, the dominating shape of the crystals is a hexagonal platelet. The crystals grown slowly (slower than 0.1 mm h
1) at smaller temperature gradients, exhibited higher crystalline quality. Typical full width at half maximum (FWHM) of the X-ray rocking curves for (0 0 4) Cu Ka. reflections are 23–32 arcsec. Probably owing to the nonuniform distribution of nitrogen in the solution across the growing crystal face, the quality of GaN crystals deteriorates with increasing growth rate (high supersaturation) and with increasing dimensions of the crystals. This is apparent especially when the size of the face becomes comparable to the size of the crucible. The deterioration of quality of 5–10 mm crystals grown at a rate of

3.2 A Primer on Conventional Substrates and their Preparation for Growth

0.5–1 mm h
1 was evidenced by the broadening of the X-ray rocking curve. GaN substrates are not available in large quantities. Extremely high N2 equilibrium pressure on GaN will most likely preclude even pseudoconventional growth methods to fall short of producing GaN substrates. Bulk GaN substrates have been prepared under high pressures (12–20 kbar) and temperatures (1200–1600  C). Despite complications and challenges, the high-pressure method is capable of growing thicker crystals, which can be sliced into platelets, as shown in Figure 3.28. 3.2.7.2 Pertinent Surfaces of GaN Improve surface morphology of epitaxial layers, metal–semiconductor contacts, processing and effect of surface features on transport, and optical properties of GaN, requires an understanding of surface structure and growth processes on an atomic scale. For the technologically relevant surfaces, that is, polar (0 0 0 1), ð0 0 0  1Þ and nonpolar ð1 1 0 1Þ, ð1 1 0 0Þ, ð1 1 2 0Þ, experimental and theoretical investigations have led to considerable understanding of their atomic geometry, the driving forces leading to these structures, and their effect on adatom kinetics, the latter is discussed in Section 3.4.2. The polar hexagonal {0 0 0 1} basal plane is the most common surface on which to grow. Even though other surfaces have been explored for growth, the structures grown on the basal plane have exhibited the best performance. Among the basal planes, the Ga-polarity surface (0 0 0 1) is more stable and can be produced with smoother surfaces than the N-polarity ð0 0 0 1Þ surface. Figure 3.29 shows the schematic representations of top view of the ideal truncated GaN(0 0 0 1) surface as presented by Neugebauer [79]. The dashed lines depict the boundary of a (2  2) unit cell. High symmetry adsorbate sites are also marked as H3 (hollow) and T4 (atop a second layer atom). The side views along the ð1 1 0 0Þm-plane and ð1 1  2 0Þa-plane are shown in Figures 3.31 and 3.32.

Figure 3.28 GaN substrate crystal, obtained by slicing of the new material, grown by directional crystallization along c-axis; (a) before slicing where the bottom ledge corresponds to the original template as shown in the cross-sectional artistic rendition and (b) one sliced template. The grid is 1 mm · 1 mm [71].

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T4

H3

Ga N

Figure 3.29 Schematic representations of top view of the ideal truncated GaN (0 0 0 1) surface. The dashed lines depict the boundary of a (2 · 2) unit cell. High-symmetry adsorbate sites are also marked as H3 (hollow) and T4 (atop a second layer atom). The darker circles depict Ga atoms, and the lighter circles show N atoms, respectively. The side views along the ð1 1 0 0Þm-plane and ð1 1  2 0Þa-plane are shown in Figures 3.31 and 3.32.

Experimentally, the most commonly observed reconstruction on (0 0 0 1) GaN is (2 · 2), which corresponds to a stable growth front, yielding high-quality thin films, as discussed in Section 3.5.6. Because this reconstruction has not been observed on ð0 0 0 1Þ GaN, its presence is considered the domain of (0 0 01) GaN leading to inversion domains, which are discussed in Section 4.1.3. Other reconstructions observed by RHEED and STM on this surface are (1  2), (4  4), (5  5), and (6  4), details of which are discussed in Section 3.5.6. The ideal truncated GaN(0 0 0 1) surface with Ga in the top surface layer is shown in Figure 3.29. Tetrahedral bonds are such that each Ga surface atom has one dangling bond and is bonded to three N atoms in the second layer below. Adding one Ga or N layer on this surface would be called a Ga or N adlayer structure and adding/removing a Ga atom would be called Ga adatom/vacancy. Theoretically, this surface has been studied by several researchers employing plane wave pseudopotential methods and ab initio tight-binding methods [79]. All studies are centered on (1  1) and (2  2) structures, specifically adatoms on H3 and T4 sites, trimers, and Ga adlayers on various sites. The calculated relative formation energies are such that the (2 · 2) H3 N adatom model (with N at the He hollow site) is energetically the most stable under N-rich conditions. The (2 · 2) Ga-vacancy model gives only slightly higher energy. In contrast, under Ga-rich conditions, the (2 · 2) T4 Ga adatom structure (Ga adatom atop the atom in the second layer below) is favored. Under N-rich conditions, the Ga-vacancy and N adatom (on an H3 site) are the stable surfaces with energies close to one another. In the case of Ga-rich conditions, the Ga adatom (on a T4 site) is the lower energy structure and is thus preferred. As for the ð0 0 0 1Þ N-face, four dominant reconstructions, (1  1), (3  3), (6  6), and c(6 · 12) listed in order of increasing surface coverage, have been identified, discussed in detail in Section 3.5.6. For N-rich conditions on this surface, a (2 · 2) H3

3.2 A Primer on Conventional Substrates and their Preparation for Growth

Ga adatom model is found to be most stable. In the case of Ga-rich conditions, a (1 · 1) adlayer structure is energetically favored. In the stable (1 · 1) model, a full monolayer (ML) of Ga atoms is situated directly atop the N atoms, with the Ga–N bond length equal to 1.97 Å (compared to 1.94 Å in bulk GaN). The Ga–Ga separation in the adlayer (3.19 Å) is considerably larger than a typical Ga–Ga separation of 2.7 Å in bulk Ga. It should be pointed out that this is a completely novel structure with no known analogue among other semiconductor surfaces. In fact, this structure violates most of the empirical rules used to describe semiconductor surfaces in that it clearly disobeys electron counting and maximizes the number of dangling bonds. One immediate conclusion that can be drawn is that the surface must be metallic, which has indeed been observed by scanning tunneling spectroscopy (STS) [79]. Regarding the (3 · 3) surface, structural models with one, two, or three additional Ga adatoms on (or in) the Ga adlayer have been considered. One additional Ga adatom is the best model for the observed (3 · 3) reconstruction where the extra Ga atom resides only 0.9 Å above the adlayer plane. If no lateral relaxation is allowed, the Ga adatom must be positioned 1.8 Å above the adlayer to preserve a reasonable Ga–Ga distance. However, the extremely large inward relaxation of the adatom is enabled by a 0.5 Å lateral relaxation of the nearest-neighbor Ga-adlayer atoms. This then allows the adatom to be situated much closer to the adlayer plane, which stabilizes the structure, adopting the nomenclature that it is an in-plane adatom model. The GaN ð1 1 0 1Þ surface has been observed as a sidewall facet in lateral epitaxial overgrowth (ELO), which is used for threading defect reduction and thus deserves discussion. The ELO method is discussed in Section 3.5.5.2. These surfaces have also been found as sidewalls in the inverted pyramid defects, which form at the termination of some threading dislocations at the GaN surface during growth of In-containing alloys [79]. Northrup et al. [80] have theoretically investigated this surface by employing first-principles calculations. The schematic representation of a stable surface structure under Ga-rich conditions is given in Figure 3.30. It consists of Ga atoms in two distinct types of sites in the surface layer. These sites have been labeled as B2 and T1 sites. The atoms in B2 sites are bonded to two N atoms in the layer below. However, the atoms in T1 sites are bonded to one N atom each in the layer below, as in the (1 · 1) Ga-adlayer structure existing on the ð1  1 0 1Þ surface [79]. The structure stable under N-rich conditions contains one Ga atom per cell bonded in an H3 site. This adatom is bonded to three N atoms in the layer below. In terms of nonpolar surfaces, the ð1 1 0 0Þ and ð1 1  2 0Þ are important as they do not exhibit the polarization endemic in layers grown on polar surfaces, see Section 2.12 for a discussion of polarization. Growth on these nonpolar surfaces has progressed well, some details of which are discussed in Section 3.5.11. These surfaces also represent the sidewalls of processed (0 0 0 1) surfaces as well as being central planes in ELO. A schematic model of the GaN ð1 1 0 0Þ surface is shown in Figure 3.31. On the ð1 1 0 0Þ surface, an equal number of threefold-coordinated Ga and N atoms exist in the top surface layer. This allows charge neutrality without changes in stoichiometry or reconstruction, hence the term nonpolar. The Ga and N atoms form an array of Ga–N dimers. The ð1 1 0 0Þ surface has been studied by Northrup et al. [81] and Filippetti et al. [82] and a review by Neugebauer [79] is available that employs density–functional

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(0 0 0 1) c-plane B 2 site

T1 site –

(1 0 1 1) facet B 2 site

Ga N

T1 site

Figure 3.30 Schematic representation of the GaN ð1101Þ Gaadlayer surface. Surface sites T1 and B2 are sites in which a Ga atom makes one or two bonds with N atoms. The darker circles depict Ga atoms and the lighter circles N atoms. Patterned after Ref. [79].

theory calculations. The two main relaxation mechanisms in effect are a contraction of the GaN bonds by 6% accompanied by a slight buckling rehybridization with N atoms. The N atoms tend to adopt a p3 configuration, whereas Ga atoms adopt a sp2 configuration. The bond rotation angle is 7% and the surface energy is 118 meV Å
2. The structure of the ð1 1 2 0Þ surface can be construed as a chain of threefoldcoordinated Ga and N atoms, as shown in Figure 3.32. In each of the unit cells, there are four surface atoms, namely two Ga and two N atoms. This surface also has been

Ga N

Figure 3.31 Schematic top view of the Wz GaN ð1 1 0 0Þ surface. Larger circles depict the atoms in the first layer and the smaller ones portray those in the second layer. The dashed lines outline the boundary of a unit cell (5.179 Å  3.171 Å). Patterned after Ref. [79].

3.2 A Primer on Conventional Substrates and their Preparation for Growth

[0 0 0 1] Ga N Figure 3.32 Schematic top view of the Wz GaN ð1 1 2 0Þ surface. The dashed lines outline the boundary of a unit cell (5.493 Å  5.179 Å). The smaller circles denote atoms in the second layer. Patterned after Ref. [79].

investigated with density–functional theory calculations [81]. The calculated Ga–N bond lengths in the surface chain are 1.85 Å (cis) and 1.87 Å (trans), representing a contraction of approximately 4–5% compared to bulk. Similar to the ð1  1 0 0Þ surface, the Ga atoms relax toward an sp2 configuration. The bond rotation angle is 7 and the surface energy is slightly larger than that for the ð1  1 0 0Þ surface by 123 meV Å
2.  Contrary to the ð1 1 0 0Þ surface, the structure formed by replacing surface N atoms with Ga atoms is not energetically favored even under Ga-rich conditions. 3.2.7.3 GaN Surface Preparation for Epitaxy A very crucial step in the homoepitaxial growth of semiconductors is the attainment of an atomically clean and smooth substrate surface, without which even severe extended defects would form at the interface between the template and epitaxial overlayer. Available methods for cleaning the surface prior to epitaxy are characterized as external and internal ones. Ex-situ processes attempt to remove any metallic and organic contaminants from the surface. However, these methods always leave a thin oxide layer on the surface (of thickness similar to the native oxide), which must be removed before growth. This is typically done in situ in the epitaxial reactor. In the vapor phase techniques, the thin oxide is removed during ramp to growth temperature under a reducing environment such as H. However, removal within the MBE UHV system (in situ) usually includes a preliminary heating for outgassing of adsorbed gases in a preparation chamber and a final heating for oxide removal and preparation of an atomically clean surface within the growth chamber [83].

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Overviews of the ex-situ and in-situ preparation of a GaN surface are found in Refs [84,85]. Because high-quality GaN epitaxy by MBE, and perhaps by other techniques as well, requires growth on GaN templates prepared by vapor phase techniques, it is necessary to discuss this issue of GaN surface cleaning in the context of MBE for the moment. Commonly, a GaN surface that has been exposed to the air is contaminated with oxygen and carbon [84–86], which must be removed before epitaxy. XPS can be used to quantify the surface contaminants. Such studies on GaN surface oxide led to the observation of two components of the O 1s peak centered at 531.3–531.6 and 532.4–532.7 eV, respectively [85–87]. The O 1s component centered at 531.3 eV, which can be removed in HCl [86], NH4OH [87], HF, and UV/O3 (ozone treatment) is attributed to stoichiometric Ga2O3, although the oxide may also contain some component of mixed oxynitride of Ga, which is also soluble in alkali solutions. The origin of the high-energy O 1s peak at 532.7 eV, which of course can be removed with ion etching, is not yet clear. Prabhakaran et al. [87] reported only that it could be removed by argon ion sputtering while the 531.3 eV O 1s peak remained. Ar ion etching was also reported to remove the C 1s peak that, concurrently with the highenergy 532.7 eV O 1s peak, paved the way for speculation that C and O are the components of C–O bonding [86]. The high-energy O peak has been attributed to OH
species (hydroxides) by King et al. [85]. This oxide component was dominant on GaN surfaces after solvent cleaning and prior to UV/O3 exposure. The carbon contamination on the GaN surface, which manifests itself with a main C 1s XPS peak at 285.7 eV, is indicative of a mixture of C–O and C–H bonding [85]. Shifting of the C 1s peak to the lower binding energy of 283.7 eV after sputter cleaning led Prabhakaran et al. [87] to suggest that this is atomic carbon and could get incorporated into the epitaxial layer. Shalish et al. [86] have observed a C 1s component at 289.9 eV, which was attributed to the carboxylic group (COOH), and a peak at 287.7 eV, which was assigned to C–Cl bonds, appearing after etching in HCl. These observations [86,87] point to the necessity of surface preparation with HCl or HF, or possibly NH4OH to remove any native oxide from the surface prior to introducing the sample into the growth system. As for C contamination, the UV/O3 oxidation treatment was found most useful [85]. After chemical preparation, the sample is still exposed to air before loading which brings us to in situ treatment. The C and O contamination could not be removed at temperatures (900–1000  C) even higher than the decomposition temperature (800 C) [84,85]. This means that annealing of GaN must be conducted in N or Ga flux to maintain the stoichiometry of the surface [85]. The difficulty associated with oxide desorption has been attributed to the strength of the Ga–N bonds [85] in that oxides probably desorb as either Ga–O or N–O species instead of O2 and this requires breaking the strong Ga–N bonds. Most of the C–O-bonded carbon was reported to desorb at temperatures between 500 and 600  C leaving behind only C–H-bonded carbon, which apparently desorbs at much higher temperatures [85]. Efficient removal of C components from the surface of GaN has been successfully conducted with NH3 and Ga fluxes at 800–900  C in the author’s laboratories and elsewhere [84,85]. King et al. [85] concluded that atomically clean surfaces obtained by annealing in NH3

3.2 A Primer on Conventional Substrates and their Preparation for Growth

at 800  C exhibited 2 · 2 reconstruction in LEED. However, Bermudez et al. [84] found only partial removal of oxygen after annealing in NH3 up to 900  C. They attributed this to the fact that the reaction Ga2 O3 þ 2NH3 ! 2GaN þ 3H2 O

ð3:2Þ

is endothermic by 0.79 eV/O atom; hence, reduction of Ga2O3 by NH3 may require temperatures above 900  C, while the reaction 3C þ 4NH3 ! 3CH4 þ 2N2

ð3:3Þ

is exothermic by about 6.7 eV/C atom. In-situ RHEED observations of GaN surfaces that have been exposed to air show the absence of any amorphous native oxide layer on the surface, in contrast to that for the conventional arsenide-based semiconductors [88]. Although the GaN surface adsorbs oxygen and other impurities, owing to the large number of unsaturated dangling bonds [89], the oxidation is self limiting due to the chemical inertness of GaN. Thus, it is possible to grow GaN overlayers by MBE after an ex situ and in situ preparation consistent with the above discussion, a recipe for which is summarized below, to the point that the interface between the epitaxial overlayer and underlying template could not even be detected with TEM. Depending on the history of the template, the procedure used at VCU consists of degreasing followed by HCl or HF : H2O treatment, which, in turn, is followed by a few minutes treatment in boiling aqua regia. After rinsing and drying, the sample is loaded into the introduction chamber of MBE. 3.2.8 Other Substrates

Other substrates such as Hf, MgO, NaCl, W, and TiO2, and so on are not discussed here as they do not represent any sustained effort. However, spinel at one point got a good deal of attention and has been discussed here. Lasers employing films grown on the c-plane of sapphire utilize etched cavities because sapphire does not cleave well. The cleaving process is further complicated in that the GaN epilayers are rotated (about 30 ) with respect to the underlying sapphire substrate making it impossible to align the cleavage plane of GaN and sapphire. For this reason, Nakamura et al. [90,91] explored lasers grown on the a-plane of sapphire where the c-plane of GaN aligns normal to the substrate surface. The facets in GaN and underlying sapphire are oriented on the a-plane and c-plane, respectively. Although the quality does not compare to that on the c-plane, improved cavity formation along the m-plane (1 0 1 0) or a-plane (1 1 2 0) outweighs its reduced material quality. Moreover, laser structures on (1 1 1) MgAl2O4 (spinel substrates), which lead to wurtzite GaN along the c-plane have also been explored for injection-laser experiments [91]. Spinel cleaves along the (1 0 0) plane, inclined to the surface with cleavage following the m-plane of GaN about where the epilayer is reached. Even though the facet quality in this scheme is the best among the aforementioned approaches, material quality degradation is too severe to pull it ahead of the other approaches. It is, however, very clear that a substrate with

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good cleavage characteristics and on which GaN can be grown without rotation is desperately needed. Considerable progress in etched facets have been made to circumvent this problem, as discussed in Volume 3, Chapter 2.

3.3 GaN Epitaxial Relationship to Substrates

Owing to the lack of native substrates for GaN epitaxy, GaN is grown on foreign substrates, often with large lattice mismatch. Consequently, internal strain minimization, among other considerations, leads to an atomic arrangement different from that of the substrate material. The degree of this difference is substrate dependent, as discussed below. 3.3.1 Epitaxial Relationship of GaN and AlN with Sapphire

Sapphire is an ubiquitous substrate on which to grow any semiconductor, and GaN is no exception. Sapphire remains the most frequently used substrate for group III nitride epitaxial growth owing to its low cost, the availability of 3-in. diameter crystals of good quality, its transparent nature, its stability at high temperatures, and a fairly mature technology for nitride growth. Before delving into the orientational relationship between the GaN epilayer and underlying substrate, taken sapphire as default, a discussion of two notations and translation between the two is warranted. Here it should be mentioned that Miller– Bravais indices containing four indices are the most common and straightforward. However, any plane and direction can be described with three indices for the hexagonal system. These two systems are referred to as Miller indices (h k l) and Miller (h j k l) indices. The latter can be obtained from the former by adding the first two indices of the former, changing the sign and assigning it as the third Miller index in the (h j k l) notation. For example, (1 1 0) in (h k l) notation would translate to ð1 1 2 0Þ in the (h j k l) notation. Similarly, (1 0 0) would translate to ð1 0  1 0Þ and (1 0 0) would translate to ð1 0 1 0Þ. Of interest are the commonly used X-ray diffraction peaks of (0 0 2) and (1 0 2), which in the (h j k l) notation would be (0 0 0 2) and ð1 0 1 2Þ. The index in the c-direction does not change, leading to a translation from (1 0 2) in the (h k l) notation to ð1 0 1 2Þ in the (h j k l) notation. In the (h j k l) notation, a dot is often placed after the second Miller index to indicate that the crystal symmetry under question is hexagonal. The orientation order of the GaN films grown on the main sapphire planes {basal, c-plane (0 0 0 1), a-plane ð1 1 2 0Þ, and R-plane ð1  1 0 2Þ} by ECR-MBE has been studied in great detail [92–94]. The epitaxial relationship between GaN and sapphire is insensitive to the method of growth in that both MBE and OMVPE layers exhibit the same relationship. A few examples of the film/substrate epitaxial relationships are (0 0 0 1) GaN || (0 0 0 1) Al2O3 with ½2 1:0 GaN || ½1  1:0 Al2O3 and ½1  1:0 GaN || ½1  2:0   Al2O3, ð2 1:0Þ GaN || (0 1.2) Al2O3 with [0 0.1] GaN || ½0 1:1 Al2O3 and ½0  1:0 GaN ||

3.3 GaN Epitaxial Relationship to Substrates Table 3.6 Crystallographic relationship between GaN films and

sapphire substrates ([13] and references therein).

Miller indices Crystal plane (h j k l) (h k l) c a m r

(0 0 0 1) ð1 1 2 0Þ ð1 0  1 0Þ ð1 0 1 2Þ

(0 0 1) (1 1 0) (1 0 0) (1 0 2)

GaN plane || sapphire GaN direction || surface plane (h j k l) sapphire direction (h j k l)

(0 0 0 1) always ð0 0 0 1Þ or ð1 0 1 0Þ ð1 0 1 3Þ or ð1 2 1 2Þ ð1 1 2 0Þ or ð1 2 1 0Þ

½1 2 1 0 jj½1 1 0 0, ½1 2 1 0 jj½1 1 0 0 ½1 1 2 0 jj ½1 0 0 0, ½1 1 2 0 jj ½0 0 0 3 ½1 2 1 0 jj½0 0 0 1, ½1 0 1 0 jj ½1 2 1 0 ½0 0 0 1 jj½1 1 0 1, ½0 0 0 1 jj½1 0 1 1, ½1 1 0 0 jj ½1 1 2 0; or ½1 1 2 0 jj ½1 1 0 2

The data on AlN on sapphire are somewhat limited, but the available data indicate AlN exhibits similar behavior to that of GaN.

½2 1:0 Al2O3 (Refs [10–12] in [95]). The aforementioned orientational relationships in the (h j k l) notation become (0 0 0 1) GaN || (0 0 0 1) Al2O3 with ½2  1 1 0 GaN || ½1 1 0 0 Al2O3 and ½1 1 0 0 GaN || [1 2 1 0] Al2O3, ð2  1 1 0Þ GaN || ð0 1  1 2Þ Al2O3 with [0 0 0 1] GaN || ½0 1 1 1 Al2O3 and ½0 1 1 0 GaN || ½2  1 1 0 Al2O3. The second set deals with a-plane GaN on r-plane sapphire, which can also be expressed as the epitaxial relationship of ð1 1 2 0Þa-plane GaN on ð1  1 0 2Þr-plane sapphire with ½1 1 2 0GaNjj½1 1 0 2sapphire, ½0 0 0 1GaNjj½ 1 1 0 1sapphire, and ½ 1 1 0 0 GaNjj½1 1 2 0sapphire owing to hexagonal symmetry. A tabular representation of epitaxial relationships of GaN grown on various planes of sapphire is given in Table 3.6. The calculated lattice mismatch between the basal GaN and the basal sapphire plane is larger than 30%. However, the actual mismatch is smaller (16%), because the small cell of Al atoms on the basal sapphire plane is oriented 30 away from the larger sapphire unit cell. This smaller lattice mismatch can be calculated by adopting the model explained in Figure 3.33. pffiffiffi 3awGaN
asapphire ¼ 0:16: ð3:4Þ asapphire It is on this plane that the best films have been grown with relatively small in-plane and out-of-plane misorientations. In general, films on this plane show either none or nearly none of the cubic GaN phase. GaN grown on the ð1 1 2 0Þa-plane turns out to be (0 0 0 1) oriented and anisotropically compressed. The effect of this uniaxial stress on the laser performance is discussed in Volume 3, Chapter 2. The in-plane relationship of GaN and sapphire is depicted in Figure 3.34 where the ½1 1 2 0 direction of GaN is aligned with the [0 0 0 1] direction of sapphire. In this orientation, the bulk positions of both the substrate and the GaN cations lie along the sapphire [0 0 0 1] direction. The mismatch between the substrate and the film is given by c sapphire
4awGaN ¼ 0:02; c sapphire

ð3:5Þ

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j 3 Growth and Growth Methods for Nitride Semiconductors

374

GaN[1 2 1 0] Sapphire[1 1 0 0]

GaN[1 0 1 0] Sapphire[1 1 2 0]

Al N GaN cell Sapphire cell Figure 3.33 Projection of bulk basal plane sapphire and GaN cation positions for the observed epitaxial growth orientation. The circles mark Al-atom positions and the dashed lines show the sapphire basal plane unit cells. The open circles mark the N-atom positions and solid lines show the GaN basal plane unit cell. The Al atoms on the sapphire plane sit at positions approximately 0. 5 Å above and below the plane position [92].

and for the GaN ½1 1 0 0 direction parallel to the sapphire ½1  1 0 0 direction by c sapphire
1:5awGaN ¼
0:005: c sapphire

ð3:6Þ

In one investigation [96], the range of the growth conditions leading to good films on the a-plane was found to be wider than that on the c-plane. Another impetus for exploring growth on the a-plane is the relative ease with which the sapphire could be cleaved along the weak single bond in the c-plane. The GaN film on the top would cleave along the ð1 1 2 0Þ a-plane or the ð1  1 0 0Þ m-plane depending on the in-plane rotation of GaN with respect to sapphire. The aforementioned discussion is somewhat academic in that GaN grown on a-plane still has [0 0 0 1] direction normal to the surface. However, growth on r-plane with OMVPE leads to a-plane GaN, the details of which are discussed in Section 3.5.11. GaN films have also been grown on the r-face f1  1 0 2g of sapphire purportedly to achieve a lattice mismatch smaller than on the c-plane sapphire. Films grown on the r-face has been reported to assume an orientation similar to f2  1 1 0g. The    arrangement in the case of the ð1102Þ face of sapphire and ð2 1 1 0Þ of GaN is depicted in Figure 3.35a. Although of no immediate impact on the topic under discussion, Figure 3.35b provides an image for us to gain some familiarity with the r-plane of GaN in relation to the basic hexagonal lattice structure. The lattice

3.3 GaN Epitaxial Relationship to Substrates

GaN[1 1 2 0] Sapphire[0 0 0 1]

GaN[1 1 0 0] Sapphire[1 1 0 0]

Al N GaN cell Sapphire cell Figure 3.34 Projection of bulk a-plane sapphire and basal plane GaN cation positions for the observed epitaxial growth orientation. The solid circles mark the Al-atom positions and the dashed lines show the sapphire a-plan unit cells. The open circles mark the N-atom positions and the solid lines show the GaN basal plane unit cell. The mismatch along the GaN ½1 1 2 0 direction

parallel to sapphire [0 0 0 1] direction is 2%. The mismatch along the GaN ½1100 direction parallel to sapphire ½1 1 0 0 direction is – 5%. The lines drawn through the N atoms are for underscoring hexagonal symmetry, not the bonds as the bonds are formed between the Ga atoms on plane above the nitrogen plane and N atoms [92].

mismatch between the ½1101 direction of sapphire and the [0 0 0 1] direction of GaN parallel to the sapphire ½1101 direction is equal to r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3c GaN
3a2sapphire þ c 2sapphire rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:01: ð3:7Þ   3a2sapphire þ c 2sapphire In the case when the ½1 1 0 0 direction of GaN is parallel to the sapphire ð0 1  2 0Þ direction, the lattice mismatch is pffiffiffi awGaN
ðasapphire = 3Þ pffiffiffi ¼ 0:16: ð3:8Þ ðasapphire = 3Þ

j375

j 3 Growth and Growth Methods for Nitride Semiconductors

376

GaN[1 1 0 0]

4.76 Å

Sapphire[1 1 2 0]

GaN[0 0 0 1]

5.12 Å N

Sapphire[1 1 0 1]

Ga Al

(a)

Figure 3.35 (a) Projection of bulk r-plane sapphire and a-plane GaN anion and cation positions for the observed epitaxial growth orientation. During growth it is assumed that the top O layer desorbs leaving behind the Al layer with its nearly square unit cells (dimensions are given in terms of angstroms). The figure should be treated with some caution as the exact placement of N atoms belonging to the GaN cell has been done somewhat arbitrarily. Calculations that can shed some light on the preferred position of Ga and N atoms with respect to Al are lacking. Because the a-plane of GaN is nonpolar, only the N and Ga atoms on the same plane are shown. The Ga atoms in the first layer would be lacking a lattice site to bond, making dangling bonds very plausible in addition to sever

distortion of the lattice as even the N atoms in the GaN unit cells not to align everywhere with the underlying Al atoms. Even though some N atoms appear to vertically align with Al below, that bond should in reality not be what is loosely referred as the long bond as that would imply the c-direction to be out of the plane, which is not the case. However, the image provides a reasonably good picture of how the a-plane GaN lattice is stacked on the r-plane sapphire. The lines connecting Al atoms represent only the unit cell not the bonds. For a 3D view of a plane GaN on r-plane sapphire, see Figure 3.161. (b) Although it is not of an immediate application, this figure illustrates the r-plane of GaN with its associated directions. In addition, the figure on the lower left indicates the placement of the r-plane in a hexagonal cell.

3.3 GaN Epitaxial Relationship to Substrates

One unit cell r-plane, GaN[1 1 0 2]

GaN[1 1 2 0] [1 1 0 1] N

c

Ga View direction Bars represent Ga–N bonds

r-plane (1 1 0 2)

c/2 b (b)

a

Figure 3.35 (Continued)

The mismatch along the [0 0 0 1] direction of GaN parallel to the ½ 1 1 0 1 direction of sapphire is 1%, which is much smaller than the 16% mismatch along ½1  1 0 0 direction of GaN parallel to the ½1 1 2 0 direction of sapphire. Growth on the r-face exhibits ridgelike features that allow relaxation of the mismatch. It is assumed, as in the case of c-plane of sapphire, the topmost O layer is desorbed and the Al layer of sapphire is then exposed. Another mechanism accomplishing the same is also implicitly assumed. A closer look, however, indicates that while the above argument about lattice mismatch would hold for only the Al–N bonds at the corners of the unit cell of sapphire, shown as rectangles in Figure 3.35a, it does not necessarily hold for the Al–N bond at the center of the unit cell. In addition, the Ga atoms that must be on the same plane as N in GaN (true for a nonpolar a-plane that is the topic of discussion here) will not have one of its bonds satisfied and would be left dangling. We must hasten to state that the representation in Figure 3.35a is a very simple one intended only to give the reader a first-order glimpse as to how the a-plane GaN might be organized on the r-plane sapphire with no consideration to energy minimization. This awaits calculations for additional insight, which is missing. More details on the r-plane sapphire can be found in Figure 3.20. The lattice mismatch between GaN and all the other substrates, inclusive of various epitaxial relationships, is tabulated in Table 3.7 for both completeness and convenience. The discussion here details the intricacies of lattice mismatch as affected by the epitaxial relationship driven by strain minimization brought about in the first place by the lattice-mismatched substrates.

j377

[0 0 0 1]//[0 0 0 1]

ð1 1 2 0Þ=ð1 1 2 0Þ (0 0 0 1)/(0 0 0 1)

Zinc blende (a ¼ 4.33)

Hexagonal (a ¼ 4.7589, c ¼ 12.991)

a-Al2O3

ð2 1 1 0Þ=ð0 1 1 2Þ

ð0 0 0 1Þ=ð2 1 1 0Þ

ð0 1 1 3Þ=ð0 1 1 0Þ

½1 0 1 0==½1 0 1 0

(0 0 0 1)/(0 0 0 1)

Wurtzite (a ¼ 3.1129, c ¼ 4.9819)

AlN

—

Zinc blende (a ¼ 4.511)

½0 0 0 1==½0 1 1 1

½0 1 1 0==½2 1 1 0

½2 1 1 0==½0 0 0 1

½0 1 1 0==½0 1 1 0

½2 1 1 0==½0 0 0 1

½0 3 3 2==½2 1 1 0

½0 1 1 0==½2 1 1 0

½21 10==½0110

—

—

—


0.18


0.06

—

—

Lattice misfit (%) Thermal strain (%) (growth temperature) (1000–25  C)

aGaN
aAlN ¼ 2:41 2.35 aAlN c GaN
c AlN ¼ 4:08 c AlN pffiffiffi 2aGaN
3aAl2 O3 pffiffiffi ¼
22:65
22.83 pffiffiffi 3aAl2 O3 3aGaN
aAl2 O3 ¼ 16:02 aAl2 O3 xaGaN
c Al2 O3 ¼
6:09; c Al2 O3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi where x ¼ c 2 þ ð2 3aÞ2 4aGaN
c Al2 O3 ¼
1:85 pffiffiffi c Al2 O3 pffiffiffi 3aGaN
3aAl2 O3 pffiffiffi ¼
33:01 3aAl2 O3 pffiffiffi 3aGaN
aAl2 O3 ¼ 16:02 aAl2 O3 4aGaN
c Al2 O3 ¼
1:85 c Al2 O3 3cGaN
x Al2 O3 ¼ 1:19; x Al2 O3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi where x ¼ c2 þ ð2 3aÞ2

—

—

—

Wurtzite (a ¼ 3.1878, c ¼ 5.185)

GaN

—

In-plane direction Lattice misfit (%) Epitaxial relationship (GaN || substrate) (room temperature)

Lattice parameter (Å)

Crystal

substrates inclusive of various epitaxial relationships between GaN and substrates mentioned.

Table 3.7 Room-temperature lattice mismatch of GaN with other III-N compounds and

378

j 3 Growth and Growth Methods for Nitride Semiconductors

Zinc blende (a ¼ 4.35997)

Hexagonal (a ¼ 3.07997, c ¼ 10.083) (0 0 0 1)/(0 0 0 1)

3C-SiC

4H-SiC

Orthorhombic (0 0 0 1)/(0 0 1) (a ¼ 5.4063, b ¼ 6.3786, c ¼ 5.0129)

Orthorhombic (a ¼ 5.428, b ¼ 5.498, c ¼ 7.710)

Zinc blende (a ¼ 5.4309)

Zinc blende (a ¼ 5.652)

b-LiGaO2

NdGaO3

Si

GaAs (0 0 1)/(0 0 1)

(0 0 0 1)/(1 1 1)

(0 0 1)/(0 0 1)

(0 0 1)/(1 1 1)

(0 0 0 1)/(1 1 1)

(0 0 0 1)/(1 0 1)

(0 0 0 1)/(0 1 1)

(0 0 0 1)/(0 0 1)

Tetragonal (a ¼ 5.169, c ¼ 6.282)

g-LiAlO2

ð1 1 0 0Þ=ð1 0 0Þ

(0 0 1)/(0 0 1)

Hexagonal (a ¼ 3.0806, c ¼ 15.1173) (0 0 0 1)/(0 0 0 1)

6H-SiC

[0 1 0]//[0 1 0]

aGaN
a6H-SiC ½1 0 1 0==½1 0 1 0 ¼ 3:48 a6H- Sic aGaN
a3C-SiC ¼ 3:46 [0 1 0]//[0 1 0] a3C-SiC aGaN
a4H--SiC ¼ 3:50 ½1 1 2 0==½1 1 2 0 a4H--SiC c GaN
aLiAlO2 [0 0 0 1]//[0 1 0] ¼ 0:31 pffiffiffi aLiAlO2 3aGaN
aLiAlO2 ¼ 6:8 ½1 0 1 0==½0 1 0 pffiffiffi aLiAlO2 3aGaN - aLiGaO2 ½1 0 1 0==½1 0 0 ¼ 0:13 aLiGaO2 ða-directionÞ p ffiffiffi 3aGaN
bLiGaO2  ¼
13:43 ½1 0 1 0==½0 1 0 bLiGaO2 ðb-directionÞ p ffiffiffi 3aGaN
aNdGaO3  ½1 0 1 0==½1 0 0 ¼ 1:72 pffiffiffi aNdGaO3 3 a
b GaN NdGaO3 ¼ 0:43 ½1 0 1 0==½0 1 0 bNdGaO pffiffiffi3 2a
2 a GaN Si pffiffiffi ¼
16:99 ½1 0 2 0==½1 1 0 2aSi a
a GaN Si ¼
16:93 ½110==½101 aSi pffiffiffi 2aGaN
2aGaAs pffiffiffi ¼
20:19 [0 1 0]//[0 1 0] 2aGaAsi aGaN
aGaAs ¼
20:19 ½1 0 2 0==½1 1 0 aGaAs

In-plane direction Lattice misfit (%) Epitaxial relationship (GaN || substrate) (room temperature)

Lattice parameter (Å)

Crystal


20.26


16.8

1.06


15.1 (b-direction)

1.59 (a-direction)


0.1

3.53

3.55

3.49

(Continued)


0.07

0.19


0.66


1.67 (b-direction)

0.54 (a-direction)


0.41

0.03

0.09

0.01

Lattice misfit (%) Thermal strain (%) (growth temperature) (1000–25  C)

3.3 GaN Epitaxial Relationship to Substrates

j379

Zinc blende (a ¼ 4.216)

Wurtzite (a ¼ 3.252, c ¼ 5.213)

MgO

ZnO

(0 0 0 1)/(1 1 1)

Mixed perovskite (a ¼ 7.730)

LSAT

aGaN
aMgO ¼ 6:99 pffiffiffi aMgO pffiffiffi 3aGaN
2aMgO pffiffiffi ¼
7:99 ½1 0 1 0==½1 1 0 2aMgO a
a GaN ZnO ¼
1:97 ½1 0 2 0==½1 0 2 0 aZnO aGaN - aScAlMgO4 ½1 0 2 0==½1 0 2 0 ¼
1:49 aScAlMgO pffiffi4ffi 4a
2 a GaN MgAl 2 O4 pffiffiffi ½1 0 2 0==½1 1 0 ¼ 11:55 2ap ffiffiffi 2 O4 MgAl 4a
2 a GaN LSAT pffiffiffi ¼ 16:64 ½1 0 2 0==½1 1 0 2aLSAT

[0 1 0]/[0 1 0]

5.8

16.68

11.61


1.64

0.04

0.06


0.15


0.21


1.19

Lattice misfit (%) Thermal strain (%) (growth temperature) (1000–25  C)


2.18

The growth temperature for each of the cases is tabulated in Table 2.36. Courtesy of J. Chaudhuri, Wichita State University.

(0 0 0 1)/(1 1 1)

Spinel (a ¼ 8.083)

(0 0 0 1)/(0 0 0 1)

(0 0 0 1)/(0 0 0 1)

(0 0 0 1)/(1 1 1)

(0 0 1)/(0 0 1)

In-plane direction Lattice misfit (%) Epitaxial relationship (GaN || substrate) (room temperature)

MgAl2O4

ScAlMgO4 Hexagonal (a ¼ 3.236, c ¼ 25.15)

Lattice parameter (Å)

Crystal

Table 3.7 (Continued)

380

j 3 Growth and Growth Methods for Nitride Semiconductors

3.3 GaN Epitaxial Relationship to Substrates

3.3.2 Epitaxial Relationship of GaN and AlN with SiC

Both AlN and GaN deposit on hexagonal SiC substrates in a simple epitaxial relationship, that is, [0 0 0 2]III-N//[0 0 0 6]SiC and ½1 1  2 0III-N ==½1 1  2 0SiC . Using  ð1 1 0 0Þ 6H-SiC substrates (not readily available commercially), Horino et al. [97] established the epitaxial relationship as ð1 1 0 0ÞGaN==ð1  1 0 0ÞSiC and ½1 1  2 0  III-N==½1 1 2 0SiC. 3.3.3 Epitaxial Relationship of GaN and AlN with Si

The epitaxial relationship of GaN on Si is GaN h0 0 0 1i||Sih1 1 1i and h2  1 1 0i GaNjjh0 1 1iSi. If the wurtzitic phase is produced on (0 0 1) Si, then the epitaxial relationship is (0 0 0 1) GaN || (0 0 1) Si and ½1 1 2 0 GaN || [1 1 0] Si. AlN on Si(1 1 1) showed two epitaxial relationships [13,98] namely AlN(0 0 0 1) ½2  1 1 0 || Si(1 1 1) ½0  2 2 for   deposition temperatures greater than 650 C and AlN(0 0 0 1) ½1 1 0 0 || Si(1 1 1) ½0 2 2. These two relationships correspond to the (4  4) and (7  7) lattices, respectively. AlN grown at low temperatures of 400–600  C by MBE shows the relationship of AlN(0 0 0 1) ½0 1 1 0 || Si(1 1 1) ½1 1 2 [99]. These results are given in Table 3.8 and the atomic distance mismatches in Table 3.9. Figure 3.36 shows the orientational relationship between wurtzite AlN and the underlying Si (0 0 1) substrate. GaN deposited on Si(1 1 1) with an AlN buffer layer shows Ga polarity, and GaN grown directly on Si(1 1 1) shows N polarity. For details, refer to Refs [13,100,101], assuming the epitaxial relationship between AlN and Si to be the same as that between GaN and Si. 3.3.4 Epitaxial Relationship of GaN with ZnO

Epitaxial relationships between GaN and ZnO are expected to be [0 0 0 1] of GaN being parallel to [0 0 0 1] of ZnO. Because the lattice mismatch between GaN and ZnO in the Table 3.8 Epitaxial relationship of Wz GaN and AlN grown on Si

when buffer conditions are such that wurtzitic form of GaN or AlN is produced. Si crystal plane Miller indices (h j k l) or (h k l)

GaN plane || Si plane in the first column

GaN direction || Si direction

(1 1 1) (0 0 1) if Wz GaN is produced (1 1 1) if Wz AlN is produced (Ts > 650  C)

(0 0 0 1) (0 0 0 1) (0 0 0 1) (0 0 0 1) (0 0 0 1)

½2 1 1 0 jj ½0 1 1 ½1 1 2 0 jj ½1 1 0 ½2 1 1 0jj½0 2 2 ½1 0 1 0jj½0 2 2 ½0 1 1 0jj½1 1 2

(1 1 1) if Wz AlN is produced (Ts ¼ 400–600  C)

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j 3 Growth and Growth Methods for Nitride Semiconductors

382

Table 3.9 Epitaxial relationship of GaN grown on Si with the

atomic distance mismatch (ADM) and the extended atomic distance mismatch (EADM).

Epilayer/substrate ð0 0  1Þh-GaN/(1 0 0)Si

(1 0 0)c-GaN/(1 0 0)Si

(0 0 1)h-GaN/(1 1 1)Si

Atomic distance (Å) of epilayer/substrate [0 1 0]Ga–Ga/[0 1 0] Si–Si 3.186 5.431 [2 1 0]Ga–Ga/[1 0 0] Si–Si 5.518 5.431 [0 1 0]Ga–Ga/[0 1 0] Si–Si 4.500 5.431 [0 1 0]Ga–Ga/[1 1 0] Si–Si 3.186 3.84

ADM (%)

EADM (%) of epilayer/substrate


41.3

(Ga–Ga) · 5/(Si–Si) · 3

1.6

2.5 (Ga–Ga) · 1/(Si–Si) · 1


17.1

1.6 (Ga–Ga) · 6/(Si–Si) · 5


17.0


0.571 (Ga–Ga) · 6/(Si–Si) · 5
0.438

For details, refer to Refs [13,100,101].

c-plane is very small, under 2%, no discernible in-plane rotation for strain reduction is expected. However, if ZnO is deposited on sapphire followed by GaN deposition, the sapphire substrate and both epitaxial layers are oriented toward each other by a 30 rotation of the unit cell. That is, the in-plane epitaxial layer and substrate are in the form of ZnO, GaN ½1 0 1 0 k Al2O3½1 1  2 0, GaN ½1  1 0 0k Al2O3½1 1  2 1 0, GaN or    ZnO ½2 1 1 0 k Al2O3½1 1 0 0, all of which indicates a 30 rotation of the epitaxial layer with respect to sapphire substrate. 3.3.5 Epitaxial Relationship of GaN with LiGaO2 and LiAlO2 and Perovskites

Some oxides have also been explored because of the small lattice misfit with GaN they provide, with LiGaO2 and LiAlO2 being the ones, particularly the latter, generating a good deal of interest. The epitaxial relationships between GaN and LiGaO2 are expected to be ½1 1 2 0 GaN || [0 1 0] LiGaO2 and [0 0 0 1] GaN || [0 0 1] LiGaO2. On the contrary, the epitaxial relationships between GaN and LiAlO2 are expected to 1 1 0 GaN//[0 0 1] LiAlO2, ð1  1 0 0Þ GaN// be ð0 1 1 0Þ GaN//(1 0 0) LiAlO2 with ½2   g-LiAlO2 (1 0 0) with ½1 1 0 0GaNjj½1 0 0LiAlO2 and ½1 1  2 0GaNjj½0 0 1LiAlO2 . The lattice misfit strain between GaN and LiAlO2 with ð1  1 0 0Þ=ð1 0 0Þ and along the [0 0 0 1]//[0 1 0] in-plane is
0.41%. The lattice structure of LiGaO2 is similar to the wurtzite structure. However, owing to Li and Ga atoms having different ionic radii, the crystal has orthorhombic structure [102]. Figure 3.37 shows the transformation of the hexagonal unit cell of GaN to an orthorhombic cell that has lattice parameters close to that of LiGaO2 (Table 3.10). Perovskite oxides have also been employed in the growth of GaN in an effort to attain a better match compared to more conventional substrates or for applications

3.3 GaN Epitaxial Relationship to Substrates

1

2

Dangling bonds

Dangling bonds Si top layer Si first underlayer [0 1 0] 2

〈2 1 1 0〉

Si second underlayer Al or N sublattice

[1 0 0] [0 0 1]//[ 0 0 0 1] 〈1 1 1 0〉

Figure 3.36 Atomic arrangement for the heteroepitaxial nucleation of 2H-AlN on the Si(0 0 1) surface. The two AlN domains with a 30 rotation are formed on neighboring terraces (1) and (2), separated by a single atomic step boundary according to the Si dangling bond directions. Refs [13,101].

where the properties of nitride semiconductors and electro-optic and nonlinear optics properties of perovskites could be offered in the same template/stack. Among the perovskite is NdGaO3 with its orthorhombic unit cell. The lattice mismatch of GaN with NdGaO3 has been calculated [102] by assuming a perovskite cell of NdGaO3 with lattice parameters a, b, and c each equal to 3.86 Å. This is followed by creating a new unit cell with a0 and b0 where a0 and b0 are the diagonals of the old perovskite cell, as shown in Figure 3.38. The c0 -axis of the new cell is naturally parallel to the c-axis of the perovskite cell but its length is doubled. Basically, this operation transforms a perovskite unit cell to a tetragonal unit cell representing GaN, as tabulated in

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GaN b a a'

b'

Figure 3.37 Transformation of a hexagonal unit cell to an orthorhombic unit cell. Courtesy of J. Chaudhuri and Ref. [102].

Table 3.11. Accordingly, (1 0 0) plane becomes ð1  1 0Þ plane and (0 0 1) becomes (0 0 1). The corresponding lattice misfit is 1.72% and misfit strain is
0.66%.

3.4 Nitride Growth Techniques

HVPE, OMVPE (inorganic VPE), RMBE, and bulk crystal growth from Ga solution are the main growth methods used for nitrides. By far the most frequently used methods are the variants of VPE methods. Although HVPE is used to produce thick GaN layers, including those thick enough to be self-supporting once peeled from the sapphire substrate, OMVPE produces sharp heterojunctions for devices. With the exception of FETs, OMVPE is the primary method employed in the investigation and production of optoelectronic devices, such as LEDs and lasers, albeit the quality of MBE films grown on HVPE buffers is slightly better than of those grown by OMVPE. Inorganic VPE was the first method used to grow epitaxial III-N semiconductors, but was nearly abandoned. The technique, however, was revived recently by growing very high quality and thick buffer layers and templates for the growth of device structures by MBE and OMVPE.

Table 3.10 Transformation of GaN hexagonal unit cell to an orthorhombic unit cell [102].

GaN original hexagonal cell with a and b lattice parameters

GaN transformed orthorhombic cell with lattice parameters a0 and b0

LiGaO2 orthorhombic cell

a ¼ 3.189 Å b ¼ 3.189 Å c ¼ 5.185 Å

pffiffiffi a ¼ 3a ¼ 5:52 Å 0 b ¼ 2b ¼ 6.38 Å c0 ¼ 5.185 · 1 Å

a ¼ 5.4063 Å b ¼ 6.3786 Å c ¼ 5.0129 Å

0

3.4 Nitride Growth Techniques

a

b'

a'

b

Figure 3.38 Transformation of a perovskite unit cell to a tetragonal unit cell. Courtesy of J. Chaudhuri and Ref. [102].

3.4.1 Vapor Phase Epitaxy

VPE has long been employed for the growth of many semiconductor structures. With ongoing source developments and improved reactor designs, this technique has become very powerful, particularly for GaN and related materials. Growth from the vapor phase is categorized on the basis of the sources used. If the sources are inorganic in nature, the term inorganic vapor phase epitaxy is used. This too can be subdivided on the basis of the sources used. For example, if a hydride source is used for the group V element, the term hydride vapor phase epitaxy is applied. If at least some of the sources are organic in nature, the terms organometallic vapor phase eptixay, organometallic chemical vapor deposition, metalorganic chemical vapor deposition, or metalorganic vapor phase epitaxy are employed. 3.4.1.1 Hydride Vapor Phase Epitaxy The genesis of the HVPE growth method can be traced to its wide use in silicon and conventional III–V semiconductors. In the early 1960s, the development of the halide precursor techniques applied to Si and Ge provided the foundation for their subsequent application to the growth of GaAs, which was coming to eminence. The HVPE method has since played an important role in the growth of III–V

Table 3.11 Transformation of a perovskite unit cell to a tetragonal unit cell.

Lattice parameters a b c

Perovskite unit cell

New unit cell

Final unit cell

3.86 Å 3.86 Å 3.86 Å

pffiffiffi a= ¼ 3:86pffiffi2ffi = b ¼ 3:86 2 c0 ¼ 3.86 · 2

a0 ¼ 5.43 Å b0 ¼ 5.43 Å c0 ¼ 7.72 Å

Courtesy of J. Chaudhuri [102].

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semiconductors. In fact, it has the unique distinction of being the first method to produce AlN [103] and, as reported by Maruska et al. [17], to produce single crystalline GaN with quality sufficient to launch the first stages of GaN technology that gained so much prominence in the 1980s. Typical thicknesses for these deposits were in the range of 50–150 mm. The advantage of this technique is that it is conducive to the growth of thick buffer layers at high growth rates on any available substrates to be used as templates for OMVPE and MBE growth, particularly the latter, of high-quality heterostructures with relatively low defect concentrations. A comprehensive treatment of this method can be found in Ref. [104]. A brief historical treatment and a succinct review will be given. After the report of Maruska et al. [17], Wickenden et al. [105] reported on GaN deposition on a-SiC and a-Al2O3, meaning wurtzitic varieties, in 1971, and Ilegems [106] obtained 100–200 mm thick single-crystalline GaN layers on sapphire substrates in 1972. Continuing on, Shintani et al. [107] investigated in 1974 the effects of the important growth parameters, such as the position of the substrate in view of the gas flow dynamics in the reactor, the reactant gas flow rate, and the substrate temperature, on the epitaxial growth rate of GaN on (0 0 0 1) sapphire substrates. Sano et al. [108] considered the influence of the surface sapphire orientation on the growth rate in 1976. In 1977, Madar et al. [109] and Jacob et al. [110] achieved doped GaN with n-type conductivity on sapphire substrates. A study on the growth rate of GaN in hydrogen as well as in inert gas ambience was undertaken by Seifert et al. [111] in 1981, which resulted in growth rates up to 800 mm h
1. Owing to difficulties associated with uniform seeding of GaN on sapphire, coalescence of islands in a timely manner and resultant high n-type background doping (typically 1019 cm
3), large defect concentrations, and inability to produce p-type GaN for light emitters, this technique was largely abandoned in the early 1980s, although Maruska et al. [112] later showed that Zn and Mg doping could be achieved by the simultaneous evaporation of the dopant source in the HCl stream. Since the first report of Maruska and Tietjen [17] many reports on GaN growth [113–124] became available in the literature. Further, several others [103,125,126] have extended this method to the growth of high-quality AlN. But when low-temperature nucleation buffer layers were employed in the context of OMVPE, as discussed in Section 3.5.5.1, followed shortly thereafter by reports of p-type conductivity in GaN, the HVPE method reappeared [127–129] because, in part, of its improved ability to grow thick GaN films with relatively low defect concentration and new techniques for nucleation layers (NLs). Further improvements resulted from lateral epitaxial overgrowth on patterned SiO2 masks [130–132], the concept of which is discussed in Section 3.5.5.2. Freestanding GaN substrates have been prepared by Kim et al. [133] and Melnik et al. [134], which eventually culminated in the production of very high quality freestanding GaN templates prepared at Samsung Advanced Institute of Technology [135], the detailed characteristics of which are discussed in Section 3.5.1.2. Despite these remarkable achievements, the nitride materials still suffer from a very high defect density because of the lattice mismatch between the nitrides and all the available foreign substrates. Recent twostep processes employing low-temperature GaN buffer layers [136] and techniques

3.4 Nitride Growth Techniques

for substrate removal [135,137] have shown good-quality materials with very promising characteristics. The development of the HVPE technique combined with other techniques for producing GaN templates may be the key to resolve the high density defect issue in the III nitride device technology. In HVPE, the group III precursors are chlorides formed by flowing hydrogen chloride gas over the liquid metal in a quartz tube. The group V precursors are hydrides fed into the reaction chamber by a separate quartz line in order to avoid premature reaction with molten source metal. For GaN growth, the chloride and hydride precursors are GaCl, which is formed by reacting Cl from HCl gas with molten Ga, and NH3, respectively. The GaCl in vapor phase is transported to the deposition zone by a carrier gas, which can be hydrogen, and/or an inert gas. The pressure inside the reaction chamber is kept at the atmospheric pressure. The reactor walls are made of high-purity quartz tube. For high-quality semiconductor films to result, the gases are of electronic quality with purity better than 1 ppm for contaminants. For nitrogen or hydrogen, each impurity is in concentration below 1 ppm. For the other gases, the total concentration of all the impurities is below 1 ppm. The metallic sources employed are of 7N, meaning they have a purity of 99.99999%. A multiple zone furnace is used as the metal source zone and substrate temperatures are different. In particular, the zone containing the metallic gallium, the central zone where the gases are homogeneously mixed, and the zone where the substrate resides and the deposition takes place are all kept at different temperatures, as shown schematically in Figure 3.39. The three-temperature zone allows one to independently set the partial pressure of each species, such as the chlorides and hydrides, and explore optimum growth conditions systematically. The influence of parametric variations in the vapor phase composition on the growth rate becomes relatively easier. The same applies to physical processes taking place during a growth. In brief, the vapor phase composition depends on the metallic source efficiency, the ammonia decomposition, and the flow of various gases introduced into the reactor. The vapor phase composition, the partial pressures of the various reactive gaseous species, and the temperature of the three zones in the reactor determine the growth rate and the solid composition of the epitaxial layer if ternaries are attempted. Obviously, the growth takes place in a thermodynamical equilibrium and a wide range of conditions can be applied. If a large flow of HCl is not introduced into the reactor, the Source zone

Mixing zone

Deposition zone

NH3

NH3

N2/H2+HCl

N2/H2+HCl GaCl

N2/H2+HCl add

Substrate

GaCl HCl H2/N2/NH3

Ga source

Bypass line

Bypass

Three-zone furnace Figure 3.39 Schematic diagram of a three-zone HVPE reactor, which utilizes Ga and ammonia sources, used for nitride growth. Patterned after Ref. [104].

Exhaust

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use of NH3 leads to initial conditions far from thermodynamical equilibrium. The growth rates from a low of 1 mm h
1 to a high of more than 100 mm h
1 can be achieved [104]. In HVPE as employed by Maruska and Tietjen [17] for GaN, HCl vapor flowing over a Ga melt causes the formation of GaCl, which is transported downstream. On the substrate surface, the GaCl reacts with NH3 and leads to GaN through the following chemical reaction: 2GaðlÞ þ 2HClðgÞ ! 2GaClðgÞ þ H2 ðgÞ; GaClðgÞ þ NH3 ðgÞ ! GaNðsÞ þ HClðgÞ þ H2 ðgÞ;

ð3:9Þ

where g, l, and s depict gaseous, liquid, and solid species, respectively. The efficiency of the first reaction had been estimated by Ban [138] to be about 99.5%. Using the GaAs analogy [139], two thermodynamic reaction pathways leading to the deposition of GaN [140] can be forwarded: GaClðgÞ þNH3 ðgÞ()GaN þ HClðgÞ þ H2 ðgÞ; 3GaClðgÞ þ 2NH3 ðgÞ()2GaN þ GaCl3 ðgÞ þ 3H2 ðgÞ:

ð3:10Þ

The gaseous species in the reactor are GaCl, GaCl3, HCl, NH3, H2, and either H2 or some other inert gas is used as carrier gas. Barin [141] used thermodynamical data to calculate the reactions of Equations 3.9 and 3.10. For additional details, see Ref. [104]. The supply of GaCl is controlled by the Ga boat temperature and the flow rates of the HCl gas and the carrier gas. The reactor in this method is heated by a two-zone resistance furnace with the region containing the Ga boat being kept at a different temperature than the region housing the substrate for reaction, as shown in Figure 3.39. The Ga zone temperature impacts the growth rate a great deal. The Ga source is held at a constant temperature between 850 and 900  C. The reaction efficiency of HCl with Ga is near unity. Although dependent on the reactor itself, the typical flow rates are about tens of sccm for HCl, 1 l min
1 for NH3 and 2 l min
1 for the carrier gas. GaN films can be grown with rates up to 1 mm h
1 on Al2O3 substrates at atmospheric pressure. However, those high growth rates deplete the Ga source material rather quickly and lead to very rough surfaces with columnar growth. The substrate zone temperature varies between 1050 and 1200  C. At lower substrate zone temperatures, the growth rate decreases exponentially owing to the decreasing pyrolysis efficiency of the GaCl and NH3. On the contrary, at higher substrate zone temperatures, thermally induced decomposition of reactants reduces the growth rate. The hydrogen ambience also aids this reduction in that competing processes such as GaHx would take place. The main issue of concern associated with HVPE and other nitride growth methods is that the initial nucleation layer on sapphire substrates determines to a larger extent the material properties of the subsequent epitaxial layer than in other methods [142–144]. The nucleation or the prelayer is typically deposited using a GaCl or NH3 pretreatment consisting of flowing GaCl or NH3 over the sapphire surface prior to the initiation of growth at high temperatures. In other cases, a ZnO wetting layer is used [127,145,146]. Nitridation of sapphire has been mentioned as a means to improve HVPE materials by several groups [147,148].

3.4 Nitride Growth Techniques

Thermal dissociation of group V species during HVPE of GaAs leads to the formation of As2 or As4 molecules, which remain volatile and chemically reactive and thus participate in the film growth. However, in the case of GaN, by-products of decomposed NH3 are N2 and H2, and N2 molecules that are stable and unreactive at the temperature of interest. Other forms of N species such as NCl3 are not considered, as they are explosive. Moreover, there is a strong thermodynamic driving force for forming parasitic gas-phase reactions, which cause deposition on the walls, making the growth mechanism difficult to unravel. The process also tends to produce large amounts of NH4Cl, GaCl3, and GaCl3-NH3, which condense and clog the exhaust lines unless they are heated to sufficiently high temperatures (>150  C). Exchange reactions with the hot quartz walls of the reactor make it difficult to use HVPE for aluminum- and magnesium-bearing compounds required for AlGaN growth and efficient p-doping, respectively. Endemic to MBE and not considered that important in the case of vapor phase deposition techniques until the advent of GaN is the concept of kinetics, involving adsorption and desorption discussed in detail for OMVPE in Section 3.4.1.3 and for MBE in Section 3.4.2. Adsorption and desorption processes depend on the kinetics of the gaseous species on the substrate surface and the diffusion kinetics of the adatoms or admolecules before arriving at the incorporation sites dubbed half-crystal or K sites (see Figure 3.40) [104]. To put it simply, kinetics consists of complex but simplified approaches that can be considered to get insight into the growth resulting from two superficial diffusion flows, namely, the NGa admolecules and the other from the NGaCl admolecules. Near the step edges, only the NGa flow directly leads to incorporation through desorption of chlorine from the NGaCl. Because GaN growth relies on GaCl, which is identical to that in GaAs, much of the GaN work benefited from systematic investigations performed for GaAs. Among the studies of GaAs growth were those reported by Shaw [149,150] in a series of articles including a systematic measure of the growth rate of GaAs on {0 0 1}, {1 1 1}A, {1 1 1}B, and {1 1 0} surfaces (using hydrogen as carrier gas and the chloride Vapor phase AB

AB+B Vapor phase followed by diffusion

Adsorption followed by diffusion

B* A

AB*

Dissociation A A*

Su bstra

A

te * Adsorbed species

Surface diffusion Figure 3.40 Kinetics processes occurring from the growth by vapor phase. Patterned after Ref. [104].

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method) as a function of temperature and the GaCl partial pressure. The nitride growth rate in HVPE exhibits an increase followed by a decrease in the growth rate with decrease in substrate temperature. The latter was found consistent with Langmuir GaCl adsorption isotherm together with a kinetic model based on GaCl adsorption [151,152]. In this treatment, the lateral interaction between GaCl adsorbed molecules and the Cl desorption by hydrogen of adsorbed GaCl molecules (dubbed the H2 mechanism) are invoked to account for the slight decrease in the GaAs growth rate at low temperatures when the GaCl partial pressure is increased. Hollan and Schiller [153,154] and Hollan et al. computed surface diffusion, a component of kinetics at play, by fitting the experimental data and computed values as a function of the substrate orientation, and of Gentner [155] on 6 off {0 0 1} GaAs substrates. In the latter study, both atmospheric and reduced pressures in hydrogen and helium carrier gases were considered. These studies were also expanded to include AsCl3 instead of HCl gas. A desorption mechanism for two adsorbed chlorine atoms by GaCl in GaCl3 was, therefore, considered with an intermediate GaCl3 adsorption step [139]. This mechanism is called the GaCl3 mechanism. Cadoret [156] developed a model involving adsorption and desorption of species on the surface of the substrates. Among the adsorbing species considered are NH3 molecules, adsorption of N atoms resulting from NH3 decomposition, and adsorption of GaCl molecules on N atoms. They follow the reactions: V þ NH3 ðgÞ()NH3 ; 3 NH3 ()N þ H2 ðgÞ; 2

ð3:11Þ

N þ GaClðgÞ()NGaCl; where (g) depicts the gas-phase species and V is a vacant site. Cadoret also considered two desorption mechanisms of chlorine: that is, desorption in HCl vapor molecules following a surface reaction with H2 and desorption in GaCl3 vapor molecules following adsorption of a GaCl molecule on two GaCl underlying molecules, which is schematically shown in Figure 3.41. The representation is based on the premise that the substrate surface, which is sapphire, is terminated with Al atoms on which N atoms bond leading to a Ga polarity sample. As in the GaAs model [139], the processes follow the reactions: 2NGaCl þ H2 ðgÞ()2NGa þ 2ClH; NGa
ClH()NGa þ HClðgÞ:

ð3:12Þ

The two mechanisms are labeled H2 and GaCl3 mechanisms. They can be treated by means of a one-monolayer model of adsorption on a (0 0 0 1) Ga or Al surface. The adsorbed species are NH3 molecules, N atoms, NGaCl, NGa–ClH, and 2N Ga–GaCl3 molecules. The one-monolayer adsorption model and the Bragg–Williams approximation are used to simplify the problem. The number of activated molecules involved in the reactions described by Equations 3.11 and 3.12 as well as possible intermediate states of hydrogen desorption from NH3 are neglected. GaCl adsorption on a Ga adatom [157,158] is assumed to be negligible because it would lead to antisite positions following chlorine desorption or would act as a simple inhibitor of the deposition

3.4 Nitride Growth Techniques

Ga N

H

H Cl

Cl

Ga

Ga

N

N

Cl Ga

NH3 N

N

N

V

(a)

Cl Ga

Ga N

N

Cl

Cl

Ga

Ga

N

Cl Ga N

NH 3 N

N

V

(b) Figure 3.41 Schematic steps of adsorption and desorption processes involved in the (a) H2 mechanism and (b) GaCl3 mechanism. Patterned after Ref. [104].

process that is not necessarily observed in GaAs. The two overall reactions corresponding to the H2 and GaCl3 growth mechanisms can be written as V þ NH3 ðgÞ þ GaClðgÞ()NGa þ HClðgÞ þ H2 ðgÞ; 2V þ 2NH3 ðgÞ þ 3GaClðgÞ()2NGa þ GaCl3 ðgÞ þ 3H2 ðgÞ:

ð3:13Þ

The adsorption and desorption processes including their flux, potential barriers associated with their activation barriers, are somewhat involved and are beyond the scope of the present treatment. An in-depth treatment of the topic can be found in Ref. [104]. In addition to adsorption and desorption, quantification dealing with growth by HVPE also involves mass transport, which is then followed by growth. In the mass transport case, partial pressures of reactant species are treated that, depending on local conditions and the substrate area, can lead to production or depletion of species. Coupled with the treatment of mass transport, the growth phase, which involves equilibrium between the vapor and substrate surface, must be treated. To get a handle on the problem, superficial flow of diffusive molecules toward step edges, which are assumed to be monomolecular in height, and surface coverage of vacant sites, their distribution over the substrate surface (assumed uniform), are considered. So is whether the growth is based on GaCl3 or H2 growth mechanisms. When the terrace width is small enough to not hinder surface diffusion, the parameters of importance are the adsorption energies of GaCl(g), HCl(g), and GaCl3(g). Further details can be found in Ref. [104]. Those who are more focused on the growth- and properties-related issues involving GaN by MBE are referred to Ref. [159].

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Summarizing, the growth process of GaN by HVPE could be analyzed by a combination of thermodynamical and kinetic considerations. The ample experimental data available in the literature regarding the growth rate on exact and misoriented {0 0 1} surfaces including those measured by Seifert [111] in He and H2 environments on 3 off (0 0.1), in the hkl configuration and (0 0 0 1) in the hjkl configuration, GaN by HVPE allowed for modeling and understanding the physical processes involved in growth. Relative kinetics considerations would indicate that the mass transfer is larger for GaN than for GaAs. Additionally, the high supersaturation, in effect, generally leads to important parasitic GaN deposition before the substrate zone, or the deposition zone, that reduces and even could potentially negate the relative supersaturation. The accuracy with which the growth rate is measured and controlled is the key to the study of the thermodynamics of the system. The variants of HVPE have also been explored and used for growth of GaN and its ternaries with AlN. Among them is a modified VPE process, dubbed the sublimation sandwich method (SSM), which was reported by Wetzel et al. [160] and Fischer et al. [161] (Figure 3.42). Initially, the GaN films were grown from metallic Ga and ammonia on (0 0 0 1) 6H-SiC, using a modification of the sandwich method described previously by Vodakov et al. [162]. In this approach, the quartz reactor contains a Ga cell for each substrate for multiwafer processing with one ammonia stream only. The gap between the substrate and the Ga source is typically about 5 mm. The ammonia flow rate through the gap is very high, 25–50 l m
1 at atmospheric pressure. Under these conditions, there is an effective mass transport of Ga vapor and nitrogen to the surface of the substrates. At growth temperatures between 1170 and 1270  C, GaN layers were obtained at growth rates of up to 0.3 mm h
1. Many other variations of this approach, including [163–171] gaseous sources such as GaCl, GaCl2, GaCl3, Ga(C2H5)2Cl, GaCl2NH3, AlC3, AlBr3, InCl3, and GaBr3 for group III element(s) as reactants for NH3, have been used. Pastrnak et al. [172] chose to react N2 with GaCl3, AlCl3, and InCl3 in their CVD process. Dryburgh [173] grew RF Coils Water

Graphite Substrate NH3

ΔT

Ga Graphite Water

Figure 3.42 Schematic diagram of a proximity HVPE vessel for the growth of GaN at very high growth rates, approaching 0.3 mm h
1. It is dubbed the SSM.

Quartz

3.4 Nitride Growth Techniques

AlN from AlSe and N2. By introducing PH3, Igarashi et al. [174] achieved several percent P incorporation in GaN. The technique became popular because of the highquality buffer layers on the freestanding GaN templates for epitaxy by heterostructure deposition systems such as OMVPE and MBE [175]. The transport and optical properties of HVPE-grown films are discussed in Volume 2, Chapters 3 and 5, respectively. 3.4.1.2 Organometalic Vapor Phase Epitaxy High-quality epitaxial III-N films and heterostructures for devices have been accomplished by OMVPE technique. Manasevit et al. [176] applied this technique to the deposition of GaN and AlN in 1971. Using triethylgallium (TEG) and ammonia (NH3) as source gases for group III and V species, respectively, the authors obtained c-axis oriented films on sapphire (0 0 0 1) and on 6H-SiC(0 0 0 1) substrates. MIS-like LEDs followed, albeit they relied on deep states induced by Zn and suffered from very low efficiencies because of their poor crystalline quality. The development of LT buffer layers addressed the quality issue some [177]. The technique improved over the years to the point that undoped GaN films with a low background carrier concentration of 5 · 1016 cm
3 and with an X-ray symmetric peak FWHM of 30 arcsec have been grown [178]. The X-ray data should be treated with caution, as the symmetric peak is not as sensitive to the edge dislocation as the asymmetric peak. OMVPE has been used for the development of LEDs [179], lasers [180], transistors [181], and detectors [182]. The best OMVPE reactors for group III nitride film growth incorporate laminar flow at high operating pressures and separate inlets for the nitride precursors and ammonia to minimize predeposition reactions. A successful, two-flow OMVPE reactor is shown in Figure 3.43 [183]. The main flow composed of reactant gases with a high velocity is directed through the nozzle parallel to a rotating substrate. The subflow gas composed of nitrogen and hydrogen is directed perpendicular to the substrate. The purpose of the flow normal to the substrate surface is to bring the reactant gases in contact with the substrate and to suppress thermal convection effects. Hydrogen is the carrier gas of choice. A rotating susceptor was used to enhance uniformity of the deposited films. If one goes with the premise that smallest rocking curve half width implies an all-around good quality, GaN films can claim this quality. These films with one of the narrowest rocking curves with FWHM values of 37 arcsec (values even under 30 arsec have been obtained) were grown with a modified EMCORE GS 3200 UTM reactor. It should be stated, however, that the X-ray data based on the symmetric diffraction peak are not a critical measure of sample quality necessarily. For a more complete analysis, one should also inspect the asymmetric peak, which is sensitive to edge dislocations. This reactor generally incorporates separate inlets for ammonia and the nitride precursor, all are normal to the substrate surface, which rotates at speeds over 1000 rpm, and a laminar flow cell to assure a uniform growth [212]. OMVPE reactors incorporating new concepts have been designed to grow layers at lower temperatures. (It should be mentioned that the motivation for lower temperature growth spawned from the perceived need to minimize the loss of nitrogen from

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Figure 3.43 A schematic representation of a vertical OMVPE system employed at Virginia Commonwealth University along with a picture of the deposition chamber (a); a photograph of the reactor chamber of the same (b). (Please find a color version of this figure on the color tables.)

the surface. However, later it became evident that high temperatures are needed to mobilize treading dislocations, as they are useful for reducing dislocation density and facilitating lateral growth. For the sake of completeness, a discussion of relatively lower temperature growth is provided.) These technologies utilize an activated form of nitrogen to lower deposition temperatures of group III nitrides. That these technologies are interesting is apparent, for example, from the deposition of polycrystalline and amorphous GaN films at temperatures lower than 350  C by plasma-enhanced CVD. Epitaxial GaN and AlN have been grown by variants of methods activating nitrogen, such as laser-assisted CVD, remote plasma enhanced CVD, atomic layer epitaxy with

3.4 Nitride Growth Techniques

NH3 cracked by a hot filament, with ammonia catalytically decomposed, photoassisted CVD, and ECR plasma-assisted CVD. However, none of these approaches has been able to produce material comparable in quality with the standard OMVPE systems and, consequently, they did not really become players in the field. We alluded to the genesis of the lower quality above. As for the mechanism involved, growth of nitride semiconductors by OMVPE relies on the transport of organometallic precursor gases, hydrides for the nitrogen source, and reacting them on or near the surface of a heated substrate. The deposition is through pyrolysis. The underlying chemical mechanisms are complex and involve a set of gas phase and surface reactions. Although OMVPE has long been assumed to be a thermodynamically equilibrium process, nitride OMVPE processes may involve kinetics as well. The fundamental understanding of the processes involved is still evolving and, as such, the reaction mechanism and the related kinetic rate parameters are poorly understood. The deposition of epitaxial nitride layers by OMVPE involves the reaction of metalcontaining In, Ga, or Al gases with ammonia, NH3. Commonly, the metal-containing gases are trimethylgallium ((CH3)3Ga), trimethylindium ((CH3)3In), or trimethylaluminum ((CH3)3Al). Radicals, reactive by most definitions, react in the gas phase with donors containing acidic hydrogen, such as NH3, and form adducts. The key here is to eliminate the unwanted radicals by forming stable molecules followed by their removal from the reaction region. The analysis of the mechanisms involved in the OMVPE process clearly indicates that any precursor must balance the requirements of volatility and stability, which often counter each other, to be transported to the surface and decomposed for deposition. To put it another way, these precursors must have appropriate reactivity to decompose thermally into the desired solid and to generate readily removable gaseous side products. Ideally, the precursors should be nonpyrophoric, water and oxygen insensitive, noncorrosive, and nontoxic. The trialkyls, trimethylgallium (TMG) [184] and triethylgallium (TEG), trimethylaluminum (TMA) [185], trimethylindium (TMI) [186], and others are usually used as III metal precursors. Ammonia (NH3) [187], hydrazine (N2H4) [142,188,189], monomethylhydrazine (CH3)N2H2 [190,191], and dimethylhydrazine (CH3)2N2H2 have all been used as nitrogen precursors with varying degrees of success. Although trialkyl compounds (TMA, TMG, TMI, etc.) are pyrophoric and extremely water and oxygen sensitive, and ammonia is highly corrosive, much of the best material grown today is produced by conventional OMVPE by reacting these compounds with NH3 at substrate temperatures close to 1000  C [192–203]. Investigators have reacted TMG [204,205], TEG, and GaCl [206,207] with NH3 plasma. Sheng et al. [208] reacted TMA and NH3 in the presence of hydrogen plasma. Wakahara et al. [209] grew InN by reacting TMI with microwave-activated N2. Eremin et al. [210] used nitrogen to transport metallic Ga to the reaction zone where it was reacted with active nitrogen. These commonly employed precursors at least satisfy the criteria of sufficient volatility and appropriate reactivity. During the growth of nitrides by employing trialkyl precursors, adduct formation between ammonia and TMA and TMG is well documented. Usually mixing at room temperature, adduct

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formation between TMG or TEG and ammonia is complete in less than 0.2 s. The resulting adduct Ga(CH3)3-NH3 has a vapor pressure of 0.92 Torr at room temperature, while the vapor pressure of Ga(C2H5)3 : NH3 is much lower. To counter early beliefs that stability of ammonia and required relatively high growth temperature the use of other more volatile nitrogen sources were explored. For example, Fujieda et al. [142] replaced NH3 with N2H4 and observed that a significantly smaller amount of N2H4 was required to maintain the same growth rate. However, they also noted that the CVD growth rate was limited by the decomposition of TMG, thus limiting the benefits of N2H4. Matloubian and Gershenzon [211] used TMA and NH3 and a substrate temperature range of 673–1473 K to grow nitrides. Single-crystal AlN films were obtained only at 1473 K. Dupuie and Gulari [212] reported that the presence of a hot filament near the substrate increased the growth rate of AlN grown with TMA and NH3 by two orders of magnitude. However, the use of a hot filament immediately raises concerns about residual contamination, most prominently oxygen, which was not addressed by the author. A case in point illustrating the elimination of unwanted radicals by forming stable molecules followed by their removal from the reaction region is AlN growth from mixtures of methyl alkyls that may proceed by the formation of an intermediate gas phase adduct (CH3AlNH3), followed by the elimination of CH4. The exact path may be that coadsorption of (CH3)3Al and NH3 at room temperature generates surface adduct species such as ((CH3)2AlNH3) and adsorbed NH3 [213]. As the substrate temperature is raised above 320  C, the appearance of vibrational bands corresponding to AlN indicates the formation of extended (Al–N) networks on the surface. These Al(NH2)2Al species finally eliminate H2 at the surface to form AlN [214]. The possible chemical reactions in the process are (a stands for adsorbate on the surface and g stands for gas-phase product):

and

2ðCH3 Þ2 Al:NH3 ðaÞ ! CH3 AlðNH2 Þ2 AlCH3 ðaÞ þ 2CH4 ðgÞ; CH3 AlðNH2 Þ2 AlCH3 ðaÞ ! AlðNHÞ2 Al þ 2CH4 ðgÞ;

ð3:14Þ

AlðNHÞ2 AlðaÞ ! 2AlNðaÞ þ H2 ðgÞ: As for GaN, investigations are relatively limited, but it would be fair to assume that processes similar to that with AlN growth are most likely in place. Adducts of Ga compounds are weaker electron acceptors than the corresponding Al adducts, and therefore these adducts may not be abundant owing to redissociation in the hot zone. It may be because of this that successful GaN growth by OMVPE requires very large V/III ratios, which favor adduct formation. Thermal stability of NH3, although low compared to that of N2, could be partially responsible for the use of high substrate temperatures, typically above 550  C for InN and above 1000  C for GaN and AlN. The high growth temperature necessitated by the process itself, associated with high nitrogen vapor pressure over GaN, lead to the inevitable nitrogen loss from the nitride film. This may also be the path to carbon contamination from the decomposition of the organic radical during metalorganic pyrolysis. The loss of nitrogen can be

3.4 Nitride Growth Techniques

alleviated by using high V/III gas ratios during the deposition, particularly for InGaN (e.g., >2000 : 1). Assuming that high substrate temperatures represent a problem in relation to ammonia, which seems reasonable particularly in early days, various alternative approaches can be and have been taken. One approach is to use alternative nitrogen precursors that are thermally less stable than NH3. Hydrazine (N2H4), which is a larger and less stable molecule, has been used in combination with TMA to deposit AlN at temperatures as low as 220  C [189]. However, hydrazine is toxic, unstable, and not as pure as NH3. Consequently, a compromise between quality and substrate temperature must be made. Researchers took the quality/purity as the primary parameter and stayed with NH3. More recently, other nitrogen sources such as tertbutylamine (t-BuNH2) [215], isopropylamine (i-PrNH2), and trimethylsilylazide (TMeSiN3) have been used with TMA or t-Bu3Al to deposit AlN films at lower substrate temperatures (400–600  C) and reduced V/III gas ratios (5 : 1–70 : 1) [216]. However, the deposited films were invariably contaminated with high levels of residual carbon (up to 11 at.%). Hydrogen and, to a lesser extent, nitrogen are predominantly used as the transport gas. They can influence the chemical reaction mechanism of Et3M or (CH3)3M in the gas phase by changing the reaction temperature of the metalorganic compounds or the concentration of reaction products. Hydrogen at the surface of the growing film can influence the growth rate and the structural properties [217,218]. To obtain a basic understanding of the role of hydrogen in GaN growth, the possible sources of hydrogen and the influence of hydrogen on the chemical reaction mechanism in the gas phase and at the surface are unique and important issues in the context of OMVPE. Pyrolysis of highly concentrated NH3 in the presence of H2 as the carrier gas results in a high concentration of molecular and atomic hydrogen near the substrate surface. Because the growth temperatures above 900  C are employed, which are higher than the decomposition temperature of the metalorganic compounds and their hydrocarbon ligands, the conditions for the desired bond breaking between the metal atom and the methyl or ethyl groups of the precursors are in place. However, the same can also lead to pyrolysis of the hydrocarbons with incorporation of hydrogen and carbon into the films. In this vein, decomposition of (CH3)3Ga (TMG) and Et3Ga in hydrogen and nitrogen atmospheres using a quadrupole mass analyzer has been investigated [219]. The decomposition reaction of the metalorganic precursors was found to be strongly affected by the presence of molecular hydrogen. The decomposition of (CH3)3Ga occurs at 400 and 500  C in H2 and N2, respectively. Similarly, the decomposition of Et3M occurs at 260 and 300  C in H2 and N2, respectively. Clearly, molecular hydrogen reduces the reaction temperature. The reaction mechanisms involve hydrolysis for (CH3)3Ga in H2, homolytic fission for (CH3)3Ga in N2, and b-elimination for Et3M in both H2 and N2. By changing the reaction temperature and the reaction mechanism of the metalorganic precursor, the partial pressure of hydrogen affects the deposition rate of GaN and therefore the structural properties of the resulting film, especially at low growth temperatures [220].

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j 3 Growth and Growth Methods for Nitride Semiconductors

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3.4.1.3 Modeling of OMVPE Growth The GaN kinetic model with the associated details, to the extent necessary, of the GaN thermal decomposition, precursor adsorption, and decomposition, Ga and N2 desorption rates, and surface diffusion rates is discussed next to identify the important chemical steps for GaN growth. This will aid in developing the kinetic model. 3.4.1.3.1 Thermal Decomposition of GaN as it Relates to Growth The rule of the thumb for modeling the growth of conventional compound semiconductors is that MBE is a nonequilibrium growth with several activation barriers in place and, as such, a kinetic model is better suited. On the contrary, OMVPE is an equilibrium process and as such, a thermodynamic model is appropriate. However, nitrides are different in that temperatures employed are high and barriers such as NH3 dissociation are in play. Consequently, a kinetics-based model for OMVPE is appropriate as reported by Koleske et al. [221]. The group III nitrides have strong chemical bonds with the Ga–N bond strength being estimated near 4.2 eV, which is comparable to C–C bond strength of 3.8 eV in diamond. In addition, the group III nitride bonds are more ionic (31–40%) (excluding BN) than those in other III–V semiconductors (<8%). Despite a strong bonding, GaN decomposes above 800  C. As in the case of the other III–V semiconductors, the heat of formation of GaN is small and similar to that of InP and AlAs, indicating that GaN decomposes easily. The GaN decomposition temperature is important for OMVPE or OMVPE GaN growth because this temperature is about 200  C less than the growth temperature. As is clear from the overall discussion of growth issues, the growth of GaN on foreign substrates is complicated in that it involves at least three steps, namely, nitridation, nucleation layer growth, and the growth of the final structure. Before we delve into the kinetic model, a few words on the GaN decomposition is warranted. GaN does not melt congruently at pressures typically used for OMVPE or MBE growth, but decomposes at above 800  C at atmospheric pressure and at lower temperature in vacuum. It was initially postulated that GaN decomposed into N2 and Ga, which was eventually observed using mass spectroscopy. In addition, the desorption of small GaN clusters has been postulated and later observed as GaNþ and Ga2N2þ ions, again with mass spectroscopy. Both thermogravimetric and mass spectroscopic techniques have been used to measure the kinetics of GaN thermal decomposition, with both measurements resulting in an activation energy (EA) of 3.1 eV, which compares with 3.6 eV determined in an MBE environment [222]. Actually, there is a wide spread in the reported activation barrier to desorption, which spans from 0.34 to 3.62 eV and depends strongly on the annealing conditions [223]. The reported activation barriers appear to group into four distinct figures, which may imply a somewhat complex desorption process and, perhaps, a lack of attention to the temperature, pressure, and environment, such as H2 or N2. GaN decomposition temperatures reported are in the range of 400–900  C, which may well be explained by pressure, temperature, and gas flow (i.e., H2 or N2), which plausibly change the GaN decomposition rates and mechanisms.

3.4 Nitride Growth Techniques

Early on, several different mechanisms were proposed to explain GaN decomposition. These included decomposition into gaseous Ga and nitrogen [224] as in Equation 3.15, liquid Ga and nitrogen as in Equation 3.16, and sublimation of GaN into a diatomic or polymeric product [225] as in Equation 3.17. 2GaNðsÞ ! 2GaðgÞ þ N2 ðgÞ

ðdecompositionÞ;

ð3:15Þ

2GaNðsÞ ! 2GaðlÞ þ N2 ðgÞ ! 2GaðgÞ þ N2 ðgÞ ðdesorptionÞ;

ð3:16Þ

2GaNðsÞ ! GaNðgÞ or ½GaNx ðgÞ ðsublimationÞ:

ð3:17Þ

As compiled by Koleske et al. [223], N2 evolution has been observed using mass spectroscopy along with GaNþ and Ga2N2þ species. Occasionally, Ga droplets 20–30 mm in size have been observed on the GaN surface, indicating that the GaN decomposition rate, kGaN, can exceed the Ga desorption rate, kGa. As mentioned above, the onset of GaN decomposition in terms of the temperature is lower in H2 as compared to inert environments, such as N2, Ar, and vacuum. Hydrogen could assist decomposition by the reverse GaN synthesis reaction, that is, the reformation of NH3 via 2GaNðsÞ þ 3H2 ! 2GaðlÞ þ 2NH3 ðgÞ:

ð3:18Þ

Measured values of the activation energy, EA, and the preexponential factor, A0, for GaN decomposition in H2 under a pressure of less than 76 Torr, in N2, and in vacuum are listed in Table 3.12. The span of EA is 2.7–3.93 eV. Surprisingly, this is lower than the EA of 4.7 eV for GaAs decomposition, despite the stronger bond strength in GaN (4.1 eV) compared to GaAs (2.0 eV). When GaN layers are annealed in H2 at higher pressures and lower temperatures, the EA decreases (Table 3.12) down to about EA of 1.8 eV. In H2 at higher pressures and higher temperatures, even lower values of EA, down to 0.38 eV have been measured. These lower EA values for decomposition in H2 suggest hydrogen aids in N removal from the surface, possibly by NH3 formation, as in Equation 3.18. Because GaN desorption is an integral part of growth and adsorption and decomposition/desorption occur simultaneously, it is important that they are treated side by side. To make sure that the GaN decomposition rates measured under vacuum conditions are applicable to OMVPE pressures, as mentioned above, Koleske et al.[221,223] measured GaN decomposition under flowing H2 at a pressure of 50 Torr. Upon heating a 2 mm thick GaN film at 1030  C for 10 min, about 40% of the GaN film was gone and the remaining GaN film was thinner and covered with Ga droplets. From this measurement, a GaN decomposition rate of about 1 · 1017–3 · 1017 cm
2 s
1 was estimated, which compares well with a calculated rate of 1 · 1017 cm
2 s
1 for the thermogravimetric determination at 1030  C, suggesting that the GaN decomposition rate is similar in vacuum and OMVPE pressures. It should also be mentioned that the activation energy, EA, for GaN decomposition is only slightly larger than the EA of 2.8 eV for Ga desorption from liquid Ga.

j399

j 3 Growth and Growth Methods for Nitride Semiconductors

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Table 3.12 Arrhenius parameters for kinetic reactions along with notes compiled in Ref. [221].

Process

Preexponent

EA (eV)

Comments

Decomposition GaAs GaN GaN

— 4 · 1029 cm
2 s
1 5 · 1028 cm
2 s
1

4.7 3.1 3.1

RHEED oscillation measurement Thermogravimetry measurement Mass spectrometry measurement

Ga desorption from Liquid Ga — GaAs — GaN — GaN — GaN 1.0 · 1028 cm
2 s
1

2.8 2.5 2.76 2.2 2.69

Review paper on liquid elements Modulated molecular beam technique RHEED study of hexagonal GaN Growth rate versus growth temperature Ga dosed, RHEED pattern recovery

N desorption from GaN 2.0 · 1044 cm
2 s
1

6.1

N dosed, RHEED pattern recovery

Surface diffusion on GaN Ga on GaAs 3.3 · 10
5 cm2 s
1 Ga on GaN 0.007 cm2 s
1 N on GaN — GaN —

1.3 2.48 >3 1.45

V/III desorption ratio 1.93 · 1016 kN/kGa

3.41

RHEED oscillations on vicinal surfaces Ga dosed, RHEED pattern recovery Estimate based on present work RHEED pattern recovery Calculation by Koleske et al. [223]

Note that a GaN decomposition activation energy of 3.6 eV has also been reported in Ref. [222].

3.4.1.3.2 Ga and N Precursor Adsorption and Decomposition Adsorption and decomposition processes are described in detail in conjunction with MBE growth of GaN in Section 3.4.2. However, the particulars of these processes, although the basics are the same, in an OMVPE environment are discussed here. Several factors, such as the precursor flux, the gas-phase diffusion rate, the sticking coefficient, the number of available surface sites, and the precursor decomposition rate, contribute to the concentration of active growth species on the surface. For the particular kinetics-based model, it will be assumed that, once adsorbed, the Ga and N precursors decompose completely to produce adsorbed Ga and N atoms and that precursor sticking coefficients are near unity. The assumption that TMG and TEG undergo full dissociation to Ga atoms is supported by several studies in that nearly complete pyrolysis of TMG for T > 500  C or TEG for T > 300  C in H2 or N2 occurs. In flowing NH3, measurable and complete TMG decompositions are observed for T > 525  C and T > 625  C, respectively. TMG and TEG have been shown to partially decompose in the gas phase but at a reduced pressure (e.g., 0.1 atm), which is parasitic. However, most of the decomposition occurs on the surface rather than in the gas phase. To ensure that the decomposition takes place on the surface, gas manifolds are designed to keep the precursors cool until they come in contact with the surface. The sticking coefficients for TMG and TEG are large because of the accommodation ratio, which increases as the molecular weight increases. TMG forms a strong adduct bond with NH3, which would further

3.4 Nitride Growth Techniques

increase the accommodation ratio and therefore the sticking probability. TMG conversion to metallic Ga has been observed on a sapphire surface at 525  C at atmospheric pressure, suggesting that metallorganic Ga precursors efficiently stick and completely dissociate to metallic Ga on GaN and sapphire surfaces. The ability to grow low temperature nucleation layers at substrate temperatures not much higher than 500  C is also indicative of this conversion. Although not much is known about NH3 adsorption and dissociation, the general assumption is that sticking of NH3 to a Ga-terminated surface should be large. Moreover, the sticking of NH3 may depend on the N surface coverage in a fashion similar to the reduced sticking of other group V hydrides on group V terminated surfaces. Once adsorbed on the surface, NH3 dissociates and forms adsorbed H and NHx<3 species. The latter either further dissociates to adsorbed N and H atoms or combines with adsorbed H to reform NH3. It has been shown that the active species that nitridates the surface is the N–H fragment, and that the large kinetic barrier associated with the decomposition of NH3 on the surface may be related to loosing the second or third H from NH3. Because H desorbs from the surface at temperatures as low as 300  C after NH3 exposure, the steady state to which H coverages at temperatures used for growth in OMVPE is small. It should also be noted that NH3 dissociation at RMBE and some OMVPE growth temperatures should be marginal, unless some sort of catalysis is in effect, in which case the rate of dissociation should depend strongly on the catalyst, such as metallic Ga on sapphire. Several investigations, in fact, showed a dramatic increase in the NH3 decomposition rate as the temperature of the catalyst was increased [226], from few percent at 900  C up to 40–50% at 1050  C. In the context of the present model, NH3 dissociation to N atoms will be assumed complete, with ramifications of this assumption given. 3.4.1.3.3 Ga and N2 Desorption from the Surface Growth is a complex process in that adsorption, dissociation, and desorption take place simultaneously at many levels. Because OMVPE does not lend itself well for the studies involving these processes, a good deal of reliance on MBE studies is called for. Studies to gain insight into these processes, particularly the desorption rates of Ga and N, are conducted by means of monitoring such processes as functions of substrate temperature. The N and Ga desorption rates, kN and kGa, respectively, can be plotted as functions of inverse temperature, from the slope of which activation energies can be obtained. Such an exercise in one study led to the Ga desorption activation barrier, EA, of 2.69 eV (among various reports, it ranged between 2.2 and 2.76 eV) as compared to the Ga desorption EA of 2.8 eV from liquid Ga and GaAs and GaN decomposition activation energy of 3.1–3.6 eV. Because the measured EA for Ga desorption from GaN, GaAs, and liquid Ga are similar and the liquid phase of Ga covers a range extending from 30 to 2403  C, it is reasonable to assume that the Ga desorption mechanism from GaN does not change for the temperatures employed in OMVPE growth. This justifies the use of kinetic parameters regarding the OMVPE growth, particularly considering the growth conditions typically employed. Using the line-of-sight quadrupole mass spectrometry as a quantitative in situ method, Koblm€ uller et al. [227] undertook an extensive investigation of Ga adsorption

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j 3 Growth and Growth Methods for Nitride Semiconductors

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and desorption on both Ga-polar (0 0 0 1) and N-polar ð0 0 0  1Þ surfaces. Monitoring the desorbing Ga atoms, two characteristic desorption regimes (exponential and steady-state) were observed that were attributed to the formation of a thin equilibrium Ga adlayer and Ga droplets on top of it. The Ga adlayer coverage differs substantially between the two surface polarities, being 1.1 monolayers on the N-polar ð0 0 0  1Þ GaN and 2.4 monolayers on the Ga-polar (0 0 0 1) GaN. Additional temperature-dependent measurements of the surface lifetime of Ga adatoms revealed fundamental differences in the adsorbate–substrate binding energetics both for the Ga adlayers on Gaand N-polar surfaces and for the Ga droplets. In the above-mentioned experiments, Ga atoms were allowed to impinge on the substrate in the absence of nitrogen being impingent on the surface. The amount of Ga adlayer was controlled by varying the length of time that the Ga effusion cell shutter remained open. After deposition, the shutter was closed and a quantitative in situ method, the line-of-sight quadrupole mass spectrometry, was used to monitor the impinging of Ga atoms. Desorption energies were determined by the dependence of the decay time of the desorption process on substrate temperature. A desorption energy of 3.1 eV was reported for Ga evaporating from Ga droplets. The desorption energies (or activation energy for desorption) for Ga from the top Ga adlayer and bottom Ga adlayer were 3.7 and 4.9 eV, respectively. In comparison, the single Ga adlayer on the ð0 0 0 1Þ GaN surface exhibited a desorption energy of 3.7 eV, which is comparable to that from the top Ga adlayer on the Ga-polar surface. Moreover, on both surface polarities, additional Ga surplus was found to lead to the accumulation of metallic Ga droplets on the top of the adlayer. Further, Ga bound to droplets was found to remain in the weakest bound state of all possible adatom surface arrangements, resulting in the lowest activation energy for Ga desorption of EA ¼ 3.1  0.2 eV. This value is consistent with decomposition activation energy of 3.1–3.6 eV mentioned above and agrees well with the evaporation energy from liquid Ga. In addition, this activation for desorption was not found to depend on the underlying GaN surface polarity or the surface roughness. Turning our attention to N, it is typical that the group V element desorbs as a dimer (V2), implying that the desorption rate is limited by the surface diffusion of two group V atoms. There are some reports that N desorption is of the first order. For N to follow first-order desorption kinetics would mean that the surface diffusion may play less of a role in the desorption mechanism than in the surface structure. For example, the first-order desorption kinetics have been measured for H2 desorption from Si(1 0 0) and found to be associated with ordering of H on the Si(1 0 0) dimers. Another plausible mechanism is that once an N atom diffuses near another N atom, the N2 molecule would immediately form and desorb from the surface because of the highly exothermic nature of the N2 formation. This would make the N2 desorption first order in terms of N coverage. However, at higher temperatures (>720  C), the kinetic order may change and this cannot be ruled out. Of paramount pertinence to OMVPE growth is that the desorption rates of N and Ga become equal at 780  C (one monolayer per second or 1.14 · 1015 atoms cm
2 s
1), and thus, at the growth temperatures commonly used in OMVPE, the N desorption rate is 1000–10 000 times that of Ga. At about 1030  C, a reasonable

3.4 Nitride Growth Techniques

OMVPE growth temperature, the estimated GaN decomposition rate is about 3 · 1017 cm
2 s
1, which coincides well with kGa. Assuming that desorption occurs only from the top layer, the desorption limited surface residence times, tD, can be calculated by dividing the surface coverage (1.14 · 1015 cm
2) by the desorption rates, kGa and kN, for Ga and N, respectively. Most studies indicate tD,Ga ¼ 1 s at temperatures in the range of 750–780  C, which is larger than for Ga on GaAs(1 1 1). 3.4.1.3.4 Ga and N Surface Diffusion At high growth temperatures and during growth where the surface diffusion length is larger than the terrace width, atoms move by surface diffusion from the terrace to the step edge for incorporation into the growing lattice, which is called the step-flow growth. As the diffusion length decreases, the surface morphology changes from step flow to islanded growth on the terraces. These two regimes have been established in MBE growth of GaAs and other conventional compound semiconductors. If thermal desorption is large, the diffusion length may be dramatically shortened. Consequently, diffusion lengths at growth temperatures must be evaluated. This can simply be done by assuming that the surface diffusion follows an Arrhenius form and that the mechanism does not change with temperature. In these descriptions and the following Einstein equation, the diffusion length, lS, is given by [228] pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:19Þ lS ¼ DS tS ¼ DS0 expð
E SD =kTÞtS ;

where DS is the diffusion coefficient,tS is the lifetime of the diffusion event, DS0 is the temperature-independent diffusion coefficient, ESD is the diffusion activation energy, and kB is the Boltzmann’s constant. Table 3.12 lists representative values of DS0 and ESD for GaN and GaAs. The values of tS can be in two limiting values, which are the lifetime before desorption (tD), which depends on the desorption rate and the lifetime before lattice incorporation (tI), which depends on the GR. The estimate lS, Ga for both limiting values of tS will be presented here. Using the DS,Ga reported in the literature [229] and tD,Ga calculated in Section 3.4.1.3.3, Koleske et al. [221] estimated lS,Ga using Equation 3.19. The temperature dependence of lS,Ga is plotted as a solid line in Figure 3.44, which indicates that lS,Ga decreases slightly as the temperature increases. This is unexpected because the temperature increases as lS increases. The decrease in lS,Ga at higher temperatures takes place because the increase in DS,Ga is counterbalanced by the Ga desorption rate, which results in a reduced tD. As shown in Figure 3.44, lS,Ga is less than 1.2 nm for temperatures ranging from 600 to 1200  C. If the Ga coverage exceeds one monolayer, lS,Ga would increase and approach the lS,Ga value for Ga on liquid Ga metal. However, Ga droplet buildup on the surface during growth is not advantageous for maintaining the desired smooth surface morphology (in contrast to MBE). Therefore, lS,Ga is about 1 nm if the Ga desorption rate from the surface is large. It must be remembered that at OMVPE growth pressures (30–770 Torr), Ga desorption appears to be suppressed, suggesting that the Ga surface lifetime, tS,Ga, may be substantially longer than the desorption limited lifetime. The limit, of course, is the incorporation lifetime (tI) into the growing lattice. The value of tI depends inversely on the growth rate, as the slower the growth rate the larger tI. To

j403

j 3 Growth and Growth Methods for Nitride Semiconductors

404

102 3

Diffusion length, (nm)

10 30

λ S,Ga = (D S,Gaτl)1/2

101

100

nm min–1

100 λ S,Ga = (D,S,GaτD)1/2

10–1 600

700

800

900

1000

1100

1200

Temperature (ºC) Figure 3.44 Estimates of the Ga diffusion length, lS,Ga, on GaN. The solid line is a calculation of lS,Ga based on tD (desorption limit) and the dashed lines are calculations of lS,Ga based on tI (growth rate limit) for growth rates of 3, 10, 30, and 100 nm min
1 [223].

show the dependence of tI on the growth rate, lS,Ga is plotted in Figure 3.44 as open segmented lines for four different GR ranging from 30 to 1000 Å min
1. For this calculation, tI was determined by dividing the thickness per monolayer (actually bilayer, which is 2.583 Å and represents one half of the lattice parameter along the c-direction) by the growth rate. Several observations can be gleaned from Figure 3.44. The term lS,Ga increases with temperature and this temperature dependence is solely owing to the temperature dependence of DS,Ga because tI is determined by the GR. As the GR decreases, both tI and lS,Ga increase, and consequently, the Ga diffusivity increases, which should aid in the growth of a more ordered lattice because the number of adatoms that incorporate is increased. Enhancements in GaN crystalline quality have been attributed to reduced growth rates for the migration-enhanced epitaxy (MEE) approach where the surface is dosed with an alternating sequence of N and Ga exposures with dwell times between exposures in an effort to increase lS,Ga. At 1050  C, the estimated values of lS,Ga using tI (no desorption regime or complete incorporation) are about 10–100 times larger than the estimated values of lS,Ga based on tS,Ga (high desorption regime or growth rate limited case). For GaN growth by OMVPE, practical growth rates are typically >150 Å min
1 (about 1 mm h
1). At a growth rate of 300 Å min
1, lS,Ga is about 11.3 nm at 1050  C. Turning our attention to the other species N, its surface diffusion length (lS,N) should be smaller than lS,Ga. Unlike Ga, no evidence for N diffusion was observed in

3.4 Nitride Growth Techniques

the RHEED intensity analysis in MBE when radio frequency (RF) activated nitrogen was used. This is not necessarily applicable to NH3 as RHEED oscillations have been observed. Assuming RF-activated nitrogen and N do not migrate outside the unit cell during the RHEED pattern, the recovery time, DS,N, must be <2.7 · 10
17 cm
2 s
1 at 700  C, which is 40 times less than DS,Ga at 700  C. If the N lifetime is similar to the Ga lifetime, as might be expected in the case of NH3 but only as an upper limit, lS,N at 700  C is about six times less than lS,Ga. It is however very likely that lS,N is much less than lS,Ga, which is supported by ab initio calculations [230]. Even at atmospheric pressure, N2 desorbs from GaN above 800  C, suggesting that for T > 800  C, once N migrates to next to an adjacent N, N2 forms and desorbs. If the surface migration of atomic Ga and N is limited, partially decomposed reactants may diffuse more readily on the surface. This may be particularly important for OMVPE growth, because partially decomposed precursors that are weakly bound to the surface may more freely diffuse across the surface. Weaker N–N or N–Ga bonds between partially dehydrogenated NH3 and the GaN surface may form with these species diffusing more readily than Ga or N atoms. Therefore, the mean diffusion rates for Ga and N species on the surface may be substantially larger than the values calculated here using diffusion constants measured by MBE in a vacuum environment. 3.4.1.3.5 Kinetic Model: Balance Between Adsorption and Desorption Using the temperature dependence of desorption rates for N and Ga, kN and kGa, reported in Ref. [229] for an MBE environment, the ratio kN/kGa can be plotted and when done, a temperature dependence of kN/kGa ¼ 1.9205 · 1016 exp[
39 563/T (K)] results. Using the kinetic values, kN/kGa is about 1000 at T ¼ 1020  C and about 0.1 at T ¼ 725  C. By choice of temperatures, these values of kN/kGa are comparable with the V/III ratios used for OMVPE and MBE growth, respectively. In addition to MBE analogue, if one also relies on the models developed for GaAs [231], one can then construct a kinetic model for GaN grown by OMVPE where activation barriers play an important role as done by Koleske et al. [221]. In this scenario, the Ga and N surface coverage, yGa and yN can be related to the incident fluxes, F, and desorption rates, k, as follows:

dqGa ¼ sGa F Ga
kGa qGa
vG ðqGa ; qN Þ for Ga; dt

ð3:20Þ

dqN ¼ sN F N
mkN qm N
vG ðqGa ; qN Þ for N; dt

ð3:21Þ

and

where s is the precursor sticking coefficient, m is the kinetic order of desorption, and vG is the growth rate. The Ga and N precursors decompose to atomic species (as discussed in Section 3.4.1.3.1 and SGa ¼ 1. For the NH3 sticking coefficient, SN, we assume that NH3 does not stick to surface sites already occupied by N. The value of SN will therefore depend on both yN and the NH3 decomposition mechanism. If two adjacent sites are necessary for the sticking and dissociation, then SN ¼ (1
yN)2; however, if only one site is required for NH3 adsorption, which is followed by

j405

j 3 Growth and Growth Methods for Nitride Semiconductors

406

dissociation, then SN ¼ (1
yN). The growth rate, vG, is then given by vG ðqGa ; qN Þ ¼ g GaN qGa qN
dGaN ð1
qN Þ;

ð3:22Þ

where g is the incorporation rate of the surface atoms into the crystal and d is the decomposition rate of atoms from the crystal to the surface (see Section 3.4.1.3.1). The incorporation rate (the first term on the right-hand side) depends on the thermal activation of the adatoms to overcome the kinetic barriers for bonding to step edges and surface diffusion. Because both the incorporation rate and the decomposition rate depend on combinations of activation barriers (kinetic factors) [231], both increase exponentially with temperature. For growth, the incorporation rate, the gGaNyGayN term, is slightly larger than the decomposition rate, the dGaN ð1
qN Þ term. For GaAs growth, the incorporation rate is maximum near yGa ¼ yN with yGa and yN both less than 0.25 ML [231]. Above the temperature where the preceding conditions apply, the desorption rate approaches the incorporation rate, resulting in a rapid decrease in vG. Clearly, having these two rates as large as possible will lead to improved ordering of the lattice, resulting in a near equilibrium growth. Under steady state, the change in surface coverage (left sides of Equations 3.20 and 3.21) is zero, which leads to sGa F Ga ¼ kGa qGa

and

sN F N ¼ mkN ðqN Þm :

ð3:23Þ

To achieve the condition where yGa ¼ yN, the ratio of incident flux and desorption flux for the two components must be equal, or sGa F Ga =kGa ¼ ðsN F N =mkN Þ1=m :

ð3:24Þ

As discussed in Section 3.4.1.3.2, N2 desorption from GaN has been observed to be a first-order process, which means that m ¼ 1. Assuming that SGa ¼ 1 and near equilibrium, and yN is small, sN ¼ ð1
qN Þx  1 results. Using these relations, Equation 3.24 becomes, F N =kN ¼ F Ga =kGa

or after rearranging

F N =F Ga ¼ kN =kGa :

ð3:25Þ

Equation 3.25 implies that for near equilibrium growth where yGa ¼ yN, the ratio of input fluxes, FN/FGa, must match the ratio of the desorption fluxes, kN/kGa. Because the N surface diffusion rate is nearly zero (see Section 3.4.2 for details regarding growth under N-rich conditions), the N incorporation rate will depend on the number of times N atoms are adsorbed and desorbed on the surface. For growth to take place, vG > 0, g GaN qGa qN > dGaN ð1
qN Þ, kN > kG, and FGa > kGa, which lead to F N =F Ga > kN =kGa :

ð3:26Þ

Because kN/kGa depends exponentially on the temperature, the choice of FN/FGa also depends exponentially on the temperature. 3.4.1.3.6 Comparison of Model with Growth Conditions for Surface Morphology It is beneficial to relate values of these terms to various growth processes. For MBE growth of GaN, growth temperature, TG ¼ 600–800  C, and the films are grown in either

3.4 Nitride Growth Techniques

Ga-rich conditions, that is,FN/FGa < 1,or nitrogen-rich conditions, that is, FN/FGa about 1–10. For HVPE and OMVPE growth, TG ¼ 800–1100  C and FN/FGa ranges from 100 to 10 000, depending on the growth temperature. It is also useful to relate the predicted condition for optimized growth, that is, FN/FGa ¼ kN/kGa to the growth temperature, TG, V/III ratio, and the eventually resultant material quality. Data for this comparison were taken from the literature from several sources to determine the applicability of the growth model forwarded [221]. Generally, the data support the model and show how the V/III ratio and TG must be selected for the growth of high-quality GaN. For OMVPE growth on sapphire with a nucleation layer, if the V/III ratio is too low or TG is too high, both of which lead to the condition FN/FGa < kN/kGa, Ga droplets form to produce a rough surface morphology. Conversely, when FN/FGa > kN/kGa or quite N rich, the film morphology is smooth. An example of the change from smooth to rough morphology as affected by the V/III ratio for a constant growth temperature is shown in Figure 3.45a, where GaN was grown at a fixed TG using different V/III ratios. The filled circles denote GaN films with a smooth morphology, while the open circles denote a film with a rough morphology. Pictures of the two surface morphologies are also shown in Figure 3.45a, along with the kN/kG line. Note that the kN/kGa line properly depicts growth conditions that result in either smooth, that is, FN/FGa > kN/kG, or rough, that is, FN/FGa < kN/kGa, surface morphology. To further illustrate this, the values of the TG and V/III ratios used by many researchers are plotted in Figure 3.45b for GaN grown on either AlN or GaN nucleation layers. Again the open squares denote growth conditions leading to rough morphology, whereas the filled circles denote growth conditions where either smooth surface morphology was obtained or optimized films, judged by electronic properties, were grown. For each of the data points, the V/III ratio and TG were assumed to be reported correctly. Note that all but three of the filled circles are above the kN/kGa line and that all of the open circles fall below the kN/kGa line. Knowledge of the growth process would pave the way for optimizing growth conditions to obtain lower free carrier concentrations and higher mobilities, narrower XRD peaks, and intense photoluminescence (PL) by Koleske et al. [221]. Optimized growth temperature values and V/III ratios taken from five different references are shown in Figure 3.46. For example, the growth conditions leading to optimum electron mobility (500 cm2 V
1 s
1) occur at TG ¼ 1040  C and a V/III ratio of 1590. This condition is very close to the kN/kGa line shown in Figure 3.46. For minimum symmetric X-ray linewidth, the conditions TG ¼ 1025  C and a V/III ratio ¼ 1240 were used, which is slightly above the kN/kGa line. However, asymmetric diffraction linewidth must be observed to draw conclusions about the quality, as this diffraction is sensitive to in-plane extended defects. If the ratio of the band edge emission and yellow emission is used as a quality indicator in terms of radiative recombination, TG ¼ 1080  C and a V/III ratio of 5130 conditions had the best figures. This too is very close to the kN/kGa line. However, the ratio monitored strongly depends on the excitation intensity and further extrapolations based on this ratio alone should be handled with caution. The kinetic model presented above brings to the foreground the importance of the growth parameters. Specifically, the control of V/III ratio and TG is essential for the

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Figure 3.45 (a) Values of the V/III ratios for a growth temperature of above 1040 C. The inset pictures represent the surface morphology for the growth conditions plotted. The filled circles denote smooth morphology and the open circle denotes a rough morphology. The line is the ratio of kN to kGa. (b) Values of the growth temperature

and V/III ratio for smooth surface morphology films or a film grown with optimized growth temperature and V/III ratio (filled circles) and rough morphology films (open squares). Data were taken from 39 sources, which are listed in Ref. [221] where the details can be found. The line is the ratio of kN to kGa. Courtesy of D. Koleske.

reproducible growth of optimized material. Available mass flow controllers (MFC) used in OMVPE systems control flows to within 0.5–1%, which translates to a V/III ratio control to within 2%. The temperature control near 1000–1100  C is more challenging, and there is a greater uncertainty in both setting TG and measuring it. Temperature nonuniformities across the substrate are common; especially for larger substrates despite great strides made in uniform heat coupling and substrate rotation. This temperature nonuniformity leads to nonuniformities in the composition and the physical properties of alloys in the following manner. For growth at 1000  C, a 1  C temperature difference shifts the kN/kGa ratio by 15 out of a total kN/ kGa of 611, which represents a 2.5% change, and at 1100  C, the kN/kGa shifts by 123 out of the total kN/kGa of 5880, representing a 2.1% change. If the wafer center is 10  C

3.4 Nitride Growth Techniques

V/III ratio

104

Conditions for optimized GaN growth

103

k N/k Ga line

102

900

1000

1100

Temperature (ºC) Figure 3.46 Optimized growth temperature and V/III ratio values taken from five different references (see Ref. [221] for details). The kN/kGa line is also plotted. Note the good correlation between the optimized growth conditions and the kN/kGa line.

cooler than the edge for TG ¼ 1040  C, the kN/kGa ratio will be 25% larger at the edge than at the center. These examples demonstrate that to have uniform material properties over the entire wafer, the TG needs to be accurately and reproducibly controlled [221]. Before we delve into the world of MBE, which is next, a discussion for historical content is provided. In spite of the early assumption that CVD process using ammonia led to N-deficient layers and plasma-activated N would lead to better films, further research and ensuing high-quality layers proved this hypothesis to be invalid. For the sake of completeness, a few words are shared here on the topic. Although trialkyl compounds (TMA, TMG, TMI, etc.) are pyrophoric and extremely water and oxygen sensitive, and ammonia is highly corrosive, much of the best material grown today is produced by conventional OMVPE by reacting these trialkyl compounds with NH3 at substrate temperatures in the vicinity of 1000  C [196,232–238]. Temperatures in excess of 800  C are required to obtain single crystalline high-quality GaN films, and the GaN films with the best electrical and optical properties are grown at or around 1050  C. At substrate temperatures exceeding 1100  C, the dissociation of GaN results in voids in the grown layer. A similar situation is observed also for AlN film growth. 3.4.2 Molecular Beam Epitaxy

In this section, a succinct overall view of growth by MBE is given followed by an indepth discussion of physical processes that take place in MBE growth environments. In MBE technique, thin films are formed in vacuum on a heated substrate through various reactions between thermal molecular beams of the constituent elements and

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the surface species on the substrate [239]. The composition of the epilayer and its doping level depend on the arrival rates of the constituent elements and dopants [240]. Therefore, MBE growth is carried out under conditions governed primarily by the kinetics, rather than by mass transfer [241]. A thorough understanding of the growth kinetics, especially the surface processes of growth, is therefore critical. MBE is an extremely versatile technique for preparing thin semiconductor heterostructures owing to the control over the growth parameters that it offers, and the inherent in-situ monitoring capability. As mentioned earlier, thin films are formed on a heated substrate through various reactions between thermal molecular beams (atomic beams in the case of RF-activated nitrogen) of the constituent elements participated by the surface species on the substrate originating from the substrate itself by surface or bulk contamination. The composition of the epilayer and its doping level depend on the arrival rates of the constituent elements and dopants, respectively. The typical growth rate of 1 mm h
1, or slightly more than one monolayer per second (ML s
1), is sufficiently low to allow for surface migration of the impinging species on the surface. In the case of growth along the h1 1 1i for cubic and c-directions for wurtzitic systems, one monolayer constitutes a bilayer. As MBE growth occurs under conditions that are governed primarily by the kinetics, rather than by mass transfer, this allows the preparation of many different structures that are otherwise not possible to attain. This is in contrast to OMVPE of conventional compound semiconductors, such as GaAs. However, in the case of GaN, even the OMVPE has elements of kinetics [242]. In nitride growth by MBE, the metal species are provided by Ga, In, and Al metal sources, the dopants are provided by pure Si for n-type and Mg for p-type using conventional Knudsen effusion cells that are heated to sufficient temperatures for the desired growth rate, composition, and doping levels. On the contrary, nitrogen gas is one of the least reactive gases because of its large molecular cohesive energy (946.04 kJ mol ¼ 9.8 eV per N2 molecule). Because of the triple bond between the two nitrogen atoms, dissociation of one molecule into two reactive nitrogen atoms requires a large amount of energy, which cannot be provided by thermal means. In a plasma environment, however, and at reduced pressures, a significant dissociation of the nitrogen molecules takes place. Atomic nitrogen is reactive even at room temperature and bonds with many metals. Consequently, group III nitrides can be grown by plasma-assisted molecular beam epitaxy, where the plasma-induced fragmentation of nitrogen molecules is combined with the evaporation of metal atoms from effusion cells. In this vein, MBE growth of GaN has been reported by electron cyclotron resonance microwave plasma assisted molecular beam epitaxy (ECR-MBE). Several laboratories in the past have attempted RMBE growth in which N2 or NH3 was decomposed on the substrate surface [243–247]. The sequence of processes taking place during growth by MBE is adsorption, desorption, surface diffusion, incorporation and decomposition. All of these processes are in effect in many ways compete with each other during growth by MBE. Adsorption can be summed as the atoms or molecules impinging on the substrate surface and sticking by overcoming an activation barrier. Desorption, on the contrary, is a process in which the species that are not incorporated into the crystal lattice leave the substrate surface by thermal vibration. During surface diffusion, imperative for

3.4 Nitride Growth Techniques

Impinging atoms

Lattice site

Desorption

Surface diffusion

Adsorption Surface diffusion Decomposition

Substrate, T s Overgrowth Incorporation into lattice

Surface nucleation (aggregation)

Interdiffusion

Figure 3.47 Surface processes during the MBE growth: adsorption, desorption, surface diffusion, lattice incorporation, and decomposition.

growth, the constituent atoms or molecules diffuse on the substrate surface to find the low-energy crystal sites for incorporation. During the incorporation phase, the constituent atoms or molecules enter the crystal lattice of the substrate or the epilayer already grown by attaching to a dangling bond, vacancy, step edge, and so on. Owing to high temperatures involved, albeit much lower than those employed in OMVPE, during decomposition, the atoms in the crystal lattice leave the surface by breaking the bond. These processes are schematically shown in Figure 3.47. Let us discuss these five processes that govern growth by MBE. 3.4.2.1 Adsorption Impinging gas molecules (atoms) condense on the surface and, depending on the strength of the interaction between the adsorbate and the surface, can be adsorped by physisorption (weak) or chemisorption (strong) [248]. Physisorption represents weak adsorbate–surface interaction because of van der Waals forces with typical binding energies on the order of 10–100 meV. Owing to the weak interaction, physisorbed atoms or molecules do not disturb the structural environment near the adsorption sites to a significant degree. In chemisorption, on the contrary, an adsorbate forms strong chemical bonds with the substrate atoms with typical binding energies on the order of 1–10 eV, thus changing the adsorbate’s chemical state.

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The adsorbate coverage characterizes the surface concentration of adsorbed species expressed in monolayer units. The coverage is a relative value associated with a given substrate. It can be converted to an absolute surface density of atoms. Considering the kinetic approach in the case of a uniform solid surface exposed to an adsorbing gas, the adsorption rate is defined as [248] p r a ¼ sf ðQÞexpð
E act =kB T s Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pmkB T s

ð3:27Þ

where p is the partial pressure of the adsorbing gas; s is the condensation coefficient responsible for the effects of the orientation and the energy accommodation of the adsorbed molecules; f(Y) is a coverage-dependent function that describes the probability of finding an adsorption site; exp(
Eact/kBT) is the temperature-dependent Boltzmann term associated with the energetics of the activated adsorption. Finite equilibrium Ga adlayer coverage has been reported for typical substrate temperatures and Ga fluxes [249]. For large Ga fluxes, up to 2.5  0.2 monolayers of Ga are adsorbed on the GaN surface. For higher Ga fluxes, Ga droplets are formed [250]. At typical growth temperatures used in MBE, Ga adlayer does not condense into a reconstruction but behaves like a liquid film [251]. The reported height of the Ga adlayer is about 3.88 Å, as measured by scanning tunneling microscopy (STM), which corresponds to 1.9 ML [252]. For lower Ga fluxes, a discontinuous transition to Ga monolayer equilibrium coverage is found, followed by a continuous decrease toward zero coverage. The practical implications of this process in terms of GaN growth are discussed in Section 3.4.1.3.1. 3.4.2.2 Desorption Desorption is a process in which the adsorbate species gain sufficient thermal energy to escape from the adsorption well and leave the surface. The probability of desorption depends on the bonding strength of the particular atom to the surface. Bonding energies are different for different materials and the strength of a bond is expressed in terms of the amount of energy needed to break it. In the kinetic approach, desorption is described in terms of a desorption rate, rdes, which represents the number of species desorbed from unit surface area per unit time, and can be expressed as [248]

r des ¼ s f ðQÞexpð
E des =kB T s Þ;

ð3:28Þ

where f (Y) describes the coverage dependence and s is the desorption coefficient standing for steric and surface mobility factors. For desorption to occur, the adsorbed species must overcome a barrier called the desorption energy, Edes. In case of activated chemisorption, the desorption energy is the sum of the binding energy in the chemisorbed state and the activation energy for adsorption, Edes ¼ Eads þ Eact. In case of nonactivated chemisorption, the desorption energy is simply the binding energy in the chemisorbed state, Edes ¼ Eads. The lifetime of an adsorbate as a function of temperature is needed for studying the desorption energy. The lifetime t is defined as the average time spent by the adsorbate on the surface marked from the time of adsorption to the time of desorption and

3.4 Nitride Growth Techniques

obeys an Arrhenius dependence in the form [250]:   E des t ¼ t0 exp : kB T s

ð3:29Þ

Equations 3.28 and 3.29 describe the desorption process to be quite sensitive to temperature. For low Ts, the lifetime of adsorbates is sufficient to point that the desorption process can be neglected. For intermediate temperatures, the growth rate is determined through the competing processes of adsorption and desorption. However, for sufficiently high Ts, the desorption rate can be greater than the deposition rate, in which case evaporation rather than deposition would take place. Specific to the case of GaN, Ga desorption has been investigated by mass spectrometry [250,253] and RHEED [254,255] with consistency lacking. The measured activation energies reported for Ga desorption are in the range of 0.4–5.1 eV [250–255]. For N, which is usually desorbed as a dimer (N2), the desorption rate is limited by the surface diffusion of two N atoms. Some reports indicate this to be a first-order process [256], meaning that the surface diffusion may play less of a role in the desorption mechanism than the surface structure. Another plausible mechanism is that once an N atom diffuses near another N atom, the N2 molecule would immediately form owing in part to the large N–N binding energy and desorbs from the surface because of the highly exothermic nature of the N2 formation [257]. The practical implications of these processes in terms of GaN growth are discussed in Section 3.4.1.3.1. 3.4.2.3 Surface Diffusion As mentioned earlier, surface diffusion describes the motion of adsorbates (atoms or molecules) on the substrate (film) surface. The motion of adsorbates that become mobile because of thermal activation is described as a random walk. When a concentration gradient is present, this random walk motion of many particles results in their net diffusion being opposite to the gradient direction in a macroscopic sense. The microscopic view of surface diffusion is an activated process that is also affected by factors such as interaction between diffusing adsorbates, formation of surface phases, and defects. For an atom on the surface to diffuse to the next lattice position, it must overcome the lattice potential between the two neighboring positions. This activation energy required for diffusion, E0, is the microscopic origin of the lattice potential. The average length of diffusion, ls, in a unit time interval is a very important parameter to characterize the diffusion process and has an exponential temperature dependence given by Equation 3.19, called the Einstein equation [248]. At high growth temperatures where the surface diffusion length is larger than the terrace width, atoms move by surface diffusion from the terrace to the step edge to be incorporated into the growing lattice, which is called the step-flow growth. As the diffusion length decreases, for example, by reduced temperature, atoms meet and nucleate new islands on the terraces before reaching the edge of an existing island or a step edge. Moreover, islands can nucleate on top of the existing islands, a process that leads to rough surface formation and 3D growth. In the intermediate region, the

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(0 0 0 1)

Top view (c-plane)

(0 0 0 1)

hcp

Diffusion path

Bridge On top

N layer

Ga2L fcc Ga1L

Ga layer

Side view (a-plane)

Ga1L N Ga2L (a)

(b)

Figure 3.48 Schematic diagram of diffusion paths on the GaN (0 0 0 1) (a) and (0 0 0 1) (b) surfaces. The side views as in the a-plane for each are also shown [256].

islands do form on terraces, but the diffusion length is sufficiently long for adsorbates to diffuse to the edges of these islands and incorporate there, which leads to smooth surface also. In this mode of growth, the RHEED intensity oscillations are observed in that the intensity is maximum when the islands expand to cover the terrace and the intensity is minimum when the coverage is 50%. Specific to the case of GaN, diffusivity for Ga and N adatoms on GaN surface is different. Zywietz et al. [256] calculated the surface potential energy for Ga and N adatoms on GaN(0 0 0 1) surface. The simulations led to the presence of two transition sites, as shown in Figure 3.48. For Ga diffusion, the lower energy transition site is the bridge position (0.4 eV) and the higher energy site is the “on-top” position (>3 eV). However, for N adatoms, the barrier for bridge diffusion is 1.4 eV and the barrier for the on-top position is similar to Ga. Significantly lower diffusion barrier for Ga relative to N results in Ga being very mobile at typical growth temperatures, whereas the diffusion of N is slower by orders of magnitude. Further, the presence of excess N strongly increases the Ga diffusion barrier from 0.4 to 1.8 eV. The very divergent surface mobilities of Ga and N adatoms have serious consequences. In the Ga-rich regime, the Ga adatoms are highly mobile and a step-flow mode results in 2D growth. Furthermore, if excess Ga adatoms are present on the surface, N adatoms can be efficiently incorporated because the probability of

3.4 Nitride Growth Techniques

fast-moving Ga adatoms capturing N atoms is high. The presence of a Ga bilayer on the surface in Ga-rich growth conditions reduces the lateral diffusion barrier from about 1.3 to about 0.5 eV, paving the way for growth of smooth layers at low temperatures [258]. For details on the Ga-bilayer issue, refer to Section 3.5.6. However, N-rich growth conditions show roughly a five times higher diffusion barrier for Ga. A rough surface can, therefore, be expected for N-rich conditions, consistent with experimental observations. 3.4.2.4 Incorporation In the incorporation process, the molecules or atoms bond to the crystal through various reactions between the constituent elements and the surface species on the substrate. This process is controlled by the interplay of thermodynamics and kinetics. The general trends in film growth are understood within the thermodynamic approach in terms of the relative surface and interface energies. However, film growth by MBE is a nonequilibrium kinetic process in which rate-limiting steps affect the growth mode [241]. There are three modes of growth/incorporation that come to bear in general, namely, Frank–van der Merve (FM), Stranski–Krastanov (SK), and Volmer–Weber (VW). These three processes with graphics are discussed in Section 3.5.16, which pertains to quantum dots. Suffice it to say, the Frank–van der Merve mode is a layer-by-layer process. Each layer is fully completed before the next layer starts to grow, and it is strictly a two-dimensional growth mode. The Volmer–Weber mode is an island growth mode. Three-dimensional islands nucleate and grow directly on the substrate surface, which is typically seen in metals unless very low – even below room temperature – deposition conditions are employed. The Stranski–Krastanov mode is a layer-plus-island growth mode and represents the intermediate case between the FM and VW growth modes. After the formation of a complete two-dimensional layer of a few monolayer thickness (the exact value of which depends on the local strain), the growth of a three-dimensional layer (islands) takes place. The occurrences of growth have their genesis in the competition between the surface desorption and the surface diffusion. Because the desorption and diffusion processes are noticeably affected by the deposition rate, the surface condition and temperature, the growth mode, and the epilayer surface can be controlled by choosing a proper III/V ratio and a substrate temperature. Several studies [259,260] have revealed that not all incident Ga atoms are incorporated into the growing GaN at the usual growth temperature in PAMBE (650–750  C), even when an excess of N flux is present (III/V flux ratio <1). Guha et al. [250] studied the incorporation ratio of Ga during GaN growth under N-rich conditions by monitoring the reflected Ga signals detected by the mass spectrometer and found that the Ga incorporation rate is sensitive to the growth temperature. At low growth temperatures, the lifetime of Ga adatoms is sufficiently long so that almost all Ga atoms encounter N adatoms and are incorporated into the crystal, resulting in a near unity incorporation ratio. As the growth temperature increases, the residence time of a Ga adatom on the surface decreases. Thus, the probability of Ga adatoms encountering N adatoms decreases and their incorporation ratio drops.

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3.4.2.5 Decomposition As motioned previously, when the desorption rate is greater than the incorporation rate, the film would begin to decompose. In compound semiconductors such as GaN, one compound splits into two (or more pieces) as the bonds between the constituents is broken. The III nitrides have chemical bonds with the Ga–N bond strength estimated near 2.2 eV [261]. In addition, III nitride bonds have a sizeable ionic bond (31–40%) nature (excluding BN) than other III–V semiconductors (<8%) [262]. A lower ionic bond would have resulted in larger covalent bond with much larger bond strength. As in the case of other III–V semiconductors, the heat of formation of GaN is small and similar to that of InP and AlAs, indicating that GaN easily decomposes. In this vein, GaN does not melt congruently at pressures typically used in MBE growth, but decomposes above 800  C at atmospheric pressure and at lower temperatures in vacuum. Guha et al. [250] reported a decomposition rate of three to four monolayers per minute at 830  C. Grandjean et al. [222] found that the decomposition rate is nearly zero below 750  C, but increases rapidly above 800  C, and reaches 1 mm h
1 at 850  C. This means that it may be impossible to grow GaN at high temperatures and that the growth temperature should be kept below 800  C in relation to growth in vacuum. Several different mechanisms have been proposed to explain GaN decomposition. These include decomposition into gaseous Ga and nitrogen, liquid Ga and nitrogen, and sublimation of GaN into a diatomic or polymeric product. One would surmise that these critical temperatures could be altered to some extent by the overpressure of N or N-containing reactive species used for growth, even though reports indicate that, to a large extent, GaN decomposition rate is similar in vacuum and at OMVPE pressures. However, a good deal of dispersion in the data allow us to assume that GaN is no different and that group V overpressure should have some affect on the rate of decomposition. In fact, GaN growth temperature with OMVPE is well in excess of 1050  C, and if the vacuum experiments regarding decomposition rates were to hold, it would be nearly impossible to achieve deposition. To make some sense of all these sometime competing processes, involving adsorption, desorption, surface diffusion, incorporation, and decomposition, it is compelling to summarize. The conventional III–V semiconductors, and nitrides are no exception, are grown under conditions (substrate temperature and system pressure) in which the III–V compounds and the gaseous environment with which they are in contact are thermodynamically stable. For high-quality compound semiconductor growth, the optimum substrate temperatures used are in the range of 1/2 to 2/3 of the melting temperature of the semiconductor. In the case of III nitrides, however, synthesis occurs, particularly with MBE, at temperatures significantly below 1/2 of the predicted melting temperature. Specific to GaN, the substrate temperature during MBE growth is about 20 to 35% (600–900  C) of the GaN melting temperature (TM  2500  C). Thus, GaN growth takes place in kinetic regime and the surface processes are critical in determining the quality. The synthesis of GaN involves a metastable growth process, which is controlled by a competition between the forward reaction (incorporation of Ga and N into the film and GaN epilayer forms) and the

3.4 Nitride Growth Techniques

reverse reaction (decomposition of GaN). The forward reaction depends on the arrival rates of Ga atoms and activated nitrogen species on the surface, as well as the substrate temperature, whereas the reverse reaction is strongly affected only by the substrate temperature. For a net growth to take place, the rate of GaN formation must be larger than the rate of decomposition. The remaining three additional processes determining the eventual interface morphology of a growing film include deposition, desorption, and surface diffusion. Their relative importance depends on the microscopic properties of the interface, the bonding energies, and the diffusion barriers. The experimentally controllable parameters are the arrival rates of group III and V compounds and their ratios and substrate temperature. By tuning these parameters, a variety of morphologies can be achieved, ranging from layer-by-layer growth with an essentially smooth interface to a rough self-affined surface. 3.4.2.6 Reflection High-Energy Electron Diffraction By far the most commonly used characterization tool in MBE growth is RHEED [263– 265]. In established materials such as GaAs, this method is capable of providing a window on processes taking place on the surface of the growing film at the atomic scale. A discussion of RHEED, which is synonymous with MBE, is therefore warranted. RHEED is a technique for probing the surface structure of solids in UHV. The method is based on diffraction of high-energy electrons from a clean and well-ordered surface. Electrons with high energy (typically 5–100 keV) and a grazing angle of incidence (typically <3 ) scatter elastically from only the first few atomic layers, making the method very surface sensitive. Compared to other diffraction methods, this glancing incidence angle geometry allows effusion cells to be placed in such a way that allows near normal radiation toward the substrates and at the same time allows real-time surface monitoring during growth [266]. The electron beam incident on the surface is also reflected following the laws of classical optics, the image of which on the phosphor screen is called the specular reflection spot. The intensity of this spot can be used to glean important information about the degree of smoothness or roughness of the surface. The growth regime in which growth initiates with islands on terraces, which eventually grow and coalesce, the intensity of the specular beam spot undergoes periodic oscillations as the growth proceeds, the period of which represents the time taken to grow one monolayer of material. This is used very successfully to monitor and even control, through a feedback loop, the growth of quantum wells and superlattices requiring ultimate thickness accuracy. As for diffraction, if electrons diffract only from the first atomic layer of a perfectly flat and ordered surface, the three-dimensional reciprocal lattice points degenerate into parallel infinite rods. In the Ewald construction that results from such an ideal case, the intersection of the Ewald sphere, whose radius is much larger than the interrod spacing for typical RHEED energies, consists only of a series of points placed on a half circle as seen in the phosphor screen depicted in Figure 3.49. In reality, however, thermal vibrations and imperfections cause the reciprocal lattice rods to have a finite thickness with additional contribution from the divergence and

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y Incident e-beam Diffracted e-beams k0

x k' Sample

z

Phosphor screen Figure 3.49 An artist’s view of the major elements of RHEED a system showing the incident and reflected beam wavevectors (k0 and k0 and resulting RHEED spots on the phosphor screen from an ideal surface wherein diffraction only from the top monolayer is assumed.

dispersion of the electron beam. Atomic steps associated with islands are the main source of broadening of the diffraction rods. In the case of GaAs, which has the subject of most of the RHEED-related investigations and ground-breaking research, the pattern starts off as a semicircle of dots and broadens into streaks as growth begins. This is not observed when the surface is rough and/or the growth mode is of the step-flow type. Therefore, even diffraction from a very flat semiconductor surface results in a diffraction pattern consisting of a series of streaks with modulated intensity rather than points expected from an ideal case. If the surface is not flat, many electrons will be transmitted through surface asperities (see the upper sketch in the inset of Figure 3.49) and scattered in different directions, resulting in a RHEED pattern comprising many spotty features. Therefore, the most obvious information gleaned from RHEED images pertains to the flatness of the surface. It also follows that diffraction from an amorphous surface such as an oxide on the semiconductor sample surface gives rise to no diffraction pattern at all and results in only a diffused background. This is pivotal in monitoring the thermal desorption of the oxide layer when the sample is initially heated prior to the commencement of the MBE growth. ! The geometric theory [265], in which diffraction of a plane wave (of wave vector k) by a single crystal is assumed, is the simplest one that can be used to describe RHEED. Before delving into RHEED, it would be instructive to review two-dimensional periodic lattice in real and k-space. Reciprocal lattice representations of periodic structures are fundamental to analytic treatments of phenomena like crystal diffraction and electron energy levels in solids. To analyze the results of this RHEED

3.4 Nitride Growth Techniques

experiment, you will need the two-dimensional version of the reciprocal lattice, which is sometimes called the reciprocal net. The surface of a crystalline material is nothing more than a two-dimensional diffraction grating in which atoms are arranged ! ! according to repetition of a unit mesh with basis vectors ! a1 ¼ ax and ! a2 ¼ ay, where ! ! x and y are the unit vectors in their respective directions and a is the spacing between the surface atoms in both the x- and y-directions. This real net has a reciprocal lattice whose basis vectors are given by !

b2 ¼

2p ! x a

!

and b2 ¼

2p ! y: a

ð3:30Þ !

!! ik r

wave e will A set of points in space located by a set of lattice vectors R and a plane ! have the periodicity condition satisfied for arbitrary wave vector k. The periodicity ! conditions are satisfied only for a set of wave vectors K , called the reciprocal lattice, which yields plane waves with periodicity of the lattice. Mathematically, the periodicity condition can be expressed as ! !

eiK ðr

!

þR Þ

!!

¼ ei k r ; !!

ð3:31Þ !

which leads to eiK R ¼ 1, which, in turn, must be must be true for all lattice vectors R, and it is this relationship that defines the reciprocal lattice. In kinematical scattering theory [267], the maximum value for scattering amplitude is obtained when the incident beam wave vector k0 and the diffracted beam wave vector k0 differ by the reciprocal lattice wave vector K of the surface atoms. In this theory, constructive interference will occur, provided the change in the wave vector is a vector of the reciprocal space, meaning diffraction results when the Laue condition [268] is satisfied, that is, !

!

!

k0
k0 ¼ K ; ! 0

! k0

ð3:32Þ

where k and are the wave vectors for the diffracted and the incident beams, ! respectively, and K is the reciprocal lattice vector. In the special case of elastic scattering, the amplitudes of the diffracted and incident electron beam wave vectors0 0 ! ! ! are equal, that is, jk j ¼ j k0 j. This condition is satisfied by an infinite number of k vectors pointing in all directions, which is the origin of the Ewald sphere. The origin ! of the Ewald sphere is same as that of the k0 , and its radius is equal to the absolute ! value of k0 . The Bragg diffraction and/or the Lau treatment indicate that the beam intensity decreases rapidly for Dk 6¼ K. Consequently, when the diffraction condition, Dk ¼ K, is satisfied, the bright spots appear on the phosphor screen, as shown in Figure 3.50. This can be visualized with the aid of a 3D Ewald sphere [268,269] of radius k, as depicted in Figure 3.51a–c. Only when the sphere passes through a reciprocal lattice point, the difference between incident and diffracted beam wave vectors is a reciprocal lattice vector K and the intensity of diffraction is the highest. Considering that the reciprocal lattice separation is inversely proportional to distances in real space, a 2D plane in real space should be a one-dimensional (1D) line in reciprocal space. As a result, a real and smooth 2D surface should be a set

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j 3 Growth and Growth Methods for Nitride Semiconductors

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Rod

k–k'

R

Origin

Possible k–k' wave vectors

k–k'

k–k' R

Plan view of 2D ordered atoms

Side view of the 2D ordered atoms

Figure 3.50 The plan view of two-dimensional ordered atoms representing the surface atoms in a RHEED experiment with a ! real surface vector of r . The figure on the right side also shows the side view of the real space network and schematic description of the origination of the lattice rods. Adapted from Ref. [270].

of infinitely long rods in the reciprocal space (Figure 3.51), leading to a streaky RHEED pattern. For a 3D rough surface, the RHEED pattern becomes spotty (Figure 3.51). The electron energy determines the wavelength and the number of Lau rings as well as rod spacing observed on the phosphor screen. The magnitude of the wave vector is given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2p 1 E ¼ 2m0 E þ ; ð3:33Þ k0 ¼ l h c where m0 is the electron rest mass and E is the accelerating energy. Equation 3.33 can be simplified and the wavelength can be expressed as [270] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  h 150:4 12:26 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q l¼ ffi ffi pffiffiffi ðlin Awhen E is in eVÞ:
6 2 Eð1þ1:95x10 Þ 2 E 2m E þðE =cÞ 0

ð3:34Þ For a primary beam energy of 5–50 KeV, l values ranging from 0.17 to 0.055 Å result [271]. The theory discussed above is good for a first-order understanding of RHEED but falls short of describing the mechanisms involved. More advanced theories [265,272] treat the interaction between the incident wave and the scatterer quantum mechanically by solving the Schr€odinger equation for the wave function of the scattered wave

3.4 Nitride Growth Techniques !

yðr Þ (this wave function is the single-electron wave function that is a superposition of incident and scattered wave functions) given an effective potential U(r): !

!

ðr2 þ Uðr Þ þ k20 Þyðr Þ ¼ 0:

ð3:35Þ

Using Green’s method, the solution for the above equation could be written as !  1 ð exp ik j !r
r! 0 j ! ! 0 ! ! ! Uðr 0 Þyðr 0 Þ dr 0 : yðr Þ ¼ exp ik0  r þ 4p jr! ?r! 0j

Figure 3.51 (a) The 3D Ewald sphere configuration; (b) a real 2D smooth surface in the reciprocal space.

ð3:36Þ

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j 3 Growth and Growth Methods for Nitride Semiconductors

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k(–1,0)

k(0,0)

k in

(0,0,0)

(b) Figure 3.51 (Continued)

The first term in the above equation represents the incident plane wave, whereas the second term represents the scattered wave. In RHEED experiments, one is interested in the value of y(r) evaluated at large distances compared0 to the dimensions !0 ! ! ! of the scatterer, that is, jr
r j  r; hence, for large jr
r j (or for r ! 1), Equation 3.36 can be written as  ! ! !  e i k 0  r ð  ! 0 ! ! ! ! ! e
i k  r Uðr 0 Þyðr 0 Þdr 0 : yðr Þ ¼ exp ik0  r þ 4pr !

So, the scattering amplitude has the form ð  ! 0 !0 
ik  r 1 ! ! ! ! 0 Fðk Þ ¼ e Uðr 0 Þyðr 0 Þdr 0 : 4p

ð3:37Þ

ð3:38Þ

3.4 Nitride Growth Techniques

To discover the intensity distribution in the diffraction pattern, the effective potential should be assumed and the scattering amplitude integral should be evaluated. If the Born approximation is used, that is, the wave function inside the crystal is assumed to be that of the incident wave, Equation 3.38 can be written as ð  !0 !  ! 0 1 ! ! ð3:39Þ FðK Þ ¼ e
i k  r Uðr 0 Þd r 0 ; 4p !

!

!

with the scattering vector k 0
k0 ¼ K . In other words, the scattering amplitude is the Fourier transform of the scattering potential. A number of theoretical calculations have been made to calculate the scattering amplitude, assuming functional forms for U(r) that take into account the periodicity of the crystal lattice. The main problem with the kinematical treatment is the oversimplified assumption that the wave function at the scatterer equals that of the incident plane wave, because this assumption overlooks the mutual interaction between the crystal and the incident electron beam. The kinematic theory ignores the ! details of the potential nearly entirely. If one substitutes for Uðr 0 Þ a sum of atomic potentials, then an equation for the relative diffracted intensity can be obtained as X ! I ¼ Fðs!Þ2 ¼ f 2 i; j exp½is  ðr!i
r!j Þ; ð3:40Þ where f is a scattering factor for identical scatterers that depends on the momentum ! transfer s . The potential is hidden in f and the position of the beams depends on the sum of the exponentials. It is kinematic (as opposed to dynamic) as the positions just depend on momentum and energy conservation. For the kinematic theory, the potential is included in the Born approximation in f, the scattering factor. However, it is just not included properly because the higher order terms of the Born approximation and, therefore, multiple scattering are neglected. Only in the case of weak surface scattering, when one can choose the diffraction condition in such a way that there is only one dominant diffracted beam, can kinematic theory be a good approximation. Because the mean free path of elastically scattered electrons in a crystal is shorter than that of inelastically scattered electrons, multiple scattering of electrons takes place. For the quantitative analysis of electron diffraction intensities, however, it is not sufficient because the strong interaction between the crystal and incident electrons changes the intensity of the total beam. A more elaborate theory, “dynamical theory,” has been introduced to deal with the diffraction problem without the oversimplification inherent to the kinematical theory. Dynamic scattering includes the potential and the main feature of dynamical theory is that it includes more terms of the Born approximation, taking multiple scattering into account. For electron scattering from a crystal, the wave within the crystal may be represented by a sum of plane waves yðr 0 Þ ¼

N X l¼0

 !0 !0  yi ðr 0 Þe
ik r :

ð3:41Þ

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j 3 Growth and Growth Methods for Nitride Semiconductors

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Substituting Equation 3.41 into Equation 3.37 and using the Laue condition, k1
k0 ¼ 2ad, the scattering amplitude could be written as f ðkÞ ¼

N X

f l ðK l Þ ¼

l¼0

N 1 X 4p l¼0

ð

yl ðr 0 Þ e
iðK
k0
2ad Uðr 0 Þ dr 0 :

ð3:42Þ

scatterer

Again, to find the scattering amplitude, the integrals in Equation 3.42 have to be solved. This difficulty is the reason why dynamical theory is seldom used for analyzing RHEED data. There are different simplified methods for dynamical calculation of the intensity of RHEED, which face similar difficulties and result in similar accuracies. All these methods decompose the wave function and the crystal potential into their Fourier components. Because real crystal surfaces include surface relaxation and reconstruction, the atomic arrangement perpendicular to the surface is not periodic, whereas the atomic arrangement parallel to the surface is. Therefore, the three-dimensional Fourier expansion near the surface is not adequate for RHEED calculations. The ! ! crystal potential Uðr Þ and the wave function yðr Þ are expanded in a two-dimensional Fourier series in the direction parallel to the surface as !

Uðr Þ ¼

N X

! !

 r== ; U n ðzÞeiR n

ð3:43Þ

n¼0

!

yðr Þ ¼

N X

yn ðzÞ e

h !  ! i ! i k0== þ Rn  r ==

;

ð3:44Þ

n¼0 !

!

where Rn is the reciprocal rod vector, k0== is the component of the incident wave vector parallel to the surface, and the direction of z is normal to the surface. Substituting Equations 3.43 and 3.44 into Equation 3.35, one can obtain a second-order differential equation in the matrix form as d2 YðzÞ þ AðzÞYðzÞ ¼ 0; dz2

ð3:45Þ !

!

where (C(z))n ¼ Cn(z) and ðAðzÞÞnm ¼ ½k20
ðk 0== þ Rn Þ2 d nm þ U n
m ðzÞ. A(z) is an N · N matrix for the N-beam case, and a key issue in dynamical theory is what number of beams must be included so that the calculation results would not change significantly if further beams were included. Further analysis involves reduction of Equation 3.45 by defining new coefficients in terms of cn(z) to a first-order homogeneous differential equation d 0 FðzÞ þ iA ðzÞFðzÞ ¼ 0; dz

ð3:46Þ

where A0 (z) is now a 2N · 2N matrix. After carefully choosing incident angles and determining diffracted beams, one can obtain the eigenvalues and eigenvectors of the transfer matrix A0 (z) and solve Equation 3.46 for the simple case in which a crystal potential is assumed to be constant in the direction normal to the surface and periodic

3.4 Nitride Growth Techniques Gate valve

Computer

Substrate CCD camera and optics

RHEED gun Video cable RHEED power supply

Camera control cable

Shutters

Liquid nitrogen shroud Shutter control

Figure 3.52 Schematic diagram of the RHEED technique in an MBE system.

in directions parallel to the surface. These results for the single-slice case may then be used for the multislice case in which the crystal is divided into multiple thin slices parallel to the surface where within each slice the potential is constant in the z-direction. By applying appropriate boundary conditions and matching them at each interface, solutions for F(z) can be obtained and RHEED intensities can be determined. Further details can be found in Refs. [264,265,270,272,273]. A typical RHEED system consists of an electron gun, a phosphor screen, and an image-processing system. Figure 3.52 shows the typical design for a RHEED system in an MBE system. Typical acceleration voltages range from 10 to 30 keV. This high energy necessary to image a reciprocal space into the relatively small solid angle of the phosphor screen has the additional advantage of reducing the influence of stray fields. The electron beam interacts with the sample at an angle that is typically in the range of 0.5–2.5 . The diffracted electron intensity pattern is converted into visible light by a phosphor screen. A charge-coupled device (CCD) camera external to the UHV system is used to record the RHEED pattern. The real-time still images and video of the CCD output can be displayed on a computer screen. The computer can also be used to analyze the RHEED pattern as well as to monit RHEED oscillations, which, in turn, can be used to control the shutters for precise growth of heterostructures. Having learned about the principle of RHEED and rods produced on the phosphor screen, we are now well motivated to learn how the image on the screen and the surface atoms correlate. If the surface atoms on a clean surface were to be situated in their bulk positions, the ensuing RHEED pattern rods, corresponding to (
1,0), (0,0), and (1,0) diffraction rods, would be separated by the reciprocal of the separation between the atomic rows (or 2p per row separation depending on deflection of

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j 3 Growth and Growth Methods for Nitride Semiconductors

426

reciprocal space). These rods are called the bulk rods, and the pattern would change when the sample is rotated about its normal. For a bcc or fcc lattice on (1 0 0) like surfaces, owing to symmetry 90 rotation later, the original image is recovered. In RHEED terms, this is called the (1 · 1) surface construction, and it is important to specify the direction of the electron beam in relation to the crystallographic directions of the lattice. In an hexagonal close packed (hcp) lattice, the symmetry considerations are such that the pattern would repeat itself after every 60 , and this too would be termed (1 · 1) surface construction. The arrangement of surface atoms in their bulk positions in hcp symmetry can be gleaned from Figure 1.7. If surface atoms, whose surface coverage can be less than a monolayer, from beyond the bulk spacing would reconstruct, with additional RHEED rods appearing between the above-mentioned bulk diffraction rods. Owing to rotational (about the normal to the crystal surface) asymmetry in reconstruction, the number of additional lines would vary if the sample is rotated. A series of commonly seen surface reconstructions for (1 · 1), c(2 · 2), and p(2 · 2) on a (1 0 0) surface in fcc and bcc lattices is shown in Figure 3.53. For example, in the 2 · 2 reconstruction, an additional rod half way between the bulk rods in both azimuths, which are normal to each other, would appear. Symmetry of different surfaces is different. As an example, p(2 · 1) and c(2 · 2) surface reconstructions on fcc (1 1 0) plane are shown in Figure 3.54. The other commonly observed surface reconstructions on fcc (1 1 0) surfaces, namely, (2 · 4) and c(2 · 4), are shown in Figure 3.55. This surface is similar to ð1 1  2 0Þ surface in the hexagonal (hcp) system.

p( 1 x 1)

c(2 x 2)

p(2 x 2)

fcc (1 0 0) and bcc (1 0 0) Figure 3.53 Various (1 0 0) surface reconstructions corresponding to (1 · 1), c(2 · 2), and p(2 · 2) for fcc and bcc lattices. For convenience, the surface atoms participating in the said reconstruction are shown as smaller and shaded solid circles and those in bulk position are shown as open circles. The heavier lines indicate, one for each, the surface atoms participating in the surface reconstruction.

3.4 Nitride Growth Techniques

p(2 x 1)

c(2 x 2) fcc (1 1 0)

Figure 3.54 Various (110) surface reconstructions corresponding to p(2 · 1) and c(2 · 2) for an fcc lattice. For convenience, the surface atoms participating in the said reconstruction are shown as smaller and shaded solid circles and those in bulk position are shown as open circles. The heavier lines indicate, one for each, the surface atoms participating in the surface reconstruction.

(2 x 4)

c(2 x 4)

fcc (1 1 0) Figure 3.55 The (2 · 4) and c(2 · 4) on (1 1 0) like surfaces for an fcc lattice. For convenience, the surface atoms participating in the said reconstruction are shown as smaller and shaded solid circles and those in bulk position are shown as open circles. The heavier lines indicate, one for each, the surface atoms participating in the surface reconstruction.

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j 3 Growth and Growth Methods for Nitride Semiconductors

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p(1 x 1)

p(2 x 2)

3x 3 r30º

fcc (1 1 1) , hcp (0 0 0 1) pffiffiffi pffiffiffi Figure 3.56 The (1 · 1), p(2 · 2), and 3  3 30 on fcc (1 1 1) and hcp (0 0 0 1) surfaces. For convenience the surface atoms participating in the said reconstruction are shown as smaller and shaded solid circles and those in bulk position are shown as open circles. The heavier lines indicate, one for each, the surface atoms participating in the surface reconstruction.

The atomic arrangement on the (1 1 1) fcc is similar to that on the (0 0 0 1) surface in the hexagonal (hcp) system. The (1 · 1) reconstruction, which is seen in Ga polarity GaN temperatures, and (2 · 2), which is seen during cooldown, and pffiffiffi at pffiffigrowth ffi 3  3 30  seen in AlN growth, discussed in detail in Section 3.5.14, are shown in Figure 3.56. A good deal of GaN growth is pursued on Si(1 1 1) substrates. The cleaning procedure for Si before epitaxy is discussed in Section 3.2.2.2. For this reason alone, the surface reconstruction of a clean Si(1 1 1) surface [274] is shown in Figure 3.57, which shows both an artistic view (a) and a scanning tunneling microscopy image (b). 3.4.2.7 Plasma-Assisted MBE (PAMBE) or RF MBE, Primarily N Source An MBE deposition system for nitrides consists essentially of a conventional MBE chamber, but with added equipment such as a compact RF source or a compact ECR source (which more or less gave way to RF relying on inductively coupled plasma (ICP)) and ammonia, as shown in Figure 3.58. These compact RF nitrogen sources are small enough to be mounted on one of the effusion cell port flanges. Some systems are equipped with both RF and ammonia sources such as the ones available in author’s laboratory for maximum versatility. RF sources operate at several frequencies such as 13.56 MHz, paving the way for compact sources, which if desired produce mostly neutral N radicals with very little if any of the ionic nitrogen. The composition of the N plasma is important and its affect on layer quality and the required growth conditions are discussed later in this section. As alluded to earlier, the reactive nitrogen can also be supplied to the surface from an ammonia

3.4 Nitride Growth Techniques

Top view

B

A C

First layer adatoms Second layer Second layer, rest atoms Third layer, including dimers Fourth layer A

Side view B

C

(a)

Figure 3.57 (a) Atomic structure of the 7 · 7 surface reconstruction seen on clean Si(1 1 1) surfaces in the dimer adatom stacking (DAS) fault model. The unit cell contains two triangles with six adatoms per triangle. The stacking sequence in the left triangular subunit is faulted and that in the other triangular region is normal. This is by far the most complicated surface reconstruction and involves four layers. In the top view the atoms closer to the surface are indicated with the largest filled circles and so on. The atoms on the fourth layer are the bulk atoms and the rest are those of the reconstructed structure. The model contains dimers (depicted

with smaller open circles along the two triangles), a stacking fault layer (indicated as second layer atoms in the left triangular subunit), and adatoms on the top layer (depicted as the first layer adatoms). In the side view, atoms on the ð1 0 1Þd lattice plane along the long diagonal of the 7  7 unit cell are represented with larger circles than those behind them. The observation of this surface reconstruction is used as pivotal test whether the Si surface is clean and ready for epitaxy. Patterned after Ref. [274]. (b) A scanning tunneling miscopy image of a clean 7  7 surface reconstruction of Si(1 1 1). Courtesy of R.M. Feenstra.

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j 3 Growth and Growth Methods for Nitride Semiconductors

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source [275,276]. Various ICP-activated RF sources, such as EPI (Veeco) Vacuum Products model Uni-Bulb, Oxford Applied Research models MPD21, CARS-25, and HD25, SVT Associates, Addon, Eiko Co., and Taisei Industry are used for PAMBE, details of which in terms of N species can be found in Ref. [83], but caution should be exercised in that the characteristic of these sources have changed continually with time owing to improvements. The PAMBE method possesses the basic nonequilibrium features of the molecular beam epitaxial growth technique and as such is well suited for growth of nitrides with large lattice mismatch with the substrate, large vapor pressure of N over the metal component, and so on. InN is one that provides the most challenge in this regard and PAMBE is reasonably well suited for its growth; refer to Section 3.5.15 for details. It should be briefly mentioned that highly In-rich conditions are used for InN growth, resulting in N-polarity. Owing to N2 activation by plasma discharge, the intensity of activated nitrogen flux does not depend on the growth temperature, TG, which is also described by Ts, which is the substrate temperature, allowing the operator to choose the optimum temperature independently in a wide range in regard to adatom surface diffusion and stoichiometry conditions required. It may also be that moderate levels of ionic N present in at least some of the plasma sources could be beneficial to growth at very low temperatures such as those used for InN. In a PAMBE system, the RF nitrogen flux through a plasma source is controlled best with a flow meter with typical flow rates giving rise to a system pressure of

RHEED gun Substrate

Liquid nitrogen shroud

Effusion cells

Optical spectrometer

RF in Cooling water in

Vent

Shutters

Mass flow controller

Phosphor screen

Manometer CCD camera

Purifier N2 in

Figure 3.58 Schematic diagram of an RF MBE system used for nitride growth at Virginia Commonwealth University.

Gate valve

3.4 Nitride Growth Techniques

anywhere between 2 · 10
6 and 1 · 10
4 Torr, the actual value being dependent on the particulars of the system in terms of pumping arrangement and system temperature. The RF plasma sources can be operated at power levels between 90 and 500 W and pressures of 10–100 Torr in the plasma chamber itself. Well-designed RF sources can have cracking efficiencies as high as 10%, though in most operating conditions, the efficiency is about 1%. In both ammonia and RF-activated growth, typical pressures are in the low 10
5 Torr regime, which can be construed as a nearly collision-free growth mode. In newer RF sources, the emission wavelength of the plasma concurs with the assertion that only the nonionic species with low energies, about 2 eV, are generated in large quantities. This energy is well below the 24 eV damage threshold energy predicted for GaN [277]. The same, however, does not necessarily hold for ECR sources in which the kinetic energy of the nitrogen ions generated can be sufficiently high and could result in surface atom displacement [278]. The plasma source in PAMBE activates chemically inert ground-state nitrogen molecules ðN2 ðX 1 Sgþ; v ¼ 0ÞÞ by converting them into a combination of atomic nitrogen and metastable nitrogen molecule and ionic nitrogen atom and molecule. Spherically, the activated species contain nitrogen atoms, N, (N ¼ 4 S; 2 P; 2 D) and vibrationally excited states (N2 ðvÞ ¼ N2 ðX 1 Sgþ ; v„0Þ), electronically excited states 00 (N 2 ¼ N 2 ðA3 Suþ ; vÞ and N2 ¼ N2 ða Sgþ Þ), ionized molecules (N2 þ ¼ N2 þ þ 2 þ ðX Su ; wÞ), ionized atoms (N ), and neutral atoms at different states, with ion energies up to tens of electron volts [279,280]. The metastable nitrogen molecules Q Q Q are also designated A3 Suþ ; B3 g ; a1 g ; C3 u [281–283]. Both N and N2 ðA3 Suþ Þ have potential energies considerably higher than the required Gibbs free energy for the GaN synthesis [283]. In short, the species in the activated nitrogen flux include nitrogen atoms (N), excited molecules (N2 þ ), and ionic species (N2 þ , N þ ) [284]. Among the N2 þ species, the N2 ðA3 Suþ Þ variety has sufficiently long lifetime to reach the substrate and is therefore considered the metastable molecular species participating in growth [285]. The cracking efficiency, the rate of transforming inert N2 into active nitrogen species, can be as high as 10% in well-designed RF nitrogen sources, though the efficiency is about 1% in most operating conditions. N2 is activated by plasma discharge, which depends on the RF plasma power and is not affected by the growth temperature Ts [286]. Nitrogen atoms have very long lifetimes in the plasma chamber, on the order of seconds, owing to the fact that the formation of a nitrogen molecule by the collision of two nitrogen atoms requires an additional interaction with a third body, such as the source walls [287]. This would lead to an enrichment of the plasma with nitrogen atoms until a steady state is reached which includes collision with the walls. Modern RF sources are indeed capable of producing reactive nitrogen-rich plasma with atomic nitrogen leading to very high GaN growth rates [288–290]. A comparison of different models of RF sources indicates that the increase in growth rates is proportional to the available amount of atomic nitrogen [288,289]. These results establish the importance of nitrogen atoms in addition to the N2 ðA3 Suþ Þ metastable

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j 3 Growth and Growth Methods for Nitride Semiconductors

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molecules in the growth process [288]. The big picture involving the RF plasma sources indicates that the temperature of the walls of the RF discharge zone dramatically affects the ion content of the plasma, which decreases to a very low level at temperatures above room temperature [289]. The ion content is also reduced with reduction in the size of the holes in the PBN aperture [282,291], whereas their number can be increased to achieve a sufficient flux of reactive nitrogen species. Finally, the experimental results suggest that the input power and gas flow may be used to control the composition of the nitrogen plasma beam. The levels of atomic nitrogen increase with increase in the input power and decrease in the N2 gas flow [292,293]. However, a similar behavior reported for ionic species [285] is explained by enhanced recombination at high nitrogen pressures in the plasma, owing to an increased probability of collisions. Given the plethora of options for sources producing reactive N through plasma activation, a number of studies have been undertaken to determine the most suitable option for PAMBE. In general, ECR sources produce mainly excited molecular species, consisting of second positive series excited N2þ (metastable molecular nitrogen) and ionized N2þ (ionized nitrogen molecule) with little atomic nitrogen. On the contrary, the early versions of RF sources typically produced high densities of atomic nitrogen and excited molecular nitrogen of the first positive series [294], as confirmed by mass spectroscopy analysis [295]. Later varieties of RF sources could indeed produce mainly metastable molecular nitrogen, which has been reported to be the preferred species for growth, as discussed in Section 3.4.2.9.3. Optical emission spectra of RF sources are very useful in determining, to a good extent, the nitrogen species responsible for growth. Shown in Figure 3.59a and b are the emission spectra from two representative RF plasma sources operating at an input RF power of 450 W and system pressures of, depending on the system employed, 5 · 10
6 Torr (for Addon source) and 5 · 10
5 Torr (for Veeco-EPI). Relative to ECR sources, the RF source emission is primarily neutral and consists of peaks associated with molecular (first-positive and second-positive series) and atomic species. The first- and second-positive molecular nitrogen emissions occur near orange–red and UV wavelengths, respectively. The atomic nitrogen related emission, however, occurs near red (infrared (IR)) wavelengths of 745, 821, and 868 nm. One particular source, labeled as 1 in Figure 3.59, exhibits the strongest emission lines associated with atomic species as well as the first-positive molecular species. This source is capable of producing GaN growth rates of more than 1 mm h
1 at substrate temperatures as high as 800  C. As mentioned previously, N atoms and excited N2 þ molecules are the species that primarily participate in the growth process. A detailed discussion of particulars of each of the available sources has been provided by Georgakilas et al. [83]. RF sources are generally void of optical emission transitions related to molecular ions. However, a N2þ transition at 391 nm was observed for low-pressure discharge. The ionized atomic N lines at approximately 776 nm (much suppressed in later varieties) and 854 nm were observed in a SVTA RF source, whereas a N þ line at 648.2 nm was observed in an Oxford HD25 RF source. Moreover, mass spectroscopy experiments showed that the atomic ion flux was typically two to three times larger than that of the molecular ions for an Oxford

3.4 Nitride Growth Techniques Atomic nitrogen

2000

750 nm

Intensity (counts)

Molecular N2

1600

Second positive molecular series

870 nm

First positive molecular series 590 nm

1200

357 nm

820 nm 580 nm

650 nm

800 400 0

-6

Addon: 450 W 6 x 10 Torr

200

400

600

(a)

800

1000

1200

1400

Wavelength (nm) Veeco RF power = 450 W

750 nm

Intensity (au)

870 nm

820 nm 650 nm 580 nm

200

(b)

400

600

800

1000

1200

Wavelength (nm)

Figure 3.59 (a) Optical spectra obtained from a nitrogen RF source manufactured by Addon using two difference detectors with overlapping spectral responsivities (190–860 nm for one and 640–1300 nm for the other), together covering the entire spectral range of interest. Note that molecular (both second and first positive series) and atomic nitrogen related transitions are seen (RF power: 450 W and the nitrogen flow is such that a pressure of 10
6 Torr is obtained).

(b) Optical spectra of a source manufactured by Veeco under similar conditions. The aperture plate is equipped with 200 holes, each about 250 mm in diameter. Note the absence of second positive molecular series and the higher intensity of the first positive molecular series being much smaller than that for atomic nitrogen. The intricacies of radicals of molecular nitrogen versus atomic nitrogen being desirable are discussed in Section 3.4.2.9.3.

CARS-25 source [298]. The reader should be cautioned that these sources continually evolve and the statements made here may or may not apply to some later varieties. By comparing the peak intensities of various species in the spectra, the relative composition of the radio frequency plasma nitrogen source can be obtained. Figures 3.60 and 3.61 show the relationships between the intensity of atomic nitrogen

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5000 745 peak

Intensity (au)

4000

868 peak

3000

821 peak

2000

1000

0

250

300

350

400

450

RF power (W)

Figure 3.60 Intensity of nitrogen RF spectrum peaks versus RF power. The system pressure is 9 · 10
6 Torr.

peaks with RF power and system pressure, respectively. These results suggest that the input power and nitrogen flow could be used to control the relative composition of the nitrogen plasma. Figure 3.60 shows that the intensity of atomic nitrogen increases linearly with input power. Figure 3.61 shows that the intensity of atomic nitrogen changes with the system pressure, which is controlled by nitrogen gas flow. As a further note, the intensity of atomic nitrogen increases with the nitrogen gas flow when the chamber background pressure is low. However, when the pressure is above 3.0 · 10
5 Torr, the intensity of atomic nitrogen peaks starts to decrease. A similar trend has also been reported for the ionic species. This behavior is explained by enhanced recombination at high nitrogen pressures in the plasma, owing to increased probability of collisions.

Intensity (au)

868 peak 745 peak

821 peak

10

20

30

40

50

–6

System pressure (10 Torr) Figure 3.61 Intensity of nitrogen RF spectrum peaks versus system pressure. The system pressure here should be treated as flow rate whose actual value depends on the system geometry and pumping speed. The RF power is 350 W.

60

3.4 Nitride Growth Techniques

Although practically not used anymore, early deposition of GaN in the realm of MBE was performed with the help of ECR sources, which are notorious for generating a good deal of ionic N species, which could cause damage. To determine the detrimental effects of the higher ion content of an ECR source on the GaN material properties, studies comparing ECR and RF-activated nitrogen sources have been undertaken [296]. In one such study, higher quality GaN layers were consistently grown with an Oxford MPD21 RF source at growth rates higher than those with an ASTeX ECR source [294]. The crystal damage in layers grown with the ECR source was reported to be comparable with that associated with ion implantation [297]. Optical admittance spectroscopy revealed distinct transitions in n-type GaN grown with an RF source as opposed to broad bands in the ECR grown samples, attributed to potential fluctuations caused by structural inhomogeneities induced by the ECR source [296]. Finally, yellow PL emission at about 2.2 eV appeared in the photoluminescence spectra of layers grown with an Irie ECR source, whereas it could be reduced in layers grown with an Eiko RF source [295]. All these GaN layers produced with ECR sources are consistent with point defects induced by plasma damage in the GaN layers [296]. Nitrogen atoms have five valence electrons and bond with all group III elements without any potential barrier [283,298,299]. The N2þ excited molecules interact with a solid-state surface through an exothermic dissociative chemisorption mechanism [300]. Therefore, N atoms and excited N 2þ molecules participate in III-N growth by RF-assisted PAMBE [281,298,301]. In contrast, ionic species of plasmaactivated nitrogen flux (N2þ , Nþ ) are responsible for ion-surface processes, such as adatom–vacancy pair generation, displacement of atoms, embedding of N2 molecules, and surface atom sputtering [302,303]. These are believed to have adverse effects on the epitaxial film quality, which in extreme cases could lead to even polycrystalline growth [304]. The damage threshold for GaN has been estimated at 18–24 eV [305]. However, the threshold ion density and doses of the energetic species for generating point defects and dislocations are not yet available. On the basis of the bond strength, one can surmise that the threshold ion energy would be larger for AlN and smaller for InN as compared to GaN. In spite of this, early attempts to grow GaN in vacuum environments utilized N2 and Nþ ions because of their availability [93]. In addition, NH3 þ ions with 30 eV energy and 3  1014 ions cm
2 s
1 current density were also utilized in ion-enhanced gas source MBE [306]. One should, however, be cautious as even a small flux (1012 cm
2 s
1) of relatively high-energy N2 ions (>20 eV) could cause surface damage. 3.4.2.8 Reactive Ion MBE Several laboratories have attempted MBE growth of the group III nitrides using nonplasma-based growth techniques. The most successful of these techniques utilized low-energy nitrogen ions from a Kaufmann ion source to grow GaN of a quality comparable with that grown with ECR sources. Several laboratories in the past have tried reactive ion molecular beam epitaxy in which N2 or NH3 was decomposed on the substrate surface [307,308]. However, decent growth rates and material quality were not obtained at MBE substrate temperatures. Two possible extensions of the

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original RMBE approach, which were thought to reduce the required substrate temperature, are the application of hydrazine and cracked NH3 as the source gas, which could be accomplished by a commercial hydride cracker. Hydrazine is believed to increase the reactivity of the nitrogen-containing species. Much effort on thermal cracking of ammonia turned out to be misguided for several reasons. Thermal cracking requires high temperatures with its expected complications that reduce this approach to a laboratory curiosity. In addition, the little atomic N produced recombines readily through collisions forming very inert and stable N2. Needless to say, this approach is not seriously pursued anymore. The use of hydrazine and the likes thereof raise safety concerns and is a source of contamination from the water vapor. Ammonia cracked only at the surface of the substrate is the successful approach and has been employed in various laboratories, as will be outlined later. High-quality layers grown on sapphire substrates by MBE permitted the fabrication of a variety of devices, such as Schottky diodes [309] and MODFETs [310,311], multiple GaN/AlGaN and InGaN/AlGaN quantum wells [312,313], and the observation of photo-pumped-stimulated emission at 300 K with a low pumping intensity [314]. In time, all MBE-grown lasers with ammonia N source on freestanding HVPE templates were prepared [315]. Growth by MBE differs greatly on the basis of whether layers are grown on a foreign substrate such as Si, sapphire, SiC, and ZnO, or on GaN, either in the form of a template or a bulk platelet. Growth on a foreign substrate generally requires high-temperature growth as low-temperature growths generally lead to good surface morphologies and large concentrations of edge dislocations propagating along the c-direction. Although substrate temperature measurements are not reliable, what is implied here is that temperature of 800  C and above would be high, whereas 650–750  C would be medium. Note that growth on sapphire and other foreign substrates may commence with high- or low-temperature nitridation and low-temperature buffer layers [316]. For growth on already existing GaN templates, the issues are very different in that the dislocation reduction measures and growth nucleation on a foreign substrate are not the main concerns. What becomes important then is the optimization of parameters such as group V/III ratio and growth temperature for optimizing properties such as mobility [317]. In this case, it is universally agreed that medium-temperature growth under appropriately Ga-rich conditions leads to best surface morphologies and mobilities [318]. 3.4.2.9 Principles of RMBE and PAMBE Growth The hallmark of the MBE approach, as applied to conventional group IV and group III–V semiconductors, is that it provides very reliable laboratory investigations of basic growth processes. The case of nitrides contradicts this long-standing status somewhat, in that little work has been published on modeling of group III-N growth. The lack of reliable data on surface migration rates, nucleation parameters, and sticking coefficients of group III and N atoms to different substrate surfaces, as well as other thermodynamic and kinetic parameters, makes quantitative estimates of growth parameters difficult. To overcome kinetic barriers in effect in the pursuit of epitaxy involving conventional compound semiconductors, the optimum substrate temperatures employed

3.4 Nitride Growth Techniques

fall in the range 1/2 to 2/3 of the melting temperature of the semiconductor (see Section 1.2.3 for details). In the case of III nitrides, however, synthesis occurs, particularly with MBE, under conditions in which the substrate temperature is much below the empirically acquired figure of 1/2–2/3 of the melting point, and thus, GaN growth is not attempted under thermodynamical equilibrium conditions [301]. MBE growth conditions employed for GaN epitaxy fall within an area of the phase diagram where the liquid Ga and molecular nitrogen gas have the lowest free energies [301]. As mentioned above, the Ga diffusion barrier makes the species mobile. The metastable growth process involves a balance between GaN formation (forward reaction such as those depicted by Equations 3.54–3.56) and GaN decomposition (reverse reaction such as those depicted by Equations 3.15–3.17) [301]. In order for net growth to take place, the rate of GaN formation must be larger than the rate of its decomposition, which is the case under the growth conditions employed, as the kinetic barrier for decomposition is large owing to the strong metal–N bonds at the surface (see Table 3.12). Practical limitations on the maximum reactive nitrogen flux attainable are such that the substrate temperature of GaN during MBE growth is about 20–35% (600–900  C) of GaN melting temperature (TM  2500  C, see Section 1.2.3 for details). Sections 3.4.2.7 and Sections 3.4.2.8 discuss the first-order guiding principles for reactive MBE growth with ammonia as the nitrogen source and PAMBE with plasma-activated nitrogen as the source. 3.4.2.9.1 Growth by RMBE Ammonia was initially thought not to be compatible with MBE systems as it reacts with metals such as copper used in the flange and valve seals, which is a simple problem and has been dealt with effectively. The other concern is that hydrogen released from ammonia makes hot metal components fragile/brittle. This has also been dealt with effectively as the amount of ammonia used is small and proper shielding methods can be employed to mitigate the potential damage. Another apprehension about NH3 is its reactivity with oxygen. This does not seem to be a problem as the vacuum systems are UHV type and the oxygen background is negligible. The technique utilizing ammonia as a nitrogen source has been termed the reactive MBE (RMBE). Basically, the RF nitrogen plasma source shown in Figure 3.58 gets replaced with an ammonia injector maintained at sufficiently high temperature to avoid ammonia condensation as the shroud walls are cold. In RMBE, ammonia is decomposed only on the surface of the substrate by pyrolysis. As such, the lower end of the temperature range where this technique can be utilized is about 600  C, albeit with very low growth rates. Temperatures somewhat below this can be used, but the ammonia flow rate must be increased by some 10-fold as done in growing initiation layers on sapphire substrates. Growth rates of about 1–3 mm h
1 can be obtained at substrate temperatures between 750 and 850  C while maintaining a high quality. This growth temperature is much lower than the growth temperature in OMVPE. Kim et al. [319] showed that RMBE can produce films of a very good quality. Although applicable to all substrates, this particular study was performed on sapphire substrates. The actual growth is preceded with a nitridation and a thin AlN buffer layer. In this approach, the substrate temperatures varied between 610 and 820  C. Growth

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Cracking efficiency of NH3 (%)

5 4 3 2 1 0 200

300

400

500

600

700

800

900

Temperature (ºC) Figure 3.62 Cracking efficiency of ammonia as a function of substrate temperature [320].

rates up to 2.9 mm h
1 were obtained for temperatures between 725 and 820  C. These growth rates are quite comparable with those produced by conventional OMVPE. Studies of the growth kinetics have been carried out by employing various V/III ratios and substrate temperatures. It was found that the layer quality increases and compressive stress decreases as the N/Ga ratio increases. It should be pointed out that incorporation of residual impurities as well as the creation of a high density of native defects is made less efficient at higher N/Ga ratios. The growth mechanism of GaN in RMBE involves thermal cracking of ammonia and surface reaction of N with Ga atoms to form GaN. Monatomic nitrogen becomes available on the film surface by pyrolysis of the ammonia in a temperature-dependent process. The efficiency of ammonia dissociation (shown in Figure 3.62) has been measured directly by using a mass spectrometer while varying the substrate temperature [320]. The growth rate as a function of ammonia flow rate was also measured at a substrate temperature of 830  C, as will be discussed shortly. The efficiency of ammonia dissociation increases slowly above 600  C being only about 4%. Not all of nitrogen of course is incorporated into the film, and above 700  C, GaN decomposition becomes a consideration contributing to the nitrogen radical flux. In light of this fact, the growth rate versus ammonia flow becomes a more important parameter. Group III–V limited growth regimes can be found from the growth rate parameter as shown above [320]. The growth rate of GaN was greatest at a nozzle temperature of 300  C and then decreased at a nozzle temperature of 400  C for a 100% ammonia beam. Seeded and anti-seeded beams did not attain as high growth rates. A nitrogen incorporation efficiency of 12% was derived for the supersonic beams [326]. Nitrogen incorporation efficiency reported by others include 16% [321], 10% [322], and 4% [320], as shown in Figure 3.62. The nitrogen incorporation efficiency for low-energy ammonia from a leak valve was derived as 26%, suggesting that dissociative chemisorption of ammonia on GaN(0 0 0 1) occurs via a precursor-mediated pathway. Because the

3.4 Nitride Growth Techniques

ammonia sticking coefficient does not increase with increasing kinetic energy, it is concluded that ammonia dissociative chemisorption is inactivated [326]. Substrate temperature and incident Ga flux affect the amount of Ga available on the surface for incorporation into the film. For constant ammonia pressure, there is a substrate temperature below which Ga will condense on the surface and above which all the incident Ga will desorb. This temperature is clearly defined and has been measured by RHEED and desorption mass spectroscopy (DMS). If a surface is exposed to ammonia in the absence of Ga and then exposed to Ga in the absence of incident ammonia, Ga will react to form GaN, and hydrogen will be desorbed. If the Ga flux is continued after all nitrogen is consumed, Ga coverage will occur. Below the condensation temperature, Ga multilayers and/or Ga droplets will form. Above the condensation temperature, less than a monolayer of weakly bound Ga (sitting on preferential sites of the underlying GaN lattice) will accumulate until the Ga flux is stopped, at which time it will desorb [323]. Adsorption of ammonia and Ga behaves differently for N- and Ga-face polarity films. N-face polarity has two types of adsorption sites: strongly bonded Ga on a nitrided surface and weakly bonded Ga on a gallided surface. Ga-face polarity only adsorbs weakly bonded Ga [324]. Ga-polarity films have been observed to give both (1 · 1) and (2 · 2) reconstructions by RHEED depending on the Ga-to-ammonia flux ratio where the transition between them is activated at 2.76 eV. It is concluded that growth on Ga-face polarity occurs by step flow because Ga adsorption is only detected when there is a coincident ammonia flux. Furthermore, no RHEED intensity oscillations could be observed for growth on Ga-face polarity [324]. Exposure of a nitrided N-face polarity film to less than a monolayer of Ga flux results in featureless steps. When this surface is exposed to ammonia again, irregularly shaped islands form [324]. The growth rate can be limited by weakly adsorbed Ga in the regime with excess Ga on the surface. A Ga-terminated surface with featureless terraces can be achieved by quenching after the Ga shutter is closed. Held et al. [325] have concluded that Ga accumulation reduces the growth rate in the ammonia-limited regime by blocking ammonia chemisorption sites. Additionally, GaN surface morphology is determined by the growth regime. Ga-limited growth displays rough highly faceted surfaces with hillocks as will be discussed shortly. On the contrary, ammonia-limited growth produces smoother, nonfaceted films. It is suggested that surface roughness under Ga-limited growth is because of a Ga adatom diffusion length of only a few atomic diameters. Stochastic roughening, originating from statistical fluctuations in the Ga arrival flux, will dominate under these conditions. In ammonia-limited conditions, the V/III ratio affects the surface morphology. As the V/III ratio is decreased, the surface roughness decreases eventually forming pits and then Ga droplets [326]. To investigate the growth kinetics [319], the substrate temperatures were varied in the range of 700–820  C with an ammonia flow rate of 45 sccm while keeping the Ga flux at about 3 · 1015 cm
2 s
1. These early investigations of growth rate and its relationship with ammonia flow were followed by additional experiments [320], all of which clearly indicate that growth rate for constant Ga flux at a given temperature increases initially with ammonia flow and saturates, as displayed in Figure 3.63. Although the scatter in the data is quite substantial, a weak trend of increase in the

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1.4 NH3-limited Ga-limited

Growth rate (µL–1)

1.2 1.0 0.8 0.6 0.4

T growth = 830ºC 0

10

20

30

40

50

NH3 flux (sccm) Figure 3.63 Variation of the growth rate with the ammonia flow at a substrate temperature of 830  C. The growth rate increases with ammonia flow rate initially and saturates, indicating of ammonia limited growth rate for lower ammonia flow rates and Ga limited growth rate for higher ammonia flow rates. After Ref. [320].

growth rate with increased ammonia flow rate before reaching saturation is apparent. This is similar to the case of plasma-assisted MBE as discussed in 3.5.b.3.ii and shown in Figure 3.69. It can be argued that, at low ammonia flow rates, the Ga desorption is responsible for the reduced growth rate, implying ammonia flow limited growth regime, which represents the Ga-rich growth conditions. It should be mentioned that these growth temperatures are sufficiently high for efficient Ga desorption unless Ga is adsorbed through bonding with N. At high ammonia flow rates, the growth rate is limited by the Ga arrival rate, which is responsible for the apparent saturation. This regime is the Ga-limited regime or N-rich growth conditions. At much lower growth temperatures, for example, 700  C, not shown, the growth rate is much smaller because of the lower ammonia cracking efficiency, supporting the Ga-desorption argument. A GaN equivalent flux can be calculated by dividing the product of the GaN density, growth rate, and Avogadro’s number by the molecular weight of GaN. Incident ammonia flux is the ammonia throughput divided by the area of the ammonia beam at the sample position. The nitrogen incorporation efficiency can be calculated by dividing the equivalent GaN flux by the incident ammonia flux [326]. The effective nitrogen flux is determined by the incident ammonia flux and the sticking coefficient for dissociative chemisorption (aNH3 ). The Ga sticking coefficient is very close to unity, so the effective Ga flux is approximately equal to the incident flux [326]. Ammonia incident kinetic energy affects the GaN surface morphology indirectly via changes in the V/III flux ratio. This has been studied by using ammonia-seeded supersonic molecular beams as a monoenergetic source of ammonia and group III precursors for MBE growth. The seeded molecular beam allows the incident kinetic energy of the precursors to be precisely controlled using a heated nozzle. The growth

3.4 Nitride Growth Techniques

rate should be enhanced if ammonia dissociative chemisorption is direct and activated. If an activation barrier to chemisorption exists, then the substrate temperature can be lowered as incident kinetic energy of the precursor is used to overcome the barrier. However, if ammonia chemisorption is inactivated and occurs via a precursor-mediated channel, the growth rate is expected to decrease with increasing ammonia kinetic energy [326]. Based on the assumption that no desorption takes place from chemisorbed states, a growth kinetics study of GaN grown by reactive ion molecular beam epitaxy (RIMBE), which is also applicable to RMBE with some modifications, was carried out by Powell et al. [93]. In this model, competitive contributions (associated with site selection) of collision-induced dissociation followed by chemisorption of nitrogen and ion-stimulated Ga desorption were considered to be the primary mechanisms that affect the growth. The model was successful in demonstrating that, at a constant substrate temperature and a constant nitrogen-to-Ga flux ratio (J N2 þ J Ga ), the growth rate decreases with increasing ion energy. However, the model could not account for the experimentally observed drop in the growth rate at high substrate temperatures. For an RMBE study, the same model can be used with some modifications in parameters and activation energies for the surface-state transitions. The equations describing the rate of change of the number of physisorbed and chemisorbed Ga atoms on the surface may be given as 0

dnGa =dt ¼ J Ga
ðnGa =tÞ
knGa ð1
qÞ þ k N s q;

ð3:47Þ

N s ðdq=dtÞ ¼ knGa ð1
qÞ
k0 N s q
f q J NH3 ;

ð3:48Þ

where k is the forward reaction rate constant from the physisorbed state to the chemisorbed state, k0 is the backward reaction rate constant from the chemisorbed state to the physisorbed state, t (describes Ns) is the average residence time of free Ga prior to thermal desorption, J NH3 is the ammonia flux, and f is the fraction of ammonia that is thermally cracked and reacts with Ga to form GaN on the growing surface. To correctly describe the nature of growth, the last term on the right-hand side of Equation 3.48 is modified to f qJ NH3 for RMBE from 2 f q J NH3 , originally formulated for RIMBE. The multiplication factor of 2 was reduced to 1 because each ammonia molecule in RMBE can provide only one nitrogen atom. The parameters used in Equations 3.47 and 3.48 are defined as k ¼ n0 expð
E p =kB TÞ;

ð3:49Þ

1=t ¼ n0 expð
E d =kB TÞ;

ð3:50Þ

0

k ¼ n0 expð
E c =kB TÞ;

ð3:51Þ

f ¼ expð
E p =kcr TÞ:

ð3:52Þ

All parameters other than those defined by Equations 3.49–3.52 are the same as those used in Powell’s model. The solution of Equations 3.47 and 3.48 (at steady state with the same values for the various parameters, as given in Powell’s model) predicts

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a decreasing growth rate as the substrate temperature increases from 650 to 850  C for a constant Ga flux and a constant V/III ratio. Another calculation for a constant Ga flux (3 · 1015 cm
2 s
1) and a constant substrate temperature (800  C) reveals an increasing growth rate as the V/III ratio becomes larger. These results are exhibited in Figure 3.64a and b. Jones et al. [327] and Evans et al. [328] reported on an empirical value for Ga desorption in the growth of GaN with gas-source MBE (GSMBE), which essentially follows the same growth scheme as used in the present experiment. The desorption 0.5 (a)

Growth rate (µm h–1)

0.4

0.3

0.2

JNH3 /JGa = 4.3

0.1

15

JGa = 3.096 x 10

0 650

700

–2 –1

cm s

750

800

850

Substrate temperature (ºC) 1.4 (b)

Growth rate (µm h–1)

1.2 1 0.8 0.6 0.4 T s = 800 ºC 15 –2 –1 JGa = 3.096 x 10 cm s

0.2 0

1

10

100 V/III ratio (JNH3/JGa)

1000

Figure 3.64 Calculated growth rate from the simple precursormediated chemisorption model. (a) Growth rate change with substrate temperature (V/III ¼ 4.3, JGa ¼ 3.096 · 1015 cm
2 s
l) (b). Growth rate change with V/III ratio (Ts ¼ 800  C, JGa ¼ 3.096 · 1015 cm
2 s
1). The solid line indicates the calculations whereas the circle represents the measured data.

10 4

3.4 Nitride Growth Techniques

energy for Ga atoms obtained by temperature-programmed desorption (TPD) was published to be 1.9  0.2 eV. In addition, Burns et al. [329] measured the thermal desorption energy of Ga on Al2O3 by means of a thermal desorption spectroscopy (TDS) technique and deduced a value of 2.05 eV. Between these two measured values, the 1.9 eV, according to Evans et al. [319] is deemed more applicable because of similarities in the growth schemes employed in the RMBE investigations. The growth rate has been calculated with 1.0 and 0.25 eV, respectively, for the parameters Ec and Ecr . The activation energy for the transition from the physisorbed state to the chemisorbed state was chosen to be zero because the sticking coefficient of Ga is generally considered to be 1. In Figure 3.64, the calculated growth rate data for 12 and 25 sccm of ammonia flow are compared with experiments. Even though there are differences in magnitude between the experimental data and the calculated values for the 12 sccm ammonia case, the calculated data follow the same qualitative trend of growth rate versus substrate temperature as the experimental data. More specifically, the calculated results also present growth rates increasing with temperature because of more efficient thermal cracking of ammonia in the temperature range where Ga desorption is not the limiting factor. Considering that the effective amount of ammonia molecules is proportional to the ammonia flux and assuming that the cracking rate depends only on temperature, it is quite reasonable to expect that the growth rate of the sample grown with the lowest ammonia flux (4 sccm) would decrease rather drastically with decreasing substrate temperature. That this is the case becomes apparent from Figure 3.64 for ammonia flow rates below 12 sccm. If the substrate temperature was not sufficiently high to produce as much or more reactive ammonia than that necessitated by the given surface concentration of Ga, the growth rate would not be determined by the ammonia flow rate. Instead, it would be determined by the Ga population on the surface and would remain constant with changes in ammonia flow rate for a given temperature, unless there is a change in Ga flux. Additional experiments carried out at 800  C point to the growth rate remaining more or less constant, independent of the ammonia flux. However, one can conclude that the growth rate is determined by the ammonia flow rate if the substrate temperature is not sufficiently high to crack substantial amounts of the ammonia supplied. In the present case, this is a temperature below 750  C. It is to be noted that Ga desorption from the surface becomes the limiting factor at higher substrate temperatures. Consequently, the growth rate decreases despite more efficient ammonia cracking. For example, a significant drop in growth rate was found at 820  C for the sample grown with 25 sccm ammonia (Figure 3.64). Furthermore, in this growth regime, the growth rate is constant at a specific temperature, regardless of the ammonia flux unless it is lower than the Ga flux. The trend in growth rates was observed using several different flow rates of ammonia. When the substrate temperature was increased to 850  C, the growth rate practically became nil. The incorporation kinetics of gallium during GSMBE of GaN using elemental Ga and NH3 gas as source materials was studied by Jenny et al. [330]. Desorption mass spectroscopy (DMS) was applied to perform in situ quantitative measurements of GaN formation, Ga desorption, and Ga surface accumulation during growth. The

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110 100

s (Ga)

NH3 (Ji) (x10–7 Torr BEP)

90 0.9

80

GaN Growth

0.6

70

0.3

60 50

GaN growth + Ga desorption

40 30

0.1

20

GaN growth + Ga desorption + Ga accumulation

10 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

–1

J i (Ga) (ML s ) Figure 3.65 Plot of the Ga incorporation ratio, s(Ga), contours as a function of Ga and NH3 fluxes. Labeled in the figure are the three primary regimes for Ga adatoms. The area above the heavy line represents the GaN growth regime. The dashed line delineates the regimes of GaN growth þ Ga desorption and GaN growth þ desorption þ accumulation. Reference [330].

rate of formation of GaN was reported as a function of incident Ga flux (0.1–0.75 ML s
1) and incident NH3 flux (1 · 10
7 to 3 · 10
5 Torr beam equivalent pressure (BEP)) at a growth temperature of 725  C. Three distinct growth regimes (Figure 3.65) were observed: (1) GaN is formed where all of the incident Ga flux is consumed; (2) part of the incident Ga atoms are consumed to form GaN, whereas the excess is desorbed by the surface; and (3) GaN formation coexists with desorption and surface accumulation of Ga. It was generally found that the Ga surface accumulation is inhibited by increasing the rate of incidence of NH3 and/or by decreasing the rate of incidence of Ga at this temperature. In addition, the order of the reaction between Ga and NH3 is determined to be unity, and this supports the validity of the Ga þ NH3 ! GaN þ ð3=2ÞH2 reaction. If the cracked-ammonia flux on the growing surface is insufficient as compared to the Ga flux, the surplus Ga will thermally desorb as substrate temperatures are high, in the vicinity of 800  C. Consequently, those Ga atoms that do not participate in the formation of GaN or desorb tend to migrate to surface sites with minimum surface energy. This could give rise to Ga clusters on the surface that appear to serve as preferred nucleation sites. It is then reasonable to argue that the larger population of Ga atoms near the cluster sites causes a locally accelerated growth. The immediate result of this is the appearance of huge hexagonal hillocks on the film surface. The proposed process is depicted schematically in Figure 3.66. The conclusion that can be drawn is that if the Ga-to-ammonia flux ratio is high, hillocks are likely to form on

3.4 Nitride Growth Techniques Large Ga flux Small NH3 flux

Growing surface

Clustering of surface Ga atoms Gathering of reactive ammonia by the concentration gradient of Ga on the surface

Hexagonal hillock formation

Appearance of highly strained area on the top of hillocks due to locally enhanced growth rate

Figure 3.66 A proposed pathway to hillock formation in the growth of GaN.

the surface. These features are confirmed with scanning electron images of GaN surfaces. A noticeable feature of hexagonal hillocks is that their tops appear to be composed of many overlapping triangles and etch pits. At higher temperatures and ammonia flow rates, the tips of the hillocks are etched giving rise to flatter surfaces. Growth on OMVPE templates with pits from screw dislocations has been shown to produce spiral hillocks, see Figure 4.11. The step edges on the hillocks have a terrace width that decreases in proportion to the supersaturation of ammonia. Terrace width was found to be independent of the Ga flux. It was found that depending on the growth rate, hillock growth can be minimized by using low ammonia supersaturation independent of the Ga flux or the desorbing H2 [331]. One growth model proposed suggests that hydrogen bonded with nitrogen adatoms suppresses their desorption from the GaN(0 0 0 1) surface. This model also shows that blocking of ammonia adsorption sites by Ga adatoms and ammonia radicals leads to unusual behavior of the growth rate as a function of substrate temperature and incident Ga flux for Garich conditions. Under this Ga-rich condition at a substrate temperature above 800  C, the surface is nearly free of adsorbed species. Under N-rich conditions at a low substrate temperature range, there was still the growth of GaN and the surface was found to be covered predominantly with ammonia radicals up to 1000  C [332]. RHEED was used to show the dependence of the substrate temperature on the growth mode. Under excess ammonia conditions giving a V/III ratio between 3 and 5, the RHEED oscillations were measured at substrate temperatures of 500, 600, 700, and 800  C. Contrary to some other experiments, well-defined oscillations were obtained at 500 and 600  C, but these were damped out at 700  C and no oscillations

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were obtained at 800  C. This shows that two-dimensional layer-by-layer growth can be obtained at low temperatures, but it changes to step-flow growth mode at higher temperatures [333]. Consistent with the notion that tips of facets can be etched for a smoother surface when high ammonia flow rates are used, Grandjean et al. [334] reported that the smoothest surface in their results had an RMS roughness value of 3.6 nm on a 5 · 5 mm2 AFM scan. This film was grown with an ammonia flow of 50 sccm corresponding to a BEP of 6.4 · 10
6 Torr. As the ammonia flow decreased, the film roughness increased. The film with the ammonia flow of 50 sccm also gave the highest PL intensity and the lowest carrier concentration. It was shown by SIMS that the oxygen and silicon concentrations might be the source of high background carrier concentrations because the concentration of these elements decreased with an increase in the ammonia flow of up to 50 sccm. Substrate temperatures can be 80–90  C higher when using ammonia as the nitrogen source rather than atomic nitrogen from plasma. Under N-rich conditions, blocking of nitrogen adsorption sites by ammonia radicals becomes essential and results in additional thermal stability of the GaN. Crystal growth carried out at a higher temperature is expected to enhance the overall properties of GaN [332]. This has been exhibited experimentally by Webb et al. [335] where optimal electron mobility was found at 900  C. The optimal substrate temperature for Ga incorporation was found to be 820  C at large ammonia overpressure owing to the competition between thermally activated pyrolysis of ammonia and re-evaporation of Ga from the surface [336]. GaN films grown by ammonia MBE on GaN buffer layers on sapphire were found to depend mostly on the temperature of the buffer. The optimal buffer growth temperature was 500  C for a 25 nm thick buffer. An annealing step of 900  C for 10 min was performed before growing 2–4 mm thick GaN films at 830  C. These conditions gave optimal PL and X-ray linewidths [337]. High-mobility GaN was achieved using substrate temperatures from 860 to 920  C and an ammonia flow rate of 50 sscm. These films were grown on an AlN buffer layer. The optimum mobility was found at a substrate temperature of 900  C and film thickness of 2 mm. It is attributed to increased grain size and defect reduction. Carbon doping was also employed to reduce the background carrier concentration to approximately 1.5 · 1017 cm
3 [335]. 3.4.2.9.2 Growth by PAMBE As mentioned in Section 3.4.1.2 in relation to OMVPE and Section 3.4.2 in relation to MBE, the growth process is a balance between incorporation and decomposition processes with conditions chosen so that the adsorption rate is greater than the desorption rate. It is a general notion that without the desorption process, if the growth temperature, for example, is very low, the quality of the resultant layers is not good. Here, this dynamic and competing process will be discussed in the framework of PAMBE. It should, however, be clear that the same also applies to RMBE. The evaporation of GaN has been studied by heating in vacuum. Guha et al. [250] reported a decomposition rate of three to four ML per minute at 830  C, whereas Grandjean et al. [222] utilized reverse RHEED oscillations as a measure of GaN evaporation to show that the decomposition rate is nearly zero below

3.4 Nitride Growth Techniques

Ga incorporation ratio

1.0 0.8 0.6 0.4 0.2 0.0 8.8

9.6

9.2

10 4/T (K –1)

10.0

10.4

Figure 3.67 Measured incorporation ratio behavior for Ga during GaN growth as a function of reciprocal growth temperature. The Ga arrival rate was 0.6 ML s
1 and the active nitrogen flux was 0.8 ML s
1 (Figure 3.2 from Ref. [250], reprinted with permission from Guha et al., Appl. Phys. Lett. 69, 2879 (1996). Copyright 1996, American Institute of Physics).

750  C, increases rapidly above 800  C, and reaches 1 mm h
1 at 850  C. On the surface, this may imply that the GaN decomposition is not an important consideration when the substrate temperature is below 750  C. However, GaN evaporation, meaning breaking already formed GaN bonds, and desorption and adsorption of impingent Ga and various N species are two different entities. As alluded to in Section 3.4.2.1, Guha et al. [250] investigated the lifetime of Ga adatoms on the GaN(0 0 0 1) surface with no impingent active nitrogen flux. A mass spectrometer mounted in direct line of sight to the wafer was used to monitor the desorbed Ga (mass 69) signal after each pulse of Ga. The resultant Ga lifetimes were in the range of 0.6–5 ns in the 685–750  C range and the activation energy for Ga desorption was E  2.2  0.2 eV, which is similar to the value of 2.5 eV for Ga desorption from a GaAs(1 1 1) surface [338] (also see Table 3.12). Incorporation ratio of Ga into the growing epitaxial GaN was also studied [250] by monitoring the reflected Ga signal detected by the mass spectrometer, the results of which are shown in Figure 3.67. The incorporation ratio was defined as 0

I½R0
RðTÞ R0 ;

ð3:53Þ

where R(T) is the reflected Ga signal at temperature T and R0 represents the total impingent Ga flux measured by observing the reflected signal at a high enough temperature where all of the incident Ga was desorbed. In Section 3.4.2, we discussed the general principles of adsorption, desorption, diffusion, and decomposition that are applicable to any growth, particularly growth in vacuum such as MBE. The implications of these processes as they apply to GaN by MBE are discussed here. The data presented in Figure 3.67 show that in the presence of active nitrogen flux, Ga adatoms have a finite surface lifetime during which they can diffuse on the surface through either incorporation or desorption [250]. It should be mentioned that this description is not applicable to growth conditions where two

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j 3 Growth and Growth Methods for Nitride Semiconductors

448

monolayers of Ga are present on the surface, in which case the Ga surface diffusion barrier is low and N surface diffusion barrier is high. Naturally for any growth to occur, the activation energy for desorption must be greater than the activation energy for surface diffusion, as discussed in Section 3.4.2.3. With increasing growth temperature, the residence time of Ga adatoms on the surface decreases. Consequently, the probability of encountering nitrogen adatoms for incorporation decreases, causing a drop in the incorporation ratio of Ga. At sufficiently low temperatures, the residence time of Ga adatoms is long enough for almost all of them to encounter nitrogen adatoms and incorporate into the epitaxial film. Although this results in a near unity incorporation ratio, the layer quality will not be as high as the selection process, which would provide a threshold for bonding, and impurity and defect incorporation is reduced. In addition to the work cited previously [250], investigation of the Ga desorption during growth of GaN has been reported by other groups [339–341]. Of particular relevance is the study by Hacke et al. [339] who studied the appearance of the 2 · 2 reconstruction on a GaN(0 0 0 1) surface with varying Ga flux and substrate temperature. Transition from 2 · 1 to 1 · 1 surface reconstruction was observed when Ga adatom density on the surface was increased. An increase in the substrate temperature required an exponential increase in the Ga flux to maintain the surface in the transitional region. This implies that the residence time of Ga adatoms on the surface is reduced; in general agreement with the data of Figure 3.67. As-determined activation energies for Ga desorption were between 2.22 and 3.25 eV with a weighted mean of 2.76 eV, which is similar to that of evaporation of metallic Ga. Although this results in a near unity incorporation ratio, the layer quality will not be as high as in the case for the selection process, which would provide a threshold for bonding and be accompanied by reduced impurity and defect incorporation. GaN growth by MBE can be conducted under Ga-limited (N-rich) conditions in which the growth rate is determined by the ratio of Ga arrival and desorption rates and under N-limited (Ga-rich) conditions in which the growth rate is determined by the ratio of arrival and desorption of reactive nitrogen on the surface at a given temperature. If one is far from the transition region, the growth rate can be made nearly independent of substrate temperature in a wide range, as has been amply discussed [342,343]. It follows then that the growth rate increases monotonically with increasing flux of reactive nitrogen species (JN) under N-limited growth conditions. In this growth regime for sufficiently low substrate temperatures (where Ga and N desorption is negligible), the fluxes of Ga and N satisfy JGa/JN > 1. Unless the excess Ga is thermally desorbed from the surface (GaN does not decompose) by employing a proper growth temperature, Ga droplets are formed on the surface, as shown in Figure 3.68. In the N-stable regime, the GaN growth rate is limited by the Ga flux, JGa, and naturally the growth rate increases monotonically with the Ga flux. This regime represents the N-rich growth conditions and occurs when JGa/JN < 1 for a given substrate temperature. This regime generally leads to surfaces with pits that gather the extended defects leading to relatively rougher surfaces as compared to that in Garich conditions. The details of the nature of defects and optical properties of each type

3.4 Nitride Growth Techniques

Figure 3.68 SEM micrograph showing the formation of Ga droplets with a density of 5 · 104 cm
2 on the surface of a GaN film that was grown under Ga-rich growth conditions.

are discussed primarily in Volume 1, Chapter 4 and Volume 2, Chapter 5, the latter in terms of the optical manifestation of defects. In an effort to gain insight into the kinetics of growth, Myoung et al. [340] have applied a precursor-mediated model with underlying physical processes remaining essentially the same as those by Guha et al. [250]. In this case, the Ga atoms arriving at the surface are assumed to be adsorbed into a mobile and weakly bound precursory state. These precursory Ga adatoms migrate on the surface, some being chemisorbed and some thermally desorbed. When the Ga atoms encounter atomic nitrogen in a chemisorbed state, the forward reaction occurs and GaN forms. Desorption from the chemisorbed state and decomposition of GaN were neglected, as the plasma sources used species lacking high kinetic energy, leading to the growth kinetics being determined by the JGa/JN ratio. Proposed model for physisorption of molecular N. A theoretical model for the MBE growth that accounts for a physisorption precursor of molecular nitrogen was proposed for the analysis of group III nitrides by Averyanova et al. [344]. The kinetics of nitrogen evaporation were found to be an essential factor influencing the MBE growth process. The high thermal stability of nitrides was explained to be related to the desorption kinetics, resulting in a low value of the evaporation coefficient. Evaporation coefficients as functions of temperature were extracted from the experimental Langmuir evaporation data of GaN and AlN. The process parameter dependent growth rate and the transition to extra liquid-phase formation during the GaN MBE were calculated. To gain insight into the critical role of the III/V flux ratio, Zywietz et al. [256] studied the diffusion of Ga and N atoms on (0 0 0 1) and GaNð0 0 0  1Þ surfaces using totalenergy density-functional theory, which led to Ga adatoms having a diffusion barrier approximately four times lower than that of N adatoms on Ga-terminated (0 0 0 1) and ð0 0 0 1Þ surfaces. This is because the adsorbed Ga atoms interact with the substrate by delocalized metallic Ga–Ga bonds, which are weak because Ga melts at 27  C, and

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j 3 Growth and Growth Methods for Nitride Semiconductors

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the adatoms assimilate the behavior of a liquid film on the surface. On the contrary, although the N adatoms could readily evaporate as N2 molecules, they can be kinetically stabilized at the surface owing to their low surface mobility because migration is required for N2 formation. Consequently, the surface could be extensively coated with excess N adatoms under N-rich growth conditions causing significant reduction in the mobility of Ga atoms because strong Ga–N bonds would have to be broken for Ga migration. Zywietz et al. [256] stated that the diffusion barriers so determined for ideal surfaces should be considered as lower limits of the effective diffusion barriers on real surfaces with defects, impurities, and steps. As expected, the III/V flux ratio and substrate temperature play an important role in determining the surface adlayer and hence the adatom mobility on the growth surface [256]. The Ga-rich growth conditions result in excess N, albeit a small amount, which is not yet incorporated on the surface and the Ga adatoms are abundant and highly mobile at the substrate temperatures employed. This results in a 2D growth by a step-flow mode and leads to planarization of the surface and a reduction in stacking faults and point defects when the films are grown on GaN templates. Abundance of Ga on the surface also leads to efficient incorporation of N adatoms because the probability of Ga atoms capturing and bonding to N atoms is much higher than N atoms forming molecules and desorbing from the surface [256]. N-rich growth conditions lead to surfaces with an abundance of N and a significantly shorter Ga diffusion length. Reduction of the diffusion length to less than the mean distance between the available bonding sites would lead to roughening of the surface. Moreover, adatoms may be “trapped” at the wrong sites, which would favor the nucleation of stacking faults. It can then be argued that slightly Ga-rich conditions are expected to be optimal for the growth of GaN. The expected influence of surface diffusion properties of Ga and N on the surface morphology of GaN grown under Gaand N-rich conditions is discussed in Section 3.4.2.9.4. The discussions above indicate that for a given Ga flux, the growth rate increases monotonically with increasing N flux until the growth rate is limited by the Ga flux and saturates, as depicted in Figure 3.69 [345] in which the GaN growth rate is plotted as a function of Ga temperature representing the Ga flux, JGa, for three different N fluxes, JN. In terms of the dilute nitrides, kinetic modeling of microscopic processes during ECR microwave plasma-assisted molecular beam epitaxial growth of GaN/GaAsbased heterostructures was done by Bandic et al. [346]. Microscopic growth processes associated with GaN/GaAs molecular beam epitaxy were examined through the introduction of a first-order kinetic model, which has been applied to ECR microwave plasma-assisted MBE (ECR-MBE) growth of a set of delta GaNyAs1
y/GaAs strainedlayer superlattices (SLS). The delta SLS consists of nitrided GaAs monolayers separated by GaAs spacers and exhibits a strong decrease in y with increasing T over the range 540–580  C. This y(T) dependence is quantitatively explained in terms of microscopic anion exchange and thermally activated N surface desorption and surface-segregation processes.

3.4 Nitride Growth Techniques -5

–1

Growth rate (μm h )

Ts = 700ºC

P = 0.7 x 10 Torr -5 P = 1.0 x 10 Torr -5 P = 1.5 x 10 Torr -5 P = 3.0 x 10 Torr

0.6

0.4

N-rich condition 0.2

Ga-rich condition 0.0 1020

1040

1060

1080

1100

1120

1140

1160

1180

Ga cell temperature (ºC) Figure 3.69 The growth rate of GaN at 700  C versus Ga cell temperature with the system pressure at 0.7 · 10
5 Torr (diamond); 1.0 · 10
5 Torr (circle); 1.5 · 10
5 Torr (triangle) and 3.0 · 10
5 Torr (square). The solid line indicates the stoichiometric condition (JGa/JN  1). The N-rich regime (JGa/JN < 1) is situated to the left side of the solid line; while the Ga-rich regime (JGa/JN > 1) is on the right side.

3.4.2.9.3 Which Species of N is Desirable? Let us now turn our attention to an issue touched upon earlier regarding the intricate nature of the extent of participation by various N species simultaneously provided by RF sources. Of particular interest to nitride growth by MBE, the effect of these N species on the growth of mainly GaN and to some extent on InN has been investigated [83,347]. Specifically, a comparative discussion of relative roles of N and N2 (likely to be the N2 ðA3 Suþ Þ species) species has been treated in a comprehensive manner in conjunction with GaN growth [348]. Although it is clear that thermodynamic equilibrium models cannot be used, quantitative information concerning the kinetics of interactions of the active nitrogen species with an evolving growth surface through different mechanisms is not available. A complex character of such interactions can be explained by a combined action of closely related chemical (such as bond breaking) and physical (such as collisions) effects, yielding a subtle intertwining of activated beam characteristics with growth parameters such as surface temperature and III/N flux ratio. The proposed chemical pathways for the growth process have been reported as follows [301]:

GaðlÞ þ 1=2N2 ! GaNðsÞ;

ð3:54Þ

where N2 represents the reactive and neutral molecules such as N2 ðA3 Suþ Þ, N2 Q Q Q ðB3 g Þ, N2 ða1 g Þ, and N2 ðC3 u Þ. Considering the lifetime only, the first species is thought to participate in growth following the reaction: GaðlÞ þ 1=2N2 þ ! GaNðsÞ;

ð3:55Þ

j451

j 3 Growth and Growth Methods for Nitride Semiconductors

452

where N2 þ ¼2

Pþ g

represents the reactive and ionized molecules, and

GaðlÞ þ N ! GaNðsÞ;

ð3:56Þ

where N ¼ P ; D represents the reactive N atoms, and 2 2

GaðlÞ þ N ! GaNðsÞ;

ð3:57Þ

where N ¼ S. All of the above processes could take place simultaneously with each contributing to the eventual growth rate determined by the relative concentration of each of these species in the plasma. Let us now attempt to treat whether the atomic N or N2 is better for growth, a topic that has been addressed by VanMil et al. [348]. Normally, determining the effect of various active nitrogen species on layer growth is complicated when RF plasma source is used because, as discussed above, these sources typically produce a complex mixture of active nitrogen superimposed on a background of presumed inert molecular nitrogen. However, there seems to be a discernible difference in the composition of N species produced by different sources. Capitalizing on this, those authors used an Oxford source, which predominantly produced atomic nitrogen and EPI source with an aperture plate having 400 holes with relatively smaller diameter, which produced predominantly radicals of molecular nitrogen. In this particular study, OMVPE grown buffer layer on (0 0 0 1) sapphire was used for the growth. These templates were mounted with In–Sn to the substrate block that was in contact with the monitoring thermocouple. A nitrogen flow rate of 0.85 sccm and RF power of 200 W employed led to a growth rate of 0.25 mm h
1 when decomposition was negligible, meaning the growth temperature was sufficiently low. The efficacy of the particular species was determined by monitoring the growth rate as a function of temperature extended to the regime where notable decomposition would occur, as displayed in Figure 3.70. The growth rate in the case of atomic nitrogen began to rapidly decrease slightly above 650  C, whereas that for molecular radicals began to drop only very slightly at about 720  C. Even at 800  C, a growth rate of about 0.14 mm h
1 was attained with molecular nitrogen, which is about 100  C higher than that for atomic nitrogen for the same growth rate. On all MBE-grown N-polarity samples, the temperature at which the growth rate began to decrease increased with a growth rate of about 0.17 mm h
1 attainable at a high temperature of 850  C. The Arrhenius-type plot of the growth rate (growth rate vs. the inverse of substrate temperature) indicated a decomposition activation energy of 3.1 eV, which is the same as that for GaN decomposition in vacuum. Of particular importance is that both ionic and neutral atomic nitrogen can participate both in the growth and in the decomposition of GaN. One decomposition pathway might be through capturing of a nitrogen atom from the GaN to form N2, which would then desorb. Although the rate constants for the reactions involving atomic nitrogen are not known, free energy considerations for such reactions are favorable [349]. This might very well be the reason for the relatively poor efficiency for growth with atomic nitrogen. In short, competition between growth, surface decomposition, and adsorbed nitrogen capture appears to limit the efficacy of atomic nitrogen. In contrast, metastable molecular nitrogen would be much less likely to 4

Growth rate (μm h–1)

3.4 Nitride Growth Techniques

0.1

Atomic N N polar Metastable N2 Ga polar N polar

650

700

750

800

850

Substrate temperature (ºC) Figure 3.70 Dependence of the GaN growth rate on the substrate temperature for different primary nitrogen species, atomic nitrogen and molecular nitrogen radicals, and surface polarity. The predicted temperature dependence based on the measured vacuum decomposition rate for GaN with an activation energy of 3.1 eV is shown with a solid line for reference. Courtesy of B.L. van Mil and T.H. Myers, Ref. [348].

result in such a decomposition pathway, hence the increased upper limit on growth temperature using metastables. The incident Ga flux required to maintain Ga-stable growth increases rapidly above 750  C for Ga-polarity GaN with a slope comparable to Ga desorption rate above liquid Ga. In the case of N-polarity samples, the temperature at which a higher Ga flux begins to be required is shifted to higher substrate temperatures, implying N-polar material reflects a tighter bonding of Ga on this surface. These data illustrate the need for increased Ga overpressure with substrate temperature for maintaining twodimensional growth, suggesting that Ga desorption may also play a role in thermal decomposition during growth. These data indicate that increase in the Ga overpressure might lead to decrease in decomposition effects in GaN at normal growth temperatures. This may very well represent an additional benefit of growth under an increased Ga flux. Studies by VanMil et al. [348] of growth rate as a function of temperature suggest that the GaN surface is most likely attacked by atomic nitrogen above 700  C while attempting to form molecular nitrogen with nitrogen from GaN. Therefore, growth with neutral metastable molecular nitrogen would results in a significantly reduced thermal decomposition rate. In contrast, more ready decomposition with atomic N might lead to N vacancies and other ailments, which would all together lead to inferior carrier transport as borne out by the data obtained in these samples.

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j 3 Growth and Growth Methods for Nitride Semiconductors

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0.35 E

0.30

Growth rate (µm h–1)

0.25 0.20

A

B

C

D

0.15 F 0.10 0.05 0.00 700

720

740

760

780

800

820

840

Tsubstrate (ºC) Figure 3.71 Growth rate as a function of growth temperature for a fixed Ga flux of 1 · 10
6 Torr BEP (beam equivalent pressure) along with electrical properties: (300 K mobility [cm2 V
1 s
1] and carrier concentration [cm
3], peak low temperature mobility [cm2 V
1 s
1]): A: semi-insulating; B: mRT ¼ 200 (mmax-LT ¼ 200) –

n ¼ 1017; C: mRT ¼ 550 (mmax-LT ¼ 1150) – n ¼ 1015; D: mRT ¼ 320 (mmax-LT ¼ 660) – n ¼ 2 · 1017; E: mRT ¼ 560 (mmax-LT ¼ 1160) – n ¼ 8 · 1016 (Ga flux was increased to retain the lower temperature value); F: semi-insulating. Courtesy of B.L. van Mil and T.H. Myers, Ref. [348].

Follow-up studies also gave insight into the transport characteristics of the GaN films grown with molecular nitrogen radicals. A series of experiments investigating the growth rate for a fixed N flux versus Ga flux and also versus substrate temperature discovered that films grown below 660  C are semi-insulating. As shown in Figure 3.71, when the substrate temperature is increased to somewhat above 730  C, the room temperature electron concentration became 1017 cm
3 and the mobility reached 200 cm2 V
1 s
1. Upon further increase in the growth temperature to 750  C, the room temperature electron concentration reduced to 1015 cm
3 and the mobility increased to 550 cm2 V
1 s
1 (the maximum mobility for this sample reached 1130 cm2 V
1 s
1 at low temperatures). Continued increase in substrate temperature to 775  C led to electron concentration of 2 · 1017 cm
3 and mobility of 320 cm2 V
1 s
1 (with a low temperature maximum of 660 cm2 V
1 s
1). A 820  C substrate temperature led to semi-insulating behavior. When the Ga flux was increased to retain the lower temperature value while growing at 810  C substrate temperature, electron concentration and mobility data of 8 · 1016 cm
3 and 560 cm2 V
1 s
1 (with a low temperature maximum of 1160 cm2 V
1 s
1), respectively, resulted. The experiments with a constant nitrogen flux and a substrate temperature of 750  C while varying the Ga flux led to the typical increase in growth rate with an increase in the Ga flux up to a point beyond which the growth rate saturated corresponding to nitrogen-limited growth. At the turn over, see Figure 3.72 where the growth rate begins to saturate, the film was semi-insulating. As the Ga flux was

3.4 Nitride Growth Techniques

0.30

0.25

Growth rate (µm h–1)

0.20

A

B

C

D

0.15

0.10

0.05

0.00 0

2

4

6

8

10

12

14

16

–7

Ga flux (10 Torr BEP) Figure 3.72 Growth rate versus Ga-flux for Gapolar growth below the onset of significant thermal decomposition along with electrical properties for layers grown at 750  C. The units are electron mobility [cm2 V
1 s
1], maximum low temperature mobility [cm2 V
1 s
1] and

carrier concentration [cm
3]): A: semi-insulating; B: mRT = 35 – n = 4 · 1016; C: mRT = 550 (mmax15 LT = 1130) – n = 10 ; D: mRT = 490 (mmax-LT = 1020) – n = 6 · 1016. Courtesy of B.L. van Mil and T. H. Myers, Ref. [348].

increased, the mobility increased reaching a maximum of 550 cm2 V
1 s
1 at room temperature. Further increase in Ga flux led to a reduction in mobility. The various growth regimes in play here are discussed in more detail in Section 3.4.2.9.4. 3.4.2.9.4 The Effect of III/V Ratio and Substrate Temperature on Surface Morphology Section 3.4.2.9.4 touches upon the role of the III/V ratio on the growth kinetics on GaN(0 0 0 1) surface. It is timely to discuss how parameters such as the flux ratio and growth temperature affect the growth and resultant layer quality. This issue has been a topic of extensive reviews [79,83,350,351]. Under Ga-stabilized conditions, layer-bylayer growth of GaN and relatively high-quality GaN growth on GaN templates have been discussed if samples are grown with RF N [317]. Typically, there are four growth regimes in terms of the resultant surface morphology, which can best be investigated if the samples are grown on templates with smooth surface morphologies such as those obtained by HVPE and OMVPE, as follows:

(a) N-rich growth regime: excess N covers the GaN surface (JGa/JN < 1). (b) Transition growth regime: presumably stoichiometric condition. The surface coverage of Ga or N can be neglected (JGa/JN  1). (c) Ga-rich growth regime: Ga adlayer covers the GaN surface (JGa/JN > 1). (d) Ga-droplet growth regime: Ga droplet formation on top of the Ga adlayer (JGa/ JN >> 1).

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j 3 Growth and Growth Methods for Nitride Semiconductors

456

4

III/V ratio

3 Ga-droplets

Ga-rich

2 Transition 1 N-rich 0

600650700750800 Growth temperature (ºC)

Figure 3.73 Surface phase diagram for PAMBE GaN homoepitaxy for a fixed N flux and varying Ga flux to vary the III/V ratio. The N rich, transition, Ga rich, and Ga-droplet regions are roughly indicated.

These four regimes are qualitatively, and somewhat subjectively, shown in Figure 3.73. Tarsa et al. [352] pointed out the significance of the III/V flux ratio (JGa/JN) to the surface morphology, structure, and optical properties of GaN layers grown on OMVPE-grown GaN “templates.” With low JGa/JN (N-stable growth), a granular surface morphology with a tilted columnar structure with a high density of stacking faults was obtained. However, with high JGa/JN (Ga-stable growth), a smooth surface morphology with characteristic spiral growth hillocks and some Ga droplets was obtained. The spiral growth hillocks have been argued to comprise bilayer height steps and terraces as a result of step-flow growth around mixed edge/screw dislocations [353,354]. Because two steps are pinned at each mixed dislocation, the spiral hillocks are comprised of two interlocking spiral ramps [355]. The RHEED studies indicate the transition between smooth and rough surface morphology in the homoepitaxial growth of GaN on GaN templates with varying flux ratio. For constant JN, there is a critical JGa at which the RHEED pattern transits between streaky (smooth surface) and spotty (rough surface) for a given growth temperature. The critical JGa is independent of substrate temperature below approximately 700  C (the measured value may change from system to system as measuring the substrate temperature accurately is not an easy task) and rises above this temperature because of Ga desorption. Figure 3.74a and b shows the typical RHEED and AFM data, respectively, for GaN grown in the N-rich regime. The Bragg spots always appear in the RHEED pattern during growth and the surface morphology is characterized by small islands in AFM images. These results indicate that the surface is very rough with an RMS roughness of 5.0 nm. The theoretical considerations of Zywietz et al. [256] show a low Ga adatom mobility on N-rich GaN(0 0 0 1) surfaces and a significantly shorter Ga diffusion length than under Ga-rich conditions. Low adatom mobility would lead

3.4 Nitride Growth Techniques

Figure 3.74 Typical (a) RHEED results and AFM (b) results for GaN grown in the N-rich regime.

to a high island nucleation rate and to multilayer growth, consistent with the rough GaN surface observed by RHEED and AFM. In the transition regime, the Bragg spots appear with weak streaky lines between them in the RHEED pattern (Figure 3.75) and the surface morphology is characterized by inverted pyramids in AFM (Figure 3.75b). The origin of these inverted pyramids is still unclear, but they appear to be similar to the “V-shaped” defects observed for GaN growth, which are defined by f1 0  1 1g facets. It is worth noting that the surface morphology of GaN layers grown in this regime strongly depends on the substrate temperature. At higher growth temperatures, the pyramids start to merge. When the substrate temperature is higher than 750  C, the surface becomes flat with deep pits. During growth under Ga-rich conditions, the RHEED pattern remains streaky (Figure 3.76). The surface morphology characterized by atomically flat terraces

Figure 3.75 Typical (a) RHEED results and AFM (b) results for GaN grown in the transition regime.

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j 3 Growth and Growth Methods for Nitride Semiconductors

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Figure 3.76 Typical (a) RHEED results and AFM (b) results for GaN grown in the Ga-rich regime.

(Figure 3.76) is much smoother than that obtained in the N-rich and transition regimes. The RMS roughness in the Ga-rich case is less than 1.0 nm. Under this growth condition, the Ga adatoms are abundant and highly mobile at the substrate temperatures employed, resulting in a 2D growth mode and a smooth surface. There are some small hexagonal pits on the surface, with a density of 108–109 cm
2, when the temperature is higher than 750  C. A higher desorption rate of Ga at this temperature may cause this phenomenon because higher substrate temperatures shift the growth toward the lower III/V regime [342]. Under the Ga-droplet regime, a weak but streaky RHEED pattern is obtained during the growth as shown in Figure 3.77. The surface morphology in this regime is smooth (Figure 3.77). In contrast to the sample grown in the Ga-rich regime, no

Figure 3.77 Typical (a) RHEED results and AFM (b) results for GaN grown in the Ga-droplet regime.

3.4 Nitride Growth Techniques

hexagonal pits are observed in large-area images. However, some spiral hillock features, attributed to bilayer height steps and terraces as a result of step-flow growth around mixed edge/screw dislocations, are visible on the surface. The typical size of the “macroscopic” Ga droplets observed on the sample is 3–5 mm at a density of 105 cm
2, as determined by optical microscopy. The issue of the Ga-adlayer coverage has been illuminated by experimental investigations [350,356] of the adsorption and desorption of Ga on the GaN (0 0 0 1) surface by analyzing the RHEED specular spot transients during Ga deposition, Ga desorption, and Ga incorporation of consumption by exposure to N. A pseudooscillatory transient with duration affected by the Ga cell temperature (TGa) was observed after opening the Ga cell shutter. This is most likely owing to sudden heat loss caused by shutter opening and concomitant effort by the temperature control system to compensate for it. This issue was surfaced in MBE of GaAs and measures dealing with sudden heat loss were undertaken to minimize the problem. After closing the Ga shutter and opening the N shutter, a RHEED transient approximately symmetric to the adsorption transient but shorter in duration was observed. The time span of the desorption transient (tdes) increased initially with the Ga deposition time (tads) but remained constant (saturated) above a certain value of tads (7 s in the experiment [356]), even a long deposition time, tads ¼ 1 h, was employed. This saturation was attributed to the formation of a dynamically stable Ga adlayer. The thickness (d) of the Ga adlayer could be determined by the expression: d¼

r GaN ; 1=tN
1=tdes

ð3:58Þ

where rGaN is the GaN growth rate under N-limited conditions and tN is the duration of the transient for Ga adlayer consumption under N flux. The measurements were made for several substrate temperatures (Ts) in the 700–740  C range and in all cases a Ga film thickness of d ¼ 2.7  0.3 ML was determined. This is comparable with the predicted [357] Ga bilayer that contains 2.3 ML of Ga, in terms of GaN surface atomic density. Likewise, Ga adlayer coverage during GaN growth has also been investigated by Adelman et al. [350] by measuring the tdes of the excess Ga after interrupting the GaN growth by shuttering off both the Ga and N fluxes. From the evolution of the RHEED transient, four growth regimes were identified, as previously shown in Figure 3.73. RHEED transients exhibiting full oscillations were observed only in regime c – the Garich region but not quite in the regime with Ga droplets. Then the four regimes were associated with different Ga adlayer coverages on the GaN surface: (a) “N-rich” regime with no Ga accumulation on the surface; (b) “slightly Ga-rich” regime with Ga coverage perhaps less than a bilayer; (c) “Ga-rich” regime with approximately a Ga bilayer present on the surface; (d) “very Ga-rich” regime with appearance of Ga droplets. Based on these results, the authors constructed a phase diagram of the Ga adlayer coverage during GaN growth as a function of impinging Ga flux and substrate temperature (Ts) for a fixed active N flux, which is shown in Figure 3.73. Within the regime c, which represents the Ga-rich case with a Ga bilayer on the surface, GaN film growth proceeds within the realm of the step-flow mode, the

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justification for which has its roots in smooth surfaces with about 40 nm terraces, the observation of spirals induced by screw dislocations and the absence of RHEED oscillations. Adelman et al. [350] pointed out that there is a growth window in which smooth GaN surface without Ga accumulation can be attained. Additional investigations along these lines that shed more light on the mechanisms likely in play are available in the literature [358–360]. The surface morphology of MBE GaN films depends on whether GaN templates or some other substrate is used. If grown on templates, the correlation between the growth conditions and surface morphology is reasonably well defined. When GaN templates are used, the films grown under N-rich conditions exhibit rough surfaces with large density of pits, whereas those grown under Ga-rich conditions are smoother with characteristic spiral hillocks. The films grown in the intermediate regime exhibit large areas of flat surfaces separated by trenches. As the Ga flux is increased, those trenches give way to pits and if the Ga flux continues to be increased, the pits give way to spiral hillocks [317,318]. When other substrates are used, the surface morphology even under Ga-rich conditions is not as smooth as those on GaN templates grown under similar conditions with Ga droplets and pit being present simultaneously. The genesis of surface pits (depressions) has been correlated to extended defects [318]. The pits in the N-rich regime correspond to pooling and termination of mixed and pure edge threading dislocations (TDs). The larger, deeper pits correlate to mixed dislocations and the shallower pits correspond to edge dislocations. Both dislocation types are also pooled at the large pits (trenches) observed in the films grown within the intermediate regime. These large pits appeared to be associated with groupings of TDs, which were predominantly edge TDs forming low angle grain boundaries. Consequently, aside from the surface morphology, N-rich conditions have the benefit of orienting and pooling the dislocations where the probability of dislocation annihilation is increased. Consequently, when grown directly on sapphire or other foreign substrates, N-rich regime leads to lower overall dislocation density [361]. Very Ga-rich regime, on the contrary, does not necessarily have this mechanism in place to this extent and the resulting films have much higher density of pure edge dislocations as compared to N-rich samples as shown in the TEM images of Figure 3.78. Note, however, that when grown on GaN templates, the extended defect structure is to a first extent determined by the template, Ga-rich conditions and lead to smooth surface morphologies, making this regime more attractive. In an effort to gain better insight into the aforementioned mechanisms, a theoretical analysis of the dislocation-mediated surface morphology (pinned steps, surface depressions, spiral hillocks) of GaN layers, grown by MBE or OMVPE, has been undertaken by Heying et al. [355], using the theories of Burton, Cabrera, and Frank. The theoretical results of Zywietz et al. [256] can be brought to bear in to shed some light on the particulars of Ga- and N-rich growth results. An increase in the Ga adlayer coverage should increase the adatom surface diffusion and reduce the surface pit density, which can be completely eliminated in the “Ga-droplet” regime if it is grown on GaN templates prepared by HVPE or OMVPE. The high-resistivity of the films grown in the N-stable regime on GaN templates could be attributed to the formation

3.4 Nitride Growth Techniques

Figure 3.78 TEM bright-field images of a 1.9 mm thick film grown with a relatively low Ga/N ratio (R  1.1). (a) Cross-sectional image revealing surface undulation with amplitude 50 nm. Pairs of threading dislocations were observed to combine into one dislocation (black arrows). Dislocations also annihilated each other by joining into a half loop (white arrows). (b) and (c) Plan view images revealing dislocations

remaining mostly at the valleys of the morphology (bright areas). Image (c) was acquired from the same sample area as in (b) but with a dynamical, two-beam condition that results in thickness contours that reflect the film morphology. Dislocation density is about 1 · 109 cm
2 at the top part of the film. Courtesy of R.M. Feenstra and Ref. [361].

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of compensating centers (point defects or point defect complexes) that trap free carriers. For reference, films grown under N-rich conditions on other substrates exhibit n-type conductivity at least at or near 1017 cm
3. An increase in the Ga-adlayer coverage, as is the case in the intermediate regime, is expected to suppress the formation of these compensating centers resulting in an increase in the mobility and electron concentration. However, the optimum properties are achieved at the highest Ga flux within the intermediate regime (at the boundary between “intermediate” and “Ga-droplet” regimes). The degradation of the electrical properties within the “Gadroplet” regime is attributed [317] to the inferior quality of the material that is grown underneath the Ga droplets [353]. Techniques such as periodic desorption of excess Ga can be employed to combat the issue of inferior quality layer growth underneath the Ga droplets. It should be mentioned that these arguments are mere plausible mechanisms that could account for the results. More extensive data and experiments are needed to illuminate the underlying mechanisms. It should also be pointed out again that the above discussion applies primarily to the growth on GaN templates prepared by variants of vapor phase epitaxy.

3.5 The Art and Technology of Growth of Nitrides

GaN samples with the best transport and structural properties have so far been produced with HVPE. TEM, Hall measurements, secondary ion mass spectroscopy, photoluminescence, and deep level transient spectroscopy are among the techniques used to quantify the critical parameters in HVPE-grown thick GaN layers, some of which are freestanding in that they have been removed by laser liftoff (LLO) from the sapphire substrates on which they were grown. The method of X-ray diffraction is commonly used for quantifying the quality of GaN layers. It is worthwhile to say a few words about the various geometries used to glean information about the lattice constant and/or the quality of the layers, and to develop a firm grasp of the data presented as well as intricacies associated with various diffraction geometries. XRD is a powerful tool for investigating the crystalline structure of materials and had an omnipresence particularly during early developments of crystalline material and dates back to 1911 when XRD patterns of rock salt were obtained [362]. Singlecrystal diffraction and powder diffraction dominated the early employment of this method [363]. Single-crystal diffraction for structural analysis reveals the crystal and molecular structure of inorganic, organic, and biological compounds. The powder diffraction method is used to determine the crystal orientation and the scattered intensity when more than one phase is present in the sample. For the case in hand, which is the III nitride semiconductor epilayers, XRD is mainly used to evaluate the quality of the film, determine the mole fraction of alloys, and investigate the thickness and fine structure of materials with superlattice structures [364,365]. X-rays used in diffraction have wavelengths of the order of 0.5–2.5 Å, which is close to the spacing of atoms in crystals. Because atoms are arranged periodically in a lattice, diffraction occurs when X-rays are coherently scattered by them and the crystal

3.5 The Art and Technology of Growth of Nitrides

structure is revealed through appropriate analyses. Bragg’s law is the simplest and most useful description of crystal diffraction: nl ¼ 2d sin q;

ð3:59Þ

where n is an integer representing the order of diffraction, l is the X-ray wavelength, d is the interplanar spacing of the diffracting planes, and y is the angle of incidence. Details about the crystal structure can be obtained from the maxima in an XRD pattern using Bragg’s law. Figure 3.79a shows the typical schematic diagram of an XRD system with a special point made in regard to the terminology used for various scattering geometries (Figure 3.79b). An X-ray source, a beam conditioner, and a detector are the necessary elements of an X-ray system. X-rays are produced in an X-ray tube by accelerating a high-energy electron beam toward a metal anode, resulting in characteristic X-ray transitions in the anode. The beam conditioners are used to collimate and monochromate the X-rays. The specimen is mounted on a sample holder that can rotate along three axes: Phi (F), Psi (C), and Omega (O). A detector is used to collect the diffracting X-rays, where the detector can move through the Bragg angle while scanning. For determining the out-of-plane lattice constant, the y
2y scan is employed, refer to Figure 3.79 for details. In this case, the magnitude of Dk varies while maintaining its orientation relative to sample normal, shown in Figure 3.80. The resulting data of intensity versus 2y show peaks for Dk values satisfying the diffraction condition. This scan mode is used to identify the lattice constant and ultimately determine the mole fraction of alloys. It is also used to investigate the thickness and fine structure of materials comprising superlattices that some use to calibrate layer thicknesses if sufficiently smooth interfaces are obtained to allow observation of the satellite peaks. For determining the layer quality, the y o scan or proverbially called rocking curve is employed. In this case, the orientation of Dk, which represents the difference between the incident wave vector k0 and the diffracted vector k0 , varies relative to the sample normal while maintaining its magnitude, as shown in Figure 3.80b. When diffraction intensity versus o (sample angle) is plotted, the full width at half maximum is a measure of sample quality. In the (0 0 2) diffraction (the hkl notation equivalent to (0 0 0 2) diffraction in the hjkl notation), the dispersion relates to the degree of tilting with respect to the basal plane. In the (1 0 4) and (1 0 2) (the hkl notation equivalent to ð1 0 1 4Þ and ð1 0 1 2Þ diffractions in the hjkl notation) diffraction geometries, the half width is sensitive to the in-plane rotation as well as perfect edge dislocations. In other words, the width of the rocking curve is a direct measure of the range of orientations present in the irradiated area of the crystal, as each subgrain of the crystal is probed while the crystal is rotated. As such, it can be manipulated by narrowing the X-ray slit widths, which serve to reduce the radiated area. Relatively thin layers in terms of HVPE standards and thick in terms of the OMVPE and MBE standards, on the order of 10 mm, show symmetric and asymmetric X-ray diffraction peaks of about the same width, near 5–10 arcmin. Because these samples

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Tilt Psi (Ψ)

Rotation Phi(φ)

2θ

s am

Detector

For rocking curve

pl e

Omega (Ω) Teta (θ)

Detector axis Beam conditioner

(a)

[0 0 0 1] Tilt

[0 0 0 1]

[0 0 0 1] Tilt

Incident beam

Tilt

Sapphire substrate

[0 0 0 1] (0 0

0 2)

Roc kin gd for ire tilt mea ction sure men t

Twist (0 0 0 1) Rocking direction for twist measurement

)

(1010) 10 0 (1

[1 0

[1 1 2 0]

10

] Twist

(b) Figure 3.79 Schematic diagram of the XRD technique (a). Schematic of an X-ray diffraction configuration for a typical GaN layer with (0 0 0 2) diffraction geometry being sensitive to the parallelinity of the basal plane and (1 0 1 0) diffraction being sensitive to twisting of the columns indicated (b).

[1 1 2 0] [1 1 2 0]

3.5 The Art and Technology of Growth of Nitrides

Δk

k'

k0

2θ

θ

(b)

Δk Sample normal

k0

k'

ω (a) Figure 3.80 Schematic configuration of XRD. (a) y/2y scan: the magnitude of Dk varies while maintaining its orientation relative to the sample normal; (b) y/o scan: the orientation of Dk varies relative to the sample normal while maintaining its magnitude.

represent a benchmark for GaN, their properties are treated, but not before a few words on X-ray diffraction analysis itself. The OMVPE and MBE grown films with very narrow out-of-plane rocking curves with FWHM as low as 37 arcsec have been reported. Conventional wisdom would imply that the X-ray rocking curve FWHM of the GaN(0 0 2) in hkl and (0 0 0 2) in hjkl convention peak is a good barometer of quality. This conclusion is based on a good deal of experience with conventional semiconductors grown homoepitaxially. In conventional semiconductor technology it is typical that the rocking curves become significantly broadened at high densities of dislocations (generally >105 cm
2).

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However, the GaN films with narrow rocking curves contain threading dislocations which are predominantly pure edge types along the c, axis. The specific threading dislocation geometry leads to distortions of only the specific crystallographic planes. Pure edge dislocations distort only the (hkl) planes with either h or k being non-zero. Rocking curves on off-axis (hkl) planes are broadened while symmetric (0 0 1) rocking curves are insensitive to the pure edge threading dislocation content in the film. Screw threading dislocations with [111] directions have a pure shear strain field that distorts all (hkl) planes with l non zero [366,367]. Therefore, the rocking-curve widths for the off-axis reflections are more reliable indicators of the structural quality of GaN films. The properties of GaN film grown on sapphire differ for the in-plane and outof-plane in terms of structural features. The measured in-plane coherence lengths are smaller and the rocking-curve widths are larger than in the plane-normal direction. It implies that optical and electrical properties are anisotropic along the film plane and plane-normal directions. The in-plane structure measurements are more closely related to the electronic mobility and optical properties than those in the planenormal direction. This premise is supported a high-resolution X-ray investigations [368]. Films with very sharp (45 arcsec) out-of-plane rocking curves show poor electrical and optical properties, which is consistent with an in-plane or asymmetric X-ray diffraction analysis. On the contrary, the films with about 5 arcmin out-of-plane rocking curves reveal much better optical and electrical properties; this is again consistent with the excellent in-plane X-ray diffraction. Judging from the out-of-plane XRD data and the cross-sectional TEM images of the same films, it appears that the planar edge dislocations do not really affect the symmetric out-of-plane diffraction and sharp out-of-plane diffraction is not necessarily a sign of enhanced overall quality. Owing, in part, to the heteroepitaxial nature of GaN growth on foreign substrates with the accompanying chemical and lattice mismatch, some kind of a buffer layer is imperative for achieving high-quality GaN. In the case of OMVPE, the buffer layer or the nucleation layer is deposited at relatively low temperatures (550–650  C) compared to that for epitaxial film growth. In the MBE case, however, AlN buffer layers are grown at higher temperatures, generally between 700 and 900  C. The low-temperature buffer layer can protect the GaAs surface at relatively high growth temperatures (>700  C) and improve the quality of the zinc blende phase in addition to enhancing GaN nucleation and initiating growth. The thickness and growth temperature of this low-temperature buffer layer have a remarkable effect on the crystalline quality of the wurtzitic and zinc blende polytypes of GaN in MBE or OMVPE growth. Lowtemperature buffer layers grown by OMVPE are of polycrystalline, mixed polytype nature, which improves upon annealing under H2 atmosphere during the ramping process to growth temperature, as discussed in Section 3.5.3 on SiC and Section 3.5.5.1 on sapphire substrates. Until the advent of buffer layers, the quality of GaN films grown directly on sapphire was very poor and the background concentrations were in some cases as high as 1020 cm
3 [5,6]. The poor crystalline quality was characterized by broad X-ray rocking curves of typically 10 arcmin and uneven surface morphologies that contained hillocks. A low-temperature AlN or GaN buffer layer improves the GaN layer quality substantially. Although the low-temperature AlN and GaN buffer layers were

3.5 The Art and Technology of Growth of Nitrides

initially used solely for OMVPE, their modified implementation in the MBE growth process proved highly beneficial too. 3.5.1 Sources

Trimethylgallium for metal and high-purity ammonia for nitrogen are the most commonly used ingredients. They are transported to the growth chamber. The gas manifold typically features fast switching of the group III elements and dopants and permits separate injection of ammonia. Nominally, c-plane [0 0 0 1] oriented sapphire substrates are used, though other orientations of sapphire and alternative substrates are sometimes employed, too. For a hydrogen flow of 40 sccm through the TMG bubbler (
10  C), an NH3 flow of 3.0 standard liter per minute (slm) and hydrogen make up the flow of 1.5 slm; the GaN growth rate was 2 mm h
1. Sources commonly employed for Al and In are TMA and TMI, whereas for n- and p-type dopants they are silane and Cp2 Mg. Before the employment of low-temperature nucleation buffer layers, to be discussed in Section 3.5.5.1, it was commonly accepted that the inability to provide sufficient quantities of nitrogen was the bottleneck deserving particular attention [369]. Consequently, much of the early efforts dealt with exploring ways to address this issue of the high vapor pressure of N2 on GaN. Several investigators substituted the more reactive hydrazine in favor of ammonia [143,370]. Moreover, attempts have been made to use Ga(C2H5)3 NH3, which already has a Ga–N bond, for the GaN source [371]. Although no evidence to justify the approach exits, plasma excitation of ammonia in a CVD growth environment was attempted. Suitable precursors are those that possess good reactivity, thorough pyrolysis, and transportability. As mentioned earlier, the precursors, ideally, should be nonpyrophoric, water and oxygen insensitive, noncorrosive, and nontoxic. TMG and TEG are very popular for Ga, though GaCl has also been tried. TMI and TMA are the commonly employed sources of In and Al, respectively. Ammonia (NH3) is the unchallenged source of nitrogen, as it is reasonably pure and stable. TMI and TMA are the commonly used sources of In and Al, respectively. TMA and NH3 have been reacted in the presence of hydrogen plasma to grow AlN. TMI has been utilized with microwave-activated N2 to grow InN. Even metallic Ga transported with nitrogen to the reaction zone where it is allowed to react with active nitrogen has been tried. During the growth of nitrides employing trialkyl precursors, adduct formation between ammonia and TMA and TMG has been well documented. Usually when mixing at room temperature, the adduct formation between TMG or TEG and ammonia is completed in less than 0.2 s. The vapor pressure of the resulting adduct Ga(CH3)3–NH3 is 0.92 Torr at room temperature, whereas that of Ga(C2H5)3 : NH3 is much lower (for a review, see Ref. [370]). Deposition of InN is nearly an intractable problem even for low-temperature deposition processes such as MBE let alone OMVPE, which is a much higher temperature process. Needless to say that InN is a very challenging problem for OMVPE because the decomposition temperature is close to the minimum temperature

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for efficient thermal activation of ammonia, about 550  C (N–H bond strength, 3.9 eV), and pyrolysis of the organometallic source [188,372]. There are two reports of alternative single-source precursors for InN. The solid-state pyrolysis of polymeric [In (NH2)3] [373] and air-stable, nonpyrophoric, and volatile single-source precursor (N3)Ga[(CH2)3N(CH3)2]2 [374]. Growth rates between 0.3 and 5 mm h
1 were obtained at the substrate temperature range of 300–450  C and a reactor pressure of 2 · 10
4 Torr [374]. As mentioned earlier in Section 3.4.2.7, plasma excitation of the nitrogen species has not proven to be necessary in CVD growth. Although trialkyl compounds (TMA, TMG, TMI, etc.) are pyrophoric and extremely water and oxygen sensitive and ammonia is highly corrosive, much of the best material grown today has been produced by conventional OMVPE by reacting these trialkyl compounds with NH3 at substrate temperatures close to 1000  C. Temperatures in excess of 800  C are required to obtain single crystalline high-quality GaN films, with the best GaN films grown at 1050  C. At substrate temperatures exceeding 1100  C, the dissociation of GaN results in voids in the grown layer. A similar situation has also been observed for AlN film growth. To overcome this difficulty, N2H4 has been employed instead of NH3, with the conclusion that a significantly smaller amount of N2H4 was required to maintain the same growth rate. 3.5.1.1 HVPE Buffer Layers and Laser Liftoff The primary types of layers grown by HVPE are GaN, though the method has been extended to include AlGaN as well, which is either directly grown on sapphire by this technique, in which case the layer quality is not very good, or a thin ZnO layer is deposited on sapphire prior to the deposition of GaN. The GaN deposition temperatures far exceed the temperature up to which ZnO is thermally stable. Sputtered deposited ZnO on sapphire in either the Ar or O2 atmosphere was successfully used, by Detchprohm et al. [375], prior to the deposition of GaN to demonstrate improved material as compared to direct deposition of GaN on sapphire. In the same vein, Molnar et al. [11,376] also used the HVPE technique to grow thick GaN templates by employing ZnO as a sacrificial buffer layer. To achieve such properties, the sapphire substrates were treated in situ with GaCl or ZnO [377]. GaN films of sufficient thicknesses to be self-supporting templates were removed from sapphire by laser liftoff [137]. Another group headed by Park et al. [21] at Samsung Advanced Institute of Technology has been producing freestanding templates with this technique as well. These layers and templates exhibit the lowest impurity concentration with the highest electron mobilities of any GaN so far. The deposition temperature is generally about 1050  C with growth rates of 15 mm h
1. The results obtained thus far indicate that high-quality GaN films and templates can be grown on sapphire and other substrates by HVPE. These layers and templates served as the basis for further growth on them by MBE and OMVPE with excellent results. As such, the technique is very useful, the extent of which could be expanded if high-resistivity films could also be obtained. In this vein, HVPE compensated with Zn exhibited high resistivities, the properties of which are discussed in Volume 2, Chapter 5 in terms of optical processes, as the electrical measurements are difficult to

3.5 The Art and Technology of Growth of Nitrides

interpret. Owing to the high temperatures employed, it has not yet been possible to grow InGaN ternary with this method. Similarly, p-type doping has so far been lacking, which means that HVPE technique alone is not in a position to produce device structures requiring p–n junctions. Although Si and O incorporation from the walls has been an issue, the high quality of the recent layers indicates that this issue is not as serious as it used to be. Gu et al. [378] extensively studied the role of ZnO while depositing GaN by the HVPE method. They observed that the thin ZnO prelayer evaporates from the sapphire surface at the GaN growth temperatures because ZnO is thermodynamically unstable, but not before a chemical reaction occurs. The XPS analysis performed by Gu et al. [378] showed traces of ZnAl2O4 after a simulated thermal cycle. One can conclude that ZnAl2O4 so formed may act as nucleation centers for GaN growth. The carrier concentration of GaN grown on ZnO (20 nm) coated sapphire was reduced by several orders of magnitude, as compared to those grown on bare sapphire, to 1 · 1016 cm
3. Elaborate buffer layers or templates are grown with the HVPE technique also, among which is the ELO for defect reduction. The topic is discussed in sufficient detail in conjunction with OMVPE in Section 3.4.1.2, the basics of which also apply to HVPE. Nevertheless, for flow and compactness, the particulars of ELO associated with HVPE are discussed in the same section. As briefly mentioned in Section 3.4.1.1, HVPE with its large growth rate is well suited for thick layers to the point where freestanding GaN can be obtained when the layer is removed from the substrate. Sacrificial buffer layers such as AlN between the substrate and the bulk of the layer followed by selective etching in, for example, KOH has not been very successful because of chemical diffusion, stagnation, and complications with growth as HVPE is much better suited for growth of GaN. Another approach that does not rely on chemical processes but makes use of residual thermal strain that serves to self-detach the layer from its sapphire substrate during cooldown has been developed [379]. However, a rather cumbersome pretreatment of the wafer and the need for lithography make the method less desirable. Although the LLO [137] method has been comparatively successful, farming of large area wafers has proved difficult. In the LLO process, a GaN layer is separated by irradiation of the substrate–film interface through the substrate with high-power pulsed laser at a wavelength to which the substrate is transparent, but is strongly absorbed in the GaN layer. The third harmonic of a Nd : YAG laser [137,380] or excimer lasers [381] meet these requirements. The absorption of such high-intensity laser pulses causes a rapid thermal decomposition of the irradiated GaN interfacial layer into metallic Ga and gaseous N2. An extensive review [382] on the topic is available and the process is described with a schematic diagram shown in Figure 3.81. As briefly touched upon, the epitaxial layers are mechanically very fragile and break into pieces during the LLO process if performed as described in Figure 3.81. This makes it imperative that the layers be supported during liftoff. The elastic modulus of the supporting layer must be compatible with that of GaN, must be flexible and easily removable, and must handle the temperature used during liftoff, that is, 650  C. To remove the veil around this suspense, strictly speaking that material is not available.

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Laser pulse Sapphire screen Thermal decomposition

Scanning Al2O3 Substrate

N2

Nitride layer

Liquid Ga

Hot plate Figure 3.81 Cartoon depicting the LLO process in which high-intensity laser pulses area allowed to enter the sample from the sapphire substrate side and thermally decompose a thin GaN-layer at the substrate interface due to strong absorption. The shock waves resulting from the

explosive generation of nitrogen gas during pulse are damped by placing the GaN-sample into sapphire powder. A hot plate can be used to raise the substrate temperature during the process to mitigate some of the accumulated thermal strain. Courtesy of M. Stutzmann, Ref. [382].

However, crack-freesamplesweredelaminatedbyfixingthe samplestotwo-component epoxy resin prior to laser liftoff. The downside of this method is that it is difficult to separate the resinfromtheGaNfilmafter sapphire removalbecause ofits highchemical stability. Although the epoxy can be dissolved with dichloromethane or in boiling water after several minutes, it still represents a complication. A more practical method, which is chronicled in Figure 3.82, has been used to separate GaN from its substrate, albeit with some cleavage lines appearing. In this process, the top surface of the film on sapphire is coated with the thermoresistant (up to 300  C) silicone to an approximately 3 mm thick support layer. The film is then affixed to a metallic plate with double-sided adhesive tape to stabilize the flexible silicone elastomer. Following laser scanning at room temperature, the sapphire substrate is removed. Then, the film with the elastometer affixed is removed with the aid of an approximately 3 mm thick transparent thermoplastic adhesive, as illustrated in Figure 3.82d. The strain in thick HVPE films, which tend to crack the wafer upon laser liftoff, or any other separation method for that matter, limits the size of the freestanding GaN available for further epitaxy. Gogova et al. [383,384] and Kasic et al. [385] have reported on the characteristics of approximately 300 mm thick crack-free GaN wafers initially grown on a 2-in. (0 0 0 1) sapphire substrate buffered with a 2 mm thick GaN layer grown by organometallic vapor phase epitaxy. What is interesting about these wafers is that during the cooldown from the growth temperature, the strain was relaxed by inducing cracks in the sapphire substrate instead, allowing strain-free bulklike GaN wafer to be attained. The in-plane homogeneity of the wafer was monitored by lowtemperature photoluminescence mapping. The position of the main near-bandgap photoluminescence line and the phonon spectra obtained from infrared spectroscopic ellipsometry showed that the 2-in. crack-free GaN is virtually strain free over a

3.5 The Art and Technology of Growth of Nitrides

Figure 3.82 Cartoons illustrating the laser liftoff process sequence of 20 GaN membranes. (a) Laser lift-off of the GaN film coated with silicone elastomer and affixed onto a support template; (b) removal of sapphire following laser scanning; (c) deposition of an approximately 3 mm thick thermoplastic adhesive layer at

120  C; (d) peeling-off of the silicone elastomer. In the last step the GaN film is fully removed by dissolving the thermoplastic adhesive in acetone. Courtesy of M. Stutzmann, Ref. [382]. (Please find a color version of this figure on the color tables.)

diameter of approximately 4 cm. The dislocation density determined from cathodoluminescence images was 2 : 0 · 107 cm
2 on the Ga-polar face. The full width at half maximum value of XRD o-scan of the freestanding GaN was 248 arcsec for the ð1 0 1 4Þ reflection. Wafer bonding in conjunction with laser liftoff has been developed, which also has the added advantage of reducing current crowding, discussed in Volume 3, Chapter 1 in the context of quasi-planar device process owing to the insulating nature of sapphire. Wong et al. [386] have used a Pd–In bonding to bond GaN to Si, whereas Jasinski et al. [387] have bonded GaN to GaAs by direct wafer fusion in which a 2 mm thick GaN film grown on sapphire was fused to a GaAs substrate in nitrogen ambience by applying uniaxial pressure of 2 MPa at temperatures of 550 and 750  C. Separation of HVPE GaN layers from sapphire produced freestanding GaN templates, paving the way for producing high-quality film growth by both OMVPE and MBE. In addition, the freestanding templates provided high-quality samples for investigating transport and optical properties of GaN with accuracy and confidence that were not possible with previous layers. The structural properties

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of the benchmark thick HVPE layers still affixed to their sapphire substrates and freestanding varieties are discussed below. 3.5.1.2 Benchmark HVPE Layers/Templates For structural analyses, transmission and cross-sectional TEM [388,389] (also with varying thicknesses to investigate the evolution of extended defects as the growth progressed) and X-ray diffraction measurements were carried out. In some cases, extended defect delineating etches were employed for quicker analysis. The dominant defects present in the layers are the threading dislocations with the density of misfit dislocations at the layer–substrate interface being about 2 · 1013 cm
2. Brightfield (BF) TEM images, recorded under multibeam conditions to image all dislocations with different Burgers vectors, were used to estimate the density of these dislocations with resultant figures to be of the order of about 1010 cm
2 near the layer–substrate interface. The density of threading dislocations gradually decreased away from the interface and for the 55 mm thick layer it reached a value of about 108 cm
2 near the surface. The threading dislocation densities are plotted as a function of distance from the interface in Figure 4.45. A gradual decrease in density of these dislocations with increase in the distance from the substrate shown on this plot indicates a gradual improvement in layer quality with thickness. The relative number of different types of threading dislocations (edge, screw, and mixed) were investigated and two types of dark field images with (0 0 0 2) and ð 2 1 1 0Þ type reflections were recorded. In the first type of images, only the screw and mixed dislocations are visible, whereas in the second ones, only the edge and mixed dislocations are observed. The results indicate that the edge, screw, and mixed dislocations are present in comparable densities. A freestanding GaN template [390] was obtained from Samsung and was grown by HVPE on sapphire to a thickness of 300 mm and separated from the sapphire substrate by laser-induced liftoff [137]. The GaN layer was then mechanically polished and dry etched on the Ga-face to obtain a smooth nearly epi-ready surface, whereas the N-face was only mechanically polished. The polarity on the two sides of the GaN template was determined by the well-established method of convergent beam electron diffraction (CBED). Because GaN is non-centro-symmetric, the difference in the intensity distribution within (0 0 0 2) and ð0 0 0 2Þ diffraction discs in the convergent beam electron diffraction pattern can be used to determine polarity of the sample, taking into account the effect of the sample thickness on the intensity distribution. The CBED patterns obtained on the side previously next to the substrate indicate that it is of ½0 0 0 1 N-polarity, which means that a long bond along the c-axis is from N to Ga. The polarity determination by CBED is consistent with chemical etching experiments in which the N-face is etched very rapidly in hot phosphoric acid (H3PO4). In addition, Schottky barriers fabricated on this surface exhibited a much reduced Schottky barrier height (0.75 eV versus 1.27 eV on the Ga-face) [391], only after some 30–40 mm of the material was removed by mechanical polishing followed by chemical etching to remove the damage caused by the first mechanical polish. An in-depth treatment of current conduction in Schottky barriers is provided in Volume 2, Chapter 1.

3.5 The Art and Technology of Growth of Nitrides

Figure 3.83 Bright field TEM micrographs of a cross-section sample near the N-face side for the g-vectors perpendicular (a) and parallel (b) to the c-axis. Note that both dislocations are visible in both images. Courtesy of J. Jasinski and Z. Liliental-Weber.

The density of threading dislocations determined from the plan view sample was estimated to be about 4  1 · 107 cm
2. These threading dislocations were observed in cross-sectional view also. Few of them are clearly visible in bright-field images as shown in Figure 3.83. The density of these dislocations determined from cross section was found to be about 3  1 · 107 cm
2, which is in good agreement, within experimental error, with that obtained from the plan view sample. For comparison, a density of about 1 · 107 cm
2 was obtained by etching the N-face in H3PO4 for 15 s at 160  C followed by counting the etch pits on several images, details of which are discussed in Section 4.2.4. Most of these threading dislocations are of mixed Burgers vectors because they are visible in bright-field images with g-vector parallel and perpendicular to the c-axis (Figure 3.83). However, caution is warranted with such a conclusion because of the very low statistics (very few dislocations observed within the electron transparent area). The plan view TEM studies for the Ga-face side revealed a surface with an estimated dislocation density of much less than 1 · 107 cm
2; however, owing to the very low statistics, there is a relatively large uncertainty for this estimation. Hot H3PO4 defect revealing process followed by several AFM images up to 50 mm · 50 mm indicated a dislocation count of about 5 · 105 cm
2. The

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significantly lower dislocation density of the Ga-face side with respect to that near the N-face was most likely because of dislocation interaction within the layer, which is remarkable because reduction through dislocation interaction has not been very successful in GaN. In standard HVPE GaN layers [392,388], the dislocation density is several orders of magnitude larger and indicates a very high structural quality of the freestanding GaN template. High-resolution X-ray rocking curves from the HVPE samples were measured by a Philips X’Pert MRD system equipped with a four-crystal Ge(2 2 0) monochromator and Cu Ka1 line of X-ray source. The instrumental resolution was better than 10 arcsec in the diffraction geometry in effect. The X-ray diffraction rocking curves from both the Ga- and N-faces were examined for different X-ray beam slit widths, from 2 down to 0.02 mm, which correspond proportionally to the spot size on the sample owing to the highly collimated nature of the beam. About 10 mm thick HVPE epitaxial layers on sapphire substrates exhibited (0 0 0 2) peaks with FWHM values ranging from 5.8 to 9.3 arcmin. The (1 0 4) in hkl and ð1 0 1 4Þ in the hjkl convention asymmetric peak FWHM values were narrower and ranged from 3.9 to 5.2. As mentioned above, the in-plane structure is more closely related to electron mobility and, to some extent, to optical properties than those in the plane-normal direction, suggesting that the asymmetric diffraction, as opposed to out-of-plane symmetric diffraction, should be weighed more heavily. A crystallographic analysis by high-resolution X-ray diffraction (HRXRD) rocking curves (o-scans) with different slit widths of a GaN template [393] shows a very narrow FWHM on the Ga-face of the freestanding template, down to 69 arcsec for (0 0 0 2) reflection (at a slit width of 20 mm), and 103 arcsec for the ð1 0  1 4Þ diffraction (at a slit width of 100 mm as the signal gets to be very small for smaller slit widths and much longer data collection times are required). The FWHM for the N-face is 160 arcsec for (0 0 0 2) direction (at a slit width of 20 mm) and 140 arcsec for the ð1 0 1 4Þ diffraction (at a slit width of 100 mm). The superior quality of the Ga-face over that of the N-face agrees well with the large deviation of extended defect densities as determined from the etching experiments. Comparing with the reported X-ray data [394,395] of HVPE GaN, the FWHMs of which are typically in the range of 5–10 arcmin for the (0 0 0 2) peak and 4–5 arcmin for the ð1 0  1 4Þ peak, the density of both types of dislocations are dramatically reduced in the freestanding GaN template. With the increase in the slit width, not only is the FWHM increased, but also a nonGaussian multipeak feature emerges. Specifically, when the slit size is increased from 0.02 to 2 mm, the FWHM of the (0 0 0 2) peak on the Ga-face increases from 69 arcsec to 20.6 arcmin; when the slit size is increased from 0.1 to 2 mm, the FWHM of the (1 0 4) (ð1 0 1 4Þ in the hjkl convention) peak increased from 103 arcsec to 24 arcmin. A more improved Samsung GaN template exhibited much better X-ray diffraction characteristics. The FWHM of (0 0 2) ((0 0 0 2) in the hjkl convention), (1 0 2) (ð1 0 1 2Þ in the hjkl convention), and (1 0 4) (ð1 0  1 4Þ in hjkl convention) peaks were 53, 137, and 54 arcsec, respectively. Source beam slit widths of 20 mm for the [0 0 2] diffraction and 50 mm for the (1 0 4) and (1 0 2) diffractions were used, as the signal intensity for these asymmetric peaks was smaller. This sort of broadening has been attributed to the tilt and twist [367] that results in submillimeter scale mosaic

3.5 The Art and Technology of Growth of Nitrides

spread and causes the observed dependence on the slit size. This seems at first not plausible as the X-ray diffraction system uses a highly collimated Cu Ka1 point source. Instead, the low defect concentration observed by TEM and defect delineating etches in these templates point to another source, namely bowing [396]. This is despite the fact that GaN was removed from sapphire and a casual observer could indeed make the assumption that it is relaxed with no cause for bowing. Although the exact genesis is not known at this time, bowing could also be caused by the surface mechanical polish, which affects both Ga- and N-faces. By assuming the FWHM at beam width of 0.02 mm to be the intrinsic broadening of GaN, an upper limit of the bowing radius of 1.20 m or perhaps slightly larger was estimated. It has been reported [137] that a bowing radius of about 0.8 m was found in the HVPE-grown GaN thick films (275 mm) before separation from sapphire substrate and a bowing radius of about 4 m after separation. As mentioned above, the sample underwent a mechanical polishing, which could be the reason for the relatively large bowing. The genesis of impurities is the chemicals used and/or introduced during the deposition process. The latter may be because of the deposition environment and impurities present in the source material used. In addition, the presence of many point defects creates ionized centers as well. Among the chemical impurities are elements such as O, Si, and C, with C receiving the attention during the early development of GaN, which gave way to O and Si as time went on. The nature of carbon in GaN is not well understood. In addition to being substitutional and well-behaved impurities, these species can also form complexes with native defects, complicating the analyses. Because the conductivity is of paramount importance for devices, it is imperative that impurity incorporation in GaN is examined. However, it is agreed that carbon compensates donors and that the resistivity of the sample can be changed by varying the carbon concentration. This is achieved in an OMVPE environment by changing the system pressure used for growth, lower pressures leading to higher carbon incorporation (therefore to higher resistivities) and higher pressures leading to lower carbon incorporation and therefore lower resistivities [397]. Secondary ion mass spectroscopy (SIMS) measurements demonstrate that it is difficult to reduce common impurities, such as C, Si, and O, much below the 1017 cm
3 level in thick GaN layers, 1016 cm
3 in freestanding templates, and somewhat worse in thin epitaxial layers grown directly on lattice-mismatched substrates. The picture is somewhat better in films grown by MBE and OMVPE on HVPE templates. The SIMS data can shed light on whether the measured donor and acceptor concentrations, by Hall measurements, are due to impurities or native defects or both. The SIMS technique is refined and calibrated to the point where chemical impurity concentrations down to about 1015 cm
3 can be quantitatively determined. In Figures 3.84 and 3.85, SIMS profiles in an HVPE layer on sapphire and a freestanding GaN template (for both Ga and N faces) are shown for O, C, and Si. In the HVPE layer, both O and Si concentrations drop rapidly away from the surface, owing in part to the artifact of the technique and in part to condensates on the surface, down to about 1017 cm
3 for Si and a high 1016 cm
3 for O. It should be mentioned that there is a rise in O concentration near the interface with the substrate owing to O out-diffusion [398].

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Figure 3.84 SIMS profiles in a HVPE GaN layer on sapphire substrate showing the depth distributions of O and Si concentrations. The average n and true n (from Hall measurements) in HVPE GaN on Al2O3 are also shown. VGa-PA and n-ECV indicate positron annihilation

Concentr ation (cm -3)

10

10

10

10

10

10

determined Ga vacancy concentration and electropolish capacitance–voltage determined electron concentration, respectively. Courtesy of Steve Novak of Evans East and D.C. Look of Wright State University.

21

N face

20

O

19

C

18

Si

17

Ga face

C O Si

16

0

1

2 Depth ( µm)

3

Figure 3.85 SIMS profiles in a HVPE GaN template showing the depth distributions of O, C, and Si concentrations from both Nand Ga- faces with Ga-face figures being below mid-1016 cm
3. Courtesy of Steve Novak of Evans East and D.C. Look of Wright State University.

3.5 The Art and Technology of Growth of Nitrides

The Ga-face profile in the Samsung template indicates levels below mid 1016 cm
3 for all three impurities. The picture is different for the N-side, however, as this side was juxtaposed to the substrate during growth and was mechanically polished after laser separation. The impurity concentration is some 1–3 orders of magnitude higher than is the case for the Ga-face. Particularly, the concentration of O and C is high, albeit some drop occurs deeper in the film. Data suggest that impurity incorporation depends on the structural quality of the film. 3.5.2 Growth on GaAs Substrates

It is abundantly clear that wurtzitic nitride polytypes dominate the wide bandgap nitride field. However, initial investigations of cubic GaN have taken place on GaAs substrates, particularly with MBE, owing to its availability and ease with which its surface can be cleaned with a combination of well-developed ex situ and in situ methods. This eventually gave way for the activity on GaAs to be primarily focused on dilute nitrides in GaAs and other conventional III–V compound hosts as discussed in Section 2.10. As for the wide bandgap cubic nitrides, the motivations are twofold. With superior electron and, particularly, hole mobilities and isotropic properties owing to the cubic symmetry and high optical gain in stimulated emission, the zinc blende phase is desirable over the wurtzitic phase. However, the sample quality is not comparable to the wurtzitic phase because of to a larger density of extended defects and their electrical activity, leading to a shift in the motivation to dilute N cubic III–V systems such as GaAsN, InGaAsN, and so on, where small quantities of N added to the lattice reduce the bandgap considerably, paving the way for GaAs-based emitters. For example, bandgap emission in the 1.3–1.5 mm range, which is very important in telecommunications, can be obtained in InGaAs : N lattice-matched or nearly lattice matched to the well-established GaAs technology. To underscore again, while a good deal of progress has been made with semiconductor laser reports in dilute GaAs : N and InGaAs : N, the zinc blende variety of GaN has been beset by the difficulty in producing low-defect content material. Several substrates discussed in this chapter such as GaAs, Si, 3C-SiC, GaP, and MgO can be used to grow zinc blende GaN, but gallium arsenide (GaAs) has emerged as the most widely used substrate for zinc blende GaN epitaxy because it is well developed and large-area substrates are commercially available. The GaAs(1 0 0) surface is used for the zinc blende phase, whereas the (1 1 1) face produces the wurtzitic phase. If the wurtzitic phase can be made pure, thick GaN layers can be grown on GaAs, which can then be selectively removed with ease to obtain freestanding GaN templates. Even a small amount of GaAs decomposition has adverse effects on zinc blende GaN epitaxy, as surface roughening or faceting enhances the onset of wurtzite growth. Because of this, MBE, which is a relatively lower temperature process, has been more commonly employed than OMVPE or HVPE. Some success has been achieved in increasing the maximum allowed temperature by encapsulating the GaAs substrate with GaN deposited at a low temperature, thereby making growth on GaAs by OMVPE and HVPE more viable.

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It is customary that the initial deposition of GaN is carried out at relatively low temperatures, typically in the range of 550–650  C, compared to that for epitaxial film growth. This GaN layer deposited at relatively lower temperatures can protect the GaAs surface during the actual growth, which takes place at relatively high growth temperatures, typically >700  C. Moreover, it does improve the quality of the zinc blende phase, in addition to enhancing GaN nucleation and initiating growth. Highresolution TEM and electron diffraction measurements show that the as-deposited GaN buffer layer at 550  C by HVPE on GaAs substrates is polycrystalline but becomes a single crystal after thermal annealing at 850  C for 10 min [13]. The thickness and growth temperature of this initial buffer layer has a pivotal role in the crystalline quality of the zinc blende GaN. In an MBE growth system with a radio frequency plasma discharge source, a two monolayer thick interfacial layer deposited at 600  C resulted in better quality without any wurtzite phase. In OMVPE experiments, a low-temperature GaN or AlN buffer layer was also grown in a temperature range of 550–575  C for several minutes up to a thickness of 20 nm, followed by growth of the main layer in a temperature range of 600–900  C. The buffer layer was also found to lead to two-dimensional GaN growth, in addition to protecting the GaAs substrate. As in the case of, and even worse than, GaN on sapphire, the large difference in lattice constants of GaN and GaAs causes a high density of extended defects, among which are the stacking faults. The MBE growth reported by Strite et al. [399] and OMVPE growth by Bae et al. [400] concluded that the majority of defects originated from disordered regions at the GaN/GaAs interface and propagated along {1 1 1} planes and their extent and nature depended on the N/Ga flux ratio with the nearstoichiometric condition bringing about the best result. The MBE method, as opposed to the OMVPE method, results in good crystal quality as expected, because it is a nonequilibrium growth method affording lower substrate temperatures to be employed, which prevent dissociation of GaAs. Although not well studied, the GaN growth temperature affects the phase of the epitaxial layer, with high temperatures (>850  C) leading to wurtzitic GaN. HVPE has been used to grow thick wurtzite GaN at high temperatures (1000  C) on GaAs(1 1 1)A with the GaN buffer layer grown at 850  C toward freestanding GaN. Although not studied in detail, atomically flat GaAs surfaces drastically suppress the wurtzite GaN phase. Nagayama et al. [401], who employed OMVPE, reported that the samples grown on GaAs(1 0 0) surfaces tilted toward ½1  1 0 showed an enhancement of the wurtzite domain on the ð1 1 1Þ face, whereas the wurtzite domain on the ð1 1 1Þ face was suppressed. The sample grown on the (1 0 0) surface tilted toward [1 1 0] showed equal levels of wurtzite domains on both the ð1  1 1Þ and ð 1 1 1Þ faces. The generation of the wurtzite domains was suggested to be suppressed by exposure of the ð1 1 1Þ or ð1 1 1Þ faces by the reduced thermal damage of the substrate surface. With a low-temperature GaN buffer layer on 2 miscut GaAs(1 0 0) by HVPE, Yang et al. [402] obtained a GaN epitaxy film containing mainly the zinc blende phase. This 2 miscut GaAs(1 0 0) without nitridation also led to a pure zinc blende GaN thin film deposited by MBE [403]. This may be owing to the arrangement of five GaN lattice spacings coinciding with four GaAs(1 0 0) lattice spacings. The zinc blende GaN

3.5 The Art and Technology of Growth of Nitrides

growth on patterned GaAs(1 0 0) substrates has also been investigated by employing OMVPE [404]. It may be that lateral epitaxial overgrowth is in play here reducing the dislocation density greatly, as has been the case on patterned sapphire and 6H-SiC substrates. The incorporation of wurtzite GaN was reduced by using ½0 1  1 oriented stripes to suppress the formation of (1 1 1)B facets. As mentioned earlier, one of the motivations for growth on GaAs is the potential for thick wurtzitic GaN followed by the removal of GaAs for a freestanding GaN template. To this end, GaAs(1 1 1) substrates are used for wurtzite GaN growth as opposed to GaAs(1 0 0) substrates, which are used for zinc blende GaN growth. Both GaAs(1 1 1) A and GaAsð1 1 1ÞB faces can result in single wurtzite GaN in MBE growth [405] and HVPE, the latter having been used for producing freestanding wurtzite GaN [406– 408]. Numerous pinholes were found to form in the GaN film when grown on a GaAs (1 1 1)B surface, possibly because of arsenic desorption, whereas a GaN layer with a mirrorlike surface was grown on a (1 1 1)A substrate [409]. 3.5.3 Growth on SiC: Nucleation Layers and GaN

As substrates, 6H- and 4H-SiC offer the advantage that the c-plane lattice constants, with <1% lattice mismatch for AlN, and 3.49 and 3.53% for 6H- and 4H varieties, respectively, for the basal plane of GaN. The thermal expansion coefficients associated with both polytypes of SiC are closer to those of GaN and AlN than in the case of sapphire, although tensile residual strain ensues, which is troublesome. Moreover, the 4H- and 6H polytypes of SiC are polar crystals in such a manner that N bond to Si is stronger on the Si-face of SiC than that of Al or Ga, which paves the way for obtaining solely Ga-polarity GaN if the surface of SiC is prepared well for epitaxy. Furthermore, the in-plane crystalline alignment, as well as the out-of-plane, of GaN is the same as that of the substrate, as opposed to the 30 in-plane rotation encountered on sapphire with the associated disorder and troublesome boundaries between the inevitable domains. Moreover, the thermal conductivity of SiC is much higher than that of sapphire and nearly twice that of GaN. The caveat with SiC is that the residual thermal mismatch is tensile, as in the case of growth on Si with all the other substrates giving rise to compressive strain (see Section 2.12.7), and GaN films tend to crack at a thickness of about 2 mm. The availability of both conducting and high-resistivity substrates bodes well for SiC to produce vertical conduction as in LEDs and lasers (avoids current crowding effects) and high-resistivity substrates as in FETs. Already, the best LEDs, lasers, and MODFETs (or HFETs) [410] are produced on SiC substrates. The early drawback of SiC surfaces being damaged during polishing has been eliminated with better SiC mechanical polishing techniques coupled with H, HCl etching techniques as discussed in detail in Section 3.2.3.2. Devices such as LEDs, injection lasers, and modulation-doped structures are reported on SiC, as detailed in Volume 3, Chapters 1–3. The early problems plaguing growth on SiC had to do with surface preparation and quality of SiC available. Associated with the surface preparation is the stacking order disparity between nitrides and SiC substrates, applicable to both 6H- and 4H-SiC

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varieties. Let us discuss the issues involved in depositing nitrides on SiC with a focus on stacking order. An examination of the stacking shows that 6H-SiC is two thirds cubic and one third wurtzitic and the 4H variety is one half cubic and one half wurtzitic, whereas nitrides are fully wurtzitic with stacking order AaBbAa along the cdirection, and in the cubic form, the stacking order is AaBbCcAa [411]. The particulars of stacking-related issues on SiC are discussed in Section 3.5.3.1. To achieve complete GaN film coverage on SiC, some sort of nucleation layer must be employed. The buffer layer takes on a new meaning also on SiC as it is also used for strain management to avoid cracking to the extent possible as discussed later in this section. The ability to form facets with somewhat more ease than with sapphire makes deposition of nucleation layers on SiC relatively easier, but again with the perennial issue of achieving complete coalescence. The nature and the deposition conditions of this nucleation layer differ significantly whether OMVPE or MBE is employed. In the former, both GaN and AlN nucleation layers can be used; the former having been attempted both at low and high temperatures, with the latter giving better results. The deposition temperature for both GaN and AlN nucleation layers in OMVPE is reasonably high (in the vicinity of 1000  C), as we will discuss in Section 3.5.3.2. However, in the case of MBE, the behavior of GaN and AlN directly on SiC is drastically different. Although AlN can be grown by MBE at very high temperatures for MBE standards (even comparable to those employed in OMVPE growth), the growth temperature of GaN directly on SiC in MBE must be kept below a certain level, about 800  C, to avoid complete desorption. 3.5.3.1 Stacking and Interfacial Relationship The discussion of nucleation and stacking of GaN on SiC critically depends on the surface atomic structure of SiC, which is discussed in Section 3.2.3.1. To gain an understanding of AlN and GaN growth, which have 2H stacking, on SiC, which has either 4H or 6H stacking, we must describe the possible surface structure of SiC and the atomic arrangement of AlN on SiC, with particular emphasis on the interface between the two. Detailed structural and electrical properties of stacking faults are discussed in Sections 4.1.4 and 4.1.6, respectively. Here, AlN is chosen because it is nearly lattice matched to SiC, thereby simplifying the picture for a clearer presentation. Unless the steps on 4H- and 6H-SiC are four bilayer and six bilayer steps, respectively, defects will form in nitride layers. In this vein, if we consider an interface step as shown in Figure 3.86, we will note that the step shown delineates two different surface terminations of the substrate, indicated as terrace 1 and 2. For a quantitative description of the step, the step height and the position of the surface termination in the stacking sequence of the substrate need to be well described [47]. The two crystals grown on the surface terminations ! associated with terrace!1 and 2 can be related to each other by a displacement vector d . If the displacement d relating the two surface terminations does not correspond to a symmetry operation of nitrides, a defect is ! generated. If so, the d vector can be reduced to the fault vector of one of the possible stacking faults. Furthermore, as shown in Figure 3.87, the epitaxy could begin at wurtzite or sphalerite positions on each of the two terraces leading to four possibilities, namely, two wurtzite interfaces, two sphalerite interfaces, and two cases with one

3.5 The Art and Technology of Growth of Nitrides

Terrace 1 d

A

C

Terrace 2

Si

Figure 3.86 Interface step and displacement vector at a (0 0 0 1) SiC surface; the area marked A under terrace 1 faces a purely wurtzite portion on terrace 2, it is considered as a faulted wurtzite sequence. Courtesy of Pierre Ruterana, Ref. [47].

2H-AlN

Interface 6H-SiC Sphalerite

Wurtzite

Si

Al

C

N

Figure 3.87 Wurtzite and sphalerite positions for the growth of AlN on (0 0 0 1) SiC. Courtesy of Pierre Ruterana, Ref. [47]. !

sphalerite and one wurtzite interface. The first two configurations do not modify d , whereas the two latter ones introduce a twin component. ! In case the displacement vector d does not correspond to a symmetry operation of epitaxial nitride layers at the interface, the area A can be considered a faulted sequence of the 2H polytype. In this case, a combination of stacking faults of the 2H polytype can result. The sum of the fault vectors of these defects can always be reduced to that of one of the three possible stacking faults, namely, I1, I2, and E of the hexagonal structure (for I1: 1/6 h3 2 2 0i, for I2: 1/3 h0 1 1 0i, or for E: 1/2 h0 00 1i), see Section 4.1.4 for a detailed description of stacking faults. Therefore, the dislocation character of all steps can always be identified as and reduced to one of these three vectors [47]. If we assume that all possible step configurations have the same probability of occurrence, we can then estimate the probability of the different defects that would occur. The results when a 2H polytype, such as GaN, is deposited on 6H-SiC, 4H-SiC, and 3C-SiC polytypes are summarized in Table 3.13 for wurtzite and general interfaces (sphalerite and/or wurtzite).

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Table 3.13 Electrical properties for Si-doped GaN films grown at 350 Torr on 6H- (on axis and vicinal) and 4H-SiC.

SiC substrate

AlN temperature ( C)

nRT (cm
3)

nCV (cm
3)

lRT (cm2 V
1 s
1)

n77 K (cm
3)

l77 K (cm2 V
1 s
1)

6H 6H off 3.5 4H

1080 1080 980

8.9 · 1016 4.6 · 1016 9.7 · 1016

7 · 1016 2 · 1016 —

876 932 884

2.6 · 1016 1.3 · 1016 2.7 · 1016

1888 2438 1895

The SiC substrate used and the AlN growth temperature are listed in columns 1 and 2. The Hall electron concentration at room temperature and 77 K are listed in columns 3 and 6, along with the electron concentration determined by CV measurements listed in column 4. The electron mobility at 300 and 77 K are listed in columns 5 and 7. Courtesy of D. Koleske [434].

Let us now discuss the 2H/6H and 2H/4H interfaces. For the former, the results show that almost one step in two is of the type I1 and 25% of the steps do not generate defects. The E type steps occur only when the sphalerite interface is taken into account; this probability is lower than 10%. For the latter, only the single and four bilayer step, tetrahedra, are considered. For wurtzite interfaces, only I1 type steps occur and one step in two does not exhibit a dislocation character. The I2 and E type steps occur only when sphalerite interfaces are present. Considering the case of a 2H nitride semiconductor on 3C-SiC, a step height of six tetrahedra is sufficient to attain coincidence between the two lattices, namely, the SiC substrate and the nitride epitaxial layer. In this case, in contrast to the 4H-SiC or 6H-SiC substrates, the position of the step in the stacking sequence of the substrate is not critical for the defect type and the only height that does not lead to a dislocation character for the step is of six tetrahedra [47]. In short, although a substrate such as SiC may present a good lattice match and the same in-plane symmetry with the deposited material, numerous defects because of the structural differences can be generated at the interface. 3.5.3.2 Nucleation Layers on SiC Although not as difficult a step as in the case of growth on sapphire in terms of initiating growth on SiC, still the nucleation layer represents a critical step in growth of nitrides on SiC as it has implications not only on the quality of the subsequent GaN layer but, more importantly, also on cracking. Typically, GaN layers of more than 2 mm thick crack upon cooling down from the reactor and/or during processing. The cracking problem, therefore, dictated the developments in nucleation layers on SiC. Relevant to the case is the somewhat lack of in situ diagnostic capabilities in OMVPE to allow in situ buffer morphology observations. Suffice it to say that thin buffer layers of GaN on SiC, particularly the high-temperature varieties, do not fully coalesce. This is also applicable to MBE growth even for AlN unless proper surface treatment procedures are employed (see Section 3.2.3.2 for surface preparation of SiC for epitaxy) and suitable growth temperatures must be employed for a two-dimensional growth, as discussed in Sections 3.5.3 and 3.2.3.2.

3.5 The Art and Technology of Growth of Nitrides

Cracking of GaN layers is caused by thermal mismatch induced tensile strain and exacerbated by the better registry, for example, lack of in-plane rotation, between GaN or AlN and SiC, as compared to sapphire, between the film and the substrate. Cracking problem is an issue also on sapphire substrates, but it occurs for thicker layers as compared to those on SiC. Applicable to growth both on sapphire and SiC is the plastic relaxation (allowing the epitaxial layer to crack) followed by regrowth that relies on epitaxial lateral overgrowth [412]. The plastic relaxation method has also been applied to AlGaN on GaN [413] wherein thick AlGaN layers also crack, which has been a difficult problem in lasers as thick cladding layers of AlGaN are required for mode confinement. Various techniques have been developed to ameliorate stress and allow deposition of crack-free >4 mm thick AlGaN layers on sapphire. LT AlGaN interlayers have proven effective in suppressing cracks during AlGaN growth on a GaN template, but the quality of low-temperature AlGaN layers is not as good as those grown at high temperatures. The coefficient of thermal expansion (CTE) mismatch between the III nitrides and SiC leads to a calculated tensile stress of approximately 0.4 GPa for GaN and approximately 0.8 GPa for AlN upon cooldown from approximately 1100  C, typical growth temperature, down to 25  C, which is typically cited as a cause for cracking of AlGaN layers on SiC [414]. Additional details of residual stress on SiC and other substrates can be found in Section 2.12.7. Owing to the smaller thermal expansion coefficient for SiC than for the Wz GaN, the GaN films on SiC typically are under tensile strain, with the biaxial stress being, depending on the growth temperature, as large as 1 GPa. Optical and Raman spectroscopy studies of OMVPE-grown and MBE-grown GaN on SiC confirm they are under tensile stress, whereas those on sapphire substrates are under compression [415]. This suggests that the mismatch of the thermal expansion coefficient is the dominant factor determining the stress in the GaN films on SiC substrate. The residual tensile biaxial strain causes a red shift of the excitonic transition energy and reduces the valence band splitting as compared to strain-free bulk GaN [416]. Raman and photoluminescence measurements [417] indicated a red shift of 2.7  0.3 cm
1/ GPa of the E 22 phonon and 20  3 meV/GPa of the excitonic transitions. The total measured stress at an arbitrary temperature is given by [418]: s ¼ si þ sth ;

ð3:60Þ

where si is the intrinsic stress, because of the lattice mismatch, and sth is the thermal stress given by sth ¼

E2 ða1
a2 ÞðT
T 0 Þ; 1
v2

ð3:61Þ

where a1 and a2 are the thermal expansion coefficients of substrate and GaN films, respectively, and E2 is the Young’s modulus, and n2 is the Poisson’s ratio of the film. Temperature T0 is a reference temperature, and for the purpose in hand, T0 is the growth temperature. Depending on the growth temperature, the tensile thermal strain Da/a should be about 0.14% (about
0.08% for Dc/c) [419] and could be a factor of 10 smaller if the growth temperature is about 1000  C. However, the lattice mismatch of GaN is not

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very large as mentioned above (about 3.1%), and for many samples, the relaxation is only partial. Perry et al. [420] and Nikitina et al. [421] reported that the a and c lattice parameters for GaN on SiC are in the range of 3.1945–3.1850 and 5.1812–5.190 Å, respectively. The former values for both a and c lattice parameters lead to an a-plane strain of 0.26% and a c-plane strain of
0.07%. The latter values for both a and c lattice parameters lead to an a-plane and c-plane strains of
0.03 and 0.09%, respectively. For the ternary compounds where the lattice mismatch depends on the composition (13.5% for InN, 1% for AlN w.r.t. SiC), the balanced nature of the thermal strain and residual mismatch strain can be different, a topic that has not been fully explored. Unlike growth on sapphire, which will be discussed below, the nitride semiconductors grown on SiC reproduce hexagonal cells of the substrate with no rotation with respect to the substrate. The strain in GaN films is also affected by the nature of extended defects; and in the spirit of Ref. [422], a strain mechanism associated with the presence of defects has been introduced by Ahmad et al. [423]. According to this study, the dominant defects that correlate to stress relaxation in GaN films are the threading dislocations, which introduce a modeled free volume into the crystal. Assuming a strain in direct proportion to n, that is, eTD ¼ L20 n;

ð3:62Þ

where L0 is a length parameter related to the range surrounding a TD defect over which stress relaxation occurs. In this model, strain relaxation because of defects affects the area of L20 . The term n is the dislocation density in question. The strain in the epilayer perpendicular to the growth c-axis is then  ð1
n2 Þ ea ¼ ð1
L20 nÞ 1 þ s
1; ð3:63Þ E2 where s is the biaxial stress. From this, and using de/dT ¼ (a1
a2), which is consistent with Equation 3.61, we obtain ds E 2 ða1
a2 Þ E2 ða1
a2 Þð1 þ L20 nÞ;  ¼ dT 1
n2 ð1
L20 nÞ 1
n2

ð3:64Þ

where the approximation holds whenever L20 n 1. As in Equation 3.61, assuming the expansion coefficients to be nearly constant over the temperature range of interest, Equations 3.63 and 3.64 reduce to the in-plane strain given by ea ¼ sð1
n2 Þ=E 2 and Equation 3.61, respectively, when n ¼ 0. The differences in the sign for ds/dT are because of the ordering of the thermal expansion coefficients discussed in relation to Equation 3.61, namely, compressive strain leading to positive sign (such as that on sapphire) and tensile strain leading to negative sign (such as that on SiC or Si). The model described accurately preserves the signs of the stress and expected stress–temperature coefficient (ds/dT) owing to the effects of either tensile or compressive stress on the open volume associated with TDs. Thus, a linear dependence between the stress–temperature coefficient and n was suggested [423]. Related to this biaxial strain, a Poisson ratio of v ¼ 0.18 was determined by Perry et al. [420]. Other reports suggest that the picture is more complex and the sign of the

3.5 The Art and Technology of Growth of Nitrides

strain may be thickness dependent [424]. Attempts have been made to explain the conflicting strain results, which have been attributed to the influence of the AlN nucleation layers on the growth mode and strain relief. Waltereit et al. [425] observed that the strain of GaN layers on SiCwas determined by the growth mode,which, in turn, was governed to a larger extent by the degree of wetting of the substrate than by the lattice mismatch. They suggested that the tensile strain, which favors the formation of cracks in the GaN layer, can be avoided by growing GaN epitaxial layers on thin and thus coherently strained AlN nucleation layers. Other buffer methods such as AlGaN and also alternating layers of GaN/AlN can be used to mitigate the cracking issue. The particulars of thenucleation layers dependon whether the interface is active for current conduction. In vertical devices, this is the case and graded AlGaN layers and/or very thin alternating layers would have to be used. Additional sources affecting stress, thus cracking, result from changes in film morphology during growth. This, however, is rather complex and its effect is not known precisely. Attempts have been made to investigate stress by in situ optical curvature measurements during OMVPE growth of GaN, AlN, and AlGaN [426,427]. In situ stress measurements during deposition of AlGaN layers on LT AlN nucleation layers on sapphire revealed stress to be initially compressive, before segueing into relatively stress-free film with increasing thickness [428]. On the contrary, AlGaN on LT GaN nucleation layers started with tensile strain, similar to growth on thick GaN. The sign of the initial stress in both cases was ascribed to epitaxial mismatch. Incorporation of LT AlGaN interlayers has been reported to reduce the tensile strain during growth of AlGaN on thick GaN [429,430] owing to a partial relaxation of strain redefining, which modifies the in-plane lattice parameter. The overall message here is that by designing the nucleation layer and structural parameters, strain state could be mitigated to allow thicker layer thicknesses to be obtained before cracking occurs. Because cracking is a more serious problem on SiC (as well as on Si and AlGaN) because of thermally induced tensile strain, much of the nucleation layer growth on SiC is discussed in the context of cracking. Even though the low-temperature GaN initiation layers in OMVPE have been attempted on SiC, they have not been all that successful in terms of quality. The same holds for lowtemperature AlN nucleation layers. In contrast, both GaN and AlN (also AlGaN) hightemperature nucleation layers in OMVPE have been successful. In MBE, the situation is different in that GaN initiation layers must be performed at moderate temperatures with MBE standards, whereas AlN initiation layers could be achieved at moderate to high temperatures. High-temperature AlN nucleation layers on SiC in OMVPE growth, and to some extent in MBE growth, may be more reproducible compared to a low-temperature AlN nucleation layer typically used on sapphire. However, as in the case of the nucleation layers on sapphire, attainment of the proper AlN or GaN NL grain structure, with the eventual goal of complete coalescence, is crucial for high-quality GaN top layers [431] and the growth of high-resistivity GaN is needed as buffer layers for FETs [432]. As mentioned in Section 3.2.3.2, unless the damage caused by mechanical polishing on SiC surface is removed, the layers grown on SiC may not be comparable with those on sapphire, as was the case until the advent of better surface preparation techniques.

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The analysis of George et al. [433] of the OMVPE-grown GaN layers on SiC and sapphire suggested that the subsequent GaN quality primarily depends the nucleation buffer layer, which, in turn, is determined by some important factors such as interfacial bonding rather than the lattice mismatch. Koleske et al. [434] found that the dependence on the substrate temperature of the nucleation layer was of paramount importance and electron mobilities of 876, 884, and 932 cm2 V
1 s
1 were obtained on 6H-SiC, 4H-SiC, and 3.5 off-axis 6H-SiC, respectively, by optimizing that temperature to 300 K. An issue of interest is the way the LT AlN layers grow on SiC. An AlN buffer layer deposited on sapphire at low temperature is likely to form large grains during the annealing temperature ramp, which improves the structural quality of the buffer layer and thus the structural quality of the subsequently deposited layer. But the sequentially deposited GaN films on AlN on SiC have inferior quality compared to those on AlN buffer layer deposited at high temperature (HT). This observation is inconsistent with layers grown on sapphire where annealed LT buffer layers result in higher GaN quality than that of HT buffer layer in the case of OMVPE [435]. Similarly, the postgrowth thermal processing of thin (200 nm) AlN films grown by MBE at high temperature, in the range of 1200–1400  C, has shown to markedly improve the quality [436]. To restate, unlike sapphire substrates, the successful nucleation layers, both AlN and GaN varieties, on SiC substrates are deposited at relatively high temperatures in an OMVPE environment. In the effort by Koleske et al. [434], the SiC substrates were subjected to an in situ treatment at 1200  C for 10 min in 2 standard liters per minute (slm) H2 at 50 Torr. Next, approximately 100 nm of AlN was grown at substrate temperatures ranging from 925 to 1190  C using 1.2 mmol min
1triethylaluminum (TEA), 1.25 slm NH3, and 1 slm H2 at a system press sure of 50 Torr. Koleske et al. [434] converged on an AlN NL deposited at 1080  C on-axis and 3.5 off-axis 6H-SiC, while an AlN NL temperature of 980  C was used for 4H-SiC.The surface topology as inspected by AFM of the AlN NL grown at 1080  C exhibited smaller AlN grains on the 6H-SiC than those on 4H-SiC, suggesting that the AlN morphology influences GaN film formation and subsequent electron mobility. Cross-sectional transmission electron microscopy (XTEM) measurements revealed the absence of screw dislocations in the AlN and a low screw dislocation density near the AlN/GaN (the layer grown on top of the AlN nucleation layer) interface, which would in part account for the high electron mobilities achieved in these films. 3.5.3.2.1 High-Temperature AlN Nucleation Layers on SiC It is a common experience that the AlN nucleation layer growth temperature influences optical properties and electron mobility (termed mRT at room temperature). Consequently, two growth temperatures, namely, 980 and 1080  C, were explored for each type of SiC substrate in the work of Koleske et al. [434]. For comparative purposes, a 100 nm thick AlN layer was grown at 1080  C on 6H- and 4H-SiC simultaneously. Clear differences are apparent in the AlN morphology as shown in the AFM images in Figure 3.88. In Figure 3.88, the AlN grain size (approximately 100 nm) grown on 6H-SiC is approximately half the size of the AlN grain size (approximately 200 nm) on 4H-SiC. Moreover, the AlN grains on

3.5 The Art and Technology of Growth of Nitrides

Figure 3.88 AFM images of the AlN film growth on (a) 6H-SiC from CREE and (b) 4H-SiC from Sterling. Both AlN films were grown to a thickness of 100 nm at 1080  C in the same growth run. The AFM vertical scale for (a) is 15 nm and (b) 10 nm per division. Courtesy of D. Koleske and after Ref. [434].

6H-SiC are more rounded and rougher (z height scale ¼ 15 nm), whereas the AlN grains on 4H-SiC have flatter tops as indicated by the smaller z height scale ¼ 10 nm in Figure 3.88. A point that was made in Section 3.2.3.2 in relation to surface preparation is in effect here also in that a major difference between the 6H- and 4H-SiC lies in the surface polish, where the 6H-SiC had visible polishing scratches, whereas the 4H-SiC had a smoother surface morphology. The epitaxial GaN layer exhibiting the highest mobility was achieved for the AlN nucleation layer grown at 1080  C (mRT ¼ 738 cm2 V
1 s
1) on 6H-SiC and at 980  C (mRT ¼ 743 cm2 V
1 s
1) on 4H-SiC [434]. The growth temperature dependence on 6H-SiC was more notable than on 4H-SiC, which in part may be a reflection of technological issues regarding substrates and their surface preparation. For lower AlN growth temperatures on 4H-SiC, the room temperature carrier concentration, nRT, is 1.5 · 1017 cm
3 and decreases gradually to 7 · 1016 cm
3 as the AlN growth temperature is increased. Because identical GaN growth and Si doping conditions were used on the AlN nucleation layers, the decrease in nRT with increase in the AlN growth temperature suggests increased compensation rather than a reduction in unintentional impurities because a decrease in mobility accompanies a decrease in electron concentration. In another report, which deals with transport but uses the FET performance and X-ray diffraction as a measure, the effect of the particulars of AlN/SiC nucleation layers on the quality of subsequent GaN/AlGaN FET structures was assessed. Green et al. [437] used high-temperature AlN nucleation layers grown between 1030 and 1080  C. Variation of the nucleation temperature, V/III ratio, and thickness were cited to have a dramatic effect on the balance between edge, screw, and mixed character dislocation densities as inferred rather indirectly from the X-ray data. The character of dislocations was established primarily through comparison of rocking curves associated with symmetric (0 0 0 2) and asymmetric ð1 0  1 2Þ and ð2 0  2 1Þ diffractions

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and, moreover, the screw dislocation component, as measured by the width of the symmetric X-ray diffraction peak widths. Green et al. [437] constructed three series of samples to optimize the AlN NL focusing on reactant flow rate, substrate temperature, and nucleation layer thickness. To probe the effect of reactant flow, three samples were grown with NH3 flow/TMA flows of 30 slm/10 sccm, 60 slm/20 sccm, and 90 slm/10 sccm, respectively, while keeping the substrate temperature at 1080  C and the layer thickness at 80 nm. The substrate temperatures of 1030, 1050, 1070, and 1080  C were employed to determine the effect of the deposition temperature, while keeping the NH3 flow/TMA flows at 30 slm/10 sccm and the layer thickness at 80 nm. The layer thicknesses of 40, 80, and 160 nm were employed to test the effect of the nucleation layer thickness while keeping the NH3 flow/TMA flows at 30 slm/10 sccm and deposition temperature at 1070  C. Increasing the NH3 flow rate decreased the growth rate of AlN, despite increasing TMA flow that points to gas-phase reactions that cause a reduction in the TMA delivery to the substrate surface. As is always the case in nucleation layer growth by OMVPE under suitable conditions, increasing the thickness of AlN layer led to smoother AlN nucleation layer surface. Decreasing both increasing AlN thickness and nucleation temperature led to an increase in the edge dislocation density, as measured by broadening of the asymmetric X-ray diffraction peak. Moreover, the screw dislocation component, as measured by the width of the symmetric X-ray diffraction peak, was relatively insensitive to the parameters studied. Further, the edge dislocation density had an impact on the trapping characteristics in FETs, which also manifested itself as a decrease in the drain saturation current under thermal stress. Among the samples with varying deposition temperature, the layer deposited at 1070  C had the smoothest surface (the NH3 flow/ TMA flows at 30 slm/10 sccm and the layer thickness is 80 nm). The lowest temperature sample, 1030  C, was very granular, the grain size of which increased when the deposition temperature was raised to 1050  C. For a deposition temperature of 1080  C, the grain size decreased when the density increased. In terms of the thickness series, the nucleation layer with a thickness of 160 nm exhibited the best surface morphology with nearly complete coalescence. The sample with 40 nm thickness was very granular, giving way to reasonably complete coalescence as the thickness was increased to 80 nm (the NH3 flow/TMA flows at 30 slm/10 sccm and deposition temperature at 1070  C). As is always the case with OMVPE, atomically smooth GaN surface resulted when several microns of the material was grown on the above-mentioned nucleation layers. This is a fluidic field and is in constant flux. The trend is to deposit the nucleation layer at as high a temperature as possible while tolerating discontinuous (incomplete) layer coverage. The subsequent layer growth conditions are then optimized to achieve coalescence rapidly. The structural and therefore electrical/optical properties thus obtained are superior to those grown on what is still called the high-temperature nucleation layers but performed at relatively lower temperatures. 3.5.3.2.2 Low-Temperature GaN Nucleation Layers on SiC As mentioned above, low-temperature AlN and GaN nucleation layers have not been as successful on SiC as the high temperature varieties. Nevertheless, attempts have been made to optimize

3.5 The Art and Technology of Growth of Nitrides

surface morphology of low-temperature GaN nucleation layers on the vicinal surfaces of 6H-SiC(0 0 0 1) in an OMVPE environment [438]. In addition, cross-sectional transmission electron microscopy, Hall measurement, and photoluminescence spectra were obtained in the subsequent GaN layers to draw conclusions about the effect of the nucleation layer. The optimum growth condition for a GaN nucleation layer on SiC(0 0 0 1) was determined to be 1 min of growth at 550  C within the scope of low-temperature GaN nucleation layers. Low-temperature GaN nucleation layers growth at 565  C on SiC attempted in author’s laboratory displayed featureless surface as opposed to a nanoporous surface typically observed for high-temperature varieties. XRD analysis led to >20 and >30 arcmin HWFM values for (0 0 2) and (1 0 2) X-ray rocking measurements, respectively. The subsequent high-temperature GaN epilayer grown on this buffer layer had good surface morphologies but exhibited only marginally improved X-ray data. It should be stated that the low-temperature buffer layer/SiC (i.e., 600  C) leads to high defective interface between the epilayer and the substrate that precludes any device application relying on the quality of that interface. Presented only for completeness, it is clear that the low-temperature GaN and also AlN nucleation layers on SiC are not successful as compared to the high-temperature varieties. 3.5.3.2.3 High-Temperature GaN Nucleation Layers on SiC Unlike the low-temperature GaN nucleation layers, high-temperature nucleation layers of the GaN, AlGaN, and AlN varieties have all been successful and which one is used depends very much on the particulars of the device performance. If GaN is grown using typical growth parameters including the substrate temperature, for example, 1080  C, the resultant GaN nucleation is 3D, which leads to the formation of large islands presenting hexagonal truncated shape with f1 1 0 1g lateral facets and a top {0 0 0 1} facet [439]. Lahreche et al. [439] developed a three-step growth process for growing high-quality mirrorlike GaN layers with only GaN layers in the structure, the growth sequence of which is as follows: During the first step, a thin 3D GaN layer (100–130 nm) is deposited at a substrate temperature in the range of 980–1080  C. The second step involves growth interruption during which the 3D GaN layer is annealed at 1080  C under an ammonia flux, which resulted in a smooth surface. The third step involves the growth of main GaN layer at 1080  C. In situ optical reflectivity (higher intensity corresponding to smoother surfaces) as well as ex-situ SEM images were used to monitor and investigate the surface morphologies of the resultant films. Figure 3.89 shows the SEM images of the as-deposited 100 nm nucleation layer whose topological features can be described as 3D growth and that of a similar nucleation layer after the annealing in ammonia flow at 1080  C. In author’s laboratory, high-temperature GaN initiation layers have been developed for growth of GaN on SiC. Figure 3.90 shows SEM images of (a) 100 nm HT GaN buffer layer grown on H2-annealed 6H-SiC and (b) 5 mm thick HT GaN epilayer grown on the 100 nm HT GaN buffer. The growth details are as follows: 3 min annealing at 950  C under H2 and NH3 and then 100 nm GaN buffer layer growth at 950  C (V/III 2000 and pressure 30 Torr) and then GaN epilayer growth at 1050  C

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Figure 3.89 SEM image of a 100 nm GaN nucleation layer as grown on 6H-SiC substrate at 980  C (a) and that of a similar layer but after annealed under ammonia flow at 1080  C during growth interruption (b). After Ref. [439].

(V/III 2000 and pressure 30 Torr). The crystalline quality of 100 nm GaN buffer layer was 5 and 13.2 arcmin for the (0 0 2) and (1 0 2) X-ray rocking FWHM, respectively. The surface of 100 nm GaN HT buffer layer shows clear signs of incomplete coalescence. The surface morphology can be controlled by the growth conditions such as pressure, temperature, V/III ratio, and thickness. The 5 mm GaN epilayer showed 4.7 and 8.5 arcmin for the (0 0 2) and (1 0 2) X-ray rocking FWHM, respectively. Figure 3.90b showed the crack generated on the GaN surface. The initiation layer of similar thickness to the one shown in Figure 3.90a has been grown at 980  C by Lahreche et al. [439] and shown not to be contiguous either, but after annealing under ammonia flow at 1080  C, the layer became smoother and importantly configured. It is therefore reasonable to conclude that this annealing occurs automatically prior to the growth of the higher temperature main GaN growth, as is the case in Figure 3.90b.

3.5 The Art and Technology of Growth of Nitrides

Figure 3.90 SEM images of (a) 100 nm HT GaN buffer grown on 6H-SiC and (b) 5 mm thick HT GaN epilayer grown on the 100 nm HT GaN buffer, which shows cracks thermal mismatch.

3.5.3.2.4 Alloy and Multiple Layer Nucleation Layers on SiC Having considered GaN nucleation layers, let us now consider AlGaN and AlN and combination of GaN, AlGaN, and AlN layers, high-temperature nucleation layers. Poisson et al. [440] used FET performance as a means of comparing GaN and AlGaN nucleation layers deposited at about 985  C to delineate as to which one is better for that particular application. A maximum drain current of approximately 1 A mm
1 has been measured for devices with GaN nucleation layers as compared to 1.3 A mm
1 obtained for devices with AlGaN nucleation layers. Moreover, in this single-step process, TEM analyses revealed that the GaN/SiC interfaces grown at 985  C are sharp with steps originating from the substrate misorientation, but not so when grown at higher growth temperatures >990  C, indicating a three-dimensional growth process. In response to the cracking of GaN films associated with growth on foreign substrates, particularly SiC substrates, because the lattice mismatch and thermal mismatch parameters are endemic and cannot be changed, Acord et al. [441] investigated the effects of growth conditions on stress evolution during OMVPE growth of AlGaN on SiC using in situ stress measurements based on wafer bowing measurements [442], dubbed multibeam optical stress sensor (MOSS) system [443]. An initial compressive stress was measured during AlGaN growth on thin AlN buffer layers, which evolved with film thickness into a tensile growth strain. The thickness at which the transition occurred was found to depend on the buffer layer growth conditions and on the Al fraction used in AlGaN. Increase in the AlN buffer layer V/III ratio from 750 to 10 600 led to an increase in the AlGaN initial compressive stress from 1.9 to 8.7 GPa, which reduced the film strain in the end being tensile. GaN was not observed to transition into tensile growth stress within the 3.8 m thickness grown, whereas increasing the Al fraction ushered in a more rapid change in tensile growth. The observed stress evolution has been explained in terms of an initial compressive growth stress that changes to a final tensile stress owing to morphological evolution during growth. Both the effects of V/II ratio and the composition of AlxGa1
xN in the range of 0 x  0.63 were investigated in terms of stress evolution. In terms of the effect of the nucleation layer/buffer layer structure, GaN grown on AlN (80 nm)/SiC exhibited compressive strain (negative slope in the plot of stress thickness product versus layer GaN thickness) with increasing GaN thickness, which

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gradually changed with the GaN top layer thickness reaching a stress thickness product value of
1 GPa mm at 4 mm GaN layer thickness. GaN grown on Al0.35Ga0.65N/GaN (70 nm)/AlN (80 nm)/SiC was under tensile strain and exhibited a thickness product value of 2 GPa mm at only 1 mm GaN layer thickness. On the contrary, GaN layers grown on Al0.38Ga0.62N/AlN (80 nm)/SiC begin with compressive stress, which gradually begins to change slope, indicating heading toward tensile strain, reaching nearly zero stress at 1 mm. The lattice mismatch between the nucleation/buffer layer and GaN epitaxial layer determines the initial sign of the stress. The implication here is that the strain balance is in effect here and might provide a means of designing a nucleation layer buffer combination with minimal stress if not zero. Acord et al. [441] also examined the effect of the V/III ratio, which was varied between 750 and 10 600, during the growth of the AlN buffer layer on the initial incremental stress and transition point (the thickness at which the incremental stress is zero) during AlGaN growth. The AlxGa1
xN epilayer composition was maintained at x ffi 0.44 (strain em ¼
1.3%), and film growth continued until a tensile incremental stress was observed. The choice of the alloy mole fraction was made to coincide with the solar-blind region (280 nm) of the optical spectrum. During Al0.44Ga0.56N growth on 80 nm AlN/SiC, transition point thickness (at which the stress is zero) increased from 0.3 mm for a V/III ratio of 750 to 1.5 mm for a V/III ratio of 10 600. The gist of this discussion is that both the layer structure and the V/III ratio, the latter affecting the topological properties of the layer surfaces, are important parameters for reducing strain/stress in the overall structure for avoiding cracks. 3.5.3.2.5 Nucleation Layers on SiC by MBE Buffer layers deposited by MBE represent a good laboratory to examine the intricacies of the initial states of growth on SiC, which is intricately dependent on SiC surface preparation before introducing the sample into the MBE chamber and in situ processes once in the vacuum system. Owing to vastly divergent surface mobility and desorption rates of Ga and Al from SiC surface, the conditions used for growth of these binaries are vastly different. It should also be mentioned that AlN on SiC has been considered as a nearly lattice-matched dielectric layer for MISEFTs and not simply a nucleation layer on which to grow [44]. Assuming atomically smooth and chemically clean SiC surface, GaN grown under Ga-rich conditions at temperatures near approximately 750  C, which varies from system to system because of inaccuracies in the substrate temperature measurements, assumes a 3D growth mode as illustrated in Figure 3.91a. If the deposition temperatures is approximately at or below 650  C, a 2D-like growth is obtained as shown in Figure 3.91b. While the GaN in the 2D mode can be assumed coherent up to the relaxation thickness, defects generate readily afterwards. At these low temperatures, which are accompanied by Ga-rich conditions, edge dislocations are generally abundant, as shown in Figure 3.78. Raising the growth temperature at the expense of walking toward the 3D mode could reduce the total density of dislocations. In addition, the 3D mode would eventually give way to coalescence at which point defects are formed and quasi-2D mode would then ensue. If the substrate temperature is raised beyond 800  C, again depending on the system owing to the complexity

3.5 The Art and Technology of Growth of Nitrides

GaN or AlN 3D nuclei Compressive strain

Defects

SiC substrate (a)

AlN 2D nuclei

Compressive strain

Coherent coalescence

SiC substrate (b)

Interface wetting

Figure 3.91 (a) Schematic representation of GaN growth on SiC (0 0 0 1) indicative of 3D islandic growth caused either by higher than optimum growth temperature in the case of GaN and silicate adsorbate in the case of AlN. Defects depicted by circles form when coalescence place; (b) A 2D growth results when the optimum

deposition temperature is used in the case of GaN and silicate free surface is obtained by Ga deposition at 600  C followed by silicate layer decomposition during ramp-up to 1000  C growth temperature in the case of AlN. Courtesy of N. Onojima.

in measuring the substrate temperature, no growth would occur. Strain also could play a role in the deposition kinetics as has been pointed out [425]. The case for AlN is different in many ways in that much higher temperatures can be used as Al evaporation and desorption are not as severe as in the case of GaN. Despite this stated difference, if the surface of SiC is not free of the silicate adsorbate layer, see Section 3.2.3.2 for details, a 3D nucleation would still occur similar to the depiction in Figure 3.91a where the lines indicate the crystal lattice and divergence in the vertical direction indicates relaxation of the epitaxial layer as it has a larger lattice constant than AlN (holds for GaN as well). However, if the surface of SiC is O free as a result of decomposition by Ga flux (Ga is typically deposited at low temperatures such as 600  C and the substrate temperature is ramped to the deposition temperature of 1000  C during which the silicate adsorbate is fully removed [44,444]), 2D growth commences in islands and coalescence occurs leading to a coherent growth that is maintained up to the critical thickness. The lattice mismatch between AlN and SiC is very small and if 2D growth conditions are used, coherent films up to the critical thickness can be obtained, beyond which the film relaxes to its natural lattice constant. This transition can be

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monitored by RHEED rod spacing as the growth progresses and/or by measuring the lattice parameter of several films below at or near and beyond the critical thickness. The transition region is usually not abrupt as many factors go into determining when the epitaxial layer should relax. For example, the growth temperature and the SiC substrate surface preparation have been shown to affect the relaxation process. About 30 and 20 nm critical thicknesses have been obtained at 900 and 1000  C, respectively. As for SiC surface preparation, c-lattice parameter measured as a function of thickness for AlN grown on SiC prepared by HCl at 1300  C in a CVD system, HF dip, prior to both the introduction into the deposition system and in situ Ga-assisted desorption of the silicate adsorbate. As shown in Figure 3.92, the HCl treatment alone led to 3D growth mode and smallest average layer thickness to reach relaxation, the HF treatment also led to 3D growth and average thickness slightly more than the critical thickness, and the Ga treatment led to 2D growth that led to a graded transition to not reaching complete relaxation even at 300 nm. The structural properties of AlN layers as determined by (0 0 0 2) diffraction, which is sensitive to screw dislocations, and ð0 1  1 4Þ diffraction, which is sensitive to edge dislocations, have also been examined. As shown in Figure 3.93, the (0 0 0 2) diffraction width, which is sensitive also to the c-plane order, gets worse with layer thickness, with HCl-only treatment leading to 3D growth showing the steepest worsening. The in situ Ga-assisted desorption of the silicate adsorbate is the best in that the degradation of the (0 0 0 2) diffraction peak is minimal with increasing thickness. The ð0 1 1 4Þ bears the same message in that HCl treatment alone is the worst, HF dip is next, and Ga-aided surface cleaning leads to narrowest peak, which remains low with increasing thickness. Measured by HRXRD

c-axis lattice constant (Å)

5.01

Coherent 5.012 Å

5.00

Ga(2D) HCl(3D)

4.99

HF(3D)

AlN bulk 4.98

4.97 0 10

4.982 Å

1

2

10 10 AlN layer thickness (nm)

3

10

Figure 3.92 Dependence of AlN c-axis lattice constant on layer thickness for three different cases; namely, 1300  C surface exposure of SiC to HCl gas, HF dip of the substrates, and in situ Gaassisted desorption of the silicate layer remaining on the surface. Courtesy of N. Onojima.

3.5 The Art and Technology of Growth of Nitrides

800

FWHM (arcsec)

XRC(0 0 0 2)

600

HCl(3D)

400

Ga(2D) 200 0 0 10

10

1

10

2

10

3

AlN layer thickness (nm)

FWHM (arcsec)

3000 XRC(0 1 1 4)

HCl(3D) 2000

1000

0 0 10 (a)

HF(3D)

Ga(2D)

10

1

10

2

10

3

AlN layer thickness (nm)

Figure 3.93 The half width at full maximum of the (0 0 0 2) diffraction, which is sensitive to screw dislocation and c-plane ordering across the wafer (a); and ð1 0 1 4Þ diffraction that is sensitive to edge dislocation for AlN (b) with varying thickness grown on SiC prepared with HCl gas, HF dip and Ga-assisted desorption of the silicate layer from the surface. Courtesy of N. Onojima.

To complete the characterization, surface morphology, as imaged by AFM, for an AlN sample grown to a thickness of 30 nm on a substrate cleaned with Ga-assisted silicate removal was taken. In addition, the same was taken for a film thickness of 290 nm, again grown on SiC subjected to Ga-assisted removal of the silicate adsorbate. Both images are shown in Figure 3.94 where the thinner films indicate fully 2D growth replicating the atomic steps on the substrate surface. The thicker layer, on the contrary, shows a step-flow growth pattern. The dark spots are associated with dislocation terminations. 3.5.3.3 Substrate Misorientation and Domain Boundaries Vicinal SiC substrates, misoriented from the (0 0 0 1) plane, can enhance twodimensional growth of epitaxial GaN by reducing the distance that atoms must diffuse on the substrate surface and adatom surface lifetime before incorporation, though results contrary to this premise have been reported [445]. This reduces the

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Figure 3.94 Initial AlN layer grown at 1000  C is coherent and within the realm of 2D growth mode replicating the atomic steps on the SiC substrate surface (a on the left), which gives way to step flow growth as the deposition progresses (b, on the right). Courtesy of N. Onojima.

adatom lifetime on the surface before incorporation as well. In addition, a misorientation by 4 can relieve the stress at the interface between the SiC substrate and GaN films [457]. In addition to edge, screw, and mixed dislocations, which propagate throughout the epitaxial layers, defects such as IDBs and double positioning boundaries (DPB) could also be present in GaN unless grown on a substrate with similar stacking order. The growth of GaN on SiC substrates will be used below as the forum to discuss IDBs and DPBs. It should be mentioned that SMB (a special kind of double positioning boundary) are inevitable because of the steps on the substrate surface. If special care was taken to initiate growth with only one species and the substrate surface did not contain any steps, then IDBs would not form. However, substrates invariably contain steps and, even with single-species initiation, the lattice inverts across each single-step substrate. For example, in 6H-SiC, the stacking sequence is AaBbCcAaCcBb, so three stacking sequences would be possible for wurtzite GaN: (1) substrate . . .AaBbCc leading to BbCcBb GaN; (2) substrate . . .BbCcAa leading to CcAaCc GaN; and (3) substrate CcBbAa leading to BbAaBb GaN. All of these leading to vertical defects, between differently stacked domains in wurtzite GaN; they are named SMB. IDBs can be avoided by ensuring that the steps on 6H-SiC and 4H-SiC are six and four bilayer steps, respectively, which can be done with H polishing and possibly followed by a slight KOH etch. A high-temperature (1300–1500  C) HCl etching to accomplish the same has been reported by Powell et al. [36] and later by Nakamura et al. [446].On 6H-SiC, SMBs would always be created for two- and four-bilayer steps, and with a 2/3 possibility for one-, three-, and five-bilayer steps, but never for sixbilayer steps [447]. The SMBs are results of individual island growth and coalescence. SMBs must be considered as inevitable during growth of GaN on nonisomorphic substrates such as SiC.

3.5 The Art and Technology of Growth of Nitrides

Consider a 6H-SiC substrate region terminated as . . .AaBbCc and an adjacent region separated by a bilayer step terminated as . . .AaBbCcAa. With subsequent wurtzite GaN growth, the stacking sequence BbCcBbCc would form over the first region, whereas the stacking sequence CcAaCcAa would form over the second region and inevitably lead to an SMB, as illustrated by the arrow labeled S1 in Figure 3.95. Two important points must be made. First, a surface bilayer step on a wurtzite substrate would not lead to SMB because subsequent wurtzite growth would have the same stacking sequence on both sides of the bilayer step. Second, 4H- and 6H-SiC are

Figure 3.95 (a) High-resolution electron micrograph of wurtzite GaN grown by plasma-enhanced MBE on 6H-SiC showing substrate steps and an associated SMB. (b) Cross-sectional ball and stick model of wurtzite GaN on 6H SiC. Steps on SiC surface such as S1 are likely to create stacking mismatch boundaries whereas others such as S2 will not.

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a mixture of cubic and hexagonal stacking and not every substrate bilayer step would lead to an SMB. For 6H-SiC, consider the substrate terminations . . .AaBbCcAa and . . .AaBbCcAaCc separated by a bilayer step. The first would lead to CcAaCcAa GaN, whereas the second would lead to AaCcAaCc GaN. Clearly, there would be no SMB at this substrate step, as illustrated by the arrow labeled S2 in Figure 3.95. The nature and particulars of SMBs and IDBs are discussed in Section 4.1.3. Wurtzite nitride growth on sapphire is more complex because the exact interface arrangement of atoms is not known and sapphire has a coordination of atoms completely different from GaN. However, if we suppose that nitrogen atoms attach in the same way to each uppermost Al atom of the substrate during the so-called nitridation of the sapphire surface, then each step on the sapphire substrate has a high probability of producing an SMB owing to the position change of Al atoms in successive basal planes. One may then suggest that every bilayer surface step, on a close-packed nonisomorphic substrate of wurtzite GaN, has a high probability of creating an SMB threading defect during wurtzite GaN epitaxy. The above-discussed viewpoint provides a fresh perspective on the notorious problem of finding a good substrate for GaN growth. For the reason of stacking-order match and close lattice match to GaN, it is tempting to state that a ZnO substrate would be superior to a ZnO buffer layer on sapphire for subsequent GaN growth. Because none of the presently known growth methods on available substrates completely eliminates the problem of SMBs in wurtzite GaN, we will briefly consider possible deleterious effects of these defects on the properties of epitaxial films. An SMB originating from a linear step is essentially two dimensional and the equivalent density of defects averaged over the GaN volume could be as high as 1018–1019 cm
3 if we estimate the linear density of these defects from TEM micrographs. The geometry of atoms and unsaturated bonds at SMB is not presently known. Extensive theoretical work is needed to understand the electronic character of these defects. 3.5.3.4 Polarity of GaN on SiC Growth on the Si-face of SiC(0 0 0 1) tends to produce Ga-polarity GaN regardless of whether AlN or GaN buffer layers are used. Polarity control afforded by SiC substrates could be a key advantage of SiC as compared to sapphire on which both Ga- and N-polarity GaN can be deposited, depending on the choice of the buffer layer. In addition, mixed polarity layers could also result, as is the case in predominantly N-polarity GaN on sapphire. The fact that the Si–N bond is stronger than any Si–Ga bond leads to GaN initiating with the N-layer first, leading to Ga-polarity GaN. The polarity of GaN on SiC has been studied by a number of techniques with some inconsistencies in the literature. The argument that Ga-face GaN grows on Si-face SiC and N-face GaN grows on C-face SiC could form the basis of argument [448]. Sasaki and Matsuoka [449] concluded that the epitaxial GaN layers on (0 0 0 1)Si and (0 0 0 1)C SiC were terminated with nitrogen and gallium, respectively. Most results are consistent with the “standard framework,” though the “antistandard” results still deserve serious attention. The stronger Si–N bond argument is supported by the electronic structure calculations of the (0 0 0 1) interface, which indicate that the stronger bonds are at

3.5 The Art and Technology of Growth of Nitrides

the Si–N interface [450]. Along the same lines, Ren and Dow [451,452] considered the lattice arrangement at the interface between SiC and GaN employing a tight-binding model. They concluded that the GaN grown on a C-terminated SiC(0 0 0 1) substrate has a local lattice mismatch of 6%, whereas the GaN grown on the Si-terminated surface has local lattice mismatch of <3%. By comparing the high-resolution electron microscopy image and simulated image of the atomic arrangements at a GaN/SiC interface, Stirman et al. [453] concluded that the atomic arrangement at the GaN/SiC interface most likely consists of N bonded with Si. However, they also argued that some Ga bonded to C, by replacing the surface Si with Ga as shown schematically in Figure 3.96, might be present to maintain the charge balance. 3.5.3.4.1 GaN Growth on SiC Despite the cost of SiC and owing to the continually improving quality, development of surface preparation methods, high thermal conductivity, and availability of high-resistivity substrates paved the way for a good deal of device structures to be explored on SiC. This imperative is complete for FETs in that all the high-performance FETs are built in nitride heterostructures on SiC substrates. Let us now discuss some general aspects and examples of GaN growth on SiC substrates with the associated optical and/or electrical properties. When AlN buffer layers are used, the large lattice mismatch between the buffer layer and the subsequent GaN is sufficient to generate large concentrations of structural defects. To ameliorate this GaN on SiC Ga

Ga

Ga

N

N Ga

Ga

N

N Si

N

N

N

Ga N

C Si C

Si

C Si

Ga N

Si

Si

C

C

Ga

C

C

N

N

C

(a) Figure 3.96 (a) With ideal SiC/GaN interface (b). Possible atomic rearrangements for GaN/SiC interface with (0 0 0 1)-oriented substrate and epilayer, with allowance for some intermixing such as N atom replaced by C atom to maintain charge balance (patterned after Ref. [453]).

N Ga N

Si

C Si

Ga

C

Si

C

N

N Ga

Ga

Si

Si

N Ga

Ga

N Interface Si

C

Ga

N

N

N

Ga

Ga

Ga

Ga

Ga

N Ga

Si

N

Ga

N

Ga

Ga

Si

C Si

C

Si C

(b)

C

C

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j 3 Growth and Growth Methods for Nitride Semiconductors

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situation by shifting more of the mismatch to the buffer substrate interface, AlxGa(1
x)N buffer layers have been employed. Lin et al. [454] found the GaN quality improved with an Al0.08Ga0.92N buffer layer grown by low-pressure OMVPE. The GaN epitaxial layer deposited on that buffer exhibited an electron mobility and carrier concentration of 612 cm2 V
1 s
1 and 1.3 · 1017 cm
3 (at 300 K), respectively. It has long been suspected that nitridation of the Si-face of SiC(0 0 0 1), if not prevented, could lead to SiN formation, which would prevent the successful growth of GaN. Consequently, growth on SiC commences with a very short exposure to N or nearly simultaneous exposure to the metal and nitrogen in the MBE growth case. Because Si–N bonds are much stronger than the Si–Ga bonds, the growth is likely to begin with N forming single bonds with underlying Si, leading to a Ga-polarity GaN. Convergent beam electron diffraction investigations do indeed indicate that growth on the Si-face of SiC results in Ga-polarity samples [455]. When nitrides are grown on SiC by OMVPE, the growth generally is initiated with AlN or AlGaN at temperatures as high as 1200  C, which requires much lower nitrogen flows than that for GaN. Most common polytypes of silicon carbide, for example, 4H and 6H, have been used for nitride heteroepitaxy as doing so provides several advantages over sapphire, among which are a smaller lattice constant mismatch (3.1%) for [0 0 0 1] oriented films and a much higher thermal conductivity (4.9 W cm
1 K
1 or lower depending on the impurity and defect concentration). With SiC, conductive substrates are available and electrical contacts to the backside of the substrate are possible, thus simplifying the vertical device structures compared to sapphire substrates. On SiC, GaN epilayers do not rotate around the c-axis and assume the in-plane orientation of the SiC substrate, making registry of planes and cleaved facet formation in lasers easier. Although the (0 0 0 1) surface is available with both carbon and silicon polarities, the Si polarity is more conducive for epitaxy and leads to Ga-polarity samples. Gallium nitride epitaxy directly on SiC is problematic in that low temperatures must be employed for continuous coverage and/or AlN AlGaN buffer layers must be used. This that appears to be a disadvantage can be used to advantage in selective epitaxy on substrates patterned with AlN. Even though the lattice constant mismatch for SiC is smaller than that for sapphire, it is still sufficiently large to cause a large density of defects to form in the GaN layers to the point that extended defect concentration may not improve. SiC’s thermal expansion coefficient is lessthan that of AlN or GaN, thus the films are typically under biaxial tensile tension at room temperature. Atomically smooth and damagefree surfaces require additional processing steps, which will be described below. What is more, the cost of silicon carbide substrates is high and currently relatively few manufacturers produce single-crystal SiC. For some heterojunction GaN/SiC devices [456], it is highly desirable to grow GaN directly on SiC without an AlN buffer layer because it will inhibit carrier injection across the AlN/SiC interface owing to perhaps the insulating nature of AlN and the large bandgap discontinuity with SiC and GaN. However, strain/stress issues and ensuing cracking problem must also be kept in full view. Direct nucleation of GaN has been applied for the growth on SiC substrate in MBE growth [457] and OMVPE growth [439,458,459]. Both moderate-temperature and low-temperature GaN nucleation accompanied by a sequential high-temperature annealing have been applied.

3.5 The Art and Technology of Growth of Nitrides

As mentioned in conjunction with Figure 3.91, a good deal of MBE experiments indicate that GaN buffer layers must be grown at relatively low temperatures for wetting. The AlN buffer layers, however, are grown at elevated temperatures such as 1000  C for best results, as discussed in Section 3.5.3.2. In OMVPE, however, the situation is different in that relatively high temperatures, compared to that used for GaN initiation layers on sapphire, can be used for growth initiation. Following initial experiments, the remaining major problem of HT GaN buffer on SiC is the unavoidable crack generation of subsequent GaN epilayer, which is assuaged somewhat by use of AlN buffer layers. Let us discuss the effect of the deposition conditions of the AlN nucleation layer on the transport properties of the subsequent GaN layers, as performed by Koleske et al. [434]. In that particular investigation, following the deposition of the hightemperature AlN nucleation layer at 1080  C, the growth temperature, Ts, was lowered to 1030  C for the growth of GaN using 21 mmol min
1 and 1.25 slm NH3 and 1 slm H2 at 250 or 350 Torr. Following the growth of 1 mm of undoped GaN, 1 mm thick Si-doped GaN layers were grown. Using electrical isolation from the n-type SiC substrates (high-resistivity SiC substrates are available albeit at a much larger cost), a 1000 Å thick AlN film was grown on 4H-SiC, which showed breakdown voltages >70 V, giving an experimental breakdown field of >700 kV mm
1, paving the way for Hall measurements, conducted at voltages below 5 V, in the subsequent GaN layers grown on AlN buffer layers on SiC. For a comparative analysis, a set of GaN layers were grown on AlN nucleation layers where the AlN nucleation layer growth temperature was varied. Specifically, the GaN films were grown at a system pressure of 250 Torr on approximately 1000 Å thick AlN nucleation layers. For the four higher AlN growth Ts, the GaN was grown on 6H- and 4H-SiC pieces simultaneously, whereas at the lower two Ts, GaN was only grown on 4H-SiC. The electrical characteristics of three additional GaN films grown at 350 Torr by Koleske et al. [434] are listed in Table 3.14. On the on-axis and 3.5 off-axis vicinal 6HSiC substrates, an AlN growth temperature of 1080  C was employed, whereas on the 4H-SiC, an AlN growth temperature of 980  C was used. To lower the electron concentration, nRT, less Si doping was also used in case it was not limited by unintentional impurities. CV measurements for GaN on 6H-SiC showed a carrier concentration, nCV, of 7 · 1016 cm
3, close to nRT of 8.9 · 1016 cm
3. On the vicinal 6H-SiC, an electron concentration, nCV, of 2 · 1016 cm
3 was measured, which is less Table 3.14 XRD and RT PL linewidths and TRPL decay constants

and amplitude ratios (at 200 mJ cm
2 excitation density) for GaN thin layer samples grown with single (s-SiN) and double (d-SiN) SiN layers on low-temperature GaN buffer layers.

Control s-SiNx d-SiNx

10 K PL D0X (meV)

XRD ð0 0 0 2Þ; ð1 0 1 2Þ ðarcminÞ

s1, s2 (ns)

A2/A1

17.7 17.1 13.0

10.7, 26 10.9, 15.6 9.6, 12

0.04, 0.15 0.09, 0.18 0.10, 0.29

0.03 0.27 0.61

The time-resolved data fit biexponential decay function: A1 exp(
t/s1) þ A2 exp(
t/s2) well.

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than the measured nRT of 4.6 · 1016 cm
3. The difference between nCV and nRT may be due to the assumed Si-doped layer thickness of 1 mm used to calculate nRT. From plan view TEM, 8 · 108 cm
2 edge dislocations and 2 · 108 cm
2 screw dislocations were deduced for the GaN film grown on 6H-SiC (first row of Table 3.14). Because the electron mobilities continue to rise as the sample temperature is lowered, reaching roughly values that are about a factor of 2 larger at 77 K than at 300 K, one can assume that ionized impurity concentration and/or scatterers caused by extended defects have been lowered in concentration [460]. TEM investigations have been conducted to directly view extended defects in GaN as affected by the AlN nucleation layer deposition temperature [434]. Both g-vector scattering configurations, parallel to ½0 1  1 0 and [0 0 0 2], were employed to discern edge and screw dislocations, respectively. For details regarding edge and screw dislocations, see Section 4.1. The general observation was that grains on SiC substrates are better aligned than on sapphire. Specifically, a strong reduction in dislocation density near the GaN/AlN interface was observed for SiC substrates as compared to sapphire substrates, inclusive of a reduction in the screw dislocation component as well. Higher magnification imaging conditions suggest that edge dislocations are likely the result of discrete AlN nuclei forming on the SiC surface. No contrast for screw dislocations was observed, which may in part explain the lower screw dislocation density in the GaN near the GaN/AlN interface. GaN or AlN nucleation layers are usually grown on SiC substrates during MBE growth of GaN-based structures [461]. A discussion of GaN and AlN in the context of nucleation layers is given in Section 3.5.3.2. Here, the emphasis is on layers grown on nucleation layers. As in any growth case, the minimal defect density, the highest mobility, and the best PL characteristics of GaN/SiC epitaxial layers are obtained on the optimized GaN buffer layers. The stress-related phenomena in GaN films grown on SiC are much smaller than those in films grown on sapphire [422]. The defect structure of GaN films grown with and without an AlN nucleation layer on 6H-SiC by ECR MBE have been studied by Smith et al. [462]. Threading defects are identified as double position boundaries originating at the substrate–film interfaces. The density of these defects has been found to correlate with the smoothness of the surface. Even early on, GaN films grown on SiC exhibited higher mobilities (as high as 580 cm2 V
1 s
1) than those grown on sapphire (230 cm2 V
1 s
1), AlN buffer layers were used in both cases [39]. The electron mobility of 580 cm2 V
1 s
1 of film grown on SiC speaks well for the potential of GaN on Si, as hot gas etching of SiC produces atomically flat and damage-free surfaces (discussed in Section 3.2.3.2). Growth of GaN by MBE is much simpler on SiC than on sapphire once proper surface cleaning procedures have been worked out. GaN on SiC has recently been grown in the author’s laboratory using both RF and ammonia for N source. Beginning with RF for N source, the SiC substrate shows a well-defined 1 · 1 pattern indicative of the efficacy of H polishing. Once the thermal desorption process is complete, the substrate temperature is reduced to the growth temperature, which is about 650  C. At that point, both Ga and N sources were directed to the substrate with fluxes clearly placing the surface under very Ga-rich conditions. During the first few seconds of the growth, the RHEED pattern is of a very well defined 3 · 3 pattern. Soon

3.5 The Art and Technology of Growth of Nitrides

afterwards, the pattern evolves to a 1 · 1 pattern, which is typical of GaN. Upon cooling after growth, the pattern gives way to a 2 · 2 pattern. Layers grown with ammonia show different RHEED images initially. At first, the pattern is spotty, indicative of 3D growth, which evolves into a typical 1 · 1 pattern in a short period. Postgrowth evaluation generally includes AFM, X-ray, and PL investigations. Generally, the surface is characterized as smooth overall, but broken steps indicate structural defects. GaN layers were also grown in H-polished SiC using ammonia. The XRD measurements for ammonia-grown GaN indicated (0 0 2) (it is (0 0 0 2) in the hjkl configuration) symmetric and (1 0 2) (it is (1 0  1 2) in the hjkl configuration) asymmetric peak FWHM values of 3.81 and 10.5 arcmin, respectively, despite the 0.23 mm thickness of the layer. The PL measurements indicated half width values of about 10 meV and an efficiency of 0.3% at 30 K, which is very good for an unintentionally doped GaN layer of this thickness. 3.5.3.5 Growth on Porous SiC With the advent of porous SiC (PSiC) substrates, which have been reported to lead to SiC epitaxial layers with better quality, as compared to on standard SiC substrates, attempts with both OMVPE and MBE have been made to utilize this method for GaN growth. Standard SiC substrates are made porous in an anodization process, which requires conducting substrates or layers in modified vessel for anodization. The motivation is that the nanosized pores would pave the way for epitaxial lateral overgrowth at the nanoscale with improved results over the conventional ELO. This approach has its genesis in SiC epitaxial growth on PSiC [463] and has been extended to GaN growth on PSiC [464]. Applicable to layers grown by both MBE and OMVPE, optical and electrical properties of SiC grown on PSiC have shown improvement over those grown on standard SiC substrates. However, GaN layers on PSiC did not show decidedly improved electrical, structural, and optical properties, though marginal improvements in all three categories have been shown using ammonia MBE. However, growth on PSiC may have fuelled the debate on nanolateral epitaxy, which will be discussed briefly here. This method has been conceptually and experimentally shown to relieve strain and may in fact cause dislocation bending as well [465]. The technique, dubbed “nanoheteroepitaxy” (NHE) has in theory been shown to provide 3D stress relief mechanisms when an epilayer is nucleated on a compliant patterned substrate with an array of nanoscale islands. Consequently, the approach can significantly reduce the strain energy in the epilayer and extend the critical thickness dramatically. Calculations show that with 10–100 nm patterning, which is achievable with advanced lithography, one can eliminate mismatch dislocations from heterojunctions that are mismatched by as much as 4.2%, which is in the realm of GaN on SiC. The details are presented in Section 3.5.5.3. The nanometer-scale pores on 6H-SiC substrate are formed by anodization in hydrofluoric acid under UV illumination [466]. In this process, platinum is used as the cathode, whereas the substrate serves as the anode. The anodized substrates usually show very few pores on the surfaces most of the pores are buried under the socalled skin layer, though this may change in that as-prepared samples may have pores on the surface. Figure 3.97a is the cross-sectional TEM image of an as-prepared PSiC

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Figure 3.97 (a) Cross-sectional TEM image showing the porous structure of anodized 6H-SiC substrate. The skin layer is present which blocks most pores from reaching the surface. The depth of the pore region is 2.8 mm, and the two different sizes of pores are 10 and 100 nm, respectively. (b) AFM surface morphology of PSiC substrate after H annealing for 1 min at 1700  C, which served to remove the skin layer.

substrate, showing such a skin layer. The pores with sizes of about 10 and 100 nm each underneath the skin layer extends 2.8 mm into the substrate. Most of the pores start from the substrate surface and penetrate into the substrate like cones, the opening angle of which depends on the anodization parameters. The skin layer can be removed by etching in wet or dry chemistry such as H annealing at 1700  C. Figure 3.97b is an AFM image of a PSiC surface after removal of the skin layer, which causes the pore dimensions to enlarge. GaN films were grown under Ga-rich conditions by MBE using NH3 as a reactive nitrogen source with four samples, A, B, C, and D investigated [467]. The first sample, A, is grown on standard (0 0 0 1) 6H-SiC substrate and the second sample, B, is grown on PSiC(0 0 0 1) substrate with a skin layer of 60 nm. The thickness of GaN in samples A and B, which were grown without any buffer layers, was small (0.11–0.15 mm). The third and fourth samples were thicker (0.77–1.40 mm) and were grown with PSiC skin layer for sample C and without PSiC skin layer for sample D.

3.5 The Art and Technology of Growth of Nitrides

Both samples C and D were grown with a thin layer of AlN buffer (40 nm) between the GaN epilayer and the PSiC substrate. The substrate of sample C had an 8 miscut toward the ð1 1 2 0Þ orientation. GaN growth temperature was about 650  C for growth on PSiC substrate and the NH3 flow rate was fixed at 10 sccm. Nominally identical growth parameters were employed, which were optimized for GaN/6H-SiC growth. H annealing at 1700  C for 1 min performs the removal of the skin layer [468]. TEM images for samples A, thin GaN layer (0.15 mm) grown on standard 6H-SiC substrate, and B, GaN layer grown on PSiC substrate with skin layer, showed dislocation densities of 1 · 1010 and 7 · 109 cm
2 near the top GaN surface, respectively. The strain condition of GaN with respect to SiC can be obtained by measuring the relative lattice constant from an electron diffraction pattern that includes both GaN and SiC reflections. The in-plane lattice mismatch for samples A and B are therefore estimated to be Da/|a| ¼ 3.06% and Da/|a| ¼ 3.25%. Considering the in-plane lattice mismatch between GaN and SiC crystals (bulk material) is Da/|a| ¼ 3.48%, we found the degree of strain relaxation to be 87 and 93% for samples A and B, respectively. It is clear that for the same thickness of GaN layer, it is easier to achieve lattice and/or thermal relaxation when grown on PSiC substrate, even with the presence of a skin layer. The effect of the skin layer is significant on the dislocation distribution, as shown in Figure 3.98. Sample C, which is GaN grown on PSiC with a skin layer, exhibits a dislocation density of about 5 · 109 cm
2 for a thickness up to 1.40 mm, as can be seen in Figure 3.98. However, when the skin layer (nonporous layer) was removed from the PSiC surface, the dislocation density near the top of a 0.77 mm GaN reduced to 1 · 109 cm
2. If the same layer was grown to a thickness of 1.40 mm, it can be predicted that the dislocation density could have been further reduced. The effect of the H annealing/etching at high temperature can be clearly observed in that many pores extend to the surface where the AlN buffer layer and GaN epilayer were grown, though the interface is much roughened because of the presence of many pores being exposed on the surface after etching. Some of the exposed pores are seen to be backfilled with Al and/or Ga droplets during the MBE growth (dark area inside pores). For samples C and D, the electron diffraction patterns indicate 100% relaxation based on the measured value of Da/|a| ¼ 3.5% for both. As discussed in Ref. [469], when Da/|a| exceeds the strain-free value (3.48%), the GaN film is likely to be slightly under tensile strain. It is also possible that the lattice constant of PSiC is slightly different from SiC, as is the case with porous Si material. A tentative growth mechanism can be suggested, though speculative at this juncture, for GaN growth on PSiC. At the beginning of growth, AlN buffer grows on the nonporous regions of SiC, including possibly the sidewalls of SiC pores, leading to columns of AlN. This is then followed by GaN growth on top of the AlN columns. This is supported by the observation of open tubes in the initial stages of GaN growth, especially on the tilted PSiC substrates [469,470]. As the growth progresses, lateral epitaxy is enhanced, and the open tubes are closed at the upper part of the GaN layer. These tubes may serve as a relief mechanism for strain caused by the lattice and thermal mismatch during growth. In addition, some threading dislocations start to bend, merge, and bury in the mid-layer.

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Figure 3.98 Cross-sectional TEM images for (a) GaN layer (1.40 mm—sample C) grown on PSiC substrate with skin layer, and (b) GaN layer (0.77 mm—sample D) grown on PSiC substrate after the skin layer is removed by high temperature (1700  C) annealing for 1 min. The insets show the electron diffraction patterns for GaN layer and the PSiC substrates.

The crystalline quality of GaN grown on PSiC substrate was assessed by HRXRD rocking curve (o-scan). The best FWHM of the (0 0 0 2) rocking curve diffraction obtained was 3.3 arcmin and that of ð1 0  1 2Þ was 13.7 arcmin. This implies that device-quality GaN film can be achieved in thin layers grown on PSiC substrate. Moreover, low-temperature PL measurements indicate good quality of GaN films grown on PSiC substrates. The PL spectra measured at 15 K for samples B, C, and D showed the best FWHM of GaN excitonic peak to be 9.5 meV for the GaN grown on PSiC with the skin layer removed (sample D), whereas GaN films with the skin layer exhibit FWHMs of 14–16 meV (samples B and C). The quantum efficiency of PL from thicker layers (samples C and D, both with and without skin layer) is quite high (about 1.6%), whereas that from the thin layer is low (about 0.1%), indicating a reduction in nonradiative defects (presumably dislocations) in thicker layers. The features of the PL spectra for all three samples are similar in that they include excitonic emission,

3.5 The Art and Technology of Growth of Nitrides

shallow donor–acceptor band with the main peak at about 3.26 eV, and a very weak yellow luminescence band at about 2.2–2.3 eV. The highest optical quality of sample D from PL spectra agrees with the lowest dislocation density (1 · 109 cm
2) of this GaN layer grown on top of the porous SiC with skin layer removed. 3.5.3.6 Zinc Blende Phase Growth Owing to the cost and size restrictions associated with SiC substrates, a substantial impetus exists to grow GaN on Si substrates. To ameliorate chemical and lattice mismatch issues, 3C-SiC(0 0 1) films are generally deposited on Si(0 0 1) before cubic GaN deposition. Specifically the lattice constant mismatch of 3.4% between 3C-SiC and zinc blende GaN and the excellent thermal stability of 3C-SiC are the driving forces. However, 3C-SiC/Si(0 0 1) substrates generally suffer from very poor crystal quality and rough surfaces, both of which tend to promote the nucleation of wurtzite GaN. Furthermore, zinc blende AlN is very difficult to stabilize. Since wurtzite AlN is strongly favored regardless of the substrate or its orientation, it is not effective as a buffer for growth of zinc blende GaN. These complications and successful device demonstrations in Wz GaN gave way to the exploration of Wz GaN on SiC, which, in turn, is formed on Si. The particular approach used for this particular combination is discussed in conjunction with Pendeo-epitaxy on Si discussed in Section 3.5.5.2.3. Getting back to the zinc blende GaN on SiC, a 3C-SiC(0 0 1) film of several microns thick was grown on Si(0 0 1) by CVD and used to grow zinc blende GaN epilayers by several groups [471]. The crystal structure of GaN in OMVPE growth is greatly affected by both the growth temperature and the V/III ratio [472]. More growths of GaN on 3C-SiC have been accomplished by MBE [471,473–475]. Daudin et al. [476] found that both the initial substrate roughness and the V/III ratio affect the wurtzite/ zinc blende ratio in MBE-deposited films, whereas a N-rich condition was found detrimental to the growth of the zinc blende phase. The observations were explained by the assumption that MBE growth of zinc blende GaN is mainly governed by the impingent and overabundant, as compared to Ga, active N flux, which directly determines the mean free path of gallium adatoms. Thin 3C-SiC film with thickness of several nanometers produced by directly carbonizing the Si(1 0 0) surface has been used in MBE growth [477,478], and it was also found, as in the case of Wz GaN growth by MBE, that a Ga-rich growth condition is beneficial for growing films with flatter surfaces [477]. The quality of zinc blende GaN films is normally worse than that of wurtzite films, as determined by X-ray diffraction, TEM, PL, and other characterization methods. More discussion of zinc blende GaN growth can be found in the section dealing with growth on GaAs substrates. 3.5.4 Growth on Si

Si substrates are very attractive not only because of their high quality, availability, and low cost but also the distant possibility of integration of Si-based electronics with wide bandgap semiconductor devices. Both zinc blende and wurtzite GaN epilayers have been grown on Si(0 0 1) by MBE [479], OMVPE [480], or HVPE [481]. It should,

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however, be mentioned that dismal quality of films along with cracking owing to the tensile residual thermal stress makes the standard growth of GaN directly on Si undesirable. Although still not as good as those on Wz SiC and sapphire, growth on 3CSiC(1 1 1) either deposited or formed on Si by carbonization and growth with the aid of epitaxial lateral overgrowth have attracted attention. In fact, very competitive AlGaN/ GaN MODFETs have been reported on Si substrates as detailed in Volume 3, Chapter 3. The lateral overgrowth method directly on Si is discussed in Section 3.5.5.2. Its variant pendeo-epitaxy on 3C SiC/(1 1 1) Si is discussed in Section 3.5.5.2.3. Nitridation as carried out on sapphire is not performed because of the possibility of either stoichiometric or nonstoichiometric SiN formation (Si3N4 is its stoichiometric form). Several growth initiation techniques can be employed. One approach utilizes the growth of a thin AlN layer directly on Si followed by GaN deposition. GaN deposition directly on Si is not generally performed in MBE as Ga evaporates rather readily from the Si surface at typical MBE deposition temperatures. The other method involves the deposition of a sub- to one monolayer of Al deposition at about 650  C followed by nitridation to convert it to AlN, which is then followed by regular deposition of AlN. Cubic GaN is generally grown on Si(1 0 0) substrates [482], whereas Si(1 1 1) substrates are employed for the wurtzitic GaN growth [483,484]. Both phases, wurtzitic and zinc blende, are frequently detected, accompanied by a large number of extended defects such as dislocations, stacking faults, and twins. GaN grown on Si (0 0 1) is predominantly cubic, but not without mixed polarity. In one experiment, a 30 nm GaN low-temperature buffer layer was employed to accommodate the 17% lattice mismatch between the film and the substrate by a combination of misoriented domains and misfit dislocations [485]. Beyond the buffer layer, highly oriented domains separated by inversion domain boundaries are found by TEM. Stacking faults, microtwins, and localized regions of wurtzitic structure are major defects in the film. The films grown on Si(1 1 1) have a predominantly wurtzitic structure with a presence of twinned cubic phase. The GaN films grown directly on Si(0 0 1) are generally phase mixed, containing both zinc blende and wurtzite modifications or only wurtzite phase and, depending on the buffer layer details, two different rotation alignments with different orientations of the wurtzite phase (with c-direction of GaN being on the (0 0 1) surface of Si or r-plane of GaN being oriented parallel to the Si surface) [486]. Owing to the large lattice mismatch, it is difficult to grow pure zinc blende phase GaN directly on the (0 0 1)Si, which is aggravated by an amorphous SixNy layer that may form at the GaN/Si interface. To mitigate this problem, a 3C-SiC thick layer produced by CVD or a thin layer produced by carbonization is commonly used. The GaN epitaxy on 3C-SiC/Si is discussed in conjunction with pendeo-epitaxy in Section 3.5.5.2.3. The utilization of other buffer layers, such as g-Al2O3, AlN, or AlxGa1
xN, resulted in the wurtzite phase growth, as will be discussed below [13]. The Si(1 1 1) orientation is generally the preferred substrate plane for GaN epitaxy. The quality of GaN on these substrates has been improved to the point where devices have been demonstrated. For example, ultraviolet LEDs grown by MBE [487] and OMVPE [488], Schottky-barrier ultraviolet detectors [489], and field-effect

3.5 The Art and Technology of Growth of Nitrides

transistors [490], in the form of Si channel with AlN as the gate dielectric [491], have been attained. Because Si is a nonisomorphic substrate to GaN, the surface step could possibly lead to an SMB, as discussed in section 3.5.3.3 on the SiC substrate. Martin et al. [492] showed that surface preparations reducing irregularities on the substrate surface greatly improved the crystalline and luminescent quality of a GaN epilayer grown by MBE. Although this was carried out on Si(1 1 1) with ECR MBE, the gist of this work applies to Si(0 0 1) also. Wide atomically flat terraces were created by etching using a solution of 7 : 1 NH4F : HF, greatly increasing the distance between surface steps and thus decreasing their density. The latter reduced the number of stacking mismatch boundaries of the epitaxial GaN films. Moreover, to overcome the problem of a large difference in lattice parameters and the strength of the Si–N bond, different buffer layers such as AlAs, oxidized AlAs, GaAs, AlN, LT GaN, ZnO, and 3C-SiC(1 1 1) have been implemented [13]. In addition, to avoid the formation of silicon nitride, it is a common practice in MBE growth to initiate the AlN buffer layer growth by exposing the Si substrate to the flux of aluminum and ammonia alternatively for a short time or by briefly exposing to RF nitrogen followed by Al. These techniques help suppress the formation of excessive SiNx at the interface and result in a 2D growth mode of GaN. Despite the 23.4% misfit in the AlN/Si system, optimized AlN films grown on a (7 · 7) reconstructed Si surface made it possible to grow GaN. In OMVPE growth, a hightemperature AlN buffer layer is much better than a low-temperature one and it is necessary to enhance 2D growth for GaN epitaxy [493,494]. In the case of MBE, the RHEED pattern obtained on GaN grown on Si followed by an AlN buffer layer displays the typical 1 · 1 pattern at the growth temperature, which transforms to a 2 · 2 pattern upon cooldown, indicative of a Ga-polarity sample, implying that N forms the long bonds to Si [495]. The XRD measurements indicated (0 0 2) symmetric and (1 0 2) asymmetric peak FWHM values of 15.9 and 10.9 arcmin, respectively. PL measurements indicated half width values of about 19 meV at 30 K. Alternative substrates, such as compliant silicon-on-insulator (SOI) substrates and porous silicon (PS) have been utilized for GaN growth both by OMVPE and MBE [13]. The premise of these substrates is that they are expected to improve the quality of epitaxial GaN layers by releasing the strain and absorbing the generated threading dislocations in the thin Si overlay of the SOI substrate in much the same way as in SiGe on Si. Moreover, selective area growth has also been applied by the use of stripepatterned and dot-patterned Si(1 1 1) by OMVPE [496] and MBE [497] in an effort to grow low-defect GaN heterostructures. Despite the progress made in all types of nitride growth on Si, the GaN layers on Si are in general inferior to those grown on sapphire, SiC, and SiC-coated Si. The advantages of Si, which are always the same no matter what the application is, are high thermal conductivity, availability in large numbers and in high quality, and Si microelectronics. Epitaxial relationship between GaN or AlN and Si is discussed in Section 3.3.3. Having discussed the issues dealing with surface preparation, interface mismatch, and so on, let us now turn our attention to thermal cracking that characterizes GaN growth on Si. The thermal expansion coefficient of GaN is much larger than that of Si, see Chapter 1. The resultant tensile strain leads to cracking when the film thickness

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exceeds a critical value (usually 1 mm) [498], as slip system is not available [413,499]. The primary cracking directions are ½1 1  2 0, ½ 12 1 0, ½ 2 1 1 0 with resultant cleavage plane being f1 1 0 0g [500]. Another difficulty involving Si substrates is that at high temperatures, a strong chemical reaction between Ga and Si takes place, namely, meltback etching, which leads to a deterioration of substrate and epilayer [501,502]. The meltback etching, once initiated, cannot be stopped and causes rough surfaces with deep voids in the substrate. A third relevant point is the nitridation of Si surface when it is exposed to ammonia, leading to the formation of amorphous Si3N4 [165]. However, this process is also reported to be self-limiting when a SixNy thickness of a few nanometers is reached [503]. The point is that special precautions must be taken to prevent surface nitridation of Si during growth of nitrides by OMVPE. As could be garnered from the above paragraph, most research relevant to GaN on Si revolves around finding ways to produce crack-free GaN on Si with thickness >1 mm. Naturally, the nucleation layers have received a great deal of attention in this regard. Typically, AlN buffer layers, grown at a high temperature of 1100  C, represent the most commonly employed scheme to grow GaN on Si [504–506]. Follstaedt et al. [504] obtained GaN epitaxy on Si(1 1 1) by using a 30 nm AlN buffer layer grown at 1080  C, with the subsequent GaN growth at 1060  C. The TEM images showed that AlN buffer layer was continuous, implying that AlN can wet silicon substrate very well. Lahreche et al. [505] found that the best AlN growth temperature is approximately 1060  C. In most cases, GaN grown using just a single hightemperature AlN buffer layer produces cracks during cooling down to room temperature. Zamir et al. [506] discovered that AlN should be grown at temperatures well above 760  C, and as close as possible to 1100  C, to get a more aligned GaN and a smooth GaN surface for short growth times, meaning thin films. Low-temperature AlN buffer has successfully been used to reduce stress in GaN on sapphire by Amano et al. [507] and Dadgar et al. [508] who used low-temperature AlN buffer to eliminate cracks in GaN on Si. The crack density was reduced to zero from an original density of 240 mm
2. Additionally, the FWHM value of X-ray rocking curve for ð2 0 2 4Þ improved from approximately 270 to 65 arcsec. Blasing et al. [509] explained the mechanism of stress reduction in GaN with a low-temperature AlN buffer layer arguing that the buffer layers grown at high temperatures are pseudomorphic, whereas those grown at lower temperatures are relaxed. Therefore, AlGaN or GaN layers grown on a low-temperature AlN interlayer grow under compressive interlayer-induced strain that compensates the thermal strain. The stress in the GaN layer depends on the growth temperature, which is likely to control the degree of AlN interlayer relaxation. To avoid surface nitridation of silicon substrate, a few monolayers of Al predeposition prior to AlN growth are usually necessary [510,511]. A lack of amorphous SixNy characterizes the interface between AlN and Si when such an AlN initiation layers is employed. Besides low-temperature AlN buffer, other methods have also been applied to obtain crack-free GaN on Si. Feltin et al. [512,513] employed GaN/AlN superlattices as a buffer layer to reduce tensile stress in the epilayer thus reduce the tendency to crack and achieved a relatively thick (2.5 mm) GaN without any cracks. A side benefit of the

3.5 The Art and Technology of Growth of Nitrides

AlN/GaN superlattice buffer composite is reduction of threading dislocation density from 1.5 · 1010 cm
2 for a sample without superlattices to 2.5 · 109 cm
2 for the best uncracked sample with four stacks of superlattices. Marchand et al. [514] reported on the growth of crack-free GaN, which was under compressive stress, using graded AlN-to-GaN buffer with a total buffer thickness of 800 nm. In contrast, the GaN buffer layers grown with a high-temperature AlN buffers showed a tensile stress and exhibited cracking in film. Additionally, HfN buffer layers have also been used to grow GaN on Si [498]. Compared to AlN, HfN is a better diffusion barrier, which can prevent diffusion of Si to epilayer, and has a closer lattice match to GaN. Crack-free GaN with a thickness of 1.2 mm has been grown on both (1 1 1) and HfN/Si(0 0 1) surfaces. Another method developed to obtain a crack-free thick GaN on Si, other than using various kinds of buffer layers, is by patterning the silicon substrate. This can be done by either masking or etching the substrate. The concept here is not to avoid cracks completely but to guide them to the masked or etched parts of the substrate. As long as the patterned regions are smaller than the average crack–crack distance, no cracking of the GaN layers should occur because the substrate is softer. A Si3N4 or SiO2 mask layer deposited directly on the substrate can be used to produce the pattern needed. Because the dielectrics are amorphous and have very long migration lengths, no GaN layer is typically grown on them as evidenced by no or a little GaN observed on such a mask. A related approach, which does not require a mask, is to etch trenches sufficiently deep to avoid the formation of continuous film during growth. In this vein, Zamir et al. [515] used patterned Si substrates, formed by reactive ion etching (RIE) square mesas, and obtained 14 mm ·14 mm crack-free GaN grown by OMVPE. Krost et al. [500] used square-patterned Si(1 1 1) aligned along Si ½1  1 0 and Si ½1 1  2 with SixNy masks, prepared by standard photolithography and wet chemical etching, to obtain crack-free 3.6 mm thick GaN in the window regions on Si. In their samples, cracks can only be observed in the SixNy stripes separating the square areas. The epitaxial lateral overgrowth method has also been applied to growth on Si substrates, as discussed in Section 3.5.5.2. Building on the earlier brief mention of this, let us now give a succinct review of GaN-based devices on Si. The FET results on Si substrates are discussed in sufficient detail in Volume 3, Chapter 3. In terms of LEDs, green InGaN LED on Si using AlN/GaN superlattice as buffer layer by MOVPE have been reported [516–518]. The turn-on voltage of the LED was 6.8 V and the operating voltage was 10.7 V at 20 mA. The electroluminescence peaked at 508 nm, with a full width at half maximum of 52 nm. An optical output power of 6 mW was achieved for an applied current of 20 mA. Dadgar et al. [519] employed in-situ SixNy mask to improve the crystalline quality of GaN and grow InGaN/GaN multiquantum well LED. By using the combination of low-temperature AlN buffer and in-situ SixNy mask, a 2.8 mm thick diode structure was grown using OMVPE. In current–voltage measurements, turn-on voltage of 2.5–2.8 V and a series resistance of 55 O were observed for a vertically contacted diode. The light output power was 152 mW at a current of 20 mA, with a wavelength being 455 nm, which is not comparable to that obtained on sapphire.

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3.5.5 Growth on Sapphire

At the time of writing this book, OMVPE is the workhorse of GaN and related materials growth, even though the growth mechanisms are not well understood. In this section, OMVPE growth and related issues will be discussed. The discussion of the growth by MBE on sapphire will be limited to the effect of buffer layers on the polarity of the resultant films. As mentioned in Section 3.4.1.1, substrate temperatures well in excess of 900  C are required to obtain single crystalline high-quality GaN films, and the GaN films with the best electrical and optical properties are grown at 1050  C or even higher. Such high temperatures in H2 and NH3 ambience provide highly formidable challenges. Metalalkyls need to be kept below pyrolysis temperatures until just before the reaction zone: ammonia and metalalkyls would have to be kept separate until just before the reaction zone as well. High growth temperatures associated with the OMVPE process require that the substrates used do not decompose at the deposition temperatures. Sapphire, Si, and SiC are among the substrates that meet this criterion. The high N/Ga flux ratio used to minimize the nitrogen loss is another issue. The high temperatures have a contradictory effect on the crystal quality. On one hand, the surface mobility of atoms and thus the quality of grown film is higher for higher substrate temperatures. On the other hand, high vapor pressure of N over Ga and particularly In brings the growth process very close to the dissociation temperature. Not as serious, but postgrowth cooling introduces more strain and thus more structural defects may be introduced during cooling. At substrate temperatures exceeding 1100  C, the dissociation of GaN results in voids in the growth layer. Use of dimethylhydrazine (DMHy) [520] as a nitrogen source permits lower deposition temperature, because DMHy decomposes at temperatures (500  C) lower than that for ammonia (800  C). The film quality is not high enough, however. Rocking curve FWHM of 150–250 arcsec and electron concentration and mobility of 5 · 019 cm
3 and 48 cm2 V
1 s
1, respectively, have been obtained. Early on, nitrogen vacancies had been widely considered to be the primary cause of large background electron concentration in GaN. However, this gave way to other concepts such as O doping. In any case, an improvement in crystal growth must involve more complete incorporation of nitrogen into the crystal lattice, as well as reducing the likelihood of impurity incorporation. In an effort to achieve this complete incorporation of nitrogen into the crystal lattice, several investigators [143,521,522] substituted the more reactive hydrazine in favor of ammonia. Andrews and Littlejohn [523] tried Ga(C2H5)3NH3, which already has a Ga–N bond, as a source material. Others used plasma excitation of nitrogen in CVD growth environment, which proved to be the most popular method of increasing the reactivity of nitrogen. In the CVD growth approach, several variations have been attempted in the presence of plasma. Suitable precursors in the metalorganic CVD (OMVPE) growth must exhibit sufficient volatility and stability to be transported to the surface. These precursors should also have appropriate reactivity to decompose

3.5 The Art and Technology of Growth of Nitrides

thermally into the desired solid and to generate readily removable gaseous side products. Ideally, the precursors should be nonpyrophoric, water and oxygen insensitive, noncorrosive, and nontoxic. In contrast to the MBE growth, plasma excitation of the nitrogen species has not proven necessary in CVD growth. In spite of the fact that trialkyl compounds (TMA, TMG, TMI, etc.) are pyrophoric and extremely water and oxygen sensitive and that ammonia is highly corrosive, much of the best material grown today is produced by conventional OMVPE by reacting these trialkyl compounds with NH3 at substrate temperatures in the vicinity of 1000–1100  C [196,524–533]. The reactive nature of constituents makes it imperative that special materials are used for any component that experiences high temperatures. This is especially true for reactors where the substrate heating is done by resistive means internal to the deposition system. Although there is a notable dependence on the reactor and associated temperature measurement methods, deposition temperatures in excess of 800  C are required just to obtain single crystalline GaN films and near or above 1050  C for high-quality GaN films. At substrate temperatures exceeding 1100  C, the dissociation of GaN results in voids in the growth layer. A similar situation is observed for AlN film growth. 3.5.5.1 OMVPE Low-Temperature Nucleation Buffer Layers As in the case of HVPE, if GaN is grown directly on sapphire, the quality of the crystals grown by OMVPE is not as good as that grown on small GaN templates because of large lattice and thermal expansion coefficient mismatch and mostly nonideal nucleation. The degradation of the quality is evident by a wide X-ray rocking curve (10 arcmin typically), rough surface (containing hillocks), high electron concentration (up to 1019 cm
3), and considerable yellow emission in the luminescence spectra. To improve the GaN layer quality, low-temperature AlN or GaN nucleation layers are usually grown following the nitridation process (which may not require a separate and deliberate step) [534,535]. Although the low-temperature AlN and GaN buffer layers (initiation or nucleation layer terminology is more descriptive and less confusing as subsequent GaN layer may also be called buffer layer in a device structure) were initially used solely for OMVPE, their use in the MBE process was also shown to be highly beneficial. The use of nucleation layers has become an indispensable step for obtaining good-quality GaN and AlGaN films. However, lowtemperature initiation layers have given way to moderately high-temperature initiation layers both on SiC and sapphire, the exact value of which depends on whether AlN or GaN initiation layer is employed. Although low-temperature layers for enhanced nucleation were explored much earlier, a milestone was achieved in 1986 when Amano et al. [536] reported the use of low-temperature buffer layers to enhance the surface morphology and electrical and optical properties of GaN. This was accomplished by depositing about 50 nm thick AlN film on c-sapphire as a nucleation layer. Until then, hexagonal GaN grown by OMVPE was deposited directly on sapphire substrates and suffered from low quality, which manifested itself with high electron concentrations approaching 1020 cm
3. In the low-temperature buffer layer approach, as reported by Amano et al., an AlN layer is first deposited at 600  C, or thereabouts, followed by a high-temperature annealing

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in H2 atmosphere, and finally, the GaN layer is grown at 1000  C. The surface morphology of the GaN film grown in this manner was improved to an RMS roughness of about 1 nm and was also free of cracks. GaN films grown on low-temperature AlN nucleation layers displayed much improved quality as measured by X-ray diffraction and photoluminescence. The FWHM of the double-crystal X-ray rocking curve (0 0 0 2), which measures ordering along the c-direction only, obtained for optimum conditions was 110 arcsec [537]. The 4 K photoluminescence spectrum measurement exhibited a sharp donor-bound exciton transition with a FWHM value of about 1 meV. This was accompanied by the observation of free exciton related transitions in some of the samples. As for the transport properties, the electron concentration was reduced to about 1017 cm
3 with an associated mobility of about 600 cm2 V
1 s
1 at room temperature, a first for that time. Such relatively low electron concentration moved the GaN layers from the degenerate state to the nondegenerate state and temperature dependency of the carrier concentration and electron mobility emerged. Consequently, the interpretation of residual donors and scattering mechanisms began. Important process parameters in the pursuit of the best nucleation layers in random order are as follows: (i) Growth rate of the nucleation layer [538–540]. (ii) Thermal annealing of the GaN buffer layer of optimal thickness [541–544]. (iii) Possible misorientation of the sapphire substrate. In one investigation, it was found that a slight misorientation of the sapphire by about 0.17 significantly improves the surface morphology [545]. (iv) Composition of the carrier gas.